Abstract We apply the theory of quantum generalized hydrodynamics (QGHD) introduced in (2020 Phys. Rev. Lett. 124 140603) to derive asymptotically exact results for the density fluctuations and the entanglement entropy of a one-dimensional trapped Bose gas in the Tonks–Girardeau (TG) or hard-core limit, after a trap quench from a double well to a single well. On the analytical side, the quadratic nature of the theory of QGHD is complemented with the emerging conformal invariance at the TG point to fix the universal part of those quantities. Moreover, the well-known mapping of hard-core bosons to free fermions, allows to use a generalized form of the Fisher–Hartwig conjecture to fix the non-trivial spacetime dependence of the ultraviolet cutoff in the entanglement entropy. The free nature of the TG gas also allows for more accurate results on the numerical side, where a higher number of particles as compared to the interacting case can be simulated. The agreement between analytical and numerical predictions is extremely good. For the density fluctuations, however, one has to average out large Friedel oscillations present in the numerics to recover such agreement.
We investigate the nonequilibrium evolution of density correlations in a thermal one-dimensional Bose gas following a trap quench. We focus on the Tonks--Girardeau limit and develop an analytical description based on a finite-temperature extension of Quantum Generalized Hydrodynamics. We benchmark our predictions against exact numerics for a quench from a double well to a harmonic trap and for a trap release, and assess their temperature range of validity. Our results capture the dynamical enhancement of correlations during the expansion stages, namely the crossover from thermally suppressed to power-law bulk correlations. For the trap release, the conformal distance associated with fixed bulk points decreases as
1/t, so the correlations progressively probe a universal regime~
∼1/x2 that is independent of temperature. We thus show that, in the hydrodynamic long-time regime, the bulk density correlations approach their zero-temperature equilibrium form independently of the initial temperature.
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 $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}({x}^{\ensuremath{'}})\ensuremath{\rangle}$ of the out-of-equilibrium TG gas in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, valid on distances $|x\ensuremath{-}{x}^{\ensuremath{'}}|$ 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.
We present a simulation study of the properties of the Tonks-Girardeau gas, a system of
N hard-core bosons confined in a one-dimensional harmonic trap. For this system the ground-state wave function is exactly known based on a Bose-Fermi mapping theorem. We employ Nelsons' interpretation of quantum mechanics in which this
N-body wave function gives rise to a set of stochastic differential equations for the positions of the
N particles, which can be simulated by standard methods. In particular, real space densities and momentum distributions can simply be determined by time averages along particle trajectories. Employing this approach we are able to significantly extend the range of particle numbers
N treated numerically compared to earlier approaches, while reproducing all exactly known results for this model. We also show that for the bosons in a harmonic trap, contrary to what has been assumed so far, the momentum distribution reflects the system size scaling of the occupation numbers of the natural orbitals; that is, it can be used to decide on the presence of a Bose-Einstein condensate.
Abstract We consider the non-equilibrium dynamics of the entanglement entropy of a one-dimensional quantum gas of hard-core particles, initially confined in a box potential at zero temperature. At t = 0 the right edge of the box is suddenly released and the system is let free to expand. During this expansion, the initially correlated region propagates with a non-homogeneous profile, leading to the growth of entanglement entropy. This setting is investigated in the hydrodynamic regime, with tools stemming from semi-classical Wigner function approach and with recent developments of quantum fluctuating hydrodynamics. Within this framework, the entanglement entropy can be associated to a correlation function of chiral twist-fields of the conformal field theory that lives along the Fermi contour and it can be exactly determined. Our predictions for the entanglement evolution are found in agreement with and generalize previous results in literature based on numerical calculations and heuristic arguments.
The scaling of the largest eigenvalue ${\ensuremath{\lambda}}_{0}$ of the one-body density matrix of a system with respect to its particle number
N defines an exponent
C and a coefficient
B via the asymptotic relation ${\ensuremath{\lambda}}_{0}\ensuremath{\sim}\mathcal{B}\phantom{\rule{0.16em}{0ex}}{N}^{\mathcal{C}}$. The case
C=1 corresponds to off-diagonal long-range order. For a one-dimensional homogeneous Tonks-Girardeau gas, a well-known result also confirmed by bosonization gives instead
C=1/2. Here we investigate the inhomogeneous case, initially addressing the behavior of
C in the presence of a general external trapping potential
V. We argue that the value
C=1/2 characterizes the hard-core system independently of the nature of the potential
V. We then define the exponents $\ensuremath{\gamma}$ and $\ensuremath{\beta}$, which describe the scaling of the peak of the momentum distribution with
N and the natural orbital corresponding to ${\ensuremath{\lambda}}_{0}$, respectively, and we derive the scaling relation $\ensuremath{\gamma}+2\ensuremath{\beta}=\mathcal{C}$. Taking as a specific case the power-law potential $V(x)\ensuremath{\propto}{x}^{2n}$, we give analytical formulas for $\ensuremath{\gamma}$ and $\ensuremath{\beta}$ as functions of
n. Analytical predictions for the coefficient
B are also obtained. These formulas are derived by exploiting a recent field theoretical formulation and checked against numerical results. The agreement is excellent.