Demonstration of Turnstiles as a Chaotic Ionization Mechanism in Rydberg Atoms
We present an experimental and theoretical study of the chaotic ionization of quasi-one-dimensional potassium Rydberg wavepackets via a phase-space turnstile mechanism. Turnstiles form a general transport mechanism for numerous chaotic systems, and this study explicitly illuminates their relevance to atomic ionization. We create time-dependent Rydberg wavepackets, subject them to alternating applied electric-field “kicks”, and measure the electron survival probability. Ionization depends not only on the initial electron energy, but also on the phase-space position of the electron with respect to the turnstile — that part of the electron packet inside the turnstile ionizes after the applied ionization sequence, while that part outside the turnstile does not. The survival data thus encode information on the geometry and location of the turnstile, and are in good agreement with theoretical predictions.
pacs:32.80.Rm, 05.45.Gg, 05.45.Ac, 45.50.Pk
Chaotic behavior appears in diverse complex systems, over an enormous range of physical scales, including the formation of weather patterns, mixing of fluids, firing of neurons, and transport in the solar system. Among these, photoabsorbtion and ionization in atomic gasses have proven to be excellent testbeds for both classical and quantum chaos. For example, oscillations in the photoabsorption spectra of atoms in applied fields have been intimately linked to chaotic electron orbits Kleppner and Delos (2001), and resonant islands have proven to be barriers to microwave ionization Koch and van Leeuwen (1995). More recently, such resonant islands have been used to trap, control, and engineer electronic Rydberg wave packets Zhao et al. (2006). Such experiments highlight the utility of highly excited Rydberg electrons as high resolution probes of chaotic phase spaces.
Whereas the above examples focused on steady state, or nearly steady state, dynamics, the present work focuses on revealing the time-dependent mechanism underlying the chaotic ionization of an electron wave packet. The experimental protocol is based on the theoretical observation Burke and Mitchell (2009) (see also Ref. Mitchell et al. (2004)) that the ionization mechanism can be explained in terms of a homoclinic tangle and its corresponding turnstile MacKay et al. (1984); Wiggins (1992). The turnstile is a structure within phase space that promotes the electron from a bound to an unbound state, and thus serves as the critical step in the ionization process. Turnstiles have been theoretically applied to chaotic transport in a wide variety of physical systems Wiggins (1992). Experimental studies, however, are significantly more limited, with notable examples to chaotic or turbulent fluids Solomon et al. (1996), including recent work on Lagrangian coherent structures Voth et al. (2002); Shadden et al. (2006), and to optical microcavities Shim et al. (2008). However, to our knowledge, their structure has not previously been experimentally measured in atomic ionization. We present experimental data on the observation of the phase-space turnstile in a system consisting of highly excited quasi-one-dimensional Rydberg atoms Dunning et al. (2009) exposed to alternating electric-field pulses, or “kicks”. The ionization probability depends not only on the electron energy, but, crucially, on the phase-space position of the electronic state with respect to the turnstile at the moment the ionization kicks are applied. We demonstrate that our measurements are a sensitive probe of the phase-space position and shape of the turnstile.
Experimental protocol and data: Potassium atoms are first excited to a high principal quantum number ( or ) quasi-one-dimensional Rydberg state using the protocol detailed in Ref. Dunning et al. (2009). A small electric field “focusing” pulse is applied to the atom that delivers an impulsive kick to the electron of scaled strength directed toward the nucleus (Fig. 1a). (Tildes denote scaled atomic units: , , .) This creates a nonstationary electronic wave packet that undergoes strong periodic focusing near the outer classical turning point. After some time delay , a sequence of three alternating positive and negative ionization kicks is applied (Fig. 1a). These are much stronger than the focusing kick, with . The fraction of Rydberg atoms that survive as a function of delay time is measured using field ionization. Figure 2 shows results for both (left column) and (right column). The delay time is increased from 0 ns to 20 ns, and the peak-to-peak duration of the ionization sequence is adjusted between 5 ns and 15 ns. The kick durations are all fixed at 600 ps.
The data in Fig. 2 show a clear periodic behavior. This is elucidated by fitting to a sinusoid and extracting the fitting parameters , , , and . The first three parameters are recorded in Table 1; the last is plotted in Fig. 3. The following features stand out. (i) The oscillation period is nearly independent of and matches the classical Kepler period of the original state; for , ns; for , ns. (ii) The parameter is the survival probability averaged over . It and the oscillation amplitude, , do not depend strongly on . (iii) The phase shift varies considerably with , as seen in the shift of the vertical lines in Fig. 2.
A classical trajectory Monte Carlo (CTMC) simulation is sufficient to reproduce the essential experimental signatures. We use a one-dimensional hydrogenic model foo (b) with the square-wave forcing shown in Fig. 1b. The fitting parameters derived from this model are included in Table 1 and Fig. 3 under the label “1D” and are in good agreement with the experimental data. As shown below, these parameters reflect the underlying phase-space geometry. The amplitude , however, depends more strongly on the details of the electronic initial state, and the experimental oscillation amplitude is somewhat less than the model predictions. As a consistency check the results of more involved three-dimensional CTMC simulations are also included.
