## Bose-Einstein condensate in an optical lattice with tunable spacing: transport and static properties

Optics Express, Vol. 13, Issue 11, pp. 4303-4313 (2005)

http://dx.doi.org/10.1364/OPEX.13.004303

Acrobat PDF (587 KB)

### Abstract

In this Letter we report the investigation of transport and static properties of a Bose-Einstein condensate in a large-spaced optical lattice. The lattice spacing can be easily tuned starting from few micrometers by adjusting the relative angle of two partially reflective mirrors. We have performed *in-situ* imaging of the atoms trapped in the potential wells of a 20 *µ*m spaced lattice. For a lattice spacing of 10 *µ*m we have studied the transport properties of the system and the interference pattern after expansion, evidencing quite different results with respect to the physics of BECs in ordinary near-infrared standing wave lattices, owing to the different length and energy scales.

© 2005 Optical Society of America

## 1. Introduction

## 2. Experimental setup

*δ*one in front of the other at a distance of ≈1 mm. As shown in Fig. 1, the multiple reflection of the laser beam from these mirrors produces, at the second order of reflection, two separate beams (of different intensities) with a relative angle 2

*δ*. These two beams, following different optical paths, are then guided by a lens system to recombine onto the condensate, where they interfere producing a periodic pattern with alternating intensity maxima and minima. The period

*d*of this lattice, that is oriented along the difference of the wavevectors, depends on the angle

*α*between the beams according to

*α=π*) the above expression reduces to the well known spacing λ/2 for a standing wave lattice. In the other limit, when the two beams are almost copropagating (α⋍

*d*may become very large. In our setup, varying the angle

*δ*between the two mirrors, it is possible to easily adjust the lattice spacing to the desired value. For the working wavelength

*λ*=820 nm the lower limit is

*d*≈8

*µ*m, corresponding to the maximum angle

*α*=25° that is possible to reach in our setup taking into account the finite size of the vacuum cell windows. Since the two beams producing the lattice do not have the same intensity, we expect the resulting interference pattern to show a reduced contrast with respect to the case in which the two beams have the same intensity. It can be shown that the intensity of the lattice

*I*, i.e. the intensity difference between constructive interference and destructive interference, is given by

_{L}*t*is the transmissivity of the partially reflecting mirrors and

*I*

_{0}is the intensity of the beam incident on them. It is easy to show that this expression has a maximum for

*t*=2/3. For this reason in the experiment we have used mirrors with

*t*⋍0.7, corresponding to the maximum lattice intensity

*I*⋍0.59

_{L}*I*

_{0}achievable with this technique.

*λ*>800 nm and transmitting in the range

*λ*<800 nm. This feature allows us to use the same imaging setup to detect both the BEC and the spatial profile of the lattice light intensity. This means that we can image in consecutive photos both the condensate and the exact potential that the condensate experiences. Indeed, since the CCD plane is conjugate to the vertical plane passing through the trap axis, the intensity profile recorded by the CCD is exactly the same (except for a magnification factor) as the one imaged onto the condensate. Furthermore, by calibrating the CCD responsivity with a reference beam of known intensity, it is possible to convert the digitized signal of each pixel into an intensity value and thus calculate the height of the potential

*V*. In the following, the lattice height and the other energy scales will be conveniently expressed in frequency units (using the implicit assumption of a division by the Planck constant

_{0}*h*).

*d*=20

*µ*m, while in Fig. 2(b) we show an absorption image of the atoms trapped in the potential wells of the same lattice. The experimental sequence used to trap the atoms in the optical lattice and to image

*in situ*the atomic distribution is the following. First we produce a cigar-shaped BEC of

^{87}Rb in a Ioffe-Pritchard magnetostatic trap, with trap frequencies

*ω*/2

_{z}*π*=8.74(1) Hz along the symmetry axis

*z*(oriented horizontally) and

*ω*=85(1) Hz along the orthogonal directions. The typical diameter of the condensates is 150

_{r}/2_{π}*µ*m axially and 15

*µ*m radially. We note that the lattice beam profile has been tailored with cylindrical lenses in order to match the condensate elongated shape (see Fig. 2(a)). After producing the BEC, still maintaining the magnetic confinement, we ramp in 100 ms the height of the lattice from zero to the final value by using an acousto-optic modulator (AOM). After the end of the ramp we wait 50 ms, then we abruptly switch off the optical lattice with the same AOM and, after a few tens of

