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arXiv:1301.5459 · Physical Review A

Revival times at quantum phase transitions

F. de los Santos, E. Romera

Published Jan 22, 2013• 18 citations
arXivHTMLPDF

Abstract

The concept of quantum revivals is extended to many-body systems and the implications of traversing a quantum phase transition are explored. By analyzing two different models, the vibron model for the bending of polyatomic molecules and the Dicke model for a quantum radiation field interacting with a system of two-level atoms, we show evidence of revival behavior for wave packets centered around energy levels as low as the fundamental state. Away from criticality, revival times exhibit smooth, nonsingular behavior and are proportional to the system size. Upon approaching a quantum critical point, they diverge as a power law and scale with the system size, although the scaling is no longer linear.

DOI
10.1103/physreva.87.013424

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The concept of quantum revivals is extended to many-body systems and the implications of traversing a quantum phase transition are explored. By analyzing two different models, the vibron model for the bending of polyatomic molecules and the Dicke model for a quantum radiation field interacting with a system of two-level atoms, we show evidence of revival behavior for wave packets centered around energy levels as low as the fundamental state. Away from criticality, revival times exhibit smooth, nonsingular behavior and are proportional to the system size. Upon approaching a quantum critical point, they diverge as a power law and scale with the system size, although the scaling is no longer linear.
We report on quantum revivals and classical periodicities of wave packets centered around the ground state within the Lipkin-Meshkov-Glick model. Special attention is paid to the behavior at first- and second-order phase transitions. In line with previous studies, we find that away from criticality, characteristic times exhibit smooth, nonsingular behavior, but upon approaching the transition points they diverge as power laws with associated critical exponents. Finite-size effects are studied and the observed phenomenology is discussed in the framework of the time-energy uncertainty relation.
The harmonic oscillator and the infinite square well are two of the simplest one-dimensional quantum systems which exhibit wave packet revivals (trivially so in the case of the oscillator.) These two potentials can be thought of as special cases of the general one-dimensional power-law potential given by V(k)(x)=V0|x/a|k (with k=2 and k→∞ for the oscillator and square well, respectively.) Using an autocorrelation function approach and the WKB approximation for the quantized energy levels in such potentials, we exhibit numerical evidence for wave packet revivals in the case of arbitrary k>0 which agree well with more analytic results. We derive expressions for the revival and collapse time scales in terms of the physical quantities of the system as well as the wave packet parameters. We find that both times scale with the power-law exponent k as |(k+2)/(k−2)|. In this way, we can explicitly exhibit the approach to the two familiar limiting cases. We also briefly consider the case of a “half” well where an infinite wall in added at the origin.
We examine the dynamical evolution of wave packets in a large class of quantum systems where two quantum numbers determine the energy spectrum and consequently the dynamical behavior. Using a generic Hamiltonian, we study the formation of coherent structures. The ratio of two time scales as well as the symmetry of the initial wave packet decisively determines the revival and fractional revivals of the system for which we give analytical results. The theory is applicable to a wide class of systems from diatomic molecules to ions in two-dimensional traps and two interacting Bose condensates.
The quantum phase transition in the U(2) vibron model is investigated in large-N limit with the coherent state method and for finite N in a numerical scheme Through analyzing the potential structure and the corresponding low-lying spectrum, it is found that the dynamical characters at the critical point can be approximately described by the E(1) critical point symmetry, but such an approximation becomes poor with the increasing of the excitation energy. In addition, the scaling behavior of the spectrum at the critical point has been also investigated in a numerical way. The results indicate that the scaling law of the spectrum at the critical point of the 2nd order phase transition may be independent of the dimension of system.
We present a generic treatment of wave-packet revivals for quantum-mechanical systems. This treatment permits a classification of certain ideal revival types. For example, wave packets for a particle in a one-dimensional box are shown to exhibit perfect revivals. We also examine the revival structure of wave packets for quantum systems with energies that depend on two quantum numbers. Wave packets in these systems exhibit quantum beats in the initial motion as well as new types of long-term revivals. As an example, we consider the revival structure of a particle in a two-dimensional box.
Abstract The out‐of‐time‐order correlators (OTOCs) is used to study the quantum phase transitions (QPTs) between the normal phase and the superradiant phase in the Rabi and few‐body Dicke models with large frequency ratio of the atomic level splitting to the single‐mode electromagnetic radiation field frequency. The focus is on the OTOC thermally averaged with infinite temperature, which is an experimentally feasible quantity. It is shown that the critical points can be identified by long‐time averaging of the OTOC via observing its local minimum behavior. More importantly, the scaling laws of the OTOC for QPTs are revealed by studying the experimentally accessible conditions with finite frequency ratio and finite number of atoms in the studied models. The critical exponents extracted from the scaling laws of OTOC indicate that the QPTs in the Rabi and Dicke models belong to the same universality class.