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arXiv:cond-mat/0211126 · Physical Review A

Finite one-dimensional impenetrable Bose systems: Occupation numbers

Peter J. Forrester, Norman E. Frankel, Timothy M. Garoni, Nicholas S. Witte

Published Apr 17, 2003• 139 citations
arXivHTMLPDF

Abstract

Bosons in the form of ultracold alkali-metal atoms can be confined to a one-dimensional (1D) domain by the use of harmonic traps. This motivates the study of the ground-state occupations ${\ensuremath{\lambda}}_{i}$ of effective single-particle states ${\ensuremath{\varphi}}_{i},$ in the theoretical 1D impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The ${\ensuremath{\lambda}}_{i}$ and ${\ensuremath{\varphi}}_{i}$ are the eigenvalues and eigenfunctions, respectively, of the one-body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed i the occupations ${\ensuremath{\lambda}}_{i}$ are asymptotically proportional to $\sqrt{N}$ in both the circular and harmonically trapped cases.

DOI
10.1103/physreva.67.043607

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Bosons in the form of ultracold alkali-metal atoms can be confined to a one-dimensional (1D) domain by the use of harmonic traps. This motivates the study of the ground-state occupations \ensuremathλi{\ensuremath{\lambda}}_{i}\ensuremathλi​ of effective single-particle states \ensuremathφi,{\ensuremath{\varphi}}_{i},\ensuremathφi​, in the theoretical 1D impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The \ensuremathλi{\ensuremath{\lambda}}_{i}\ensuremathλi​ and \ensuremathφi{\ensuremath{\varphi}}_{i}\ensuremathφi​ are the eigenvalues and eigenfunctions, respectively, of the one-body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed i the occupations \ensuremathλi{\ensuremath{\lambda}}_{i}\ensuremathλi​ are asymptotically proportional to N\sqrt{N}N​ in both the circular and harmonically trapped cases.
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.
The exact N-particle ground-state wave function for a one-dimensional condensate of hard-core bosons in a harmonic trap is employed to obtain accurate numerical results for the one-particle density matrix, occupation number distribution of the natural orbitals, and momentum distribution. Our results show that the occupation of the lowest orbital varies as N0.59,{N}^{0.59},N0.59, in contrast to N0.5{N}^{0.5}N0.5 for a spatially uniform system, and N for a true Bose-Einstein condensate.
The main work of this thesis involves the calculation, using the Bethe ansatz, of two of the signature quantities of the one-dimensional delta-function Bose gas. These are the density matrix and concomitantly its Fourier transform the occupation numbers, and the correlation function and concomitantly its Fourier transform the structure factor. The coefficient of the delta-function is called the coupling constant; these quantities are calculated in the finite-coupling regime, both expansions around zero coupling and infinite coupling are considered. Further to this, the density matrix in the infinite coupling limit, and its first order correction, is recast into Toeplitz determinant form. From this the occupation numbers are calculated up to 36 particles for the ground state and up to 26 particles for the first and second excited states. This data is used to fit the coefficients of an ansatz for the occupation numbers. The correlation function in the infinite coupling limit, and its first order correction, is recast into a form which is easy to calculate for any N, and is determined explicitly in the thermodynamic limit.
A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue λ 0 of the one-body-density matrix scales as λ 0 ∼ N , where N is the total number of particles.Putting λ 0 ∼ N C to define the scaling exponent C, then C = 1 corresponds to ODLRO and C = 0 to the single-particle occupation of the density matrix orbitals.When 0 < C < 1, C can be used to quantify deviations from ODLRO.In this paper we study the exponent C in a variety of one-dimensional bosonic and anyonic quantum systems at T = 0.For the 1D Lieb-Liniger Bose gas we find that for small interactions C is close to 1, implying a mesoscopic condensation, i.e. a value of the zero temperature "condensate" fraction λ 0 /N appreciable at finite values of N (as the ones in experiments with 1D ultracold atoms).1D anyons provide the possibility to fully interpolate between C = 1 and 0. The behaviour of C for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.[2] A. Colcelli, J. Viti, G. Mussardo, and A. Trombettoni, Universal off-diagonal longrange-order behavior for a trapped Tonks-Girardeau gas, Phys.Rev. A 98, 063633 (2018).
We present a simulation study of the properties of the Tonks-Girardeau gas, a system of NNN 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 NNN-body wave function gives rise to a set of stochastic differential equations for the positions of the NNN 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 NNN 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.
We describe the properties of a pair of ultracold bosonic atoms in a one-dimensional harmonic trapping potential with a tunable zero-ranged barrier at the trap center. The full characterization of the ground state is done by calculating the reduced single-particle density, the momentum distribution, and the two-particle entanglement. We derive several analytical expressions in the limit of infinite repulsion (Tonks-Girardeau limit) and extend the treatment to finite interparticle interactions by numerical solution. As pair interactions in double wells form a fundamental building block for many-body systems in periodic potentials, our results have implications for a wide range of problems.