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arXiv:1006.0198 · New Journal of Physics

An eccentrically perturbed Tonks–Girardeau gas

John Goold, M Krych, Zbigniew Idziaszek, Thomás Fogarty, Thomas Busch

Published Sep 24, 2010• 17 citations

Abstract

We investigate the static and dynamic properties of a Tonks–Girardeau gas in a harmonic trap with an eccentric δ-perturbation of variable strength. For this, we first find the analytic eigensolution of the single particle problem and then use this solution to calculate the spatial density and energy profiles of the many-particle gas as a function of the strength and position of the perturbation. We find that the crystal nature of the Tonks state is reflected in both the lowest occupation number and the momentum distribution of the gas. As a novel application of our model, we study the time evolution of the spatial density after the sudden removal of the perturbation. The dynamics exhibits collapses and revivals of the original density distribution, which occur in units of the trap frequency. This is reminiscent of the Talbot effect from classical optics.

DOI: 10.1088/1367-2630/12/9/093041

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We investigate the static and dynamic properties of a Tonks–Girardeau gas in a harmonic trap with an eccentric δ-perturbation of variable strength. For this, we first find the analytic eigensolution of the single particle problem and then use this solution to calculate the spatial density and energy profiles of the many-particle gas as a function of the strength and position of the perturbation. We find that the crystal nature of the Tonks state is reflected in both the lowest occupation number and the momentum distribution of the gas. As a novel application of our model, we study the time evolution of the spatial density after the sudden removal of the perturbation. The dynamics exhibits collapses and revivals of the original density distribution, which occur in units of the trap frequency. This is reminiscent of the Talbot effect from classical optics.

We determine the exact many-body properties of a bosonic Tonks-Girardeau gas confined in a harmonic potential with a tunable $\ensuremath{\delta}$-function barrier at the trap center. This is done by calculating the reduced single-particle density matrix, the pair-distribution function, and the momentum distribution of the gas as a function of barrier strength and particle number. With increasing barrier height we find that the ground-state occupation in a diagonal basis diverges from the

\sqrt{N}

behavior that is expected for the case of a simple harmonic trap. In fact, the scaling of the occupation number depends on whether one has an even or odd number of particles. Since this quantity is a measure of the coherence of our sample we show how the odd-even effect manifests itself in both the momentum distribution of the Bose gas and interference fringe visibility during free temporal evolution.

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 In this paper we study the dynamics of Tonks–Girardeau (TG) gases in a harmonic potential driven by a Gaussian pulse, which is in correspondence with the excitation dynamics of electrons in matter driven by an ultrashort laser pulse. The evolving dynamics of the TG gas are obtained with a Bose–Fermi mapping method combined with numerical techniques. We calculate the evolving dynamics of the occupation distribution of single-particle energy levels, the density distribution and the momentum distribution of the system. It is shown that the system arrives at a dynamically stable state at the end of driving. In the high-frequency regime, TG gases return back to the ground state, while in the low-frequency regime the population inversion exhibits that all atoms occupy high levels.

Using a scaling transformation we exactly determine the dynamics of an harmonically confined Tonks-Girardeau gas under arbitrary time variations of the trap frequency. We show how during a one-dimensional expansion a ``dynamical fermionization'' occurs as the momentum distribution rapidly approaches an ideal Fermi gas distribution, and that under a sudden change of the trap frequency the gas undergoes undamped breathing oscillations displaying alternating bosonic and fermionic character in momentum space. The absence of damping in the oscillations is a peculiarity of the truly Tonks regime.