Quantum theory of bright matter-wave solitons in harmonic confinement

Abstract

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. This paper investigates bright quantum-matter-wave solitons beyond the Gross-Pitaevskii equation (GPE). As proposals for interferometry and creating nonlocal quantum superpositions have been formed, it has become necessary to investigate effects not present in mean-field models. We investigate the effect of harmonic confinement on the internal degrees of freedom, as the ratio of zero-point harmonic oscillator length to classical soliton length, for different numbers of atoms. We derive a first-order energy correction for the addition of a harmonic potential to the many-body wave function and use this to create a variational technique based on energy minimization of this wave function for an arbitrary number of atoms, and include numerics based on diagonalization of the Hamiltonian in a basis of harmonic oscillator Fock states. Finally we compare agreement between a Hartree product ground state and the Bethe ansatz solution with a Gaussian envelope localizing the center of mass and show a region of good agreement.

Controlled formation and reflection of a bright solitary matter-wave

“…Solitary waves realised experimentally typically contain ≲1,000 atoms, placing them well outside of the thermodynamic limit and potentially outside the reach of the mean-field description. Several theoretical studies of bright solitary waves beyond the mean-field description have now been performed, either including effects of quantum noise using the truncated Wigner method3 or using approximate analytic and numerical methods to simulate the full many-body problem4520. These generate results potentially in conflict with the behaviour predicted by the GPE treatment.…”

Collision dynamics and entanglement generation of two initially independent and indistinguishable boson pairs in one-dimensional harmonic confinement

“…We consider an effective 1D system (taken to be reduced from a 3D configuration where the radial degrees of freedom are strongly confined by a harmonic trapping potential) of structureless bosons subject to attractive or repulsive contact interactions V (|x 1 − x 2 |) = g 1D δ(x 1 − x 2 ), i.e., a LiebLiniger-(McGuire) gas [31][32][33], with the addition of an axial harmonic confining potential. In second-quantized form, this can be described by the following Hamiltonian:…”

“…IV B4, predicted to be most significant for g < 0. We can write our wave function at any point in time as |ψ(t) = c 2,2 (t)|ψ 2,2 (t) + c 1,3 (t)|ψ 1,3 (t) + c 0,4 (t)|ψ 0,4 (t) , (32) with |ψ n,N−n (t) normalized wave functions that are superpositions of states with n and N − n atoms to the left, and vice versa, and {c n,N−n } a set of complex constants, the modulus squares of which are the probabilities to find n or N − n atoms on either side. In order to qualitatively predict the incremental changes to {c n,N−n (t)} from before to after a collision, we use time-dependent perturbation theory, assuming |g 1 is a small parameter and neglecting any contribution from c 0,4 (t) (specifically at the time of collisions).…”

“…In the absence interactions, no real entanglement can be generated between the two subsystems (although our chosen measure is only meaningful when the subsystems are well separated). Additionally, with strong interactions the effect of the confinement may be diminished and the integrability of the free system [31] may also effect entanglement generation, particularly in the attractive case where a new length scale is introduced that can be smaller than the confinement [32].…”

Bright-soliton quantum superpositions: Signatures of high- and low-fidelity states

“…Initially the many-particle ground state is prepared in the harmonic trap [44]: throughout this work, we assume that the internal degrees of freedom are described by the Lieb-Liniger soliton (9) while the CoM motion is determined by the harmonic confinement. The center of the trap is then (quasi-)instantaneously shifted and the scattering potential in the middle of the trap is switched on.…”