We consider a set of two level atoms interacting with a single quantized bosonic mode governed by the Dicke model. In this model it is well known that under a critical value of the light-matter coupling a spontaneous radiation process takes place. In the present work, we investigate the dynamics of the system and we study the Wigner distribution function to visualize the effect of the minimal coupling on the ground state wavefunction from the normal phase to the superradiant one. We show also that the entanglement of Bi-partite model is limited by the presence of the diamagnetic term.
We consider a set of two level atoms interacting with a single quantized bosonic mode governed by the Dicke model. In this model it is well known that under a critical value of the light-matter coupling a spontaneous radiation process takes place. In the present work, we investigate the dynamics of the system and we study the Wigner distribution function to visualize the effect of the minimal coupling on the ground state wavefunction from the normal phase to the superradiant one. We show also that the entanglement of Bi-partite model is limited by the presence of the diamagnetic term.
The controllability of current quantum technologies allows one to implement spin-boson models where two-photon couplings are the dominating terms of light-matter interaction. In this case, when the coupling strength becomes comparable with the characteristic frequencies, a spectral collapse can take place, i.e., the discrete system spectrum can collapse into a continuous band. Here, we analyze the thermodynamic limit of the two-photon Dicke model, which describes the interaction of an ensemble of qubits with a single bosonic mode. We find that there exists a parameter regime where two-photon interactions induce a superradiant phase transition, before the spectral collapse occurs. Furthermore, we extend the mean-field analysis by considering second-order quantum fluctuation terms, in order to analyze the low-energy spectrum and compare the critical behavior with the one-photon case.
Dicke model describes a collective interaction between the two-level atoms and the light cavity and has been predicted to show a peculiar quantum phase transition, which is a second-order phase transition from a normal phase (in a weak-coupling strength) to a superradiant phase (in a strong-coupling strength). The model plays an important role in illustrating the quantum ground-state properties of many-body macroscopic quantum states. In the experiment, Dicke quantum phase transition in an open system could be formed by a Bose-Einstein condensate coupled to a high-finesse optical cavity. This experiment on the Bose-Einstein condensate trapped in the optical cavity have opened new frontiers, which could combine the cold atoms with quantum optics and makes it possible to enter into the strongly coupled regime of cavity quantum electrodynamics. In strong coupled regime, the atoms exchange the photons many times before spontaneous emission and cavity losses set in. It has become a hot research topic in recent years and plays an important role in many fields of modern physics, such as condensed matter physics, nuclear physics, etc. It can be applied to the manipulation of the geometric phase and entanglement in quantum information and computing. Quantum phase transition has been widely studied for the Dicke model as a typical example. Many different research methods about the mean-field approximation have been used to analyze the ground state properties of the Dicke model. In this paper, we study the ground state properties of two-component Bose-Einstein condensate in a single-mode cavity. Meanwhile, the associated quantum phase transition is described by the spin-coherent-state variational method, whose advantage is that the ground state energy and wave function can be obtained without the thermodynamic limit. By taking the average in the boson coherent state, we obtain an equivalent effective pesudospin Hamiltonian, which will be diagonalized by using the spin coherent state. Finally, we can obtain the energy functional, which is the basics of the variation to obtain the numerical solution of photon number and the expression of the atomic number and the ground state energy. This paper presents a rich phase diagram, which can be manipulated by changing the atom-field coupling imbalance between two components and the atom-field frequency detuning. While in the single-mode Dicke model there exist only the normal phase and the superradiation phase. When the frequency of one component atom is zero or the frequency of the two component atoms are equal in optical cavity, the system returns to the standard Dicke model, in which there occurs the second-order phase transition from the normal phase to the superradiant phase by adjusting the atom-field coupling. In conclusion, we discover that the stimulated radiation comes from the collective state of atomic population inversion, which does not exist in the single-mode Dicke model. Meanwhile, the new stimulated-radiation state S and S, which can only be produced by one component of the atom, are observed in the two-component Bose-Einstein condensates in the single-mode optical cavity. By adjusting the atom-field coupling imbalance and the atom-field frequency detuning (the blue or red detuning), the order of the superradiation state and the stimulated-radiation states can be exchanged between the two components of the atom.
Interactions among emitters provide a powerful means of controlling collective light--matter phenomena, yet their role in superradiant criticality has not been thoroughly investigated. Here we construct a minimal model that can yield analytical insights --- a Dicke model with two interacting atoms coupled to a single mode cavity --- to study such interaction effects. We show that interaction changes the phase boundary, and may even completely suppress the atom-photon coupling threshold for superradiance and change the universality class of the phase transition. We further study the dissipative phase transition and quantum dynamics in the presence of dissipation channels such as photon loss and spin relaxation. Our results demonstrate important roles played by the atom-atom interaction and how it can be engineered to control atom-photon coupling.
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 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.
The full Dicke model is composed of a single bosonic mode and an ensemble of N identical two-level atoms. In the model, the coupling between the bosonic mode and the atoms generates resonant and nonresonant processes. We also consider a dipole-dipole interaction between the atoms, which is able to generate entangled states in the atomic system. By assuming thermal equilibrium with a reservoir at temperature ${\ensuremath{\beta}}^{\ensuremath{-}1}$, the transition from fluorescent to superradiant phase and the quantum phase transition are investigated. It is shown that the critical behavior of the full Dicke model is not modified by the introduction of the dipole-dipole interaction.
The Dicke model describes an ensemble of N identical two-level atoms (qubits) coupled to a single quantized mode of a bosonic field. The fermion Dicke model should be obtained by changing the atomic pseudo-spin operators by a linear combination of Fermi operators. The generalized fermion Dicke model is defined introducing different coupling constants between the single mode of the bosonic field and the reservoir, g1 and g2 for rotating and counter-rotating terms respectively. In the limit N → ∞, the thermodynamic of the fermion Dicke model can be analyzed using the path integral approach with functional method. The system exhibits a second order phase transition from normal to superradiance at some critical temperature with the presence of a condensate. We evaluate the critical transition temperature and present the spectrum of the collective bosonic excitations for the general case (g1 ̸ = 0 and g2 ̸ = 0). There is quantum critical behavior when the coupling constants g1 and g2 satisfy g1 + g2 = (ω0 Ω) 1 2, where ω0 is the frequency of the mode of the field and Ω is the energy gap between energy eigenstates of the qubits. Two particular situations are analyzed. First, we present the spectrum of the collective bosonic excitations, in the case g1 ̸ = 0 and g2 = 0, recovering the well known results. Second, the case g1 = 0 and g2 ̸ = 0 is studied. In this last case, it is possible to have a superradiant phase when only virtual processes are introduced in the interaction Hamiltonian. Here also appears a quantum phase transition at the critical coupling g2 = (ω0 Ω) 1 2, and for larger values for the critical coupling, the system enter in this superradiant phase with a Goldstone mode.