A Fuhrmanek

A M Lance

C Tuchendler

P Grangier

Y R P Sortais
email: yvan.sortais@institutoptique.fr

A Browaeys

Imaging a single atom in a time-of-flight experiment

Keywords: numbers: 37.10.De, 37.10.Gh, 37.10.Vz

We perform fluorescence imaging of a single 87 Rb atom after its release from an optical dipole trap. The time-of-flight expansion of the atomic spatial density distribution is observed by accumulating many single atom images. The position of the atom is revealed with a spatial resolution close to 1 µm by a single photon event, induced by a short resonant probe. The expansion yields a measure of the temperature of a single atom, which is in very good agreement with the value obtained by an independent measurement based on a release-and-recapture method. The analysis presented in this paper provides a way of calibrating an imaging system useful for experimental studies involving a few atoms confined in a dipole trap.

Introduction

Time-of-flight imaging of ultra-cold atomic gases in expansion is a common way to study their properties. It provides a direct measurement of the momentum distribution and is therefore routinely used to extract the temperature of cold thermal samples [1]. It can also give access to spatial density or momentum correlations in atomic ensembles. These features have, for instance, enabled the observation of bunching (anti-bunching) with bosonic (fermionic) atoms [2,3]. They also enable the study of condensed matter phenomena that emerge when confining matter waves in periodic optical potentials [4,5]. Ultimately, one would like to observe the atoms individually in these mesoscopic systems, not only when the atoms are confined but also when they move or are released from the trap in order to access out-of-equilibrium properties. While fluorescence imaging is widely used in experiments to detect single trapped atoms [6,7,8], and sometimes spatially resolve them [9,10,11], fluorescence imaging of freely propagating single atoms has been demonstrated only recently [12]. In that experiment, cold atoms are released from a trap and fall under the gravity through a sheet of light, which is imaged on an intensified CCD camera using efficient collection optics. The presence of an atom is revealed by an individual spot corresponding to the detection of many fluorescence induced photons. The detection efficiency of a single atom is close to unity and the spatial resolution ( 10 µm) is set by the motion of the atom in the light sheet.

In this paper, we present a complementary approach where we demonstrate timeof-flight fluorescence imaging of a single Rb atom in free space, based on single photon detection, with a spatial resolution of ∼ 1 µm. A single atom is first trapped in a microscopic dipole trap and then released in free space where it evolves with its initial velocity. To detect the atom and locate it with the best accuracy possible, we illuminate it with a very short pulse of resonant light and collect the fluorescence on an image intensifier followed by a CCD camera. The presence of the atom is revealed by a single photon event and, for probe pulses as short as 2 µs, the probability to detect an atom in a single shot of probe light is 4.4%. We repeat the experiment until the spatial distribution of the atom is reconstructed with a sufficient signal-to-noise ratio; the accumulation of successive single atom images yields an average result that exhibits the same features as would a single experiment with many non-interacting atoms. Average images recorded for increasing time-of-flights allow us to measure the root mean square (abbreviated rms) velocity of the atomic expansion, and thus the temperature of a single atom. Our method allows us to measure temperatures over a wide range.

The analysis of the time-of-flight of a single atom released from our microscopic dipole trap also serves as a calibration of our imaging system. This calibration will be used in future experiments where we plan to study the behavior of a cloud of a few tens of cold atoms held in the microscopic trap. In this regime the cloud is very dense and light scattering of near-resonant light used for diagnostic purposes may exhibit a collective behavior (see e.g. [13]). It is therefore important to understand the optical response of the imaging system in the single atom case to interpret the images in the multi-atom regime where collective effects may come into play. Moreover, because the atoms can be illuminated right after their release from the dipole trap, our method allows us to explore the properties of the momentum distribution of such a gas in the near-field regime.

The paper is organized as follows: in Sect. 2 we describe the experimental setup. In Sect. 3 we explain the requirements to perform single atom time-of-flight imaging. Section 4 describes the experimental sequence and shows images of a single atom taken by a CCD camera after a variable time-of-flight. We extract the temperature using a standard fit based on an expansion model. Section 5 details the effects that contribute to the spatial resolution of our imaging system. Section 6 re-analyzes the data by using a Monte Carlo simulation taking into account the effects described in Sect. 5. The temperature result is also compared to an independent measurement based on a releaseand-recapture technique. Finally, Sect. 7 examines the noise sources in our imaging system.

Experimental setup

Our experimental setup has been described elsewhere [9] and is summarized in figure 1. Briefly, a single rubidium 87 atom is trapped in a tight optical dipole trap. The dipole trap is produced by focusing a laser beam (λ trap = 850 nm) down to a spot with waist w 0 = 1.1 µm, using a high numerical aperture aspheric lens (NA = 0.5). The trap depth can be as large as 20 mK for a laser power of 80 mW. We use the same lens to collect the fluorescence light (λ fluo = 780 nm), which is sent onto an avalanche photodiode (APD) and an image intensifier † followed by a low noise CCD camera ‡ (see below for more details).

In order to image the atom, we illuminate it with probe light, which consists of two counter-propagating beams (to avoid radiation pressure force) in a σ + -σ -configuration, and is resonant with the (5 2 S 1/2 , F = 2) to (5 2 P 3/2 , F ′ = 3) transition. The saturation parameter of the probe light is s = I/I sat ∼ 1 for each beam. We also superimpose repumping light on the probe beams, tuned to the (5 2 S 1/2 , F = 1) to (5 2 P 3/2 , F ′ = 2) transition.

Requirements for time-of-flight imaging of a single atom

The principle of a time-of-flight experiment is to measure the position of atoms after a period of free expansion. From the rms positions of the atoms, one extracts the rms velocity σ v of the atoms. In our experiment, we measure the position of the atom by illuminating it with resonant laser light and collecting its fluorescence. This method requires that the position of the atom change by less than the resolution of the imaging † Model C9016-22MGAAS from Hamamatsu. The phosphor screen of the intensifier is then imaged onto the CCD camera through a 1 : 1 relay lens. ‡ Model Pixis 1024 from Princeton Instruments. system during the light pulse. In our case, the imaging system is diffraction limited with a resolution σ diff = 0.5 µm. This imposes a pulse duration of τ < σ diff /σ v . Typically, for a rubidium atom at the Doppler temperature (T Doppler ≃ 150 µK), this yields probe pulses as short as 4 µs. For a collection efficiency of ∼ 1% and a scattering rate R ≈ Γ/2 ≃ 2 × 10 7 s -1 (Γ is the line width of the optical transition), the number of detected photons per pixel would approach unity in single shot, which is well below the capabilities of our CCD camera. We solved this issue by inserting a light intensifier in front of the CCD camera (see figure 1). The intensifier acts as a fast shutter (opened during the probe pulse only), and amplifies a single photon event to a level about two orders of magnitude above the noise level of the CCD camera. Using this intensifier, the presence of one atom is revealed by one single photon event (the case of detecting more than one photon emitted by a single atom during the probe pulse is very unlikely).

