## Precise shaping of laser light by an acousto-optic deflector |

Optics Express, Vol. 21, Issue 21, pp. 24837-24846 (2013)

http://dx.doi.org/10.1364/OE.21.024837

Acrobat PDF (796 KB)

### Abstract

We present a laser beam shaping method using acousto-optic deflection of light and discuss its application to dipole trapping of ultracold atoms. By driving the acousto-optic deflector with multiple frequencies, we generate an array of overlapping diffraction-limited beams that combine to form an arbitrary-shaped smooth and continuous trapping potential. Confinement of atoms in a flat-bottomed potential formed by a laser beam with uniform intensity over its central region confers numerous advantages over the harmonic confinement intrinsic to Gaussian beam dipole traps and many other trapping schemes. We demonstrate the versatility of this beam shaping method by generating potentials with large flat-topped regions as well as intensity patterns that compensate for residual external potentials to create a uniform background to which the trapping potential of experimental interest can be added.

© 2013 OSA

## 1. Introduction

1. M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. **56**, 243 (2007). [CrossRef]

2. M. D. Barrett, J. A. Sauer, and M. S. Chapman, “All-optical formation of an atomic Bose-Einstein condensate,” Phys. Rev. Lett. **87**, 010404 (2001). [CrossRef] [PubMed]

3. G. D. Bruce, S. L. Bromley, G. Smirne, L. Torralbo-Campo, and D. Cassettari, “Holographic power-law traps for the efficient production of Bose-Einstein condensates,” Phys. Rev. A **84**, 053410 (2011). [CrossRef]

4. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. **81**, 3108 (1998). [CrossRef]

5. J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature **453**, 891–894 (2008). [CrossRef] [PubMed]

8. C. Weitenberg, S. Kuhr, K. Mølmer, and J. F. Sherson, “Quantum computation architecture using optical tweezers,” Phys. Rev. A **84**, 032322 (2011). [CrossRef]

4. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. **81**, 3108 (1998). [CrossRef]

9. V. L. Campo, K. Capelle, J. Quintanilla, and C. Hooley, “Quantitative determination of the Hubbard model phase diagram from optical lattice experiments by two-parameter scaling,” Phys. Rev. Lett. **99**, 240403 (2007). [CrossRef]

10. H. Xiong and B. Wu, “Atomic quantum corrals for Bose-Einstein condensates,” Phys. Rev. A **82**, 053634 (2010). [CrossRef]

*U*(

_{dip}*r⃗*) ∝

*I*(

*r⃗*)/

*δ*with

*I*(

*r⃗*) the spatially dependent laser intensity and

*δ*the detuning of laser light from resonance. The dipole force therefore depends on the intensity gradient of the laser light. Any phase gradient affects only the scattering force, negligible under detuning far from the atomic resonance due to its

*I/δ*

^{2}dependence in comparison to the

*I/δ*dependence of the dipole force.

11. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express **15**, 8619–8625 (2007). [CrossRef] [PubMed]

12. L. Amico, A. Osterloh, and F. Cataliotti, “Quantum many particle systems in ring-shaped optical lattices,” Phys. Rev. Lett. **95**, 063201 (2005). [CrossRef] [PubMed]

13. K. Henderson, C. Ryu, C. MacCormick, and M. G. Boshier, “Experimental demonstration of painting arbitrary and dynamic potentials for Bose-Einstein condensates,” New J. Phys. **11**, 043030 (2009). [CrossRef]

14. N. Houston, E. Riis, and A. S. Arnold, “Reproducible dynamic dark ring lattices for ultracold atoms,” J. Phys. B **41**, 211001 (2008). [CrossRef]

15. R. A. Williams, J. D. Pillet, S. Al-Assam, B. Fletcher, M. Shotter, and C. J. Foot, “Dynamic optical lattices: two-dimensional rotating and accordion lattices for ultracold atoms,” Opt. Express **16**, 16977–16983 (2008). [CrossRef] [PubMed]

16. S. Al-Assam, R. A. Williams, and C. J. Foot, “Ultracold atoms in an optical lattice with dynamically variable periodicity,” Phys. Rev. A **82**, 021604 (2010). [CrossRef]

17. M. Pasienski and B. DeMarco, “A high-accuracy algorithm for designing arbitrary holographic atom traps,” Opt. Express **16**, 2176–2190 (2008). [CrossRef] [PubMed]

*π*/256, a Boulder Nonlinear Systems nematic liquid crystal SLM is capable of a far more versatile range of truly arbitrary potentials, but the switching frequency of hundreds of Hz could limit their applicability to dynamic manipulation of optical trapping potentials. AODs have an update frequency on the order of 10 MHz, facilitating almost seamless switching between dynamic frames, thus combining the versatility and switching rate necessary for an arbitrary dynamic manipulation sequence.

