## Dynamics of two coupled Bose-Einstein Condensate solitons in an optical lattice

Optics Express, Vol. 14, Issue 8, pp. 3594-3601 (2006)

http://dx.doi.org/10.1364/OE.14.003594

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### Abstract

The characteristics of two coupled Bose-Einstein Condensate (BEC) bright solitons trapped in an optical lattice are investigated with the variational approach and direct numerical simulations of the Gross-Pitaevskii equation. It is found that the optical lattice can be controllably used to capture and drag the coupled BEC solitons. Its effect depends on the initial location of the BEC solitons, the lattice amplitude and wave-number, and the amplitude of the coupled BEC solitons. The effective interaction between the two coupled solitons is the attractive effect.

© 2006 Optical Society of America

## 1. Introduction

^{[11. M. H. Anderson, J. R. Matthews, and C. E. Wieman, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science 269,198 (1995) [CrossRef] [PubMed] ]}, their study has experienced enormous experimental and theoretical advancements. The observation of the bright solitons and the quantum interference phenomena between two coupled BEC have yielded a stunning new demonstration of the wave-like behavior of atoms and provided us with the perspective of getting a better understanding of the complicated behavior of quantum many-body systems and with the hope of realizing novel concrete applications of quantum mechanics, such as atom interferometers and atom lasers

^{[22. K. E. Strecker, G. B. Partridge, and A. G. Truscott, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150(2002) [CrossRef] [PubMed] ]}.

^{[33. L. Khaykovich, F. Schreck, and G. Ferrari, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290(2002) [CrossRef] [PubMed] ]}(OL) and, ever since, this field has attracted considerable attention. A periodic OL potential is created along the waveguide axis (

*x*axis) by interference patterns from multiple laser beams

^{[44. I. Carusotto, D. Embriaco, and G. C. La Rocca, “Nonlinear atom optics and bright-gap-soliton generation in finite optical lattices,” Phys. Rev. A 65, 053611(2002) [CrossRef] ]}. It can be modeled by

*V*

^{opt}(

*x*)=

*V*

_{0}cos(

*kx*). The longitudinal envelope of the OL potential is determined by the profile of the laser beam waist and the lattice wave-number

*k*can be tuned by the geometry of laser beams. The great flexibility of an OL potential arises from the fact that the above parameters can be tuned experimentally, providing precise control over the shape and time-variation of the external potential. OL potentials are, therefore, of particular interest to a number of BEC applications—ranging from matter-wave optics to precision measurements and quantum information processing.

^{[55. Y. S. Cheng, H. Li, and R. Z. Gong, “Effects of parameters on the stationary state and self-trapping of three coupled Bose-Einstein Condensate solitons,” Chin. Opt. Lett. 3, 715(2005), 66. P. Massignan and M. Modugno, “One-dimensional model for the dynamics and expansion of elongated Bose-Einstein condensates,” Phys. Rev. A 67, 023614(2003) [CrossRef] ]}

## 2. Formalism

^{[1515. R. Carretero-Gonzalez and K. Promislow, “Localized breathing oscillations of Bose-Einstein condensates in periodic traps,” Phys. Rev. A 66, 033610(2002) [CrossRef] ]}

*u*≡

*u*(

*x*,

*t*) is the mean-field BEC wave-function, and the nonlinearity coefficient

*g*=±1 accounts for repulsive(+) and attractive(-) interatomic interactions, respectively, and relates to the s-wave scattering length as. In this paper, we study the coherent evolution of bright solitons (for

*g*=-1).

