## Axicon lens for coherent matter waves

Optics Express, Vol. 14, Issue 20, pp. 8947-8957 (2006)

http://dx.doi.org/10.1364/OE.14.008947

Acrobat PDF (368 KB)

### Abstract

We have realized a conical matter wave lens. The repulsive potential of a focused laser beam was used to launch a Bose-Einstein condensate into a radially expanding wavepacket whose perfect ring shape was ensured by energy conservation. In spite of significant interactions between atoms, the spatial and velocity widths of the ring along its radial dimension remained extremely narrow, as also confirmed by numerical simulations. Our results open the possibility for cylindrical atom optics without the perturbing effect of mean-field interactions.

© 2006 Optical Society of America

## 1. Introduction

1. C. S. Adams, M. Sigel, and J. Mlynek, “Atom optics,” Phys. Reports **240(3)**, 143–210 (1994). [CrossRef]

2. M. Morinaga, M. Yasuda, T. Kishimoto, and F. Shimizu, “Holographic Manipulation of a Cold Atomic Beam,” Phys. Rev. Lett. **77**, 802–805 (1996). [CrossRef] [PubMed]

6. G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. **69**, 1636 (1992). [CrossRef] [PubMed]

7. J. J. McClelland, R. E. Scholten, E. C. Palm, and R. J. Celotta, “Laser-focused atomic deposition,” Science **262**, 877–880 (1993). [CrossRef] [PubMed]

8. C. Henkel, J. Schmiedmayer, and C. Westbrook, and eds., “Special Issue-Atom chips: manipulating atoms and molecules with microfabricated structures,” Eur. Phys. J. D **35(1)**, 1–2 (2005). [CrossRef]

9. J. G. Kalnins, J. M. Amini, and H. Gould, “Focusing a fountain of neutral cesium atoms with an electrostatic lens triplet,” Phys. Rev. A (**72**, 043,406 (2005). [CrossRef]

10. M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. v. d. Zouw, and A. Zeilinger, “Wave-particle duality of C_{60} molecules,” Nature **401**, 680–682 (1999). [CrossRef]

11. D. A. Kokorowski, A. D. Cronin, T. D. Roberts, and D. E. Pritchard, “From Single-to Multiple-Photon Deco-herence in an Atom Interferometer,” Phys. Rev. Lett. **86**, 2191–2195 (2001). [CrossRef] [PubMed]

12. S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature **392**, 151–154 (1998). [CrossRef]

13. E.W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, “AWell-Collimated Quasi-Continuous Atom Laser,” Science **283**, 1706–1709 (1999). [CrossRef] [PubMed]

14. N. P. Robins, C. Figl, S. A. Haine, A. K. Morrison, M. Jeppesen, J. J. Hope, and J. D. Close, “Achieving Peak Brightness in an Atom Laser,” Phys. Rev. Lett. **96**, 140,403 (2006). [CrossRef]

15. A. E. Leanhardt, T. A. Pasquini, M. Saba, A. Schirotzek, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, “Cooling Bose-Einstein condensates below 500 picokelvin,” Science **301(5639)**, 1513–1515 (2003). [CrossRef]

16. G. Roati, E. d. Mirandes, F. Ferlaino, H. Ott, G. Modugno, and M. Inguscio, “Atom Interferometry with Trapped Fermi Gases,” Phys. Rev. Lett. **92**, 230,402 (2004). [CrossRef]

## 2. Experimental description

^{6}sodium atoms in a purely linear quadrupole potential formed by a pair of anti-Helmholtz coils following the procedure in our earlier work [17

17. D. S. Naik, S. R. Muniz, and C. Raman, “Metastable Bose-Einstein condensate in a linear potential,” Phys. Rev. A **72**, 051,606 (2005). [CrossRef]

*e*

^{2}waist of 40

*µ*m to prevent Majorana transitions near the magnetic field zero [18

18. E. Majorana, “Orientated atoms in a variable magnetic field,” Nuovo Cimento **9**, 43–50 (1932). [CrossRef]

19. D. S. Naik and C. Raman, “Optically plugged quadrupole trap for Bose-Einstein condensates,” Phys. Rev. A **71**, 033,617 (2005). [CrossRef]

20. K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. v. Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett. **75**, 3969–3973 (1995). [CrossRef] [PubMed]

*Rρ*=18

*µ*m and Rz=10 mm, respectively.

