Non-dispersive wave packets in periodically driven quantum systems

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430 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 It is always possible to nd a set of canonically conjugate phase space coordinates adapted to the dynamics of the system. These are the action-angle coordinates (I; ), whose existence is guaranteed by the Liouville–Arnold theorem [3], with 0 6 I; (47) 0 6 6 2 ; (48) { ; I} =1 (49) and {:;:} the usual Poisson brackets, Eq. (23). A fundamental property is that the Hamilton function in these coordinates depends on I alone, not on : H0 = H0(I) : (50) As a consequence of Hamilton’s equations of motion, I is a constant of motion, and = t + 0 (51) evolves linearly in time, with the angular velocity (I)= 9H0 (I) ; (52) 9I which depends on the action I. The period of the motion at a given value of I reads T = 2 : (53) (I) In simple words, the action I is nothing but the properly “rescaled” total energy, and the angle just measures how time evolves along the (periodic) orbits. In a one-dimensional system, the action variable can be expressed as an integral along the orbit, see Eq. (36). Suppose now that the system is exposed to a periodic driving force, such that the Hamilton function, in the original coordinates, writes H = H0(p; z)+ V (p; z) cos !t ; (54) with ! the frequency of the periodic drive and some small parameter which determines the strength of the perturbation. For simplicity, we choose a single cosine function to de ne the periodic driving. For a more complicated dependence on time [86], it is enough to expand it in a Fourier series, see Section 3.5. The equations become slightly more complicated, but the physics is essentially identical. We now express the perturbation V (p; z) in action-angle coordinates. Since Eq. (48), we obtain a Fourier series: is 2 -periodic, +∞ V (I; )= Vm(I) exp(im ) : (55) m=−∞ Note that, as evolves linearly with time t for the unperturbed motion (and therefore parametrizes an unperturbed periodic orbit), the Vm can also be seen as the Fourier components of V (t) evaluated along the classical, unperturbed trajectory. Furthermore, since the Hamilton function is real, V−m=V ∗ m.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 431 Again for the sake of simplicity, we will assume that both are real and thus equal. The general case can be studied as well, at the price of slightly more complicated formulas. Plugging Eq. (55) into Eq. (54) results in the following Hamilton function in action-angle coordinates: +∞ H = H0(I)+ Vm(I) exp(im ) cos !t ; (56) m=−∞ which (assuming V−m = Vm—see above) can be rewritten as +∞ H = H0(I)+ Vm(I) cos(m − !t) : (57) m=−∞ For su ciently small, the phase space trajectories of the perturbed dynamics will remain close to the unperturbed ones (for short times). This means that m − !t evolves approximately linearly in time as (m −!)t (see Eq. (51)), while I is slowly varying. It is therefore reasonable to expect that all the terms Vm(I) cos(m −!t) will oscillate rapidly and average out to zero, leading to an e ective approximate Hamiltonian identical to the unperturbed one. Of course, this approach is too simple. Indeed, close to a “resonance”, where (s −!) is small, the various terms Vm cos(m −!t) oscillate, except for the m = s term which may evolve very slowly and a ect the dynamics considerably. For simplicity, we restrict the present analysis to the principal resonance such that !. The extension to higher resonances (with s !) is discussed in Section 5. Our preceding remark is the basis of the “secular approximation” [3,18]. The guiding idea is to perform a canonical change of coordinates involving the slowly varying variable − !t. Because of the explicit time dependence, this requires rst the passage to an extended phase space, which comprises time as an additional coordinate. The Hamilton function in extended phase space is de ned by H = Pt + H (58) with Pt the momentum canonically conjugate to the new coordinate—time t. The physical time t is now parametrized by some ctitious time, say . However, 9H = 9Pt dt =1; d (59) i.e., t and are essentially identical. H, being independent of , is conserved as evolves. The requested transformation to slowly varying variables − !t reads 6 ˆ = − !t ; (60) Î = I; (61) ˆPt = Pt + !I ; (62) 6 This canonical change of coordinates is often refered to as “passing to the rotating frame”. It should however be emphasized that this suggests the correct picture only in phase space spanned by the action-angle coordinates (I; ). In the original coordinates (p; z), the transformation is usually very complicated, and only rarely a standard rotation in con guration space (see also Section 3.4).
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Non-dispersive wave packets in periodically driven quantum systems

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