Entanglement and quantum correlations between atoms are not usually considered key ingredients of the superradiant phase transition. Here we consider the Tavis−Cummings model, a solvable system of two-levels atoms, coupled with a single-mode quantized electromagnetic field. This system undergoes a superradiant phase transition, even in a finite-size framework, accompanied by a spontaneous symmetry breaking, and an infinite sequence of energy level crossings. We find approximated expressions for the ground state, its energy, and the position of the level crossings, valid in the limit of a very large number of photons with respect to that of the atoms. In that same limit, we find that the number of photons scales quadratically with the coupling strength, and linearly with the system size, providing a new insight into the superradiance phenomenon. Resorting to novel multipartite measures, we then demonstrate that this quantum phase transition is accompanied by a crossover in the quantum correlations and entanglement between the atoms (qubits). The latters therefore represent suited order parameters for this transition. Finally, we show that these properties of the quantum phase transition persist in the thermodynamic limit.
Entanglement and quantum correlations between atoms are not usually considered key ingredients of the superradiant phase transition. Here we consider the Tavis−Cummings model, a solvable system of two-levels atoms, coupled with a single-mode quantized electromagnetic field. This system undergoes a superradiant phase transition, even in a finite-size framework, accompanied by a spontaneous symmetry breaking, and an infinite sequence of energy level crossings. We find approximated expressions for the ground state, its energy, and the position of the level crossings, valid in the limit of a very large number of photons with respect to that of the atoms. In that same limit, we find that the number of photons scales quadratically with the coupling strength, and linearly with the system size, providing a new insight into the superradiance phenomenon. Resorting to novel multipartite measures, we then demonstrate that this quantum phase transition is accompanied by a crossover in the quantum correlations and entanglement between the atoms (qubits). The latters therefore represent suited order parameters for this transition. Finally, we show that these properties of the quantum phase transition persist in the thermodynamic limit.
Entanglement and quantum correlations between atoms are not usually considered key ingredients of the superradiant phase transition. Here we consider the Tavis-Cummings model, a solvable system of two-levels atoms, coupled with a single-mode quantized electromagnetic field. This system undergoes a superradiant phase transition, even in a finite-size framework, accompanied by a spontaneous symmetry breaking, and an infinite sequence of energy level crossings. We find approximated expressions for the ground state, its energy, and the position of the level crossings, valid in the limit of a very large number of photons with respect to that of the atoms. In that same limit, we find that the number of photons scales quadratically with the coupling strength, and linearly with the system size, providing a new insight into the superradiance phenomenon. Resorting to novel multipartite measures, we then demonstrate that this quantum phase transition is accompanied by a crossover in the quantum correlations and entanglement between the atoms (qubits). The latters therefore represent suited order parameters for this transition. Finally, we show that these properties of the quantum phase transition persist in the thermodynamic limit.
Quantum phase transition in Ξ-configuration Tavis-Cummings model driven by an electromagnetic field for a finite number of atoms
The quantum phase transitions of a three-level Tavis-Cummings model in the \Xi Ξ -configuration driven by a single-mode electromagnetic field in a QED cavity are studied for n_a=1,\,2 n a = 1 , 2 , and 4 4 atoms. Although conventional observables like photon number and energy level populations suffice to locate quantum phase transitions, they often lack sensitivity to finite-size effects. Here, we show that the expectation values of the quadratic Casimir invariants associated with the atomic subsystems offer a more robust measure, exhibiting sharp signatures of criticality that are validated by fidelity calculations. Additionally, the von Neumann entropy is employed to map the entanglement landscape between matter and the quantized field. A key finding is that the external driving field acts as a control parameter that shrinks the normal phase region; as the driving intensity increases, the separable ground state vanishes, leading to the dominance of field-matter entanglement across the parameter space.
Abstract In this paper we analyze properties of the phase transition that appears in a set of quantum optical models; Dicke, Tavis–Cummings, quantum Rabi, and finally the Jaynes–Cummings model. As the light-matter coupling is increased into the deep strong coupling regime, the ground state turns from vacuum to become a superradiant state characterized by both atomic and photonic excitations. It is pointed out that all four transitions are of the mean-field type, that quantum fluctuations are negligible, and hence these fluctuations cannot be responsible for the corresponding vacuum instability. In this respect, these are not quantum phase transitions. In the case of the Tavis–Cummings and Jaynes–Cummings models, the continuous symmetry of these models implies that quantum fluctuations are not only negligible, but strictly zero. However, all models possess a non-analyticity in the ground state in agreement with a continuous quantum phase transition. As such, it is a matter of taste whether the transitions should be termed quantum or not. In addition, we also consider the modifications of the transitions when photon losses are present. For the Dicke and Rabi models these non-equilibrium steady states remain critical, while the criticality for the open Tavis–Cummings and Jaynes–Cummings models is completely lost, i.e. in realistic settings one cannot expect a true critical behaviour for the two last models.
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.
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.
We analyze multipartite correlations in a generalized Dicke model involving two optical modes interacting with an ensemble of two-level atoms. In particular, we examine correlations beyond the standard bipartite entanglement and derive exact results in the thermodynamic limit. The model presents two superradiant phases involving the spontaneous breaking of either a Z2 or a U(1) symmetry. The latter is characterized by the emergence of a Goldstone excitation, found to significantly affect the correlation profiles. Focusing on the correlations between macroscopic subsystems, we analyze both the mutual information and the entanglement of formation for all possible bipartitions among the optical and matter degrees of freedom. It is found that while each mode entangles with the atoms, the bipartite entanglement between the modes is zero, and they share only classical correlations and quantum discord. We also study the monogamy of multipartite entanglement and show that there exists genuine tripartite entanglement, i.e., quantum correlations that the atoms share with the two modes but that are not shared with them individually, only in the vicinity of the critical lines. Our results elucidate the intricate correlation structures underlying superradiant phase transitions in multimode systems.