*z*-axis and had a 70 µm 1/

*e*

^{2}radius (roughly 4 times larger than the cloud radius) and variable power. The plug beam was turned on using an acousto-optic modulator (AOM), with typical sub-microsecond switching time, while the quadrupole field was switched off within a few tens of microseconds, much faster than the evolution of the atomic motion. Therefore the entire condensate was instantaneously located on the top of an azimuthally symmetric potential hill with a steep gradient and began to accelerate radially outwards, as illustrated schematically in Fig. 1. This strong radial acceleration caused the atom cloud to form an extremely thin, nearly perfectly shaped ring that propagated outward (see Fig. 2(a) for the ring structure and associated movie).

*z*-direction using a 200 microsecond probe pulse resonant with the transition from the

*F*=1 ground level to the

*F*

^{′}=2 excited level. After a few milliseconds, the atoms had been expelled from the region of the laser focus. After this point, neither the external potential nor the interactions played a significant role (the expansion had lowered the density to the point where interaction effects were negligible). The subsequent dynamics of the cloud was thereafter purely ballistic.

*z*-axis). Since atoms gain exactly the same kinetic energy independent of which direction they were expelled, the nearly perfect circular shape was completely independent of minor misalignments of the plug laser or deviations from cylindrical symmetry of the laser focus.

_{f}, which we have plotted in Fig. 2(d) as a function of the laser intensity, proportional to the peak AC Stark shift

*U*, which was determined from the measured laser power and beam width. The velocity fits well to a square root function v

_{f}∝ U

^{1/2}with the expected prefactor for energy conservation.

*refractive element*for atom optics and by neglecting atom-atom interactions, at least initially. The inhomogeneous dipole force shapes the transverse velocity distribution of the atoms in the following manner. For a laser beam with 1/

*e*

^{2}intensity radius

*W*the AC Stark shift potential is given by

*U*(

*ρ*)=

*U*

_{0}

*e*

^{-2ρ2/W2}. An initially stationary atom which is located at transverse position

*ρ*satisfying 0<

*ρ*<

*R*

_{ρ}, where Rρ is the Thomas-Fermi radius of the condensate in the

*x-y*plane, will be accelerated to an asymptotic radial velocity given by the energy conservation relation:

*R*

_{ρ}≪

*W*, to lowest order each atom will be given the same positive radial velocity

*thin*lenses, where the change in trajectory during the interaction with the lens element can be neglected. Kaenders et al. have realized a positive axicon for a slow atomic beam by exploiting the linear magnetic potential of a quadrupole magnet, which resulted in a uniform transverse force across the sample that caused the beam to focus on a line [21

21. W. G. Kaenders, F. Lison, I. Muller, A. Richter, R. Wynands, and D. Meschede, “Refractive components for magnetic atom optics,” Phys. Rev. A **54**, 5067 (1996). [CrossRef] [PubMed]

*during*the interaction with the beam. Thus we have realized a

*thick*atom lens, which does not require the force to be constant in space as it was for the thin lens implemented by Kaenders et al. The conical feature of the final velocity distribution arises simply from energy conservation and an extremely narrow spatial distribution of atoms

*R*

_{ρ}≪

*W*, and thus does not depend on the details of the forces which are applied. The implementation of thick lenses is greatly facilitated using Bose-Einstein condensates since the atomic dynamics are intrinsically slow. It is therefore straightforward to engineer a long interaction time with the lens element, which opens up new possibilities for realizing refractive elements for atom optics [22

22. I. Bloch, M. Kohl, M. Greiner, T. W. Hansch, and T. Esslinger, “Optics with an Atom Laser Beam,” Phys. Rev. Lett. **87**, 030,401 (2001). [CrossRef]

23. A. S. Arnold, C. MacCormick, and M. G. Boshier, “Diffraction-limited focusing of Bose-Einstein condensates,” J. Phys. B **37**, 485 (2004). [CrossRef]

*k*

_{B}×6.5

*µ*K. Before application of the plug, the Thomas-Fermi density profile of the atoms is given by ni

*n*

_{0}is the peak density. After the interaction, the final density profile is given in cylindrical coordinates by

*n*

_{f}(

*ρ,ϕ,z*), where

*ϕ*is the azimuthal angle. We measure the column density

*ñf*(

*ρ,ϕ*)=∫

*n*

_{f}(

*ϕ,ϕ,z*)

*dz*. Due to the cylindrical symmetry of the problem, we can define an azimuthally averaged density profile

*ñ*

_{f}(

*ρ*)

*dρ=N*results, where N is the number of atoms.

