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arXiv:1012.0776 · Physical Review E

Thermodynamics of quantum jump trajectories in systems driven by classical fluctuations

Adrián A. Budini

Published Dec 3, 2010• 21 citations
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Abstract

The large-deviation method can be used to study the measurement trajectories of open quantum systems. For optical arrangements this formalism allows to describe the long time properties of the (nonequilibrium) photon counting statistics in the context of a (equilibrium) thermodynamic approach defined in terms of dynamical phases and transitions between them in the trajectory space [J. P. Garrahan and I. Lesanovsky, Phys. Rev. Lett. 104, 160601 (2010)]. In this paper, we study the thermodynamic approach for fluorescent systems coupled to complex reservoirs that induce stochastic fluctuations in their dynamical parameters. In a fast modulation limit the thermodynamics corresponds to that of a Markovian two-level system. In a slow modulation limit, the thermodynamic properties are equivalent to those of a finite system that in an infinite-size limit is characterized by a first-order transition. The dynamical phases correspond to different intensity regimes, while the size of the system is measured by the transition rate of the bath fluctuations. As a function of a dimensionless intensive variable, the first and second derivatives of the thermodynamic potential develop an abrupt change and a narrow peak, respectively. Their scaling properties are consistent with a double-Gaussian probability distribution of the associated extensive variable.

