There is widespread interest in many-body quantum systems that exhibit limit-cycle or time-crystalline behaviour. An ideal quantum limit cycle would be realized using fully coherent driving (to minimize noise) and also have a continuous internal symmetry (to ensure generation of monochromatic radiation). While these two requirements may seem incompatible, we introduce in this work a large class of multi-mode bosonic limit cycle models based on coherent parametric driving which possess an O(N) continuous symmetry. Surprisingly, the full quantum dissipative steady state of these models can be found exactly. They exhibit rich physics, including steady state entanglement, reduced phase diffusion and the possibility of realizing quantum limit tori. The basic mechanism we identify provides a unified way to understand how coherent parametric driving can yield symmetry-enriched limit cycles, and also helps us understand related models where the relevant symmetries are weakly broken. The models we study are compatible with a range of different experimental platforms, including quantum optical setups and superconducting quantum circuits.
There is widespread interest in many-body quantum systems that exhibit limit-cycle or time-crystalline behaviour. An ideal quantum limit cycle would be realized using fully coherent driving (to minimize noise) and also have a continuous internal symmetry (to ensure generation of monochromatic radiation). While these two requirements may seem incompatible, we introduce in this work a large class of multi-mode bosonic limit cycle models based on coherent parametric driving which possess an O(N) continuous symmetry. Surprisingly, the full quantum dissipative steady state of these models can be found exactly. They exhibit rich physics, including steady state entanglement, reduced phase diffusion and the possibility of realizing quantum limit tori. The basic mechanism we identify provides a unified way to understand how coherent parametric driving can yield symmetry-enriched limit cycles, and also helps us understand related models where the relevant symmetries are weakly broken. The models we study are compatible with a range of different experimental platforms, including quantum optical setups and superconducting quantum circuits.
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.
Time crystals are a nonequilibrium phase of matter that extend fundamental spontaneous symmetry breaking into the temporal dimension, typically requiring external driving for their realization. Here, we explore the nonequilibrium phase diagram of a two-dimensional dissipative Heisenberg spin system without external coherent or incoherent driving. Through numerical analysis of spin dynamics, we identify nonstationary steady states, some of which are limit cycles with persistent periodic oscillations, while others exhibit chaotic, aperiodic behavior. The emergence of limit cycle steady states breaks the continuous time-translation symmetry of this time-independent many-body system, classifying them as continuous time crystals. We further validate these oscillatory behaviors by testing their stability against local perturbations and assess the robustness of the emergent continuous time crystals by introducing isotropic Gaussian white noise. This work provides insights into the intricate interplay between the dissipation and spin interaction, and opens possibilities for realizing dissipation-induced, heating-immune time crystals. Time crystals represent a nonequilibrium phase of matter without an equilibrium counterpart. The authors investigate the nonequilibrium steadystate phase diagram of a two-dimensional dissipative Heisenberg model in the absence of driving, and identify a continuous time crystalline phase exhibiting persistent oscillations in time.
We present an exact solution in arbitrary dimensions for the steady states of a class of quantum driven-dissipative bosonic models, where a set of modes is subject to arbitrary two-photon driving, single-photon loss, and a global Hubbard (or Kerr)-like interaction. Our solutions reveal a wealth of striking phenomena, including the emergence of dissipative phase transitions, nontrivial mode competition physics and symmetry breaking, and the stabilization of many-body SU(1,1) pair-coherent states. Our exact solutions enable the description of spatial correlations, and are fully valid in regimes where traditional mean-field and semiclassical approaches break down.
We propose a general mechanism for generating limit cycle (LC) oscillations by coupling a linear bosonic mode to a dissipative nonlinear bosonic mode. By analyzing the stability matrix, we show that LCs arise due to a supercritical Hopf bifurcation. We find that the existence of LCs is independent of the sign of the effective nonlinear interaction. The bosonic model can be realized in three-level systems interacting with a quantized light mode as realized in atom-cavity systems. Using such a platform, we experimentally observe LCs in an atom-cavity system with attractive optical pump lattice, thereby confirming our theoretical predictions for the minimal model and interactions needed to generate LCs for a class of driven-dissipative systems.
Time crystals are classified as discrete or continuous depending on whether they spontaneously break discrete or continuous time translation symmetry. Although discrete time crystals have been extensively studied in periodically driven systems, the experimental realization of a continuous time crystal is still pending. We report the observation of a limit cycle phase in a continuously pumped dissipative atom-cavity system that is characterized by emergent oscillations in the intracavity photon number. The phase of the oscillation was found to be random for different realizations, and hence, this dynamical many-body state breaks continuous time translation symmetry spontaneously. Furthermore, the observed limit cycles are robust against temporal perturbations and therefore demonstrate the realization of a continuous time crystal.
Collective bosonic excitations are a fascinating aspect of broken-symmetry correlated phases. A wealth of such phases emerged in tailored moiré heterostructures, where, in addition, new direct knobs of control exist. Our work explores how the associated collective bosonic modes can be directly manipulated and amplified via parametric driving. As we show, parametric driving is closely related to many-body quantum geometry, as it reveals information about the fidelity susceptibility of the underlying ground state with respect to changes in the control knobs of the interacting system. The parametric response hinges on the tunability of the hierarchy of competing ground states, which we show manifests as squeezing of the bosonic mode vacuum. Furthermore, parametrically-driven modes can produce a high-amplitude modulation in the system that could be easily observed, and could also be used to produce a new non-equilibrium prethermal states with different symmetries. We derive a general framework for parametric amplification of collective bosonic modes in broken-symmetry phases, and provide case-studies of current relevance. Elucidating the connection between the dynamics of these excitations and microscopic electronic observables is key to harnessing their immense potential.