Abstract

We demonstrate an experimental technique for high-resolution, high-speed spatial manipulation of atom clouds. By combining holographically engineered laser beams from a spatial light modulator with off-axis shear mode acousto-optic deflectors, we manipulate 1 × 3 arrays of cold atoms with individual site addressability. Additionally, we demonstrate smooth 2-dimensional motion of atomic ensembles, and the ability to guide multiple atomic ensembles independently.

©2007 Optical Society of America

1. Introduction

Spatial control of atoms has recently attracted considerable interest for possible uses in a variety of applications. The successes of fountain and beam-based atom interferometry for measuring rotations and gravity gradients [1] have encouraged prospects for compact, guided atom interferometry analogous to optical fiber sensing devices. To this end, integrated atom optical approaches employing current carrying wires on substrates to guide atoms in magnetic fields have been demonstrated [2-4]. Optical techniques have also been considered as candidates for atom guides and for quantum computing applications. Micro optical devices can confine atoms in static arrays of addressable optical dipole traps [5-6]. Optical lattice geometries offer large numbers of regularly spaced trap sites [7], but control of individual sites is difficult. Time-dependent traps for controlling atom-atom interactions have been proposed [8], and reconfiguration of optical traps for atoms using spatial light modulators (SLMs) has recently been discussed [9]. These devices have been used to create hollow atom guides and traps [10-13] and to update optical potentials on the millisecond time scale [14-16], but because of interframe artifacts, they must be used with additional algorithms for smooth operation [17]. SLMs are capable of generating complex intensity profiles and arrays of traps, but optical interference between trap sites may produce undesired effects.

Dynamic spatial control of atoms using acousto optics or SLMs has been of interest recently. Shin et al. [18] used an acousto optic modulator (AOM) driven by two RF frequencies simultaneously to create a double well potential for condensates in a coherent manner for a subsequent interference measurement. Multiple frequency AOMs have also been used to create optical box potentials for Bose Einstein condensates [19] and compensate for irregularities in optical potentials for atoms [20]. Recently, Boyer et al. [15] showed dynamic splitting of a condensate with a ferroelectric SLM with kilohertz refresh rate. Bergamini et al. [13] used a nematic SLM to create single-atom traps that in principle could be dynamically moved. Yavuz et al. used an AOM to steer resonant light to two separated atom traps to control interactions between them [21].

In this paper, we demonstrate, for the first time to the best of our knowledge, a technique that allows smooth, multidimensional, dynamic control of the spatial locations of multiple atom clouds. To accomplish this we combine the holographic capabilities of an SLM with the high-speed and resolution of acousto-optic deflectors (AODs) to control arrays of atoms confined in distinct hollow optical potentials in a smooth manner. Although 1D atom guidance in static hollow beams using SLMs has recently been demonstrated [10-12], we greatly extend this capability by manipulating multiple atom traps in parallel, splitting an atom cloud into multiple sites, and moving atom clouds along 2D trajectories. We demonstrate this technique with crossed hollow laser beams, which confine atoms to regions of low light intensity. Such confinement lowers photon scattering rates, light-assisted collisional losses, and energy level shifts relative to red-detuned traps [23].

The use of AODs allows the formation of multiple traps that are frequency shifted from each other, reducing interference effects. In addition, since AODs are driven with RF sources, the intensities and locations of the deflected beams can be controlled with high resolution and dynamic range exceeding the capabilities of liquid crystal SLMs. An AOD that operates in slow-shear mode is especially promising for atom manipulation, as these can have high deflection angles and large active apertures (∼1 cm) with flicker-free, ∼50kHz update rates.

2. Hollow beam deflection with AODs

AODs are generally used in beam scanning applications with standard Gaussian laser beams. When a collimated beam passes through an AOD, it is diffracted by an acoustic RF wave in the AOD crystal and up- or down-shifted in frequency by integral units of the RF drive frequency. The angular deflection is proportional to the RF drive frequency, so that when the deflected beam is focused through a lens, the focused spot can be moved in real time simply by scanning the RF drive frequency. Additionally, if multiple RF drive frequencies are used simultaneously, multiple spots are formed in the focal plane, and AODs have long been used for spatially dispersing RF spectra. AODs are often specified by the number of resolvable focused Gaussian spots over the RF bandwidth. This number can vary from fewer than ten to several thousand depending on the type of acousto optic crystal used and the RF bandwidth of the device, etc. For this work, we have used an AOD with 1200 resolvable spots (Crystal Technology 4075-S). At 780 nm, light is deflected by approximately 1 mrad / MHz.

AODs can also deflect shaped laser beams that have more spatial frequency content, preserving the intensity profile of the incident laser beam. In this case, the number of resolvable beams at a lens focus will be reduced as the spatial frequency bandwidth of the beam is increased. For beams with very high spatial frequency content, the deflected beam will become distorted if the Bragg condition cannot be met for all spatial frequency components simultaneously. To the best of our knowledge, this work also reports the first use of AODs for the manipulation of beams with high spatial frequency content.

