This thesis is about the theory of open quantum many-body physics with a particular focus on driven-dissipative photonic and spin lattices. After a review of the main physical platforms and theoretical concepts, we describe our original results. In Chapter 2, we present a theory of the single-mode Kerr model with a time-dependent pump. This model describes a single-mode optical cavity with a third-order optical nonlinearity. In the regime of parameters where the semiclassical analysis shows bistability, we find that a dynamic hysteresis loop appears in the exact solution. The hysteresis area as a function of the sweep time shows a double power-law decay where the second exponent is independent of the system parameters. We show how these effects are related to the emergence of a dissipative phase transition. We also describe concisely the experimental results which have recently confirmed such theoretical predictions. In Chapter 3, we present the corner-space renormalization method. In order to obtain the steady-state density matrix of the lattice, we solve the Lindblad master equation in a subspace of the Hilbert space (the "corner"). The states spanning the corner space are selected iteratively using eigenvectors of the density matrix of smaller lattice systems, merging in real space two lattices at each iteration and selecting M pairs of states by maximizing their joint probability. The accuracy of the results is then improved by increasing the dimension M of the corner space until convergence is reached. The method has been benchmarked on a two-dimensional Bose-Hubbard model with coherent driving. The strength and limitations of the method are critically discussed.In Chapter 4, we investigate a dissipative phase transition in the two-dimensional anisotropic Heisenberg XYZ model. Using the corner-space renormalization method we present a finite-size analysis of steady-state observables. In particular, we show the critical behaviour of the magnetic susceptibility, the entropy growth and the entanglement witnesses, providing a first evaluation of the critical exponents characterizing the transition. A study of the dynamics of finite-size systems is also consistent with a critical slowing down. For comparison, we present the corresponding analysis for one-dimensional arrays, showing the absence of criticality due to the reduced dimension.In Chapter 5, we explore the non-equilibrium photonic phases of a dissipative Bose-Hubbard model with incoherent pumping of coupled two-level systems. Within a Gutzwiller mean-field approach, we determine the steady-state phase diagram of the system. We predict a second-order phase transition between an incompressible Mott-like phase and a coherent delocalized phase.
This thesis is about the theory of open quantum many-body physics with a particular focus on driven-dissipative photonic and spin lattices. After a review of the main physical platforms and theoretical concepts, we describe our original results. In Chapter 2, we present a theory of the single-mode Kerr model with a time-dependent pump. This model describes a single-mode optical cavity with a third-order optical nonlinearity. In the regime of parameters where the semiclassical analysis shows bistability, we find that a dynamic hysteresis loop appears in the exact solution. The hysteresis area as a function of the sweep time shows a double power-law decay where the second exponent is independent of the system parameters. We show how these effects are related to the emergence of a dissipative phase transition. We also describe concisely the experimental results which have recently confirmed such theoretical predictions. In Chapter 3, we present the corner-space renormalization method. In order to obtain the steady-state density matrix of the lattice, we solve the Lindblad master equation in a subspace of the Hilbert space (the "corner"). The states spanning the corner space are selected iteratively using eigenvectors of the density matrix of smaller lattice systems, merging in real space two lattices at each iteration and selecting M pairs of states by maximizing their joint probability. The accuracy of the results is then improved by increasing the dimension M of the corner space until convergence is reached. The method has been benchmarked on a two-dimensional Bose-Hubbard model with coherent driving. The strength and limitations of the method are critically discussed.In Chapter 4, we investigate a dissipative phase transition in the two-dimensional anisotropic Heisenberg XYZ model. Using the corner-space renormalization method we present a finite-size analysis of steady-state observables. In particular, we show the critical behaviour of the magnetic susceptibility, the entropy growth and the entanglement witnesses, providing a first evaluation of the critical exponents characterizing the transition. A study of the dynamics of finite-size systems is also consistent with a critical slowing down. For comparison, we present the corresponding analysis for one-dimensional arrays, showing the absence of criticality due to the reduced dimension.In Chapter 5, we explore the non-equilibrium photonic phases of a dissipative Bose-Hubbard model with incoherent pumping of coupled two-level systems. Within a Gutzwiller mean-field approach, we determine the steady-state phase diagram of the system. We predict a second-order phase transition between an incompressible Mott-like phase and a coherent delocalized phase.