17. M. Pasienski and B. DeMarco, “A high-accuracy algorithm for designing arbitrary holographic atom traps,” Opt. Express **16**, 2176–2190 (2008). [CrossRef] [PubMed]

## 2. Beam shaping using an acousto-optic deflector

### 2.1. Compensation potential

18. C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J. **44**, 76–86 (1916). [CrossRef]

19. A. J. den Dekker and A. van den Bos, “Resolution: a survey,” J. Opt. Soc. Am. A **14**, 547–557 (1997). [CrossRef]

*a*between adjacent beams to achieve a flat-topped composite potential is therefore chosen such that: In one dimension, our intensity distribution

*f*(

*x*) is the sum of the

*N*constituent Gaussian beams, with 1/

*e*

^{2}waist

*w*, and relative amplitudes

*A*and positions

_{n}*x*:

_{n}*a*= 0.527

*w*, with

*w*the beam waist. After optimisation, the experimental error is 1.4% over the flat region of the intensity profile and 2.3% over the full distribution. In this and subsequent figures, the corrugations visible on the compensation potential arise from dust specks on the imaging camera rather than being features of the potential itself.

*x*

^{10}. Whilst the example shown is for a single row of Gaussians, this principle can be readily extended to constructing arbitrary-shaped two-dimensional potentials by deflecting the beam in both x- and y-directions with separations calculated as above (see the Experimental Techniques section for discussion of the dual-axis AOD). The only subtlety arising from additional rows of beams is that if the frequency spacing is equal in both x- and y-directions, then beams lying along diagonal lines have the same frequency and thus interfere. Such undesirable interference is easily avoided by using different frequency spacings and elliptical spots to fulfill the Sparrow criterion in both directions; elliptical distributions occur anyway in the focal plane of an optical system with a high numerical aperture and linearly polarised light [20]. This extension is simple in comparison to a similar extension of the target output of a computer-generated hologram. With an IFTA used to improve the range and versatility of accessible patterns, hologram calculation becomes more complicated for large continuous potentials due to the appearance of optical vortices in the calculation process [21

21. P. Senthilkumaran, F. Wyrowski, and H. Schimmel, “Vortex stagnation problem in iterative Fourier transform algorithms,” Opt. Laser. Eng. **43**, 43–56 (2005). [CrossRef]

*w*, whereas an eighth-order super-Lorentzian as demonstrated in [22

22. J. Liang, J. Kohn, M. F. Becker, and D. J. Heinzen, “High-precision laser beam shaping using a binary-amplitude spatial light modulator,” Appl. Optics **49**, 1323–1330 (2010). [CrossRef]

*w*. The accuracy of exotic patterns calculated using an IFTA can be improved by incorporating Helmholtz propagation into hologram calculation [23

23. A. L. Gaunt and Z. Hadzibabic, “Robust digital holography for ultracold atom trapping,” Sci. Rep. **2**, 721 (2012). [CrossRef] [PubMed]

22. J. Liang, J. Kohn, M. F. Becker, and D. J. Heinzen, “High-precision laser beam shaping using a binary-amplitude spatial light modulator,” Appl. Optics **49**, 1323–1330 (2010). [CrossRef]

*n*, the order we want to cancel: We fit the functional form of this potential using a sum of equal-width Gaussians by adjusting their relative positions and amplitudes. The accuracy increases with the number of beams used, and depends on the complexity of the target distribution. Along the same axis this AOD-generated composite potential has the form: where

_{max}*N*is the number of constituent beams,

*a*the amplitude of each beam,

_{i}*s*the displacement along the x-axis and

_{i}*V*(

_{beam}*x*) the Gaussian function produced by each constituent deflected beam. The optimal set of parameters

*a*,

_{i}*s*to cancel the external potential are determined using an optimisation routine.

_{i}*𝒪*(2) term, with matching performed using the Taylor expansion of the potentials. These beams would have a blue frequency detuning to create a repulsive potential. For this example, the experimental error over the entire pattern is 1.8%.