For the manipulation experiments we have used hollow beams that have been generated by multiplying the wavefront of a Gaussian laser beam by an azimuthal phase exp(inθ), where n is the charge number of the resulting beam, using a nematic liquid crystal SLM. Figure 1(a) shows Gaussian and hollow beams with n = 0, 1, 2, 4, and 8. By sending these beams through an AOD driven by four simultaneous frequencies, 1 × 4 arrays are generated [Fig. 1(b)]. The beams are imaged onto a CCD camera with a 500 mm focal length lens and are separated by 0.9 mm with a drive frequency difference of 2 MHz. For these values of n, the AOD preserves the intensity profile of the incident beam. The deflection makes the beams slightly compressed in the deflected direction, which is compensated prior to the AOD by a small modification to the SLM phase profile. For comparison, we show a 1 × 4 array generated solely by the SLM. The phase profile applied to the SLM was generated using a Gerchberg-Saxton algorithm [22]. Because each site is generated with the same polarization and frequency of light, interference effects do not wash out with time.

 figure: Fig. 1.

Fig. 1. (a). Hollow beams formed by the SLM. (b). 1 × 4 arrays formed by driving the AOD with four frequencies simultaneously. c) 1 × 4 array created solely by the SLM.

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In addition to statically generated arrays of beams, each beam in a 1 × N array generated by the AOD can be independently and dynamically controlled using voltage-controlled oscillators (VCO). By stacking two AODs orthogonally, we can move the 1 × N arrays in the orthogonal direction. In Fig. 2 we demonstrate movement of a 1 × 4 array of n = 1 beams, with two of the sites controlled independently with triangle waves of periods of 20 and 40 ms. The RF drive frequency required to separate two n = 1 beams from one another is a few hundred kilohertz. The interference resulting from the RF frequency difference effect will be noticeable only when the beams are directly overlapped and the RF frequency difference approaches 0.

 figure: Fig. 2.

Fig. 2. 1 × 4 array of n=1 hollow laser beams. The two central beams are modulated by a triangle wave with periods of 20 and 40 ms as the 1 × 4 array is scanned from left to right across the frame. Time between frames is ∼3 msec. (Movie: 0.28 Mb) [Media 1]

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3. Experiments with Atoms

The potential energy U(r) of a two-level atom in an intensity distribution I(r) is given by

U(r)=ħΔ2ln[1+I(r)/Io1+4(Δ/Γ)2]

where Δ=ωL - ωo denotes the laser frequency (ωL) detuning from the atomic resonance (ωo) in rad/sec. For 85Rb, the saturation intensity I o = 1.6 mW/cm2 and the natural linewidth Γ=2π × 6.1 MHz. When Δ/Γ ≫ [I( r)/I 0]1/2, Eq. (1) can be approximated as

U(r)=ħΓ8[(ΓΔ)I(r)Io]

The scattering rate Γsp of an atom in this field is approximated as

Γsp(r)=Γ8[(ΓΔ)2I(r)Io]

By blue-detuning the optical potential, the atoms seek low intensity and the scattering rate is reduced substantially. For this demonstration, we have used higher charge number beams for trap formation, because this maximizes the use of available laser power while keeping scattering rates low [10-12]. By crossing hollow beams with high charge number, atoms are confined in a “box” potential in which they primarily sample only the field-free region. Although box potentials could, in principle, be formed with red-detuned beams, a large volume trap for atoms would require high laser power in addition to a large detuning to compensate for the atom confinement in the high intensity portion of the beam. The low scattering rates in blue-detuned traps have allowed moderate laser powers (< 100 mW) and small (∼1-10 GHz) detunings to guide cold atoms along blue-detuned hollow beams with high charge number [10-12].

3.1 Optical Layout

A collimated Gaussian laser beam with 1/e 2 radius ω0=1.7mm is modified by a reflective SLM (Boulder Nonlinear Systems) to have azimuthal phase exp(inθ) and is passed through an achromat with focal length f 0 = 200 mm located f 0 away from the SLM. The lens is situated 180 mm away from the center of the vacuum chamber. For hollow beams generated in this manner, the plane of peak intensity occurs prior to the focal plane [10]. Figure 3 shows the layout of the crossed beams.

After passing through the chamber and trapping region, the hollow beam is relayed with a series of achromats and mirrors to intersect with itself. For robust manipulation of this intersection point and ease of alignment, we image the MOT region back onto itself using an 8f imaging arrangement, shown in Fig. 3(a). This arrangement consists of two pairs of lenses of focal lengths f 1 = 200 mm and f 2 = 80 mm. The distance between the first and second lenses (and between the third and fourth lenses) is f 1+ f 2 = 280 mm; the distance between the second and third lens is 2f 2. The first and last lenses are each located f1 away from the MOT region. This layout produces a 1:1 reproduction of the intensity and phase profile at the MOT location. With this arrangement, once the 8f relay is aligned so that the input and output beams overlap, they remain overlapped with any transverse motion of the input beam. Furthermore, the trap size can be made larger or smaller simply by moving the focus of the single lens before the chamber, and to accommodate colder, denser, or smaller atom number ensembles, the trap size can also be reduced by lowering n. The crossed beam geometry allows traps of controllable size with nearly unity aspect ratio.

 figure: Fig. 3.

Fig. 3. Optical layout for the crossed hollow beams. (a). Setup for single beam deflection (Movie: 0.24 Mb) [Media 2] (b). Multiple drive frequencies (Movie: 0.38 Mb) [Media 3]

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To move the intersection point in a dynamic manner, an AOD is inserted in the imaging relay, as shown in Fig. 3(b). By modulating the rf of the AOD, we can move the trap along the Z-axis for 1-dimensional trajectories. Additionally, we can move the trap along 2-dimensional trajectories by using a second AOD preceding the chamber. In principle, motion in the third dimension can be controlled by using 2 AODs before the chamber that determine X-Y location of the incident beam at the MOT. In principle, 2D control of trap locations is possible with red-detuned Gaussian beams and orthogonal AOMs, but the trap aspect ratio is only close to unity for small traps and very tight focusing.

