C Tuchendler

A M Lance

A Browaeys

Y R P Sortais

P Grangier

Energy distribution and cooling of a single atom in an optical tweezer

Keywords: numbers: 37.10.Jk, 42.50.Ct, 03.67.-a

We investigate experimentally the energy distribution of a single 87 Rb atom trapped in a strongly focused dipole trap under various cooling regimes. Using two different methods to measure the mean energy of the atom, we show that the energy distribution of the radiatively cooled atom is close to thermal. We then demonstrate how to reduce the energy of the single atom, first by adiabatic cooling, and then by truncating the Boltzmann distribution of the single atom. This provides a non-deterministic way to prepare atoms at low microKelvin temperatures, close to the ground state of the trapping potential.

I. INTRODUCTION

In the last thirty years, the manipulation of single quantum objects has been the subject of considerable attention. After the pioneering experiments of H. Dehmelt with a single electron in a Penning trap [1], a single ion was trapped in a Paul trap [2], which led to the observation of quantum jumps [3,4]. Several techniques were then developed to cool the single trapped particle, such as optical sideband cooling, which was experimentally demonstrated in 1989 [5]. More recently, the interest in the manipulation of single quantum objects has grown with the recognition that they can carry quantum information, and that they therefore can be a resource for quantum information processing [START_REF] Nielsen | Quantum computation and quantum information[END_REF]. In this context, single trapped laser cooled ions have accumulated an impressive amount of results in recent years (see e.g. [START_REF] Benhelm | [END_REF]).

The trapping of single neutral atoms, on the other hand, is more recent, owing in part to their weaker interaction with the electromagnetic field. So far, the trapping of neutral atoms has been realized in tight magnetooptical traps [8,9] or optical dipole traps [10,11,12,13]. As the trapping potentials are typically less than a mil-liKelvin deep, laser cooling is the usual starting point of the manipulation of single atoms. As is the case for ions, it is tempting to further reduce the energy of the trapped atom, to ultimately reach the ground vibrational state of the trapping potential. In the framework of quantum computing, for example, the entanglement of two atoms via controlled collisions usually requires ground state cooling (see e.g. [14]).

The knowledge and control of the energy of a single atom, trapped in a sub-micron optical dipole trap, is important for many purposes. For instance, in a recent experiment [15], we have used two single rubidium atoms, trapped in two optical tweezers separated by a few microns, as single-photon sources. When the two photons * Electronic address: andrew.lance@institutoptique.fr (each photon coming from one atom) are superimposed on a beamsplitter, we observe a "coalescence" effect : the two (indistinguishable) photons copropagate from the same output of the beamsplitter, due to a two-photon interference effect. We have shown that the ultimate visibility of this effect is limited by the spread in frequencies of the emitted photons, which in turn is due to the spread in energy of the emitting trapped atoms. In another experiment [16], using Raman transitions we have prepared a single atom in a superposition of two internal states of the hyperfine manifold, and we have studied the dephasing of this quantum bit. In this case, due to the fact that successive atoms have different energies, each realization of the qubit evolves differently with respect to a local oscillator. This results in a loss of coherence, that we were nevertheless able to compensate for by applying rephasing "spin-echo" techniques. For this application, a lower energy spread of the trapped atom would also increase the dephasing time, thus avoiding the extra "spin-echo" sequence.

In the present paper, we analyze in more detail the methods used in these previous works to determine the mean energy of a single atom in the optical tweezer [15,16,17]. Similar or complementary techniques have been reported by other groups [12,18]. We then exploit these temperature measurements to characterize several cooling methods to further decrease the temperature of the atoms, and we show how to approach the ground state level in the two radial directions of the trap, which exhibit the strongest confinement.

The paper is organized as follows. We describe in section II the experimental setup. In section III, we present a "release and recapture" technique used to measure the temperature of the atoms. In section IV we develop a cooling sequence used to laser-cool the atom in the tweezer. In section V, we perform the spectroscopy of the energy distribution of the atom in the tweezer and find that this distribution is very close to a Boltzmann distribution. Section VI describes how the mean energy of the single atom can be decreased by truncating this Boltzmann distribution. This provides a non-deterministic A beam of 87 Rb atoms, originating from an oven, is slowed using a Zeeman coil (not shown) and counter propagating slowing (Ω1) and repumping (Ω2) lasers at ∼ 780 nm. An optical molasses is formed at the intersection of six counter propagating lasers (Ω3) and repumpers (Ω4) operating also at ∼ 780 nm. An aspherical lens is used to tightly focus a ∼ 850 nm laser (Ω5) in the region of the optical molasses. The fluorescence of the atom, due to the molasses and Zeeman lasers, is collected using the same aspherical lens. A dichroic mirror together with other spatial and spectral filters are used to filter the fluorescence signal, which is detected using singlephoton counting avalanche photodiode (APD).

method to prepare a single atom close to the ground state of the trapping potential. In section VII we reduce further the temperature of the atoms by adiabatically lowering the potential trap. Finally, we conclude by discussing some implications of these results.

II. EXPERIMENTAL SETUP

Figure 1 shows a schematic of the experimental setup. A beam of 87 Rb atoms is slowed using a Zeeman slowing technique [19]. The counter-propagating laser beam at ∼ 780 nm (denoted Ω 1 ) is 18.7 Γ red detuned from the F = 2 → F ′ = 3 cycling transition, and combined with a repumping laser (Ω 2 ) that is 3.7 Γ red detuned from the F = 1 → F ′ = 2 repumping transition. Here, Γ ≈ 2π × 6 × 10 6 rad/s is the linewidth of the D 2 transition.

We form an optical molasses at the intersection of six counter-propagating cooling lasers that are 4.2 Γ red detuned from the F = 2 → F ′ = 3 transition (denoted Ω 3 ), together with six repumping lasers (Ω 4 ) resonantly tuned to the F = 1 → F ′ = 2 transition. This optical molasses is used as a reservoir of cold atoms from which the dipole trap is loaded.

We produce the optical dipole trap by focusing a ∼ 850 nm laser in the center of the molasses, using a large numerical aperture (NA = 0.5) aspherical lens [17]. The optical dipole trap has a measured optical waist of w = 1.03 ± 0.01 µm. A comprehensive characterization of the optical system and of the dipole trap itself is presented in detail elsewhere [17]. Assuming a Gaussian beam profile in the transverse direction for the dipole trap and a Lorentzian profile along the propagation axis, we calculate the radial and axial frequencies and find ν ⊥ ∼ 160 kHz and ν ∼ 30 kHz respectively for 10 mW of laser power. The potential trap depth, proportional to the laser power, is U ∼ 2.8 mK for a 10 mW of laser power.

This optical tweezer allows us to trap single Rb 87 atoms via a collisional blockade mechanism, which prevents two or more atoms from being trapped simultaneously due to inelastic collisions [11,17,20]. The ∼ 780 nm fluorescence of the trapped single atoms is detected using the same aspherical lens that is used to create the dipole trap. This fluorescence is filtered using a dichroic mirror, together with other spatial and spectral filters, and then detected using a single photon counter (Avalanche Photodiode), with an overall collection and detection efficiency that is estimated to be ∼ 0.5%. From the output signal of the single-photon counter we distinguish between the presence (or absence) of an atom in the tweezer, within a 10 ms acquisition period, with a confidence better than 99%. Each experimental sequence that will be presented in this paper is triggered on a detected single atom in the dipole trap.

Our goal in the following sections will be to investigate the energy distribution and temperature of the atoms trapped in the optical tweezer, under various cooling conditions.

III. RELEASE AND RECAPTURE TECHNIQUE

In this section we will describe a "release and recapture" method that we use to determine the temperature of the atoms in the dipole trap [15,16]. We emphasize that what we mean by the temperature of the atoms is a temperature extracted by averaging the energy over many realizations of the same experiment with a single atom. The premise of this method is that information can be obtained about the energy distribution of a single atom by switching off the dipole trap for a variable time ∆t (as illustrated in Fig. 2(a)) and then determining the probability of recapturing the atom, which is denoted as P rr (∆t). On average, an atom with a high energy is more likely to escape during the release time ∆t, compared to an atom with a low energy, as is well-known in the context of laser cooling of atomic samples [21,22]. In the single atom regime used here, this method is particularly robust, since on each repetition of the experimental sequence the result is binary : either the atom has escaped the trapping region, or it is recaptured. This binarization of the measurement outcome makes it immune to various noise sources, in particular to fluctuations in the fluorescence level of the atom, which is used to decide on the presence of the atom.

Figure 2(a) illustrates a schematic of the "release and recapture" sequence, which goes as follows. (i) After the atom is trapped, the cooling and loading lasers (Ω 1 , Ω 2 , Ω 3 and Ω 4 ) are switched off. (ii) The dipole trap laser (Ω 5 ) is then turned off for a time ∆t. (iii) Finally, we perform a fluorescence detection by switching all the cooling and loading lasers back on, thus determining whether the atom has escaped or not. Figure 2 shows typical experimental results for the "release and recapture" technique, where the probability of recapturing the atom is plotted as a function of the release time ∆t (more details about this experiment will be presented in the following section).

