We introduce a group-theoretical extension of the Dicke model, which describes an ensemble of two-level atoms interacting with a finite radiation field. The latter is described by a spin model whose main feature is that it possesses a maximum number of excitations. The approach adopted here leads to a nonlinear extension of the Dicke model that takes into account both the intensity dependent coupling between the atoms and the radiation field and an additional nonlinear Kerr-like or Pösch–Teller-like oscillator term, depending on the degree of nonlinearity. We use the energy surface minimization method to demonstrate that the extended Dicke model exhibits a quantum phase transition, and we analyze its dependence upon the maximum number of excitations of the model. Our analysis is carried out via three methods: through mean-field analysis (i.e., by using the tensor product of coherent states), by using parity-preserving symmetry-adapted states (using the critical values obtained in the mean-field analysis and numerically minimizing the energy surface), and by means of the exact quantum solution (i.e., by numerically diagonalizing the Hamiltonian). Possible connections with the qp-deformed algebras are also discussed.
We introduce a group-theoretical extension of the Dicke model, which describes an ensemble of two-level atoms interacting with a finite radiation field. The latter is described by a spin model whose main feature is that it possesses a maximum number of excitations. The approach adopted here leads to a nonlinear extension of the Dicke model that takes into account both the intensity dependent coupling between the atoms and the radiation field and an additional nonlinear Kerr-like or Pösch–Teller-like oscillator term, depending on the degree of nonlinearity. We use the energy surface minimization method to demonstrate that the extended Dicke model exhibits a quantum phase transition, and we analyze its dependence upon the maximum number of excitations of the model. Our analysis is carried out via three methods: through mean-field analysis (i.e., by using the tensor product of coherent states), by using parity-preserving symmetry-adapted states (using the critical values obtained in the mean-field analysis and numerically minimizing the energy surface), and by means of the exact quantum solution (i.e., by numerically diagonalizing the Hamiltonian). Possible connections with the qp-deformed algebras are also discussed.
Two-level atoms interacting with a one-mode cavity field at zero temperature have order parameters which reflect the presence of a quantum phase transition at a critical value of the atom–cavity coupling strength. Two popular examples are the number of photons inside the cavity and the number of excited atoms. Coherent states provide a mean field description, which becomes exact in the thermodynamic limit. Employing symmetry-adapted (SA) SU (2) coherent states the quantum crossover, precursor of the critical behavior, can be described for a finite number of atoms. A variation after projection treatment, involving a numerical minimization of the SA energy surface, associates the quantum crossover with a discontinuity in the order parameters, which originates from competition between two local minima in the SA energy surface. Although this discontinuity is not present in finite systems, it provides a good description of 1/ N effects in the observables.
A system of N two-level atoms, Tavis-Cummings Dicke (TC-Dicke) model, interacting with a one-mode electromagnetic radiation field in the presence of the Stark shifts is studied, which is expected to predict new phenomena that are not explored in the original TC-Dicke model. We obtained the potential energy surface of the system using a trial state the direct product of coherent states in each subspace. In the frame of mean-field approaches, the variational energy is evaluated as the expectation value of the Hamiltonian for this state. The order of the quantum phase transitions is determined explicitly and numerically. We estimate the ground-state energy and the macroscopic excitations in the superradiant phase. Moreover, we investigated the critical properties of the TC-Dicke model in the classical spin limit and coherent state. We observed that in the thermodynamic limit, the energy surface takes a simple form a direct description of the phase transition. Moreover, it is found that when the microwave amplitude changes the new phase transition occurs with the Stark shift. The analytical solutions and numerical results, which appear in this paper are agreement with our paper which published recently in Int. J. Mod. Phys. B when we studied the same model using a different coherent state.