3.2Experiment

For these demonstrations, the experiment proceeds as follows. First, 107 85Rb atoms are loaded in 5 seconds into a standard vapor cell magneto-optical trap (MOT). After loading, they are cooled in a 5 msec molasses phase to 5 μK, during which the MOT repump and cooling laser powers are linearly reduced to zero and the MOT laser detuning is linearly increased from 3Γ to 6Γ. The magnetic fields are shut off at the beginning of the molasses phase. Mechanical shutters block both the MOT and repump beams to ensure that no resonant light enters the vacuum chamber during the confinement time.

At the end of the molasses phase, the blue detuned hollow beams are switched on. These beams are derived from an extended cavity diode laser (New Focus Vortex TLB-7000), which is amplified to 200 mW by a tapered amplifier stage (Eagleyard Photonics EYP-TPA-0780-01000-3006-CMT03). 100 mW is coupled into polarization maintaining fiber, whose output is collimated to a 1/e2 intensity waist of 1.7 mm. The diffraction efficiency of the reflective SLM is ∼50% at 780 nm. We have used a detuning of +12 GHz from the F=3 to F’=4 transition, determined by measuring the beat signal between the MOT laser and the blue-detuned light with a fast photodiode. The deflection efficiencies of the AODs are ∼50%, and they each shift the laser frequency by ∼75 MHz. For purely 1D motion, we use only 1 AOD inserted into the 8f relay to increase the available laser power.

At the end of the guiding time T, the blue-detuned light is extinguished and the MOT and repump beams are switched on for 5 msec to image the remaining atoms onto a CCD camera. For all demonstrations with atoms, we have used hollow beams with n=4. An image of the beams at the MOT location is shown in Fig. 4(a). The full-width at half-maximum wall thickness of the beam is ∼70 μm, and the diameter of the central ring measured between wall maxima is ∼440 μm. For 20 mW of laser power, the barrier height is ∼ħΓ/4, or ∼75 μK. The scattering rate of an atom at the peak intensity is Γsp = 2π ∙ 750 Hz. As described above, the use of blue-detuned potentials, which confine the atoms to the dark regions, reduces this scattering rate substantially [23-24].

In Fig. 4(b), we demonstrate oscillatory motion of the confined atoms so that they follow a trajectory x(t) = 0.5A[1-cos(2πt/T)], with A = 0.9 mm, and T = 50 ms. The total frequency change applied to the AOD is ∼4 MHz. The loss of atom number over this period is primarily due to the atoms that were initially captured near the edges of the trap. These atoms will boil out of the trap quickly due to the small detuning. We have compared the loss rate in a static trap (A = 0) and moving trap (A = 0.9 mm) (Fig. 5) by fitting the remaining atom number to a decaying exponential, and found equivalent decay constants. With trap oscillation, the 1/e decay constant is 21 +/- 2 msec; in the static case it is 23 +/- 2 msec. Because the MOT is initially larger than the trap volume, we have used only the data after 10 msec expansion for the fit.

 figure: Fig. 4.

Fig. 4. (a). Image of n=4 hollow beam at the MOT location and cross sectional profile. (b). Fluorescence images of cold 85Rb atoms confined in an n=4 crossed hollow beam that is moving sinusoidally with period 50 ms. Time between frames is 2.5 ms, and the deflection amplitude is 0.9 mm.

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 figure: Fig. 5.

Fig. 5. Atoms remaining in the crossed blue detuned trap as a function of time. Fits use data after 10 msec to eliminate atoms starting outside the trap. For clarity, error bars are not displayed, but are approximately 10% of the measured values, arising from variations in initial MOT number.

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In addition to linear motion of one atom trap, we have demonstrated 2D motion of the trapped atoms, as well as control of three distinct ensembles. For 2D motion, we added an AOD prior to the chamber entrance to provide vertical motion. This second VCO is driven π/2 out of phase with the first to generate the circular motion. Figure 6 shows the atoms moving in a 0.5mm diameter circle in a period of 50 msec.

 figure: Fig. 6.

Fig. 6. 2D motion of atoms. Circle diameter is ∼500 microns. Time between frames is 10 msec. (Movie: 0.6 MB) [Media 4]

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Finally, we have demonstrated 1×3 linear arrays of traps, and independent motion of each trap site by driving a single AOD with three simultaneous RF frequencies (Fig. 7). In the 1 × 3 array, all frequencies start out at 75 MHz. At time T=0, two hollow beams begin moving with equal and opposite sinusoidal motion, and the third remains stationary. The clouds are separated up to ∼1 mm apart from each other and rejoined 50 ms later. Atom loss is higher in this case, because the trapping power at each site is further reduced by a factor of 3. Some of this loss can be observed along the vertical direction in Fig. 7. We note that the spacing between traps and the trap size will scale linearly with the focal length f 0.

Splitting an atom cloud with hollow beams in this manner is clearly not ideal, since the atoms are forced to climb up and down a large optical potential during the splitting process. The atoms then slosh back and forth in their respective potentials; this behavior can be seen in the central site of the 1 × 3 array in Fig. 7. This was done as a matter of technical convenience, as the goal of this report has been to demonstrate the smooth spatial control over trapping sites offered by the AODs.

 figure: Fig. 7.

