Analysis of circular wave packets generated by pulsed electric fields.

We demonstrate that circular wave packets in high Rydberg states generated by a pulsed electric field applied to extreme Stark states are characterized by a position-dependent energy gradient that leads to a correlation between the principal quantum number n and the spatial coordinate. This correlation is rather insensitive to the initial state and can be seen even in an incoherent mix of states such as is generated experimentally allowing information to be placed into, and extracted from, such wave packets. We show that detailed information on the spatial distribution of a circular wave packet can be extracted by analyzing the complex phase of its expansion coefficients.

Introduction

A wave packet is a coherent superposition of non-degenerate stationary states. Because of interference, such superpositions can form spatially localized states whose position is sensitive to the complex phase of the expansion coefficients and evolves in time [1-3]. With current technology it is possible to experimentally create localized electronic wave packets by exciting an electron in an atom [4-6]. One such wave packet that has been recently realized mimics the Bohr model of an atom [7], i.e., a classical electron traveling in a near-circular orbit [4]. Given the very high , at which these studies were undertaken, the diameter of the excited atom is about . Despite its macroscopic size the wave packet still displays quantum characteristics. Its coherence can be maintained on time-scales [8] allowing observation of collapses and revivals in its localization [9]. Because the suppression of decoherence prevents the loss of stored information, the coherently evolving Bohr wave packet can carry both quantum and classical information [10]. In this paper we analyze how information can be placed into, and extracted from, such wave packets [11]. In particular, we show that even an incoherent ensemble (as is typically generated experimentally) can be used to store information by coherently exciting all members of the ensemble using a pulsed electric field which transforms the incoherent ensemble into a partially coherent wave packet (full quantum calculations for Rydberg wave packets with are still out of reach and, therefore, simulations are done for in this paper).

Bohr wave packet

Circular states are not directly accessible from low- ground states by absorption of a few photons. Instead a method based on Stark precession [12] can be employed which transforms strongly oriented quasi-one-dimensional low angular momentum Rydberg states with (accessible from the ground state by a single photon) to circular states [13]. In this protocol, a transverse electric field F exerts a torque on the oriented states transferring angular momentum. By switching off the field after an appropriate time, (atomic units are used throughout), the transfer of angular momentum can be adjusted to generate near-circular states. When the switch-on/off of the transverse field is adiabatic, the product state is a superposition within a single n manifold and is stationary. However, sudden non-adiabatic switch-on/off results in a superposition that involves several n levels (see Fig. 1a) [11] and the creation of a wave packet moving in a near-circular Bohr-like orbit.

Fig. 1

Expectation value of the principal quantum number n evaluated locally at each position , (see Eq. (5)). The transverse pump pulse (duration: 2.5 ns, magnitude: 346 mV/cm) is applied along the +y axis to an extreme parabolic state initially aligned along the x axis. The pump process is simulated by numerically solving the time-dependent Schrödinger equation. The distribution is plotted (a) immediately after the pump pulse is switched off, (b) after one orbital period , and (c) after two orbital periods . Scaled coordinates ,  are used.