*θ*about the ring center, which could be readily determined to within the pixel size, about 5

*µ*m. For each slice, we folded the data about the center to obtain two data sets corresponding to angles

*θ*and

*θ*+

*π*, both starting at

*ρ*=0. The process was repeated 180 times with 1 degree rotation increments and the final radial distribution for each image was the sum of all the data sets. For all times-of-flight, the distribution clearly has an asymmetric shape, with a long tail extending toward smaller radii and a relatively sharp cutoff at larger radii. We quantified the asymmetry by defining the half-width at half-maximum (HWHM) for the distribution with respect to its peak, defined as

*ρ=ρ*

_{0}. This allowed us to define a backward and forward HWHM,

*W*

_{back}and

*W*

_{f}, respectively, corresponding to

*ρ*<

*ρ*

_{0}and

*ρ*>

*ρ*

_{0}, respectively. These are plotted in Fig. 3(b), and clearly show that

*W*

_{back}>

*W*

_{f}for the majority of the data points.

*ρ*close to 0 and group B located at

*ρ*close to

*R*

_{ρ}, the Thomas-Fermi radius of the BEC. At

*t*=0, atoms in group A experience a negligible force since the gradient in the laser intensity is close to zero. Therefore, they remain momentarily stationary. However, atoms in group B experience an immediate force due to the finite gradient, and begin moving radially outward. This leads to a momentary spreading of the wavepacket. This spreading is aided by atom-atom interactions, which act to reduce the atom density

*n*further through the positive mean-field pressure ∝

*n*

^{2}[24]. However, for longer times, group A atoms begin to move outward as well. They acquire more kinetic energy than group B atoms due to their larger initial potential energy, as expressed in Eq. 1. Eventually, group A will catch up with group B at some intermediate time

*τ*

_{f}, which we denote the

*focusing time*, when the spreading is reversed. This focusing time is, typically, considerably larger than the average transit time of atoms

*τ*

_{trans}through the beam. The key is that for times

*τ*

_{trans}<

*t*≤

*τ*

_{f}the atoms’ velocity is close to the asymptotic value given by Eq. 1, but their

*spatial*distribution has not yet equilibrated. Moreover, by this time the axicon has reduced the atom density by spreading the atoms onto a ring, and interactions have no further effect on the propagation, which is governed purely by linear optics. Finally, at very long times

*t*≫

*τ*

_{f}, the wavepacket should defocus. That is, it will resemble a map of the asymptotic velocity distribution, with fast atoms in front, and slow atoms behind.

*t*~

*τ*

_{f}. Thus we observe no increase in the forward width

*W*

_{f}and a reduction in the backward width

*W*

_{back}. The latter provides evidence of the temporal focusing of the wavepacket. Further evidence was provided by classical simulations which we have performed based on the specific parameters of our experiment. We numerically computed

*n*

_{max}single atom trajectories in the presence of the plug for times 0<

*t*<

*t*

_{sim}. We chose various starting locations 0<

*ρi,k*

*<R*ρ at time

*t*=0, where

*k*=1,2, …

*n*

_{max}, and computed a sequence of final locations

*ρf,k*at

*t=t*

_{sim}. The resulting classical “wavepacket” was constructed by assigning to each

*ρf,k*a probability density weighted in proportion to the initial number of atoms located at

*ρi,k*, i.e.,∝ 2

*πρ*

_{i}

*,k ∫n*

_{i}(

*ρ*

_{i}

*,k,z*)

*dz*, the column density in the trap. The integration limits used are

*n*

_{max}=200 trajectories and times

*t*

_{sim}=0,5,10,15,20 ms. The simulated wavepacket shows a strong resemblance to the data in Fig. 3(a). In particular, we observe an asymmetric distribution with a shrinking tail, i.e.,

*W*

_{back}decreases with time. A close examination of the final classical wavepacket shows that it is in fact multi-valued at the leading edge, an indication that many classical trajectories have crossed one another, an effect confirmed by a detailed examination of the individual trajectories used to construct the wavepacket. At this point the classical analysis begins to break down, and one requires a full quantum treatment. Nonetheless, this simple simulation confirms the temporal focusing of the axicon with a time on the order of, or close to 20 ms.

