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No-go theorem for photon condensation: a non-perturbative extension to first-order phase transitions

Gian Marcello Andolina, F. M. D. Pellegrino, Alberto Mercurio, Omar Di Stefano, Marco Polini, Salvatore Savasta

Published Apr 19, 2021• 5 citations
OpenAlex

Abstract

Equilibrium phase transitions between a normal and a photon condensate state (also known as superradiant phase transitions) are a highly debated research topic, where proposals for their occurrence and no-go theorems have chased each other for the past four decades. Previous no-go theorems have demonstrated that gauge invariance forbids second-order phase transitions to a photon condensate state when the cavity-photon mode is assumed to be spatially uniform. First-order phase transitions were previously considered as possible paths to bypass the no-go theorem. In particular, it has been theoretically predicted that a collection of three-level systems coupled to light can display a first-order phase transition to a photon condensate state. Here, we lay down a general no-go theorem, which forbids first-order as well as second-order superradiant phase transitions in a spatially-uniform cavity field. By using the tools of lattice gauge theory, we then derive a fully gauge-invariant model for three-level systems coupled to light. In agreement with the general theorem, our model does not display photon condensation.

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Equilibrium phase transitions between a normal and a photon condensate state (also known as superradiant phase transitions) are a highly debated research topic, where proposals for their occurrence and no-go theorems have chased each other for the past four decades. Previous no-go theorems have demonstrated that gauge invariance forbids second-order phase transitions to a photon condensate state when the cavity-photon mode is assumed to be spatially uniform. First-order phase transitions were previously considered as possible paths to bypass the no-go theorem. In particular, it has been theoretically predicted that a collection of three-level systems coupled to light can display a first-order phase transition to a photon condensate state. Here, we lay down a general no-go theorem, which forbids first-order as well as second-order superradiant phase transitions in a spatially-uniform cavity field. By using the tools of lattice gauge theory, we then derive a fully gauge-invariant model for three-level systems coupled to light. In agreement with the general theorem, our model does not display photon condensation.
Whether or not superradiant phase transitions (photon condensates) can occur in equilibrium systems is a long-debated open question. The realization of photon condensation in a spatially uniform cavity field is forbidden by a ``no-go'' theorem stemming from gauge invariance. Here, bypassing the theorem by considering a spatially varying electromagnetic field, the authors derive a series of criteria for the occurrence of photon condensation. The cases of both three-dimensional and two-dimensional electronic systems are analyzed. This work paves the way for the realization of equilibrium photon condensation in solid-state systems.
In spite of decades of work it has remained unclear whether or not superradiant quantum phases, referred to here as photon condensates, can occur in equilibrium. In this Rapid Communication, we first show that when a nonrelativistic quantum many-body system is coupled to a cavity field, gauge invariance forbids photon condensation. We then present a microscopic theory of the cavity quantum electrodynamics of an extended Falicov-Kimball model, showing that, in agreement with the general theorem, its insulating ferroelectric and exciton condensate phases are not altered by the cavity and do not support photon condensation.
The possibility of a superradiant phase transition in light-matter systems is the subject of much debate, due to numerous apparently conflicting no-go and counter no-go theorems. Using an arbitrary-gauge approach we show that a unique phase transition does occur in archetypal many-dipole cavity QED systems, and that it manifests unambiguously via a macroscopic gauge-invariant polarization. We find that the gauge choice controls the extent to which this polarization is included as part of the radiative quantum subsystem and thereby determines the degree to which the abnormal phase is classed as superradiant. This resolves the long-standing paradox of no-go and counter no-go theorems for superradiance, which are shown to refer to different definitions of radiation.
Abstract We investigate the ground state of two physically motivated modifications of the Dicke model. The first modification corresponds to particles whose phase space contains only two states, for example, artificially created qubits. The second modification describes two-level systems that arise as a result of truncating the full Hilbert space of atoms to two levels that are in resonance with the electromagnetic field and are described by the gauge-invariant Dicke model. We demonstrate that the behavior of these systems is qualitatively distinct in both cases. In particular, in the first scenario, a superradiant phase transition (SPT) into a state with a non-zero amplitude of the classical field is possible, while in the second case, the so-called order parameter η = 〈 a ˆ 〉 which would indicate condensation into a coherent state remains zero, meaning that the SPT is forbidden. It resolves the long-standing discussion about the SPT in the Dicke model. At the same time, the average number of photons n ¯ = 〈 a ˆ † a ˆ 〉 ≠ 0 , and the collective excitation in the system manifests as a non-classical ‘squeezed’ state of the field. We analyze the observable characteristics of both systems in a wide range of variation of their parameters.
We consider a model of free fermions in a ladder geometry coupled to a non-uniform cavity mode via Peierls substitution. Since the cavity mode generates a magnetic field, no-go theorems on spontaneous photon condensation do not apply, and we indeed observe a phase transition to a photon condensed phase characterized by finite circulating currents, alternatively referred to as the equilibrium superradiant phase. We consider both square and triangular ladder geometries, and characterize the transition by studying the energy structure of the system, light-matter entanglement, the properties of the photon mode, and chiral currents. The transition is of first order and corresponds to a sudden change in the fermionic band structure as well as the number of its Fermi points. Thanks to the quasi-one dimensional geometry we scrutinize the accuracy of (mean field) cavity-matter decoupling against large scale density-matrix renormalization group simulations. We find that light-matter entanglement is essential for capturing corrections to matter properties at finite sizes and for the description of the correct photon state. The latter remains Gaussian in the the thermodynamic limit both in the normal and photon condensed phases.
We investigate the equilibrium behavior of a superconducting circuit QED system containing a large number of artificial atoms. It is shown that the currently accepted standard description of circuit QED via an effective model fails in an important aspect: it predicts the possibility of a superradiant phase transition, even though a full microscopic treatment reveals that a no-go theorem for such phase transitions known from cavity QED applies to circuit QED systems as well. We generalize the no-go theorem to the case of (artificial) atoms with many energy levels and thus make it more applicable for realistic cavity or circuit QED systems.