*µ*s, we flash the imaging beam for the detection phase. The latter time interval is necessary not to perturb the imaging with lattice light coming onto the CCD, but is small enough not to let the atoms expand from the lattice sites once the optical confinement is released. The combination of the CCD electronic shutter and an interferential bandpass filter placed in front of the camera (centered around

*λ*=780 nm) allows a complete extinction of the lattice light at the time of acquisition. In Fig. 2(c) and 2(d) we show, respectively, the power spectrum of the two-dimensional Fourier transform of the distributions shown in Figs. 2(a) and 2(b). As one can see, both the distributions are characterized by sharp peaks in momentum space: from the position of these peaks it is possible to precisely measure the spatial period of the observed structures.

## 3. Static properties

*d*. After releasing the atoms from the trapping potential one expects to observe a periodic interference pattern with spacing

*m*is the atomic mass,

*t*is the expansion time and

_{exp}*d*is the spacing of the optical lattice [16

16. M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of Interference Between Two Bose Condensates,” Science **275**, 637–641 (1997). [CrossRef] [PubMed]

*d*=10

*µ*m, producing after

*t*=28 ms an interference pattern with fringe spacing

_{exp}*d*

*′*=12.8

*μ*m, easily detectable with our imaging setup. We note that for this lattice spacing the recoil energy

*E*=

_{R}*h*, the natural energy scale for measuring the lattice height, is only 6 Hz, almost 600 times smaller than the recoil energy for a regular standing-wave lattice with

^{2}/8md^{2}*d*=0.4

*µ*m spacing.

*k*is the modulus of the wavevector and

*n*-th source from the detection point. If one assumes that

*N*=20 sources identified by three different sets of variables {

*z*}, in which we release one at a time the hypothesis of identical phases and uniform spacing. The diagrams on the left show the displacement of each source

_{n},ϕ_{n}*δz*-nd from the regular lattice position and its phase

_{n}=z_{n}*φ*. The graphs on the right show the field intensity |

_{n}*A*(

*z*′)|

^{2}calculated with Eq. (6).

*d*between the sources. In optics, this is the intensity distribution produced by a diffraction grating illuminated by coherent light. In matter-wave optics, a similar interferogram is observed in the superfluid regime after the expansion of BECs released from optical lattices produced with near-infrared standing waves [14

14. P. Pedri, L. Pitaevskii, S. Stringari, C. Fort, S. Burger, F. S. Cataliotti, P. Maddaloni, F. Minardi, and M. Inguscio, “Expansion of a Coherent Array of Bose-Einstein Condensates,” Phys. Rev. Lett. **87**, 220401 (2001). [CrossRef] [PubMed]

*zn=nd*. The interferogram on the right corresponds to the randomly generated set of displacements shown on the left. While in the case of random phases and uniform spacing a periodic interference pattern is still visible in a single shot, in the case of random spacing no characteristic structure is visible at all, even if all the sources emit coherently. This model reproduces the density distribution of a BEC released from a deep optical speckle potential, where a broad gaussian density profile, without any internal structure, has been experimentally observed [17].

## 4. Transport properties

*z*along the trap axis, as described in [2

2. 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]

*z*=32

*µ*m and different lattice heights. The black points refer to the regular undamped oscillation of the condensate in the pure harmonic potential at the trap frequency

*ν*=(8.74±0.03) Hz.

_{z}*V*

_{0}=5 kHz=830

*E*, we observe that the center-of-mass motion is blocked and the atomic cloud stays at a side of the displaced harmonic trap. Indeed, in this regime the height of the periodic potential becomes larger than the chemical potential of the condensate and the BEC is split into an array of condensates located at the different sites. Differently from the standing wave lattice [2

_{R}2. 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]

3. 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]

12. Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J. Dalibard, “Interference of an Array of Independent Bose-Einstein Condensates,” Phys. Rev. Lett. **93**, 180403 (2004). [CrossRef] [PubMed]

## 5. Conclusions

*µ*m spacing. In particular, we have evidenced the presence of interference fringes after expansion even in the insulating regime in which inter-well tunnelling is heavily suppressed and the center-of-mass dynamics is inhibited. The appeal of this system is that, by increasing the lattice spacing to 20

*µ*m or more, it becomes possible to optically resolve

*in situ*the single lattice sites. This possibility could be important for the implementation of quantum computing schemes, where addressability is a fundamental requirement. As an extension of this work, the following step could be made in the direction of manipulating the single sites either optically or with the application of radiofrequency/microwave transitions coupling different internal levels.

