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).
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).
In this paper, the quantum phase transition of cold atoms in a two-mode photomechanical cavity with nonlinear coupling between the optical field (mode 1) and the mechanical oscillator is studied on the basis of the two-mode Dicke model. The functional of the ground state energy of the system is obtained by spin coherent states and variational method. By solving and judging the stability, the phase transformation point and ground state phase diagram are obtained. It is found that there are bistable state of normal phase and reverse normal phase, coexistent state of superradiation phase and reversed normal phase, and reversed normal phase that exists alone. The different interaction strengths between atoms and two-mode light fields greatly affect the value of the phase transition point. There is a quantum phase transition from a normal phase through a phase transition point to a superradiant phase. The light-phonon nonlinear coupling has no effect on the phase transition point, but induces the collapse of the superradiant phase. There is a turning point through which the quantum phase transition from the superradiant phase to the reversed normal phase can be realized. The region of the superradiation phase decreases with the increase of the photon-phonon coupling, and it shrinks to zero at the critical value of the coupling, that is, the turning point and the phase transition point coincide, and there may be a reversal of the atomic population between the two normal phases. The nonlinear coupling of the light-phonon also produces an unstable non-zero photon state, which corresponds to the superradiation state. In the absence of mechanical oscillators, the results of the two-mode Dicke model are returned.
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.
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.
In this paper, the quantum phase transitions caused by the interactions between light and atoms, as well as between light and mechanical oscillators, are discussed theoretically in a coupled optomechanical cavity containing two optical field modes (N atoms and mechanical oscillator). By using Holstein-Primak off transformation method, new translational boson operators and four parameters are assumed. The ground state energy functional of the system and a set of equations composed of four parameters are given. The correctness of the assumed translation boson operators is proved by two special cases. In the case of resonance, the characteristics of the obtained solutions are shown by solving the equations, graphical method and Hessian matrix judgment. The stable zero solution is called the normal phase, the unstable zero solution is named the unstable vacuum macroscopic phase, and the stable non-zero solution is referred to as the superradiation phase. The phase can transition from normal phase to superradiation phase, and the point of phase transition can be changed by adjusting the coupling intensity of the two cavity light fields. When the nonlinear photon-phonon interaction caused by radiation pressure is considered, the phase diagram of the system is expanded from the original two phase regions to three phase regions, including the coexistent normal phase and superradiation phase, the bistable superradiation phase, and the unstable vacuum macroscopic phase region, where the bistable superradiation phase is similar to the optical bistable phenomenon. At the same time, there is also a turning point curve, which overlaps with the phase transition point curve, indicating the existence of multiple quantum phase transitions in the system. These predictions can be detected by measuring the average number of photons. The coupled optomechanical cavity that we studied, when considering the coupling of the two-mode optical field and the atomic ensemble but no mechanical oscillator, reflects the interaction between the two-mode optical field and the atom, thus concluding that the transformation point is small and the quantum phase transition is easy to occur. When the coupling between the mechanical oscillator and the two-mode optical field is not considered, the interaction between the single-mode optical field and the atom is reflected, returning to the quantum phase transition of the Dicke model under rotating wave approximation.
In this paper we explore the rich structure of macroscopic many-particle quantum states for a Bose-Einstein condensate in an optical cavity with a tunable nonlinear atom-photon interaction [K. Baumann et al., Nature (London) 464, 1301 (2010)]. Population inversion, bistable normal phases, and the coexistence of normal-superradiant phases are revealed by adjusting the experimentally realizable interaction strength and pump-laser frequency. For the negative (effective) cavity frequency we observe, remarkably, an inverted quantum phase transition (QPT) from the superradiant to the normal phases with an increase in atom-field coupling, which is just opposite the QPT in the normal Dicke model. Bistable macroscopic states are derived analytically in terms of the spin-coherent-state variational method by taking into account both normal and inverted pseudospin states.
We investigate in this paper the macroscopic quantum states of multiple atoms in an optical cavity with non-Hermitian coupling motivated by recent experiments. The tunable atom-photon nonlinear interaction is included to demonstrate the variety of quantum phases in the realistic realization of Dicke phase transition with cold atoms. The energies of macroscopic quantum states are obtained in terms of the spin-coherent-state variational method, in which the nonunitary operator of the similarity transformation is used to diagonalize the effective non-Hermitian Hamiltonian, which is pseudo-Hermitian below the exceptional point (EP). The superradiant phase (SP) and related quantum phase transition (QPT) disappear completely for the non-Hermitian Dicke model (DM) because of imaginary coupling, which leads to instability of states with nonzero photons. The situation is dramatically changed by including the nonlinear atom-photon interaction generated by the pump field. It is found that the second EP emerges at a higher value of coupling. The spectrum is complex between the two EPs and real otherwise. Phase diagrams are obtained for red and blue cavity pump field detunings, showing explicitly the revival of the SP and QPT, respectively. However, the QPT from the SP to the normal phase with the increase of coupling is opposite to the transition in the normal DM. The critical property of the Berry phase indicates the first-order nature of the QPT, which again is different from the DM, where the QPT is known as the second order.