Keywords:

Table of contents

Acknowledgments.

During this thesis period I have had the support of many people whom I would like to thank. First of all I would like to thank my supervisor Denis Vion for his incredible eort to this work and guidance during the thesis, as well as interesting discussions about life in general and many a ride back to the train station. My co-supervisors Daniel Esteve and Patrice Bertet were also very helpful in solving particular problem as well as planning general course of the experiments. Many a thanks goes to Simon Bernon who worked as a post-doc on the project with me during the rst 10 months. Our circuit-QED implementation of the quantum Zeno dynamics (QZD) is to a large extent his idea. Helene le Sueur was a great help when she joined the project during the tomography measurements of the QZD and contributed a lot to the implementation Maximum Likelihood. We beneted from a parametric amplier made by Xin Zhou which made measurement a lot easier.

I had very good technical support from Pief Orla especially with matters concerning the clean-rooms, and from Pascal Senat designing the cavities and other mechanical parts, and from our mechanical workshop I would like to thank Dominique Duet and Vincent Padilla. The support from Patrick Pari and Philippe Forget at the cryogenics lab at SPEC was extremely valuable when we had problems with our dilution refrigerator.

From outside the lab I would like to thank Pierre Rouchon for his help in developing the Maximum Likelihood algorithm we used to process our data and Mazyar Mirrahimi for fruitful discussions concerning measurement strategies and help in modeling the QZD numerically.

I very much like to thank my oce mate Audrey Bienfait for good times and bad, countless lunches brought for me when I couldn't go to the canteen and rides back to Paris. My gratitude goes out to the rest of the Quantronics group for their help and company.

I would like to thank my roommates a during the period Coline Laugraud, Yann Bermond and David Gössmann for fantastic years, laughs and parties. And Finally I would like to thank my family and my parents Júlíus and Svanhvít for their love and support throughout it all.

Chapter 1

Introduction and summary

One of the fundamental postulates of quantum mechanics is the back action of a measurement on a quantum system. It is a non-unitary process which projects the system onto an eigenspace of the observable and the measurement outcome is the corresponding eigenvalue. Measurements are usually considered to be instantaneous and temporally separate events in between which the system evolves. In 1977 however, Misra and Sudarshan [START_REF] Misra | The Zeno's paradox in quantum theory[END_REF] put forth the theoretical framework for a continuously monitored quantum system and found that it suppresses the evolution of measurement outcomes. They named the phenomenon the quantum Zeno eect(QZE) in reference to a paradox by the Greek philosopher Zeno of Elea on blocked motion in classical dynamics.

Continuing from there but shifting the focus from measurement observables to the dynamics of the monitored system, initial theoretical work has been done by Facchi and Pascazio introducing the quantum Zeno dynamics(QZD). An experimental proposal put forth by Raimond et al. [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF] for a cavity-QED system showed interesting dynamics in multidimensional subspaces created by the QZD. The work presented here addresses the experimental observation of QZD in a cavity-QED system based on superconducting circuits known as 3D circuit-QED.

This chapter rst sets the framework for the experiments presented by tracing the history of quantum mechanics, the development of circuit-QED and its current state and outlook. The subsequent sections introduce the content of the thesis in an executive summary covering the experimental concept, technical building blocks, results, analysis and conclusions.

Quantum mechanics: From theory to experiments on single quantum systems

At the beginning of the 20th century experiments revealed that classical physics could not fully explain the behavior of atoms and their interaction with light. In response, quantum mechanics was created and by the 1930s a comprehensive theoretical framework had emerged for describing the behavior of atoms and sub-atomic particles. The most common interpretation of the formalism, known as the Copenhagen interpretation, contains some strange features that become even more surprising when considering individual quantum systems. Early on, observing these features was beyond experimental techniques and some physicists even believed that quantum theory was incomplete. One of the best known attempts at demonstrating that is the so called EPR paradox by Einstein, Podolsky, and Rosen [17]. In order to complete quantum theory, the skeptics proposed an incorporation of local hidden variables. In 1964 J.S. Bell published a theorem which proposes a way to test this local hidden variable hypothesis experimentally [9]. He introduced a correlation measure between two particles initially entangled, whose value has an upper bound which is lower in the hidden variable theory than in quantum theory. Improvements such as reformulations of Bell's theory and better experimental techniques on single quantum systems, led to an experiment by Aspect et al. in 1982 on photon pairs [3] where the value of the correlation measure exceeded that allowed in the hidden variable theory.

During the following decades, the measurement and control of individual quantum systems became a fruitful research eld as acknowledged by the Nobel prize awarded to David Wineland and Serge Haroche in 2012. Wineland and his group managed to trap arrays of ions in an RF electric eld trap, cool them down to their ground state and coherently manipulate them with lasers. The Haroche group on the other hand developed superconducting microwave cavities which could store photons for up to 130 ms. By letting highly excited Rydberg atoms pass through a cavity and interact with the photons they could perform a non-destructive measurement of a photon in the the cavity. This eld is known as cavity quantum electro dynamics (cavity-QED).

The canonical example of a cavity-QED system is that of a two level system(TLS, also regarded as an eective spin 1/2) coupled to the electromagnetic eld of the cavity (regarded as a single mode harmonic oscillator). When the interaction rate between the two systems Kristinn Júlíusson is stronger than their loss rates (so-called strong coupling regime) the cavity can probe the spin state in a non-destructive manner and vice versa.

Experiments in the single quantum system regime that started out with Rydberg atoms in cavities and trapped ions have since spread out to a variety of platforms for quantum mechanics such as semiconductor quantum dots, mechanical oscillators, cold atoms, or impurities in solids. One of the most successful implementation has been with superconducting circuits where the prospect of quantum information processing(QIP) has been a big incentive. These circuits are based on the LC-oscillator where the linear inductance is replaced by an element called a Josephson junction (JJ). The nonlinear inductance of the JJ breaks up the equidistant level spacing of a harmonic LC-oscillator, thus allowing two states to be isolated as a TLS which can represent a quantum bit (qubit). The TLS excitation frequency is usually in the few GHz range and operations such as transition between levels can be below 10 ns. The rst superconducting qubit was demonstrated by Nakamura et al. [START_REF] Nakamura | Coherent control of macroscopic quantum states in a single-Cooper-pair box[END_REF] in 1999, and had a nanosecond. In 2002 Vion et al. [START_REF] Vion | Manipulating the Quantum State of an Electrical Circuit[END_REF] reported lifetimes of 500 ns with a circuit partly immune to decoherence and tted with single shot qubit readout, thus truly opening measurements on individual quantum circuits. In 2004 a successful adaptation of cavity-QED to superconducting circuits was reported by Wallra et al. [START_REF] Wallraff | Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics[END_REF] following a proposal by Blais et al. [6] of a superconducting qubit coupled to a transmission line microwave resonator, thereby creating the eld of circuit-QED. In comparison with Rydberg atoms, these man made superconducting qubits have the benet of parameter tailoring which allowed for stronger coupling than possible in cavity-QED. The ability to tailor the qubit design proved to be useful again when a new design called a transmon solved the debilitating problem of the qubit sensitivity to uctuating charges in its surroundings (Koch et al. [28]).

The next signicant improvement in the eld of circuit-QED came in 2011 when Paik et al. redesigned the transmon qubit in order to couple it to the electric eld inside a bulk 3D cavity resonator [START_REF] Hanhee Paik | Observation of High Coherence in Josephson Junction Qubits Measured in a Three-Dimensional Circuit QED Architecture[END_REF]. The reduced importance of lossy dielectrics in this 3D circuit-QED design yielded a sizable improvement coherence times, with both T 1 and T 2 around 100 s [START_REF] Rigetti | Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms[END_REF] in single qubit circuits.

Recent experiments in 3D circuit-QED focus on the dynamics of coherent states in cavities, including superpositions thereof, both for fundamental physics experiments and Quantum information processing [27] [START_REF] Mirrahimi | Dynamically protected cat-qubits: a new paradigm for universal quantum computation[END_REF]. As proposed by Raimond et al. [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF] QZD provides a powerful tool to manipulate individual components of coherent states. In this thesis, the quantum Zeno dynamics is explored in a 3D circuit-QED system tailored for that purpose.

Thesis summary

The work presented in this thesis aims at exploring the quantum Zeno dynamics in a harmonic oscillator evolving under a coherent resonant drive.

We rst discuss the general dynamics in an arbitrary Hilbert space before introducing a special case proposed for a cavity-QED experiment analog to Raimond et al. [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF]. Thereafter the development of the experimental building blocks is described along followed by the sample characterization. Next a measurement technique called Wigner tomography is introduced followed by the experimental observation of the QZD. Numerical simulations of the experimental results are used to analyze the experimental imperfections before giving nal conclusions.

General QZD

One of the fundamental postulates of quantum mechanics is the measurement postulate: The eect of a measurement on a quantum system is to project it onto the eigenstate corresponding to the measured eigenvalue. Let us consider an initial measurement giving the eigenvalue i corresponding to the measurement projector P i , and a Hamiltonian H acting on the system that does not commute with P i (i.e. [H ; P i ] = / 0). Immediately after the measurement the probability of measuring i is i (0) = 1 but after non-zero time t the unitary evolution U(t) = exp(¡i H t) yields generically i (t) < 1: If the measurement however is repeatedly performed at intervals that are much shorter than the characteristic evolution time, the evolution of i away from unity is hindered and in the limit ! 0 + it is completely blocked. This is known as the quantum Zeno eect (QZE). Facchi et al. have in 3 papers [19] [18] [20] explored the QZE in the context of the evolution of the projected system as it divides Hilbert space into subspaces. The system then evolves within each subspace, which gives rise to the quantum Zeno dynamics (QZD). Facchi et al. explain that in the ! 0 + limit, the measurements are a unitary process, and they extend the blocking eect to other types of unitaries coupling the initial system to external degrees of freedom. In a proposal by Raimond et al., the QZD of a coherently driven harmonic oscillator is explored in the context of cavity-QED [START_REF] Raimond | Quantum Zeno dynamics of a eld in a cavity[END_REF] [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF]. The strong coupling of the electric eld in the cavity to Rydberg atoms enables selective manipulation of any Fock states jni, the eigenstates of the operator measuring the number n of excitation or photons in the cavity. By either a projective measurement or a unitary evolution addressing only a specic Fock state jn = n 0 i one can keep the population probability of this state at zero. This creates an exclusion circle (EC) of radius = n 0 p in phase space and splits the Hilbert space H = H n<n0 H n=n0 H n>n0 into three parts. Although H n<n0 and H n>n0 both evolve under the same drive their dierent boundary conditions lead to dierent dynamics. Details are given in Chapter 2.

Designing a QZD setup for 3D circuit-QED

Implementing the experiment required both a conceptual and technical adaptation to circuit-QED, which required innovative solutions.

Theoretical design

We describe now how we met the challenge of combining Fock state selective manipulation while introducing minimal non-linearities in the harmonic oscillator, by using a transmon qubit.

A transmon is a multilevel superconducting circuit that we simplify here to a 3-level system (3LS) with levels jg i; jei andjf i and excitation frequencies ! ge and ! e f = ! ge + . When coupled to a harmonic oscillator with resonance frequency ! 0 (see g. 1.2), the system is described by a generalized Jaynes-Cummings Hamiltonian H JC3 = ~(! 0 a y a + ! ge jeihej + (! ge + ! ef )jf ihf j + g 0 (a y a q + a q y a) );

where a and a y are the annihilation and creation operators for the harmonic system, a q and a q y the analogs for the transmon and g 0 (or vacuum Rabi splitting) between the two. In the dispersive case with detuning ge = ! 0 ¡ ! ge much larger than g 0 , a residual Kerr-nonlinearity K = g 0 2 g e 2 + g e 2 is inherited by the harmonic oscillator when the 3LS is in jg i. This causes undesired dephasing between the Fock states as explained in chapter 2. For Fock state selective excitation of the Transmon while keeping K low, we devise a scheme where ! e f = ! 0 . This gives rise to new dressed states j; ni = je; ni jf ; n ¡ 1i 2 p ¡ jq; ni = jqi jni ;

with frequencies shifted to ! ;n , where ! ;n = ! ge 2 p g 0 n p . Consequently, Fock state dependent excitation of the Transmon is possible at frequency ! +;n , which can be used for manipulating Fock state jni or mapping its occupation probability p(n) onto the transmon excitation probability. This ! 0 = ! ef scheme yields a sucient separation ! ;n+1 ¡ ! ;n = 2 p g 0 ( n + 1 p ¡ n p ) between adjacent states even with a low g 0 , which in return yields a small non linearity K when the 3LS is left in jg i.

Figure 1.2 illustrates three protocols for implementing the QZD mentioned above, i.e blocking Fock state jn = n 0 i. The rst two (see c and d) are described in [START_REF] Raimond | Quantum Zeno dynamics of a eld in a cavity[END_REF], where a coherently driven cavity undergoes a unitary evolution U (t) = exp(¡iHt/~) interrupted N times by either projective measurements or unitary kicks considered to be instantaneous. MHz is weak enough to preserve the harmonic oscillator when the atom is in its ground state but strong enough to induce an anharmonic oscillator when the atom is excited. This allows for photon number dependent manipulations of the atom. QZD can be induced by alternating a coherent drive pulse at ! 0 and blockade on Fock state jn 0 i by either c) applying a pulse at ! +;n0 followed by a projective readout of the 3LS, d) applying a 2 pulse inducing a selective phase (¡1) n;n 0 jni: e) The third method of inducing the QZD is by applying a continuous drive at ! 0 and another one at ! +;n0 which splits up the jn 0 i level rendering it o resonant from ! s .

Kristinn Júlíusson

The third method is not presented in ref. [START_REF] Raimond | Quantum Zeno dynamics of a eld in a cavity[END_REF] but has been used in our experiment and in [8] [START_REF] Signoles | Conned quantum Zeno dynamics of a watched atomic arrow[END_REF]. It consists in applying a continuous tone at frequency ! +;n 0 and amplitude block to block jn = n 0 i when its population probability is zero. Photons from this tone dress the states jg; n 0 i by separating them by 2 block . Once the blocking tone has been turned on, another continuous signal is applied at ! s which drives the harmonic oscillator. Exciting the jg; n 0 i state is however o-resonant and thus leaves the dynamics conned to n < n 0 and n > n 0 . Details are given in chapter 2.

Physical implementation

We adapted the ideas described above to a 3D circuit-QED platform described in details in chapter 5. The 3LS is represented by the lowest three levels of a Transmon qubit with anharmonicity / 2 = ¡260 MHz, and the harmonic oscillator by a high Q mode (TE 120) of a superconducting parallelepiped cavity in aluminium made of two parts. This mode devoted to storing the quantum eld is called storage mode. In order to maximize its internal quality factor Q 0 we use high purity aluminium (>99.99%) and avoid putting obstacles across the supercurrents lines. To achieve the resonance condition ! ef = ! 0 the transmon junction has a SQUID geometry making it frequency-tunable with a magnetic eld. Since magnetic eld cannot penetrate the superconducting aluminium, the SQUID of the transmon is left outside the superconducting cavity. For this end, a part of the transmon chip sticks out of the cavity through a slit milled in a designated thin wall (1mm-thick), with the SQUID located inside a small copper cap (see g. 1.3). The transmon geometry and location on the chip is designed such that it couples to the electric eld inside the cavity. A Helmholtz coil is then used to control the magnetic ux threading the SQUID loop. For transmon readout, we use the third mode of the cavity (TE210) dispersively coupled to the transmon. As depicted in g. 1.3, the resonance frequency ! r of this readout mode depends on the transmon state jg i, jei, or jf i. For readout a microwave pulse at the resonance frequency corresponding to jg i is sent to the cavity. The transmitted amplitude then discriminates jg i from the other states (not jg i). The storage mode requires as long a life time as possible. It also requires a relatively weak coupling of g 0 / 2 = 8 MHz to achieve a low Kerr non-linearity K / 2 = 500 Hz nevertheless strong enough to resolve at least 15 photon number states. On the contrary, the readout mode is strongly coupled to the transmon with g r / 2 = 150 MHz for good readout contrast, and has a low Purcell limited quality factor Q r = 1.2 ¡ 1. 8 10 4 for fast enough readout. These parameters are achieved by placing the cavity slit (and thereby the transmon) close to a node of the storage mode, and to an antinode of the readout mode, given the eld distribution of these modes inside the cavity, analogous to ref [START_REF] Leek | Cavity Quantum Electrodynamics with Separate Photon Storage and Qubit Readout Modes[END_REF]. Following the same principle, we couple very weakly the storage mode to the ports, leaving its quality factor limited by internal losses, while coupling strongly the readout mode. Moreover the coupling is asymmetrical such that the Q-factor of the readout mode is mostly 1), (2). A transmon qubit on a sapphire substrate is inserted into the cavity through a slit in the wall such that it is partially outside the cavity. It is covered with a Cu cap allowing a magnetic eld to penetrate the SQUID. b) A photograph of one half of the cavity with the transmon and Cu cap mounted. c) A lumped element representation of the system. Microwave signals arrive at (1) and leave via (2). Ports a are weakly coupled to the oscillator (internal loss limited) and strongly coupled to the Readout mode (external loss limited) in a asymmetric manner, favoring photons leaving via (2). The frequency-tunable transmon is strongly coupled to the readout mode (g r /2 = 150MHz) but weakly coupled to the oscillator g 0 /2 = 8 MHz. d) Transmission through the readout mode when the transmon has been prepared in jg i(red), jei(green) and jf i(black).

limited by its coupling to the output port. For quantitative determination of the design parameters we use a 3D nite-element microwave simulation software called CST [53] (see chapter 6.2). The simulated design is shown in g. 1.4. We start with an empty cavity consisting of a 3D volume and bounded by a perfect conductor. Ports are then dened as the external boundaries of the coupling pins. To characterize an empty CST simulates the S-parameter matrix in a chosen frequency range or retrieves directly the eigenmode frequencies and Q-factors.

The sapphire chip is added can be added to the structure as well as the transmon in the form of 2D metallic sheets and a port at the location of the SQUID [START_REF] Nigg | Black-Box Superconducting Circuit Quantization[END_REF]. The admittance of the xed part of the coupled system Y (!) = Z ¡1 (!) is then simulated as a 1 port device. Exporting the simulated admittance and adding in parallel the tunable Josephson inductance of the SQUID yields the total admittance Y total (!), and the resonance frequencies ! res of the coupled system are determined by Im[Y total (! res )] = 0 as depicted in g. 1.4e. Varying the added inductance simulates a magnetic eld variation applied to the SQUID and yields the anticrossing of the transmon with the various modes, determining g 0 and g r at the smallest resonance separation as shown in g. 1.4f. The behavior of the total admittance at low frequencies ! ! res is Im[Z total (!)] ' ¡1 ! C which gives and estimate on C and consequently E C . In this manner we tune the shape and location of the transmon to reach the desired parameters. We found that our simulations underestimate C, g 0 and g r by about 20%.

Sample fabrication and sample characterization

Getting the setup together requires development of a few individual elements. First we made and analyzed a 3D-transmon with state of the art lifetimes. That transmon design was then modied to t a cavity in a sample optimized for the QZD.

Fabrication and characterization of a 3D transmon

At the beginning of the PhD period the eld of 3D circuit-QED was quite new and no 3D qubit had been measured in the group. The rst task was therefore to make a 3D transmon, as reported in details in chapter 5. We were particularly interested in the energy relaxation time T 1 and coherence time T 2 because of the improvements reported by early experiments in 3D circuit-QED. Frequency tunability being an advantage for characterizing the transmon decoherence sources, we designed this transmon with a SQUID and put it in a copper cavity transparent to magnetic eld. This transmon has similar dimensions to the ones already reported by Paik et al. [START_REF] Hanhee Paik | Observation of High Coherence in Josephson Junction Qubits Measured in a Three-Dimensional Circuit QED Architecture[END_REF] and is fabricated on a high resistivity Si substrate (>4000cm). Its anharmonicity is /2 = ¡358 MHz and its coupling to the rst cavity mode is g r /2 = 90 MHz.

In our experiment the low quality factor Q7000 of the Cu cavity at ! 0 / 2=7 GHz happens to be the main relaxation channel for the transmon. In the dispersive regime, this relaxation via the cavity mode , known as Purcell relaxation, is given by the Purcell decay rate ¡ Purcell = (g 0 / ge ) 2 , with = ! / Q the cavity decay rate. As depicted in g. 1.5, T 1 increases at low ! ge where an increased detuning ge lowers ¡ Purcell . Relaxation times T 1 > 40s

¡ 2 = ¡ 1 2 + ¡ ';Th + ¡ '; :
The rst term is the contribution of relaxation that sets the upper limit T 2 6 2T 1 , and the two other ones are dephasing rates, ¡ ';Th due to the residual thermal population of photons in the storage mode and ¡ ';

due to the ux noise in the SQUID loop. Both eects induce noise on transmon frequency, the degree of which varies with the transmon frequency in dierent ways. The photon noise aects the cavity frequency pull = g 0 2

(1/ ge ¡ 1/ ef ) per photon in the cavity and has a greater noise contribution at small detunings ge . In addition to the thermal population, the photon number noise also determined by the mode decay rate(i.e. ¡ ';th (; n th ; )). A thermal photon population of 0.22% was measured in the cavity. The amount of ux noise in the SQUID loop translates into frequency noise and scales as |@! ge / @|, hence having no noise contribution at the maximal frequency. The ux noise is characterized by its spectral density and was found to be predominantly 1/f noise with 8 0 / Hz p at 1 Hz.

Another Transmon was made on a sapphire substrate, which is more compatible with high-Q cavities as it has a lower loss tangent than silicon. The parameters were very similar to the ones discussed and well suited for the QZD experiment, as discussed in chapter 5. b) The expected behavior of the excitation frequency transmon with a SQUID(blue line) under a ux bias is tted to the measured excitation frequencies(red crosses). The green line indicates the resonance frequency of the cavity. Inset) T 1 and T 2 at dierent frequencies. The Purcell decay subside as detuning from the cavity increases at the same time as the sensitivity to ux noise increases and lowers T 2 . c) The measured T 1 (red dot) compared with the Purcell limit (blue line). The frequency is maximized at / 0 = ¡0.31 and is lowered, crossing the resonator (green lines), when varying the bias in either direction. b) and c) contain data from two different cooldowns of the same sample.

Kristinn Júlíusson

Building blocks and parameters of the setup

An aluminium cavity designed after a CST simulations was machined by a workshop using a digital mill. The two measured cavities had a storage mode with Q-factor up to Q 0 = 2 10 6 and Q 0 = 1 10 7 , the latter having a slit parallel to the supercurrents of the storage mode. Further description is found in chapter 6. The transmon consists of two Al pads with dimensions of hundreds of microns separated by a few micron wide wire. The rst design, as measured in the Cu cavity, had a pad separation of 50 m. In order to have a SQUID outside the cavity, the separation was drastically increased to 3mm when the new cavity was redesigned (see g. 1.6c).

f 0 =6.43GHz Q = 2 Million

Sample characterization

The sample was cooled down to 30 mK in a setup described in chapter 3.

Pulses are generated with a microwave signal IQ-mixed and shaped with an arbitrary waveform generator to both manipulate and readout the sample. A Helmholtz coil generates a magnetic eld to tune the transmon frequency. Figure 1.7 shows a summary of spectroscopic characterization of the sample: The couplings of the transmon to the storage and readout modes are g 0 /2 = 8 MHz and g r /2 = 150 MHz, respectively, and the anharmonicity is /2 = -260 MHz. The resonance condition ! ef = ! 0 can be reached and it is possible to resolve the photon number splitting up to more than 15 photons. This experiment gives shorter T 1 than for the test transmon in the Cu cavity: 5 s intrinsic T 1 reduced to 1.5-2.5 s by the Purcell eect at the operating point. This is however long enough to go forth with the quantum Zeno experiment. With the long-Kristinn Júlíusson lived storage mode, it enabled us to implement the Zeno blockade and observe the quantum Zeno dynamics (QZD) by Wigner tomography.

Q R = 1 3 k f R = 7.33 GHz f O = 6.43GHz (d) (e) (c) (b) f ef f ge Q  = 2 . 2 M frequency (GHz) / 0 (a) 2 x (g O = 151MHz) 2 x (g R = 7.45MHz) f gf /2 = +12 +10 +8 +6 +4

Wigner tomography, experimental results and analysis

To present the experimental results we rst introduce the Wigner function, which is a common representation of the quantum eld in a harmonic oscillator. Next we explain the protocol for creating and measuring the QZD, and describe the data obtained for three types of evolutions. Lastly, we discuss the experimental imperfections and decoherence of the Zeno dynamics.

QZD analysis with the Wigner tomography

The Wigner function W () is a quasi-probability distribution bijective to the density matrix , and can be represented as W () = Tr[D ^()P ^D ^(¡)], where D ^() = exp(a ^y ¡ a ^) is the coherent displacement operator, P ^is the photon number parity operator, and is the complex-valued amplitude of the eld. The Wigner function is convenient to represent coherent states j i = D ^()j0i, which are the eigenstates of the coherent drive Hamiltonian, as well as their quantum superpositions. Furthermore, using the property that W > 0 for all classical states can be used to identify non-classical states.

There are several ways of experimentally reconstructing the Wigner function in a tomographic manner with the use of a Transmon and the ones relevant are discussed in more detail in chapter 6.

If one can perform the parity measurement hP ^i then the Wigner function can be mapped following the mathematical formulation given above.

In our setup, we cannot perform a global parity measurement but we can Fock state-selectively excite the transmon up to a truncation jn max i, which are used to implement truncated parity measures P ^0 and P ^00 depicted in g. 1.8d, 1.8e and 1.8f. Figure 1.8a shows a reconstructed Wigner function using the truncated parity measure P ^00 . It consists in applying vn max / 2 simultaneous pulses exciting the transmon for any odd state up to n max (in our case n max =10) and thus maps the truncated parity P ^00 = n=odd j+; nihg; nj onto the transmon.

Kristinn Júlíusson

This is however an inecient method limited to a small part of Hilbert space. We therefore switched to a numerical method called maximum likelihood (maxlike) where, instead of measuring W () directly, is rst reconstructed and then used to calculate W (). The method takes as an input a set of measurement observables and their experimentally measured values, and outputs the density matrix that is most likely to give those same values. The Wigner function can then easily be calculated once is known. The freedom of choosing a set of observables alleviates the truncation restriction and we measure the photon number distribution in combination with displacement pulses as depicted in g. 1.8b and 1.8c. Reconstructing with maxlike requires much fewer measurement than mapping the Wigner function pixel by pixel and thus is much more ecient. In comparison, the pixel-by-pixel reconstruction in g. 1.8a took 100 times longer to measure than using maxlike as done in g. 1.8b.

QZD: Conned, accelerated and elongated

Protocol

The quantum Zeno dynamics with continuous tone blockade at n 0 is performed with the pulse sequence described in g. 1.9. Starting in the vacuum state, a rst optional step is to prepare a dierent coherent state j = / 0i. Then, two tones are applied simultaneously to induce the Zeno dynamics: (i) a blockade tone with frequency ! +n0 and amplitude block , and (ii) a drive tone with frequency ! s , amplitude ", and zero phase . To record the evolution, we let it unfold for increasing durations t Z , and measure the observables for the maxlike reconstruction of with a displacement pulse, Fock state selective excitation pulse and readout.

We have applied this protocol to observe three variations of the QZD. Snapshots of the evolutions can be seen in g. 1.10.

Conned Zeno dynamics inside the exclusion circle

The quantum Zeno dynamics of a system conned to the Hilbert space H n<n 0 is observed for n 0 =3 as seen in g. 1.10a and n 0 =4 as described in chapter 7. Starting in the vacuum state the system is driven along the positive real axis, undergoes a phase ip as it hits the blockade, creating a cat-like state before returning to the vacuum state. In an evolution over 2 such periods lasting a total of 26 s the height of the peak has lowered by half and its shape become distorted. The frames taken at 6.5 and 19.5 s show that the cat-like states take greater extremes at negative values of Im[] and demonstrate nonclassical dynamics with negative values of W . The Kerr factor has no visible eect on the evolution and the long lifetime of the cavity and the amount of photon number splitting allows us to comply with the inequality 0 " block j! n ¡ ! n+1 j, where 0 = ! 0 / Q 0 .

Accelerated Zeno dynamics: going through an exclusion disk

The favorable sample parameters allowed us to explore the QZD further in the subspace H n>n 0 , as proposed in [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF]. This requires a coherent state preparation outside the exclusion circle, now set at n 0 = 2, and subsequently applying the blockade and drive. In g. 1.10b the initial state = ¡ 7 p is crashed into the middle of the exclusion circle where it undergoes a phase shift and continues its motion along the real axis from the opposite side of the exclusion circle. This has the appearance of an acceleration in phase space and creates a cat-like state midway through the acceleration with fringes inside the EC. As the state as reemerged and continues its motion a part of it is left inside the EC. p with the bottom part of the exclusion circle causing it to crash tangentially into it, yielding an elongated state with large and small extensions along the circle tangent and radius, respectively.

Sqeezing on exclusion circle

:!start in!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!block!n 0 =!2,!and!drive!coherently ideal Zeno case experiment (a) (b) (c) 0 ! 7 ! " 5 2 i ! " " Im# $ Re# $ Im# $ Re# $ Im# $ Re# $

Squeezing from Zeno dynamics

Decoherence analysis in QZD

In order to understand our experimental results and why they deviate from the ideal simulations seen in g. 1.10 we seek to reproduce the experimental data with numerical simulations, taking into account dephasing and relaxation of the system as well as the non-ideal implementation of the blockade. For that end we use a quantum optics toolbox in Python called QuTiP [25]. An in-depth discussion is found in chapter 7. To simulate the ideal Zeno dynamics of a harmonic oscillator in g. 1.10 we follow the path laid out by Raimond [START_REF] Raimond | Quantum Zeno dynamics of a eld in a cavity[END_REF] with N stroboscopic alternations between a coherent drive U (t/N ) and unitary kicks U K :

(t) = U Z y (t) 0 U Z (t) Kristinn Júlíusson with U Z (t; N ) = [U K U (t/N )] N .
In simulating our experiments, the storage mode is modeled as a harmonic oscillator truncated at 20 -40 levels, coupled to a 3-level system representing the transmon and considered as quasi-harmonic (same creation operator as for a harmonic oscillator despite the anharmonicity). The input parameters ! ef = ! 0 , ! ge and g 0 as well as the decay rates 0 , ¡ 1 , and ¡ 2 are those measured in the experiment. The coupled undriven Hamiltonian is diagonalized to obtain the eigenvalues and eigenstates. Then the photon number resolved qubit frequencies ! n are calculated, both for the symmetric and antisymmetric superpositions j; ni, as well as the Rabi frequencies of each transition as seen in g. 1.11a. The ! n match perfectly the data whereas Rabi frequency are only semi-quantitative due to the crudeness of the harmonic approximation for the transmon.

The conned Zeno dynamics is then simulated during a bit more than two periods: the coherent cavity drive and the Zeno blockade drive at frequencies ! 0 and ! n=n0 , respectively, are added to the total Hamiltonian. The photon number probabilities p(n) are nally plotted in g. 1.11c. The simulated decay parameters T 1 , T 2 and s match those measured in the experiment but the simulated drives are up to 30% stronger. We furthermore use the model to estimate the inuence of the dierent experimental imperfections such as leakage of the Zeno barrier, the eect of T 1 and and more. As explained in chapter 8, we nd that the Zeno blockade tone dephases the Fock states adjacent to the one being blocked. The simulations also show that the Transmon is non-negligibly excited in state j+n 0 i, thereby making the system sensitive to the relatively short T 1 of the transmon. The rates s and ¡ 1 are found to be the main sources of decoherence, both contributing a similar amount. The leakage through the blockade was found to be about 7% per period.

Conclusions and outlook

During this PhD period we have built a 3D circuit-QED setup with parameters optimized for observing the quantum Zeno dynamics (QZD) of the electromagnetic eld in a superconducting microwave cavity. We succeeded in making cavities with quality factors of up to 10 million at ! 0 / 2 s 6.5GHz. Selective Fock state manipula-tion was achieved by partially inserting a tunable transmon qubit in the cavity and leaving its SQUID loop outside, in order to tune its frequency magnetically and match the resonance condition ! 0 = ! e f . The transmon was placed close to the electric eld node of the storage mode, therefore yielding a relatively weak cavity-transmon coupling g 0 /2 = 8 MHz and a low Kerr nonlinearity below 500 Hz. Despite this small g s , the resonance condition yields however a large separation of 11 MHz ( n + 1 p ¡ n p ) between the line used to address Fock states jni and the adjacent line. The occupation probability p(n) of any jni is measured by exciting the transmon at ! +n conditionally to the occupation of jni, and measuring the microwave transmission of another cavity mode with a low Q 15k coupled to the transmon with g r /2 = 150MHz.

