Non-dispersive wave packets in periodically driven quantum systems

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440 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 with Eqs. (40) and (65), dEn dn = ˝! : (87) n=n0 Note that, by this de nition, n0 is not necessarily an integer. In the semiclassical limit, where n0 is large, the WKB approximation connects n0 to the center of the classical resonance island, see Eq. (39), n0 + 4 = Î 1 ˝ ; (88) En0 = H0(Î 1) ; (89) with being the Maslov index along the resonant trajectory. Furthermore, for n close to n0, we can expand the unperturbed energy at second order in (n − n0), En En0 +(n− n0)˝! + 1 d 2 2En dn2 (n − n0) 2 ; (90) n0 where the second derivative d2En=dn 2 is directly related to the classical quantity H ′′ 0 , see Eq. (67), within the semiclassical WKB approximation, Eq. (39). Similarly, the matrix elements of V are related to the classical Fourier components of the potential [7], Eq. (55), 〈 n|V | n+1〉 〈 n+1|V | n+2〉 V1(Î 1) ; (91) evaluated at the center Î = Î 1—see Eq. (65)—of the resonance zone. With these ingredients and r = n − n0, Eq. (85) is transformed in the following set of approximate equations: E − H0(Î 1)+! Î 1 − ˝ − 4 ˝2 ′′ H 0 (Î 1)r 2 2 dr = V1(dr+1 + dr−1) ; (92) where dr ≡ cn0+r;n0+r : (93) Note that, because of Eq. (87), the r values are not necessarily integers, but all have the same fractional part. The tridiagonal set of coupled equations (92) can be rewritten as a di erential equation. Indeed, if one introduces the following function associated with the Fourier components dr: f( )= exp(ir )dr ; (94) Eq. (92) can be written as − ˝2 2 r H ′′ 0 (Î 1) d2 d 2 + H0(Î 1) − ! Î 1 − ˝ + V1 cos 4 f( )=Ef( ) ; (95) which is nothing but the quantum version of the pendulum Hamiltonian, Eq. (66). Thus, the present calculation is just the purely quantum description of the non-linear resonance phenomenon. The
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 441 dummy variable introduced for convenience coincides with the classical angle variable ˆ . In general, r is not an integer, so that the various exp(ir ) in Eq. (94) are not periodic functions of . However, as all r values have the same fractional part, if follows that f( ) must satisfy “modi ed” periodic boundary conditions of the form f( +2 ) = exp(−2i n0)f( ) : (96) The reason for this surprising boundary condition is clear: r =n−n0 is the quantum analog of Î −Î1. In general, the resonant action Î 1 is not an integer or half-integer multiple of ˝—exactly as n0 is not an integer. The semiclassical quantization, Eq. (79), which expresses the ˆ periodicity of the eigenstate, applies for the Î variable. When expressed in terms of the variable Î − Î 1, it contains the additional phase shift present in Eq. (96). Few words of caution are in order: the equivalence of the semiclassical quantization with the pure quantum approach holds in the semiclassical limit only, when the quantum problem can be mapped on a pendulum problem. In the general case, it is not possible to de ne a quantum angle variable [94]. Hence, the quantum treatment presented here is no more general or more powerful than the semiclassical treatment. They both rely on the same approximations and have the same limitations: perturbative regime (no overlap of resonances) and semiclassical approximation. Finally, Eq. (95) can be written in its standard form, known as the “Mathieu equation” [95]: d2y +(a− 2q cos 2v)y =0: (97) dv2 The correspondence with Eq. (95) is established via =2v; (98) a = 8[E − H0(Î 1)+!(Î 1 − ˝=4)] ˝2H ′′ 0 (Î 1) 4 V1 q = ˝2H ′′ 0 (Î 1) ; (99) : (100) The boundary condition, Eq. (96), is xed by the so-called “characteristic exponent” in the Mathieu equation, = −2n0 (mod 2) : (101) The Mathieu equation has solutions (for a given characteristic exponent) for a discrete set of values of a only. That implies quantization of the quasi-energy levels, according to Eq. (99). The quantized values a ( ; q) depend on q and , and are labeled 12 by a non-negative integer . They are well known—especially asymptotic expansions are available both in the small and in the large q regime—and can be found in standard handbooks [95]. For example, Fig. 9(a) shows the rst 12 In the standard text books as [95], the various solutions of the Mathieu equation are divided in odd and even solutions, and furthermore in - and 2 -periodic functions. In our case, only the “a2p” and “b2p” (in the language of [95]) are to be considered.
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Non-dispersive wave packets in periodically driven quantum systems

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