One-particle density matrix and momentum distribution of the out-of-equilibrium 1D Tonks-Girardeau gas: Analytical results at large $N$

Abstract

In one-dimensional (1D) quantum gases, the momentum distribution (MD) of the atoms is a standard experimental observable, routinely measured in various experimental setups. The MD is sensitive to correlations, and it is notoriously hard to compute theoretically for large numbers of atoms $N$, which often prevents direct comparison with experimental data. Here we report significant progress on this problem for the 1D Tonks-Girardeau (TG) gas in the asymptotic limit of large $N$, at zero temperature and driven out of equilibrium by a quench of the confining potential. We find an exact analytical formula for the one-particle density matrix $⟨{\hat{\Psi }}^{†}(x)\hat{\Psi }(x^{\prime })⟩$ of the out-of-equilibrium TG gas in the $N\to \infty$ limit, valid on distances $|x-x^{\prime }|$ much larger than the interparticle distance. By comparing with time-dependent Bose-Fermi mapping numerics, we demonstrate that our analytical formula can be used to compute the out-of-equilibrium MD with great accuracy for a wide range of momenta (except in the tails of the distribution at very large momenta). For a quench from a double-well potential to a single harmonic well,which mimics a `quantum Newton cradle' setup, our method predicts the periodic formation of peculiar, multiply peaked, momentum distributions.

arXiv comment13pages, 6 figures. v2: minor changes; v3: fixed layout issues in appendices

References in corpus (75)

• One dimensional Bosons: From Condensed Matter Systems to Ultracold Gases
• Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents
• Bragg spectroscopy of a Bose-Einstein condensate
• Regimes of quantum degeneracy in trapped 1D gases
• Emergent hydrodynamics in integrable quantum systems out of equilibrium
• Fermi gases in one dimension: From Bethe Ansatz to experiments
• Bosonizing one-dimensional cold atomic gases
• Measurement of interaction energy near a Feshbach resonance in a 6Li Fermi gas
• Generalized HydroDynamics on an Atom Chip
• Extension of Bogoliubov theory to quasi-condensates
• Pair correlations in a finite-temperature 1D Bose gas
• Verification of universal relations in a strongly interacting Fermi gas
• Short-Distance Correlation Properties of the Lieb-Liniger System and Momentum Distributions of Trapped One-Dimensional Atomic Gases
• Conformal Field Theory for Inhomogeneous One-dimensional Quantum Systems: the Example of Non-Interacting Fermi Gases
• Momentum spectroscopy of 1D phase fluctuations in Bose-Einstein condensates
• Exact coherent states of a harmonically confined Tonks-Girardeau gas
• Large-scale description of interacting one-dimensional Bose gases: generalized hydrodynamics supersedes conventional hydrodynamics
• Lecture notes on Generalised Hydrodynamics
• Generalized hydrodynamics in strongly interacting 1D Bose gases
• Quantum Generalized Hydrodynamics
• One-particle dynamical correlations in the one-dimensional Bose gas
• Fermionization in an expanding 1D gas of hard-core bosons
• Finite one dimensional impenetrable Bose systems: Occupation numbers
• Breakdown of time-dependent mean-field theory for a one-dimensional condensate of impenetrable bosons
• Higher-Order Hydrodynamics in 1D: a Promising Direction and a Null Result
• Bose-Einstein condensation into non-equilibrium states studied by condensate focusing
• Observation of Dynamical Fermionization
• Finite temperature correlations in the Lieb-Liniger 1D Bose gas
• High-momentum tail in the Tonks gas under harmonic confinement
• Generalized Hydrodynamics in the 1D Bose gas: theory and experiments
• Universal properties of hard-core bosons confined on one-dimensional lattices
• Hydrodynamics of the interacting Bose gas in the Quantum Newton Cradle setup
• Correlation functions of integrable models: a description of the ABACUS algorithm
• Momentum distribution of a Fermi gas of atoms in the BCS-BEC crossover
• Emergence of curved light-cones in a class of inhomogeneous Luttinger liquids
• A Numerical Renormalization Group for Continuum One-Dimensional Systems
• Universal correlations of trapped one-dimensional impenetrable bosons
• Momentum distribution dynamics of a Tonks-Girardeau gas: Bragg reflections of a quantum many-body wavepacket
• Exact prefactors in static and dynamic correlation functions of 1D quantum integrable models: applications to the Calogero-Sutherland, Lieb-Liniger and XXZ models
• Momentum-resolved study of an array of 1D strongly phase-fluctuating Bose gases
• Universal contact for a Tonks-Girardeau gas at finite temperature
• Ground-state properties of hard core bosons in one-dimensional harmonic traps
• Nonuniversal prefactors in correlation functions of 1D quantum liquids
• Momentum distribution and correlation function of quasicondensates in elongated traps
• Conformal field theory on top of a breathing one-dimensional gas of hard core bosons
• Quantum ripples over a semi-classical shock
• High-momentum tails as magnetic structure probes for strongly-correlated $SU(\kappa )$ fermionic mixtures in one-dimensional traps
• Free expansion of impenetrable bosons on one-dimensional optical lattices
• Universal scaling of density and momentum distributions in Lieb-Liniger gases
• Sudden Expansion of a One-Dimensional Bose Gas from Power-Law Traps
• Yang-Yang thermometry and momentum distribution of a trapped one-dimensional Bose gas
• Free expansion of a Lieb-Liniger gas: Asymptotic form of the wave functions
• Breakdown of Tan's relation in lossy one-dimensional Bose gases
• Momentum distribution of 1D Bose gases at the quasi-condensation crossover: theoretical and experimental investigation
• Locally quasi-stationary states in noninteracting spin chains
• Momentum-space correlations of a one-dimensional Bose gas
• The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases
• Tan's Contact for Trapped Lieb-Liniger Bosons at Finite Temperature
• One-particle density matrix of trapped one-dimensional impenetrable bosons from conformal invariance
• Exact nonequilibrium dynamics of finite-temperature Tonks-Girardeau gases
• Fermi distribution of semicalssical non-eqilibrium Fermi states
• Quantum Shock Waves and Population Inversion in Collisions of Ultracold Atomic Clouds
• Experimental Observation of Curved Light-Cones in a Quantum Field Simulator
• Strongly interacting trapped one-dimensional quantum gases: an exact solution
• Exact entanglement growth of a one-dimensional hard-core quantum gas during a free expansion
• Exact hydrodynamic solution of a double domain wall melting in the spin-1/2 XXZ model
• Quantum quench and thermalization of one-dimensional Fermi gas via phase space hydrodynamics
• Rapidity and momentum distributions of 1D dipolar quantum gases
• Momentum distribution of a freely expanding Lieb-Liniger gas
• One-particle density matrix of a trapped Lieb-Liniger anyonic gas
• Universal off-diagonal long-range order behaviour for a trapped Tonks-Girardeau gas
• General finite-size effects for zero-entropy states in one-dimensional quantum integrable models
• Exact dynamics of a one dimensional Bose gas in a periodic time-dependent harmonic trap
• Correlations of zero-entropy critical states in the XXZ model: integrability and Luttinger theory far from the ground state
• Tailoring Dynamical Fermionization: Delta kick cooling of a Tonks-Girardeau gas

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