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arXiv:1111.5686 · New Journal of Physics

Quantum dynamics of atomic bright solitons under splitting and recollision, and implications for interferometry

A D Martin, J Ruostekoski

Published Apr 27, 2012• 71 citations
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Abstract

We numerically study the classical and quantum dynamics of an atomic bright soliton in a highly elongated one-dimensional harmonic trap with a Gaussian barrier. In the regime of the recent experiment by Dyke et al the system realizes a coherent nonlinear soliton beam-splitter and interferometer whose accuracy we analyze. In the case of tighter radial trap confinement and enhanced quantum fluctuations, a split soliton can represent a spin-squeezed, or alternatively, a fragmented condensate with reduced phase coherence that can be measured by interfering the split soliton by the barrier. We also find large quantum mechanical uncertainties in the soliton's position and momentum due to nonlinear interaction with the barrier.

DOI
10.1088/1367-2630/14/4/043040

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We numerically study the classical and quantum dynamics of an atomic bright soliton in a highly elongated one-dimensional harmonic trap with a Gaussian barrier. In the regime of the recent experiment by Dyke et al the system realizes a coherent nonlinear soliton beam-splitter and interferometer whose accuracy we analyze. In the case of tighter radial trap confinement and enhanced quantum fluctuations, a split soliton can represent a spin-squeezed, or alternatively, a fragmented condensate with reduced phase coherence that can be measured by interfering the split soliton by the barrier. We also find large quantum mechanical uncertainties in the soliton's position and momentum due to nonlinear interaction with the barrier.
We numerically study the classical and quantum dynamics of an atomic bright soliton in a highly elongated one dimensional harmonic trap with a Gaussian barrier. In the regime of the recent experiment by Dyke et al the system realizes a coherent nonlinear soliton beam-splitter and interferometer whose accuracy we analyze. In the case of tighter radial trap confinement and enhanced quantum fluctuations, a split soliton can represent a spin-squeezed, or alternatively, a fragmented condensate with reduced phase coherence that can be measured by interfering the split soliton by the barrier. We also find large quantum mechanical uncertainties in the soliton’s position and momentum due to nonlinear interaction with the barrier.
Bose–Einstein condensates with attractive interactions have stable 1D solutions in the form of bright solitary-waves. These solitary waves behave, in the absence of external potentials, like macroscopic quantum particles. This opens up a wide array of applications for the testing of quantum mechanical behaviours and precision measurement. Here we investigate these applications with particular focus on the interactions of bright solitary-waves with narrow potential barriers. We first study bright solitons in the Gross–Pitaevskii equation as they are split on Gaussian and δ-function barriers, and then on Gaussian barriers in a low energy system. We present analytic and numerical results determining the general region in which a soliton may not be split on a finite width potential barrier. Furthermore, we test the sensitivity of the system to quantum fluctuations. We then study fast-moving bright solitons colliding at a narrow Gaussian potential barrier. In the limiting case of a δ-function barrier, we show analytically that the relative norms of the outgoing waves depends sinusoidally on the relative phase of the incoming waves, and determine whether the outgoing waves are bright solitons. We use numerical simulations to show that outside the high velocity limit nonlinear effects introduce a skew to the phase-dependence. Finally, we use these results to analyse the process of soliton interferometry. We develop analyses of both toroidal and harmonic trapping geometries for Mach–Zehnder interferometry, and then two implementations of a toroidal Sagnac inter- ferometer, also giving the analytical determination of the Sagnac phase in such systems. These results are again verified numerically. In the Mach–Zehnder case, we again probe the systems sensitivity to quantum fluctuations.
