Non-dispersive wave packets in periodically driven quantum systems

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476 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Hamiltonian (184), as a time-independent operator, has some energy levels and corresponding eigenstates. Its spectrum is not !-periodic, although the unitary transformation assures that there is a one-to-one correspondence between its spectrum (eigenstates) and the Floquet spectrum of Eq. (164). 18 It was observed [143] that Hamiltonian (184) allows for the existence of a stable xed (equilibrium) point in a certain range of the microwave amplitude F. Later on, it was realized [34] that wave packets initially localized in the vicinity of this xed point will not disperse (being bound by the fact that the xed point is stable) for at least several Kepler periods. In the laboratory frame, these wave packets (also called “Trojan states” [44,45,34,47]) appear as wave packets moving around the nucleus along the circular trajectory, which is nothing but the periodic orbit at the center of the resonance island discussed in Section 3.4.2. In the original formulation [34] and the discussion which followed [44,45,47,55], great attention was paid to the accuracy of the harmonic approximation (see below). This was of utmost importance for the non-spreading character of Gaussian-shaped Trojan wave packets considered in [44,45,34,47,55]. As soon pointed out in [49], however, the accuracy of this approximation is immaterial for the very existence of the wave packets, which are to be identi ed, as shown above, with well-de ned Floquet states. Let us recapitulate the xed-point analysis of [34,143] in the rotating frame. Inspection of the classical version of the Hamiltonian Hrot, Eq. (184), shows that, due to symmetry, one may seek the xed point at z = y = 0. The condition for an equilibrium ( xed) point, i.e., d˜r=dt =0; d˜p=dt =0, yields immediately that pz;eq = px;eq =0, py;eq = !xeq, with the subscript “eq” for “equilibrium”. The remaining equation for dpx=dt gives the condition − F + ! 2 xeq − |xeq| x3 =0; (185) eq that de nes the position of the xed point as a function of F. Following [34] let us introduce the dimensionless parameter q = 1 ! 2 : (186) |xeq| 3 One may easily express the xed point position, the microwave eld amplitude, as well as the corresponding energy in terms of q and !. Explicitly: 1 xeq1 = q1=3 ; ! 2=3 1 − q F = q1=3 !4=3 ; 2=3 1 − 4q ! Eeq = 2 q (187) and xeq2 = − 1 q1=3 ; ! 2=3 q − 1 F = q1=3 !4=3 ; 2=3 1 − 4q ! Eeq = 2 q : (188) For F =0;q= 1 in Eqs. (187) and (188). For F¡0, (i.e., q¿1 in Eq. (187)), xeq1 is an unstable xed point, while for moderately positive F (i.e., 8=9 ¡q¡1 [34,143]) it is stable. Stability of the second equilibrium point xeq2 is achieved by changing the sign of F. For moderate elds (q close to unity), the stable and the unstable xed points are located on opposite sides of the nucleus, and at 18 In fact, if | i〉 is an eigenstate of Hrot with energy Ei, then U † | i〉 is a Floquet eigenstate with quasi-energy Ei, while states shifted in energy by k! are of the form exp(ik!t)U † | i〉. For a more detailed discussion of this point, see [68].
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 477 almost the same distance from it. As the whole analysis is classical, it has to obey the scaling laws discussed in Section 3.2.5. Hence, all quantities describing the equilibrium points in the preceding equations scale as powers of the microwave frequency. A consequence is that there is a very simple correspondence between the scaled microwave amplitude and the dimensionless parameter q: F0 = F! 4=3 = Fn 4 0 = 1 − q q 1=3 : (189) The parameter q can thus be thought of as a convenient parametrization (leading to simpler algebraic formula) of the scaled microwave amplitude. A xed point in the rotating frame corresponds to a periodic orbit with exactly the period T = 2 =! of the microwave driving eld in the original frame. The stable xed point (periodic orbit) thus corresponds to the center of the resonance island, and to the stable equilibrium point of the pendulum in the secular approximation. Similarly, the unstable xed point corresponds to the unstable equilibrium point of the pendulum. Note that the stable xed point approaches xeq =! −2=3 =Î 2 1 when F → 0, i.e., the radius of the circular classical Kepler trajectory with frequency !. Thus, the stable xed point smoothly reaches the location of the circular state of the hydrogen atom, with a classical Kepler frequency equal to the driving microwave frequency. Its energy Eeq = −3! 2=3 =2 is the energy of the circular orbit in the rotating frame. Since the non-dispersive wave packets are localized in the immediate vicinity of the stable xed point in the rotating frame, an expansion of the Hamiltonian around that position is useful. Precisely at the xed point, all rst-order terms (in position and momentum) vanish. At second order, ˜˜p 2 Hrot Hharmonic = Eeq + 2 − !(˜x ˜p y − ˜y ˜p x )+!2 q ˜y 2 2 − !2q ˜x 2 + !2q ˜z 2 ; (190) 2 where ( ˜x; ˜y; ˜z)=(x−xeq;y−yeq;z−zeq) (and accordingly for the momenta) denotes the displacement with respect to the xed point. Thus, in the harmonic approximation the motion in the z direction decouples from that in the x–y plane and is an oscillation with frequency ! √ q. The Hamiltonian for the latter, up to the additive constant Eeq, can be expressed in the standard form for a 2D, rotating anisotropic oscillator H = ˜p2 x +˜p2 y 2 + !2 (a ˜x 2 + b ˜y 2 ) 2 − !(˜x ˜p y − ˜y ˜p x ) ; (191) where the two parameters a and b are equal to −2q and q, respectively. This standard form has been studied in textbooks [93]. It may be used to describe the stability of the Lagrange equilibrium points in celestial mechanics (see [34,93] and references therein). Because this Hamiltonian mixes position and momentum coordinates, it is not straightforward to determine the stability at the origin. The result is that there are two domains of stability: and a; b ¿ 1 (192) − 3 6 a; b 6 1 with (a − b) 2 +8(a + b) ¿ 0: (193) For the speci c CP case, where a = −2q and b = q; only the second stability region is relevant, and the last inequality implies 8=9 6 q 6 1 for the xed point to be stable.
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Non-dispersive wave packets in periodically driven quantum systems

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