Driven-dissipative collective spin systems can host several long-lived metastable states; rare switching between them governs slow relaxation and first-order dissipative phase transitions associated with changes in the dominant stationary state. In the companion Letter [arXiv:2604.15091], we introduced a quantum instanton framework based on exact quasiprobability dynamics to describe this switching and illustrated it with a minimal model possessing a simplifying symmetry. Here we develop its numerical implementation, combining collocation and shooting methods, and apply it to two microscopically motivated systems: the Lipkin-Meshkov-Glick model with purely collective dissipation and the cooperative resonance fluorescence with collective and local dissipation. By determining the leading exponential scaling of switching rates between metastable attractors, we predict the stationary magnetization, the first-order dissipative phase-transition point, and the exponential scaling of slow relaxation rates. These predictions agree with quantum master-equation calculations and an exact analytical result for the second model without dephasing. We also examine how the choice of quasiprobability representation affects numerical convergence and find that the alternative semiclassical Wigner approach, based on Gaussian truncation, has varying accuracy across models and parameter regimes. These results substantiate the framework's applicability to experimentally relevant collective spin systems with both collective and local dissipation.
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23 pages, 11 figures. Companion paper to arXiv:2604.15091
Driven-dissipative collective spin systems can host several long-lived metastable states; rare switching between them governs slow relaxation and first-order dissipative phase transitions associated with changes in the dominant stationary state. In the companion Letter [arXiv:2604.15091], we introduced a quantum instanton framework based on exact quasiprobability dynamics to describe this switching and illustrated it with a minimal model possessing a simplifying symmetry. Here we develop its numerical implementation, combining collocation and shooting methods, and apply it to two microscopically motivated systems: the Lipkin-Meshkov-Glick model with purely collective dissipation and the cooperative resonance fluorescence with collective and local dissipation. By determining the leading exponential scaling of switching rates between metastable attractors, we predict the stationary magnetization, the first-order dissipative phase-transition point, and the exponential scaling of slow relaxation rates. These predictions agree with quantum master-equation calculations and an exact analytical result for the second model without dephasing. We also examine how the choice of quasiprobability representation affects numerical convergence and find that the alternative semiclassical Wigner approach, based on Gaussian truncation, has varying accuracy across models and parameter regimes. These results substantiate the framework's applicability to experimentally relevant collective spin systems with both collective and local dissipation.
Collective spin systems -- driven-dissipative spin ensembles whose unitary and dissipative processes act uniformly on all spins, allowing their dynamics to be described as a single macrospin -- play an important role in atomic and solid-state physics. Their intrinsic nonlinearity gives rise to multiple long-lived metastable states that ultimately relax to a unique most probable state. This dominant state can change with a control parameter, leading to first-order phase transitions. We develop a real-time instanton approach based on quantum quasiprobability dynamics that captures the stationary state in the large-spin limit and the asymptotic scaling of relaxation rates. We further show that these features are not accurately described by the previously applied semiclassical Wigner approach due to its neglect of non-Gaussian fluctuations.
Metastability in many-body quantum systems and its associated exponentially-long timescales have been the subject of considerable recent interest. Here, we focus on a class of driven-dissipative many-body open quantum systems described by a Lindbladian having hidden time-reversal symmetry (a form of quantum detailed balance). Examples include boundary-driven interacting spin chains, bosonic lattice models and driven-dissipative collective spin models. We suggest that for such systems, slow timescales in the vicinity of a dissipative first-order phase transition can be analytically predicted using a special purification of the non-equilibrium steady state. We show the accuracy of our conjecture through detailed studies of a dissipative transverse-field Ising model with collective and local decay, and a driven-dissipative nonlinear cavity model. Our results allow quantitative insights into metastability and slow dynamics for a range of systems, including cases where semiclassical or path-integral instanton approaches are intractable.
