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arXiv:1810.05888 · Journal of the Physical Society of Japan

Dynamical Properties of the Finite-size Dicke Model Coupled to a Thermal Reservoir

Ryosuke Imai, Y. Yamanaka

Published Jan 11, 2019• 2 citations
arXivHTMLPDF

Abstract

We investigate the dynamical properties of the finite-size Dicke model coupled to a photon reservoir in the dispersive regime. The system-reservoir coupling in our Hamiltonian includes counter-rotating terms, which are relevant in the strong atom-cavity coupling. Because the dispersive regime is considered, the dynamics of low-energy states are described sufficiently accurately within the finite-dimensional subspace of the dressed states. Using the separation of the time scales between the system and the reservoir, we derive the Markovian quantum master equation in the subspace without ignoring the counter-rotating terms. The temporal evolution of the expectation of the cavity mode shows that the bifurcation of the long-lived state corresponds to the superradiant transition in the isolated model. The master equation explicitly gives the steady state solution. The numerical results for the first-order correlation function on the steady state indicate that the strong atom-cavity coupling enhances the coherence and softens the dephasing in the superradiant region.

DOI
10.7566/jpsj.88.024401

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We investigate the dynamical properties of the finite-size Dicke model coupled to a photon reservoir in the dispersive regime. The system-reservoir coupling in our Hamiltonian includes counter-rotating terms, which are relevant in the strong atom-cavity coupling. Because the dispersive regime is considered, the dynamics of low-energy states are described sufficiently accurately within the finite-dimensional subspace of the dressed states. Using the separation of the time scales between the system and the reservoir, we derive the Markovian quantum master equation in the subspace without ignoring the counter-rotating terms. The temporal evolution of the expectation of the cavity mode shows that the bifurcation of the long-lived state corresponds to the superradiant transition in the isolated model. The master equation explicitly gives the steady state solution. The numerical results for the first-order correlation function on the steady state indicate that the strong atom-cavity coupling enhances the coherence and softens the dephasing in the superradiant region.
In this study, the energy spectrum and thermal equilibrium states of the finite-size Dicke-Stark model were numerically obtained within the extended coherent state space by solving the dressed master equation for strongly coupled light-atom systems. The critical point of the superradiant phase transition in the infinite-size Dicke-Stark model was analytically derived using the mean-field approach and confirmed with numerical calculation. Under thermal equilibrium conditions, analyses of the negativity, zero-time-delay two-photon correlation function, and atom-spin squeezing parameters in the finite-size Dicke-Stark model reveal that as the coupling strength increases, the light field undergoes a transition from photon bunching to anti-bunching and then back to bunching. The Stark field can modulate both the maximum and minimum values of the two-photon correlation function and their corresponding coupling strengths. At low temperatures, the system exhibits entanglement and spin squeezing. As temperature rises, entanglement gradually diminishes, while strong coupling facilitates the preservation of entanglement in the system state. Atom-spin squeezing spin squeezing is highly sensitive to temperature and vanishes rapidly with increasing temperature. This work contributes to the fundamental understanding of quantum phenomena in Dicke-Stark systems.
The Dicke model famously exhibits a phase transition to a superradiant phase with a macroscopic population of photons and is realized in multiple settings in open quantum systems. In this paper, we study a variant of the Dicke model where the cavity mode is lossy due to the coupling to a Markovian environment while the atomic mode is coupled to a colored bath. We analytically investigate this model by inspecting its low-frequency behavior via the Schwinger-Keldysh field theory and carefully examine the nature of the corresponding superradiant phase transition. Integrating out the fast modes, we can identify a simple effective theory allowing us to derive analytical expressions for various critical exponents including the dynamical exponent. We find excellent agreement with previous numerical results when the non-Markovian bath is at zero temperature; however, contrary to these studies, our low-frequency approach reveals that the same exponents govern the critical behavior when the colored bath is at finite temperature unless the chemical potential is zero. Furthermore, we show that the superradiant phase transition is classical in nature, while it is genuinely nonequilibrium. We derive a fractional Langevin equation and conjecture the associated fractional Fokker-Planck equation that captures the system's long-time memory as well as its nonequilibrium behavior. Finally, we consider finite-size effects at the phase transition and identify the finite-size scaling exponents, unlocking a rich behavior in both statics and dynamics of the photonic and atomic observables.
