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arXiv:cond-mat/0402080

Quantized Hydrodynamic Theory of One-Dimensional Hard Core Bosons

Tarun Kanti Ghosh

Published Feb 3, 2004• 1 citation
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Abstract

We present a quantized hydrodynamic theory and its applications of one-dimensional hard-core bosons in a harmonic trap. Quantizing the Hamiltonian of a trapped hard-core bosons and diagonalize it in terms of the phase and density fluctuations associated with the Bose field operator. As an applications, we calculate discrete energy spectrum, dynamic structure factor, momentum transferred by a two-photon Bragg pulse, single-particle density matrix, momentum distribution, and two-particle correlation function. The dynamic structure factor has multiple peaks due to the discrete nature of the energy spectrum which can be observed by a two-photon Bragg pulse with long duration. The coherence length at zero temperature is very small due to the "fermionization" of the system. We also compute the momentum distributions which has oscillatory behavior at large momentum. The pair distribution function shows that there are many deep valleys at various relative separations which implies shell structure due to the Pauli blocking in real space.

DOI
10.48550/arxiv.cond-mat/0402080

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We present a quantized hydrodynamic theory and its applications of one-dimensional hard-core bosons in a harmonic trap. Quantizing the Hamiltonian of a trapped hard-core bosons and diagonalize it in terms of the phase and density fluctuations associated with the Bose field operator. As an applications, we calculate discrete energy spectrum, dynamic structure factor, momentum transferred by a two-photon Bragg pulse, single-particle density matrix, momentum distribution, and two-particle correlation function. The dynamic structure factor has multiple peaks due to the discrete nature of the energy spectrum which can be observed by a two-photon Bragg pulse with long duration. The coherence length at zero temperature is very small due to the "fermionization" of the system. We also compute the momentum distributions which has oscillatory behavior at large momentum. The pair distribution function shows that there are many deep valleys at various relative separations which implies shell structure due to the Pauli blocking in real space.
We analytically study effects of density and phase fluctuations of quasi-one-dimensional degenerate atomic Bose gases in the mean-field as well as in the hard-core bosons regimes. We obtain the analytic expressions for dynamic structure factors in both the regimes. We also calculate single-particle density matrix and momentum distribution by taking care of the phase fluctuations upto fourth-order term, the density fluctuations as well as the non-condensate density in both the regimes. In the mean-field regime, there is a long-tail in the momentum distributions at large temperature, which can be used to identify the quasi-condensate from a pure condensate. The single-particle correlation functions of hard-core bosons is almost zero even at zero temperature due to the fermionization of the bosonic systems. Two-particle correlation function in the hard-core bosons regime shows many deep valleys at various relative separations. These valleys at various relative separations imply shell structure due to the Pauli blocking in real space.
The one-particle density matrices for hard-core bosons in a one-dimensional harmonic trap are computed numerically for systems with up to 160 bosons. Diagonalization of the density matrix shows that the many-body ground state is not Bose-Einstein condensed. The ground-state occupation, the amplitude of the lowest natural orbital, and the zero momentum peak height scale as powers of the particle number, and the corresponding exponents are related to each other. Close to its diagonal, the density matrix for hard-core bosons is similar to that of noninteracting fermions.
We calculate the static and dynamic single-particle correlation functions in one-dimensional (1D) trapped Bose gases and discuss experimental measurements that can directly probe such correlation functions. Using a quantized hydrodynamic theory for the low-energy excitations, we calculate both the static and dynamic single-particle correlation functions for a 1D Bose gas that is a phase-fluctuating quasicondensate. For the static (equal-time) correlation function, our approximations and results are equivalent to those of Petrov, Shlyapnikov, and Walraven. The Fourier transform of the static single-particle correlation function gives the momentum distribution, which can be measured using Doppler-sensitive Bragg scattering experiments on a highly elongated Bose gas. We show how a two-photon Raman out-coupling experiment can measure the characteristic features of the dynamic or time-dependent single-particle correlation function of a 1D Bose quasicondensate.
Abstract For a harmonically trapped system consisting of two bosons in one spatial dimension with infinite contact repulsion (hard core bosons), we derive an expression for the one-body density matrix ρB\rho _\mathrm{B}ρB​ ρ B in terms of center of mass and relative coordinates of the particles. The deviation from ρF\rho _\mathrm{F}ρF​ ρ F , the density matrix for the two fermions case, can be clearly identified. Moreover, the obtained ρB\rho _\mathrm{B}ρB​ ρ B allows us to derive a closed form expression of the corresponding momentum distribution nB(p)n_\mathrm{B}(p)nB​(p) n B ( p ) . We show how the result deviates from the noninteracting fermionic case, the deviation being associated with the short-range character of the interaction. Mathematically, our analytical momentum distribution is expressed in terms of one and two variables confluent hypergeometric functions. Our formula satisfies the correct normalization and possesses the expected behavior at zero momentum. It also exhibits the high momentum 1/p41/p^4 1/p4 1 / p 4 tail with the appropriate Tan’s coefficient. Numerical results support our findings.
Hydrodynamization refers to the remarkably rapid process in relativistic heavy-ion collisions by which hydrodynamic descriptions become applicable. Following the observation of analogous behavior in ultracold one-dimensional (1D) Bose gases, hydrodynamization has been conjectured to be a universal dynamical phenomenon in quantum systems following high-energy quenches. Theoretical studies in this cold-atom setting have so far been restricted to quenches from ground states. Here we study how nonzero temperatures affect hydrodynamization. Specifically, using a homogeneous 1D gas of hard-core bosons, we explore how the initial temperature affects the timescales associated with hydrodynamization and prethermalization following a Bragg-pulse quench. We find that while the hydrodynamization coherence time remains unchanged, increasing temperature shortens both the damping time of the hydrodynamization oscillations and the prethermalization time. We argue that this is mainly the result of the broadening of the initial rapidity distribution, and introduce a nonzero-temperature dephasing time defined in terms of the extent of the rapidity distribution.
Using the exact N-particle ground state wave function for a one-dimensional gas of hard-core bosons in a harmonic trap we develop an algorithm to compute the reduced single-particle density matrix and corresponding momentum distribution. Accurate numerical results are presented for up to N=8N=8N=8 particles, and the momentum distributions are compared to a recent analytic approximation.