*V*(

*x*) is the normalized external confining potential. It is convenient to decompose the potential

*V*(

*x*) as follows

*V*

_{con}describes the magnetic trap(MT). The periodic potential,

*V*

_{opt}, is formed by an OL,

*k*is the wave-number and

*V*

_{0}is the lattice depth measured in units of the lattice recoil energy,

*E*

_{rec}. The parameter Ω = ω

_{x}/ω

_{⊥}, where

*ω*

_{x}and

*ω*

_{⊥}are the confining frequencies of MT in the axial and transverse directions. In Eq. (1),

*t*is the dimensionless time measured in terms of

*x*is the longitudinal spatial variable measured in units of the transverse harmonic-oscillator length,

*a*

_{⊥}(

*h*/

*mω*

_{⊥})

^{1/2}. Accordingly, |

*u*|

^{2}corresponds to the rescaled population density of the condensate measured in units of

*mω*

_{⊥}/4π

*ħ*|

*a*

_{s}|, and

*m*is the atomic mass. In practice, the MT has a flat central portion supporting several hundreds of periods of the periodic potential. The whole potential is sufficient to satisfy the desired near-periodic potential. It is also possible to load the condensate onto

*V*(

*x*) and then adiabatically remove

*V*

_{con}, leaving only the periodic potential. In this work, we let

*V*

_{con}= 0 and focus on optical lattices,

*viz*.

*u*=

*u*

_{1}+

*u*

_{2}and assuming mutual coupling only through intensity overlap in the |

*u*|

^{2}term we find the following coupled equations

*u*

_{i}(

*i*=1,2) is the wave-function of the two BECs respectively and

*u*

_{i}is normalized to the number of particles in the

*i*-th BECs.

*N*=

*N*

_{1}+

*N*

_{2}=

*u*

_{1}|

^{2}

*dx*+∫+∞-∞|

*u*

^{2}|

^{2}

*dx*is the rescaled total number of atoms(a conserved quantity). In Eq. (4), we have neglected the fast varying term

*i*=1,2), where

^{*}denotes the complex conjugate. Compared with Ref. [7

7. C. Lee, W. Hai, and X. Luo, “Quasispin model for macroscopic quantum tunneling between two coupled Bose-Einstein condensates,” Phys. Rev. A **68**, 053614(2003) [CrossRef]

16. C. Pare and M. Florjariczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A **41**, 6287–6295(1990) [CrossRef] [PubMed]

## 3. Computation and discussion

*A*

_{i}represents the amplitude (and inverse width),

*x*

_{0i}and

*ẋ*

_{oi}=

*dx*

_{0i}/

*dt*represent the location and the velocity of the soliton center, and ϕ

_{i}is the phase. During the evolution of the two coupled BEC solitons, we assume that the wave-function ui retains the Sech-shape given by Eq. (5), but

*A*

_{i},

*x*

_{0i},ϕ

_{i}are the functions of time as a result of the weak interaction. The condition of the normalization is

*N*=

*u*

_{1}|

^{2}

*dx*+

*u*

_{1}|

^{2}

*dx*= 2(

*A*

_{1}+

*A*

_{2}).

^{[17–1917. H. Li, Y. S. Cheng, and D. X. Huang, “Effects of traps on dynamics of two coupled Bose-Einstein condensates,” Opt. Commun. 258, 306–314 (2006) [CrossRef] ]}

_{i}and the amplitude

*A*

_{i}once the other parameters are known so that the equations are self-consistent. We assume that the system of the BEC solitons is symmetric,

*A*

_{1}=

*A*

_{2}=

*A*,

*x*

_{01}= Δ/2 and

*x*

_{02}= -Δ/2,

*viz*. Δ =

*x*

_{01}-

*x*

_{02}is the separation between the BEC solitons. From Eq. (8b), we can obtain

*V*

_{eff2}versus the separation Δ is plotted in Fig. 1 which indicates that the interaction between the two coupled BEC solitons forms a potential well. The separation within which the two solitons can interact is finite, and the effective potential is approximately equal to zero beyond the special separation (see Fig. 1). This is because that the wave-functions of the two BEC solitons do not overlap in that case. We can also conclude that the interaction of two BEC solitons is attractive, and that the solitons will move down the potential slope towards

*x*=0, an equilibrium point, and oscillate around the equilibrium point. Beyond the potential field, the two solitons are in neutral equilibrium. From Fig. 1, we find that the width and depth of the effective potential well are very sensitive to the amplitude

*A*of a soliton. For example, the effective potential well becomes deeper and narrower as

*A*increases. The separation where the two solitons can interact becomes smaller. This is because that the amplitude

*A*reflects the number of atoms in a BEC soliton, and the bigger the number of atoms is, the greater the ability of the self-focusing is. On the other hand, in the absence of mutual coupling, the OL leads to a periodical potential field

*V*

_{eff1}. The periodical potential is relative to the parameters of the OL and solitons. At the same time, the wave-number

*k*has a greater effect on the amplitude of the effective potential

*V*

_{eff1}. The larger the wave-number is, the smaller the amplitude becomes.