Experimental sequence and results

The experimental sequence is summarized in figure 2. It starts with loading and cooling a single atom in the dipole trap. Under illumination by the cooling beams, photons scattered by the trapped single atom are partially collected by the same aspheric lens and directed towards the APD. This light is used as a trigger signal for the subsequent time-of-flight sequence. The cooling beams are switched off immediately upon detection of the atom. The single atom is kept in the dipole trap for an extra 30 ms where the trapped atom can be further cooled by adiabatically ramping down the trap depth [14]. We also use this 30 ms interval to let the atoms in the molasses spread out, with all cooling beams having been switched off. This precaution is taken in order to minimize light scattered by the background molasses during the subsequent probe pulse.

After the single atom is trapped and cooled, the dipole trap is switched off and the single atom time-of-flight experiment takes place. We let the single atom fly for a variable time t TOF and then illuminate it by a 2 µs pulse of probe light. At the same time, the intensifier is switched on for 2 µs and the probe-induced fluorescence is collected by the intensified CCD camera. The loading sequence is then started again, in order to prepare for the next time-of-flight experiment. The acquisition of one image for a given time-of-flight is performed by repeating the experimental sequence described above, with a cycle rate of ∼ 0.5 -2 s -1 and accumulating the total fluorescence light on the CCD. When a sufficient number of photons have been detected (typically 100), the CCD chip is read out and the image is displayed. Note that, for each sequence, the CCD receives light only during the 2 µs the intensifier is on. In this way the intensifier also serves as a fast switch, preventing stray light from reaching the CCD during the cooling and trapping phases.

Figure 3 shows typical images taken for time-of-flights as long as 50 µs. The longer the time-of-flight, the lower the peak signal, and the larger the number of accumulations required. For a measured rms size σ ≃ 1 µm (corresponding to the time-of-flight t T OF = 1 µs of image i)), we perform ∼ 3400 sequences, corresponding to 3400 single trapped atoms, and detect 150 photons (this number of photons is extracted from an independent calibration of the intensifier response to a single photon event). This means that the probability to detect a single atom in a single realization of the experiment is 4.4 × 10 -2 when using a 2 µs-probe.

The images are well fitted by a 2D Gaussian model. Within the error bars the images are isotropic. We plot the rms size σ of the expanding "cloud" along one axis versus the time-of-flight t TOF . We fit the data shown in figure 3 by the general form

σ(t TOF ) = σ(0) 2 + σ 2 v t 2 TOF ( 1
)
that gives the rms position of a particle after a time-of-flight t TOF when the initial position and the velocity are taken from distributions with standard deviations σ(0) and σ v . We find σ(0) = 1. of a single atom in the trap being a thermal Maxwell Boltzmann distribution [14], this translates into a temperature T = mσ 2 v /k B = 21(2) µK (m is the mass of a rubidium atom and k B is the Boltzmann constant).

Let us now compare the result for σ(0) to the expected rms radial position of an atom in equilibrium and trapped in a harmonic potential with depth U and transverse size w 0 (at 1/e 2 ), i.e.

σ ⊥ (0) = k B T mω 2 ⊥ , (2)
where ω ⊥ is the radial oscillation frequency of the atom in the trap. With ω ⊥ ≈ 2π × 26 kHz and T = 20 µK, we find σ ⊥ (0) = 0.3 µm, below the diffraction limit of the imaging system. Taking the latter into account, we should thus expect a rms size of 0.6 µm at null time-of-flight, i.e. a factor 1.8 below the actual data.

Spatial resolution of our imaging system

In order to understand the size at t TOF = 0, we investigated experimentally the effects that contribute to the loss in resolution of our imaging system and lead to the measured value σ(0). These effects are listed in table 1 and sum up quadratically to yield a value of 1.1 µm, in agreement with the measure of σ(0) obtained in Sect. 4. The dominant contribution comes from the loss of resolution of the imaging system due to the intensifier. This contribution was measured by imaging an object with a sharp edge and characterizing the blurred edge in the image obtained. The second largest contribution comes from the diffraction limit of the imaging optics, which is due to the numerical aperture of the aspheric lens, and was tested by removing the intensifier and illuminating a trapped atom for 100 ms. In this case, the atom acts as a point source for the imaging system and the associated response on the CCD is well fitted by a Gaussian shape with a size σ diff = 0.5(1) µm.

The thermal distribution contributes for 0.3 µm due to the transverse size of the distribution σ ⊥ (0), and 0.1 µm due to the effect of depth of focus associated to the longitudinal size of the distribution σ (0). We measured this effect of the depth of focus by imaging a single trapped atom for various positions of the trap along the optical axis of the imaging system. Fig. 4 shows the rms size of a Gaussian fit to the data, although for large values of the defocus δz they slightly deviate from a Gaussian. The results tend asymptotically to the expected rms value of a disc with uniform intensity

σ defocus = 1 2 δz × tan α ( 3
)
where α is related to the numerical aperture by sin α = NA. Finally, we analyze the contribution of the movement of the atom during the probe pulse. Firstly, the photons scattering by the probe induces a random walk of the atom, leading to a rms position in the plane perpendicular to the probe beam

σ τ,scatter = 1 3 v rec √ R τ 3/2 (4)
where v rec is the recoil velocity, R is the spontaneous emission rate, and τ is the duration of the probe pulse [15]. Secondly, the atom moves during the probe pulse due to the thermal velocity. An analytical calculation of the associated rms displacement yields

σ τ,thermal = σ v τ / √ 3 (5)
Both contributions ( 4) and ( 5) broaden the image of a single atom when the duration of the probe τ is increased. We tested this effect by increasing τ up to 30 µs, as shown in figure 5. Although negligible for 2 µs probe pulses and atoms at 150 µK (as is the case in figure 5), this effect alone would be comparable to the intensifier response if we were using pulses as long as 20 µs in the perspective of scattering more photons per shot and thus detect single atoms with a larger efficiency.

