Recently, strong-coupling regimes of superconducting qubits or quantum dots inside a microwave circuit cavity and BEC atoms inside an optical cavity were achieved experimentally. The strong-coupling regimes in these systems were described by the Dicke model. Here, we solve the Dicke model by a $1/N$ expansion. In the normal state, we find a $\sqrt{N}$ behavior of the collective Rabi splitting. In the superradiant phase, we identify an important Berry phase term that has dramatic effects on both the ground state and the excitation spectra of the strongly interacting system. The single photon excitation spectrum has a low-energy quantum phase diffusion mode in imaginary time with a large spectral weight and also a high-energy optical mode with a low spectral weight. The photons are in a number squeezed state that may have wide applications in high sensitive measurements and quantum-information processing. Comparisons with exact diagonalization studies are made. Possible experimental schemes to realize the superradiant phase are briefly discussed.
Recently, strong-coupling regimes of superconducting qubits or quantum dots inside a microwave circuit cavity and BEC atoms inside an optical cavity were achieved experimentally. The strong-coupling regimes in these systems were described by the Dicke model. Here, we solve the Dicke model by a 1/N expansion. In the normal state, we find a N behavior of the collective Rabi splitting. In the superradiant phase, we identify an important Berry phase term that has dramatic effects on both the ground state and the excitation spectra of the strongly interacting system. The single photon excitation spectrum has a low-energy quantum phase diffusion mode in imaginary time with a large spectral weight and also a high-energy optical mode with a low spectral weight. The photons are in a number squeezed state that may have wide applications in high sensitive measurements and quantum-information processing. Comparisons with exact diagonalization studies are made. Possible experimental schemes to realize the superradiant phase are briefly discussed.
Recently, several experiments successfully achieved the strong coupling of a BEC of $ N \sim 10^{5} $ $ ^{87}Rb $ atoms to the photons inside an ultrahigh-finesse optical cavity. The strong coupling regime was also achieved with artificial atoms such as superconducting qubits inside micro-wave circuit cavity. These systems are described by the Dicke model where a single mode of photons coupled to an assembly of $ N $ atoms with the same coupling strength $ g $. In this paper, by solving the Dicke model by $ 1/ N $ expansion, we show that at the thermodynamic limit $ N = \infty $, one can achieve the Bose-Einstein Condensation (BEC) of the cavity photons at sufficiently strong coupling $ g $. At a finite $ N $, we identify an emergent quantum phase diffusion mode inside the BEC phase and also work out many remarkable experimental consequences of this mode such as its low frequency, photon number squeezing properties and photon statistics. The photons from the BEC phase are in a number squeezed state with much enhanced signal/noise ratio which may have wide applications in the field of high resolution, high sensitive measurement and also in quantum information processing. The effects of dissipations due to leaking photons out of the cavity are also briefly discussed. The difference with a Laser beam is stressed. Several experimental schemes to realize the BEC of the photons and detect the phase diffusion mode are proposed.
The Dicke model was proposed by Dicke in 1954 to describe a single mode of photon coupled to an assembly of N atoms with the same strength. It was found that there is a quantum phase transition from a normal phase at weak coupling to a super-radiant phase at strong coupling at the thermodynamic limit N =∞. Here, we solve the model at N ≥ 1 and also discuss its possible experimental implementations inside a cavity. By studying the Dicke model by 1/N expansion, we identify an emergent quantum phase diffusion mode inside the super-radiant phase and also work out many remarkable experimental consequences of this mode such as its low frequency, photon number squeezing properties and photon statistics. The energy of the phase diffusion modecan be continuously tuned into many frequency ranges from Micro-wave to Terahertz (THz) to Infra-red. The photons from the super-radiant phase are in a number squeezed state with much enhanced signal/noise ratio which may have wide applications in quantum information processing. The photon statistics is strong sub-Poissonian. The effects of dissipations due to leaking photons out of the cavity are also discussed. The connections with the recent experiments of the strong coupling of a BEC of N ~ 105 87Rb atoms to the photons inside an ultrahigh-finess optical cavity are carefully analyzed. Several experimental schemes to detect the phase diffusion mode are proposed.
