## Linear and nonlinear dynamics of matter wave packets in periodic potentials

Optics Express, Vol. 12, Issue 1, pp. 11-18 (2004)

http://dx.doi.org/10.1364/OPEX.12.000011

Acrobat PDF (376 KB)

### Abstract

We investigate experimentally and theoretically the nonlinear propagation of ^{87}Rb Bose Einstein condensates in a trap with cylindrical symmetry. An additional weak periodic potential which encloses an angle with the symmetry axis of the waveguide is applied. The observed complex wave packet dynamics results from the coupling of transverse and longitudinal motion. We show that the experimental observations can be understood applying the concept of effective mass, which also allows to model numerically the three dimensional problem with a one dimensional equation. Within this framework the observed slowly spreading wave packets are a consequence of the continuous change of dispersion. The observed splitting of wave packets is very well described by the developed model and results from the nonlinear effect of transient solitonic propagation.

© 2004 Optical Society of America

## 1. Introduction

2. F.S. Cataliotti, S. Burger, S. C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Ingusio, “Josephson Junction Arrays with Bose-Einstein Condensates”, Science **293**843 (2001). [CrossRef] [PubMed]

3. A. Trombettoni and A. Smerzi, “Discrete Solitons and Breathers with Dilute Bose-Einstein Condensates,” Phys. Rev. Lett. **86**2353 (2001). [CrossRef] [PubMed]

7. V.V. Konotop and M. Salerno, “Modulational instability in Bose-Einstein condensates in optical lattices,” Phys. Rev. A **65**021602 (2002). [CrossRef]

## 2. Effective mass and dispersion concept

*E*

_{n}(

*q*) shown in Fig. 1(a). This relation is well known in the context of electrons in crystals [8] and exhibits a band structure. It shows the eigenenergies of the Bloch states as a function of the quasi-momentum

*q*. The modified dispersion relation leads to a change of wavepacket dynamics due to the change in group velocity

*v*

_{g}(

*q*)=1/

*ℏ*

*∂E*/

*∂q*(see Fig. 1(b)), and the group velocity dispersion described by the effective mass

*m*

_{eff}=

*ℏ*

^{2}(

*∂*

^{2}

*E*/

*∂q*

^{2})

^{-1}(see Fig. 1(c)), which is equivalent to the effective diffraction introduced in the context of light beam propagation in optically-induced photonic lattices [9]. In our experiment only the lowest band is populated, which is characterized by two dispersion regimes, normal and anomalous dispersion, corresponding to positive and negative effective mass. A pathological situation arises at the quasimomentum

*v*

_{g}(

*q*) is extremal, |

*m*

_{eff}| diverges and thus the dispersion vanishes.

10. B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion Management for Atomic Matter Waves,” Phys. Rev. Lett. **91**060402 (2003). [CrossRef] [PubMed]

## 3. Experimental setup

^{87}Rb Bose-Einstein condensate (BEC). The atoms are collected in a magneto-optical trap and subsequently loaded into a magnetic time-orbiting potential trap. By evaporative cooling we produce a cold atomic cloud which is then transferred into an optical dipole trap realized by two focused Nd:YAG laser beams with 60

*µm*waist crossing at the center of the magnetic trap (see Fig.2(a)). Further evaporative cooling is achieved by lowering the optical potential leading to pure Bose-Einstein condensates with 1·10

^{4}atoms in the |

*F*=2,

*mF*=+2〉 state. By switching off one dipole trap beam the atomic matter wave is released into a trap acting as a one-dimensional waveguide with radial trapping frequency ω

_{⊥}=2

*π*·100

*Hz*and longitudinal trapping frequency ω

_{‖}=2

*π*·1.5

*Hz*. It is important to note that the dipole trap allows to release the BEC in a very controlled way leading to an initial mean velocity uncertainty smaller than 1/10 of the photon recoil velocity.

