## Real-time control of the periodicity of a standing wave: an optical accordion

Optics Express, Vol. 16, Issue 8, pp. 5465-5470 (2008)

http://dx.doi.org/10.1364/OE.16.005465

Acrobat PDF (1148 KB)

### Abstract

We report an experimental method to create optical lattices with real-time control of their periodicity. We demonstrate a continuous change of the lattice periodicity from 0.96 *µ*m to 11.2 *µ*m in one second, while the center fringe only moves less than 2.7 *µ*m during the whole process. This provides a powerful tool for controlling ultracold atoms in optical lattices, where small spacing is essential for quantum tunneling, and large spacing enables single-site manipulation and spatially resolved detection.

© 2008 Optical Society of America

## 1. Introduction

1. O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. **78**, 179–215 (2006). [CrossRef]

2. G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. **82**, 1060–1063 (1999). [CrossRef]

3. O. Mandel, M. Greiner, A Widera, T. Rom, T.W. Hänsch, and I. Bloch, “Controlled collisions for multi-particle entanglement of optically trapped atoms,” Nature **425**, 937–940 (2003). [CrossRef] [PubMed]

4. J. V. Porto, S. Rolston, B. Laburthe Tolra, C. J. Williams, and W. D. Phillips, “Quantum information with neutral atoms as qubits,” Phil. Trans. R. Soc. Lond. A **361**, 1417–1427 (2003). [CrossRef]

5. F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson junction arrays with Bose-Einstein condensates,” Science **293**, 843–846 (2001). [CrossRef] [PubMed]

6. K. Henderson, H. Kelkar, B. Gutiérrez-Medina, T. C. Li, and M. G. Raizen, “Experimental study of the role of atomic interactions on quantum transport,” Phys. Rev. Lett. **96**, 150401 (2006). [CrossRef] [PubMed]

7. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature **415**, 39–44 (2002). [CrossRef] [PubMed]

8. F. Gerbier, S. Fölling, A. Widera, O. Mandel, and I. Bloch, “Probing number squeezing of ultracold atoms across the superfluid-Mott insulator transition,” Phys. Rev. Lett. **96**, 090401 (2006). [CrossRef] [PubMed]

9. B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature **429**, 277–281 (2004). [CrossRef] [PubMed]

10. T. Kinoshita, T. Wenger, and D. S. Weiss, “Observation of a one-dimensional Tonks-Girardeau gas,” Sceince **305**, 1125–1128 (2004). [CrossRef]

11. R. Scheunemann, F. S. Cataliotti, T. W. Hänsch, and M. Weitz, “Resolving and addressing atoms in individual sites of a CO2-laser optical lattice,” Phys. Rev. A **62**, 051801(R) (2000). [CrossRef]

12. K. D. Nelson, X. Li, and D. S. Weiss, “Imaging single atoms in a three-dimensional array,” Nature Phys. **3**, 556–560 (2007). [CrossRef]

13. S. Peil, J. V. Porto, B. Laburthe Tolra, J. M. Obrecht, B. E. King, M. Subbotin, S. L. Rolston, and W. D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A **67**, 051603(R) (2003). [CrossRef]

15. 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]

16. K. Venkatakrishnan, N. R. Sivakumar, C. W. Hee, B. Tan, W. L. Liang, and G. K. Gan, “Direct fabrication of surface-relief grating by interferometric technique using femtosecond laser,” Appl. Phys. A **77**, 959–963 (2003). [CrossRef]

17. B. Tan, N. R. Sivakumar, and K. Venkatakrishnan, “Direct grating writing using femtosecond laser interference frings formed at the focal point,” J. Opt. A: Pure Appl. Opt. **7**, 169–174 (2005). [CrossRef]

18. X. S. Xie, M. Li, J. Guo, B. Liang, Z. X. Wang, A. Sinitskii, Y. Xiang, and J. Y. Zhou, “Phase manipulated multi-beam holographic lithography for tunable optical lattices,” Opt. Express **15**, 7032–7037 (2007). [CrossRef] [PubMed]

*µ*m while the lattice spacing is changed from 0.96

*µ*m to 11.2

*µ*m in one second. This stability allows one to change the lattice spacing in real time while keeping atoms trapped. This is also useful in many other applications of optical lattices, such as sorting microscopic particles [19

