L R Schreiber
email: schreiber@tudelft.nl

F R Braakman

T Meunier

V Calado

J Danon

J M Taylor

W Wegscheider

L M K Vandersypen

Coupling molecular spin states by photon-assisted tunneling

Artificial molecules containing just one or two electrons provide a powerful platform for studies of orbital and spin quantum dynamics in nanoscale devices. A well-known example of these dynamics is tunneling of electrons between two coupled quantum dots triggered by microwave irradiation. So far, these tunneling processes have been treated as electric dipole-allowed spin-conserving events.

Here we report that microwaves can also excite tunneling transitions between states with different spin. In this work, the dominant mechanism responsible for violation of spin conservation is the spinorbit interaction. These transitions make it possible to perform detailed microwave spectroscopy of the molecular spin states of an artificial hydrogen molecule and open up the possibility of realizing full quantum control of a two spin system via microwave excitation. * l.r.

In recent years, artificial molecules in mesoscopic systems have drawn much attention due to a fundamental interest in their quantum properties and their potential for quantum information applications. Arguably, the most flexible and tunable artificial molecule consists of coupled semiconductor quantum dots that are defined in a 2-dimensional electron gas using a set of patterned electrostatic depletion gates. Electron spins in such quantum dots exhibit coherence times up to 200 µs [START_REF] Bluhm | Long coherence of electron spins coupled to a nuclear spin bath[END_REF], about 10 4 -10 6 times longer than the relevant quantum gate operations [START_REF] Petta | Coherent manipulation of coupled electron spins in semiconductor quantum dots[END_REF][START_REF] Koppens | Driven coherent oscillation of a single electron spin in a quantum dot[END_REF], making them attractive quantum bit (qubit) systems [START_REF] Hanson | Spins in few-electron quantum dots[END_REF].

The molecular orbital structure of these artificial quantum objects can be probed spectroscopically by microwave modulation of the voltage applied to one of the gates that define the dots [START_REF] Van Der Wiel | Electron transport through double quantum dots[END_REF]. In this way, the delocalized nature of the electronic eigenstates of an artificial hydrogen-like molecule was observed [START_REF] Oosterkamp | Microwave spectroscopy of a quantum-dot molecule[END_REF][START_REF] Petta | Manipulation of a single charge in a double quantum dot[END_REF]. More recently, electrical microwave excitation was used for spectroscopy of single spins [START_REF] Nowack | Coherent control of a single electron spin with electric fields[END_REF][START_REF] Laird | Hyperfine-mediated gate-driven electron spin resonance[END_REF][START_REF] Pioro-Ladriere | Electrically driven singleelectron spin resonance in a slanting Zeeman field[END_REF] and coherent single-spin control [START_REF] Nowack | Coherent control of a single electron spin with electric fields[END_REF][START_REF] Pioro-Ladriere | Electrically driven singleelectron spin resonance in a slanting Zeeman field[END_REF], via electric dipole spin resonance (EDSR).

Here, we perform microwave spectroscopy [START_REF] Oosterkamp | Microwave spectroscopy of a quantum-dot molecule[END_REF][START_REF] Petta | Manipulation of a single charge in a double quantum dot[END_REF][START_REF] Petersson | Microwave-driven transitions in two coupled semicondoctor charge qubits[END_REF] on molecular spin states in an artificial hydrogen molecule formed by a double quantum dot (DD) which contains exactly two electrons. In contrast to all previous PAT experiments, we observe not only the usual spin-conserving tunnel transitions, but also transitions between molecular states with different spin quantum numbers. We discuss several possible mechanisms and conclude from our analysis that these transitions become allowed predominantly through spin-orbit (SO) interaction. The possibility to excite spin-flip tunneling transitions lifts existing restrictions in our thinking about quantum control and Fig. 1a displays a scanning electron micrograph of a sample similar to that used in the experiments. It shows the metal gate pattern that electrostatically defines a DD and a quantum point contact (QPC) within a GaAs/(Al,Ga)As two-dimensional electron gas. An on-chip Co micro-magnet (µmagnet) indicated in blue in Fig. 1a generates an inhomogeneous magnetic field across the DD, which adds to the homogeneous external in-plane magnetic field B (see the appendix A for more sample details), but is not needed for the molecular spin spectroscopy. The sample was mounted in a dilution refrigerator equipped with high-frequency lines. The gate voltages are set so that the DD can be considered as a closed system (the interdot tunneling rates are 10 4 times larger than the dot-to-lead tunneling rates), and the tilt of the DD potential is tuned by the dc-voltages V L and V R , applied to the left and right side gates. Working near the turn-on of the first conductance plateau, the current through the QPC, I QP C , depends upon the local charge configuration and provides a sensitive meter for the absolute number of electrons (n L , n R ) in the left and right dot, respectively [START_REF] Field | Measurements of coulomb blockade with a noninvasive voltage probe[END_REF][START_REF] Johnson | Triplet-singlet spin relaxation via nuclei in a double quantum dot[END_REF].

First, we excite the DD as indicated in the top panel of Fig. 1b, by adding to V R continuous-wave microwave excitation at fixed frequency ν = 11 GHz. When the photon energy of the microwaves matches the energy splitting between the ground state and a state with a different charge configuration, a new steady-state charge configuration results, which is visible as a change in the QPC current, ∆I QP C . The excitation is on-off modulated at 880 Hz and lock-in detection of ∆I QP C reveals the microwave-induced change of the charge configuration (see the appendix A for further experimental details). The lower panel of Fig. 1b shows ∆I QP C as a function of V L and V R near the (1, 1) to (0, 2) boundary of the charge stability diagram. Sharp red (blue) lines indicate microwave-induced tunneling of an electron from the right to the left dot (left to right), labeled as ∆n = +1 (∆n = -1), respectively (see Fig. 1c). Sidebands can result from multi-photon absorption. At the boundaries with the (0, 1) and (1, 2) charge states, no energy quantization is observed, since here electrons tunnel to and from the electron-state continuum of the leads. At first sight, the observations in Fig. 1b thus appear to be well explained by the usual spin-conserving PAT processes.

Surprisingly, the position in gate voltage of the resonant lines exhibits a striking dependence on the in-plane magnetic field, B. This is clearly seen in Figs. 2a and2b, which display the measured PAT spectrum along the DD detuning ε axis (dashed black arrow in Fig. 1b) as a function of B for 20 GHz and 11 GHz excitation, respectively. Since the gate constitutes an open-ended termination of the transmission line, the excitation produces negligible AC magnetic fields at the DD, and is therefore expected to give rise to only electric-dipole allowed spin-conserving transitions, with no B dependence. Furthermore, there is a pronounced asymmetry between the position of the red and blue PAT lines.

In these figures, the detuning axis was calibrated for all magnetic fields by introducing a reference line (see also Fig. 1d, black arrow) that facilitates interpretation of the spectra despite residual orbital effects of the magnetic field. This line was produced by interspersing the microwaves every 5 µs with 200 ns gate voltage pulses along the detuning axis (see top panel in Fig. 1d), leading to singlet-triplet mixing as described in Ref. [START_REF] Petta | Coherent manipulation of coupled electron spins in semiconductor quantum dots[END_REF]. The short gate voltage pulses do not noticeably alter the position of the PAT lines (compare Figs. 1b and1d). The reference peaks visible at around ε = 200 µeV in Figs. 2a and 2b were aligned by shifting all data points at a given B by the same amount in detuning (see the appendix for the full details of this post-processing step).

INTERPRETATION OF THE PHOTON-ASSISTED TUNNELING SPECTRA

The complexity of the PAT spectra shown in Fig. 2a,b can be understood in detail if we allow for non-spin conserving transitions. The two diagrams in Fig. 2c show the energies of all relevant DD-states (four (1, 1)-states and one singlet S(0, 2)-state) as a function of ε for two different (fixed) magnetic fields, i.e. the spectrum of the DD along the two horizontal dotted lines in Fig. 2b [START_REF] Koppens | Control and detection of singlettriplet mixing in a random nuclear field[END_REF]. Note that the only difference between the two diagrams is the splitting between the three triplet T (1, 1)-states.

First we explain the resonances observed along the upper dotted line in Fig. 2b (B = 2.5 T). In the corresponding (upper) diagram in Fig. 2c, we plot the ground state energy for all ε with a thick line. If the microwave excitation is off-resonance with all transitions, the system will be in this ground state. For instance, at ε = 150 µeV, there is no state available 11 GHz above the S(0, 2) ground state (gray arrow) and the system stays in S(0, 2). However, when decreasing the detuning, at some point T + (1, 1) becomes energetically accessible (red arrow) and, since we allow for non-spin-conserving transitions, is populated due to the microwave excitation. For this PAT transition, the spin projection on the quantization axis is changed by ∆m = +1. The resulting change of steady-state charge population (increased population of (1, 1), or ∆I QP C > 0) is detected by the QPC and yields the red peak in Fig. 2b. Decreasing ε further, there are two more resonances detectable: (i) the S(0, 2)-S(1, 1) transition (dotted red arrow, ∆m = 0), although the signal will be weakened due to the fact that S(0, 2) is not unambiguously the ground state anymore. Note that a transition S(0, 2)-T 0 (1, 1) could appear in nearly the same detuning position, as will be discussed below. (ii) the T + (1, 1)-S transition (blue arrow), where S stands for the hybridized S(0, 2) -S(1, 1) singlet, results in a negative (blue, ∆I QP C < 0,∆m = -1) signal from the charge detector since the ground state is now (1, 1) and the excited state is (0, 2). We see that this simple analysis explains both the positions and the signs of the resonances observed in the data.

A similar analysis can be made for other magnetic fields. For instance, for the spectrum plotted in the lower diagram of Fig. 2c we find two resonances with ∆I QP C > 0 (red arrows), and one with ∆I QP C < 0 (blue arrow). Note that the 'blue' transition now connects the ground state to the other branch of the hybridized S compared to the high magnetic field case. Indeed, the singlet anti-crossing is directly probed, resulting in the two blue curved lines observed in the data around ε = 0 (Fig. 2b). The fading out of the blue signal at low fields can be understood from pumping into the metastable state S(1, 1): the microwaves excite the system from T + (1, 1) to S(0, 2), from where it relaxes quickly to S(1, 1). However, relaxation from S(1, 1) back to the ground state is slow due to the small energy difference of this transition and the small phonon density of states at low energies [START_REF] Johnson | Triplet-singlet spin relaxation via nuclei in a double quantum dot[END_REF][START_REF] Meunier | Experimental signature of phononmediated spin relaxation in a two-electron quantum dot[END_REF]. This pumping weakens the detector signal, since S(1, 1) has the same charge configuration as the ground state.

