Metastability separates apparent equilibration from true convergence: a dynamical system can remain near an intermediate state long before reaching its stationary state. In quantum dynamics used for optimization, such a plateau can delay progress toward a target concentrated near the global minimizers of a nonconvex potential. In this paper, we initiate the study of the general mechanisms by which coherent evolution and dissipation shape metastability, using quantum Langevin dynamics (QLD) as a concrete example [CLW+25], where the objective function is encoded in the potential and coherent transfer between wells interacts with tunable dissipation. To quantify this interaction, we solve a two-state model precisely and derive a spectral-separation criterion in terms of tunneling frequency, dissipation strength, and damping imbalance. Extending the analysis to multi-state models, we show that sufficiently weak tunneling between wells can separate slow population transfer from fast relaxation within each well. In addition, we establish conditions under which changing the structure of dissipation shortens the time needed to approach a prescribed target state. Numerical experiments on double-well and multi-well potentials demonstrate these improvements while keeping the target and accuracy fixed and accounting for the time spent on control.
Metastability separates apparent equilibration from true convergence: a dynamical system can remain near an intermediate state long before reaching its stationary state. In quantum dynamics used for optimization, such a plateau can delay progress toward a target concentrated near the global minimizers of a nonconvex potential. In this paper, we initiate the study of the general mechanisms by which coherent evolution and dissipation shape metastability, using quantum Langevin dynamics (QLD) as a concrete example [CLW+25], where the objective function is encoded in the potential and coherent transfer between wells interacts with tunable dissipation. To quantify this interaction, we solve a two-state model precisely and derive a spectral-separation criterion in terms of tunneling frequency, dissipation strength, and damping imbalance. Extending the analysis to multi-state models, we show that sufficiently weak tunneling between wells can separate slow population transfer from fast relaxation within each well. In addition, we establish conditions under which changing the structure of dissipation shortens the time needed to approach a prescribed target state. Numerical experiments on double-well and multi-well potentials demonstrate these improvements while keeping the target and accuracy fixed and accounting for the time spent on control.
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 stabilizing effect of quantum fluctuations on the escape process and the relaxation dynamics from a quantum metastable state are investigated. Specifically, the quantum dynamics of a multilevel bistable system coupled to a bosonic Ohmic thermal bath in strong dissipation regime is analyzed. The study is performed by a non-perturbative method based on the real-time path integral approach of the Feynman-Vernon influence functional. We consider a strongly asymmetric double well potential with and without a monochromatic external driving, and with an out-of-equilibrium initial condition. In the absence of driving we observe a nonmonotonic behavior of the escape time from the metastable region, as a function both of the system-bath coupling coefficient and the temperature. This indicates a stabilizing effect of the quantum fluctuations. In the presence of driving our findings indicate that, as the coupling coefficient γ increases, the escape time, initially controlled by the external driving, shows resonant peaks and dips, becoming frequency-independent for higher γ values. Moreover, the escape time from the metastable state displays a nonmonotonic behavior as a function of the temperature, the frequency of the driving, and the thermal-bath coupling, which indicates the presence of a quantum noise enhanced stability phenomenon. Finally, we investigate the role of different spectral densities, both in sub-Ohmic and super-Ohmic dissipation regime and for different cutoff frequencies, on the relaxation dynamics from the quantum metastable state. The results obtained indicate that, in the crossover dynamical regime characterized by damped intrawell oscillations and incoherent tunneling, the spectral properties of the thermal bath influence non-trivially the short time behavior and the time scales of the relaxation dynamics from the metastable state.
The spontaneous switching of a quantum particle between the wells of a double-well potential is a phenomenon of general interest to physics and chemistry. It was broadly believed that the switching rate decreases steadily as a function of the size of the energy barrier. This view was challenged by a recent experiment on a driven superconducting Kerr nonlinear oscillator (often called the Kerr-cat qubit or the Kerr parametric oscillator), whose energy barrier can be continuously increased by ramping up the drive. Remarkably, as the drive amplitude increases, the switching rate exhibits a steplike decrease termed the ``staircase.'' The view challenged by the experimental staircase demands a deep review of our understanding of the role of quantum effects in double wells. In this work, we use a Lindbladian model of dissipation to derive a semianalytical formula for the switching rate, resolving a continuous transition between tunneling-dominated dynamics and dissipation-dominated dynamics. These two dynamics are observed respectively in the flat part and the steep part of each step in the staircase. Our formula exposes two distinct dissipative processes that limit tunneling: the dephasing of interwell superpositions freezes tunneling via the quantum Zeno effect, and the decay from one excited state to another limits the time during which tunneling can be attempted. This physical understanding allows us to pinpoint the critical drive amplitude that separates the flat and steep part of each step using an equation involving the rates of the two dissipative processes and the tunnel splitting. In addition, analyzing the transition matrix elements in the formula shows that in the regime of a few ($\ensuremath{\lesssim}10$) states in the well and under moderate to low temperatures, highly excited states are populated predominantly via cascaded and direct thermal heating rather than quantum heating. At very low temperatures, we find that the perturbation induced by the non-Hermitian Hamiltonian part of the Lindbladian model becomes increasingly important and facilitates a form of quantum heating that has not been identified before. We numerically map the activation mechanism as a function of drive amplitude, damping rate, and temperature. Our theory deepens the understanding of switching dynamics between metastable quantum states, highlights the importance of a general interplay between tunneling and dissipation, and identifies a novel quantum regime in activated transitions.
Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring the properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time.
Open quantum systems can rapidly lose most microscopic information, leaving only a few degrees of freedom to govern their long-time dynamics. Classical Markov state models (MSMs) describe such metastable dynamics as transitions among a few representative phases and are widely used to reduce complex-system dynamics in condensed matter and chemical physics. In quantum systems, however, phase labels alone are insufficient when coherence persists between metastable states. Even when the slow modes are known, their spectral projection need not produce valid quantum states. We construct quantum Markov state models (QMSMs) that describe the surviving information and its evolution on a small physical state space containing classical sectors and quantum matrix blocks. We quantify metastability by assuming that the evolution channel C changes little when applied a second time, with sufficiently small defect η=∥C2−C∥⋄, and that the number of slow degrees of freedom is bounded independently of the full system size. Under these assumptions, we construct compression and reconstruction channels whose composition recovers every reduced state exactly, while the reverse composition gives an exactly idempotent channel approximating C, answering Kitaev's exact-rounding question [Kit25]. Our constructions give an optimal diamond-norm bound of O(η1/3) on all input states, as well as an improved bound of O(η1/2) on metastable states prepared by C, with constants depending only on the slow dimension. The reduced transition channel can be iterated to predict the microscopic dynamics with controlled error. We illustrate the QMSM through a weakly driven dissipative spin chain supporting either a metastable logical qubit or long-lived classical phases.
Abstract. Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time. Submitted to: New J. Phys. † Author to whom any correspondence should be addressed. ar