## Dynamic optical lattices: two-dimensional rotating and accordion lattices for ultracold atoms

Optics Express, Vol. 16, Issue 21, pp. 16977-16983 (2008)

http://dx.doi.org/10.1364/OE.16.016977

Acrobat PDF (177 KB)

### Abstract

We demonstrate a novel experimental arrangement which can rotate a 2D optical lattice at frequencies up to several kilohertz. Ultracold atoms in such a rotating lattice can be used for the direct quantum simulation of strongly correlated systems under large effective magnetic fields, allowing investigation of phenomena such as the fractional quantum Hall effect. Our arrangement also allows the periodicity of a 2D optical lattice to be varied dynamically, producing a 2D accordion lattice.

© 2008 Optical Society of America

## 1. Introduction

10. S. Tung, V. Schweikhard, and E. A. Cornell, “Observation of Vortex Pinning in Bose-Einstein Condensates,” Phys. Rev. Lett. **97**, 240402 (2006) [CrossRef]

13. L. Fallani, C. Fort, J. E. Lye, and M. Inguscio, “Bose-Einstein condensate in an optical lattice with tunable spacing: transport and static properties,” Opt. Express **13**, 4303–4313 (2005) [CrossRef] [PubMed]

14. T. C. Li, H. Kelkar, D. Medellin, and M. G. Raizen, “Real-time control of the periodicity of a standing wave: an optical accordion,” Opt. Express **16**, 5465–5470 (2008) [CrossRef] [PubMed]

*µ*m) after the Mott insulator had been achieved. Detecting and manipulating atoms at single lattice sites are important criteria for quantum information processing schemes with neutral atoms [16

16. D. Jaksch, “Optical Lattices, Ultracold Atoms and Quantum Information Processing,” Contemp. Phys. **45**, 367–381 (2004). [CrossRef]

## 2. Experimental arrangement

*λ*/

**2**another often-used technique is having two beams intersecting at an angle of

**2**

*θ*, producing a lattice with a spacing of

*d*=

*λ*/(

**2**sin

*θ*). Two parallel beams incident on a lens will form a 1D optical lattice in the focal plane of the lens where they intersect. Such an arrangement was used recently to realize a 1D accordion lattice [14

14. T. C. Li, H. Kelkar, D. Medellin, and M. G. Raizen, “Real-time control of the periodicity of a standing wave: an optical accordion,” Opt. Express **16**, 5465–5470 (2008) [CrossRef] [PubMed]

**2**D rotating lattice as shown in Fig. 1.

**2**, which introduced a parallel displacement about the optical axis between the two beams. By generating this second, displaced beam from the first we ensure both beams have the same polarization and frequency and thus will interfere with each other. This would not be the case if two individual AODs were used to rotate two independent beams. The entire setup had cylindrical symmetry about the optical axis of the system and hence rotation could be realized.

**2**andM

**2**in the long arm were chosen such that the output beams from the short and long arms had the same waist and convergence despite the unequal path lengths. This can be seen most easily by realizing that the long arm of the arrangement in Fig. 2 contains a conventional 4f optical imaging system, that is, a one-to-one telescope, mapping the waist and wavefront curvature of the incoming beam onto the beam leaving the long arm [17]. The beam incident on the BS did not therefore need to be collimated to ensure that the two output beams had identical Gaussian characteristics. The unequal path lengths of the different arms mean that the laser must have a coherence length greater than the path difference, however this requirement is easily satisfied by lasers typically used for creating optical lattices and by a reasonable choice for L

**2**. For L

**2**we used a 10cm focal length achromatic doublet, while a Titanium-doped Sapphire laser operating at 830nm (coherence length≫1m) was used for the lattice beams.

**r**

*designates coordinates in the back focal plane of L3,*

_{F}*F*is the focal length of lens L3,

**k**

_{1}and

**k**

_{2}are the wavevectors of the intersecting beams as shown in Fig. 3 and

*ω*is the angular frequency of the laser beams. The resultant intensity thus has the form

**k**

_{1},

**k**

_{2}can be expressed as

**k**

*=(*

_{i}**2**

*π*/

*λ*)(-

*x*

_{i}**e**

*-*

_{x}*y*

_{i}**e**

*+*

_{y}*F*

**e**

*)/(*

_{z}*x*

^{2}

*+*

_{i}*y*

^{2}

*+*

_{i}*F*

^{2})

^{1/2}, where

*x*(

_{i}*t*),

*y*(

_{i}*t*) are the coordinates of the beams in the front focal plane of L3. However high numerical aperture objective lenses have a finite thickness and are designed to obey the Sine Condition [18] to ensure good imaging of off-axis objects. A consequence of obeying the Sine Condition is that the wavevectors

**k**

_{1},

**k**

_{2}make a different angle to the optical axis, with

**k**

*=(*

_{i}**2**

*π*/

*λ*)(-

*x*

_{i}**e**

*-*

_{x}*y*

_{i}**e**

*+*

_{y}*F*

**e**

*)/*

_{z}*F*. The optical system of Fig. 2 created two beams in the front focal plane of L3 such that