In order to verify this interpretation, we calculate in Figs. 2d,e the position and intensity of the spectral lines at fixed microwave frequency, based on the energy level diagram of Fig. 2c. In the simulations, all single-photon transitions between the ground state and the excited states are allowed by including a matrix element |T ± (1, 1) S(0, 2)| (see appendix A). The input parameters for the calculation of the resonant positions are the interdot tunnel coupling t c , the absolute electron g-factor |g|, a magnetic-field contribution b 0 from the µmagnet parallel to B as well as a magnetic field gradient ∆ B = (∆B ⊥

x , 0, ∆B ) between the dots (in Fig. 3, we show how t c , |g| and b 0 can be extracted from the experimental spectra). The color scale represents the calculated steady-state ∆n that results from microwave excitation, orbital hybridization and phonon absorption and emission at 100 mK. All the PAT transitions visible in the simulation also appear in the experiment, with excellent agreement in both the position and relative intensity of the spectral lines. Especially, the vanishing signal due to spin pumping is also predicted by the simulations which include phonon relaxation.

When we zoom in on the boxed region of Fig. 2b, we see an additional horizontal blue feature at B ≈ 2 T (Fig. 2f) that also appears in the calculated spectra of Fig. 2e. This feature is due to a triplet resonance from T + (1, 1) to T 0 (1, 1) that becomes detectable by relaxation into the meta-stable S(0, 2) state. In the detuning range where this line appears, the S(0, 2) state lies energetically only slightly above the T + (1, 1) state, so relaxation back to the T + (1, 1) ground state is suppressed, again by the small phonon density of states at low energies (see Fig. S2a of the appendix). The triplet resonance is expected to appear at E z = gµ B (B + b 0 ) = hν, where h is Planck's constant and µ B the Bohr magneton. The inset of Fig. 2f shows the magnetic fields corresponding to the center of the measured triplet resonance line for three excitation frequencies (see Figs. S2b,c of the appendix for the spectra), which are in good agreement with the expected positions (black dashed lines in Fig. 2f) based on the values |g| and b 0 determined in the next section from other features of the PAT spectra. Surprisingly, the measured triplet resonance exhibits a finite slope in the B(ε) spectra. A longitudinal magnetic field gradient ∆B gives rise to such a detuning dependence, but the ∆B required in our simulations to reproduce the observed slope is ∆B 80 mT/50 nm, an order of magnitude larger than the gradient we calculate for the µmagnet. The magnitude of the slope remains a puzzle.

EXTRACTING ARTIFICIAL MOLECULE PARAMETERS

We now show how |g|, t c and b 0 , the parameters used for all simulations, can be extracted independently from the experimental spin-flip PAT spectra. For this analysis we only use the relative distance ∆ε between PAT lines at fixed magnetic field, in order to be independent from the calibration of the detuning axis by means of the reference line. Fig. 3a shows ∆ε ± as a function of B using the 20 GHz data. ∆ε ± is defined as the difference in detuning between the red ∆m = ±1 and ∆m = 0 PAT lines. For a fixed ν, both ∆ε ± increase linearly with the Zeeman energy and therefore allow fitting of |g|. (Note that for ∆ε + the linearity is only exact for sufficiently large B, at which the singlet anti-crossing does not affect the T + (1, 1) energy; see the appendix for a detailed discussion). A least-squares fit to the ∆ε + data gives |g| = 0.382 ± 0.004 (Fig. 3a). From the linear behavior of ∆ε + , we also deduce that there is negligible dynamic nuclear polarization in the experiment. Knowing |g| precisely, we make use of the blue anticrossing in Figs. 2b and 2f in order to determine t c (and b 0 ). Fig. 3b shows the difference in detuning ∆ε ′ between the blue ∆m = -1 and the red ∆m = 0 lines in Fig. 2f. Assuming the ∆m = 0 line corresponds to the S(0, 2)-S(1, 1) transition (as shown below), then

∆ε ′ = t 2 c -(hν -gµ B (B + b 0 )) 2 hν -gµ B (B + b 0 ) -(hν) 2 -(2t c ) 2 ,
(1) where the first term is the detuning position of the T + (1, 1)-S transition and the second the one of the S(0, 2)-S(1, 1) transition. The best fits are obtained with t c = 8.7 ± 0.1 µeV and b 0 = 109 ± 16 mT. This value for b 0 matches very well our simulations of the stray field of the µmagnet at the DD location (see appendix A).

An important question left open so far is whether the red ∆m = 0 line involves predominantly transitions from S(0, 2) to S(1, 1) or to T 0 (1, 1). The transition to S(1, 1) does not require a change in the (total) spin and is thus expected to be excited more strongly than that to T 0 (1, 1). However, relaxation from S(1, 1) back to S(0, 2) will be stronger as well, so it is not obvious what steady-state populations will result in either case. Furthermore, given the small energy difference between S(1, 1) and T 0 (1, 1), the two transitions are not resolved in Fig. 2. Fig. 3a helps to answer this question: The observation that ∆ε + > ∆ε -indicates that the ∆m = 0 line originates from the transition to S(1, 1) and not to T 0 (1, 1). For the former we expect ∆ε ± = E z ± J, with J(ε, t c ) =

t 2 c ε + O(t 4
c ) the exchange energy, whereas the latter would result in ∆ε + ∆ε -(in both scenario's, b 0 causes an additional fixed offset in both ∆ε ± , but it does not contribute to their difference). This interpretation is consistent with the increase of ∆ε + with larger interdot tunnel coupling, hence larger J (Fig. 3a inset; note that the slopes are not affected). It is further supported by the data in Fig. S3b.

So far only single-photon processes were considered, but at higher microwave power, also multi-photon lines emerge (Fig. 4a), mostly for the S(0, 2)-S(1, 1) transition (green dashed lines in Fig. 4a). Like the singlephoton S(0, 2)-S(1, 1) line, their position in detuning is B-independent (see appendix for details).

IDENTIFICATION OF THE SPIN-FLIP MECHANISMS

Having shown the power of spin-flip PAT for detailed molecular spin spectroscopy, we now discuss the mechanisms responsible for this process as confirmed by our simulations. As a first possibility, the transitions from S(0, 2) to the triplet (1, 1) states can take place through a virtual process involving S(1, 1): the state S(0, 2) is coupled to S(1, 1) by the interdot tunnel coupling, and an (effective) magnetic field gradient ∆ B = (∆B ⊥ , 0, ∆B ) across the DD couples the spin part of all the (1, 1) states to each other [START_REF] Johnson | Triplet-singlet spin relaxation via nuclei in a double quantum dot[END_REF][START_REF] Taylor | Relaxation, dephasing and quantum control of electron spins in double quantum dots[END_REF]. Here ∆ B has a contribution from the effective nuclear field and from the µmagnet. The transition matrix element from S(0, 2) to

T ± (1, 1) is ∝ t c ∆B ⊥ B , assuming E z ≫ J.
In the following, we use the B-dependence of this process as a fingerprint and focus on the red ∆m = +1 line in Figs. 2a and2b, as we can follow it over the entire magnetic field range. The intensity of this line is constant in B and even if the microwave amplitude E is varied, we observe no B-dependence in the area under this peak (Fig. 4b). Before we conclude that the transition rate is magnetic field independent, we recall that the observed PAT lines reflect the steady-state change in the charge configuration resulting from stimulated photon emission and absorption and spontaneous relaxation. In order to rule out that a field-independent steady state is reached from a field dependence of relaxation and excitation that cancel each other, we verify that the spontaneous relaxation rate is field-independent as well (see appendix). These observations suggest that the coupling mechanism is magnetic field independent and thus virtual processes involving S(1, 1) do not give a strong contribution to the transition rates. More recently, two mechanisms were considered that provide a direct, B-independent matrix element between S(0, 2) and the (1, 1) triplet states: (i) the hyperfine contact Hamiltonian is of the form j δ(r -r j ) I j • S. Thus, nuclear spins, I j , in the barrier regions, where the S(0, 2) spatially overlaps with each of the (1, 1) triplet states, can flip-flop with the electron spin, S, simultaneously with charge tunneling [START_REF] Stopa | Inhomogeneous nuclear spin flips: Feedback mechanism between electronic states in a double quantum dot and the underlying nuclear spin bath[END_REF]. (ii) The SO Hamiltonian is of the form p x,y S x,y and can directly couple states which differ in both orbital and spin [START_REF] Nadj-Perge | Disentangling the effects of spinorbit and hyperfine interactions on spin blockade[END_REF][START_REF] Danon | Pauli spin blockade in the presence of strong spin-orbit coupling[END_REF]. (When the orbital part of the initial and final state are the same, the SO Hamiltonian does not provide a direct matrix element and the transition rate becomes B-dependent [START_REF] Nowack | Coherent control of a single electron spin with electric fields[END_REF][START_REF] Khaetskii | Spin-flip transitions between zeeman sublevels in semiconductor quantum dots[END_REF][START_REF] Fujisawa | Allowed and forbidden transitions in artificial hydrogen and helium atoms[END_REF][START_REF] Golovach | Electric-dipoleinduced spin resonace in quantum dots[END_REF][START_REF] Golovach | Spin relaxation at the singlet-triplet crossing in a quantum dot[END_REF].) The ratio of the SO mediated rate and the hyperfine mediated rate can be estimated as E0 √ N 4A d λSO , which is a few thousand in the experiment (see the appendix). Here E 0 is the single-dot level spacing, N the number of nuclei in contact with one dot, A the hyperfine coupling strength, d the interdot distance and λ SO the spin-orbit length. We therefore believe that SO interaction is the dominant spin-flip mechanism for the observed PAT transitions. The presence of a magnetic field independent matrix element between S(0, 2) and the (1, 1) triplets is confirmed by the observation that the intensity of the reference signal shows no field dependence.

Finally, we extract from Fig. 4a the fitted linewidth as a function of driving power (Fig. 4c). For small E, we find a width 1 GHz, similar to that observed earlier for spin-conserving PAT processes [START_REF] Oosterkamp | Microwave spectroscopy of a quantum-dot molecule[END_REF]. For stronger driving, both the ∆m = 0 and the ∆m = +1 lines are further broadened, up to E ≈ 1.5 mV. If these lines were power broadened, their width would imply transition rates in excess of 1 GHz. However, we do not believe that this is the case, since in measurements with short microwave bursts the populations saturated only on a long (10 µs) timescale (data not shown). Presumably charge or gate voltage noise is responsible for the broadening instead.

We have shown that in our DD system, all (1, 1) spin states have at least weakly allowed electric dipole transitions to S(0, 2). In materials with high SO interaction like InAs, the effect of the non-spin conserving PAT will be even stronger. In materials with weak SO interaction, a strong gradient magnetic field can be used to facilitate spin-flip tunneling transitions. In all cases, this opens the possibility of realizing full quantum control of the (1, 1) spin space via off-resonant (microwave) Raman transitions through the excited S(0, 2) state, which enables a variety of new approaches to manipulating and even defining qubits in DDs. Furthermore, such control enables new measurement techniques that do not rely on Pauli spin blockade [START_REF] Ono | Current rectification by Pauli exclusion in a weakly coupled double quantum dot system[END_REF]. An example is a measurement that distinguishes parallel from anti-parallel spins while acting non-destructively on the S(1, 1) -T 0 (1, 1) subspace, by coupling resonantly the T + (1, 1) and T -(1, 1) states to S(0, 2) followed by charge readout. This constitutes a partial Bell measurement and leads to a new method for producing and purifying entangled spin states [START_REF] Taylor | Solid-state circuit for spin entanglement generation and purification[END_REF].