We confirmed the analysis above by a Monte Carlo simulation that takes into account all the effects mentioned above. It reproduces accurately our experimental data (see figure 4 and figure 5). Here, we note that the simulation indicates a significant deviation from a Gaussian shape for long probe durations or large values of the defocus. Because of the presence of noise in our imaging system, we did not consistently observe significant deviations on the real images and could not calculate any reliable value for the rms size of the images. We thus fitted our images with a Gaussian model and compared it to a Gaussian fit of our simulation. The discrepancy between the analytical expression (5) and the results in figure 5 is an indication of the error that we make by doing so. Note also that we have not included in the model the potential effect of the cooling of the atom by the counter-propagating probe beams.

Temperature results and comparison to release-and-recapture experiments

We now come back to the temperature result obtained by fitting the data using equation ( 1). Among all the effects degrading the resolution, the depth of focus is the (Color Online) rms size of the image of a single trapped atom for various positions of the trap along the optical axis of the imaging system. For small values of the defocus, the image size is given essentially by the response of the intensifier. For large values of the defocus, the main contribution comes from the depth of focus, which scales linearly with defocus according to equation (3) (dashed line). Our data (red circles), which agree well with a Monte Carlo simulation (blue dotted line), result from the convolution of both the optical response of the imaging system and the longitudinal profile of the thermal distribution (blue filled curve) with a width σ (0) = 1.6 µm. The fact that the contribution of equation ( 5) is larger than both experimental and simulated results for long probe durations is due to the significant deviation of the distributions from a Gaussian model. 16) µK (red circles). The simulated results are also shown (blue and red dotted lines respectively). The dashed lines (respectively blue and red) show the results of a simple model neglecting all broadening effects except for the transverse dimension of the thermal distribution, σ ⊥ (0). only one that varies with the time-of-flight as the atom can fly in the direction parallel to the optical axis. Therefore, equation ( 1) is not strictly valid in our case. We now use the Monte Carlo simulation mentioned above to fit the data shown in figure 3. The starting point of this simulation is a thermal distribution with a temperature that we adjust in order to reproduce the data. We find T = 20(2) µK. Not surprisingly, this result is in good agreement with the rough analysis mentioned in Sect. 4, since the effect of the depth of focus is small.

To cross check the temperature measurement, we use an independent method based on a release-and-recapture technique described in detail in reference [14]. This method uses an APD to detect the presence or the absence of the atom in the trap after release for a variable time. Figure 7 shows the results obtained by the release-and-recapture method. A fit to the data yields a temperature T = 19(2) µK in good agreement with the results of the time-of-flight method.

The two methods presented above can be extended to higher temperatures. For example, figure 6 and figure 7 show results obtained for a temperature T ∼ 150 µK, achieved by leaving the dipole trap depth unchanged after loading it with a single atom. Our detection method is therefore applicable over a large range of temperatures with no anticipated limitation in the low temperature range.

Analysis of the noise of the imaging system

We now address the issue of the noise of our imaging system. The peak signal in the time-of-flight image shown in figure 3(i) is 30600 adu in 1 pixel § and corresponds to the detection of ∼ 150 single atoms after 3400 shots of 2 µs probe pulses. Normalized to one shot, the mean peak signal is thus 9 adu in 1 pixel. This should be compared to the background noise, which results from three contributions shown in figure 8: read-out noise from the CCD camera , a background noise contribution from the probe light, and a background noise contribution from spurious light (other than probe light).

We have measured the read-out noise of the CCD camera and found 12 adu in one image. This noise is independent of the number of shots N seq performed to acquire the image, since the CCD is read out only once after the probe pulses have illuminated the atom and the associated scattered light has fallen on the CCD. Normalized to one shot, the read-out noise of the CCD thus scales as 12 adu/N seq . By contrast, the contributions, per shot, of the probe light and spurious light, scale as 1/ N seq . The three contributions were measured independently and add up quadratically to yield the data shown in figure 8. The signal to noise ratio is by far limited by the probe light contribution, which is due in part to scattering by atoms of the Rb beam intersecting the trapping region, and in part to scattering by the optics mounts under vacuum.

Figure 8 allows us to extract the number of sequences necessary to reach a given § 1 adu (analog-to-digital unit) corresponds to the digitization step of the analog signal acquired by the CCD camera. For the measurements presented in this paper, the electron-to-adu conversion factor was 1 e -/adu.

The dark count of the CCD being ∼ 0.001 adu/pixel/s is negligible for the parameters of our experiment.

Figure 8. (Color Online) Noise contributions, normalized to one shot, of our imaging system, versus the number of shots performed to acquire an image. The total rms noise (solid lines) is the quadratic sum of three contributions that were measured independently : the CCD read-out noise (red dash dotted line) and the background contributions due to spurious light (red dotted line) and probe light (dashed lines). The latter is larger when the atomic beam is switched on (case (2): black dashed line) than when it is off (case (1): red dashed line), due to the scattering by atoms of the atomic beam intersecting the probe beam. In both cases, the quadratic sum of the three contributions is in good agreement with the measured rms values of the total noise (black triangles and red circles, respectively). The peak signal, normalized to one shot (blue horizontal lines) is shown for two values of the time-of-flight, t TOF = 1 µs and t TOF = 50 µs, corresponding to images shown in figure 3(i) and (iii). The vertical dashed line indicates the minimum number of shots required to detect one atom.

signal-to-noise ratio. As explained at the end of Sect. 4, the probability to detect one photon (and therefore one atom) in single shot is 4.4 ×10 -2 using a 2 µs-duration probe, meaning that 23 shots are necessary to detect on average one photon. For a time-offlight image to be correctly fitted, we have found that we need typically 100 detected photons, which implies 2300 sequences. For instance, in the case of the image shown in figure 3(i), the signal to noise ratio is ∼ 20, while it is ∼ 9 for figure 3(iii).

Conclusion

In conclusion, we have demonstrated fluorescence imaging of a single atom in free flight by accumulating many images containing a single photon event corresponding to a single atom. We used this time-of-flight technique to measure the temperature of the atom after release from a microscopic optical dipole trap. This temperature measurement was confirmed by an independent method based on a release-and-recapture technique.

The large numerical aperture of our imaging system and the extreme confinement of the atoms in the trap allow a high spatial resolution on the order of ∼ 1 µm. The low noise level of our imaging system yields images showing ∼ 150 atoms with a very good signal to noise ratio (∼ 20). These measurements have been performed in conditions where the atomic motion during the probe pulse can be completely neglected (see table 1). We thus obtained a very accurate characterization of the optical performance of our system. In a next step, it will be possible to increase the single atom detection efficiency, by increasing the probe pulse duration (to the expense of a somehow reduced spatial resolution and signal-to-noise ratio). Finally, the measurements and the analysis presented in this work provide a calibration of our imaging system for future time-of-flight experiments where many atoms are confined in a microscopic dipole trap, and where interactions may play a central role.