With this setup we managed to observe several versions of the QZD under a coherent drive using a continuous blocking tone: We conned the evolution to the Hilbert space H n<n0 by blocking n 0 =3 or 4. By preparing a coherent state outside the exclusion circle in H n>2 and subsequently driving the eld toward it we could observe the acceleration of the movement through the circle. Finally we produced a squeezed-like state by crashing a coherent state into the side of the exclusion circle.

Our experimental setup is a good candidate to continue toward implementing QZD in a stroboscopic manner and demonstrating the phase space tweezers proposed in [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF]. A few improvements could be made: First, the Fock state selectivity could be improved with a larger g 0 giving larger photon number splitting without suering from signicant non-linearities. Second, relaxation during operations could be reduced with Optimal control pulses allowing fast manipulations while minimizing the o-resonant drive of the adjacent transitions. A longer T 1 would help by making possible to use longer, more selective pulses with smaller decay during manipulations as well as readout.

Chapter 2 Theoretical background

In this chapter the theoretical background behind the experiments is presented. First the quantum Zeno eect (QZE) and quantum Zeno dynamics (QZD) 2.1 are introduced on a conceptual level for an arbitrary system with a discrete energy spectrum.

In this thesis we induce and observe the QZD of the electromagnetic eld inside a cavity. This is done by using a superconducting transmon qubit inserted into the cavity. This particular circuit QED system is presented in second section.

The last section describes the implementation of QZD in such a cavity QED system. It illustrates how the QZD can be used to tailor the eld in a cavity using 'phase space tweezers'.

The quantum Zeno Dynamics

We discuss the theory of the quantum Zeno eect and its generalization, the quantum Zeno dynamics, following refs [18][19] [20] .

Let us rst recall some of the fundamental behavior of quantum system in (discrete) Hilbert space H represented by a density matrix . This matrix has eigenvalues i with corresponding eigenvectors j i i forming an orthonormal basis in H and can be written as

= X i i j i ih i j;
(2.1) with the normalization condition i i = 1. When only one i is nonzero the system is in a pure state j i i. Under the action of a Hamiltonian H the system evolves according to the Schrödinger equation

i~d dt = [H ; ] (2.2)
leading to a unitary evolution operator U (t):

(t) = U (t)(0)U y (t): (2.3)
Any measurement on the system is described by an observable operator O ^= j j P ^j;

(2.4)

where f j g are the possible measurement outcomes, and P ^j are the projectors onto the corresponding eigenspaces, satisfying P j = P j y , P j P j y = P j 2 = P j and j P j = 1. The outcome j of a measurement at time t is obtained with probability

j (t) = Tr ¡ (t)P ^j : (2.5)
After receiving a measurement outcome j the system updated density matrix is:

= j = P ^jP ^j j : (2.6)

QZD with projective measurements

The quantum Zeno eect (QZE) arises when frequent enough projective measurements suppress the evolution of the measured eigenvalues, as shown by Misra and Sudarshan [START_REF] Misra | The Zeno's paradox in quantum theory[END_REF]. Let us assume that a time independent Hamiltonian H induces a unitary evolution U (t) = exp(¡i H t), that an observable O ^with [H ^; O ^] = / 0 is repeatedly measured. The rst measurement after a time gives i with the probability

i ( ) = Tr[P i U ( )P i (0) P i U y ( )P i ]; (2.7)
which in general is less than unity and leaves the system in i as given by eq. 2.6. We now interrupt the unitary evolution with a succession of N measurements during the period t. We consider the particular case when all measurements give i , which leads to the following state

i (N ) (t) = V N (t)(0)V N y (t) i (N ) (t) V N (t) [P i U (t/N )P i ] N : (2.8)
The denominator

i (N ) (t) = Tr[V N (t)(0)V N y (t)]; (2.9)
Kristinn Júlíusson which ensures the normalization of the density matrix, represents the probability of measuring i in every instance. By expanding U (t) and taking the limit N ! 1 we get

V N (t) = [P i (1 ¡ iHt/N + O(1/N 2 )P i )] N = P i [1 ¡ iHt/N + O(1/N 2 )] N (2.10) ! N !1 P i exp(¡iP i HP i t) U Z (t);
where

H Z P i HP i ; (2.11)
denes a Zeno Hamiltonian.

When the system is continuously observed, the probability of nding the same eigenvalue i throughout a time period t is

i (t) = lim N !1 i (N ) (t) = Tr[U Z (t) i U Z y (t)] = Tr[ i P i ] = 1: (2.12)
In words, the QZE freezes the evolution of the observable O, such that it always yields the initial outcome i :

Facchi and Pascazio [19] pointed out that, although the QZE freezes i ; it does not prohibit the evolution of i (t) within the subspace H i corresponding to i . Indeed an (uninterrupted) unitary evolution U ^(t) due to H ^can lead to (t) spreading out of H i (if H ^; P ^i = / 0). However, the repeated measurements result in the unitary evolution U Z (t) due to H Z , and conne to H i with no leak of probability into

H i? (H = H i H i? ).
The result in eqs. 2.8, 2.9 and 2.10 is generalized in [20] to multiple Zeno subspaces H i that span the entire Hilbert space H = L i H i .

QZD with unitary kicks

Facchi and Pascazio [20][18] show that the same dynamics as obtained in the last section can be induced by replacing the non-unitary measurements with instantaneous unitary processes (unitary kicks) : When interrupting the evolution under U ^(t) N times with an instantaneous unitary kick U K , the expression of the total unitary becomes

U K (N ) (t) = [U K U (t/N )] N :
(2.13) To derive the evolution of eq. 2.10 let us consider an auxiliary unitary

W N (t) U K yN U K (N ) (t) ¡! N !1 U (t) = exp(¡iH Z t); (2.14)
which N ! 1 limit (derived in ref. [18]) gives rise to a Zeno Hamiltonian

H Z = X j P j HP j ;
(2.15)

with P j the eigenprojectors of U K (U K P j = e ij P j ). From eq. 2.13 and 2.14 the stroboscopic evolution can be expressed as

U K (N ) (t) = [U K U (t/N )] N U K N e ¡iHZt (2.16)
for N ! 1. The Hamiltonian in eq. 2.15 is the generalization of eq. 2.11 and illustrates that the evolution under U K (N ) takes place in each subspace H j = P ^jH independently.

QZD with continuous coupling

In addition to projective measurements and the unitary kicks the QZD can also be induced by coupling the system to an auxiliary one. The coupling Hamiltonian KH C where K is the coupling strength yields the system Hamiltonian (omitting the Hamiltonian of the auxiliary system)

H K = H + KH C :
(2.17)

Kristinn Júlíusson

The resulting unitary operator U ^K(t) = exp(¡iH K t): Similarly to eq. 2.14, an auxiliary unitary gives the Zeno dynamics

exp(iKH C t)U K (t) ¡! K!1 U(t) = exp(¡iH Z t); (2.18)
where

H Z = X j P j HP j (2.19)
is the Zeno Hamiltonian and P n are the eigenprojectors of H C , with the corresponding eigenvalues n and eigensubspaces H n :

H C = X j j P j ; (2.20)
From this one can conclude that the unitary evolution due to H K is

U K exp(¡iKH C t) U (t) = exp ¡ ¡i j ¡ K j P ^j + P ^jHP ^j t (2.21)
which yields the Zeno subspaces.

The seemingly dierent operations presented in the last thee sections indeed give the same dynamics given by eqs. 2.10, 2.16 and 2.21: The separation of Hilbert space into Zeno subspaces with independent unitary dynamics. This separation is essentially a consequence of the destruction of phase coherence between the subspaces.

Deviations from the ideal case occurs when N and K in eq. 2.10, 2.16 and 2.21 are nite, or when the projections or unitary kicks are not instantaneous. This leads to probability leakage between Zeno subspaces, established by the underlying unitary drive U ^, that is not fully blocked by the one of the three methods mentioned above. The characteristics of the leakage depend on the blockade method and will be discussed in the context of cavity QED in section 2.3.

Cavity QED

The experiments in this thesis use cavity QED, implemented in superconducting circuits, to observe the QZD. A transmon qubit is placed inside a 3D cavity resonator, where the electric dipole moment of the transmon couples to the electric eld of the cavity.

Theoretical background

In this section the basics of cavity QED will be introduced in three parts: First the physics of a cavity resonator is explored, and an equivalent circuit model is presented and quantized. Then, based on the quantized circuit, superconducting qubits are derived by introducing the Josephson junction. Starting from the Cooper pair box, the basics of the transmon are described followed by a section on the general TLS. The last part covers the coupling of the harmonic cavity with a TLS in the Jaynes-Cummings model, and an extension to a three level system. This leads to a model well suited for our QZD experiments.

Harmonic oscillator

Here we outline the physics of a cavity resonator by rst describing the electric elds and deriving an equivalent circuit model for each mode. The coupling to two ports is then added to the model and the key features of the impedance, admittance and scattering matrices are presented. Lastly the circuit is quantized followed by an introduction on Fock states and coherent states

Cavity resonator

The resonators used in these experiments are approximately rectangular cuboids carved into a metal block. The electric elds are derived from Maxwell's equations for a box with boundary conditions of a perfect conductor [START_REF] David | Microwave Engineering[END_REF] [13].

Let us consider an empty box with the dimensions (d x ,d y ,d z ) along the (x,y,z) directions as shown in g. 2.2. The electric eld inside the cavity is E ~= (E x ; E y ; E z ) with

E n = E n0 cos(k n n) m= / n sin(k m m) e i!t : (2.22)
Here E n0 are the electric eld amplitudes and the wave vector projections

k n = l n /d n dene the mode indices l n 2 N 0 ( n l n > 2).
The eld oscillates at a frequency ! (lx;ly;lz) = k (lx;ly;lz) c with k (lx;ly;lz

) = k x 2 + k y 2 + k z 2 q ; (2.23)
and c the speed of light. Each mode is thus a standing wave with the mode index l n indicating the number of anti-nodes along coordinate n.

The electric and magnetic elds of each mode can be written as

E ~= k 2 q" ~and B ~=

kq_h Kristinn

Júlíusson with q the generalized mode amplitude having the generalization of a charge, and " ~and h ~are the vectors of the electric and magnetic elds respectively. Integrating the energy density of the elds over the volume gives

E el = " 0 2 Z E 2 dV = q 2 2C and E ma = 1 2 0 Z H 2 dV = L q_ 2 2 ; (2.24)
which denes an eective inductance L and capacitance C [START_REF] Louis | Radiation Laboratory Series[END_REF]:

L (l x ;l y ;l z ) =
Q int = R/Z r : (2.27)
In our experiment one dimension of the cavity is much shorter than the other two, such that the lowest frequencies have the corresponding mode index zero. Taking d a; b means l z = 0 for the lowest modes, which have only one electric eld component E z .

Coupling to two ports

To couple energy in and out of the cavity, two antenna protrude into the cavity through a hole in the wall as shown in g. 2.2a. The coupling strength depends on the electric eld strength at location of the hole and the length of the antenna.

In the circuit model in g. 2.2b, the ports are represented by two capacitors C 1 and C 2 coupled to two resistors representing the transmission lines to which they are attached.

LCR-circuit representation and S-parameters

We now use the circuit model to characterize a mode of the cavity, including its coupling to the to two ports, and see how these characteristics can be measured in practice. The circuit in g. 2.2 b can be transformed into the LCR-model shown in g. 2.2 c using the Norton-Thevenin theorem and approximating the series combination of C i and Z 0;i (i = 1; 2) to a parallel one when ! ! 0 .

Z 0,1 Z 0,2 (a) (b) R C L P 1 P 2 C 2 C 1 Z 0,1 Z 0,2 (c)
This results in modied resistance R 0 and C 0 . R 0 is interpreted as the real part of the admittance of the total circuit Re(Y ) = Re(Z ¡1 ):

1

R 0 ' 1 R + 1 R 1 + 1 R 2 ; (2.28)
where for each port i

R i Z 0;i = 1 (Z 0;i ! 0 C i ) 2 + 1;
(2.29)

and C 0 is given by

C 0 = C + X i=1;2 C i (Z 0;1 ! 0 C i ) 2 + 1 : (2.30)
This gives rise to a modied resonance frequency ! 0 0 = 1/ LC 0 p and quality factor Q tot = R 0 /Z r which can be decomposed into

Q tot ¡1 = Q int ¡1 + Q 1 ¡1 + Q 2 ¡1 : (2.31)
For each component of Q tot a coupling rate is dened as = ! 0 0 / Q. These rates add up: tot = int + 1 + 2 . Both 1 and 2 represent the energy transfer rate through the ports while int is the energy loss rate inside the resonator.

Scattering parameters

The lines coupling to the two cavity ports are coaxial cables with waves traveling through them. At the port the voltage and currents are

V i = V i + + V i ¡ (2.32) I i = I i + ¡ I i ¡ ;
(2.33)

with V i + (I i + ) and V i ¡ (I i ¡
) represent the voltage(current) of the incident and reected waves o port i respectively. These parameters are used to dene three 2x2 matrices: The impedance matrix [Z];

Z ij = V i I j I i= / j =0
(port i open)

the admittance matrix [Y];

Y ij = I i V j V i= / j =0
(port i shorted)

and the scattering matrix [S];

S ij = V i ¡ V j + V i= / j + =0
(all ports terminated with a load):

The matrices [Z] and [Y] characterize the internal microwave network represented by the blue and green elements in g. 2.2 b, while [S] characterizes the full circuit in the gure. The parameters of [S] are the reection (i = j) and transmission (i = / j) coecients of the device, and their value around a resonance frequency ! 0 0 of a mode are

Theoretical background 2 1 tot i Q Q        n n ( ) Arg S 2 n n | | S 1/ 2 2 1 2 4 / tot Q QQ 0 / tot f Q  2  2 n m | | S n m ( ) Arg S
S nn = ¡ 1 ¡ 2 Qtot Qn + i2Q tot ! ! 0 0 ¡ 1 1 + iQ tot ! ! 0 0 ¡ 1 ; (2.34) S nm;n= / m = 2 Qtot QnQm p 1 + i2Q tot ! ! 0 0 ¡ 1 ; (2.35) arg(S nm;n= / m ) = ¡atan 2 Q tot ! ! 0 0 ¡ 1 ; (2.36)
and are plotted in g. 2.3. The energy transmission jS ij j 2 takes a Lorentzian shape in frequency and the full width at half maximum (FWHM) of the peak is the decay rate tot =! 0 / Q tot . The individual Q i (and i ) components can then be obtained from eqs. 2.25, 2.26, 2.27.

Kristinn Júlíusson

The cavity used in the experiment is not exactly a rectangular cuboid. The circuit model can however still be used with lumped element parameters extracted from numerical simulations, using the following method:

We choose two points P 1 and P 2 on one of the ports as shown in g. 2.2 a, one on the inner conductor and the other on the wall of the port (ground). Figure 2.2 b,c show the location of these point in the circuit model. The simulator calculates the admittance

Y P 2 P 1 (!) = Z P1P1 ¡1 (!) = 1 R 0 + i!C 0 ¡ i !L (2.37)
as seen from these two points. The individual elements and ! 0 0 can be obtained using the following identities

Im[Y (! 0 0 )] = 0; C 0 = Im[Y 0 (! 0 0 )] 2 ; R 0 = 1 Re[Y (! 0 0 )] : (2.38)

Quantizing the LC-circuit

To describe a resonator quantum mechanically we consider the RCL circuit presented in the last section, but disregard its resistance and coupling to the two ports. Analogous to the energy of a classical circuit given in eq. 2.19 the quantum Hamiltonian is

H ^= Q ^2 2C + ^2 2L (2.39)
where the charge and ux operators Q ^and ^are canonical conjugate variables [Q ^; ^] = i~. By dening the bosonic creation and annihilation operators

a y ^= 1 2~Z r r ( ^+ iZ r Q ^) (2.40) and a ^= 1 2~Z r r ( ^¡ iZ r Q ^);
(2.41)

the Hamiltonian can be written in second quantization as

H ^= ~!0 a ^ya ^+ 1 2
:
(2.42)

Theoretical background

Here ~!0 is the energy 2.2 per photon, n ^= a ^ya ^is the photon number operator, and the 1/2 represent the vacuum uctuation which will from here on be omitted.

Coherent states and Fock states

This section covers two types of quantum states that are central to the physics in this thesis: The eigenstates of the photon number operator n ^, called photon number states or Fock states, and the states generated by a classical sinusoidal source called coherent states, which are also the eigenstates of the annihilation operator a Fock states The eigenvalue spectrum of n ^is non-degenerate and made of nonnegative integers n. The associated Fock states are represented by [ni such that n ^jni=njni and they form an innite orthonormal basis in Hilbert space. In light of eq. 2.31, the Fock states are also the eigenstates of the harmonic oscillator Hamiltonian where the eigenvalues are

E n = hnjH ^jni = n~! 0 (2.43)
and consist of n energy quanta of ~!0 called photons, hence the term photon number operator for n ^. The photon number spectrum is often represented by a ladder as shown in g. 2.4. As these operators add or remove (a ^y or a ^) one elementary excitation, i.e. a ^yjni = n + 1 p jn + 1i and a ^jni = n p jn ¡ 1i;

(2.44) their application can be seen as going a step up or down the ladder, respectively. This intuitively describes the creation of a Fock state jni from the vacuum as climbing up the ladder in n steps, which is formally expressed using eq 2.30

jni = a ^yn n! p j0i:

(2.45)

Coherent states

Coherent states arise from the applying a coherent drive, described in a frame rotating at the drive frequency, by the Hamiltonian [

D ^() ¡1 = D ^() y = D ^(¡) D ^( 1 )D ^( 2 ) = D ^( 1 + 2 )e (1 2 ¡ 1 2 )/2 D ^(0) = 1:
(2.49) The representation of a coherent state in the Fock state basis is

If
j i = X n c n jni = e ¡j j 2 /2 X n n n! p jni (2.53)
which reveals a Poissonian photon number probability distribution P (n) = jc n j 2 = e ¡j j 2 j j 2n n! ;

(2.54) depicted in g. 2.5b.

Superconducting qubit

Superconducting qubits are lumped element circuits composed of inductors, capacitors and elements called Josephson junctions (JJ) which can be regarded as non-linear inductors. The non-linear character of the JJ leads to non equidistant energy level of the circuit such that two levels can be used as a qubit.

The dierent types of superconducting qubits vary in the number and conguration of JJs as well as the relative energies in the dierent components of the circuit. The various qubit types (phase qubit, ux qubit, uxonium, Cooper pair box etc) are discussed in ref [15]. The qubit used in this thesis is the transmon which is a derived from the Cooper pair box (CPB).

Kristinn Júlíusson

The Josephson junction

A Josephson junction consists of two superconductors separated by an insulating barrier. The charge carriers in the superconductors on either side are described by a macroscopic wave functions with well dened phases ' 1 and ' 2 as shown in g. 2.6a. Josephson predicted that the phase = ' 1 ¡ ' 2 would result in Cooper pairs tunneling through the barrier, giving rise to a supercurrent and a voltage [26]

I = I c sin() (2.55) V = 0 @ @t : (2.56)
Here the critical current I c is the maximal current the junction can support and 0 = ~/2e is the ux quantum. From eq 2.41 and 2.42 one can derive an inductance

L J () = 0 I c cos() L J 0 1 + 2 2 + O( 4 ) (2.57)
with the Josephson inductance L J0 and a series exposing a non-linear behavior, which is a key attribute of the JJ circuits. The energy of the junction is

S S I     = (a) (b)  ext (c) Vg Cg EJ = E J E C E J ( ext )
E = E J (1 ¡ cos()) (2.58)
where E J = I c 0 is the Josephson energy. Taking into account the energy from the capacitance between the two superconductors C J , the quantized Hamiltonian of the junction is

H ^JJ = E C + E J (1 ¡ cos( ^)) (2.

59)

Theoretical background

where the E C = Q ^2/(2C J ) is the same as in eq(2.28) and the second is the Josephson Hamiltonian with quantum phase ^.

DC-SQUID

A JJ with a tunable E J can be obtained by placing two JJ in parallel, which forms a loop known as a superconducting quantum interference device (SQUID) [14], and is shown in g. 2.6c. Assuming perfect symmetry between the junction I c = I c1 = I c2 , the phase difference across each junction are 1 and 2 , and are equal in the absence of external ux. However, if a magnetic ux ext penetrates the SQUID it induces a dierence between the two phases

2 ¡ 1 = ext / ' 0 [mod 2]
(2.60) with 0 = 2 ' 0 = h / 2e the quantum of magnetic ux. The critical current of the SQUID now depends on ext :

I c;SQ ( ext ) = 2I c cos ext 0 :
As a result, both the Josephson inductance L J0 and Josephson energy E J become tunable with ext :

L SQ0 ( ext ) = 0 I c ( ext )
(2.61) By tuning the gate voltage V g one can control the number of Cooper pairs in the island. V g can be expressed as a reduced charge number

E SQ ( ext ) = 2 0 I c cos ext 0 : (2.
n g = V g C g e 2 :
(2.64)

The conjugate operator to n ^CP is the phase operator of the JJ ând the Hamiltonian of the CPB is

H ^CPB = 4E C (n ^CP ¡ n g ) 2 ¡ E J cos( ^):
(2.65)

H ^CPB can be written in the phase basis as

H ^CPB = 4E C 1 i @ @ ¡ n g 2 ¡ E J cos( ^): (2.66)
This Hamiltonian can be solved exactly and its eigenenergies are [12]

E k>0 = E C M A n + 1 ¡ (k + 1)[mod 2] + 2n g (¡1) k ; ¡ E J 2E C ; (2.67)
where M A is the characteristic value of the even Mathieu function.

The lowest energy levels of the CPB as a function of n g are shown in g. 2.8 for dierent ratios E J /E C . They form parabolas that anticross and yield energy bands bands E k . Note that at the degeneracy point E k (n g = 1 / 2) the CPB is less sensitive to noise on n g as @(E 1 ¡ E 0 )/@n g = 0. Operating there was a key to the long coherence time of the Quantronium CPB [START_REF] Vion | Manipulating the Quantum State of an Electrical Circuit[END_REF]. The dispersion of band k is

k = jE k (n g = 0) ¡ E k (n g = 1/2)j:
(2.68) and decreases with E J / E C . By increasing the value of E J / E C above 50 the CPB enters the phase regime and is called a transmon. The lowest energy levels become almost completely at and insensitive to n g while some sensitivity remains in the higher levels. In return the separation between the levels becomes increasingly similar.

Transmon

To reach the transmon regime a shunt capacitor C S is placed in parallel with the JJ as shown in g. 2.9 a, which is equivalent to g. 2.8a with C g being replaced by C g + C S . Usually the electrodes on either side of the JJ in a transmon are of a similar size and not grounded, giving resemblance to a dipole in an external eld E, as shown in g. 2.9 b.

In the transmon regime approximate values of the energies are [28]

E k ' ¡E J + 8E C E J p k + 1 2 ¡ E C 12 (6k 2 + 6k + 3): (2.69)
The transition energies, dened as

E nm = E m ¡ E n , give E ge = 8E C E J p ¡ E C : (2.70)
The price to pay for decreasing the noise sensitivity with a shunt capacitor is a lower anharmoniciy E 12 ¡ E 01 ' ¡E C : The lowest levels of the transmon can therefore be seen approximately as a weakly anhamonic ladder, with the steps decreasing by at each step.

The charge dispersion dened in eq. 2.57 scales as which allows for a signicant increase of E J /E C compare to the CPB before suering from crowding of the transition frequencies E nm /~.

n / exp ¡ ¡ 2E J /E C p (2.71) (a) (b) E J V g 2C g 2C g C S R +Q -Q JJ E

Transmon as a two level system

As any multi level system, the Hamiltonian of the transmon can be written as

H ^Tr = X l ~!l jlihlj (2.72)
where the levels, starting from the ground state, are denoted as jg i; jei; jf i; jhi::: A transmon is often regarded as a TLS (qubit) by excluding all but the jg i and jei levels. Moreover, for a multilevel system the two level formalism can be applied to the transition between any two levels. It is therefore useful to introduce some of the notation developed for qubits.

Bloch sphere representation

The Hilbert space of a qubit is two dimensional and its eigenstates can be represented as vectors

jg i = 0 1
and jei = 1 0

; (2.73)
and the Hamiltonian as

H ^qb = ~!ge 2 ^z; (2.74)
where ! ge is the excitation frequency.

The general representation of a qubit state is a 2x2 density matrix expressed as with ~= ( ^x; ^y; ^z) a vector of the Pauli matrices

= 1 2 (1 + a ~ ~); ( 2
^x = 0 1 1 0 ^y = 0 ¡i i 0 ^z = 1 0 0 ¡1 : (2.76)
The Bloch vector a ~allows for a visual representation of the qubit state on the Bloch sphere shown in g 2.10 a. A pure state jqi = cos 2

jg i + e i' sin 2 jei (2.77)
satises |a ~|=1 and can be represented as a point on the surface of the Bloch sphere, dened by its polar and azimuthal angles and '.

A mixed state is a point inside the Bloch sphere (|a ~|<1 in eq, 2.75) and is reduced to the origin |a ~|=0 for a maximally mixed state.

Coherent driving

The Bloch vector rotates under a coherent drive at a frequency ! dr

H ^dr = ~Adr 2 cos(! dr t + ' dr ) ^x (2.78)
with A dr the amplitude and ' dr the phase of the drive. The joint qubitdrive Hamiltonian in the rotating wave approximation (RWA):

H ^qb+dr RWA = 2 [ ^z + (cos(' dr ) ^x + sin(' dr ) ^y)];
(2.79)

with the detuning = ! ge ¡ ! dr and / A dr the Rabi frequency. As depicted in g 2.10b, the Bloch vector is driven around an axis d ~dened by the phase of the drive ' dr and the ratio of the detuning and the Rabi frequency . In the resonant case = 0 the rotation axis d ~is in the equatorial plane leading to a rotation along a meridian at the Rabi frequency . In the o-resonant case = / 0 the Bloch vector rotates at a modied Rabi frequency 0 = 2 + 2 p and furthermore, the phase ' between jg i and jei oscillates at a frequency .

Relaxation and dephasing

Under free evolution the qubit interaction with its environment gives a relaxation rate ¡ 1 and free induction decay rate ¡ 2 . These two eects are calculated by separating the Hamiltonian into its longditudinal and transverse parts H ^qb = h ~qb ~= (h qb;x x + h qb;y y + h qb;z z ). The sensitivity of the Hamiltonian to a noise source in an environment variable is then treated as a perturbation H ^qb

() = ¡ ~/ 2 D ^ , where D ^ 1 / ~¡@h ~qb / @
. Detailed discussions on relaxation and decay can be found in ref. [24].

The transversal part D ^;? (?=x; y) represents noise involving exchange of energy with the environment. Since ~!ge k B T this is dominantly an emission of energy from the qubit at a rate ¡ rel , but a small thermal excitation rate ¡ ex can still be present summing up to G 1 =G rel +G ex . The individual rates follow from Fermi's golden rule:

¡ rel = 2 D ^;? 2 S (! ge ); (2.80) ¡ ex = 2 D ^;? 2 S (¡! ge ); (2.81)
with S (! ge ) the noise spectral density of around ! ge . Assuming a weak coupling to a short-correlated and cold (¡ rel ¡ ex ) environment, the qubit excitation probability p e decays as p e (t) = p e (0)e ¡¡1t :

(2.82)

The free induction decay rate is composed of two parts; a contribution from relaxation and one from dephasing:

¡ 2 = ¡ 1 2 + ¡ ': (2.83) Theoretical background
The dephasing rate ¡ ' is due to the low frequency uctuations of the transition frequency ! ge (), which depend on the noise spectral density S of and on the sensitivity coecient D ^;z .

In the case of a at noise spectrum

¡ ';W = S !ge (! = 0) = D ^;z 2 S (! = 0): (2.84)
This leads to an exponential decay exp(¡¡ ';W t) of the o diagonal elements of the density matrix.

In the case of 1/f noise with spectral density S (!) = A 1/f /j!j, the decay takes a Gaussian form exp(¡¡ ';1/f

2 t 2 ) with ¡ ';1/f 2 = A 1/f D ^;z 2 ln 1 ! ir t + O(1)
(2.85) depending weakly on t and ! ir a protocol dependent cut-o frequency.

Cavity and circuit QED

Historically cavity QED is the physics of cavities made of two highly reective mirrors with an atom trapped between them as depicted in g. 2.11 a. This in a nutshell is the physics of a harmonic oscillator coupled to a spin. The same situation can be created with superconducting circuits, and thus circuit-QED can be seen as a subbranch of cavity-QED.

In this section the canonical example of cavity-QED, a harmonic oscillator coupled to a spin-1/2, will be discussed rst in the form of the Jaynes-Cummings Hamiltonian. We specically look at the two cases; a spin resonant with the cavity, and a spin far o-resonant from it. Thereafter the eects of additional levels, as present in the transmon, modify the system. The particular case of where the jei ! jf i transition is resonant with the harmonic oscillator ! ef = ! 0 is explicitly considered.

The rst section follows an outline presented in detail in ref. [23] and the second one is based on ref. [28].

Jaynes-Cummings model

The Jaynes-Cummings Hamiltonian is composed of three terms representing the harmonic oscillator H ^HO , the spin-1/2 (qubit) H ^qb and their interaction H ^I. H ^HO and H ^qb are expressed in eq. 2.42 and eq. 2.65 while H I will be introduced here.

Kristinn Júlíusson

Harmonic oscillator transmon qubit

   ge  ef g  C c (b) (a) C' C s +C g L I C Figure 2
.11. a) Schematic of a cavity QED system with a cavity coupled to a TLS. On resonance, the energy oscillates between the two systems. b) Equivalent circuit representation of a cavity mode coupled to a transmon with a coupling strength g 0 .

The interaction arises from the spin dipole d ^coupling to the electric eld E ^HO of the cavity, and writes in the frame rotating at the frequency of the eld

H I = ¡d ^ E ^HO = ¡[d ^¡ + d ^+] i[ a ^+ a ^y]: (2.86)
Expanding the product gives four terms, two of which are proportional to ^¡a ^and ^+a ^y, and play a minor role in the dynamics when the harmonic oscillator frequency ! 0 and the spin frequency ! qb are close.

In the RWA these terms are neglected, yielding

H I = ¡i~g 0 [a ^ ^+ + a ^y ^¡]; (2.87)
where 2g 0 = d / ~is the vacuum Rabi frequency, the frequency at which the two systems exchange a photon when resonant. In terms of the circuit in g 2.11 b, the coupling g 0 = 2eV rms

0 hg jn ^CP jei with = C C / (C C + C s + C g ) and V rms 0 = ~!0 /2C 0 p
being the zero-point voltage uctuations of the mode [28] (see also sections 2.2.1 and 2.2.2).

Considering for a moment the uncoupled system the eigenenergies are ~(n! 0 ! ge /2) and the eigenstates are noted jg; ni and je; ni. In the case of a small detuning ge = ! ge ¡ ! 0 the states form doublets of je; ni and jg; n + 1i, each one having n + 1 elementary excitations. By including the coupling H I again the Hamiltonian restricted to the doublet n is

H ^n = ~!0 n1 + 2 [ g e ^z + sg n ^y];
(2.88)

Theoretical background with g n = g 0 1 + n p : By diagonalizing H ^n one gets the eigenenergies

E n = ~!0 n 2 ge 2 + 4g n 2 q (2.89)
and the corresponding eingenstates

j+; ni = cos( n /2)je; ni + i sin( n /2)jg; n + 1i j¡; ni = sin( n /2)je; ni ¡ i sin( n /2)jg; n + 1i (2.90)
with the mixing angle n = arctan(2 g n / ge ). These 'dressed' eigenstates can thus according to eq. 2.61 be represented on a Bloch sphere with ' = /2. Two regimes will now be discussed: the resonant regime with ge = 0 and the dispersive one with j ge j g 0 :

Resonant regime

In the uncoupled resonant regime the eigenstates are given by eq. 2.90 with n = / 2, and the eigenenergies are degenerate. The coupling lifts that degeneracy according to eq. 2.72, separating the levels by 2~g n .

In the undressed basis the evolution of the quantum state is as follows: Starting from j (0)i = je; ni and switching on the coupling, the evolution under the Hamiltonian 2.71 is j (t)i = cos(g n t)je; ni + sin(g n t)jg; n + 1i

(2.91)

and corresponds to an oscillation between jg; n + 1i and je; ni at frequency 2g n .

