We develop an approach for understanding the dynamics of open quantum systems by analyzing individual quantum trajectories in the eigenbasis of the Liouvillian superoperator. From trajectory-eigenstate overlaps, we construct a quasiprobability distribution that characterizes the degree of localization of the trajectories in the Liouvillian eigenbasis. Contrary to the common wisdom that late-time dynamics are governed solely by the steady state and the slowest-decaying modes, we show that trajectories can remain well spread over transient eigenstates deep within the bulk of the Liouvillian spectrum even at late times. We demonstrate this explicitly using numerical simulations of interacting spin chains and bosonic systems. Moreover, we find that the delocalization of the trajectory strongly correlates with the purity of the trajectory-averaged steady state, establishing a further link between the trajectory and ensemble pictures of open quantum dynamics.
We develop an approach for understanding the dynamics of open quantum systems by analyzing individual quantum trajectories in the eigenbasis of the Liouvillian superoperator. From trajectory-eigenstate overlaps, we construct a quasiprobability distribution that characterizes the degree of localization of the trajectories in the Liouvillian eigenbasis. Contrary to the common wisdom that late-time dynamics are governed solely by the steady state and the slowest-decaying modes, we show that trajectories can remain well spread over transient eigenstates deep within the bulk of the Liouvillian spectrum even at late times. We demonstrate this explicitly using numerical simulations of interacting spin chains and bosonic systems. Moreover, we find that the delocalization of the trajectory strongly correlates with the purity of the trajectory-averaged steady state, establishing a further link between the trajectory and ensemble pictures of open quantum dynamics.
Efficient simulation of Markovian open quantum systems remains a central challenge because the density-matrix description grows exponentially with system size. Quantum trajectory methods provide an alternative by replacing mixed-state evolution with stochastic pure-state realizations. Here we investigate this framework for normal Lindblad generators, whose orthogonal eigenoperator decomposition precludes transient amplification. By decomposing the Lindbladian into deterministic smooth and stochastic jump contributions, we derive an exact steady-state balance relation that identifies the interplay between these processes as the mechanism underlying Liouvillian normality. We further show that normal Lindbladians exclude exceptional points and that, although individual quantum trajectories generally exhibit stochastic coupling between Liouvillian eigenmodes, these couplings cancel upon ensemble averaging, recovering independent orthogonal relaxation modes. These results provide a trajectory-level interpretation of Liouvillian normality and clarify how a global property of the Lindblad generator is realized through stochastic quantum dynamics.
In closed quantum dynamics, energy eigenstates represent physical excitations which, for systems with a classical limit, localize on quantized orbits in phase space. Such a picture is lacking in open systems: Liouvillians have distinct right and left eigenstates that decay with time and are difficult to interpret. We recover a physical correspondence by introducing a gauge-invariant quasiprobability measure combining right and left eigenstates that interprets a Liouvillian mode as a set of coherences, allowing one to visualize the modes as quasiprobability distributions in a ``doubled'' phase space. Using this measure we show that, remarkably, a damped oscillator retains its quantized ``orbits'' for a large class of linear and nonlinear loss, with measurable signatures. A thermal bath broadens these orbits, similar to energy levels, leading to a characteristic multistage relaxation. This provides an experimentally accessible open-system analog of the eigenstate-orbit correspondence underlying semiclassical quantization. For quadratic systems, our results yield dissipative analogs of invariant tori - a central concept in Hamiltonian systems.
Metastability in open quantum systems is usually inferred from spectral separation in the Liouvillian, which governs unconditional, ensemble-averaged dynamics. We show that this diagnosis is incomplete at the level of individual quantum trajectories: conditioned realizations of the same unconditional dynamics can bypass, transiently access, or operationally preserve a metastable memory, depending on the monitored channel and the observed record. We demonstrate these mechanisms in a driven-dissipative nonlinear oscillator realizing a quantum associative memory, by comparing spectra, target fidelities, and phase-space distributions. The non-Hermitian dynamics obtained by post-selecting on the absence of detected events supports metastable retrieval, but with a distinct long-time fate: normalization selects the least-decaying mode of the non-Hermitian spectrum rather than the addressed memory branch. In contrast, stochastic jump trajectories can preserve retrieval over extended times when the post-measurement update remains compatible with the coherent memory structure. Thus trajectory-level metastability is not determined by the averaged generator alone; it requires compatibility between the monitored channel, the measurement record, and the metastable manifold.
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
The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. We demonstrate the efficiency of our method on several examples of coupled bosonic systems.
Stochastic switching is a central phenomenon in dissipative many-body systems, offering a key probe of metastability across classical and quantum regimes. Here, we unravel the connection between switching dynamics and quantum metastability through the lens of spectral decomposition, quantum-jump simulations and the large deviation principles. By establishing a direct correspondence between classical fixed points and quantum metastable states, we distinguish trajectory-level, noise-induced metastability from spectrum-level, deterministic metastability in a Markovian open quantum system with bistability. The Liouvillian gap, the steady-state occupation ratio and the observed switching rates of the metastable states all exhibit exponential scaling with system size, giving rise to a quantum analogue of the Arrhenius law, with the inverse system size serving as an effective temperature. These results provide a unified picture of quantum bistability and clarify the relaxation processes of strongly interacting, dissipative quantum systems far from the thermodynamic limit.