Non-dispersive wave packets in periodically driven quantum systems

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500 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 and its area Î =4 Vs(Î s) H ′′ 0 (Î s) As( )= 16 s (234) Vs(Î s) H ′′ 0 (Î : (235) s) Again, the dependence of As( )on | | implies that even small perturbations may induce signi cant changes in the phase space structure, provided the perturbation is resonant with a harmonic of the unperturbed classical motion. 27 The construction of a non-dispersive wave packet is simple once the s-resonance structure is understood: indeed, any set of initial conditions trapped in one of the s-resonance islands will classically remain trapped forever. Thus, a quantum wave packet localized initially inside a resonance island is a good candidate for building a non-dispersive wave packet. There remains, however, a di culty: the wave packet can be initially placed in any of the s resonance islands. After one period of the driving, it will have jumped to the next island, meaning that it will be far from its initial position. On the other hand, the Floquet theorem guarantees the existence of states which are strictly periodic with the period of the drive (not the period of the resonant internal motion). The solution to this di culty is to build eigenstates which simultaneously occupy all s-resonance islands, that is, which are composed of s wave packets each localized on a di erent resonance island. After one period of the drive, each individual wave packet replaces the next one, resulting in globally T -periodic motion of this “composite” Floquet state. If the system has a macroscopic size (i.e., in the semiclassical limit), individual wave packets will appear extremely well localized and lying far from the other ones while maintaining a well-de ned phase coherence with them. For s =2, the situation mimics a symmetric double well potential, where even and odd solutions are linear combinations of nonstationary states, each localized in either one well [7]. In order to get insight in the structure of the Floquet quasi-energy spectrum, it is useful to perform the semiclassical EBK quantization of the secular Hamiltonian (229). Quantization of the motion in (Î; ˆ ), see Section 3.1.3, provides states trapped within the resonance islands (librational motion), and states localized outside them (rotational motion). As usual, the number of trapped states is given by the size, Eq. (235) of the resonance island: Number of trapped states 8 Vs(Î s) ˝s H ′′ 0 (Î : (236) s) The quantization can be performed along the contours of any of the s clones of the resonance island, giving of course the same result. However, this does not result in a s-degeneracy of the spectrum: 27 The situation is very di erent in the opposite case, when ! is the sth SUB-harmonic [93] of the internal frequency. A resonance island may then exist but it is typically much smaller as it comes into play only at order s in perturbation theory, with a size scaling as | | s=2 :
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 501 indeed, the s clones belong to the same torus in phase space (see above) and do not generate s independent states. If the number of trapped states is su ciently large, the harmonic approximation to the pendulum (or secular) Hamiltonian can be used, with the frequency of the harmonic motion around the stable resonant orbit given by !harm = s | Vs(Î s)H ′′ 0 (Î s)| : (237) In order to get the complete semiclassical Floquet spectrum, we additionally have to perform the semiclassical quantization in the (t; ˆPt) plane, giving 1 ˆPt dt = 2 0 ˆPt 2 = s ˆPt ! = j + ˝ (238) 4 with j integer. This nally yields the semiclassical Floquet levels (in the harmonic approximation): EN;j = H0(Î s) − ! s Î s + j + ˝ 4 ! ′′ − sign(H 0 (Î s)) | Vs(Î s)|− N + s 1 ˝!harm (239) 2 with N a non-negative integer. The wave packet with optimum localization in the (Î; ˆ ) plane, i.e., optimum localization along the classical unperturbed orbit, is the N =0 state. According to Eq. (239), the semiclassical quasi-energy spectrum has a periodicity ˝!=s; whereas the “quantum” Floquet theory only enforces ˝! periodicity. Thus, inside a Floquet zone of width ˝!, each state appears s times (for 0 6 j¡s), at energies separated by ˝!=s. Note that this property is a direct consequence of the possibility of eliminating the time dependence of H in Eq. (228) by averaging over , leading to the time-independent expression (229) for Hsec. Therefore, it will be only approximately valid for the exact quantum Floquet spectrum. In contrast, the ˝! periodicity holds exactly, as long as the system Hamiltonian is time-periodic. We will now recover the ˝!=s periodicity in a quantum description of our problem, which will provide us with the formulation of an eigenvalue problem for the wave-packet eigenstates anchored to the s-resonance, in terms of a Mathieu equation. In doing so, we shall extend the general concepts outlined in Section 3.1.4 above. Our starting point is Eq. (85), which we again consider in the regime where the eigenenergies En of the unperturbed Hamiltonian H0 are locally approximately spaced by ˝ . The resonance condition (225) implies dEn dn n=n0 = ˝ ! s ; (240) where, again, n0 is not necessarily an integer, and is related to the resonant action and its associated Maslov index through Î s = n0 + ˝ : (241) 4 When the resonance condition is met, the only e cient coupling in Eq. (85) connects states with the same value of n − sk. In other words, in the secular approximation, a given state (n; k) only couples to (n + s; k + 1) and (n − s; k − 1). We therefore consider a given ladder of coupled states labeled
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