Non-dispersive wave packets in periodically driven quantum systems

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432 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 which transforms H into ˆH = ˆPt + H0(Î) − !Î + +∞ m=−∞ Vm(Î) cos(m ˆ +(m − 1)!t) : (63) This Hamilton function does not involve any approximation yet. Only in the next step, we average ˆH over the fast variable t, i.e., over one period of the external driving. This has the e ect of canceling all oscillating terms in the sum, except the resonant one, de ned by m = 1. Consequently, we are left with the approximate, “secular” Hamilton function: Hsec = ˆPt + H0(Î) − !Î + V1(Î) cos ˆ : (64) The secular Hamilton function no longer depends on time. Hence, ˆPt is a constant of motion and we are left with an integrable Hamiltonian system living in a two-dimensional phase space, spanned by (Î; ˆ ). The above averaging procedure is valid at rst-order in . Higher-order expansions, using, e.g., the Lie algebraic transformation method [3], are possible. 7 Basically, the interesting physical phenomena are already present at lowest non-vanishing order, to which we will restrain in the following. The dynamics generated by the secular Hamilton function is rather simple. At order zero in , Î is constant and ˆ evolves linearly with time. As we can read from Eq. (64), a continuous family (parametrized by the value of 0 6 ˆ ¡2 ) of xed points exists if d ˆ =dt = 9Hsec=9Î vanishes, i.e., at actions Î 1 such that (Î 1)= 9H0 9I (Î 1)=!: (65) Thus, unperturbed trajectories that are resonant with the external drive are xed points of the unperturbed secular dynamics. This is precisely why slowly varying variables are introduced. Typically, Eq. (65) has only isolated solutions—we will assume that in the following. Such is the case when 9 2 H0=9I 2 does not vanish—excluding the pathological situation of the harmonic oscillator, where all trajectories are simultaneously resonant. Hence, if 9 2 H0=9I 2 is positive, the line (Î = Î 1; 0 6 ˆ ¡2 , parametrized by ˆ ) is a minimum of the unperturbed secular Hamilton function Hsec; if9 2 H0=9I 2 is negative, it is a maximum. At rst order in , the xed points of the secular Hamiltonian should have an action close to Î 1. Hence, it is reasonable to perform a power expansion of the unperturbed Hamiltonian in the vicinity of Î = Î 1. We obtain the following approximate Hamiltonian: with Hpend = ˆPt + H0(Î 1) − !Î 1 + 1 ′′ H 2 0 (Î 1)(Î − Î 1) 2 + V1(Î 1) cos ˆ (66) H ′′ 0 = 92 H0 9I 2 : (67) 7 An example is given in [87], in a slightly di erent situation, where the perturbation is not resonant with the internal frequency.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 433 Action I ∆ I I=I 1 Resonance island 0 1 2 θ/π Fig. 8. Isovalues of the Hamiltonian Hpend of a pendulum in a gravitational eld. This Hamiltonian is a good approximation for the motion of a periodically driven system when the driving frequency is resonant with the internal frequency of the system. The stable equilibrium point of the pendulum is surrounded by an island of librational motion (shaded region). This de nes the resonance island of the periodically driven system, where the internal motion is locked on the external driving. This non-linear phase-locking phenomenon is essential for the existence of non-dispersive wave packets. Consistently at lowest order in , it is not necessary to take into account the dependence of V1 on Î. As already anticipated by the label, Hpend de ned in Eq. (66) describes a usual, one-dimensional pendulum: ˆ represents the angle of the pendulum with the vertical axis, Î − Î 1 its angular velocity, 1=H ′′ 0 (Î 1) its momentum of inertia and V1(Î 1) the gravitational eld. This equivalence of the secular Hamilton function with that of a pendulum, in the vicinity of the resonant action Î 1, is extremely useful to gain some physical insight in the dynamics of any Hamiltonian system close to a resonance. In particular, it will render our analysis of non-dispersive wave packets rather simple. Fig. 8 shows the isovalue lines of Hpend in the (Î; ˆ ) plane, i.e., the classical phase space trajectories in the presence of the resonant perturbation. In the absence of the resonant perturbation, these should be horizontal straight lines at constant Î. We observe that the e ect of the resonant perturbation is mainly to create a new structure, called the “resonance island”, located around the resonant action Î 1. To characterize this structure, let us examine the xed points of Hamiltonian (66). They are easily calculated (imposing 9Hpend=9Î = 9Hpend=9 ˆ = 0), and located at and Î = Î 1; ˆ = 0 with energy H0(Î 1) − !Î 1 + V1(Î 1) (68) Î = Î 1; ˆ = with energy H0(Î 1) − !Î 1 − V1(Î 1) ; (69) respectively. If H ′′ 0 (Î 1) (and thus the “kinetic energy” part in Hpend) is positive, the minimum of the potential V1(Î 1) cos ˆ corresponds to a global minimum of Hpend, and thus to a stable equilibrium point. The maximum of V1(Î 1) cos ˆ is a saddle point of Hpend, and thus represents an unstable equilibrium point, as the standard intuition suggests. For H ′′ 0 (Î 1) ¡ 0 the situation is reversed—and less intuitive
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Non-dispersive wave packets in periodically driven quantum systems

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