## Instabilities of a Bose-Einstein condensate in a periodic potential: an experimental investigation

Optics Express, Vol. 12, Issue 1, pp. 4-10 (2004)

http://dx.doi.org/10.1364/OPEX.12.000004

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### Abstract

By accelerating a Bose-Einstein condensate in a controlled way across the edge of the Brillouin zone of a 1D optical lattice, we investigate the stability of the condensate in the vicinity of the zone edge. Through an analysis of the visibility of the interference pattern after a time-of-flight and the widths of the interference peaks, we characterize the onset of instability as the acceleration of the lattice is decreased. We briefly discuss the significance of our results with respect to recent theoretical work.

© 2004 Optical Society of America

## 1. Introduction

1. C. Menotti, A. Smerzi, and A. Trombettoni, “Superfluid dynamics of a Bose-Einstein condensate in a periodic potential,” New J. Phys. **5**, 112 (2003). [CrossRef]

2. Biao Wu and Qian Niu, “Superfluidity of Bose-Einstein condensate in an optical lattice: Landau-Zener tunneling and dynamical instability,” New J. Phys. **5**, 104 (2003). [CrossRef]

3. Pearl J. Y. Louis, Elena A. Ostrovskaya, Craig M. Savage, and Yuri S. Kivshar, “Bose-Einstein condensates in optical lattices: Band-gap structure and solitons,” Phys. Rev. A **67**, 013602 (2003). [CrossRef]

4. O. Morsch, J.H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, “Bloch oscillations and mean-field effects of Bose-Einstein condensates in optical lattices,” Phys. Rev. Lett. **87**, 140402 (2001). [CrossRef] [PubMed]

5. M. Cristiani, O. Morsch, J.H. Müller, D. Ciampini, and E. Arimondo, “Experimental properties of Bose-Einstein condensates in one-dimensional optical lattices: Bloch oscillations, Landau-Zener tunneling, and mean-field effects,” Phys. Rev. A **65**, 063612 (2002). [CrossRef]

6. 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 (London) **415**, 6867 (2002). [CrossRef]

2. Biao Wu and Qian Niu, “Superfluidity of Bose-Einstein condensate in an optical lattice: Landau-Zener tunneling and dynamical instability,” New J. Phys. **5**, 104 (2003). [CrossRef]

7. F. S. Cataliotti, L. Fallani, F. Ferlaino, C. Fort, P. Maddaloni, and M. Inguscio, “Superfluid current disruption in a chain of weakly coupled BoseEinstein condensates,” New J. Phys. **5**, 71 (2003). [CrossRef]

8. Yuri S. Kivshar and Mario Salerno, “Modulational instabilities in the discrete deformable nonlinear Schrödinger equation,” Phys. Rev. E **49**, 3543 (1994). [CrossRef]

9. V.V. Konotop and M. Salerno, “ Modulational instability in Bose-Einstein condensates in optical lattices,” Phys. Rev. A **65**, 021620(R) (2002). [CrossRef]

10. Karen Marie Hilligsøe, Markus K. Oberthaler, and Karl-Peter Marzlin, “Stability of gap solitons in a Bose-Einstein condensate,” Phys. Rev. A **66**, 063605 (2002). [CrossRef]

11. R.G. Scott, A.M. Martin, T.M. Fromhold, S. Bujkiewicz, F.W. Sheard, and M. Leadbeater, “Creation of Solitons and Vortices by Bragg Reflection of Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. **90**, 110404 (2003). [CrossRef] [PubMed]

12. J.R. Anglin, “ Second-quantized Landau-Zener theory for dynamical instabilities,” Phys. Rev. A **67**, 051601(R) (2003). [CrossRef]

13. Yu. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves”, Phys. Rep. **331**, 117 (2000). [CrossRef]

14. Jason W. Fleischer, Mordechai Segev, Nikolaos K. Efrimidis, and Demetrios N. Christodoulides, “Observation of two-dimensional discrete solitions in optically induced nonlinear photonic lattices,” Nature **422**, 147 (2003). [CrossRef] [PubMed]

## 2. Experimental setup and procedure

^{87}

*Rb*atoms is described in detail in [17

17. J.H. Müller, D. Ciampini, O. Morsch, G. Smirne, M. Fazzi, P. Verkerk, F. Fuso, and E. Arimondo, “Bose-Einstein condensation of rubidium atoms in a triaxial TOP trap,” J. Phys. B:At. Mol. Opt. Phys. **33**, 4095 (2000). [CrossRef]

*ν*:

_{x}*ν*:

_{y}*ν*in the ratio 2 : 1 : √2. Our trap is, therefore, almost isotropic. The consequences for our experiment of this near isotropy will be discussed at the end of this paper.

