P. Vasseur Cédric Demonceaux

Pascal Vasseur

Claude Pégard

UAV Attitude Computation by Omnidirectional Vision in Urban Environment

Attitude is one of the most important parameters for a UAV during a flight. Attitude computation methods based vision generally use the horizon line as reference. However, the horizon line becomes an inadequate feature in urban environment. We then propose in this paper an omnidirectional vision system based on straight lines (very frequent in urban environment) that is able to compute the roll and pitch angles. The method consists in finding bundles of horizontal and vertical parallel lines in order to obtain an absolute reference for the attitude computation. We also develop here a new and efficient method for line extraction and bundle of parallel line detection. An original method of horizontal and vertical plane detection is also provided. We show experimental results on different images extracted from video sequences.

I. INTRODUCTION

The attitude of an autonomous or tele-operated UAV is a very important parameter during a flight. However, inertial sensors which are useful for this measurement, are sensitive to drift and expensive if a high accuracy is required. The use of vision sensors becomes then an interesting option which allows not only to compute the attitude but also to perform higher level tasks (see figure 1(a)). In this case, most of the proposed approaches are based on the extraction of the horizon line from the image in order to compute the roll and pitch angles [START_REF] Demonceaux | Omnidirectional Vision on UAV for Attitude Computation[END_REF] [START_REF] Demonceaux | Robust Attitude Estimation with Catadioptric Vision[END_REF] [6] [START_REF] Todorovic | Sky/Ground Modeling for Autonomous MAV Flight[END_REF] [START_REF] Wang | Complete Pose Determination for Low Altitude Unmanned Aerial Vehicle Using Stereo Vision[END_REF]. Thus, in [START_REF] Ettinger | Vision-guided Flight Stability and Control for Micro Air Vehicles[END_REF] and [START_REF] Todorovic | Sky/Ground Modeling for Autonomous MAV Flight[END_REF], the authors use a perspective camera and the horizon is considered as a straight line. They define a criterion based on color and on texture which has to be optimized in order to separate the ground from the sky. In [START_REF] Woo | Robust Horizon and Peak Extraction for Vision-based Navigation[END_REF], an infrared camera is used in order to find the horizon. With this sensor, the authors are able to treat images in which steep reliefs are present. In [START_REF] Wang | Complete Pose Determination for Low Altitude Unmanned Aerial Vehicle Using Stereo Vision[END_REF], a stereoscopic system detects a known beacon on the ground for a complete pose determination. However, such an approach radically limits the evolution space of the UAV. In [START_REF] Demonceaux | Omnidirectional Vision on UAV for Attitude Computation[END_REF] and [START_REF] Demonceaux | Robust Attitude Estimation with Catadioptric Vision[END_REF], attitude computation is performed from catadioptric images. These images present three advantages in comparison with perspective images. First, the horizon line is always present in the image, even if partially. This characteristic ensures a possible estimate for every image. Second, the horizon can be detected accurately even in the presence of occultations. Finally, contrary to the perspective case which supposes that the altitude is known, the pitch can be estimated without prior knowledge. In this way, the attitude can be completely and accurately FirstName.Name@u-picardie.fr computed with a catadioptric image. In both approaches, the estimation of the angles is performed according from the horizon line extracted in [START_REF] Demonceaux | Omnidirectional Vision on UAV for Attitude Computation[END_REF] thanks to a robust method based on color difference between the sky and the ground and in [START_REF] Demonceaux | Robust Attitude Estimation with Catadioptric Vision[END_REF] thanks to an optimization function also based on the color. However, all these methods provide good results only in a free space where the horizon is largely visible. Indeed, in the case of a low altitude flight in urban environment, you quickly realize that these methods are inadequate because the horizon is almost non-existent, as shown in figure 1(b). Thus, in [START_REF] Li | Estimating Head Orientation Based upon Sky-ground Representation[END_REF] the authors propose to use accelerometers in association with omnidirectional vision in order to compute attitude. Another approach without any addition of supplementary sensors consists in using adequate features for urban environment such as straight lines. Indeed, if the system is able to detect horizontal and vertical lines in the image, it is then able to compute roll and pitch angles. The use of a perspective camera permits a fast detection of lines but does not ensure the collect of enough significative information for an accurate estimation. In this way, we propose in this paper a system based on a central catadioptric sensor which displays several advantages for our problem. First, the field of view ensures the presence of a large number of lines. Next, geometric characteristics of central sensors allow to compute directly and accurately the roll and the pitch by using the sphere of equivalence [START_REF] Geyer | Catadioptric Projective Geometry[END_REF]. Finally, the sphere of equivalence allows to find horizontal and vertical planes which are useful for the attitude computation. The use of omnidirectional vision in urban environment for a six degree of freedom robot is not really new. Indeed, in [START_REF] Hrabar | Omnidirectional Vision for an Autonomous Helicopter[END_REF], the authors propose an omnidirectional sensor associated with two perspective cameras for navigation in a urban canyon. In [START_REF] Bosse | Vanishing Points and 3D Lines from Omnidirectional Video[END_REF], an algorithm of structure from motion based on parallel lines is proposed. The approach presented in this paper contains two distinct contributions. First, we propose a new method of line detection valid for any calibrated central catadioptric sensor. We show that this method allows a fast and accurate detection of the lines in the image. This permits to subsequently detect bundles of parallel lines by using antipodal vanishing points. The second contribution concerns the vertical and horizontal planes of the scene thanks to the detection of the sky in the image. Finally, the estimation of the attitude is naturally deduced from these planes.

