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arXiv:1303.1388 · Physical Review A

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

David I. H. Holdaway, Christoph Weiß, S. A. Gardiner

Published Apr 26, 2013• 9 citations
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Abstract

We investigate finite-number effects in collisions between two states of an initially well-known number of identical bosons with contact interactions, oscillating in the presence of harmonic confinement in one dimension. We investigate two $N/2$ (interacting) ground states, which are initially displaced from the trap center, and the effects of varying interaction strength. The numerics focus on the simplest case of $N=4$. In the noninteracting case, such a system would display periodic oscillation with a half harmonic oscillator period (due to the left-right symmetry). With the addition of contact interactions between the bosons, collisions generate entanglement between each of the states and distribute energy into other modes of the oscillator. We study the system numerically via an exact diagonalization of the Hamiltonian with a finite basis, investigating left-right number uncertainty as our primary measure of entanglement. Additionally, we study the time evolution and equilibration of the single-body von Neumann entropy for both the attractive and repulsive cases. We identify parameter regimes for which attractive interactions create behavior qualitatively different from that of repulsive interactions, due to the presence of bound states (quantum solitons), and explain the processes behind this.

DOI
10.1103/physreva.87.043632

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We investigate finite-number effects in collisions between two states of an initially well-known number of identical bosons with contact interactions, oscillating in the presence of harmonic confinement in one dimension. We investigate two N/2N/2N/2 (interacting) ground states, which are initially displaced from the trap center, and the effects of varying interaction strength. The numerics focus on the simplest case of N=4N=4N=4. In the noninteracting case, such a system would display periodic oscillation with a half harmonic oscillator period (due to the left-right symmetry). With the addition of contact interactions between the bosons, collisions generate entanglement between each of the states and distribute energy into other modes of the oscillator. We study the system numerically via an exact diagonalization of the Hamiltonian with a finite basis, investigating left-right number uncertainty as our primary measure of entanglement. Additionally, we study the time evolution and equilibration of the single-body von Neumann entropy for both the attractive and repulsive cases. We identify parameter regimes for which attractive interactions create behavior qualitatively different from that of repulsive interactions, due to the presence of bound states (quantum solitons), and explain the processes behind this.
We investigate finite number effects in collisions between two states of an initially well defined number of identical bosons with attractive contact interactions, oscillating in the presence of harmonic confinement in one dimension. We investigate two N/2N/2N/2 atom bound states, which are initially displaced (symmetrically) from the trap center, and then left to freely evolve. For sufficiently attractive interactions, these bound states are like those found through use of the Bethe ansatz (quantum solitons). However, unlike the free case, the integrability is lost due to confinement, and collisions can cause mixing into different bound-state configurations. We study the system numerically for the simplest case of N=4N=4N=4, via an exact diagonalization of the Hamiltonian within a finite basis, investigating left-right number uncertainty as our primary measure of entanglement. We find that for certain interaction strengths, a phase-matching condition leads to resonant transfer to different bound-state configurations with highly non-Poissonian relative number statistics.
In this thesis we investigate the properties of ultra-cold quantum gases in reduced dimension and the effects of harmonic confinement on soliton-like properties. We study regimes of agreement between mean-field and many-body theories the generation of entanglement between initially independent finite sized atomic systems. Classical solitons are non-dispersing waves which occur in integrable systems, such as atomic Bose-Einstein condensates in one dimension. Bright and dark solitons are possible, which exist as peaks or dips in density. Quantum solitons are the bound-state solutions to a system satisfying quantum integrability, given via the Bethe Ansatz. Such integrability is broken by the introduction of harmonic confinement. We investigate the equivalence of the classical field and many-body solutions in the limit of large numbers of atoms and derive numerical and variational approaches to examine the ground state energy in harmonic confinement and the fidelity between a Hartree-product solution and a quantum soliton solution. Soliton collisions produce no entanglement between either state and result only in an asymptotic position and phase shift, however external potentials break integrability and thus give the possibility of entangling solitons. We investigate the dynamical entanglement generation between two atomic dimers in harmonic confinement via exact diagonalisation in a basis of Harmonic oscillator functions, making use of the separability of the centre-of-mass component of the Hamiltonian. We show repulsive states show complex dynamics, but with an overall tendency towards states of larger invariant correlation entropy, whereas attractive states resist entanglement unless a phase matching condition is satisfied. This phase matching condition could in theory be used to generate states with highly non-Poissonian number superpositions in atomic systems with controlled number.
