Matter wave interference pattern in the collision of bright solitons

doi:10.1016/j.physleta.2009.09.056

Physics Letters A

Volume 373, Issue 47, 23 November 2009, Pages 4381-4385

https://doi.org/10.1016/j.physleta.2009.09.056 Get rights and content

Abstract

We investigate the dynamics of Bose–Einstein condensates in a quasi one-dimensional regime in a time-dependent trap and show analytically that it is possible to observe matter wave interference patterns in the intra-trap collision of two bright solitons by selectively tuning the trap frequency and scattering length.

Introduction

When a gas of massive bosons is cooled to a temperature very close to absolute zero in an external potential, a large fraction of the atoms collapse into the lowest quantum state of the external potential forming a condensate known as a Bose–Einstein condensate (BEC) [1], [2], [3], [4]. Bose–Einstein condensation is an exotic quantum phenomenon observed in dilute atomic gases and has made a huge turnaround in the fields of atom optics and condensed matter physics. The experimental realization of BECs in weakly interacting gases [5] has really kickstarted the upsurge in this area of research leading to flurry of activities in this direction while the observation of dark [6] and bright solitons [7], periodic waves [8], vortices and necklaces [9] has given an impetus to the investigation of this singular state of matter.

The dynamics of BECs is governed by an inhomogeneous nonlinear Schrödinger (NLS) equation called the Gross–Pitaevskii (GP) equation and the behaviour of the condensates depends on the scattering length (binary interatomic interaction) and the trapping potential. Eventhough the GP equation is in general nonintegrable, it has been recently investigated for specific choices of scattering lengths and trapping potentials using Darboux [10], [11] and gauge transformation approach [12], [13], [14].

It is known that a BEC comprises of coherent matter waves analogous to coherent laser pulses and all the atoms are in phase. In other words, the atoms occupy the same volume of space, move at identical speeds, scatter light of the same color and so on. Hence, it looked fundamentally impossible to distinguish them by any measurement. This quantum degeneracy arising out of high degree of coherence has been recently exploited in an experiment by Andrews et al. [15] and Ketterle's group [16] showing that when two separate clouds of BECs overlap under free ballistic expansion, the result is a fringe pattern of alternating constructive and destructive interference just as it occurs with two intersecting laser beams. Javanainen et al. [17] have shown that when two independent condensates are dropped on top of each other, one also observes similar interference pattern with or without phase. In other words, when BECs were made to collide upon release from the trap, de Broglie wave interference pattern containing stripes of high and low density were clearly observed. These experiments which underlined the high degree of spatial coherence of BECs led to the creation of atom laser [18]. Can one observe the same matter wave interference pattern by allowing the bright solitons which are condensates themselves to collide in a trap? Motivated by this consideration, we investigate the collisional dynamics of condensates in a time-dependent trapping potential.

Section snippets

Gross–Pitaevskii equation and Lax-pair

At the mean field level, the time evolution of macroscopic wavefunction of BECs is governed by the Gross–Pitaevskii (GP) equation, $i ℏ \frac{\partial Ψ (\vec{r}, t)}{\partial t} = (\frac{- ℏ^{2}}{2 m} \nabla^{2} + g {| Ψ (\vec{r}, t) |}^{2} + V) Ψ (\vec{r}, t)$ where $Ψ (\vec{r}, t)$ represents the condensate wave function normalized by the particle number $N = \int d r {| Ψ |}^{2}$ , $g = 4 π ℏ^{2} a_{s} (t) / m$ , $V = V_{0} + V_{1}$ , $V_{0} (x, y) = m ω_{⊥}^{2} (x^{2} + y^{2}) / 2$ , $V_{1} = m ω_{0}^{2} (t) z^{2} / 2$ . In the above equation, $V_{0}$ and $V_{1}$ represent atoms in a cylindrical trap and time-dependent trap along z-direction respectively. The time-dependent trap could be

Bright soliton interaction and matter wave interference

To investigate the collisional dynamics of condensates in the presence of a time-dependent trap, we now consider the two bright soliton of the following form, $ψ^{2} (z, t) = \sqrt{\frac{1}{γ (t)}} \frac{A_{1} + A_{2} + A_{3} + A_{4}}{B_{1} + B_{2}} e^{- \frac{i}{2} c (t) z^{2}}$ where $A_{1} = {- 2 β_{2} [{(α_{2} - α_{1})}^{2} - (β_{1}^{2} - β_{2}^{2})] - 4 i β_{1} β_{2} (α_{2} - α_{1})} e^{(θ_{1} + i ξ_{2})},$ $A_{2} = - 2 β_{2} [{(α_{2} - α_{1})}^{2} + (β_{1}^{2} + β_{2}^{2})] e^{(- θ_{1} + i ξ_{2})},$ $A_{3} = {- 2 β_{1} [{(α_{2} - α_{1})}^{2} + (β_{1}^{2} - β_{2}^{2})] + 4 i β_{1} β_{2} (α_{2} - α_{1})} e^{(i ξ_{1} + θ_{2})},$ $A_{4} = - 4 i β_{1} β_{2} [(α_{2} - α_{1}) - i (β_{1} - β_{2})] e^{(i ξ_{1} - θ_{2})},$ $B_{1} = - 4 β_{1} β_{2} [\sinh (θ_{1}) \sinh (θ_{2}) + \cos (ξ_{1} - ξ_{2})],$ $B_{2} = 2 \cosh (θ_{1}) \cosh (θ_{2}) [{(α_{2} - α_{1})}^{2} + (β_{1}^{2} + β_{2}^{2})]$ and $θ_{1} = 2 β_{1} z - 4 \int_{0}^{t} (α_{1} β_{1}) d t^{'} + 2 δ_{1},$ $ξ_{1} = 2 α_{1}$

Conclusion

In conclusion, we have shown how the concept of matter wave interference is manifested in the collisional dynamics of two bright solitons for suitable choices of scattering length $γ (t)$ and trap frequency $M (t)$ . We also observe that the phase difference between the condensates oscillates with time giving a measure of the coherence of the condensates. We believe that the occurrence of matter wave interference in the intra-trap collision of bright solitons in certain parametric domains is a new

Acknowledgements

V.R. wishes to thank DST for offering a Senior Research Fellowship. R.R. acknowledges the financial assistance from DST and UGC in the form of major and minor research projects. Authors thank the referee for his critical comments on the manuscript.

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Matter wave interference pattern in the collision of bright solitons

Abstract

Introduction

Section snippets

Gross–Pitaevskii equation and Lax-pair

Bright soliton interaction and matter wave interference

Conclusion

Acknowledgements

Phys. Rev. Lett.

Phys. Rev. Lett.

Phys. Lett. A

New J. Phys.

Z. Phys.

Sitzber. Kgl. Preuss. Akad. Wiss.

Sitzber. Kgl. Preuss. Akad. Wiss.

Rev. Mod. Phys.

Bose–Einstein Condensation in Dilute Gases

Science

Phys. Rev. Lett.

Phys. Rev. Lett.

Phys. Rev. Lett.

Science

Nature

Phys. Rev. Lett.

Phys. Rev. Lett.

Phys. Rev. Lett.

Phys. Rev. Lett.