In the following of the paper, we first describe the line detection algorithm with a presentation of the image formation model that we use. In section III, the detection of bundles of parallel lines is presented. Horizontal and vertical planes extraction is described in section IV with a new sky detection algorithm and the attitude computation. We present experimental results in section V followed by a conclusion and perspectives in section VI.

II. LINE DETECTION

A. Introduction

Line detection in catadioptric images is not a trivial problem. Indeed, the projection of any 3D real line is a conic that may be degenerate. Thus, in the case of an uncalibrated sensor, it is necessary to estimate five parameters for each line while only two parameters are necessary for a calibrated sensor. Nevertheless, even in the calibrated case the detection is still difficult and requires important computation times. Thus, in [START_REF] Vasseur | Central Catadioptric Line Detection[END_REF] and [START_REF] Ying | Catadioptric Line Features Detection using Hough Transform[END_REF], the authors propose a detection based on an adaptation of the Hough transform. These methods permit a good detection but with important computation times for our application. In [START_REF] Barreto | Direct Least Square Fitting of Paracatadioptric Line Images[END_REF], the approach, valid uniquely for paracatadioptric sensors, consists in determining the conic equation in giving only two points on the line. This method, useful in the uncalibrated case, does not present possible extensions to the other central sensors and does not propose any automatic selection of the two initial points. In this paper, our method is valid for any central catadioptric sensor and also uses two points for each line. However, the selection of these points is totally automatic and the detection is based on a prediction/verification algorithm.

B. Algorithm Overview

The algorithm is composed of four main steps which respectively consist in detecting the edges in the image, in chaining the connected edge pixels, in selecting significant chains and finally in verifying if each chain is the projection of a real 3D line. Edge detection is performed by a Canny edge detector and we only keep chains of edgels (pixels of the edges) with a length greater than N bP ixels. The third step aims to verify if the detected chains do not contain ruptures of direction. In this way, we compute the difference of gradient orientation for each pair of connected pixels of each chain. If two consecutive pixels have a difference of orientation greater than a threshold, the chain is then divided. The final aim is to obtain chains which are probably the projection of a single 3D edge. The last operation consists then in verifying which chains are the images of 3D lines.

C. Line Projection Model

As mentioned previously, we suppose that the sensor is calibrated. To do this, we use the method proposed in [START_REF] Barreto | General Central Projection Systems: Modeling, Calibration and Visual Servoing[END_REF]. Thus, we first propose in this section the image formation model that we use in our algorithm. The projection of a 3D point is performed by the way of the unitary sphere (figure 2). Thus, we define oriented projective ray P 1 passing by 3D point x w and the center of the sphere. This ray intersects the surface of the sphere in x s . We then consider oriented projective ray P 2 passing by x s and a point situated on the z-axis between the center of the sphere and the north pole. This point is at distance ξ from the center of the sphere and only depends on the mirror geometric characteristics. P 2 intersects the plane at infinity in point x i . Finally, homography H defined between the plane at infinity and the catadioptric image plane projects point x i into point x c . H includes intrinsic parameters of the camera, possible rotations between the sphere frame and the camera frame, and finally the parameters of the mirror. According to this model, we can develop the projection of a 3D line into the catadioptric image plane (figure 3). We consider plane Π R which contains the real 3D line and the center of the sphere. This plane intersects the sphere and then defines a great circle onto its surface. The set of oriented projective rays passing through the points of the great circle and point O 2 then define a cone which intersects the plane at infinity into conic C i . Finally, homography H transforms C i into conic C c in the catadioptric image plane. In the plane at infinity, we know that the equation of conic C i is equal to :

C i =   n 2 x (1 -ξ 2 ) -n 2 z ξ 2 n x n y (1 -ξ 2 ) n x n z n x n y (1 -ξ 2 ) n 2 y (1 -ξ 2 ) -n 2 z ξ 2 n y n z n x n z n y n z n 2 z  
(1) with (n x , n y , n z ) the vector, which describes the normal to plane P w which contains the 3D line. We obtain the equation of conic C c in image plane thanks to the following relation :

C c = H -T C i H -1 (2)
It appears that if the normal to the plane which contains the 3D line is known, we can easily obtain the equation of the conic in the image plane. Thus, if two points of the great circle are known, we obtain the normal vector by the cross product of these two points expressed in frame O 1 . Since the sensor is calibrated, any point of the image plane can be back-projected on the sphere by :

x i = H -1 x c ( 3
)
and then by computing the intersection between the sphere and the projective ray defined by x i and the center of the sphere. The algorithm of line detection then consists in selecting two significant points for each chain of edges and in backprojecting them onto the sphere. Two points are considered as significant if they are close to the endpoints of the chain In fact, this step corresponds to the prediction of the position of the projection of the 3D line into the image. The verification then consists in counting the number of edgels of the chain which belong to the predicted conic. In this way, we consider that an edgel is on the conic if the error computed by x T c C x x c is less than a tolerance. The chain is considered as the projection of a 3D line if and only if at least 90% of its edgels verify the previous tolerance. Figure 4 presents different steps of the detection algorithm.

