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arXiv:2302.09964 · Physical Review Research

Photon condensation, Van Vleck paramagnetism, and chiral cavities

Alberto Mercurio, Gian Marcello Andolina, F. M. D. Pellegrino, Omar Di Stefano, Pablo Jarillo‐Herrero, Claudia Felser, Frank H. L. Koppens, Salvatore Savasta, Marco Polini

Published Mar 19, 2024• 15 citations
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Abstract

We introduce a gauge-invariant model of planar, square molecules coupled to a quantized spatially varying cavity electromagnetic vector potential Â(r) . Specifically, we choose a temporally cavity hosting a uniform magnetic field B̂ , as this is the simplest instance in which a transverse spatially varying Â(r) is at play. We show that when the molecules are in the Van Vleck paramagnetic regime, an equilibrium quantum phase transition to a photon condensate state occurs. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.013303

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We introduce a gauge-invariant model of planar, square molecules coupled to a quantized spatially varying cavity electromagnetic vector potential Â(r) . Specifically, we choose a temporally cavity hosting a uniform magnetic field B̂ , as this is the simplest instance in which a transverse spatially varying Â(r) is at play. We show that when the molecules are in the Van Vleck paramagnetic regime, an equilibrium quantum phase transition to a photon condensate state occurs. Published by the American Physical Society 2024
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.
We investigate a minimal molecular model consisting of square planar plaquettes hosting multiple electrons, whose dynamics is governed by a tight-binding Hamiltonian supplemented by on-site Hubbard repulsion. By coupling this system to a spatially nonuniform cavity mode, we analyze the emergence of a magnetostatic instability, namely photon condensation, originating from the paramagnetic Van Vleck mechanism. The global behavior of the system is analyzed for different electronic filling factors, and we find that, except for the special cases of half-filling and a single electron, where the transition, if it occurs, is necessarily a second-order phase transition, the global system may also undergo a first-order transition because of the action of the electron-electron interaction. The polaritonic excitation energies are analyzed, providing clear spectroscopic signatures of the magnetostatic instability and of its order
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.
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.
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.
We review the recent developments and the current status in the field of quantum-gas cavity QED. Since the first experimental demonstration of atomic self-ordering in a system composed of a Bose–Einstein condensate coupled to a quantized electromagnetic mode of a high-Q optical cavity, the field has rapidly evolved over the past decade. The composite quantum-gas-cavity systems offer the opportunity to implement, simulate, and experimentally test fundamental solid-state Hamiltonians, as well as to realize non-equilibrium many-body phenomena beyond conventional condensed-matter scenarios. This hinges on the unique possibility to design and control in open quantum environments photon-induced tunable-range interaction potentials for the atoms using tailored pump lasers and dynamic cavity fields. Notable examples range from Hubbard-like models with long-range interactions exhibiting a lattice-supersolid phase, over emergent magnetic orderings and quasicrystalline symmetries, to the appearance of dynamic gauge potentials and non-equilibrium topological phases. Experiments have managed to load spin-polarized as well as spinful quantum gases into various cavity geometries and engineer versatile tunable-range atomic interactions. This led to the experimental observation of spontaneous discrete and continuous symmetry breaking with the appearance of soft-modes as well as supersolidity, density and spin self-ordering, dynamic spin-orbit coupling, and non-equilibrium dynamical self-ordered phases among others. In addition, quantum-gas-cavity setups offer new platforms for quantum-enhanced measurements. In this review, starting from an introduction to basic models, we pedagogically summarize a broad range of theoretical developments and put them in perspective with the current and near future state-of-art experiments.