Fig. 7. Control of multiple atoms clouds with the AOD. Maximum distance between top and bottom cloud is 2 mm. Time between frames is ~3 msec. (Movie 0.3 MB) [Media 5]

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4. Summary and conclusions

This work represents a step towards using purely optical potentials for rapid dynamical control of atom traps. We have demonstrated high-speed cold atom manipulation using a combination of spatial and acousto-optic light modulation. We have used crossed hollow beams with high charge number to smoothly move atom clouds along multidimensional trajectories, and controlled multiple atom clouds independently and simultaneously. The AOD offers flicker-free high-speed control of the atoms with no interframe artifacts. The techniques are potentially of use for atom waveguides and interferometers, and for controlled collisions of atoms.

Acknowledgments

This work was funded in part by the Office of Naval Research and by the Defense Advanced Research Projects Agency.

References and Links

1. J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, “Large area light-pulse Atom Interferometry,” Phys. Rev. Lett. 85,4498–4501 (2000). [CrossRef]   [PubMed]  

2. N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000). [CrossRef]   [PubMed]  

3. Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005). [CrossRef]   [PubMed]  

4. C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, “Interference of Bose-Einstein condensates split with an atom chip,” Phys. Rev. A 72,021604(R) (2005).

5. R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer , “Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits,” Phys. Rev. Lett. 89,097903 (2002). [CrossRef]   [PubMed]  

6. R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, “Interferometer-type structures for guided atoms,” Phys. Rev. Lett. 89,220402 (2002). [CrossRef]   [PubMed]  

7. A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, and I. Bloch, “Coherent collisional spin dynamics in optical lattices,” Phys. Rev. Lett. 95,190405 (2005). [CrossRef]   [PubMed]  

8. T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61,022304(2000). [CrossRef]  

9. D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light odulators in atom ptics,”gOpt. Express 11,158–166 (2003). [CrossRef]  

10. F. K. Fatemi and M. Bashkansky, “Cold atom guidance using a binary spatial light modulator,” Opt. Express 14,1368–1375 (2006). [CrossRef]   [PubMed]  

11. N. Chattrapiban, E. A. Rogers, I. V. Arakelyan, R. Roy, and W. T. Hill, “Laser beams with embedded vortices: tools for atom optics,” J. Opt. Soc. Am. B 23,94–103 (2006). [CrossRef]  

12. D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation,” J. Mod. Optics 53,547–556 (2006). [CrossRef]  

13. S. Bergamini, B. Darquie, M. Jones, L. Jacubowiez, A. Browaeys, and P. Grangier “Holographic generation of microtrap arrays for single atoms by use of a programmable phase modulator,” J. Opt. Soc. Am. B 21,1889–1894 (2004). [CrossRef]  

14. F. K. Fatemi and M. Bashkansky, “Generation of hollow beams by using a binary spatial light modulator,” Opt. Lett. 31,864–866 (2006). [CrossRef]   [PubMed]  

15. V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006). [CrossRef]  

16. W. J. Hossack, E. Theofanidou, J. Crain, K. Heggarty, and M. Birch, “High-speed holographic optical tweezers using a ferroelectric liquid crystal microdisplay,” Opt. Express 11,2053–2059 (2003). [CrossRef]   [PubMed]  

17. V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, “Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms,” J. Mod. Opt. 51,2235 (2004). [CrossRef]  

18. Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt , “Atom Interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. Lett. 92,050405 (2002). [CrossRef]  

19. C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, “Direct observation of sub-Poissonian number statistics in a degenerate bose gas,” Phys. Rev. Lett. 95260403, (2005). [CrossRef]  

20. T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, “Bose-Einstein condensate in a box,” Phys. Rev. A 71,041604R (2005). [CrossRef]  

21. D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006). [CrossRef]   [PubMed]  

22. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35,237–246 (1972).

23. R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59,R1750–R1753 (1999). [CrossRef]  

24. T. Kuga, Y. Torii, Shiokawa N., and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. v. Lett. 78,4713–4716 (1997).

References

  • View by:

