Perturbative results on localization for a driven two-level system

Abstract

Using perturbation theory in the strong coupling regime, that is, the dual Dyson series, and renormalization group techniques to re-sum secular terms, we obtain the perturbation series of the two-level system driven by a sinusoidal field till second order. The third order correction to the energy levels is obtained proving how this correction does not modify at all the localization condition for a strong field as arising from the zeros of the zero-th Bessel function of integer order. A comparison with weak coupling perturbation theory is done showing how the latter is contained in the strong coupling expansion in the proper limits. The strong coupling expansion we obtain proves to be accurate in the regime of high-frequency driving field. This computation gives an explicit analytical form to Floquet eigenstates and quasi-energies for this problem, for high-frequency driving fields, supporting recent theoretical and experimental findings for quantum devices expected to give a representation for qubits in quantum computation.

Third-order correction to localization in a two-level driven system

“…5 we note as the series we obtained in it for the Floquet quasi-energies till third order was not correct [eqs. (29), (30) and (31) in Ref.…”

“…(31) in Ref. 5 . We have given explicitly the first term with J 0 (z) that recovers the small z limit.…”

Polynomial scheme for time evolution of open and closed quantum systems

“…[12], however, the Master equation description is not universally valid for all the models of environment and fragile in some systems [14].…”

Quantum NOT operation and integrability in two-level systems

“…Notice that we may have dynamical localization (ω 0 = 0) for values of B 3 and ω that produce zeros in J 0 [25,26]. Equations above were obtained by expanding the (co)sine in terms of Bessel functions sums.…”

“…Therefore, by (26) with Ω = B 3 − ω 2 and φ = 0, the explicit contour for the separatrix at stroboscopic instant is given by…”