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 present a theoretical method to study driven-dissipative correlated quantum systems on lattices with two spatial dimensions (2D). The steady-state density matrix of the lattice is obtained by solving the master equation in a corner of the Hilbert space. The states spanning the corner space are determined through an iterative procedure, using eigenvectors of the density matrix of smaller lattice systems, merging in real space two lattices at each iteration and selecting M pairs of states by maximizing their joint probability. The accuracy of the results is then improved by increasing the dimension M of the corner space until convergence is reached. We demonstrate the efficiency of such an approach by applying it to the driven-dissipative 2D Bose-Hubbard model, describing lattices of coupled cavities with quantum optical nonlinearities.
This thesis is devoted to the theoretical study of driven-dissipative many-body quantum systems with a particular focus on advanced computational methods. After reviewing several state-of-the-art numerical methods to determine the steady-state and dynamics of open quantum systems described by a Lindblad master equation for the density matrix, this thesis manuscript presents two original methods: (i) an optimal unraveling of stochastic quantum trajectories for disordered systems; (ii) a variational neural network method. The stochastic methods have been applied to investigate the driven-dissipative Bose-Hubbard lattice model that can for instance describe lattices of coupled nonlinear electromagnetic resonators such those obtained with semiconductor microcavities or superconducting quantum circuits. Our study has shown that a critical slowing down of the dynamics occurs for some parameters in dimensions higher than one. The phenomena is linked to a first-order dissipative phase transition and to lattice-level collective bistability. Furthermore, we reveal the role of disorder on the spatial correlation properties of these phenomena. In the last part of the thesis, we present the strong connection between fundamental concepts of machine learning and variational approaches in quantum physics. We present a novel computational method for open quantum systems based on neural network ans\"{a}tze that is compatible with a stochastic minimisation of a cost function through Monte Carlo sampling. As a first application, we have applied this method to the dissipative transverse quantum Ising model. Exciting perspectives and possible developments are also drawn in the concluding remarks.
Present-day experiments have started to couple traditional simulations of ultracold atom experiments with quantum light-fields in cavities. This has provided a wealth of op- portunities to enlarge the number of interaction potentials: cavity mediated long-range interactions compete with kinetic energies, longitudinal fields, short-ranged collisional or magnetic spin-spin interactions. The intracavity many-body lattice models often have to be maintained far from equilibrium through the presence of external driving lasers that help to boost and engineer the various interaction potentials. The steady influx of energy is compensated by a steady stream of energy out of the atom-cavity system for example by photon losses or atomic spontaneous emission. Several experiments have demonstrated that such environments can give rise to new competing quantum phases. \nBut this long-standing ambition to push for models with tailorable interaction potentials can bring with it also considerable challenges in their theoretical description, since sponta- neous symmetry breaking transitions in many body lattice systems coupled to dynamical light-fields with single-photon character occur in the presence of drive and dissipation for the photonic force carriers. This clearly calls for model systems where the above men- tioned interplay of interactions, drive, dissipation and cooperative many-body behaviour can be theoretically studied to provide simple, experimentally verifiable predictions. \nThe Dicke model is, through its simplicity (an exactly solved ferromagnet with infinite range atom-atom interactions mediated by a single cavity mode), an exceptionally well- suited candidate. As the generic model for atom-light interactions, it has been experimen- tally realized in a variety of modern quantum optical systems, highlighting its relevance for present-day research. The Dicke model is also highly versatile itself. It has been extended into the dissipative realm, was promoted to account for multiple optical light modes and was used to describe multiple, coupled single-mode cavity structures. It was adapted to treat spin-selective coupling to a cavity to describe superradiance phase tran- sitions in multi-level atomic systems. Moreover, it was realised in electronic circuits where the dipole coupling of real atoms to single mode fields is replaced by a capacitive cou- pling of artificial atoms to a resonator mode. This illustrates that the Dicke model and its extended variants are ’future-proof’ and continue to be of relevance for fundamental light-matter interactions and for driven-dissipative phase