  1. J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, “Large area light-pulse Atom Interferometry,” Phys. Rev. Lett. 85,4498–4501 (2000).
    [Crossref] [PubMed]
  2. N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000).
    [Crossref] [PubMed]
  3. Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
    [Crossref] [PubMed]
  4. C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, “Interference of Bose-Einstein condensates split with an atom chip,” Phys. Rev. A 72,021604(R) (2005).
  5. R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer , “Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits,” Phys. Rev. Lett. 89,097903 (2002).
    [Crossref] [PubMed]
  6. R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, “Interferometer-type structures for guided atoms,” Phys. Rev. Lett. 89,220402 (2002).
    [Crossref] [PubMed]
  7. A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, and I. Bloch, “Coherent collisional spin dynamics in optical lattices,” Phys. Rev. Lett. 95,190405 (2005).
    [Crossref] [PubMed]
  8. T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61,022304(2000).
    [Crossref]
  9. D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light odulators in atom ptics,”gOpt. Express 11,158–166 (2003).
    [Crossref]
  10. F. K. Fatemi and M. Bashkansky, “Cold atom guidance using a binary spatial light modulator,” Opt. Express 14,1368–1375 (2006).
    [Crossref] [PubMed]
  11. N. Chattrapiban, E. A. Rogers, I. V. Arakelyan, R. Roy, and W. T. Hill, “Laser beams with embedded vortices: tools for atom optics,” J. Opt. Soc. Am. B 23,94–103 (2006).
    [Crossref]
  12. D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation,” J. Mod. Optics 53,547–556 (2006).
    [Crossref]
  13. S. Bergamini, B. Darquie, M. Jones, L. Jacubowiez, A. Browaeys, and P. Grangier “Holographic generation of microtrap arrays for single atoms by use of a programmable phase modulator,” J. Opt. Soc. Am. B 21,1889–1894 (2004).
    [Crossref]
  14. F. K. Fatemi and M. Bashkansky, “Generation of hollow beams by using a binary spatial light modulator,” Opt. Lett. 31,864–866 (2006).
    [Crossref] [PubMed]
  15. V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006).
    [Crossref]
  16. W. J. Hossack, E. Theofanidou, J. Crain, K. Heggarty, and M. Birch, “High-speed holographic optical tweezers using a ferroelectric liquid crystal microdisplay,” Opt. Express 11,2053–2059 (2003).
    [Crossref] [PubMed]
  17. V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, “Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms,” J. Mod. Opt. 51,2235 (2004).
    [Crossref]
  18. Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt , “Atom Interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. Lett. 92,050405 (2002).
    [Crossref]
  19. C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, “Direct observation of sub-Poissonian number statistics in a degenerate bose gas,” Phys. Rev. Lett. 95260403, (2005).
    [Crossref]
  20. T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, “Bose-Einstein condensate in a box,” Phys. Rev. A 71,041604R (2005).
    [Crossref]
  21. D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006).
    [Crossref] [PubMed]
  22. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35,237–246 (1972).
  23. R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59,R1750–R1753 (1999).
    [Crossref]
  24. T. Kuga, Y. Torii, Shiokawa N., and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. v. Lett. 78,4713–4716 (1997).

2006 (6)

F. K. Fatemi and M. Bashkansky, “Cold atom guidance using a binary spatial light modulator,” Opt. Express 14,1368–1375 (2006).
[Crossref] [PubMed]

N. Chattrapiban, E. A. Rogers, I. V. Arakelyan, R. Roy, and W. T. Hill, “Laser beams with embedded vortices: tools for atom optics,” J. Opt. Soc. Am. B 23,94–103 (2006).
[Crossref]

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation,” J. Mod. Optics 53,547–556 (2006).
[Crossref]

F. K. Fatemi and M. Bashkansky, “Generation of hollow beams by using a binary spatial light modulator,” Opt. Lett. 31,864–866 (2006).
[Crossref] [PubMed]

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006).
[Crossref]

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006).
[Crossref] [PubMed]

2005 (5)

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, “Direct observation of sub-Poissonian number statistics in a degenerate bose gas,” Phys. Rev. Lett. 95260403, (2005).
[Crossref]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, “Bose-Einstein condensate in a box,” Phys. Rev. A 71,041604R (2005).
[Crossref]

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, “Interference of Bose-Einstein condensates split with an atom chip,” Phys. Rev. A 72,021604(R) (2005).

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, and I. Bloch, “Coherent collisional spin dynamics in optical lattices,” Phys. Rev. Lett. 95,190405 (2005).
[Crossref] [PubMed]

2004 (2)

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, “Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms,” J. Mod. Opt. 51,2235 (2004).
[Crossref]

S. Bergamini, B. Darquie, M. Jones, L. Jacubowiez, A. Browaeys, and P. Grangier “Holographic generation of microtrap arrays for single atoms by use of a programmable phase modulator,” J. Opt. Soc. Am. B 21,1889–1894 (2004).
[Crossref]

2003 (2)

W. J. Hossack, E. Theofanidou, J. Crain, K. Heggarty, and M. Birch, “High-speed holographic optical tweezers using a ferroelectric liquid crystal microdisplay,” Opt. Express 11,2053–2059 (2003).
[Crossref] [PubMed]

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light odulators in atom ptics,”gOpt. Express 11,158–166 (2003).
[Crossref]

2002 (3)

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt , “Atom Interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. Lett. 92,050405 (2002).
[Crossref]

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer , “Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits,” Phys. Rev. Lett. 89,097903 (2002).
[Crossref] [PubMed]

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, “Interferometer-type structures for guided atoms,” Phys. Rev. Lett. 89,220402 (2002).
[Crossref] [PubMed]

2000 (3)

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61,022304(2000).
[Crossref]

J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, “Large area light-pulse Atom Interferometry,” Phys. Rev. Lett. 85,4498–4501 (2000).
[Crossref] [PubMed]

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000).
[Crossref] [PubMed]

1999 (1)

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59,R1750–R1753 (1999).
[Crossref]

1997 (1)

T. Kuga, Y. Torii, Shiokawa N., and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. v. Lett. 78,4713–4716 (1997).

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35,237–246 (1972).

Anderson, D. Z.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

Arakelyan, I. V.

Bashkansky, M.

Bergamini, S.

Birch, M.

Birkl, G.