In its simplest form, the Bohr wave packet can be expressed as a superposition of circular states [9,14],where and are the moduli and phases of the expansion coefficients. The quantum numbers n extend over a narrow range that is much smaller than their mean value . During field-free evolution after the pump pulse, the phases evolve as with . The spatial distribution of the wave packet expanded in this basis setis a sum of cross-terms between pairs of circular states. Because the overlap between two circular states decreases rapidly with increasing , the terms with small provide the dominant contribution to the spatial distribution. Since circular states do not have any nodal structures in the radial r and directions , the term forms a probability density localized on a circle with radius confined within the xy plane. For with , the spatial distribution is narrow in r and (see, for example, Fig. 2b) but smoothly distributed in azimuthal angle . In consequence, the time dependence introduced by the interference terms ( with ) is most clearly detectable in the azimuthal angle . Thus, in the following we focus on the spatial distribution projected onto the space. The term with yields the distributionwhere and with . Each cosine term in the summation enhances the probability density formed by at and suppresses it at . Thus, a distribution localized at results. Since , the time dependence indicates that the localized wave packet travels along a circular orbit much like a classical particle. Indeed is approximately equal to the classical orbital frequency . With the summation over n, the distribution can thus be interpreted as an ensemble of “classical particle”-like components. The spatial distribution localizes strongly when all relative phases are nearly the same for all components and is broad for randomly distributed phases. The n-dependent angular velocity leads to dephasing and rephasing and induces a series of collapses and revivals in the localization. This dynamical behavior can be visualized by considering the application of a transverse pulsed field (duration: 2.5 ns, magnitude: 346 mV/cm) along the +y axis to an extreme parabolic state with , which is initially aligned along the x axis. The evolution during this “pump” pulse is simulated by numerically solving the time-dependent Schrödinger equation. Fig. 2b shows the probability densityevaluated immediately after the pump field is switched off. While the probability density is nearly uniformly distributed around a circle, the wave packet has an energy gradient along the axis of the pump field, i.e., along the y axis. This can be clearly seen in the expectation value of n evaluated at each position , i.e.,In practice, the integral can be evaluated by first calculating the wave function weighted by , i.e.,at each position and then integrating its absolute square . The correlation between the principal quantum number n and the angle (shown in Fig. 1a) is key to analyzing the field-free evolution following the pump pulse. After one orbital period (with ), the component with returns to its original position. The higher (lower) n levels, however, move back (advance) in phase as evidenced by the clockwise (counterclockwise) shift in their densities seen in Fig. 1b. At , they move back (advance) even further and the majority of the components of the wave packet collapse at the angle (see Fig. 1c) resulting in the strongly focused probability density seen in Fig. 2c. The evolution of the spatial distribution can be monitored by calculating the expectation value of a Cartesian coordinate, for example (see Fig. 2a). The initial uniform distribution yields a nearly vanishing . When the probability density is strongly localized, such as at undergoes large periodic oscillations. As each component of the wave packet evolves with a different rate , it undergoes dephasing (Fig. 2d and e) and rephasing (Fig. 2f) leading to collapses and revivals in the spatial focusing and, hence, in the oscillations in [8].

Fig. 2

(a) Time evolution of the expectation value for the same wave packet as in Fig. 1. (b–f) Snapshots of the probability density (Eq. (4)) plotted at scaled times of (b) 0, (c) 2, (d) 6, (e) 25 and (f) 54.

So far, only the interference terms with have been considered. Additional information on the spatial distribution can be obtained by extending the analysis to pairs with . While the contributions from these pairs are typically small, they can become non-negligible if a certain n level has a nearly vanishing whence the corresponding interference terms with are negligible. The distribution with is a summation of cosine terms, each of which enhances the probability density at two azimuthal angles and where . Since both peak positions evolve with the same angular velocity with , they remain apart during the whole evolution. In Fig. 1, the states with are initially concentrated at and . Considering that a pair of circular states with yields a double-peaked distribution (Eq. (7)) while that with yields a single-peaked distribution (Eq. (3)), the double-peaked distribution evident in Fig. 1 for hints a significant contribution. In fact, the quantum simulations reveal that the populations of and are about 10 times larger than that of . Accordingly, the pair ( and 150, or and 151) contributes little to due to the selection rule of the dipole operator while the pair ( and 151) can be detected by probing observables, such as .

Ensemble of near-circular wave packets

In the previous section, the Bohr wave packet is considered to be a fully coherent superposition of circular states. Such a superposition can be generated through Stark precession by applying a transverse pump field (along the y axis) to a pure extreme parabolic state . Because of both the small oscillator strength for excitation of extreme Stark states and Doppler broadening, in practice photoexcitation generates an ensemble of near-extreme Stark states [15] each of which can be transformed to a near-circular state. Here we show that the pump pulse coherently modifies all the Stark states in the ensemble imprinting on them the spatial character discussed above. Therefore, information can be stored and retrieved even using a partially coherent ensemble of states.