*W/cm*

^{2}. The chosen detuning of 2 nm to the blue off the D2 line leads to a spontaneous emission rate below 1

*Hz*. The standing light wave and the waveguide enclose an angle of

*θ*=21° (see Fig. 2(b)). The frequency and phase of the individual laser beams are controlled by acousto-optic modulators driven by a two channel arbitrary waveform generator allowing for full control of the velocity and amplitude of the periodic potential. The light intensity and thus the absolute value of the potential depth was calibrated independently by analyzing results on Bragg scattering [11

11. M. Kozuma, L. Deng, E.W. Hagley, J. Wen, R. Lutwak, K. Helmerson, S.L. Rolston, and W.D. Phillips, “Coherent Splitting of Bose-Einstein Condensed Atoms with Optically Induced Bragg Diffraction,” Phys. Rev. Lett. **82**871 (1999). [CrossRef]

12. B.P. Anderson and M.A. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays,” Science **282**1686 (1998); [CrossRef] [PubMed]

13. O. Morsch, J. 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]

14. C.F. Bharucha, K.W. Madison, P.R. Morrow, S.R. Wilkinson, Bala Sundaram, and M.G. Raizen, “Observation of atomic tunneling from an accelerating optical potential,” Phys. Rev. A **55**R857 (1997) [CrossRef]

*n*(

*x, t*), are obtained by integrating the absorption images over the transverse dimension.

## 4. Dynamics in reciprocal space

*θ*, a change of the transverse velocity leads to a shift of the central quasimomentum of the wave packet. The coupling between the transverse motion in the waveguide and the motion along the standing light wave gives rise to a nontrivial motion in reciprocal (see Fig. 2(e,f)) and real space.

*M** is a mass tensor describing the directionality of the effective mass. We deduce the time dependent quasimomentum

*q*

_{c}(

*t*) in the direction of the periodic potential by identifying

*ℏq̇*

_{c}=

*F*

_{x}

*̂*and

*x̂*see Fig. 2(b)). Subsequently we can solve the one dimensional NPSE (non-polynomial nonlinear Schrödinger equation)[15

15. L. Salasnich, A. Parola, and L. Reatto, “Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates,” Phys. Rev. A **65**043614 (2002). [CrossRef]

*q*

_{c}(

*t*). Thus the transverse motion is taken into account properly for

*narrow*momentum distributions. We use a split step Fourier method to integrate the NPSE where the kinetic energy contribution is described by the numerically obtained energy dispersion relation of the lowest band

*E*

_{0}(

*q*). It is important to note, that this description includes all higher derivatives of

*E*

_{0}(

*q*), and thus goes beyond the effective mass approximation.

*Acceleration scheme I*: After the periodic potential is adiabatically ramped up to

*V*

_{0}=6

*E*

_{rec}it is accelerated within 3ms to a velocity

*v*

_{pot}=cos

^{2}(

*θ*)1.5

*v*

_{rec}. Then the potential depth is lowered to

*V*

_{0}=0.52

*E*

_{rec}within 1.5ms and the periodic potential is decelerated within 3ms to

*v*

_{pot}=cos

^{2}(

*θ*)

*v*

_{rec}subsequently.

*V*

_{0}and

*v*

_{pot}are kept constant during the following propagation. The calculated motion in reciprocal space

*q*

_{c}(

*t*) is shown in Fig. 2(e).

*Acceleration scheme II*: The periodic potential is ramped up adiabatically to

*V*

_{0}=0.37

*E*

_{rec}and is subsequently accelerated within 3ms to a final velocity

*v*

_{pot}=cos

^{2}(

*θ*)×1.05

*v*

_{rec}. The potential depth is kept constant throughout the whole experiment. Fig. 2(f) reveals that in contrast to the former acceleration scheme the quasimomentum for the initial propagation is mainly in the negative effective mass regime.

## 5. Experimental and numerical results

### 5.1. Preparation I

18. R.G. Scott, A.M. Martin, T.M. Fromholz, 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]

*V*

_{0}=6

*E*

_{rec}) over the critical negative mass regime. While the real space distribution does not change during this process, the momentum distribution broadens due to self phase modulation [16, 17]. The subsequent propagation in the positive mass regime leads to a further broadening in momentum space and real space (t=4-9ms).

### 5.2. Preparation II

## 6. Conclusion

## Acknowledgment

## References and links

1. | “Bose-Einstein condensation in atomic gases,” ed. by M. Inguscio, S. Stringari, and C. Wieman, (IOS Press, Amsterdam1999) |

2. | F.S. Cataliotti, S. Burger, S. C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Ingusio, “Josephson Junction Arrays with Bose-Einstein Condensates”, Science |

3. | A. Trombettoni and A. Smerzi, “Discrete Solitons and Breathers with Dilute Bose-Einstein Condensates,” Phys. Rev. Lett. |

4. | M. Steel and W. Zhang, “Bloch function description of a Bose-Einstein condensate in a finite optical lattice,” cond-mat/9810284 (1998). |

5. | P. Meystre, |

6. | The experimental realization in our group will be published elsewhere. |

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

8. | N. Ashcroft and N. Mermin, |

9. | A.A. Sukhorukov, D. Neshev, W. Krolikowski, and Y.S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically-induced lattices,” nlin.PS/0309075. |