DOI
10.1103/physreve.82.061106

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The large-deviation method can be used to study the measurement trajectories of open quantum systems. For optical arrangements this formalism allows to describe the long time properties of the (nonequilibrium) photon counting statistics in the context of a (equilibrium) thermodynamic approach defined in terms of dynamical phases and transitions between them in the trajectory space [J. P. Garrahan and I. Lesanovsky, Phys. Rev. Lett. 104, 160601 (2010)]. In this paper, we study the thermodynamic approach for fluorescent systems coupled to complex reservoirs that induce stochastic fluctuations in their dynamical parameters. In a fast modulation limit the thermodynamics corresponds to that of a Markovian two-level system. In a slow modulation limit, the thermodynamic properties are equivalent to those of a finite system that in an infinite-size limit is characterized by a first-order transition. The dynamical phases correspond to different intensity regimes, while the size of the system is measured by the transition rate of the bath fluctuations. As a function of a dimensionless intensive variable, the first and second derivatives of the thermodynamic potential develop an abrupt change and a narrow peak, respectively. Their scaling properties are consistent with a double-Gaussian probability distribution of the associated extensive variable.
We apply the large-deviation method to study trajectories in dissipative quantum systems. We show that in the long time limit the statistics of quantum jumps can be understood from thermodynamic arguments in terms of dynamical phases and transitions between them in trajectory space. We illustrate our approach with three simple examples: a driven 2-level system where we find a particular scale invariance point in the ensemble of trajectories of emitted photons; a blinking 3-level system, where we argue that intermittency in the photon count is related to a crossover between distinct dynamical phases; and a micromaser, where we find an actual first-order phase transition in the ensemble of trajectories.
We apply a large-deviation method to study the diffusive trajectories of the quadratures of light emitted from open quantum systems. We formulate the study of quadrature trajectories in terms of characteristic operators and show that, in the long-time limit, the statistics of such trajectories obey a large-deviation principle. We take our motivation from homodyne detection schemes which allow the statistics of light quadratures to be measured. We illustrate our approach with four examples of increasing complexity: a driven two-level system, a ``blinking'' three-level system, a pair of weakly coupled two-level driven systems, and the micromaser. We discuss how quadrature operators can serve as alternative order parameters for the classification of dynamical phases, which is particularly useful in cases where the statistics of quantum jumps cannot distinguish such phases. The formalism we introduce also allows us to analyze the properties of the light emitted in quantum-jump trajectories which deviate far from the typical dynamics.
In this paper, we develop a quantum-jump approach for describing the photon emission process of single fluorophore systems coupled to complex classically fluctuating reservoirs. The formalism relies on an open quantum system approach where the dynamics of the system and the reservoir fluctuations are described through a density matrix whose evolution is defined by a Lindblad rate equation. For each realization of the photon-measurement processes it is possible to define a conditional system state (stochastic density matrix) whose evolution depends on both the photon detection events and the fluctuations between the configurational states of the reservoir. In contrast to standard fluorescent systems the photon-to-photon emission process is not a renewal one, being defined by a (stochastic) waiting time distribution that in each recording event parametrically depends on the conditional state. The formalism allows calculating experimental observables such as the full hierarchy of joint probabilities associated with the time intervals between consecutive photon recording events. These results provide a powerful basis for characterizing different situations arising in single-molecule spectroscopy, such as spectral fluctuations, lifetime fluctuations and light-assisted processes.
We analyze dynamical large deviations of quantum trajectories in Markovian open quantum systems in their full generality. We derive a quantum level-2.5 large deviation principle for these systems, which describes the joint fluctuations of time-averaged quantum jump rates and of the time-averaged quantum state for long times. Like its level-2.5 counterpart for classical continuous-time Markov chains (which it contains as a special case), this description is both explicit and complete, as the statistics of arbitrary time-extensive dynamical observables can be obtained by contraction from the explicit level-2.5 rate functional we derive. Our approach uses an unraveled representation of the quantum dynamics which allows these statistics to be obtained by analyzing a classical stochastic process in the space of pure states. For quantum reset processes we show that the unraveled dynamics is semi-Markovian and derive bounds on the asymptotic variance of the number of quantum jumps which generalize classical thermodynamic uncertainty relations. We finish by discussing how our level-2.5 approach can be used to study large deviations of nonlinear functions of the state, such as measures of entanglement.
This thesis investigates the dynamical properties of equilibrium and nonequilibrium systems, both quantum and classical, under the guise of a thermodynamic formalism. Large deviation functions associated with the generating functions of time-integrated observables play the role of dynamical free energies and thus determine the trajectory phase structure of a system. The 1d Glauber-Ising chain is studied using the time-integrated energy as the dynamical order parameter and a whole curve of second order trajectory transitions are uncovered in the complex counting field plane. The leading dynamical Lee-Yang zeros of the associated generating function are extracted directly from the time dependent high order cumulants. Resolving the cumulants into constituent contributions the motion of each contribution’s leading Lee-Yang zeros pair allows one to infer the positions of the trajectory transition points. Contrastingly if one uses the full cumulants only the positions of those closest to the origin, in the limit of low temperatures, can be inferred. Motivated by homodyne detection schemes this thermodynamic approach to trajectories is extended to the quadrature trajectories of light emitted from open quantum systems. Using this dynamical observable the trajectory phases of a simple “blinking” 3-level system, two weakly coupled 2-level systems and the micromaser are studied. The trajectory phases of this observable are found to either carry as much information as the photon emission trajectories or in some cases capture extra dynamically features of the system (the second example). Finally, the statistics of the time-integrated longitudinal and transverse magnetization in the 1d transverse field quantum Ising model are explored. In both cases no large deviation function exists but the generating functions are still calculable. From the singularities of these generating functions new transition lines emerge. These were shown to be linked to: (a) the survival probability of an associated open system, (b) PT-symmetry, (c) the temporal scaling of the cumulants and (d) the topology of an associated set of states.
Abstract We consider quantum stochastic processes and discuss a level 2.5 large deviation formalism providing an explicit and complete characterisation of fluctuations of time-averaged quantities, in the large-time limit. We analyse two classes of quantum stochastic dynamics, within this framework. The first class consists of the quantum jump trajectories related to photon detection; the second is quantum state diffusion related to homodyne detection. For both processes, we present the level 2.5 functional starting from the corresponding quantum stochastic Schrödinger equation and we discuss connections of these functionals to optimal control theory.