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arXiv:1608.06293 · Physical Review A

Dicke phase transition without total spin conservation

Emanuele G. Dalla Torre, Yulia E. Shchadilova, Eli Y. Wilner, Mikhail D. Lukin, Eugene Demler

Published Dec 27, 2016• 50 citations
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Abstract

We develop a fermionic path-integral formalism to analyze the phase diagram of open nonequilibrium systems. The formalism is applied to analyze an ensemble of two-level atoms interacting with a single-mode optical cavity, described by the Dicke model. While this model is often used as the paradigmatic example of a phase transition in driven-dissipative systems, earlier theoretical studies were limited to the special case when the total spin of the atomic ensemble is conserved. This assumption is not justified in most experimental realizations. Our approach allows us to analyze the problem in a more general case, including the experimentally relevant case of dissipative processes that act on each atom individually and do not conserve the total spin. We obtain a general expression for the position of the transition, which contains as special cases the two previously known regimes: (i) nonequilibrium systems with losses and conserved spin and (ii) closed systems in thermal equilibrium and with the Gibbs-ensemble averaging over the values of the total spin. We perform a detailed study of different types of baths and point out the possibility of a surprising nonmonotonic dependence of the transition on the baths' parameters.

DOI
10.1103/physreva.94.061802

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We develop a fermionic path-integral formalism to analyze the phase diagram of open nonequilibrium systems. The formalism is applied to analyze an ensemble of two-level atoms interacting with a single-mode optical cavity, described by the Dicke model. While this model is often used as the paradigmatic example of a phase transition in driven-dissipative systems, earlier theoretical studies were limited to the special case when the total spin of the atomic ensemble is conserved. This assumption is not justified in most experimental realizations. Our approach allows us to analyze the problem in a more general case, including the experimentally relevant case of dissipative processes that act on each atom individually and do not conserve the total spin. We obtain a general expression for the position of the transition, which contains as special cases the two previously known regimes: (i) nonequilibrium systems with losses and conserved spin and (ii) closed systems in thermal equilibrium and with the Gibbs-ensemble averaging over the values of the total spin. We perform a detailed study of different types of baths and point out the possibility of a surprising nonmonotonic dependence of the transition on the baths' parameters.
We investigate nonequilibrium phase transitions for driven atomic ensembles interacting with a cavity mode and coupled to a Markovian dissipative bath. In the thermodynamic limit and at low frequencies, we show that the distribution function of the photonic mode is thermal, with an effective temperature set by the atom-photon interaction strength. This behavior characterizes the static and dynamic critical exponents of the associated superradiance transition. Motivated by these considerations, we develop a general Keldysh path-integral approach that allows us to study physically relevant nonlinearities beyond the idealized Dicke model. Using standard diagrammatic techniques, we take into account the leading-order corrections due to the finite number NNN of atoms. For finite NNN, the photon mode behaves as a damped classical nonlinear oscillator at finite temperature. For the atoms, we propose a Dicke action that can be solved for any NNN and correctly captures the atoms' depolarization due to dissipative dephasing.
The Dicke model describes an ensemble of N identical two-level atoms (qubits) coupled to a single mode of a bosonic field. The fermion Dicke model should be obtained by changing the atomic pseudo-spin operators by a linear combination of Fermi operators. The generalized fermion Dicke model is defined introducing different coupling constants between the single mode of the bosonic field and the reservoir. In the thermodynamic limit, the fermion Dicke model can be analized using the path integral approach with functional method. The system exhibits a second order phase transition from normal to superrandiance at some critical temperature with the presence of a condensate. We evaluate the critical transition temperature and present the spectrum of the collective bosonic excitations. There is quantum phantum critical behavior when the coupling constants satisfy an especific condition. Two particular situations are analyzed. First, we present the spectrum of the collective bosonic excitations in the case using the rotating-wave approximation, recovering the well know results. Second, the case only considering virtual processes. In the last case, it is possible to have a superradiance phase when only virtual processes are introduced in the interaction Hamiltonian. Here also appears a quantum phase transition at some critical coupling, and for larger values for the critical coupling, the system enter in this superradiant phase with a Goldstone mode.
Abstract The Dicke model describes the coupling between a quantized cavity field and a large ensemble of two‐level atoms. When the number of atoms tends to infinity, this model can undergo a transition to a superradiant phase, belonging to the mean‐field Ising universality class. The superradiant transition was first predicted for atoms in thermal equilibrium and was recently realized with a quantum simulator made of atoms in an optical cavity, subject to both dissipation and driving. This progress report offers an introduction to some theoretical concepts relevant to the Dicke model, reviewing the critical properties of the superradiant phase transition and the distinction between equilibrium and nonequilibrium conditions. In addition, it explains the fundamental difference between the superradiant phase transition and the more common lasing transition. This report mostly focuses on the steady states of atoms in single‐mode optical cavities, but it also mentions some aspects of real‐time dynamics, as well as other quantum simulators, including superconducting qubits, trapped ions, and using spin–orbit coupling for cold atoms. These realizations differ in regard to whether they describe equilibrium or nonequilibrium systems.
We consider the full Dicke spin-boson model composed by a single bosonic mode and an ensemble of N identical two-level atoms with different couplings for the resonant and anti-resonant interaction terms, and incorporate a dipole–dipole interaction between the atoms. Assuming that the system is in thermal equilibrium with a reservoir at temperature β −1 , we compute the free energy in the thermodynamic limit N → ∞ in the saddle-point approximation to the path integral and determine the critical temperature for the super-radiant phase transition. In the zero temperature limit, we recover the critical coupling of the quantum phase transition, presented in the literature.
The Dicke model describes the collective behavior of a subwavelength-size ensemble of two-level atoms (i.e., spin-1/2) interacting identically with a single quantized radiation field of a cavity. Across a critical coupling strength it exhibits a zero-temperature phase transition from the normal state to the superradiant phase where the field is populated and the collective spin acquires a nonzero x component, which can be imagined as ferromagnetic ordering of the atomic spins along x. Here we introduce a variant of this model where two subwavelength-size ensembles of spins interact with a single quantized radiation field with different strengths. Subsequently, we restrict ourselves to a special case where the coupling strengths are opposite (which is unitarily equivalent to equal-coupling strengths). Because of the conservation of the total spin in each ensemble individually, the system supports two distinct superradiant states with x-ferromagnetic and x-ferrimagnetic spin ordering, coexisting with each other in a large parameter regime. The stability and dynamics of the system in the thermodynamic limit are examined using a semiclassical approach, which predicts nonstationary behaviors due to the multistabilities. At the end, we also perform small-scale full quantum-mechanical calculations, with results consistent with the semiclassical ones.
The Dicke model describing an ensemble of two-state atoms interacting with a single quantized mode of the electromagnetic field (with omission of the ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{2}$ term) exhibits a zero-temperature phase transition at a critical value of the dipole coupling strength. We propose a scheme based on multilevel atoms and cavity-mediated Raman transitions to realize an effective Dicke model operating in the phase transition regime. Optical light from the cavity carries signatures of the critical behavior, which is analyzed for the thermodynamic limit where the number of atoms is very large.