_{z}*λ*, as described in detail in [4

4. O. Morsch, J.H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, “Bloch oscillations and mean-field effects of Bose-Einstein condensates in optical lattices,” Phys. Rev. Lett. **87**, 140402 (2001). [CrossRef] [PubMed]

5. M. Cristiani, O. Morsch, J.H. Müller, D. Ciampini, and E. Arimondo, “Experimental properties of Bose-Einstein condensates in one-dimensional optical lattices: Bloch oscillations, Landau-Zener tunneling, and mean-field effects,” Phys. Rev. A **65**, 063612 (2002). [CrossRef]

*ν*between them. The resulting periodic potential has a lattice constant

*d*=

*λ*/2=0.39

*µ*m, and the depth of the potential (depending on the laser intensity and detuning from the atomic resonance of the rubidium atoms) can be varied between 0 and

*a*=0.3ms

^{-2}to

*a*=5ms

^{-2}.

^{4}atoms, we adiabatically relax the magnetic trap frequency to

*ν*=42Hz. Thereafter, the intensity of the lattice beams is ramped up from 0 to a value corresponding to a lattice depth of ≈2

_{x}*E*. The ramping time is of the order of several milliseconds in order to ensure adiabaticity [18

_{rec}18. Y.B. Band and M. Trippenbach, “Bose-Einstein condensates in time-dependent light potentials: Adiabatic and nonadiabatic behavior of nonlinear wave equations,” Phys. Rev. A **65**, 053602 (2002). [CrossRef]

*t*. Finally, both the magnetic trap and the optical lattice are switched off, and the condensate is observed after a time-of-flight of 21ms by absorption imaging.

## 3. Results: visibility and radial width

4. O. Morsch, J.H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, “Bloch oscillations and mean-field effects of Bose-Einstein condensates in optical lattices,” Phys. Rev. Lett. **87**, 140402 (2001). [CrossRef] [PubMed]

*=2*

_{prec}*ℏk*with

_{L}*k*=2

_{L}*π*/

*λ*), as can be seen in Figs. 1 (a) and (c). The shape of the interference pattern in the transverse direction is shown in Figs. 1 (b) and (d).

2. Biao Wu and Qian Niu, “Superfluidity of Bose-Einstein condensate in an optical lattice: Landau-Zener tunneling and dynamical instability,” New J. Phys. **5**, 104 (2003). [CrossRef]

8. Yuri S. Kivshar and Mario Salerno, “Modulational instabilities in the discrete deformable nonlinear Schrödinger equation,” Phys. Rev. E **49**, 3543 (1994). [CrossRef]

11. R.G. Scott, A.M. Martin, T.M. Fromhold, S. Bujkiewicz, F.W. Sheard, and M. Leadbeater, “Creation of Solitons and Vortices by Bragg Reflection of Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. **90**, 110404 (2003). [CrossRef] [PubMed]

12. J.R. Anglin, “ Second-quantized Landau-Zener theory for dynamical instabilities,” Phys. Rev. A **67**, 051601(R) (2003). [CrossRef]

**5**, 104 (2003). [CrossRef]

*a*=0.3ms

^{-2}. Here, the condensate has reached the same point close to the Brillouin zone edge as in Figs. 1 (a) and (b), but because of the longer time it has spent in the unstable region, the interference pattern is almost completely washed out. It is also evident that the radial expansion of the condensate is considerably enhanced when the Brillouin zone is scanned with a small acceleration.

*perpendicular*to the optical lattice direction, we obtain a two-peaked curve (see Fig. 1 (a)) for which we can define a visibility (in analogy to spectroscopy) reflecting the phase coherence of the condensate (visibility close to 1 for perfect coherence, visibility →0 for an incoherent condensate). In order to avoid large fluctuations of the visibility due to background noise and shot-to-shot variations of the interference pattern, we have found that a useful definition of the visibility is as follows:

*h*is the mean value of the two peaks (both averaged over 1/10 of their separation symmetrically about the positions of the peaks). By averaging the longitudinal profile over 1/3 of the peak separation symmetrically about the midpoint between the peaks, we obtain

_{peak}*h*. For instance, applying this definition to the profiles shown in Figs. 1 (a) and (c), we obtain

_{middle}*visibility*=0.98 and

*visibility*=0.6, respectively. Owing to fluctuations in the background and hence the definition of the zero point of the longitudinal profile, the visibility thus measured can slightly exceed unity, in which case we define it to be 1. The second observable is the width of a Gaussian fit to the interference pattern integrated

*along*the lattice direction over the extent of one of the peaks (see Fig. 1 (b) and (d)).