Figure 1 .

 1 Figure1. (Color Online) Optical setup of our trapping (red) and imaging (green) systems. A single atom is trapped at the focal point of the aspheric lens and emits fluorescence photons when illuminated with probe beams (perpendicular to the plane of the figure). A fraction of the emitted fluorescence is sent to a fiber-coupled avalanche photodiode (APD) in a photon counting mode, which is used to trigger the timeof-flight sequence. The remaining fluorescence is also detected by an intensified CCD camera (I-CCD), with an efficiency η d = 2.2 × 10 -3 , taking into account the solid angle of the aspheric lens (6.7%), the transmission of the optics (33%) and the measured quantum efficiency of the intensifier photocathode (η intensifier = 10%).

Figure 2 .

 2 Figure 2. (Color Online) Time sequence for a time-of-flight experiment. Note that the time axis is not drawn to scale. The loading sequence lasts ∼ 1 s, while the adiabatic cooling, time-of-flight, and probing sequences are much shorter.

Figure 3 .

 3 Figure 3. (Color Online) Results of a typical single atom time-of-flight experiment.The atom is released from a trap with depth U ≈ 0.08 mK. The rms size of the "cloud" is plotted versus the time-of-flight of the single atom after it is released from the dipole trap. The dashed line is a fit to the data (circles), using Eq. (1). We show a typical error bar obtained by repeating the same experiment several times. Insets show images and associated cross-sections of the data for three particular time-of-flights. Each image results from the detection of a large number of successively trapped single atoms. The rms size of the cloud is thus the rms position of a single atom after a given time-offlight. Image i) corresponds to 3400 sequences and 150 detected photons, and therefore 150 detected atoms. Image iii) corresponds to 12000 accumulations and 520 detected atoms.

Figure 4 .

 4 Figure 4. (Color Online) rms size of the image of a single trapped atom for various positions of the trap along the optical axis of the imaging system. For small values of the defocus, the image size is given essentially by the response of the intensifier. For large values of the defocus, the main contribution comes from the depth of focus, which scales linearly with defocus according to equation (3) (dashed line). Our data (red circles), which agree well with a Monte Carlo simulation (blue dotted line), result from the convolution of both the optical response of the imaging system and the longitudinal profile of the thermal distribution (blue filled curve) with a width σ (0) = 1.6 µm.

Figure 5 .

 5 Figure 5. (Color Online) Effect on the measured spatial distribution of a single atom as a result of its movement during a probe pulse of variable length, after 1 µs time of flight. The temperature of the atom is 150 µK. Data (red circles) are in good agreement with the Monte Carlo simulation results (blue dotted line). At low pulse durations, the image size is limited by the intensifier response. Also shown : analytical rms contributions of equation (4) (dash dotted line) and of equation (5) (dashed line). The fact that the contribution of equation (5) is larger than both experimental and simulated results for long probe durations is due to the significant deviation of the distributions from a Gaussian model.

Figure 6 .

 6 Figure 6. (Color Online) Time-of-flight measurements of a single atom for two different temperatures : T = 20(2) µK (blue diamonds) and T = 150(16) µK (red circles). The simulated results are also shown (blue and red dotted lines respectively). The dashed lines (respectively blue and red) show the results of a simple model neglecting all broadening effects except for the transverse dimension of the thermal distribution, σ ⊥ (0).

Figure 7 .

 7 Figure 7. (Color Online) Release-and-recapture measurements of a single atom for the same loading and cooling parameters as in figure6. The fit (dotted lines) yields T = 19(2) µK and 149(15) µK (data points are in diamonds and circles, respectively).

Table 1 .

 1 Spatial resolution budget of our system, when the light source is a single atom with temperature T = 20 µK illuminated by a probe pulse with duration τ = 2 µs (see text). The rms size of the global response is the quadratic sum of the different rms contributions.

	Effect	rms size (µm)
	Intensifier	0.9(2)
	Diffraction (σ diff )	0.5(1)
	Atomic thermal distribution (σ ⊥ (0))	0.3
	Depth of focus (σ (0))	0.1
	Atomic displacement during τ (σ τ,thermal )	0.04
	Atomic random walk (σ τ,scatter )	0.02
	Quadratic sum	1.1(2)

1(1) µm and σ v = 45(2) mm.s -1 . The energy distribution

Acknowledgments

We acknowledge support from IARPA, the Institut Francilien de Recherche sur les Atomes Froids (IFRAF), PPF "Manipulation d'atomes froids par des lasers de puissance" and the European Union through the Integrated Project SCALA. A M Lance was supported by an E.U. Marie Curie fellowship and A. Fuhrmanek is partially supported by the DAAD Doktorandenstipendium.


Y R P Sortais
email: yvan.sortais@institutoptique.fr

H Marion

C Tuchendler

A M Lance

M Lamare

P Fournet

C Armellin

R Mercier

G Messin

A Browaeys

P Grangier

Diffraction limited optics for single atom manipulation

Keywords: numbers: 32.80.Lg, 32.80.Pj, 42.15.Eq

We present an optical system designed to capture and observe a single neutral atom in an optical dipole trap, created by focussing a laser beam using a large numerical aperture (N.A. = 0.5) aspheric lens. We experimentally evaluate the performance of the optical system and show that it is diffraction limited over a broad spectral range (∼ 200 nm) with a large transverse field (±25 µm). The optical tweezer created at the focal point of the lens is able to trap single atoms of 87 Rb and to detect them individually with a large collection efficiency. We measure the oscillation frequency of the atom in the dipole trap, and use this value as an independent determination of the waist of the optical tweezer. Finally, we produce with the same lens two dipole traps separated by 2.2 µm and show that the imaging system can resolve the two atoms.

I. INTRODUCTION

The observation and manipulation of a few individual particles are at the core of many present experiments in atomic and molecular physics, quantum optics, quantum information, as well as in biology and chemistry. Quite often, these experiments rely on high numerical aperture optics which collect a very weak fluorescence signal emitted by the particles. These optics generally operate at the diffraction limit, in order to image small objects with a high spatial resolution. As a few examples, the objects may be single ions in a Paul trap [1], single neutral atoms in microscopic optical dipole traps [2], Bose-Einstein condensates in a double well potential [3], or single fluorophores in a biological membrane [4]. The high resolution of the imaging optics allows the observation of periodic chains of ions [5,6] or arrays [7] of microscopic objects, with distances between them as small as a few microns.

