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Optics Letters

Statistical mechanics of weakly nonlinear optical multimode gases

Konstantinos G. Makris, Fan O. Wu, Paweł S. Jung, Demetrios N. Christodoulides

Published Feb 7, 2020• 66 citations
Publisher

Abstract

By utilizing notions from statistical mechanics, we develop a general and self-consistent theoretical framework capable of describing any weakly nonlinear optical multimode system involving conserved quantities. We derive the fundamental relations that govern the grand canonical ensemble through maximization of the Gibbs entropy at equilibrium. In this classical picture of statistical photo-mechanics, we obtain analytical expressions for the probability distribution, the grand partition function, and the relevant thermodynamic potentials. Our results universally apply to any other weakly nonlinear multimode bosonic system.

DOI
10.1364/ol.387863

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By utilizing notions from statistical mechanics, we develop a general and self-consistent theoretical framework capable of describing any weakly nonlinear optical multimode system involving conserved quantities. We derive the fundamental relations that govern the grand canonical ensemble through maximization of the Gibbs entropy at equilibrium. In this classical picture of statistical photo-mechanics, we obtain analytical expressions for the probability distribution, the grand partition function, and the relevant thermodynamic potentials. Our results universally apply to any other weakly nonlinear multimode bosonic system.
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The past few years have witnessed a resurgence of interest in multimode structures, predominantly driven by the ever-increasing demand for higher information capacities. This renaissance, in turn, incited a flurry of activities in the general area of nonlinear multimode optics. The sheer complexity associated with the presence of hundreds or thousands of nonlinearly interacting modes that collectively act as a many-body system, has led to new opportunities in observing a multitude of novel optical effects that would have been otherwise impossible in single mode settings. In this talk, a thermodynamic theory capable of describing complex, highly multimoded, nonlinear optical systems is presented. It is shown that the mode occupancies in such nonlinear multimode arrangements follow a universal behavior that always tends to maximize the system’s entropy at steady-state. This thermodynamic response takes place irrespective of the type of nonlinearity involved and can be utilized to either heat or cool an optical multimode system. Aspects associated with adiabatic compressions and expansions will be discussed along with the possibility for all-optical Carnot cycles.
During the evolution of coupled nonlinear oscillators on a lattice, with dynamics dictated by the discrete nonlinear Schrödinger equation (DNLSE systems), two quantities are conserved: system energy (Hamiltonian) and system density (number of particles). If the number of system oscillators is large enough, a significant portion of the array can be considered to be an "open system", in intimate energy and density contact with a "bath" - the rest of the array. Thus, as indicated in previous works, the grand canonical formulation can be exploited in order to determine equilibrium statistical properties of thermalized DNLSE systems. In this work, given the values of the two conserved quantities, we have calculated the necessary values of the two Lagrange parameters (typically designated β,μβ,μβ,μ) associated with the grand canonical partition function in two different ways. One is numerical and the other is analytic, based on a published approximate entropy expression. In addition we have accessed a purposely-derived approximate PDF expression of site-densities. Applying these mathematical tools we have generated maps of temperatures, chemical potentials, and field correlations for DNLSE systems over the entire thermalization zone of the DNLSE phase diagram, subjected to all system-nonlinearity levels. The end result is a rather complete picture, characterizing equilibrated large DNLSE systems.
We analyze the general relation between canonical and grand canonical ensembles in the thermodynamic limit. We begin our discussion by deriving, with an alternative approach, some standard results first obtained by Kac and coworkers in the late 1970s. Then, motivated by the Bose-Einstein condensation (BEC) of trapped gases with a fixed number of atoms, which is well described by the canonical ensemble and by the recent groundbreaking experimental realization of BEC with photons in a dye-filled optical microcavity under genuine grand canonical conditions, we apply our formalism to a system of non-interacting Bose particles confined in a two-dimensional harmonic trap. We discuss in detail the mathematical origin of the inequivalence of ensembles observed in the condensed phase, giving place to the so-called grand canonical catastrophe of density fluctuations. We also provide explicit analytical expressions for the internal energy and specific heat and compare them with available experimental data. For these quantities, we show the equivalence of ensembles in the thermodynamic limit.
Predicting the stationary behavior of observables in isolated many-body quantum systems is a central challenge in quantum statistical mechanics. While one can often use the Gibbs ensemble, which is simple to compute, there are many scenarios where this is not possible and one must instead use another ensemble, such as the diagonal, microcanonical or generalized Gibbs ensembles. However, these all require detailed information about the energy or other conserved quantities to be constructed. Here we propose a general and computationally easy approach to determine the stationary probability distribution of observables with few outcomes. Interpreting coarse measurements at equilibrium as noisy communication channels, we provide general analytical arguments in favor of the applicability of a maximum entropy principle for this class of observables. We show that the resulting theory accurately predicts stationary probability distributions without detailed microscopic information like the energy eigenstates. Extensive numerical experiments on 7 non-weakly interacting spin-1/2 Hamiltonians demonstrate the broad applicability and robustness of this framework in both quantum integrable and chaotic models.
The equilibrium and quasi-equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. For equilibrium, the model of Bogolyubov is generalized to finite temperature by using the grand partition function. The thermodynamic properties and the pair-correlation function are calculated. The statistical mechanics for moving systems is then developed and applied to the problem of a rotating fluid. For quasi-equilibrium, general transport equations are derived from first principles, independent of statistics and model. For the Bogolyubov model, the familiar two-fluid hydrodynamics is then derived, leading to the phenomena of first and second sound.