Non-dispersive wave packets in periodically driven quantum systems

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414 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 spread, due to the accumulation of relative phases of the di erent contributions to the sum in Eq. (13). Whereas the classical dynamics in a one-dimensional binding potential V (˜r) are described by periodic orbits (at any energy), the quantum dynamics are in general not periodic. A return to the initial state is only possible if all the phases exp(−iEnt=˝) simultaneously take the same value. This implies that all the energy levels En (with cn = 0) are equally spaced, or that all the level spacings are integer multiples of some quantity. In practice, this is realized only for the harmonic oscillator (in any dimension), and for tops or rotors where the Hamiltonian is proportional to some component of an angular momentum variable. Another possibility is to use the linear Stark e ect in the hydrogen atom which produces manifolds of equally spaced energy levels. However, experimental imperfections (higher-order Stark e ect and e ect of the ionic core on non-hydrogenic Rydberg atoms) break the equality of the spacings and consequently lead to dispersion [4]. The equality of consecutive spacings has a simple classical interpretation: since all classical trajectories are periodic with the same period, the system is exactly back in its initial state after an integer number of periods. In other words, in those special cases, there is no wave-packet spreading at long times. However, for more generic systems, the energy levels are not equally spaced, neither are the spacings simply related, and a wave packet will spread. For a one-dimensional, time-independent system, it is even possible to estimate the time after which the wave packet has signi cantly spread (this phenomenon is also known as the “collapse” of the wave packet [5,6]). This is done by expanding the various energies En around the “central” energy En0 of the wave packet: En En0 +(n− n0) dEn 2 (n − n0) d (n0)+ dn 2 2En dn2 (n0) : (15) The wave packet being initially localized, its energy is more or less well de ned and only a relatively small number nn0 of the coe cients cn have signi cant values. At short times, the contribution of the second order term in Eq. (15) to the evolution can be neglected. Within this approximation, the important energy levels can be considered as equally spaced, and one obtains a periodic motion of the wave packet, with period: 2 ˝ Trecurrence = : (16) (dEn=dn)(n0) In the standard semiclassical WKB approximation (discussed in Section 2.1) [7], this is nothing but the classical period of the motion at energy En0, and one recovers the similarity between the quantum motion of the wave packet and the classical motion of a particle. At longer times, the contributions of the various eigenstates to the dynamics of the wave packet will come out of phase because of the second-order term in Eq. (15), resulting in spreading and collapse of the wave packet. A rough estimate of the collapse time is thus when the relevant phases have changed by 2 . One obtains Tcollapse 1 ( n) 2 2˝ (d 2 En=dn 2 )(n0) : (17) Using the standard WKB approximation, one can show that this expression actually corresponds to the time needed for the corresponding classical phase space density to signi cantly spread under the in uence of the classical evolution.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 415 At still longer times, a pure quantum phenomenon appears, due to the discrete nature of the energy spectrum. Since the (n−n0) 2 factors in Eq. (15) are all integers, the second-order contributions to the phase are all integer multiples of the phase of the n − n0 = 1 term. If the latter is an integer multiple of 2 , all the second-order contributions will rephase, inducing a revival of the wave packet in its original shape. A re ned estimation of the revival time actually shows that this analysis overestimates the revival time by a factor two. 2 The correct result is [8–11] 2 ˝ Trevival = (d2En=dn 2 : (18) )(n0) Based on the very elementary considerations above, we can so far draw the following conclusions: • An initially localized wave packet will follow the classical equations of motion for a nite time t ∼ Trecurrence; • its localization properties cannot be stationary as time evolves; • in general, the initial quasi-classical motion is followed by collapse and revival, with the corresponding time-scales Trecurrence ¡Tcollapse ¡Trevival. In the sequel of this report, we will show how under very general conditions it is indeed possible to create wave packets as single eigenstates of quantum systems, i.e., as localized ground states in an appropriately de ned reference frame. The most suitable framework is to consider quantum evolution in classical phase space, that provides a picture which is independent of the choice of the basis and allows for an immediate comparison with the classical Hamiltonian ow. In addition, such a picture motivates a semiclassical interpretation, which we will expand upon in Section 2. The appropriate technical tool for a phase space description are quasiprobability distributions [12] as the Wigner representation of the state | (t)〉, W (˜r; ˜p)= 1 (2 ˝) f ∗ ˜r + ˜x ˜r − 2 ˜x ˜x · ˜p exp i d 2 ˝ f ˜x; (19) where f is the number of degrees of freedom. The Wigner density W (˜r; ˜p) is real, but not necessarily positive [12,13]. Its time evolution follows from the Schrodinger equation [12]: 9W (˜r; ˜p; t) ˝ ˝ = −2H(˜r; ˜p; t) sin W (˜r; ˜p; t) ; (20) 9t 2 where → ∇˜p ∇˜r − ← → ∇˜r ∇˜p = ← and the arrows indicate in which direction the derivatives act. Eq. (20) can serve to motivate the semiclassical approach. Indeed, the sin function can be expanded in a Taylor series, i.e., a power expansion in ˝. At lowest non-vanishing order, only terms linear in contribute and 2 It must also be noted that, at simple rational multiples (such as 1 3 , 1 2 (21) 2 , ) of the revival time, one observes “fractional 3 revivals” [8–11], where only part of the various amplitudes which contribute to Eq. (13) rephase. This generates a wave function split into several individual wave packets, localized at di erent positions along the classical orbit.
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Non-dispersive wave packets in periodically driven quantum systems

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