Dispersive regime

In the dispersive regime j ge j g 0 the energies are only slightly shifted from those of an uncoupled system and we keep the same names for the eigenstates. Mathematically, a unitary transformation that eliminated the qubit-cavity interaction to lowest order followed by an expansion in 0 = g 0 / ge yields the eective dispersive Hamiltonian

[6] 2.3 H ^e = ~h ! 0 a ^ya ^+ ! ge 2 + 0 a ^ya ^ ^z i (2.92) = ~h (! 0 + 0 ^z)a ^ya ^+ ! ge 2 ^z i : (2.93)
Grouping terms by ^z or a ^ya ^in eqs. 2.92 and 2.93 emphasizes that one subsystem can probe the other: The second term in eq. 2.92 shows that for each photon in the cavity the qubit frequency is shifted by 2 0 , thus allowing one to probe spectroscopically the probability distribution of the Fock states p(n) by mapping it onto the qubit excitation. The rst term in eq. 2.93 shows that the resonance frequency of the harmonic oscillator is shifted by +(¡) 0 when the qubit is in jei (jg i). A probe signal at ! 0 0 reveals thus the state of the qubit.

An important point is that the coupling of the qubit to the harmonic oscillator with a nite Q tot provides a new relaxation channel for the qubit at a rate [28]

Purcell = tot g 0 2 ge 2 : (2.94)
This is know as the Purcell eect.

Circuit QED for a transmon

Most of the results derived in the last section for a TLS are also valid when higher levels are included. This section cover the essential dierences (see [28] for details).

The Hamiltonian is now a generalization of the Jaynes-Cummings Hamiltonian:

H ^Tr = X k ~!k jkihkj + ~!0 a ^ya ^+ ~X l;m g l m jlihmj(a ^+ a ^y): (2.95)
The coupling energies g lm are negligible except for adjacent levels (l = m 1). In that case

g k+1k / hk + 1jn ^CP jki k + 1 2 r E J 8E C 1/4 ; (2.96)
and transmon couplings thus scales as

g k+1k k + 1 p g 0 : (2.97)
In the dispersive regime g 0 j ge j; j e f j the eective Hamiltonian in eq. 2.93 is still valid but with where

ge = g 0 2 ge and ef = ¡ 2 p g 0 2 ef (2.99)
represent the shift of the uncoupled eigenenergies due to the jg i ! jei and jei ! jf i transitions. This means that the shifts are no longer symmetric with respect to the uncoupled levels as shown in g. 2.12.

A non-linearity (named Kerr non-linearity) of the harmonic oscillator emerges when the generalized Jaynes-Cummings Hamiltonian is expanded in the dispersive limit to higher order than done in eq. 2.93. This introduces an additional term

H ^Kerr = K 2 a ^ya ^ya ^a ^(2.100)
to the oscillator Hamiltonian eq. 2.42, with (ref [START_REF] Nigg | Black-Box Superconducting Circuit Quantization[END_REF] and eq. 2.98)

K = 1 0 2 2 = 4 g 0 2 ge 2 + ge ! 2 ¡! ge 4 g 0 ge 4 : (2.101)
Using the addressability of individual Fock states has been a key element in experiments on manipulating the eld of a 3D cavity dispersively coupled to a transmon [27][48]. This requires a large coupling g 0 and a moderate detuning ge for adequate separation 2 0 between adjacent Fock state dependent excitation frequencies. This large separability comes however with a price of an increased Kerr-nonlinearity eq. 2.101. This causes dephasing between the Fock states jni and thus limits the lifetime of coherent states.

Kristinn Júlíusson

e-f/cavity resonance condition

In order to reduce the Kerr non-linearity while still having the Fock state addressability we device a new scheme depicted in g 2. 13. The general idea is to have ! ef resonant with ! 0 to maximize the separation between Fock state dependent transmon frequencies for a xed g 0 . Conversely a g 0 smaller than in the dispersive case can be used reach the same order of magnitude for the separation. This reduced g 0 together with the resonant condition yields a much smaller non-linearity K as shown below. 13. An 'ef-resonant' cavity-transmon system. a) The harmonic oscillator is coupled to a frequency tunable transmon. b) The levels of the uncoupled system laid out as three harmonic oscillators corresponding to jg i; jei and jf i, showing that the levels je; n + 1i and jf ; ni degenerate. When the coupling is included the the degeneracy is lifted by 2n p g 0 . The levels jg; ni are o-resonant by and thus almost unaected. c) The eigenstates jg; ni form a quasi harmonic ladder which can be excited to j; ni at Fock state dependent frequencies ! n; .

O O ef E ! 0 n 1 n 2 n … g e f g O ! " # g O e E (a) (c) 0 g 1 g 2 g … 0 e

Theoretical background

An exact calculation of the energy levels is found in appendix A but an intuitive picture of the system can be obtained by considering jei and jf i as a TLS resonantly coupled to the cavity mode with a coupling strength 2 p g 0 . As discussed above, this forms dressed states

j; ni = je; n + 1i jf ; ni 2 p ; (2.102)
with eigenenergies (n + 1)! 0 2n p g 0 . As a result the excitation frequency

! ;n = ! g e 2n p g 0 (2.103)
of the transition +n jg; ni ! j; ni becomes Fock-state dependent with a separation

;n = ! ;n+1 ¡ ! ;n 2 p g 0 ( n + 1 p ¡ n p ) (2.104)
that decreases with n. We thus refer to the two non-linear ladders j+; ni and j¡; ni shown in g. 2.13.

The inherited Kerr non-linearity expressed in eq 2.100 is not valid here as it assumes g 0 ef , but direct diagonalization gives

K ef = 2 g 0 4 : (2.105)
It is interesting to compare the separation of the photon number peaks given by eq. 2.103 and the Kerr factor in eq. 2.105 to the separation 2 0 2(g 0 / ge ) 2 and Kerr factor eq. 2.101 in far dispersive regime ge . The separation of the Fock state dependent transmon excitation frequencies can be made of the same order in both cases, i.e. 2 0 ;n for small n, by choosing a coupling g 0 0 much smaller than g 0 . The non-linearity is strongly reduced in the ef-resonant case by a factor K / K ef = ( g e / g 0 ) 4 /32 with ge g 0 . As shown in chapter 6 we reach a value of K e f = ¡350Hz in our experiment.

The system resulting from the ! 0 = ! e f resonance is thus a quasiharmonic oscillator with individual Fock state addressability. The next section illustrates how the QZD can be implemented in such a system.

Kristinn Júlíusson

QZD in cavity QED

In ref. [START_REF] Raimond | Quantum Zeno dynamics of a eld in a cavity[END_REF] and [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF] J.M Raimond et.al discuss theoretically the QZD of a coherently driven harmonic oscillator in the context of cavity QED. They choose projectors fP i g i = fP s ; P s? g, where P s = jsihsj is the projector onto the Fock state jn = si and P s;? = 1 ¡ jsihsj, to create the Zeno subspaces H n=s and H n= / s (H = H n=s H n= / s ). The Zeno subspace H n= / s is a harmonic oscillator with the level jn = si removed such that a transition between H <s and H >s is suppressed. This separation of H <s and H >s can be depicted as an exclusion circle (EC) in phase space as shown in g. 2.15. Note that though H n= / s = H <s H >s is a single Zeno subspace, individually H <s and H >s are not.

In section 2.1 three ways of inducing the QZD were discussed: Projective measurement, unitary kicks and continuous coupling to an external degree of freedom.

To implement the rst two in cavity QED, as proposed by Raimond et.al , a small displacement pulse D() alternates with proper Rabi rotations on the jg; si ! j+; si transition as shown in g 2.14. A rotation followed by a readout of the transmon implements the projective measurement, where as a 2 Rabi rotation implements a unitary kick reversing the phase of jsi. For an arbitrary Rabi angle the kick becomes non-unitary. Raimond et.al show however that for a non-zero and a large the evolution is close to being QZD. This protocol can be regarded of a stroboscopic version of continuous coupling to an external degree of freedom presented in section 2.1.3.

Experimental demonstrations of QZD so far (including this thesis) implement it by coupling continuously to an external degree of freedom [START_REF] Signoles | Conned quantum Zeno dynamics of a watched atomic arrow[END_REF] [8]. In our scheme this means driving the jg; si ! j+; si transition continuously at a rate implementing the coupling strength K from eq. 2.17. From a dierent perspective, this way of blocking can also be described by the Autler-Townes eect.

N ! "#,s R ! "#,s R ! "#,s R … Step

Theoretical background

This section covers three instances of QZD implemented with unitary kicks and a section on the evolution when using non-unitary pulses as simulated in ref. [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF] which illustrates the robustness of the process. A particularly interesting example of using the QZD to manipulate superposed coherent states, the phases space 'tweezers' [START_REF] Raimond | Quantum Zeno dynamics of a eld in a cavity[END_REF], is then summarized. Finally we illustrate how the Autler Towns eect leads to QZD.

QZD in cavity QED using unitary kicks

As discussed in section 2.1.2, the QZD can be induced by interrupting underlying unitary evolution U ^(t) N times with a unitary kick U ^K (eq. 2.11). We consider U ^(t) to be a coherent drive at frequency ! 0 , as dened in eq. 2.45 and when interrupted N times, U ^(t / N ) becomes the displacement operator D ^() where = ¡it/(N~).

The unitary kick U ^K is implemented as a 2 Rabi rotation on the jg; si ! j+; si transition such that

U ^K = 1 ¡ 2jsihsj:
(2.106)

The steps of the unitary evolution operator are thus U ^KD ^(). Three examples of evolutions in H n= / s were simulated in ref. [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF] for s=6 (EC of radius 6 p ), and N=49 and = 0.1 (int(2 6 p / )=N. Figure 2.15 shows snapshots of the evolutions using Wigner's representation (see chapter 4).

The top row, g 2.15 a, shows the connement inside the EC: Starting in j0i the amplitude increases along the positive real axis until it reaches 2 when it collides with the EC at N = 15. This causes a rapid phase shift, giving an amplitude ¡2 at N = 35. Midway through the shift, at N = 25, a coherent superposition state of the two phases is created. At N = 45 the state has almost returned to the vacuum.

In g 2.15 b and c the initial states are j = ¡5i and = ¡4 + i 6 p , respectively, following otherwise the same evolution. In the g 2.15 b the eld collides with the EC at N = 20 and undergoes a phase shift, creating a superposition state at N = 25, and emerging as a coherent state on the right side of the EC. This has the appearance of the state accelerating its movement in phase space over the EC.

QZD in cavity QED using non-unitary kicks

In section 2.1 the QZD is discussed in the limit N !1 and K ! 1. This is unattainable in practice leading to imperfect QZD. These imperfections are studied in ref. [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF] for the case of the connement by an EC at s = 6 described in the previous section, but with an arbitrary Rabi angle 0 6 6 2 and variable displacement per step, keeping N = 2 6 p : When = / 2 the rotation is no longer a unitary in H. However, by considering the total cavity-transmon Hilbert space H tot = H H tr , the kick U ^K = e ¡i/2 P ^+ + e i/2 P ^¡ + P ^?

(2.107) is unitary with

P ^ = ju ihu j; ju i = jg; si j+; si 2 p ;
(2.108) and P ^? = 1 ¡ P ^+ ¡ P ^¡. In order to study the imperfections of the QZD when the displacement per step are not innitely small and the kicks are limited in size Raimond et.al. simulate the evolution for 0.05 6 6 1 and 0 6 6 2. The 'transmission' T through the EC is dened as [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF]:

T = w >s 2 w <s 2 + w >s 2 ; (2.109)
where w >s 2 and w <s 2 represent the weight of the wave function outside and inside the EC. T is shown in g. 2.17. The graph illustrates that T is low for a moderate values of and , and that the QZD is a robust process.

Phase space tweezers

In ref. [START_REF] Raimond | Quantum Zeno dynamics of a eld in a cavity[END_REF] and [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF] J.M Raimond et.al also propose a clever and versatile way to use the QZD for manipulating coherent states in phase space, named the phase space 'tweezers', which is worth summarizing here: This enables the displacement of individual components of superposed coherent states

j i = c 1 j 1 i + c 2 j 2 i
(2.110)

.17. Method to manipulate a superposition of coherent states with the phase space 'tweezers'.

provided that the overlap is small (j 1 ¡ 2 j 1). As shown in g. 2.17, in order to displace j 2 i but not j 1 i one rst applies D ^(¡ 1 ) creating

D ^(¡ 1 )j i = c 1 j0i + c 2 j 2 ¡ 1 i: (2.111)
Placing the EC now on s = 1, e.g. with a unitary kick, and applying a coherent drive displaces the 'free' component j 1 ¡ 2 i while j0i is kept in place. Removing the EC and applying a displacement D ^( 1 ) restores j0i to j 1 i.

Repeating this sequence allows for almost arbitrary tailoring of coherent states in phase space.

QZD in cavity QED via continuous coupling

In our cavity QED experiment the QZD is implemented with continuous coupling as presented in section 2.1.3. This done by driving continuously the jg; si ! j+; si transition at a Rabi frequency . With the notations of section 2.1.3 the driving Hamiltonian

KH C 0 = 2 jsihsj ¡ j+ihg j + jg ih+j (2.112)
in H H tr reduces to

KH C = 2 jsihsj (2.113)
in H, which leads to the projectors P s and P s? , and Hilbert spaces H n=s and H n= / s already mentioned.

Theoretical background

To shed a dierent light on the physics we also describe below the eect of the continuous drive in terms of dressed states and the Autler-Townes eect.

Applying a continuous drive at frequency ! d ' ! +;s drives the transition jg; si ! j+; si. By considering these two levels as a TLS the Hamiltonian of the driven system in the rotation frame is given by eq.

E = 2 + 2 p 2 ;
(2.115) creating an anticrossing as shown in g. 2.18a 2.4 . This is known as the Autler-Townes doublet [11]. Going back to the ef-resonant system and considering the case of a resonant drive ! d = ! +;s , the levels around jg; si are depicted in g. 2.18 b; The energy levels of both jg; si and j+; si are split by such that the transition between them independent of , and thus still ! +;s . However, the levels split from jg; si are not resonant with the harmonic ladder. This eectively removes the jsi Fock state from the ladder creating an EC at n = s.

!,s , g s , 1 g s ! 0 +s 0 ± !" # - + - + (a) (b) !" # , 1 g s
Ideally the drive is perfectly selective; inuencing only the jg; si state while leaving the jg; n = / si states unaected. The nite separation between ! +;n=s and ! +;n= / s however results in a non-zero oresonant shift of the ! +;n= / s transition, the inuence being greatest with the neighboring transitions at ! +;s+1 and ! +;s¡1 . This spoils the harmonicity of the ladder and the selectivity of the EC. These parasitic eects limit the practical values as discussed in chapter 7.

Theoretical background

Chapter 3 Measurement setup, instruments and techniques

This chapter describes the various technical aspects of the experiment. We start of by explaining the transmon fabrication process and give the specic recipes, then we introduce the working principle of a dilution refrigerator used to cool the sample and lastly discuss the microwave signal generation and detection techniques used for the measurements.

Qubit fabrication

The transmons are fabricated with a technique known as double angle evaporation and is well suited for making Josephson junctions. It is based on suspending a mask, made by electron-beam lithography (EBL), over a substrate as depicted in g. 3.1. Evaporating Al at two dierent angles sequentially with an oxidation step in between creates a partial overlap of the two Al depositions separated by an insulating Al 2 O 3 layer creating a Josephson junction. The mask is made with two layers of dierent resists, the bottom one being more sensitive to the electron beam leading to a void beneath the top layer, called an undercut. Here we describe the general principles of the different fabrication steps while the recipes can be found in section 3.5.

Substrate choice and resist deposition

The transmons are fabricated on a low loss substrates, either high resistivity Si (r Si >4000 Wcm) or sapphire (Al 2 O 3 ). Both come in the form of commercial monocrystalline 2 inch diameter wafers, 330 mm thick with a c-axis crystal orientation in the case of Sapphire and 300 mm thick with a 50 nm SiO 2 layer and a [100] orientation for Si. The SiO 2 is removed with hydrouoric acid (HF). A Si substrate was used for the rst sample because a fabrication recipe was at hand. Later, a recipe for sapphire fabrication was developed because it has a lower loss tangent than Si.

Various steps can be taken to ensure a clean wafer. Their application to the dierent samples are found in the recipe.

Acetone to remove grease. O 2 plasma either in an asher or a reactive ion etcher (RIE) is used to burn o organic residues.

Commercial resist remover (remover 1165 from Shipley).

Machine rinsing the wafer with high pressure water jets.

HF bath removes various materials e.g. SiO 2 of Si.

Kristinn Júlíusson

After a water or an acetone rinse, the wafer is rinsed in isopropanol (IPA) which is the easy to dry by blowing N 2 . Lastly, just before depositing the bilayer, the accumulated water vapor can be evaporated from the wafer by placing it on a hotplate and letting it cool after.

The mask is made out of two polymer layers (i.e. bilayer) mixed with a solvent, which is evaporated during baking. First is a layer of copolymer MMA-MAA (methyl methacrylate -methacrylic acid) and then on top of that a layer of PMMA (polymethyl methacrylate). PMMA has the role of a mask and should be as thin as possible, representing a 2D mask, down to the limit of breaking. The copolymer suspends the PMMA o the substrate without blocking the Al deposition and its thickness depends on the Al evaporation angles and shift of the mask pattern needed.

For deposition each resist layer the wafer is placed onto a spinner and a few drops of the resist placed at its center, such that it covers most of the wafer. During the spinning the resist spreads out to a layer whose thickness depends on spinning speed and the viscosity of the resist. Spinning at a higher speed for a few seconds at the end can remove the thick resist bump that tends to accumulate at the edge of the wafer. After spinning, the resist is baked on a hot plate and becomes hard as solvents evaporate from it. The thickness of the layer is measured by optical interferometry using Filmetrics F20 Thin Film Measurement System. During EBL the accumulation of charge must be avoided. I the case of an insulating substrate like sapphire a thin layer of Al is Evaporated on top of the bilayer.

Measurement setup, instruments and techniques

Having prepared the wafer, it is diced into chips of the appropriate dimensions for each experiment. With silicon this can be done by using a diamond scriber along the crystal orientations of the wafer and cleaving afterwards. Scraping and breaking is more challenging with sapphire due to its non-orthogonal crystal symmetry and are therefore diced with a dicing saw. After dicing the chips are ready for EBL.

Electron beam lithography

The eect of an electron beam on the resist is to break the polymer molecules into smaller pieces, making them soluble in a developer. The higher sensitivity of the lower layer to this eect, as well as the scattering of the incoming electrons o the substrate, produces a volume called an undercut, where the copolymer is removed from underneath the PMMA. The undercut can be enhanced by an undercut box, where an area around the structure is exposed with a weak dose such that the PMMA is minimally aected as depicted in g. 3.1. Where two undercuts overlap a bridge is formed in the PMMA and is a key feature in producing Josephson junctions, as will be discussed in the forthcoming.

The equipment used for electron-beam lithography is a Raith e-line masker with a few nanometer precision interferometric stage. The chip is mounted on a stage where clamps hold it in place. The electrons striking the sample discharge through the substrate and the stage or via the charging layer and the clamps. The corners of the chip are gently scratched to facilitate focusing. The stage is subsequently loaded into the SEM via loadlock and reaches an operating pressure of < 6 10 ¡6 mbar.

The sample is exposed with a 25 keV beam at a current varying with the beam aperture and is measured with a Faraday cup. For the transmon pads we use about 4.5 nA and around 18 pA. Before starting the exposure however, the tilt of the chip is calibrated for with focusing on 3 corners of the sample for subsequent planar interpolation of the focus, and the geometric aberrations of the beam deections are calibrated for using the interferometric stage enabling automatic correction. After synchronizing the software coordinate system to the location of the chip, a macro automatically runs the exposure process. The exposure le (GDSII) of the two dierent transmon geometries is shown in g. 3.3. The structures away from the junctions are dosed Kristinn Júlíusson strongly to ensure all the resist will be removed. Around the junction the dose has been tuned down to ensure the ne structure of the mask remain intact. A rectangular area around the SQUID, exposed at a very low dose, is the undercut box. Details are given in the recipe.

Measurement setup, instruments and techniques

After e-beam exposure the sample is developed by immersing the chip in various liquids and shaking it gently. If a charging layer is in place it is rst removed in a 20 g/mol solution of KOH, rinsed in deionized water and dried with N 2 . It is important to dry the sample well and prevent water droplets from evaporating on the surface of the as it leaves dirt behind. When the bilayer is uncovered the exposed parts are dissolved in MIBK(methyl isobuty ketone) solution, 25% MIBK 75% IPA, rinsed in IPA and thoroughly dried with N 2 .

After developing some resist residues may still remain on the substrate below the mask. They are removed with O 2 -plasma ashing or RIE at a high O 2 pressure and low acceleration voltage.

Metal evaporation

With the mask prepared the transmon is made by double angle evaporation using either a Plassys MEB 550S e-gun or Plassys MEB 450 egun. In both cases the sample is clamped onto a stage that can rotate around an axis perpendicular to the evaporation direction. We also cover the chip with a metallic mask which prevents deposition on the edges of the chip. After mounting, the chamber is pumped down to 10 ¡6 mbar or below.

Before the Al deposition, the substrate is cleaned in-situ by ion milling with a neutralized beam of Ar + ions at 500 eV at both deposition angles. Thereafter the rst layer of Al of 25nm is deposited, followed by an oxidation step where a few mbars of the Ar/O 2 mixture lls the camber for around 5 to 10 min. Lastly the second Al layer of 35 to 60 nm is deposited. Care must thus be taken with electrostatic discharge when handling the sample from this point forward. Finally the remaining resist is removed in a lifto procedure described in the recipe. The oxidation of the junction continues for some time after the sample has been exposed to air. Therefore we saturate the oxidation by heating the sample on a hot plate.

Lifto, characterization and mounting

The Josephson energy E J of the SQUID can be calculated from its normal state resistance R n ; using Ambegaokar Barato reection [2]

E J = h (T = 0) 8R n e 2 ;
where h is Plancks constant, (T = 0) is the superconducting gap of Al at T=0, and e is the electric charge. The room temperature resistance between the two pads is therefore measured with a probe station, a multimeter and a circuit designed to protect the sample from electrical discharge when placing the probes. Essentially, a tunable resistance is lowered to shunt the sample while the probes are being lowered or raised. Then, to measure it, the tunable resistance is increased, passing all the current through the sample. The junction resistance is then calculated from the two resistance tunings, taking into account the resistance of the Al wires of the transmon. The room temperature resistance is known to increase by 20% at 4K. Scaling by that amount allows us to calculate E J .

If the calculated E J is close to the targeted value the transmon is mounted in the cavity. For the rst experiments, the chip was placed in a designated groove in in the Cu cavity with small pieces of compressed indium as seen in g. 3.5a. A thin indium wire is placed around the outside of a rim which is then compressed when the cavity is closed with its other half. For the other experiments the transmon is rst mounted in a Cu cap and blocked with a manually tightened screw to avoid breaking the chip. The part of the chip now protruding from the Cu cap is then inserted into the slit of the cavity (which has already been closed with an indium seal). Finally a Cu bar is placed on top of the cap and fastened to keep it in place.

Fabrication recipe

The following table contains the recipes for the three qubits used in this thesis. Sample 1 and 2 were fabricated on Si and sapphire, respectively, and are both transmons of the rst type as shown in g. 3.3a. Sample 3 was used in the QZD and is a transmon of the type shown in g. 3.3b.

Cryogenic setup

In our experiments we cool the samples down to low temperature T fr 30mK such that the system is in its ground state and thermal excitation is negligible compared to the Transmon excitation energy: k B T fr ~!ge , where k B is Boltzmanns constant. Aluminium used in the sample and the NbTi used in the low temperature cabling have critical temperatures T C T fr . In order to reach these low temperatures the sample and Kristinn Júlíusson a large part of the electrical setup is placed inside a dilution refrigerator.

Dilution refrigeration

Dilution refrigerators are machines with several stages at dierent temperatures down to T fr 10 to 50mK. There are two cooling steps, rst cooling from room temperature to a few K where a second process takes over and cools down to T fr . The rst step divides the refrigerator into either wet fridges, which use a liquid He 4 reservoir to cool down to 4 K, or dry fridges that use a pulse tube, a machine that pumps He 4 through a membrane in a closed cycle loop to cool down to 3-5 K.

The second cooling step is called dilution shown in g. 3.6a is based on closed loop circulation of a He 3 / He 4 mixture. The mixture, already being in thermal contact with the 3-5 K reservoir, is compressed into a mixing chamber through a high impedance line where it condenses and is pumped out through a low impedance line with a still, before being re-injected, forming a closed loop. Pumping at the still evaporates mainly He 3 as it has a much higher partial pressure than He 4 , cools the still to 0.7-1 K. Thermally coupling the still to the injection line cools the re-injected helium and once the mixture goes below the triple point at 870mK it splits up into two phases, a He 3 concentrated and a He 3 dilute phase. The pumping of He 3 through the still is from the dilute phase which requires that the concentration equilibrium be regained with He 3 from the concentrated phase. This process costs energy, which is supplied by the environment and thus cools it.

Mounting the sample

We used an in-house made wet refrigerator for the rst experiments with the rst type of transmon in a Cu cavity, and an also home-made dry refrigerator prototype for the QZD experiment. The setups being similar, only that for the dry fridge is presented in detail below unless noted otherwise.

As shown in g. 3.6b, the cavity is mounted to a Cu plate and sits inside a Helmholtz coil. Semi-ex SMA cable connect to the coupling ports and a Cu braid tightened onto the cavity is used for improved thermalization. The setup is placed in a -metal can to shield the sample from stray magnetic elds. The closed -metal can is then mounted onto the mixing chamber plate of the dilution refrigerator.

Kristinn Júlíusson

Fridge circuitry

A diagram of the electrical circuitry is shown in g. 3.7. It includes a bilar dc line A1 for biasing the Helmholtz coils, and ve coaxial lines: the main circuit is made of the input microwave line C3 connecting the microwave sources to the input port of the cavity, and the main output microwave line D2 connecting the cavity output port to the measurement setup at room temperature. This output line includes a parametric amplier, which also uses three secondary input lines for microwave pumping (C4), dc biasing (B1), and optionally testing the amplier (C2) without passing the signal through the cavity.

All lines go from room temperature to 30 mK through three other stages at 70K, 4K, and 1K. The coaxial ones are 2.2 mm in diameter and have SMA connectors between the dierent stages. The material is chosen with the perspective of having thermal conductivity suciently low to limit the thermal ow between successive stages, while keeping the electrical conductivity high enough to limit the microwave losses: CuNi rigid cables with an attenuation of 4 dB at 7 GHz are used from room temperature to 4K. Then superconducting NbTiN rigid cables with an attenuation lower than 1 dB are used from 4K to the mixing chamber plate. At the mixing chamber semi-exible Cu cables connect the dierent components.

The electromagnetic noise along the lines is thermalized using a combination of attenuators, lters, and isolators: Attenuators are placed only along input microwave lines C2, C3 and C4 because input signals can always be increased whereas the small output signal on D2 should be preserved. Then two pairs of dissipative low pass lter (IR) and non-dissipative 4-8 GHz bandpass lters are used on C3 and D2 to protect the cavity-transmon sample from radiation outside the 4-8GHz window. Finally a double isolator is added on output line D2 to protect the sample from the noise propagating backward from the amplication chain.

The output signal on D2 rst passes through the lters and the double isolator mentioned above, through the direct path of a directional coupler (see explanation below), and is then routed with a double circulator to a quantum limited 17dB gain parametric amplier (JPA) operated in reexion (see 3.3.4 for details). The returned amplied signal is routed by the same circulator to a 32 dB gain HEMT (high electron mobility transistors) amplier placed at 4K. On the JPA side, the dc bias line B1, whose function is to align the frequency of the JPA with the cavity readout mode, is RC-low pass ltered at 4K. This dc line and the pumping line C4 are both connected to the paramp ux port with a bias tee (see 3.3.4). The C2 line devoted to testing the paramp, is coupled to it with a 20 dB directional coupler. That has the advantage of attenuating the input noise by 20 dB without attenuating the output signal.

Microwave measurements

In this section the dierent measurement setups and equipment are described.

Kristinn Júlíusson

Vectorial Network Analyzer measurements

A vectorial network analyzer (VNA) measures the 2x2 complex scattering matrix fS ij=1;2 g of a two-port 'device under test' as a function of frequency. For our experiments, two microwave VNAs (Anritsu 37397C and Keysight N5232A) have been used to characterize individual components or the whole electrical setup inside the refrigerator or at room temperature, tune the coupling of each cavity port pin from S 11 and S 22 , determine quality factors of the readout and storage cavity modes by S 21 measurements, and map their resonance frequency as a function of the magnetic ux applied to the transmon (anticrossings).

Signal generation

For spectroscopic measurement involving more than one microwave tone or microwave pulses rather than continuous waves, the home-made microwave setup shown in g. 3.8 is used. This setup on one hand generates control signals made up of pulses and continuous signals at dierent frequencies, and on the other detects signals coming from the sample in a process known as homodyne detection; where a probe signal is sent to the sample and the returning signal is mixed with a local oscillator at the same frequency, yielding a DC signal that is easy to measure with an analog-to-digital converter (ADC).

Signal generation

The input signals are generated with microwave sources mixed with signals from an arbitrary waveform generator (AWG). For microwave sources we used Agilent E8257D and Anritsu MG3692A, Tektronix 5014C for AWG, Yokogawa for DC voltage generators and HMC-C041 and HMC-C009 IQ-mixers from Hittite. All RF equipment are phase locked to a 10 MHz clock generated by the AWG. In addition, a 1 GHz reference phase locking is used between the two generators for the readout tone and the paramp pump (see 3.3.4 and 4.1.1 for further information). After IQ-mixing the amplitude of the signal is ne tuned with a voltage variable attenuator (VVA) and subsequently combined with similarly generated signal, high-pass ltered and split up. One part goes to the fridge and one to an oscilloscope or a spectrum analyzer for and monitoring.

As mentioned above, the signal is created by IQ-mixing which will now be explained. The microwave source sends a signal LO(t) = I 0 cos(! LO t) to the LO port of the mixer and the AWG send sig- nals I(t); Q(t) to the in-phase and quadrature-phase port I and Q.

The resulting signal at the RF output is then

RF(t) = I(t)cos(! LO t) + Q(t)sin(! LO t):
For the control signal we do heterodyne modulation where quadrature signals are at a frequency ! IF in a pulse envelope s(t),

I(t) = s(t) cos (! IF t ¡ ) and Q(t) = s(t) sin (! IF t ¡ ). The resulting signal is Kristinn Júlíusson then at ! LO ¡ ! IF , RF(t) = s(t) cos[(! LO ¡ ! IF ) t + ]:
This allows for exible pulse generation with the AWG as s(t) is the shape function of the pulse, ! IF sets the frequency within 400MHz range from ! LO and can be set arbitrarily for dierent pulses and even vary with time.

Signal detection

The returning signal from the fridge passes through an amplier chain as depicted in g. 3.8. Filters, attenuators and circulators are used to reduce reections between the components and lter out noise at frequencies dierent from the detection signal as it can saturate the ampliers. This amplier chain was located 2-3 meters away from the signal generation setup to minimize amplication of stray radiation.

The GHz frequency signal is too fast for our data acquisition card so we demodulate it. The process is the reverse from the modulation described above with ! IF = 0. The probe signal is split up, one part goes to the LO of the mixer while the other part is sent to the fridge, comes back through the amplier chain and to the RF of the mixer. This creates signals at the I and Q ports, one at 2! LO and on DC, which after low-pass ltering become I(t) = s(t) cos() and Q(t) = s(t) sin(). Finally each quadrature is amplied before going to two ports of an analog-to-digital data acquisition card Acqiris xc200.

Mixer calibration

In reality the mixers are imperfect and do not follow exactly the behavior just described above. When no signal is applied to the I and Q ports a signal at ! LO leaks through to the RF output. This leakage can be minimized by tuning small dc voltages on the I and Q ports. The eect of the remaining leak is small if no transition in the experiment is at ! LO . In addition, the I and Q ports are not exactly / 2 out of phase and have a small amplitude imbalance. This causes small signals to appear at the opposite sideband frequency ! LO + ! IF (and other harmonic frequencies of the modulation). By iteratively tuning the amplitude and phase of the I and Q RF signals to compensate for the mixer imperfections, the opposite sideband and harmonics are minimized.

Measurement setup, instruments and techniques

Parametric amplication

As described in section 3.3.2 a Josephson parametric amplier (JPA) rst amplies the output signal at 30mK. This JPA described in detail in [START_REF] Zhou | High-gain weakly nonlinear ux-modulated Josephson parametric amplier using a SQUID array[END_REF] adds the minimum noise allowed by quantum mechanics to the signal. Here we give a brief description of its principle and how it is operated.