In order to extract the temperature from these measurements, we perform Monte-Carlo simulations of the trajectories of the single atoms, to determine the probability that the atoms are recaptured after a release time ∆t. This simulation requires the knowledge of the energy distribution of the atom in the trap. We assume that we are in the harmonic approximation of the trapping potential, and that the position-velocity distribution of the atom follows a thermal Maxwell-Boltzmann law. The standard deviations of the position of the atom in the axial and radial directions of the trapping potential are ∆x = k B T /mω 2 and ∆x ⊥ = k B T /mω 2 ⊥ respectively, while the standard deviation of the velocity is ∆v = k B T /m, where T is the temperature. The parameters of the optical trapping potential have been determined previously [17], and take into account the tilt due to gravity, which creates a local maximum in the potential along the gravity axis. The effective trap depth is defined as the height of this potential barrier with respect to the minimum of the potential.

The simulation goes as follows:

using the Maxwell-Boltzmann distribution we randomly generate the three-dimensional position-velocity vector (x i , y i , z i , v x,i , v y,i , v z,i ) of the atom in the potential for a given temperature. We then calculate the trajectory of each atom during the release time ∆t in a single time-step calculation. The position of the atom after this release time is

(x i + v x,i ∆t, y i + v y,i ∆t -g∆t 2 /2, z i + v z,i ∆t),
where the potential gradient due to gravity along the y-axis has been included. We finally determine the energy of each atom when the trap is turned back on. If the total energy of the atom after the time-of-flight is smaller than the effective trap depth, we consider that the atom is recaptured, otherwise it is considered to have escaped.

By simulating many trajectories of the single atoms for each release time ∆t, we can numerically determine the probability of recapturing the atom as a function of this release time. These simulations are repeated for a range of different temperatures. We calculate the weighted least square value (χ 2 ) between the experimental data and the simulated results for each temperature. The chisquare value is defined as

χ 2 = i {[f i -P rr (∆t i )] 2 /σ 2
i } where P rr (∆t) is the experimental data that has an uncertainty σ and f is the value predicted by the model. The best-fit temperature corresponds to the Boltzmann distribution that minimizes this least square value.

We performed the "release and recapture" technique to determine the temperature of the atoms just after they were loaded in the trap. Figure 2(b) shows the Monte-Carlo simulation that best fits the corresponding experimental data. In this example, the best fit temperature is T = 168 ± 6 µK in a trap depth of U ∼ 2.8 mK. The ratio between the trap depth and the temperature of the atoms is 15, indicating that we are approximately in the harmonic regime of the trapping potential.

Due to experimental miscounts and losses, the probability of recapturing an atom for zero release time was less than unity, being instead typically 0.95. We scaled the Monte-Carlo simulation by this factor to achieve a better fit to the experimental results. The uncertainty on the temperature that minimized the chi-square value is defined as σ T = 2(∂ 2 χ 2 /∂T 2 ) -1 , assuming that the χ 2 function varies harmonically around this minimum point [START_REF] Bevington | Data Reduction and Error Analysis for the Physical Sciences[END_REF]. Furthermore, we have included in the error budget for the temperature, the statistical uncertainty in fitting a parabola function to this chi-square minimum.

We will use this "release and recapture" technique throughout the rest of the paper as a diagnostic to measure the temperature of the atoms in the optical tweezer.

IV. RADIATIVE COOLING OF A SINGLE ATOM

In this Section, we discuss the laser cooling of a single atom after it has been captured in the dipole trap. For that purpose, the cooling sequence was optimized by controlling the molasses lasers parameters (cooling time, frequency detuning, intensity), as well as the trapping laser intensity. This was done by maximizing the recapture probability of the single atom after a "release and recapture" experiment with ∆t = 10 µs.

Figure 3 shows the optimized laser cooling sequence. Initially, all lasers are on and the molasses cooling lasers are detuned by ∆ 1 = -4.5Γ to maximize the loading rate of the molasses and optical dipole trap. (i) The sequence is triggered on the detection of an atom, where the dipole trap laser power is ∼ 10 mW. (ii) The Zeeman laser and the Zeeman repumper laser are switched off. At the same time, the cooling lasers are linearly detuned from ∆ 1 = -4.5Γ to ∆ 2 = -2Γ in ∼ 1.2 ms. During this time, the intensities of the molasses lasers are reduced from I 1 ∼ 2 mW/beam, to approximately one third of their initial intensity. After this cooling period the cooling and repumping lasers are switched off.

Using this optimized laser-cooling sequence we were able to significantly reduce the temperature of the atoms. Figure 2(b) shows the "release and recapture" experi-mental results after the laser-cooling, together with the best-fit simulation results. This corresponds to a temperature of 31 ± 1 µK for a trap depth of ∼ 2.5 mK, which yields the ratio U/T ∼ 81. This represents more than a factor 5 reduction in the temperature with respect to the temperature of the atoms directly loaded from the optical molasses.

We note that the optimized cooling sequence is achieved when the laser detuning is brought closer to resonance with respect to the atomic transition. This fact may seem inconsistent with the sub-Doppler theory of laser cooling (see e.g. [START_REF] Drewsen | [END_REF]), which predicts that the temperature is lower for larger detuning. However one has to take into account that when the atom is trapped in the tweezer, the resonance frequency is light shifted by about 10Γ with respect to the free space case. Therefore the detuning experienced by the trapped atom varies only from -14.5Γ to -12Γ. The extra cooling effect is then attributed to the decrease (by a factor 3) in the intensity of the laser beams, rather than to the frequency change, now in reasonable agreement with sub-Doppler laser cooling.

This optimized cooling technique is the starting point for each experiment presented throughout the rest of the paper. We will now turn our attention to validating the assumption about the thermal distribution. In addition, we would like to cross-check the temperature results using an alternative method.

V. ENERGY DISTRIBUTION OF A SINGLE ATOM IN AN OPTICAL TWEEZER

In this section we determine the energy distribution of the single atom in the dipole trap. Provided that this distribution is indeed thermal, the corresponding temperature can be evaluated. To achieve this, we apply a type of spectroscopy method proposed and experimentally demonstrated by Alt et al. in ref. [18].

This method involves the adiabatic lowering of the trap depth to a point where the atom can potentially escape. As shown in Fig. 4(a), by lowering the trap depth, an atom with initial energy E i in a trap of depth U i will eventually have an energy E esc that is equal to the final trap depth U esc and hence the atom will escape the trap. Thus, measuring the depth at which the atom escapes the trap yields information about the initial energy of the atom.

Figure 4(b) shows a schematic of the experimental sequence for this method, which goes as follows: (i) The single atom is trapped and laser cooled. (ii) The dipole trap depth is then adiabatically lowered from U i ∼ 2.8 mK to U min . The lowering time is approximately ∼ 2.5 ms. The trap depth is held constant for a duration of ∼ 20 ms, to allow an atom with an energy greater than the trap depth sufficient time to escape [30]. (iii) After this waiting time, the trap depth is adiabatically raised to the initial trap depth U i in ∼ 2.5 ms. Finally, the laser beams are switched back on to determine if the atom has escaped or not. We leave to section VII the discussion about the adiabaticity of this process. Here we assume that the hypothesis for the adiabatic change of the potential is fulfilled. Figure 4 (c) shows the experimental probability of recapture for various minimum trap depths U min normalized to the initial trap depth U i [31].

It is now necessary to map the minimum trap depth U min back to the corresponding energy in the initial trap E i . To perform this mapping we calculate the onedimensional action that can be expressed as the integral

S(E, U ) = xmax 0 2m[E -U (x)]dx
, where E is the energy, U (x) is the potential and x max is the position where the atom has zero kinetic energy. In the adiabatic limit, the action of the atom in the trap is conserved as the trap depth is adiabatically lowered [START_REF] Landau | Mechanics[END_REF]. We solve the constant action equation S(E i , U i ) = S(U esc , U esc ) in the radial direction of our potential trap that includes the gravity potential gradient. The numerical results of the U esc /U i → E i /U i mapping is plotted in Figure 4 (e). We subsequently apply the mapping U esc /U i → E i /U i to our experimental data. Figure 4 (d) shows the percentage of atoms that have an energy less than E i in the initial trap of depth U i plotted as a function of the normalized energy E i /U i .

We now wish to compare the measured energy distribution to a thermal one. In the harmonic trap limit (i.e. when the ratio of the trap depth to the temperature of the atoms satisfies U i /k B T ≫ 1), we calculate the Boltzmann energy distribution for a temperature T . We assume that the normalized energy distribution of the atom is

f th (E) = 1 2(k B T ) 3 E 2 e -E/kBT (1)
where E 2 corresponds to the three dimensional density of states. The probability that an atom has an energy less than E is defined by P surv (E) = E 0 f th (E ′ ) dE ′ and is given by

P surv (E) = 1 -1 + η + 1 2 η 2 e -η (2)
where η = E/k B T . Equation ( 2) can be interpreted as the survival probability of the atom remaining in the dipole trap after the truncation of the Boltzmann distribution. Figure 4(d) shows a fit of the integrated Boltzmann distribution P (E i /U i ) to the experimental data using Eq. (2). To achieve a better fit to the data, the maximum survival probability is rescaled to 0.95, to account for non-ideal recapture probability of our experiment when U min = U i . We find that the measured energy distribution is well fitted by a thermal distribution. This fact shows that over many realizations of the experiments, the energy of successive atoms after laser cooling is well described by a Boltzmann distribution, in agreement with several previous theoretical and experimental studies [START_REF] Lett | [END_REF]. The Boltzmann distribution in Figure 4(d) has a corresponding temperature of T = 33 ± 2 µK for a trap depth U i ∼ 2.8 mK, where the error in the temperature corresponds the statistical uncertainty in the fit. This temperature is in good agreement with the "release and recapture" temperature results presented in the previous section.

