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Physical Review A

Periodic orbit quantization of a weakly interacting two-body system using perturbed symmetry-broken trace formulas

Jamal Sakhr, Randall S. Dumont, Niall D. Whelan

Published Nov 3, 2006• 1 citation
Publisher

Abstract

The semiclassical limit of the quantum few-body problem has not been studied in general terms from the point of view of periodic orbit theory. In a previous paper, we studied noninteracting two-body systems [Phys. Rev. A 62, 042109 (2000)] and discussed the fact that the periodic orbits occur in continuous families. Interactions destroy the periodic orbit families leaving a discrete set of isolated periodic orbits. In this paper, we consider the effect of weak two-body interactions, which can be thought of as symmetry-breaking perturbations and can thus be analyzed using a theory developed by Creagh [Ann. Phys. (N.Y.) 248, 1 (1996)]. The P\"oschl-Teller two-body system confined in a square well is analyzed to illustrate the use of the formalism. It is shown that the effect of the interaction can be evaluated for all two-particle periodic orbits, and that the coarse-grained quantum density of states can be fully reproduced from simply summing the perturbed contributions of each periodic orbit family. Good numerical estimates of the quantum singlet energies can actually be obtained, but it is found that that perturbed trace formulas cannot reproduce the multiplet splittings predicted from quantum mechanics. Several interesting properties are observed depending on the range of the interaction and on whether the interaction is attractive or repulsive.

