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Journal of Modern Optics

Quantum Fluctuations in Equilibrium Superradiance

M. Trache

Published Dec 1, 1991• 0 citations
Publisher

Abstract

The role of the quantum fluctuations of the order parameter in a second-order phase transition is discussed, for a superradiant model. The mean field equations and the equation of the critical temperature are obtained. The change of the value of the critical exponent of the specific heat α is connected with the supplementary dimension introduced by the frequency dependence of the field variables. The functional integral method is used, most of the calculations being performed within the Gaussian approximation.

DOI
10.1080/09500349114552521

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The role of the quantum fluctuations of the order parameter in a second-order phase transition is discussed, for a superradiant model. The mean field equations and the equation of the critical temperature are obtained. The change of the value of the critical exponent of the specific heat α is connected with the supplementary dimension introduced by the frequency dependence of the field variables. The functional integral method is used, most of the calculations being performed within the Gaussian approximation.
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This paper investigates the role of statistical fluctuations about the equilibrium value of the order parameter in the vicinity of a phase-transition temperature. The systems under consideration are assumed to be of finite size, but the behavior in the thermodynamic limit is also described. Our objective is to examine the simplest exactly solvable model of non-Gaussian static fluctuations for samples of finite size. To this end a Landau form is employed for the model Hamiltonian and various expansions are analyzed in order to describe first- and second-order phase transitions, field-induced transitions, and liquid-vapor critical phenomena. The approach differs from previous treatments primarily in the application of a quartic non-Gaussian model for statistical fluctuations which redresses some of the shortcomings of the standard Gaussian approximation. In particular, it is found that for finite values of V (volume) no singularities exist even at the critical temperature. The treatment is generalized by examining a class of non-Gaussian distribution functions, and particular emphasis is placed on strict adherence to the uniform convergence criterion so that general conclusions can be drawn.
The existence of zero-dimensional superradiant quantum phase transitions seems inconsistent with conventional statistical physics. This work clarifies this apparent inconsistency. We demonstrate the corresponding effective field theories and finite-temperature properties of light-matter interacting systems, and show how this zero-dimensional quantum phase transition occurs. We first focus on the Rabi model, which is a minimum model that hosts a superradiant quantum phase transition. With the path-integral method, we derive the imaginary-time action of the photon degrees of freedom. We also define a dynamical critical exponent as the rescaling between the temperature and the photon frequency, and perform dimensional analysis to the effective action. The dynamical critical exponent shows that the effective theory of the Rabi model is a free scalar field, where a true second-order quantum phase transition emerges. These results are also verified by numerical simulations of imaginary-time correlation functions of the order parameter. Furthermore, we also generalize this method to the Dicke model. Our results make the zero-dimensional superradiant quantum phase transition compatible with conventional statistical physics, and pave the way to understand it in the perspective of effective field theories. Published by the American Physical Society 2024
The critical phenomenon of the second order phase transition is considered for temperatures above the critical point. A general phenomenological approach is developed. The generalized free energy F[{η q }: T ] is introduced. It is a functional of the Fourier component η q of the local order parameter η( r ). A general property of the correlation function <η q η - q > is established rigorously. The equilibrium free energy and the wave-number dependent susceptibility χ q are expressed in terms of the linked clusters, large clusters corresponding to highr order fluctuation of the order parameter. The general rule for calculating the contribution of each cluster is given. The rule is shown to be very simple. It is found that the specific heat is expressed as a sum of the temperature derivatives of χ q 's. As an example of the application of the linked cluster method, the susceptibility is calculated in the vicinity of the critical point by summing up the clusters that are most divergent when approaching the criticat point. The linked cluster expansion developed here could be a clue to the whole mystery of the singularity at the critical point.
The concept of a quantum fluctuation phase transition (QFPT) proposes that the collective behavior of quantum fluctuations can itself undergo a non‑analytic change as a control parameter is varied, independently of any underlying order parameter associated with a conventional symmetry breaking. In this work we formulate a minimal theoretical framework that captures fluctuation‑driven criticality, derive mean‑field equations for the free energy as a function of a generalized fluctuation field, and employ a renormalization group (RG) analysis to identify the universality class of the transition. The resulting critical exponents differ from those of standard quantum phase transitions, reflecting the dominance of higher‑order fluctuation terms. We discuss possible experimental signatures in ultracold atomic gases, spin‑oriented systems, and superconducting qubits operating at nanokelvin temperatures, and argue that the observation of a QFPT would provide a new route to discovering exotic quantum phases and could attract Nobel‑prize level attention.