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arXiv:0707.2196 · Physical Review A

Single-particle density matrix and the momentum distribution of dark “solitons” in a Tonks-Girardeau gas

Hrvoje Buljan, Karlo Lelas, Robert Pezer, Marinko Jablan

Published Oct 9, 2007• 5 citations
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Abstract

We study the reduced single-particle density matrix (RSPDM), the momentum distribution, and natural orbitals and their occupancies of dark ``soliton'' (DS) states in a Tonks-Girardeau gas. DS states are specially tailored excited many-body eigenstates, which have a dark solitonic notch in their single-particle density. The momentum distribution of DS states has a characteristic shape with two sharp spikes. We find that the two spikes arise due to the high degree of correlation observed within the RSPDM between the mirror points ($x$ and $\ensuremath{-}x$) with respect to the dark notch at $x=0$; the correlations oscillate rather than decay as the points $x$ and $\ensuremath{-}x$ are being separated.

DOI
10.1103/physreva.76.043609

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We study the reduced single-particle density matrix (RSPDM), the momentum distribution, and natural orbitals and their occupancies of dark ``soliton'' (DS) states in a Tonks-Girardeau gas. DS states are specially tailored excited many-body eigenstates, which have a dark solitonic notch in their single-particle density. The momentum distribution of DS states has a characteristic shape with two sharp spikes. We find that the two spikes arise due to the high degree of correlation observed within the RSPDM between the mirror points (xxx and \ensuremath−x\ensuremath{-}x\ensuremath−x) with respect to the dark notch at x=0x=0x=0; the correlations oscillate rather than decay as the points xxx and \ensuremath−x\ensuremath{-}x\ensuremath−x are being separated.
The momentum distribution (MD) dynamics of a Tonks-Girardeau (TG) gas is studied in the context of Bragg reflections of a many-body wave packet. We find strong suppression of a Bragg reflection peak for a large and dense TG wave packet; our observation illustrates the dependence of the MD on the interactions and wave function symmetry. The MD is calculated from the reduced single-particle density matrix (RSPDM). We develop a method for calculating the RSPDM of a TG gas, which is operative for a large number of particles, and does not depend on the external potential and the state of the system. The method is based on a formula expressing the RSPDM via a dynamically evolving single-particle basis.
The relations among the occupation number of the lowest natural orbital (ONLNO), momentum distributions (MD) and off-diagonal long-range element (ODLRE) of the reduced single-particle density matrix (RSPDM) are studied while Tonks-Girardeau gas in one dimensional periodic potential is in the ground state. For NNN-body systems of large enough, RSPDM and its lowest natural orbital do not vary with NNN in overlapped areas in commensurate and incommensurate cases correspondingly. In commensurate case, the ODLRE is exponential attenuation with NNN, which results in that the ONLNO and MD are invariant with NNN. While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to N\sqrt{N}N​, which results in the different behavior of the ONLNO and MD.
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 NNN, 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 NNN, 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.
In momentum space, we investigate the correlation properties of the ground state of Tonks–Girardeau gases. With Bose–Fermi mapping method, the exact ground state wavefunction in coordinate space can be obtained based on the wavefunction of spin-polarized Fermions. By Fourier transformation we obtain the ground state wavefunction in momentum space, and therefore the pair correlation and the reduced one-body density matrix (ROBDM) in momentum space, whose diagonal part is the momentum distribution. The ROBDM in momentum space is the Fourier transformation of the ROBDM in coordinate space and the pair correlation in momentum space is the Fourier transformation of the reduced two-body density matrix in coordinate space. The correlations in momentum space display larger values only in small momentum region and vanish in most other regions. The lowest natural orbital and occupation distribution in momentum space are also obtained.
We study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the N-particle and (N−1)-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations, we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons.