We numerically study the dynamics and stationary states of a spin ensemble strongly coupled to a single-mode resonator subjected to loss and external driving. Employing a generalized cumulant expansion approach we analyze finite-size corrections to a semiclassical description of amplitude bistability, which is a paradigm example of a driven-dissipative phase transition. Our theoretical model allows us to include inhomogeneous broadening of the spin ensemble and to capture in which way the quantum corrections approach the semiclassical limit for increasing ensemble size $N$. We set up a criterion for the validity of the Maxwell-Bloch equations and show that close to the critical point of amplitude bistability even very large spin ensembles consisting of up to ${10}^{4}$ spins feature significant deviations from the semiclassical theory.
We numerically study the dynamics and stationary states of a spin ensemble strongly coupled to a single-mode resonator subjected to loss and external driving. Employing a generalized cumulant expansion approach we analyze finite-size corrections to a semiclassical description of amplitude bistability, which is a paradigm example of a driven-dissipative phase transition. Our theoretical model allows us to include inhomogeneous broadening of the spin ensemble and to capture in which way the quantum corrections approach the semiclassical limit for increasing ensemble size N. We set up a criterion for the validity of the Maxwell-Bloch equations and show that close to the critical point of amplitude bistability even very large spin ensembles consisting of up to 104 spins feature significant deviations from the semiclassical theory.
We present a theoretical study of the nonlinear dynamics and stationary states of an inhomogeneously broadened spin ensemble coupled to a single-mode cavity driven by an external drive with constant amplitude. Assuming a sizable number of constituents within the ensemble allows us to use a semiclassical approach and to formally reduce the theoretical description to the Maxwell-Bloch equations for the cavity and spin amplitudes. We explore the critical slowing-down effect, quench dynamics, and asymptotic behavior of the system near a steady-state dissipative phase transition accompanied by a bistability effect. Some of our theoretical findings have recently been successfully verified in a specific experimental realization based on a spin ensemble of negatively charged nitrogen-vacancy centers in diamond strongly coupled to a single-mode microwave cavity [see A. Angerer et al., Sci. Adv. 3, e1701626 (2017)].
Spin ensembles inside a cavity act as light-matter interfaces that play an eminentrole in the development of future quantum devices. An apparent asset of ensemble-based systems is the collective enhancement of the coupling strength, facilitating astrong interaction between the light and matter components. Particularly promisingin this regard, due to their convenient handling and integrability, are solid statespin ensembles such as nitrogen-vacancy centers in diamond, phosphorous donors in silicon, or rare earth doped crystals. Such ensemble-based setups, however, are almost inevitably inhomogeneously broadened as the different spins of the ensemble typically have different local environments leading to a spectral distribution in their transition frequencies. This so-called inhomogeneous broadening is a major source for decoherence in solid state systems and a major challenge to be addressed for future technical applications.In this thesis, we aim to provide the theoretical framework necessary to describeand understand various effects of inhomogeneously broadened spin ensembles coupled to a single-mode cavity—a system that has been realized in numerous different quantum optical setups and in very different parameter regimes.For macroscopic spin ensembles, i.e., ensembles consisting of a vast number ofspins, quantum fluctuations of the individual spins can typically be neglected, allowing a semiclassical description based on Maxwell-Bloch equations. Here, we study two prominent effects in the strongly nonlinear regime namely optical bistability, resembling a first-order phase transition, and critical slowing down. In particular, we show that the onset of bistability not only depends on the width of the spectral spin distribution but also strongly depends on its exact shape. Furthermore, we analyze the transient times of the cavity amplitude which show a power-law divergence when an external parameter is changed in the vicinity of the phase transition.This critical slowdown is demonstrated in collaboration with the group ofJörg Schmiedmayer at the Atominstitut of TU Wien. In the experiment, negativelycharged nitrogen vacancy centers in diamond are strongly coupled to a superconducting microwave resonator showing transient times of up to 11 hours.In mesoscopic spin ensembles with moderate numbers of constituents, quantumfluctuations gain importance, and semiclassical methods fail. We study thiscrossover between the semiclassical and the quantum regime using a cumulant expansion approach to study the semiclassical-to-quantum boundary for inhomogeneously broadened spin ensembles inside a cavity. In addition, we develop a time-adaptivevariational renormalization group method to calculate the full quantumdynamics of spin ensembles containing up to one hundred spins. With the aid ofthis method, we examine mesoscopic spin ensembles whose spin distributions form a comb-shaped structure. These so-called atomic frequency combs have promising properties for the storage of quantum information. On the one hand, we show that arbitrary multi-photon states are absorbed by the comb-shaped spin ensembleand re-emitted into the cavity at periodic time intervals. On the other hand, wedemonstrate that by means of such a spin ensemble a coherent input pulse can beconverted into periodic pulses of non-classical light.Finally, we address spin echoes, which are a fundamental part in nuclear magneticresonance as well as in electron spin resonance. Specifically, we transfer the Hahnecho to the strong coupling regime, where we find a new dynamical phenomenonfeaturing a periodic sequence of self-sustained echoes after a conventional Hahn echo pulse. In collaboration with the group of Hans Hübl at the Technical UniversityMunich, a first demonstration of this intriguing multiecho signature is presented inan electron spin resonance experiment with phosphorus donors in silicon.
