Non-dispersive wave packets in periodically driven quantum systems

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416 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 one obtains 9W (˜r; ˜p; t) = {H; W (˜r; ˜p; t)} ; (22) 9t where {:;:} denotes the classical Poisson bracket [3]: 3 {f; g} = i=1:::f 9f 9ri 9g 9pi − 9f 9pi 9g 9ri : (23) Eq. (22) is nothing but the classical Liouville equation [3] which describes the classical evolution of a phase space density. Hence, in the “semiclassical limit” ˝ → 0, the Wigner density evolves classically. Corrections of higher power in ˝ can be calculated systematically. For example, the next order is ˝3H 3W=24 in Eq. (20), and generates terms which contain third-order derivatives (in either position and=or momentum) of the Hamiltonian. Therefore, for a Hamiltonian of maximal degree two in position and=or momentum, all higher-order terms in Eq. (20) vanish and the Wigner distribution follows the classical evolution for an arbitrary initial phase space density, and for arbitrarily long times. The harmonic oscillator is an example of such a system [2,16], in agreement with our discussion of Eq. (15) above. Now, once again, why does a wave packet spread? At rst sight, it could be thought that this is due to the higher-order terms in Eq. (20), and thus of quantum origin. This is not true and spreading of a wave packet has a purely classical origin, as illustrated by the following example. Let us consider a one-dimensional, free particle (i.e., no potential), initially described by a Gaussian wave function with average position z0, average momentum p0 ¿ 0, and spatial width : 1 (z; t =0)= √ exp i 1=4 p0z 2 (z − z0) − ˝ 2 2 : (24) The corresponding Wigner distribution is a Gaussian in phase space: W (z; p; t =0)= 1 ˝ exp 2 2 (z − z0) (p − p0) − − 2 2 ˝2 : (25) As the Hamiltonian is quadratic in the momentum, without potential, this distribution evolves precisely alike the equivalent classical phase space density. Hence, the part of the wave packet with p¿p0 will evolve with a larger velocity than the part with p¡p0. Even if both parts are initially localized close to z0, the contribution of di erent velocity classes implies that their distance will increase without bound at long times. The wave packet will therefore spread, because the various classical trajectories have di erent velocities. Spreading is thus a completely classical phenomenon. This can be seen quantitatively by calculating the exact quantum evolution. One obtains W (z; p; t)= 1 ˝ exp − (z − z0 − pt=m) 2 2 − 2 (p − p0) 2 3 We choose here the most common de nition of the Poisson bracket. Note, however, that some authors [14,15] use the opposite sign. ˝ 2 (26)
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 417 Momentum p 15 10 5 t=0 t=0.5 t=1 0 −5 0 5 Position z 10 15 Fig. 1. Evolution of the Wigner density of a wave packet for a free particle moving in a one-dimensional con guration space, compared to the classical evolution of a swarm of classical particles with the same initial probability density. While the uncertainty in momentum does not vary with time, the Wigner density stretches along the position coordinate and loses its initial minimum uncertainty character. This implies spreading of the wave packet. This has a purely classical origin, as shown by the classical evolution of the swarm of particles, which closely follow the quantum evolution. The contour of the Wigner density is chosen to contain 86% of the probability. for the Wigner distribution, and (z; t)= 1 1=4 ei ˝ 2 + 2t2 exp i 1=4 m2 2 p0z ˝ − (z − z0 − p0t=m) 2 2 2 +2i˝t=m for the wave function (ei is an irrelevant, complicated phase factor). The former is represented in Fig. 1, together with the evolution of a swarm of classical particles with an initial phase space density identical to the one of the initial quantum wave packet. Since the quantum evolution follows exactly the classical one, the phase space volume of the wave packet is preserved. However, the Wigner distribution is progressively stretched along the z-axis. This results in a less and less localized wave packet, with z(t)=√ 2 p(t)= ˝ √ 2 : 1+ ˝2t2 ; m2 4 The product z p, initially minimum (˝=2), continuously increases and localization is eventually lost. 1.2. Gaussian wave packets—coherent states We have already realized above that, for the harmonic oscillator, the second derivative d 2 En=dn 2 in Eq. (15) vanishes identically, and a wave packet does not spread, undergoing periodic motion. (27) (28)
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Non-dispersive wave packets in periodically driven quantum systems

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