This thesis covers research in three separate but interlinked areas: superfluidity in driven-dissipative polariton systems, path integral methods for the study of metastability in Markovian systems, and phase space methods for spin master equations. The unifying theme of the three works is their non-equilibrium nature, but other overlaps are also present. The first two topics share a primary analytical tool, the Feynman-Vernon path integral, with the first applying it to quantum field theory and the second to quantum mechanics. The first and third both utilise the TruncatedWigner method, albeit for very different purposes and rather different phase spaces, while the second and third illustrate the twin approaches of geometric and deformation quantisation. The first part is subdivided into three, dealing with different sub-projects undertaken under the broad umbrella of superfluidity. These are, in order, the absence of it in coherently driven polaritons, the application of TruncatedWigner to its study in Markovian polariton systems, and the analysis of whether non-linear dissipation can cause a non-interacting bosonic gas to exhibit it. Non-equilibrium functional integral methods are extensively used for all three but with varying emphasis: the first features extensive diagrammatic calculations, the second focuses on the stochastic semi-classical limit of the integral, while the third foregoes Feynman diagrams for a generating function approach. Conclusive results are obtained in the first project2, which completes a calculation by a predecessor in the group to show that no regime of coherently driven polaritons exhibits superfluidity; in the second I derive a method to extract the linear response tensor used for identifying superfluid components via the Truncated Wigner method derived from the functional integral, which should allow the numerical study of systems too complicated for the analytical tools employed in the first sub-part; in the final, third part I analyse a non-interacting bosonic system with non-linear dissipation, serving as a minimal model of a dye cavity photon Bose-Einstein condensate, to find that at mean-field level it is predicted to exhibit superfluidity. I then perform a significant portion of the associated fluctuation correction calculation, though difficulties with model regularisation prevent a conclusive answer being given at this time. Part II is devoted to calculations of Lindbladian spectra via the coherent state path integral, the initial aim of which was the study of metastability of systems governed by such Markovian superoperators. A working perturbation theory based on the exact propagator for a coherently driven-dissipative harmonic oscillator is derived and tested on some toy models. This perturbation theory in principle can be applied to systems not covered by the existing methods of third quantisation3 and exact expressions for truncated Fock bases. The part also includes a review of the two pre-eminent approaches to the coherent state path integral, emphasizing its links to holomorphic polarisation and highlighting the difficulties associated with it and consequently instanton calculations performed using it. Finally, moving away from bosonic systems, I develop a stochastic Truncated Wigner approach to spin degrees of freedom of arbitrary magnitude. This builds on earlierwork constructing the Stratonovich-Weyl representation of spin as functions on the 2-sphere S2 by calculating the explicit form of the series expansion of the associated star product to higher order than previously available in the literature. This is then used to study nearest-neighbour and long-range anisotropic XY models in two dimensions, obtaining promising results, in particular the direct observation of vortices associated with the BKT transition that destroy the staggered XY phase in the first model. This method differs from existing discrete Truncated Wigner methods and those specialised to spin-1 2, in particular being based around an expansion in a small parameter proportional to ℏ. This allows it to be easily integrated with bosonic phase space methods, since a truncation can then be carried out at a consistent order in ℏ.
Many-body quantum-optical systems, where a collection of emitters interacts through a common electromagnetic reservoir, exhibit rich out-of-equilibrium behavior and hold promise for applications in quantum technologies. However, exact numerical simulations of their dynamics quickly become unfeasible due to the exponential growth of the Hilbert space with system size. Semiclassical, phase-space approaches -- such as the Truncated Wigner approximation (TWA) -- provide computationally efficient alternatives by capturing leading-order quantum fluctuations. In this paper, we present a comprehensive overview of how to tackle problems in many-body quantum optics using phase-space methods. We derive the exact partial differential equation governing many-body dissipative evolution in any phase-space representation and discuss the approximations that yield the dissipative TWA proposed by Mink and Fleischhauer [SciPost Phys. 15, 233 (2023)]. We find that P and Q distributions are generally suboptimal for many-body quantum optics. Additionally, we extend the formalism to calculate multi-time correlation functions, thereby broadening the scope of phase-space simulations of open spin systems to include coherence and spectral properties, as well as directional correlations of collectively radiating emitters. These developments provide valuable tools for investigating exotic light sources driven by collective dissipation, driven-dissipative phase transitions, and a wealth of many-body phenomena arising in state-of-the-art experimental platforms.
We apply a recently developed theory for metastability in open quantum systems to a one-dimensional dissipative quantum Ising model. Earlier results suggest this model features either a nonequilibrium phase transition or a smooth but sharp crossover, where the stationary state changes from paramagnetic to ferromagnetic, accompanied by strongly intermittent emission dynamics characteristic of first-order coexistence between dynamical phases. We show that for a range of parameters close to this transition or crossover point the dynamics of the finite system displays pronounced metastability, i.e., the system relaxes first to long-lived metastable states before eventual relaxation to the true stationary state. From the spectral properties of the quantum master operator we characterize the low-dimensional manifold of metastable states, which are shown to be probability mixtures of two, paramagnetic and ferromagnetic, metastable phases. We also show that for long times the dynamics can be approximated by a classical stochastic dynamics between the metastable phases that is directly related to the intermittent dynamics observed in quantum trajectories and thus the dynamical phases.
We propose a semiclassical framework for solving open quantum dynamics in driven-dissipative spin systems. Our method consists of generalized spin-wave approximations tailored to describing quantum trajectories unravelled from the master equation, and generically applies to regimes beyond the reach of conventional spin-wave theories, including short-range interactions and local quantum jumps, enabling the efficient simulation of large-scale interacting spins. We illustrate the versatility of our framework by studying a variable-range driven-dissipative Ising model on a 2D lattice. When the dissipation acts along the drive axis, we find a continuous phase transition breaking the Z2 symmetry, and demonstrate that the interaction range, when tuned from fully-connected to nearest-neighbour, profoundly alters the universality class of the criticality. With the dissipation along the interaction axis, we show the emergence of a first-order transition. Demonstrated with both state-diffusion and quantum-jump types of trajectory dynamics, our framework provides a powerful toolbox for the efficient semiclassical description of non-equilibrium dynamics and many-body phases in spin systems.