Abstract In this article, we use the spin coherent state transformation and the ground state variational method to theoretically calculate the ground function. In order to consider the influence of the atom-atom interaction on the extended Dicke model’s ground state properties, the mean photon number, the scaled atomic population and the average ground energy are displayed. Using the self-consistent field theory to solve the atom-atom interaction, we discover the system undergoes a first-order quantum phase transition from the normal phase to the superradiant phase, but a famous Dicke-type second-order quantum phase transition without the atom-atom interaction. Meanwhile, the atom-atom interaction makes the phase transition point shift to the lower atom-photon collective coupling strength.
Abstract In this article, we use the spin coherent state transformation and the ground state variational method to theoretically calculate the ground function. In order to consider the influence of the atom-atom interaction on the extended Dicke model’s ground state properties, the mean photon number, the scaled atomic population and the average ground energy are displayed. Using the self-consistent field theory to solve the atom-atom interaction, we discover the system undergoes a first-order quantum phase transition from the normal phase to the superradiant phase, but a famous Dicke-type second-order quantum phase transition without the atom-atom interaction. Meanwhile, the atom-atom interaction makes the phase transition point shift to the lower atom-photon collective coupling strength.
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.
In this paper, we reveal a second-order phase transition for the Dicke model with the dipole–dipole interaction between the atoms. By means of Holstein–Primakoff transformation, the energy spectra for both the normal and superradiant phases are explicitly obtained and therefore the scaling behaviour near the critical transition point can be given clearly. It is also shown that the dipole–dipole interactions between the atoms deeply affect the critical transition point, the ground-state and excitation energies and the scaled atomic inversion in the ground state.
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).
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.
The multi-mode Dicke model, which describes many atoms interacting with the multi-mode photons, has attracted much attention; it not only exhibits rich quantum phases, but also has an important application in quantum information. In this paper, we explore the ground-state properties of the two-mode Dicke model by the Holstein-Primakoff transformation and Boson expansion method, and theoretically predict a new first-order quantum phase transition. In the experiment, this quantum phase transition could be detected by measuring the mean-photon number or the atom population.
When an atomic ensemble is coupled with a cavity field governed by the well-known Dicke model , an important quantum phase transition (QPT) from the normal phase to the superradiant phase occurs, which was predicted more than 30 years ago. In this paper, both the atom-photon nonlinear interaction and the driving field are added to the Dicke's Hamiltonian. With the method of Holstein-Primafoff transformation, the ground state energy is given by the theoretical calculation, and the rich phase figures are presented. Meanwhile, these properties are observed experimentally. We mainly dicuss the effects of the atom-photon nonlinear interaction and the driving field on QPT.