Many of these experiments rely on the ability not only to observe, but also to trap and manipulate the particles. Large numerical aperture optics, when diffraction limited, can focus laser beams down to sub-micron spots that can be used as sharp optical tweezers [8]. In atomic physics, this strong confinement can be used to trap exactly one atom in the tweezer [2,9], with high oscillation frequencies due to the sharp focussing. Using independently controlled optical tweezers simultaneously, one can control the collision between two (or more) individual particles [10]. This approach has already been implemented to investigate interactions between moving biological objects [11], and offers an interesting perspec-tive to realize quantum logic operations between few cold neutral atoms or ions [5,12].

In the case of atoms or ions, the design of large numerical aperture optics requires to take into account the ultra-high vacuum environment that is necessary to produce and manipulate them. The optics may be either inside the vacuum chamber -and then must be bakable and vacuum compatible -or outside -but then the focussed beam must go through a vacuum window, which generally creates significant aberrations. Another constraint arises from the arrangement of the trapping system surrounding the particles (e.g. electrodes for an ion trap or laser cooling beams), which usually requires a long enough working distance of at least a few millimeters. These constraints add up to the requirements of large numerical aperture and diffraction limited performance, and often lead to a rather complicated design and manufacturing of vacuum compatible custom objectives.

In this article, we describe and characterize a simple optical system, based on the combination of a large numerical aperture aspheric lens placed inside the vacuum chamber, and a few standard lenses placed outside. This system combines the powerful techniques of optical tweezers and confocal microscopy to trap, manipulate and observe single ultra-cold 87 Rb atoms. The simplicity and low cost of the design compare favorably with custom objectives based on spherical lenses [13,14] and suggest broad applicability to other fields of research where excellent spatial resolution is critical. The numerical aperture is N.A. = 0.5 and, for a fixed working distance on the order of a centimeter, it performs at diffraction limit over a large spectral range, from 700 nm up to 880 nm [START_REF]The numerical aperture is N.A. = n sin α where n is the refraction index of the medium in which the lens is working, and α is half of the angular aperture of the lens. Here n = 1 since our lens operates under vacuum[END_REF]. We take advantage of this property to both focus the tweezer beam at a wavelength of 850 nm, and collect fluorescence light at 780 nm through the same lens.

The paper is organized as follows. Section II details the requirements of the system and describes the optical setup. In Section III the performance of the tweezer is characterized using optical techniques. Section IV ex-plains how to trap and detect a single atom at the focal point of the tweezer, and Section V how to measure the oscillation frequency of one atom in the trap. This method provides a way to probe the light field locally, and to perform an independent measurement of the laser beam waist at the focal spot.

II. REQUIREMENTS AND DESIGN OF THE OPTICAL SYSTEM

Trapping a single atom in an optical tweezer leads to the following requirements for our objective lens. First, controlling the evolution of the atom, or its interactions with another atom, requires that the trap size be smaller than a few microns, i.e. that the objective be diffraction limited while having a numerical aperture as large as possible. In particular, our method for single atom trapping relies on a "collisional blockade mechanism" [START_REF] Schlosser | [END_REF] : in a very small trap, the two-body loss rate is so high that if an atom is already trapped, a second atom entering the trap leads to a fast inelastic collision and, eventually, to the loss of the two atoms. Second, loading of the tweezer is performed by focussing the laser beam into a reservoir of laser cooled atoms, an optical molasses in our case. In practice, this implies that the focussing lens has a large enough working distance to allow for optical access of the cooling beams, typically at least 5 to 10 mm is desirable. Third, the laser used to produce the optical tweezer is far off-resonance at 850 nm, in order to avoid heating and decoherence of the trapped atom due to spontaneous emission. On the other hand, the atom is probed by exciting the D 2 transition and collecting the fluorescence at 780 nm, with a dichroic plate separating the two radiations. Since it is convenient to have both beams going through the same aspheric lens, a third requirement is that the system be diffraction limited over a broad spectral range while keeping the working distance constant [17]. Fourth, in view of future experiments using several single atoms trapped in adjacent tweezers, it is also desirable that the objective remains diffraction limited off axis, which requires a large enough transverse field. For instance, a field of view of ±25 µm and a resolution of 2 µm (see below) allows one to easily address several hundreds of traps. Designing arbitrary arrays could be achieved for example by combining our large N.A. lens and a spatial phase modulator placed on the incoming laser beam [7], or by using appropriate optical lattices [18].

The requirements enumerated above can be fulfilled by combining a sufficient number of spherical lenses. This is the case for the microscope objective that we designed for our first generation experiment dedicated to single atom manipulation [13,19]. This objective consists of nine spherical lenses and has a numerical aperture N.A. = 0.7. Tight mechanical constraints and careful alignment of the lenses resulted in a diffraction limited performance with spatial resolution of 0.7 µm. The transverse field is ±10 µm. The working distance is 10 mm and the objec- tive operates under ultra-high vacuum.

For simplification and scalability purposes, we have built a second generation apparatus that uses a single commercial molded aspheric lens [START_REF]Model 350240 from[END_REF] with N.A. = 0.5, a working distance of ∼ 5.7 mm and a focal length of 8 mm. This lens is manufactured by LightPath Technologies, Inc. [21] and is diffraction limited at 780 nm for an object plane at infinite distance, with a 0.25 mm thick glass plate between the lens and the image focal plane. The absence of this glass plate results in an enhanced spherical aberration when operating the lens at an infinite conjugate ratio. However, we noticed that aberration-free operation can be restored by slightly defocussing the aspheric lens, and operating with weakly non-collimated beams. Low numerical aperture lenses can provide such beams without introducing aberrations. This beam shaping, which is crucial for our system, was optimized by using an optical design software [22].

The optical setup is shown in Fig. 1. The source used for the tweezer is a single mode laser diode operating at 850 nm. We focus this light into a single mode polarization maintaining fiber with an output N.A. = 0.11. The beam at the output of the fiber is shaped by using a triplet [START_REF]Model 09 LAM 126 from Melles Griot[END_REF] outside the vacuum chamber that provides a slightly converging beam. Then it is injected into the vacuum chamber and onto the lens mounted under ultrahigh vacuum [START_REF]The vacuum chamber was baked up to 130 • C for two days[END_REF]. The same aspheric lens is then used as an imaging system, to collect the fluorescence emitted by the atoms trapped in the optical tweezer. The collection solid angle is Ω/4π = 0.076. A polarization independent dichroic plate separates fluorescence light at 780 nm from the tweezer light at 850 nm. Imaging of the atomic samples is performed by using a doublet and a singlet [START_REF]From Melles Griot : doublet model 01 LAO 226[END_REF] outside the vacuum chamber. The overall transverse magnification of the imaging system is ∼ 25, which allows imaging of our 1 µm diameter dipole trap onto a 2 pixels×2 pixels area of a CCD camera (pixel size is 13 µm×13 µm). We also use a polarization beamsplitter to collect part of the fluorescence onto an avalanche photodiode (APD).

