Non-dispersive wave packets in periodically driven quantum systems

More documents

Recommendations

Info

434 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 since the “kinetic energy” is negative—and the maximum of Hpend is now a stable equilibrium point, as the reader may easily check by standard linear stability analysis in the vicinity of the xed point. Thus, in compact form, if V1(Î 1)H ′′ 0 (Î 1) is positive, ˆ = is a stable equilibrium point, while ˆ = 0 is unstable. If V1(Î 1)H ′′ 0 (Î 1) is negative, the stable and unstable points are interchanged. There are two qualitatively di erent types of motion: • Close to the stable equilibrium point of the pendulum, ˆ oscillates periodically, with an amplitude smaller than . This is the “librational motion” of the pendulum inside the resonance island. Any trajectory started within this region of phase space (the shaded area in Fig. 8) exhibits librational motion. It should be realized that the resonance island con nes the motion to nite intervals in Î and ˆ , and thereby strongly a ects all trajectories with action close to the resonant action. According to Eqs. (68) and (69), the resonance island is associated with the energy range [H0(Î 1) − !Î 1 −| V1(Î 1)|;H0(Î 1) − !Î 1 + | V1(Î 1)|]. • For any initial energy outside that energy range the pendulum has su cient kinetic energy to rotate. This is the “rotational motion” of the pendulum outside the resonance island, where ˆ is an unbounded and monotonous function of time. Far from the center of the island, the motion occurs at almost constant unperturbed action Î, with an almost constant angular velocity in ˆ , tending to the unperturbed motion. This illustrates that the e ect of the perturbation is important for initial conditions close to the resonance island, but negligible for non-resonant trajectories. The size of the resonance island can be simply estimated from Eq. (66) and Fig. 8. The extension in ˆ is 2 , its width in Î (which depends on ˆ )is Î =4 V1(Î 1) H ′′ 0 (Î ; (70) 1) and the total area [3] A( )=16 V1(Î 1) H ′′ 0 (Î : (71) 1) The dependence of A( ) on | | implies that even a small perturbation may induce signi cant changes in the phase space structure, provided the perturbation is resonant. The above picture is valid in the rotating frame de ned by Eqs. (60)–(62). If we go back to the original action-angle coordinates (I; ), the stable (resp. unstable) xed point of the secular Hamiltonian is mapped on a stable (resp. unstable) periodic orbit whose period is exactly equal to the period of the driving perturbation, as a consequence of Eq. (65). Any trajectory started in the vicinity of the stable periodic orbit will correspond to an initial point close to the xed point in the rotating frame, and thus will remain trapped within the resonance island. In the original coordinate frame, it will appear as a trajectory evolving close to the stable periodic orbit forever. In particular, the di erence in between the stable periodic orbit and any orbit trapped in the resonance island remains bounded within (− ; + ), for arbitrarily long times. This means that the phase of any trapped trajectory cannot drift with respect to the phase of the periodic orbit. As the latter evolves at the driving frequency, we reach the conclusion that the phase of any trajectory
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 435 started in the resonance island will be locked on the phase of the driving eld. This is the very origin of the phase locking phenomenon discussed in Section 1.4 above. A crucial point herein is that the resonance island covers a signi cant part of phase space with nite volume: it is the whole structure, not few trajectories, which is phase locked. This is why, further down, we will be able to build quantum wave packets on this structure, which will be phase locked to the classical orbit and will not spread. The classical version of a non-spreading wave packet thus consists of a family of trajectories, trapped within the resonance island, such that this family is invariant under the evolution generated by the pendulum Hamiltonian. The simplest possibility is to sample all trajectories within the “energy” range [H0(Î 1) − !Î 1 −| V1(Î 1)|;H0(Î 1) − !Î 1 + | V1(Î 1)|] of the pendulum. In real space, this will appear as a localized probability density following the classical stable periodic orbit, reproducing its shape exactly after each period of the drive. So far, to derive the characteristics of the resonance island, we have consistently used rst-order perturbation theory, which is valid for small . At higher values of , higher-order terms come into play and modify the shape and the precise location of the resonance island. However, it is crucial to note that the island itself considered as a structure is robust, and will survive up to rather high values of (as a consequence of the KAM theorem [3]). Since the size of the resonance island grows with | |, Eq. (71), the island may occupy a signi cant area in phase space and eventually interact with islands associated with other resonances, for su ciently large | |. The mechanism of this “resonance overlap” is rather well understood [88,89]: in general, the motion close to the separatrix (where the period of the classical motion of the pendulum tends to in nity) is most sensitive to higher-order corrections. The general scenario is thus the non-integrable perturbation of the separatrix and the emergence of a “stochastic” layer of chaotic motion in phase space, as | | is increased. At still larger values of | |, chaos may invade large parts of phase space, and the resonance island may shrink and nally disappear. While considering realistic examples later on, we shall enter the non-perturbative regime. Let us, however, consider rst the quantum perturbative picture. 3.1.2. Quantum dynamics As shown in the previous section, the dynamics of a one-dimensional system exposed to a weak, resonant, periodic driving is essentially regular and analogous to the one of a pendulum, Eq. (67) (in the rotating frame, Eqs. (60)–(62)). In the present section, we will show that the same physical picture can be employed in quantum mechanics, to construct non-dispersive wave packets. They will follow the stable classical trajectory locked on the external drive, and exactly reproduce their initial shape after each period. Our starting point is the time-dependent Schrodinger equation associated with Hamiltonian (54): 8 d| (t)〉 i˝ dt =(H0 + V cos !t)| (t)〉 : (72) Since Hamiltonian (54) is periodic in time, the Floquet theorem 9 guarantees that the general solution of Eq. (72) is given by a linear combination of elementary, time-periodic states—the so-called 8 For simplicity, we use the same notation for classical and quantum quantities, the distinction between them will become clear from the context. 9 The Floquet theorem [90] in the time domain is strictly equivalent to the Bloch theorem for potentials periodic in space [91].
Page 1 and 2: Abstract Physics Reports 368 (2002)
Page 3 and 4: A. Buchleitner et al. / Physics Rep
Page 25: A. Buchleitner et al. / Physics Rep
Page 43 and 44: → ; → ; A. Buchleitner et al. /
Page 77 and 78:
A. Buchleitner et al. / Physics Rep
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
|ψ (z)| 2 |ψ (z)| 2 0.3 0.2 0.1 0
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
γ /∆ σ/∆ 10 -1 10 -2 10 -3 0.
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139:
show all

Non-dispersive wave packets in periodically driven quantum systems

Create successful ePaper yourself

Delete template?

Save as template?