In a parametric amplier a signal at ! S is amplied by a transfer of energy from a pump tone at ! P . In our superconducting circuits the transfer medium is a lumped element LC-resonator with a resonance frequency ! R . Having the inductance L (partially or entirely) made up of a SQUID or, as in our case, a SQUID array makes it tunable in magnetic ux, which allows for the parametric pumping. We pump at twice the signal frequency ! P ' 2! S ' 2! R , on a dedicated line distinct form the signal line. This pumping induces three wave mixing ! P = ! S + ! I , where ! I is the frequency of a complimentary idler signal. In the non-degenerate mode ! p = / 2!s resulting in ! S = / ! I , the gain of the reected signal is

jG S j 2 = 1 + jG I j 2 = 1 + 4 r 2 [1 ¡ r 2 + r 2 ¡ r 2 ] 2 + 4 r 2 ;
where r = (! P / 2 ¡ ! R ) / ¡ a is the pump-resonance detuning, r = (! S ¡ ! p / 2) / ¡ a is the signal-pump detuning, r = ! R a P / ¡ a is the pumping strength, ¡ a is the amplitude decay rate and a P is the relative pump strength. When ! S = ! I the signal and the idler are degenerate and the gain becomes sensitive to the phase between the signal and the pump:

jG S ;deg j 2 () = 1 + 4 r 2[ r ¡ r cos(2)] + (1 + r 2 + r 2 ) sin(2) (1 ¡ r 2 + r 2 ) 2 :
For our experiments, the JPA is operated in this degenerate mode. First the JPA resonator frequency is tuned with a dc ux bias to the frequency of the signal to be amplied ! S ' ! R . Then a pump tone is applied at ! P = 2! S ; the power is increased until either amplication or deamplication of the signal is clearly seen. At this stage the phase of the pump tone is tuned to maximize the amplication. Finally the pump power is adjusted to reach a gain of 18dB.

Measurement techniques

This chapter covers some of the measurement procedures used to characterize the cavities and the transmons used in this thesis. The rst section addresses the characterization of a transmon qubit, and the second one how we characterize the quantum state of a cavity mode.

Transmon

The coupling strength g 0 of the transmon to a cavity mode at frequency ! 0 is determined by sweeping the ! ge frequency of the transmon around ! 0 with an external magnetic ux. Measuring with a VNA the transmission spectrum of the cavity around ! 0 results in an anticrossing separated by 2 g 0 when ! 0 = ! ge .

The transmon is characterized in the dispersive regime where j! 0 ¡ ! ge j g 0 .

Resonator spectroscopy

An example of an anticrossing obtained by VNA measurement of the transmission of one of our cavity modes with the transmon in jg i is shown if g. 4.1a. When the transmon frequency ! ge is swept across ! 0 using a ux bias, it yields two resonance peaks with the minimal peak separation giving the coupling strength g 0 . This behavior is observed at low power i.e. the single photon regime. In g. 4.1b taken at / 0 ' 0.4 for instance, the resonance at low power is indeed at the dispersively shifted frequency ! 0 ¡ ge . At high power the resonance frequency shifts back to that of the bare cavity ! 0 . This eect is attributed to the saturation of the transmon by the probe eld [START_REF] Reed | High-Fidelity Readout in Circuit Quantum Electrodynamics Using the Jaynes-Cummings Nonlinearity[END_REF]. Knowing ge , g 0 and ! 0 one can in principle calculate the frequency ! ge of the transmon.

Readout signal

The transmon state is measured in the low power regime (typically 10-100 photons). , the transmitted amplitude A (g) = / A (e) = / A (f ) provides a measurement of the state in a single shot and projects it. In practice however the overlap between the peaks is so small that A (e) A (f ) 0 and only a clear discrimination can be made between jg i and jg i=not jg i. Figure 4.2b shows 3 measured spectra, averaged over thousands of repetitions, after preparing separately the states jg i, jei and jf i. The non-zero transmission A 0 at ! 0 (g) for jei and jf i is because of partial relaxation during readout. The probability of the transmon state ending in jg i is thus

P g = A A (g) : (4.1)
The initial probability of being in jgi, correcting for relaxation, can be deduced following a calibration presented in section 4.2.1.

To construct the amplitude signal A we use the homodyne detection setup described in chapter 3. A microwave pulse at frequency ! is sent to the cavity and the I and Q quadratures of the eld are measured with the ADC. The sampling rate ¡ samp varies from 0.5 to 0.025 GS/s and the samples I(t i ) and Q(t i ) (i = 0; :::; n samp ) form time traces consisting of n sanp = 50 to 1000 points spanning a time t samp = n samp /¡ samp between 100 ns and 40 s long. Each time trace (see g. 4.2c)has a Gaussian noise larger than the signal, even at maximum transmission ! = ! 0 (g) . By averaging N = 10 2 to 10 5 traces the noise is reduced such that the pulse can be detected in

I (t i ) = ( n I n (t i )) / N and Q (t i )( n Q n (t i )) / N ,
as shown in g. 4.2d,e. Finally a time average is taken over I (t i ) and Q (t i ) giving two scalars, I = ( i= j i=k

I (t i )) / (k ¡ j) and Q = ( i= j i=k Q (t i )) / (k ¡ j),
which are translated to an amplitude and a phase

A = I 2 + Q 2 p and ' = atan Q I : (4.2)
Measurement techniques

Removing I and Q osets

Osets I o and Q o on the measured quadratures I and Q are minimized but are in practice never zero. As a result the measured quadratures are I(!) = A cos (!x / c) + I oset and Q(!) = A sin(!x / c) + Q oset , where x is the total length of the microwave circuit, c the speed of light in the cables, and ! the frequency of the measured homodyne signal. Consequently, when sweeping ! in spectroscopy of a cavity mode the amplitude is

A 0 = A 2 + I oset 2 + Q oset 2 + 2A(I oset cos(!x/c) + Q oset sin(!x/c))
q A 0 has both a constant oset and oscillating terms causing ripples in the background of the spectrum. By measuring the quadratures with and without the probe signal we can subtract these eects and obtain A.

Optimal readout pulse length and integration time

In most experiments the transmon is manipulated with a sequence of pulses, and then readout. Immediately after the manipulation the transmon starts to relax, diminishing the signal A = A (g) ¡ A (e) in time. This suggests it is therefore better to measure the transmon state immediately after the manipulation and integrate over a short time period. However integrating over longer times t int increases the signalto-noise ratio (SNR) as 1/ t int p . The two eects combined yield which has a maximum at t int = 1.2T 1 and was typically used in the QZD experiments presented later in this thesis.

Fast loading of the readout probe signal

To minimize the loss of the readout signal due to transmon relaxation during the rise time of the eld, it is important to load the mode as quickly as possible.

At a constant amplitude at the input port the cavity amplitude increases as (1 ¡ e ¡tin ) as shown in g. 4.2 d. A two-amplitudestep readout pulse as depicted in g. 4.2 e can be used to 'load' the cavity faster. The short kick at the beginning of the pulse is typically 3 times higher in amplitude than the rest and lasts until the amplitude has reached the steady state value of the lower step. Normally the two steps have the same phase but in cases when the cavity mode is slightly non-linear a compensation phase is applied '. This sort of fast loading is explored further in ref [START_REF] Mcclure | Rapid Driven Reset of a Qubit Readout Resonator[END_REF].

Qubit spectroscopy

The transmon frequencies are obtained with two-tone spectroscopy, either pulsed or continuous. The corresponding protocols are shown in g. 4.4:

Pulsed spectroscopy consists in applying a pulse whose frequency ! is swept and whose length well exceeds T 1 and T 2 . Thereafter a readout pulse is applied at ! 0 (g)

. When ! = ! ge the transmon is driven to the maximal mixture of jg i and jei, causing a dip in the transmitted amplitude. When ! = ! gf / 2 the jg i ! jf i transition is driven in a two photon process from which the anharmonicity = 2! ge ¡ ! gf can be determined. Because this is a second order process it typically requires about 10 dB more drive power than the jg i ! jei transition. Spectroscopy on the jei ! jf i transition is done by preparing the transmon in state jei (explained below) before the spectroscopy pulse is applied, and by mapping the population of jei to the ground state jg i afterwards for a clear readout signal.

Sweeping a continuous tone has the benet of the transmon state not decaying during readout. Another consequence is that photons of the readout pulse are present in the cavity while the qubit is being manipulated. This is a way to probe the photon number splitting (see section 2.2.3.1) but is less suitable than pulsed spectroscopy for precise determination of transmon levels. b) In continuous spectroscopy the signal is stable throughout the readout pulse. This scheme also reveals the photon number peaks.

Rabi oscillations of the transmon

Once a transition frequency between two transmon levels has been determined, the Rabi rotations on this transition can be calibrated: a resonant pulse with an amplitude (t) induces a rotation of an angle / R (t)dt proportional to the area of the pulse. In practice square or Gaussian pulse shapes are used, whose amplitude or the duration is varied. The protocol and measurements of the latter case is depicted in g. rotations. Note that in the case of frequency crowding, unwanted oresonant driving of other transitions may occur. This can be reduced by optimal pulse shaping.

Energy relaxation of the transmon

To measure the energy relaxation time T 1 we use the calibrated Rabi rotations to prepare the transmon in jei with a Rabi angle = . By varying the delay time before readout one can map the exponential decay shown in g. 4.6. The measured data points are tted with a function f ( ) = a + be ¡ /T1 : Note that in the case of a frequency tunable transmon, T 1 varies with the transmon-cavity detuning due to the Purcell eect discussed in chapter 2.

Measurement techniques

Free induction decay of the transmon

The free induction decay time is measured by a procedure called Ramsey interferometry. The protocol is depicted in g. 4.7. It consists of a / 2 rotation driven at a small detuning = ! d ¡ ! ge . The detuning frequency / 2 is typically in the 0.5 -2 MHz range. This prepares a state 1/ 2 p (jgi + jei) on the equator of the Bloch sphere, which precesses around the Z-axis at a frequency . After a time a phase ' = has accumulated yielding 1 / 2 p (jg i + e i' jei). When a second / 2 pulse is applied the probability to measure measure jg i becomes g = cos 2 (' / 2). This induces oscillations, called Ramsey fringes, at frequency / 2 when the precessing time is varied as shown in g. 4.7. This gives the transition frequency ! ge /2 with a precision of 0.1MHz which is typically better than the precision from spectroscopy.

As discussed in section 2.2.2 a free induction decay rate ¡ 2 is comprised of the relaxation rate ¡ 1 and the pure dephasing rate ¡ ' , as show in eq. 2.83. This results in a decaying amplitude of the Ramsey fringes. The decay is exponential when the ¡ 2 is dominated by ¡ 1 or if the noise spectral density S ' is white. This is usually the case and the measured data shown in g. 4.7 can thus be tted with f (t) = a + b cos( /2)e ¡ /T2 . In case 1/f noise the decay becomes Gaussian and requires dierent protocol dependent analysis as discussed in chapter 5.

Kristinn Júlíusson

Hahn echo refocusing

A 'refocusing' protocol, called Hahn echo, can be applied to dynamically reduce the dephasing rate ¡ ' . It consists in adding a pulse midway between the two / 2 pulses in the Ramsey protocol with = 0 depicted in g. 4.8. By considering the low frequency part of S ' as quasi-static over , the total phase ' = ' s + ' f accumulated can be decomposed into a static part ' s and a high frequency noise contribution ' f . The evolution over the sequence is:

jg i + jei 2 p wait /2 jg i + e i'1/2 jei 2 p pulse jei + e i'1/2 jg i 2 p wait /2 e i'2/2 jei + e i'1/2 jg i 2 p :
The static part of ' 1 and ' 2 is the same and cancel, leading to a random phase involving only the high frequency part ('

f ;1 +' f ;2 )/2.
This leads to an exponentially decaying echo signal f (t) = a + be ¡ /T !"

R # !" R #!"

Measurement techniques

Cavity eld characterization

In our QZD experiment manipulation and characterization of the quantum state of a cavity mode is needed. This is done with microwave pulses resonant with the cavity mode and mapping the state of the mode onto the transmon in a tomographic manner. The transmon is read out with a dierent cavity mode as discussed in chapter 6.

Here a calibration method of the displacement pulses and photon number distribution is presented. Then we discuss the Wigner function and two tomographic methods to measure it. The principal method uses a Maximum Likelihood algorithm which is presented at the end of the chapter.

Mapping Fock state occupation to transmon excitation

The photon number distribution of a cavity mode is measured by mapping the probability p(n) for each n individually onto the transmon excitation probability by applying a pulse at f +;n (see g. 2.13b) and subsequently reading out the transmon state. In our experiment this procedure needs to be calibrated due to partial relaxation during the pulse an the readout.

Calibrating Fock state probability measurements and displacements

The displacement of the eld amplitude in the cavity mode at frequency ! 0 is calibrated with respect to the length and amplitude of a coherent pulse at ! 0 . The scaling between the two is approximately linear for drive times t T cav . The calibration is done by varying the amplitude of the applied pulse and analyzing the variation of the photon number distribution:

In the ef-resonant system, the Rabi frequencies n and decay times vary for the dierent jg; ni ! j+; ni transitions. The frequencies n can be measured (see chapter 6) but the amplitude A n of each of the photon number peak must be scaled with a calibration factor c n compensating for transmon relaxation during readout, to give the photon number probability p(n) = A n / c n . To determine c n we assume that the distribution p(n) from a displacement pulse is Poissonian P n (n) with an expectation value n / 2 varying linearly with the area of the pulse, as expected for a coherent drive. The drive eld increment @ and scaling factors c n are extracted from a collective t to measurements of the photon number peaks for various displacements. Figure 4.9a,b,c show the raw data, the calibrated probabilities, as well as the residual error between model and the calibrated data. This error is below 2.5% and is homogeneously distributed, which conrms the validity of the model and the quality of the calibration.

As a self consistency check the rescaled data p(n) = A n / c n as a function of is compared with the Poissonian distribution value P n (n) obtained from @.

Wigner function and Wigner tomography

In this thesis the quantum state of a cavity mode dened by it density matrix is graphically represented in phase space, i.e. the plane of the complex eld amplitude , by the so-called Wigner function -A coherent state j i appears as a 2D Gaussian centered at = with a width 1 / 2 p representing the quantum uctuations on both quadratures.

W () =
-The Wigner function also clearly discriminates between a statistical mixture and a coherent superposition of two coherent states j 1 i and j 2 i: In both cases two Gaussians are centered at 1 and 2 while the coherent superposition has additional fringes between them.

The Wigner function can also be expressed as [23] W

() = 2 Tr[D ^()P ^D ^(¡)] (4.6)
where P ^is the photon number parity operator with the expectation value

hP ^i = X p(n) n even ¡ X p(n) n odd : (4.7)
The value of W () at a specic point in phase space is measured by applying a displacement pulse D ^(¡) and measuring hP ^i. Measuring W () using displacements and parity measurements for a multiple values of spanning a window in phase space is called Wigner tomography.

In cavity and circuit QED several methods can be used to map directly the parity of the eld onto the excitation of a real or an articial atom. In the dispersive limit where the photon number splitting is equidistant for all n, this can be achieved with Ramsey interferometry as explained in [23][45] [START_REF] Vlastakis | Deterministically Encoding Quantum Information Using 100-Photon Schrödinger Cat States[END_REF]. Due to the non-equidistant separation of the photon number peaks in our ef-resonant scheme, two alternative methods of mapping the parity onto the transmon state are used: rst, the photon number probabilities in eq. 4.7 can be measured individually as explained in section 4.2.1. Second, the parity can be mapped directly onto the transmon state with a composite pulse made of the superposition of pulses at all f +;n odd (up to a truncation), as will be demonstrated in chapter 6. However, the decreasing separation +;n Kristinn Júlíusson between adjacent f +;n forces a truncation at low n, which limits the mappable window in phase space. We thus switch to a more standard tomographic method involving measurements of p(n) for only the lowest few n.

Standard quantum eld tomography

In this method we reconstruct the density matrix in the Hilbert space H tr truncated to n tr Fock states, with a procedure called maximum likelihood [22][21], and subsequently calculate the Wigner function.

The truncation n tr is chosen suciently large such that the anticipated has a trace T () ' 1 in H tr , but not at larger values as it is futile and time consuming. Now an n tr x n tr complex valued hermitian matrix with unit trace is characterized by n tr 2 ¡ 1 real numbers, which requires in principle sampling at least n tr 2 ¡ 1 dierent measurement observables. In our case of a cavity eld, the measured observables {p n; } correspond to m dis dierent displacement pulses D ^(), each followed by measurements of p(n) for the lowest m Fock Fock states. The displacements cover either the full Hilbert space or a smaller area in phase space containing the full weight of the anticipated W . As an example see g. 4.10a (black dots). The number m Fock < n tr of measured Fock states (4 to 7 typically) is in practice limited by the decreasing +;n for high n as discussed in section 2.2.3.3. These m dis m Fock measurements make up a data set fp n;a g that is used to reconstruct as will be explained in section 4.2.4.1.

As discussed in section 4.2.2 the data fp n;a g have a Gaussian noise with a standard deviation < 1%. By oversampling m dis m Fock = k n tr 2 (k > 1) precision of the reconstruction of can be improved.

Maximum Likelihood treatment

The density matrix is reconstructed from the data set fp n; g with a maximum likelihood method. To explain the essence of the method we consider a complete set of M projectors P ^i corresponding to measurements on N identically prepared states giving the data set fn i g, where n i counts the number of outcomes i . From the data set a frequency f i = n i / N is dened to estimate the probability i = Tr P ^i . We consider now the likelihood functional

L() = C Y i Tr ¡ P ^i n i ; (4.8)
Measurement techniques dened as the probability of getting fn i g for a given . Here C is a combinatory factor irrelevant for what follows. The density matrix ML maximizing L is by denition the state that gives fn i g most likely.

Global maximization of L() is in general not ecient for traditional numerical methods. An alternative algorithm is based on a xed point method iteratively applying the operator

R() = X i f i P i Tr[P i ^]; (4.9)
i.e. i+i = R( i ) i R( i ), starting from an initial guess matrix 0 . We see that as ! ML , f i ! Tr[P i ^] and R ! 1, such that R( ML ) ML = ML . More details on Maximum Likelihood can be found in refs [22][21], including incomplete sets of projectors and cases where the errors due to the imperfect measurements are incorporated.

As mentioned above our data set has a Gaussian noise, a case not covered in refs [22][21], so that the formalism was adapted in collaboration with Pierre Rouchon at Mines-ParisTech. The essence of the method is presented here below and a more complete description and the optimization algorithm are given in appendices B and C.

The projectors fP i g we use, denoted now fE n; g, are a combination of displacement pulses and Fock state projectors

E n; = D ^(¡ )jnihnjD ^(),
= R D exp(f())P 0 ()d R D exp(f ())P 0 ()d ; Kristinn Júlíusson
where D is the convex set of density operators, P 0 () is some prior probability law of , and

f() = ¡ X n; (Tr[E n; ] ¡ p n, ) 2 2 n; 2 (4.11)
is the logarithm of the likelihood function to be maximized. When ML , it is known that ML is a good approximation of BM .

We not that the targeted ML obeys [START_REF] Six | Quantum state tomography with noninstantaneous measurements, imperfections and decoherence[END_REF] the following necessary and sucient conditions that there exists a scalar ML such that:

[ ML ; rf( ML )] = 0 and ML P ML 6 rf ML 6 ML 1;

(4.12)

where

rf () = ¡ X n; Tr[E n; ] ¡ p n; n;a 2 E n; : (4.13)
is the gradient of f at and P ML is the orthogonal projector on the range of ML . The maximization of f can then be achieved by a gradient algorithm with orthogonal projection on D. A single iteration is expressed as

k+1 = [ k + grf( k )] (4.14)
where g > 0 is a normalization parameter limiting the Hessian kr 2 f k such that 1 . g kr 2 f k . 10, and is the orthogonal projection onto the set of physical density matrices, i.e. Hermitian non-negative matrices with unit trace. Figure 4.10 shows the reconstructed Wigner function from articially generated data. In absence of noise the reconstructed density matrix is identical to the original one independent of the initial guess 0 . In g. 4.10 b a Gaussian noise with = 1% has been added to the data. This results in ripples in the Wigner function of the reconstructed matrix.

More details can be found in appendix B as well as a method to estimate the eect of n; on the Wigner function.

Kristinn Júlíusson

Chapter 5

Reaching long lifetimes in 3D circuit-QED

The rst experiments performed during this thesis were aimed at achieving long lifetimes (several tens of s) of a transmon as reported in [37][43]. In contrast to these works we used a magnetically tunable transmon inside a Cu cavity, through which an external magnetic eld can be applied. The lower cavity lifetime compared to a superconducting cavity results in a shorter Purcell limit of the transmon relaxation time (see section 2.2.3.1), but enables a decoherence analysis at dierent frequencies.

Design

Copper cavity

In the experiment the lowest mode of the Cu cavity is coupled to the transmon. The design is based on the design in ref. [START_REF] Hanhee Paik | Observation of High Coherence in Josephson Junction Qubits Measured in a Three-Dimensional Circuit QED Architecture[END_REF]: the cavity is roughly a cuboid of dimensions 27336 mm 3 with rounded edges to facilitate fabrication. Two small holes are pierced in one of the walls, through which electric eld is coupled to the cavity using proper SMA connectors playing the roles of two antennas. This design, shown in g. 5.1a, is simulated in a nite element simulation software, as will be discussed in chapter 6, and gives a frequency f 0 = 7.15 GHz of the fundamental mode. A Cu cavity based on the simulated design was fabricated by the in-house workshop and is shown in g. 5.1b. It consists of two blocks which are screwed together with Cu screws to form the cavity. One part has a rim around the edge of the cavity with a groove for the transmon chip. The other part has a groove around the edge of the cavity, into which the rim of the other part ts. An In seal is placed around the rim before closing and is then compressed as the two halves are pressed together. This ensures good connection and eliminated radiation through the cut between the two halves. The pins used to couple to the cavity are made from Cu but are gold coated. They are led down the length to give the right coupling. Measuring the transmission spectrum of an empty cavity at 77 K gives a resonance frequency f 0 = 7.358 GHz, but with a Si chip at 30 mK the bare resonance is shifted to 7.078 GHz as the relative permittivity of Si is " r Si = 11.68.

Transmon qubit

The transmon design has been described in chapter 3. It is based on the design of the transmon that I used in my master thesis and in ref [1], where two 500250 m 2 pads are separated by 130 m and give a coupling g 0 = 180 MHz. Here, we reduce the pad size to 400200 m 2 and the separation to 50 m to get roughly half the coupling.

Kristinn Júlíusson

The transmon is mounted in the cavity with a small piece of In pressed on the corner of the chip to hold it in place. An external coil is then aligned with the transmon SQUID and attached to the cavity. The setup is then mounted and well thermalized with Cu at the lowest stage of a wet dilution refrigerator, and nally cooled down to 30 mK.

A note on the setup

A key element in achieving long lifetimes is to make sure that all stray elds and radiations are eliminated. As a part of that, the absorptive infra-red and microwave band-pass lters from section 3.2.3 are added to the setup. In addition the innermost copper shield of the dilution stage is doubled with a ¡ metal shield painted inside with absorptive paint ; a Cu mesh grid covered in a 5 mm thick layer of absorptive Eccosorb rubber is also attached to the bottom of the shield.

Characterization

Two samples with long T 1 were measured: KJ6ch10 fabricated on Si and KJ15ch11 fabricated on sapphire.

Transmon on Si substrate

Spectroscopic characterization and T 1 measurements

Once the sample has been cooled the transmon is characterized using the standard methods described in section 4.1.

With the VNA the bare resonator is characterized by measuring the transmission spectrum in the high-power regime shown in g. 5.2a, yielding a frequency f 0 = 7.079 GHz and a quality factor Q 0 = 7000 limited mostly by internal losses (Q ext = 47000, Q int = 8200). Switching to lower spectroscopy powers, the magnetic ux penetrating the SQUID is swept yielding the anti crossing shown in g. 5.2b. At resonance f 0 = f ge the peak separation gives g 0 = 90 MHz as shown in g. 5.3c. The spectrum is measured with a few tens of photons such that the bare resonance is also visible in the middle.

We switch to pulsed two-tone spectroscopy to determine the transmon spectrum. Figure 5.3a shows the measured transmon frequency f ge (red dots) and the dispersively shifted resonator frequency Reaching long lifetimes in 3D circuit-QED VNA transmission spectra showing the anticrossing of the transmon and the resonator as the ux penetrating the SQUID is swept. c) Spectrum around f 0 at / 0 = 0.160 when f 0 = f g e . The separation of the hybridized transmon-resonator peaks indicated yields the coupling strength g 0 . The presence of a few tens of photons results in the intrinsic resonance frequency being also visible in the middle, as well as multi-photon peaks [5].

f ~0 (green dots) as a function of the ux penetrating the SQUID. The back line f ge max cos(/ 0 ) p with f ge max = 7.565 GHz the maximum transmon frequency corresponds to the expected behavior of a tunable transmon according to eqs. 2.62 and 2.70. The transmon spectrum in g. 5.3b is taken at / 0 = ¡0.26 and shows both f ge = 6.252 GHz and f gf /2 = 6.093 GHz, yielding the anharmonicity = 2(f gf /2 ¡ f ge ) = ¡318 MHz. Taking E C = ¡ and using eq 2.70 yields the Josephson energy E J max = 23.131 GHz at zero ux.

We measure the transmon relaxation rate ¡ 1 = 1 / T 1 at dierent frequencies as shown in g. 5.3c (black dots). It follows the Purcell limit (red line) from eq. 2.94 when the detuning = f ge ¡ f 0 is small but deviates from it as the detuning increases. The blue line is a calculated rate ¡ 1 = ¡ 1;int + ¡ Purcell with an intrinsic relaxation rate

! " # 1 # 1,int + # Purcell # Purcell # 1 [MHz]
6.1 6.2 6.3 6.4 Transmon frequency f ge (red dots) and resonator frequency f 0 (green dots) as a function of the ux penetrating the SQUID. The black line is the calculated transmon frequency f g e max cos(/ 0 ) p . b) Transmon spectrum at / 0 = 0.258 showing both f g e and f g f /2 . c) Relaxation rate ¡ 1 measurements (black dots) as a function of f g e . The Purcell limit (red line) is plotted for comparison as well as the calculated ¡ 1 = ¡ 1;int + ¡ Purcell (blue line) with ¡ 1;int = 0.0154 MHz, yielding a good t with the data. ¡ 1;int = 0.0154 MHz corresponding to T 1;int = 65 s, and ts well to the measured data.

Thermal excitations

The energy fluctuations of the electromagnetic environment (either thermal or out-of-equilibrium) can cause non-zero photon population in the resonator and excitations of the transmon.

Reaching long lifetimes in 3D circuit-QED

To estimate this photon population in the resonator we measure the relative amplitude of the photon number peaks corresponding to jn = 0i and jn = 1i as shown in g. 5.4, using a T p = 12 s long spectroscopic pulse with amplitude corresponding to a Rabi frequency R /2=1.11 MHz. The transmon excitation probability p e;0 = 0.50 at f ge;n=0 = 6.329 GHz (corresponding to Fock state jn = 0i) is estimated using the Bloch equations to be : A t of the whole spectrum by two Lorentzians with the same width yields the excitation probability p e;1 = 0.013 at f ge;n=1 = 6.322 GHz, from the relative comparison of the two peak amplitudes. To calculate the thermal population n th yielding this value of p e;1 we consider that the photon number inside the cavity uctuates with rising and lowering rates + and ¡ (where ¡ is the cavity decay rate ). From the low value of p e;1 compared to p e;0 we anticipate a very small number n th and thus a low + ' n th ¡ . We therefore only consider the case of only one photon entering (and possibly exiting) the cavity during the pulse.

Kristinn Júlíusson

The measured excitation p e;1 is simply the product of the excited population sin 2 ( f R ) acquired during the overlap of the Rabi pulse and the presence of one photon, by the probability 1 ¡ n th to have zero photon at t = 0, the probability + e ¡¡t1 dt 1 to have one photon entering the cavity between t 1 and t 1 + dt 1 , the probability ¡ e ¡¡ d to have the photon leaving after , the probability e ¡+(Tp¡t1¡ ) that a second photon does not enter, and the probability e ¡¡1(Tp¡t1¡ ) that the transmon has not relaxed at the end of the Rabi pulse, integrated over t 1 < T p and < T p ¡ t 1 .

From the more detailed discussion of appendix C we get

p e;1 ' k 2 2(1 + k 2 )(u + u 2 ) 1 + k 2 k 2 + (1 + u) 2 ¡ e ¡¡1Tp (1 + u) e ¡¡n thTp n th (5.1)
with k = R / ¡ and u = ¡ 1 / ¡ . Solving this equation numerically yields a thermal population n th = 0.22%. This corresponds to a photon temperature T ph = 56 mK calculated from the Bose-Einstein distribution.

Symmetrically, we try to estimate the thermal population of the transmon from the amplitudes A e and A g of the resonator peaks at frequencies f 0 (e) and f 0 (g)

, corresponding to the transmon states jei and jg i respectively, with A e / A g = p e / p g . First the resonator frequencies f 0 (g) and f 0 (e) are located by pulsed spectroscopy as shown in g. 5.5a, where the red spectrum is measured right after preparing the transmon in state jei. The magenta spectrum averaged using a VNA with the transmon at thermal equilibrium, shows a main Lorentzian peak at f 0 (g) , which is tted and subtracted from the measurement. This yields the background shown in green g. 5.5b. No second peak is detected at f 0 (e) beyond the noise level yielding an upper bound on the thermal transmon excitation p e;th of 0.5%.

Dephasing

We measure the free induction decay rate ¡ 2 from Ramsey oscillations at a few transmon frequencies f ge , as shown in g. 5.6b (green dots), as well as the the decay rate ¡ 2 E (magenta dots) from Hahn echo sequences (see section 4.2.6), which cancels the dephasing due to low frequency noise. These values are not necessarily true rates, since the Ramsey and echo signals decay non exponentially at certain magnetic uxes; and jei (red) using the pulsed measurement setup. For improved SNR and to be in the single photon regime, the spectrum corresponding to jgi is averaged over night using a VNA (magenta). b) An interval of two linewidths is used to t the VNA curve by a Lorentzian on top of a polynomial background (red lines). Subtracting the tted Lorentzian from the measured curve gives the background (green), possibly containing a Lorentzian contribution at f 0 (e) (dashed vertical line).

consequently, they have to be understood as the inverse of the time at which the signal has decayed by 1/e. At / 0 = 0, where f ge = f ge max , we measure the same value ¡ 2 = ¡ 2 E = 0.154 MHz. At lower frequencies we observe an increase in ¡ 2 but a decrease in ¡ 2 E . To explain this behavior we consider three contributions to ¡ 2 and ¡ 2 E : the contribution ¡ 1 / 2 from transmon relaxation, the dephasing rate ¡ th due to thermal photons in the resonator, and the dephasing rate ¡ sq due to ux noise in the SQUID:

The contribution ¡ 1 / 2 on ¡ 2 and ¡ 2 E (see eq. 2.83) is shown as a blue line in g. 5.6b, taking the tted ¡ 1 from g. 5.3b.

The transmon dephasing is attributed to thermal photon noise in the resonator and to ux noise in the transmon squid:

The dephasing rate [START_REF] Rigetti | Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms[END_REF]

¡ th = 2 Re 2 4 1 + 4i 2 + 16i n th s ¡ 1 3 5 (5.2)
is due to thermal photons randomly entering and exiting the cavity, and hence stark shifting f ge . The expression eq 5.2 is plotted in g. 5.6b (purple line) but note that its value is only valid in the dispersive regime. are plotted for a comparison with ¡ 2 E and ¡ 2 respectively. b) Ramsey fringes measured (blue dots) at / 0 = 0.268, and tted to oscillations in a combined exponential and Gaussian decay envelope f (t) = a + b cos(t)e ¡¡ 0 t¡(¡ sq;1/f t) 2 , yielding ¡ 0 = 0.078 MHz and ¡ sq;1/f = 0.146 MHz.

The last contribution, due to the ux noise spectral density S , is in the case of a white noise (see section 2.2.2)

¡ sq;w = 4 3 @f ge @ 2 S (f = 0); (5.3)
and in the case of a 1/ f noise with S = A 1/f / f and A 1/f the spectral density at 1 Hz

¡ sq;1/f 2 = @f ge @ 2 A 1/f ln 1 ! ir ; (5.4)
Reaching long lifetimes in 3D circuit-QED

where the Ramsey and echo decay e ¡(¡ sq;1/f t) 2 is Gaussian. To estimate S (f = 0) and A 1/f in the above equations, we measure Ramsey fringes at f ge = 6.157 GHz, where the ux noise starts to be the dominant contribution, and t it with f (t) = a + b cos( t)e ¡¡ 0 t¡(¡ sq;1/f t) 2 as shown in g. 5.6c. We nd ¡ 0 = 78 kHz w¡ 1 / 2 + ¡ th = 88 kHz (see magenta line in g. 5.6b), which means the white ux noise is a minor contribution that we neglect. The t gives also ¡ sq;1/f 2 yielding A 1/f = 8.3 0 / Hz p , by taking = 12 s (the longest measurement sequence) and ! ir / 2 = 1kHz (the repetition rate of the experiment). This value is comparable to the ux noise observed in SQUIDS all over the world for the past 30 years [START_REF] Frederick | Lowfrequency noise in dc superconducting quantum interference devices below 1 K[END_REF], and yields the cyan line plotted in g. 5.6b. Because this last contribution is Gaussian , we'll 'sumup' all the contributions at the experimental time

T 2 = 1 / ¡ 2 as ¡ 1 /2 + ¡ th + ¡ sq;1/f 2 /¡ 2 ,
for a valid comparison with the measured ¡ 2 (same 1/e decay). Let's now discuss the results:

At / 0 = 0 the sensitivity to ux noise is zero, the contribution from ux noise is thus zero, and the calculated value ¡ 1 / 2 + ¡ th (magenta line) ts the measured value of ¡ 2 . Moreover the contributing noise is white, such that the echo cannot cancel a signicant fraction of it, and ¡ 2 ' ¡ 2 E . When the frequency is decreased, the sensitivity to ux noise increases, the corresponding contribution increases quickly, and makes the total ¡ 2 increase by a factor 3. As this noise is essentially at low frequency, the echo cancels a large fraction of it, so that ¡ 2 ' ¡ 1 /2 + ¡ th within a factor 1.7 . At low frequency, where the ux noise contribution becomes dominant, the ¡ sq;1/f deduced from the t at intermediate frequency (cyan line) seems to underestimate the eect (it is far from the bottom green circles): hence the discrepancy of up to a factor 3 between the measured ¡ 2 (green circles) and the total calculated ¡ 2 (green triangles). The reason for this underestimation remains not understood.

