Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1203.6006 · Physical Review A

Quantum theory of bright matter-wave solitons in harmonic confinement

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

Published May 15, 2012• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.85.053618

Related papers

Many body effects in one-dimensional attractive Bose gases

David I. H. Holdaway

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.

•Jan 1, 2013•0 citations

Scattering bright solitons: Quantum versus mean-field behavior

Bettina Gertjerenken, T. P. Billam, Lev Khaykovich +1 more

We investigate scattering bright solitons off a potential using both analytical and numerical methods. Our paper focuses on low kinetic energies for which differences between the mean-field description via the Gross-Pitaevskii equation (GPE) and the quantum behavior are particularly large. On the $N$-particle quantum level, adding an additional harmonic confinement leads to a simple signature to distinguish quantum superpositions from statistical mixtures. While the nonlinear character of the GPE does not allow quantum superpositions, the splitting of GPE solitons takes place only partially. When the potential strength is increased, the fraction of the soliton which is transmitted or reflected jumps noncontinuously. We explain these jumps via energy conservation and interpret them as indications for quantum superpositions on the $N$-particle level. On the GPE level, we also investigate the transition from this stepwise behavior to the continuous case.

arXiv:1208.2941•Physical Review A•Sep 7, 2012•50 citations

Dynamics of bright solitons in Bose–Einstein condensates: investigations of soliton behaviour in the vector Gross–Pitaevskii equation and applications to enhanced matter-wave interferometry

Callum L. Grimshaw

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.

•Jan 1, 2021•0 citations

Progress towards quantum-enhanced interferometry with harmonically trapped quantum matter-wave bright solitons

Bettina Gertjerenken, Timothy P. Wiles, Christoph Weiß

We model the dynamics of attractively interacting ultracold bosonic atoms in a quasi-one-dimensional wave guide with additional harmonic trapping. Initially, we prepare the system in its ground state and then shift the zero of the harmonic trap and switch on an additional narrow scattering potential near the center of the trap. After colliding with the barrier twice, we propose to measure the number of atoms opposite the initial condition. Quantum-enhanced interferometry with quantum bright solitons allows us to predict detection of an offset of the scattering potential with considerably increased precision as compared to single-particle experiments. In a future experimental realization this might lead to measurement of weak forces caused, for example, by small horizontal gradients in the gravitational potential---with a resolution of several micrometers given essentially by the size of the solitons. Our numerical simulations are based on the rigorously proved effective potential approach developed in previous papers [Phys. Rev. Lett. 102, 010403 (2009) and Phys. Rev. Lett. 103, 210402 (2009)]. We choose our parameters such that the prerequisite of the proof (that the solitons cannot break apart, for energetic reasons) is always fulfilled, thus exploring a parameter regime inaccessible to the mean-field description via the Gross-Pitaevskii equation due to Schr\"odinger-cat states occurring in the many-particle quantum dynamics.

arXiv:1508.00656•Physical Review A•Nov 30, 2016•10 citations

Splitting bright matter-wave solitons on narrow potential barriers: Quantum to classical transition and applications to interferometry

J. L. Helm, S. J. Rooney, Christoph Weiß +1 more

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.

arXiv:1401.7202•Physical Review A•Mar 10, 2014•50 citations

Internal oscillations of a dark-bright soliton in a harmonic potential

Majed O D Alotaibi, Lincoln D. Carr

Abstract We investigate the dynamics of a dark-bright soliton in a harmonic potential using a mean-field approach via coupled nonlinear Schrödinger equations appropriate to multi-component Bose–Einstein condensates. We use a modified perturbed dynamical variational Lagrangian approximation, where the perturbation is due to the trap, taken as a Thomas–Fermi profile. The wave function ansatz is taken as the correct hyperbolic tangent and secant solutions in the scalar case for the dark and bright components of the soliton, respectively. We also solve the problem numerically with pseudo-spectral Runge–Kutta methods. We find, analytically and numerically, for weak trapping the internal modes are nearly independent of center of mass motion of the dark-bright soliton. In contrast, in tighter traps the internal modes couple strongly to the center of mass motion, showing that for dark-bright solitons in a harmonic potential the center of mass and relative degrees of freedom are not independent. This result is robust against noise in the initial condition and should, therefore, be experimentally observable.

arXiv:1805.03339•Journal of Physics B Atomic Molecular and Optical Physics•Sep 7, 2018•9 citations
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.
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 investigate scattering bright solitons off a potential using both analytical and numerical methods. Our paper focuses on low kinetic energies for which differences between the mean-field description via the Gross-Pitaevskii equation (GPE) and the quantum behavior are particularly large. On the NNN-particle quantum level, adding an additional harmonic confinement leads to a simple signature to distinguish quantum superpositions from statistical mixtures. While the nonlinear character of the GPE does not allow quantum superpositions, the splitting of GPE solitons takes place only partially. When the potential strength is increased, the fraction of the soliton which is transmitted or reflected jumps noncontinuously. We explain these jumps via energy conservation and interpret them as indications for quantum superpositions on the NNN-particle level. On the GPE level, we also investigate the transition from this stepwise behavior to the continuous case.
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 model the dynamics of attractively interacting ultracold bosonic atoms in a quasi-one-dimensional wave guide with additional harmonic trapping. Initially, we prepare the system in its ground state and then shift the zero of the harmonic trap and switch on an additional narrow scattering potential near the center of the trap. After colliding with the barrier twice, we propose to measure the number of atoms opposite the initial condition. Quantum-enhanced interferometry with quantum bright solitons allows us to predict detection of an offset of the scattering potential with considerably increased precision as compared to single-particle experiments. In a future experimental realization this might lead to measurement of weak forces caused, for example, by small horizontal gradients in the gravitational potential---with a resolution of several micrometers given essentially by the size of the solitons. Our numerical simulations are based on the rigorously proved effective potential approach developed in previous papers [Phys. Rev. Lett. 102, 010403 (2009) and Phys. Rev. Lett. 103, 210402 (2009)]. We choose our parameters such that the prerequisite of the proof (that the solitons cannot break apart, for energetic reasons) is always fulfilled, thus exploring a parameter regime inaccessible to the mean-field description via the Gross-Pitaevskii equation due to Schr\"odinger-cat states occurring in the many-particle quantum dynamics.
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.
Abstract We investigate the dynamics of a dark-bright soliton in a harmonic potential using a mean-field approach via coupled nonlinear Schrödinger equations appropriate to multi-component Bose–Einstein condensates. We use a modified perturbed dynamical variational Lagrangian approximation, where the perturbation is due to the trap, taken as a Thomas–Fermi profile. The wave function ansatz is taken as the correct hyperbolic tangent and secant solutions in the scalar case for the dark and bright components of the soliton, respectively. We also solve the problem numerically with pseudo-spectral Runge–Kutta methods. We find, analytically and numerically, for weak trapping the internal modes are nearly independent of center of mass motion of the dark-bright soliton. In contrast, in tighter traps the internal modes couple strongly to the center of mass motion, showing that for dark-bright solitons in a harmonic potential the center of mass and relative degrees of freedom are not independent. This result is robust against noise in the initial condition and should, therefore, be experimentally observable.