Theory of classical metastability in open quantum systems Katarzyna Macieszczak,1 Dominic C. Rose,2, 3 Igor Lesanovsky,4, 2, 3 and Juan P. Garrahan2, 3 1TCM Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom 2School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 3Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 4Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany We present a general theory of classical metastability in open quantum systems. Metastability is a consequence of a large separation in timescales in the dynamics, leading to the existence of a regime when states of the system appear stationary, before eventual relaxation toward a true stationary state at much larger times. In this work, we focus on the emergence of classical metastability, i.e., when metastable states of an open quantum system with separation of timescales can be approximated as probabilistic mixtures of a finite number of states. We find that a number of classical features follow from this approximation, for the manifold of metastable states, long-time dynamics between them, and symmetries of the dynamics. Namely, those states are approximately disjoint and thus play the role of metastable phases, the relaxation toward the stationary state is approximated by a classical stochastic dynamics between them, and weak symmetries correspond to their permutations. Importantly, the classical dynamics is observed not only on average, but also at the level of individual quantum trajectories: We show that time coarse-grained continuous measurement records can be viewed as noisy classical trajectories, while their statistics can be approximated by that of the classical dynamics. Among others, this explains how first-order dynamical phase transitions arise from metastability. Finally, to verify the presence of classical metastability in a given open quantum system, we develop an efficient numerical approach that delivers the set of metastable phases together with the effective classical dynamics. Since the proximity to a first-order dissipative phase transition manifests as metastability, the theory and tools introduced in this work can be used to investigate such transitions—which occur in the large size limit—through the metastable behavior of many-body systems of moderate sizes accessible to numerics. CONTENTS 1. Long-time dynamics 12 2. Classical generator 12 I. Introduction3 3. Classical system dynamics 13 4. Hierarchy of classical long-time II. Metastability in open quantum systems5 dynamics 13 A. Dynamics of open quantum systems5 5. Classical observable dynamics 14 B. Spectral theory of metastability5 B. Classical characteristics of quantum C. Quantitative approach6 trajectories 14 D. Dissipative phase transitions7 1. Statistics of quantum trajectories 14 2. Classical tilted generator 15 III. Classical metastability in open quantum 3. Classical cumulants 15 systems7 4. Classical dynamics of quantum A. Definition of classical metastability7 trajectories 17 B. Test of classicality8 5. Classical metastability and dynamical C. Figures of merit9 phase transitions 18 IV. Classical metastable phases9 VI. Classical weak symmetries 19 A. Physical representation of metastable A. Symmetry and general metastability 20 arXiv:2006.01227v2 [cond-mat.stat-mech] 19 Jul 2021 manifold 10 1. Symmetry of metastable manifolds 20 B. Classical degrees of freedom 10 2. Symmetry of long-time dynamics 20 C. Approximate disjointness of metastable B. Symmetry and classical metastability 20 phases 11 1. Approximate symmetry of metastable D. Classical hierarchy of metastable phases 11 phases 20 2. No continuous symmetries 20 V. Classical long-time dynamics 11 3. Symmetric set of metastable phases 21 A. Classical average dynamics of system and 4. Symmetry of classical long-time observables 12 dynamics 21 2 5. Symmetric test of classicality 23 2. Trace-norm vs. L1-norm in classical metastable manifolds 42 VII. Unfolding classical metastability numerically 23 a. Distance between metastable states 42 A. Metastable phases from master operator b. Distance between metastable phases 42 spectrum 23 c. Norms of long-time generator 43 1. Metastable phases construction 23 d. Relaxation time 43 2. Construction for hierarchy of metastable 3. Orthogonality and disjointness of phases in manifolds 25 classical metastable manifolds 44 3. Construction for metastable manifolds a. Bounds on scalar products of metastable with symmetries 25 phases 44 4. Construction utilizing order parameters 25 b. Proof of Eqs. (31)-(33) in the main text 45 B. Metastable phases from biased quantum 4. Nonuniqueness of phases in classical trajectories 26 metastable manifolds 