III. PARALLEL LINE BUNDLE DETECTION

Once the lines have been detected in the image, we have to determine which lines are respectively horizontal and vertical. In this way, we first aim to find bundles of parallel lines and we use the following property : Property : If d 1 and d 2 are two parallel lines with unit vector -→ u , then their equivalent great circles C 1 and C 2 on the unitary sphere intersect into two antipodal points I 1 and I 2 . These points have the property of being independent from the position of d 1 and d 2 in the 3D scene and only depend on direction -→ u . We then have --→ I 1 I 2 = -→ u (figure 5). Proof : Consider P 1 and P 2 , planes which contain respectively d 1 and d 2 and also O 1 . It is obvious that these planes intersect into line d which includes O 1 . This demonstrates that two great circles C 1 and C 2 intersect into two antipodal points I 1 and I 2 . Let note -→ n 1 (respectively -→ n 2 ) be the normal of plane P 1 (respectively P 2 ). However,

d 1 ∈ P 1 and d 2 ∈ P 2 , then -→ n 1 . -→ u = -→ n 2 . -→ u = 0 and -→ u = -→ n 1 × -→ n 2 .
In the same way, --→

I 1 I 2 ∈ P 1 ∩ P 2 , then --→ I 1 I 2 . -→ n 1 = --→ I 1 I 2 . -→ n 2 = 0 and --→ I 1 I 2 = -→ n 1 × -→ n 2 = -→ u .

This property has then two consequences :

• The set of parallel lines with unit vectors -→ u provides a set of great circles which intersect into two antipodal points I 1 and I 2 , and --→

I 1 I 2 = -→ u .
• Consider two bundles of parallel lines with unit vector -→ u and -→ v . Notice I i 1 and I i 2 the intersection points of each bundle (i = 1, 2). If these two bundles are perpendicular, we obtain

--→ I 1 1 I 1 2 .
--→ I 2 1 I 2 2 = 0 Thus, in order to detect bundles of parallel lines, we use an exhaustive algorithm which computes the intersection points of each pair of lines. If at least three lines have the same intersection points, we consider them as a bundle. Finally, we verify which bundles are perpendicular (figure 6). In the best case, if the scene is made of several buildings such as in figure 7, we obtain three perpendicular bundles. Thanks to their unit vectors -→ u 1 , -→ u 2 , -→ u 3 , these bundles define an orthogonal frame with respectively one vertical direction and two horizontal directions. Otherwise, if only two perpendicular bundles are detected, we obtain the frame with

-→ u 1 , -→ u 2 , -→ u 3 = -→ u 1 × -→ u 2 .
The attitude can then be estimated thanks to this orthogonal frame. However, it is necessary to determine which vector of this frame represents the vertical unit vector beforehand. This is the aim of the next section.

IV. ATTITUDE COMPUTATION

Once the orthogonal bundles of the 3D scene are obtained, we can not differentiate the vertical and horizontal directions. However, in order to estimate the attitude of the UAV, it is necessary to know the vertical direction in the scene. For this purpose, we propose to detect the sky in the image in order to find the vertical line bundle.

A. Sky detection

The sky detection in the images is a subject which has often been treated for outdoor robotics [START_REF] Li | Estimating Head Orientation Based upon Sky-ground Representation[END_REF] but also for image orientation [START_REF] Luo | Automatic Image Orientation Detection via Confidence-Based Integration of Low-Level and Semantic Cues[END_REF]. Most of these systems use color and/or texture information in order to manage the different appearances of the sky (clear, gray, cloudy,. . . ). In our approach, in order to obtain a fast algorithm, we propose to only use the brightness information. With that aim, we suppose that the sky represents the brightest part of the image, which is a reasonable assumption. The second suggested hypothesis is that the sky is almost always in the part close to the omnidirectional image border. Except if the camera is turned upside down, this assumption is always true and we will see that even this case can be easily resolved. The algorithm can then be described in the following way:

1) Finding the point of maximum brightness in the image.

2) Considering a circle centered on the image center and with a radius Rmax -5, where Rmax is the radius of the omnidirectional image perimeter (see figure 8(a))

3) Normalizing by the maximum brightness, the brightness of the pixels which belong to this circle. 7) Using a region growing process by using initial seeds determined from the complementary arc of circle previously detected (see figure 8(d)).

In the case where the sky is not detected, we apply the same algorithm in considering an inner circle, rather than an outer circle as described previously. However, this happens only when the UAV is flying upside down.

B. Horizontal Plane Detection

The trihedron -→ u 1 , -→ u 2 , -→ u 3 permits to define 3 planes, P ij which contain O 1 , -→ u i and -→ u j , i = j. These 3 planes define 3 different partitions of the sphere. horizontal plane is in fact plane P ij which partitions the spherical image without separating points of the sky (Fig 9). However, let us note that in few cases, it can happen that two P ij planes can be candidate. In this case, we cannot find the horizontal plane and we need to make a tracking of this plane in the sequence in order to clear up this ambiguity.

Once the horizontal plane has been detected, the vertical direction of the scene is simply vector -→ u i which is not contained in the horizontal plane.

