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Cavity Quantum Electrodynamics with Spin Ensembles: From the Semiclassical to the Quantum Regime

Matthias Zens

Published Nov 1, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.34726/hss.2021.80222

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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.
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)].
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 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 NNN. 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{10}^{4}104 spins feature significant deviations from the semiclassical theory.
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.
This thesis focuses on two problems in circuit quantum electrodynamics. We first investigate theoretically the coupling of a resonator to a continuous distribution of inhomogeneously broadened emitters. Studying this formalism is strongly motivated by recent proposals to use collections of emitters as quantum memories for individual excitations. Such systems benefit from the collective enhancement of the interaction strength, while keeping the relaxation properties of a single emitter. We discuss the influence of the emitters inhomogeneous broadening on the existence and on the coherence properties of the polaritonic peaks. We find that their coherence depends crucially on the shape of the distribution and not only on its width. Taking into account the inhomogeneous broadening allows to simulate with a great accuracy a number of pioneer experimental results on a ensemble of NV centers. The modeling is shown to be a powerful tool to obtain the properties of the spin ensembles coupled to a resonator. We also suggest an original Josephson qubit readout method based on a dc-SQUID with high loop inductance. This system supports a diamond-shape artificial atom where we define logical and ancilla qubits coupled through a cross-Kerr like term. Depending on the logical qubit state, the ancilla is resonantly or dispersively coupled to the resonator, leading to a large contrast in the transmitted microwave signal amplitude. Simulations show that this original method can be faster and have higher fidelity than methods currently used in circuit QED.
When in a quantum optical system the coupling between matter and cavity mode becomes comparable to the bare excitation frequency, we enter a non-perturbative coupling regime, as perturbation theory fails describing the system’s dynamics. While recent advances in Cavity Quantum Electrodynamics allowed to achieve very high values of the coupling strength, thanks to the resonators properties optimization and the employment of solid-state devices, an ever growing interest has been shown about the possibility of significantly modify materials’ properties. It has been demonstrated that the chemical structure of molecules strongly coupled to a photon mode can be altered, which opens the possibility to manipulate and control chemical reactions. The aim of this thesis is to explore non-perturbative regimes on several quantum systems, and to investigate the effects of the coupling upon their properties, such as internal degrees of freedom or electronic states structure. I first developed a novel theory to determine the polariton spectrum of a dipolar ensamble in which a Ising-like dipole-dipole interaction in the 2 non-perturbative regime is considered. A further important focus is the investigation of the saturation effects due to the inclusion of the inter-dipole interaction, and the interplay between the latter and light-matter coupling strength. I also explored specifically the influence of the coupling on the rotational degrees of freedom of an ensemble of two-dimensional freely rotating dipoles, all coupled to a single cavity mode, finding that they are driven by the collective light-matter coupling to undergo a crossover between an isotropic and an aligned phase. I then investigated the case of cavity-embedded doped quantum wells, demonstrating that not only it is possible to couple a discrete cavity mode and bound-to-continuum transitions, but also that a novel bound exciton state appears, induced by the coupling strength. This results shows how light–matter coupling can be used to tune both optical and electronic properties of semiconductor heterostructures beyond those permitted by mere crystal properties. Finally, I explored the physics of an array of THz metamaterial resonators coupled to cyclotron resonances of a two-dimensional electron gas, developing a multiple-mode theory that takes in account the interaction between multiple photon modes mediated by the electrons. My results show that this cross-interaction, due to the strong two-dimensional geometry of the optically active medium, leads to the hybridization of different uncoupled photon modes, and manifests as a visible change of the distribution of the coupled electromagnetic field.