*A*=1, and

*V*

_{0}=0.5. The solid, dashed and dotted lines correspond to the wave-number

*k*=0.1, 0.5, 1.5 respectively. We find that the wave-number

*k*plays an important role. It deforms the shape of the effective potential, and changes the dynamic characteristics of the system.

*V*

_{eff1}is dominant and its shape is fluted with

*V*

_{eff2}around the point

*x*=0. Apart from the flute, the whole effective potential is similar to the case without mutual coupling. The system is in a stable equilibrium at the minimum of the effective potential or in a vibration around the stable equilibrium point.

*V*

_{eff2}is dominant within the separation where the two solitons can interact because the amplitude of

*V*

_{eff1}is smaller and two small-peak values appear whose amplitude depends on the amplitude of the OL and the slope of

*V*

_{eff2}. When the peaks of the effective potential are low, the BEC solitons may cross over the peaks and meet at the bottom of the effective potential if the solitons possess larger initial energy. In that situation, the separation within which the two solitons can meet by mutual attraction becomes much wider. We notice that the OL potential will be considered as a perturbation parameter if the OL period is much smaller than the width of the BEC soliton. Then the variational approach is non-effective and the perturbation theory will be employed

^{[2020. D. J. Frantzeskakis, G. Theocharis, and F. K. Diakonos, “Interaction of dark solitons with localized impurities in Bose-Einstein condensates,” Phys. Rev. A 66, 053608(2002) [CrossRef] ]}. Hence our investigations are valid when the OL is assumed to be smooth and slowly varying on the soliton scale.

*u*

_{i}(

*t*=0)=

*A*sech(

*x*-

*x*

_{0i}) and the other parameters are

*k*=0.5, and

*A*=1 for Fig. 3(a) and

*A*=2 for Fig. 3(b, c, d). The initial locations of the motionless BEC soliton center

*x*

_{0i}(

*i*=1, 2) and the lattice depth of the OL

*V*

_{0}are indicated in the figures. When Δ is smaller, the two solitons will meet at

*x*=0 because of the attractive interaction as shown in Fig. 3(a). In the case of a larger Δ, the interaction between the two solitons is sufficiently small, and each of the two solitons evolves independently. When we initialize the solitons at the respective minimum of the potential, the solitons remain steady and are in stable equilibrium, which is indicated in Fig. 3(b). If the initial position of the soliton center is around the bottom of the potential, the corresponding soliton will vibrate around the minimum, and the frequency of the vibration is relative to

*V*

_{0}and

*A*(see Fig. 3(c) and (d)).

*k*. Hence, we can change the separation of the two coupled BEC solitons by slowly changing the OL, as long as the OL tuning is slow enough to ensure adiabatic change. Fig. 4 shows that a slowly tuning OL drags two BEC solitons and takes them to another position. The initial input pulse is

*u*

_{i}(

*t*=0)=

*A*sech(

*x*-

*x*

_{0i}), the parameters of the system are selected as

*A*=2,

*V*

_{0}=0.5, and

*k*=1.5-0.01

*t*, and the initial location of the motionless BEC soliton center is

*x*

_{0i}=±6.28. We notice that a moving localized attractive impurity or a moving OL (a total OL is moved) may drag a stationary soliton

^{[2020. D. J. Frantzeskakis, G. Theocharis, and F. K. Diakonos, “Interaction of dark solitons with localized impurities in Bose-Einstein condensates,” Phys. Rev. A 66, 053608(2002) [CrossRef] , 2121. R. G. Scott, A. M. Martin, and S. Bujkiewicz, “Transport and disruption of Bose-Einstein condensates in optical lattices,” Phys. Rev. A 69, 033605(2004) [CrossRef] ]}, but here our aim is to drag the soliton by tuning the wave-number

*k*of an OL. Apparently, this issue is important for applications of the BEC.