*W*

_{f}in our measurements remains stationary-atoms simply pile up from behind, and do not spread out until a much longer time

*t*≫

*τ*

_{f}, when the different momentum components separate again. Unfortunately, we could not observe the ring at larger TOF as it exceeded the camera sensor size2. Although our experiments do not measure the asymptotic momentum distribution, we can make a rough estimate of the M

^{2}, or quality factor for our atom beam, which measures the phase space volume occupied. From the reduction of

*W*

_{back}with time in Fig. 3 we estimate a velocity width of order

*δ*

_{vr}~2 mm/s. The measured spatial width near the temporal focus is no larger than 20

*µ*m, and therefore,

25. J. F. Riou, W. Guerin, Y. L. Coq, M. Fauquembergue, V. Josse, P. Bouyer, and A. Aspect, “Beam Quality of a Nonideal Atom Laser,” Phys. Rev. Lett. **96**, 070,404 (2006). [CrossRef]

*slowly*from a BEC, where the inhomogeneous mean field interaction between outcoupled atoms and the stationary condensate affects the spatial profile of the laser. For example, RF coupling results in a roughly constant energy but a broad transverse distribution due to the mean-field [26

26. M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Output coupler for Bose-Einstein condensed atoms,” Phys. Rev. Lett. **78**, 582–585 (1997). [CrossRef]

27. Y. Le Coq, J. H. Thywissen, S. A. Rangwala, F. Gerbier, S. Richard, G. Delannoy, P. Bouyer, and A. Aspect, “Atom Laser Divergence,” Phys. Rev. Lett. **87**, 170,403 (2001). [CrossRef]

25. J. F. Riou, W. Guerin, Y. L. Coq, M. Fauquembergue, V. Josse, P. Bouyer, and A. Aspect, “Beam Quality of a Nonideal Atom Laser,” Phys. Rev. Lett. **96**, 070,404 (2006). [CrossRef]

*z*-direction was almost completely unaffected. Due to the long confocal parameter of 2.8 mm, there is a negligible variation of the laser intensity and therefore no force exerted along that direction over the relevant experimental timescales. In Fig. 4 we have observed the atoms after 10 and 15 ms TOF imaged along one of the radial directions. The ring of atoms appears in this view as a band with a considerable axial width, comparable to that of an ordinary BEC expansion in the absence of the plug, as shown in Figs. 4(b,d).

## 3. Theory

*ρ*and

*z*, taking advantage of the azimuthal symmetry of the problem. We describe the condensate by the order parameter

*ψ*(

*ρ,z,t*), normalized such that

*n*(

*ρ,z,t*)=

*N*|

*ψ*(

*ρ,z,t*)|

^{2}. The condensate dynamics is governed by the Gross-Pitaevskii equation

*N*is the number of atoms,

*g=4π̄h*

^{2}

*α/m*is the mean field interaction strength,

*m*is the atomic mass,

*a*is the scattering length, and

*V*(

*ρ,z,t*) is the combined external potential applied by the magnetic trap and the plug beam.We assume the magnetic trap is turned off and the plug beam is turned on instantaneously at time

*t*=0. Explicitly, we write the potential as

*µ*

_{M}

*≈2p̄h×*0.7 MHz/G is the atomic magnetic moment,

*U*

_{0}is the AC Stark shift at the center of the plug, and

*W*is the 1/

*e*

^{2}radius of the plug. At

*t*=0, we assume the condensate is in the ground state of the magnetic trap with the plug turned off. The ground state is itself calculated by evolving Eq. 2 in imaginary time. The dynamics induced by the plug is then found for

*t*>0 by integrating Eq. 2 in real time. The grid size, spatial resolution, and time step were varied to ensure convergence. To ease the computational burden, the simulation was done for a smaller atom number than the experiment but with the nearly same magnetic field gradient (see Table 1). We used approximately the same values of the ratio of plug potential to chemical potential (

*µ*

_{a}), and the ratio of plug radius to Thomas-Fermi radius of the condensate as in the experiment.

^{2}radius of the plug potential. Therefore, the subsequent expansion of the cloud should be mostly ballistic, with only small corrections to the momentum distribution expected.