In the last couple of years much effort has been dedicated to the development and the investigation of hybrid quantum devices, which combine the advantages of very dissimilar quantum systems for the realization of efficient quantum computation and communication technologies. Particularly interesting in this regard are solid-state quantum memories based on ensembles of spins (e.g. nitrogen vacancy defects in diamond or rare-earth doped crystals) coupled to superconducting microwave cavities. Here the spin-ensemble acts as a robust memory where the collective coupling to the cavity mode allows for the coherent transfer of quantum information. The major downside of ensembles inside solid-state systems is their natural tendency to exhibit inhomogeneous broadening of the transition frequencies, which makes experiments as well as their theoretical description very challenging. On the theoretical side it is primarily the computational complexity due to the exponential growth of the associated Hilbert space with the system size that inevitably calls for approximation schemes. To date, the theoretical description of large spin-cavity systems is mainly limited to weak driving fields, where the number of spin-excitations in the ensemble is negligible, corresponding to a Holstein-Primakoff approximation, or to mean-field approaches, where correlations within the system are neglected. In this thesis we aim to develop a model that accurately accounts for the dynamics of large inhomogeneously broadened spin ensembles, coupled to a single cavity mode, even in the case of strong driving fields. In particular we employ the generalized cumulant expansion method to account for correlations within the system, which allows us to go beyond the Holstein-Primakoff and mean-field approximations. As a first step, we demonstrate the applicability of the cumulant expansion technique to the well-studied Jaynes-Cummings model and investigate its main limitations. We then move on to the Tavis-Cummings model with multiple spins inside the cavity and generalize our approach to ensembles containing a very large number of spins (ca. 10 12). Our model aims not only to provide a proper description of ensemble-based quantum memories, but also to serve as a novel tool for the investigation of interesting new physics arising from cooperative effects in inhomogeneous ensembles.
We investigate two coupled nonlinear cavities that are coherently driven in a dissipative environment. We perform semiclassical, numerical, and analytical quantum studies of this dimer model when both cavities are symmetrically driven. In the semiclassical analysis, we find steady-state solutions with different photon occupations in two cavities. Such states can be considered analogs of the closed system double-well symmetry-breaking states. We analyze the occurrence and properties of these localized states in the system parameter space and examine how the symmetry-breaking states, in the form of a bistable pair, are associated with the single-cavity bistable behavior. In a full quantum calculation of the master equation dynamics that includes quantum fluctuations, the symmetry-breaking states and bistability disappear due to the quantum fluctuations. In the quantum trajectory picture, we observe enhanced quantum jumps and switching, which indicate the presence of the underlying semiclassical symmetry-breaking states. Finally, we present a set of analytical solutions for the steady-state correlation functions by using the complex P representation and discuss its regime of validity.
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.
We theoretically investigate the critical properties of a single driven-dissipative nonlinear photon mode. In a well-defined thermodynamical limit of large excitation numbers, the exact quantum solution describes a first-order phase transition in the regime where semiclassical theory predicts optical bistability. We study the behavior of the complex spectral gap associated with the Liouvillian superoperator of the corresponding master equation. We show that in this limit the Liouvillian gap vanishes exponentially and that the bimodality of the photon Wigner function disappears. The connection between the considered thermodynamical limit of large photon numbers for the single-mode cavity and the thermodynamical limit of many cavities for a driven-dissipative Bose-Hubbard system is discussed.