19. M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature **426**, 421–424 (2003). [CrossRef] [PubMed]

## 2. Theory and experiment

*D*are brought together by a lens to produce an optical lattice at the focal plane of the lens. If the beams at the focus are plane waves, the lattice spacing will be

*d*=

*λ*/(2sin(

*θ*/2)), where

*λ*denotes the laser wavelength, and

*θ*is the angle between the two beams at the focus. For a thin lens, if the beams are symmetric with respect to the axis of the lens, the angle will be approximately

*θ*≈2tan

^{-1}(

*D*/2

*f*). Thus

*f*is the focal length of the lens. We can tune the lattice spacing by changing

*D*. Eq. (1) is commonly used for calculating the lattice spacing [16

16. K. Venkatakrishnan, N. R. Sivakumar, C. W. Hee, B. Tan, W. L. Liang, and G. K. Gan, “Direct fabrication of surface-relief grating by interferometric technique using femtosecond laser,” Appl. Phys. A **77**, 959–963 (2003). [CrossRef]

*θ*is large.

*U*(

*x*

_{1},

*y*

_{1}, -

*f*), then the field at the back focal plane of the lens

*U*

_{0}(

*x*,

*y*,

*f*) is the Fourier transform of

*U*(

*x*

_{1},

*y*

_{1}, -

*f*). For a Gaussian beam, the resulting intensity distribution

*I*

_{0}(

*x*,

*y*,

*f*)=

*U**

_{0}(

*x*,

*y*,

*f*)

*U*

_{0}(

*x*,

*y*,

*f*) is still Gaussian and can be calculated easily. If the incident beam is shifted horizontally by +

*D*/2 or -

*D*/2, the resulted field at the back focal plane will become

*θ*is large. From Eq. (5), we get the angle between the two beams at the focus to be

*θ*=2sin

^{-1}(

*D*/2

*f*).

*λ*=532 nm) through two lenses (L1, L2), one pinhole (P1) and one iris. A

*λ*/2 waveplate is used to change the orientation of the polarization of the beam, which controls the relative power of the two parallel beams. The s-polarized component of the beam is reflected by BS2 and the p-polarized component passes through both beamsplitters. After passing through the

*λ*/4 waveplate twice, the p-polarized component becomes s-polarized and is reflected by BS1. These two parallel beams are brought together by an achromatic doublet lens L3 to create the optical lattice. Then the optical lattice is magnified by an objective and imaged by a CCD camera. We tried two lenses with different focal lengths (30.0 and 80.0 mm) for L3. The optical lattice is formed at the focal plane of L3. Its periodicity can be changed in real time by movingM1 with a stepper motor, and its fringe contrast can be close to 100% by tuning the

*λ*/2 waveplate.

## 3. Results and discussion

*µ*m. It was created from two parallel beams separated by 19.25 mm, which interfered at the focal plane of a

*f*=30.0 mm lens. The waists of the two beams at the focus are 36

*µ*m and 40

*µ*m, which were measured by a scanning knife edge. Although the beams were close to the edge of the lens and the angle was large (

*θ*≈38 °), the interference fringes were very straight even at the edge of the beams. This agrees with Eq. (5).

*µ*m/pixel, where the uncertainty comes from the measurements of beam waists. It can also be calibrated by fitting the data with Eq. (5), which gives 0.0855±0.0001

*µ*m/pixel. We use 0.0853

*µ*m/pixel for the data shown in the figure. Equation (5) agrees with all data points, and clearly fits the data better than Eq. (1) when

*D*is large.

*µ*m and 6.20

*µ*m. They are formed at the focal plane of a

*f*=80.0 mm lens. The change of spacing is achieved by movingM1.