*a*=5,1,0.5 and 0.3ms

^{-2}) we measured the visibility and radial width of the interference pattern as a function of the quasimomentum of the condensate. For large accelerations, the quasimomentum can be simply calculated from

*a*and the duration of the lattice acceleration, whereas for small accelerations the restoring force of the magnetic trap has to be taken into account as the spatial motion of the condensate becomes appreciable. In these cases, we derived the quasimomentum reached in the experiment from a numerical integration of the semi-classical equations of motion of the condensate in the presence of the periodic potential (giving rise to Bloch oscillations due to the dispersion relation of the lowest energy band) and of the magnetic trap.

^{-2}, both the visibility and the radial width of the interference pattern remain reasonably stable when the edge of the Brillouin zone is crossed. In contrast, for

*a*=0.5ms

^{-2}and

*a*=0.3ms

^{-2}one clearly sees a drastic change in both quantities as the quasimomentum approaches the value 1. For those accelerations, the condensate spends a sufficiently long time in the unstable region of the Brillouin zone and hence loses its phase coherence, resulting in a sharp drop of the visibility. At the same time, the radial width of the interference pattern increases by a factor between 1.5 and 1.7. This increase is evidence for an instability in the transverse direction and may, for instance, be due to solitons in the longitudinal direction decaying into vortices via a snake instability [11

11. R.G. Scott, A.M. Martin, T.M. Fromhold, S. Bujkiewicz, F.W. Sheard, and M. Leadbeater, “Creation of Solitons and Vortices by Bragg Reflection of Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. **90**, 110404 (2003). [CrossRef] [PubMed]

*a*=0.5ms

^{-2}and

*a*=0.3ms

^{-2}, the interference patterns for quasimomenta larger than 1 were so diffuse that it was not possible to measure the visibility nor the radial width in a meaningful way.

## 4. Comparison with a 1-D numerical simulation

19. D. Choi and Q. Niu, “Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. **82**, 2022 (1999). [CrossRef]

*n*

_{0}is the density of the condensate and

*a*=5.4nm is the s-wave scattering length of

_{s}^{87}

*Rb*. For the experimental parameters used in the experiment described here, the value of

*C*was ≈0.008.

*C*is set to 0 in the numerical simulation, the visibility remains unaltered when the zone edge is crossed, whereas for finite values of

*C*the visibility decreases as a quasimomentum of 1 is approached. Furthermore, the larger the value of

*C*, the more pronounced the decrease in visibility near the band edge. For

*C*=0.008, corresponding to the value realized in our experiment, the onset of the instability is located just below a quasimomentum of 0.8. Experimentally, we find that the visibility starts decreasing consistently beyond a quasimomentum of ≈0.6–0.7, agreeing reasonably well with the results of the simulation.

## 5. Discussion and outlook

**90**, 110404 (2003). [CrossRef] [PubMed]

^{-2}, for which the condensate spends more than 2ms in the critical region around the edge of the Brillouin zone (having an extension of around 1/10 of the BZ [2]), indicates that the growth rate of the instability should be of the order of 500s

^{-1}. This agrees reasonably well with a rough estimate of ≈300s

^{-1}derived from a recent work by Wu and Niu [2

**5**, 104 (2003). [CrossRef]

*C*through the condensate density by increasing the trap frequency, as this would result in an even stronger restoring force along the lattice direction, making it impossible to cross the BZ edge with a constant (and small) acceleration. In order to overcome this problem, it would be advantageous to use, for instance, a dipole trap which allows one to increase the radial trapping frequency whilst still maintaining a small longitudinal frequency. Another interesting question to address is whether the fact that the direction of our optical lattice is at a small angle (a few degrees) to the axis of the magnetic trap might lead to chaotic motion, contributing to the increase in radial width of the interference pattern we observe [20].