We deal with the effects of an anharmonic trap on an interacting two-boson system in one dimension. Our primary focus is on the role of the induced coupling between the center of mass and the relative motion as both anharmonicity and the (repulsive) interaction strength are varied. The ground state reveals a strong localization in the relative coordinate, counteracting the tendency to fragment for stronger repulsion. To explore the quantum dynamics, we study the system's response upon (i) exciting the harmonic ground state by continuously switching on an additional anharmonicity and (ii) displacing the center of mass, this way triggering collective oscillations. The interplay between collective and internal dynamics materializes in the collapse of oscillations, which are explained in terms of few-mode models.
We study systems of two identical dipolar particles confined in quasi one-dimensional harmonic traps. Numerical results for the dependencies of the entanglement on the control parameters of the systems are provided and discussed in detail. In the limit of a strong interaction between the particles, the occupancies and the von Neumann entropies of the bosonic and fermionic ground states are derived in closed analytic forms by applying the harmonic approximation. The strong correlation regimes of the system with the dipolar bosons and the system with the charged ones are compared with each other in regard to aspects of their entanglement.
We have investigated quantum entanglement for two interacting ultracold bosonic atoms in one-dimensional harmonic traps. The effective potential is modeled by delta interaction. For this two-atom system, we have investigated quantum entanglement properties, such as von Neumann entropy and linear entropy for its ground state and excited states. Using a computational scheme that is different from previously employed, a total of the lowest 16 states are studied. Here we show the dependencies of entanglement properties under various interacting strengths. Comparisons for the ground state entanglement are made with earlier results in the literature. New results for the other 15 excited states are reported here.
Abstract The out-of-equilibrium quantum dynamics of an interacting Bose gas trapped in a one-dimensional asymmetric double-well potential is studied by solving the many-body Schrödinger equation numerically accurately. We examine how the gradual loss of symmetry of the confining trap affects the macroscopic quantum tunneling dynamics of the system between the two wells. In an asymmetric double well, the two wells are not equivalent anymore, say, the left well is deeper than the right one. Accordingly, we analyze the dynamics by initially preparing the condensate in both the left and the right wells. The dynamics of the system is characterized by the time evolution of a few physical quantities of increasing many-body complexity, namely, the survival probability, depletion and fragmentation, and the many-particle position and momentum variances. In particular, we have examined the frequencies and amplitudes of the oscillations of the survival probabilities, the time scale for the development of fragmentation and its degree, and the growth and oscillatory behavior of the many-particle position and momentum variances. There is an overall suppression of the oscillations of the survival probabilities in an asymmetric double well, except for resonant values of asymmetry for which the one-body ground state energy in the right well matches with one of the one-body excited states in the left well, thereby resulting in resonantly enhanced tunneling from the right well ground state. Overall, depending on whether the condensate is initially prepared in the left or right well, the repulsive inter-atomic interactions affect the survival probabilities differently. For a sufficiently strong repulsive interaction, the system is found to become fragmented. The degree of fragmentation depends both on the asymmetry of the trap and the initial well in which the condensate is prepared in a non-trivial manner. Furthermore, we show that the phenomenon of resonantly enhanced tunneling can be accompanied by a large degree of fragmentation (depletion) for the strong (weak) interaction. The many-particle position and momentum variances follow the density oscillations of the system in the asymmetric double well and bears prominent signatures of the degree of depletion or fragmentation, depending on the strength of the interactions. These quantities further exhibit a fine structure signifying a breathing-mode oscillation. Finally, a universality of fragmentation for systems made of different numbers of particles but the same interaction parameter is also found and its dependence on the asymmetry is investigated. The phenomenon is robust despite the asymmetry of the junction and admits a macroscopically-large fragmented condensate characterized by a diverging many-particle position variance. This is as far as one can get from the dynamics of the density in the junction.