Appendix A: Methods 1. Sample fabrication 30 nm thick TiAu gates are fabricated on a 90 nm deep (Al 0.3 ,Ga 0.7 )As/GaAs two-dimensional electron gas (2DEG) by means of ebeam lithography. The double dot axis is aligned along the [110] GaAs crystal direction (z-direction), which is parallel to the external magnetic field direction B (Fig. 1a). The 2DEG is Si δdoped (40 nm away from the hetero-interface), exhibits an electron density of 2.05 × 10 11 cm -2 and a mobility of 2.06 × 10 6 cm 2 /Vs at 1 K in the dark. The grounded, 275 nm thick, 2 µm wide and 10 µm long Co µmagnet is evaporated on top of a 80 nm thick dielectric layer, aligned along B (magnetic easy axis) and placed ≈ 400 nm away from the closest dot center. We calculate [START_REF] Goldman | Magnet designs for a crystal-lattice quantum computer[END_REF] that at the double dot position the µmagnet adds b 0 ≈ 110 mT to B z and generates a magnetic field gradient of ∆B ≈ 6 mT/50 nm and a transverse gradient of ∆B ⊥ ≈ -6 mT/50 nm at saturation (B z 2 T).

Measurement

The sample is mounted in an Oxford KelvinOx 300 dilution refrigerator at 30 mK. Left and right side gate voltages, V L and V R , are set by low-pass filtered dc lines and ≈ 60 dB attenuated coaxial lines combined with biastees with a cutoff frequency of 30 Hz. The pre-amplified current through the quantum-point contact is read out by a lock-in amplifier locked to the 880 Hz on-off modulation of the microwaves. The bias across the double dot is set to 0 µV. Voltage pulses to the left and right side gates are generated with a Sony Textronix AWG520. The microwaves are generated with a HP83650A and combined with the pulses to the right side gate. Microwave bursts and detuning pulses are synchronized to ensure that the microwave excitation is switched off during the detuning pulses that generate the reference signal (see Fig. 1d).

Simulation

The Hamiltonian describing the two-spin system near the (1, 1)-(0, 2) transition is taken to be a five-state system, with four (1, 1) spin states and a (0, 2) spin singlet [START_REF] Taylor | Relaxation, dephasing and quantum control of electron spins in double quantum dots[END_REF] in the presence of an external magnetic field B and a magnetic field gradient ∆ B, which includes both the quasi-static nuclear field and the field from the µmagnet. This is given by

H = gµ B P 11 ( S 1 • ( B + ∆ B) + S 2 • ( B - ∆ B
))P 11 -ε|S(0, 2) S(0, 2)|+H t , were P 11 is the projector onto the (1, 1) subspace, ε is the detuning due to the difference in gate potentials from the left and right gates and the tunnel coupling

H t = t c (|S(1, 1) S(0, 2)|) + t SO (|T + (1, 1) S(0, 2)|) + t SO (|T -(1, 1) S(0, 2)|) + h.c.
with t c the spin-conserving tunnel coupling and t SO the spin-orbit coupling set to 5 % of t c .

To find the signal we expect theoretically from the experiment, we add a weak, rapidly oscillating term to the Hamiltonian: ε → ε 0 + Ω cos(νt). We diagonalize H with Ω = 0, then make a rotating frame transformation in which levels are grouped into bands n (defined by a projector P n ) where the states in a band n are much closer in energy than ν, while the energy difference between states in band n and n + 1 are within 2/3rds of hν. Each band rotates at a rate nν, and we can then make a rotating wave approximation, keeping terms due to δ that couple adjacent bands, i.e., our perturbation in the rotating frame and rotating wave approximation is V = Ω n P n |(0, 2)S (0, 2)S|P n+1 + h.c.. Next, we add dissipation and dephasing by including relaxation due to coupling of the electron charge to piezoelectric phonons in a two-orbital (Heitler-London-like) model.

Appendix B: Calibration of the detuning axis

The photon-assisted tunneling (PAT) spectra in Figs. 2a and2b of the main article are measured both with a high energy resolution along the double dot (DD) detuning axis and over a wide external magnetic field range. In the experiment, we observe a monotonous, reproducible drift of the stable charge regions predominantly along the right side gate voltage as we change the magnetic field. Changing the voltages applied to the left and to the right side gate accordingly, we partially compensate for this drift. We then record an 11 GHz PAT spectrum as displayed in Fig. S1a. In order to precisely calibrate the detuning axis for all B, a reference signal is generated together with the PAT spectrum by interspersing the microwaves every 5 µs by a 200 ns detuning pulse with amplitude P ε towards negative detuning (Fig. S1b). 200 ns are found to be sufficient to mix the S(0, 2) state entirely with the T + (1, 1) state at their anti-crossing ε ST + , so that a Pauli spin blocked T + (1, 1) signal is observed at a detuning ε = ε ST + + P ε . The magnitude of P ε is chosen such that the reference signal appears at a detuning position far away from the PAT signal. The pulses do not alter the detuning position of the PAT resonances. In addition to ST + mixing, mixing of the S with T 0 (1, 1) and T -(1, 1) is observed due to the pulsing. The former gives rise to a positive ∆I QP C background on the left of the reference signal in Fig. S1b. The latter generates a weak second reference line that overlaps with the ST + reference line for B → 0 T, but shifts towards negative detuning as B is increased.

In a post-processing step, we separately fit the position of the ST + peaks for all B and shift every row of the spectrum, such that the peaks are vertically aligned at ε * = P ε as shown in Fig. S1c. Thus, ε * = 0 is at the ST + anti-crossing, by definition. All data points at a given B are shifted by the same amount in detuning. This ε * -detuning axis is well-defined but 'moves' with respect to the ε-detuning axis as a function of B, since ε * = ε -ε ST + (B). The lever arm for the voltage to energy conversion is read from the voltage distance of the second and third ∆m = 0 PAT line at ν = 11 GHz (see Fig. 4a), which equals the photon energy hν [START_REF] Petta | Manipulation of a single charge in a double quantum dot[END_REF]. For the analysis of all spectra we used the same lever arm.

For better readability of the PAT spectra, we finally 1d). A red reference line appears at positive DD detuning, when the detuning pulse reaches exactly the ST + anti-crossing. Also a weak line due to mixing at the ST - anti-crossing is observed. Both lines are marked by arrows. The more red background is due to ST 0 mixing. c, PAT spectrum with calibrated detuning axis. The horizontal lines of the spectrum are shifted so that the reference pulse appears at the detuning position, which equals the pulse amplitude Pε converted into energy (see text) along the detuning axis. Here ε * = 0 is equal to the ST + anti-crossing for every magnetic field. d, To convert the ε * scale of the detuning axis into the ε scale, where ε = 0 equals the S(1, 1)-S(0, 2) anti-crossing, the spectrum is sheared by the Zeeman-energy using the electron g-factor that is independently determined as explained in the main article. At low magnetic fields this transformation of scales is wrong and gives rise to a curvature of the spectrum, which is a function of the tunnel coupling.

convert the ε * -scale to the ε-scale found in literature, for which ε = 0 is defined by the S(1, 1) to S(0, 2) anticrossing. To do so, we additionally shift all data points at a given B by |g|µ B |B| towards positive detuning (Fig. S1d). However, since ε ST + = |gµ B B| holds true only for |gµ B B| ≫ t c , where t c is the interdot tunnel coupling, the detuning axis conversion fails for low magnetic fields. As a result the PAT resonance-lines bend towards positive ε for B → 0 T in Fig. S1d and in the spectra shown in the main article (Fig. 2a,b). ∆ε + 0 = ∆ε + (B → 0) by a linear fit at a sufficiently high magnetic field range. This procedure turned out to be impractical with ∆ε - 0 (B), because the ∆m = -1 line fades out at high magnetic fields. Note that the linear extrapolation of the S(0, 2)/T + (1, 1) transition to zero field (red arrow) exhibits a different detuning position than the S(0, 2)/T 0 (1, 1) transition (violet arrow), because the linear extrapolation follows the dashed blue line in the inset of Fig. S3b,i.e. the linear extrapolated detuning position is not affected by the hybridization of the singlets. The detuning position of the S(0, 2)/T 0 (1, 1) PAT transition, however, is affected by the singlet hybridization, since it lowers the energy of the initial state S(0, 2). In summary, the value of ∆ε + 0 is always larger then zero, but also depends upon the nature of the ∆m = 0 PAT resonance.

In the inset of Fig. S3b, ∆ε + 0 for the S(0, 2)/S(1, 1) transition is drawn. Obviously, ∆ε + 0 is considerably smaller, if the S(0, 2)/T 0 (1, 1) PAT resonance dominates over the S(0, 2)/S(1, 1) resonance. The analysis of ∆ε + 0 is complicated by the remanence of the µmagnet, which is not exactly known, but should be smaller than the fully magnetized field of b 0 = 109 mT. Due to the remanence, the linear extrapolation of the S(0, 2)/T + (1, 1) transition towards zero external magnetic field, leaves an additional offset on ∆ε + 0 . This offset, however, is independent from ν and t c . In Fig. S3b, the extrapolated ∆ε + 0 values are plotted as a function of the microwave frequency ν for two tunnel couplings (filled circles and open squares). Fitting the filled circles with the well-known t c = 8.7 µeV, we determine a reasonable fit by assuming the ∆m = 0 PAT resonance to be purely singlet (green line). The fit with a potential S(0, 2)/T 0 (1, 1) transition (violet line) fails at ν = 5 GHz. The only fit parameter used here is the remanence of the µmagnet, which was found to be ≈ 70 mT for the fit function assuming a S(0, 2)/S(1, 1) transition, and ≈ 140 mT assuming a S(0, 2)/T 0 (1, 1) transition. The latter is very unlikely, since only a maximum magnetization of b 0 = 109 mT was found at an external magnetic field of 2 T.

As a final check, we take ∆ε + 0 values into account, which were determined when the DD was tuned to a smaller t c = 4 µeV (open squares). These data points cannot be fitted by the fit function that assumes a S(0, 2)/T 0 (1, 1) transition at all, since the ∆ε + 0 /gµ B values observed are already smaller than the remanence of ≈ 140 mT, which would stay valid for the altered tunnel coupling. Obviously, this leads to a contradiction, since all ∆ε + 0 would become negative after subtracting the remanence. Only the assumption of a purely singlet ∆m = 0 PAT transition in combination with the smaller remanence of ≈ 70 mT, as fitted above, allows reasonable fitting. Our conclusion from the main article is therefore further supported.