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

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

### 2.2. Arbitrary continuous potentials

25. S. Hunn, K. Zimmermann, M. Hiller, and A. Buchleitner, “Tunneling decay of two interacting bosons in an asymmetric double-well potential: A spectral approach,” Phys. Rev. A **87**, 043626 (2013). [CrossRef]

*w*. The well depth is controlled by a low-amplitude Gaussian midway between the reservoir and a spatially separated single well. The parameters defining the potential and the interplay between reservoir and single well are sufficiently flexible that the intensity distribution can be easily and precisely modified, allowing dynamic real-time manipulation of trapped atoms. The illustrated experimental realisation has an error of 2.1% over the entire pattern region.

## 3. Conclusion

13. K. Henderson, C. Ryu, C. MacCormick, and M. G. Boshier, “Experimental demonstration of painting arbitrary and dynamic potentials for Bose-Einstein condensates,” New J. Phys. **11**, 043030 (2009). [CrossRef]

## A. Experimental techniques

*u*≈ 610 ms

_{s}^{−1}is the speed of sound in the TiO

_{2}crystal of the AOD, Δ

*f*the frequency deviation and

*λ*the wavelength of light used: 780 nm in section 2.1 and 830 nm in section 2.2. We determined

*u*from the measured angular separation of the diffracted beams for given input frequencies. The resultant linear separation of spots corresponding to different diffraction angles is fixed by a lens placed at a focal length from the AOD, serving to focus the spatially separated beam components onto a CCD camera (Unibrain Fire-i 521b) as well as ensuring parallel propagation to this point. For imaging purposes, this focussing lens was of focal length

_{s}*f*= 400 mm in section 2.1 and

*f*= 500 mm in section 2.2; with lens diameters of 25.4 mm, these yield diffraction-limited beam sizes of 162 μm and 203 μm respectively. To apply this technique to dipole force traps requires significantly smaller diffraction-limited beam sizes; for example, an objective lens with a focal length of 40 mm, diameter 25.4 mm and corresponding diffraction-limited beam size of 1.6 μm with wavelength 830 nm produces a trapping frequency of a few kHz for

^{87}Rb atoms. This is well below the hundreds of kHz acoustic deflection frequency difference between constituent beams of the composite potential such that any beating between neighbouring spots will not adversely affect atoms with the trapping frequencies considered here.

*A*and image-plane positions (

_{n}*x*,

_{n}*y*) of the multiple spots comprising the sculpted beam are computationally determined by fitting a function corresponding to Eq. 3, a sum of Guassian beams, to a target intensity array with the Sparrow criterion providing a useful starting point for flat-topped beams. A small amount of manual optimisation is performed on the experimental output, although the results are likely to benefit from an automated optimisation routine as has been demonstrated in conjunction with an IFTA for holographic beam shaping [26

_{n}26. G. D. Bruce, J. Mayoh, G. Smirne, L. Torralbo-Campo, and D. Cassettari, “A smooth, holographically generated ring trap for the investigation of superfluidity in ultracold atoms,” Phys. Scripta **T143**, 014008 (2011). [CrossRef]

*a*of its constituent frequency components

_{i}*f*, the AWS output resembles a multiplexed signal of the form:

_{i}*n*according to the internal clock frequency

*f*= 125 MHz:

_{clock}*a*is the amplitude associated with each frequency

_{i}*f*, with the total amplitude less than 2048, and

_{i}*L*the total packet length. The packet length is the number of total number of points per period multiplied by the total number of periods in each wave segment, and must be less than 5320. The number of points per period is a compromise between being a multiple of 8 and a multiple of the time period associated with the lowest beating frequency in the signal as a fraction of the internal clock period. This latter condition minimises flicker at the start of each wave segment loop. The number of deflecting frequencies output through the AWS at a single time can limit the complexity of the composite intensity profile. However, the examples illustrated above demonstrate that even a restricted number of deflected beams can be used to accurately reproduce a range of potentials.

