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Nonclassical states and quantum synchronization of dissipative nonlinear oscillators

Martin Koppenhöfer

Published Jan 1, 2020• 5 citations
Publisher

Abstract

In recent years, significant progress has been made to push micro-, nano-, and optomechanical systems into the quantum regime. The common goal is to demonstrate and control quantum effects in these systems, which enable applications in quantum metrology and quantum information processing. This process is hampered by dissipation, i.e., the interaction of these systems with their environment. \n \n \nIn this thesis, we focus on two different aspects of dissipative nonlinear systems in the quantum regime. In the first part, we study how states with genuinely quantum properties can be generated by a continuous measurement of the interaction between the quantum system and its environment. This approach turns dissipation into a useful tool to generate nonclassical states of light and matter, which have been identified as important resources for quantum-enhanced sensing, quantum communication, and quantum error processing. We discuss the generation of mechanical states with a sub-Poissonian phonon-number distribution in an optomechanical phonon laser beyond the resolved-sideband regime, and we propose a heralded protocol to generate nonclassical states by photon-counting measurements. We apply this protocol to a Kerr nonlinear oscillator and show that it enables the creation of states with a negative Wigner function although the steady-state Wigner function of this system is strictly positive. \n \n \nIn the second part of this thesis, we focus on self-sustained oscillators in the quantum regime. If a weak perturbation is applied to a self-sustained oscillator, the oscillator can adjust its frequency of oscillation. This effect is called synchronization and has been identified as a universal feature of many different complex classical systems, e.g., electrical circuits, biological systems, and power grids. In recent years, several theoretical proposals have been put forward to study synchronization in the quantum regime. However, an experimental demonstration of quantum effects in synchronization has still been missing. \n \n \nWe develop an analytical framework to study the synchronization of a quantum self-sustained oscillator to an external signal. This framework establishes a unified description of the above-mentioned proposals and allows us to identify the quantum-mechanical resource of synchronization. Based on these findings, we discover a novel interference-based quantum synchronization blockade effect and we derive a bound on the maximum degree of synchronization that can be achieved in the quantum regime. The framework also reveals a large freedom in tailoring a quantum system that is able to synchronize. Taking advantage of this freedom, we propose alternative implementations of quantum self-sustained oscillators that reduce the experimental challenges. \n \n \nFinally, we use digital quantum simulation to implement a quantum self-sustained oscillator on a current quantum computer. Applying an external signal to the oscillator, we verify typical features of quantum synchronization, and we demonstrate interference-based quantum synchronization blockade. Our results are the first experimental demonstration of genuinely quantum effects in synchronization and they show that state-of-the-art noisy intermediate-scale quantum computers are powerful enough to implement and study realistic dissipative quantum systems. \n