Appendix E: Spin flip-tunneling mechanism

As noticed in the main text, there is a direct matrix element between the S(0, 2) and the (1, 1) triplet states. This can occur due to the nuclear spins in the barrier between the dots and due to spin-orbit (SO) interaction. We look at a toy model to examine the relative importance of these two processes. Specifically, we consider the hopping matrix element for a single electron with spin S between two wavefunctions associated with an electron on the left (|L ) and on the right (|R ) via the perturbation:

V = m * λ SO [-α(S z p ỹ -S ỹ p z ) + β(p ỹ S ỹ + p z S z )] + +Av 0 j δ(r -r j ) I j • S (E1)
where we have absorbed the Rashba (α) and Dresselhaus (β) terms into a single spin-orbit interaction with a characteristic spin-orbit length λ SO ∼ 10 µm. We recall that m * is the effective electron mass, A ≈ 100µeV, v 0 is the unit cell volume and I j is the nuclear spin at r j . We now wish to estimate the spin-flip tunneling for the single electron case, given by averaging over the orbital dipole:

R| V |L (E2)
This can be evaluated explicitly for |L(R) = exp(-(z ± a/2) 2 /4σ 2 )/(2πσ 2 ) 1/4 φ(x, y) where z is the inter-dot axis (at an angle θ with the axis z from the spin-orbit interaction in Eq. E1) and φ(x, y) is the transverse-longitudinal wavefunction. As tunneling occurs only along the z-axis, matrix elements with p y are zero. We find two tunneling matrix elements: We remark that the rms value for B nuc,f is given by

t SO =
gµ B | B nuc,f | 2 = A v 2 0 j |ψ L | 2 |ψ R | 2 I(I + 1) ≈ e -a 2 /8σ 2 A v 2 0 j |ψ L | 4 (E7)
That is, it is the rms value for a single dot, A/ √ N , multiplied by exp(-a 2 /8σ 2 ). Also, the size of the single-particle wavefunction, σ, is related to the orbital energy scale of a single dot by ∆ ≈ 2 m * σ 2 . Thus, the relative strength of the two tunneling terms (including spin flip) is

|t SO | |t nuc | = ∆ A/ √ N 3| n × B ext | 2|B ext | a 4λ SO (E8)
where N is the number of spins in a single quantum dot. We explore briefly how this ratio varies with dot size σ and spacing a. Specifically, ∆ √ N ∝ σ -1 , so a larger dot reduces the strength of spin-orbit tunneling compared to hyperfine-assisted tunneling. On the other hand, increasing the distance a increases the relative strength of spinorbit tunneling to hyperfine-assisted tunneling. Setting in A = 100 µeV, N = 4 × 10 6 , a = 75 nm, ∆ = 1000 µeV and λ SO = 10 µm, we calculate the ratio of the matrix elements |tSO| |tnuc| to be ≈ 60. We remark that in external field parallel to n prevents any SO spin-charge flips. All spin charge flips occur only via t nuc .

In the bases (T -(1, 1), | ↓↑ , | ↑↓ , T + (1, 1), S -(0, 2)) the two electron Hamiltonian that ignores the small t nuc term has the form where, e.g., the Larmor precession frequency of an electron spin in the left dot is B z -∆B .

        B z ∆B ⊥ x -i∆B ⊥ y 2 - ∆B ⊥ x -i∆B ⊥ y 2 0 -t SO,y / √ 2 ∆B ⊥ x +i∆B ⊥ y 2 -∆B 0 - ∆B ⊥ x -i∆B ⊥ y 2 (-it

1 FIG. 1 :

 11 FIG. 1: Photon-assisted tunneling in a 2-electron double quantum dot. a, Scanning-electron micrograph top view of the double dot gate structure with Co micromagnet (blue). The voltages applied to the left VL and right VR side gates (red) control the detuning ε of the double dot potential. The double dot charge state is read out by means of the current IQP C running through a nearby quantum point contact (white arrow). b, Charge stability diagram around the 2electron regime at B = 1.5 T. (nL, nR) indicate the absolute numbers of electrons in the left and right dot, respectively. During measurements, 11 GHz microwaves with 880 Hz onoff modulation are applied to the right side gate. The top panel displays schematically one cycle of the ac signal applied to the right side gate. Along the detuning axis (dashed arrow) PAT-lines are observed. c, In the conventional picture of PAT, the first sidebands seen in b should appear when the detuning of the (0, 2) and (1, 1) states matches the photon energy and interdot tunneling is induced. Further sidebands are then interpreted as multi-photon transitions. µF is the chemical potential of the left and right electron reservoir. d, Same as in b, but the microwaves are interrupted every 5 µs by a 200 ns, Pε = 2 mV detuning pulse applied to the left and right side gates (see the schematic in the top panel). The pulses generate a reference line (black arrow) due to mixing at the ST + anti-crossing.

FIG. 2 :

 2 FIG. 2: Photon-assisted-tunneling spectra and simulations. a,b, Microwave induced change of the QPC current ∆IQP C as a function of the double dot detuning ε and the external magnetic field B for 20 GHz (a) and 11 GHz (b) frequency, respectively. singlet-triplet mixing due to 2 mV detuning pulses generates a reference signal that is used to calibrate the detuning axis (see lower panel in Fig. 2f). c, Eigenenergies vs. double dot detuning ε of the 2-electron spin states in the (1, 1) and (0, 2) charge regime for two external magnetic fields B = 2.5 T (upper panel) and B = 1.5 T (lower panel), respectively. S(0, 2), S(1, 1) and T (1, 1) character of the eigenstates is indicated by blue, green and red color, respectively. The molecular spin ground state is indicated by thick lines. The vertical arrows indicate PAT transitions for a constant microwave frequency involving spin flips. The transition indicated by a dashed arrow is suppressed, because the initial state lies above the ground state. The red circle in the lower panel indicates the detuning position of the reference signal, that is generated by a detuning pulse with amplitude Pε to the ST + anti-crossing. d,e, Simulated PAT spectra for 20 GHz (a) and 11 GHz (b) frequency, respectively. The color indicates the change of the population of the steady-state charge state ∆n as would be observed in an on-off lock-in detection. A finite temperature of 100 mK and spontaneous relaxation via the phonon bath are taken into account. f, IQP C scanned with higher resolution in the anti-crossing region (black rectangle in Fig. 2b) at 11 GHz excitation. The green dashed lines indicate the expected detuning positions of the PAT transitions. The horizontal black dashed line indicates the magnetic field, at which the electron spin resonance condition is fulfilled Ez = gµB(B + b0) = hν. The graph is concatenated from two scans that overlap at 1.9 T. The inset displays three magnetic fields (blue dots), at which the center of the horizontal blue triplet resonance line is observed, as a function of the microwave frequency ν. The dashed black line gives the expected position of the triplet resonance.

FIG. 3 :

 3 FIG.3: Analysis of the photon-assisted-tunneling spectra. a, The detuning difference ∆ε between the spin conserving line ∆m = 0 and the ∆m = ±1 line is plotted as a function of the external magnetic field B at 20 GHz (see Fig2a), in order to fit the absolute effective electron g-factor g from the slopes of the linear fits (solid lines). inset, ∆ε for the blue ∆m = +1 line at 11 GHz for different tunnel couplings. The offset of the curves increases with increasing tc (red to black circles) while g remains constant. b, ∆ε between the ∆m = 0 line and the ∆m = -1 (1, 1) to (0, 2) transition from Fig.2c. The fit of the anti-crossing (red line) allows for a precise determination of tc.

FIG. 4 :

 4 FIG. 4: Power-dependence of the photon-assisted tunneling spectra and spontaneous relaxation. a ∆IQP C as a function of the double dot detuning ε and the microwave amplitude E for 11 GHz and B = 1.0 T (middle panel) and B = 2.5 T (lower panel). The excitation scheme from Fig. 1d is employed with reference pulse amplitude Pε = 1.5 mV. The green (red) dashed lines mark the multi-photon ∆m = 0 (∆m = +1) PAT transitions. The reference signal stemming from the pulse is marked by orange arrows. The voltage amplitude E is measured at the end of the coaxial lines at room temperature. The uppermost panel displays a linecut measured at 1 T. The red line is a least-squares fit by a sum of four lorentzian peaks to the data. b The fitted Lorentzian peak area of the ∆m = +1 line from a is plotted as a function of microwave amplitude E and magnetic field B. c The fitted Lorentzian linewidth at half maximum (FWHM) of the ∆m = +1 and the lines ∆m = 0 from a is plotted as a function of microwave amplitude E for B = 1 T. The error bars in b,c and are determined from the Lorentzian least-squares fit (see the inset of Fig. 4a).

Fig. S 1 :

 1 Fig. S 1: Calibration of the detuning axis for the PAT spectra. a, PAT spectrum as measured without pulses (see inset of Fig. 1b) with a microwave frequency of 11 GHz. The voltage applied to the right side gate VR and left side gate VL is swept simultaneously to follow the detuning axis indicated in Fig 1b. b, same PAT spectrum, but with 200 ns pulses that intersperse the microwave bursts every 5 µs (see top panel of Fig.1d). A red reference line appears at positive DD detuning, when the detuning pulse reaches exactly the ST + anti-crossing. Also a weak line due to mixing at the ST - anti-crossing is observed. Both lines are marked by arrows. The more red background is due to ST 0 mixing. c, PAT spectrum with calibrated detuning axis. The horizontal lines of the spectrum are shifted so that the reference pulse appears at the detuning position, which equals the pulse amplitude Pε converted into energy (see text) along the detuning axis. Here ε * = 0 is equal to the ST + anti-crossing for every magnetic field. d, To convert the ε * scale of the detuning axis into the ε scale, where ε = 0 equals the S(1, 1)-S(0, 2) anti-crossing, the spectrum is sheared by the Zeeman-energy using the electron g-factor that is independently determined as explained in the main article. At low magnetic fields this transformation of scales is wrong and gives rise to a curvature of the spectrum, which is a function of the tunnel coupling.

Fig. S 2 :

 2 Fig. S 2: Triplet resonance in the PAT spectra. a, Energy eigenstates of the two-electron spin states plotted with the same color code as in Fig. 2c of the main article. Microwaves are resonant to the T + (1, 1)/T 0 (1, 1) transition. Spontaneous relaxation from T 0 (1, 1) to the metastable S(0, 2) enables the detection via ∆IQP C . b, The PAT spectrum as measured with a microwave frequency of 7.5 GHz exhibits a nearly horizontal blue feature as visible in Fig. 2f. Also at this microwave frequency the line starts from the magnetic field, at which we expect the electron spin resonance condition to be fulfilled (black dashed line) hν = gµ(B + b0). c, PAT spectrum recorded at 13 GHz with corresponding line indicating electron spin resonance.

Fig. S 3 :

 3 Fig. S 3: Analysis of the ∆m = 0 PAT resonance. a, ∆ε + and ∆ε -calculated for the parameters |g| = 0.382, tc = 8.7 µeV, b0 = 109 mT, ν = 20 GHz and two possible scenarios: The ∆m = 0 line is resonant between the S(0, 2) ground state S(1, 1) (red and orange line) and T 0 (1, 1) (green and blue line). Zoom-in for low-magnetic fields in the inset.b, The detuning offset ∆ε + 0 between the transition ∆m = +1 for B → 0 mT (red arrow in the inset) and the ∆m = 0 transitions from S(0, 2) to S(1, 1) (green arrow) and to T 0 (1, 1) (violet arrow), respectively, is plotted as a function of the microwave frequency ν. The error bars are determined from the linear extrapolation B → 0 mT. Shown are offsets measured at tc = 8.7 µeV (filled circles) and at tc ≈ 4 µeV (open rectangles). The green and violet lines are least-squares fits to the measured offsets assuming a pure S(0, 2) to S(1, 1) and to T 0 (1, 1) transition, respectively.