Once the working distance between the aspheric lens and the focal point has been fixed, the distances between the lenses shown in Fig. 1 can be varied along the opti-cal axis by a few centimeters, whilst keeping the system diffraction limited. Likewise, the transverse alignment of the lenses and the fiber source is tolerant within ∼ 1 mm.

III. OPTICAL CHARACTERIZATION OF THE TWEEZER

In practice, aberrations of the optical system deform the wave front and result in energy being spread away from the central lobe of the diffraction image, thus leading to a peak intensity attenuation and to a shallower dipole trap. We quantify the amount of aberrations of the optical system by considering the ratio of the peak intensity I aberr in presence of aberrations to the intensity I stig which would be obtained if the optical system were perfectly stigmatic (i.e. free of aberrations). This ratio is called the Strehl ratio [START_REF] Born | Principles of Optics[END_REF] and is related to the deformation of the wave front as follows :

S = I aberr I stig ≃ 1 -4π 2 ∆ 2 λ 2
Here, ∆ denotes the root-mean-square departure of the actual wave front with respect to the ideal one, and λ is the wavelength of the radiation propagating through the system. A practical criteria is that the peak intensity attenuation S is larger than the arbitrary value 0.8 for the amount of aberrations to be acceptable (S ≥ 0.8). This sets an upper limit to the amount of aberrations that we tolerate in our system (∆ ≤ λ/14). For a system free of aberrations, ∆ = 0 and S = 1. Experimentally, the optical tweezer was characterized in three steps. First, we tested the performance of the triplet and the aspheric lens separately by using a wavefront analyzer (Fizeau interferometer [START_REF] Born | Principles of Optics[END_REF]). We find that both exhibit diffraction limited performance : ∆ ≤ λ/40 for the triplet (N.A. = 0.12) and ∆ ≤ λ/30 for the aspheric lens (N.A. = 0.5), the latter being remarkable for such a large numerical aperture. The residual deformation of the wave front is due mainly to a small spherical aberration.

For the second step in characterizing the optical tweezer, we aligned the triplet and the aspheric lens and evaluated the Strehl ratio S of the ensemble by analyzing point spread functions of the system and comparing them to the ideal case, i.e. with no aberrations present. In order to record the point spread function, we used a back-illuminated sub-micron pinhole in the focal plane of the aspheric lens, in place of the cold atoms. The pinhole being much smaller than 1.22 λ/N.A. = 2 µm, its intensity diagram is flat over the full aperture of the aspheric lens. The optical response of the system to this point source illumination was observed on a linear 16 bits CCD camera after magnification by a factor 50 by an aberration-free objective. Figure 2(a) shows a crosssection of the point spread function measured on axis in the best focus plane, corrected for the (×50) magnification of the observation objective and for the (×4) magnification of the (triplet, aspheric lens) system. The corresponding CCD image is shown on Fig. 3(a). Reversal of light propagation ensures that this actually is the point spread function of the system, had it been illuminated by an isotropic point source in the focal plane of the triplet (see Fig. 1). It should be compared to the (2J 1 (ζ)/ζ) 2 Airy function predicted by diffraction theory. Here ζ = 2πr × N.A./λ, where r is the radial coordinate, N.A. = 0.5, and λ = 850 nm. The full width at half maximum of the measured point spread function is FWHM ⊥ = 0.9 µm and the radius of the first dark ring is 1.06 µm, which is in agreement with diffraction theory within a few percent. Normalizing the peak intensity of the Airy function to 1, we measure a peak intensity of 0.93 for our system (the total flux is kept constant in both cases), which is a measure of the Strehl ratio S on axis. For a full characterization of the light potential that the atomic samples will explore, we also scanned the magnifying objective along the axis around the best focus position. Figure 4 shows the measured on-axis intensity variation, corrected for the longitudinal magnification of the objective and (triplet, aspheric lens) system. The onaxis intensity vanishes symmetrically 7 µm away from the best focus position, and bright rings appear off-axis, in place of the previous dark rings (contrast inversion). The full width at half maximum of the longitudinal intensity distribution is FWHM = 6.3 µm, which again is in good agreement with diffraction theory.

Our final step in characterizing our optical tweezer was investigating the off-axis performance of our system by moving the pinhole perpendicular to the optical axis. Figure 3 shows the point spread functions measured for a pinhole respectively on-axis and off-axis by 30 µm. The latter displays the characteristic V-shaped flare of a comatic aberration. Since our system is nearly free of aberrations on axis, we determine its Strehl ratio by directly comparing the peak intensities of images taken with an off-axis and on-axis pinhole, provided the total flux is kept constant [see Fig. 2(b)]. The performance of the optical tweezer can be equivalently evaluated by plotting its optical modulation transfer function (MTF) that characterizes the attenuation of the various spatial frequencies present in a test object, due to the optics [27]. Figure 5 shows cross-sections of the 2D-Fourier transform of the images shown in Fig. 3, and compares them to the frequency response of an aberration-free system with similar numerical aperture : the 2D-autocorrelation function of a circular aperture. The measurements show that our system displays a field of ±25 µm over which S ≥ 0.8.

It should be noted here that the results described above were obtained by illuminating the system with spherical wavefronts, issued from a sub-micron pinhole. The situation is slightly different for our single atom tweezer, because the system is illuminated with a gaussian beam provided by an optical fiber, with a beam waist equal to the aspheric lens radius. This has two consequences : first, the diffraction rings structure shown above is significantly damped, due to the gaussian apodization effect. Second, the point spread function becomes slightly broa- der, with a FWHM ⊥ increased by 9%, and is neither an Airy function nor purely gaussian.

IV. DETECTION OF A SINGLE ATOM TRAPPED IN AN OPTICAL TWEEZER

The optical tweezer described above is a powerful tool for manipulating single neutral atoms. The alignment of the optical tweezer onto the center region of an optical molasses of cold 87 Rb atoms indeed results in the trapping of a single atom in the microscopic dipole trap as shown below. The optical molasses [START_REF] Lett | [END_REF]29] is produced by six counter-propagating cooling beams at ∼ 780 nm and a Zeeman slowed beam of Rb atoms [30], providing a large reservoir of cold atoms (with typical size ∼ 1 mm) surrounding the microscopic optical tweezer.