DOI
10.1103/physreva.74.052104

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The semiclassical limit of the quantum few-body problem has not been studied in general terms from the point of view of periodic orbit theory. In a previous paper, we studied noninteracting two-body systems [Phys. Rev. A 62, 042109 (2000)] and discussed the fact that the periodic orbits occur in continuous families. Interactions destroy the periodic orbit families leaving a discrete set of isolated periodic orbits. In this paper, we consider the effect of weak two-body interactions, which can be thought of as symmetry-breaking perturbations and can thus be analyzed using a theory developed by Creagh [Ann. Phys. (N.Y.) 248, 1 (1996)]. The P"oschl-Teller two-body system confined in a square well is analyzed to illustrate the use of the formalism. It is shown that the effect of the interaction can be evaluated for all two-particle periodic orbits, and that the coarse-grained quantum density of states can be fully reproduced from simply summing the perturbed contributions of each periodic orbit family. Good numerical estimates of the quantum singlet energies can actually be obtained, but it is found that that perturbed trace formulas cannot reproduce the multiplet splittings predicted from quantum mechanics. Several interesting properties are observed depending on the range of the interaction and on whether the interaction is attractive or repulsive.
Semiclassical periodic orbit theory is used in many branches of physics. However, most applications of the theory have been to systems that involve only single-particle dynamics. In this paper, we develop a semiclassical formalism to describe the density-of-states for two noninteracting particles. This includes accounting properly for particle exchange symmetry. As specific examples, we study two identical particles in a disk and in a cardioid. In each case, we demonstrate that the semiclassical formalism correctly reproduces the quantum densities.
The thesis addresses the fundamental interplay between indistinguishability and interactions in few- and many-body quantum systems, treated in two separate approaches, both analytic in nature, using semiclassical methods. The few-body sector, considering a strictly fixed number of particles, is addressed within the framework of first quantization. By sticking to descriptions as (micro-)canonical ensembles the approach overcomes the inappropriate use of virial expansions. It is based on cluster expansions where quantum statistics are treated exactly and interparticle forces are described non-perturbatively. After a full formal development the cluster expansions are supplemented with short-time dynamical information that is equated with the description of smooth spectral properties or equivalently temperatures above strong quantum degeneracy, which, in the single-particle context, is the essential ingredient to what is known as semiclassical Weyl expansions. This reduction allows for the fully analytical description of universal features of large classes of systems by either truncating to a specific class of allowed collision processes, i.e., two-body collisions, or by extracting higher-order interaction kernels from related integrable models. The consistent combination with scaling considerations, minimal ground-state information and strong-coupling expansions leads to a toolbox providing most of the thermodynamic and spectral properties of one-dimensional systems with short-range interactions, all in terms of the lowest-order interaction kernels. The analytical results are in excellent qualitative and quantitative agreement with extensive numerical simulations for both integrable and non-integrable models of experimental relevance in cold atom physics. The many-body sector is in contrast treated in the second quantized formulation of quantum mechanics making use of the well-known collection of semiclassical methods that are subsumed under the name periodic orbit theory. As these methods apply in a highly system specific manner, the attractive Bose gas on a ring is chosen as exemplary model to show a case where the semiclassical quantization of mean-field dynamics, here considered as the classical analogue, can be used to obtain analytical non-trivial information on correlation-related many-body effects, especially related to quantum critical behavior in this model. After a simplification by truncating to the lowest single-particle momentum modes the consistent application of Einstein-Brillouin-Keller quantization yields a description valid for large numbers of bosons. Besides analytically reproducing numerical spectra to high accuracy, the approach enables one to extract asymptotic scaling laws for level spacings around quantum criticality, representing properties. These (finite) energy differences are intrinsically tied to inter-particle correlations and therefore elude a more conventional description by means of a mean-field theory extended with Bogoliubov quasi particles. An application to the scrambling time of the system, related to a pseudo quantum butterfly effect that stems from a local instability, underlines the importance of the method in the otherwise inaccessible regime of extreme particle numbers.
In this article we extend previous semiclassical studies by including more general perturbative potentials of the harmonic oscillator in arbitrary spatial dimensions. Our starting point is a radial harmonic potential with an arbitrary even monomial perturbation, which we use to study the resulting U(D)\mathrm{U}(D)U(D) to O(D)\mathrm{O}(D)O(D) symmetry breaking. We derive the gross structure of the semiclassical spectrum from periodic orbit theory, in the form of a perturbative (ℏ→0\hbar \rightarrow 0ℏ→0) trace formula. We then show how to apply the results to even order polynomial potentials, possibly including mean-field terms. We have drawn the conclusion that the gross structure of the quantum spectrum is determined from only classical circular- and diameter-orbits for this class of systems.
Abstract One of the fundamental results of semiclassical theory is the existence of trace formulae showing how spectra of quantum mechanical systems emerge from massive interference among amplitudes related with time-periodic structures of the corresponding classical limit. If it displays the properties of Hamiltonian integrability, this connection is given by the celebrated Berry–Tabor trace formula, and the periodic structures it is built on are tori supporting closed trajectories in phase space. Here we show how to extend this connection into the domain of quantum many-body systems displaying integrability in the sense of the Bethe ansatz, where a classical limit cannot be rigorously defined due to the presence of singular potentials. Formally following the original derivation of Berry and Tabor (1976 Proc. R. Soc. A 349 101), but applied to the Bethe equations without underlying classical structure, we obtain a many-particle trace formula for the density of states of N interacting bosons on a ring, the Lieb–Liniger model. Our semiclassical expressions are in excellent agreement with quantum mechanical results for <?CDATA N=2,3N = 2,3N=2,3?> N = 2 , 3 and 4 particles. For N = 2 we relate our results to the quantization of billiards with mixed boundary conditions. Our work paves the way towards the treatment of the important class of integrable many-body systems by means of semiclassical trace formulae pioneered by Michael Berry in the single-particle context.
We consider the many-body spectra of interacting bosonic quantum fields on a lattice in the semiclassical limit of large particle number N. We show that the many-body density of states can be expressed as a coherent sum over oscillating long-wavelength contributions given by periodic, nonperturbative solutions of the, typically nonlinear, wave equation of the classical (mean-field) limit. To this end, we construct the semiclassical approximation for both the smooth and oscillatory parts of the many-body density of states in terms of a trace formula starting from the exact path integral form of the propagator between many-body quadrature states. We therefore avoid the use of a complexified classical limit characteristic of the coherent state representation. While quantum effects such as vacuum fluctuations and gauge invariance are exactly accounted for, our semiclassical approach captures quantum interference and therefore is valid well beyond the Ehrenfest time where naive quantum-classical correspondence breaks down. Remarkably, due to a special feature of harmonic systems with incommensurable frequencies, our formulas are generically valid also in the free-field case of noninteracting bosons.
For the periodic orbit quantization of a system of particles confined in a two-dimensional disk billiard, the proliferation of multiparticle periodic orbits with length precludes standard length truncation of the trace formula and alternative truncation schemes are necessary. An amplitude ordering of the single-particle periodic orbits reveals a quantum-classical correspondence for the angular momentum of a particle in the disk which is useful for the quantization of all the low energy states of the multiparticle problem. We explicitly show that a simple amplitude truncation of the trace formula better approximates the semiclassical density of states and in particular resolves closely spaced levels that are otherwise unresolved using the standard approach.