*x*

_{2}(

*t*)=-

*x*

_{1}(

*t*),

*y*

_{2}(

*t*)=-

*y*

_{1}(

*t*). Equation 3 then becomes

*x*

_{1}(

*t*) and

*y*

_{1}(

*t*) by steering the beam with the AOD, e.g. for a lattice rotating at an angular frequency of Ω,

*x*

_{1}(

*t*)=(

*D*/2)cosΩ

*t*,

*y*

_{1}(

*t*)=(

*D*/2) sinΩ

*t*, where

*D*is the separation of the two beams in the front focal plane of L3. Substituting this into Equation 4 one finds

*d*=

*λF*/

*D*is the periodicity of the lattice. Li

*et al*[14

14. T. C. Li, H. Kelkar, D. Medellin, and M. G. Raizen, “Real-time control of the periodicity of a standing wave: an optical accordion,” Opt. Express **16**, 5465–5470 (2008) [CrossRef] [PubMed]

*D*, the separation of the beams in the front focal plane of L3.

*D*is determined by the angle of deflection of the beam from the optical axis (

*ϕ*in Fig. 2) and the focal length of L1,

*D*=2

*f*

_{1}tan

*ϕ*. The smallest lattice spacing achievable,

*d*

_{min}, is set by the maximum intersection angle of the beams and hence the numerical aperture (N.A.) of objective lens L3,

*d*

_{min}=

*λ*/(2N.A.). The minimum lattice periodicity possible is of considerable importance in experiments with cold atoms as quantum tunnelling and the mutual interaction energy of two atoms on the same site become negligible at larger lattice spacings. High N.A. objective lenses have been successfully interfaced with cold atoms, for example in [19

19. N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, “Sub-poissonian loading of single atoms in a microscopic dipole trap,” Nature **411**, 1024–1027 (2001) [CrossRef] [PubMed]

*d*

_{min}≈600nm. The largest possible lattice period was limited by the Gaussian profile of the laser beams, that is, to achieve a reasonable number of fringes in the central, uniform intensity region of the beam profile

*d*is required to be smaller than the beam waist.

20. S. K. Schnelle, E. D. van Ooijen, M. J. Davis, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Versatile two-dimensional potentials for ultracold atoms,” Opt. Express **16**, 1405–1412 (2008) [CrossRef] [PubMed]

*π*/2 out of phase with the first. These two rotating beams can then be combined at a beam splitter cube before both entering the Michelson interferometer-like optics. In order to prevent interference between the two one-dimensional lattices one pair of lattice beams must be detuned by MHz relative to the other pair. This can be achieved by operating the two AODs about different center frequencies which are separated by MHz. In this case the rf signal applied to each AOD is modulated sinusoidally, with the center frequency for each AOD separated by a frequency larger than twice the modulation amplitude. Alternatively an acoustic-optic modulator can be used to shift the frequency of a beam before it enters an AOD.

## 3. Demonstration of two-dimensional rotating and accordion lattices

*µ*m to 18

*µ*m, details of which will be published in future work.

## 4. Conclusion

21. G. Milne, D. Rhodes, M. MacDonald, and K. Dholakia, “Fractionation of polydisperse colloid with acousto-optically generated potential energy landscapes,” Opt. Lett. **32**, 1144–1146 (2007) [CrossRef] [PubMed]

## Acknowledgements

## References and links

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

2. | J. K. Jain, |

3. | S. Viefers, “Quantum Hall physics in rotating Bose-Einstein condensates,” J. Phys.: Condens. Matter |

4. | A. S. Sorenson, E. Demler, and M. D. Lukin, “Fractional Quantum Hall States of Atoms in Optical Lattices,” Phys. Rev. Lett. |

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

6. | R. Bhat, M. Krämer, J. Cooper, and M. J. Holland, “Hall effects in Bose-Einstein condensates in a rotating optical lattice,” Phys. Rev. A |

7. | D. Jaksch and P. Zoller, “Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms,” New J. Phys. |

8. | M. Polini, R. Fazio, A. H. MacDonald, and M. P. Tosi, “Realization of Fully Frustrated Josephson Junction Arrays with Cold Atoms,” Phys. Rev. Lett. |

9. | K. Kasamatsu, “Uniformly frustrated bosonic Josephson junction arrays,” arXiv:0806.2012 |

10. | S. Tung, V. Schweikhard, and E. A. Cornell, “Observation of Vortex Pinning in Bose-Einstein Condensates,” Phys. Rev. Lett. |

11. | K. Kasamatsu and M. Tsubota, “Dynamical vortex phases in a Bose-Einstein condensate driven by a rotating optical lattice,” Phys. Rev. Lett. |

12. |
“Optical Lattices and Quantum degenerate |

13. | L. Fallani, C. Fort, J. E. Lye, and M. Inguscio, “Bose-Einstein condensate in an optical lattice with tunable spacing: transport and static properties,” Opt. Express |