## Acknowledgments

## References and links

1. | M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. |

2. | M. D. Barrett, J. A. Sauer, and M. S. Chapman, “All-optical formation of an atomic Bose-Einstein condensate,” Phys. Rev. Lett. |

3. | G. D. Bruce, S. L. Bromley, G. Smirne, L. Torralbo-Campo, and D. Cassettari, “Holographic power-law traps for the efficient production of Bose-Einstein condensates,” Phys. Rev. A |

4. | D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. |

5. | J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature |

6. | M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature |

7. | R. N. Palmer and D. Jaksch, “High-field fractional quantum Hall effect in optical lattices,” Phys. Rev. Lett. |

8. | C. Weitenberg, S. Kuhr, K. Mølmer, and J. F. Sherson, “Quantum computation architecture using optical tweezers,” Phys. Rev. A |

9. | V. L. Campo, K. Capelle, J. Quintanilla, and C. Hooley, “Quantitative determination of the Hubbard model phase diagram from optical lattice experiments by two-parameter scaling,” Phys. Rev. Lett. |

10. | H. Xiong and B. Wu, “Atomic quantum corrals for Bose-Einstein condensates,” Phys. Rev. A |

11. | S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express |

12. | L. Amico, A. Osterloh, and F. Cataliotti, “Quantum many particle systems in ring-shaped optical lattices,” Phys. Rev. Lett. |

13. | K. Henderson, C. Ryu, C. MacCormick, and M. G. Boshier, “Experimental demonstration of painting arbitrary and dynamic potentials for Bose-Einstein condensates,” New J. Phys. |

14. | N. Houston, E. Riis, and A. S. Arnold, “Reproducible dynamic dark ring lattices for ultracold atoms,” J. Phys. B |

15. | R. A. Williams, J. D. Pillet, S. Al-Assam, B. Fletcher, M. Shotter, and C. J. Foot, “Dynamic optical lattices: two-dimensional rotating and accordion lattices for ultracold atoms,” Opt. Express |

16. | S. Al-Assam, R. A. Williams, and C. J. Foot, “Ultracold atoms in an optical lattice with dynamically variable periodicity,” Phys. Rev. A |

17. | M. Pasienski and B. DeMarco, “A high-accuracy algorithm for designing arbitrary holographic atom traps,” Opt. Express |

18. | C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J. |

19. | A. J. den Dekker and A. van den Bos, “Resolution: a survey,” J. Opt. Soc. Am. A |

20. | R. Dorn, S. Quabis, and G. Leuchs, “The focus of light - linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. |

21. | P. Senthilkumaran, F. Wyrowski, and H. Schimmel, “Vortex stagnation problem in iterative Fourier transform algorithms,” Opt. Laser. Eng. |

22. | J. Liang, J. Kohn, M. F. Becker, and D. J. Heinzen, “High-precision laser beam shaping using a binary-amplitude spatial light modulator,” Appl. Optics |

23. | A. L. Gaunt and Z. Hadzibabic, “Robust digital holography for ultracold atom trapping,” Sci. Rep. |

24. | T. P. Meyrath, F. Schreck, J. L. Hanssen, C. Chuu, and M. G. Raizen, “Bose-Einstein condensate in a box,” Phys. Rev. A |

25. | S. Hunn, K. Zimmermann, M. Hiller, and A. Buchleitner, “Tunneling decay of two interacting bosons in an asymmetric double-well potential: A spectral approach,” Phys. Rev. A |

26. | G. D. Bruce, J. Mayoh, G. Smirne, L. Torralbo-Campo, and D. Cassettari, “A smooth, holographically generated ring trap for the investigation of superfluidity in ultracold atoms,” Phys. Scripta |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(020.7010) Atomic and molecular physics : Laser trapping

(230.1040) Optical devices : Acousto-optical devices

(020.1475) Atomic and molecular physics : Bose-Einstein condensates

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: July 24, 2013

Revised Manuscript: September 13, 2013

Manuscript Accepted: September 17, 2013

Published: October 10, 2013

**Citation**

Dimitris Trypogeorgos, Tiffany Harte, Alexis Bonnin, and Christopher Foot, "Precise shaping of laser light by an acousto-optic deflector," Opt. Express **21**, 24837-24846 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-24837