47 VIII. Conclusions and outlook 26 E. Classical long-time dynamics 47 1. Classical stochastic dynamics 47 Acknowledgments 27 a. Positivity and probability conservation 47 b. Stochastic trajectories 48 References 27 c. Weak symmetries 48 Supplemental Material 31 2. Classical dynamics of average system state 49 a. Best classical stochastic approximation of A. Classical metastability in proximity to long-time dynamics generator 49 dissipative phase transition at finite size 31 b. Derivation of Eq. (37) in the main text 49 1. Example in Figs.1–5 of the main text 31 c. Derivation of Eq. (39) in the main text 50 a. Model 31 d. Derivation of Eq. (41) in the main text 50 b. Dissipative phase transition 31 e. Approximation of dynamics resolvent 51 c. Plot parameters and numerical results 31 f. Classical discrete approximation of 2. General case 32 long-time dynamics 52 a. Dissipative phase transition and its 3. Classical statistics of quantum trajectories 53 proximity 32 a. Activity in quantum trajectories 53 b. Perturbation theory 33 b. Homodyne current in quantum trajectories 54 B. Metastability in open quantum systems 35 c. Time-integrals of system observables in 1. Projection on low-lying modes 35 quantum trajectories 56 ~ a. Derivation of Eq. (12) in the main text 35 d. Corrections in approximations of Ws, ~ ~ b. Bound on metastability of states closest Wh, and Wr 57 to Eq. (4) in the main text 35 e. Rates of average and fluctuations in 2. Metastable regime 35 quantum trajectories after initial 3. Relaxation times 36 relaxation 59 a. Definitions 36 f. Rates of average and fluctuations in b. Relation to master operator spectrum 37 quantum trajectories during metastable c. Relation to metastable regime 37 regime 65 g. Asymptotic rates of fluctuations in C. Classical metastability in open quantum quantum trajectories 69 systems 38 h. Multimodal distribution of quantum 1. Definition of classical metastability 38 trajectories 70 2. Test of classicality 38 a. Distance of barycentric coordinates to F. Classical hierarchy of metastabilities 75 probability distributions 38 1. Hierarchy of metastabilities 75 b. Derivation of Eq. (22) in the main text 39 2. Hierarchy of classical metastable manifolds 75 c. Optimality of test of classicality 39 3. Hierarchy of classical metastable phases 77 a. Supports and basins of attraction 77 D. Classical metastable phases 40 b. Decay subspace 78 1. Properties of dual basis in classical 4. Hierarchy of classical long-time dynamics 78 metastable manifolds 40 a. Hierarchy of continuous approximations a. Properties of dual basis 40 of classical long-time dynamics 78 b. Distance to POVM 41 b. Hierarchy of discrete approximations of c. Cross-correlations of dual basis 41 classical long-time dynamics 80 3 G. Classical weak symmetries 81 however, more sophisticated (albeit still approximate) 1. Symmetries of low-lying eigenmodes 81 techniques such as variational approaches [26–28], per- 2. Symmetries of classical metastable turbative expansions in lattice connectivity [25, 29], infi- manifolds 81 nite tensor network simulations [13] or a field-theoretical a. Discrete symmetries of classical analysis [16] can still indicate a unique stationary state. metastable manifolds 82 While it is unusual to see phase transitions at finite b. No nontrivial continuous symmetries of system sizes [30–33], first-order phase transitions in sta- classical metastable manifolds 82 tionary states manifest at large enough finite system c. Symmetric set of metastable phases 83 sizes [34] through the occurrence of metastability, i.e., dis- 3. Symmetries of classical long-time dynamics 84 tinct timescales in the evolution of the system statistics: a. Derivation of Eq. (77) in the main text 84 classically, in the probability distribution over configura- b. Classical dynamics of symmetric degrees tion space [7, 22, 35, 37, 39]; quantum mechanically, in of freedom 85 the density matrix [5, 11]. The statistics of such systems 4. Example of classicality test with weak at long times can be understood in terms of metastable symmetry 85 phases which generally correspond to the phases on ei- ther side of the transition being distinct from the unique H. Quantitative analysis of algorithm in Sec. VII A stationary state for a given set of parameters. Therefore, of the main text 86 already at a finite system size the structure of a possi- 1. Extreme eigenstates of dynamics ble first-order dissipative phase transition can be fully eigenmodes for metastable phases 86 determined by investigating metastable states of the sys- 2. Rotations of eigenmodes to expose tem [11, 42], which is of particular importance for many-