We point out that the adiabatic lowering of the dipole trap depth is a powerful tool in manipulating the energy distribution of the atom. For example, at very low trap depths the more energetic atoms will escape with a higher probability. This adiabatic lowering indeed acts as a filter such that only the coolest atoms will remain in the trap, as we will now show.

VI. REDUCTION OF MEAN ENERGY BY TRUNCATING THE BOLTZMANN DISTRIBUTION

We have shown in Sections IV and V that we can cool the atoms down to a temperature of approximately T = 33 µK using laser cooling techniques. It is possible to further reduce the mean energy of the trapped atom by truncating the Boltzmann distribution at an energy U trunc . Here, we perform this truncation by adiabatically lowering the trapping potential, to a point where atoms with a higher energy are likely to escape, while atoms with a lower energy remain trapped. This filtering via the lowering of the trap potential appears to be similar to evaporative cooling, but a crucial difference is that there is no collision-induced rethermalization, since the experiment is done at the single atom level [27].

Figure 5(a) shows a schematic of the experimental sequence that we use to investigate the energy distribution of the truncated Boltzmann distribution. This experimental sequence is identical to the one described in section V, but we now perform a "release and recapture" sequence directly after it. The entire temporal sequence is shown in the inset of figure 5(b). The additional "release and recapture" sequence allows us to determine the temperature of the atoms after the truncation process and after the trap depth has been returned to the initial depth.

To determine the truncation energy U trunc in the trap of potential depth U f = U i , we first calculate the truncation energy in the shallow trap of potential depth U min (using the measured minimum optical dipole trap power). We then use the numerical results that map U min → E i (see Fig. 4(e)) to determine the truncation energy U trunc in the trap of potential depth U f .

In the case of a truncated Boltzmann distribution, it is not sufficient to assume that the position-velocity vector of the atom follows a Gaussian Maxwell Boltzmann distribution, as we did for the "release and recapture" experiments in Sections III and IV. To account for this effect, in the following simulations we consider that the energies of the atoms are drawn from a truncated Boltz- mann distribution corresponding to a temperature T and a truncation energy U trunc . The Boltzmann distribution is discretized into N bins with energies E j equally spaced by ∆U = U trunc /N , for ∆U/2 ≤ E j ≤ (U trunc -∆U/2). For each distribution, the number of simulations with atoms with an energy E j is weighted according to this discretized Boltzmann distribution. For an atom with energy E j , we randomly distribute the energies among the three directions given by the cartesian axes. The period of motion of the atom is then randomly chosen for each of the three cartesian axes (ensuring that the total energy of the atom remains E j ). Finally, the trajectories of the atoms are simulated during the time of flight as described in Section III. We typically use several hundred atom trajectories for each release time ∆t, and we discretize the Boltzmann distribution into 10 bins. When the truncation energy is very large with respect to the temperature (i.e. the distribution is approximately not truncated), this new model yields the same results as the previous release and recapture model.

The experimental results are shown in Figure 5. We repeated the experiment several times for different minimum trap depths U min . The data were recorded over a span of several days, where all other parameters were kept approximately constant. For each minimum trap depth we measured the recapture probability, P rr (∆t = 0), the temperature, T , of the truncated Boltzmann distribution and we calculated the associated mean energy, E , of this distribution. The small day to day variations in the experimental parameters result in a small spread of the temperatures for a given value of U min . Nevertheless, Figure 5(c) shows that the temperature of the Boltzmann distribution remains approximately constant down to very small trap depths, even when the distribution becomes truncated.

In contrast, the mean energy is reduced markedly as the distribution becomes more truncated, as we will now describe. The mean energy of a truncated Boltzmann distribution in a three dimensional harmonic potential can be expressed as:

E = Utrunc 0 E 3 e -E/kBT dE Utrunc 0 E 2 e -E/kBT dE = 3k B T 1 -1 + η + 1 2 η 2 + 1 6 η 3 e -η 1 -1 + η + 1 2 η 2 e -η (3)
where η = U trunc /k B T . In the limit where the Boltzmann distribution is approximately non-truncated, i.e. η ≫ 1, the mean energy is given by E = 3k B T . However, as shown in Eq. ( 3) the mean energy is reduced significantly as the truncation is increased, i.e. η → 0. The probability that the atom survives this truncation process is given by Eq. ( 2) as P surv (U trunc ). Naturally, this probability drops quickly as η → 0, thus reducing the efficiency (or effective duty cycle) of this scheme. Figure 5(d) shows the mean energy calculated using Eq. ( 3) with the measured values of the temperature and the truncation energy U trunc . The solid line is a theoretical prediction using Eq. ( 3) for a temperature T = 33 µK.

For minimum trap depths above 10 µK, i.e. when the energy distribution is approximately non-truncated, the mean energy is E ∼ 100 µK. For minimum trap depths below 10 µK, we see that the mean energy is significantly reduced. For example, for a minimum trap depth U min ∼ 0.4 µK, we measure a mean energy of the atom at the end of this filtering sequence of E ∼ 30 µK, where the final trap depth is U f ∼ 2.8 mK. The truncation of the Boltzmann distribution thus leads to a reduction of the mean energy by a factor of 100µK / 30µK ∼ 3.3. At the same time, the probability of recapture is reduced to P rr (∆t = 0) ∼ 0.1.

In this section we have shown that if one is willing to compromise on the probability of recapturing the atom (or correspondingly, the duty cycle of the experiment), truncating the Boltzmann distribution is a good way to reduce the mean energy of the atom.

VII. COOLING A SINGLE ATOM BY ADIABATICALLY LOWERING THE TRAPPING POTENTIAL

Let us consider an atom in a trapping potential. By adiabatically lowering the trap depth the occupation probabilities of the vibrational levels are preserved [START_REF] Reif | Fundamentals of Statistical and Thermal Physics[END_REF]. For our optical dipole trap this conservation of the vibrational number results in T / √ U being a constant. In this section we will experimentally demonstrate that by slowly lowering the trap depth, the temperature of the atoms follows this relation, thus reinforcing our assumptions about adiabaticity.

To test the adiabaticity of our lowering scheme we perform the same experimental sequence as in section VI, but we now implement the "release and recapture" sequence when the trap is shallowest, i.e. we measure the temperature of the atoms at the minimum trap depth U min [32]. This is in contrast to section VI, where the temperature of the atoms was measured after the trap depth was returned to its initial value.

A schematic of the experimental sequence is shown in Figure 6, together with the recapture probability, P rr (∆t = 0), and the temperature, T , measured for various minimum trap depths U min . The data in figure 6(c), shows evidence that the temperature of the Boltzmann distribution follows closely an adiabatic lowering behavior. For comparison, we have plotted the adiabatic scaling law T / √ U = const, where the constant is equal to T i / √ U i with T i = 33 µK and U i = 2.8 mK. The criteria for adiabaticity states that the rate of change of the oscillation frequency ω must be lower than ω 2 (ω is the oscillation frequency) at all times during the evolution [START_REF] Landau | Mechanics[END_REF]. Care must be taken to fulfill the adiabaticity criteria when the trapping potential is very small, corresponding to a small oscillation frequency. The rate of change of the oscillation frequency must be suffi- ciently small in this case. In our experiment, this is done by smoothly ramping up and down the intensity of the dipole trap laser with acousto-optic modulators. The behavior of the measured temperature in Figure 6(c) gives us a strong indication that indeed the adiabaticity criteria is fulfilled when the trapping depth is small. Figure 6(b) shows the probability of recapture for zero release time P rr (∆t = 0) plotted as a function of the minimum trap depth U min , together with a theoretical fit. We reemphasize that for minimum trap depths less than U min ∼ 10 µK, the Boltzmann distribution becomes truncated. This implies that we are no longer in the three-dimensional harmonic approximation, and hence the density of states is no longer described by ρ(E) ∝ E 2 . Nevertheless, for clarity in Figure 6, we have presented the results assuming that ρ(E) ∝ E 2 for all data points. As a typical result, we measure a temperature of T = 1.69 ± 0.02 µK for the atoms in a trap that has been adiabatically lowered to a depth of U min ∼ 12 µK. In this case, the Boltzmann distribution is only slightly truncated, implying that the quadratic density of states should still be valid. This temperature corresponds to a one hundred fold decrease in the temperature of the atoms compared to those that were directly loaded from the molasses.

VIII. DISCUSSION

We have shown that the single atom cooling via filtering techniques is quite efficient. Starting from an initial mean energy of 100 µK after the laser cooling phase, we get a final mean energy of the atom of 30 µK with a probability of ∼ 0.1. On our experiment, the loading rate of single atoms is on the order of 1 per second. If we are willing to comprise our duty cycle, we could obtain on average one atom every 10 seconds with a mean energy of 30 µK in a ∼ 2.8 mK deep trap, which corresponds to a ratio of the trap depth to mean energy of ∼ 93.