## Acknowledgments

## References and links

1. | C. Menotti, A. Smerzi, and A. Trombettoni, “Superfluid dynamics of a Bose-Einstein condensate in a periodic potential,” New J. Phys. |

2. | Biao Wu and Qian Niu, “Superfluidity of Bose-Einstein condensate in an optical lattice: Landau-Zener tunneling and dynamical instability,” New J. Phys. |

3. | Pearl J. Y. Louis, Elena A. Ostrovskaya, Craig M. Savage, and Yuri S. Kivshar, “Bose-Einstein condensates in optical lattices: Band-gap structure and solitons,” Phys. Rev. A |

4. | O. Morsch, J.H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, “Bloch oscillations and mean-field effects of Bose-Einstein condensates in optical lattices,” Phys. Rev. Lett. |

5. | M. Cristiani, O. Morsch, J.H. Müller, D. Ciampini, and E. Arimondo, “Experimental properties of Bose-Einstein condensates in one-dimensional optical lattices: Bloch oscillations, Landau-Zener tunneling, and mean-field effects,” Phys. Rev. A |

6. | 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 (London) |

7. | F. S. Cataliotti, L. Fallani, F. Ferlaino, C. Fort, P. Maddaloni, and M. Inguscio, “Superfluid current disruption in a chain of weakly coupled BoseEinstein condensates,” New J. Phys. |

8. | Yuri S. Kivshar and Mario Salerno, “Modulational instabilities in the discrete deformable nonlinear Schrödinger equation,” Phys. Rev. E |

9. | V.V. Konotop and M. Salerno, “ Modulational instability in Bose-Einstein condensates in optical lattices,” Phys. Rev. A |

10. | Karen Marie Hilligsøe, Markus K. Oberthaler, and Karl-Peter Marzlin, “Stability of gap solitons in a Bose-Einstein condensate,” Phys. Rev. A |

11. | R.G. Scott, A.M. Martin, T.M. Fromhold, S. Bujkiewicz, F.W. Sheard, and M. Leadbeater, “Creation of Solitons and Vortices by Bragg Reflection of Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. |

12. | J.R. Anglin, “ Second-quantized Landau-Zener theory for dynamical instabilities,” Phys. Rev. A |

13. | Yu. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves”, Phys. Rep. |

14. | Jason W. Fleischer, Mordechai Segev, Nikolaos K. Efrimidis, and Demetrios N. Christodoulides, “Observation of two-dimensional discrete solitions in optically induced nonlinear photonic lattices,” Nature |

15. | Dragomir Neshev, Andrey A. Sukhorukov, Yuri S. Kivshar, and Wieslaw Krolikowski, “Observation of transverse instabilities in optically-induced lattices,” nlin.PS/0307053. |

16. | Andrey A. Sukhorukov, Dragomir Neshev, Wieslaw Krolikowski, and Yuris S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” nlin.PS/0309075. |

17. | J.H. Müller, D. Ciampini, O. Morsch, G. Smirne, M. Fazzi, P. Verkerk, F. Fuso, and E. Arimondo, “Bose-Einstein condensation of rubidium atoms in a triaxial TOP trap,” J. Phys. B:At. Mol. Opt. Phys. |

18. | Y.B. Band and M. Trippenbach, “Bose-Einstein condensates in time-dependent light potentials: Adiabatic and nonadiabatic behavior of nonlinear wave equations,” Phys. Rev. A |

19. | D. Choi and Q. Niu, “Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. |

20. | Mark Fromhold, University of Nottingham (U.K.), private communication. |

**OCIS Codes**

(000.2190) General : Experimental physics

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

**ToC Category:**

Focus Issue: Cold atomic gases in optical lattices

**History**

Original Manuscript: November 7, 2003

Revised Manuscript: December 22, 2003

Published: January 12, 2004

**Citation**

M. Cristiani, Oliver Morsch, N. Malossi, M. Jona-Lasinio, M. Anderlini, E. Courtade, and E. Arimondo, "Instabilities of a Bose-Einstein condensate in a periodic potential: an experimental investigation," Opt. Express **12**, 4-10 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-4