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### References

- M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys.56, 243 (2007). [CrossRef]
- M. D. Barrett, J. A. Sauer, and M. S. Chapman, “All-optical formation of an atomic Bose-Einstein condensate,” Phys. Rev. Lett.87, 010404 (2001). [CrossRef] [PubMed]
- G. D. Bruce, S. L. Bromley, G. Smirne, L. Torralbo-Campo, and D. Cassettari, “Holographic power-law traps for the efficient production of Bose-Einstein condensates,” Phys. Rev. A84, 053410 (2011). [CrossRef]
- D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett.81, 3108 (1998). [CrossRef]
- J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008). [CrossRef] [PubMed]
- M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature415, 39–44 (2002). [CrossRef] [PubMed]
- R. N. Palmer and D. Jaksch, “High-field fractional quantum Hall effect in optical lattices,” Phys. Rev. Lett.96, 180407 (2006). [CrossRef] [PubMed]
- C. Weitenberg, S. Kuhr, K. Mølmer, and J. F. Sherson, “Quantum computation architecture using optical tweezers,” Phys. Rev. A84, 032322 (2011). [CrossRef]
- V. L. Campo, K. Capelle, J. Quintanilla, and C. Hooley, “Quantitative determination of the Hubbard model phase diagram from optical lattice experiments by two-parameter scaling,” Phys. Rev. Lett.99, 240403 (2007). [CrossRef]
- H. Xiong and B. Wu, “Atomic quantum corrals for Bose-Einstein condensates,” Phys. Rev. A82, 053634 (2010). [CrossRef]
- S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express15, 8619–8625 (2007). [CrossRef] [PubMed]
- L. Amico, A. Osterloh, and F. Cataliotti, “Quantum many particle systems in ring-shaped optical lattices,” Phys. Rev. Lett.95, 063201 (2005). [CrossRef] [PubMed]
- K. Henderson, C. Ryu, C. MacCormick, and M. G. Boshier, “Experimental demonstration of painting arbitrary and dynamic potentials for Bose-Einstein condensates,” New J. Phys.11, 043030 (2009). [CrossRef]
- N. Houston, E. Riis, and A. S. Arnold, “Reproducible dynamic dark ring lattices for ultracold atoms,” J. Phys. B41, 211001 (2008). [CrossRef]
- R. A. Williams, J. D. Pillet, S. Al-Assam, B. Fletcher, M. Shotter, and C. J. Foot, “Dynamic optical lattices: two-dimensional rotating and accordion lattices for ultracold atoms,” Opt. Express16, 16977–16983 (2008). [CrossRef] [PubMed]
- S. Al-Assam, R. A. Williams, and C. J. Foot, “Ultracold atoms in an optical lattice with dynamically variable periodicity,” Phys. Rev. A82, 021604 (2010). [CrossRef]
- M. Pasienski and B. DeMarco, “A high-accuracy algorithm for designing arbitrary holographic atom traps,” Opt. Express16, 2176–2190 (2008). [CrossRef] [PubMed]
- C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J.44, 76–86 (1916). [CrossRef]
- A. J. den Dekker and A. van den Bos, “Resolution: a survey,” J. Opt. Soc. Am. A14, 547–557 (1997). [CrossRef]
- R. Dorn, S. Quabis, and G. Leuchs, “The focus of light - linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt.50, 1917–1926 (2003).
- P. Senthilkumaran, F. Wyrowski, and H. Schimmel, “Vortex stagnation problem in iterative Fourier transform algorithms,” Opt. Laser. Eng.43, 43–56 (2005). [CrossRef]
- J. Liang, J. Kohn, M. F. Becker, and D. J. Heinzen, “High-precision laser beam shaping using a binary-amplitude spatial light modulator,” Appl. Optics49, 1323–1330 (2010). [CrossRef]
- A. L. Gaunt and Z. Hadzibabic, “Robust digital holography for ultracold atom trapping,” Sci. Rep.2, 721 (2012). [CrossRef] [PubMed]
- T. P. Meyrath, F. Schreck, J. L. Hanssen, C. Chuu, and M. G. Raizen, “Bose-Einstein condensate in a box,” Phys. Rev. A71, 041604 (2005). [CrossRef]
- S. Hunn, K. Zimmermann, M. Hiller, and A. Buchleitner, “Tunneling decay of two interacting bosons in an asymmetric double-well potential: A spectral approach,” Phys. Rev. A87, 043626 (2013). [CrossRef]
- G. D. Bruce, J. Mayoh, G. Smirne, L. Torralbo-Campo, and D. Cassettari, “A smooth, holographically generated ring trap for the investigation of superfluidity in ultracold atoms,” Phys. ScriptaT143, 014008 (2011). [CrossRef]

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