*z*-direction. Moreover, in the radial direction, the atoms are being forced outward and form a steep front at

*ρ*≅32

*µ*m. However, a spatial tail of the distribution still exists for smaller

*ρ*. Our data in Fig. 3(a) for 5 ms and at short TOF bears a striking similarity to the spatial distribution in Fig. 5(b). Figure 5(c) shows the velocity distribution of the atoms and confirms that the majority of the atoms occupy a very narrow peak in momentum space near 38 mm/s, corresponding to a kinetic energy of 1.95

*µ*K, about 90% of the peak plug potential. This is consistent with our observations in Fig. 3(b) of an extremely narrow velocity spread for the expanding ring. The velocity width (HWHM) computed from the simulation is about 600

*µ*m/s, about a factor of 3 larger than the Heisenberg limited width of 215

*µ*m/s for a cloud of radius

*R*

_{ρ}=6.4

*µ*m. For comparison we have also shown the momentum distribution of a mean-field driven expansion in the absence of the plug. To understand the role that mean-field effects play on the ring formation, we performed similar calculations for an expansion time of 1.3 ms using the same initial wavefunction but with the mean-field term in the Gross-Pitaevskii equation set to zero. This represents the purely “linear” evolution of the atomic wavepacket. The resulting momentum distribution is also shown in Fig. 5(c), and is qualitatively very similar to the nonlinear evolution. A similar sharp peak with a narrow velocity width is formed, demonstrating that the dynamics are largely insensitive to the effects of interactions.

*ρ*=0. These atoms will catch up with the peak of the wavepacket only at later times. Moreover, it is intriguing to note the presence of smaller, “satellite” momentum peaks in both simulations which appear to be interference phenomena. To explore these further, we extended the non-interacting calculation to the longer time of 1.8 ms. This calculation proved easier than the fully interacting case due to the larger grid size used. Due to the similarity between the curves in Fig. 5(c), the linear evolution provides a useful guide to the dynamics. An expanded view of the momentum peaks is shown in Fig. 5(d) at both 1.3 and 1.8 ms for the linear evolution. The interference peaks appear to increase in amplitude with time, and coincide with a reduction of the low velocity tail. It is clear that the radial phase profile is not flat, and there may be differences between the interacting and non-interacting cases at longer times. Future work will explore these interferences in detail, with an eye toward understanding their effect on atom-optical applications of the axicon lens, where the phase profile is of importance.

## 4. Conclusion

## Acknowledgment

## Footnotes

1 | It should be noted that there is one difference-the transverse velocity change is only constant for an initially stationary Bose-Einstein condensate, whereas an optical axicon is insensitive to the incident angle of the light rays as well as their position, with the change in angle being a constant in the paraxial approximation. For our atoms, this would correspond to a non-zero radial momentum before the plug is applied. |

2 | Demagnification of the image would have allowed us to observe the ring at larger TOF, but at the cost of reduced spatial resolution, which would compromise the width measurement. |

## References and links

1. | C. S. Adams, M. Sigel, and J. Mlynek, “Atom optics,” Phys. Reports |

2. | M. Morinaga, M. Yasuda, T. Kishimoto, and F. Shimizu, “Holographic Manipulation of a Cold Atomic Beam,” Phys. Rev. Lett. |

3. | “Special Issue on Atom Optics,” Quantum and Semiclassical Optics: J. Eur. Opt. Soc. Part B8, 495 (1996). |

4. | P. R. Berman, ed., |

5. | P. Meystre, |

6. | G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. |

7. | J. J. McClelland, R. E. Scholten, E. C. Palm, and R. J. Celotta, “Laser-focused atomic deposition,” Science |

8. | C. Henkel, J. Schmiedmayer, and C. Westbrook, and eds., “Special Issue-Atom chips: manipulating atoms and molecules with microfabricated structures,” Eur. Phys. J. D |

9. | J. G. Kalnins, J. M. Amini, and H. Gould, “Focusing a fountain of neutral cesium atoms with an electrostatic lens triplet,” Phys. Rev. A ( |

10. | M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. v. d. Zouw, and A. Zeilinger, “Wave-particle duality of C |

11. | D. A. Kokorowski, A. D. Cronin, T. D. Roberts, and D. E. Pritchard, “From Single-to Multiple-Photon Deco-herence in an Atom Interferometer,” Phys. Rev. Lett. |

12. | S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature |

13. | E.W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, “AWell-Collimated Quasi-Continuous Atom Laser,” Science |

14. | N. P. Robins, C. Figl, S. A. Haine, A. K. Morrison, M. Jeppesen, J. J. Hope, and J. D. Close, “Achieving Peak Brightness in an Atom Laser,” Phys. Rev. Lett. |

