Non-dispersive wave packets in periodically driven quantum systems

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418 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 ψ (z) (a) 1 0.5 0 −0.5 −1 −10 −5 0 Position z 5 10 Momentum p (b) 15 10 5 0 −10 −5 0 Position z 5 10 Fig. 2. (a) Coherent state of the harmonic oscillator [2]. The probability density and the modulus of the wave function (——-) have a Gaussian distribution. The real (- - -) and the imaginary (···) part of the wave function show, in addition, oscillations which re ect the non-vanishing momentum. Whereas the envelope of the wave function preserves its shape under time evolution, the frequency of the oscillations exhibits the same time dependence as the momentum of the corresponding classical particle. (b) Contour of the corresponding Wigner distribution which shows localization in both position and momentum. The isovalue for the contour is chosen to enclose 86% of the total probability. For this speci c system, one can de ne a restricted class of wave packets, which are minimum uncertainty states (i.e., z p = ˝=2), and remain minimal under time evolution [2]. Nowadays, these states are known as “coherent” states of the harmonic oscillator [16], and are frequently employed in the analysis of simple quantum systems such as the quantized electromagnetic eld [17,18]. They have Gaussian wave functions, see Fig. 2(a), given by Eq. (24), and characterized by an average position z0, an average momentum p0, and a spatial width ˝ = ; (29) m! where ! is the classical eigenfrequency of the harmonic oscillator. The corresponding Wigner distribution, Eq. (25), also has Gaussian shape. The properties of coherent states are widely discussed in the literature, see [19,20]. In the “naturally scaled”, dimensionless coordinates z m!=˝ and p= √ m!˝, the classical trajectories of the harmonic oscillator are circles, and the Wigner distribution is an isotropic Gaussian centered at (z0;p0), see Fig. 2(b). Under time evolution, which follows precisely the classical dynamics, its isotropic Gaussian shape is preserved. An important point when discussing wave packets is to avoid the confusion between localized wave packets (as de ned above) and minimum uncertainty (coherent or squeezed [18]) states. The latter are just a very restricted class of localized states. They are the best ones in the sense that they have optimum localization. On the other hand, as soon as dynamics is considered, they have nice properties only for harmonic oscillators. In generic systems, they spread exactly like other wave packets. Considering only coherent states as good semiclassical analogs of classical particles is in our opinion a too formal point of view. Whether the product z p is exactly ˝=2 or slightly larger is certainly of secondary relevance for the semiclassical character of the wave packet. What counts is that, in the semiclassical limit ˝ → 0, the wave packet is asymptotically perfectly localized in all
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 419 directions of phase space. This was Schrodinger’s original concern, without reference to the actual value of z p [2]. Finally, for future applications, let us de ne the so-called Husimi representation of the quantum wave function [21]. It is the squared projection of a given quantum state over a set of coherent states. Let us denote the gaussian wavefunction of Eq. (24) (with given by Eq. (29)) as |Coh(z0;p0)〉. Then the Husimi representation of (z) is de ned as Hus(z; p)= 1 |〈Coh(z; p)| 〉| 2 ; (30) where the factor 1= is due to the resolution of unity in the coherent states basis [19] and is often omitted (to confuse the reader). Alternatively, the Husimi function may be looked upon as a Wigner function convoluted with a Gaussian [12]. 1.3. A simple example: the one-dimensional hydrogen atom We now illustrate the ideas discussed in the preceding sections, using the speci c example of a one-dimensional hydrogen atom. This object is both, representative of generic systems, and useful for atomic systems to be discussed later in this paper. We choose the simplest hydrogen atom: we neglect all relativistic, spin and QED e ects, and assume that the nucleus is in nitely massive. The Hamiltonian reads: H = p2 e2 − 2m z ; (31) where m is the mass of the electron, e 2 =q 2 =4 0, with q the elementary charge, and z is restricted to the positive real axis. The validity of this model as compared to the real 3D atom will be discussed in Section 3.3. Here and in the rest of this paper, we will use atomic units, de ned by m, e 2 and ˝. The unit of length is the Bohr radius a0=˝ 2 =me 2 =5:2917×10 −11 m, the unit of time is ˝ 3 =me 4 =2:4189×10 −17 s, the unit of energy is the Hartree me 4 =˝ 2 =27:2eV,twice the ionization energy of the hydrogen atom, and the unit of frequency is me 4 =2 ˝ 3 =6:5796 × 10 16 Hz [22]. With these premises, the energy levels are 4 En = − 1 2n 2 for n ¿ 1 : (32) Clearly, the levels are not equally spaced, and therefore (see Eq. (15)) any wave packet will spread. Fig. 3 shows the evolution of a wave packet built from a linear combination of eigenstates of H, using a Gaussian distribution of the coe cients cn in Eq. (13). The distribution is centered at n0 = 60, with a width n =1:8 for the |cn| 2 . The calculation is done numerically, but is simple in the hydrogen atom since all ingredients—energy levels and eigenstates—are known analytically. At time t = 0, the wave packet is localized at the outer turning point (roughly at a distance 2n 2 0 from the origin), and has zero initial momentum; its shape is roughly Gaussian. After a quarter of a classical Kepler period Trecurrence, it is signi cantly closer to the nucleus, with negative velocity, 4 The present analysis is restricted to bound states of the atom. Continuum (i.e., scattering) states also exist but usually do not signi cantly contribute to the wave-packet dynamics. If needed, they can be incorporated without any fundamental di culty [23,24].
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Non-dispersive wave packets in periodically driven quantum systems

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