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Progress of Theoretical Physics

Quantum Molecular Dynamics Study of a Two-Level Model with a Periodic External Field

Masayoshi Inoue, Shuichi Shiraishi

Published Nov 1, 1982• 1 citation
Publisher

Abstract

Periodic and quasi-periodic response of a two-level model to an external field is studied numerically and analytically, where the methods of (1) Fourier analysis, (2) stroboscopic representation and (3) projection of an orbit are used. In general, the model shows a quasi-periodic motion. However, when the intensity of the external field E takes a specific value, the motion becomes a periodic motion whose periodicity is N times bigger than that of the external field. The structure of this synchronization phenomenon for E≲0.5 is expressed by N[(ω-Ω)2 + (µE/2)2]1/2 = nω; (n = 1,2,3,···), where ω is the angular frequency of the external field, Ω is the Bohr frequency and µ is the electric moment. We find that the motion is described approximately by three modes for a weak external field. This approximate solution is better than Rabi's solution based on the rotating-wave approximation.

DOI
10.1143/ptp.68.1470

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Periodic and quasi-periodic response of a two-level model to an external field is studied numerically and analytically, where the methods of (1) Fourier analysis, (2) stroboscopic representation and (3) projection of an orbit are used. In general, the model shows a quasi-periodic motion. However, when the intensity of the external field E takes a specific value, the motion becomes a periodic motion whose periodicity is N times bigger than that of the external field. The structure of this synchronization phenomenon for E≲0.5 is expressed by N[(ω-Ω)2 + (µE/2)2]1/2 = nω; (n = 1,2,3,···), where ω is the angular frequency of the external field, Ω is the Bohr frequency and µ is the electric moment. We find that the motion is described approximately by three modes for a weak external field. This approximate solution is better than Rabi's solution based on the rotating-wave approximation.
We considered the semiclassical Rabi model, that is, a two-level system interacting with a single-mode, classical field. In resonance and for an arbitrary initial state of the system, we obtained a simple, approximate, and analytic solution that takes into account the counterrotating terms and that is accurate as long as the Rabi frequency is smaller than or equal to 1/2 (1/4) of the angular frequency of the field and 1 (10) Rabi oscillations are considered. In addition, the approximate solution has the same level of complexity as that obtained with the rotating-wave-approximation (RWA) and allows one to describe the evolution of the Bloch vector of the system in terms of a \textit{slow-precessional} and a \textit{fast-nutational} motion similar to that of a symmetric top with one point fixed. Finally, the approximate solution also leads to a simple criterion that allows one to determine the accuracy of the solution in the RWA.
We study the dynamics of a two-level quantum system interacting with an external electromagnetic field periodic and quasiperiodic in time. The quantum evolution is described exactly by the classical equations of motion of a gyromagnet in a time-dependent magnetic field. We prove that this classical system is integrable as a consequence of the underlying unitary quantum dynamics. As a consequence, for the periodic case, (i) rigorous assessment of the validity of the rotating-wave approximation (RWA) becomes possible even beyond the assumptions of resonance and weak coupling (the latter conditions are also shown to follow from the method of averaging); (ii) we determine conditions for the realization of the quantum NOT operation beyond the RWA, by means of classical stroboscopic maps. The results bear upon areas as diverse as quantum optics, nuclear magnetic resonance, and quantum computation.
We present the time evolution of the electronic two-level model in rotating wave approximation and calculate the Aharonov–Anandan phase of the vector states with cyclic evolution either within the period of an external monochromatic electric field or the period corresponding to the generalized Rabi frequency. The Aharonov–Anandan phase in the most general case is shown to be dependent on the initial vector state, unless the system evolves in the adiabatic regime; in the latter case, the Aharonov–Anandan phase recovers Berry's phase. In the quasi-resonant regime of the electronic two-level model, the different relations between Γ and the frequency ω of the electric field correspond to distinct regimes of the coupling between this external field and matter.
The extended quantum Rabi models make a significant contribution to understanding of the quantum nature of the atom-light interaction. We transform two kinds of extended quantum Rabi model, the anisotropic Rabi model and the asymmetric Rabi model, into the rotating frame and regard them as periodically driven quantum systems. The analytical solutions of the quasienergy spectrum as well as the Floquet modes for both models are constructed by applying the Floquet theory and the high-frequency expansion, which is applied to the nonstroboscopic dynamics of physical observables such as atomic inversion, transverse magnetization, and atom-field correlation. For the anisotropic Rabi model, the quasienergy fits well with the numerical results even when the rotating-wave coupling is in the deep strong-coupling regime $g\ensuremath{\simeq}\ensuremath{\hbar}\ensuremath{\omega}$ if the counter-rotating term is small enough compared to the driving frequency. Avoided level crossing may occur for quasienergy with the same parity when the positive branch spectrum lines for the total excitation number NNN cross the negative branch lines for N+2N+2N+2, while the high-frequency expansion fails to predict this due to the conservation of the total excitation number. Furthermore, we present analytical and numerical studies of the long-time evolution of population and conclude that the analytical method is credible for the population dynamics. For the asymmetric Rabi model, we find that the external bias field which breaks the parity symmetry of the total excitation number tends to cluster the upper and lower branches into two bundles, and the detuning induced gap in the first temporal Brillouin zone shows a quadratic dependence on the bias. Fourier analysis is applied to extract the frequency composition and two-frequency driving behavior is revealed. Varying the bias strength will change the time-averaged value of the oscillation, which shows how the bias competes with the detuning and atom-field coupling in the driving dynamics. Both models prove that treating the Hamiltonian in the rotating frame by Floquet theory gives an alternative tool in the study of interaction between atom and light.
We study phase-synchronization in a driven two-level system coupled to a non-Markovian bosonic reservoir. The dynamics is described by treating the system-bath coupling and the coherent drive without invoking the rotating-wave approximation, and simulated using the numerically exact hierarchical equations of motion. We observe that a robust phase-locking develops and that the corresponding synchronization measure rapidly acquires a finite value when the system is tuned to what we identify as a resonant-ratio condition, namely when the ratio between the drive amplitude and its frequency coincides with a zero of the Bessel function J0J_0J0​. We provide an explanation for this phenomenon by means of a static approximation derived from a Fourier analysis of the periodically driven Hamiltonian.