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:0901.4437 · Physical Review A

Momentum distribution of a freely expanding Lieb-Liniger gas

Dario Jukić, Bruno Klajn, Hrvoje Buljan

Published Mar 12, 2009• 27 citations
arXivHTMLPDF

Abstract

We numerically study free expansion of a few Lieb-Liniger bosons, which are initially in the ground state of an infinitely deep hard-wall trap. Numerical calculation is carried out by employing a standard Fourier transform, as follows from the Fermi-Bose transformation for a time-dependent Lieb-Liniger gas. We study the evolution of the momentum distribution, the real-space single-particle density, and the occupancies of natural orbitals. Our numerical calculation allows us to explore the behavior of these observables in the transient regime of the expansion, where they are nontrivially affected by the particle interactions. We derive analytically (by using the stationary phase approximation) the formula which connects the asymptotic shape of the momentum distribution and the initial state. For sufficiently large times, the momentum distribution coincides (up to a simple scaling transformation) with the shape of the real-space single-particle density (the expansion is asymptotically ballistic). Our analytical and numerical results are in good agreement.

DOI
10.1103/physreva.79.033612

Related papers

Free expansion of a Lieb-Liniger gas: Asymptotic form of the wave functions

Dario Jukić, Robert Pezer, Thomas Gasenzer +1 more

The asymptotic form of the wave functions describing a freely expanding Lieb-Liniger gas is derived by using a Fermi-Bose transformation for time-dependent states, and the stationary phase approximation. We find that asymptotically the wave functions approach the Tonks-Girardeau (TG) structure as they vanish when any two of the particle coordinates coincide. We point out that the properties of these asymptotic states can significantly differ from the properties of a TG gas in a ground state of an external potential. The dependence of the asymptotic wave function on the initial state is discussed. The analysis encompasses a large class of initial conditions, including the ground states of a Lieb-Liniger gas in physically realistic external potentials. It is also demonstrated that the interaction energy asymptotically decays as a universal power law with time, ${E}_{\mathrm{int}}\ensuremath{\propto}{t}^{\ensuremath{-}3}$.

arXiv:0804.2580•Physical Review A•Nov 3, 2008•58 citations

Dynamical fermionization in a one-dimensional Bose-Fermi mixture

Ovidiu I. Pâţu

After release from the trap the momentum distribution of an impenetrable gas asymptotically approaches that of a spinless noninteracting Fermi gas in the initial trap. This phenomenon is called dynamical fermionization and, very recently, has been experimentally confirmed in the case of the Lieb-Liniger model in the Tonks-Girardeau regime. We prove analytically and confirm numerically that following the removal of axial confinement the strongly interacting Bose-Fermi mixture exhibits dynamical fermionization and the asymptotical momentum distribution of each component has the same shape as its density profile at $t=0$. Under a sudden change of the trap frequency to a new nonzero value the dynamics of both fermionic and bosonic momentum distributions presents characteristics which are similar to the case of single component bosons experiencing a similar quench. Our results are derived using a product representation for the correlation functions which, in addition to analytical considerations, can be implemented numerically very easily with complexity which scales polynomially in the number of particles.

arXiv:2108.05155•Physical Review A•Jun 9, 2022•10 citations

Sudden Expansion of a One-Dimensional Bose Gas from Power-Law Traps

A. S. Campbell, Dimitri M. Gangardt, Karen V. Kheruntsyan

We analyze free expansion of a trapped one-dimensional Bose gas after a sudden release from the confining trap potential. By using the stationary phase and local density approximations, we show that the long-time asymptotic density profile and the momentum distribution of the gas are determined by the initial distribution of Bethe rapidities (quasimomenta) and hence can be obtained from the solutions to the Lieb-Liniger equations in the thermodynamic limit. For expansion from a harmonic trap, and in the limits of very weak and very strong interactions, we recover the self-similar scaling solutions known from the hydrodynamic approach. For all other power-law traps and arbitrary interaction strengths, the expansion is not self-similar and shows strong dependence of the density profile evolution on the trap anharmonicity. We also characterize dynamical fermionization of the expanding cloud in terms of correlation functions describing phase and density fluctuations.

