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Physical Review A

Encoding and decoding phase information in high- n circular wave packets

S. Yoshida, Carlos O Reinhold, Joachim Burgdörfer, B. Wyker, F. B. Dunning

Published Jun 30, 2010• 5 citations
Publisher

Abstract

We demonstrate theoretically and experimentally the extraction of detailed information on the density matrix of very-high-$n$ ($>300)$ near-circular Rydberg wave packets through Fourier analysis of the quantum beat and quantum revival signals. The remarkably long coherence times associated with circular wave packets facilitate the preservation and read-out of phase information encoded in this matrix. We illustrate the power of the method by determining the angular localization of the components of a wave packet.

DOI
10.1103/physreva.81.063428

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We demonstrate theoretically and experimentally the extraction of detailed information on the density matrix of very-high-nnn (>300)>300)>300) near-circular Rydberg wave packets through Fourier analysis of the quantum beat and quantum revival signals. The remarkably long coherence times associated with circular wave packets facilitate the preservation and read-out of phase information encoded in this matrix. We illustrate the power of the method by determining the angular localization of the components of a wave packet.
High angular momentum Rydberg wave packets are studied. Application of carefully tailored electric fields to low angular momentum, high- n ( n ∼ 300) Rydberg atoms creates coherent superpositions of Stark states with near extreme values of angular momentum, [cursive l]. Wave packet components orbit the parent nucleus at rates that depend on their energy, leading to periods of localization and delocalization as the components come into and go out of phase with each other. Monitoring survival probability signals in the presence of position dependent probing leads to observation of characteristic oscillations based on the composition of the wave packet. The discrete nature of electron energy levels is observed through the measurement of quantum revivals in the wave packet localization signal. Time-domain spectroscopy of these signals allows determination of both the population and phase of individual superposition components. Precise manipulation of wave packets is achieved through further application of pulsed electric fields. Decoherence effects due to background gas collisions and electrical noise are also detailed. Quantized classical trajectory Monte-Carlo simulations are introduced and agree remarkably well with experimental results.
We demonstrate that the quantum coherence of mesoscopic very-high- n , n ∼305, Rydberg wave packets travelling along nearly circular orbits can be maintained on microsecond time scales corresponding to hundreds of classical orbital periods. The coherence is probed through collapses and revivals of periodic oscillations in the average electron position. The temporal interferences of spatially separated Schrödinger cat-like wave packets are also observed. A novel hybrid quantum–classical trajectory method is employed to simulate the wave packet dynamics.
A semiclassical propagator is applied to describe the dynamics of circular-orbit wave packets which are initially well localized in three dimensions. A sum over classical Kepler trajectories for the wave-packet autocorrelation function is obtained which is extremely accurate well past the classical regime of the evolution regime of the wave packet. The nonclassical nature of the wave-packet evolution at long times is associated with the interference between amplitudes from classical paths at different energies. The time scale of the wave-packet revivals is related to the shearing rate of the corresponding classical ensemble.
We examine the long-term time dependence of Gaussian wave packets in a circular infinite well (billiard) system and find that there are approximate revivals. For the special case of purely m=0m=0m=0 states (central wave packets with no momentum) the revival time is ${T}_{\mathrm{rev}}^{(m=0)}=8\ensuremath{\mu}{R}^{2}/\ensuremath{\Elzxh}\ensuremath{\pi},$ where $\ensuremath{\mu}$ is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that ${T}_{\mathrm{rev}}^{(m\ensuremath{\ne}0)}=({\ensuremath{\pi}}^{2}{/2)T}_{\mathrm{rev}}^{(m=0)}\ensuremath{\approx}{5T}_{\mathrm{rev}}^{(m=0)}$ and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of $m\ensuremath{\ne}0$ states is increased. The dependence of the revival structure on the initial position, energy, and angular momentum of the wave packet and the connection to the energy spectrum is discussed in detail. The results are also compared to two other highly symmetrical two-dimensional infinite well geometries with exact revivals, namely, the square and equilateral triangle billiards. We also show explicitly how the classical periodicity for closed orbits in a circular billiard arises from the energy eigenvalue spectrum, using a WKB analysis.
We show that the time frequency analysis of the autocorrelation function is, in many ways, a more appropriate tool to resolve fractional revivals of a wave packet than the usual time domain analysis. This advantage is crucial in reconstructing the initial state of the wave packet when its coherent structure is short-lived and decays before it is fully revived. Our calculations are based on the model example of fractional revivals in a Rydberg wave packet of circular states. We end by providing an analytical investigation which fully agrees with our numerical observations on the utility of time-frequency analysis in the study of wave packet fractional revivals.
Quantum revivals in very-high-n (n ~ 300) high-l Rydberg wave packets generated from parent np states are used to examine decoherence induced by the application of 'coloured' noise from a random pulse generator and by collisions. In the absence of external perturbations, the high-l wave packets maintain their coherence for periods ~ 1 µs, i.e. for many hundreds of orbits. This coherence can be destroyed on sub-microsecond timescales by the application of even very small amounts of electrical noise at a rate that depends markedly on the spectral characteristics of the noise. In contrast, measurements over similar timescales with CO2 target gas densities of ~ 1011 cm−3 provided no evidence of collisional dephasing. The mechanisms responsible for decoherence are discussed with the aid of classical and quantum simulations. The results of these simulations are in good accord with the experimental data.