*µ*m to 11.2

*µ*m, and back to 0.96

*µ*m in real time is presented in Fig. 6. The optical lattice is formed at the focal plane of a

*f*=80.0 mm lens. Mirror M1 is moved horizontally by 20.0 mm to change

*D*from 43.81 mm to 3.79 mm. This figure is constructed from real-time images (30 frames/second) similar to Fig. 5. It shows that our optical lattice is very stable. The center fringe (marked by a solid line) moved less than 2.7

*µ*m during the whole process, which is only quarter of the final lattice spacing. There is no apparent difference in the vibration of the lattice whether M1 is moving or not. This means that the vibration due to translating M1 does not transfer to the optical lattice. In Fig. 6(b), we change the lattice spacing from 0.96

*µ*m to 11.2

*µ*m in one second, wait for half a second, and change the spacing back to 0.96

*µ*m in another one second. The life time of ultracold atoms in the optical lattices can be longer than 10 seconds [12

12. K. D. Nelson, X. Li, and D. S. Weiss, “Imaging single atoms in a three-dimensional array,” Nature Phys. **3**, 556–560 (2007). [CrossRef]

*D*by moving BS1 and BS2 together horizontally, which was similar to Ref. [17

17. B. Tan, N. R. Sivakumar, and K. Venkatakrishnan, “Direct grating writing using femtosecond laser interference frings formed at the focal point,” J. Opt. A: Pure Appl. Opt. **7**, 169–174 (2005). [CrossRef]

*µ*m in the perpendicular direction, the difference between the optical path lengths of the two beams will change by 4.26

*µ*m. Thus the center fringe of the optical lattices will shift (or vibrate) by 8 fringes, which is 80

*µ*m when the lattice spacing is 10

*µ*m.

## 4. Conclusion

*µ*m while the lattice spacing is changed by one order of magnitude. Such accordion lattices can work as magnifiers or compressors for many applications.

## Acknowledgments

## References and links

1. | O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. |

2. | G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. |

3. | O. Mandel, M. Greiner, A Widera, T. Rom, T.W. Hänsch, and I. Bloch, “Controlled collisions for multi-particle entanglement of optically trapped atoms,” Nature |

4. | J. V. Porto, S. Rolston, B. Laburthe Tolra, C. J. Williams, and W. D. Phillips, “Quantum information with neutral atoms as qubits,” Phil. Trans. R. Soc. Lond. A |

5. | F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson junction arrays with Bose-Einstein condensates,” Science |

6. | K. Henderson, H. Kelkar, B. Gutiérrez-Medina, T. C. Li, and M. G. Raizen, “Experimental study of the role of atomic interactions on quantum transport,” Phys. Rev. Lett. |

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

8. | F. Gerbier, S. Fölling, A. Widera, O. Mandel, and I. Bloch, “Probing number squeezing of ultracold atoms across the superfluid-Mott insulator transition,” Phys. Rev. Lett. |

9. | B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature |

10. | T. Kinoshita, T. Wenger, and D. S. Weiss, “Observation of a one-dimensional Tonks-Girardeau gas,” Sceince |

11. | R. Scheunemann, F. S. Cataliotti, T. W. Hänsch, and M. Weitz, “Resolving and addressing atoms in individual sites of a CO2-laser optical lattice,” Phys. Rev. A |

12. | K. D. Nelson, X. Li, and D. S. Weiss, “Imaging single atoms in a three-dimensional array,” Nature Phys. |

13. | S. Peil, J. V. Porto, B. Laburthe Tolra, J. M. Obrecht, B. E. King, M. Subbotin, S. L. Rolston, and W. D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A |

14. | J. H. Huckans, “Optical lattices and quantum degenerate |

15. | 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 |

16. | K. Venkatakrishnan, N. R. Sivakumar, C. W. Hee, B. Tan, W. L. Liang, and G. K. Gan, “Direct fabrication of surface-relief grating by interferometric technique using femtosecond laser,” Appl. Phys. A |

17. | B. Tan, N. R. Sivakumar, and K. Venkatakrishnan, “Direct grating writing using femtosecond laser interference frings formed at the focal point,” J. Opt. A: Pure Appl. Opt. |

18. | X. S. Xie, M. Li, J. Guo, B. Liang, Z. X. Wang, A. Sinitskii, Y. Xiang, and J. Y. Zhou, “Phase manipulated multi-beam holographic lithography for tunable optical lattices,” Opt. Express |