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, “Interferometer-type structures for guided atoms,” Phys. Rev. Lett. 89,220402 (2002).
[Crossref] [PubMed]

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer , “Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits,” Phys. Rev. Lett. 89,097903 (2002).
[Crossref] [PubMed]

Bloch, I.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, and I. Bloch, “Coherent collisional spin dynamics in optical lattices,” Phys. Rev. Lett. 95,190405 (2005).
[Crossref] [PubMed]

Boyer, V.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006).
[Crossref]

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, “Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms,” J. Mod. Opt. 51,2235 (2004).
[Crossref]

Bright, V. M.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

Browaeys, A.

Buchkremer, F. B. J.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer , “Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits,” Phys. Rev. Lett. 89,097903 (2002).
[Crossref] [PubMed]

Calarco, T.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61,022304(2000).
[Crossref]

Cassettari, D.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006).
[Crossref]

Chandrashekar, C. M.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006).
[Crossref]

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, “Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms,” J. Mod. Opt. 51,2235 (2004).
[Crossref]

Chattrapiban, N.

Chuu, C.-S.

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, “Direct observation of sub-Poissonian number statistics in a degenerate bose gas,” Phys. Rev. Lett. 95260403, (2005).
[Crossref]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, “Bose-Einstein condensate in a box,” Phys. Rev. A 71,041604R (2005).
[Crossref]

Cirac, J. I.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61,022304(2000).
[Crossref]

Cornell, E. A.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

Crain, J.

Darquie, B.

Davidson, N.

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59,R1750–R1753 (1999).
[Crossref]

Deb, A. B.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006).
[Crossref]

Dekker, N. H.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000).
[Crossref] [PubMed]

Dholakia, K.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation,” J. Mod. Optics 53,547–556 (2006).
[Crossref]

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light odulators in atom ptics,”gOpt. Express 11,158–166 (2003).
[Crossref]

Diot, Q.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

Drndic, M.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000).
[Crossref] [PubMed]

Dumke, R.

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, “Interferometer-type structures for guided atoms,” Phys. Rev. Lett. 89,220402 (2002).
[Crossref] [PubMed]

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer , “Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits,” Phys. Rev. Lett. 89,097903 (2002).
[Crossref] [PubMed]

Ertmer, W.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer , “Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits,” Phys. Rev. Lett. 89,097903 (2002).
[Crossref] [PubMed]

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, “Interferometer-type structures for guided atoms,” Phys. Rev. Lett. 89,220402 (2002).
[Crossref] [PubMed]

Fatemi, F. K.

Folling, S.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, and I. Bloch, “Coherent collisional spin dynamics in optical lattices,” Phys. Rev. Lett. 95,190405 (2005).
[Crossref] [PubMed]

Foot, C. J.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006).
[Crossref]

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, “Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms,” J. Mod. Opt. 51,2235 (2004).
[Crossref]

Freegarde, T.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation,” J. Mod. Optics 53,547–556 (2006).
[Crossref]

Gerbier, F.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, and I. Bloch, “Coherent collisional spin dynamics in optical lattices,” Phys. Rev. Lett. 95,190405 (2005).
[Crossref] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35,237–246 (1972).

Gericke, T.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, and I. Bloch, “Coherent collisional spin dynamics in optical lattices,” Phys. Rev. Lett. 95,190405 (2005).
[Crossref] [PubMed]

Gherardi, D. M.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation,” J. Mod. Optics 53,547–556 (2006).
[Crossref]

Godun, R. M.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006).
[Crossref]

Grangier, P.

Hanssen, J. L.

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, “Direct observation of sub-Poissonian number statistics in a degenerate bose gas,” Phys. Rev. Lett. 95260403, (2005).
[Crossref]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, “Bose-Einstein condensate in a box,” Phys. Rev. A 71,041604R (2005).
[Crossref]

Heggarty, K.

Henage, T.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006).
[Crossref] [PubMed]

Hill, W. T.

Hinds, E. A.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61,022304(2000).
[Crossref]

Hirano, T.

T. Kuga, Y. Torii, Shiokawa N., and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. v. Lett. 78,4713–4716 (1997).

Hossack, W. J.

Jacubowiez, L.

Jaksch, D.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61,022304(2000).
[Crossref]

Jo, G.-B.

C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, “Interference of Bose-Einstein condensates split with an atom chip,” Phys. Rev. A 72,021604(R) (2005).

Johnson, T. A.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006).
[Crossref] [PubMed]

Jones, M.

Kasevich, M. A.

J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, “Large area light-pulse Atom Interferometry,” Phys. Rev. Lett. 85,4498–4501 (2000).
[Crossref] [PubMed]

Ketterle, W.

C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, “Interference of Bose-Einstein condensates split with an atom chip,” Phys. Rev. A 72,021604(R) (2005).

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt , “Atom Interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. Lett. 92,050405 (2002).
[Crossref]

Khaykovich, L.

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59,R1750–R1753 (1999).
[Crossref]

Kishimoto, T.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

Kuga, T.

T. Kuga, Y. Torii, Shiokawa N., and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. v. Lett. 78,4713–4716 (1997).

Kulatunga, P. B.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006).
[Crossref] [PubMed]

Laczik, Z. J.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006).
[Crossref]

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, “Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms,” J. Mod. Opt. 51,2235 (2004).
[Crossref]

Leanhardt, A. E.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt , “Atom Interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. Lett. 92,050405 (2002).
[Crossref]

Lee, C. S.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000).
[Crossref] [PubMed]

Livesey, J.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation,” J. Mod. Optics 53,547–556 (2006).
[Crossref]

Lorent, V.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000).
[Crossref] [PubMed]

Mandel, O.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, and I. Bloch, “Coherent collisional spin dynamics in optical lattices,” Phys. Rev. Lett. 95,190405 (2005).
[Crossref] [PubMed]

McGloin, D.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation,” J. Mod. Optics 53,547–556 (2006).
[Crossref]

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light odulators in atom ptics,”gOpt. Express 11,158–166 (2003).
[Crossref]

McGuirk, J. M.

