Semiclassical many-mode Floquet theory. IV. Coherent population trapping and SU(3) dynamical evolution of dissipative three-level systems in intense bichromatic fields
The many-mode Floquet theory (MMFT) recently developed by authors is extended to incorporate the irreversible damping mechanisms for the nonperturbative treatment of the dynamical evolution of dissipative three-level systems at two-photon or multiphoton coherent resonance trapping conditions induced by two strong linearly polarized monochromatic fields. It has been recently shown by several workers that under the rotating-wave approximation (RWA), population may be permanently trapped in the three-level system if the coherent monochromatic fields are exactly two-photon resonant with the initial and final states, decoupled from the intermediate decaying level. In practice, the inclusion of the non-RWA terms necessarily modifies the resonant trapping conditions and behavior. In this paper we extend the generalized Van Vleck (GVV) nearly degenerate perturbation theory to an analytical treatment of the non-Hermitian two-mode Floquet Hamiltonian. This reduces the infinite-dimensional time-independent non-Hermitian Floquet Hamiltonian to a 3\ifmmode\times\else\texttimes\fi{}3 effective Hamiltonian, from which essential properties of the coherent population-trapping behavior as well as the dynamical evolution of the dissipative SU(3) coherence vector S\ensuremath{\rightarrow}(t) can be readily obtained and expressed in terms of only three complex quasienergy eigenvalues and eigenvectors. The MMFT-GVV studies show that the RWA two-photon resonant trapping condition is substantially modified by the effects of non-RWA terms, and that the system can be ``quasitrapped'' for only a finite amount of time characterized by a small imaginary energy (width) associated with a coherent superposition state of the initial and final levels. Furthermore, it is found that the initially eight-dimensional coherence vector S\ensuremath{\rightarrow}(t) evolves predominantly to a one-dimensional scalar at the two-photon or multiphoton resonant quasitrapping conditions. Detailed results and pictorial representations of the population trapping and SU(3) dissipative dynamical evolution are presented.
The many-mode Floquet theory (MMFT) recently developed by authors is extended to incorporate the irreversible damping mechanisms for the nonperturbative treatment of the dynamical evolution of dissipative three-level systems at two-photon or multiphoton coherent resonance trapping conditions induced by two strong linearly polarized monochromatic fields. It has been recently shown by several workers that under the rotating-wave approximation (RWA), population may be permanently trapped in the three-level system if the coherent monochromatic fields are exactly two-photon resonant with the initial and final states, decoupled from the intermediate decaying level. In practice, the inclusion of the non-RWA terms necessarily modifies the resonant trapping conditions and behavior. In this paper we extend the generalized Van Vleck (GVV) nearly degenerate perturbation theory to an analytical treatment of the non-Hermitian two-mode Floquet Hamiltonian. This reduces the infinite-dimensional time-independent non-Hermitian Floquet Hamiltonian to a 3\ifmmode\times\else\texttimes\fi{}3 effective Hamiltonian, from which essential properties of the coherent population-trapping behavior as well as the dynamical evolution of the dissipative SU(3) coherence vector S\ensuremath{\rightarrow}(t) can be readily obtained and expressed in terms of only three complex quasienergy eigenvalues and eigenvectors. The MMFT-GVV studies show that the RWA two-photon resonant trapping condition is substantially modified by the effects of non-RWA terms, and that the system can be ``quasitrapped'' for only a finite amount of time characterized by a small imaginary energy (width) associated with a coherent superposition state of the initial and final levels. Furthermore, it is found that the initially eight-dimensional coherence vector S\ensuremath{\rightarrow}(t) evolves predominantly to a one-dimensional scalar at the two-photon or multiphoton resonant quasitrapping conditions. Detailed results and pictorial representations of the population trapping and SU(3) dissipative dynamical evolution are presented.
The SU(3) dynamical evolution of three-level systems at two-photon resonance induced by two strong linearly polarized monochromatic fields is studied exactly by means of the semiclassical many-mode Floquet theory (MMFT) recently developed by the authors. Within the rotating-wave approximation (RWA), Hioe and Eberly have recently shown that the eight-dimensional SU(3) coherent vector S\ensuremath{\rightarrow} characterizing the time evolution of three-level systems can be factored into three independent vectors of dimensions three, four, and one, at appropriate two-photon resonance conditions. In practice, however, if the laser-atom interactions occur away from the two-photon resonance, or if the RWA is not valid, etc., this Gell-Mann--type SU(3) dynamical symmetry will be broken. It is shown in this paper that instead of solving the time-dependent generalized Bloch equations, the SU(3) dynamical evolution of the coherent vector S\ensuremath{\rightarrow} as well as various symmetry-breaking effects can be expediently studied by the use of the MMFT and expressed in terms of a few time-independent quasienergy eigenvalues and eigenvectors. Furthermore, we have extended the generalized Van Vleck (GVV) nearly degenerate perturbation theory to an analytical treatment of the two-mode Floquet Hamiltonian. This reduces the infinite-dimensional time-independent Floquet Hamiltonian into a 3\ifmmode\times\else\texttimes\fi{}3 effective Hamiltonian, from which useful analytical properties of the SU(3) coherent vector can be easily obtained. The combination of the MMFT and the GVV method thus greatly facilitates the study of the dynamical evolution. Pictorial comparison of the exact and the RWA results of the time evolution of the eight-dimensional coherent vector under several different physical conditions is presented and discussed at length.
