Non-dispersive wave packets in periodically driven quantum systems

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446 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 describes the unperturbed atomic part, with bound energy spectrum: En = − 1 ; 2n2 The external driving n¿ 1 : (108) V =˜r · ˜F(t) (109) is characterized by the amplitude F of the microwave eld, and by its frequency !. We shall consider in detail the case of linear polarization, V = Fz cos !t ; (110) in Section 3.4, and that of circular polarization V = F(x cos !t + y sin !t) ; (111) in Section 3.4. The general case of elliptic polarization will be studied in Section 3.5. In the treatment of the perturbed Coulomb problem, Eq. (106), there is an apparent di culty with the singularity of the Coulomb potential at the origin, especially for the restricted one-dimensional model of the atom. This led several authors, see, e.g., [99–104], to consider unphysical potentials without singularity, for example of the type (r 2 + a 2 ) −1=2 . This is completely unnecessary and potentially dangerous. Indeed, such a potential breaks the Coulomb degeneracy which is responsible for the closed character of elliptical Kepler orbits (in the classical world), and for the degeneracy of the energy levels (in the quantum world). Such an unphysical symmetry breaking strongly modi es the structure of the non-linear resonance island, and a ects the existence and properties of non-dispersive wave packets outlined in Section 3.3.2. The Coulomb singularity can be rigorously regularized (in any dimension), both in classical and in quantum mechanics. In classical mechanics, this is made possible through the well-known Kustaanheimo–Stiefel transformation [105], used by various authors for perturbed Coulomb problems, see, e.g., [106–110]. In quantum mechanics, one may use a basis set of non-orthogonal functions, known as the Sturmian functions. Ultimately, the whole analysis relies on the dynamical symmetry properties of the Coulomb interaction and the associated SO(4,2) group [111–114]. It not only allows to treat the Coulomb singularity properly, but also to de ne a basis set of Sturmian functions extremely e cient for numerical calculations. The most common set are “spherical” Sturmian functions characterized by three quantum numbers, L and M for the angular structure (the associated wave functions are the usual spherical harmonics), and the positive integer n, for the radial part. As discussed in Section 3.1.2, the quantum properties of the non-dispersive wave packets are encoded in the spectrum of the Floquet Hamiltonian (75), which acts in con guration space extended by the time axis. The temporal properties are completely independent of the spatial dimension, and any Floquet eigenstate can be expanded in a Fourier series indexed by the integer k, as in Eq. (84). The Schrodinger equation for the Floquet Hamiltonian is tridiagonal in k, as in Eq. (85). When the spatial part of the wave function is expressed in a Sturmian basis, one nally obtains a generalized eigenvalue problem (A − EB)| ¿= 0, where both, A and B, are sparse matrices, the elements of which are known analytically and obey the following selection rules: | M| 6 1; | L| 6 1; | n| 6 2; | k| 6 1 : (112)
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 447 When a static electric or magnetic eld is added, see Section 4, some additional non-zero matrix elements exist, but sparsity is preserved. The eigenvalues can then be calculated using an e cient diagonalization routine such as the Lanczos algorithm [115–118]. Because—in the presence of a microwave eld—the system is unbounded, there are in general no exact bound states but rather resonances. Using Sturmian functions, the properties of the resonances can be calculated directly using the complex rotation technique [119–125]. The price to pay is to diagonalize complex symmetric matrices, instead of real symmetric ones. The advantage is that the resonances are obtained as complex eigenvalues En − i n=2 of the complex rotated Hamiltonian, En being the position of the resonance, and n its width. All essential properties of resonances can be obtained from complex eigenstates [126]. 3.2.3. Simpli ed 1D and 2D models Because explicit calculations for the real 3D hydrogen atom may be rather complicated, it is fruitful to study also simpli ed 1D and 2D approximations of the real world. Let us rst consider the simpli ed restriction of the atomic motion to one single dimension of con guration space, H0 = p2 2 − 1 z with the external driving along z, with z¿0 ; (113) V = Fz cos !t : (114) The energy spectrum of H0 is identical to the spectrum in 3D [127]: E (1D) n = − 1 with n ¿ 1 : (115) 2n2 Such a one-dimensional model allows to grasp essential features of the driven atomic dynamics, and provides the simplest example for the creation of non-dispersive wave packets by a near-resonant microwave eld. The classical dynamics live on a three-dimensional phase space, spanned by the single dimension of con guration space, the canonically conjugate momentum, and by time. This is the lowest dimensionality for a Hamiltonian system to display mixed regular–chaotic character [3]. For a circularly (or elliptically) polarized microwave, a 1D model is of course inadequate. One can use a two-dimensional model where the motion of the electron is restricted to the polarization plane. The energy spectrum in two dimensions is: E (2D) 1 n = − with n ¿ 0 : (116) 2(n +1=2) 2 It di ers from the 3D (and 1D) energy spectrum by the additional 1=2 in the denominator, due to the speci c Maslov indices induced by the Coulomb singularity. 3.2.4. Action-angle coordinates In order to apply the general theory of non-linear resonances and non-dispersive wave packets derived in Section 3.1, we need the action-angle coordinates for the hydrogen atom. For the simpli ed 1D model, the result is simple: the principal action I and the canonically conjugate angle are de ned
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Non-dispersive wave packets in periodically driven quantum systems

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