Another parameter characterizing the external motion of the atom in the trap is the mean vibrational number along one axis, defined as the ratio of the mean energy of the atom along this axis to the energy difference between vibrational states. As the trap has two different oscillation frequencies along the axial and radial directions, this corresponds to two different mean vibrational numbers along these directions. If we assume the equipartition of the mean energy among the three axis, we get for the mean vibrational number along the radial direction n ⊥ = E /3hν ⊥ . For the mean energy of 100 µK that we obtained after laser-cooling the atom, this leads to n ⊥ ∼ 4.3. By contrast, the lowest mean energy of 30 µK that we obtained after filtering leads to n ⊥ ∼ 1.3, which is 3.3 times smaller than the mean vibrational number before filtering. This mean number, close to 1 along the radial direction, raises some questions about the validity of our classical treatment for the temperature measurement for the lowest trap depths. This analysis, albeit interesting, is beyond the scope of this paper. This also places us in a good position to further cool the atom down to the ground state, for example by using Raman sideband cooling [5].

In the framework of quantum information, there has been proposed a protocol to entangle two atoms based on the emission of a single photon by one of the two trapped atoms [29]. A crucial feature of this proposal is the ability to localize each atom within a distance smaller than the wavelength of the emitted photon, the so-called Lamb-Dicke regime. Reference [29] shows that for atoms described by a thermal distribution at temperature T , this condition becomes η th = η LD k B T / ω ≪ 1, with ω being the mean oscillation frequency of the atom in the trap, and η LD = E r / ω being the Lamb-Dicke parameter (E r is the recoil energy). In our case, for a trap depth of ∼ 2.8 mK, we find η LD = 0.20 and η th = 0.56 for a temperature of 33 µK, as measured above. We are therefore in the right regime to apply this entanglement protocol. We finally note that η th can be lowered when increasing the trap depth adiabatically. In this case η th is related to the trap depth U by η th ∝ U -1/4 .

IX. CONCLUSION

As a summary of this paper, we have investigated the energy of a single atom trapped in a tight optical dipole trap. We have found that after applying laser cooling of the atom in the tweezer, the energy of the trapped atom follows a Boltzmann distribution. We have extracted the temperature of the atoms from the measured energy distribution. We have also described a "release and recapture" method to measure the temperature of the atoms assuming a truncated Boltzmann distribution. This second method is in good agreement with the first one.

We have shown that by adiabatically lowering the trap depth we have reduced the temperature of the atoms down to several microKelvin. Finally, we have cooled the atom by filtering the hottest atoms over many realizations of the same experiment, the equivalent of evaporative cooling but at the single atom level. This leads to an atom close to the ground state in the radial direction of the trap.

[29] C. Cabrillo, J. I. Cirac, P. Garcia-Fernadez and P. Zoller, Phys. Rev. A 59 1025 (1999).

[30] We assume that an atom with an energy greater than the trap depth has sufficient time to exit the trap provided the following condition is fulfilled: 1/τ ≪ ν , where τ is the time the atom is in the trap and ν is the trap frequency in the axial direction. This condition is satisfied down to a final trap depth U f /Ui ∼ 10 -4 , where τ × ν ∼ 10 ≫ 1.

[31] For the effective minimum trap depth, Umin, taking grav-ity into account turns out to be important for trap depths less than ∼ 10 µK because the gravity potential acts to tilt the trapping potential, thus lowering the potential barrier along the gravity axis.

[32] After the "release and recapture" sequence we needed to bring the trap depth close to its initial value to measure if the atom is still present in the trap, otherwise the lasers sent on the atom would kick it out of the shallow trap too quickly to measure its presence.

FIG. 1 :

 1 FIG. 1: (Color online) Schematic of the experimental setup.A beam of 87 Rb atoms, originating from an oven, is slowed using a Zeeman coil (not shown) and counter propagating slowing (Ω1) and repumping (Ω2) lasers at ∼ 780 nm. An optical molasses is formed at the intersection of six counter propagating lasers (Ω3) and repumpers (Ω4) operating also at ∼ 780 nm. An aspherical lens is used to tightly focus a ∼ 850 nm laser (Ω5) in the region of the optical molasses. The fluorescence of the atom, due to the molasses and Zeeman lasers, is collected using the same aspherical lens. A dichroic mirror together with other spatial and spectral filters are used to filter the fluorescence signal, which is detected using singlephoton counting avalanche photodiode (APD).

FIG. 2 :

 2 FIG. 2: (Color online) Measurement of the temperature of the atoms using the "release and recapture" method. (a) Schematic of the experimental sequence (see text). (b) Experimental results showing the probability of recapturing the atom, Prr(∆t), as a function of release time ∆t. (i) (Circles) Temperature of the atoms just after the loading in the dipole trap. Each data point is the accumulation of 100 sequences.Superimposed on this data is a fit by the Monte-Carlo simulation of the "release and recapture" method, which is the average of 200 trajectories for each release time. The temperature is 168 ± 6 µK for a trap depth of ∼ 2.8 mK. (ii) (Diamonds) For comparison we have also plotted the results for the temperature measurement using the same method, but after the atom has been laser-cooled in the tweezer. Each data point is the accumulation of 200 sequences. The best-fit temperature is 31 ± 1 µK for a trap depth of ∼ 2.5 mK.

FIG. 3 :

 3 FIG. 3: (Color online) Schematic of the temporal sequence for the laser cooling sequence. The details are given in the body of the text. The box labeled Temperature Experiments shows the point in the sequence where we can manipulate the intensity of the dipole trap laser in order to measure the temperature of the atoms. (See Sections V, VI and VII).

FIG. 4 :

 4 FIG. 4: (Color online) Spectroscopy of the energy distribution of a single atom using the adiabatic lowering method. (a) Schematic showing that due to the adiabatic lowering of the dipole trap, an atom with initial energy Ei in a trap depth Ui eventually has an energy equal to the trap depth, i.e. Emin = Umin = Uesc. (b) Schematic of the experimental sequence (see text). (c) Experimental results of the survival probability for various minimum trap depths Umin. Each data point is the accumulation of 200 sequences, with error bars due to the binomial statistics. Solid line is a theoretical fit to the data. (d) Reconstruction of the Boltzmann distribution in the initial trap of depth Ui. Plot showing the percentage of atoms with an energy less than Ei (in the initial trap of depth Ui) versus the normalized energy Ei/Ui. Solid line is a theoretical fit to the data (see text). Dashed line represents the corresponding Boltzmann distribution (e) Numerical results for solving the constant action equation S(Ei, Ui) = S(Uesc, Uesc).

FIG. 5 :

 5 FIG. 5: (Color online) Investigation of a truncated Boltzmann distribution using the "release and recapture" method. (a) Schematic of the experimental sequence. (b) Probability of recapture for zero release time Prr(∆t = 0) plotted as a function of the minimum trap depth Umin. Prr(∆t = 0) is defined as the mean of the first four recapture probabilities with the smallest release times of a given data set (e.g. see Fig. 2(b)), where the error bars are due to normal statistics. Solid line is a theoretical fit to the data. Inset shows a schematic of the temporal experiment sequence. (c) The temperature T plotted as a function of Umin. Solid line corresponds to a constant temperature of 33 µK. Grey bar corresponds to the standard deviation of the spread in the measured temperatures for the minimum trap depth Umin ∼ 2.8 mK. (d) The calculated mean energy E of the truncated Boltzmann distribution shown as a function of Umin. The solid line is a theoretical prediction of the mean energy of a truncated Boltzmann distribution with a temperature of 33 µK.

FIG. 6 :

 6 FIG. 6: (Color online) Investigation of the adiabaticity during the change of the optical trap depth. (a) Schematic of the experimental sequence (see text). (b) Probability of recapture for zero release time Prr(∆t = 0), as a function of the minimum trap depth Umin. The solid line is a theoretical fit to the data. Inset shows a schematic of the temporal experimental sequence. (c) Temperature T of the Boltzmann distribution plotted as a function of Umin. The solid line is a theoretical prediction of the adiabatic scaling law T / √ U = const. The dashed line is the extension of the solid line to minimum trap depths at which the density of states deviates from that of a three dimensional harmonic trap.

Acknowledgments

We acknowledge support from the Integrated Project 'SCALA' which is part of the European IST/FET/QIPC Program, from Institut Francilien des Atomes Froids (IFRAF), from ARDA/DTO, and from PPF-ENS. Y. R. P. Sortais was supported by IFRAF and CNRS fellowship, and A. M. Lance also by IFRAF and by a Marie Curie fellowship.


J Beugnon

C Tuchendler

H Marion

A Gaëtan

Y Miroshnychenko

Y R P Sortais

A M Lance

M P A Jones

G Messin

A Browaeys

P Grangier

Two-dimensional transport and transfer of a single atomic qubit in optical tweezers

Quantum computers have the capability of outperforming their classical counterparts for certain computational problems [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]. Several scalable quantum computing architectures have been proposed. An attractive architecture is a large set of physically independant qubits, arranged in three spatial regions where (i) the initialized qubits are stored in a register, (ii) two qubits are brought together to realize a gate, and (iii) the readout of the qubits is performed [START_REF] Kielpinski | Architecture for a large-scale ion-trap quantum computer[END_REF][START_REF] Calarco | Quantum computations with atoms in optical lattices: Marker qubits and molecular interactions[END_REF]. For a neutral atombased architecture, a natural way to connect these regions is to use optical tweezers to move qubits within the system. In this letter we demonstrate the coherent transport of a qubit, encoded on an atom trapped in a sub-micron tweezer, over a distance typical of the separation between atoms in an array of optical traps [START_REF] Bergamini | Holographic generation of microtrap arrays for single atoms using a programmable phase modulator[END_REF][START_REF] Dumke | Micro-optical Realization of Arrays of Selectively Addressable Dipole Traps: A Scalable Configuration for Quantum Computation with Atomic Qubits[END_REF][START_REF] Miroshnychenko | Quantum engineering: An atom-sorting machine[END_REF]. Furthermore, we transfer a qubit between two tweezers, and show that this manipulation also preserves the coherence of the qubit.

