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arXiv:1209.5320 · Physical Review A

Excited-state phase transition leading to symmetry-breaking steady states in the Dicke model

Ricardo Puebla, Armando Relaño, J. Retamosa

Published Feb 19, 2013• 60 citations
arXivHTMLPDF

Abstract

We study the phase diagram of the Dicke model in terms of the excitation energy and the radiation-matter coupling constant $\ensuremath{\lambda}$. Below a certain critical value ${\ensuremath{\lambda}}_{c}$, all the energy levels have a well-defined parity. For $\ensuremath{\lambda}>{\ensuremath{\lambda}}_{c}$ the energy spectrum exhibits two different phases separated by a critical energy ${E}_{c}$ that proves to be independent of $\ensuremath{\lambda}$. In the upper phase, the energy levels have also a well-defined parity, but below ${E}_{c}$ the energy levels are doubly degenerated. We show that the long-time behavior of appropriate parity-breaking observables distinguishes between these two different phases of the energy spectrum. Steady states reached from symmetry-breaking initial conditions restore the symmetry only if their expected energies are above the critical. This fact makes it possible to experimentally explore the complete phase diagram of the excitation spectrum of the Dicke model.

DOI
10.1103/physreva.87.023819

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We study the phase diagram of the Dicke model in terms of the excitation energy and the radiation-matter coupling constant \ensuremathλ\ensuremath{\lambda}\ensuremathλ. Below a certain critical value \ensuremathλc{\ensuremath{\lambda}}_{c}\ensuremathλc​, all the energy levels have a well-defined parity. For \ensuremathλ>\ensuremathλc\ensuremath{\lambda}>{\ensuremath{\lambda}}_{c}\ensuremathλ>\ensuremathλc​ the energy spectrum exhibits two different phases separated by a critical energy Ec{E}_{c}Ec​ that proves to be independent of \ensuremathλ\ensuremath{\lambda}\ensuremathλ. In the upper phase, the energy levels have also a well-defined parity, but below Ec{E}_{c}Ec​ the energy levels are doubly degenerated. We show that the long-time behavior of appropriate parity-breaking observables distinguishes between these two different phases of the energy spectrum. Steady states reached from symmetry-breaking initial conditions restore the symmetry only if their expected energies are above the critical. This fact makes it possible to experimentally explore the complete phase diagram of the excitation spectrum of the Dicke model.
We study the thermodynamics of the full version of the Dicke model, including all the possible values of the total angular momentum j, with both microcanonical and canonical ensembles. We focus on both the excited-state quantum phase transition, appearing in the microcanonical description of the maximum angular momentum sector, j=N/2, and the thermal phase transition, which occurs when all the sectors are taken into account. We show that two different features characterize the full version of the Dicke model. If the system is in contact with a thermal bath and is described by means of the canonical ensemble, the parity symmetry becomes spontaneously broken at the critical temperature. In the microcanonical ensemble, and despite that all the logarithmic singularities which characterize the excited-state quantum phase transition are ruled out when all the j sectors are considered, there still exists a critical energy (or temperature) dividing the spectrum into two regions: one in which the parity symmetry can be broken, and another in which this symmetry is always well defined.
We study the phase transition properties of the ground state of the Dicke model by means of a variational procedure together with the catastrophe formalism.The proposed variational state is the tensorial product of the SU(2) and Weyl coherent states, and the expectation value of the Dicke Hamiltonian with respect this state is calculated.The stability properties of the energy surface exhibit a phase transition from the normal to the super-radiant behavior of the two-level atoms, when ωa ωF = ±4 γ 2 .At the same time, the analytic form of the ground state in terms of the minimum critical points of the system is obtained.The Dicke Hamiltonian is invariant under transformations of the point group C 2 = {I, exp (iπ Λ)}, with Λ the excitation number operator.To restore this parity symmetry, we separate the variational proposed states into orthogonal even and odd components, which give rise to analytic expressions for the ground and first excited states.The fidelity of these projected variational states is verified by studying their overlap with the exact quantum results.