C. Pitch and roll estimation

Finally, knowing the vertical direction of scene -→ N = (n x , n y , n z ), we deduce roll ρ and pitch ψ angles as follows :

ψ = n x | n x | × arccos |n z | n 2 x + n 2 z (4) ρ = -n y | n y | × arccos   |n z | n 2 y + n 2 z   (5)
V. EXPERIMENTAL RESULTS

In this section, we show results on different images taken with a paracatadioptric sensor. We present in figures 10(a), 10(b), 11(a), 11(b) the detected lines on these images. Figures 10 11,(c) to (f) present two orthogonal bundles (in red lines and in blue lines) of each scene. We can see that vertical and horizontal lines are correctly detected. Moreover, we can note that in each case, the horizontal plane is detected without any ambiguity because the sky represents an important part of the image. This property is often checked, it is one of the main advantages when we use omnidirectional sensors.

Since we do not have ground truth values of the roll and pitch angles for these images, we only propose a visual validation of the results. Nevertheless, these results show an interesting stability between the consecutive images and appear to be coherent with the images. For the first image (

VI. CONCLUSION AND PERSPECTIVES

In this paper, we have developed a new method of attitude estimation for a UAV in urban environment by catadioptric vision. We propose to detect bundles of parallel lines in the image in order to detect horizontal and vertical planes. The attitude can then be easily estimated. The main contributions deal with a new line detection algorithm for calibrated central catadioptric sensor and with an original approach in order to detect the horizontal plane and the vertical direction based on the sky detection in the image. Experimental results display very promising results which have to be compared with measures obtained from an inertial sensor. In our future work, we propose to improve the line detection method by including the chain division directly in the prediction/verification process in order to avoid the heuristic step. In a more interesting way, we plan to use direction information of the parallel lines in order to obtain the orientation of the 3D planes and thus to obtain a 3D estimation of the scene.

Fig. 1 .

 1 Fig. 1. Omnidirectional images captured from a UAV, (a) High altitude (horizon line is largely visible), (b) Low altitude in urban environment (horizon line is almost non-existent).

Fig. 2 .Fig. 3 .

 23 Fig. 2. Image formation model. Example of projection via the unitary sphere for a 3D point.

Fig. 4 .

 4 Fig. 4. (a) Original image, (b) edge detection, (c) chains of edgels (d) detected lines

Fig. 5 .

 5 Fig. 5. Parallel line projection onto the surface of the unitary sphere.

Fig. 6 .

 6 Fig. 6. Two orthogonal bundles of parallel lines.

Fig. 7 .

 7 Fig. 7. Scene with three buildings and the obtained orthogonal frame.

4 )Fig. 8 .

 48 Fig. 8. (a) Original image with the circle of points used for sky detection, (b) binarized histogram of the circle, (c) probable position of the sky with the initial seeds for the region growing (d) detected sky after region growing process

Fig. 9 .

 9 Fig.9. Horizontal plane detection.

  Fig 10 left row), we obtain ψ = 19.7 • and ρ = 15.2 • , for the second image (Fig 10 right row), we obtain ψ = -37.5 • and ρ = 17.8 • , third image (Fig 11 left row) ψ = 37.5 • and ρ = -20 • . Let us note that the image (Fig 11 right row), corresponds to the sensor placed horizontally and we obtain : ψ = 2 • and ρ = -1 • .

Fig. 10 .

 10 Fig. 10. Attitude estimation, left row : first image, right row : second image. (a),(b) Line detection, (c)-(f) orthogonal bundles

This work has been supported by GIP-ANR, CaViAR Project


Sébastien Garcia
email: sgarcia@phys.ethz.ch

Jakob Reichel

Romain Long
email: long@lkb.ens.fr

Improving the lifetime in optical microtraps by using elliptically polarized dipole light

Tightly focused optical dipole traps induce vector light shifts ("fictitious magnetic fields") which complicate their use for single-atom trapping and manipulation. The problem can be mitigated by adding a larger, real magnetic field, but this solution is not always applicable; in particular, it precludes fast switching to a field-free configuration. Here we show that this issue can be addressed elegantly by deliberately adding a small elliptical polarization component to the dipole trap beam.

In our experiments with single 87 Rb atoms laser-cooled in a chopped trap, we observe improvements up to a factor 11 of the trap lifetime compared to the standard, seemingly ideal linear polarization. This effect results from a modification of heating processes via spin-state diffusion in state-dependent trapping potentials. We develop Monte-Carlo simulations of the evolution of the atom's internal and motional states and find that they agree quantitatively with the experimental data. The method is general and can be applied in all experiments where the longitudinal polarization component is non-negligible.

I. INTRODUCTION

When a linearly polarized electromagnetic beam is focussed so tightly that the paraxial approximation breaks down, its polarization near the focus is no longer purely transverse, but acquires a longitudinal component that increases with the confinement of the field and leads to polarization gradients in the strongly confined region. The effect has been known for a long time [1], but currently gains importance in the context of light-matter interfaces with ultracold atoms. While similar effects play an enabling role in the new field of chiral quantum optics [2], they complicate the use of strongly focussed, far-detuned laser beams for trapping and manipulating single atoms. Such single-atom tweezers [3] are a powerful tool with many applications. They have been used, for example, to prepare a single trapped atom by collisional blockade [3], to entangle single atoms by Rydberg blockade [4,5], to laser cool a single atom to its vibrational ground state [6,7], and to couple an atom to a near-field optical structure [8]. When realized with a pigtailed optical fiber, a single-atom tweezer also enables an attractively simple source of narrowband single photons [9].

