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Nonlinear dynamics in quantum synchronization and topological transport

Talitha Weiß

Published Jan 1, 2017• 6 citations
OpenAlex

Abstract

In this thesis we investigate two different aspects of nonlinear dynamical behaviour: First, we study quantum synchronization which is based on nonlinear, so-called limit-cycle oscillators. Second, we investigate the influence of nonlinear interactions on topological transport in bosonic systems. We begin with the investigation of the synchronization of two optomechanical systems in the presence of quantum noise. With numerical simulations of the full quantum and a semiclassical model, we identify different phase synchronization regimes, although exact phase locking is prevented due to the inevitable quantum noise. Our main results are noise-induced transitions between different synchronization states, as well as noise-induced bistability, i.e., the appearance of a second synchronization state in the quantumregime, even if in the classical, noiseless limit only a single stable synchronization state exists. We give an overview about the classical-to-quantumtransition and also compare our findings to synchronization in the presence of classical, thermal noise. Subsequently, we study the quantum synchronization dynamics of a quantum Van der Pol oscillator coupled to an external periodic driving. This paradigm system for limit-cycle oscillators allows us to derive an effective, analytical quantum model and identify the different dynamical regimes, like overdamped, underdamped, and notably also quantum-coherent phase motion. We explore the distinct signatures of these different dynamical regimes, e.g., in the spectrum or the squeezing. Moreover, we show that in the remarkable regime of quantum-coherent phase dynamics quantum states with negative Wigner densities (like Schrödinger cat states) are preserved for many oscillations of the system. Our effective model gives insight into how to achieve quantum coherence in synchronization dynamics, while numerical simulations of the full model confirm the predicted behaviour. Finally, we turn to the classical dynamics of a Chern topological insulator in a nonlinear bosonic system. To this end, we study the half Bernevig-Hughes-Zhang model on a square lattice in the presence of a local Kerr nonlinearity. We investigate the stability of a linear edge state in the nonlinear model and present a linear stability analysis to gain insight into the unstable scattering processes enabled by the nonlinearity. In a time evolution of the full nonlinear model, we can show that these processes lead to a spatially periodic modulation of the edge channel on an intermediate timescale. However, eventually we observe that strong nonlinear processes lead to more complicated behaviour and a significant radiation into the bulk.

