Out‐of‐Time‐Order Correlators and Quantum Phase Transitions in the Rabi and Dicke Models

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.

The Jaynes-Cummings model and its descendants

“…As a simple, but still experimentally relevant, non-integrable system, the Dicke model has attracted much attention in recent years regarding OTOC and related issues [850][851][852][853][854][855]. In exploring properties of the evolution of complex quantum systems one is usually interested in generic and universal features, and one expects the OTOC F(t) to not depend strongly on the particular choice of V and Ŵ .…”

“…In exploring properties of the evolution of complex quantum systems one is usually interested in generic and universal features, and one expects the OTOC F(t) to not depend strongly on the particular choice of V and Ŵ . For the Dicke model different choices have been considered; V = Ŵ = â † â [854,855], V = p and Ŵ = q [852], and V = ρ(0), V = σi and Ŵ = σj or V = Ŵ = x [853], and Ŵ = exp(iδφx) [851]. In the latter case, the V operator is a projection onto the initial state of the system, and for pure states the OTOC becomes F(t) = | ψ(0)|e −i Ĥt e iδφx e i Ĥt |ψ(0) | 2 .…”

Proposal for measuring out-of-time-ordered correlators at finite temperature with coupled spin chains

“…The decay time and nature of spreading (e.g., diffusive or ballistic) can depend on parameters in the Hamiltonian [30][31][32][33][34][35] or state properties like the temperature (T ) [36][37][38]. In addition to characterizing the nature of scrambling, OTOCs can also be used to diagnose quantum phase transitions [39][40][41][42][43][44][45][46][47]. Due to their fundamental importance in information scrambling and other applications listed above, there has been enormous interest in measuring OTOCs in recent years.…”

Dynamical scaling laws of out-of-time-ordered correlators

“…On the one hand, in the context of holographic models dual to classical gravity, it was recently conjectured that the butterfly velocity is maximum at quantum critical point (QCP) [11]; however, whether this conjecture holds universally true for such models are yet to be understood [54][55][56][57]. On the other hand, a longtime average of OTOC shows distinctive features between a normal and broken-symmetry phase in Ising-type spin models [58] and it has been proposed that the long-time average (or the long-time saturated value [59]) of OTOC can be used to detect QPTs [58,60]. The latter approach is however limited to detect the presence of broken-symmetry phase and does not allow one to extract a broad phenomenology of QPTs such as universality and scaling as well as its connection to butterfly effect from OTOC; moreover, as it relies on asymptotic dynamics on a long-time scale, a long coherence time is required in experiments, which is in general a challenging requirement for quantum many-body systems.…”

“…OTOC as a probe of QPT.-It has been recently proposed that the long-time average of OTOC [58,60] or the longtime saturating value of OTOC [59] can be used to detect QPTs. Here, we shall show that the dynamical scaling laws for OTOC of global operators can be utilized to locate the QCP and extract the critical exponents ν and z of QPTs using the transient dynamics of OTOC.…”