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Journal of Physics B Atomic and Molecular Physics

Non-resonant effects in the fluorescent Dicke model. I. Exact steady state analysis

S. V. Lawande, Ravinder Rupchand Puri, Saad S. M. Hassan

Published Nov 14, 1981• 43 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/14/21/028

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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.

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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.

•Jan 1, 2009•0 citations
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.
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 authors consider a system comprised of a collection of identical two-level atoms interacting with the electromagnetic field in the dipole approximation and with an externally applied steady-state coherent driving field. The atoms are considered to interact with each other only via the electromagnetic field and are assumed to be contained within a volume much smaller than a resonance wavelength. The system is treated as a quantum-statistical ensemble in the rotating frame, rotating at the carrier frequency of the externally applied coherent field. The Hamiltonian in this frame is explicitly time independent and the exact analog of a spin temperature is defined. The thermodynamic-Green's-function method is used to determine the thermodynamic equilibrium properties of the system in the rotating frame. This results in a nonlinear relation between the applied field and the field in the volume containing the atoms. The expression is a nonlinear function of the effective or "spin" temperature as well, and has the form of a first-order phase transition with the macroscopic atomic polarization as order parameter. In the low-"spin"-temperature limit, and for perfect tuning, the expression reduces to the exact form of the nonlinear relation for optical bistability derived by Bonifacio and Lugiato and relating the incident and transmitted fields for atoms in a ring cavity in the mean-field approximation. The results contain absorptive as well as dispersive contributions. These results predict essentially different behavior as a function of off tuning compared with behavior predicted by others using statistical steady-state models far from thermodynamic equilibrium. In the limit of zero value for the applied field amplitude, the results reduce to the conditions for the well-known second-order superradiant" phase transition. The condition for the existence of the second-order phase transition is shown to depend on the cavity tuning and the photon escape rate. The authors also show why, in light of the experiments in optical bistability, this second-order phase is difficult to observe.
The exact steady-state solution is presented for the reduced atomic density operator describing the cooperative resonance fluorescence from a system of two atoms separated by an arbitrary distance in an off-resonance laser field, with the dipole-dipole near-field interaction between the atoms included. It is shown that the destructive effect of the dipole-dipole interaction on the formation of squeezing in two-atom resonance fluorescence found recently by Ficek et al. can be compensated to a very large extent by an appropriate choice of the detuning between the laser frequency and the single-atom transition frequency.
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.