We observe the fluorescence of the trapped atoms with an avalanche photodiode (APD) used in single photon counting mode, as well as a CCD camera with low readout noise (see Fig. 1). We use an interference filter (centered at 780 nm, with 10 nm bandwidth) and a 400 µm diameter pinhole in confocal configuration to reduce the background signal. The residual scattering of the 780 nm cooling beams in the vacuum chamber and on the optical table contributes for 81% of this background signal, the remaining 19% coming from the background fluorescence at 780 nm by the molasses itself. We estimate the overall collection efficiency of our system to be ∼ 1%, taking into account the angular collection efficiency of the lens (7.6%), the transmission of the optics and the quantum efficiency of the APD. inelastic collision with another entering atom. The collision process occurs because it is light-assisted by the molasses beams at 780 nm (collisional blockade mechanism) and results in the two atoms being ejected from the trap. The absence of two-atom steps is by itself an indication that the waist of the trapping beam is no more than a few microns, since this condition is required for the collisional blockade mechanism [START_REF] Schlosser | [END_REF]. In that case, the loss rate due to two-body collisions becomes huge when the trapping volume is very small, and pairs of atoms are never observed.

For the data of Fig. 6, we use 5.6 mW of trapping light at 850 nm. Assuming a waist of 1 µm (see Sec. V below), we calculate a trap depth of 1.5 mK, or equivalently a lightshift of 32 MHz. The cooling beams that induce the fluorescence of the atom are red-detuned with respect to the atomic transition by 5Γ, where Γ = 2π × 6 × 10 6 rad.s -1 is the linewidth of the D 2 transition. The fluorescence signal exhibits steps of 2.7 × 10 4 photons.s -1 on top of a background signal of 1.3 × 10 4 photons.s -1 . The analysis of the corresponding histograms shown in Fig. 6(b) indicates that both background and step signals are shot-noise limited. This, together with the step height of 2.7 × 10 4 photons.s -1 , allows us to discriminate between the presence and the absence of an atom within 10 ms with a confidence better than 99%. In the presence of the cooling laser beams, the storage time of the atom is limited by the light-assisted collision with a second atom. It can be varied from 100 ms up to about 10 s, depending on the density of the molasses cloud.

We have also measured the lifetime of the atom in the absence of the cooling light and found a 1/e decay time of ∼ 10 s. This was measured by varying the duration T off during which the cooling beams are switched off, and measuring the probability to detect the atom fluorescence again right after T off . This lifetime may be limited by collisions with the background gas (pressure in the 10 -9 Pa range) as well as heating mechanisms.

Finally, we briefly address the issue of addressability and resolution of our imaging system. For this purpose, we produce two tweezers by sending a second trapping beam at 850 nm at a small angle in the aspheric lens. An angle of 0.25 mrad, together with the focal length of 8 mm of the aspheric lens, results into a 2 µm distance between the two traps. The loading of the two traps is not deterministic but we can easily find periods of time during which two atoms are present at the same time in the two optical tweezers. Figure 7 shows a cross-section of a CCD image taken when such an event occurs. A gaussian fit to the peak signal produced by each single atom indicates a waist of 0.9 ± 0.2 µm, which validates the performance of the tweezer system and the performance of the imaging system altogether. We emphasize that the two atoms are always present during the 100 ms integration time. Therefore this data corresponds to realistic conditions for "reading out" a quantum register [31] with two qubits separated by 2.2 µm. This indicates that our imaging system may resolve two atoms with separation as small as 2 µm, within a time as short as 10 ms.

surement of the trap dimension. Knowing the power P trap of the trapping laser, and assuming that the trapping beam is gaussian, we can calculate from the oscillation frequency the waist w 0 of the optical tweezer :

w 0 = hΓ mω 2 r P trap πI sat Γ 3δ 1 + 2Γ 3δ 2 1/4
where I sat ≃ 1.67 mW/cm 2 is the saturation intensity of the D 2 transition, ω r is the oscillation frequency of the atom measured in the radial plane (i.e. perpendicular to the tweezer optical axis), and δ 1 ≃ 2π ×2.4×10 13 rad.s -1 (resp. δ 2 ≃ 2π × 3.2 × 10 13 rad.s -1 ) is the frequency detuning of the tweezer beam relative to the D 1 (resp. D 2 ) transition [START_REF] Grimm | Advances in Atomic, Molecular and Optical Physics[END_REF].

In order to measure the oscillation frequency ω r , we followed the procedure described in [START_REF] Reymond | [END_REF]34]. Once a single atom is loaded in the trap, the cooling beams are switched off for a time T off = 50 ms. Meanwhile, the dipole trap beam is switched off and on twice, as shown in the time sequence of Fig. 8(a). The first "off" pulse (duration δt 1 ) increases the initial amplitude of the oscillation, once the trap is turned back on (see details below). After a variable period of time ∆t, we switch the potential off again for a fixed duration δt 2 . After this time, we turn the molasses beams on again and determine whether the atom is still present or not. We then repeat this procedure on about 100 atoms, and measure the probability to keep the atom at the end of this sequence for various ∆t. We obtain the curve shown in Fig. 8(b).

This curve shows oscillations that we can understand in the following way. Assume the atom is oscillating in the trap. If the atom reaches the bottom of the trap when the second "off" pulse occurs, it will most likely leave the trap during the time δt 2 , because its velocity is maximal at this point. Alternatively, if the atom reaches the apogee of its oscillation when the second "off" pulse occurs, it will most likely be recaptured in the trap after time δt 2 , because its velocity is null at the apogee. Due to the symmetry of the motion, the probability of keeping the atom oscillates at twice the oscillation frequency when ∆t is varied. The recapture probability, which we measure when the cooling beams are switched on again (i.e. at t = T off ), also depends on duration δt 2 , which is adjusted to optimize the contrast at the beginning of the oscillation.

The role of the first "off" period is crucial in this measurement. Since each data point shown in Fig. 8(a) is averaged over 100 single atoms, all these atoms must oscillate in the trap with the same phase if one wants to see any oscillations at all. The first pulse precisely fulfills this function, as explained in Fig. 9. In this figure, we consider the phase space defined by the position x and the velocity v x in the radial plane, assuming a harmonic motion with the frequency ω r . We suppose that the initial distribution of atoms follows a gaussian distribution, both in position and velocity. The width of this distribution in the phase space (x, v x /ω r ) depends on the mean energy of the atoms. During the free flight of duration δt 1 , the initial isotropic distribution evolves towards an elliptical distribution (Fig. 9), because the velocities of the atoms remain constant when the trap is switched off. In phase space, the ellipse is pulled along the x axis and its long axis makes an angle θ that decreases with time : the longer the duration δt 1 , the "flatter" the ellipse, keeping a constant area in phase space. For an oscillation frequency around 119 kHz and δt 1 = 1.3 µs, we calculate that the angle of the ellipse is θ ≃ 32 • and the ratio between the lengths of the long axis and the small axis is ∼ 2.6. When the trap is turned back on at the end of the free flight period, the atoms that are still in the trap oscillate with a larger amplitude, and they are now almost all in phase with each other. We note that for the chosen δt 1 , only the radial oscillation frequency is excited, because along the longitudinal axis the free expansion remains quite small compared to the initial size of the atomic "cloud".