10. | B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion Management for Atomic Matter Waves,” Phys. Rev. Lett. |

11. | M. Kozuma, L. Deng, E.W. Hagley, J. Wen, R. Lutwak, K. Helmerson, S.L. Rolston, and W.D. Phillips, “Coherent Splitting of Bose-Einstein Condensed Atoms with Optically Induced Bragg Diffraction,” Phys. Rev. Lett. |

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

13. | O. Morsch, J. 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. |

14. | C.F. Bharucha, K.W. Madison, P.R. Morrow, S.R. Wilkinson, Bala Sundaram, and M.G. Raizen, “Observation of atomic tunneling from an accelerating optical potential,” Phys. Rev. A |

15. | L. Salasnich, A. Parola, and L. Reatto, “Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates,” Phys. Rev. A |

16. | G.P. Agrawal, |

17. | G.P. Agrawal, |

18. | R.G. Scott, A.M. Martin, T.M. Fromholz, 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. |

**OCIS Codes**

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

(270.5530) Quantum optics : Pulse propagation and temporal solitons

(350.4990) Other areas of optics : Particles

**ToC Category:**

Focus Issue: Cold atomic gases in optical lattices

**History**

Original Manuscript: November 11, 2003

Revised Manuscript: December 21, 2003

Published: January 12, 2004

**Citation**

Th. Anker, M. Albiez, B. Eiermann, M. Taglieber, and M. Oberthaler, "Linear and nonlinear dynamics of matter wave packets in periodic potentials," Opt. Express **12**, 11-18 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-11

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### References

- �??Bose-Einstein condensation in atomic gases,�?? ed. by M. Inguscio, S. Stringari, and C. Wieman, (IOS Press, Amsterdam 1999)
- F.S. Cataliotti, S. Burger, S. C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Ingusio, �??Josephson Junction Arrays with Bose-Einstein Condensates�??, Science 293 843 (2001). [CrossRef] [PubMed]
- A. Trombettoni and A. Smerzi, �??Discrete Solitons and Breathers with Dilute Bose-Einstein Condensates,�?? Phys. Rev. Lett. 86 2353 (2001). [CrossRef] [PubMed]
- M. Steel and W. Zhang, �??Bloch function description of a Bose-Einstein condensate in a finite optical lattice,�?? cond-mat/9810284 (1998).
- P. Meystre, Atom Optics (Springer Verlag, New York, 2001) p 205, and references therein.
- The experimental realization in our group will be published elsewhere.
- V.V. Konotop, M. Salerno, �??Modulational instability in Bose-Einstein condensates in optical lattices,�?? Phys. Rev. A 65 021602 (2002). [CrossRef]
- N. Ashcroft and N. Mermin, Solid State Physics (Saunders, Philadelphia, 1976).
- A.A. Sukhorukov, D. Neshev, W. Krolikowski, and Y.S. Kivshar, �??Nonlinear Bloch-wave interaction and Bragg scattering in optically-induced lattices,�?? nlin.PS/0309075.
- B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, �??Dispersion Management for Atomic Matter Waves,�?? Phys. Rev. Lett. 91 060402 (2003). [CrossRef] [PubMed]
- M. Kozuma, L. Deng, E.W. Hagley, J.Wen, R. Lutwak, K. Helmerson, S.L. Rolston, andW.D. Phillips, �??Coherent Splitting of Bose-Einstein Condensed Atoms with Optically Induced Bragg Diffraction,�?? Phys. Rev. Lett. 82 871(1999). [CrossRef]
- B.P. Anderson, and M.A. Kasevich, �??Macroscopic Quantum Interference from Atomic Tunnel Arrays,�?? Science 282 1686 (1998); [CrossRef] [PubMed]
- O. Morsch, J. M¨uller, 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]
- C.F. Bharucha, K.W. Madison, P.R. Morrow, S.R.Wilkinson, Bala Sundaram, and M.G. Raizen, �??Observation of atomic tunneling from an accelerating optical potential,�?? Phys. Rev. A 55 R857 (1997) [CrossRef]
- L. Salasnich, A. Parola, and L. Reatto, �??Effective wave equations for the dynamics of cigar-shaped and diskshaped Bose condensates,�?? Phys. Rev. A 65 043614 (2002). [CrossRef]
- G.P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
- G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).
- R.G. Scott, A.M. Martin, T.M. Fromholz,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]

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