Transmon on Sapphire substrate

The transmon KJ15ch11 discussed here had the same dimensions as the transmon in section 5.2.1 but was fabricated on a sapphire substrate with a low loss tangent, which is important for future experiments in high-Q cavities.

Kristinn Júlíusson

The purpose of this sample was on one hand to verify that we could reproduce a long ¡ 1 with the new recipe developed for sapphire substrate. On the other hand we wanted to improve ¡ 2 at the optimal cavity-transmon detuning for readout by having the maximum transmon frequency well below the resonator frequency: indeed at high frequencies f ge , the sensitivity to ux noise j@f ge / @ j is low, and ¡ 2 E and ¡ 2 are only bounded by ¡ 1 / 2, which is xed by the detuning. We thus reduce E J max to achieve this goal. Since all other parameters are kept the same, we will only discuss these new goals. ! " The transmon and resonator frequencies f ge and f 0 are mapped in g. 5.7a. The maximum transmon frequency f ge max = 6.62 GHz gives E J max = 18.9 GHz, E C being the same as for the previous sample.

(# 1,int +# Purcell )/2 # 2 on Si # E 2 on Si # E 2 on sapph. # 2 on sapph. (# 1,int +# Purcell ) # 1 # [MHz]
The transmon relaxation and free induction decay are measured at three to four dierent frequencies as shown in g. 5.7b. The measured ¡ 1 (blue dots) are very similar to those in the previous sample and a calculated ¡ 1 = ¡ 1;int + ¡ Purcell (blue line) with an intrinsic ¡ 1;int = 0.013 MHz ts the data.

Reaching long lifetimes in 3D circuit-QED 117 This sample yields values of ¡ 2 (green dots) and ¡ 2 E (magenta dots) a factor 1.5 to 2 lower than the previous sample (green and magenta squares), as expected. The rate ¡ 2 increases with lower f ge while ¡ 2 E is close to ¡ 1 /2 (blue dashed line), illustrating once more the eect of the echo sequence on low frequency noise.

Conclusion

We successfully fabricated and characterized a ux tunable transmon qubits, fabricated both on Si or sapphire substrates, coupled to the fundamental mode of a 3D Cu cavity resonator. We measured an upper bound of 0.5% on the thermal excitation of the transmon and a thermal population of in the cavity n th = 0.22%, which is lower than achieved in similar experiments [8][43].

The relaxation time T 1 reaches 40 s at high transmon-resonator detuning but is still limited by the Purcell eect. The intrinsic lifetimes, estimated by removing the Purcell contribution, are 65 s and 77 s for the Si and sapphire substrates respectively, which is of the same order as the longest lifetimes observed today [START_REF] Rigetti | Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms[END_REF][37] [16]. Measurements of T 2 E indicate that T 2 could be close to 2T 1 when the 1 / f ux noise is negligible (even at high values T 1 40 s). This can be achieved for a small frequency range f ge near / 0 = 0 by choosing E J max appropriately.

As a conclusion the parameters as well as an appropriate E J max are well suited for QZD experiments. However, our implementation required a re-design of the transmon geometry for reasons explained in chapter 6. We observed a lower T 1 , probably for the following reason:

In the new design the pads are further apart (3 mm) and connected by a few m wide wire, causing a high concentration of the electric eld. Such narrow structures have been shown to have shorter relaxation times than broader ones [16].

Chapter 6 Design and characterization of an sample suited for QZD

This chapter discusses the design and characterization of the sample used for the QZD experiments presented in chapter 7.

Design and fabrication

We designed a 3D circuit-QED system that realizes the ef-resonant scheme presented in chapter 2, where a high-Q mode of a cavity resonator is coupled to a transmon with a weak coupling constant. Another cavity mode, with a low-Q and a strong coupling constant, is dispersively coupled to the transmon and used to readout its state. We refer to the high-Q mode as the 'storage mode' or 'oscillator', and the low-Q mode as the 'readout mode'. A circuit diagram of the system is depicted in g. 6.1.

Q 0 Q R oscillator f 0 tunable transmon readout f R g 0 g R 1 2 Figure 6.1.
A circuit diagram of two modes of a cavity coupled to a transmon. The transmon and the so-called storage mode at frequency f 0 represent the harmonic oscillator and 3LS in the ef-resonant scheme discussed in chapter 2. The additional mode at f R is used to readout the transmon state.

To estimate the target parameters well-suited for QZD experiments we consider a cavity approximated by a cuboid with dimensions [x=44, y=56, z=6] mm. The storage mode is the TE 120 mode at frequency ! 0 / 2 6.4 GHz and the readout mode, TE 210, is at frequency ! R / 2 7.3 GHz, as shown in g. 6.2. We try to obtain the highest Q int possible for the storage mode, expecting at least 1 million as seen in [START_REF] Hanhee Paik | Observation of High Coherence in Josephson Junction Qubits Measured in a Three-Dimensional Circuit QED Architecture[END_REF]. The readout mode is strongly overcoupled at Q tot 10k for fast readout and should have a much larger Q int . In order to overcouple the readout mode and undercouple the storage mode simultaneously, we place the coupling pins close to an electric eld node of storage mode and an antinode of the readout mode.

To induce a photon number splitting

;n g 0 ¡ 2(n + 1) p ¡ 2n p > 2 MHz (see section 2.2.

Kristinn Júlíusson

of the transmon with respect to the elds nodes and antinodes of both modes using the same logic as for the coupling pins.

Transmon fabrication is not precise enough to make a non-tunable transmon that fullls the ef-resonant condition. Consequently the transmon has to include a SQUID to be tunable with magnetic ux. This poses a challenge since achieving a high-Q of the cavity requires it to be superconducting and thereby screening all magnetic eld. We solve the problem by inserting the transmon only partially into the cavity such than one pad is inside and couples to the eld modes, whereas the other pad as well as the SQUID are outside the cavity, where a magnetic eld can be applied. The transmon is inserted through a slit made in one of the cavity walls that has been made particularly thin (1mm).

Cavity

Finite element microwave simulations were used to precisely determine the cavity geometry. Two cavities were then machined and assembled, before being tested to verify that they were suited for QZD experiments.

3D simulations

The actual cavity design is essentially a cuboid, with the same dimensions as in the last section, where some of the edges have been rounded for the purpose of practical machining. It is simulated in a 3D simulation software named CST microwave studio [53]. The two cavity ports are located 0.5 mm away from the storage mode node plane as shown in g. 6.3 a. They are made of 3 mm long and 2 mm diameter holes, added to the geometry as vacuum cylinders, and 0.6 mm diameter coaxial Cu pins (cylinders) whose length can be varied.

The slit, the sapphire chip and the Cu block are added to the model as shown in g. 6.3b-d. The slit is place on the wall opposite to the pins and centered 0.45 mm away from the storage node plane. The shape of the slit is the same in the cavity wall as in the Cu block shown in g. 6.3d. The slit edges are 0.5 mm wide to guide the 0.33 mm thick chip while the center is wider (0.7 mm) to avoid scratching the transmon and lessen the coupling to potentially lossy metals. The sapphire chip is extends from the bottom wall of g. 6.3b, throughout the slit, and into the Cu block. In CST the coupling pins and Cu block are modeled as normal metal with the room temperature resistivity of Cu, the sapphire chip a lossless block with relative permittivity " r = 9.3, the rest of as vacuum with lossless walls

The simulated transmission spectrum between the two ports is shown in g. 6.3 e. It shows resonances of the rst three modes of the cavity at 4,4 GHz (fundamental mode TE 110), 6.4 GHz (storage mode TE 120) and 7.3 GHz (readout mode TE 210). Note that the frequency sampling of the simulator is not sucient to resolve the resonance line shapes. Figure 6.4 shows the electric eld and surface current distributions of the modes. The presence of the sapphire chip induces a notable asymmetry between the two antinodes of the readout mode.

The cavity is carved out of two blocks of metal which are then put together as shown in the next section. With that in mind we attempt to obtain the highest-Q possible for the storage mode by cutting the cavity along the node of the readout mode such that the supercurrents of the storage mode are parallel to the cut and thus not perturbed.

Realization and testing

First, two identical cavities are made by an external workshop according to the design described above, but without the slit and the Cu block, which are added later by the in-house workshop. The bulk of the cavity is made of a high purity aluminium (N4) and additional assembling parts are made of copper, with some being goldened to prevent oxidation. The coupling pins are SMA feedthroughs with a Cu wire soldered into the hole on one end. The SMA throughs are then screwed into the cavity wall, until the Cu pin enters into the cavity through a small hole, and is then fastened with a nut. This way the penetration of the pin into the cavity can be easily adjusted. By measuring the S-matrix at room temperature the coupling of each pin can be tuned in-situ as explained in chapter 2. The coupling pins and the two nal designs of the cavity, varying by the orientation of the slit, are shown in g. 6.5. The internal quality factors Q int of the dierent modes, at dierent steps towards the nal design, are estimated from measurements of the external quality factors Q 1;2 at room temperature (see section 2.2.1.2) and from the total quality factor Q tot measured at 30 mK. Anticipating Q int of order 1 million, we choose Q 1;2 100k on the readout mode in order to overcouple the readout mode without undercoupling excessively the storage mode: Like this the sensitivity to estimate Q int;R is enough while the transmitted signal at the storage mode frequency is suciently strong.

The rst preparation step after machining the cavities was to clean them in an ultrasonic bath of acetone and to chemically polish them for 15 hours in a bath of {4H 3 PO 4 ,4CH 3 COOH,1HNO 3 ,1H 2 O}, removing roughly 20 m of material. Measurements at 34 mK gave Q 0;int 5M and Q R;int 20k for both cavities. Thereafter one of the cavities was developed towards its nal state. The Q int 's at each step are listed in table 6.1: The slit lowers Q 0;int to 0.6M . Three steps followed after that; the slit was covered with a Cu block, a sapphire chip was inserted in it and an indium seal between the two cavity parts was added, with no clear eect on Q 0;int . On the contrary, Q R;int remained low compared to the other modes until the In seal was added, indicating that the poor connectivity between the two cavity halves was dominating losses.

The transmission measurements of the three modes after the last development step is shown in g. 6.6. With these Q-factors we proceed with towards the QZD experiment, overcoupling the readout mode (Q R;tot 10-20k). The storage mode remains strongly undercoupled with Q 0;int ' Q 0;tot in the range from 0.7 -2.2M at 30 mK, depending on the quality of the In seal.

The cavity design was slightly modied for the other cavity where the slit was aligned parallel to the supercurrents of the storage mode. After removing another 20 m of material with chemical polishing it was measured to have Q 0;tot = 10M. This cavity design was however implemented late in the thesis period and we never managed to couple it successfully to a transmon.

Transmon qubit

The transmon design for the QZD experiment is signicantly modied from the design presented in chapter 5. The mode couplings and anharmonicity is estimated from CST simulations while the lifetimes T 1 and T 2 are tested experimentally.

3D simulations

Inspired by refs. [43][35] the transmon-cavity system is simulated to estimate the couplings g 0 , g R and the anhamonicity . This is done by adding to the previous simulation the two transmon electrodes modeled as 2D sheets of perfect conductors on the sapphire block ( see g. 6.7a). The pin ports are removed and the SQUID is replaced by an internal port.

Design and characterization of an sample suited for QZD The total admittance Y total (!) of the transmon-cavity system is the sum of the admittance Y S (!) seen from the port in the simulation and of the SQUID admittance Y J (!) ' 1/iL J !.

The resonance frequencies given by Im[Y total ] = 0 (see section 2.2.1.2) now includes the transmon mode. By varying L J this mode can be brought close to a cavity mode. These two modes (cavity and transmon) repel each other, as shown in g. 6.7 b, which reproduces the anticrossing with the minimal separation 2g, as shown in g. 6.7 c. The anharmonicity is estimated from the charging energy E C , with ' E C = e 2 / 2C (eq 2.63). C is here the simulated total capacitance C S = 1 / i!Y S (!) seen from the port, at frequencies ! much lower than all cavity resonances. We thus simulate Y S (!) at ! /2 = 1 GHz and get E C ' . In the simulations the bottom pad is placed 0.5 mm below the top of the Cu block, with the port 50 m above the pad (see g. 2.3b). A 1 m wide wire extends from the bottom pad to the top pad located inside the cavity. The couplings g 0 , g R are varied by changing the Design and characterization of an sample suited for QZD pad separation and the dimensions of the top pad as shown in g. 6.8a,b. The value of g 0 is governed by its distance to the storage mode node. After the values of g 0 and g R have been determined, the width of the bottom pad can be varied to get the targeted E C with only a small eect on the couplings.

A geometry with pad dimension 200x200 m 2 (top) and 200x400 m 2 (top), separated by 2.9 mm and located 0.78 mm away from the node of the storage mode, we get g R = 120 MHz, g 0 = 7.5 MHz and E C = 493 MHz, with which we proceed to experimental testing.

Realization and testing of preliminary samples

We fabricated and tested four preliminary transmons with the dimensions described above (see g. 6.9) using a similar recipe as for sample KJ19ch1 (see section 3.1.5).

The rst three transmons had problems such as a very short T 1 = 500 ns, being shorted, and having unexplained instable behavior. The fourth sample gave the rst impression of being suitable for the QZD experiments: From a room temperature measurement of the SQUID , estimated as twice the separation between f ge and f gf / 2, varies from 230 MHz at / 0 = 0 to 500 MHz at / 0 = 0.4. We attribute this variation partly to the strong coupling to the readout mode that shifts the jei and jf i levels to a dierent extent, and partly to the possible change of the transmon capacitance C between 8 and 5 GHz. .10 b shows relaxation-and dephasing time measurements at / 0 = 0, where f ge =8.01 GHz, giving T 1 = 1.5 s and T 2 = 2.8 s (see protocol in section 4.1). T 1 was measured for dierent f ge and compared with the estimated Purcell limit due the its coupling to the readout mode. Figure 6.10 c shows an intrinsic T 1 = 3 s but at the efresonant condition ! g e = ! 0 + = 2 6.76 GHz, it is reduced to 1.8 s because of the Purcell eect.

At this ef-resonant working point, where = ¡320MHz, we discovered a problem making this seemingly usable sample ill-suited for the QZD experiments. The energy of the level jf i happens to be unstable. This is observed by tuning the transmon frequency to 50 MHz below the ef-resonance, i.e. f ge = 6.71 GHz, exciting the transmon to jei and performing spectroscopy of the jei ! jf i transition. Figure 6.11 a shows the monitoring of the resonance peak over 17 hours and g. 6.11 b shows a few selected traces. The resonance varies from being a single peak at 6.378 GHz to two peaks separated by up to 7MHz. This behavior studied in ref [START_REF] Ristè | Millisecond charge-parity uctuations and induced decoherence Bibliography in a superconducting transmon qubit[END_REF] is due to uctuating charges: As described in chapter 2 (see g. 2.8), the energy levels of the transmon are modulated as a function of the reduce charge n g , with a period of one Cooper pair 2e, and with an amplitude that increases with E C . Two types of n g uctuations are here in play: Electrons in dielectrics in the vicinity of the transmon electrodes move around, yielding a drift in n g . In addition the number of out-of-equilibrium quasi-particles, with charge e, uctuates rapidly which induces jumps n g = 0.5 [START_REF] Ristè | Millisecond charge-parity uctuations and induced decoherence Bibliography in a superconducting transmon qubit[END_REF], and therefore two branches of each energy band. The two peaks shown in g. 6.11 correspond to the two branches of the jf i level, whose separation varies with n g . For n g = 0.25 [2e] the energies are degenerate for the two parities, resulting in a single peak, while at n g = 0 the separation is maximal and equal to the dispersion of the energy band f (eq. 2.68). The dispersion f = 7 MHz observed here corresponds to E C = 427 2 MHz, according to calculations of the exact Hamiltonian of the transmon coupled to the readout modem using the measured couplings and transition frequencies.

This results shows that a new transmon geometry with smaller E C is needed, keeping similar coupling parameters g 0 and g R .

Characterizing the sample used in the QZD experiment.

The fth fabricated transmon sample (KJ19ch1) was well suited for the QZD experiment.

Its pad dimensions are modied to achieve the same couplings to the modes but a 25% lower E C . This results in a 5 m wide wire and pad sizes to 360x200 m 2 and 200x315 m 2 , as shown in chapter 3. Simulations give E C = 385 MHz, g 0 = 7.5 MHz and g R = 117 MHz, and fabrication as described in chapter 3 yields E J ;Max = 30 GHz, estimated from room temperature resistance of the SQUID.

The coupling pins to the cavity were asymmetrically tuned for better detection eciency, with Q in =116k and Q out = 20k on the input and output ports. This amounts to Q ext = 17 k.

Design and characterization of an sample suited for QZD

Spectroscopic characterization

Once the sample is at 34 mK the transmission spectrum is measured with a vector network analyzer (VNA) as a function of the applied ux . The hybridized frequencies f ~R and f ge extracted from the spectra are shown in g. 6.12a (red dots). At / 0 = 0.23 and 0.31 the frequency f ge anticrosses with the readout and storage modes giving

-0.2 0.0 0.2 0. 4 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7
f R = 7.33 GHz Q R = 13k f O = 6.43 GHz Q = 2.2M (d) (c) ef-resonant 3LS
/ 0 (a)

2 x (g 0 = 7.45MHz) frequency f [GHz] 2 x (g R = 149MHz) f e0=>+/-1 = -261MHz
Figure 6.12. Spectroscopic characterization of the system. a) Measured transition frequencies as a function of the magnetic ux applied though the transmon SQUID. Blue and purple horizontal dashed lines correspond to the readout and storage mode frequencies f R and f 0 , red dots to the hybridization of f g e and f R and the green line to f e f . The 'ef-resonant' condition is obtained at f 0 = f R . b) Transmission spectrum at f R = f ge determining the transmon coupling g R to the readout mode. c) Transmon jei ! jf i (magenta) spectroscopy at f 0 = f e f . The double peak corresponding to the transitions je0i ! j1i shows the ef-resonant working point. Spectroscopy on jgi ! jei (red) showing f ge yields the anharmonicity . d) Transmission spectrum at f 0 = f g e determining the transmon coupling g 0 to the storage mode.

Kristinn Júlíusson

g R = 149 MHz and g 0 = 7.45 MHz, respectively (g. 6.12b,d). The second excitation frequency f ef is obtained by pulsed jei ! jf i spectroscopy as for g. 6.11 described in the previous section. This time it is stable and is displayed as the green line in g. 6.12a. Figure 6.12c shows the transmon spectrum at / 0 ' 0.291, where the ef-resonance f 0 ~= f ef occurs (f 0 ~= 6.4328 GHz is the dispersively shifted frequency -see appendix A). The red resonance line shows f ge = 6.694 GHz, and the magenta line the jei ! jf i spectroscopy. It reveals two peaks at f 0 ~ 10.4 MHz, representing the transitions je0i ! j¡1i and je0i ! j+1i. The green vertical dashed line midway between the two magenta peaks indicates f 0 ~= f e f , yielding an anharmonicity = ¡261 MHz, which corresponds to E C = 238 MHz. At / 0 = 0.5, f g e is almost zero and transmons eect on the cavity modes is negligible. Figures 6.13a,b show the resonance peaks of the storage and readout modes respectively at that point, revealing Q 0 = 2.2 million and Q R = 12 thousand. This entails that the lifetime of the storage mode is T 0 = 54 s. The sample was warmed up to room Design and characterization of an sample suited for QZD temperature due to technical reasons a three times during the measurement period, but the sample was never disturbed. A degradation in Q 0 was observed each time: 2.2 ! 1.9 ! 1.6 ! 1.2 million. We speculate that the repeated cycle of thermal contraction and expansion degraded the quality of the In seal.

The relaxation time of the transmon was measured when f ge = 6.18 GHz, giving T 1 = 4.1 s and is shown in g. 6.15 a. Subtracting the Purcell eect gives an intrinsic lifetime T 1 = 6.2 s.

Characterization at ef-resonance

The sample is now characterized further at the ux bias / 0 = 0.291, where the sample is at the ef-resonance.

Fine tuning the ef-resonance

The ef-resonant point is reached by ne tuning the ux bias using the following method: The storage mode is populated with a coherent eld with n ' 1 using a weak 150 ns pulse. Spectroscopy around f ge is then performed. With n ' 1 the jn = 0i and jn = 1i Fock state population probabilities are p(0) ' p(1) ' 0.36 such that the three transitions jg0i ! je0i, jg1i ! j¡1i and jg1i ! j+1i are observed. The ux bias is then tuned such that the frequency separation of the peaks is equal, i.e.

¡;0 = +;0 , as shown in g. 6.14, where ¡;0 =10,4 MHz and +;0 = 10; 4 MHz with an error of 0.1MHz. This is in accordance with the predicted value of 2 p g 0 = 10.5 MHz.

6.680 6.685 6.690 6.695 6.700 6.705 6.710 0.10 0.11 0.12 0.13 0.14

!"# amplitude [V] Frequency f [GHz] !"#

Spectroscopy of the photon number splitting

The Fock state dependent transitions f ;n are found by repeating the same type of spectroscopy, varying the lling n to populate the dierent Fock states. Figure 6.16a shows the peaks at frequencies f n for n = ¡1; 0; 1; ::::; 16, resulting from three values of n , and reproduce approximately the Poissonian distribution as will be discussed in the next section. In g. 6.16b the experimentally measured frequencies (orange dots) are compared to the analytical calculations (including 3 transmon levels) from appendix A and to the numerical diagonalization of an eective oscillator-transmon Hamiltonian including also the fourth level of the transmon jhi.

Note that the dierence between the analytical calculations and numerical diagonalization of the Hamiltonian with three levels is indistinguishable. A fair agreement is observed between the experiment and the eective four level transmon model where the values of f 0 , g 0 , g R , as well asf ge , are extracted from measurements, and the energy 3f ge ¡ 848 MHz of level jhi is calculated from g R and f ~R. This shows that the photon number splitting is greater for the + ladder than for the -ladder, which is a consequence of the non-resonant transmon levels jg i and jhi.

Measuring the Fock state probabilities

This section contains three calibrations needed to measure accurately the occupation probabilities p(n) of the dierent Fock states. The height of the photon number peaks when a pulse is applied need to Kristinn Júlíusson be mapped to p(n). In addition to calibrating pulses with Rabi oscillation, we apply two calibrations. One that scales the readout signal, which varies with the storage mode lling, to S n 2 [0; 1] and another that maps the signals S n to p(n), taking into account the relaxation of the transmon during the pulse or the readout.

Calibrating pulses on the +; n transitions

The Rabi rotation on the +n transitions are calibrated at frequencies f +;n for n = 0; :::; 10 and a lling n ' n for each f +;n . The Rabi oscillations shown in g. 6.17a are induced with Gaussian-shaped pulses with a standard deviation = 140 ns and a total length of 6, and are varied in amplitude . The amplitudes of the oscillations decrease with n as expected from the Poissonian distribution of the coherent state j n p i. The Rabi frequencies ;n are show in g. 6.17b(dots). The Rabi frequencies for n > 0 are 2 p slower than for n = 0 because of the hybridized character of the former. For n = +1; ::::; +10 the Rabi frequency increases slightly. The numerical calculations (circles) capture the change in Rabi frequency between n = 0 and n > 0. It however predicts decreasing Rabi frequency with n for the + ladder and an increasing Rabi frequency for the -ladder. We have not managed to explain this discrepancy. The left inset conrms the linear increase of with and yields the top scale of panel a. Residual error between data and r are homogeneously distributed over the whole dataset ( similar to the example shown in chapter 4) in a Gaussian manner with a standard deviation = 0.6% and a shift of 0.1% (right inset). Lower inset is the reconstruction of the Wigner function of a targeted state = ¡ 5 p + i 2 p using the tomography and maximum likelihood method.

With the c n coecients giving access to p(n), a full characterization of the mode can be performed using one of the quantum state tomography methods of section 4.2.2.

Non-linearities of the storage mode

We now demonstrate experimentally the low Kerr factor at the efresonance. With g 0 = 7.45 MHz and = ¡261 MHz eq. 2.105 gives a Kerr factor K e f = ¡346Hz per photon. However, when diagonalizing numerically the Hamiltonian of the system including jhi, the Kerr eect is found to be even smaller, depend on n, and to chancel and revers sign at about 20 photons. Because this ultra-small Kerr eect is much slower than the relaxation time T 0 = 54 s it is very dicult to measure dynamically, by either recording the trajectory of a eld in phase space as in [START_REF] Yi Yin | Dynamic quantum Kerr eect in circuit quantum electrodynamics[END_REF][27], or the power dependence of the resonance line shape at short times as in the supplementary material of [27]. Consequently, we simply measure the steady-state power transmission spectrum P 2 (f ) around f 0 ~for several input powers P 1 , using the VNA with measuring bandwidth of 1kHz 1/ T 0 . The corresponding curves are shown in g. 6.20 a (dots) in arbitrary units of the transmitted power P 2 (left scale).

Analyzing quantitatively the dataset requires a precise knowledge of the average photon number n in the storage mode as a function of ?) P 1 = ¡149.6 dBm (n = 1) at resonance}. The non-linearity in frequency calculated with 4 transmon levels (magenta line) is smaller that the -346 Hz/photon Ker constant (dark blue line). A large non-linearity in power is however observed (see text). b) Calibration of P 1 versus n . Red: Spectroscopy of the jg0i ! je0i transition with a calibrate pulse and no lling in the storage mode. Blue: measurement of the transmission signal at f g e (vertical blue dashed line), indicating the occupation probability p(0), as the storage mode is lled with a continuous drive at increasing power P 1 . The average photon number n reaches 1 when P 1 = ¡149.6dBm (thin blue dashed line). c) Simulated photon number distributions p 1 (n) at the top of each simulated resonance of panel a. Note the 0.25 scaling factor for n = 1 f and P 1 . This precise knowledge is dicult to obtain, rst because of the very weak coupling between the input and output ports, and second due to the large systematic errors of 1.5 dB and 3 dB on the transmission of the input and output lines respectively. Therefore further insitu calibration and data analysis is performed: using the modulationdemodulation setup (see chapter 3), a continuous tone is applied at f 0 ãt varying powers P 1 and the occupation probability p(0) of the vacuum Design and characterization of an sample suited for QZD measured, as shown in g. 6.20b. Assuming a Poissonian distribution, n = 1 when p(0) = 0.365 which corresponds to a nominal value P 1;1 = ¡149.6 dBm.

Then quantum simulations of the storage mode coupled to the effective 4 level transmon are performed using the QuTiP python toolbox [25], the already mentioned parameters and the experimentally calibrated P 1 . The solid lines on g. 6.20a show the simulated transmission spectra for values of n corresponding to P 1 , using the steady-state solver of QuTiP. By scaling vertically the experimental curves such that the simulation and experiment match for P 1 = ¡135 dBm, we obtain a fair agreement for all curves, leading to the following results: First, the resonance lines display indeed a very small Kerr eect, with a shift towards lower frequency with increasing P 1 that is signicantly smaller than K ef (oblique dark blue line), and changing sign between n = 15 and 25, in good agreement with the calculated shift (magenta line). A second and unanticipated result is that although the non-linearity in frequency is small, a large non-linear in input power is observed, with P 2 and n increasing by only a factor 3 when P 1 is increased by a factor 8.

Simulating the time evolution of the system initialized in its ground state and then driven coherently at f 0 ~reveals the cause of this nonlinearity: due to the small hybridization of the storage mode with the transmon (see appendix A), and the nite lifetimes (T 1 ; T 2 ) of the latter, the eld starts perfectly coherent at the beginning of the dynamics and slowly becomes incoherent when approaching the steady state. The decoherence speed increases with n as illustrated in g. 6.20 c where the photon number distribution p P 1;1 (n) obtained from the steady state solver at the top of each resonance curve of g. 6.20 a: although p P 1;1 (n) corresponds almost exactly to the Poissonian distribution at times scales relevant for the calibration discussed in the last section, the distribution becomes increasingly non-Poissonian for larger n . This means that large coherent states can only be obtained at times shorter than a few tens of T 2 , which was the case in the previous section where the total operation sequence (create state+tomography) was t ' 3T 2 / 5.

Measuring the Wigner function

As mentioned in section 4.2.2 the Wigner function at point in phase space can be directly measured by a displacement D(¡) and a parity Kristinn Júlíusson measurement of the eld (eq. 4.4) using two dierent methods. These method are here implemented with a truncation at 10 photons i.e. with truncated parity operators

P 0 = X n=10 p(n) n=even ¡ X n=10 p(n) n=odd ( 6.1)
and

P 00 = 1 ¡ 2 X n=10 p(n) n=odd : (6.2)
In the rst case the p(n) are measured separately using the method described above in section 6.2.3. In the second case the odd photon number states are excited simultaneously up to n=9 with superimposed Gaussian pulses. The selectivity of these pulses is limited by their Gaussian frequency distribution and o-resonant driving. We test this selectivity by driving Rabi oscillations on jg0i ! je0i with the storage mode in jn = 0i, while at the same time applying the oresonant superimposed pulses at frequencies f +;n for n 2 f1; 3; 5; 7; 9g. The two truncated-parity measures are tested on the vacuum state as shown in g. 6.21a, where dots are the measured values and the lines are calculated values from eqs 6.1 and 6.2. The truncated parity measures follow the expected Gaussian decrease in n p with a halfwidth at half-maximum of 1/ 2 p , but only up to n p ' 2.5 where errors occur.

We use the parity measure P 00 to map the Wigner function of the vacuum state, pixel by pixel, as shown in g. 6.21b. This measurement consisted in 2500 measurements in phase space which is signicant over sampling for state dened by 63 real numbers in an 8 dimensional Hilbert space. The yellow color at the corners of the graph shows the eect of the truncation. As a comparison the Wigner function in g. 6.21c is reconstructed using the maximum likelihood method discussed in section 4.2.4, with 5 displacements and n 2 [0; 4]. The density matrix , in a Hilbert space truncated to 10 photons using the maximum likelihood algorithm.

These two method will also be used in chapter 7 to observe the quantum Zeno dynamics of the eld The two parity measure P 0 (blue) and P 00 (red) applied to the vacuum state.

The measurements (dots) behave as calculations predict (lines) and give correct measures of the parity up to n p 2.5, where the truncation starts to have an eect. b) Measurement of the Wigner function using P 00 . c) Reconstruction of the Wigner function from measurements using the maximum likelihood method.

Kristinn Júlíusson

Chapter 7

Experimental observation of quantum Zeno dynamics

With all the tools described in the previous chapters in place, we show in this last chapter the QZD in our ef-resonant system, where a continuous blocking tone at f +;n for n 2 f2; 3; 4g compartmentalizes Hilbert space into H <n and H >n . This separation is represented by an an exclusion circle (EC) in phase space. We investigate the dynamics under a continuous drive of the storage mode at frequency f 0 . The rst section shows how the drive strengths at f 0 and f +;n are chosen. The second section shows QZD conned inside the exclusion circle (EC) corresponding to n = 3 or n = 4, and compares the experimental results with numerical simulations taking into account the dierent decoherence channels. The last section shows two cases of a state being prepared outside the EC at n = 2, and colliding with it either head on or tangentially. Lastly, we summarize some of the experimental work on QZD done by other groups during the thesis period.

Probing the Zeno blockade strength

Our QZD protocol consists in applying simultaneously a Zeno blocking tone at a particular frequency f +;n , and a coherent drive to the storage mode at f 0 ~, as shown in g. 7.1. Square pulses with Gaussian rise and fall are used to limit their frequency spread. The Gaussian ends are truncated at 3, with = 80 ns and 10 ns for the blocking and lling pulses, respectively. The pulses are aligned such that the plateaus are reached simultaneously, which ensures the storage mode is never driven in the absence of a blocking tone.

