Abstract
Phase transitions of thermal systems and the laser threshold were first connected more than forty years ago. Despite the nonequilibrium nature of the laser, the Landau theory of thermal phase transitions, applied directly to the Scully-Lamb laser model (SLLM), indicates that the laser threshold is a second-order phase transition, associated with a $U(1)$ spontaneous symmetry breaking (SSB). To capture the genuine nonequilibrium phase transition of the SLLM (i.e., a single-mode laser without a saturable absorber), here we employ a quantum theory of dissipative phase transitions. Our results confirm that the $U(1)$ SSB can occur at the lasing threshold but, in contrast to the Landau theory and semiclassical approximation, they signal that the SLLM "fundamental" transition is a different phenomenon, which we call Liouvillian spectral collapse; that is, the emergence of diabolic points of infinite degeneracy. By considering a generalized SLLM with additional dephasing, we witness a second-order phase transition, with a Liouvillian spectral collapse, but in the absence of symmetry breaking. Most surprisingly, the phase transition corresponds to the emergence of dynamical multistability even without SSB. Normally, bistability is suppressed by quantum fluctuations, while in this case, the very presence of quantum fluctuations enables bistability. This rather anomalous bistability, characterizing the truly dissipative and quantum origin of lasing, can be an experimental signature of our predictions, and we show that it is associated with an emergent dynamical hysteresis.
Highlights
The study and experimental realization of the lasing transition is a milestone in quantum optics
While hysteresis [40,41,42] and slowing-down properties [43,44,45] have been characterized for first-order dissipative transitions, here we demonstrate a dynamical hysteresis for a second-order phase transition with or without spontaneous symmetry breaking (SSB)
We analyzed the Scully-Lamb laser model (SLLM) transition in the weak-gain saturation (WGS) regime, within the open quantum system framework provided by the spectral properties of the corresponding Liouvillian superoperator
Summary
The study and experimental realization of the lasing transition is a milestone in quantum optics. We demonstrate that the SLLM second-order phase transition is characterized by a Liouvillian spectral collapse. [30], it is explicitly shown that there is a closure of the spectral gap (to be associated with the emergence of a critical timescale in the photon-number evolution) in the limit of a vanishing saturation rate These results corroborate the validity of the Liouvillian analysis of the SLLM transition. This approach enables us to describe more deeply the interplay between the U(1) symmetry and quantum fluctuations in triggering critical timescales In this regard, our analysis shows a novel behavior of dissipative phase transitions [33,34,35,36,37,38,39]. [24], this leads to a second-order phase transition in the absence of SSB Both the semiclassical theory and the Landau approach fail to describe this model. In Appendix B, we provide additional details concerning the lack of SSB in the SLLM using quantum trajectories
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