For standard dipole traps, the polarization is commonly chosen to be linear to ensure an equal light shift for all magnetic sublevels of the ground state [10]. In single-atom tweezer experiments, where tight focussing is mandatory, the atom experiences an elliptical polarization with spatial variations on the wavelength scale even for a linearly polarized input field. The result is a vector light shift, which depends on the magnetic quantum number and can be described by a an effective, "fictitious" magnetic field [11]. Fictitious magnetic field gradients in combination with diffusion between Zeeman sub-states of the atoms induced by some near-resonant light are detrimental as they lead to heating mechanisms that prevent efficient cooling and reduce trap lifetime [6,7]. While an actual compensation of fictitious magnetic field gradients has been achieved for nanofiber-based dipole traps using counter-propagating fields [12,13], this is difficult to realize in other optical microtraps. The approach that has been followed so far with optical tweezers is to mitigate the effect by adding a stronger, real magnetic field in a direction orthogonal to the fictitious field in order to reduce the field gradient [6,7]. It is however not always possible to add strong real magnetic fields, whose switching dynamics are usually slow.

In this article, we experimentally demonstrate that a properly chosen non-linear input polarization can be used instead of a real magnetic field to mitigate the damaging effects of vector light shift gradients. Our singleatom tweezer is chopped at rather high frequencies up to 4 MHz. In this way, spectroscopy, optical cooling and single-photon generation can be performed on an unperturbed atom during the field-free periods, while still providing a strong effective trapping potential [9,14,15]. As we show, a slightly elliptical polarization for the input field of the chopped tweezer can improve the lifetime of the trapped and laser-cooled atom by more than an order of magnitude. This is against the common belief that linear polarization is the best choice for an optical dipole trap [16], which actually holds only as long as the longitudinal component of the electromagnetic field and Zeeman sub-states diffusion can be neglected. We analyze how the experimental lifetime of the trapped atom depends on the trap chopping frequency and on the dipole trap light polarization. We then develop a model based on trap modifications due to quantum jumps between different Zeeman sub-states of the atom and perform Monte-Carlo simulations in very good agreement with the experimen- tal data.

II. MEASUREMENT OF SINGLE ATOM LIFETIME FOR DIFFERENT POLARIZATIONS

In our experimental setup (see Fig. 1) described in detail in Ref. [9], we have developed a miniature and robust fiber-pigtailed optical tweezer, where the same fiber is used to trap a single 87 Rb atom and to read-out its fluorescence. We have also demonstrated that, once loaded, the pigtailed tweezer can be used as a singlephoton source at 780 nm. To load the tweezer, we produce a laser-cooled atoms cloud in a magneto-optical trap (MOT) at the focus of a far off-resonance optical dipole trap with a wavelength of 810(1) nm and a measured waist of 1.4(1) µm (the number in parentheses is the numerical value of standard uncertainty referred to the corresponding last digits of the quoted result). To eliminate trap-induced light shifts, which reduce the efficiency of the MOT cooling, constitute a source of spectral broadening, and to avoid the generation of 780 nm photons by anti-Stokes Raman scattering of the trapping light inside the fiber, we chop the dipole trap light with a 50% duty cycle using an acousto-optical modulator (see Fig. 1

.d) .

If the chopping frequency is much larger than twice the largest trap oscillation frequency, the atom is expected to stay trapped in this attractive time-averaged potential. When the dipole trap light is on, the trap depth is 5.6(8) mK, and the calculated transverse and longitudinal trap frequencies are f t = 167(24) kHz and f l = 22(4) kHz.

We have investigated the effect of the chopping frequency on the lifetime of the atom inside the trap (see Fig. 2) in the presence of the MOT beams used to load the dipole trap. The lifetime is obtained by an exponential fit of the survival probability of the atom versus the duration of the trapping. To be mainly limited by background gas collisions, we work in the weak loading regime, where loss due to a second atom entering the trap is negligible. For linearly polarized dipole trap light, the lifetime was limited by background gas collisions at a value of about 500 ms for chopping frequencies above 3 MHz. Below 3 MHz, the lifetime decreases with the chopping frequencies and no single-atom trapping signal could be detected below 500 kHz. So, with this polarization, the chopping frequency needs to be much larger, by about one order of magnitude, than the largest calculated trap frequency. For chopping frequencies between 750 kHz and 1.5 MHz, we observe a strong improvement of the lifetime by adding a small circular component to the polarization of the dipole trap light. The lifetime is found optimal for an elliptical polarization characterized by the angle α opt = arctan(E b /E a ) = 7.0(2) deg, where E a and E b are the semi-major and semi-minor axes of the electric-field ellipse, respectively. At a chopping frequency of 750 kHz, the lifetime is improved by a factor 11.

These results cannot be explained by the chopping of the dipole trap light, which is expected to be negligible when the trap chopping frequency f m becomes higher than the largest parametric heating frequency (2f t ) of the trap [17]. Moreover, this effect applies to all polarization configurations and so cannot explain the improved lifetime observed for slightly elliptical polarization. So, we can deduce from these results that additional heating mechanisms reduce the lifetime for a linearly dipole trap light and that this effect can be compensated by using an elliptical polarization.