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•Jul 19, 2018•0 citations
In this thesis we investigate two different aspects of nonlinear dynamical behaviour: First, we study quantum synchronization which is based on nonlinear, so-called limit-cycle oscillators. Second, we investigate the influence of nonlinear interactions on topological transport in bosonic systems. We begin with the investigation of the synchronization of two optomechanical systems in the presence of quantum noise. With numerical simulations of the full quantum and a semiclassical model, we identify different phase synchronization regimes, although exact phase locking is prevented due to the inevitable quantum noise. Our main results are noise-induced transitions between different synchronization states, as well as noise-induced bistability, i.e., the appearance of a second synchronization state in the quantumregime, even if in the classical, noiseless limit only a single stable synchronization state exists. We give an overview about the classical-to-quantumtransition and also compare our findings to synchronization in the presence of classical, thermal noise. Subsequently, we study the quantum synchronization dynamics of a quantum Van der Pol oscillator coupled to an external periodic driving. This paradigm system for limit-cycle oscillators allows us to derive an effective, analytical quantum model and identify the different dynamical regimes, like overdamped, underdamped, and notably also quantum-coherent phase motion. We explore the distinct signatures of these different dynamical regimes, e.g., in the spectrum or the squeezing. Moreover, we show that in the remarkable regime of quantum-coherent phase dynamics quantum states with negative Wigner densities (like Schrödinger cat states) are preserved for many oscillations of the system. Our effective model gives insight into how to achieve quantum coherence in synchronization dynamics, while numerical simulations of the full model confirm the predicted behaviour. Finally, we turn to the classical dynamics of a Chern topological insulator in a nonlinear bosonic system. To this end, we study the half Bernevig-Hughes-Zhang model on a square lattice in the presence of a local Kerr nonlinearity. We investigate the stability of a linear edge state in the nonlinear model and present a linear stability analysis to gain insight into the unstable scattering processes enabled by the nonlinearity. In a time evolution of the full nonlinear model, we can show that these processes lead to a spatially periodic modulation of the edge channel on an intermediate timescale. However, eventually we observe that strong nonlinear processes lead to more complicated behaviour and a significant radiation into the bulk.
Recently, several studies have investigated synchronization in quantum-mechanical limit-cycle oscillators. However, the quantum nature of these systems remained partially hidden, since the dynamics of the oscillator's phase was overdamped and therefore incoherent. We show that there exist regimes of underdamped and even quantum-coherent phase motion, opening up new possibilities to study quantum synchronization dynamics. To this end, we investigate the Van der Pol oscillator (a paradigm for a self-oscillating system) synchronized to an external drive. We derive an effective quantum model which fully describes the regime of underdamped phase motion and additionally allows us to identify the quality of quantum coherence. Finally, we identify quantum limit cycles of the phase itself.
Synchronization is a collective phenomenon in which interacting nonlinear oscillators develop a persistent relation between their phases and frequencies. In quantum systems, its description requires connecting the classical dynamics of limit-cycle oscillators with quantum fluctuations, dissipation, and the notion of phase. In this tutorial review, we offer a systematic framework for analyzing synchronization in classical and quantum dynamical systems. Starting from amplitude-phase dynamics, we introduce successively topics of relevance, such as phase locking, the Adler equation, Arnold tongues, and the Kuramoto model, and use these concepts to analyze coupled oscillators and their stability. We then formulate the corresponding description for quantum limit-cycle oscillators using Lindblad master equations and phase-space methods, with explicit examples of two and many coupled oscillators. We further discuss noise, synchronization measures, and the extreme quantum regime, including synchronization in spin systems. The emphasis throughout is on analytical derivations and a practical methodology for identifying synchronized states, determining their stability, and characterizing synchronization in quantum systems. The review is designed as a self-contained guide from which we hope that students can learn the essential concepts and techniques, while researchers can readily find the analytical tools and methodological insights needed to approach new synchronization problems.
Limit-cycle oscillators are the basic building blocks for synchronization; yet, the notion of a quantum limit cycle has remained unclear. Here, we study quantum limit cycles and synchronization in the presence of continuous heterodyne measurement. The resulting quantum trajectories, i.e., time evolutions of the quantum state conditioned on the measurement outcome, make the quantum limit cycles apparent. We focus on the paradigmatic model of the quantum van der Pol oscillator and on two-level systems. Our work provides insights into limit cycles in quantum systems, emphasizing their similarity to classical limit cycles subject to noise. Additionally, we connect theoretical measures of quantum synchronization to quantities experimentally accessible via heterodyne detection.
Parametrically driven Kerr nonlinear quantum oscillator plays a central role in quantum optics, superconducting circuits, and in engineered light-matter interactions. For the development of future quantum technologies, it is crucial to understand the behavior of this quantum system under realistic dissipative conditions. The objective of this thesis is to investigate the phase-space dynamics of a dissipative Kerr nonlinear oscillator under different driving regimes and corresponding dissipation channels, with a particular focus on the emergence of nonclassical features in phase space. The aim is also to understand how nonlinearity and different driving mechanisms influence quantum properties such as coherence, nonclassicality, and phase-space structure. The analysis was carried out using the Wigner phase-space formalism and numerical simulations of the Lindblad master equation. The results show that in the case of finite nonlinearity the coherence time is prolonged compared to the harmonic case. In the single photon regime, the interplay between the photon drive and nonlinearity can lead to coherence revivals after which the system relaxes into a squeezed steady state. In the two photon and dissipation regime, the system exhibits parity-protected dynamics, leading to very long coherence times.
We study quantum van der Pol oscillator in which single-photon pumping competes with a nonlinear two-photon dissipative channel to produce a quantum limit cycle, and the question of how two such oscillators with non-identical intrinsic frequencies synchronize in the sense of phase locking when coupled dissipatively to one another. For a single oscillator, starting from a Lindblad description of the dynamics, we derive the exact Wigner-function evolution equation, and show that its Fokker--Planck approximation admits in the appropriate regime an equivalent classical Itô stochastic description that reproduces remarkably well the full Lindblad dynamics. Notably, the stochastic approach provides an efficient means of capturing system dynamics in the semiclassical regime, where a large number of energy eigenstates contribute and direct numerical simulations become computationally demanding. Restricting to the three lowest Fock states and using a parity symmetry, we obtain in closed form the stationary-state density matrix and its Wigner-function representation that displays pronounced nonclassical features. For two non-identical oscillators, we study the full Lindblad dynamics numerically and characterize synchronization through the stationary-state order parameter as a function of frequency mismatch and dissipative coupling. We find that increasing the effective nonlinear dissipative strength suppresses the synchronized region. Restricting the two oscillators to the three-level qutrit subspace, a parity symmetry reduces the stationary-state problem to an exact 414141-dimensional linear system, allowing to compute the synchronization order parameter. In this reduced description, results in the limit of large dissipation agree closely with those from the full dynamics, while exhibiting in the limit of small dissipation a double-peak synchronization structure accompanied by a phase reversal.
Synchronisation has been explored extensively for classical systems from the mid-1600s, and is a phenomenon widely understood in many diverse systems. In the age of technology miniaturisation, there are great incentives to develop an understanding of quantum synchronisation, where quantum fluctuations become the dominant source of noise and non-classical features become important. This thesis considers different types of synchronisation in systems that can display highly non-linear, quantum behaviours. Phase synchronisation for a micromaser coupled to an external drive is initially considered, and it is found that for low photon numbers a semiclassical approximation underestimates phase synchronisation, but a perturbation theory can be accurate for sufficiently weak coupling in either a quantum or semiclassical regime. When a small detuning between the drive and micromaser is introduced, frequency entrainment is also found to exist. Then two micromasers are coupled with each other, in two different configurations, and synchronisation through a relative phase preference is found for each. A semiclassical approximation once again underestimates the phase synchronisation. Entanglement and mutual information are found to behave differently to the relative phase preference depending on each system's parameters. Finally, a mean-field theory is considered for many coupled van der Pol oscillators and micromasers respectively. As the coupling between the oscillators is increased, a critical point is reached and the systems undergo a transition to a phase synchronised state. This bistability appears via a saddle-node bifurcation, and in addition to examining a variety of properties before and after the transition the critical scaling behaviour was investigated close to the transition.