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Physical Review Letters

Observation of Moving Wave Packets Reveals Their Quantum State

Ulf Leonhardt, Michael G. Raymer

Published Mar 18, 1996• 90 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.76.1985

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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 data analysis scheme is proposed for determining the quantum state of a one-dimensional wave packet describing atomic translational motion in free space. The Whittaker-Shannon sampling theorem is applied to reconstruct exactly a certain portion of the momentum-basis density matrix from Fourier transforms of measured position-probability densities. The sampling theorem is valid if the spatial wave correlation function equals zero ouside some finite range. It is most useful in cases where the range of available data is insufficient to allow a complete reconstruction of the state by phase-space tomography, as in recent experimental studies of atomic-beam state reconstruction by Pfau and Kurtsiefer.
The density matrix of a nonrelativistic wave packet in an arbitrary, one-dimensional, and time-dependent potential can be reconstructed by measuring hydrodynamical moments of the Wigner distribution. An nnnth order Taylor polynomial in the off-diagonal variable is obtained by measuring the probability distribution at n+1n+1n+1 discrete time values.
We discuss nonspreading wave packets in one dimensional Schrödinger equation. We derive general rules for constructing nonspreading wave packets from a general potential V(x,t)\textmd{V}(x,t)V(x,t). The essential ingredients of a nonspreading wave packet, the shape function f(x)f(x)f(x), the motion d(t)d(t)d(t), the phase function ϕ(x,t)ϕ(x,t)ϕ(x,t) are derived. Since the form of the shape of a nonspreading wave packet does not change in time, the shape equation should be time independent. We show that the shape function f(x)f(x)f(x) is the eigenfunction of the time independent Schrödinger equation with an effective potential VeffV_{\textmd{eff}}Veff​ and an energy EeffE_{\textmd{eff}}Eeff​. We derive nonspreading wave packets found by Schrödinger, Senitzky, and Berry and Balazs as examples. We show that most stationary potentials can only support stationary nonspreading wave packets. We show how to construct moving nonspreading wave packets from time dependent potentials, which drive nonspreading wave packets into an arbitrary motion.
We show how to reconstruct the quantum state of one-dimensional wave packets in the continuous part of the spectrum. We assume that the position probability distribution is measured for a sufficiently long time interval and that the potential is known, but otherwise arbitrary. Our paper fills the gap between the tomographic state reconstruction based on free evolution [J. Bertrand and P. Bertrand, Found. Phys. 17, 397 (1987)] and the state determination of moving bound states [U. Leonhardt and M. G. Raymer, Phys. Rev. Lett. 76, 1985 (1996)]. Our result may be relevant with respect to the state determination of atomic beams [Ch. Kurtsiefer, T. Pfau, and J. Mlynek, Nature (London) 386, 150 (1997)] moving in arbitrary potentials.