We study the steady-state behavior of the open Dicke model, which describes the collective interaction of NNN spin-1/21/21/2 particles with a lossy, quantized cavity mode and exhibits a superradiant phase transition above a critical light-matter coupling. While the standard model conserves total spin, Kirton and Keeling [Phys. Rev. Lett. 118, 123602 (2017)] demonstrated that even infinitesimal homogeneous local dephasing destroys this phase transition and that local atomic decay can restore it. We analyze this interplay using degenerate perturbation theory across subspaces of fixed total spin, SSS. For coupling strengths above the threshold, there exists a critical spin value Sc{S}_{c}Sc​ such that the superradiant phase transition occurs only for S>ScS>{S}_{c}S>Sc​. The perturbative approach captures how weak dephasing and decay induce mixing between different SSS subspaces, yielding a steady-state spin distribution whose width scales as 1/N1/\sqrt{N}1/N​. This framework requires only the first and second moments and can be implemented via different methods that can yield these two moments (for example, the second-cumulant approach), circumventing the need for full density matrix calculations. These results bridge the quantum Rabi model and Dicke physics, elucidate the roles of dephasing and decay in collective quantum effects, and apply broadly to open quantum systems with degenerate steady states.
We study the steady-state behavior of the open Dicke model, which describes the collective interaction of NNN spin-1/21/21/2 particles with a lossy, quantized cavity mode and exhibits a superradiant phase transition above a critical light-matter coupling. While the standard model conserves total spin, Kirton and Keeling \cite{PhysRevLett.118.123602} demonstrated that even infinitesimal homogeneous local dephasing destroys this phase transition, and that local atomic decay can restore it. We analyze this interplay using degenerate perturbation theory across subspaces of fixed total spin, SSS. For coupling strengths above the threshold, there exists a critical spin value ScS_cSc​ such that the superradiant phase transition occurs only for S>ScS>S_cS>Sc​. The perturbative approach captures how weak dephasing and decay induce mixing between different SSS-subspaces, yielding a steady-state spin distribution whose width scales as 1/N1/\sqrt{N}1/N​. This framework requires only the first and second moments and can be implemented via different methods that can yield these two moments (for example, the 2nd-cumulant approach), circumventing the need for full density matrix calculations. These results bridge the quantum Rabi model and Dicke physics, elucidate the roles of dephasing and decay in collective quantum effects, and apply broadly to open quantum systems with degenerate steady states.
We propose that a broad class of excited-state quantum phase transitions (ESQPTs) gives rise to two different excited-state quantum phases. These phases are identified by means of an operator C[over ^], which is a constant of motion in only one of them. Hence, the ESQPT critical energy splits the spectrum into one phase where the equilibrium expectation values of physical observables crucially depend on this constant of motion and another phase where the energy is the only relevant thermodynamic magnitude. The trademark feature of this operator is that it has two different eigenvalues ±1, and, therefore, it acts as a discrete symmetry in the first of these two phases. This scenario is observed in systems with and without an additional discrete symmetry; in the first case, C[over ^] explains the change from degenerate doublets to nondegenerate eigenlevels upon crossing the critical line. We present stringent numerical evidence in the Rabi and Dicke models, suggesting that this result is exact in the thermodynamic limit, with finite-size corrections that decrease as a power law.
We report a kind of phase transition which takes place in isolated quantum systems with a discrete symmetry. Different phases are separated by a critical energy and recognized by means of an order parameter, which can be experimentally measured. Below the critical energy, the symmetry is broken and the equilibrium state is non-thermal, keeping large amounts of memory of the initial condition. Above, the symmetry is restored and a thermal equilibrium state is reached. We identify two universality classes, one showing a second-order phase transition and represented by the Lipkin-Meshkov-Glick model, and another showing a first-order phase transition and represented by the Dicke model. This work is dedicated to the memory of Joaquín Retamosa .