We present an analytical and numerical study of the fluorescence spectra of a bistable driven system by means of the Keldysh diagram technique in pseudoparticle representation. The spectra exhibit smooth transition between the ultraquantum and the quasiclassical limits and indicate the threshold value of the external field when changing of the most probable stable state occurs. The analysis of the fluorescence spectra also allows us to determine the most probable stable state of the system. It was also shown that, at integer and half-integer detuning-nonlinearity ratios, multiphoton resonance leads to abrupt changes in fluorescence spectra. It was also revealed that the fluorescence spectra are symmetric in the limit of zero environment temperature. The predicted features of the spectra could be observed in experiments with ultrahigh-quality resonators in either microwave or optical domains.
We present an analytical and numerical study of the fluorescence spectra of a bistable driven system by means of the Keldysh diagram technique in pseudoparticle representation. The spectra exhibit smooth transition between the ultraquantum and the quasiclassical limits and indicate the threshold value of the external field when changing of the most probable stable state occurs. The analysis of the fluorescence spectra also allows us to determine the most probable stable state of the system. It was also shown that, at integer and half-integer detuning-nonlinearity ratios, multiphoton resonance leads to abrupt changes in fluorescence spectra. It was also revealed that the fluorescence spectra are symmetric in the limit of zero environment temperature. The predicted features of the spectra could be observed in experiments with ultrahigh-quality resonators in either microwave or optical domains.
The nonequilibrium statistics and kinetics of a simple bistable system (resonantly driven nonlinear oscillator coupled to reservoir) have been investigated by means of master equation for the density matrix and quasiclassical Fokker--Planck equation in quasienergy space. We found that the system's statistical and kinetic properties drastically change when the quasienergy states become nearly degenerate and the occupation of the most excited state is strongly enhanced. It has been revealed that, in nearly degenerate case, a critical quasienergy parameter emerges. Below the critical quasienergy value the eigenstates are superpositions of the quasiclassical states from different phase-space regions, while above this value the eigenstates correspond to only one particular region of the phase space. We have also generalized Keldysh theory for ionization of atoms in the electromagnetic field for bistable systems. It has been demonstrated that Keldysh parameter in the bistability region is large when pumping intensity is smaller than the critical value. It has been shown by direct calculations that multiphoton transition amplitude coincides with the tunneling amplitude. So, multiphoton transitions and tunneling between the regions of the phase space are just the same effects. We also demonstrated that, for bistable systems, the Keldysh parameter logarithmically depends on the external field amplitude.
The topic of this thesis is a theoretical study of nonequilibrium transitions and quantum statistical properties of nonlinear quantum optical systems driven by an external radiation source. In certain limiting cases, a comparison is made between these transitions and the phase transitions found in equilibrium physical systems. In chapters one and two, the mathematical tools are introduced. In operator terms, the time development is described by a Markovian master equation in the interaction picture. This is equivalent to a corresponding time development equation or Fokker-Planck equation in a vector space of c-numbers. In order to deal with the type of Fokker-Planck equation that results, we introduce a quantum classical correspondence resulting in a distribution function over a complex phase-space, which is a generalisation of the Glauber-Sudarshan P-function. In chapter three this is applied to a model of a coherently driven mode with nonlinear dispersion and absorption. We find in the limit of zero temperature, that the spectrum is symmetric relative to the input frequency, and an exact solution is obtained for the distribution function. For a detuned driving field and nonlinear dispersion, optical bistability can occur. In chapter four a model of sub/second harmonic generation is introduced. This has several non-equilibrium transitions, including dispersive optical bistability, and bistable behaviour with coherent phase-locked input to both modes. Exact solutions occur in the limit of zero temperature and adiabatic elimination of one mode. In both chapters three and four, steady-state photon anti-bunching occurs with an absorptive nonlinearity. In chapter five we include interactions between the radiation mode and a fluorescent atomic system. In this case different behaviour occurs depending on the relative decay rates of the individual atoms and of the radiation mode. In the case of a high-Q interferometer, the atomic variables can be adiabatically eliminated. Both dispersive and absorptive bistability can occur. We show by analytic and numerical calculations (in the case of inhomogeneous broadening) that dispersive operation has advantages in requiring a lower atomic density and input field to observe bistability. When the input field has Gaussian (rather than coherent) photon statistics, there is no bistability, but enhanced photon bunching occurs. Finally, there is a different type of behaviour when the field mode decays rapidly and can be adiabatically eliminated. Within the cooperation lifetime the system can be described by a J²-invariant Hamiltonian, giving the special case of only collective damping. The result is a new type of critical point transition in the thermodynamic limit, with the appearance of a family of solutions like Lotka-Volterra cycles for a coherent driving field above threshold.