The Dicke model describes the coherent interaction of a laser-driven ensemble of two-level atoms with a quantized light field. It is realized within cavity QED experiments, which in addition to the coherent Dicke dynamics feature dissipation due to, e.g., atomic spontaneous emission and cavity photon loss. Spontaneous emission supports the uncorrelated decay of individual atomic excitations as well as the enhanced collective decay of an excitation that is shared by NNN atoms and whose strength is determined by the cavity geometry. We derive a many-body master equation for the dissipative Dicke model including both spontaneous emission channels and analyze its dynamics on the basis of Heisenberg-Langevin and stochastic Bloch equations. We find that the collective loss channel leads to a region of bistability between the empty and the superradiant state. Transitions between these states are driven by nonthermal Markovian noise. The interplay between dissipative and coherent elements leads to a genuine nonequilibrium dynamics in the bistable regime, which is expressed via a nonconservative force and a multiplicative noise kernel appearing in the stochastic Bloch equations. We present a semiclassical approach, based on stochastic nonlinear optical Bloch equations, which for the infinite-range Dicke model become exact in the large-NNN-limit. The absence of an effective free-energy functional, however, necessitates the inclusion of fluctuation corrections with O(1/N)O(1/N)O(1/N) for finite $N<\ensuremath{\infty}$ to locate the nonthermal first-order phase transition between the superradiant and the empty cavity.
The Dicke model can exhibit quantum phase transition between the normal and the superradiant phases when the strength of the light-matter coupling exceeds the ultrastrong-coupling regime. However, it is challenging to observe this phase transition in practical systems due to limited coupling strength or finite two-photon A2{A}^{2}A2 terms. Here we show that by applying a periodic modulation to the frequency of the two-level systems in a standard Dicke model in the strong-coupling regime, an anisotropic Dicke model with tunable rotating and counter-rotating terms in the ultrastrong-coupling regime can be achieved. We calculate the ground state and the excitation spectrum of this model in terms of the modulation parameters. Our result shows that the superradiant phases can be observed in cavity- or circuit-quantum electrodynamics systems with strong coupling.
We study the effect of spontaneous emission and incoherent atomic pumping on the nonlinear semiclassical dynamics of the unbalanced Dicke model---a generalization of the Dicke model that features independent coupling strengths for the co- and counter-rotating interaction terms. As well as the ubiquitous superradiant behavior the Dicke model is well-known for, the addition of spontaneous emission combined with the presence of strong counter-rotating terms creates a laserlike behavior termed counter-lasing. These states appear in the semiclassical model as stable periodic orbits. We perform a comprehensive dynamical analysis of the appearance of counter-lasing in the unbalanced Dicke model subject to strong cavity dissipation, such that the cavity field can be adiabatically eliminated to yield an effective Lipkin-Meshkov-Glick (LMG) model. If the coupling strength of the corotating interactions is small, then the counter-lasing phase appears via a Hopf bifurcation of the deexcited state. We find that if the rate of spontaneous emission is small, this can lead to resurgent superradiant pulses. However, if the corotating coupling is larger, then the counter-lasing phase must emerge via the steady-state superradiant phase. Such a transition is the result of the competition of the coherent and incoherent processes that drive superradiance and counter-lasing, respectively. We observe a surprisingly complex transition between the two, associated with the formation of a chaotic attractor over a thin transitional parameter region.
In this paper, we investigate the finite-temperature properties and phase transition of the Dicke model. Converting the atomic pseudo-spin operator to the two-mode Fermi operators, we obtain the partition function in terms of the imaginary-time path integral. The atomic population and average photon number as analytic functions of the atom-photon coupling strength are found from the thermodynamic equilibrium equation, which leads to the stationary state at a finite temperature and is determined by the variation in an extremum-condition of the Euclidean action with respect to the bosonic field. In particular we study the phase transition from normal to superradiation phase at a fixed low-temperature, in which the phase transition is dominated by quantum fluctuations. The phase transition induced by the variation of the atom-photon coupling strength indeed obeys the Landau continuous phase-transition theory, in which the average photon-number can serve as an order parameter with non-zero value that characterizes the superradiation phase. In the zero temperature limit our results recover exactly all those obtained from the quantum phase transition theory at zero temperature. In addition, we discuss the thermodynamic properties and compare the difference between finite-temperature phase transition and zero-temperature quantum phase transition. It is discovered that the average photon-number and mean energy in the low-temperature stationary state coincide with the corresponding values of zero-temperature in the strong coupling region. The entropy of the superradiation phase decays rapidly to zero with the increase of coupling strength.