In the quest for an implementation of a quantum computer, scalability is a major concern. In the trapped ion approach (see e.g. [START_REF] Chiaverini | Implementation of the semiclassical quantum Fourier transform in a scalable system[END_REF]), a lot of effort is being devoted to building arrays of small ion traps [START_REF] Seidelin | Microfabricated Surface-Electrode Ion Trap for Scalable Quantum Information Processing[END_REF], and to moving ion qubits whilst avoiding heating and decoherence [START_REF] Rowe | Transport of quantum states and separation of ions in a dual RF ion trap[END_REF]. Neutral atoms also offer promising properties for the realization of large quantum registers. For example, one-or two-dimensional adressable arrays of dipole traps have been demonstrated using holographic techniques [START_REF] Bergamini | Holographic generation of microtrap arrays for single atoms using a programmable phase modulator[END_REF], micro-fabricated elements [START_REF] Dumke | Micro-optical Realization of Arrays of Selectively Addressable Dipole Traps: A Scalable Configuration for Quantum Computation with Atomic Qubits[END_REF], or active rearrangement of single atoms [START_REF] Miroshnychenko | Quantum engineering: An atom-sorting machine[END_REF][START_REF] Schrader | Neutral Atom Quantum Register[END_REF]. An alternative approach is to use the Mott insulator transition to initialize a three-dimensional register by loading a Bose-Einstein condensate into an optical lattice [START_REF] Greiner | Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms[END_REF]. Recent progress has shown subwavelength addressability in such a system [START_REF] Lee | Subwavelength addressability and spindependent transport in a double-well optical lattice[END_REF]. To perform quantum computations, however, an additional key feature is the ability to perform the gate between two arbitrary qubits of the register.

Here we demonstrate a scheme where a neutral atom qubit is transfered between two moving tweezers ("register" to "moving head"), and then transported towards an interaction zone where the two-qubit gate should be implemented [START_REF] Jaksch | Entanglement of Atoms via Cold Controlled Collisions[END_REF][START_REF] Brennen | Quantum Logic Gates in Optical Lattices[END_REF][START_REF] Jaksch | Fast Quantum Gates for Neutral Atoms[END_REF][START_REF] Dorner | Quantum logic via optimal control in holographic dipole traps[END_REF]. We show that these manipulations of the external degrees of freedom preserve the coherence of the qubit, and do not induce any heating. This transport in a moving tweezer is a promising alternative to the recently demonstrated transport of qubits in "optical conveyor belts" [START_REF] Miroshnychenko | Quantum engineering: An atom-sorting machine[END_REF][START_REF] Kuhr | Coherence Properties and Quantum State Transportation in an Optical Conveyor Belt[END_REF], or in state-dependent moving optical lattices [START_REF] Mandel | Coherent Transport of Neutral Atoms in Spin-Dependent Optical Lattice Potentials[END_REF]. Altogether, these results pave the way towards a scalable neutral atom quantum computing architecture.

In our experiment, we trap a single rubidium 87 atom in an optical dipole trap created by a tightly focused laser beam [START_REF] Schlosser | Collisional Blockade in Microscopic Optical Dipole Traps[END_REF][START_REF] Sortais | Diffraction-limited optics for single-atom manipulation[END_REF]. As described in detail in [START_REF] Jones | Fast quantum state control of a single trapped neutral atom[END_REF], the qubit is encoded onto the |0 = |F = 1, M = 0 and |1 = |F = 2, M = 0 hyperfine ground states separated by ω hf ≈ 6.8 GHz. We initialize the qubit in state |0 by optically pumping the atom. We drive single-qubit operations by a Raman transition using two phased-locked laser beams, one of which being the dipole trap. The internal dephasing time of the qubit, measured by Ramsey interferometry, is ≈ 630 µsec. This time is mostly limited by the residual motion of the atom in the trap that leads to a fluctuation of the frequency of the qubit transition. This dephasing can be reversed by applying a spin echo technique where a π pulse is inserted between the two π/2 pulses of the Ramsey sequence. Using this technique we measure an irreversible dephasing time of 34 ms.

The experimental setup for the moving tweezer is represented in figure 1. We reflect the dipole trap beam off a mirror affixed to a tip-tilt platform prior to the large numerical aperture lens. The platform is actuated by piezo-electrical transducers and can rotate with a maximal angle of 2.5 mrad in both the horizontal and vertical directions. We have measured the position of the dipole trap for different angles of the platform by observing the position of the atom on the CCD camera. The maximal angle corresponds to a total displacement of the tweezer of 18 ± 1 µm. This motion is two-dimensional, as demonstrated in figure 1.

We first analyze the influence of a displacement of the tweezer on the external degrees of freedom of the qubit. For this purpose, we measure the temperature of the single atom in the tweezer using a release and recapture technique [START_REF] Lett | Observation of Atoms Laser Cooled below the Doppler Limit[END_REF] (see Methods). In the absence of motion, the temperature of the atom is 56.0 ± 1.4 µK. We repeat this measurement after moving the tweezer by a total distance of 360 µm, consisting of 20 round trips of 18 µm, along the z-axis. Each round trip lasts a time of 6 ms. We do not find any measurable loss due to this transport. After the motion, we measure a temperature of 54.8 ± 1.6 µK. As the energy difference between two vibrational quanta in the radial direction is 4 µK, this temperature is compatible with no change in the radial vibrational state. This absence of motional heating is a crucial feature for entanglement schemes based on controlled collisions [START_REF] Dorner | Quantum logic via optimal control in holographic dipole traps[END_REF][START_REF] Mandel | Controlled collisions for multi-particle entanglement of optically trapped atoms[END_REF] and results from the adiabaticity of the displacement. A motion is adiabatic if the acceleration a fulfills maσ ≪ hΩ (Ω is the oscillation frequency of the atom, m its mass and σ the extension of the ground state wave function [START_REF] Browaeys | Transport of atoms in a quantum conveyor belt[END_REF]). This gives a maximum acceleration of ≈ 10 4 m/sec 2 , much larger than the experimentally measured ≈ 15 m/sec 2 .

We secondly study the influence of the motion on the coherence of the qubit. As the duration of the transport is larger than the dephasing time of the qubit (630 µs), we apply the spin echo sequence to rephase the qubit [START_REF] Jones | Fast quantum state control of a single trapped neutral atom[END_REF]. The time sequence of the experiment is shown in figure 2. Figure 3(a) shows the amplitude of the spin-echo fringes for various trap displacements along the y-axis. This amplitude is constant when we scan the tweezer over all the transverse field of the objective. This demonstrates that the motion does not affect the internal coherence of the qubit. We observe the same behavior when we move the tweezer along the z-axis.

We also observe a phase shift of the spin-echo fringes, as shown in figure 3(b). This is a signature that the two states of the qubit dephase with respect to each other during the motion (in a reproducible way), despite the presence of the rephasing π pulse. We attribute this phase shift to the asymmetry of the trajectory during the first and second part of the round trip displacement. We have modelled this effect and found a good agreement with the data (see Methods section). This understanding of the phase evolution of the qubit during the motion is crucial for a possible implementation in a quantum computer where qubit phases need to be controlled.

With the idea of transfering an atom from the "register" to the "moving head", we have investigated the transfer of a qubit from one tweezer to a second one. For this experiment, the two traps are superimposed and the positions of both tweezers are fixed. The experimental sequence is shown in figure 4(a). We load an atom in the first tweezer, transfer it to a second tweezer, and transfer it back to the first tweezer, with no measurable loss. When the two traps have the same depth, we measure a temperature of the atom after the double transfer of 56.3 ± 1.8 µK, while the temperature with no transfer is 53.4 ± 1.4 µK. Therefore, the transfer does not induce any significant motional heating. We analyze the influence of the transfer on the coherence of the qubit by inserting the double transfer between the two π/2 pulses of a Ramsey sequence, as shown in figure 4(a). Figure 4 presents the amplitude and the phase of the Ramsey fringes after this sequence for depths of the second tweezer ranging from 0.2 mK to 0.6 mK. This transfer does not affect the amplitude of the Ramsey signal when the depth of the second trap is varied, thus showing that the coherence is robust against the transfer between the two traps.

Figure 4(c) shows that the phase of the Ramsey fringes, varies linearly with respect to the depth of the second trap. This is explained by the differential potential experienced by the two states, which is proportional to the depth of the trap U . If the depths of the two traps differ by ∆U , the Ramsey fringes are shifted after a holding time T by a phase proportional to ∆U T /h, with respect to the situation where no transfer is applied. This phase is thus a useful tool to make sure that the two traps are identical.