We discuss a mean-field quantum-mechanical model which describes the dynamics of a homogeneously broadened system of two-level atoms contained in a pencil-shaped resonant cavity and driven by a coherent resonant field EI. The model is treated in the semiclassical approximation. The model is justified on the basis of Maxwell-Bloch equations with two coherently coupled directions of propagation, with boundary conditions for the fields taken into account. Above a suitable critical density of atoms the system exhibits a bistable behavior including both the stationary situation and the transient, both the light transmitted in the forward direction and the fluorescent light. Bistability is shown to be a consequence of atomic cooperation. We give a simple description of optical bistability leading to new predictions for the transient behavior of the transmitted light and for the spectrum of the fluorescent light. The damping constant which characterizes the rate of appraoch to the stationary situation exhibits a hysteresis cycle. In the low-transmission regime the approach is monotonic, whereas in the highly transmitting situation the approach is oscillatory. One finds a critical slowing down in correspondence with the values EI(+), ${{E}_{I}}^{(\ensuremath{-})}$ of the incident field where the transmitted field (as well as the total fluorescence intensity, the rate of approach to the stationary state, etc.) changes discontinuously. On the basis of the regression hypothesis we give a qualitative description of the spectrum of the fluorescent light. This is shown to undergo hysteresis and discontinuous changes at the same values EI(+), ${E}_{I}^{(\ensuremath{-})}$ of the incident field. Below the critical density of atoms one recovers the usual picture of resonance fluorescence, with a continuous transition from a single-line spectrum to a three-peaked structure when the Rabi frequency becomes equal to the natural linewidth. Above the critical density the triplet appears only when the Rabi frequency becomes equal to the cooperative linewidth of pure super-fluorescence (i.e., for EI=EI(+)). Moreover, it appears discontinuously: when EI crosses the value EI(+) the spectrum changes from a single narrow line to a triplet with well-separated sidebands.
We study the fluorescence spectrum of a strongly driven two-level system (TLS) with modulated transition frequency, which is a bichromatically driven TLS and has multiple resonance frequencies. We are aiming to provide a reliable description of the fluorescence in a regime that is difficult to tackle with perturbation theory and the rotating-wave approximation (RWA), and illustrate the spectral features of the fluorescence under off- and multiphoton-resonance conditions. To go beyond the RWA, we use a semianalytical counter-rotating-hybridized rotating-wave method that combines a unitary transformation and Floquet theory to calculate the two-mode Floquet states and quasienergies for the bichromatically driven TLS. We then solve the master equation accounting for the spontaneous decay in the bases of the two-mode Floquet states, and derive a physically transparent fluorescence spectrum. In comparison with the numerically exact spectrum from the generalized Floquet-Liouville approach, the present spectrum is found to be applicable in a wide range of the parameters where the RWA and the secular approximation may break down. We find that the counter-rotating (CR) terms of the transverse field omitted in the RWA have non-negligible contributions to the spectrum under certain conditions. Particularly, at the multiphoton resonance the width of which is comparable with the Bloch-Siegert shift, the RWA and non-RWA spectra markedly differ from each other because of the CR-induced shift. We also analyze the symmetry of the spectrum in terms of the transition matrix elements between the two-mode Floquet states. We show that the strict symmetry of the spectrum cannot be expected without the RWA but the almost symmetric spectrum can be obtained at the single-photon resonance that takes the Bloch-Siegert shift into account if the driving is moderately strong and at the multiphoton resonance with a sufficiently weak transverse field.
Resonance fluorescence spectra of a driven Kerr nonlinear resonator is investigated both theoretically and experimentally. When the Kerr nonlinear resonator is driven strongly such that the induced Rabi frequency is comparable to or larger than the Kerr nonlinearity, the system cannot be approximated as a two-level system. We theoretically derive characteristic features in the fluorescence spectra such as the decrease of the center-peak intensity and the asymmetric sideband peaks in the presence of finite dephasing. Those features are consistently explained by the population of the initial dressed state and its transition matrix element to the final dressed state of the transition corresponding to each peak. Finally, we experimentally measure the resonance fluorescence spectra of a driven superconducting Kerr nonlinear resonator and find a quantitative agreement with our theory.
We study the semiclassical mean-field theory of absorptive optical bistability in the limit of a fully developed hysteresis cycle, i.e., in the limit of large values of the bistability parameter C. We show by multiple time-scale perturbation analysis that it is possible to describe the transition between the two stable branches by simple equations. Near the low-transmission branch, the behavior of the system is governed by an equation for the atomic population, whereas, near the high-transmission branch, it is an equation for the field amplitude which determines the long-time evolution of the system. We then present an analytic study, completed by numerical results whenever necessary, of the time-dependent response of the system when the input field is swept across the bistable region. We study the influence of the initial conditions, the sweeping velocity, and the ratio of the atomic relaxation times on the dynamical hysteresis as well as its relation with the stationary hysteresis.