Dissipative phase transitions in $n$-photon driven quantum nonlinear resonators

Fabrizio Minganti1,2, Vincenzo Savona1,2, and Alberto Biella3

1Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
2Center for Quantum Science and Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
3Pitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Università di Trento, I-38123 Trento, Italy

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We investigate and characterize the emergence of finite-component dissipative phase transitions (DPTs) in nonlinear photon resonators subject to $n$-photon driving and dissipation. Exploiting a semiclassical approach, we derive general results on the occurrence of second-order DPTs in this class of systems. We show that for all odd $n$, no second-order DPT can occur while, for even $n$, the competition between higher-order nonlinearities determines the nature of the criticality and allows for second-order DPTs to emerge only for $n=2$ and $n=4$. As pivotal examples, we study the full quantum dynamics of three- and four-photon driven-dissipative Kerr resonators, confirming the prediction of the semiclassical analysis on the nature of the transitions. The stability of the vacuum and the typical timescales needed to access the different phases are also discussed. We also show a first-order DPT where multiple solutions emerge around zero, low, and high-photon numbers. Our results highlight the crucial role played by $strong$ and $weak$ symmetries in triggering critical behaviors, providing a Liouvillian framework to study the effects of high-order nonlinear processes in driven-dissipative systems, that can be applied to problems in quantum sensing and information processing.

Phase transitions are ubiquitous in nature. They can be triggered by thermal fluctuations competing with energy minimization, leading to abrupt changes in the system's thermodynamic properties. In quantum systems, phase transitions can occur even at zero temperature, where they are characterized by an abrupt change of the system's ground state as a parameter is varied. This concept holds true even when a quantum system is driven away from thermal equilibrium and interacts with its environment. What makes these dissipative phase transitions distinctive is that multiple factors compete to determine the system's phase: driving fields, dissipation, and interactions. In this context, numerous essential questions persist, including how and whether dissipative phase transitions can be observed and the role of driving fields and dissipation in determining their features. In our work, we study the physics of non-linear, driven-dissipative quantum resonators – a paradigmatic model in this field. Motivated by the recent technological advances in the engineering and control of this class of systems, we consider driving and dissipation mechanisms that inject and dissipate a specific number $n$ of photons. We derive the general conditions upon which dissipative phase transitions emerge and describe their main features through a full quantum analysis. We show how the type of driving and dissipation, and in particular the number of photons $n$, determine the nature of the transition and highlight the role that the underlying symmetries of the system play in determining its critical properties. Our findings hold significance both in advancing fundamental knowledge and in the development of quantum information technologies that rely on nonlinear quantum resonators.

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