We fit the data shown in figure 8(b) with a damped sine function and measure a radial oscillation frequency ω r /2π = 119 ± 3 kHz. The damping of the oscillation is attributed to the anharmonicity of the trap potential explored by the atom after the first "off" period [START_REF] Reymond | [END_REF]34]. Since the trapping beam power was measured to be P trap = 5.6 ± 0.1 mW and assuming a gaussian intensity distribution at the focal point of the trap beam, we infer a beam waist w 0 = 1.03 ± 0.01 µm. From this result we calculate a trap depth U 0 = 1.5 mK and a longitudinal oscillation frequency ω z /2π = 22 kHz. The waist w 0 extracted from the data is in good agreement with the value of 0.9 ± 0.2 µm, obtained by imaging a single atom on a CCD camera, as presented in Sec. IV.

VI. CONCLUSION

In conclusion, we have demonstrated a diffraction limited optical system with a large numerical aperture (N.A. = 0.5) that acts as a sharp optical tweezer used to trap a single atom. The large collection efficiency of this same lens allows single atom detection with confidence better than 99% within 10 ms, and resolves atoms at the microscopic level. The resolution is good enough that we can resolve two atoms trapped in two tweezers separated by less than 2 µm. This system is based on low cost commercial lenses and is relatively tolerant to small misalignments. It also provides a large field of view (±25 µm) and a large spectral range over which it remains diffraction limited. We believe that this system is thus a valuable tool for experiments manipulating and observing single particles. It should be useful for applications in quantum computing using neutral atoms and also for addressing strings of individual ions.

Fig. 1 :

 1 Fig. 1: (Color Online) Optical set-up of our trapping (solid line) and imaging (dotted lines) systems. For clarity, schematic is not to scale.

Fig. 2 :

 2 Fig. 2: (Color Online) (a) Cross-section of the point spread function of the tweezer system, measured at 850 nm on axis (crosses). We show the Airy function (solid line) for comparison. The measured peak intensity is S = 0.93 on axis. When the point source is moved off-axis by r0 = 30 µm, the point spread function gets distorted and the peak intensity is attenuated (S = 0.77). (b) Cross-section along the direction of transverse displacement of the point source (crosses) and the Airy function (solid line). Total flux is kept constant in both cases.

Fig. 3 :

 3 Fig. 3: (Color Online) Images of the point spread function of our tweezer system, taken with a linear CCD camera at 850 nm, for a sub-micron pinhole source on axis (a) and offaxis by 30 µm (b). Cross-sections shown on Fig. 2 were taken along the horizontal axis of these images, across the center, where intensity is maximum.

Fig. 4 :

 4 Fig. 4: (Color Online) On-axis intensity variation close to the focal point, as a function of the position along the optical axis. The prediction from diffraction theory is shown for comparison, with the intensity normalized as in Fig. 2.

Figure 6 (

 6 a) shows a time sequence of the fluorescence signal collected on the APD observed in 10 ms time bins. For this figure, the molasses beams and the dipole trap are continuously on. The sudden jumps in the fluorescence signal correspond to a single atom entering the optical tweezer, while the sudden drops of the fluorescence signal correspond to the atom leaving the trap due to an

Fig. 5 :

 5 Fig. 5: (Color Online) Modulation transfer function (MTF) of the tweezer system. MTF measured on-axis (circles) and off-axis by 30 µm (crosses) are compared to the theoretical frequency response of an aberration-free system (solid line). Strehl ratio is S = 0.93 on-axis, and S = 0.77 off-axis.

Fig. 6 :

 6 Fig. 6: (a) Fluorescence of a single atom measured by the APD. Each point corresponds to a 10 ms time bin. (b) Histogram of the measured fluorescence recorded over 100 s. The two Poisson distributions correspond to the presence and the absence of a single atom in the dipole trap. The arrow indicates the threshold that we use to discriminate between the two cases.

Fig. 7 :

 7 Fig. 7: Cross-section of a CCD image showing two single atoms trapped in two adjacent optical tweezers, corrected for the magnification (×25) of the imaging system. The distance between the two optical tweezers is 2.2 ± 0.1 µm. Each peak is fitted by a gaussian model (dashed lines) and exhibits a waist w = 0.9 ± 0.2 µm. The solid line represents the sum of the fits of the two fluorescence signals emitted by each single atom. Vertical bars represent the intensity measured by each pixel of the CCD camera during a time window of 100 ms.

Fig. 8 :

 8 Fig. 8: (Color Online) Oscillations of a single atom in the dipole trap. (a) shows the time sequence used for this measurement. Dipole trap is switched off twice, during δt1 ∼ 1.3 µs and δt2 ∼ 6.2 µs. In the time interval ∆t separating these two "off" pulses, the dipole trap is switched on again. (b)shows the probability to keep the atom after this time sequence. Each point corresponds to 100 successful events (i.e. with one single atom at the beginning of the sequence) for a given time delay ∆t. Error bars are statistical. Solid line is a damped sine fit to the data, showing that the atom oscillates with frequency ωr/2π = 119 ± 3 kHz.

Fig. 9 :

 9 Fig. 9: Monte Carlo simulation of the evolution of 2000 atoms in the radial plane of the tweezer, using phase space representation. (a) The atoms are distributed with an initial symmetric gaussian distribution before the first "off" period of duration δt1. Circle represents one standard deviation of the distribution. (b) The distribution evolves towards an ellipse with angle θ during the free flight of duration δt1. As a consequence, the atoms oscillate in phase when they are recaptured in the trap.

Once the atom is trapped, we switch off the cooling beams, whilst the optical tweezer is kept on. The atom oscillates at the bottom of the dipole trap, and measurement of its oscillation frequency provides an in-situ mea-

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), and from ARDA/DTO. Y. R. P. Sortais and H. Marion were supported by IFRAF and CNRS fellowships, and A.M. Lance also by IFRAF.