Phase and amplitude dynamics of quantum self-oscillators

Self-oscillators form a special class of oscillators, generating and maintaining a periodic motion while having some (or complete) independence of the frequency spectrum of oscillations from the spectrum of their power source. Pendulum clocks, brain neurons, fireflies, and cardiac pacemaker cells, are all examples of self-oscillators. Self-oscillations are not limited to the regime of classical physics, but are seen in the quantum regime as well. In both regimes, self-oscillators may demonstrate two intriguing phenomena: (1) Synchronization, a phenomenon in which self-oscillators adjust their rhythm due to weak coupling to a drive or to another self-oscillating systems; (2) Amplitude death, a phenomenon in which two or more coupled self-oscillators approach a stable rest-state. In the work presented in this thesis, we have mostly investigated these phenomena in quantum self-oscillators.
Chapter 2 tries to answer the question ``Are there quantum effects in the synchronization phenomenon, which cannot be modeled classically?" Using a quantum model of a self-oscillator with nonlinearity in its energy spectrum, we have answered this question in the affirmative. We have demonstrated that the anharmonic, discrete energy spectrum of the oscillator leads to multiple resonances in both phase locking and frequency entrainment.
Coupling two quantum anharmonic self-oscillators, we show in Ch. 3 that genuine quantum effects are also expected in the amplitude death phenomenon. This is apparent in the multiple resonances of the mean phonon number of the oscillators, reflecting their quantized nature.
Chapter 4 is concerned with the investigation of the synchronization phenomenon in an experimental system, an optomechanical cell coupled to a drive. In the classical parameter regime, we derive analytical Adler equations describing the synchronization of the optomechanical cell to two different drives: (1) an optical drive and (2) a mechanical drive. We demonstrate numerically that synchronization should also be observed in the quantum parameter regime.
In Ch. 5 we describe our work in the field of Cooper pair splitters, a device consisting of two quantum dots side-coupled to a conventional superconductor. In this work, we go beyond the standard approximation of assuming the quantum dots to have a large charging energy. We derive a low-energy Hamiltonian describing the system, and suggest a scheme for the generation of a spin triplet state shared between the quantum dots, therefore extending the capabilities of the Cooper pair splitter to create entangled nonlocal electron pairs.