19. | M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature |

**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: February 19, 2008

Revised Manuscript: April 1, 2008

Manuscript Accepted: April 1, 2008

Published: April 3, 2008

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5465

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### References

- O. Morsch and M. Oberthaler, "Dynamics of Bose-Einstein condensates in optical lattices," Rev. Mod. Phys. 78, 179-215 (2006). [CrossRef]
- G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, "Quantum logic gates in optical lattices," Phys. Rev. Lett. 82, 1060-1063 (1999). [CrossRef]
- O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Höansch, and I. Bloch, "Controlled collisions for multi-particle entanglement of optically trapped atoms," Nature 425, 937-940 (2003). [CrossRef] [PubMed]
- J. V. Porto, S. Rolston, B. Laburthe Tolra, C. J. Williams, and W. D. Phillips, "Quantum information with neutral atoms as qubits," Phil. Trans. R. Soc. Lond. A 361, 1417-1427 (2003). [CrossRef]
- F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson junction arrays with Bose-Einstein condensates," Science 293, 843-846 (2001). [CrossRef] [PubMed]
- K. Henderson, H. Kelkar, B. Guti’errez-Medina, T. C. Li, and M. G. Raizen, "Experimental study of the role of atomic interactions on quantum transport," Phys. Rev. Lett. 96, 150401 (2006) [CrossRef] [PubMed]
- M. Greiner, O. Mandel, T. Esslinger, T.W. H¨ansch, and I. Bloch, "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms," Nature 415, 39-44 (2002). [CrossRef] [PubMed]
- F. Gerbier, S. F¨olling, A. Widera, O. Mandel, and I. Bloch, "Probing number squeezing of ultracold atoms across the superfluid-Mott insulator transition," Phys. Rev. Lett. 96, 090401 (2006) [CrossRef] [PubMed]
- B. Paredes, A. Widera, V. Murg, O. Mandel, S. F¨olling, I. Cirac, G. V. Shlyapnikov, T. W. H¨ansch, and I. Bloch, "Tonks-Girardeau gas of ultracold atoms in an optical lattice," Nature 429, 277-281 (2004). [CrossRef] [PubMed]
- Q1. T. Kinoshita, T. Wenger, and D. S. Weiss, "Observation of a one-dimensional Tonks-Girardeau gas," Science 305, 1125-1128 (2004) [CrossRef]
- R. Scheunemann, F. S. Cataliotti, T. W. Hänsch, and M. Weitz, "Resolving and addressing atoms in individual sites of a CO2-laser optical lattice," Phys. Rev. A 62, 051801(R) (2000). [CrossRef]
- Q2. K. D. Nelson, X. Li, and D. S. Weiss, "Imaging single atoms in a three-dimensional array," Nat. Phys. 3, 556-560 (2007). [CrossRef]
- S. Peil, J. V. Porto, B. Laburthe Tolra, J. M. Obrecht, B. E. King, M. Subbotin, S. L. Rolston, and W. D. Phillips, "Patterned loading of a Bose-Einstein condensate into an optical lattice," Phys. Rev. A 67, 051603(R) (2003). [CrossRef]
- J. H. Huckans, "Optical lattices and quantum degenerate 87Rb in reduced dimensions, " Ph.D. dissertation, University of Maryland, College Park (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]
- Q3. K. Venkatakrishnan, N. R. Sivakumar, C. W. Hee, B. Tan, W. L. Liang, and G. K. Gan, "Direct fabrication of surface-relief grating by interferometric technique using femtosecond laser, " Appl. Phys. A 77, 959-963 (2003). [CrossRef]
- B. Tan, N. R. Sivakumar, and K. Venkatakrishnan, "Direct grating writing using femtosecond laser interference frings formed at the focal point," J. Opt. A: Pure Appl. Opt. 7, 169-174 (2005) [CrossRef]
- X. S. Xie, M. Li, J. Guo, B. Liang, Z. X. Wang, A. Sinitskii, Y. Xiang, and J. Y. Zhou, "Phase manipulated multi-beam holographic lithography for tunable optical lattices," Opt. Express 15, 7032-7037 (2007). [CrossRef] [PubMed]
- M. P. MacDonald, G. C. Spalding, and K. Dholakia, "Microfluidic sorting in an optical lattice," Nature 426, 421-424 (2003). [CrossRef] [PubMed]

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