J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, “Large area light-pulse Atom Interferometry,” Phys. Rev. Lett. 85,4498–4501 (2000).
[Crossref] [PubMed]

Melville, H.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation,” J. Mod. Optics 53,547–556 (2006).
[Crossref]

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light odulators in atom ptics,”gOpt. Express 11,158–166 (2003).
[Crossref]

Meyrath, T. P.

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, “Bose-Einstein condensate in a box,” Phys. Rev. A 71,041604R (2005).
[Crossref]

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, “Direct observation of sub-Poissonian number statistics in a degenerate bose gas,” Phys. Rev. Lett. 95260403, (2005).
[Crossref]

Muther, T.

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, “Interferometer-type structures for guided atoms,” Phys. Rev. Lett. 89,220402 (2002).
[Crossref] [PubMed]

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer , “Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits,” Phys. Rev. Lett. 89,097903 (2002).
[Crossref] [PubMed]

N., Shiokawa

T. Kuga, Y. Torii, Shiokawa N., and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. v. Lett. 78,4713–4716 (1997).

Ozeri, R.

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59,R1750–R1753 (1999).
[Crossref]

Pasquini, T. A.

C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, “Interference of Bose-Einstein condensates split with an atom chip,” Phys. Rev. A 72,021604(R) (2005).

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt , “Atom Interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. Lett. 92,050405 (2002).
[Crossref]

Prentiss, M.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000).
[Crossref] [PubMed]

Price, G. N.

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, “Direct observation of sub-Poissonian number statistics in a degenerate bose gas,” Phys. Rev. Lett. 95260403, (2005).
[Crossref]

Pritchard, D. E.

C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, “Interference of Bose-Einstein condensates split with an atom chip,” Phys. Rev. A 72,021604(R) (2005).

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt , “Atom Interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. Lett. 92,050405 (2002).
[Crossref]

Proite, N.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006).
[Crossref] [PubMed]

Raizen, M. G.

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, “Bose-Einstein condensate in a box,” Phys. Rev. A 71,041604R (2005).
[Crossref]

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, “Direct observation of sub-Poissonian number statistics in a degenerate bose gas,” Phys. Rev. Lett. 95260403, (2005).
[Crossref]

Rhodes, D. P.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation,” J. Mod. Optics 53,547–556 (2006).
[Crossref]

Rogers, E. A.

Roy, R.

Saba, M.

C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, “Interference of Bose-Einstein condensates split with an atom chip,” Phys. Rev. A 72,021604(R) (2005).

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt , “Atom Interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. Lett. 92,050405 (2002).
[Crossref]

Saffman, M.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006).
[Crossref] [PubMed]

Sanner, C.

C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, “Interference of Bose-Einstein condensates split with an atom chip,” Phys. Rev. A 72,021604(R) (2005).

Saravanan, R. A.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35,237–246 (1972).

Schmiedmayer, J.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61,022304(2000).
[Crossref]

Schreck, F.

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, “Bose-Einstein condensate in a box,” Phys. Rev. A 71,041604R (2005).
[Crossref]

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, “Direct observation of sub-Poissonian number statistics in a degenerate bose gas,” Phys. Rev. Lett. 95260403, (2005).
[Crossref]

Segal, S. R.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

Shin, Y.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt , “Atom Interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. Lett. 92,050405 (2002).
[Crossref]

Sibbett, W.

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light odulators in atom ptics,”gOpt. Express 11,158–166 (2003).
[Crossref]

Smirne, G.

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006).
[Crossref]

Smith, S. P.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000).
[Crossref] [PubMed]

Snadden, M. J.

J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, “Large area light-pulse Atom Interferometry,” Phys. Rev. Lett. 85,4498–4501 (2000).
[Crossref] [PubMed]

Spalding, G.

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light odulators in atom ptics,”gOpt. Express 11,158–166 (2003).
[Crossref]

Theofanidou, E.

Thywissen, J. H.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000).
[Crossref] [PubMed]

Torii, Y.

T. Kuga, Y. Torii, Shiokawa N., and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. v. Lett. 78,4713–4716 (1997).

Urban, E.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006).
[Crossref] [PubMed]

Volk, M.

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer , “Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits,” Phys. Rev. Lett. 89,097903 (2002).
[Crossref] [PubMed]

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, “Interferometer-type structures for guided atoms,” Phys. Rev. Lett. 89,220402 (2002).
[Crossref] [PubMed]

Walker, T. G.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006).
[Crossref] [PubMed]

Wang, Y.-J.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

Westervelt, R. M.

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000).
[Crossref] [PubMed]

Widera, A.

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, and I. Bloch, “Coherent collisional spin dynamics in optical lattices,” Phys. Rev. Lett. 95,190405 (2005).
[Crossref] [PubMed]

Wu, S.