In order to probe the blocking strength, we start with the storage mode in the vacuum state, (no step 1) and displace the dimensionless eld in the IQ-plane (see section 4.2.3) at a constant rate _ = 0.27 s ¡1 for up to 30 s, while applying a blocking tone at n = 3 with a Rabi frequency +;3 / 2 varying from 0 to 2 = _ 0.54 MHz. The Fock state occupation probabilities p(n) for n 2 f0; :::; 4g are then measured, see g. 7.2. At low Rabi frequency the monitored photon number states are gradually populated with maximas at times 0, 4, 6, 7.5, 8.5 s and remain empty after that. This evolution of the Fock state populations is expected in the absence of Zeno blockade. At Rabi frequency +;3 / 2 ' _ = 0.27 s ¡1 , an oscillation in the population of the Fock states j0i, j1i and j2i with a period of about 15.5 s gradually appears, meanwhile the population of j3i and j4i is suppressed. The oscillation and suppression become gradually clearer as the Rabi frequency reaches +;3 / 2 = 2 _ . The period of the oscillations decreases towards the asymptotic Zeno period T z = 2 / 3 p _ ' 14 s expected from a coherent drive in a Hilbert space truncated to three levels. Experimental observation of quantum Zeno dynamics is reduced down to 7% when blocking at the maximum strength +;3 / 2 = 0.54 MHz, as shown in g. 7.2b. This transmission decreases down to 7%, a value close to the 5% obtained from simulations described later in this chapter. Using a higher Rabi frequency would further reduce the transmission and improve the blockade of Fock state j3i, but at the expense of also blocking j2i partially, as we discuss now.

The f +;2 and f +;4 transition frequencies are +;2 = 3.9 MHz and +;3 = 3.3 MHz away from the blocking frequency f +;3 (see g. 6.16). These transitions are thus o-resonantly driven around a vector of the {jgni; j+ni} Bloch sphere at an angle n = arctan( +;3 / +;n ) with the z-axis (see section 2.2.2.3). These angles are 8 and 9 for n = 2 and 4, and would correspond to a population transfer from jg2i to j+2i up to 14% in absence of cavity drive ( _ = 0). In presence of drive during the QZD, the population of j+2i is however much lower (a few percent) as simulations will show.

To emphasize this off-resonant drive effect, we perform another experiment exaggerating the eect. Instead of varying the amplitude of the blocking tone, we keep it xed at the same maximum value as before and sweep its frequency f between f +;2 and f +;4 (see g. 7.3). The white dashed lines indicate the Zeno blockades at the exact frequencies f +;nblock with n block 2 f2; 3; 4g, with increasing Zeno periods. In between, when blocking at intermediate frequency (f +;2 + f +;3 ) / 2 or (f +;3 + f +;4 ) / 2, the blockade is suppressed substantially but not completely: oscillations in p(n) remain up to 30% due to oresonant driving (and partial blocking) of the adjacent transitions. The o-resonant driving eect at these intermediate frequencies, although exacerbated, conrms that it is detrimental to increase the blocking strength further.

The lling rate _ = 0.27 s ¡1 used here yields about two Zeno period in 30 s, a time of the order T cav . Since slowing down the lling would lead to fewer Zeno periods during the cavity lifetime, we choose to keep this lling rate for the next experiment. We also keep the maximum Zeno blocking strength R / 2 = 0.54 MHz, since it balances the few percent o-resonant population of j+2i against the 5-7% leak through the EC. at half a period.

Experimental observation of quantum Zeno dynamics

The QZD with a blockade on n=4 is shown in g. 7.6. The same drive amplitude is used as for the blockade on n = 3 and gives approximately the same Rabi frequency +;3 ' +;4 . At half integer periods the state is in a superposition of amplitudes = 1.5 and fringes with three minima lie between them. At half a Zeno period the minimas all reach negative values but are strongly reduced one period later.

In prevision of comparing the measured dynamics with simulation, we record the data at smaller time intervals for the same conned dynamics as in g. 7.5 (blockade at n=3). Instead of measuring the full density matrix, which is time consuming, we only measure the photon number probabilities p(n) for n = f0; 1; 2; 3; 4g. The results, plotted in g. 7.7 (dots), show a Zeno period of about 13.5 s and a 5% transmission per period.

Kristinn Júlíusson

Simulating the QZD of our transmon oscillator system

To understand the dynamics and identify the eect of the dierent decoherence channels we now simulate the QZD.

The model

We model our system as a harmonic oscillator truncated to 30 levels coupled to a three level transmon with the transition ! ef resonant with the oscillator. The Hamiltonian of the system is

H = H 0 + H c + H dr ; (7.1)
where the three terms are the free, coupling, and drive Hamiltonians respectively. In the lab frame these three Hamiltonians write

H 0 = f 0 a y a + (f 0 ¡ )jeihej + (2f 0 ¡ )jf ihf j H c = g 0 [a q a y + a q y a]; (7.2)
H dr = " ll cos(2f 0 t)(a + a y ) + " s cos(2f +;s t)(a q + a q y ) with a q = jg ihej + 2 p jeihf j and f +;s = f ¡ + 2s p g 0 . Going to the rotating frame applying the unitary transform U = exp(¡iH 0 0 t / ~), with H 0 0 = f 0 a y a + (f +;s )jeihej + (f 0 ¡ f +;s )jf ihf j; (7.3) the terms of the Hamiltonian become

H 0 rot = ¡ 2s p g 0 (jeihej + jf ihf j); H c rot = ¡ 2 p g 0 (jeihf ja y + jf iheja) (7.4) H dr rot = " ll 2 (a + a y ) + " block 2 (a q + a q y )
where terms rotating at 2f 0 , f 0 + f s and f 0 ¡ f s and averaging to zero have been neglected, as well as the far o-resonant coupling to the jg i ! jei transition. blocking strength is doubled at a constant ratio " ll /" blcok (top-center panel), and by the lower value when the blocking strength is divided by 2 (top-right panel). The dynamics shown in the top-right panel is closer to the ideal QZD but operating at these parameters would have required a much longer cavity lifetime.

Sticking to the drive strengths used in the experiment, we now

Experimental observation of quantum Zeno dynamics include the dierent decay channels individually (see middle row). A decay in the oscillations is observed, with a damping in p(g0) of 32, 33 and, 12% after two periods due to the cavity decay, transmon relaxation and transmon dephasing, respectively. We deduce that the contributions of cavity losses and transmon relaxation to QZD decoherence are about the same in our experiment. The vulnerability to transmon relaxation is due to the spurious population p(+3) mentioned above, and is thus reduced with a stronger blockade, as conrmed by simulation (not shown).

Lastly, the evolution is simulated with all decay channels (bottom panel) and ts well the measurements, as shown in g. 7.7.

Wigner function of the simulated QZD

The Wigner function of the simulated evolution is also reconstructed at every quarter-period as for the data in section 7.2.1, as shown in g. 7.9 (middle and bottom row). Qualitative agreement is found between the simulations and the experiments except for interference fringes extending along the positive I axis, indicating a coherent superposition with the small part of the wave function transmitted through the Zeno blockade. Note that the already mentioned asymmetry in Q is captured in the simulation, although with a smaller amplitude than in the experiment. Overall the model seems to underestimate the eect of dephasing.

We also compare the data and full model simulations to the case of ideal QZD in an oscillator Hilbert space truncated to 3 levels (j0i, j1i and j2i) and the result is shown in g. 7.9 (top row). As expected, the state at every integer period is the same. No asymmetry in Q is observed, and the cross sections shown on the side in g. 7.9 shows negative values reaching -0.4, two times greater than measured in the experiment (see g. 7.9 experiment cross section: blue, Q>0).

Finally, as for the occupation probabilities p(q n), the eect of blocking strength is investigated. The Wigner function after half a period is simulated without any decay for drive strengths 2 " block , " block and 0.5 " block , keeping the ratio " ll / " block constant. The results are shown in g. 7.10. At the weakest blocking the Wigner function is almost symmetric in Q, quite similar to the ideal case, Kristinn Júlíusson and has great negative values at both negative and positive Q. As the blockade strength is increased the Q asymmetry re-appears and the negativity for Q < 0 becomes more shallow. This indicates that the asymmetry is induced by the blockade tone slightly shifting the adjacent jg2i ! j+2i transitions (see section 2.3.4), and that the eect can be reduced by applying a weaker blockade. ). The measurement (middle row) is compared with the simulations of the full model (top row) and with the ideal case (bottom row). The evolution of the Wigner function is well captured by the simulation using the measured decoherence parameters. The asymmetry in Q at half integer periods, which does not occur in the ideal case, is captured by the full model simulations. The white dashed lines correspond to the cross sections shown on the left side after 0.5 (blue) and 1.5 (red) periods. In the ideal case the two cross sections are identical.

Experimental observation of quantum Zeno dynamics Inducing the QZD with stroboscopic unitary kicks as discussed in section 2.3.1 could also reduce this eect as no blockade tone would be present while the cavity is being lled, and thus the levels remain unshifted.

-2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0

Kristinn Júlíusson

QZD of a coherent state collision with an EC

Here we consider the dynamics of the eld prepared in a coherent state outside the EC and colliding with it, as discussed in section 2.3.1. The blockade at n = 2 is obtained with the same drive amplitude as before, yielding roughly the same Rabi frequency on the jg2i ! j+2i transition, and with the same eld displacement rate _ = 0.27 s ¡1 .

The evolution now takes place in a larger Hilbert space than in the conned case. For the maxlike reconstruction of an 18 18 density matrix, 1608 dierent observables are measured, consisting in 201 displacements indicated by dots in g. 7.11, and eight photon number probabilities (n = 0; :::; 7) measured at each displacement.

Head-on collision

For the rst experiment the coherent state = 7 p is prepared before the blockade and displacement is applied. The amplitude of the state is chosen as the smallest possible such that it is still suciently unaected by the EC, more precisely with only p(2)=2.2% and p(1) + p(0) = 0.7%. Figure 7.12a shows the measured evolution of the Wigner function at 3 s intervals. As the state collides with the exclusion circle, the maximum of the Wigner function diminishes and the amplitude starts to spread along the EC and re-appears on the other side. After 15 s the state is in a superposition of two components centered at = 1.6. The fringes that appear between the two components form a

Kristinn Júlíusson

V-shape rotated by 90 located in the EC. The state that emerges after the collision is dominantly 'coherent-state like', but with about 1/3 of the weight trapped inside the EC due to imperfect blockade (partial transmission through the EC) and photon decay. Figure 7.12b shows the calculated evolution of the same experiment, but without decay, and induced by applying to j i the operator U 2 = exp(¡iH 2 t / ~) where the drive Hamiltonian

H 2 = " ll 0 (a 2 y + a 2 ) (7.6)
involves an annihilation operator

a 2 = a ¡ 2 p j1ih2j ¡ 3 p j2ih3j (7.7)
that forbids jn = 1i $ jn = 2i and jn = 2i $ jn = 3i transitions, and a lling rate " ll 0 adjusted to reproduce the experimental displacement rate. Under this drive the Hilbert space is compartmentalized into H = H n<2 H n=2 H n>2 realizing the perfect QZD as discussed in section 2.2.1 and veried in g. 7.14 below. Apart from the lack of decay, the spread of the state around the outside EC is captured in the calculation. The interference fringes that appear inside the EC are visible in the experiment but considerably sharper in the calculation. Since the last experimental frame is still quite close to the EC we calculate the state 10 s later (i.e. at 40 s), which shows the interference fringes between the dominant part of the wavefunction transmitted around the EC and the small part trapped inside it.

Tangential collision

The second experiment follows an identical protocol with a dierent initial state = ¡ 5 p ¡ i 2 p . The coherent state amplitude j j is the same as before but its phase is tuned to get a tangential collision with the EC. This entails choosing jIm[]j p = 2. The evolution of the Wigner function, measured at 3 s intervals, is shown in g. 7.13a. When the state is driven along the positive I axis, the part of the state at higher Q collides with the EC rst, causing it to phase shift earlier than the rest of the state. This happens gradually as the state is displaced further on to the EC, giving the appearance of the state rounding itself around the EC. The Wigner function of the state emerging from the collision has an elongated shape. The simulated evolution, using the same method as in g. 7.12b, is shown in g. 7.13b.

As for the head-on collision, the calculation highlights the interference fringes that appear inside the EC during the collision, and after it between the two components inside and outside the EC.

To investigate the elongated shape resulting form this tangential collision we plot in g. 7.15 the cross sections of the Wigner function after 30 s evolution, along the line indicated in the inset. For comparison we also plot the cross section of a coherent state (represented by a Gaussian of height 2/ and half width at half maximum = 1/ 2 p ). Due to the aforementioned experimental imperfections the height of the cross section extracted from measurements is 1/2.3 of the height of the coherent state. When comparing to a scaled prole of a coherent state, the mesured state has a smaller width at half maximum (0.4 compared to 0.57). Note that the small asymmetry of the measured peak is due to small weights inside the EC as can be seen on the measurement curve at low values of j j. The calculated cross section of the perfect QZD has a slightly lower maxima than the coherent state due to the population Experimental observation of quantum Zeno dynamics k = 50i, corresponding to our harmonic oscillator. Starting in the state jn e ; k = 0i, a radio frequency eld at frequency 230.15 MHz populates these k levels at a rate of 152 kHz. The Zeno blockade is induced by driving the transition jn e ; k z i ! jn g ; k z i with a continuous microwave tone at a Rabi frequency 3 MHz and for k z = 4 and 5.

The monitored populations of the levels jn e ; ki bounce o a 'wall' at k = k z + 1 in presence of the Zeno blockade with 25% transmission through the wall. The Husimi Q-function [23] is also measured, showing the periodic behavior of conned QZD (see section 2.3.1), with a period of 1.46 s. The Wigner function at roughly half a period shows fringes indicating a coherent superposition of the two amplitudes with opposite phase.

QZD in a Rb ensemble

In ref [4] Barontini et al. entangle N = 36 Rb atoms at the antinode of a eld in an optical microcavity using the QZD. The ensemble of atoms with two hyperne levels j0i and j1i in their ground state form a collective spin with N + 1 Dicke states jn N i (corresponding to Fock states in our oscillator), with n the total number of atoms in j1i. Driving the j0i ! j1i transition induces Rabi oscillation of the collective spin, which plays the role of the coherent drive of our harmonic oscillator. The Zeno blockade is applied on j0 N i by driving an optical Experimental observation of quantum Zeno dynamics transition. Starting in the state jN N i and driving towards j0 N i yields a state close to the highly entangled W state, j1 N i = 1 / N p (j10; ::::; 0i + j01; ::::; 0i + :::: + j10; ::::; 1i).

QZD in dispersive circuit-QED

The conned QZD was also observed in circuit-QED by Bretheau et al. [8] in 2015. 7.1 Similar to the platform used in this thesis, they used a transmon in a superconducting 3D cavity. The dierence lies in the transmon not being frequency tunable, but xed at 5.622 GHz, and dispersively coupled to the fundamental mode of the cavity at 7.804 GHz, used both for storing the quantum eld and measuring the transmon. The photon number split frequencies f (n) f ge ¡ n are thus equidistant with a separation = 4.63 MHz. The transmon and cavity lifetimes are T 1 = 11.5 s, T 2 = 8.9 s, and T cav = 1.3 s, respectively.

The QZD is induced by both a continuous lling at a rate _ = 3 s ¡1 and continuous blocking of n z = 2; 3; 4 and 5 at a Rabi frequency R / 2 = 6.24 MHz. The photon number probability p(n) up to n z is monitored during 3 s of evolution showing 1.5 to 2 Zeno periods of the conned dynamics. Wigner functions are measured to produce snapshots of the evolution of the eld over one period for each n z , clearly showing negative values.

The main dierence with the experiment in this thesis is the shorter cavity lifetime by a factor 30-40. This makes the authors choose a lling rate and a blockade strength 10 times higher (same ratio R / 2 / _ = 2) in order to observe a few Zeno periods. This stronger blockade yields a ratio / R =1.3 about 8 times higher (here corresponds to +;4 in the ef-resonant scheme), and consequently a strong o-resonant drive of the unblocked Fock states. These states thus undergo a rather strong phase shift which is partially compensated by the negative Kerr eect. The dynamics observed is thus farther from the ideal QZD, although the essence of it is perfectly captured. The decay of the QZD oscillations is only 2-3 times faster than in the present thesis due to the fast drive compensating the short cavity time. Unlike in our experiment, the contribution of the transmon decay channels to this QZD decay is very likely negligible according to our analysis of decoherence, given the T 1 about 5 times longer than in this thesis and about 7-9 times longer than T cav .

7.1. State connement to the lowest levels (up to 7) of a superconducting anharmonic oscillator have also shown a superposition of two coherent-like states [10] Kristinn Júlíusson

Experimental observation of quantum Zeno dynamics

Chapter 8 Perspectives

The quantum Zeno dynamics (QZD) has been an active domain of research during this PhD period. Some of the proposed experiments in refs. [START_REF] Raimond | Quantum Zeno dynamics of a eld in a cavity[END_REF] and [START_REF] Raimond | Phase Space Tweezers for Tailoring Cavity Fields by Quantum Zeno Dynamics[END_REF] have been implemented in this thesis work and by others ( [45][8][4]). However, manipulation of states using the phase space 'tweezers' (see section 2.3. 3) is yet to be demonstrated.

In terms of perspective for QZD experiments, we foresee the following possible improvements: A longer lifetime of the cavity is an important step towards reducing decoherence during the QZD and implementing the tweezers. A cavity with lifetimes T cav 5-10 times longer (0.25 ms) than used in our demonstration of QZD has been obtained at the end of this thesis. Even longer cavity lifetimes T cav ' 10ms have been reported [START_REF] Reagor | Reaching 10 ms single photon lifetimes for superconducting aluminum cavities[END_REF] for new cavity geometries in circuit-QED, which gives good prospects for the future. The other main decoherence source for the QZD found in our work is the transmon relaxation. It could be improved beyond 10 s by adapting the cavity dimensions to increase the transmon-readout mode detuning, thus reducing the Purcell eect, and by using better transmons as reported in chapter 5. A sucient increase in T cav alone would already allow for slower displacement of the cavity eld, which in return would make the blocking more ecient, and the dynamics less sensitive to transmon relaxation. In our setup the coupling strength g 0 could also be increased by up to a factor 2 to increase the photon number splitting by the same amount, at the cost of a 16 times larger Kerr non-linearity K ( from 300 Hz to 4.8 kHz), which would still be acceptable and is anyway an over estimate as shown in chapter 6.

With the sample reported in this thesis, the QZD using unitary kicks (see section 2.3.1) could be attempted. Using the unitary kicks would enable the implementation of exclusion circles with partial transmission without dephasing the transmitted component of the wave function, allowing for example the generation of a complex superpo-sitions of coherent states. This would require optimal pulse shaping of the blocking pulse in order to minimize the o-resonant driving of the adjacent transitions but would leave the levels of the harmonic oscillator unperturbed during displacement of the cavity eld.

In a more general perspective, as the QZD can be calculated from well established quantum theory, I think that QZD experiments do not answer any fundamental questions. However, its application in manipulating superposed coherent states, namely using the tweezers, could be used in attempts at quantum simulations and quantum information processing [START_REF] Mirrahimi | Dynamically protected cat-qubits: a new paradigm for universal quantum computation[END_REF] [START_REF] Ofek | Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information[END_REF].

The Bayesian Mean estimate BM from the measurement data Y is dened by the very general relation (see, e.g., [7]):

BM = R D P(Y j)P 0 ()d R D P(Y j)P 0 ()d ;
where D is the convex set of density operators (here the underlying Hilbert space is of nite dimension) and P 0 is some prior probability law of (e.g., Gaussian unitary ensemble [START_REF] Madan | Random Matrices[END_REF]).

Here BM is given by

BM = R D exp(f())P 0 ()d R D exp(f ())P 0 ()d ; where f (), the log-likelihood function, reads f () = ¡ X n; (s n; ¡ Tr[E n; ]) 2 2 n; 2
When the support of the likelihood function is mainly concentrated around its maximum at ML , it is known that ML is a good approximation of BM . Following [START_REF] Six | Quantum state tomography with noninstantaneous measurements, imperfections and decoherence[END_REF], we can also compute an estimation of the Bayesian variance of Tr( ML A), denoted by ML 2 (A) for any observable A and depending on the second-order derivatives of the log-likelihood function at its maximum ML . This transforms the variance n; on the data into variance on the MaxLike estimate ML . In particular when we take, for any complex number , A = 2 exp(a y ¡ a) exp(ia y a) exp(¡a y + a)

we get via ML 2 (A) an estimation of the variance of the MaxLike estimate of the Wigner function at position in the phase space.

Following [START_REF] Six | Quantum state tomography with noninstantaneous measurements, imperfections and decoherence[END_REF], ML is characterized by the following necessary and sucient conditions: there exists a nonnegative scalar ML such that:

[ ML ; rf ( ML )] = 0; ML P ML = P ML rf ML and rf ML 6 ML 1 Kristinn Júlíusson where P ML is the orthogonal projector on the range of P ML and rf ML is the gradient at ML of the log-likelihood:

rf () = ¡ X n; s n; ¡ Tr[E n; ] n; 2 E n;
We have also

ML 2 (A) Tr(A jj R ¡1 (A jj ))
where

B jj = B ¡ Tr(BPML)
Tr(PML) P ML ¡ (1 ¡ P ML )B(1 ¡ P ML ) is the orthogonal projector of any Hermitian operator B on the tangent space at ML to the submanifold of Hermitian operators with trace one and with ranks equal to the rank of ML . Here above, the linear super-operator R reads for any Hermitian operator X,

R(X) = X n; Tr ¡ xE n; jj n; 2 E n; jj + 1 2 ( ML 1 ¡ rf Ml )X ML + + 1 2 ML + X( ML 1 ¡ rf Ml )
with ML + the Moore-Penrose pseudo-inverse of ML .

The maximization of f can be achieved by a gradient algorithm with orthogonal projection on D. The computation of k+1 from k , the value of at step k, reads k+1 = ( k + g rf( k )) where g > 0 is a normalization parameter such that g kr 2 f k is not too large (order 1 to 10 in general). A rough estimation of size of the Hessian kr 2 f k is given by n;

Tr 2 (En;) n; 2 = n; 1 n; 2 A 7 ! (A) 2 D
is the orthogonal projection of the Hermitian matrix A on the convex set D, the set of Hermitian non negative matrices of unit trace. From rst order stationary conditions attached to any unitary change of frames, (A) and A commute and thus are diagonal in the same basis. Consequently denoting by V 2R n +1 , the vector of eigenvalues of A, the eigenvalues of (A), denoted by V 2R n +1 are simply derived from V by its orthogonal projection in the real Euclidean space R n +1 on the simplex S = (x 0 ; :::

:; x n ) 8n; x n > 0; X n x n = 1 :
Maximum likelihood algorithm adapted to Gaussian errors

= n th (1 ¡ n th )( ¡ ) 2 e ¡¡ATp Z t1=0 Tp e ¡Bt1 Z =0 Tp¡t1 sin 2 ( R 2 )e ¡B d d t 1 = n th (1 ¡ n th ) 2 ( ¡ ) 2 ¡ B 2 + R 2 ( R 2 (¡ B 2 + R 2 )e (¡B+¡1)Tp ¡ 1 (¡ B + ¡ 1 ) (¡ B + ¡ 1 ) 2 + R 2 ¡ e ¡ B Tp ¡ B R 2 ¡ ¡ 1 ¡ B 2 + (1 ¡ e ¡BTp ) R 2 ¡ B ¡ 1 (¡ B + ¡ 1 ) ¡ ¡ B (¡ + ¡ B ) 2 ¡ R 2 (¡ + ¡ B ) 2 + R 2 cos( R T p ) ¡ (¡ + 2¡ B ) R (¡ + ¡ B ) 2 + R 2 cos( R T p ) ) with ¡ A = + + ¡ 1 and ¡ B = ¡ ¡ + ¡ ¡ 1 .
The term e ¡¡Tp reaches zero rapidly as soon as T p reaches a few cavity lifetimes 1 / ¡ , which makes the oscillating terms in the curly bracket vanish. Introducing k = R / ¡ and u = ¡ 1 / ¡ , and assuming ¡ + ; ¡ 1 , we nd

p e;1 ' k 2 2(1 + k 2 )(u + u 2 ) e ¡¡n thTp n th (C.2) 1 + k 2 k 2 + (1 + u) 2 ¡ e ¡¡1Tp (1 + u) :
Given ¡ T p 1 and ¡ 1 T p 1, we have neglected in the above reasoning the smaller contribution of a single photon entering before the Rabi pulse starts.

Kristinn Júlíusson

The contribution of scenario 2 with a second photon entering the cavity during the Rabi pulse after the rst one has left is also negligible for the following reason: After the rst excitation of the transmon and the rst photon decay, the transmon frequency is shifted by and the transmon state jei accumulates a deterministic phase ' = (t 2 ¡ t 1 ¡ ) at a rate faster than + so that at the entrance of the second photon at time t 2 , the phase has been randomized and the new Rabi oscillation can increase or decrease p e;1 with comparable probabilities. Consequently p e;1 is well approximated by eq. C.2, provided it yields a very small n th . It varies proportionally to e ¡¡n thTp n th , with a geometrical factor involving the ratios of the secondary rates to the main rate ¡ . This exponential scaling makes the method particularly sensitive, but only valid at small n th .

Estimating thermal photon population in the resonator

The harmonic oscillator being an exactly solvable system with a single degree of freedom, is often taken as a model system in many areas of physics. In particular, for demonstrating coherent control of simple quantum systems, physicists have used cold electromagnetic resonators in the quantum regime, with their quantum states controlled by a single atom, be it a real one in cavity-QED [1] or an articial one in circuit-QED [2,3].

For instance, the preparation of a single photon Fock state by passing an atom through a cavity [4], or of an arbitrary quantum state using the Law and Eberly protocol [57] in circuit-QED, are landmark results. In circuit-QED relevant to this work, the resonator has a frequency ν O in the microwave domain, and the articial atom is a superconducting quantum bit that can be regarded as an ancillary multilevel system (MLS) with states |g ,|e ,|f ,|h . . . possibly tunable in energy. The MLS can be used resonantly [6,7], or o-resonantly [8] when its coupling to the resonator is suciently strong to split the transition frequency ν ge into dierent lines at frequencies ν n that depend on the photon number n [9].

In this so-called photon number split regime, any particular Fock state |n is addressed at frequency ν n , and symmetrically, the resonator can be driven coherently at a frequency ν O,|x that depends on the MLS state |x . This allows for both manipulating and measuring the oscillator eld in various ways, for instance by MLS-state conditional cavity phase shift [10], photon-number selective phase gates [11] that could lead to universal control of the oscillator eld [12], or by Wigner and quantum state tomography of the eld [13,14].

This o-resonant method has been demonstrated with transmon qubits [15] coupled dispersively to the resonator, i.e. with detunings |∆ xy = ν xy -ν O | between any ν xy transition of the transmon and ν O much larger than the coupling frequency g xy between this transition and the cavity eld. In this dispersive regime, the frequencies ν n = ν ge + nχ are equidistant and separated by the so-called dispersive shift χ 2αε 2 [15], with α = ν ef -ν ge the transmon anharmonicity, ε = g O /∆, g O ≡ g ge the transmon-oscillator coupling, and ∆ = ν ge -ν O their detuning. A drawback of this scheme is to transfer part of the transmon MLS anharmonicity to the oscillator [16,17], which can drastically perturb its coherent dynamics [14,18,19] and necessitate careful design [16] and correction protocols [11]. This non-linearity [14,15,17] results in a shift of the oscillator frequency ν O (or Kerr non-linearity) of about K = αε 4 per photon. K scaling as χ 2 /α cannot be minimized (at xed α) without losing the selectivity between Fock states. In this work, we propose a dierent transmon-oscillator coupling scheme (see Fig. 1) that yields a much smaller Kerr non-linearity for the same Fock state selectivity. The idea consists in having the ef transition resonant with ν O , i.e. ∆ = -α, to displace signicantly the qubit excited levels even at small coupling g O , while at the same time staying in the dispersive regime for the rst transition ν ge to get a small non-linearity. We explain in details in the next section this 'ef-resonant' condition when the MLS is a slightly anharmonic three level system, for which analytic results can be obtained. Then, we describe in section II our implementation of the proposal, and characterize it experimentally in section III.

I. THE 'EF-RESONANT' COUPLING SCHEME FOR A HARMONIC OSCILLATOR

To explain the interest of our coupling scheme in the simplest way, we rst consider the case of a harmonic oscillator (O) with frequency ν O and annihilation operator a, coupled to a three-level system (3LS) with eigenstates {|g , |e , |f } and lowering operator a q . We also assume a weak anharmonicity α = ν ef -ν ge ν ge , such that a q can be approximated by the annihilation operator of a harmonic oscillator restricted to three levels. The two subsystems obey the resonant condition ν ef = ν O (detuning ∆ = ν ge -ν O = -α ) and are subject to an exchange interaction with coupling frequency g O α, yielding the Hamiltonian

H = H O + H 3LS + H coupling , H O = hν O a † a, H 3LS = h ν O a † q a q -α |e e| -α |f f | , H coupling = hg O a † a q + aa † q .
(1)

This coupling makes the Hamiltonian matrix blockdiagonal in the basis |xn ≡ |x ⊗ |n (x = g, e, f ), with subsequent blocks of size 1, 2, 3, 3, 3... Using the reduced coupling ε = -g O /α, these blocks write

[0] |g0 , h ν O I 2 -α 0 ε ε 1 |g1 ,|e0 , h   nν O I 3 -α   0 √ nε 0 √ nε 1 2 (n -1)ε 0 2 (n -1)ε 1     Bn , (2)
with I k the identity matrix of dimension k and B n = {|gn , |e (n -1) , |f (n -2) } the basis for n ≥ 2. The diagonalization of each block yields analytical eigenenergies and vectors, which for n ≥ 2 are functions of the three real solutions of the cubic equation x 3 -2x 2 + 1 + 2 2 -3n 2 x + nε 2 = 0. To shed light on the physics, we expand these analytical quantities in the small parameter ε. As shown in Fig. 1, the levels form three distinct energy ladders{| gn }, {|-n } and {|+n }:

The ladder{| gn } n≥0 corresponds to the almost unperturbed oscillator when the 3LS is left in its ground state.

With eigenenergies and eigenvectors

E(| gn ) = nh [ ν O + (n -1)K/2] + o( 5 ) | gn = 1 -nε 2 /2, - √ nε, 2n (n -1)ε 2 Bn + o( 3 ) , (3)

{E(|±n

)} = {0, h ν O -α 1 + ε 2 + o(ε 3 ) , ..., h nν O -α 1 ± 2 (n -1)ε + nε 2 /2 + o(ε 3 ) } {|±n } = |g0 , ε 1 -ε 2 2 , ...,     n 2 ± n n-1 7n-8 8 ε 2 1 √ 2 ± n 8 √ n-1 ε + (33n-32)n 64 √ 2(n-1) ε 2 ± 1 √ 2 -n 8 √ n-1 ε + n 2 64 √ 2(n-1) ε 2     Bn + o(ε 3 ) . ( 4
)
For n ≥ 2 the zeroth-order approximation in of these eigenvectors are simply the symmetric and anti- 3D microwave simulator and methods adapted from [17].

ge n   O g ge  n    O  0 n  1 n  2 n  … g e f g O O ge      O ef    O  O K    O  O K    2 O K    O nK     dressed states 0 g 1 g 2 g … gn ~2  3  4    1 n   0  0 1 e   Figure
Attention is paid to keep the transmon's charging energy (one electron) below 300 MHz × h in order to avoid variations of the level f energy due to charge parity uctuations [21]. For better measurement eciency, output port 2 is about 6 times more strongly coupled to each mode than input port 1.

The equivalent circuit of the system is shown in Fig. 2c.

The transmon-cavity system is mounted inside a coil placed in a mu-metal shield, and is attached to the cold plate of a cryofree dilution refrigerator with base tem- Microwave pulses at the cavity (qubit) frequencies νO,R (ν+n) are obtained by single sideband mixing of a continuous microwave (LO) with an intermediate frequency modulated pulse generated by two channels of an arbitrary waveform generator (AWG). All pulses travel along an attenuated and ltered line to cavity port 1. The readout signal transmitted at port 2 is ltered, isolated from backward propagating noise, amplied with a parametric amplier (JPA) in reection, a high electron mobility transistor (HEMT), and room temperature ampliers, then demodulated to produce two quadratures, which are nally ltered, amplied and digitized (ADC). [ 14,18], or the power dependence of the resonance line shape at short time as in the supplementary information of [14]. Consequently, we simply measure the steadystate transmitted power P 2 (ν) of mode O as a function of the excitation frequency ν at several input powers P 1 , using the VNA with a narrow enough measuring bandwidth of 1 kHz. The corresponding curves are shown in Fig. 5a (dots) in arbitrary units of the output power P 2 (left scale).