The four standard quantum optics model in cavity QED such as Rabi, Dicke, Jaynes-Cummings ( JC ) and Tavis-Cummings (TC) model were proposed many decades ago. Despite their relative simple forms and many previous theoretical works, their solutions at a finite $ N $ inside the superradiant regime remain unknown. In view of recent remarkable experimental advances in several experimental systems such as cold atoms inside a cavity or superconducting qubits inside a microcavity to realize these models, it becomes topical and important to understand what would be the new phenomena in these models at a finite $ N $, especially inside the superradaint regime. In this work, we resolved this outstanding problem. We use three independent methods such as $ 1/J $ expansion, the strong coupling expansion and exact diagonization (ED) to study $ U(1)/Z_2 $ Dicke model at a finite $ N $ and different anisotropy parameters $ 0 \leq \beta \leq 1 $ as the atom-photon interaction strength increases. This model include the four standard quantum optics model as its special limit. We identify 4 different regimes: the normal, $ U(1) $, strong quantum tunneling (QT) and weak QT regime. We map out the evolution of energy level structures, compute all kinds of photon correlation functions and find nearly perfect agreements between the strong coupling expansion and the ED not only in the weak and strong QT regimes, but also in the $ U(1) $ regime not too close to the critical point at $ N=\infty $.
In view of recent proposals for the realization of anisotropic light-matter interaction in such platforms as (i) nonstationary or inductively and capacitively coupled superconducting qubits, (ii) atoms in crossed fields, and (iii) semiconductor heterostructures with spin-orbital interaction, the concept of a generalized Dicke model, where coupling strengths of rotating wave and counter-rotating wave terms are unequal, has attracted great interest. For this model we study photon fluctuations in the critical region of normal-to-superradiant phase transition when both the temperatures and numbers of two-level systems are finite. In this case, the superradiant quantum phase transition is changed to a fluctuational region in the phase diagram that reveals two types of critical behaviors. These are regimes of Dicke model (with discrete Z2 symmetry), and that of anti-Tavis-Cummings and Tavis-Cummings U(1) models. We show that squeezing parameters of photon condensate in these regimes show distinct temperature scalings. Besides, relative fluctuations of a photon number take universal values. We also find a temperature scale below which one approaches a zero-temperature quantum phase transition where quantum fluctuations dominate. Our effective theory is provided by a non-Goldstone functional for condensate mode and by Majorana representation of Pauli operators. We also discuss the Bethe ansatz solution for integrable U(1) limits.
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.
Four standard quantum optics models, that is, the Rabi, Dicke, Jaynes-Cummings, and Tavis-Cummings models, were proposed by physicists many decades ago. Despite their relative simple forms and many previous theoretical works, their physics at a finite N, especially inside the superradiant regime, remain unknown. In this work, by using the strong-coupling expansion and exact diagonalization (ED), we study the Z2-U(1) Dicke model with independent rotating-wave coupling g and counterrotating-wave coupling ${g}^{\ensuremath{'}}$ at a finite N. This model includes the four standard quantum optics models as its various special limits. We show that in the superradiant phase, the system's energy levels are grouped into doublets with even and odd parity. Any anisotropy $\ensuremath{\beta}={g}^{\ensuremath{'}}/g\ensuremath{\ne}1$ leads to the oscillation of parities in both the ground and excited doublets as the atom-photon coupling strength increases. The oscillations will be pushed to the infinite coupling strength in the isotropic Z2 limit $\ensuremath{\beta}=1$. We find nearly perfect agreement between the strong-coupling expansion and the ED in the superradiant regime when $\ensuremath{\beta}$ is not too small. We also compute the photon correlation functions, squeezing spectrum, and number correlation functions that can be measured by various standard optical techniques.