2 m * σ 2 a

 22 4λ SO e -a 2 /8σ 2 n • S (E3)t nuc = gµ B B nuc,f • S (E4) n = -cos(θ)[(α -β) cos(θ) + (α + β) sin(θ)]ẑsin(θ)[(β -α) sin(θ) + (α + β) cos(θ)]ŷ (E5) gµ B B nuc,f = Av 0 j |ψ L (r j )||ψ R (r j )| I j (E6)

Fig. S 5 :

 5 Fig. S 5: Power-dependence of the PAT spectra for different magnetic fields. a-e, Microwave induced change of the QPC current as a function of the double dot detuning ε and the microwave amplitude E at the sample for 11 GHz.From a to e, the external magnetic field is increased by 500 mT. The detuning axis is calibrated so that ε * = 0 mV is equal to the detuning position at the S(1, 1)-T + (1, 1) anti-crossing (see Fig.1cfor the calibration method). The pulse amplitude of the reference line is Pε = 1.5 mV.

  SO,z -t c )/ √ 2

				
				
	--t SO,y / ∆B ⊥ x +i∆B ⊥ y 2 0 √ 2 (it SO,z -t c )/ 0 -∆B ⊥ x +i∆B ⊥ y 2 √ 2 (it SO,z + t c )/ ∆B ∆B ⊥ y 2 x +i∆B ⊥ √ 2 -t SO,y / ∆B ⊥ x -i∆B ⊥ y -B z 2 √ 2	-t SO,y -ε (-it SO,z + t c )/	√ 2	     	(E9)

Acknowledgments

We gratefully acknowledge discussions with S. M. Frolov, M. Laforest, D. Loss, Yu. V. Nazarov, K. C. Nowack, M. Shafiei and thank H. Keijzers for help with the sample fabrication and R. Schouten, A. van der Enden and R. G. Roeleveld for technical support. This work is supported by the 'Stichting voor Fundamenteel Onderzoek der Materie (FOM)' and a Starting Investigator grant of the 'European Research Council (ERC)'.

L.R.S, F.R.B., V.C. and T.M. performed the experiment, W.W. grew the heterostructure, T.M. fabricated the sample, L.R.S., J.D., J.M.T and L.M.K.V. developed the theory, J.M.T. did the simulations, all authors contributed to the interpretation of the data and commented on the manuscript, and L.R.S., J.D., J.M.T. and L.M.K.V. wrote the manuscript.

Appendix C: Triplet spin resonance

In Fig. 2f of the main article, a PAT feature is observed that is due to a transition from the T + (1, 1) ground state to the T 0 (1, 1) excited state. The T 0 (1, 1) state can relax via spontaneous phonon emission to the singlet bonding state, a superposition of S(1, 1) and S(0, 2). This state is metastable, since the spontaneous phonon relaxation is suppressed by the small energy difference to the T + (1, 1) ground state (see Fig. S2a), which makes the transition detectable by ∆I QP C . A peculiarity of the T + (1, 1) to T 0 (1, 1) resonance is its slope in the PAT spectrum, which might be a result from a gradient magnetic field along the magnetic field direction as discussed in the main article. Note that in the same detuning range, we observe also direct PAT transitions from the T + (1, 1) ground state to the singlet anti-bonding state and singlet bonding state at lower and higher magnetic fields, respectively.

Here we investigate the position of the T + (1, 1) to T 0 (1, 1) resonance, as a function of the microwave frequency ν. The Figs. S2b and In the main article, we discuss whether the red (∆m = 0, ∆n = 1) PAT resonance is dominantly due to a transition from the S(0, 2) ground state to the S(1, 1) state or to the T 0 (1, 1) state. It is difficult to spectroscopically resolve these transitions, since they differ only by the exchange energy J(ε). In Fig 3a In both scenarios, the ∆ε -(B) increases non-linearly towards high B, since the resonance ∆m = -1 becomes sensitive to the curved singlet anti-crossing at E z ≈ hν-t c , where E z is the Zeeman energy. This curvature is not observed in the experiment as the ∆m = -1 transition fades out at B ≈ 1.5 T. In both scenarios, the ∆ε + (B) functions are linear, which holds true only if E z ≪ t c -hν. The choice of a high ν and an appropriate B-range allows to extract |g| from ∆ε + (B) by a simple linear least-squares fit as demonstrated in the main article.

As a final step, we analyze the splitting of the ∆ε + (B) and ∆ε -(B) function quantitatively. Their offsets depend upon the exchange energy J(ε, t c ), which we can experimentally vary by t c or indirectly by ν. For hν → 2t c , the detuning position of the ∆m = 0 resonance becomes strongly affected by J. As shown in the inset of Fig. S3b, the S(0, 2)/S(1, 1) transition (green arrow) shifts more towards negative detuning ε than the potential S(0, 2)/T 0 (1, 1) transition (violet arrow). We measure ∆ε + (B) for various ν and determine the Appendix F: Simulations of the PAT spectrarelaxation

The simulated spectra in Figs. 2c,d of the main article include the effect of the phonon-mediated relaxation. In addition to the explanations of the simulations in appendix A, we continue here on the coupling of the electron spin to the phonon bath. We thereby neglect deformation phonons as the energy scales examined in the experiment [START_REF] Petta | Manipulation of a single charge in a double quantum dot[END_REF][START_REF] Nowack | Coherent control of a single electron spin with electric fields[END_REF][START_REF] Laird | Hyperfine-mediated gate-driven electron spin resonance[END_REF][START_REF] Pioro-Ladriere | Electrically driven singleelectron spin resonance in a slanting Zeeman field[END_REF][START_REF] Petersson | Microwave-driven transitions in two coupled semicondoctor charge qubits[END_REF][START_REF] Field | Measurements of coulomb blockade with a noninvasive voltage probe[END_REF][START_REF] Johnson | Triplet-singlet spin relaxation via nuclei in a double quantum dot[END_REF][START_REF] Koppens | Control and detection of singlettriplet mixing in a random nuclear field[END_REF][START_REF] Meunier | Experimental signature of phononmediated spin relaxation in a two-electron quantum dot[END_REF][START_REF] Taylor | Relaxation, dephasing and quantum control of electron spins in double quantum dots[END_REF][START_REF] Stopa | Inhomogeneous nuclear spin flips: Feedback mechanism between electronic states in a double quantum dot and the underlying nuclear spin bath[END_REF][START_REF] Nadj-Perge | Disentangling the effects of spinorbit and hyperfine interactions on spin blockade[END_REF][START_REF] Danon | Pauli spin blockade in the presence of strong spin-orbit coupling[END_REF][START_REF] Khaetskii | Spin-flip transitions between zeeman sublevels in semiconductor quantum dots[END_REF][START_REF] Fujisawa | Allowed and forbidden transitions in artificial hydrogen and helium atoms[END_REF][START_REF] Golovach | Electric-dipoleinduced spin resonace in quantum dots[END_REF] are much smaller than the characteristic frequency scale of a phonon on the length scale of the dot c ph /l dot ∼ 60 -120 GHz. To determine the coupling, we take as an ansatz for the electronic wavefunctions the Fock-Darwin states, given by Gaussians, and calculate the coupling after orthogonalizing the states with the perturbation V ph = e,k f (k z ) ρω e,k e i k• r β e (a e,k -a † e,-k ) [START_REF] Kittel | Quantum Theory of Solids[END_REF], where f (k z ) ≈ 1 for the energy scales we are working with. We then use Fermi's golden rule to calculate excitation and relaxation from thermal and spontaneous emission of phonons. Finally, we numerically solve the superoperator for the steady state and compare the expectation value of |S(0, 2) S(0, 2)| with and without the excitation Ω, mimicking the effect of the lock-in detection.

Appendix G: Measurement of spontaneous relaxation

In the main article, we state that the spontaneous relaxation rate from T + (1, 1) to S(0, 2) is found to be magnetic field independent. Relevant for the PAT process is the spontaneous relaxation at a fixed energy difference between the T + (1, 1) and S(0, 2) state, which is set by the photon energy. Thus, when changing the magnetic field, the detuning ε has to be changed accordingly. The measurement of the spontaneous relaxation Normalized QPC current averaged over τ = 5µs immediately after full mixing at the ST + anticrossing for various magnetic fields at zero microwave power. The spontaneous spin relaxation after mixing is measured at a distance Pε to the ST + mixing point (see scheme in Fig. 2c). The dependence on the averaged time interval τ is displayed in the inset for B = 1.5 T and Pε = 2 mV together with a least-squares fit (red solid line).

rate is done as follows: Starting from S(0, 2), we populate the T + (1, 1) by 50 % via a 200 ns detuning pulse [START_REF] Barthel | Rapid single-shot measurement of a singlet-triplet qubit[END_REF] with amplitude P ε in the absence of microwaves, and monitor the decay back to S(0, 2). The relaxation rate Γ s can be extracted from the time averaged lockin signal ∆I QP C (τ ) = ∆I0 τ τ 0 exp (-tΓ s ) dt, where τ is the time spent in Pauli blockade between the pulses. ∆I 0 = ∆I QP C (0), which is independent from B, is extracted from the fit in the inset of Fig. S4. In order to cover various values of the detuning and the magnetic field, we next fix τ = 5 µs and record ∆I QP C as a function of P ε for three different magnetic fields. We observe that ∆I QP C (5 µs) and thus Γ s are essentially independent of B (Fig. S4). This holds true for all P ε and hence for all T + (1, 1)-S(0, 2) energy splittings. Note that regardless of B, this energy splitting is given by P ε alone. This reflects exactly the situation in the PAT experiment, for which the microwave frequency alone sets the energy splitting and thus also the required phonon energy for the spontaneous relaxation process.

Appendix H: Power dependence

In Fig. 4a of the main article, the power dependence of the PAT sidebands is shown for two magnetic fields. In Fig. S5a-e, ν = 11 GHz-spectra measured with a series of magnetic field values are plotted, to ease keeping track of the resonances as they shift in detuning with the external magnetic field. Here, the spectra are plotted such that ε * = 0 is the ST + anti-crossing for all B (compare Fig. S1c). The reference peaks due to pulsing to the ST + anti-crossing are all aligned at ε * = P ε = 1.5 mV, which is the pulse amplitude. The amplitude of the microwaves E is estimated from the attenuation of the high-frequency circuit at room temperature.

PAT sidebands at larger detuning appear when E is increased as expected for PAT. Oscillation of the PAT amplitude as a function of E is hardly visible, because we cannot reach sufficiently high E and because the PAT lines become broadened as a function of power. One sideband emerges at high ε * ≈ 0.5 mV already at low E. This resonance stays at a constant ε * for all B and is therefore a transition from S(0, 2) to T + (1, 1) (∆m = 1). As B is increased, the other transitions move towards lower ε * while keeping the distance along ε * . They are due to PAT transition with ∆m = 0. At B ≈ 2 T, the ∆m = 1 PAT transition overlaps with the 2-photon ∆m = 0 PAT transition. This is expected, since ν = 11 GHz equals Zeeman energy at this magnetic field in our double dot.


