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arXiv:1107.4323 · Physical Review A

Critical exponent of a quantum-noise-driven phase transition: The open-system Dicke model

D. Nagy, Gergely Szirmai, P. Domokos

Published Oct 24, 2011• 141 citations
arXivHTMLPDF

Abstract

The quantum phase transition of the Dicke model has been observed recently in a system formed by motional excitations of a laser-driven Bose-Einstein condensate coupled to an optical cavity [Baumann et al., Nature (London) 464, 1301 (2010)]. The cavity-based system is intrinsically open: photons leak out of the cavity where they are detected. Even at zero temperature, the continuous weak measurement of the photon number leads to an irreversible dynamics toward a steady state. In the framework of a generalized Bogoliubov theory, we show that the steady state exhibits a dynamical quantum phase transition. We find that the critical point and the mean field are only slightly modified with respect to the phase transition in the ground state. However, the critical exponents of the singular quantum correlations are significantly different in the two cases. There is also a drastic modification of the atom-field entanglement, since the divergence of the logarithmic negativity of the ground state at the critical point is suppressed and a finite entanglement is found in the steady state.

DOI
10.1103/physreva.84.043637

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The quantum phase transition of the Dicke model has been observed recently in a system formed by motional excitations of a laser-driven Bose-Einstein condensate coupled to an optical cavity [Baumann et al., Nature (London) 464, 1301 (2010)]. The cavity-based system is intrinsically open: photons leak out of the cavity where they are detected. Even at zero temperature, the continuous weak measurement of the photon number leads to an irreversible dynamics toward a steady state. In the framework of a generalized Bogoliubov theory, we show that the steady state exhibits a dynamical quantum phase transition. We find that the critical point and the mean field are only slightly modified with respect to the phase transition in the ground state. However, the critical exponents of the singular quantum correlations are significantly different in the two cases. There is also a drastic modification of the atom-field entanglement, since the divergence of the logarithmic negativity of the ground state at the critical point is suppressed and a finite entanglement is found in the steady state.
Recent experiments have demonstrated an open system realization of the Dicke quantum phase transition in the motional degrees of freedom of an optically driven Bose–Einstein condensate in a cavity. Relevant collective excitations of this light–matter system are polaritonic in nature, allowing access to the quantum critical behavior of the Dicke model through light leaking out of the cavity. This opens the path to using photodetection-based quantum optical techniques to study the dynamics and excitations of this elementary quantum critical system. We first discuss the photon flux observed at the cavity face and find that it displays a different scaling law near criticality than that obtained from the mean-field theory for the equivalent closed system. Next, we study the second-order correlation measurements of photons leaking out of the cavity. Finally, we discuss a modulation technique that directly captures the softening of polaritonic excitations. Our analysis takes into account the effect of the finite size of the system, which may result in an effective symmetry-breaking term.
A laser-driven Bose-Einstein condensate of ultracold atoms loaded into a lossy high-finesse optical resonator exhibits critical behavior and, in the thermodynamic limit, a phase transition between stationary states of different symmetries. The system realizes an open-system variant of the celebrated Dicke model. We study the transition for a finite number of atoms by means of a Hartree-Fock-Bogoliubov method adapted to a damped-driven open system. The finite-size scaling exponents are determined and a clear distinction between the nonequilibrium and equilibrium quantum criticality is found.
We demonstrate that criticality in a driven-dissipative system is strongly influenced by the spectral properties of the bath. We study the open-system realization of the Dicke model, where a bosonic cavity mode couples to a large spin formed by two motional modes of an atomic Bose-Einstein condensate. The cavity mode is driven by a high-frequency laser and it decays to a Markovian bath, while the atomic mode interacts with a colored bath. We reveal that the soft mode fails to describe the characteristics of the criticality. We calculate the critical exponent of the superradiant phase transition and identify an inherent relation to the low-frequency spectral density function of the colored bath. We show that a finite temperature of the colored bath does not modify qualitatively this dependence on the spectral density function.
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.
Ground state properties and level statistics of the Dicke model for a finite number of atoms are investigated based on a progressive diagonalization scheme (PDS). Particle number statistics, the entanglement measure and the Shannon information entropy at the resonance point in cases with a finite number of atoms as functions of the coupling parameter are calculated. It is shown that the entanglement measure defined in terms of the normalized von Neumann entropy of the reduced density matrix of the atoms reaches its maximum value at the critical point of the quantum phase transition where the system is most chaotic. Noticeable change in the Shannon information entropy near or at the critical point of the quantum phase transition is also observed. In addition, the quantum phase transition may be observed not only in the ground state mean photon number and the ground state atomic inversion as shown previously, but also in fluctuations of these two quantities in the ground state, especially in the atomic inversion fluctuation.