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### References

- C. Menotti, A. Smerzi, and A. Trombettoni, �??Superfluid dynamics of a Bose-Einstein condensate in a periodic potential,�?? New J. Phys. 5, 112 (2003). [CrossRef]
- BiaoWu and Qian Niu, �??Superfluidity of Bose-Einstein condensate in an optical lattice: Landau-Zener tunneling and dynamical instability,�?? New J. Phys. 5, 104 (2003). [CrossRef]
- Pearl J. Y. Louis, Elena A. Ostrovskaya, Craig M. Savage, and Yuri S. Kivshar, �??Bose-Einstein condensates in optical lattices: Band-gap structure and solitons,�?? Phys. Rev. A 67, 013602 (2003). [CrossRef]
- O. Morsch, J.H. Muller, M. Cristiani, D. Ciampini, and E. Arimondo, �??Bloch oscillations and mean-field effects of Bose-Einstein condensates in optical lattices,�?? Phys. Rev. Lett. 87, 140402 (2001). [CrossRef] [PubMed]
- M. Cristiani, O. Morsch, J.H. M¨uller, D. Ciampini, and E. Arimondo, �??Experimental properties of Bose-Einstein condensates in one-dimensional optical lattices: Bloch oscillations, Landau-Zener tunneling, and mean-field effects,�?? Phys. Rev. A 65, 063612 (2002). [CrossRef]
- 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 (London) 415, 6867 (2002). [CrossRef]
- F. S. Cataliotti, L. Fallani, F. Ferlaino, C. Fort, P. Maddaloni and M. Inguscio, �??Superfluid current disruption in a chain of weakly coupled BoseEinstein condensates,�?? New J. Phys. 5, 71 (2003). [CrossRef]
- Yuri S. Kivshar and Mario Salerno, �??Modulational instabilities in the discrete deformable nonlinear Schrodinger equation,�?? Phys. Rev. E 49, 3543 (1994). [CrossRef]
- V.V. Konotop and M. Salerno, �??Modulational instability in Bose-Einstein condensates in optical lattices,�?? Phys. Rev. A 65, 021620(R) (2002). [CrossRef]
- Karen Marie Hilligsøe, Markus K. Oberthaler, and Karl-Peter Marzlin, �??Stability of gap solitons in a Bose-Einstein condensate,�?? Phys. Rev. A 66, 063605 (2002). [CrossRef]
- R.G. Scott, A.M. Martin, T.M. Fromhold, S. Bujkiewicz, F.W. Sheard, and M. Leadbeater, �??Creation of Solitons and Vortices by Bragg Reflection of Bose-Einstein Condensates in an Optical Lattice,�?? Phys. Rev. Lett. 90, 110404 (2003). [CrossRef] [PubMed]
- J.R. Anglin, �??Second-quantized Landau-Zener theory for dynamical instabilities,�?? Phys. Rev. A 67, 051601(R) (2003). [CrossRef]
- Yu. S. Kivshar and D. E. Pelinovsky, �??Self-focusing and transverse instabilities of solitary waves,�?? Phys. Rep. 331, 117 (2000). [CrossRef]
- Jason W. Fleischer, Mordechai Segev, Nikolaos K. Efrimidis, and Demetrios N. Christodoulides, �??Observation of two-dimensional discrete solitions in optically induced nonlinear photonic lattices,�?? Nature 422, 147 (2003). [CrossRef] [PubMed]
- Dragomir Neshev, Andrey A. Sukhorukov, Yuri S. Kivshar, andWieslaw Krolikowski, �??Observation of transverse instabilities in optically-induced lattices,�?? nlin.PS/0307053.
- Andrey A. Sukhorukov, Dragomir Neshev, Wieslaw Krolikowski, and Yuris S. Kivshar, �??Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,�?? nlin.PS/0309075.
- J.H. Muller, D. Ciampini, O. Morsch, G. Smirne, M. Fazzi, P. Verkerk, F. Fuso, and E. Arimondo, �??Bose-Einstein condensation of rubidium atoms in a triaxial TOP trap,�?? J. Phys. B: At. Mol. Opt. Phys. 33, 4095 (2000). [CrossRef]
- Y.B. Band and M. Trippenbach, �??Bose-Einstein condensates in time-dependent light potentials: Adiabatic and nonadiabatic behavior of nonlinear wave equations,�?? Phys. Rev. A 65, 053602 (2002). [CrossRef]
- D. Choi and Q. Niu, �??Bose-Einstein Condensates in an Optical Lattice,�?? Phys. Rev. Lett. 82, 2022 (1999). [CrossRef]
- Mark Fromhold, University of Nottingham (U.K.), private communication.

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