Non-dispersive wave packets in periodically driven quantum systems

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444 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 call this situation “optimal resonance”. From Eq. (101), it is associated with an integer value of n0. On the contrary, a half-integer value of n0 corresponds to = 1 and the least optimal case. 3.2. Rydberg states in external elds 3.2.1. Rydberg atoms In order to construct non-dispersive wave packets, a quantum system subject to periodic driving with classically non-linear dynamics is needed. The latter requirement rules out the harmonic oscillator, and all its variants. The simplest periodic driving is certainly provided by an externally applied, monochromatic electromagnetic eld. Extremely stable, tunable and well controlled sources exist over a wide range of frequencies. Furthermore, incoherent processes which destroy the phase coherence of the quantum wave function have to be minimized. Otherwise, they will spoil the localization properties of the non-dispersive wave packets and—in the worst case—destroy them completely. Therefore, the characteristic time scales of the incoherent processes should be at least much longer than the period of the driving eld. In this respect, atomic electrons appear as very good candidates, since—given suitable experimental conditions—atoms can be considered as practically isolated from the external world, with spontaneous emission of photons as the only incoherent process. Spontaneous emission is usually a very slow mechanism, especially for highly excited states: the spontaneous lifetime of typical atomic states is at least four or ve orders of magnitude longer than the classical Kepler period (typically nanoseconds vs. femtoseconds for weakly excited states [22]). The Coulomb interaction between the nucleus and the electrons is highly non-linear, which is very favorable. The e ciency of the coupling with an external electromagnetic eld is known to increase rapidly with the degree of excitation of the atom [22]. As we have seen in the preceeding Sections 3.1.1 and 3.1.2, non-dispersive wave packets are the quantum mechanical counterparts of non-linear resonances in periodically driven Hamiltonian systems, where the period of the drive matches some intrinsic time-scale of the unperturbed Hamiltonian dynamics. Due to the immediate correspondence between the classical Kepler problem and the hydrogen atom, the relevant time-scale in this simplest atomic system is the unperturbed classical Kepler period, which—compare Eq. (40)— coincides with the inverse level spacing between neighbouring eigenstates of the unperturbed atom, for large quantum numbers. Hence, the driving eld frequency has to be chosen resonant with an atomic transition in the Rydberg regime, typically around the principal quantum number n0 = 60. This is the microwave domain, where excellent sources exist. Thus, we believe that atomic Rydberg states are very well suited for the experimental preparation of non-dispersive wave packets 13 . In most cases, the energy scale involved in the dynamics of Rydberg electrons is so small that the inner electrons of the ionic core can be considered as frozen and ignored. Thus, we will consider mainly the hydrogen atom as the simplest prototype. Multi-electron e ects are discussed in Section 6.2. Alternative systems for observing non-dispersive wave packets are considered in Section 5.2. 13 Note, however, that this is a speci c choice. Any driven quantum system with a su ciently high density of states and mixed regular–chaotic classical dynamics will exhibit non-dispersive wave packets. Since a mixed phase space structure is the generic scenario for dynamical systems, non-dispersive wave packets are expected to be a completely general and ubiquitous phenomenon.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 445 Compared to our simple one-dimensional model introduced in Section 3.1 above, an atom displays a couple of additional features: • A real hydrogen atom is a three-dimensional (3D) system. However, it is a degenerate system, because the energy depends on the principal quantum number n only, but not on the angular or magnetic quantum numbers L and M, respectively. Thus, the structure of the energy levels, which is crucial for the properties of the non-dispersive wave packet, see Section 3.1.2, is identical in the 1D and 3D cases. Before discussing the properties of 3D wave packets in Sections 3.3.2–4, we will consider a simpli ed 1D model of the hydrogen atom in Section 3.3.1. • Although spontaneous emission is a weak incoherent process, it nonetheless limits the life time of the non-dispersive wave packets which may decay to lower lying states, losing the phase coherence of the electronic wave function. Non-dispersive wave packets exhibit speci c spontaneous decay properties which are studied in Section 7.2. • The electron in a hydrogen atom is not necessarily bound. It may ionize, especially when the atom is exposed to a microwave eld. This is a coherent decay process where the ionized electron keeps its phase coherence. There are no exact bound states in the system, but rather resonances. From a quantum point of view, the Floquet spectrum is no longer discrete but continuous and we actually deal with an open system. In a more elementary language, the atom can successively absorb several photons so that its energy exceeds the ionization threshold. If initially prepared in a wave-packet eigenstate, this is a pure quantum phenomenon, since the classical dynamics remain trapped within the resonance island forever. The multi-photon ionization may then be considered as a tunneling process from inside the resonance island to the non-resonant part of phase space, where the Rydberg electron eventually escapes to in nity. This picture is elaborated in Section 7.1. 3.2.2. Hamiltonian, basis sets and selection rules In the presence of a microwave eld, the dipole approximation [18,94] can be used to describe the atom– eld interaction. Di erent gauges can be used, the physics being of course independent of the choice of gauge. The most common choices are the length and the velocity gauges. For simplicity, in our discussion, we shall use the length gauge, although actual quantum calculations are usually a bit easier in the velocity gauge [96–98]. The Hamiltonian reads H = ˜p2 q2 − 2m 4 0r − q˜r · ˜F(t) ; (105) where q is the (negative) charge of the electron, m its mass, and ˜F(t) the microwave electric eld acting on the atom. We neglect here all relativistic, spin, QED e ects, etc., and assume an in nitely massive nucleus. Later, unless speci ed otherwise, we shall use atomic units, where |q| ≡4 0 ≡ m ≡ ˝ ≡ 1, and the Hamiltonian reduces to Here, H = ˜p2 2 H0 = ˜p2 2 − 1 r +˜r · ˜F(t) : (106) − 1 r (107)
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Non-dispersive wave packets in periodically driven quantum systems

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