As a conclusion we have shown that we can move and transfer a single qubit between two tweezers with no measurable motional heating. We have also shown no loss of coherence of the atomic qubit under transfer and displacement. In combination with our holographic array of dipole traps [START_REF] Bergamini | Holographic generation of microtrap arrays for single atoms using a programmable phase modulator[END_REF], and efficient single-qubit operation and readout [START_REF] Jones | Fast quantum state control of a single trapped neutral atom[END_REF], we have made a first step towards designing a scalable architecture of a quantum computer based on neutral atoms.

Methods:

Temperature measurement:

After trapping a single atom, we switch off the dipole trap for a time adjustable between 1 and ∼ 30 µsec. We then turn the trap back on and check for the presence of the atom. We repeat this sequence 100 times for each release time and calculate the probability to recapture the atom after the corresponding time of flight. We compare our data with a 3-dimensional Monte Carlo simulation, taking into account the potential produced by the Gaussian trapping beam, and assuming a thermal distribution of the position and the velocity of the atom at the beginning of the time of flight. The error bar of this fitted temperature corresponds to one standard deviation in the least square-based fit.

Phase shift during the motion:

The hyperfine splitting of 6.8 GHz means that the dipole trap detuning is slightly larger for |0 than for |1 , giving rise to a small differential lightshift. Therefore the qubit transition frequency is ω hf + ηU/h, with U the depth of the dipole trap and η = 7 × 10 -4 for our trap. If the tweezer is not moved, the π-pulse compensates for the phase accumulated during the two parts of the motion. When the tweezer is moved off axis, the waist of the beam increases sligthly, resulting in a shallower trap. Figure 2(a) shows that with the tweezer starting on axis, the atom spends more time far from the axis where the dipole trap is shallower, whereas on the way back it spends more time around the axis where the dipole trap is stronger. The average depth is then different for the two parts of the motion, and so are the phases. As the phase of the spin-echo signal is the difference of the phases accumulated during the two periods of the motion, it is expected to vary as we move the tweezer further away off axis. We have modelled this effect by calculating the dephasing accumulated during the transport, taking into account the actual displacement of the tweezer from the sensor curve and the measured Rabi frequencies for different positions of the tweezer off axis. The result of this model is shown as circles in figure 3(b) and is consistent with the data.

FIG. 1 :

 1 FIG.1: Experimental setup. A large numerical aperture lens focuses two independent dipole trap beams at 810 nm each to a size of 0.9 micrometers. An optical power of 400 µW results into a trap depth of 500 µK and oscillation frequencies of 81 kHz and 15 kHz, in the radial and axial directions respectively. The two trapping lasers have the same linear polarization and their frequencies are separated by 10 MHz to avoid interferences. The moving tweezer is displaced by rotating a tip-tilt platform. The same large numerical aperture lens is used to collect the fluorescence light at 780 nm from the atom. This fluorescence light is separated from the trapping light by the dichroic mirror and sent to a singlephoton counter module and a CCD camera. The insert shows a fluorescence picture of an atom moved along an elliptical trajectory in the y-z plane. The picture is a summation of 5 images taken at different times during the motion.

FIG. 2 :

 2 FIG. 2: Principle of the moving qubit experiment. (a) Position of the atom when the pulses of the spin echo sequence are applied. Starting at the initial position, the first pulse prepares the atom in a superposition (|0 + |1 )/ √ 2. The tweezer is then moved along the y-axis to the target position and the π pulse is applied. Finally, the tweezer is brought back to its initial position and the coherence is checked by applying a second π/2 pulse and measuring the state of the qubit. (b) Example of displacement of the tweezer versus time. The signal is obtained from the sensor attached to the tip-tilt platform, which is converted into a distance travelled by the tweezer. The π/2 and π pulses, being 2 and 4 µs long respectively, are not to scale.

FIG. 3 :

 3 FIG. 3: Results of the moving qubit experiment. Figure (a) shows the amplitude of the spin echo signal versus the amplitude of the displacement. The error bars are the root mean square (RMS) uncertainty obtained from the fit of the fringes. The 60% contrast is a result of the damping of the fringes after 6 ms [21]. The dashed line represents the average of the data for no trap displacement. Figure (b) shows the phase shift versus the amplitude of the displacement. The triangles are the data and the circles are the calculated value of the dephasing based on the model described in the Methods section.

FIG. 4 :

 4 FIG. 4: Experiment on the transfer of the qubit between two tweezers. Figure (a) details the time sequence. A π/2 pulse is applied when the atom is initially in trap 1, which has a depth of 500 µK. The atom is transferred to trap 2 in a time of 20 µs. After a time of 160 µs, the atom is transferred back to trap 1 and a second π/2 pulse is applied. Figure (b) shows the amplitude of the spin echo signal as a function of the depth of the trap the atom is transferred into. The error bars are the RMS uncertainty obtained from the fit of the fringes.Figure (c) displays the phase of the Ramsey oscillations for different depths of the second trap. The solid line is a linear fit to the data. The dashed lines in (b) and (c) correspond to no transfer.

  FIG. 4: Experiment on the transfer of the qubit between two tweezers. Figure (a) details the time sequence. A π/2 pulse is applied when the atom is initially in trap 1, which has a depth of 500 µK. The atom is transferred to trap 2 in a time of 20 µs. After a time of 160 µs, the atom is transferred back to trap 1 and a second π/2 pulse is applied. Figure (b) shows the amplitude of the spin echo signal as a function of the depth of the trap the atom is transferred into. The error bars are the RMS uncertainty obtained from the fit of the fringes.Figure (c) displays the phase of the Ramsey oscillations for different depths of the second trap. The solid line is a linear fit to the data. The dashed lines in (b) and (c) correspond to no transfer.

Acknowledgments: We would like to thank W.D. Phillips, T. Porto, I. Deutsch, P. Jessen for stimulating discussions. We acknowledge financial support from IFRAF, ARDA/DTO and the European Integrated project SCALA. LCFIO is CNRS UMR8501. M.P.A. Jones and A.M. Lance are supported by Marie Curie Fellowships. A. Gaëtan is supported by a DGA Fellowship.


Flavia Bonomo
email: fbonomo@dc.uba.ar

Ivo Koch
email: ivo.koch@gmail.com

Pablo Torres
email: ptorres@fceia.unr.edu.ar

Mario Valencia-Pabon

k-tuple chromatic number of the cartesian product of graphs *

Keywords: k-tuple colorings, Cartesian product of graphs, Kneser graphs, Cayley graphs, Hom-idempotent graphs

A k-tuple coloring of a graph G assigns a set of k colors to each vertex of G such that if two vertices are adjacent, the corresponding sets of colors are disjoint. The k-tuple chromatic number of G, χ k (G), is the smallest t so that there is a k-tuple coloring of G using t colors. It is well known that χ(G H) = max{χ(G), χ(H)}. In this paper, we show that there exist graphs G and H such that χ k (G H) > max{χ k (G), χ k (H)} for k ≥ 2. Moreover, we also show that there exist graph families such that, for any k ≥ 1, the k-tuple chromatic number of their cartesian product is equal to the maximum k-tuple chromatic number of its factors.

Introduction

A classic coloring of a graph G is an assignment of colors (or natural numbers) to the vertices of G such that any two adjacent vertices are assigned different colors. The smallest number t such that G admits a coloring with t colors (a t-coloring) is called the chromatic number of G and is denoted by χ(G). Several generalizations of the coloring problem have been introduced in the literature, in particular, cases in which each vertex is assigned not only a color but a set of colors, under different restrictions. One of these variations is the k-tuple coloring introduced independently by Stahl [START_REF] Stahl | n-Tuple colorings and associated graphs[END_REF] and Bollobás and Thomason [START_REF] Bollobás | Set colourings of graphs[END_REF]. A k-tuple coloring of a graph G is an assignment of k colors to each vertex in such a way that adjacent vertices are assigned distinct colors. The k-tuple coloring problem consists into finding the minimum number of colors in a k-tuple coloring of a graph G, which we denote by χ k (G). The cartesian product G H of two graphs G and H has vertex set V (G)×V (H), two vertices being joined by an edge whenever they have one coordinate equal and the other adjacent. This product is commutative and associative up to isomorphism. There is a simple formula expressing the chromatic number of a cartesian product in terms of its factors:

χ(G H) = max{χ(G), χ(H)}. (1)
The identity (1) admits a simple proof first given by Sabidussi [START_REF] Sabidussi | Graphs with given group and given graph-theoretical properties[END_REF]. The Kneser graph K(m, n) has as vertices all n-element subsets of the set [m] = {1, . . . , m} and an edge between two subsets if and only if they are disjoint. We will assume in the rest of this work that m ≥ 2n, otherwise K(m, n) has no edges. The Kneser graph K(5, 2) is the well known Petersen Graph. Lovász [START_REF] Lovász | Kneser's conjecture, chromatic number and homotopy[END_REF] showed that χ(K(m, n)) = m -2n + 2. The value of the k-tuple chromatic number of the Kneser graph is the subject of an almost 40-year-old conjecture of Stahl [START_REF] Stahl | n-Tuple colorings and associated graphs[END_REF] which asserts that: if k = qn -r where q ≥ 0 and 0 ≤ r < n, then χ k (K(m, n)) = qm -2r. Stahl's conjecture has been confirmed for some values of k, n and m [START_REF] Stahl | n-Tuple colorings and associated graphs[END_REF][START_REF] Stahl | The multichromatic numbers of some Kneser graphs[END_REF]. An homomorphism from a graph G into a graph H, denoted by G → H, is an edgepreserving map from V (G) to V (H). It is well known that an ordinary graph coloring of a graph G with m colors is an homomorphism from G into the complete graph K m . Similarly, an n-tuple coloring of a graph G with m colors is an homomorphism from G into the Kneser graph K(m, n).