Dicke model, as an important many-body model in quantum optics, describes the interaction between multiple identical two-level atoms and a quantized electromagnetic field. This spin-boson model shows collective phenomena in light-matter interaction systems and can undergo a second-order quantum phase transition from a normal phase to a superradiant phase when the coupling strength between the two-level atoms and the optical field exceeds a critical value. Dicke model embodies unique many-body quantum theories. And it has been widely studied and obtained many significant research results in quantum information, quantum process, and other quantum systems. Meanwhile, Dicke model also has wide applications in quantum optics and condensed matter physics. The extended Dicke model, describing the interaction of a Bose-Einstein condensate in an optical cavity, provides a remarkable platform for studying extraordinary quantum phase transitions in theory and experiment. Based on the recent experiment on non-Hermitian coupling between two long-lived atomic spin waves in an optical cavity, in this work we use spin-coherent-state variational method and present the macroscopic quantum-state energy of the non-Hermitian Dicke model. The spin coherent state variational method has an advantage in the theoretical research of macroscopic quantum states, especially in the normal and the inverted pseudospin states. In the variational method, optical coherent states and atomic extremum spin coherent states are used as the trial wave functions. A Hermitian transformation operator is proposed to diagonalize the non-Hermitian Hamiltonian, which is different from the ordinary quantum mechanics where the transformation operator must be unitary. Herein, the energy function is not necessarily real in the entire coupling region. Beyond an exceptional point, the spectrum becomes complex and introducing biorthogonal sets of atomic extremum states is necessary to evaluate the average quantities. The normal phase (for the zero average photon number) possesses real energy and atomic population. The non-Hermitian interaction destroys the superradiant phase (for the stable nonzero average photon number) and leads to the absence of quantum phase transition. However, the introduced atom-photon interaction, which is induced by the pump field experimentally, can change the situation, dramatically. The pump field can balance the loss by the non-Hermitian atom-photon interaction to achieve the superradiant phase. An interesting double exceptional point are observed in the energy functional. There is the real spectrum below the first exceptional point and beyond the second exceptional point, while there is a complex spectrum between these two exceptional points. The superradiant phase appears only beyond a critical value, which is related to the nonlinear interaction and the pump laser. A new and inverted quantum phase transition from the superradiant phase to the normal phase, is observed by modulating the atom-field coupling strength. The superradiant phase of the population inversion state appears for a negative effective frequency and a large atom-photon interaction. The influence of the dissipative coupling may be observed in cold atom experiment in an optical cavity. All the parameters adopted in this work are the actual experimental parameters.
Quantum phase transition is an important subject in the field of quantum optics and condensed matter physics. In this work, we study the quantum phase transition of the two-mode Dicke model by using the nonlinear atom-light interaction introduced into the interaction between one mode light field and atom. The spin coherent variational method is used to study macroscopic multi-particle quantum systems. Firstly, the pseudo spin operator is diagonalized to obtain the variational fundamental state energy functional by means of spin coherent state transformation under the condition of coherent state light field. The energy functional is used to find the extreme value of the classical field variable, and the second derivative is determined to find the minimum value, and finally the exact solution of the ground state energy is given. Four different proportional relationships are used to study the two-mode optical field, and the rich structure of macroscopic multi-particle quantum states is given by adjusting atom-optical nonlinear interaction parameters under the experimental parameters. The abundant ground state properties such as bistable normal phase, coexisting normal-superradiation and atomic population inversion under blue and red detuning are presented. The nonlinear atom-light interaction causes blue detuning, and there is also a second-order quantum phase transition from the normal phase to the superradiation phase in the standard two-mode Dicke model. In the case of red detuning, a novel and stable reversed superradiation phase also appears. With the increase of the coupling coefficient, the reversed superradiation phase is transformed into the reversed normal phase. The nonlinear interaction between atoms and light and the different ratio of two modes of light field have great influence on the phase boundary of quantum phase transition, and the region of quantum state, as shown in Fig. (a)–(d). When the nonlinear interaction takes two definite values, the curve of the ground state physical parameters changing with the coupling parameters of atoms and light also reflects the novel second-order inverse quantum phase transition from the reversed superradiation phase to the reversed normal phase in red detuning, as shown in Fig. (a1)–(d3).
We study the quantum phase transition of an N two-level system ensemble interacting with an optical degenerate parametric process, which can be described by the finite size Dicke Hamiltonian plus counter-rotating and quadratic field terms. Analytical closed forms of the critical coupling value and their corresponding separable ground states are derived in the weak and strong coupling regimes. The existence of bipartite entanglement between the two-level system ensemble and photon field as well as between ensemble components for moderate coupling is shown through numerical analysis. Given a finite size, our results also indicate the co-existence of squeezed fields and squeezed atomic ensembles.
The energy spectrum of Dicke Hamiltonians with and without the rotating wave approximation for an arbitrary atom number is obtained analytically by means of the variational method, in which the effective pseudo-spin Hamiltonian resulting from the expectation value in the boson-field coherent state is diagonalized by the spin-coherent-state transformation. In addition to the ground-state energy, an excited macroscopic quantum-state is found corresponding to the south- and north-pole gauges of the spin-coherent states, respectively. Our results of ground-state energies in exact agreement with various approaches show that these models exhibit a zero-temperature quantum phase transition of the second order for any number of atoms, which was commonly considered as a phenomenon of the thermodynamic limit with the atom number tending to infinity. The critical behavior of the geometric phase is analyzed.