In recent years, the cavity quantum photomechanics has been developed rapidly, and played a very important role in quantum information processing, quantum basic principle verification, and high-precision measurement. The kinds of quantum mechanical behaviors have also been explored and discovered in the study of cavity mechanics. By placing the Kerr medium in the system, quantum nonlinearity is introduced into the optomechanical system. Quantum phase transition is a relatively important part in the research of condensed matter physics. Since Dicke quantum phase transition was successfully observed experimentally, the problem of quantum phase transition in the optical cavity has attracted more attention. The spin-coherent-state variation method and the Holstein-Primakoff transformation are used to theoretically calculate the ground state energy functional, and the rich structure of the macroscopic multi-particle quantum state is given by adjusting the parameters. The quantum phase transition evolution equation describes the relationship between each phase and the time of generating a new phase when reaching the critical phase transition point. At the same time, the mode squeezing of multi-mode hybrid optomechanical system has also became one of the basic problems of quantum mechanical behavior in cavity quantum dynamics. In this article, we explore the quantum dynamics of optomechanical devices including single-cavity and dual-cavities. We find that the system will undergo a dynamic phase transition, which is similar to the Dicke-Hepp-Lieb superradiant type phase transition, and a new dynamic critical point appears in the coupling between the momentum quadratures of the two optical fields. By manipulating the coupling parameters, we can achieve selective energy exchange between any two modes and the critical coupling point corresponds to selective energy exchange. Mode squeezing, which is easy to measure by applying the quantum uncertainty relationship, is also revealed and consistent with selective energy exchange. The study of coordinate and momentum variances gives us the revelation that the compressed orthogonal variables are the most suitable for measurement because of the small quantum noise. In fact, phononic modes can store energy in a longer duration, while photonic modes can transfer energy in a long distance. This phenomenon makes the hybrid optomechanical cavities useful in the next-generation quantum communications and quantum information processing units.
In recent years, the cavity quantum photomechanics has been developed rapidly, and played a very important role in quantum information processing, quantum basic principle verification, and high-precision measurement. The kinds of quantum mechanical behaviors have also been explored and discovered in the study of cavity mechanics. By placing the Kerr medium in the system, quantum nonlinearity is introduced into the optomechanical system. Quantum phase transition is a relatively important part in the research of condensed matter physics. Since Dicke quantum phase transition was successfully observed experimentally, the problem of quantum phase transition in the optical cavity has attracted more attention. The spin-coherent-state variation method and the Holstein-Primakoff transformation are used to theoretically calculate the ground state energy functional, and the rich structure of the macroscopic multi-particle quantum state is given by adjusting the parameters. The quantum phase transition evolution equation describes the relationship between each phase and the time of generating a new phase when reaching the critical phase transition point. At the same time, the mode squeezing of multi-mode hybrid optomechanical system has also became one of the basic problems of quantum mechanical behavior in cavity quantum dynamics. In this article, we explore the quantum dynamics of optomechanical devices including single-cavity and dual-cavities. We find that the system will undergo a dynamic phase transition, which is similar to the Dicke-Hepp-Lieb superradiant type phase transition, and a new dynamic critical point appears in the coupling between the momentum quadratures of the two optical fields. By manipulating the coupling parameters, we can achieve selective energy exchange between any two modes and the critical coupling point corresponds to selective energy exchange. Mode squeezing, which is easy to measure by applying the quantum uncertainty relationship, is also revealed and consistent with selective energy exchange. The study of coordinate and momentum variances gives us the revelation that the compressed orthogonal variables are the most suitable for measurement because of the small quantum noise. In fact, phononic modes can store energy in a longer duration, while photonic modes can transfer energy in a long distance. This phenomenon makes the hybrid optomechanical cavities useful in the next-generation quantum communications and quantum information processing units.
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.
