The exact steady-state solution for the reduced atomic density operator for a model system of N identical two-level atoms confined to a single site (the Dicke model) and driven by a totally incoherent broad-band (chaotic) field is derived through the equivalent Fokker-Planck equation of the system. Steady-state values of the atomic observables, higher moments and atomic fluctuations are calculated for an arbitrary N and an arbitrary field strength. It is shown that when the chaotic field is a thermal (black-body) field the atomic system is driven into a steady relative occupation number that obeys the Boltzmann distribution law only for N=1. (The same result for N>1 atoms can be reached only if the cooperative interactions between the atoms are ignored.) In the thermodynamic limit N to infinity there is no critical behaviour in contrast to the resonant coherently driven case. It is also shown that semiclassical (direct) factorisation of the equations of motion does not lead to the correct results for the N to infinity limit of the exact quantum case. Within the exact theory numerical results for finite N (<or=100) are presented for the time-dependent expectation values of the atomic operators, the intensity and the intensity autocorrelation function. Superradiant pulses are emitted by N initially inverted atoms but as the incident intensity is increased this superradiant behaviour is obscured. The steady-state fluorescence spectrum is a single Lorentzian of width proportional to N in the weak-field case. In the strong-field case the spectrum is insensitive to the number of atoms and the result is similar to the one-atom situation. Photon bunching and antibunching effects are investigated for the weak- and strong-field limits. When the chaotic driving field is simply a single-mode thermal field the approach to atomic saturation is delayed in comparison with the case of the broad-band chaotic field.
The exact steady-state solution for the reduced atomic density operator for a model system of N identical two-level atoms confined to a single site (the Dicke model) and driven by a totally incoherent broad-band (chaotic) field is derived through the equivalent Fokker-Planck equation of the system. Steady-state values of the atomic observables, higher moments and atomic fluctuations are calculated for an arbitrary N and an arbitrary field strength. It is shown that when the chaotic field is a thermal (black-body) field the atomic system is driven into a steady relative occupation number that obeys the Boltzmann distribution law only for N=1. (The same result for N>1 atoms can be reached only if the cooperative interactions between the atoms are ignored.) In the thermodynamic limit N to infinity there is no critical behaviour in contrast to the resonant coherently driven case. It is also shown that semiclassical (direct) factorisation of the equations of motion does not lead to the correct results for the N to infinity limit of the exact quantum case. Within the exact theory numerical results for finite N (<or=100) are presented for the time-dependent expectation values of the atomic operators, the intensity and the intensity autocorrelation function. Superradiant pulses are emitted by N initially inverted atoms but as the incident intensity is increased this superradiant behaviour is obscured. The steady-state fluorescence spectrum is a single Lorentzian of width proportional to N in the weak-field case. In the strong-field case the spectrum is insensitive to the number of atoms and the result is similar to the one-atom situation. Photon bunching and antibunching effects are investigated for the weak- and strong-field limits. When the chaotic driving field is simply a single-mode thermal field the approach to atomic saturation is delayed in comparison with the case of the broad-band chaotic field.
The exact steady state solution for the reduced atomic density operator for a model of N identical two-level atoms occupying the same site (the Dicke model) and driven by a CW off-resonant laser field is presented. Exact expressions are given for the atomic observables and atomic correlation functions of arbitrary order. Exact thermodynamic analysis (N to infinity ) shows that the critical phase transition behaviour found in the resonant case is destroyed due to non-resonant effects. For large N and increasing detuning parameter, quantum fluctuations tend to vanish and hence the atomic observables tend to assume their semiclassical values. Also it is found that for a fixed arbitrary detuning parameter and increasing N these fluctuations disappear for the whole range of the driving field strength. The authors explain why it is not possible in the limit N to infinity to recover the correct asymptotic limit for the resonant case from the results of the dispersive case.
For pt.I see ibid., vol.14, p.4171 (1981). Semiclassical and quantal time-dependent results are presented for a model of N identical two-level atoms occupying the same site (the Dicke model) and driven by a CW off-resonance laser field. The exact semiclassical time-dependent solutions are derived from an analysis of the Fokker-Planck equation for the system in the atomic coherent states representation. Steady-state behaviour predicted by these solutions is in agreement with that obtained from the exact quantal steady-state analysis in the thermodynamic limit N to infinity . The semiclassical theory predicts (in the steady state) one type of fluorescent spectrum, which is a single sharp line ( delta function) centered at the driving field frequency. This is like the case at exact resonance and below threshold. Quantum mechanically, analytical results are obtained in the strong-field limit and within the secular approximation of the master equation: the fluorescence spectrum is the usual dynamical Stark triplet, apart from a cooperative factor N 2 . The absorption spectrum and the second-order intensity autocorrelation function are also calculated. Numerical results are also presented for finite N(N<or approximately=5). The effect of detuning on the additional sidebands (occurring at higher harmonics) in both the fluorescent and adsorption spectrum are examined.