Analyzing quantitatively the dataset requires a precise knowledge of the average photon number < n > in the resonator as a function of ν and P 1 . In this aim, we perform the following additional in-situ calibration and data analysis. We rst use the transmon to determine experimentally the input power P 1,1 = -149.6 dBm that populates the cavity with < n >= 1, which corresponds to p β=1 (n = 0) = 0.365 (see Fig. 5b) assuming a coherent steady state ρ 1 = |β = 1 β = 1|. Then quantum simulations of the oscillator O coupled to the eective 4 level transmon are performed using the QuTiP Python toolbox [25], the already mentioned measured parameters, and the calibrated P 1 . Solid lines in Fig. 5a show the resonance lines < n > (ν) obtained with the steady state solver of QuTiP. By scaling vertically the experimental curves so that simulation and experiment match for P 1 = -135 dBm, we obtain a fair agreement for all curves, leading to the following results: First the resonance lines display indeed a very small Kerr eect, with a shift towards lower frequency with increasing P 1 signicantly smaller than K (oblique dark blue line), and changing sign between < n >= 15 and 25, in good agreement with the calculated shift (magenta line).

A second and unanticipated eect is that although the non-linearity in frequency is small, a large non-linearity in input power is observed, with P 2 and < n > increasing by a factor of only 3 when P 1 is increased by a factor 8 (see extreme curves in Fig. 5a). Simulating the time evolution of the system initialized in its ground state and driven coherently at ν O reveals the cause of this nonlinearity in P 1 : the small hybridization of the oscillator with the transmon (see Eq. 3) that has nite coherence times (T 1 , T * 2 ), progressively induces Fock state dephasing. The increasing eld perfectly coherent at the beginning of the dynamics slowly becomes incoherent when ap-proaching the steady state,which reduces its amplitude. This is illustrated in Fig. 5c by the photon number distributions p P 1,t=∞ (n) obtained from the steady state solver at the top of each resonance curve of panel a: although p P 1,t=∞ (n) corresponds almost exactly to the Poisson distribution for ρ 1 (which validates the calibration of P 1 in the previous section), the other distributions for larger < n > are less and less Poissonian. Large coherent states can nevertheless be obtained at times shorter than a few tens of T * 2 , as observed in the previous section for a du- ration of the coherent drive t T * 2 /8. Note that the transmon-induced cavity non-linearity in power observed and simulated here also exists in the dispersive regime, and is an eect that would deserve a theoretical evaluation.

IV. CONCLUSION

We have described a way to manipulate the quantum state of a harmonic oscillator by coupling it to a anharmonic multilevel system (MLS), without paying the price of a large Kerr non-linearity of the oscillator inherited from the MLS. We have demonstrated our 'ef-resonant' scheme using a 3D circuit-QED setup, in a new geome-try involving a tunable transmon qubit partially inserted inside a single multimode superconducting cavity. Fock state manipulation was demonstrated by quantum state tomography of a coherent eld in the cavity. The nonlinearity was measured to be very small, provided the total eld manipulation time is not much longer than the qubit coherence time. Our setup and coupling scheme provide a new platform for manipulating at will mesoscopic quantum elds inside a harmonic resonator, and producing non-classical states in various ways. In particular, the ef-resonant scheme would reduce the Kerr non-linearity of the promising platform proposed and developed [8,26,27] for encoding quantum information in Schrödinger cat states of the cavity eld. We plan to use this scheme for demonstrating the quantum Zeno dynamics of the cavity eld as proposed in [28].

1

  Introduction and summary . . . . . . . . . . . . . . . . . . . . 1.1 Quantum mechanics: From theory to experiments on single quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Thesis summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 General QZD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Designing a QZD setup for 3D circuit-QED . . . . . . . . . . . 1.4.1 Theoretical design . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Physical implementation . . . . . . . . . . . . . . . . . . . . 1.5 Sample fabrication and sample characterization . . . . . . . . 1.5.1 Fabrication and characterization of a 3D transmon . . 1.5.2 Building blocks and parameters of the setup . . . . . . . 1.5.3 Sample characterization . . . . . . . . . . . . . . . . . . . . 1.6 Wigner tomography, experimental results and analysis . . . . 1.6.1 QZD analysis with the Wigner tomography . . . . . . . 1.6.2 QZD: Conned, accelerated and elongated . . . . . . . . Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conned Zeno dynamics inside the exclusion circle . Accelerated Zeno dynamics: going through an exclusion disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Squeezing from Zeno dynamics . . . . . . . . . . . . . 1.6.3 Decoherence analysis in QZD . . . . . . . . . . . . . . . . 1.7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . 2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . 2.1 The quantum Zeno Dynamics . . . . . . . . . . . . . . . . .

Figure 1 . 1 .

 11 Figure 1.1. Schematic representation of the quantum Zeno dynamics. The QZD can be used to compartmentalize Hilbert space into subspaces, each one having dierent dynamics.

Figure 1 . 2 .

 12 Figure 1.2. a) Harmonic oscillator is coupled to a 3 level atom, resonant with the jei ! jf i transition at ! 0 / 2 = ! e f / 2 = 6.4GHz. The jgi ! jei transition is o-resonant by / 2 = 260 MHz. b) A coupling strength g 0 / 2 = 8MHz is weak enough to preserve the harmonic oscillator when the atom is in its ground state but strong enough to induce an anharmonic oscillator when the atom is excited. This allows for photon number dependent manipulations of the atom. QZD can be induced by alternating a coherent drive pulse at ! 0 and blockade on Fock state jn 0 i by either c) applying a pulse at ! +;n0 followed by a projective readout of the 3LS, d) applying a 2 pulse inducing a selective phase (¡1) n;n 0 jni: e) The third method of inducing the QZD is by applying a continuous drive at ! 0 and another one at ! +;n0 which splits up the jn 0 i level rendering it o resonant from ! s .

Figure 1 . 3 .

 13 Figure 1.3. a) Illustration of the 3D circuit QED implementation. An Al cavity with two ports (1),(2). A transmon qubit on a sapphire substrate is inserted into the cavity through a slit in the wall such that it is partially outside the cavity. It is covered with a Cu cap allowing a magnetic eld to penetrate the SQUID. b) A photograph of one half of the cavity with the transmon and Cu cap mounted. c) A lumped element representation of the system. Microwave signals arrive at (1) and leave via(2). Ports a are weakly coupled to the oscillator (internal loss limited) and strongly coupled to the Readout mode (external loss limited) in a asymmetric manner, favoring photons leaving via (2). The frequency-tunable transmon is strongly coupled to the readout mode (g r /2 = 150MHz) but weakly coupled to the oscillator g 0 /2 = 8 MHz. d) Transmission through the readout mode when the transmon has been prepared in jg i(red), jei(green) and jf i(black).

Figure 1 . 4 .

 14 Figure 1.4. Design and simulation of the system. a) Schematic of the geometric distribution of the storage and readout mode within the cavity. Slit, the input port and are shown at the node/antinode of the storage/readout mode. b) A screen shot of the of the nite-element simulation cavity. c) A zoom-in on the simulated transmon with a port at the location of the SQUID. d) The simulated admittance Im[Y (!)] after adding the SQUID inductance. The roots Im[Y (! res )]=0 represent resonance frequencies. e) The anticrossing is simulated varying the SQUID inductance. The minimum separation is the coupling strength g 0 .

Figure 1 . 5 .

 15 Figure 1.5. a) A Transmon fabricated on a Si substrate in a Cu cavity.b) The expected behavior of the excitation frequency transmon with a SQUID(blue line) under a ux bias is tted to the measured excitation frequencies(red crosses). The green line indicates the resonance frequency of the cavity. Inset) T 1 and T 2 at dierent frequencies. The Purcell decay subside as detuning from the cavity increases at the same time as the sensitivity to ux noise increases and lowers T 2 . c) The measured T 1 (red dot) compared with the Purcell limit (blue line). The frequency is maximized at / 0 = ¡0.31 and is lowered, crossing the resonator (green lines), when varying the bias in either direction. b) and c) contain data from two different cooldowns of the same sample.

Figure 1 . 6 .

 16 Figure 1.6. Building blocks of the setup. a) Aluminium cavity with a Cu cap and a transmon on a sapphire chip inside. The two halves are closed with an In seal. b) A Lorentzian with a resonance frequency ! 0 / 2 = 6.43 GHz and a quality factor Q 0 = 2 10 6 is tted to the transmission of the storage mode. c) Schematic of the transmon used in the QZD experiments.

Figure 1 . 7 .

 17 photon number

Figure 1 . 8 .

 18 Figure 1.8. Reconstruction of the Wigner function. a) Wigner function of the vacuum state measured pixel by pixel using the truncated parity operator P 00shown in f). b) Wigner function of the vacuum state calculated from a density matrix reconstructed with the maximum likelihood method. c) Reconstructed Wigner function using maxlike of a coherent state j = ¡7i in a Hilbert space truncated to 18 photons. The black dots represent the displacements f i g in the tomography sequence. d) The parity can be reconstructed from the photon number distribution but requires a separate measurement for every Fock state. e) All the odd photon number state can be simultaneously excited with a multiplexed pulse consequently mapping the parity onto the Transmon state and hence require only one measurement. f) The measured parity of the vacuum state using the methods in f) and e). Diminished Fock state resolution leads us to truncate your parity operators to 10 photons which causes errors in our parity estimate already at 4 photons.

Figure 1 . 9 .

 19 Figure 1.9. Protocol used for all QZD experiments. 1) A displacement pulse D() is optionally applied to initialize the eld in a coherent state j = / 0i. 2) Evolution under QZD. The continuous Zeno blockade tone is switched on as well as the coherent cavity drive. 3 and 4) After evolution over a time t Z tomography is performed: the eld is displaced by D() for several values of . After each displacement the photon number distribution is measured from n = 0 to n max .

  » generation :!start in!!!!!!!!!!!!!,!block!n 0 =!3,!and!drive!forbidden region:!start in!!!!!!!!!!!!!!!!!,!block!n 0 =!2,!and!drive!coherently

Figure 1 . 10 .

 110 Figure 1.10. Three examples of quantum Zeno dynamics: connement, tunneling and squeezing. For each set we compare the data in the bottom row to the ideal case in the top row ( see section 1.2.9). a) QZD blocked at n 0 = 3. showing the reection on the exclusion circle with negativity clearly visible at half a period and still present at 1.5 periods. b) Acceleration through the exclusion circle. As it is driven to the right the state smears out at the border and reappears on the opposite side. Midway through it resembles a Schrödinger cat state. c) By starting o center from the origin and driving tangentially into the exclusion circle, a squeezed-like state is produced. Black dots represent the displacements used for the tomography.

Figure 1 . 11 .

 111 photon number n

Figure 2 . 2 .

 22 Figure 2.2. 3D cavity resonator. a) Schematic of a cavity coupled to s source and a load with coaxial cables. The loss in the walls is represented by resistances. b) Circuit model representation of a mode of the cavity c) LCRcircuit equivalent to b)

Figure 2 . 3 .

 23 Figure 2.3. Scattering parameters of transmission and reection around a resonance frequency f 0 . The width of the Lorentzian transmission peak gives Q tot . Components Q i corresponding to individual ports can be determined by the depth of the reection, leaving the internal Q int to be determined by subtraction from Q tot .

Figure 2 . 5 .

 25 Figure 2.5. Coherent state. a) The representation of a coherent state in the IQ phase space of the electric eld in the cavity. The state can be seen as analogous to a classical eld, characterized with an amplitude and a phase. b) Three examples of Fock state distributions for coherent states with amplitudes = 1; 2 and 8 p .

Figure 2 . 6 .

 26 Figure 2.6. Josephson junction. a) Two superconductors separated by an insulator, with a phases ' 1 and ' 2 . b) The circuit representation of a junction is a cross. The box is drawn around it represents the capacitance. c) Two JJ in parallel form a SQUID loop which can be seen as a single junction tunable with an external magnetic ux through the loop.

Figure 2 . 9 .

 29 Figure 2.9. Transmon qubit. a) Circuit diagram of a transmon as a SQUID with a shunt capacitor in parallel. b)Antenna representation of a transmon

Figure 2 . 10 .

 210 Figure 2.10. Bloch sphere is a sphere with radius 1. a)State of a qubit can be represented as a vector a ~on the Bloch sphere. ja ~j = 1 for a pure state. b)Applying a coherent drive causes the Bloch vector to rotate around an axis d ~.

Figure 2 . 12 .

 212 Figure 2.12. Dispersive shift of a harmonic oscillator. a) Coupling to a TLS leads to symmetric shift around the bare resonance frequency ! 0 . b) Adding a third level destroys the symmetry and the shift depends on the anharmonicity (see text).

Figure 2 .

 2 Figure 2.13. An 'ef-resonant' cavity-transmon system. a) The harmonic oscillator is coupled to a frequency tunable transmon. b) The levels of the uncoupled system laid out as three harmonic oscillators corresponding to jg i; jei and jf i, showing that the levels je; n + 1i and jf ; ni degenerate. When the coupling is included the the degeneracy is lifted by 2n p g 0 . The levels jg; ni are o-resonant by and thus almost unaected. c) The eigenstates jg; ni form a quasi harmonic ladder which can be excited to j; ni at Fock state dependent frequencies ! n; .

Figure 2 . 14 .

 214 Figure 2.14. Protocol for implementing the QZD with unitary kicks decomposed into N steps. Each steps consists of a displacement by and a Rabi rotation on the jg; si ! j+; si transition at ! +;s .

Figure 2 . 15 .

 215 Figure 2.15. Snapshots of numerical simulations of the cavity state undergoing the QZD with = 0.1 and s = 6, at every fth step. a) Starting in the vacuum state, the dynamics is periodic inside H <6 , with a period of 50 steps. b) The initial state j = ¡5i is in H >6 . It collides with the EC from the outside and creates a superposition state at N = 25 steps. c) Starting in = ¡4 + i 6 p the state collides tangentially with the EC resulting in an squeezed state. Figure reproduced from ref. [40]©.

Figure 2 .

 2 Figure 2.15 c shows a tangential collision with the EC resulting in an squeezed state with uctuations perpendicular to the tangent smaller than the vacuum uctuations.

Figure 2 . 16 .

 216 Figure 2.16. Two measures of the robustness of the QZD. The quantum state is evaluated after a total displacement of 2 6 p for varying Rabi angles and numbers of interruptions N = int ¡ 2 6 p / . The transmission of the EC is obtained by comparing the weight of the wave function on either side of the EC. The green line indicates 50% transmission. Figure from ref [40]©

Figure 2 . 18 .

 218 Figure 2.18. Energy levels shift due to strong coherent driving. a) Single energy level of a TLS (horizontal dotted line) anticrosses with the energy of a drive eld (increasing dotted line) at 0, forming dressed states E + and E ¡ . b) Schematic of part of the transmon-cavity level structure when driving at ! +;s (i.e. = 0). The excitation spectrum from jg; si has split into a doublet and is no longer resonant with ! 0 , thus implementing an EC at jn = si.

Figure 3 . 1 .

 31 Figure 3.1. Transmon fabrication process. a) The two resist layers and an (optional) Al layer prepared on a substrate. b) Electron beams shot at a high dose (black) break up the resist molecules. A low dose bombardment at the surrounding mainly aects the lower layer. c) The Al layer is removed with KOH and subsequently the exposed resist is dissolved in MIBK. d) The rst layer of Al is deposited at an angle. d) Exposure to O 2 creates an insulating Al 2 O 3 layer. f) A second layer of Al is deposited at an opposite angle where the overlap with the rst Al layer forms a Josephson junction.

Figure 3 . 3 .

 33 Figure 3.3. Electron beam lithography pattern of the transmons. a) and b) are snap shots of the GDSII exposure les controlling the shapes traced out by the electron beam. The color represents the dose in reference to a 250 mC/cm 2 nominal dose with 10%(blue), 100%(green) and 190%(red). a) The rst transmon design used in a Cu cavity while the one in b) is used in the QZD experiment. c) Bilayer mask after exposure and development. We see the frame of the undercut box around the main pattern. d) Bridge in the PMMA: The bending of the structure is due to the stress induced by the imaging electron beam. c) and d) are test structure for tuning the doses of the undercut boxes and the main pattern.

Figure 3 . 4 .

 34 Figure 3.4. Images of the fabricated transmon. a) and b) optical images. c) SEM image of the squid. d) SEM image of a Josephson junction. e) The circuit used to protect and measure the SQUID resistance R n .

Figure 3 . 5 .

 35 Figure 3.5. a) A transmon of the rst type mounted in a groove of a Cu cavity. b) The QZD transmon mounted in a Cu cap. c) The cavity closed with an indium seal and the Cu cap secured with a pusher bar. d) The Transmon mounted in the open cavity.

Figure 3 . 6 .

 36 Figure 3.6. Our experiments are performed in dilution refrigerators. a) Dilution refrigeration principle. The He 3 / He 4 mixture separates into He 3 concentrated and dilute phases. Pumping on the diluted phase perturbs the phase equilibrium and the system uses the energy from its environment to restore it. b) The inside of a dilution fridge of the dry type. c) The sample inside a metal can, mounted onto the coldest stage of the fridge. d) The sample mounted between two coils and connected to semi-ex SMA cables.

Figure 3 . 7 .

 37 Figure 3.7. Schematics of the electrical circuitry inside the refrigerator. We have three RF signal input lines: C3 to send signals to the experiment, C2 to probe the paramp and C4 to pump the paramp. D2 is the signal detection line. Lines A1 and B1 are for the DC bias of the sample and paramp respectively

Figure 3 . 8 .

 38 Figure 3.8. Room temperature microwave setup. Signals are generated by IQ-mixing a continuous microwave tone with pulse shaped IF signals from an AWG. The VVA sets the power range of the signals that are then combined, ltered and nally enter the fridge. We split up the readout tone into a probe signal and a local oscillator signal (LO) signal for homodyne detection. The signal returning from the fridge is amplied and demodulated before entering the ADC data acquisition card.

Figure 3 . 9 .

 39 Figure 3.9. a) Lumped element circuit model of a paramp. A one port LCresonator has a tunable inductance by using a squid array. The input signal ! S is amplied and reected when a pump tone ! P ' 2! S inductively pumps the squid array via the separate pump line. b) Optical image of the paramp. c) Gain curves for varying pump power from a paramp of the same design. d) When operated in the degenerate mode ! P = 2! S , the gain jG S ;deg j depends on the phase between ! P and ! S .

Figure 4 . 1 .

 41 Figure 4.1.Transmission measurement of a cavity resonator coupled to a tunable transmon. a) Anticrossing when sweeping ! g e across ! 0 using a ux bias. The two resonance peaks, separated by 2 g 0 when ! g e = ! 0 , give the coupling strength. b) Transmission around ! 0 in the dispersive limit as a function of input power. At low power the resonance frequency is shifted due to the coupling to the transmon. When the power is increased the resonance frequency eventually switches back to ! 0 .

Figure 4 . 2 .

 42 a shows a schematic illustration of typical cavity mode resonances for the transmon prepared in jg i, jei and jf i. When sending a probe signal at the frequency ! 0 (g)

Figure 4 . 2 .

 42 Figure 4.2. Dispersive readout. a) Schematic illustration of the resonator peaks corresponding to the transmon states jgi, jei and jf i. The transmon is read out by applying a probing pulse at ! 0 (g) . b) Measured transmission spectrum when the transmon is prepared in jgi, jei or jf i. c) Continuous I and Q signals (see g. 3.7 and 3.8 for setup) at ! 0 (g) when all the signal is put on one quadrature. Signal-to-noise rations (SNR) are indicated for a single trace, averaging 2 10 5 traces, and adding a paramp. d) and e) Comparison of normal and 'fast-load' pulses. The fast-load pulse reaches the nal amplitude three times faster than the normal one, thus recording the transmon state sooner after its preparation.

Figure 4 . 3 .

 43 Figure 4.3. The signal-to-noise ratio (SNR) of the readout signal as a function of integration time. The maximum SNR is at 1.2T 1 :

Figure 4 . 4 .

 44 Figure 4.4. Transmon spectroscopy: protocols and example measurements a) Pulsed spectroscopy shows a dip in the transmission amplitude at ! ge (red). The blue curve shows the two-photon jgi ! jf i transition at ! g f / 2. b) In continuous spectroscopy the signal is stable throughout the readout pulse. This scheme also reveals the photon number peaks.

Figure 4 . 5 .

 45 Figure 4.5. Rabi oscillations: protocol and example measurement. When varying the duration of a square pulse the excitation probability of the transmon oscillates at Rabi frequency /2 = 9.26 MHz.

Figure 4 . 6 .

 46 Figure 4.6. Relaxation time: protocol and example measurement. By exciting the transmon and varying the time before its state is read out, one can map the exponential decay. Here T 1 = 39s.

Figure 4 . 7 .

 47 Figure 4.7. Ramsey fringes: protocol and example measurement. Two / 2 pulses at a frequency ! g e + , separated by a varying time , induce oscillations in the transmon excitation probability at frequency / 2=0.83 MHz The oscillations decay with a free induction decay rate ¡ 2 , giving T 2 = 6.5 s.

Figure 4 . 8 .

 48 Figure 4.8. Hahn echo:: protocol and example measurement. The Hahn echo protocol dynamically decreases the dephasing rate. With = 0, a pulse after / 2 leads to partial refocusing of the phase information at . Here T 2 E = 20.5 s.

  photon number n

Figure 4 . 9 .

 49 Figure 4.9. Calibration of the photon number probability measurements and the cavity lling rate. a) Raw signal amplitude of the rst 11 photon number peaks. b) After a calibration procedure the amplitudes are translated into probabilities and the lling pulse amplitude into cavity eld amplitude. c) Dierence between the experimental calibrated probabilities and theoretical Poissonian distributions. d) Theoretical Poissonian distributions and calibrated photon number probabilities for a few llings.

Figure 4 . 10 .

 410 Figure 4.10. Reconstruction of the Wigner function of j = 0i using our Maximum likelihood on articially generated data fp n; g with n=0-8 and 72 displacement pulses . a) The method reconstructs the vacuum state perfectly. The white circle shows the truncation of the Hilbert for the numerical processing and the black dots show the displacements used in the tomography. b) Ripples appear in the reconstructed Wigner function when a Gaussian noise with = 1% is added to the data. c) The dierence between the initial Wigner functions in a) and b) shows the ripples more clearly.

Figure 5 . 1 .

 51 Figure 5.1. Sample and assembly parts. a) Transmon on a Si chip placed in a Cu cavity before it is closed. b) SMA coupling pin. c) Cu cavity closed and mounted on a coil, the axis of which is aligned with the transmon SQUID.

Figure 5 . 2 .

 52 Figure 5.2. Resonator spectra. a) Spectrum in the high power regime yields the intrinsic resonance frequency f 0 and quality factor of the resonator. b)VNA transmission spectra showing the anticrossing of the transmon and the resonator as the ux penetrating the SQUID is swept. c) Spectrum around f 0 at / 0 = 0.160 when f 0 = f g e . The separation of the hybridized transmon-resonator peaks indicated yields the coupling strength g 0 . The presence of a few tens of photons results in the intrinsic resonance frequency being also visible in the middle, as well as multi-photon peaks[5].

2 Figure 5 . 3 .

 253 Figure 5.3. Spectroscopic and relaxation time characterization. a)Transmon frequency f ge (red dots) and resonator frequency f 0 (green dots) as a function of the ux penetrating the SQUID. The black line is the calculated transmon frequency f g e max cos(/ 0 ) p . b) Transmon spectrum at / 0 = 0.258 showing both f g e and f g f /2 . c) Relaxation rate ¡ 1 measurements (black dots) as a function of f g e . The Purcell limit (red line) is plotted for comparison as well as the calculated ¡ 1 = ¡ 1;int + ¡ Purcell (blue line) with ¡ 1;int = 0.0154 MHz, yielding a good t with the data.

Figure 5 . 4 .

 54 Figure 5.4. Transmon spectroscopy.The transmon excitation frequency f ge shifts by ¡ / = ¡7 MHz per photon. A big peak is observed at f ge;n=0 = 6.329 GHz, corresponding to no photons in the resonator, and a small peak is observed at f ge;n=1 = 6.322 GHz corresponding to a single photon in the resonator. The peaks, tted with two Lorentzians with a width of 2.35 MHz, have amplitudes 0.12 and 4.43 mV.

Figure 5 . 5 .

 55 Figure 5.5. Resonator spectra for determining the thermal excitation of the transmon. a) Resonance spectra measured with the transmon in jgi (blue)and jei (red) using the pulsed measurement setup. For improved SNR and to be in the single photon regime, the spectrum corresponding to jgi is averaged over night using a VNA (magenta). b) An interval of two linewidths is used to t the VNA curve by a Lorentzian on top of a polynomial background (red lines). Subtracting the tted Lorentzian from the measured curve gives the background (green), possibly containing a Lorentzian contribution at f 0

Figure 5 . 6 .

 56 Figure 5.6. Dephasing rates at dierent frequencies f g e . a) Fitted transmon frequency f g e (red line) and indication of the resonator frequency f 0 (dark blue line). b) Measured free induction rate ¡ 2 (green full circles) and echo rates ¡ 2 E (magenta full circles). The dierent calculated contributions ¡ 1 /2 (blue line), ¡ th (purple line) from thermal resonator photons, and ¡ sq (cyan line) from 1/f ux noise are shown. The summed up values ¡ 1 / 2 + ¡ th (magenta line) and ¡ 1 / 2 + ¡ th + ¡ sq 2 / ¡ 2 (green triangle+line)

Figure 5 . 7 .

 57 Figure 5.7. Spectroscopic and decoherence characterization. a) Transmon frequency f g e (red dots) and resonator frequency f 0 (orange dots) as a function of the ux penetrating the SQUID. The black line is the calculated transmon frequency f g e max cos(/ 0 ) p . b) Relaxation rate ¡ 1 (blue dots) ts the calculated rate ¡ 1;int + ¡ Purcell with ¡ 1;int = 0.013 MHz. The free induction decay rate ¡ 2 (green dots) and Hahn echo rate ¡ 2E (magenta dots), as a function of f g e , are compared to the corresponding values of the previous sample on Si (green and magenta squares). The lower limit (¡ 1;int + ¡ Purcell ) / 2 of ¡ 2 and ¡ 2 E is shown as a blue dashed line. Neither thermal photon number n th nor the ux spectral density S were measured in this experiment.

  3.3) up to n = 10 we target g 0 10 MHz. We target ' E C = 450 MHz that yields a small Kerr eect K ef = 220Hz, which corresponds to a shift of 1/3 of the storage mode linewidth at 10 photons (assuming Q = 1M ). Minimum transmon lifetimes T 1 and T 2 of 1s are needed, and a coupling g R > 100 MHz to the readout mode is targeted. For the Josephson energy in the absence of a ux bias we target E J ;Max 20 GHz, yielding f ge = 8 GHz, which is high enough to measure the anticrossing with the readout mode. The dierent couplings to the two modes is achieved by careful placement

Figure 6 . 2 .

 62 Figure 6.2. Schematic representation of the eld distribution of the TE120 storage mode (blue) and TE210 readout mode (green), having orthogonal node planes. A slit for inserting the transmon (yellow) and the two ports (black dots) are placed close to a storage mode node and to a readout mode antinode.

Figure 6 . 3 .

 63 Figure 6.3. Cavity-chip design and corresponding simulations. a) Two coaxial cylindrical ports are placed on the top wall cavity. b) Cross section view of the port (top), cavity (center), Cu block (bottom) with a sapphire chip. c) Port and chip locations d) Slit geometry. e) Simulated transmission spectrum between the two ports showing the frequencies of the rst 3 resonances of the cavity.

Figure 6 . 4 .

 64 Figure 6.4. Electric eld and current distributions of the modes. The absolute value of the electric eld, which is almost completely along the z axis (in and out of the plane). The presence of the sapphire chip with " r = 9.3 distorts the distribution of the eld.

Figure 6 . 5 .

 65 Figure 6.5. The Al cavities and assembling parts. a) The two parts of the cavity. One wall of the cavity is only 1 mm thick and has a slit through which the transmon is inserted. A part of the transmon, including the SQUID, remains outside the thin cavity wall. The opposite wall has two holes for coupling pins. b) The two parts of the cavity are closed with an indium seal to ensure good conductivity between the two halves. A Cu block covering the part of the transmon protruding out of the cavity is pushed down by a gold-coated Cu bar. c) Cavity pins are made by soldering a Cu wire into one end of an SMA trough. d) A second generation of the cavity design where the slit has been rotated gave Q 0 = 10 million.

Figure 6 . 6 .

 66 Figure 6.6. Resonances of the cavity after the last preparation step with the estimated total, internal and external quality factors Q T ; Q i and Q e . All internal Q-factors are over 0.7 million.

Figure 6 . 7 .

 67 Figure 6.7. Simulating the transmon coupling to cavity mode (readout mode taken as an example). a)The transmon geometry is added to the cavity with the SQUID replaced by an internal port. b) Total admittance of the transmon-cavity system at a particular SQUID inductance L J , showing the bare readout frequency and the two hybridized transmon-readout modes on either side c) Separation between two hybridized modes as a function of L J . The minimum yields the coupling constant g.

Figure 6 . 8 .

 68 Figure 6.8. Determining transmon geometry from CST simulation to reach targeted couplings and charging energy. a) Inuence of pad separation on the coupling g R to the readout mode. b) inuence of the dimensions of the top pad (the one inside the cavity) on g R . c) Charging energy E C as a function of the bottom pad width. The red marks indicate the nal set of parameters.

Figure 6 . 9 .

 69 Figure 6.9. Transmon KJ19ch1 used in the QZD experiments. a) Two pads are separated by a 3 mm long wire with a SQUID located 50 m above the bottom pad. b) The SQUID is made up of two 200x200nm 2 junctions in a loop with an area of 1.1x2.6m 2 . c) Transmon fabricated on a sapphire chip and mounted in the Cu block.

Design and characterization of an sample suited for QZD 129 -Figure 6 . 10 .

 129610 Figure 6.10. Characterization of the fourth preliminary transmon qubit. a) Mapping of f g e , f gf /2 and f R ~as a function of ux . b) Relaxation and Ramsey fringe measurements at = 0, yielding T 1 = 1.5 s and T 2 = 2.8 s. c) Measured T 1 (red dots) for dierent values of f ge . Black line shows the Purcell limit of T 1 whereas the green dots show the intrinsic T 1 after removing the Purcell contribution.

Figure 6

 6 Figure 6.10 b shows relaxation-and dephasing time measurements at / 0 = 0, where f ge =8.01 GHz, giving T 1 = 1.5 s and T 2 = 2.8 s (see protocol in section 4.1). T 1 was measured for dierent f ge and compared with the estimated Purcell limit due the its coupling to the readout mode. Figure6.10 c shows an intrinsic T 1 = 3 s but at the efresonant condition ! g e = ! 0 + = 2 6.76 GHz, it is reduced to 1.8 s because of the Purcell eect.

Figure 6 . 11 .

 611 Figure 6.11. Spectroscopy of the jei ! jf i transition. a) Repeated spectroscopy over a 17-hour period showing two resonances with a frequency separation varying from 0 to 7 MHz. b) Three cross section from a.

Figure 6 . 13 .

 613 Figure 6.13. Bare resonance peaks of a) the storage mode with f 0 = 6.43306 GHz and Q 0 = 2.2 million, and b) the readout mode with f R = 7.328 GHz and Q R = 12 thousand.

Figure 6 . 14 .Figure 6 . 15 . 6 . 2 . 2 . 2 Figure 6 .

 61461562226 Figure 6.14. The equal separation ¡;0 = +;0 of f ;1 from f g e indicates the ef-resonance at / 0 = 0.291.

Figure 6 . 16 .

 616 Figure 6.16. Photon number splitting at ef-resonance. a) Transmon excitation spectra around f g e for three dierent llings of the storage mode (horizontally shifted for clarity), showing the transitions for f n with n 2 f¡2; ¡1; 0; +1; ::::; +17g. b) Transition frequencies f n (orange dots) are compared with calculations from appendix A (blue + and -signs) and numerical calculations which include the transmon level jhi (green, circled + and -signs). The dotted line shows f ge 2n p g 0:

Figure 6 . 17 .Figure 6 . 19 .

 617619 Photon number n

Figure 6 . 20 .

 620 photon number n

Figure 6 .

 6 21 shows the Rabi oscillations with and without the o-resonant tones. The additional tones disturb the Rabi oscillations signicantly after 1.5 periods. However, by having alternate phases, i.e. a phase for n 2 f3; 7g, the disturbance after a pulse is small. Since the parity should not reveal the photon number and the photon number signals S +;n are dierent, we use dierent Rabi angles n on each transition +n to ensure the indistinguishability. More precisely the n are adjusted to give equal signals S +;n ( n ) = min n (S +;n ).