III. EFFECT OF THE POLARIZATION ON THE OPTICAL TRAPS FOR THE DIFFERENT ZEEMAN SUB-LEVELS

The modification of the lifetime induced by polarization finds its origin in the dipole potential of the ground state which depends on the Zeeman sub-level the atom occupies. For a dipole trap light frequency close to the D1 and D2 transition frequencies of an alkali atom, this potential is given by [7,18]:

U dip (r) = U dip,0 (r) 1 + δ 1 -δ 2 2δ 1 + δ 2 C (r) . g F h F (1)
with U dip,0 (r) = hΓ 24

Γ δ 1 + 2 Γ δ 2 I (r) I S (2)
the scalar dipole potential. In these equations, Γ is the natural linewidth of the excited P levels, δ i = ω L -ω i are the detunings between the laser and the D1 and D2 transition frequencies (i = 1, 2 respectively), I (r) the light intensity and I S the saturation intensity. We note F the atom angular momentum operator (norm F = 2 here) and

g F = [F (F + 1) -I(I + 1) + J(J + 1)]/[F (F + 1)]. C (r) = e p (r) × e p (r)
* is a vector characterizing the ellipticity and direction of the polarization represented by a unit vector e p (r). For a polarization e p = i sin(α)e x + cos(α)e y , we have C = sin(2α)e z : for linear polarization C = 0, and for a circular polarization C = 1. When C is non-zero, the different m F sublevels experience different light-shifts that are analogous to energy shifts induced by a fictitious magnetic field given by:

B fict (r) = -U dip,0 (r) |δ 1 -δ 2 | 2δ 1 + δ 2 g F µ B g L,F C (r) , (3)
where µ B is the Bohr magneton and g L,F is the Landé g-factor, which is equal to g F in the usual limits where the electron spin, electron orbital and nuclear Landé gfactors are approximated to g L,S 2, g L,L 1 and g L,I 0, respectively.

In our optical tweezer, a Gaussian beam is strongly focussed down to a waist w 0 = 1.4(1) µm. For such a diverging beam, a purely transverse incident light (which is linearly-polarized along the y-axis) generates at the focus plane strong variations of the local polarization due to interferences of the different focused light rays. The local C (r) characteristic vector is calculated via vector Debye integral [1] and we plot the product C (r) |U dip,0 (r) |, which is opposite of the fictitious magnetic field in our case, on Fig. 3.a. The effect is well approximated by a gradient C = 0.267(36) µm -1 of the C (r) = (C y, 0, 0) vector. Figure 3.c presents the different potentials of the five m F sub-levels along the y-axis in our trap by choosing the quantization axis of angular momentum as the unit vector of x-axis. At first order, the traps are displaced with state-dependent trap centers:

y c (m F ) = w 2 0 4 δ 1 -δ 2 2δ 1 + δ 2 C g F m F = -m F × 16.7(8) nm .
(4) We can see on Fig. 3.a that an atom crossing the x-axis while maintaining its spin will see a flip of the orientation of the fictitious magnetic field. This situation is analogous to the evolution of atom spins crossing the center of a linear magnetic trap. In our linearly polarized dipole trap, the scalar potential provides an additional confinement which leads only to a displacement of the trap centers instead than spin flip losses.

If a large circular component is present in the polarization of the dipole trap light before the focus, it will induce a strong uniform z-component of the C vector, which will dominate over the linear polarization gradients, making them negligible. The dipole potential is then given by :

U dip (r, m F ) = U dip,0 (r) 1 + δ 1 -δ 2 2δ 1 + δ 2 g F m F C (r) (5)
where the quantization axis of the angular momentum is chosen in the direction of the local C vector. The traps of the different sub-levels have the same center, but differ in trap depth and oscillation frequencies, as shown by Fig. 3.d. This effect increases with the weight of the circular component of the polarization. As can be seen in Fig. 3.b, an atom crossing the y-axis will experience a continuous evolution of the orientation of the fictitious magnetic field whose variation decreases when the circular component increases. For a large enough circular component, the atom spin will be able to follow adiabatically the orientation of the field.

The natural question is then to determine what is the minimal amount of circular component (i.e. the smallest value of the z-component C z of the C vector) one needs to add to obtain an adiabatic evolution and thus mitigate the polarization gradient effects. We can estimate a lower bound for C z by using the criterion for adiabatic evolution of the spin given by ω L |dθ/dt| where ω L is the Larmor precession angular frequency and |dθ/dt| is the change of the orientation angle θ of the field during the atom motion. Considering that C = (C y, 0, C z ) define the angle θ and using an harmonic approximation for the trap, one finds that the maximum of angle variation is obviously in the center of the trap where the atom speed v y is maximum and is given by ponent fulfills:

|C z | h|C ||v y | U dip,0 (0) g F δ1-δ2 2δ1+δ2 . ( 6
)
For our optical tweezer, we find that |C z | 0.018 by estimating the maximum speed v y at the center of an harmonic trap with by v y = 2k B T D /m Rb , where m Rb is the 87 Rb atom mass and T D 140 µK the Doppler temperature. We find indeed experimentally that for our optimal polarization |C z | = 0.243 (5). This is one order of magnitude larger than the estimated lower bound, ensuring an adiabatic evolution of the spin, that minimizes the effect of state-dependent trap centers induced by the polarization gradient.

