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International Journal of Quantum Chemistry

Generalized Rabi oscillations in a three-level metastable system under periodic and quasiperiodic perturbations

Satrajit Adhikari, D. N. Nath, S. P. Bhattacharyya

Published Jan 1, 1996• 1 citation
Publisher

Abstract

The quantum dynamics of prototypical three-level metastable system under external perturbations, both time-independent and time-dependent, are studied numerically as well as analytically. The regular Rabi oscillations generated in the system under a monochromatic perturbation are shown to develop additional features when the system has a metastable state. The autocorrelation function of ψ(t) reveals nondecaying, almost quasiperiodic, behavior in such systems. The Rabi oscillations turn more complex when a bichromatic perturbation with two incommensurate frequencies is allowed to interact with the three-level system, irrespective of the presence or absence of a metastable state. A rapid decay in the autocorrelation function of the wavefunction [ψ(t)] is observed in such cases. © 1996 Wiley & Sons, Inc.

DOI
10.1002/(sici)1097-461x(1996)58:1<3::aid-qua1>3.0.co;2-1

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We have found quantum chaos in a system consisting of two levels coupled through a time-dependent perturbation, when this perturbation is quasiperiodic with two incommensurate frequencies. This contrasts with the case of a monochromatic perturbation where the oscillations of the physical observables are nonchaotic, because of the Floquet theorem. The existence of chaos is confirmed by the observation of rapidly decreasing autocorrelation functions, and by a continuous Fourier spectrum for time-dependent fluctuations of physical observables.
Abstract The interplay of coherence and decoherence is played out in a three-level quantum system, in which the third level is incoherently coupled to the second one which itself is in coherent interaction with the first level. The study is based on a stochastic scenario in which the coherent, unitary evolution of the system is randomly interrupted by a Poisson-driven pulse sequence. In the absence of an external pulse, the system undergoes coherent, unitary evolution restricted to the subspace spanned by the first level (level 1) and the second level (level 2). The application of a pulse induces transitions between the second and the third level (level 3), thereby introducing non-unitary effects that perturb the otherwise isolated two-level dynamics. The pulses are assumed to have infinitesimal duration, with strengths modeled as random variables that are uncorrelated across different pulses. A representative model for the stochastically-averaged transition (super)operator mimicking the dynamics induced by the application of pulses allows for an analytical derivation of the matrix elements of the averaged density operator, especially, the element pertaining to the first level. When the system is initially in level 1, we obtain in particular the temporal behavior of the stay-put probability, that is, the probability P 1 ( t ) that the system is still in level 1 at time t. As a function of time, the quantity P 1 ( t ) exhibits a coherence-to-decoherence crossover behavior. At short times t ≪ 1 / λ , where λ is the average frequency at which pulses are applied to the system, coherent dynamics dominate. Consequently, P 1 ( t ) displays pronounced Rabi-like oscillations. At long times t ≫ 1 / λ , decoherence effects prevail, leading to an exponential decay of the form P 1 ( t ) − 1 / 3 ∼ exp ⁡ ( − λ t ) . The results are relevant in the context of well-established Rabi oscillations, applicable to molecular beams and magnetic resonance experiments, as well as in more contemporary frameworks including quantum information processing, quantum resetting protocols, and the theory of quantum trajectories.
We analyze the dynamics of a superconducting qubit and the phenomenon of multiorder Rabi oscillations in the presence of a time-modulated external field. Such a field is also presented as a bichromatic field consisting of two spectral components, which are symmetrically detuned from the qubit resonance frequency. This approach leads to obtaining qualitative quantum effects beyond those for the case of monochromatic excitation of qubits. We calculate Floquet states and quasienergies of the composite system ``superconducting qubit plus time-modulated field'' for various resonant regimes. We analyze the dependence of quasienergies from the amplitude of an external field, demonstrating the zeros of difference between quasienergies. We show that, as a rule, populations of qubit states exhibit aperiodic oscillations, but we demonstrate the specific important regimes in which dynamics of populations becomes periodically regular.
We present a numerical study of the interaction of a two-level atom with a frequency-modulated cavity mode described by the Jaynes-Cummings Hamiltonian. In particular, we explore how the population of the atomic levels changes for monochromatic and bichromatic modulations of the cavity field. The primary goal is to find sets of boundary conditions for which the evolution of the system exhibits nonlinear dynamic behavior, by solving the equations of motion of the atomic system using embedded and semi-implicit numerical algorithms. We find that the system shows quasiperiodicity when the monochromatic frequencies or ratio of the bichromatic frequencies of the field are irrational (incommensurate) numbers. In addition, when the modulated frequencies are overtones of the Rabi frequency a nearly single dominant frequency becomes apparent for the evolution of the atomic level population of the system.
The question of the behavior of quantum systems in time-dependent fields whose classical counterparts exhibit chaotic behavior is addressed. For any nondissipative bounded quantum system under the influence of polychromatic (i.e., quasiperiodic) fields, it is proved by means of the many-mode Floquet theory1 that the autocorrelation function will recur infinitely often in the course of time, indicating no strict quantum stochasticity is possible.2 In particular, for an N-level quantum system undergoing multiphoton transitions, its dynamic behavior is described by the quasiperiodic motion of an (N2 – 1)-dimensional coherence vector S in accord with the SU(N) dynamic symmetries. On the other hand, for any dissipative quantum system, SU(N) symmetries are broken, and chaotic behavior is observed as the coherence vector Sevolves from an initially (N2 – 1)-dimensional space to a lower-dimensional space. The recurrence and chaotic phenomena are illustrated for two- and three-level quantum systems driven by intense bichromatic laser fields.2
Evolution of quantum fidelity for two-level systems is studied in the context of periodic echo. From a general treatment for time independent case, we obtain a simple condition on the governing Hamiltonians under which the systems display periodic quantum echo. For a specific time dependent problem the quantum fidelity is shown to exhibit Rabi oscillation. This may be considered as a simple mechanism to generate periodic echo, except for a specific initial superpositional state in which case the fidelity remains invariant.