Characterization of Dynamical Phase Transitions in Quantum Jump Trajectories Beyond the Properties of the Stationary State

Abstract

We describe how to characterize dynamical phase transitions in open quantum systems from a purely dynamical perspective, namely, through the statistical behavior of quantum jump trajectories. This approach goes beyond considering only properties of the steady state. While in small quantum systems dynamical transitions can only occur trivially at limiting values of the controlling parameters, in many-body systems they arise as collective phenomena and within this perspective they are reminiscent of thermodynamic phase transitions. We illustrate this in open models of increasing complexity: a three-level system, a dissipative version of the quantum Ising model, and the micromaser. In these examples dynamical transitions are accompanied by clear changes in static behavior. This is however not always the case, and in general dynamical phase behavior needs to be uncovered by observables which are strictly dynamical, e.g. dynamical counting fields. We demonstrate this via the example of a class of models of dissipative quantum glasses, whose dynamics can vary widely despite having identical (and trivial) stationary states.Recent experimental progress in quantum optics and cold atomic physics has stimulated a great interest in the study of open many-body quantum systems [1][2][3][4][5][6][7]. Currently, much effort is dedicated to the understanding and classification of dynamical phases and transitions among them, for example in the Dicke Model [7][8][9], in lattice bosons subject to engineered dissipation [1], and in spin systems [3][4][5]10]. Typically, dynamical phase transitions are detected and analyzed through changes in static order parameters, such as superfluid density or spin polarization, calculated in the system's stationary state.The aim of this work is to characterize dynamical phases exclusively through time-correlations within quantum jump trajectories, and not through static order parameters. We do so by building on an elegant connection between open quantum systems -described by a Lindblad master equation -and matrix product states [11] (MPS) put forward in [12]. We pursue the usual dual description of open system dynamics: On the one hand any individual realization of the dynamics is stochastic. In the case of an open quantum system this is represented by a stochastic wave function corresponding to a specific sequence of quantum jumps, with the whole ensemble of these stochastic trajectories being encoded in a MPS. On the other hand, the evolution of probabilities is deterministic and is derived from the evolution of the density matrix under the action of a quantum master operator (QMO). The dynamical phase structure of an open system is given by the low lying spectrum of the QMO [5]. More specifically, dynamical phases are characterized by how the spectrum responds to changes in physical parameters [3][4][5] or counting fields [13,14]. When this response is non-analytic we have the signature of a dynamical phase transition [13,14]. This analytic structure is, however, also mirrored in the ensemble o...

Classical stochastic dynamics and continuous matrix product states: gauge transformations, conditioned and driven processes, and equivalence of trajectory ensembles

“…This approach is a classical version of the known connection between continuous MPSs and open quantum dynamics [11][12][13]. It allows to describe in a compact way conditioned trajectory ensembles and demonstrate ensemble equivalences, in analogy with what can be done in the quantum case [15,16]. The key property of cMPSs is that of gauge invariance from which the equivalences follow.…”

“…It is possible in this way to represent the whole ensemble of quantum trajectories of the open system as a cMPS. This MPS approach can be then extended for the application of large-deviation methods to describe and characterise open quantum dynamics [15,16].…”

Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables

“…An important observation is that the bounds on fluctuations of a counting observable and its FPTs are controlled by the average dynamical activity, in analogy to the role played by the entropy production in the case of currents [1,13]. We hope these results will add to the growing body of work applying large deviation ideas and methods to the study of dynamics in driven systems [29][30][31][32][33][34][35][36][37][38][39][40][41], glasses [25,[42][43][44][45][46][47], protein folding and signaling networks [48][49][50][51], open quantum systems [52][53][54][55][56][57][58][59][60][61][62][63][64], and many other problems in nonequilibrium [65][66][67][68][69][70][71][72].…”

Dynamical phase transitions as a resource for quantum enhanced metrology

“…First, it provides improved precision due to the fact that the effective "size" of the system and output is now Nt, where t is the observation time and N is the system size. The second advantage arises from the fact that open systems can feature dynamical phase transitions (DPTs) [17][18][19], which, in contrast to static transitions, are characterized by singular changes in observables on the whole dynamical evolution and not just on the state of the system. We show that at a first-order DPT [18,19] the quantum Fisher information (QFI) of the system-and-output state may become quadratic in t giving rise to Heisenberg scaling.…”

“…We overcome the problem of mixed states by considering the combined state of the system and output. This is a pure quantum state-actually a matrix product state (MPS) [13,14,16,17]-which encodes the state of the system as well as the record of emissions for the whole observation time. This allows us to find the best estimation precision using the system-output state as a resource.…”