14. | T. C. Li, H. Kelkar, D. Medellin, and M. G. Raizen, “Real-time control of the periodicity of a standing wave: an optical accordion,” Opt. Express |

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

16. | D. Jaksch, “Optical Lattices, Ultracold Atoms and Quantum Information Processing,” Contemp. Phys. |

17. | B.E.A Saleh and M.C. Teich, |

18. | M. Born and E. Wolf, |

19. | N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, “Sub-poissonian loading of single atoms in a microscopic dipole trap,” Nature |

20. | S. K. Schnelle, E. D. van Ooijen, M. J. Davis, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Versatile two-dimensional potentials for ultracold atoms,” Opt. Express |

21. | G. Milne, D. Rhodes, M. MacDonald, and K. Dholakia, “Fractionation of polydisperse colloid with acousto-optically generated potential energy landscapes,” Opt. Lett. |

**OCIS Codes**

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

(020.7010) Atomic and molecular physics : Laser trapping

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: September 2, 2008

Revised Manuscript: October 7, 2008

Manuscript Accepted: October 7, 2008

Published: October 9, 2008

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16977

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### References

- M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(de), and U . Sen, "Ultracold atomic gases in optical lattices: Mimicking condensed matter physics and beyond," Adv. Phys. 56, 243-379 (2007). [CrossRef]
- J. K. Jain, Composite Fermions (OUP, 2007). [CrossRef]
- S. Viefers, "Quantum Hall physics in rotating Bose-Einstein condensates," J. Phys.: Condens. Matter 20,123202.1-123202.14 (2008). [CrossRef]
- A. S. Sorenson, E. Demler, and M. D. Lukin, "Fractional Quantum Hall States of Atoms in Optical Lattices," Phys. Rev. Lett. 94, 086803 (2005). [CrossRef]
- R. N. Palmer and D. Jaksch, "High field fractional quantum Hall effect in optical lattices," Phys. Rev. Lett. 96, 180407 (2006). [CrossRef] [PubMed]
- R. Bhat, M. Kramer, J. Cooper, and M. J. Holland, "Hall effects in Bose-Einstein condensates in a rotating optical lattice," Phys. Rev. A 76, 043601 (2007). [CrossRef]
- D. Jaksch and P. Zoller, "Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms," New J. Phys. 5, 56.1-56.11 (2003). [CrossRef]
- M. Polini, R. Fazio, A. H. MacDonald and M. P. Tosi, "Realization of Fully Frustrated Josephson Junction Arrays with Cold Atoms," Phys. Rev. Lett. 95, 010401 (2005) [CrossRef] [PubMed]
- K. Kasamatsu, "Uniformly frustrated bosonic Josephson junction arrays," arXiv:0806.2012
- S. Tung, V. Schweikhard, and E. A. Cornell, "Observation of Vortex Pinning in Bose-Einstein Condensates," Phys. Rev. Lett. 97, 240402 (2006). [CrossRef]
- K. Kasamatsu and M. Tsubota, "Dynamical vortex phases in a Bose-Einstein condensate driven by a rotating optical lattice," Phys. Rev. Lett. 97, 240404 (2006). [CrossRef]
- J. H. Huckans, "Optical Lattices and Quantum degenerate 87Rb in reduced dimensions," PhD thesis, University of Maryland (2006).
- L. Fallani, C. Fort, J. E. Lye, and M. Inguscio, "Bose-Einstein condensate in an optical lattice with tunable spacing: transport and static properties," Opt. Express 13, 4303-4313 (2005). [CrossRef] [PubMed]
- T. C. Li, H. Kelkar, D. Medellin, and M. G. Raizen, "Real-time control of the periodicity of a standing wave: an optical accordion," Opt. Express 16, 5465-5470 (2008). [CrossRef] [PubMed]
- M. Greiner, O. Mandel, T. Essingler, T. W. H¨ansch and I. Bloch, "Quantum phase transition from a superfluid to a Mott insulator transition in a gas of ultracold atoms," Nature 415, 39-44 (2002).
- D. Jaksch, "Optical Lattices, Ultracold Atoms and Quantum Information Processing," Contemp. Phys. 45, 367-381 (2004). [CrossRef]
- B. E. A Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991) pp. 136-139.
- M. Born and E. Wolf, Principles of Optics, 7th Edition (CUP, 1999) pp. 178-180.
- N. Schlosser, G. Reymond, I. Protsenko and P. Grangier, "Sub-poissonian loading of single atoms in a microscopic dipole trap," Nature 411, 1024-1027 (2001). [CrossRef] [PubMed]
- S. K. Schnelle, E. D. van Ooijen, M. J. Davis, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Versatile twodimensional potentials for ultracold atoms," Opt. Express 16, 1405-1412 (2008). [CrossRef] [PubMed]
- G. Milne, D. Rhodes, M. MacDonald, and K. Dholakia, "Fractionation of polydisperse colloid with acoustooptically generated potential energy landscapes," Opt. Lett. 32, 1144-1146 (2007). [CrossRef] [PubMed]

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