In the presence of the pump field , the Hamiltonian (within a single manifold) can be written as [13]where is the Stark frequency and are the y components of the pseudospins (: angular momentum vector and  = : Runge–Lenz vector). Similar to a spin in a magnetic field, precess about the y axis with angular velocities and , respectively. For example, the extreme parabolic state is described by an anti-parallel pair of pseudospins (i.e., and ). Each spin rotates in the opposite direction and, after a quarter of the precession period, the two become parallel (i.e., ) forming a circular state ( and ). Similarly, a near-extreme Stark state will be transformed to another Stark state oriented along the −z axis. Due to the non-adiabatic switch-off of the pump field, the final quantum numbers n and are distributed within a finite range with with , i.e.,The sudden switch-off imprints a similar energy gradient parallel to the pump field as seen in Fig. 1. We confirm this in Fig. 3a and b which display calculated for the wave packets created from two different Stark states ( and  − 133 with ). Since the energy gradient is nearly independent of the initial quantum number , it can survive averaging over an ensemble of near-circular (Stark) states (Fig. 3c) whose density matrix is given bywhere is the excitation probability of an initial Stark state with . Such an ensemble is generated experimentally by transforming an ensemble of near-extreme Stark states with different values of using a transverse pump field. The spatial wave function of this near-circular state (Eq. (9)) has the form with (independent of n) which has the same -dependence as the ideal wave packet [Eq. (1)]. When the spatial distribution is projected onto the coordinate, we can employ the same analysis as for the near-circular states. The (geometrical) angle of localization for each component wave packet can be mapped to the relative phases and [see Eqs. (3) and (7)]. Since the distribution is similar for all wave packets generated from different initial values of , the relative phases and determining the energy gradient are independent of and depend only on n. The interference patterns along the direction thus survive even for an ensemble of many near-circular states. Note that, unlike circular states, near-circular states have nodal structures in the r and directions and their coherent superpositions yield additional interference patterns as can be seen in Fig. 3d and e. As the number of nodes depends on the initial value of (or, equivalently, on the final value of ), an incoherent mix of wave packets with different values of destroys (partially) the nodal structures and the averaged spatial distribution becomes rather smooth (Fig. 3f).

Fig. 3

(a–c) Expectation value (Eq. (5)) for the wave packets generated by applying a pump pulse (duration: 2.5 ns, magnitude: 346 mV/cm) to a near-extreme parabolic state with (a) and (b) , and (c) to an ensemble of parabolic states with and . (d–f) Corresponding probability density distributions (Eq. (4)). The pump process is simulated by numerically solving the time-dependent Schrödinger equation. The distributions are plotted immediately after the pump pulse.

Summary and outlook

Bohr wave packets generated by pulsed electric field have a strong correlation between the principal quantum number n and the azimuthal angle . Such correlation can be realized by the sudden switching on/off of a pump field and exploiting the energy gradient along its polarization axis. As the energy gradient is essentially independent of the initial state, this n– mapping survives even for an incoherent initial ensemble such as is typically generated experimentally. Detailed information on the spatial distribution can be extracted by decomposing it into a sum of interference terms . While corresponds to the number of peaks in the distribution for each individual n component, each peak evolves approximately with the classical angular velocity . The total distribution can be viewed as an ensemble of “classical particle-like” states. The dynamical behavior of can be monitored by evaluating the expectation value . This can be easily extended for higher orders of to monitor through . Moreover, the Fourier transform of these time evolving signals provides the amplitudes and the relative phases of the expansion coefficients (Eq. (1)) [11]. The expectation values are also accessible experimentally providing opportunities to probe the spatial distribution of wave packets [16].