Bright solitons in a quasi-1D Bose–Einstein condensate can be used to enhance precision in matter-wave interferometry, due to their inherent robustness and support against dispersion. Such a soliton interferometer typically relies on a potential barrier used to split a single soliton into two smaller coherent solitons which can then be recombined on the same barrier. In this thesis we examine two extensions to this scheme. Firstly, we investigate a binary BEC system consisting of two bright solitons which are coupled through a mutual nonlinear interaction term. We derive a set of conditions under which the two components can be separated on a potential barrier and use numerical simulations to probe the regimes beyond which this mathematical treatment is applicable. We then use the numerical simulations to look at the effect of the nonlinear coupling on the dynamics of the binary solitons interacting with the barrier. We also look at the interference behaviour found by doubling the simulation time in either a ring trap or a harmonic trapping potential (to ensure recombination on the barrier); as well as the case where the solitons start spatially separated on either side of the barrier in order to find conditions under which the solitons will combine on the barrier. We find a good agreement between the analytical predictions and the results of simulations. Beyond the regions of parameter space where the predictions are expected to hold, we find complex transmission and interference behaviour as a result of nonlinear effects. The second part of this thesis consists of an examination of the prospect of using a subwavelength barrier scheme in a soliton interferometry experiment. This involves using two resonant coupling beams in a Λ-system with a spatially varying intensity. Under certain conditions, this can be used to form an effective potential barrier with a width which is not diffraction-limited. We look at suitable parameter regimes for such a barrier to split and recombine solitons in an interferometer and probe the effects of possible complications such as misalignment in the beams and different scattering lengths in the different states. We simulate the soliton interferometer using the full three component GPE as well as the single component analogue with the effective potential in order to characterise the soliton behaviour and the dependence of the interferometer sensitivity on the system parameters. We find a trend towards an idealised soliton interferometer with a decreasing value of the parameter, w, controlling the barrier characteristics. Also, we demonstrate an agreement between the relevant three component GPE and the analogous single component GPE, in the limit of strong coupling fields.
We study bright solitons in the Gross-Pitaevskii equation as they are split and recombined in a low-energy system. We present analytic results determining the general region in which a soliton may not be split on a potential barrier and confirm these results numerically. Furthermore, we analyze the energetic regimes where quantum fluctuations in the initial center-of-mass position and momentum become influential on the outcome of soliton splitting and recombination events. We then use the results of this analysis to determine a parameter regime where soliton interferometry is practicable.
We investigate the dynamics of Bose-Einstein condensate bright solitons made of alkali-metal atoms with negative scattering length and under harmonic confinement in the transverse direction. Contrary to the one-dimensional (1D) case, the 3D bright soliton exists only below a critical attractive interaction that depends on the extent of confinement. Such a behavior is also found in multisoliton condensates with box boundary conditions. We obtain numerical and analytical estimates of the critical strength beyond which the solitons do not exist. By using an effective 1D nonpolynomial nonlinear Schr\"odinger equation, which accurately takes into account the transverse dynamics of cigarlike condensates, we numerically simulate the dynamics of the ``soliton train'' reported in a recent experiment [Nature (London) 417, 150 (2002)]. Then, analyzing the macroscopic quantum tunneling of the bright soliton on a Gaussian barrier, we find that its interference in the tunneling region is strongly suppressed with respect to nonsolitonic case; moreover, the tunneling through a barrier breaks the shape invariance of the matter wave. Finally, we show that the collapse of the soliton is induced by the scattering on the barrier or by the collision with another matter wave when the density reaches a critical value, for which we derive an accurate analytical formula.
We consider a matter-wave bright soliton interferometer composed of a harmonic potential trap with a Rosen-Morse barrier at its center on which an incident soliton collides and splits into two solitons. These two solitons recombine after a dipole oscillation in the trap at the position of the barrier. We focus on the characterization of the splitting process in the case in which the reflected and transmitted solitons have the same number of atoms. We obtain that the velocity of the split solitons strongly depends on the nonlinearity and on the width of the barrier and that the reflected soliton is in general slower than the transmitted one. Also, we study the phase difference acquired between the two solitons during the splitting and we fit semianalytically the main dependences with the velocity of the incident soliton, the nonlinearity, and the width of the barrier. The implementation of the full interferometer sequence is tested by means of the phase imprinting method.