A graph G is said hom-idempotent if there is a homomorphism from G G → G. We denote by G → H if there exists no homomorphism from G to H.

The clique number of a graph G, denoted by ω(G), is the maximum size of a clique in G (i.e., a complete subgraph of G). Clearly, for any graphs G and H, we have that χ(G) ≥ ω(G) (and so, χ k (G) ≥ χ k (K ω(G) ) = kω(G)) and, if there is an homomorphism from G to H then, χ(G) ≤ χ(H) (and so, χ k (G) ≤ χ k (H)).

A stable set S ⊆ V is a subset of pairwise non adjacent vertices of G. The stability number of G, denoted by α(G), is the largest cardinality of a stable set in G. Let m ≥ 2n.

An element i ∈ [m] is called a centre of a stable set S of the Kneser graph K(m, n) if it lies in each n-set in S. Lemma 1 (Erdős-Ko-Rado [4]). If m > 2n, then α(K(m, n)) = m-1 n-1 . An independent set of K(m, n) with size m-1
n-1 has a centre i, for some i ∈ [m]. Lemma 2 (Hilton-Milner [START_REF] Hilton | Some intersections theorems for systems of finite sets[END_REF]). If m ≥ 2n, then the maximum size of an stable set in K(m, n) with no centre is equal to

1 + m-1 n-1 -m-n-1 n-1
.

A graph G = (V, E) is vertex transitive if its automorphism group acts transitively on V , that is, for any pair of distinct vertices of G there is an automorphism mapping one to the other one. It is well known that Kneser graphs are vertex transitive graphs [START_REF] Godsil | Algebraic graph theory[END_REF].

Lemma 3 (No-Homomorphism Lemma, Albertson-Collins [START_REF] Albertson | Homomorphisms of 3-chromatic graphs[END_REF]). Let G, H be graphs such that

H is vertex transitive and G → H. Then, α(G)/|V (G)| ≥ α(H)/|V (H)|.
In this paper, we show that equality (1) does not hold in general for k-tuple colorings of graphs. In fact, we show that for some values of k ≥ 2, there are Kneser graphs K(m, n)

for which χ k (K(m, n) K(m, n)) > χ k (K(m, n))
. Moreover, we also show that there are families of graphs for which equality (1) holds for k-tuple colorings of graphs for any k ≥ 1. As far as we know, our results are the first ones concerning the k-tuple chromatic number of cartesian product of graphs.

2 Cartesian products of Kneser graphs Lemma 4. Let G be a graph and let k > 0. Then,

χ k (G G) ≤ kχ(G). Proof. Clearly, χ k (G G) ≤ kχ(G G).
However, by equality (1) we know that χ(G G) = χ(G), and thus the lemma holds.

Corollary 1. χ k (K(m, n) K(m, n)) ≤ kχ(K(m, n)) = k(m -2n + 2).
Larose et al. [START_REF] Larose | On normal Cayley graphs and Hom-idempotent graphs[END_REF] showed that no connected Kneser graph

K(m, n) is hom-idempotent, that is, for any m > 2n, there is no homomorphism from K(m, n) K(m, n) to K(m, n). Lemma 5 ([8]). Let m > 2n. Then, K(m, n) K(m, n) → K(m, n).
Concerning the k-tuple chromatic number of some Kneser graphs, Stahl [START_REF] Stahl | n-Tuple colorings and associated graphs[END_REF] showed the following results.

Lemma 6 ([11]). If 1 ≤ k ≤ n, then χ k (K(m, n)) = m -2(n -k). Lemma 7 ([11]). χ k (K(2n + 1, n)) = 2k + 1 + k-1 n , for k > 0. Lemma 8 ([11]). χ rn (K(m, n)) = rm, for r > 0 and m ≥ 2n.
By using Lemma 8 we have the following result.

Lemma 9. Let m > 2n. Then, χ n (K(m, n) K(m, n)) > χ n (K(m, n)).
Proof. By Lemma 8 when r = 1, we have that

χ n (K(m, n)) = m. If χ n (K(m, n) K(m, n)) = m, then there exists an homomorphism from the graph K(m, n) K(m, n) to K(m, n) which contradicts Lemma 5.
By Lemma 6, Lemma 9 and by using Corollary 1, we have that,

Corollary 2. Let n ≥ 2. Then, 2n+2 ≤ χ n (K(2n+1, n) K(2n+1, n)) ≤ 3n.
In particular, when n = 2, we have that χ 2 (K(5, 2) K(5, 2)) = 6.

In the case k = 2 we have by Lemma 9, Lemma 6 and by Corollary 1, the following result.

Corollary 3. Let q > 0. Then, q + 4 ≤ χ 2 (K(2n + q, n) K(2n + q, n)) ≤ 2q + 4.

By Corollary 3, notice that in the case when k = n = 2 and q ≥ 1, we must have that χ 2 (K(q + 4, 2) K(q + 4, 2)) > q + 4, otherwise there is a contradiction with Lemma 5. This provides a gap of one unity between the 2-tuple chromatic number of the graph K(q + 4, 2) K(q + 4, 2) and the graph K(q + 4, 2). In the next Lemma 11, we will show that such a gap can be as large as desired. However, first we need to introduce the following.

It is well known that the chromatic index of a complete graph K 2n (i.e. the minimum number of colors needed to color the edges of K 2n such that any two incident edges be assigned different colors) on 2n vertices is equal to 2n -1 (see [START_REF] Berge | Graphs and Hypergraphs[END_REF]), where each color class i (i.e. the subset of pairwise non incident edges colored with color i) has size n. Therefore, using this fact, we obtain the following result.

Lemma 10. Let q ≥ 1. Then, the set of vertices of the Kneser graph K(2q + 4, 2) can be partitioned into 2q + 3 disjoint cliques, each one with size q + 2.

Proof. Notice that there is a natural bijection between the vertex set of K(2q + 4, 2) and the edge set of the complete graph K 2q+4 with vertex set [2q + 4]: each vertex {i, j} in K(2q + 4, 2) is mapped to the edge {i, j} in K 2q+4 . Now, there is a (2q + 3)-edge coloring of K 2q+4 where each class color is a set of pairwise non incident edges with size q + 2. Notice that two edges e, e ∈ K 2q+4 are non incident edges if and only if e ∩ e = ∅. Therefore, a class color of the edge-coloring of K 2q+4 represents a clique of K(m, n).

Lemma 11. Let q > 0. Then, χ 2 (K(2q + 4, 2) K(2q + 4, 2)) ≥ 2q + 2 3 q + 5. Proof. First, recall that a stable set X in K(2q + 4, 2) has size at most 2q + 3 if X has centre (see Lemma 1) and |X| ≤ 1 + (2q + 4 -1) -(2q + 4 -2 -1) = 3 if X has no centre (see Lemma 2). Besides, by Lemma 10, observe that the vertex set of K(2q + 4, 2) can be partitioned in 2q + 3 sets S 1 , . . . , S 2q+3 such that each S i induces a K q+2 for i = 1, . . . , 2q + 3. Consider the subgraph H i of K(2q + 4, 2) K(2q + 4, 2) induced by S i × V (K(2q + 4, 2)) for i = 1, . . . , 2q + 3. Let I be a stable set in K(2q + 4, 2) and

I i = I ∩ H i for i = 1, . . . , 2q + 3. Then I v i = I i ∩ ({v} × V (K(2q + 4, 2)
)) is a stable set in K(2q + 4, 2) for each v ∈ S i and i = 1, . . . , 2q + 3. Now, assume w.l.o.g that r (r ≤ q +2) stable sets I 1 i , . . . , I r i have distinct centre j 1 , . . . , j r , respectively (the case when two of these stable sets have the same centre can be easily reduced to this case). Let W be the set of subsets with size two of {j 1 , . . . , j r }. Therefore, for all m ∈ {1, . . . , r}, I m i -W has size at most 2q + 3 -(r -1) = 2q + 4 -r since each centre j m belongs to r -1 elements in W . Besides, each element of W belongs to exactly one set I m i for m ∈ {1, . . . , r}, since S i induces a complete subgraph. Then,

|I 1 i ∪ . . . ∪ I r i | ≤ ( r m=1 |I m i -W |) + |W | ≤ r(2q + 4 -r) + r(r-1) 2 . Next, each remaining stable set (if exist) I r+1 i , . . . , I q+2 i has no centre, then |I d i | ≤ 3 for all d ∈ {r+1, . . . , q+2}. Thus, |I i | ≤ r(2q+4-r)+ r(r-1) 2 +3(q+2-r) = -r 2 2 +r(2q+ 1
2 )+3(q+2). Since the last expression is non decreasing for r ∈ {1, . . . , q + 2}, we have that

|I i | ≤ -(q+2) 2 2 + (q + 2)(2q + 1 2 ) + 3(q + 2) = (q + 2)( 3 2 q + 5 2 ).
Therefore, |I i | ≤ (q + 2)( 3 2 q + 5 2 ) for every i = 1, . . . , 2q + 3. Since |I| = 2q+3 i=1 |I i |, it follows that |I| ≤ (2q + 3)(q + 2)( 3 2 q + 5 2 ) and thus, α(K(2q + 4, 2) K(2q + 4, 2)) ≤ (2q + 3)(q + 2)( 3 2 q + 5 2 ).

