We explore the steady-state critical behavior of the finite-size open Dicke model—a model that incorporates spontaneous emission decay of the collective atomic spin states and decay of the cavity field. From the perspective of quantum information theory, we can often better characterize the quantum phase transition. In this paper, we characterize the super-radiant phase transition of the steady state of the open Dicke model by numerically calculating the quantum Fisher information (QFI). We calculate the QFI for the atomic state and the cavity field state, as well as their derivatives. We find that the QFI of the cavity field state is more sensitive to atomic decay, and is suppressed more severely in the presence of atomic decay. In contrast, the QFI of the atomic state is less sensitive to the photon loss of the cavity field.
We explore the steady-state critical behavior of the finite-size open Dicke model—a model that incorporates spontaneous emission decay of the collective atomic spin states and decay of the cavity field. From the perspective of quantum information theory, we can often better characterize the quantum phase transition. In this paper, we characterize the super-radiant phase transition of the steady state of the open Dicke model by numerically calculating the quantum Fisher information (QFI). We calculate the QFI for the atomic state and the cavity field state, as well as their derivatives. We find that the QFI of the cavity field state is more sensitive to atomic decay, and is suppressed more severely in the presence of atomic decay. In contrast, the QFI of the atomic state is less sensitive to the photon loss of the cavity field.
doi:10.1088/1367-2630/16/6/063039 The single-mode Dicke model is well known to undergo a quantum phase transition from the so-called normal phase to the superradiant phase (hereinafter called the ‘superradiant quantum phase transition’). Normally, quantum phase transitions can be identified by the critical behavior of quantities such as entanglement, quantum fluctuations, and fidelity. In this paper, we study the role of the quantum Fisher information (QFI) of both the field mode and the atoms in the ground state of the Dicke Hamiltonian. For a finite but large number of atoms, our numerical results show that near the critical atom-field coupling, the QFI of the atomic and the field subsystems can surpass their classical limits, due to the appearance of nonclassical quadrature squeezing. As the coupling increases far beyond the critical point, each subsystem becomes a highly mixed state, which degrades the QFI and hence the ultimate phase sensitivity. In the thermodynamic limit, we present the analytical results of the QFI and their relationship with the reduced variances of the field mode and the atoms. For each subsystem, we find that there is a singularity in the derivative of the QFI at the critical point, a clear signature of the quantum criticality in the Dicke model.
The single-mode Dicke model is well known to undergo a quantum phase transition from the so-called normal phase to the superradiant phase (hereinafter called the 'superradiant quantum phase transition'). Normally, quantum phase transitions can be identified by the critical behavior of quantities such as entanglement, quantum fluctuations, and fidelity. In this paper, we study the role of the quantum Fisher information (QFI) of both the field mode and the atoms in the ground state of the Dicke Hamiltonian. For a finite but large number of atoms, our numerical results show that near the critical atom-field coupling, the QFI of the atomic and the field subsystems can surpass their classical limits, due to the appearance of nonclassical quadrature squeezing. As the coupling increases far beyond the critical point, each subsystem becomes a highly mixed state, which degrades the QFI and hence the ultimate phase sensitivity. In the thermodynamic limit, we present the analytical results of the QFI and their relationship with the reduced variances of the field mode and the atoms. For each subsystem, we find that there is a singularity in the derivative of the QFI at the critical point, a clear signature of the quantum criticality in the Dicke model.
Motivated by recent efforts in simulating nonequilibrium scenarios of the Dicke model in quantum-gas cavity QED, we investigate direct probing of the normal-to-superradiant quantum phase transition via Quantum Fisher Information (QFI). This transition represents a paradigmatic example of spontaneous symmetry breaking in quantum optics, where the system’s Z2 symmetry is broken in the superradiant phase. At zero temperature, we derive analytical expressions for the QFI in the limit where the atomic transition frequency—scaled by the cavity frequency—tends to infinity. Furthermore, we analyze the impact of finite temperature on the QFI in both the thermodynamic limit and the regime of a finite but large number of atoms. All results demonstrate that the QFI exhibits a singularity as the coupling crosses the critical point—a clear signature of quantum criticality associated with spontaneous symmetry breaking. The divergent behavior of the QFI across the quantum phase transition directly relates to measuring dynamic susceptibilities using experimentally accessible Bragg spectroscopy tools and resources.
We study the steady-state behavior of the open Dicke model, which describes the collective interaction of N spin-1/2 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, S. For coupling strengths above the threshold, there exists a critical spin value Sc such that the superradiant phase transition occurs only for S>Sc. The perturbative approach captures how weak dephasing and decay induce mixing between different S-subspaces, yielding a steady-state spin distribution whose width scales as 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 study the steady-state behavior of the open Dicke model, which describes the collective interaction of N spin-1/2 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, S. For coupling strengths above the threshold, there exists a critical spin value Sc such that the superradiant phase transition occurs only for S>Sc. The perturbative approach captures how weak dephasing and decay induce mixing between different S subspaces, yielding a steady-state spin distribution whose width scales as 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.
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.