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arXiv:1504.01620 · New Journal of Physics

Exact quantum decay of an interacting many-particle system: the Calogero–Sutherland model

Adolfo del Campo

Published Jan 19, 2016• 49 citations
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Abstract

The exact quantum decay of a one-dimensional Bose gas with inverse-square interactions is presented. The system is equivalent to a gas of particles obeying generalized exclusion statistics. We consider the expansion dynamics of a cloud initially confined in a harmonic trap that is suddenly switched off. The decay is characterized by analyzing the fidelity between the initial and the time-evolving states, also known as the survival probability. It exhibits early on a quadratic dependence on time that turns into a power-law decay, during the course of the evolution. It is shown that the particle number and the strength of interactions determine the power-law exponent in the latter regime, as recently conjectured. The nonexponential character of the decay is linked to the many-particle reconstruction of the initial state from the decaying products.

DOI
10.1088/1367-2630/18/1/015014

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The exact quantum decay of a one-dimensional Bose gas with inverse-square interactions is presented. The system is equivalent to a gas of particles obeying generalized exclusion statistics. We consider the expansion dynamics of a cloud initially confined in a harmonic trap that is suddenly switched off. The decay is characterized by analyzing the fidelity between the initial and the time-evolving states, also known as the survival probability. It exhibits early on a quadratic dependence on time that turns into a power-law decay, during the course of the evolution. It is shown that the particle number and the strength of interactions determine the power-law exponent in the latter regime, as recently conjectured. The nonexponential character of the decay is linked to the many-particle reconstruction of the initial state from the decaying products.
While exponential decay is ubiquitous in nature, deviations at both short and long times are dictated by quantum mechanics. Nonexponential decay is known to arise due to the possibility of reconstructing the initial state from the decaying products. We discuss the quantum decay dynamics by tunneling of a many-particle system, characterizing the long-time nonexponential behavior of the nonescape and survival probabilities. The effects of contact interactions and quantum statistics are described. It is found that, whereas for noninteracting bosons the long-time decay follows a power law with an exponent linear in the number of particles NNN, the exponent becomes quadratic in NNN in the fermionic case. The same results apply to strongly interacting many-body systems related by the generalized Bose-Fermi duality. The faster fermionic decay can be traced back to the effective hard-core interactions between particles, which are as well the decaying products, and exhibit spatial antibunching which hinders the reconstruction of the initial unstable state. The results are illustrated with a paradigmatic model of quantum decay from a trap allowing leaks by tunneling, whose dynamics is described exactly by means of an expansion in resonant states.
One-dimensional fractional statistics is studied using the Calogero-Sutherland model (CSM) which describes a system of non-relativistic quantum particles interacting with inverse-square two-body potential on a ring.The inversesquare exchange can be regarded as a pure statistical interaction and this system can be mapped to an ideal gas obeying the fractional exclusion and exchange statistics.The details of the exact calculations of the dynamical correlation functions for this ideal system is presented in this paper.An effective low-energy one-dimensional "anyon" model is constructed; and its correlation functions are found to be in agreement with those in the CSM; and this agreement provides an evidence for the equivalence of the first-and the second-quantized construction of the 1D anyon model at least in the long wave-length limit.Furthermore, the finite-size scaling applicable to the conformally invariant systems is used to obtain the complete set of correlation exponents for the CSM.
We show that the Calogero-Sutherland (C-S) gas, a famous exact soluble one-dimensional system with an inverse square long range interaction, can be realized by dimension reduction in a cold Bose atom system with a dipole-dipole interaction. Depending on the orientation of the dipoles, the effective interaction is either attractive or repulsive. The low-lying effective theory may be a Luttinger liquid when the exclusion statistics parameter λλλ may be well-defined. We hope that the C-S gas can be realized experimentally and the Luttinger liquid character can be observed.
We consider N particles interacting pair-wise by an inverse square potential in one dimension (Calogero-Sutherland-Moser model). When trapped harmonically, its classical canonical partition function for the repulsive regime is known in the literature. We start by presenting a concise re-derivation of this result. The equation of state is then calculated both for the trapped and the homogeneous gas. Finally, the classical limit of Wu’s distribution function for fractional exclusion statistics is obtained and we re-derive the classical virial expansion of the homogeneous gas using this distribution function. PACS numbers: 03.65.Sq, 05.30.Pr Submitted to: J. Phys. A: Math. Gen. The virial expansion of a classical interacting system 2
We consider N particles interacting pairwise by an inverse square potential in one dimension (Calogero–Sutherland–Moser model). For a system placed in a harmonic trap, its classical partition function for the repulsive regime is recognised in the literature. We start by presenting a concise re-derivation of this result. The equation of state is then calculated both for the trapped and the homogeneous gas. Finally, the classical limit of Wu's distribution function for fractional exclusion statistics is obtained and we re-derive the classical virial expansion of the homogeneous gas using this distribution function.
We consider a quantum quench of the trap frequency in a system of bosons interacting through an inverse-square potential and confined in a harmonic trap (the harmonic Calogero model). We determine exactly the initial state in terms of the postquench eigenstates and derive the time evolution of simple physical observables. Since this model possesses an infinite set of integrals of motion that allow its exact solution, a generalized Gibbs ensemble (GGE), i.e., a statistical ensemble that takes into account the conservation of all integrals of motion, can be proposed in order to describe the values of local physical observables long after the quench. Even though, due to the presence of the trap, physical observables do not exhibit equilibration but periodic evolution, such a GGE may still describe correctly their time-averaged values. We check this analytically for the local boson density and find that the GGE conjecture is indeed valid, in the thermodynamic limit.