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Journal of Physics B Atomic Molecular and Optical Physics

Floquet states and quasienergies of a periodically driven two-level system

Qiongtao Xie

Published May 8, 2009• 7 citations
Publisher

Abstract

We investigate the Floquet states and quasienergies of a two-level quantum system driven by a high-frequency driving field. This quantum system is mapped onto a classical one in a function transformation. The Floquet states and quasienergies are calculated in a multiple scale expansion in the inverse of the frequency (ω) to order ω−2. It is shown that the coherent destruction of tunnelling is a second-order effect.

DOI
10.1088/0953-4075/42/10/105501

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We investigate the Floquet states and quasienergies of a two-level quantum system driven by a high-frequency driving field. This quantum system is mapped onto a classical one in a function transformation. The Floquet states and quasienergies are calculated in a multiple scale expansion in the inverse of the frequency (ω) to order ω−2. It is shown that the coherent destruction of tunnelling is a second-order effect.
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.
We investigate the quasienergies and Floquet states of two weakly coupled Bose-Einstein condensates driven by a periodic force. The quasienergies and Floquet states of this system are computed within two different theoretical frameworks: the mean-field model and the second-quantized model. The mean-field approach reveals a triangular structure in the quasienergy band. Our analysis of the corresponding Floquet states shows that this triangle signals the onset of a localization phenomenon, which can be regarded as a generalization of the well-known phenomenon called coherent destruction of tunneling. With the second-quantized model, we find also a triangular structure in the quantum quasienergy band, which is enveloped by the mean-field triangle. The close relation between these two sets of quasienergies is further explored by a semiclassical method. With a Sommerfeld rule generalized to time-dependent systems, the quantum quasienergies are computed by quantizing semiclassically the mean-field model and they are found to agree very well with the results obtained directly with the second-quantized model.
We investigate nonlinear Floquet states and quasienergies of a Bose-Einstein condensate in a double-well potential subject to a high-frequency driving field. A multiple time scale method is used to obtain the nonlinear Floquet states and quasienergies in both weak and strong nonlinearity regimes. It is shown that there are more nonlinear Floquet states without linear counterparts in the strong nonlinearity regime than in the weak nonlinearity regime, and they are phase-locked to an integer multiple of the external driving frequency. The validity of the effective description in the two different regimes is discussed. The stability criterion for these nonlinear Floquet states is found analytically and confirmed numerically.
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The quantum dynamics of a quartic double well, subjected to a harmonically oscillating field, is studied in the framework of the Floquet formalism. The modifications of the familiar tunneling process due to driving and dissipation are investigated numerically and explained in terms of the quasienergy spectrum. In absence of dissipation, there is a onedimensional manifold in the parameter space spanned by amplitude and frequency of the driving force, where tunneling is almost completely suppressed by the coherent driving. The influence of dissipation is described on basis of the reduced density matrix in the Floquet representation. In particular, we consider the effect of weak Ohmic damping. In the classical limit, this system corresponds to a damped bistable Duffing oscillator
We consider the dynamics of a two-level system (qubit) driven by strong and short resonant pulses in the framework of Floquet theory. First we derive analytical expressions for the quasienergies and Floquet states of the driven system. If the pulse amplitude varies very slowly, the system adiabatically follows the instantaneous Floquet states, which acquire dynamical phases that depend on the evolution of the quasienergies over time. The difference between the phases acquired by the two Floquet states corresponds to a qubit state rotation, generalizing the notion of Rabi oscillations to the case of large driving amplitudes. If the pulse amplitude changes very fast, the evolution is nonadiabatic, with transitions taking place between the Floquet states. We quantify and analyze the nonadiabatic transitions during the pulse by employing adiabatic perturbation theory and exact numerical simulations. We find that, for certain combinations of pulse rise and fall times and maximum driving amplitude, a destructive interference effect leads to a remarkably strong suppression of transitions between the Floquet states. This effect provides the basis of a quantum control protocol, which we name Floquet interference efficient suppression of transitions in the adiabatic basis (FIESTA), that can be used to design ultrafast high-fidelity single-qubit quantum gates.