Non-dispersive wave packets in periodically driven quantum systems

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426 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 only during the last 10 years [43,32–34,44–75], after the recent major developments of non-linear dynamics. 2. Semiclassical quantization In this section, we brie y recall the basic results on the semiclassical quantization of Hamiltonian systems, which we will need for the construction of non-spreading wave packets in classical phase space, as well as to understand their properties. This section does not contain any original material. 2.1. WKB quantization For a one-dimensional, bounded, time-independent system, the Hamilton function (equivalent to the total energy) is a classical constant of motion, and the dynamics are periodic. It is possible to de ne canonically conjugate action-angle variables (I; ), such that the Hamilton function depends on the action alone. The usual de nition of the action along a periodic orbit (p.o.) writes: I = 1 2 p dz ; (36) p:o: where p is the momentum along the trajectory. The WKB (for Wentzel, Kramers, and Brillouin) method [76] allows to construct an approximate solution of the Schrodinger equation, in terms of the classical action-angle variables and of Planck’s constant ˝: (z)= 1 i √ exp p ˝ p dz as an integral along the classical trajectory. This construction is possible if and only if the phase accumulated along a period of the orbit is an integer multiple of 2 . This means that the quantized states are those where the action variable I is an integer multiple of ˝. This simple picture has to be slightly amended because the semiclassical WKB approximation for the wave function breaks down at the turning points of the classical motion, where the velocity of the classical particle vanishes and, consequently, expression (37) diverges. This failure can be repaired [76] by adding an additional phase =2 for each turning point. This leads to the nal quantization condition I = 1 p dz = n + ˝ ; (38) 2 p:o: 4 where n is a non-negative integer and —the “Maslov index”–counts the number of turning points along the periodic orbit ( = 2 for a simple 1D periodic orbit). Thus, the WKB recipe is extremely simple: when the classical Hamilton function H(I) is expressed in terms of the action I, the semiclassical energy levels are obtained by calculating H(I) for the quantized values of I: En = H I = n + ˝ : (39) 4 (37)
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 427 Finally, as a consequence of Eqs. (38) and (39), the spacing between two consecutive semiclassical energy levels is simply related to the classical frequency of the motion, En+1 − En dEn dn = ˝dH dI = ˝ (I) ; (40) a result which establishes the immediate correspondence between a resonant transition between two quantum mechanical eigenstates in the semiclassical regime, and resonant driving of the associated classical trajectory. In the vicinity of a xed (equilibrium) point, the Hamilton function can be expanded at secondorder (the rst-order terms are zero, by de nition of the xed point), leading to the “harmonic approximation”. If the xed point is stable, the semiclassical WKB quantization of the harmonic approximation gives exactly the quantum result, although the semiclassical wave function, Eq. (37), is incorrect. This remarkable feature is not true for an unstable xed point (where classical trajectories escape far from the xed point), and the WKB approximation fails in this case. 2.2. EBK quantization For a multi-dimensional system, it is a much more complicated task to extract the quantum mechanical eigenenergies from the classical dynamics of a Hamiltonian system. The problem can be solved for integrable systems, where there are as many constants of motion as degrees of freedom [77]. This is known as EBK (for Einstein, Brillouin, and Keller) quantization [78], and is a simple extension of the WKB quantization scheme. Let us choose two degrees of freedom for simplicity, the extension to higher dimensions being straightforward. If the system is integrable, the Liouville– Arnold theorem [3] assures the existence of two pairs of canonically conjugate action-angle variables, (I1; 1) and (I2; 2), such that the classical Hamilton function depends only on the actions: H = H(I1;I2) : (41) The classical motion is periodic along each angle (the actions being constants of the motion) with frequencies 1 = 9H ; (42) 9I1 2 = 9H : (43) 9I2 In the generic case, these two frequencies are incommensurate, such that the full motion in the four-dimensional phase space is quasi-periodic, and densely lls the so-called “invariant torus” dened by the constant values I1 and I2. The semiclassical wave function is constructed similarly to the WKB wave function. Turning points are now replaced by caustics [29,78,79] of the classical motion (where the projection of the invariant torus on con guration space is singular), but the conclusions are essentially identical. The single-valued character of the wave function requires the following quantization of the actions: I1 = ˝ ; (44) n1 + 1 4
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Non-dispersive wave packets in periodically driven quantum systems

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