Nonclassical states and quantum synchronization of dissipative nonlinear oscillators

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. In 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. In 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. We 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. Finally, 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