The authors present an exact treatment of two- and multiquantum transitions in three-level systems driven by two intense linearly polarised monochromatic fields based on the semiclassical many-mode Floquet theory developed recently. Further they extend the almost degenerate perturbation theory of Salwen (1955-6) to the two-mode Floquet Hamiltonian and obtain approximate analytical formulae for multiphoton transition probabilities, resonance bichromatic shifts and widths, and absorption lineshapes beyond the conventional rotating-wave approximation (RWA). Detailed comparison of the analytical, the RWA, and the exact results is given. Several novel features of multiphoton lineshape characteristics are pointed out.
For pt.I see Chem. Phys. Lett. vol.96, p.464 (1983). The non-linear multiphoton excitation dynamics of two-level systems under the influence of two strong linearly polarised monochromatic fields is studied exactly for the first time along the semiclassical many-mode Floquet theory. The present approach relaxes the restrictions of the conventional generalised rotating-wave approximation (GRWA) and allows correct treatment of various non-linear features such as power broadening, resonance shift, asymmetry in absorption lineshape, etc. By extending the nearly degenerate perturbative method of Salwen (1955), approximate analytical expressions for transition probabilities are obtained for multiphoton resonance processes for which (n+1) photons of one field are absorbed, whereas n photons of the other field are emitted, where n is an arbitrary integer. Detailed comparison of the analytical, the GRWA and the numerically exact results is given. Also presented is a case study of the time evolution of spin 1/2 systems in multi-quantum NMR transitions driven by pulsed bichromatic radiofrequency fields.
A Floquet-Liouville supermatrix (FLSM) approach is presented for nonperturbative treatment of the time development of the density-matrix operator of atoms and molecules exposed to intense polychromatic fields. By extending the many-mode Floquet theory (MMFT) recently developed, the time-dependent Liouville equation for the density matrix of quantum systems undergoing relaxations (due to radiative decays and collisional dampings, etc.) can be transformed into an equivalent time-independent non-Hermitian FLSM eigenvalue problem. This yields a numerically stable and computationally efficient approach for the unified treatment of nonresonant and resonant, one- and multiple-photon, steady-state and transient phenomena in nonlinear optical processes, much beyond the conventional rotating-wave-approximation (RWA) method. Connections of the FLSM approach to the MMFT in the limit of zero relaxations are also made, providing the understanding of the physical significance of FLSM supereigenvalues and eigenvectors. In addition to the exact FLSM formalism, we have also presented higher-order perturbative results, based on the extension of the generalized Van Vleck (GVV) nearly degenerate perturbation theory, appropriate for somewhat weaker fields and near-resonant processes, but beyond the RWA limit. The implementation of the GVV method in the time-independent Floquet-Liouvillian allows the reduction of the infinite-dimensional FLSM into a finite-dimensional GVV-Liouville matrix, from which essential analytical results are readily obtained. As an illustration of the usefulness of the new formalism, we extend both the FLSM and the GVV methods to a formal study of the multiphoton-induced resonance fluorescence spectra of two-level systems subject to purely radiation relaxations. Both the time-averaged power spectrum and the time-dependent physical spectrum are exploited in details, and novel new features in intense fields are pointed out.
We study driven and open quantum systems described by a time-periodic Lindblad master equation. In closed systems, the stroboscopic dynamics can always be described by an effective time-independent Floquet Hamiltonian; this idea is the basis of Floquet engineering. However, in the presence of dissipation, the existence of an effective time-independent Floquet Lindbladian is not guaranteed due to the non-unitary nature of the evolution. Using Floquet theory, we construct a well-defined time-independent Floquet Lindbladian in an extended Sambe-Liouville space, transforming the initial time-dependent problem to a static and non-Hermitian eigenvalue problem. For harmonic driving, we introduce a matrix continued fraction method to nonperturbatively resum multiphoton processes and construct an effective Floquet Lindbladian acting only on the physical Liouville space. Compared to other high-frequency expansions, this method has the advantage of providing the whole infinite series expansion at once. Using a resolvent formalism, we show how to obtain a spectral Floquet representation of correlation functions of an open quantum system. As an application, we consider a dissipating two-level system in a linearly polarized field and calculate its resonance fluorescence spectrum. Furthermore, we consider a parametrically driven quantum dot with pump and loss for which we calculate its spectral function and current-voltage characteristics.
We present a practical nonperturbative method for exact treatment of intensity-dependent generalized nonlinear optical susceptibilities χ(ω) in intense polychromatic fields, valid for arbitrary laser intensities, detunings, and relaxation. By means of the many-mode Floquet theory, the time-dependent Liouville equation can be transformed into an equivalent time-independent infinite-dimensional Floquet–Liouville supermatrix (FLSM) eigenvalue problem. It is then shown that the nonlinear optical susceptibilities χ(ω) can be completely determined simply from the supereigenvalues and eigenfunctions of the Floquet–Liouvillian L̂F. In addition to this exact FLSM approach, we have also presented higher-order perturbative results, based on the extension of the Salwen’s nearly degenerate perturbation theory, appropriate for somewhat weaker fields and near-resonant multiphoton processes, but beyond the conventional perturbative or rotating wave approximation (RWA). In the case of two-level systems, for example, the implementation of Salwen’s method in the time-independent L̂F allows the reduction of the infinite-dimensional FLSM into a 4×4 dimensional effective Hamiltonian, from which essential analytical formulas for intensity-dependent χ(ω) can be obtained. These methods are applied to a detailed study of intensity-dependent spectral line shapes (such as hole burning and extra resonance peaks at the line center, and the effects of saturation, detuning, and radiative and collisional damping, etc.) and subharmonic structures in nonlinear multiple wave mixings χ[(m+1)ω1−mω2] for two-level systems in intense linearly polarized bichromatic fields.