arXiv:1501.01896•Physical Review Letters•Mar 27, 2015•65 citations

Fermi-Bose Transformation for the Time-Dependent Lieb-Liniger Gas

Hrvoje Buljan, Robert Pezer, Thomas Gasenzer

Exact solutions of the Schrödinger equation describing a freely expanding Lieb-Liniger gas of delta-interacting bosons in one spatial dimension are constructed. We demonstrate that for any interaction strength the system enters a strongly correlated regime during such expansion. The asymptotic form of the wave function is shown to have the form characteristic for "impenetrable-core" bosons. Exact solutions are obtained by transforming a fully antisymmetric (fermionic) time-dependent wave function that obeys the Schrödinger equation for a free gas. This transformation employs a differential Fermi-Bose mapping operator that depends on the strength of the interaction and the number of particles.

arXiv:0709.1444•Physical Review Letters•Feb 29, 2008•73 citations

Lieb-Liniger gas in a constant-force potential

Dario Jukić, S. Galić, Robert Pezer +1 more

We use Gaudin's Fermi-Bose mapping operator to calculate exact solutions for the Lieb-Liniger model in a linear (constant-force) potential (the constructed exact stationary solutions are referred to as the Lieb-Liniger-Airy wave functions). The ground-state properties of the gas in the wedgelike trapping potential are calculated in the strongly interacting regime by using Girardeau's Fermi-Bose mapping and the pseudopotential approach in the $1/c$ approximation ($c$ denotes the strength of the interaction). We point out that quantum dynamics of Lieb-Liniger wave packets in the linear potential can be calculated by employing an $N$-dimensional Fourier transform as in the case of free expansion.

arXiv:1005.3660•Physical Review A•Aug 12, 2010•8 citations

Single-particle density matrix for a time-dependent strongly interacting one-dimensional Bose gas

Robert Pezer, Thomas Gasenzer, Hrvoje Buljan

We derive a $1/c$ expansion for the single-particle density matrix of a strongly interacting time-dependent one-dimensional Bose gas, described by the Lieb-Liniger model ($c$ denotes the strength of the interaction). The formalism is derived by expanding Gaudin's Fermi-Bose mapping operator up to $1/c$ terms. We derive an efficient numerical algorithm for calculating the density matrix for time-dependent states in the strong coupling limit, which evolve from a family of initial conditions in the absence of an external potential. We have applied the formalism to study contraction dynamics of a localized wave packet upon which a parabolic phase is imprinted initially.

