We present a new convergent strong coupling expansion for two-level atoms in external periodic fields, free of secular terms. As a first application, we show that the coherent destruction of tunnelling is a third-order effect. We also present an exact treatment of the high-frequency region, and compare it with the theory of averaging. The qualitative frequency spectrum of the transition probability amplitude contains an effective Rabi frequency. 02.30.Mv, 31.15.Md, 73.40.Gk The advent of strong laser pulses has stimulated interest in strong-coupling expansions in quantum optics and quantum electrodynamics. Such expansions are also of considerable general conceptual interest in several branches of physics. However, particularly in the case of periodic and quasi-periodic perturbations, the usual series, e.g., the Dyson series, are plagued by secular terms, leading to a violation of unitarity when the expansion is truncated at any order. In addition, small denominators appear in the quasi-periodic case (see the discussion in the introduction in [1]). These problems have been formally solved in a nice letter by W. Scherer [2] and in the papers which followed [3,4]. The main shortcoming in these works is that convergence was not controlled, an admittedly difficult enterprise. By writing an Ansatz in exponential form, and "renormalizing" the exponential inductively, we were able to eliminate completely the secular terms and to prove convergence in the special case of a two-level atom subject to a periodic perturbation, described by the Hamiltonian [5]The corresponding Schrödinger equation isadopting = 1 for simplicity. Above f (t) is of the formwith F n = F −n , since f is real, and σ i are the Pauli matrices satisfying [σ 1 , σ 2 ] = 2iσ 3 plus cyclic permutations. Assuming F n of order one, the situation where ǫ is "small" characterizes the strong coupling domain. It is convenient to perform a time-independent unitary rotation of π/2 around the 2-axis in (1), replacing H 1 (t) byand the Schrödinger equation bywith Φ(t) = exp(−iπσ 2 /4)Ψ(t).The following result was proved in [1]. Let f be continuously differentiable and g be a particular solution of the generalized Riccati equationThen the function Φ : R → C 2 given bywhereand S(t) ≡ t 0 R(τ ) −2 dτ , is a solution of the Schrödinger equation (5) with initial value Φ(0) = φ + (0) φ − (0) . A simple computation [1] shows that the components φ ± of Φ(t) satisfy a complex version of Hill's equationIn [1] we attempted to solve (10) using the Ansatz φ(t) = exp −i t 0 (f (τ ) + g(τ ))dτ , from which it follows that g has to satisfy the generalized Riccati equation (6). A similar idea was used by F. Bloch and A. Siegert in [6]. For ǫ ≡ 0 a solution of (6) is given by exp −i t 0 f (τ )dτ . Thus, in the above Ansatz we are searching for solutions in terms of an "effective external field" of the form f + g, with g vanishing for ǫ = 0. It is thus natural to posewhere 1