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

Yavuz, D. D.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006).
[Crossref] [PubMed]

Zoller, P.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61,022304(2000).
[Crossref]

gOpt. Express (1)

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light odulators in atom ptics,”gOpt. Express 11,158–166 (2003).
[Crossref]

J. Mod. Opt. (1)

V. Boyer, C. M. Chandrashekar, C. J. Foot, and Z. J. Laczik, “Dynamic optical trap generation using FLC SLMs for the manipulation of cold atoms,” J. Mod. Opt. 51,2235 (2004).
[Crossref]

J. Mod. Optics (1)

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre-Gaussian light beams formed by spatial light modulation,” J. Mod. Optics 53,547–556 (2006).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Express (2)

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35,237–246 (1972).

Phys. Rev. A (5)

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59,R1750–R1753 (1999).
[Crossref]

T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, “Bose-Einstein condensate in a box,” Phys. Rev. A 71,041604R (2005).
[Crossref]

V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, “Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator,” Phys. Rev. A 73, 031402(R) (2006).
[Crossref]

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61,022304(2000).
[Crossref]

C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, “Interference of Bose-Einstein condensates split with an atom chip,” Phys. Rev. A 72,021604(R) (2005).

Phys. Rev. Lett. (9)

R. Dumke, M. Volk, T. Muther, F. B. J. Buchkremer, G. Birkl, and W. Ertmer , “Micro-optical realization of arrays of selectively addressable dipole traps: a scalable configuration for quantum computation with atomic qubits,” Phys. Rev. Lett. 89,097903 (2002).
[Crossref] [PubMed]

R. Dumke, T. Muther, M. Volk, W. Ertmer, and G. Birkl, “Interferometer-type structures for guided atoms,” Phys. Rev. Lett. 89,220402 (2002).
[Crossref] [PubMed]

A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, and I. Bloch, “Coherent collisional spin dynamics in optical lattices,” Phys. Rev. Lett. 95,190405 (2005).
[Crossref] [PubMed]

J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, “Large area light-pulse Atom Interferometry,” Phys. Rev. Lett. 85,4498–4501 (2000).
[Crossref] [PubMed]

N. H. Dekker, C. S. Lee, V. Lorent, J. H. Thywissen, S. P. Smith, M. Drndic, R. M. Westervelt, and M. Prentiss, “Guiding neutral atoms on a chip,” Phys. Rev. Lett. 84,1124–1127 (2000).
[Crossref] [PubMed]

Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson Interferometer on a chip using a Bose-Einstein Condensate,” Phys. Rev. Lett. 94,090405 (2005).
[Crossref] [PubMed]

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt , “Atom Interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. Lett. 92,050405 (2002).
[Crossref]

C.-S. Chuu, F. Schreck, T. P. Meyrath, J. L. Hanssen, G. N. Price, and M. G. Raizen, “Direct observation of sub-Poissonian number statistics in a degenerate bose gas,” Phys. Rev. Lett. 95260403, (2005).
[Crossref]

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96,063001 (2006).
[Crossref] [PubMed]

Phys. v. Lett. (1)

T. Kuga, Y. Torii, Shiokawa N., and T. Hirano, “Novel optical trap of atoms with a doughnut beam,” Phys. v. Lett. 78,4713–4716 (1997).

Supplementary Material (5)

Media 1: AVI (48 KB)     
Media 2: AVI (297 KB)     
Media 3: AVI (335 KB)     
Media 4: AVI (109 KB)     
Media 5: AVI (85 KB)     

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Figures (7)

Fig. 1.
Fig. 1. (a). Hollow beams formed by the SLM. (b). 1 × 4 arrays formed by driving the AOD with four frequencies simultaneously. c) 1 × 4 array created solely by the SLM.
Fig. 2.
Fig. 2. 1 × 4 array of n=1 hollow laser beams. The two central beams are modulated by a triangle wave with periods of 20 and 40 ms as the 1 × 4 array is scanned from left to right across the frame. Time between frames is ∼3 msec. (Movie: 0.28 Mb) [Media 1]
Fig. 3.
Fig. 3. Optical layout for the crossed hollow beams. (a). Setup for single beam deflection (Movie: 0.24 Mb) [Media 2] (b). Multiple drive frequencies (Movie: 0.38 Mb) [Media 3]
Fig. 4.
Fig. 4. (a). Image of n=4 hollow beam at the MOT location and cross sectional profile. (b). Fluorescence images of cold 85Rb atoms confined in an n=4 crossed hollow beam that is moving sinusoidally with period 50 ms. Time between frames is 2.5 ms, and the deflection amplitude is 0.9 mm.
Fig. 5.
Fig. 5. Atoms remaining in the crossed blue detuned trap as a function of time. Fits use data after 10 msec to eliminate atoms starting outside the trap. For clarity, error bars are not displayed, but are approximately 10% of the measured values, arising from variations in initial MOT number.
Fig. 6.
Fig. 6. 2D motion of atoms. Circle diameter is ∼500 microns. Time between frames is 10 msec. (Movie: 0.6 MB) [Media 4]
Fig. 7.
Fig. 7. Control of multiple atoms clouds with the AOD. Maximum distance between top and bottom cloud is 2 mm. Time between frames is ~3 msec. (Movie 0.3 MB) [Media 5]

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

U ( r ) = ħ Δ 2 ln [ 1 + I ( r ) / I o 1 + 4 ( Δ / Γ ) 2 ]
U ( r ) = ħ Γ 8 [ ( Γ Δ ) I ( r ) I o ]
Γ sp ( r ) = Γ 8 [ ( Γ Δ ) 2 I ( r ) I o ]

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