arXiv:0907.4608•Physical Review A•Nov 19, 2009•10 citations
We numerically study free expansion of a few Lieb-Liniger bosons, which are initially in the ground state of an infinitely deep hard-wall trap. Numerical calculation is carried out by employing a standard Fourier transform, as follows from the Fermi-Bose transformation for a time-dependent Lieb-Liniger gas. We study the evolution of the momentum distribution, the real-space single-particle density, and the occupancies of natural orbitals. Our numerical calculation allows us to explore the behavior of these observables in the transient regime of the expansion, where they are nontrivially affected by the particle interactions. We derive analytically (by using the stationary phase approximation) the formula which connects the asymptotic shape of the momentum distribution and the initial state. For sufficiently large times, the momentum distribution coincides (up to a simple scaling transformation) with the shape of the real-space single-particle density (the expansion is asymptotically ballistic). Our analytical and numerical results are in good agreement.
The asymptotic form of the wave functions describing a freely expanding Lieb-Liniger gas is derived by using a Fermi-Bose transformation for time-dependent states, and the stationary phase approximation. We find that asymptotically the wave functions approach the Tonks-Girardeau (TG) structure as they vanish when any two of the particle coordinates coincide. We point out that the properties of these asymptotic states can significantly differ from the properties of a TG gas in a ground state of an external potential. The dependence of the asymptotic wave function on the initial state is discussed. The analysis encompasses a large class of initial conditions, including the ground states of a Lieb-Liniger gas in physically realistic external potentials. It is also demonstrated that the interaction energy asymptotically decays as a universal power law with time, ${E}_{\mathrm{int}}\ensuremath{\propto}{t}^{\ensuremath{-}3}$.
After release from the trap the momentum distribution of an impenetrable gas asymptotically approaches that of a spinless noninteracting Fermi gas in the initial trap. This phenomenon is called dynamical fermionization and, very recently, has been experimentally confirmed in the case of the Lieb-Liniger model in the Tonks-Girardeau regime. We prove analytically and confirm numerically that following the removal of axial confinement the strongly interacting Bose-Fermi mixture exhibits dynamical fermionization and the asymptotical momentum distribution of each component has the same shape as its density profile at t=0t=0t=0. Under a sudden change of the trap frequency to a new nonzero value the dynamics of both fermionic and bosonic momentum distributions presents characteristics which are similar to the case of single component bosons experiencing a similar quench. Our results are derived using a product representation for the correlation functions which, in addition to analytical considerations, can be implemented numerically very easily with complexity which scales polynomially in the number of particles.
We analyze free expansion of a trapped one-dimensional Bose gas after a sudden release from the confining trap potential. By using the stationary phase and local density approximations, we show that the long-time asymptotic density profile and the momentum distribution of the gas are determined by the initial distribution of Bethe rapidities (quasimomenta) and hence can be obtained from the solutions to the Lieb-Liniger equations in the thermodynamic limit. For expansion from a harmonic trap, and in the limits of very weak and very strong interactions, we recover the self-similar scaling solutions known from the hydrodynamic approach. For all other power-law traps and arbitrary interaction strengths, the expansion is not self-similar and shows strong dependence of the density profile evolution on the trap anharmonicity. We also characterize dynamical fermionization of the expanding cloud in terms of correlation functions describing phase and density fluctuations.
Exact solutions of the Schrödinger equation describing a freely expanding Lieb-Liniger gas of delta-interacting bosons in one spatial dimension are constructed. We demonstrate that for any interaction strength the system enters a strongly correlated regime during such expansion. The asymptotic form of the wave function is shown to have the form characteristic for "impenetrable-core" bosons. Exact solutions are obtained by transforming a fully antisymmetric (fermionic) time-dependent wave function that obeys the Schrödinger equation for a free gas. This transformation employs a differential Fermi-Bose mapping operator that depends on the strength of the interaction and the number of particles.
We use Gaudin's Fermi-Bose mapping operator to calculate exact solutions for the Lieb-Liniger model in a linear (constant-force) potential (the constructed exact stationary solutions are referred to as the Lieb-Liniger-Airy wave functions). The ground-state properties of the gas in the wedgelike trapping potential are calculated in the strongly interacting regime by using Girardeau's Fermi-Bose mapping and the pseudopotential approach in the 1/c1/c1/c approximation (ccc denotes the strength of the interaction). We point out that quantum dynamics of Lieb-Liniger wave packets in the linear potential can be calculated by employing an NNN-dimensional Fourier transform as in the case of free expansion.
We derive a 1/c1/c1/c expansion for the single-particle density matrix of a strongly interacting time-dependent one-dimensional Bose gas, described by the Lieb-Liniger model (ccc denotes the strength of the interaction). The formalism is derived by expanding Gaudin's Fermi-Bose mapping operator up to 1/c1/c1/c terms. We derive an efficient numerical algorithm for calculating the density matrix for time-dependent states in the strong coupling limit, which evolve from a family of initial conditions in the absence of an external potential. We have applied the formalism to study contraction dynamics of a localized wave packet upon which a parabolic phase is imprinted initially.