IV. HEATING MECHANISMS

In order to get a better understanding, we identify different heating processes of the trapped single atom that compete with the cooling provided by the MOT beams. For the linear and elliptic polarization configurations, they originate from the change of spin sub-level m F due to the excitation by the MOT beams and desexcitation to the ground state. The evolution of an atom is schematically represented by Fig. 4, for the linear and elliptic polarizations, respectively. When the dipole trap is on, light shifts are so large (117 (18) MHz at the center) that the excitation by the MOT beams can be neglected, and the atom oscillates freely in the trap while conserving its spin state. When the dipole trap is switched off, the MOT beams become almost resonant and provide both cooling and spin state transitions. When the dipole trap laser is switched on again, the quantum jumps of the spin-state may have led the atom to a different m F -state and so to experience a different trap. We have then three different heating mechanisms :

• for any polarization: when the dipole trap is off, the atom can escape by spatial diffusion. This process is negligible if the chopping frequency is much larger than twice the largest trap oscillation frequency.

• for linear polarization: the spin-state diffusion gives rise to a random trap shaking due to the displacement of the state-dependent trap centers.

• for elliptical polarization: the spin-state diffusion leads to heating by a random modulation of the trap frequencies.

The different heating effects are larger for lower chopping frequencies, because the atom evolves longer in the trap and in the MOT between two changes, which gives larger energy differences. An optimal elliptic polarization exists mitigating the heating by random trap shaking in the linearly polarized beam and the heating by random trap-frequency modulation for moderate elliptic polarization, while the modulation amplitude of the latter increases with the weight of the circular component.

In dipole traps with large waist, polarization gradients are negligible and only trap-frequency modulation induced by elliptical polarization limits the lifetime. The best polarization is then the linear one that cancels this effect [16]. If polarization gradients cannot be neglected, the two last heating mechanisms will be present whenever spin-state diffusion occurs even for static trap. This is specially the case for every experiment where cooling of the atoms by near-resonant light is undertaken [6,7]. In this case, a non-zero polarization minimizing the heating exists. Our experimental situation emphasizes the effect of spin-state diffusion by using a chopped trap where the cooling light is nearly resonant light when the trap is switch off.

V. SIMULATION OF SINGLE ATOM LIFETIME

To confirm the above explanation of the observed lifetime evolution with the trap modulation frequency in terms of trap modifications implied by quantum jumps of spin state, we perform Monte-Carlo simulations of the atom motion in our pulsed trap. For a given polarization, the simulation considers the evolution of several thousands of atoms subjected to about a million cycles of alternating trapping and cooling phases. In order to obtain realistic computational times, we make some approximations that simplify the calculations while keeping the main physical features of the experimental situation.

The first approximation consists in limiting the simulation to two dimensions given by the strong axes of the trap (transverse to the dipole trap laser propagation axis) which have oscillation frequencies an order of magnitude higher than the weak axis one. Thus, the energy change during one modulation cycle is much more important along these axes. We also consider truncated harmonic traps which provide a good approximation of the Gaussian potential in the center where the atom mostly evolves, and which have a depth limited by the maximum dipole trap light shift. For a linear polarization, the state dependent traps are shifted along the y-axis by y c (m F ) (cf. Eq 4). For an elliptic polarization, we change the trap depth and frequencies according to Eq. 5.

The exact treatment of the cooling in 6 nearly resonant MOT beams implies to know the 5 relative phases between the beams and to resolve the 78 coupled optical Bloch equations. We simplify this description by taking an usual classical damping force -m Rb γ v v where γ v is the characteristic damping coefficient. In our setup, the cooling on the y-axis is stronger. Due to geometrical hindrance, two pairs of MOT beams have indeed a 10 deg deviation from this axis, so we use here a damping coefficient 2 cos(10 deg)γ v . The Doppler cooling limit is considered by canceling this force if the atom kinetic energy is lower than the thermal energy k B T D at the Doppler temperature when the cooling phase starts. To approximately describe the change in m F spin state of the atom, we calculate, from the Clebsch-Gordan coefficients, the probabilities of excitation to the different sub-levels of the 5 2 P 3/2 F = 3 excited state in an isotropic light field and the probabilities of spontaneous emission down to the 5 2 S 1/2 F = 2 sub-levels. It gives us the probabilities to change from a state to another per photon scattering event. For a given initial state, the final state probabilities after the cooling phase are then given by its duration τ 1/2 = 1/(2f m ), the average scattering rate Γ ch and the analytic solution of the coupled equations of state population evolution. These two parameters γ v and Γ ch of the cooling phase are difficult to estimate carefully in our system because we need intense MOT beams (50 mW/cm 2 ) to provide efficient trap loading, thus we can not apply the usual low saturation limit. Nevertheless, an heuristic equation established in Ref. [19] (Eq. 4) and the measured rate of spontaneously emitted photons in the detection respectively give some approximate initial values of γ v and Γ ch . These values are then adjusted to fit the experimental data. We find γ v = 3.3 kHz which is 30% lower that the initial value, and with Γ ch = 35 MHz that is roughly three time higher than the observed spontaneous emission. The latter result can be explained by the fact that we are using beam intensities well above the saturation, where stimulated emission processes dominate.