transitions. \nIn this thesis, we investigate magnetic phase transitions in driven-dissipative atomic en- sembles interacting with quantum light. We present three research projects on variants of cooperative radiation of an ensemble of laser driven two-level atoms in a single mode optical cavity, as described by the Dicke model. \nThroughout the chapters 2,3 and 4 that contain the main body of research of this thesis, we investigate phase transitions between non-equilibrium stationary states in engineered quantum-optical systems each of which extends the conventional Dicke model physics. As a starting point, we map the quantum equations of motion onto a set of semiclassical nonlinear stochastic equations and analyse their stationary states and instabilities with master equations for atomic spin and photon mean-field amplitudes. These are used to obtain experimentally relevant parameters such as critical atom-light couplings for phase transitions, phase diagrams and properties of stationary non-equilibrium states in addition to cavity output spectra that identify the imprint of magnetic correlations in the light-field. \nIn chapter 2, we help resolve a discrepancy between earlier experimental investigations of the critical atom-light coupling strength for the superradiance transition in the Dicke model: higher external pumping strengths than theoretically predicted were needed to observe a coherent, superradiant state of the light field in an optical cavity. By including incoherent spontaneous emission of atomic excitations, we extend the dissipative Dicke model to a two loss channel variant containing both photon leakage and atomic decay that reproduces the experimentally observed critical atom-light coupling. \nRecent experiments have started to interface quantum many body lattice models with coherent cavity fields, thereby allowing to realize new quantum phases through competing atom-cavity and atom-atom interactions. In chapter 3, we consider a simplified model for such a set-up where a single quantized mode of the light-field interacts with an ensemble of Rydberg-dressed atoms inside a high finesse optical cavity. This model provides a base case for further studies of quantum magnets in optical cavities. At the heart of this model is a competition of short- (dipolar atom-atom) and long-range (atom-light) interactions at the Hamiltonian level in the presence of both spontaneous emission and photon leak- age through the cavity mirrors. We show that different magnetic phases can coexist with coherent atomic radiation and provide clear experimental signatures to identify the mag- netic structure and intra-cavity dynamics. We suggest an experimental level-scheme for a quantum optical implementation of our model. \nIn chapter 4 we consider a generic, collective decay for many-body excitations in the paradigmatic Dicke model. This extension drastically enriches the dynamics as it induces a bicritical point and a bistable regime dominated by true non-equilibrium fluctuations that induce a dissipative first-order phase transition that can only be resolved by including finite fluctuation corrections with the help of stochastic Langevin equations. We investigate the hysteretic response to time-dependent ramps of the atom-light coupling. Discontinuous first-order phase transitions where metastable states coexist in a hysteresis domain have been investigated in recent dissipative quantum-optical experiments. We review noise- activation far from thermal equilibrium in chapter 5.
The Bose-Hubbard model provides a foundational paradigm for exploring quantum phase transitions in many-body systems. This thesis presents a systematic investigation into this model and its extensions, designed to provide a clear path for understanding both the rich underlying physics and the sophisticated numerical methods required for its study. Our approach is sequential: we begin with the standard one-dimensional Bose-Hubbard model, introduce complexity via the extended model with long-range interactions, and conclude by analyzing the impact of dissipation in a cavity-coupled open quantum system.To navigate these regimes, we implement and apply a suite of powerful computational techniques, including Density Matrix Renormalization Group (DMRG), finite-temperature Quantum Monte Carlo (QMC), and Mean-Field Theory (MFT). A significant component of this work is the detailed exposition of the implementation and logic of these numerical methods. Through their comparative application, we precisely characterize the equilibrium phase diagrams and benchmark the capabilities of each method. Crucially, our analysis of the open-system dynamics reveals that dissipation replaces the equilibrium phase diagram with one governed by non-equilibrium steady states (NESS), where the system evolves to either a stable solution or a persistent limit cycle.As such, this thesis is intended to serve a dual purpose: it is both a rigorous physical study of quantum phase transitions in a condensed matter system and a selfcontained pedagogical guide for researchers beginning to explore the phenomenology of quantum phase transitions through numerical schemes.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.