Non-dispersive wave packets in periodically driven quantum systems

More documents

Recommendations

Info

518 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 |Energy Shift| (atomic units) Width (Ionization Rate) (atomic units) 10 -8 10 -9 10 -10 10 -8 10 -10 10 -12 10 -14 39 40 41 42 Effective Principal Quantum Number n0 = ω −1/3 Fig. 51. Typical uctuations of the width (ionization rate) and of the energy (with respect to its averaged, smooth behavior) of the non-dispersive wave packet of a two-dimensional hydrogen atom in a circularly polarized microwave eld. The data presented are obtained for small variations of the e ective principal quantum number n0 = ! −1=3 around 40, and a scaled microwave electric eld F0 =0:0426. To show that the uctuations cover several orders of magnitude, we use a logarithmic vertical scale, and plot the absolute value of the shift. on extremely long time-scales and is completely negligible in atomic systems. In practice, ionization of the wave-packet is essentially mediated by a pure quantum process, exponentially unlikely in the semiclassical limit. As we shall see below, this tunneling process has quite interesting properties which may be quantitatively described for microwave-driven atoms. More details can be found in [50,65,66]. Due to the initial tunneling step, the lifetimes of non-dispersive wave packets will typically be much longer than those of Floquet states localized in the chaotic sea surrounding the island [145]. Moreover, since the ionization mechanism involves chaotic di usion, many quantum mechanical paths link the initial wave packet to the nal continuum. Thus, the lifetime of the wave packet will re ect the interferences between those di erent possible paths, and will sensitively depend on parameters such as the microwave frequency or amplitude, that a ect the interfering paths through the chaotic sea. These uctuations, reported rst in [50], are perfectly deterministic and resemble the conductance uctuations observed in mesoscopic systems [178]. In Fig. 51, we show the uctuations of the ionization rate (width) of the non-dispersive wave packet of the two-dimensional hydrogen atom in a circularly polarized microwave eld. The energy levels and widths are obtained as explained in Section 3.2, by numerical diagonalization of the complex rotated Hamiltonian. All the data presented in this section have been obtained in the regime where the typical ionization rate is smaller than the mean energy spacing between consecutive levels, so that the ionization can be thought as a small perturbation acting on bound states. The width (although very small) displays strong uctuations over several orders of magnitude. Similarly, the real part of the energy (i.e., the center of the atomic resonance) displays wild uctuations. The latter can be observed only if the smooth variation of the energy level with the control parameter (following approximately the semiclassical prediction given by Eq. (174)) is substracted. Therefore, we tted the numerically obtained energies by a smooth function and substracted this t to obtain the displayed uctuations. Note that these uctuations are (a) (b)
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 519 so small that an accurate t is needed. 34 This can be easily seen in Fig. 30 where, on the scale of the mean level spacing, these uctuations are invisible by eye (the level appears as a straight horizontal line). The explanation for the uctuations is the following: in a quantum language, they are due to the coupling between the localized wave packet and states localized in the chaotic sea surrounding the resonance island. While the energy of the wave packet is a smooth function of the parameters F and !, the energies of the chaotic states display a complicated behavior characterized by level repulsion and large avoided crossings. It happens often that—for some parameter values—there is a quasi-degeneracy between the wave-packet eigenstate and a chaotic state, see the numerous tiny avoided crossings in Fig. 30. There, the two states are e ciently mixed, the wave packet captures some part of the coupling of the chaotic state to the continuum and its ionization width increases (see also [145]). This is the very origin of the observed uctuations. Simultaneously, the chaotic state repels the wave-packet state leading to a deviation of the energy from its smooth behavior, and thus to the observed uctuations. This mechanism is similar to “chaos-assisted tunneling”, described in the literature [179–189] for both, driven one-dimensional and two-dimensional autonomous systems. There, the tunneling rate between two symmetric islands—which manifests itself through the splitting between the symmetric and antisymmetric states of a doublet—may be strongly enhanced by the chaotic transport between the islands. We have then a “regular” tunneling escape from one island, a chaotic di usive transport from the vicinity of one island to the other (many paths, leading to interferences and resulting in large uctuations of the splitting), and another “regular” tunneling penetration into the second island. In our case, the situation is even simpler—we have a “regular” tunneling escape supplemented by a chaotic di usion and eventual ionization. Thus, instead of the level splitting, we observe a shift of the energy level and a nite width. Since these uctuations stem from the coupling between the regular wave-packet state and a set of chaotic states, it is quite natural to model such a situation via a Random Matrix model [65], the approach being directly motivated by a similar treatment of the tunneling splitting in [187]. For details, we refer the reader to the original work [65]. It su ces to say here that the model is characterized by three real parameters: —which characterizes the mean strength of the coupling between the regular state and the chaotic levels, —which measures the decay of the chaotic states (due to ionization; direct ionization transitions from the wave-packet state to the continuum are negligible), and —which is the mean level spacing of chaotic levels. The two physically relevant, dimensionless parameters are = and = . In the perturbative regime ( = ; = 1) it is possible to obtain analytical [65] predictions for the statistical distribution of the energy shifts P(s) (of the wave-packet’s energy from its unperturbed value) and for the distribution of its widths P( ). P(s) turns out to be a Cauchy distribution (Lorentzian), similarly to the tunneling splitting distribution found in [187]. The distribution of the widths is a bit more complicated (it is the square root of which is approximately Lorentzian distributed). The perturbative approach fails for the asymptotic behavior of the tails of the distributions, where an exponential cut-o is expected and observed in numerical studies [65,187]. By tting the predictions of the Random Matrix model to the numerical data of Fig. 51, we may nally extract the values of = , the strength of the decay, and of = , the coupling between the regular and the chaotic states. An example of such a t is shown in Fig. 52. The numerical data are collected around some mean values of n0 and F0, typically 1000 data points 34 In particular, the semiclassical expression is not su ciently accurate for such a t.
Page 1 and 2:
Abstract Physics Reports 368 (2002)
Page 3 and 4:
A. Buchleitner et al. / Physics Rep
Page 5 and 6:
Page 7 and 8:
Page 9 and 10:
Page 11 and 12:
Page 13 and 14:
Page 15 and 16:
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
→ ; → ; A. Buchleitner et al. /
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60: A. Buchleitner et al. / Physics Rep
Page 97 and 98: |ψ (z)| 2 |ψ (z)| 2 0.3 0.2 0.1 0
Page 109: A. Buchleitner et al. / Physics Rep
Page 113 and 114: γ /∆ σ/∆ 10 -1 10 -2 10 -3 0.
Page 139: A. Buchleitner et al. / Physics Rep
show all

Non-dispersive wave packets in periodically driven quantum systems

Create successful ePaper yourself

Delete template?

Save as template?