Non-dispersive wave packets in periodically driven quantum systems

More documents

Recommendations

Info

454 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 The quantum quasi-energy levels are then given by Eq. (102) which reads E ;k=0 = − 3 2n2 − 0 3 8n4 a ( ; q) : (149) 0 As discussed in Section 3.1.2, the non-dispersive wave packet with maximum localization is associated with = 0 and is well localized inside the non-linear resonance island between the internal Coulombic motion and the external driving, provided the parameter q is of the order of unity (below this value, the resonance island is too small to support a localized state). The minimum scaled microwave amplitude is thus of the order of F0;trapping = Ftrappingn 4 0 1 n2 ; (150) 0 which is thus much smaller—by a factor n2 0 , i.e., three orders of magnitude in typical experiments— than the electric eld created by the nucleus. This illustrates that a well chosen weak perturbation may strongly in uence the dynamics of a non-linear system. From the experimental point of view, this is good news, a limited microwave power is su cient to create non-dispersive wave packets. We have so far given a complete description of the dynamics of the resonantly driven, onedimensional Rydberg electron, from a semiclassical as well as from a quantum mechanical point of view, in the resonant approximation. These approximate treatments are now complemented by a numerical solution of the exact quantum mechanical eigenvalue problem described by the Floquet equation (75), with H0 and V from Eqs. (113) and (114), as well as by the numerical integration of the classical equations of motion derived from Eq. (137). Using this machinery, we illustrate some of the essential properties of non-dispersive wave packets associated with the principal resonance in this system, whereas we postpone the discussion of other primary resonances to Section 5.3. Fig. 10 compares the phase space structure of the exact classical dynamics generated by the Hamilton function (137), and the isovalue curves of the pendulum dynamics, Eq. (144), for the case n0 = 60, at scaled eld strength F0 = Fn4 0 =0:01. The Poincare surface of section is taken at phases !t = 0 (mod 2 ) and plotted in (I; ) variables which, for such times, coincide with the (Î; ˆ ) variables, see Eqs. (60) and (61). Clearly, the pendulum approximation predicts the structure of the invariant curves very well, with the resonance island surrounding the stable periodic orbit at (Î ≈ 60; ˆ = ), the unstable xed point at (Î ≈ 60; ˆ = 0), the separatrix, and the rotational motion outside the resonance island. Apparently, only tiny regions of stochastic motion invade the classical phase space, which hardly a ects the quality of the pendulum approximation. It should be emphasized that—because of the scaling laws, see Section 3.2.5—the gure depends on the scaled eld strength F0 only. Choosing a di erent microwave frequency with the same scaled eld leads, via Eq. (139), to a change of n0 and, hence, of the scale of I. Fig. 11 compares the prediction of the Mathieu approach, Eq. (149), for the quasi-energy levels of the Floquet Hamiltonian to the exact numerical result obtained by diagonalization of the full Hamiltonian, see Section 3.2. Because of the ˝! periodicity of the Floquet spectrum, the sets of energy levels of the pendulum, see Fig. 9, are folded in one single Floquet zone. For states located inside or in the vicinity of the resonance island—the only ones plotted in Fig. 9a—the agreement is very good for low and moderate eld strengths. Stronger electromagnetic elds lead to deviations between the Mathieu and the exact result. This indicates higher-order corrections to the pendulum approximation.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 455 Fig. 10. Poincare surface of section for the dynamics of a 1D hydrogen atom driven by an external oscillatory electric eld, see Eq. (137). The driving frequency is chosen as ! =1=60 3 , such that the non-linear resonance island is centered at principal action (or e ective principal quantum number) n0=60. The scaled external eld amplitude is set to F0=Fn 4 0=0:01. Although the driving eld is much weaker than the Coulomb eld between the electron and the nucleus, it su ces to create a relatively large resonance island which supports several non-dispersive wave packets. Energy (10 −4 a.u.) −1.372 −1.382 −1.392 −1.402 −1.412 0 0.02 0.04 0.06 0 0.02 0.04 0.06 F 0 Fig. 11. Comparison of the exact quasi-energy spectrum of the 1D hydrogen atom driven by a linearly polarized microwave eld (right), Eq. (137), with the prediction of the pendulum approximation (left), Eq. (149). Because of the ˝! periodicity of the Floquet spectrum, the energy levels described by the pendulum approximation are folded inside one Floquet zone. The agreement between the exact quantum result and the pendulum approximation is very good. The lled circle shows the most localized non-dispersive wave packet N = 0 shown in Figs. 13–15, while the lled square represents the hyperbolic non-dispersive wave packet partly localized in the vicinity of the unstable equilibrium point of the pendulum, shown in Figs. 17 and 18. The open circle and square compare the exact location of the respective quasienergy values with the Mathieu prediction, which is considerably better for the ground state as compared to the separatrix state.
Page 1 and 2: Abstract Physics Reports 368 (2002)
Page 3 and 4: A. Buchleitner et al. / Physics Rep
Page 43 and 44: → ; → ; A. Buchleitner et al. /
Page 45: A. Buchleitner et al. / Physics Rep
Page 97 and 98:
|ψ (z)| 2 |ψ (z)| 2 0.3 0.2 0.1 0
Page 99 and 100:
A. Buchleitner et al. / Physics Rep
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?