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Physical Review A

Measurement of the quantum state of electronic wave packets

R. R. Jones, M. B. Campbell

Published Dec 8, 1999• 11 citations
Publisher

Abstract

A technique for determining the quantum state of electronic wave packets from probability distribution measurements is presented. Specifically, a measurement of the fractional probability in each of a wave packet's constituent eigenstates can be used in conjunction with a measurement of the wave packet's time-dependent probability distribution to recover the complex amplitude of each constituent eigenstate. The method is demonstrated for radial Rydberg wave packets using previously measured time-dependent probability distributions in coordinate and momentum space. By exploiting delay imaging techniques, it should be possible to retrieve the quantum state of wave packets in real time.

DOI
10.1103/physreva.61.013403

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A technique for determining the quantum state of electronic wave packets from probability distribution measurements is presented. Specifically, a measurement of the fractional probability in each of a wave packet's constituent eigenstates can be used in conjunction with a measurement of the wave packet's time-dependent probability distribution to recover the complex amplitude of each constituent eigenstate. The method is demonstrated for radial Rydberg wave packets using previously measured time-dependent probability distributions in coordinate and momentum space. By exploiting delay imaging techniques, it should be possible to retrieve the quantum state of wave packets in real time.
Ultrashort, nearly unipolar electric field pulses are used to produce novel dynamic states in Rydberg atoms. A single 500 fsec field pulse redistributes essentially all of the population from a single stationary Rydberg state to hundreds of different final states. The dynamic evolution of the nonstationary wave packet is monitored using a second, time-delayed pulse. The time-dependent momentum-space probability distribution of the wave packet can be retrieved experimentally using a classical impulse approximation.
We present a method for performing quantum state holography, with which we completely characterize the amplitude and phase of an attosecond electron wave packet. Our approach is an extension of a recent publication [J. Mauritsson et al., Phys. Rev. Lett. 105, 053001 (2010)] in which we demonstrated experimentally that the energies and amplitudes of an attosecond electron wave packet can be characterized using attosecond electron interferometry. Here we show theoretically that attosecond electron interferometry can be extended to retrieve the phases of all the states that make up the wave packet. We demonstrate the feasibility of our method by analyzing a wave packet created by a shake-up process. We show that our method can successfully retrieve arbitrary phases and/or lifetimes added to the component eigenstates.
We show how to infer the quantum state of a wave packet from position probability distributions measured during the packet's motion in an arbitrary potential. We assume a nonrelativistic one-dimensional or radial wave packet. Temporal Fourier transformation and spatial sampling with respect to a newly found set of functions project the density-matrix elements out of the probability distributions. The sampling functions are derivatives of products of regular and irregular wave functions. We note that the ability to infer quantum states in this way depends on the structure of the Schr\"odinger equation.
A wave-packet interference technique is proposed to measure exotic atomic Rydberg wave functions. For example, the subject wave packet is excited by an arbitrary pulse or pulse sequence. The resulting wave function is probed by a well-characterized pulse that shares a phase relationship with the excitation pulse(s). The observable is the population in each eigenstate after the probe pulse. By making a series of such measurements, varying the phase between excitation and probe pulses, the wave function created by the excitation pulse can be inferred. The method is experimentally feasible for atomic systems, and the extension of the method to molecular systems also appears possible.
We show that wave-packet interference techniques can reconstruct the quantum state of wave packets excited by strong laser pulses. The measurement may be accomplished even when the strength of the excitation is sufficient to significantly deplete the ground state. The probe of the wave packet can still remain perturbative. An inversion algorithm is also developed that is a generalization of the standard Ramsey fringe probe of a wave packet. The normalized sums and differences of the populations of two of the Rydberg states display Ramsey fringes and from these fringes the relative phase between these two states can be found. The algorithm offers advantages in efficiency over previous interferometric algorithms. The algorithm is not sensitive to the precise nature of the distribution of the phase between the pump and probe pulses.