## Doctoral Thesis

Optomechanical (om) systems are characterized by their nonlinear light-matter interaction. This is responsible for unique dynamic properties and allows the detection of a variety of classical and quantum mechanical phenomena on a microscopic as well as on a macroscopic scale. In this work we have studied the dynamic behavior of two laser-driven om systems, the single om cell ("cavity optomechanics / membrane-in-the-middle setup") and a two-dimensional hexagonal array of these cells ("om graphene"). The first case was motivated by the possibility to detect the transition from quantum mechanics to classical mechanics directly on the basis of the dynamic behavior. For this we focus on multistability effects of the optical and mechanical degrees of freedom, that are modeled by harmonic oscillators. Our description is based on the quantum optical master equation, which takes into account the environmental interaction assuming a vanishing temperature. As a consequence of decoherence, the dynamics occur near the semiclassical limit, i.e. it is characterized by quantum fluctuations. The quantum-to-classical transition is realized formally by rescaling the equations of motion. In the classical limit, quantum fluctuations disappear and the mean field equations were evaluated by analytical and numerical methods. We found that classical multistability is characterized by stationary signatures on the route to chaos, as well as by the coexistence of single-periodic orbits for the mechanical degree of freedom. The latter point was extensively evaluated by means of a self-consistent approach. For the dynamics in the quantum regime quantum fluctuations cannot be neglected. For this purpose, the master equation was solved by means of a numerical implementation of the Quantum State Diffusion (QSD) method. Based on Wigner and autocorrelation functions, we were able to show that quantum multistability is a dynamic effect: chaotic dynamics is suppressed and there is a time-dependent distribution of the phase space volume on classical simple-periodic orbits. The results can be interpreted within a semiclassical picture, which makes use of the single QSD quantum trajectory. Accordingly, the quantum-classical transition is explained as a time-scale effect, which is determined by tunneling probabilities in an effective mean-field potential. The subject of the second part of the work is the transport of low-energy Dirac quasiparticles in om graphene, propagating as light and sound waves. For this purpose, we investigated the scattering of a plane light wave by laser-induced photon-phonon coupling planar and circular barriers. The starting point is the om Dirac equation, which results from the continuum approximation of the Hamiltonian description of the two-dimensional array near the semiclassical limit. This work was motivated by the rich and interesting relativistic transport and tunneling phenomena found for electrons in graphene, which now appear in a new way. The reason is the presence of the new spin degree of freedom, which distinguishes the optical and mechanical excitations. In this spin space, the om interaction can be understood as a potential, which in our analysis consists of a time-independent and a time-dependent sinusoidal part. For the first case of a static barrier, the transport is elastic and is characterized by stationary scattering signatures. After solving the scattering problem via continuity conditions we were able to identify different scattering regimes depending on scattering parameters. In addition to relativistic phenomena such as Klein tunneling, simple parameter variation allows to use the barrier as a resonant light-sound interconverter and angle-dependent emitter. For the oscillating barrier, the transport is inelastic and is characterized by dynamic scattering signatures. To solve the time-periodic scattering problem, we have applied the Floquet theory for an effective two-level system. As a result of the barrier oscillation, photons and phonons can get and give away energy portions in the form of integer multiples of the oscillation frequency. The interference of short (classical) and long-wave (quantum) components leads to mixing of the scattering regimes. This allows to use the barrier as a time-periodic light-sound interconverter with interesting radiation characteristics. In addition, we have argued that the oscillating barrier provides the necessary energetic conditions for detecting zitterbewegung.

Modern cavity QED and cavity optomechanical systems realize the interaction of light with mesoscopic devices, which exhibit discrete (atom-like) energy spectra or perform micromechanical motion. In this thesis we have studied the crossover from the quantum regime to the classical limit of two prototypical models, the Dicke model and the generic optomechanical model. The physical problems considered in this approach range from a ground state phase transition, its dynamical response to general nonequilibrium dynamics including Hamiltonian and driven dissipative chaotic motion. The classical limit of these models follows from the classical limit of at least one of its subsystems. The classical equations of motion result from the respective quantum equations through the application of the semiclassical approximation, i.e., the neglect of quantum correlations. The approach of the results from quantum mechanics to the prediction of the classical equations can be obtained by subsequently decreasing the respective scaling parameter. In order to obtain exact results we have utilized advanced numerical methods, e.g., the Lanczos diagonalization method for ground state calculations, the Kernel Polynomial Method for dynamical response functions, Chebyshev recursion for time propagation, and quantum state diffusion for open system dynamics. We have studied the quantum phase transition of the Dicke model in the classical oscillator limit. Our work shows that in this limit the transition occurs already for finite spin length but with the same critical behavior as in the classical spin limit. We have derived an effective model for the oscillator degrees of freedom and have discussed the differences of both classical limits with respect to quantum fluctuations around the mean-field ground state and spin-oscillator entanglement. In this thesis we have proposed a variational ansatz for the Dicke model which extends the mean-field description through the inclusion of spin-oscillator correlations. The ansatz becomes correct in the limit of large oscillator frequency and in the limit of a large spin. For the latter it captures the leading quantum corrections to the classical limit exactly including the spin-oscillator entanglement entropy. We have studied the dynamics of spin and oscillator coherent states in the nonresonant Dicke model at weak coupling. In this regime periodic collapses and revivals of Rabi oscillations occur, which are accompanied by the buildup and decay of atom-field entanglement. The spin-oscillator wave function evolves into a superposition of multiple field coherent states that are correlated with the spin configuration. In our work we provide a description of the underlying dynamical mechanism based on perturbation theory. Our analysis shows that collapse and revival at nonresonance is distinguished from the resonant case treated within the rotating wave approximation by the appearance of two time scales instead of one. We have extended our study of the Dicke dynamics to the case of increasing spin length, as the system approaches the classical spin limit. We described the emergence of collective excitations above the ground state that converge to the coupled spin-oscillator oscillations observed in the classical limit. With increased spin length the corresponding Green functions thus reveal quantum dynamical signatures of the quantum phase transition. For the dynamics at larger coupling and energy, classical phase space drift and quantum diffusion hinders the direct comparison of quantum and classical observables. As we show in our work, signatures of classical quasiperiodic orbits can be identified in the Husimi phase-space functions of the propagated wave function and individual eigenstates with energies close to that of the quasiperiodic orbits. The analysis of the generic optomechanical system complements our study of cavity QED systems by a quantum dissipative system. In this thesis we have shown for the first time, how the route to chaos in the classical optomechanical system takes place, given as a sequence of consecutive period doubling bifurcations of self-induced cantilever oscillations. In addition to the semiclassical dynamics we have analyzed the possibility of chaotic motion in the quantum regime. Our results showed that quantum mechanics protects the optomechanical system against irregular dynamics. In sufficient distance to the semiclassical limit simple periodic orbits reappear and replace the classically chaotic motion. In this way direct observation of the dynamical properties of an optomechanical system makes it possible to pin down the crossover from quantum to classical mechanics.