A system of N identical superimposed two-level atoms (the Dicke model) interacting with one another and driven by a fluctuating laser field of finite phase and amplitude bandwidths is considered. An exact master equation for the phase-averaged reduced atomic density operator is derived using the theory of multiplicative stochastic processes. It is shown that in the absence of amplitude fluctuations, this equation admits an exact steady-state solution. The exact solution is further used to obtain and analyse the behaviour of the phase-averaged atomic observables and atomic correlation functions in the steady state. The principal results of our analysis is that the phase fluctuations tend to smooth the discontinuities in the observables and the correlation functions found in the non-fluctuating case. Further, including the amplitude fluctuations as well we derive in the high-field limit a Markovian master equation for the density operator averaged over the ensembles of both phase and amplitude fluctuations. The quantum regression theorem is used to derive and discuss exact analytical expressions for the fluorescent spectrum and the second-order intensity correlation function for a system of two interacting atoms.
We study the spontaneous emission of a multilevel equidistant excited quantum oscillator (nuclei, atoms, molecules) in the case in which the symmetry of the dynamical transitions between the levels is submitted to the SU (1,1) symmetry group and compare our results with those well known for the SU (2) symmetry. It is shown that the equidistant radiator in the process of spontaneous decay in the case of SU (1,1) symmetry exhibits either only acceleration or only a decrease in spontaneous emission rates, while in the case of SU (2) symmetry the radiation pulse is similar to that for the Dicke super-radiance. The exact steady-state solution for the atomic density operator for an equidistant radiator driven by a totally incoherent broad-band (chaotic) field for a SU (1,1) symmetry group is obtained using the Fokker-Planck equation. It is found that for a strong-field limit in the case of SU (1,1) symmetry the fluctuations in the atomic population increase with increasing intensity of the external field and do not achieve their maximal value as in the case of SU (2) symmetry. Moreover, in the case of SU (1,1) transition symmetry we find that there is a superbunching effect for certain values of group parameter I , while in the case of SU (2) symmetry only bunching and antibunching occur. It is found that for vanishing values of the external field intensity the second-order intensity correlation function achieves negative values.
In this thesis, we study the nonequilibrium dynamics of the time dependent single mode Dicke model in the thermodynamic limit in the context of the Liouville-von Neumann canonical method. We report an exact solution for this model under a non-equilibrium situation, generated by time dependent control parameters. We compute the correlations: in the atom-field ground state by means of the density operator for the pure state of the universe, and the corresponding reduced density operator for the atoms when the field is in resonance with the atomic ensemble. We obtain a closed relationship between the correlations in the ground state, the localisation of the atom¿s Wigner function in phase space, and the localisation of the wave function of the universe. As a measure of bipartite entanglement, we calculate the linear entropy from which we study the atom-field entanglement dynamics. In particular, we find a strong relationship between the stability of the auxiliary dynamical parameters, the reported entanglement, and the dynamics of certain physical observables. Furthermore, we use the Liouville-von Neumann technique to study the influence of quantum and thermal fluctuations on the dynamics of the normal and the superradiant phase. In addition, we calculate the density matrix for the quantum mechanical evolution of an initial thermal equilibrium state in the case of an instantaneous atom-field coupling and time independent ω and ω0.
Now, many different approaches have been presented to study the different semi-classical models derived from the Dicke Hamiltonian, which reflect a fact that the quantum-mechanical spin possesses no direct classical analog. The Hartree-Fock-type approximation is one of the widely used approaches, with which we derive the Heisenberg equations of motion for the system and replace the operators in these equations with the corresponding expectation values. In the present paper, we investigate the role of quantum phase transition in determining the chaotic property of the time-dependent driven Dicke model. The semi-classical Hamiltonian is derived by evaluating the expectation value of the Dicke Hamiltonian in a state, which is a product state of photonic and atomic coherent states. Based on the inverse of the relations between the position-momentum representation and the Bosonic creation-annihilation operators, the Hamiltonian is rewritten in the position-momentum representation and it undergoes a spontaneous symmetry-breaking phase transition, which is directly analogous to the quantum phase transition of the quantum system. In order to depict the Poincaré sections, which are used to analyze the trajectories through the four-dimensional phase space, we give the equations of motion of system from the derivatives of the semi-classical Hamiltonian for a variety of different parameters and initial conditions. According to the Dicke quantum phase transition observed from the experimental setup , we study the effect of a monochromatic non-adiabatic modulation of the atom-field coupling in Dicke model (i.e., the driven Dicke model) on the system chaos by adjusting the pump laser intensity. The change from periodic track to chaotic figure reflects the quantum properties of the system, especially the quantum phase transition point, which is a key position for people to analyse the shift from periodic orbit to chaos. In an undriven case, the system reduces to the standard Dicke model. We discover from the Poincaré sections that the system undergoes a change from the classical periodic orbit to a number of chaotic trajectories and in the superradiant phase area, the whole phase space is completely chaotic. When the static and driving coupling both exist, the system shows rich chaotic motion. The ground state properties are mainly determined by the static coupling, while the orbit of the system is adjusted by the driving coupling. If the static coupling is greater than the critical coupling, the system displays completely chaotic images in the Poincaré sections, and the periodic orbits in the chaos can also be adjusted by the strong driving coupling. While the static coupling is less than the critical coupling, the system can also show the chaotic images by adjusting the driving coupling strength and oscillation frequency. Moreover, by tuning the oscillation frequency, the Poincaré sections may change from the classical orbits to the chaos, and back to the classical orbits in the normal phase of the system.