## 4. Conclusions

## Acknowledgments

## References

1. | M. H. Anderson, J. R. Matthews, and C. E. Wieman, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science |

2. | K. E. Strecker, G. B. Partridge, and A. G. Truscott, “Formation and propagation of matter-wave soliton
trains,” Nature |

3. | L. Khaykovich, F. Schreck, and G. Ferrari, “Formation of a Matter-Wave Bright Soliton,” Science |

4. | I. Carusotto, D. Embriaco, and G. C. La Rocca, “Nonlinear atom optics and bright-gap-soliton generation in finite optical lattices,” Phys. Rev. A |

5. | Y. S. Cheng, H. Li, and R. Z. Gong, “Effects of parameters on the stationary state and self-trapping of three coupled Bose-Einstein Condensate solitons,” Chin. Opt. Lett. |

6. | P. Massignan and M. Modugno, “One-dimensional model for the dynamics and expansion of elongated Bose-Einstein condensates,” Phys. Rev. A |

7. | C. Lee, W. Hai, and X. Luo, “Quasispin model for macroscopic quantum tunneling between two coupled Bose-Einstein condensates,” Phys. Rev. A |

8. | E. A. Ostrovskaya and Y. S. Kivshar, “Localization of Two-Component Bose-Einstein Condensates in Optical Lattices,” Phys. Rev. Lett. |

9. | A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, “Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. |

10. | S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, “Coherent oscillations between two weakly coupled
Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping,” Phys. Rev. A |

11. | G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic Bose-Einstein
condensate in a double-well potential,” Phys. Rev. A |

12. | J. Williams, R. Walser, J. Cooper, E. Cornell, and M. Holland, “Nonlinear Josephson-type oscillations of a
driven, two-component Bose-Einstein condensate,” Phys. Rev. A |

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

14. | D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, “Dynamics of Component Separation in a Binary Mixture of Bose-Einstein Condensates,” Phys. Rev. Lett. D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Measurements of Relative Phase in Two-Component
Bose-Einstein Condensates,” Phys. Rev. Lett. |

15. | R. Carretero-Gonzalez and K. Promislow, “Localized breathing oscillations of Bose-Einstein condensates in periodic traps,” Phys. Rev. A |

16. | C. Pare and M. Florjariczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A |

17. | H. Li, Y. S. Cheng, and D. X. Huang, “Effects of traps on dynamics of two coupled Bose-Einstein condensates,” Opt. Commun. |

18. | S. Raghavan and G. P. Agrawal, “Switching and self-trapping dynamics of Bose-Einstein solitons,” J. Mod. Opt. |

19. | P. Ohberg and L. Santos, “Dark Solitons in a Two-Component Bose-Einstein Condensate,” Phys. Rev. Lett. |

20. | D. J. Frantzeskakis, G. Theocharis, and F. K. Diakonos, “Interaction of dark solitons with localized impurities in Bose-Einstein condensates,” Phys. Rev. A |

21. | R. G. Scott, A. M. Martin, and S. Bujkiewicz, “Transport and disruption of Bose-Einstein condensates in optical lattices,” Phys. Rev. A |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(270.5530) Quantum optics : Pulse propagation and temporal solitons

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: February 15, 2006

Revised Manuscript: April 3, 2006

Manuscript Accepted: April 11, 2006

Published: April 17, 2006

**Citation**

Yongshan Cheng, Rongzhou Gong, and Hong Li, "Dynamics of two coupled Bose-Einstein Condensate solitons in an optical lattice," Opt. Express **14**, 3594-3601 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3594

