Recently, several studies involving open quantum systems which possess a strong symmetry have observed that every individual trajectory in the Monte Carlo unravelling of the master equation will dynamically select a specific symmetry sector to ‘freeze’ into in the long-time limit. This phenomenon has been termed ‘dissipative freezing’, and in this paper we argue, by presenting several simple mathematical perspectives on the problem, that it is a general consequence of the presence of a strong symmetry in an open system with only a few exceptions. Using a number of example systems we illustrate these arguments, uncovering an explicit relationship between the spectral properties of the Liouvillian in off-diagonal symmetry sectors and the time it takes for freezing to occur. In the limiting case that eigenmodes with purely imaginary eigenvalues are manifest in these sectors, freezing fails to occur. Such modes indicate the preservation of information and coherences between symmetry sectors of the system and can lead to phenomena such as non-stationarity and synchronisation. The absence of freezing at the level of a single quantum trajectory provides a simple, computationally efficient way of identifying these traceless modes.
Recently, several studies involving open quantum systems which possess a strong symmetry have observed that every individual trajectory in the Monte Carlo unravelling of the master equation will dynamically select a specific symmetry sector to ‘freeze’ into in the long-time limit. This phenomenon has been termed ‘dissipative freezing’, and in this paper we argue, by presenting several simple mathematical perspectives on the problem, that it is a general consequence of the presence of a strong symmetry in an open system with only a few exceptions. Using a number of example systems we illustrate these arguments, uncovering an explicit relationship between the spectral properties of the Liouvillian in off-diagonal symmetry sectors and the time it takes for freezing to occur. In the limiting case that eigenmodes with purely imaginary eigenvalues are manifest in these sectors, freezing fails to occur. Such modes indicate the preservation of information and coherences between symmetry sectors of the system and can lead to phenomena such as non-stationarity and synchronisation. The absence of freezing at the level of a single quantum trajectory provides a simple, computationally efficient way of identifying these traceless modes.
In driven-dissipative systems, the presence of a strong symmetry guarantees the existence of several steady states belonging to different symmetry sectors. Here we show that when a system with a strong symmetry is initialized in a quantum superposition involving several of these sectors, each individual stochastic trajectory will randomly select a single one of them and remain there for the rest of the evolution. Since a strong symmetry implies a conservation law for the corresponding symmetry operator on the ensemble level, this selection of a single sector from an initial superposition entails a breakdown of this conservation law at the level of individual realizations. Given that such a superposition is impossible in a classical stochastic trajectory, this is a a purely quantum effect with no classical analog. Our results show that a system with a closed Liouvillian gap may exhibit, when monitored over a single run of an experiment, a behavior completely opposite to the usual notion of dynamical phase coexistence and intermittency, which are typically considered hallmarks of a dissipative phase transition. We discuss our results on a coherently driven spin ensemble with a squeezed superradiant decay, a simple model that presents a wealth of nonergodic dynamics.
In dissipative quantum systems, strong symmetries can lead to the existence of conservation laws and multiple steady states. The investigation of such strong symmetries and their consequences on the dynamics of the dissipative systems is still in its infancy. In this work we investigate a strong symmetry for bosonic atoms coupled to an optical cavity, an experimentally relevant system, using adiabatic elimination techniques and numerically exact matrix product state methods. We show the existence of multiple steady states for ideal bosons coupled to the cavity. We find that the introduction of a weak breaking of the strong symmetry by a small interaction term leads to a direct transition from multiple steady states to a unique steady state. We point out the phenomenon of dissipative freezing, the breaking of the conservation law at the level of individual realizations in the presence of the strong symmetry. For a weak breaking of the strong symmetry we see that the behavior of the individual trajectories still shows some signs of this dissipative freezing before it fades out for a larger symmetry breaking terms.
Strong dynamical symmetries provide an algebraic description of persistent oscillations in open quantum systems governed by the Lindblad equation. For finite-dimensional systems with a full-rank stationary state, such symmetries are necessary and sufficient for non-decaying oscillatory modes to appear in the Liouvillian's spectrum. These modes can be directly constructed by acting on stationary states with the identified symmetry operators. Here we establish an asymptotic counterpart of this algebra for collective spins. Taking a single spin S in a magnetic field, subject to dissipation through linear jump operators, we identify the exact conditions on the field and dissipative couplings under which non-decaying oscillatory modes emerge as S→∞. We then write down a simple, closed form for the corresponding `symmetry' operator which can be applied to the steady state to construct these modes, which are eigenvectors of the Liouvillian, with imaginary eigenvalues, up to O(1/S) corrections. A second operator, an emergent conserved quantity, completes the algebra, and its slow relaxation determines how quantum noise selects the stationary state. The construction can be directly extended to a pair of interacting spins sharing a common reservoir, with the resulting quantum algebra making the entanglement during a quench analytically accessible.
Abstract In nature, instances of synchronisation abound across a diverse range of environments. In the quantum regime, however, synchronisation is typically observed by identifying an appropriate parameter regime in a specific system. In this work we show that this need not be the case, identifying conditions which, when satisfied, guarantee that the individual constituents of a generic open quantum system will undergo completely synchronous limit cycles which are, to first order, robust to symmetry-breaking perturbations. We then describe how these conditions can be satisfied by the interplay between several elements: interactions, local dephasing and the presence of a strong dynamical symmetry—an operator which guarantees long-time non-stationary dynamics. These elements cause the formation of entanglement and off-diagonal long-range order which drive the synchronised response of the system. To illustrate these ideas we present two central examples: a chain of quadratically dephased spin-1s and the many-body charge-dephased Hubbard model. In both cases perfect phase-locking occurs throughout the system, regardless of the specific microscopic parameters or initial states. Furthermore, when these systems are perturbed, their nonlinear responses elicit long-lived signatures of both phase and frequency-locking.
Open quantum systems with nearly degenerate energy levels have been shown to exhibit long-lived metastable states in the approach to equilibrium, even when modelled with certain Lindblad-form quantum master equations. This is a result of dramatic separation of timescales due to differences between Liouvillian eigenvalues. These metastable states often have nonzero coherences that die off only in the long-time limit once the system reaches thermal equilibrium. We examine two distinct situations that give rise to this effect: one in which dissipative dynamics couple together states only within a nearly degenerate subspace, and one in which they give rise to jumps over finite-energy splittings, between nearly degenerate subspaces. We find, in each case, that a change of basis can often lead to a representation that more naturally captures the impact of the system-bath interaction than does the energy eigenbasis, revealing that separate timescales are associated with separate processes (e.g., decoherence into a nonenergy eigenbasis, decay of population correlations to the initial state). This approach is paired with the inspection of quantum trajectories, which further provide intuition as to how open system evolution is characterized when coherent oscillations, thermal relaxation, and decoherence all occur simultaneously. Published by the American Physical Society 2024
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, the micromaser, and a dissipative version of the quantum Ising model. In these examples dynamical transitions are accompanied by clear changes in static behavior. This is however not always the case, and, in general, dynamical phases need 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.