## Acknowledgments

## References and links

1. | B. P. Anderson and M. A. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays,” Science |

2. | 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 |

3. | 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 |

4. | G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum Logic Gates in Optical Lattices,” Phys. Rev. Lett. |

5. | D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of Atoms via Cold Controlled Collisions,” Phys. Rev. Lett. |

6. | 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 |

7. | D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral Atom Quantum Register,” Phys. Rev. Lett. |

8. | R. Scheunemann, F. S. Cataliotti, T. W. Hänsch, and M. Weitz, “Resolving and addressing atoms in individual sites of a CO |

9. | R. Dumke, M. Volk, T. Müther, 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. |

10. | O. Morsch, J. H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, “Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices”, Phys. Rev. Lett. |

11. | 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 |

12. | Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J. Dalibard, “Interference of an Array of Independent Bose-Einstein Condensates,” Phys. Rev. Lett. |

13. | M. Greiner, I. Bloch, O. Mandel, T.W. Hänsch, and T. Esslinger, “Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates,” Phys. Rev. Lett. |

14. | P. Pedri, L. Pitaevskii, S. Stringari, C. Fort, S. Burger, F. S. Cataliotti, P. Maddaloni, F. Minardi, and M. Inguscio, “Expansion of a Coherent Array of Bose-Einstein Condensates,” Phys. Rev. Lett. |

15. | W. Zwerger, “MottHubbard transition of cold atoms in optical lattices,” J. Opt. B |

16. | M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of Interference Between Two Bose Condensates,” Science |

17. | J. E. Lye, L. Fallani, M. Modugno, D. Wiersma, C. Fort, and M. Inguscio, “A Bose-Einstein condensate in a random potential,” preprint arXiv:cond-mat/0412167 (2004). |

**OCIS Codes**

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

(020.7010) Atomic and molecular physics : Laser trapping

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 3, 2005

Revised Manuscript: May 17, 2005

Published: May 30, 2005

**Citation**

Leonardo Fallani, Chiara Fort, Jessica Lye, and Massimo Inguscio, "Bose-Einstein condensate in an optical lattice with tunable spacing: transport and static properties," Opt. Express **13**, 4303-4313 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4303

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### References

- B. P. Anderson and M. A. Kasevich, �??Macroscopic Quantum Interference from Atomic Tunnel Arrays,�?? Science 282, 1686�??1689 (1998). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, �??Entanglement of Atoms via Cold Controlled Collisions,�?? Phys. Rev. Lett. 82, 1975�??1978 (1999). [CrossRef]
- 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]
- D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, �??Neutral Atom Quantum Register,�?? Phys. Rev. Lett. 93, 150501 (2004). [CrossRef] [PubMed]
- 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]
- R. Dumke, M. Volk, T. Müther, 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]
- O. Morsch, J. H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, �??Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices�??, Phys. Rev. Lett. 87, 140402 (2001). [CrossRef] [PubMed]
- 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]
- Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J. Dalibard, �??Interference of an Array of Independent Bose-Einstein Condensates,�?? Phys. Rev. Lett. 93, 180403 (2004). [CrossRef] [PubMed]
- M. Greiner, I. Bloch, O. Mandel, T.W. Hänsch, and T. Esslinger, �??Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates,�?? Phys. Rev. Lett. 87, 160405 (2001). [CrossRef] [PubMed]
- P. Pedri, L. Pitaevskii, S. Stringari, C. Fort, S. Burger, F. S. Cataliotti, P. Maddaloni, F. Minardi, and M. Inguscio, �??Expansion of a Coherent Array of Bose-Einstein Condensates,�?? Phys. Rev. Lett. 87, 220401 (2001). [CrossRef] [PubMed]
- W. Zwerger, �??MottHubbard transition of cold atoms in optical lattices,�?? J. Opt. B 5 S9�??S16 (2003). [CrossRef]
- M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, �??Observation of Interference Between Two Bose Condensates,�?? Science 275, 637�??641 (1997). [CrossRef] [PubMed]
- J. E. Lye, L. Fallani, M. Modugno, D. Wiersma, C. Fort, and M. Inguscio, �??A Bose-Einstein condensate in a random potential,�?? preprint arXiv:cond-mat/0412167 (2004).

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