DOI
10.5451/unibas-007231633

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In recent years, significant progress has been made to push micro-, nano-, and optomechanical systems into the quantum regime. The common goal is to demonstrate and control quantum effects in these systems, which enable applications in quantum metrology and quantum information processing. This process is hampered by dissipation, i.e., the interaction of these systems with their environment. \n \n \nIn this thesis, we focus on two different aspects of dissipative nonlinear systems in the quantum regime. In the first part, we study how states with genuinely quantum properties can be generated by a continuous measurement of the interaction between the quantum system and its environment. This approach turns dissipation into a useful tool to generate nonclassical states of light and matter, which have been identified as important resources for quantum-enhanced sensing, quantum communication, and quantum error processing. We discuss the generation of mechanical states with a sub-Poissonian phonon-number distribution in an optomechanical phonon laser beyond the resolved-sideband regime, and we propose a heralded protocol to generate nonclassical states by photon-counting measurements. We apply this protocol to a Kerr nonlinear oscillator and show that it enables the creation of states with a negative Wigner function although the steady-state Wigner function of this system is strictly positive. \n \n \nIn the second part of this thesis, we focus on self-sustained oscillators in the quantum regime. If a weak perturbation is applied to a self-sustained oscillator, the oscillator can adjust its frequency of oscillation. This effect is called synchronization and has been identified as a universal feature of many different complex classical systems, e.g., electrical circuits, biological systems, and power grids. In recent years, several theoretical proposals have been put forward to study synchronization in the quantum regime. However, an experimental demonstration of quantum effects in synchronization has still been missing. \n \n \nWe develop an analytical framework to study the synchronization of a quantum self-sustained oscillator to an external signal. This framework establishes a unified description of the above-mentioned proposals and allows us to identify the quantum-mechanical resource of synchronization. Based on these findings, we discover a novel interference-based quantum synchronization blockade effect and we derive a bound on the maximum degree of synchronization that can be achieved in the quantum regime. The framework also reveals a large freedom in tailoring a quantum system that is able to synchronize. Taking advantage of this freedom, we propose alternative implementations of quantum self-sustained oscillators that reduce the experimental challenges. \n \n \nFinally, we use digital quantum simulation to implement a quantum self-sustained oscillator on a current quantum computer. Applying an external signal to the oscillator, we verify typical features of quantum synchronization, and we demonstrate interference-based quantum synchronization blockade. Our results are the first experimental demonstration of genuinely quantum effects in synchronization and they show that state-of-the-art noisy intermediate-scale quantum computers are powerful enough to implement and study realistic dissipative quantum systems. \n
Classical self-sustained oscillators, which generate periodic motion without periodic external forcing, are ubiquitous in science and technology. The realization of nonclassical self-oscillators is an important goal of quantum physics. We here present the experimental implementation of a quantum van der Pol oscillator, a paradigmatic autonomous quantum driven-dissipative system with nonlinear damping, using a single trapped atom. We demonstrate the existence of a quantum limit cycle in phase space in the absence of a drive and the occurrence of quantum synchronization when the nonlinear oscillator is externally driven. We additionally show that synchronization can be enhanced with the help of squeezing perpendicular to the direction of the drive and, counterintuitively, linear dissipation. We also observe the bifurcation to a bistable phase-space distribution for large squeezing. Our results pave the way for the exploration of self-sustained quantum oscillators and their application to quantum technology.
In a nutshell, a cavity optomechanical system consists of a mechanical oscillator inside an optical cavity. Cavity optomechanics studies the interplay between the light-matter radiation-pressure interaction and the change in the cavity frequency induced by mechanical motion. By leveraging the tools of quantum optics and quantum measurement, cavity quantum optomechanics enables the engineering of nonclassical quantum states of bulk matter. In this thesis, I present protocols for engineering such nonclassical quantum states in Part II. First, I introduce a scheme to ‘grow’ a quantum kitten state of a mechanical oscillator into a macroscopic Schrödinger cat state using a series of nonlinear pulsed optomechanical interactions. Next, I detail our proposed schemes to generate and verify entanglement between two mechanical oscillators. Moving on from pulsed optomechanics, I describe the theory behind an experiment that achieved multi-phonon subtraction from a mechanical oscillator in a continuously driven Brillouin optomechanical system. In the final chapter of Part II, I provide a theoretical preview of an experiment that investigates the behaviour of the mechanical oscillator when no light is detected at the output. In Part III, I introduce a new framework for nonlinear cavity quantum optomechanics that incorporates the intrinsic nonlinearity of the radiation-pressure interaction, the nonlinearity from the cavity enhancement, and higher-order mechanical position terms in the expansion of the cavity frequency. I then propose a scheme for measuring mechanical position using general-dyne detection, applicable to pulsed and continuous measurements. This predicts a new regime of operation, the “stochastic Gaussian” regime. Finally, I discuss a protocol for deterministic mechanical squeezing that utilizes the cavity response.
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.
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.
Can synchronization -- the widespread spontaneous emergence of coordinated dynamics in classical nonlinear systems -- also occur in the quantum regime? This question has recently sparked intense research into collective behavior and temporal self-organization in quantum systems. Typical quantum features such as the linearity of time evolution and the presence of quantum noise seem to hinder the appearance of synchrony at the microscopic level. At the same time, quantum coherence and quantum correlations may provide novel mechanisms for enhancing synchronization beyond its classical counterpart. We here survey recent theoretical and experimental advances in quantum synchronization, ranging from the characterization of synchronous oscillations and genuinely nonclassical forms of synchrony to many-body synchronization on quantum networks.
Entanglement had played a crucial role in developing frontier technologies as a critical resource, for instance, in quantum teleportation and quantum sensing schemes. Notably, thanks to the ability to cool down the vibrational modes of mechanical oscillators to its quantum regime, entanglement between mechanical modes and the production of nonclassical mechanical states have emerged as central resources for quantum technological applications. Thus, proposing deterministic schemes to achieve those tasks is of paramount importance. While the dominant scheme for bipartite mechanical entanglement involves Gaussian optomechanical interactions (linearized regime) to generate two-mode squeezed vacuum states, entangling two-modes exploiting the bare non-Gaussian optomechanical interaction (nonlinear strong single-photon regime) remains less covered. This work proposes an on-demand scheme to engineer phononic non-Gaussian bipartite entanglement in the nonlinear regime by exploiting cavity dissipation. Interestingly, our protocol (operating in the resolved sideband and photon blockade regime) renders the possibility of achieving a high degree of steady-state entanglement. We further show that our deterministic scheme is robust in the presence of decoherence and temperature within state-of-the-art optomechanics, along with the required conditions to obtain non-Gaussianity of the achieved bipartite mechanical steady-state.