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### References

- M. H. Anderson, J. R. Matthews and C. E. Wieman, "Observation of Bose-Einstein Condensation in a dilute atomic vapor," Science 269,198 (1995) [CrossRef] [PubMed]
- K. E. Strecker, G. B. Partridge and A. G. Truscott, "Formation and propagation of matter-wave soliton trains," Nature 417,150 (2002) [CrossRef] [PubMed]
- L. Khaykovich, F. Schreck and G. Ferrari, "Formation of a matter-wave bright soliton," Science 296,1290 (2002) [CrossRef] [PubMed]
- I. Carusotto, D. Embriaco and G. C. La Rocca, "Nonlinear atom optics and bright-gap-soliton generation in finite optical lattices," Phys. Rev. A 65, 053611 (2002) [CrossRef]
- Y. S. Cheng, H. Li, and R. Z. Gong, "Effects of parameters on the stationary state and self-trapping of three coupled Bose-Einstein Condensate Solitons," Chin. Opt. Lett. 3,715 (2005)
- P. Massignan and M. Modugno, "One-dimensional model for the dynamics and expansion of elongated Bose-Einstein Condensates," Phys. Rev. A 67,023614 (2003) [CrossRef]
- C. Lee, W. Hai and X. Luo, "Quasispin model for macroscopic quantum tunneling between two coupled Bose-Einstein Condensates," Phys. Rev. A 68,053614 (2003) [CrossRef]
- E. A. Ostrovskaya and Y. S. Kivshar, "Localization of two-component Bose-Einstein Condensates in optical lattices," Phys. Rev. Lett. 92,180405 (2004) [CrossRef] [PubMed]
- A. Smerzi, S. Fantoni, S. Giovanazzi and S. R. Shenoy, "Quantum coherent atomic tunneling between two trapped Bose-Einstein Condensates," Phys. Rev. Lett. 79,4950 (1997) [CrossRef]
- S. Raghavan, A. Smerzi, S. Fantoni and S. R. Shenoy, "Coherent oscillations between two weakly coupled Bose-Einstein Condensates: Josephson effects, oscillations, and macroscopic quantum self-trapping," Phys. Rev. A 59, 620 (1999) [CrossRef]
- G. J. Milburn, J. Corney, E. M. Wright and D. F. Walls, "Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential," Phys. Rev. A 55,4318 (1997) [CrossRef]
- J. Williams, R. Walser, J. Cooper, E. Cornell and M. Holland, "Nonlinear Josephson-type oscillations of a driven, two-component Bose-Einstein Condensate," Phys. Rev. A 59,R31 (1999) [CrossRef]
- B. P. Anderson and M. A. Kasevich, "Macroscopic quantum interference from atomic tunnel arrays," Science 282, 1686 (1998) [CrossRef] [PubMed]
- D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman and E. A. Cornell, "Dynamics of component separation in a binary mixture of Bose-Einstein Condensates," Phys. Rev. Lett. 81,1538 (1998),
- D. S. Hall, M. R. Matthews, C. E. Wieman and E. A. Cornell, "Measurements of relative phase in two-component Bose-Einstein Condensates," Phys. Rev. Lett. 81,1543 (1998) [CrossRef]
- R. Carretero-Gonzalez and K. Promislow, "Localized breathing oscillations of Bose-Einstein condensates in periodic traps," Phys. Rev. A 66,033610 (2002) [CrossRef] [PubMed]
- C. Pare and M. Florjariczyk, "Approximate model of soliton dynamics in all-optical couplers," Phys. Rev. A 41,6287-6295(1990) [CrossRef]
- H. Li, Y. S. Cheng and D. X. Huang, "Effects of traps on dynamics of two coupled Bose-Einstein Condensates," Opt. Commun. 258, 306-314 (2006)
- S. Raghavan and G. P. Agrawal, "Switching and self-trapping dynamics of Bose-Einstein Solitons," J. Mod. Opt. 47,1155 (2000) [CrossRef] [PubMed]
- P. Ohberg and L. Santos, "Dark Solitons in a two-component Bose-Einstein Condensate," Phys. Rev. Lett. 86,2918-2921(2001) [CrossRef]
- D. J. Frantzeskakis, G. Theocharis and F. K. Diakonos, "Interaction of dark solitons with localized impurities in Bose-Einstein Condensates," Phys. Rev. A 66,053608 (2002) [CrossRef]
- R. G. Scott, A. M. Martin and S. Bujkiewicz, "Transport and disruption of Bose-Einstein condensates in optical lattices," Phys. Rev. A 69,033605 (2004)

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