15. | A. E. Leanhardt, T. A. Pasquini, M. Saba, A. Schirotzek, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, “Cooling Bose-Einstein condensates below 500 picokelvin,” Science |

16. | G. Roati, E. d. Mirandes, F. Ferlaino, H. Ott, G. Modugno, and M. Inguscio, “Atom Interferometry with Trapped Fermi Gases,” Phys. Rev. Lett. |

17. | D. S. Naik, S. R. Muniz, and C. Raman, “Metastable Bose-Einstein condensate in a linear potential,” Phys. Rev. A |

18. | E. Majorana, “Orientated atoms in a variable magnetic field,” Nuovo Cimento |

19. | D. S. Naik and C. Raman, “Optically plugged quadrupole trap for Bose-Einstein condensates,” Phys. Rev. A |

20. | K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. v. Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of Sodium Atoms,” Phys. Rev. Lett. |

21. | W. G. Kaenders, F. Lison, I. Muller, A. Richter, R. Wynands, and D. Meschede, “Refractive components for magnetic atom optics,” Phys. Rev. A |

22. | I. Bloch, M. Kohl, M. Greiner, T. W. Hansch, and T. Esslinger, “Optics with an Atom Laser Beam,” Phys. Rev. Lett. |

23. | A. S. Arnold, C. MacCormick, and M. G. Boshier, “Diffraction-limited focusing of Bose-Einstein condensates,” J. Phys. B |

24. | L. Pitaevskii and S. Stringari, |

25. | J. F. Riou, W. Guerin, Y. L. Coq, M. Fauquembergue, V. Josse, P. Bouyer, and A. Aspect, “Beam Quality of a Nonideal Atom Laser,” Phys. Rev. Lett. |

26. | M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Output coupler for Bose-Einstein condensed atoms,” Phys. Rev. Lett. |

27. | Y. Le Coq, J. H. Thywissen, S. A. Rangwala, F. Gerbier, S. Richard, G. Delannoy, P. Bouyer, and A. Aspect, “Atom Laser Divergence,” Phys. Rev. Lett. |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(140.3320) Lasers and laser optics : Laser cooling

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: June 22, 2006

Manuscript Accepted: August 30, 2006

Published: October 2, 2006

**Citation**

S. R. Muniz, S. D. Jenkins, T. A. B. Kennedy, D. S. Naik, and C. Raman, "Axicon lens for coherent matter waves," Opt. Express **14**, 8947-8957 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-8947

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### References

- C. S. Adams, M. Sigel, and J. Mlynek, "Atom optics," Phys. Reports 240, 143-210 (1994). [CrossRef]
- M. Morinaga, M. Yasuda, T. Kishimoto, and F. Shimizu, "Holographic Manipulation of a Cold Atomic Beam," Phys. Rev. Lett. 77, 802-805 (1996). [CrossRef] [PubMed]
- "Special Issue on Atom Optics," Quantum and Semiclassical Optics: J. Eur. Opt. Soc. Part B 8, 495 (1996).
- P. R. Berman, ed., Atom Interferometry (Academic Press, San Diego, 1997).
- P. Meystre, Atom optics, Springer series on atomic, optical, and plasma physics; 33 (AIP Press/Springer, New York :, 2001).
- G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, "Using light as a lens for submicron, neutral-atom lithography," Phys. Rev. Lett. 69, 1636 (1992). [CrossRef] [PubMed]
- J. J. McClelland, R. E. Scholten, E. C. Palm, and R. J. Celotta, "Laser-focused atomic deposition," Science 262, 877-880 (1993). [CrossRef] [PubMed]
- C. Henkel, J. Schmiedmayer, C. Westbrook, and eds., "Special Issue - Atom chips: manipulating atoms and molecules with microfabricated structures," Eur. Phys. J. D 35(1), 1-2 (2005). [CrossRef]
- J. G. Kalnins, J. M. Amini, and H. Gould, "Focusing a fountain of neutral cesium atoms with an electrostatic lens triplet," Phys. Rev. A 72, 043,406 (2005). [CrossRef]
- M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. v. d. Zouw, and A. Zeilinger, "Wave-particle duality of C60 molecules," Nature 401, 680-682 (1999). [CrossRef]
- D. A. Kokorowski, A. D. Cronin, T. D. Roberts, and D. E. Pritchard, "From Single- to Multiple-Photon Decoherence in an Atom Interferometer," Phys. Rev. Lett. 86, 2191-2195 (2001). [CrossRef] [PubMed]
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