We initiate a simulation by picking an atom in m F = 0 state and randomly distributed in phase space with a Gaussian density probability characterized by the Doppler temperature T D and the trap frequency. We start the first cycle by the trapping phase and calculate the position and speed after a chopping period half τ 1/2 of evolution. Then, we let the atom move under the cooling force for another τ 1/2 . Afterward, we randomly change the internal state according to the calculated probabilities and reckon the atom energy from its state, position and speed in the dipole trap. We repeat this cycle until the atom energy reaches a value higher than the trap depth (which means that it escapes from the trap) or until the time exceeds a random value from an exponential distribution. The latter takes into account the background gas collisions which limit the atom lifetime to 550 ms according to the experimental data. By using the simulation without spin state changes, we first verified that the heating due to the chopping of the dipole trap is negligible for chopping frequencies higher than 400 kHz, as expected since 2f t 340 kHz. For the different polarization configurations in presence of spinstate diffusion, the lifetimes extracted from the survival probabilities of 500 simulated atoms are presented on Fig. 5. The results show the improvement of the life-time that was observed experimentally for an elliptic polarization with a small circular component compared to a linear polarization, and agree quantitatively well with the experimental data. As a larger circular polarization component of the dipole trap light is added (|C z | increasing), the lifetime decreases and almost cancels for a pure circular polarization even for large modulation frequencies. This explains why we were not able to detect any atom for large circular components in the polarization of the dipole trap light. These simulations confirm the existence of an optimal elliptic polarization that limits heating due to atom spin state quantum jumps in high NA tweezers.

VI. CONCLUSION AND OUTLOOK

We have shown how some deleterious effects of polarization gradients in strongly focused optical tweezers can be mitigated by using elliptical polarization of the incoming dipole trap light. This method is general and can be applied in any experiment where the light field is tightly confined so that the longitudinal component of the polarization is non-negligible and where spin-state diffusion occurs due to the presence of near-resonant light, required for example to cool the atoms inside the traps. Besides strongly focused optical tweezers, this also includes optical nanofiber [20] and optical microcavities [21,22].

FIG. 1 .

 1 FIG. 1. (a) Experimental setup: the fiber-pigtailed optical tweezer is placed inside a cubic 25 mm size all-glass cell. To load the dipole trap, we produce a cloud of laser-cooled atoms in a MOT with three retro-reflected 1 mm diameter beams, two of them crossing at an angle of 20 deg to avoid clipping at the lens. (b) Image by fluorescence of a 87 Rb atomic cloud in the MOT. For this image, the rubidium pressure inside the cell has been strongly increased to make the cloud visible. (c) With the same high-level pressure and by sending resonant light at 780 nm in the fiber, we can see the position of the tweezer focus by imaging the fluorescence of the vapor background atoms. As the focus at 810 nm is 2 µm apart, the dipole trap light is well focussed at the center of the MOT. (d) The optical tweezer collects also the resonant photons from trapped single atoms. An acousto-optic modulator (AOM) chops with 50% duty cycle the dipole trap laser whose polarization is controlled by two waveplates .

FIG. 2 .

 2 FIG. 2. Observed lifetimes as function of trap modulation frequency for linear polarized dipole trap light (black squares) and for an optimized elliptic polarization with αopt = 7.0(2) deg corresponding to |Cz| = 0.243(5) (red dots). The error bars represent the standard deviation of fitted lifetimes for random exponential distributions with same lifetime and same number of trapped atoms (on the order of 10 3 per point) than the data. Inset: example of lifetime extraction by exponential fit (green) of experimental survival probability decay (black dots) as function of time (∆t) elapsed since the atom entered in the trap, for a linear polarization with 3 MHz chopping frequency.

5 FIG. 3 .

 53 FIG. 3. Dipole potential in the fiber-pigtailed tweezer: (a) Representation of the vector C (r) |U dip,0 (r) |, which is opposite of the fictitious magnetic field B fict by blue vectors in the xy focal plane (the beam propagates towards increasing values of z) for a linear polarization aligned with the y-axis. The red density plot represents the scalar dipole potential. (b) Same as in (a) for optimal-lifetime elliptic polarization. (c) Dipole potentials of the five Zeeman sub-levels from mF = -2 (red) to mF = 2 (blue) of the F = 2 ground state for a linear polarization aligned with the y-axis. The traps are mainly displaced by a regular interval (the trap centers of the extreme states are indicated by dashed vertical lines). (d) Same as in (c) for the case of the optimal-lifetime elliptic polarization. In this case, the trap depths and the oscillation frequencies are modified.

FIG. 4 .

 4 FIG.4. Simplified schemes (with only two spin states represented in red and blue) of the cycles followed by the trapped atom. The MOT cooling when the dipole trap is off competes with the heating of trap potential fluctuations induced by the spin-state quantum jumps.

1 FIG. 5 .

 15 FIG.5. Results of lifetime as function of trap pulsing frequency given by atomic motion simulations for different dipole trap light polarization with 500 simulated atoms for each point. The error bars represent the 4.7% relative standard deviation for 500 events of an exponential distribution.

ACKNOWLEDGMENTS

We acknowledge funding from the ANR SAROCEMA project, grant ANR-14-CE32-0002 of the French Agence Nationale de la Recherche and from Émergence-UPMC-2009 research program. We thank D. Maxein and L. Hohmann for contributions in the early stage of the experiment.