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.
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.
This thesis investigates the physics of strongly interacting driven–dissipative bosonic systems in one dimension, combining field-theoretical approaches with various numerical simulation techniques. While weakly interacting condensates are accurately described by a generalized Gross–Pitaevskii equation, the strongly correlated regime of the same microscopic models remains far less understood, particularly out of equilibrium. This thesis aims to bridge this gap by developing controlled quantum-many-body simulations for open systems governed by Lindblad dynamics, as well as investigating the corresponding action under the functional renormalization group.In the first part, we formulate a field-theoretical description of driven–dissipative one-dimen-sional bosons using the Schwinger–Keldysh formalism and the functional renormalization group (fRG).Two models are analyzed: (i) a driven-dissipative condensate whose phase dynamics is expected to exhibit emergent Kardar–Parisi–Zhang (KPZ) behavior, for which we search the emergent KPZ fixed point starting from the bosonic field theory; and (ii) a two-photon–driven system with two-body losses realizing dissipative cat qubits. For the latter, we determine the phase diagram and identify a continuous phase transition between symmetry-broken and symmetric phases, which motivates the construction of an fRG Ansatz capturing this critical behavior.The second part of the thesis focuses on the open system dynamics using exact diagonalization and tensor-network methods. For the Bose–Hubbard model with incoherent pump and loss, we characterize the steady state, and study spectral functions across the full parameter space. We show that key ground-state features are absent under dissipation, while the high-temperature unitary spectrum captures aspects of the open-system dynamics. In an extended model with additional two-body and non-local losses, we establish the semi-classical emergence of KPZ scaling and extract the KPZ dynamical exponent from the spectral function. A finite-size analysis exposes the current limitations of the tensor-network methods for reaching the high occupancy regime in the full quantum evolution.Finally, a last project develops a two-loop perturbative RG analysis of a spin-1 chain at its continuous symmetry breaking quantum critical point, revealing critical exponents distinct from the Ising universality class.Together, these results advance the understanding of strongly interacting open quantum systems, elucidating their symmetries, effective field theories, and dynamical properties, and outlining key directions for future theoretical and numerical work.
We review recent results on the coherence and superfluidity of driven dissipative condensates, i.e., systems of weakly interacting nonconserved bosons, such as polariton condensates. The presence of driving and dissipation has dramatically different effects depending on dimensionality and anisotropy. In three dimensions, equilibrium behaviour is recovered at large scales for static correlations, while the dynamical behaviour is altered by the microscopic driving. In two dimensions, for an isotropic system, drive and dissipation destroy the algebraic order that would otherwise exist; however, a sufficiently anisotropic system can still show algebraic phase correlations. We discuss the consequences of this behaviour for recent experiments measuring phase coherence and outline potential measurements that might directly probe superfluidity. Introduction This chapter is dedicated to superfluidity and its relation to Bose-Einstein condensation (BEC), a topic with a long history. Many reviews of the concepts of condensation and superfluidity in thermal equilibrium can be found; see for example Refs. [1, 2, 3, 4]. The focus of this chapter is on how these concepts apply (or fail to apply) to driven dissipative condensates – systems of bosons with a finite lifetime, in which loss is balanced by continuous pumping. We focus entirely on the steady state of such systems, neglecting transient, time-dependent behaviour. Experimentally, the most studied example of a driven dissipative condensate has been microcavity polaritons, an overview of which is given in Ref. [5] and Chapter 4. However, similar issues can arise in many other systems, most obviously photon condensates [6] (see also Chapter 19), magnon condensates [7] (see also Chapters 25–26), and potentially exciton condensates (although typical exciton lifetimes are much longer than for polaritons). Even experiments on cold atoms (see, e.g., Chapter 3) could be driven into a regime in which such physics occurs, when considering continuous loading of atoms balancing threebody losses [8] or atom laser setups [9, 10, 11]. Experiments on polaritons are two-dimensional, and in two dimensions it is particularly important to clearly distinguish three concepts often erroneously treated as equivalent: superfluidity, condensation, and phase coherence. This is because no true Bose-Einstein condensate exists in a homogeneous two-dimensional system.
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.
The study of many-body quantum physics has been recently challenged by the appearance of an increasing number of experimental platforms where genuine quantum phenomena take place notwithstanding the presence of an environment and of dissipation. Exciton-polaritons, lossy atomic and molecular gases, cavity QED arrays, arrays of trapped ions and Rydberg atoms, are only few prominent examples of a long list. The present work is ideally divided into two parts, the former is devoted to methodological results; the latter to the theoretical characterization of experimental data. Whereas the notion of equilibrium has been a fruitful guide to the development of standard many-body physics, these setups are inherently out of equilibrium and their modelisation requires the introduction of new theoretical tools. Several methods for addressing this dissipative out-of-equilibrium dynamics have been proposed, based for instance on quantum trajectories, truncated Wigner expansions, tensor networks, machine learning, but the solution of many-body physics for open quantum systems remains a formidable task. Clearly, techniques developed in the framework of Hamiltonian closed systems are a continuous source of inspiration for novel developments, and in this work we present the generalization of one such technique, the so-called flow equations, to the dissipative framework where the generator of the dynamics is a linear non-Hermitian operator; notable examples include non-Hermitian Hamiltonians and Lindblad master equations. In the second part of this work we focus on two-body lossy dynamics in strongly-correlated cold atomic gases. In particular, we derive an effective description for the long-time dynamics of a spin ½ fermionic system by extending a pre-existing theory for the bosonic case.
In this thesis we explore different phenomena occurring in open quantum many-body systems. This is a research field which is becoming increasingly more important due to the experimental progress towards realization of quantum-simulators and quantum computers using a variety of platforms, from ultracold atoms to superconducting qubit arrays. These systems are inherently driven and open, and it is an ongoing effort to develop theoretical tools and approaches to study their non-equilibrium physics. In the first part of this thesis we explore the different non-equilibrium condensate phases of resonantly interacting bosons in the presence of coherent pump and incoherent losses. This study might be of relevance to existing Rydberg-polariton setups. We derive an effective non-equilibrium field-theory of this model and study its resulting phase diagram. We find a rich phase diagram in- cluding a phase where particles form a condensate of tightly bound molecules. By changing the detuning from the scattering resonance it is possible to drive an Ising phase transition from the molecule condensate phase to a more standard atom condensate phase. In the second part of this thesis we explore the many-body localization (MBL) transition in an open-driven system. Typically, coupling to a bath is expected to destroy localization and transform the sharp MBL transition to a crossover. Here we show how one can use the couplings to non- equilibrium baths in order to detect sharp signatures of the transition, including the divergence of the dynamical exponent in the Griffith regime of the ergodic phase. This is done by solving for the steady-state of the Lindblad quantum-master equation using matrix-product operator tech- niques. Our work here suggests a new scalable numerical approach to study the MBL transition. In the third part of this thesis we study the entanglement properties of measurement trajectories in a free-fermion system subject to dephasing noise. These trajectories describe the evolution of an open system when photons exiting the system are continuously monitored. We are inspired by recent works which studied the entanglement dynamics in hybrid quantum circuits consisting of random-unitary gates and projective measurements. There it was found that a phase transition between a phase with volume-law scaling of entanglement and an area-law phase occurs at a critical value of the measurement rate. In our case we find a new regime, where at weak noise rate the system exhibits logarithmic entanglement scaling, similar to that of (1 + 1)d conformal field- theories. For some measurement protocols, we find a transition to an area-law phase at a critical noise strength.
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.