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.
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.
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.
Hybrid quantum systems, which combines the advantages of different quantum systems, play an essential role in the realisation of quantum computation, especially as quantum memory. This thesis discusses the very promising hybrid quantum system based on the ensemble of negatively charged nitrogen-vacancy centers (NV-centers) in diamond and superconducting microwave cavities. The decoherence of this system, due to the inhomogeneous spin broadening, seems to be the major obstacle of utilising the NV-centers as a quantum information storage. This works focuses on the study of the dynamics of our NV-center spin ensemble strongly coupled to a superconducting resonator driven by pulse sequences with a carrier frequency equaling the cavity frequency. A precise knowledge of the spin broadening, which has a non-Lorentzian distribution, led to the observation of the so called 'cavity protection effect'. Furthermore this work demonstrates the enhancement of the damped Rabi oscillations between the spin ensemble and the cavity mode through appropriate chosing of the microwave pulses, which could change in the order of two magnitudes. Last we also confirmed the theoretical analysis of the crossover from markovian to non-markovian dynamics for our system, which is achieved for increasing coupling strength.
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)].
The object of this PhD Thesis is light and matter quantum interaction including studies on cavity and circuit quantum electrodynamics (CQED and cQED), quantum information theory and entanglement dynamics in open systems. In recent years both theoretical and experimental efforts have been devoted to study problems regarding the transfer of quantum correlations, quantum memories, entanglement protection against decoherence and new regimes accessible in cQED. One of the fundamental model underlying these topics is the well known Jaynes-Cummings (JC) one for light-matter interaction, introduced in quantum optics already in 1963. This paradigmatic model describes the coherent exchange of a single excitation between a qubit (a two-level system) and a mode of a quantized harmonic oscillator. The JC model describes a lot of purely quantum effects such as Rabi oscillations and generation of entangled quantum states, which were confirmed by several sophisticated experiments in CQED. This model has been then applied in many other frameworks. Analytical results can be derived only under the so-called rotating wave approximation (RWA), which allows to neglect the energy contributions coming from the counter-rotating terms, which describe the simultaneous excitation (or de-excitation) of the qubit and the field mode. In this Thesis a first interesting investigation, based on the JC model, is related to the realization of quantum memories. The main aspect is the transfer of generic quantum entangled states of a propagating radiation to some qubits, such as two-level atoms placed in separated optical cavities. Generalizing previous results for bipartite entangled systems, I investigated the more complex case of tripartite systems, where the very quantification of entanglement is still an open problem. This issue is broadened to open systems in contact with a common environment which induces decoherence effects unavoidable in realistic implementations. The main result of this analysis is that it is possible to map in an optimal way a quantum state from radiation to qubits in the unitary dynamics, and even in the presence of a dissipative environment a significative amount of entanglement can be transferred. This study has been carried on with either theoretical calculations or numerical simulations performed adopting the powerful Monte Carlo wave function method. A second important problem for quantum memories faced in this PhD Thesis, regards the protection of qubits entanglement for long enough times in order to implement quantum information tasks. A possible way to obtain this goal is to add an external coherent field driving two or more two-level atoms interacting with a common mode of a cavity electromagnetic field. In fact it is possible to demonstrate that, under certain conditions, an effective Hamiltonian of the whole system involves, together with the familiar JC terms, also the counter-rotating ones. The main feature that comes out from this analysis is the possibility to freeze, during system dynamics, peculiar quantum entangled states of the atomic qubits in so called decoherence-free subspaces. This important property allows to store the entanglement in matter qubits, isolating the quantum state from environment induced decoherence. Though the system includes dissipation, analytical results have been obtained for multipartite entanglement and decoherence by solving the master equation through a projection on a suitable basis and using the method of characteristics. In particular it is possible to monitor the rate of decoherence via measurements of joint atomic probabilities, shading light on the fascinating border between the quantum and the classical world. A last topic investigated in this Thesis concerns the recently developed and quite promising implementation of the JC model well outside CQED, that is cQED. Here superconductive circuits, which play the role of artificial atoms, interact on-chip with transmission line resonators, instead of the usual electromagnetic cavities. These systems have the advantage of being quite easily engineered exploiting the most advanced available technologies. Moreover they allow a great flexibility in handling the relevant parameters, since the inter-system coupling is here of capacitive or inductive nature, instead of the usual electric dipole interaction in CQED. In this way a strong and ultrastrong coupling regime have become available, giving a direct access to the physics beyond the JC model and its possible applications. I first performed a detailed characterization of system dynamics in order to test the main approximations usually introduced to describe superconductive circuits as qubits. Then I described, both analytically and numerically, the dissipative dynamics of a superconductive qubit coupled to a resonator mode in the deep strong regime.
We study the dynamics of a spin ensemble strongly coupled to a single-mode resonator driven by external pulses. When the mean frequency of the spin ensemble is in resonance with the cavity mode, damped Rabi oscillations are found between the spin ensemble and the cavity mode which we describe very accurately, including the dephasing effect of the inhomogeneous spin broadening. We demonstrate that a precise knowledge of this broadening is crucial both for a qualitative and a quantitative understanding of the temporal spin-cavity dynamics. On this basis we show that coherent oscillations between the spin ensemble and the cavity can be enhanced by a few orders of magnitude, when driving the system with pulses that match special resonance conditions. Our theoretical approach is tested successfully with an experiment based on an ensemble of negatively charged nitrogen-vacancy centers in diamond strongly coupled to a superconducting coplanar single-mode waveguide resonator.