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arXiv:cond-mat/0612376 · Physical Review A

Expansion of a mesoscopic Fermi system from a harmonic trap

Pavel Nagornykh, Victor M. Galitski

Published Jun 6, 2007• 3 citations
arXivHTMLPDF

Abstract

We study the dynamics of time-of-flight expansion of an atomic Fermi system with a finite number of particles $N$ after it is released from a harmonic trapping potential. We consider two different initial states: the Fermi sea state and the paired state. In the former case, we derive exact and simple analytic expressions for the dynamics of the particle density and density-density correlation functions, taking into account the level quantization and possible anisotropy of the trap. To describe the paired state, we use the projection of the grand-canonical BCS wave function onto the subspace with a fixed number of particles, and obtain analytic expressions for the density and its correlators in the leading order with respect to the ratio of the trap frequency and the superconducting gap (the ratio is assumed small). We discuss several dynamic features, such as the time evolution of the peak due to pair correlations, which may be used to distinguish between the Fermi sea and the paired state.

DOI
10.1103/physreva.75.065601

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We study the dynamics of time-of-flight expansion of an atomic Fermi system with a finite number of particles NNN after it is released from a harmonic trapping potential. We consider two different initial states: the Fermi sea state and the paired state. In the former case, we derive exact and simple analytic expressions for the dynamics of the particle density and density-density correlation functions, taking into account the level quantization and possible anisotropy of the trap. To describe the paired state, we use the projection of the grand-canonical BCS wave function onto the subspace with a fixed number of particles, and obtain analytic expressions for the density and its correlators in the leading order with respect to the ratio of the trap frequency and the superconducting gap (the ratio is assumed small). We discuss several dynamic features, such as the time evolution of the peak due to pair correlations, which may be used to distinguish between the Fermi sea and the paired state.
This work deals with the key constituent behind the existence of superfluid states in ultracold fermionic gases confined in a harmonic trap in 2D, namely, the formation of Cooper pairs in the presence of a Fermi sea in inhomogeneous confinement. For a set of finite-range models representing particle–particle interaction, we first ascertain the simultaneity of the emergence of bound states and the divergence of the s-wave scattering length in 2D as a function of the interaction potential parameters in free space. Then, through the analysis of two particles interacting in 2D harmonic confinement, we evaluate the energy shift with respect to the discrete harmonic oscillator levels for both repulsive and attractive cases. All of these results are the basis for determining the energy gaps of Cooper pairs arising from two particles interacting in the presence of a Fermi sea consisting of particles immersed in a 2D harmonic trap.
We investigate the entanglement properties of the nonequilibrium dynamics of one-dimensional noninteracting Fermi gases released from a trap. The gas of NNN particles is initially in the ground state within a hard-wall or a harmonic trap, and then it expands after dropping the trap. We compute the time dependence of the von Neumann and R\'enyi entanglement entropies and the particle fluctuations of spatial intervals around the original trap in the limit of a large number NNN of particles. The results for these observables apply to one-dimensional gases of impenetrable bosons as well. We identify different dynamical regimes at small and large times, depending also on the initial condition, i.e., whether it is that of a hard-wall or a harmonic trap. In particular, we show analytically that the expansion from hard-wall traps is characterized by the asymptotic small-time behavior $S\ensuremath{\approx}(1/3)\mathrm{ln}(1/t)$ of the von Neumann entanglement entropy and the relation $S\ensuremath{\approx}{\ensuremath{\pi}}^{2}V/3$ (where VVV is the particle variance), which are analogous to the equilibrium behaviors whose leading logarithms are essentially determined by the corresponding conformal field theory with central charge c=1c=1c=1. The time dependence of the entanglement entropy of extended regions during the expansion from harmonic traps has a remarkable property in that it can be expressed as a global time-dependent rescaling of the space dependence of the initial equilibrium entanglement entropy.
One-particle properties of noninteracting fermions in a one-dimensional harmonic trap and at zero temperature are studied. Exact expressions and asymptotic results for a large fermion number N are given for the particle density distribution n0(z,N).{n}_{0}(z,N).n0​(z,N). For large N and near the classical boundary at the Fermi energy, the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of the density distribution is calculated exactly. It displays a small but characteristic hump near 2kF,{2k}_{F},2kF​, with kF{k}_{F}kF​ being a properly defined Fermi wave number. This is due to Friedel oscillations, which are identified and discussed. These quantum effects are missing in the semiclassical approximation. Momentum distributions are also evaluated and discussed. As an example of a time-dependent one-particle problem, we calculate exactly the evolution of the particle density when the trap is suddenly switched off, and find a simple scaling behavior, in agreement with recent general results.
Pairing lies at the heart of superfluidity in fermionic systems. Motivated by recent experiments in mesoscopic Fermi gases, we study up to six fermionic atoms with equal masses and equal populations in two different spin states, confined in a quasi-two-dimensional harmonic trap. We couple a stochastic variational approach with the use of an explicitly correlated Gaussian basis set, which enables us to obtain highly accurate energies and structural properties. Utilising two-dimensional two-body scattering theory with a finite-range Gaussian interaction potential, we tune the effective range to model realistic quasi-two-dimensional scattering. We calculate the excitation spectrum, pair correlation function, and number of pairs as a function of increasing attractive interaction strength. For up to six fermions in the ground state, we find that opposite spin and momentum pairing is maximised well below the Fermi surface in momentum space. By contrast, corresponding experiments on twelve fermions have found that pairing is maximal at the Fermi surface and strongly suppressed beneath [M. Holten et al., Nature 606, 287-291 (2022)]. This suggests that the Fermi sea — which acts to suppress pairing at low momenta via Pauli blocking — emerges in the transition from six to twelve particles.
We study the expansion of an interacting atomic system at zero temperature, following its release from an isotropic three-dimensional harmonic trap and calculate the time dependence of its density and momentum distribution, with special focus on the behavior of the contact parameter. We consider different quantum systems, including the unitary Fermi gas of infinite scattering length, the weakly interacting Bose gas, and two interacting particles with highly asymmetric mass imbalance. In all cases analytic results can be obtained, which show that the initial value of the contact, fixing the 1/k41/{k}^{4}1/k4 tail of the momentum distribution, disappears for large expansion times. Our results raise the problem of understanding the recent experiment of R. Chang et al. [Phys. Rev. Lett. 117, 235303 (2016)] carried out on a weakly interacting Bose gas of metastable 4He^{4}\mathrm{He}4He atoms, where a 1/r41/{r}^{4}1/r4 tail in the density distribution was observed after a large expansion time, implying the existence of the 1/k41/{k}^{4}1/k4 tail in the asymptotic momentum distribution.