Figure 6 . 21 .

 621 Figure 6.21. Inuence of oresonant driving. Rabi oscillations with a single tone at f g e drives the jg0i ! je0i transition (red). Adding frequency tones at f +;n for n 2 f1; 3; 5; 7; 9g disturbs the Rabi oscillations (blue). The o resonant eect is reduced by having an opposite phase for n 2 f3; 7g (green.)

Figure 6 . 22 .

 622 Figure 6.22. Truncated parity measures and measured Wigner functions. a)The two parity measure P 0 (blue) and P 00 (red) applied to the vacuum state. The measurements (dots) behave as calculations predict (lines) and give correct measures of the parity up to n p 2.5, where the truncation starts to have an eect. b) Measurement of the Wigner function using P 00 . c) Reconstruction of the Wigner function from measurements using the maximum likelihood method.

Figure 7 . 1 .

 71 Figure 7.1. Pulse sequence for inducing the QZD and subsequent tomography of the eld in the storage mode. The dierent steps are indicated on the graph. The empty pulses in colors represent the pulse envelopes. The dense blue lling represents the product of the envelopes with the IF frequency used to generate the actual microwave pulses using heterodyne modulation.

  The 'transmission' T = 1 ¡ n=0 2 p(n) across the EC after one period

Figure 7 . 2 .

 72 Figure 7.2. Probing the Zeno blocking strength on Fock state jn = 3i. a) Occupation probabilities p (n) for n = 0; :::; 4 as a function of the lling time at a xed lling rate _ = 0.27 s ¡1 , and of the blocking strength +;3 at frequency f +;3 . b) Loss of population inside the EC n=0 2 p(n) after one period ('transmission'), as a function of blocking strength +;3 .

Figure 7 . 3 .Figure 7 . 4 .Figure 7 . 5 .Figure 7 . 6 .

 73747576 Figure 7.3. Occupation probabilities p (n) for n = 0; :::; 4 as a function of the lling time at a xed lling rate _ = 0.27s ¡1 , and blocking frequency f at a xed amplitude (yielding +;3 /2 = 2 _ ). The white dashed lines indicate the oscillations in p(n) when blocking at the proper frequencies f +;2 , f +;3 or f +;4 . Midway between these frequencies the blockade does not go to zero and the population of the vacuum state returns once to 30%.

Figure 7 . 7 .

 77 Figure 7.7. Evolution of the measured (dots) and simulated (lines) probabilities p(n) during the QZD of the eld initially in the vacuum state, and conned inside the exclusion circle with radius 3 p (blockade on n=3).

Figure 7 . 8 .

 78 Figure 7.8. Simulation of the occupation probabilities p(qn) as indicated, for the conned QZD with blockade at n = 3. Simulations based on the experimental parameters are performed without decay channels (top row), individual decay channels (middle row), and the full model (bottom panel). The rst row compares the blocking and lling strengths used in the experiment (left) with stronger (center) ore weaker values.

Figure 7 . 9 .

 79 Figure 7.9. Wigner function snapshots every quarter period of the QZD of the eld initially inside the vacuum state, and conned in the exclusion circle of radius 3 p (blockade on n=3). The measurement (middle row) is compared with the simulations of the full model (top row) and with the ideal case (bottom row). The evolution of the Wigner function is well captured by the simulation using the measured decoherence parameters. The asymmetry in Q at half integer periods, which does not occur in the ideal case, is captured by the full model simulations. The white dashed lines correspond to the cross sections shown on the left side after 0.5 (blue) and 1.5 (red) periods. In the ideal case the two cross sections are identical.

Figure 7 . 10 .

 710 Figure 7.10. Wigner function of the cavity state after simulating the QZD up to half a Zeno period when the blocking strength is a) doubled, b) the same, or c) halved compared to the experiment, keeping the ratio " ll /" block constant.

Figure 7 . 11 .Figure 7 . 12 .

 711712 Figure 7.11. Wigner function of the state = ¡ 7 p reconstructed from the measurement of 1608 observables (201 displacements shown as black dots) using the maxlike algorithm. The white dashed circle shows the limit of the 18 18 density matrix used for the reconstuction.

Figure 7 . 13 .Figure 7 . 14 .

 713714 Figure 7.13. QZD of a tangential collision with an EC of a radius 2 p . a) The coherent state = ¡ 5 p ¡ i 2 p is prepared and is subsequently driven towards the EC. b) Calculated of the perfect QZD.

Figure 7 . 16 .

 716 Figure 7.16. Figure reproduced from ref [45]© demonstrating the QZD in Rydberg atoms with spin J = 25 blocked at k z = 4. a) Experimental Wigner function, W (; ), obtained from the reconstructed density matrix of the spin state after QZD at the phase inversion time t 1 = 0.76s. b) Results of the numerical simulation of the experiment. The delity of the calculated density matrix c with is Tr 2 p c p p = 0.93.

Figure 7 . 17 .

 717 Figure 7.17. One period of the QZD both measured (top) and simulated (bottom) with a block on n = 3 and n = 4. Clear negativities are observed in both cases and a good agreement is between the experiment and simulations.Figure from [8]©.

Figure C. 1 .

 1 Figure C.1. Schematic illustration of the protocol and induced dynamics. a) Protocol of a spectroscopy pulse exciting the transmon in the presence of a photon. b) General scenario allowing arbitrary photon numbers. c) Scenario starting with no photons and considering a single excitation during T p d) Same as c) except it allows for a repeated excitation after the previous one has decayed.

  this eective oscillator Õ has a shifted frequency ν O = ν O + α ε 2 -ε 4 , and a small Kerr non-linearity K = 2αε 4 inherited from the 3LS. The two other ladders {|±n } (extended down to n = 0 by |+0 ≡ |-0 ≡ g0 and |+1 ≡ |-1 ≡ e0 ) have energies and eigenvectors

ν

  ±n do not vary linearly with n as in the usual dispersive case, but as √ n. Selective addressing of | gn requires the separation ∆ν ±n = ν ±(n+1) -ν ±n to be larger than the transition linewidth. A second condition is that the driving strength of the ν ±n transition is low enough to avoid driving o-resonantly the neighboring transitions at ν ±(n+1) . It is now interesting to compare the Kerr non-linearity K = 2α (g O /∆) 4 obtained here with the value K =

2

 2

  1. 'MLS ef-resonant' scheme for manipulating Fock states |n of a quantum harmonic oscillator. Left: the oscillator with angular frequency νO is coupled with a coupling frequency gO to a multilevel system (MLS) with eigenstates{|g , |e , |f , ...}, the |e ↔ |f transition of which is resonant with νO, whereas the |g ↔ |e transition is detuned by α. Right: The resulting energy diagram consists of a quasi-harmonic ladder {| gn } when the 3LS is left unexcited, and of two anharmonic ladders of levels |±n that correspond approximately to symmetric and anti-symmetric superpositions of |e (n -1) and |f (n -2) states for n ≥ 2. The | gn ↔ |± (n + 1) transitions can be driven at dierent frequencies νge ± 2 (n -1)gO to manipulate selectively any | gn . ∆ g O /∆ 4 that would be obtained for a two-level system or the value K" = α " g " O /∆ " 4 obtained in perturbation for a transmon in the far dispersive regime g " O , α " ∆ " [17], keeping the same separation S = √ 2g O = 2α " g " O /∆ " 2 between the rst two Fock state dependent excitation frequencies. With respect to the far dispersive case, the new non-linearity is thus reduced by a factor K /K = α/α (α/S) 2 /2 that can be made large easily. This reduction factor, which reaches several hundreds (at xed transmon anharmonicity α = α " ) in our implementation of section II, is what makes our 'efresonant' scheme interesting. What we show here with a simple ef-resonant 3-LS is that getting out of the perturbation regime α " ∆ " reduces drastically the Kerr nonlinearity. However considering only three levels makes the argumentation only qualitative for a transmon at large number of photons in the oscillator, and a quantitative evaluation requires taking into account at least the fourth transmon level as we do in section III. II. EXPERIMENTAL IMPLEMENTATION We implement the proposed 'ef-resonant' scheme in a three-dimensional circuit-QED setup [20] combining a cavity with input (1) and output (2) ports and a tunable transmon qubit [15] (see Fig. 2a-b). To be superconducting at low temperature and have a high internal quality factor, the cavity is made of two blocks of pure aluminum, which are milled, pierced, polished, and chemically etched over about 20 µm. The transmon is fabricated on sapphire by double-angle evaporation of Al and oxidation, through a suspended shadow mask made by ebeam lithography. It has two pads connected by a 2.6 mm long wire including a magnetic ux tunable Josephson junction with a SQUID geometry located 50 µm above the bottom pad. This enables tuning the transmon energy spectrum and reaching the 'ef resonant' condition. The transmon is only partly inserted in the cavity so that the SQUID remains about 0.1 mm outside, in the applied external magnetic eld. More precisely, the bottom part of the transmon with the SQUID is held and protected by a copper block, the other side being inserted in the cavity through a slit in the bottom wall. The two halves of the cavity are then pressed one against the other with an indium seal in-between. In our design the TE120 cavity mode is used as the quantum oscillator O at frequency ν O ∼ 6.4 GHz whereas mode TE210 at frequency ν R ∼ 7.3 GHz is used for reading the quantum state of the transmon dispersively [2, 3]. The transmon and the ports are thus placed very close to a node of mode O so that the corresponding coupled quality factor Q O is dominated by the internal losses, and the transmon-oscillator coupling g O is about 10 MHz. They are also placed at an antinode of mode R to get a strong transmon-readout coupling g R ∼ 150 MHz and thus a large enough dispersive shift χ R ∼ 10 MHz, and a low coupled quality factor Q R ∼ 15 × 10 3 allowing fast readout of the transmon. The exact positions as well as the precise transmon geometry are determined using the CST

Figure 2 .

 2 Figure 2. Circuit-QED implementation of the 'ef-resonant' scheme. (a) The harmonic oscillator O and the MLS are the mode 120 of a superconducting Al cavity and a tunable transmon qubit with a SQUID. The transmon chip is inserted only partly in the cavity, the SQUID being exposed to a dc magnetic eld -→ B . The transmon is weakly coupled to O and strongly to the cavity mode 210 used for dispersive readout of the transmon state. The two modes have quality factors QO = 2 × 10 6 and QR = 15 × 10 3 . (b): Picture of one halfcavity with chip and Cu cap. (c) Equivalent electric circuit of the system with relevant frequencies fR,O, quality factors QR,O, and coupling frequencies gR,O. O is driven coherently and resonantly through port 1 (purple pulse). Fock state |gn are manipulated by driving the | gn → |+n transition (red pulse). A projective measurement on Fock state | gn is obtained by a π pulse at ν+n followed by a readout pulse (in blue). (d) Electrical setup at room temperature (300K) and inside the dilution refrigerator: Cavity resonances are measured with continuous waves using a vectorial network analyzer (VNA) whereas pulsed experiments use heterodyne modulation and demodulation.Microwave pulses at the cavity (qubit) frequencies νO,R (ν+n) are obtained by single sideband mixing of a continuous microwave (LO) with an intermediate frequency modulated pulse generated by two channels of an arbitrary waveform generator (AWG). All pulses travel along an attenuated and ltered line to cavity port 1. The readout signal transmitted at port 2 is ltered, isolated from backward propagating noise, amplied with a parametric amplier (JPA) in reection, a high electron mobility transistor (HEMT), and room temperature ampliers, then demodulated to produce two quadratures, which are nally ltered, amplied and digitized (ADC).

Fig

  Fig. 2d.

5 -Figure 3 .Figure 4 .Figure 5 .

 5345 photon number n

  . . . 2.1.1 QZD with projective measurements . . . . . . . . . . . . . 2.1.2 QZD with unitary kicks . . . . . . . . . . . . . . . . . . . . 2.1.3 QZD with continuous coupling . . . . . . . . . . . . . . . . 2.2 Cavity QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.1 Harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.1 Cavity resonator . . . . . . . . . . . . . . . . . . . . . .

62) 2.2.2.2 Cooper pair box

  4E C . In the so called charge regime E C & E J , the eigenstates are almost pure charge states jn CP i, corresponding to an exact number n CP of excess Cooper pairs (hence the name charge qubit).

	A Cooper pair box (CPB) is an electrical circuit as shown in g. 2.7, where a small superconducting island is connected to a superconducting electrode by a JJ, and to a gate voltage through a capacitor. The charging energy of one electron in the CPB is E C = e 2 2C (2.63) with C = C g + C J the total capacitance of the island. Electrons tunnel through JJ in Cooper pairs so the energy needed to V g C g Figure 2.7. Cooper pair box. Superconducting island (dotted line) is connected to the ground via a JJ and to add/remove one pair to/from the island is Kristinn Júlíusson E J a voltage gate.

Theoretical background Figure 2.8. Energy

  levels of a charge qubit. For the CPB E J & E C the energy levels are sensitive to variations in n g . As E J / E C increases the levels become more at although the n g sensitivity is greater in higher levels.

Table 3 .1.

 3

	Sample	KJ6ch27		KJ15ch11	KJ19ch1
	Wafer	SiO2 50 nm/	Al2O3 330m	Al2O3 330m
		Si 300 m
	Wafer	Dip	in	HF	to
	preparation	remove

SiO 2 Rinse in water Rinse in IPA Dry Remover 1165@60 C 10min + 1min in ultra- sound. Twice in sep- arate baths. Water jet machine 180s, dry 140s, heater 9, rinse in IPA. O 2 plasma ashing at 0.2 mbar (17cc) and 100W for 1 min. Acetone @45 C for 2 min in ultrasound. IPA @45 C for 2 min in ultrasound Dry O2 plasma reactive ion etching @ 0.3mbar(50cc) 50W for 1 min Buer HF 2 min Rinse in water Rinse in IPA Dry Dry vapor @110 C for 2 min, and cool afterwards.

		Measurement setup, instruments and techniques
	Spinning	MMA/

MAA 8.5 EL 10: drops put in the center of the wafer. Spin at 2000 rpm for 60 s + 6000 rpm for 2 s Bake @170 C for 120 s Measure thickness of 545 nm PMMA 950k A3 (AFM/STM) drops put in the center of the wafer. Spin at 4000 rpm for 60 s + 8000 rpm for 2 s Bake @170 C for 15 min Measure thickness of 113 nm MMA/MAA 8.5 EL 10: drops put in the center of the wafer. Spin at 2000 rpm for 60 s + 6000 rpm for 2 s Bake @180 C for 5 min Measure thickness of 593 nm PMMA 950k A3 (AFM/STM) drops put in the center of the wafer. Spin at 4000 rpm for 60 s + 6000 rpm for 2 s Bake @180 C for 15 min Measure thickness of 120 nm MMA/MAA 8.5 EL 10: drops put in the center of the wafer. Spin at 1250 rpm for 60 s + 8000 rpm for 4 s Bake @180 C for 5 min Measure thickness of 709 nm PMMA 950k A3 (AFM/STM) drops put in the center of the wafer. Spin at 3250 rpm for 60 s + 8000 rpm for 4 s Bake @180 C for 15 min Measure thickness of 139 nm Discharging layer None Evaporate 10 nm of Al Evaporate 10 nm of Al

	Dicing	Scribe pack side	Spin	UV3	resist
		with	a	diamond	2000rpm,	bake
		scriber.		@120 C
		Break into chips

for 2 min Dice 150 m deep on the back side with a dicing saw. Remove UV3 resist in IPA bath for 1 min Break into chips

	Spin	UV3	resist
	2000rpm,	bake
	@130 C

for 1 min Spin UV3 resist 2000rpm, bake @75 C for 15 s Measure thickness of 1000 nm Dice from the back with a dicing saw, 75 m per pass. Remove UV3 resist in IPA bath for 1 min

	Kristinn Júlíusson
			Beam current:
			Big structures
			5.24 nA
			Small structures
			23 pA
			Undercut boxes
			23 pA
	Develop-	Remove	exposed
	ment	resist MIBK in IPA
		(1:

3) for 50s Rinse in IPA Dry well Remove charging layer 20g/mol in KOH for 27 s Rinse in water Dry well Remove exposed resist MIBK in IPA (1:3) for 60s Rinse in IPA Dry well Remove charging layer 20g/mol in KOH for 35 s Rinse in water Dry well Remove exposed resist MIBK in IPA (1:3) for 60s Rinse in IPA Dry well

	Mask	pre-	None	None	O 2 plasma reactive
	cleaning				ion	etching	@
					0.3mbar(

50cc), 66V for 12s Evaporation Pump vacuum 6.7 10 ¡8 mbar Ar ion milling 5mA for 2s at two angles 27 and -27 Evaporate 25 nm of Al at 1 nm/s at an angle of 27 Oxidize for 3:10 min 30 s @6.7 mbar Evaporate 35 nm of Al at 1 nm/s at an angle of 27 Pump vacuum 1.7 10 ¡7 mbar Ar ion milling 5mA for 2s at two angles 28 and -30 Evaporate 25 nm of Al at 1 nm/s at an angle of 28 Oxidize for 3:10 min 30 s @6.7 mbar Evaporate 60 nm of Al at 1 nm/s at an angle of -30 Pump vacuum 2.5 10 ¡6 mbar Ar ion milling 5mA for 2s at two angles -27 and 23 Evaporate 25 nm of Al at 1 nm/s at an angle of -27 Oxidize for 10 min 30 s @13.7 mbar Evaporate 60 nm of Al at 1 nm/s at an angle of 23

		Measurement setup, instruments and techniques
	Lift o	Acetone @60 C for	Same	Same
		3 min + squirting
		jets with pipette
		Rinse in IPA
		Bake sample on a
		hot plate @100 C
		for 1 min to satu-
		rate oxidation
	Junction	Make sure probes,
	resistance	sample	stage,
	measure-	tweezers	and
	ment	sample box

are all grounded Place the sample on the stage and turn of the lights to reduce conduction through the sub- strate.

  while the data set Y fp n; g veries p n; = Tr[E n; ] + n; , with n; being a Gaussian noise with a standard deviations n; .

	For Y the probability of having all p n; 2 [s n; ; s n; + ds n; ] knowing reads
	P 0	Y 2	n;	[s n; ; s n; + ds n; ] 1	=
	exp @ ¡	X n;	(s n; ¡ Tr[E n; ]) 2 2 n; 2	A n;	ds n; n; 2 p :	(4.10)
	Using this expression, the Bayesian Mean estimate of the density
	matrix is given by
				BM

Table 6 . 1 .

 61 Table of internal quality factors measured at cryogenic temperatures for dierent preparations of the cavity.

	Preparation	Fundamental	Storage	Readout
	Cleaned and chemically
	etched, no In seal	>1.5M	5.5M	20k
	Slit made	4.0M	0.6M	3.6k
	Slit covered with Cu
	block	2.3M	1.0M	17k
	Slit with sapphire chip
	and Cu block	1.6M	1.2M	10k
	Slit with sapphire chip
	and Cu block; cavity
	closed with In seal	2.1M	0.8M	750k

  symmetric superposition of |e (n -1) and |f (n -2) . A particular Fock state | gn of Õ is selectively manipula-

ble by addressing the | gn → |± (n + 1) transitions to the hybridized oscillator-transmon states, at frequencies ν ±n = ν ge ± √ 2ng O + (3n + 1)g O ε/2 + o(ε 2 ) (with a Rabi frequency about √ 2 slower for all n > 0 than for the pure |g → |e transition). Note that these frequencies

Kristinn Júlíusson

Table of contents

2.1. The eect is named after the ancient philosopher Zeno of Elea and refers to his paradox described in ref.[20]

2.2. We speak equally of energies and frequencies as they are related simply by ~.Kristinn Júlíusson

2.3. Note that the omitted vacuum energy ~!0 / 2 gives rise to a (Lamb) shift of ! g e : Kristinn Júlíusson

= ge ¡ e f /2; (2.98) Theoretical background

2.4. The rate is only approximately the same for the two levels as explained in refKristinn Júlíusson

ACKNOWLEDGMENT

We gratefully acknowledge discussions within the Quantronics group, technical support from P. Orla, P. Senat, J.C. Tack, D. Duet, and V. Padilla, as well as nancial support from the European research contracts CCQED and ScaleQIT.

Calibrating the transmon readout

We refer to the transmitted amplitude A (g) at the top of the readout mode when the transmon is in jg i as the 'base line' (see section 4.1.2). When the population of the storage mode is increased a decrease of the base line is observed, as shown in g. 6.18. This decrease is due to a cross-Kerr eect making the resonance frequency of the readout mode depend on the state of the storage mode. In any experiment involving both lling of the storage mode and driving of the transmon, this eect is taken into account by measuring rst the base line A (g) leaving the transmon in jg i, before measuring A in the complete sequence. The signal S = 1 ¡A / A (g) corresponds thus to the probability of the transmon being excited.

Calibrating the photon number probabilities

The signals S +;n following pulses on the +n transition are mapped to Fock state occupation probabilities p(n) using the calibration method discussed in section 4.2.1. A 300 ns square pulse is used to create a coherent state j i, where n = 2 is varied over 26 values by changing the pulse amplitude , followed by a pulse on the frequencies f +;n for n = 0; ::::; 10. A waiting time 6T 0 is taken between each repetition of the measurements to let the eld in the storage mode decay. Each point (n; ) is averaged over 40000 sequences.

From the calibration we get the lling speed k = / = 2.29V ¡1 and the calibration coecients c n = S +;n / p(n)=f0.764; 0.835; 0.847; 0.846; 0.833; 0.854; 0.846; 0.834; 0.847; 0.832; 0.841g, for n=0,...,10. The calibrated probabilities p(n) are shown in g. 6.19 as a function of A. The bottom panel shows the Poissonian distributions P n (n) with n = (kA) 2 and probabilities p(n) for a few values of A. The discrepancy between the two is within the 2 error bars with the standard deviation 0.6%.

QZD inside an exclusion circle

The full description of the cavity eld evolution will be presented here in the form of the Wigner function. Then we will analyse the evolution of p(n) by simulating the evolution numerically.

Wigner function snapshots of conned QZD

Having chosen the blocking and lling strengths, we induce the quantum Zeno dynamics of the storage mode eld conned inside the EC with a radius j j = 3 p . The Wigner function of the eld is obtained at dierent times through out a 30 s period (two Zeno periods), using Wigner tomography with the truncated parity measure P 00 discussed in section 6.3. Figure 7.4a shows snapshots of the evolution every quarterperiod The amplitude increases along the positive real axis until it reaches the EC and a sharp phase inversion occurs. Thereafter the state returns back to the origin, completing one period. At half-integer periods a 'cat-like' state is observed with two components located at at ' 1.1, with fringes in the center indicating a coherent superposition of the two components. Cross sections of the Wigner function along the white dashed lines in g. 7.4a are plotted in g. 7.4b. These curves show the negative parts of the Wigner function, reaching ¡ 0.23 and ¡0.19 after half a period, and ¡0.09 and ¡0.03 after 1.5 period. After two periods the amplitude maximum is again centered at the origin but with a smaller value, and a bigger spread of the state. We attribute this deformation to limited coherence times T cav = 47 s (Q 0 = 1.9 M), T 1 = 1.9 s, and T ' = 9 s in this second experimental run.

Artifacts due to the truncated parity measure P 00 appear as articially high values of W in the corners of the frames in g. 7.4a. We thus turn to the maximum likelihood method (maxlike) presented in section 4.2.4. By the time this was implemented, the experiment was in its third run with T cav = 39 s (Q 0 = 1.6 M), T 1 = 1.6 s, and T ' = 9 s.

Figure 7.5 shows a repetition of the same experiment, with the new sample parameters and using the maxlike method. The shorter lifetimes result in a faster decay of the Zeno dynamics, shown by the diminished negative values of the Wigner function: Compared to the ideal Zeno dynamics shown in section 2.3.1, we also observe an asymmetry of the Wigner function, with a higher amplitude at Q > 0 visible

Kristinn Júlíusson

The Hamiltonian is encoded in the tensorial basis jq i jni with the transmon levels q 2 fg; e; f g and oscillator Fock states n < 30. We model the dynamics of the density matrix under H with the Lindblad master equation [23]

where

a q are the collapse operators corresponding to the cavity decay, the transmon relaxation and the transmon dephasing.

Simulated evolution of p(n) using QuTiP

The Lindblad equation is integrated using the mesolver (master equation solver) of the Python quantum toolbox QuTiP [25]. Prior to integration, the eigenstates of the Hamiltonian H 0 + H c are obtained and identied as the jq ni with q 2 fg; +; ¡g (eigenstates of our ef-resonance scheme). During integration QuTiP follows either the full density matrix or the occupation probabilities p(qn) Tr(jqnihqnj) of the jqni states. These simulations best t the data when using the parameters g 0 / 2 = 7.5 MHz, " ll = 0.28 s ¡1 and " block = 2 p 0.54 MHz, very close or equal to the measured values g 0 = 7.45 MHz, " ll = 0.27 s ¡1 and " block = 2 p 0.54 MHz. For the decay parameters we take the run 3 values T 1 = 1.6 s and T ' = 9 s mentioned above, and a cavity life time T cav = 35 s directly measured at the ef-resonance. The evolution is also simulated without any decay, as well as with each decay channel individually (i.e. nite T cav , T 1 or T ' ).

All simulations of the fp(qn)g for blockade at Fock state jn = 3i are shown in g. 7.8. In absence of any of the three decay channels (top left panel), the p(qn) oscillate as expected for ideal QZD except for a spurious population of p(+3) of 7% at half integer periods. This is due to the imperfect blockade allowing a small population of jg3i, which is then immediately transfered to j+3i. By increasing the ratio " ll / " blcok by 2 (top right panel) the population p(+3) is reduced to 1.7%. Another discrepancy with the ideal QZD is that p(g1) does not reach zero at half integer periods. This is due to o-resonant driving of the jg2i ! j+2i transition. This is evidenced by p(g1) being much larger when the blocked in H n<2 H n=2 . Compared to the width of 0.57 of the coherent state, the calculated QZD induces a squeezed state with a width of 0.25.

Even though the peak in the prole from the measurement has a smaller width than a coherent state one cannot say it is a squeezed state since the peak amplitude is signicantly lower and large parts of the state are located away from the peak. The simulation however shows that a squeezing of more than half can be achievedusing this method.

QZD experiments by other groups

QZD in Rydberg atoms

Experimental realization of the quantum Zeno dynamics was rst reported by Signoles et al. [START_REF] Signoles | Conned quantum Zeno dynamics of a watched atomic arrow[END_REF] in 2014.

The authors used Rydberg atoms prepared with principal quantum number n e = 51 and with the highest angular momentum J = 25. The angular momentum projection on the quantization axis of the generalized Bloch sphere can take integer values J ¡ k with k 2 [0; 2J], and forms a Hilbert space of 51 nearly-equidistant levels jn e ; k = 0i,...,jn e ; Kristinn Júlíusson

Appendix A Energy spectrum of the ef-resonance scheme

We consider the case of a harmonic oscillator with frequency f 0 and annihilation operator a, coupled to a three-level system (3LS) with eigenstates fjg i; jei; jf ig and lowering operator a q . We also assume a weak anharmonicity = f ef ¡ f ge f ge , such that a q can be approximated by the annihilation operator of a harmonic oscillator restricted to three levels. The two subsystems obey the resonant condition ef = 0 (detuning = f ge ¡ f 0 = ¡ ) and are subject to an exchange interaction with coupling frequency g 0 , yielding the Hamiltonian

H coupling = h f 0 (a y a q + a a q y ):

This couplings make the Hamiltonian matrix diagonal by block in the tensorial basis jx ni = jxi jni (x = g; e; f ), with subsequent blocks of size 1, 2, 3, 3, 3... Using the reduced coupling = ¡g 0 /, these blocks write

with I k the identity matrix of dimension k and B n = fjg ni; je (n ¡ 1)i; jf (n ¡ 2)ig the basis for n 2. The diagonalization of each block yields analytical eigenenergies and vectors, which for n 2 are functions of the three real solutions of the cubic equation

To shed light on the physics, we expand these analytical quantities in the small parameter ". As shown in Fig., the levels form three distinct energy laddersfjg nig, fj¡nig and fj+nig: The ladderfjg nig n0 corresponds to the almost unperturbed oscillator when the 3LS is left in its ground state. With eigenenergies and eigenvectors 5)

this eective oscillator O ~has a shifted frequency f ~0 = f 0 + ( 2 ¡ 4 ), and a small Kerr non-linearity K = 2 4 inherited from the 3LS. The two other ladders fjnig (extended down to n = 0 by j+0i j¡0i jg 0i and j+1i j¡1i je 0i) have energies and eigenvectors For n 2 the zeroth-order approximation in of these eigenvectors are simply the symmetric and anti-symmetric superposition of je (n ¡ 1)i and jf (n ¡ 2)i. A particular Fock state jg ni of selectively manipulable by addressing the jg ni ! j(n + 1)i transitions to the hybridized oscillator-transmon states, at frequencies f n = f ge 2 n p g 0 + (3 n + 1) g 0 /2 + o( 2 ) (with a Rabi frequency about 2 p slower for all n > 0 than for the pure jg i ! jei transition).

Kristinn Júlíusson

Appendix B Maximum likelihood algorithm adapted to Gaussian errors

This note is derived by Pierre Rouchon after discussions with Daniel Esteve, Kristinn Juliusson, Helene le Sueur and Denis Vion.

Take an integer a collection of n > 0 and K = f 1 ; ::::; K g a collection of k complex amplitudes.

We consider the following somehow idealized tomography of the density operator of an harmonic oscillator from the following measurement data set:

fp n; g n2f0;::::;n g;2K :

For According to KKT, the component w n of W are characterized by the following system depending on the components v n of V 8n 2 f0; :::; n g: w n > 0; p n > 0 and w n p n = 0 8n 2 f0; :::; n g:

where p n > 0 is the multiplier associated to the inequality constraint x n > 0 and q 2 R to the equality constraint n w n = 1. These conditions are obtained from the Lagrangian L(w 0 ; ::::; w n ; p 0 ; ::::;

and its minimization versus (w 0 ; ::::; w n ; p 0 ; ::::; p n ; q) on the convex set where, for all n; w n > 0 and p n > 0.

Kristinn Júlíusson

Appendix C Estimating thermal photon population in the resonator

This appendix calculates how to translate the relative amplitudes of the photon splitted ge spectroscopic peaks (see section 1.2.1.2) of the transmon into population of Fock state 1 (assuming it is small). We consider a Rabi pulses of duration T p at the transmon excitation frequency f ge;n=1 in presence of 1 photon in the cavity, which induces Rabi oscillation on the jg i ! jei transition at the Rabi frequency f R = R / 2 . The spectroscopic signal at f ge;n=1 (see g. 5.4 in chapter 5) is proportional to the probability p e;1 of having the transmon excited at the end of the pulse, which we evaluate now. During the Rabi pulse, the number of thermal photons in the cavity uctuates as shown in g. C.1, with rising and lowering rates + and ¡ (here ¡ = is the cavity decay rate). However, we anticipate from the experiment a very small average photon number n th and thus a low + = ¡ n th . The general scenario of g. C.1b (several photons entering and exiting the cavity during the pulse, several photons present at the same time, etc) is thus very unlikely, and we can focus on simpler scenari corresponding to a single appearance of one photon during T (scenario1 = 1 entrance + 1 possible exit), or two photons (scenario 2 = rst entrance, rst exit, second entrance, and possibly second exit).

In scenario 1, p e is simply the product of the excited population sin 2 ( f R ) acquired during the overlap of the Rabi and 1 photon pulse, by the probability 1 ¡ n th to have zero photon at t = 0, the probability + e ¡¡t1 dt 1 to have one photon entering the cavity between t 1 and t 1 + dt 1 , the probability ¡ e ¡¡ d to have the photon leaving after , the probability e ¡+(Tp¡t1¡ ) that a second photon does not enter, and the probability e ¡¡1(Tp¡t1¡ ) that the transmon has not relaxed at the end of the Rabi pulse, integrated over t 1 < T p and < T p ¡ t 1 :

Appendix D

Published article

185

Manipulating Fock states of a harmonic oscillator while preserving its linearity (Dated: July 18, 2016)

We present a new scheme for controlling the quantum state of a harmonic oscillator by coupling it to an anharmonic multilevel system (MLS) with rst to second excited state transition frequency onresonance with the oscillator. In this scheme that we call 'ef-resonant', the spurious oscillator Kerr non-linearity inherited from the MLS is very small, while its Fock states can still be selectively addressed via an MLS transition at a frequency that depends on the number of photons. We implement this concept in a circuit-QED setup with a microwave 3D cavity (the oscillator, with frequency 6.4 GHz and quality factor QO = 2 × 10 6 ) embedding a frequency tunable transmon qubit (the MLS). We characterize the system spectroscopically and demonstrate selective addressing of Fock states and a Kerr non-linearity below 350 Hz. At times much longer than the transmon coherence times, a non-linear cavity response with driving power is also observed and explained.