Connection to turnstile geometry: We now demonstrate how the experimental data reveal the presence of a phase-space turnstile and how the turnstile geometry provides a qualitative and quantitative framework for understanding trends in the data. Consider first the electron dynamics when subjected to periodic forcing, in which the single square-wave forcing cycle of duration (Fig. 1) is replaced by a periodic repetition (Fig. 4a). A stroboscopic picture of the dynamics is defined by the Poincaré map , which takes the radial position and momentum of the electron at a given time and returns their values one forcing period later. This map has a fixed point at Burke and Mitchell (2009) to which stable and unstable manifolds are attached (Fig. 4b). (A stable/unstable manifold / consists of those phase-space points that converge to the fixed point in the positive/negative time direction Wiggins (1992).) The stable and unstable manifolds together form a homoclinic tangle Wiggins (1992)(a “broken separatrix”), which defines the inner gray zone (roughly the “bound” electron states) and the outer white zone (the “ionized” states.) The tangle also defines regions called lobes, which fall into two categories, those that govern electron capture () and those that govern electron escape/ionization (). Under the forcing dynamics, and . The critical step for ionization is the transition as this promotes electron states from bound to ionized. An analogous process governs capture, and collectively this mechanism is called a phase-space turnstile MacKay et al. (1984). Being only interested in ionization here, we call the “turnstile lobe”. Its size, shape, and position govern the electron survival probability foo (c).
Following the focusing kick, the electronic state is no longer stationary. The corresponding classical ensemble has a narrow energy distribution (Fig. 6) centered at with . It forms a partially localized, or “focused”, ensemble that hugs and moves along the shell. (Unless otherwise noted, the remainder uses the case.) The observed oscillations in survival probability (Fig. 2) can now be understood as the ensemble passing in and out of the lobe—survival is higher when the bulk is outside and smaller when inside. Figure 5 illustrates this with snapshots of the ensemble at a sequence of times: b) Immediately after the focusing kick ; the ensemble roughly corresponds to the energy shell shifted slightly downward. c) The first survival maximum; the ensemble is almost entirely outside , resulting in negligible ionization. d) The first survival minimum; the ensemble has reflected once off the nucleus and now lies almost entirely within , resulting in large ionization. e) The second maximum. Since the ensemble trajectories lie close to the original energy shell, the survival oscillation period is approximately the Kepler period .
Next consider the average survival probability ; can be interpreted as the fraction of time a trajectory spends inside , averaged over the focused ensemble. To visualize this, it is easiest to work in the canonical coordinates (electron energy ) and (time to reach the nucleus). Consider Fig. 6a. The bottom left curve is the negative Kepler period, forming the left phase-space boundary. It is physically identified with the right vertical boundary line , which represents nuclear impact. Time evolution in these coordinates consists of uniform motion rightward along horizontal lines; when a trajectory reaches , it jumps back to . The lobe intersects the energy shell in one large segment. (See Fig. 6b, ns, for intersection segments.) The lobe has two “horns”, which stretch downward, intersecting the left boundary. From here, they re-emerge as two tendrils at the right boundary (taking advantage of the periodic boundary conditions), which reach up and again intersect in several short segments. In energy-time coordinates, the fraction of time a trajectory of energy spends inside equals the total length of the intersection between and the line of constant , divided by . Since for the focused ensemble is small, we need only consider , with . Thus, , where is the total length of the intersection segments in Fig. 6b, ns. Thereby, , as recorded in Table 1 column 1D’, and in excellent agreement with the experimental value .
Previous work Burke and Mitchell (2009) showed that as increases (with kick strength and duration fixed), the lobe in Fig. 6a shifts leftward. This is reflected in Fig. 6b, which shows the intersections of with the line for increasing values of . In particular, the primary wide segment shifts left with increasing . However, another critical effect is the increase in number and length of the shorter segments on the right. As above, these segments are the intersections with the horns, which have wrapped around. Physically, the wide segment contains trajectories that strike the nucleus once during the ionization sequence, whereas the shorter segments contain trajectories that strike the nucleus multiple times during ionization. As more time elapses between ionization pulses, there is more time for trajectories to strike the nucleus, and the relative importance of these ionization pathways increases. By , 24% of the ionizing trajectories have multiple nuclear impacts. The corresponding values, computed from the interval lengths in Fig. 6b, are recorded in Table 1 column 1D’ for varying . These agree well with the 1D and experimental data. Were ionization via multiple impacts not included, this agreement would be notably worse, especially for .
The leftward shift of the intervals in Fig. 6b with increasing explains the phase shift in Fig. 3. The large dots in Fig. 6b are the average positions of the escaping points (where the average is taken using the repeated segments on the left.) To account for the initial position of the focused ensemble, the values of are plotted as in Fig. 3, where they track the changes in phase seen as T increases. Ionizing trajectories that impact the nucleus multiple times again play an important role, as they shift the average dot leftward in Fig. 6b.
Conclusions: Present work identifies the turnstile lobe as the critical mechanism for promoting bound electronic states to ionization, the turnstile geometry providing a convenient framework for explaining the experimental results. More broadly, the results demonstrate that kicked atomic systems provide a convenient laboratory to explore the turnstile mechanism, common to a wide variety of physical systems exhibiting chaotic transport.
We thank Jeff Mestayer for initial assistance; K.B. acknowledges support of the UCOP Presidential Dissertation Year Fellowship; K.M. acknowledges the support of NSF grant PHY-0748828; and F.B.D. acknowledges the support from the Robert A. Welch Foundation under grant C-0734 and the NSF under grant PHY 0964819.
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