Mark A Hughes
email: m.a.hughes@salford.ac.uk

Dr Naitik A Panjwani

Dr Matias Imanol Urdampilleta

Nafsika Theodoropoulou

Dr Ilana Wisby

Kevin P Homewood

Ben Murdin

Dr Tobias Lindström

Dr J David Carey

J David

Carey Dr

Prof P Kevin

Prof B Homewood

Dr J David Murdin

Carey

Coupling of Erbium-Implanted Silicon to a Superconducting Resonator

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Introduction

The optical fibre telecommunications network makes telecoms wavelength photons at 1.5 μm by far the best candidate for transferring quantum information over distance. Er 3+ intra 4fshell transitions can be optically addressed at telecoms wavelengths which would allow transfer of quantum information over distance. The use of rare earth (RE) ions is well suited to overcome a paradox of quantum technology (QT) platform requirements: sufficient decoupling from the environment to avoid decoherence, but a strong enough interaction with the environment to allow addressing, readout and gating. The advantage of RE ions arises as they possess a partially filled 4f shell which is shielded from the environment by the outer 5s and 5p shells leading to extraordinary coherence times of 6 hours [1] and 4.4 ms [2] for optically detected nuclear spin and electron dipole transitions, respectively; however, even with their atomic scale shielding, long lived entanglement between RE dopants in a solid matrix has been observed, [START_REF] Bartolomé | Molecular Magnets: Physics and Applications[END_REF] and entanglement between internal degrees of freedom of single RE ions can still exist up to thousands of K, making this one of the most stable known entanglements. [4] A practical quantum computing architecture developed in Si will move from a one-off device to production far quicker than for any other candidate, and features can be patterned in Si on the scale required for many quantum device architectures. Ion implantation of Si is a well understood technology in IC fabrication, and commercial adoption of new technologies tend to favour those based on established fabrication platforms and techniques. Recently, increases in coherence times by several orders of magnitude have been demonstrated in donor impurities in silicon by using isotopically pure 28 Si. [5] However, these donor impurities do not interact with light at telecommunications wavelengths, which is critical for many quantum communication schemes. Given expected improvements in T2 by using 28 Si, optimising processing for the appropriate Er-related centre [6] and reducing Er concentration, Er implanted Si is potentially the only know QT platform with telecoms addressability, long T2 and IC tooling compatibility. The spin state of a single Er ion implanted into a silicon single electron transistor has been optically addressed and electrically readout, [7] whereas, the spin state of a single Er ion in Y2SiO5, coupled to a silicon nanophotonic cavity, can be readout optically with a single shot. [8] This demonstrates that Er implantation is compatible with a potential quantum computing architecture and that Er could be integrated in to quantum communication and information processing schemes.

Arguably the most sophisticated quantum computers to-date use superconducting (SC) circuits. Using Er spin ensembles can also provide a telecommunications interface for SC resonator qubits, [9] and a single ensemble could be used to store many qubits by using holographic encoding. [10] The NV centre in diamond is perhaps unique among solid state defects in that it can be optically spin polarized to give an effective spin temperature of ~ 0 K and coherence times (T2) of a few ms, while the surrounding diamond lattice is at room temperature. [11] All the other viable solid-state quantum technology platforms must be cooled to ~ 0 K to have adequate coherence times. However, the lack of IC processing capability in diamond makes it far less viable for large scale quantum computers, and its visible wavelength operation makes it less viable for quantum communication schemes.

Here we report the first coherence measurements of Er implanted Si, or, in fact, any implanted Er, in the form of spin echo measurements, we demonstrate the first of coupling between Er implanted in Si and a superconducting resonator. We also report a previously unreported Er related defect state which leads to an energy level scheme analogous to that of the diamond NV centre, and gives a mechanism for optical spin polarisation of the Zeeman ground state which could lower the effective spin temperature and increase the working temperature at which reasonably long coherence times can be obtained.

Experimental

Sample preparation

Sample 1, with 10 19 cm -3 Er and 10 20 cm -3 O, was prepared by implanting Er with a total areal dose of 2.6×10 15 cm -2 and O with a total areal dose of 1.3×10 16 cm -2 at 77 K into one face of a <100> oriented 8000 ± 500 Ωcm Si wafer of 500 µm thickness supplied by Topsil. Sample 1 was annealed with recipe a, which consisted of a 450°C for 30 min anneal to smooth the crystalline-amorphous interface, a 620°C for 180 min anneal to recrystallize the amorphized region and a 850°C for 30 s anneal to activate the Er. It was previously found that annealing at 850°C significantly increased the ESR signal strength. [12] Sample 2, with 3×10 17 cm -3 Er and 10 20 cm -3 O, and sample 3, with 10 17 cm -3 Er and 10 20 cm -3 O, were prepared by implanting at room temperature into both faces of a <100> oriented 5000-9999 Ωcm Si wafer of 50 µm thickness supplied by Si-Mat. Samples 2 and 3 were annealed with recipe b (750°C for 2 min) and recipe c (600°C for 167 min, then 850°C for 30 s), respectively. For all samples, O and Er ions were implanted at a range of energies to give a flat concentration profile down to a depth of around 1.5 µm, see supplementary Figure S1. Isotope specific implantation was used so that only the zero nuclear spin 166 Er was implanted.

ESR and spin echo measurements

CW and pulsed ESR measurements were performed in a Bruker E580 ESR spectrometer. All ESR measurements were recorded with the magnetic field, B0, parallel to the [001] direction of the wafer with an uncertainty of ±5°. When using Er concentrations ≤ 3×10 17 cm -3 and recipe a, no spin echo signal could be observed. We switched to 50 µm thick wafers in a effort to maximise the ESR resonator's Q factor. We also tried two different recrystallization strategies: shorter time, higher temperature and longer time, lower temperature. These were 750°C for 2 min (recipe b) and 550°C for 335 min (recipe d), with no activation anneal; recipe d resulted in no measurable ESR resonances. However, recipe b resulted in strong, narrow ESR resonance lines, and a measurable echo signal. With samples 1 and 2, the Q factor of the ESR resonator was ~600 and ~9000, respectively, due to metallic doping at the high Er concentrations in sample 1 and from higher sample volume. For pulsed measurements the Q factor was detuned to ~100 for both samples.

Optically Modulated Magnetic Resonance (OMMR)

For OMMR measurements we used a Bruker EMX ESR spectrometer, incorporating a super high-Q resonator with optical access. A 20 mW external cavity laser tuneable from ~1490 to 1620 nm, was fibre coupled to a C-band erbium doped fibre amplifier (EDFA) with a gain bandwidth of ~1510-1600 nm and an output of up to ~150 mW. The output of the EDFA was sent through a linear polarizer, parallel to the [11 ̅ 0] direction of the Er implanted Si wafer, then modulated at ~30 Hz with a mechanical chopper. The output signal from the ESR spectrometer's on-board lock-in amplifier was used as the input to an external lock-in amplifier (SRS830) and referenced to the mechanical chopper. OMMR spectra were produced by sweeping the external cavity laser wavelength and reading the SRS830 lock-in signal to give a spectrum of the ESR signal that has been modulated by the laser. A microwave power of 2.1 mW, a field modulation of 100 kHz, a time constant of 5 ms and a microwave frequency of 9.37 GHz was used for all OMMR measurements.

PL measurements

PL spectra were obtained by cooling the sample in a cold finger LH2 cryostat at 15 K. Excitation was by a 462 nm 50 mW laser diode and the generated fluorescence was dispersed in a Bentham TMc300 monochromator and detected with an IR PMT coupled with standard phase sensitive detection. All spectra were corrected for the system response.

Superconducting resonator coupling

Superconducting resonator coupling measurements were performed in a dilution refrigerator, fitted with a vector magnet, at 20 mK. A superconducting lumped element micro-resonator was fabricated by sputtering 200 nm of NbN, patterned by standard e-beam lithography, onto an R-cut sapphire substrate. Sample 3 was placed face down on the microresonator and the microresonator was placed in a magnetic field that was stepped from zero to 93 mT. The power in the resonator was ~3 pW. At each magnetic field the microwave transmission coefficient, S21, was measured using a vector network analyser (VNA). This was repeated for magnetic field orientations between 0° and 160° in steps of 5°, with 0° corresponding to B0 parallel to the face of the resonator and sample. The magnetic field was rotated around the [110] crystal axis of the sample. Numerical fitting of the S21 response of the microresonator was used to extract the total measured loss tangent tan δtot = 1/Qtot, where Qtot is the total measured Q factor and tan δtot = tan δc + tan δdiel + tan δB + tan δions, are the loss tangents due to coupling to the transmission line, dielectric losses, the external magnetic field and the Er ions, respectively. Numerical fitting was then used to extract tan δions.

Results and discussion

Spin Echo Measurements

We used three different annealing recipes-a, b and c and isotopically selective implantation of 166 Er to fabricate samples with Er concentrations between 10 17 and 10 19 cm -3 for electron spin resonance (ESR), spin echo and SC resonator coupling measurements. When implanted into Si, Er exists in its usual 3+ oxidation state. [13] Oxygen was co-implanted to a concentration of 10 20 cm -3 for all samples and is required to generate narrow Er-related ESR [14] and photoluminescence (PL) [15] lines by the creation of various O coordinated Er (Er-O) centres. Previous measurements of the angular dependence of the Er-related ESR lines in Er implanted Si have identified a number of different Er-O ESR centres: three monoclinic centres labelled OEr-1, OEr-1' and OEr-3, and three trigonal centres labelled OEr-2, OEr-2'and OEr-4. [14,16,17] None of the ESR centres are optically active. [12] Also, Zeeman measurements of molecular-beam epitaxy (MBE) grown Er doped Si have identified an orthorhombic Er-O centre, [18] which is not ESR active. [12] The g tensors of these centres are given in Table 1.

Table 1 The g tensors of Er doped Si determined by ESR and Zeeman measurements. The ESR centres OEr-1, OEr-2 etc are referred to as 1,2 etc. in subsequent ESR spectra.

Centre Symmetry gx gy gz

Ref.

ESR

OEr-1 Monoclinic C1h 0.8 5.45 12.6 [14] OEr-1' Monoclinic C1h 0.8 5.45 12.55 [14] OEr-3 Monoclinic C1h 1.09 5.05 12.78 [14] OEr-4 Trigonal C3v 2.0 6.23 6.23 [14] OEr-2 Trigonal C3v 0.45 3.46 3.22 [14] OEr-2' Trigonal C3v 0.69 3.24 3.24 [14] Zeeman Er-1 Orthorhombic C2v 0 0 18.4 [18] By comparison to these angular dependencies we can attribute the CW ESR line at 963 G from the sample with 10 19 cm -3 Er and 10 20 cm -3 O with annealing recipe a (sample 1), shown in Figure 1a, to the OEr-3 monoclinic centre. These Er-related ESR lines have been attributed to an Er 3+ centre based on similar g tensors to Er doped Y2O3, which has the same crystal structure as Er2O3, and EXAFS measurements of Er and O implanted Si which found a similar Er-O bond length to that in Er2O3. [14,17] and was also present in empty tube measurements, we therefore treated the off-resonance echo as a background and subtracted it, as illustrated in supplementary Figure S2. Figure 1b shows the integrated intensity of the on-resonance echo peak as a function of the delay time between the π/2 and π pulses (t12). Fitting to an exponential decay gives an estimated T2 of 180 ± 80 for sample 1.

