The Floquet Theory of the Two-Level System Revisited

Abstract

We reconsider the periodically driven two level system and especially the Rabi problem with linear polarization. The Floquet theory of this problem can be reduced to its classical limit, i. e. , to the investigation of periodic solutions of the classical Hamiltonian equations of motion in the Bloch sphere. The quasienergy is essentially the action integral over one period and the resonance condition due to J. H. Shirley is shown to be equivalent to the vanishing of the time average of a certain component of the classical solution. This geometrical approach is applied to obtain analytical approximations to physical quantities of the Rabi problem with linear polarization as well as asymptotic formulas for various limit cases.

Periodic thermodynamics of the Rabi model with circular polarization for arbitrary spin quantum numbers

“…We will make heavy use of a procedure which allows one to transfer a solution to the Schrödinger equation for a spin with spin quantum number s = 1/2 in a timedependent external field to a solution of the corresponding Schrödinger equation for general s, see also Ref. [31]. This procedure essentially rests on the fact that a spin-s state can be represented as a direct symmetrized product of 2s spin-1/2 states, as exposed by Landau and Lifshitz [32].…”

Dynamics of quantum systems driven by time-varying Hamiltonians: Solution for the Bloch-Siegert Hamiltonian and applications to NMR

“…Determining the dynamics of two-level systems driven by time-dependent Hamiltonians is still an open problem, which continues to be a very active area of research [8,[20][21][22][23][24][25][26][27][28][29]. This is because of the experimental relevance of such systems, their role as theoretical models, and the need to master the internal evolution of qubits undergoing quantum gates [27,30].…”

On the geometry of the classical Rabi problem

“…and satisfies the transformed equation of motion ( 16), as can be easily confirmed by direct computation. This solution has been used to obtain results for the case of a monochromatic, linearly polarized magnetic field in the limit of a vanishing constant field component, see, e. g., [3]. We note the curiosity that the system (S, H) of two curves that is derived from a solution of (5) in the way described above can be considered as a kind of "clock" in the sense that it allows, at least locally, the reconstruction of the original parametrization τ → s(τ ) and τ → h(τ ).…”

“…We will slightly generalize the initial point of time to τ = τ 0 . Since scalar products are invariant under time evolution according to (5) it follows that the fundamental solutions will form a right-handed orthonormal frame for all τ ∈ Ê, i. e., R(τ, τ 0 ) ≡ s (1) (τ ), s (2) (τ ), s (3)…”

“…According to the preceding remarks it seems that the solutions of (4) provide the information about the corresponding solutions of (2) up to a (time-depending) phase factor. But it can be shown [3] that also this phase factor can be reconstructed from periodic solutions of (4) by means of certain integrals. Here we encounter the rare case where a quantum problem and the corresponding classical problem are essentially equivalent.…”