Let t < 2q 2 +18q+24 3q+5 .

Assume that exists a 2-tuple coloring of the graph K(2q + 4, 2) K(2q + 4, 2)) with 2q + t colors. Therefore, there exists an homomorphism from K(2q + 4, 2) K(2q + 4, 2) to K(2q + t, 2). Now, from the well known no-homomorphism Lemma 3, we have that,

α(K(2q + 4, 2) K(2q + 4, 2)) ≥ α(K(2q+t,2))|V (K(2q+4,2) K(2q+4,2))| |V (K(2q+t,2))| . Then, α(K(2q + 4, 2) K(2q + 4, 2)) ≥ (2q+t-1).(2q+4) 2 (2q+3) 2 2 4(2q+t)(2q+t-1) = (2q+4) 2 (2q+3) 2 2(2q+t) . Let us see that (2q+4) 2 (2q+3) 2 2(2q+t) is greater than (2q+3)(q+2)( 3 2 q+ 5
2 ), which is a contradiction.

To this end, observe that if t < 2q 2 +18q+24 3q+5 , then

(2q + 3)(q + 2)( 3 2 q + 5 2 )(2q + t) = (2q + 3)(q + 2)((3q 2 + 5q) + ( 3 2 q + 5 2 )t) < (2q + 3)(q + 2)((3q 2 + 5q) + (q 2 + 9q + 12)) = (2q + 3)(q + 2)(4q 2 + 14q + 12)) = (2q+4) 2 (2q+3) 2 2 .
Therefore, χ 2 (K(2q + 4, 2) K(2q + 4, 2)) ≥ 2q + t + 1.

Claim 1. Let q > 0 be an integer. Then, 2q 2 +18q+24 3q+5 is not an integer.

Proof. By polynomial division, we have that 2q 2 + 18q + 24 = (3q + 5)( 23 q + 44 9 ) -4 9 . If (2q 2 + 18q + 24)/(3q + 5) is an integer, then k = ( 23 q + 44 9 ) -4 9(3q+5) is also an integer. Multiplying by 9 both terms in the last equality we have, 6q + 44 = 9k + 4 3q+5 . As q > 0 then, 0 < 4 3q+5 < 1 contradicting the assumption that k is an integer.

Finally, since 2 3 q + 4 < 2q 2 +18q+24 3q+5 and, by previous Claim 1, 2q 2 +18q+24 3q+5 is not an integer,

then 2 3 q + 4 < 2q 2 +18q+24 3q+5
and thus, we have finally that, 2q + 2 3 q + 5 ≤ χ 2 (K(2q + 4, 2)). As a corollary of Lemma 11 and by Corollary 1, we obtain the following result.

Corollary 4. χ 2 (K(6, 2) K(6, 2)) = 8. Theorem 1. Let k > n and let t = χ k (K(m, n) K(m, n)), where m > 2n. Then, either t > m + 2(k -n) or t < m + (k -n).
Proof. Suppose that m+(k-n) ≤ t ≤ m+2(k-n). Therefore, there exists an homomorphism K(m, n) K(m, n) → K(t, k). Now, Stahl [START_REF] Stahl | n-Tuple colorings and associated graphs[END_REF] showed that there is an homomorphism K(m, n) → K(m -2, n -1) whenever n > 1 and m ≥ 2n. Moreover, it's easy to see that there is an homomorphism K(m, n) → K(m -1, n -1). By applying the former homomorphism t -(m + (k -n)) times to the graph K(t, k) we obtain an homomorphism

K(t, k) → K(2(m + k -n) -t, 2k + m -n -t).
Finally, by applying 2k + m -t -2n times the latter homomorphism to the graph K(2(m + k -n) -t, 2k + m -n -t) we obtain an homomorphism K(2(m+k -n)-t, 2k +m-n-t) → K(m, n). Therefore, by homomorphism composition, K(m, n) K(m, n) → K(m, n) which contradicts Lemma 5.

We can also obtain a lower bound for the k-tuple chromatic number of the graph K(m, n) K(m, n) in terms of the clique number of K(m, n). In fact, notice that

ω(K(m, n) K(m, n)) = ω(K(m, n)) = m
n . Thus, we have the following result.

Theorem 2. Let k > n. Then, χ k (K(m, n) K(m, n)) ≥ kω(K(m, n)) = k m n . In particu- lar, if n divides m then, χ k (K(m, n) K(m, n)) ≥ m + (k -n) m n .
3 Cases where

χ k (G H) = max{χ k (G), χ k (H)} Theorem 3. Let G and H be graphs such that χ(G) ≤ χ(H) = ω(H). Then, χ k (G H) = max{χ k (G), χ k (H)}.
Proof.

(G H) = χ k (G) = χ k (H).
In the case when G is not a bipartite graph, we have the following results.

An automorphism σ of a graph G is called a shift of G if {u, σ(u)} ∈ E(G) for each u ∈ V (G) [START_REF] Larose | On normal Cayley graphs and Hom-idempotent graphs[END_REF]. In other words, a shift of G maps every vertex to one of its neighbors.

(G H) = max{χ k (G), χ k (H)}. Proof. Let A ∪ B be a bipartition of the vertex set of H. Let f be a k-tuple coloring of G with χ k (G) colors. Clearly, χ k (G) ≥ χ k (H). We define a k-tuple coloring ρ of G H with χ k (G) colors as follows: for any vertex (u, v) of G H with u ∈ G and v ∈ H, define ρ((u, v)) = f (u) if v ∈ A, and ρ((u, v)) = f (σ(u)) if v ∈ B.
We may also deduce the following direct result.

Theorem 5. Let G be an hom-idempotent graph an let H be a subgraph of G. Thus,

χ k (G H) = max{χ k (G), χ k (H)} = χ k (G).
Let A be a group and S a subset of A that is closed under inverses and does not contain the identity. The Cayley graph Cay(A, S) is the graph whose vertex set is A, two vertices u, v being joined by an edge if u -1 v ∈ S. If a -1 Sa = S for all a ∈ A, then Cay(A, S) is called a normal Cayley graph. Lemma 12 ([6]). Any normal Cayley graph is hom-idempotent.

Note that all Cayley graphs on abelian groups are normal, and thus hom-idempotents. In particular, the circulant graphs are Cayley graphs on cyclic groups (i.e., cycles, powers of cycles, complements of powers of cycles, complete graphs, etc). By Theorem 5 and Lemma 12 we have the following result. Theorem 6. Let Cay(A, S) be a normal Cayley graph and let Cay(A , S ) be a subgraph of Cay(A, S), with A ⊆ A and S ⊆ S. Then, χ k (Cay(A, S) Cay(A , S )) = max{χ k (Cay(A, S)), χ k (Cay(A , S ))}.

Definition 1. Let G be a graph with a shift σ. We define the order of σ as the minimum integer i such that σ i is equal to the identity permutation.

Theorem 7. Let G be a graph with a shift σ of minimum odd order 2s + 1 and let C 2t+1 be a cycle graph, where t ≥ s. Then, χ k (G C 2t+1 ) = max{χ k (G), χ k (C 2t+1 )}.

Proof. Let {0, . . . , 2t} be the vertex set of C 2t+1 , where for 0 ≤ i ≤ 2t, {i, i + 1 mod n} ∈ E(C 2t+1 ). Let G i be the i th copy of G in G C 2t+1 , that is, for each 0 ≤ i ≤ 2t, G i = {(g, i) : g ∈ G}. Let f be a k-tuple coloring of G with χ k (G) colors. We define a k-tuple coloring of G C 2t+1 with χ k (G) colors as follows: let σ 0 denotes the identity permutation of the vertices in G. Now, for 0 ≤ i ≤ 2s, assign to each vertex (u, i) ∈ G i the k-tuple f (σ i (u)). For 2s + 1 ≤ j ≤ 2t, assign to each vertex (u, j) ∈ G j the k-tuple f (u) if j is odd, otherwise, assign to (u, j) the k-tuple f (σ 1 (u)). It's not difficult to see that this is in fact a proper k-tuple coloring of G C 2t+1 .

Theorem 4 .

 4 Let G be a non bipartite graph having a shift σ ∈ AU T (G), and let H be a bipartite graph. Then, χ k

  Let t = ω(H) and let {h 1 , . . . , h t } be the vertex set of a maximum clique K t in H with size t. Clearly, χ k (G) ≤ χ k (H) = χ k (K t ). Let ρ be a k-tuple coloring of H with χ k (H) colors. By equality (1), there exists a t-coloring f of G H. Therefore, the assignment of the k-set ρ(h f ((a,b)) ) to each vertex (a, b) in G H defines a k-tuple coloring of G H with χ k (K t ) colors.Notice that if G and H are both bipartite, then χ k

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