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arXiv:1208.4984 · Journal of Physics B Atomic Molecular and Optical Physics

Nonlocal quantum superpositions of bright matter-wave solitons and dimers

Bettina Gertjerenken, Christoph Weiß

Published Aug 8, 2012• 16 citations
arXivHTMLPDF

Abstract

The scattering of bright quantum solitons at barrier potentials in one-dimensional geometries is investigated. Such protocols have been predicted to lead to the creation of nonlocal quantum superpositions. The centre-of-mass motion of these bright matter-wave solitons generated from attractive Bose-Einstein condensates can be analysed with the effective potential approach. An application to the case of two particles being scattered at a delta potential allows analytical calculations not possible for higher particle numbers as well as a comparison with numerical results. Both for the dimer and a soliton with particle numbers on the order of N = 100, we investigate the signatures of the coherent superposition states in an interferometric setup and argue that experimentally an interference pattern would be particularly well observable in the centre-of-mass density. Quantum superposition states of ultra-cold atoms are interesting as input states for matter-wave interferometry as they could improve signal-to-noise ratios.

DOI
10.1088/0953-4075/45/16/165301

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The scattering of bright quantum solitons at barrier potentials in one-dimensional geometries is investigated. Such protocols have been predicted to lead to the creation of nonlocal quantum superpositions. The centre-of-mass motion of these bright matter-wave solitons generated from attractive Bose-Einstein condensates can be analysed with the effective potential approach. An application to the case of two particles being scattered at a delta potential allows analytical calculations not possible for higher particle numbers as well as a comparison with numerical results. Both for the dimer and a soliton with particle numbers on the order of N = 100, we investigate the signatures of the coherent superposition states in an interferometric setup and argue that experimentally an interference pattern would be particularly well observable in the centre-of-mass density. Quantum superposition states of ultra-cold atoms are interesting as input states for matter-wave interferometry as they could improve signal-to-noise ratios.
Scattering quantum bright solitons off barriers has been predicted to lead to nonlocal quantum superpositions, in particular the NOON state. The focus of this paper lies on signatures of both high- and low-fidelity quantum superposition states. We numerically demonstrate that a one-dimensional geometry with the barrier potential situated in the middle of an additional---experimentally typical---harmonic confinement gives rise to particularly well-observable signatures. In the elastic-scattering regime we investigate signatures of NOON states on the NNN-particle level within an effective potential approach. We show that removing the barrier potential and subsequently recombining both parts of the quantum superposition leads to a high-contrast interference pattern in the center-of-mass coordinate for narrow and broad potential barriers. We demonstrate that the presented signatures can be used to clearly distinguish quantum superposition states from statistical mixtures and are sufficiently robust against experimentally relevant excitations of the center-of-mass wave function to higher-lying oscillator states. For two-particle solitons we extend these considerations to low-fidelity superposition states: even for strong deviations from the two-particle NOON state we find interference patterns with high contrast.
We consider a one-dimensional matter-wave bright soliton, corresponding to the ground bound state of N bosonic particles of mass m having a binary attractive delta potential interaction on the open line. For a full N -body quantum treatment, we derive several results for the scattering of this quantum soliton on a short-range, bounded from below, external potential, restricting to the low energy, elastic regime where the centre-of-mass kinetic energy of the incoming soliton is lower than the internal energy gap of the soliton, that is the minimal energy required to extract particles from the soliton.
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.
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.
We discuss an interferometric scheme employing interference of bright solitons formed as specific bound states of attracting bosons on a lattice. We revisit the proposal of Castin and Weiss [Phys. Rev. Lett. vol. 102, 010403 (2009)] for using the scattering of a quantum matter-wave soliton on a barrier in order to create a coherent superposition state of the soliton being entirely to the left of the barrier and being entirely to the right of the barrier. In that proposal, it was assumed that the scattering is perfectly elastic, i.e. that the center-of-mass kinetic energy of the soliton is lower than the chemical potential of the soliton. Here we relax this assumption: By employing a combination of Bethe ansatz and DMRG-based analysis of the dynamics of the appropriate many-body system, we find that the interferometric fringes persist even when the center-of-mass kinetic energy of the soliton is above the energy needed for its complete dissociation into constituent atoms.