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Physica Scripta

Exact quasienergy crossing in a driven two-level dynamics

Xian-Geng Zhao

Published Jul 1, 1995• 0 citations
Publisher

Abstract

The dynamic effect of particles in a periodically driven two-level system is studied with the help of SU(2) symmetry. General closed-form solutions for the quasienergy and the Floquet states are obtained, from which it is found that the avoided quasienergy crossing is confined in the range (−ω/2, ω/2), where ω is the frequency of the driving fields. Of particular interest are high-frequency driving fields, of which it is demonstrated that the exact quasienergy crossings occur at certain values of amplitude and frequency of driving fields, and this phenomenon of dynamic localization should be determined by experiments.

DOI
10.1088/0031-8949/52/1/004

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The dynamic effect of particles in a periodically driven two-level system is studied with the help of SU(2) symmetry. General closed-form solutions for the quasienergy and the Floquet states are obtained, from which it is found that the avoided quasienergy crossing is confined in the range (−ω/2, ω/2), where ω is the frequency of the driving fields. Of particular interest are high-frequency driving fields, of which it is demonstrated that the exact quasienergy crossings occur at certain values of amplitude and frequency of driving fields, and this phenomenon of dynamic localization should be determined by experiments.
The quantum behaviour of electrons in a two-level system under the influence of an AC-DC field is investigated. The closed-form solution of quasienergy is obtained, from which the exact level crossing is found to appear at a fan structure in the section (E, E 0 ) through the parameter space (E, E 0 , omega ), where E is the magnitude of the AC field, E 0 is the magnitude of the DC field, and omega is the frequency of the AC field. It is demonstrated that the emergence of the exact crossing of quasienergies is coincident with the onset of coherent destruction of tunnelling.
We investigate the Floquet spectrum of a detuned, driven two-level system and show that it exhibits exact quasienergy crossings when the detuning is an integer multiple of the energy quantum of the driving field. This behavior can be explained by a hidden time-nonlocal parity, which allows the Floquet modes to be classified as even or odd. Then a generic feature is the emergence of exact crossings between quasienergies of different parity. A constructive proof of the existence of the symmetry is based on a scalar recurrence relation. Moreover, we present a general scheme for its numerical computation, which can be applied to models beyond the two-level system. Analytical results are illustrated with numerical data.
We investigate the Floquet spectrum and excitation properties of a two-ultracold-atom system with periodically driven interaction in a three-dimensional harmonic trap. The interaction between the atoms is changed by varying the s-wave scattering length in two ways, the cosine and the square-wave modulations. It is found that as the driving frequency increases, the Floquet spectrum exhibits two main features for both modulations, the accumulating and the spreading of the quasienergy levels, which further lead to different dynamical behaviors. The accumulation is associated with collective excitations and the persistent growth of the energy, while the spread indicates that the energy is bounded at all times. The initial scattering length, the driving frequency and amplitude can all significantly change the Floquet spectrum as well as the dynamics. However, the corresponding relation between them is valid universally. Finally, we propose a mechanism for selectively exciting the system to one specific state by using the avoided crossing of two quasienergy levels, which could guide preparation of a desired state in experiments.
The quasienergy spectrum of a strongly driven two-level system as a function of the driving parameters exhibits conical intersections, which are enabled by hidden time-nonlocal symmetries. We show that each such crossing carries a quantized topological charge: the Floquet--Berry phase acquired along an adiabatic loop around a cone is equal to a Z2\mathbb{Z}_2Z2​-valued charge. We further identify a second family of degeneracies that occurs at vanishing driving amplitude, when the level splitting matches mmm energy quanta of the field. Along the Stark-shifted resonance line, the minimum quasienergy gap opens as ∣A∣m|A|^m∣A∣m, and the charge is nontrivial only for odd mmm. We analytically derive both results from a perturbative reduction to a spin-1/21/21/2 in an effective two-dimensional magnetic field and confirm them numerically through the Bargmann invariant. Moreover, we propose a chirality-based protocol that cancels the dynamical phase to isolate the geometric one, and an ancilla-based Ramsey readout that renders the topological charge directly observable.
We study the dynamics of a two-level quantum system under the influence of sinusoidal driving in the intermediate-frequency regime. Analyzing the Floquet quasienergy spectrum, we find combinations of the field parameters for which population transfer is optimal and takes place through a series of well-defined steps of fixed duration. We also show how the corresponding evolution operator can be approximated at all times by a very simple analytical expression. We propose this model as being specially suitable for treating periodic driving at avoided crossings found in complex multilevel systems, and thus show a relevant application of our results to designing a control protocol in a realistic molecular model.