Figure 1d

shows the CW ESR and field dependent echo for the sample with 3×10 17 cm -3 Er and 10 20 cm -3 O with annealing recipe b (sample 2). There are two main resonances at 867 and 934 G, respectively, with weaker resonances at 892 and 964 G. Using the well described angular dependence of Er implanted Si ESR, [14,16,17] these resonances are also attributed to the OEr-3 and OEr-1 centres, as shown; all of these resonances are visible in the echo spectrum. Figure 1d shows the echo intensity as a function of B0 for various t12 between 0.14 and 2.24 μs for sample 2. The on-resonance echo signal disappears below the detection limit then reappears with increasing t12, indicating the presence of very strong electron spin echo envelope modulation (ESEEM). 1.

In Figure 2a we show the echo decay profile, at fixed B0, for sample 2. The background echo signal was subtracted as shown in supplementary Figure S3 to give an on-off resonance echo decay. As can be seen from Figure 1c, a subtraction of the echo decay at 850 G from that at 867 G, should leave only the echo component attributable to implanted Er. The echo decay displays strong superimposed oscillations from the ESEEM effect [19] caused by superhyperfine coupling with neighbouring nuclear spins; similar oscillations were observed in Er:CaWO4, but were significantly weaker than those seen here. [20] Since the 4f wavefunction is highly localised, the superhyperfine coupling between a RE and a neighbouring nuclear spin is usually regarded as magnetic dipole-dipole only. [21] The echo decay of an isolated Er 3+ ion (effective electron spin S = ½) in proximity to a 29 Si nuclei (nuclear spin I = ½) can be described as follows, [19] 𝐼(𝑡 12 ) = 𝐼 0 𝑒 -(

2𝑡 12 𝑇 2 ) 𝑥 [1 -2𝑘sin 2 (2𝜋𝜈 𝛼 𝑡 12 /2)sin 2 (2𝜋𝜈 𝛽 𝑡 12 /2)]. (1)
The first term is the empirical Mims equation, [22] , which describes the echo decay in the absence of nuclear coupling where T2 is the spin coherence time, and x is an exponential stretch factor which is determined by spin dynamics. The term in square brackets describes the ESSEM modulation, where k is the modulation index, να and νβ are the 29 Si nuclear resonance frequencies for the two possible Er 3+ electron spin orientations (S = ±½). Equation The modulation of the on-off resonance decay is very strong with fitting yielding k = 0.45.

The fit also yielded a T2 of 9 ± 3 μs. Our value of T2 of ~10 µs at 5 K compares to ~5 µs at 5 K (~50 µs at 2.5 K) for ~10 16 cm -3 Er doped CaWO4. [20] and 1.6 µs at 1.9 K for ~2×10 18 cm -3

Er doped Y2SiO5. [23] This is a promising comparison given the difficulty of removing defects after implantation that could lead to decoherence. Further optimisation of the recrystallization process, reductions in Er concentration and isotopic purification of the Si may lead to coherence times applicable to quantum communication and computation.

The saturation recovery profile is shown in Figure 2c gives a spin relaxation time, T1, at 5 K, ~1 ms; the background subtraction is shown in supplementary Figure S4. Given that T1>>T2 , it is concluded that T2 is limited by local field fluctuations. [22] For both donors in silicon and RE doped transparent crystals, the ESEEM effect is thought to be caused largely by nuclear spins in very close vicinity to the echo producing centre. For example in both P [24,25] and Bi [26] doped Si, the ESEEM effect is attributed to the four nearest Si lattice positions, in Ce 3+ doped CaWO4, the ESEEM of Ce 3+ can be accurately modelled using the closest ten lattice positions of surrounding W atoms. [19] Spectral diffusion can be caused by various electron [27] and nuclear [28] spin flip-flop process; in Er implanted Si, nuclear induced spectral diffusion is most likely since 29 Si nuclear spins are present and for the electron spins T1>>T2.

The nuclear spins involved in ESEEM, which experience large hyperfine fields, cannot flip flop their spins and contribute to spectral diffusion due to conservation of energy, whereas the nuclear spins involved in spectral diffusion must experience very weak hyperfine fields to allow flip-flops, and therefore consist of a separate, larger, group of nuclei that are further from the echo producing centre. [24] The observation that fitting yielded x ~ 1 in Figure 2a, is somewhat unexpected and indicates no significant spectral diffusion occurs. [2] Due to our isotope specific implantation, the only nuclear spins in our sample are from 29 Si.

Photoluminescence

Figure 3a shows the PL spectrum from sample 1 at 15 K. PL peaks from this sample that we have previously demonstrated originate from the orthorhombic centre [12] are indicated.

Figure 3b shows the Arrhenius plot of the PL intensity of the main peak at 6505 cm -1 . The temperature quenching from Er implanted Si typically displays two activation energies, with slow quenching of the PL at temperatures up to ~150 K and rapid quenching after ~150 K, due to Auger and back-transfer quenching mechanisms, respectively, which are explained elsewhere, [29] and can be described by Equation ( 2), below, but without the 𝐴 1 𝑒 (-𝐸 𝑁𝑇 /𝑘 𝐵 𝑇) term. However, Figure 3b shows an increase in PL intensity as temperature is increased from ~150 K. This negative thermal quenching has not been reported before in Er implanted Si, to the best of our knowledge, and implies population of the initial state, in this case the Er 3+ 4 I13/2 state, by thermal excitation from a middle state, which we will refer to as the Er-O deep state, between the initial and final states, and is described by Equation (2) for the case of one middle state. [30] There is an excellent fit to Equation (2) which gives Auger quenching (EAQ), back transfer (EBT) and negative thermal quenching (ENT) activation energies of 900 (110), 2600 (320) and 1520 (190) cm -1 (meV), respectively. This compares to previous reports of EAQ and EBT of 160 (20) and 1210 (150) cm -1 (meV), respectively. [13] The PL lifetime of sample 1 was less than the system response of ~1 µs, whereas the PL lifetime of Er implanted Si is usually ~1 ms. [31] This decrease in lifetime could be caused by clustering, but it is also consistent with the Er-O deep state indicated by Arrhenius plot in Figure 3a which could facilitate non-radiative decay. 𝐼(𝑇) = 𝐼 0 1+𝐴 1 𝑒 (-𝐸 𝑁𝑇 /𝑘 𝐵 𝑇) 1+𝐶 1 𝑒 (-𝐸 𝐵𝑇 /𝑘 𝐵 𝑇) +𝐶 2 𝑒 (-𝐸 𝐴𝑄 /𝑘 𝐵 𝑇) (2) PL from samples similar to our sample 1 have been previously reported, [14,31] the only differences to our processing was that the Er implant chain went up to 5 MeV, rather than our 4 MeV, the final activation anneal was 900°C, rather than our 850°C and the P-doped FZ wafer had a resistivity of ~220 Ωcm, rather than our wafer with a resistivity of 8000 ± 500 Ωcm. The PL spectrum was similar to ours; however, the Arrhenius plot of the PL showed no negative thermal quenching and EAQ and EBT of 160 and 1210 cm -1 , respectively. [31] Time decay of the PL revealed fast (80 μs) and slow (800 μs) lifetime components. [14,31] This very similar sample therefore has no evidence of an Er-O deep state, unlike our sample 1. The most likely reason is the significantly lower P concentration of our wafer since many Er-O defect states are known to form in Er implanted Si and their presence and energy depends strongly on the presence of other impurities, including P. [13] This higlights how sensitive the energy level structure and optical properties of the Er-O centres are to processing conditions. The fraction of implanted Er that becomes optically active varies between 1 and 10% for different studies; [13] while relatively low, this large range of yields is also consistent with a high sensitivity to processing conditions, which leaves a lot of scope for optimising the processing conditions. There are no estimates of the implantation yield for ESR active centres.

Figure 3c shows the PL mechanism for sample 1, which is based on the wellestablished PL mechanism for Er implanted Si, [13] where above bandgap energy excitation generates excitons which are trapped at Er coordinated O (Er-O) donor traps which introduces a state, EDT, ~1210 cm -1 below the conduction band, as measured by deep level transient spectroscopy (DLTS). This value of EDT and our EBT from fitting in Figure 3b indicates that the bottom of the Er 3+ 4 I15/2 manifold is offset by and energy ,Eoff, of ~1500 (185) cm -1 (meV) below the top of the valence band, whereas previously reported values of EBT usually put the 4 I15/2 manifold just above the VB. [13]

Optically modulated magnetic resonance

Figure 4a shows a contour plot built from many OMMR spectra at various magnetic fields.

The various OMMR spectra appear similar, with four broad bands at 6230, 6380, 6530 and 6610 cm -1 , which vary in intensity with the magnetic field. Figure 4a shows a broad optically generated ESR resonance appears which appears as two bands at 850 and 1170 G, both of which are positive since the second lock-in gives the absolute value of the modulated ESR signal. Because of the lock-in technique used in ESR spectrometers, ESR spectra have zero signal at the maximum of the magnetic centre's absorption; for the same reason, the OMMR spectra are minimised at ~ 960 G, which we refer to as the zero-crossing region. Figure 4b show a close-up of the zero-crossing region which shows how the four broad bands evident in Auger quenching E AQ ~ 900 cm -1 Carrier generation

E DT ~1210 cm -1 E BT ~2600 cm -1 Er-O deep state NR relaxation Offset E off ~ 1500 cm -1 E NT ~1520 cm -1 H F H CF
the OMMR spectra from various magnetic fields inside (951 and 969 G) and outside (1180 G) the zero-crossing region. It is clear how the narrow peaks inside the zero-crossing region align with broad peaks outside the zero-crossing region. The reason the OMMR peaks broaden outside the zero-crossing region is probably an artefact of the measurement system since the zero-crossing region is particular to the lock-in technique used in ESR spectrometers. Peaks that we previously demonstrated must originate from transitions to the 4 I13/2 excited state manifold of the orthorhombic Er-O centre [12] are highlighted. We previously [12] proposed a model for the OMMR mechanism involving transitions from the Si valence band (VB) to the 4 I13/2 manifold which relax back to the Zeeman ground state of the Figure 4d shows OMMR spectra at 10 K and 4 K, both at magnetic fields outside the zero-crossing region. The broad peaks can be aligned by adding 80 cm -1 to the 10 K spectrum, showing that Eoff at 4 K is 197 cm -1 (80 cm -1 less than at 10 K), indicating that Eoff has a temperature dependence. An

Eoff which increases with temperature is consistent with the model from PL temperature quenching in Figure 3b where Eoff ~1500 cm -1 was derived from EBT which came from fitting the temperature quenching at >200 K. Peaks at 6285 and 6360 cm -1 are yet to be identified but could originate from another Er-O center with a higher Eoff than the orthorhombic Er-O center. When the temperature was reduced from 10 K to 4 K the OMMR intensity increased and there were no narrow peaks visible at the zero crossing region, indicating that a low OMMR signal intensity is required to observe the narrow OMMR peaks and could therefore lead to a very low effective spin temperature. Various schemes have been used previously to spin polarise REs to the Zeeman excited state, such as using stimulated emission in Er 3+ :Y2SiO5, [32] circularly polarised light in Ce 3+ :YAG, [33] and by tuning the branching ratio between Zeeman states in ground and excited state manifolds in Nd 3+ :

YVO4. [34] However, these schemes don't have the same analogy to the NV centre mechanism that the Er-O mechanism in Figure 5b has.