Optomechanical cavity is a powerful connection between a nanomechanical oscillator and a quantized electromagnetic field. In this system, a novel photon-phonon nonlinear interaction arising from the nanomechanical oscillation is produced through the radiation pressure. Now this nonlinear photon-phonon interaction has become an important resource for implementing high-precision measurements and processing quantum information. Motivated by T. Esslinger group’s experiment, it is very meaningful to explore the exotic quantum phenomena when a ultra-cold BEC is trapped in an optomechanical cavity. In this paper, we mainly investigate phase transition and the finite-temperature thermodynamic properties of a Bose-Einstein condensate in an optomechanical cavity. It’s worth mentioning that at zero temperature many different mean-field approximate methods have been used to analyze the ground state properties of a Bose-Einstein condensate in an optomechanical cavity. Two common methods are Holstein-Primakoff transformation and spin coherent state variation. In this paper, an interesting imaginary-time path integral approach has been introduced to study finite temperature thermodynamic properties and phase transition of a Bose-Einstein condensate in an optomechanical cavity. First, we obtained system's partition function by taking imaginary-time path integration. Meanwhile, an effective action has been obtained by means of this method, which is the basic of the variation to get the numerical solution of photon number and the expression of the atomic number. At zero temperature, these results are consistent with what we have obtained by Holstein-Primakoff transformation or spin coherent state variational method. By adjusting the atom-field coupling strength and other parameters the second-order phase transition from the normal phase to the superradiant phase has been revealed. Meanwhile, a new unstable superradiant state was also found. And we found that in addition to the normal phase and superradiation phase, there exists an un-solution region of the mean photon number. Meanwhile, we find that the nonlinear photon-phonon interaction does not affect the normal phase. However, in the superradiant phase, the nonlinear photon-phonon interaction can enhance the macroscopic collective excitations. At the same time, the thermodynamic properties of the system are also discussed. According to the obtained distribution function, we can derive the analytical expression of the average energy and the free energy. Furthermore, the expression of entropy at finite temperature can also be obtained. we find the nonlinear photon-phonon interaction does not affect the average energy in the normal phase, but the average energy in the superradiant phase can deeply deviate in the large nonlinear photon-phonon interaction. It’s worth mentioning that the mean photon number and average energy in the finite-temperature tend to be consistent with the case in absolute zero temperature in the strong coupling region, while the entropy in the superradiant phase is rapidly reduced to zero as the atom-field coupling strength increases. In other words, strongly coupled collective excited states are highly ordered and are not affected by thermal fluctuations in the temperature range we are considering. The thermodynamic properties, such as the entropy and corresponding specific heat, characterize the Dicke phase transition.
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.
Abstract Based on spin-coherent-state variational method, we mainly study the multiple stable macroscopic quantum states and quantum phase transitions of a Bose-Einstein Condensate in an optomechanical dual-cavity by modulating the dual-cavity interaction and the nonlinear phonon-photon interaction. Especially, the collapse of superradiant phase can be tuned in the existing nonlinear photon-phonon interaction, but the critical quantum phase transition point or boundary hasn’t been inflfluenced. As a result, the system occurs an additional phase transition from the superradiant phase to the inversely atomic populated state. Moreover, when dual-cavity coupling interaction increases to a certain value, a new quantum phase transition from the normal phase to the inversely atomic populated state will appear. Finally, the superradiant phase completely collapses and the normal phase also collapses into an unstable macroscopic vacuum state for a strong dual-cavity coupling interaction.
The A2 term presents a fundamental challenge to realizing the superradiant phase transition (SPT) in cavity quantum electrodynamics. Here, we propose a hybrid quantum system enabling SPT regardless of the presence of the A2 term. The system consists of a qubit, a mechanical mode, and an optical cavity, where the qubit and mechanical mode constitute a quantum Rabi model, while the mechanical mode and cavity form an optomechanical system. Crucially, the auxiliary cavity introduces a switchable A2 term that effectively counteracts or even fully eliminates the original A2 effect. This allows controllable observation of SPT, diagnosed via the second-order equal-time correlation function g(2)(0) of phonons. Furthermore, the auxiliary cavity exponentially reduces the critical coupling strength, significantly relaxing experimental requirements. In addition, we show that phonons in the normal phase display bunching but are coherent in the superradiant phase. Interestingly, higher-order squeezing is found in both phases, with near-perfect higher-order squeezing achieved at the SPT point, establishing it as a probe for SPT behavior. Our work demonstrates that hybridizing optomechanics and cavity quantum electrodynamics provides a promising route to accessing SPT physics in the presence of the A2 term.