Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
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A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 419<br />
directions of phase space. This was Schrod<strong>in</strong>ger’s orig<strong>in</strong>al concern, without reference to the actual<br />
value of z p [2].<br />
F<strong>in</strong>ally, for future applications, let us de ne the so-called Husimi representation of the <strong>quantum</strong><br />
<strong>wave</strong> function [21]. It is the squared projection of a given <strong>quantum</strong> state over a set of coherent states.<br />
Let us denote the gaussian <strong>wave</strong>function of Eq. (24) (with given by Eq. (29)) as |Coh(z0;p0)〉.<br />
Then the Husimi representation of (z) is de ned as<br />
Hus(z; p)= 1 |〈Coh(z; p)| 〉| 2 ; (30)<br />
where the factor 1= is due to the resolution of unity <strong>in</strong> the coherent states basis [19] and is often<br />
omitted (to confuse the reader). Alternatively, the Husimi function may be looked upon as a Wigner<br />
function convoluted with a Gaussian [12].<br />
1.3. A simple example: the one-dimensional hydrogen atom<br />
We now illustrate the ideas discussed <strong>in</strong> the preced<strong>in</strong>g sections, us<strong>in</strong>g the speci c example of a<br />
one-dimensional hydrogen atom. This object is both, representative of generic <strong>systems</strong>, and useful<br />
for atomic <strong>systems</strong> to be discussed later <strong>in</strong> this paper. We choose the simplest hydrogen atom: we<br />
neglect all relativistic, sp<strong>in</strong> and QED e ects, and assume that the nucleus is <strong>in</strong> nitely massive. The<br />
Hamiltonian reads:<br />
H = p2 e2<br />
−<br />
2m z<br />
; (31)<br />
where m is the mass of the electron, e 2 =q 2 =4 0, with q the elementary charge, and z is restricted to<br />
the positive real axis. The validity of this model as compared to the real 3D atom will be discussed<br />
<strong>in</strong> Section 3.3.<br />
Here and <strong>in</strong> the rest of this paper, we will use atomic units, de ned by m, e 2 and ˝. The unit of<br />
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,<br />
the unit of energy is the Hartree me 4 =˝ 2 =27:2eV,twice the ionization energy of the hydrogen atom,<br />
and the unit of frequency is me 4 =2 ˝ 3 =6:5796 × 10 16 Hz [22].<br />
With these premises, the energy levels are 4<br />
En = − 1<br />
2n 2 for n ¿ 1 : (32)<br />
Clearly, the levels are not equally spaced, and therefore (see Eq. (15)) any <strong>wave</strong> packet will spread.<br />
Fig. 3 shows the evolution of a <strong>wave</strong> packet built from a l<strong>in</strong>ear comb<strong>in</strong>ation of eigenstates of<br />
H, us<strong>in</strong>g a Gaussian distribution of the coe cients cn <strong>in</strong> Eq. (13). The distribution is centered at<br />
n0 = 60, with a width n =1:8 for the |cn| 2 . The calculation is done numerically, but is simple<br />
<strong>in</strong> the hydrogen atom s<strong>in</strong>ce all <strong>in</strong>gredients—energy levels and eigenstates—are known analytically.<br />
At time t = 0, the <strong>wave</strong> packet is localized at the outer turn<strong>in</strong>g po<strong>in</strong>t (roughly at a distance 2n 2 0<br />
from the orig<strong>in</strong>), and has zero <strong>in</strong>itial momentum; its shape is roughly Gaussian. After a quarter of<br />
a classical Kepler period Trecurrence, it is signi cantly closer to the nucleus, with negative velocity,<br />
4 The present analysis is restricted to bound states of the atom. Cont<strong>in</strong>uum (i.e., scatter<strong>in</strong>g) states also exist but usually<br />
do not signi cantly contribute to the <strong>wave</strong>-packet dynamics. If needed, they can be <strong>in</strong>corporated without any fundamental<br />
di culty [23,24].