Superconducting resonator coupling

The implanted face of sample 3 (10 17 Er cm -3 and 10 20 O cm -3 , annealing recipe c) was placed in contact with the superconducting NbN lumped-element microresonator on R-cut Al2O3 shown in Figure 6a, which had a centre frequency ωr/2π = 3.04 GHz, see experimental section. Figure 6b shows the loss tangents due to coupling to Er ions (tan δions) as function of B0 and orientation. There is a single narrow resonance, with a FWHM of 50±10 G, that varies smoothly between 740 and 870 G depending on the B0 orientation. There is also a very broad resonance centred at 500 G and at B0 // [001] (0 o orientation), the resonance shifts to 600 G at 50 o B0 orientation; we simulated the angular dependence of the six ESR centres (three trigonal, three monoclinic) previously identified Er and O implanted Si system [14] , see Table 1, but found no correspondence with this broad resonance. The narrow resonance had a remarkable correspondence with the trigonal OEr-2' centre identified in ref. [14] with g|| = 0.69 and g┴ = 3.24, which is shown in the simulation in This could be due to preferential coupling of the trigonal centre, or the trigonal centre has a shorter T1 than the other centres at 20 mK, which prevented saturation.

The Q factor of a resonator coupled to an ensemble of spins can be modelled as a single mode harmonic oscillator according to modelling for the trigonal OEr-2' centre identified by Carey et al. [14] with gǁ = 0.69 and g┴ = 3.24. d),

𝑄 𝑡𝑜𝑡 = Δ 2 +𝛾 2 2𝑔 𝑐𝑜𝑙 2 𝛾+𝜅(Δ 2 +𝛾 2 ) 𝜔 𝑟 , (3)
Simulated angular dependent ESR extended to higher B0 to show the positions of the three expected ESR resonances with trigonal symmetry. The microwave frequency was 3.04 GHz for all microresonator measurements and simulations.

A promising scheme to develop a hybrid quantum computer is to link the long coherence time of silicon-based qubits with the ability to entangle superconducting qubits, which tend to have shorter coherence times. This requires the coupling of silicon and superconducting qubits which can be achieved by the exchange of a microwave photon. [36] These hybrid quantum circuits would exhibit long coherence times while allowing quantum state manipulation, and a quantum transducer could be developed using the ability of REs coupled to superconducting resonators to coherently convert optical photons to microwave photons using RE microwave Zeeman transitions as intermediaries. This would have applications in networking quantum signal processors, quantum key distribution and quantum metrology.

Conclusions

Er implanted Si is shown to be a promising platform for the development of QTs and is potentially highly scalable since it can utilise the silicon and ion implantation technology used in the IC industry. Er implanted Si can also exploit the atomic scale barrier to decoherence that is intrinsic to REs, and the recently developed ultra-low spin environment of isopically purified 28 Si. Whereas the Er component itself is compatible with telecommunications wavelength photons and could be utilised for quantum communications schemes. We report the first coherence measurement of implanted Er in the form of a spin coherence time of ~10 μs at 5 K for 3×10 17 cm -3 Er. The origin of this echo is an Er centre surrounded by six O atoms with monoclinic site symmetry. The spin echo decay profile had superimposed modulations due strong superhyperfine coupling with 29 Si.

The temperature quenching of PL from a 10

  Figure 1a also shows that the spin echo peaks at the same B0 as the Er-related ESR resonance at 963 G, which has a full width at half maximum (FWHM) of 7 G. Note that a small angular deviation in the [11 ̅ 0] axis can cause significant shifting of the ESR resonances. A spin echo signal was also present off-resonance,

Figure 1 a

 1 Figure 1 a) CW ESR and field dependent echo signals for 10 19 cm -3 Er and 10 20 cm -3 O with annealing recipe a (sample 1); limitations of the measurement system meant that delays shorter than 130 ns could not be used. The inset illustrates the alignment of the sample 1 and 2 during ESR measurements. b), Integrated intensity of the echo signal as a function of delay time for sample 1. c), CW ESR and field dependent echo signal for 3×10 17 cm -3 Er and 10 20 cm -3 O with annealing recipe b (sample 2). d), Contour plot showing the echo intensity as a function of magnetic field at various t12 for sample 2. All CW ESR measurements were made at 10 K, all echo measurements at 5 K, and the microwave frequency was 9.61 GHz. The ESR resonances are attributed to the previously identified ESR centres in Table1.

( 1 )

 1 encompasses weaker sum (ν+ = να + νβ) and difference (ν-= νανβ) frequency components. In Figure2afitting to Equation (1) yields να of 0.70, which is close to the Larmor frequency (νL) of 0.76 MHz for29 Si at this magnetic field, and νβ of 1.63 MHz; these frequencies can be seen in the FFT in Figure2b. The FFT peaks are broadened by the relatively short sampling time. The deviation of νβ from νL indicates stronger super-hyperfine coupling between Er electron spins and29 Si nuclear spins. The FFT of the fitted decay show the ν+ and νcomponents; these cannot be resolved on the FFT of the measured decay because of insufficient signal-to-noise.

Figure 2

 2 Figure 2 a) Spin echo decay profile at a B0 of 867 G fitted to Equation (1). b) FFT of measured and fitted decay profile. c) saturation recovery at 867 G fitted with a single exponential fit to give T1 of ~1 ms. All measurements were on sample 2 (3×10 17 cm -3 Er and 10 20 cm -3 O, with annealing recipe b) at 5 K, and the microwave frequency was 9.61 GHz.

Figure 3 a

 3 Figure 3 a) Photoluminescence spectrum of 10 19 cm -3 Er and 10 20 cm -3 O with annealing recipe a (sample 1) at 15 K excited at 462 nm. b) Arrhenius plot of the PL intensity from sample 1 at various temperatures fitted to Equation (2). c) Proposed PL mechanism for sample 1.

Figure

  Figure 4anarrow and split into narrower bands in the zero-crossing region. Figure4c shows

4

  Figure3a, should have the same energy as the lowest energy OMMR transition (equivalent to absorption) in Figure4c, yet there is a difference Eoff = 227 cm -1 . Figure4dshows

Figure 4 a)

 4 Figure 4 a) Contour plot created from various OMMR spectra at magnetic fields between 480 and 2300 G at 10 K. b) close up of the zero-crossing region at 10 K. c) OMMR spectra at various magnetic fields inside and outside the zero-crossing region, arrows indicate the peaks associated with transitions to the 4 I13/2 manifold, temperature was 10 K. d) OMMR spectra at 4 K and 10 K, shifted by 80 cm -1 to illustrate the alignment of peaks, arrows indicate unidentified OMMR peaks. B0 was approximately aligned with the [110] axis in panels.

Figure 5a shows theFigure 5a )

 5a Figure5ashows the unilluminated ESR spectrum of sample 1 with the well-known narrow

Figure 6c .Figure 6d .

 6c6d Figure 6d. Only one previously identified Er centre is evident in the hybrid measurements.

Figure 6 a

 6 Figure 6 a), Image of the superconducting micro-resonator that was coupled to sample 3 (10 17 Er cm -3 and 10 20 O cm -3 , annealing recipe c). b), Angular dependent micro-resonator ESR measurement of sample 3 at 20 mK. c), simulated angular dependent ESR spectrum using EASYSPIN numerical

  19 cm-3 Er sample displayed previously unreported negative thermal quenching which implied the presence of an Er-O defect state below 4 I13/2 Er 3+ excited state, the unusually low PL lifetime of <1 μs is consistent with nonradiative decay enabled by this defect state. Fitting the temperature quenching of PL indicated that the 4 I15/2 Er 3+ ground state was buried in the valence band; this was also indicated by OMMR measurements. An ESR resonance under illumination at 1.5 μm that was three orders of magnitude stronger than unilluminated ESR resonances also implied the presence of an Er-O defect state below 4 I13/2 Er 3+ excited state to enable spin selective nonradiative decay to the Zeeman ground state. This mechanism is analogous to the optical spin polarisation mechanism of NV centres in diamond, and if correct could mean high temperature operation of Er qubits in Er implanted Si is possible.

We observed the first coupling between a superconducting resonator and Er implanted Si with gcol = 1MHz and gi > 6 Hz, which provides a basis for future networks of hybrid quantum processors that exchange quantum information over the telecommunication network. Out of six known Er-related ESR centres, only one trigonal centre coupled to the SC resonator at 20 mK.

Acknowledgements

This work was supported by the UK EPSRC grants EP/R011885/1 and EP/H026622/1. We acknowledge the European Research Council for financial support under the FP7 for the award of the ERC Advanced Investigator Grant SILAMPS 226470. We would like to thank Prof. John Morton for helpful discussions.

Data availability

The datasets generated during the current study are available in the Mendeley Data repository http://dx.doi.org/10.17632/s73x8nb8dn.1

where Δ is the detuning from the spin resonance peak, γ is the spin linewidth, κ is the cavity linewidth = 2πωr/Qtot = 0.56 MHz for the 0° orientation and was independently measured away from the resonance for each B0 orientation, Qtot is the total measured cavity Q, and gcol is the collective coupling strength. Supplementary Figure S5 shows the fitting of Equation (3) to the Qtot for the 0° orientation which gives gcol/2π = 1 MHz and γ/2π = 80 MHz. The average for all B0 orientations was gcol/2π = 1.1 ± 0.3 MHz and γ/2π = 85 ± 25 MHz. The coupling strength of an individual spin to the SC resonator is given by gi = gcol/√N, where N is the number of spins coupled to the resonator; using the number of Er ions above the microresonator (~3.7×10 10 ) gives a lower limit for gi of ~ 6 Hz, since the implantation yield is unknown for this ESR centre. This compares to gi ~ 2 Hz for Gd implanted Al2O3 [35] and ~ 70 Hz for Er implanted Y2SiO5 crystal. [9] We observed no change in ωr as B0 was swept through the Er spin resonance, indicating the system is operating in the weak coupling regime. Our micro-resonator measurement represents the first reported coupling of a SC resonator to a RE ensemble implanted in Si. The strong coupling regime could be attained by optimization of the annealing recipe to produce only one ESR centre and by operating at higher centre frequencies.

Supporting information

Supporting Information is available from the Wiley Online Library or from the author.

Competing financial interests:

The authors declare no competing financial interest

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Erbium implanted silicon is a promising quantum technology platform. This work shows the first coherence measurement of any implanted erbium to be ~10 µs at 5K. Three independent measurements all suggest that erbium implanted silicon has an energy level structure analogous to NV centres in diamond, which could lead to optical spin polarisation and high temperature operation of erbium qubits.