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Physical Review A

State reconstruction in one-dimensional quantum mechanics: The continuous spectrum

Ulf Leonhardt, Stephan Schneider

Published Oct 1, 1997• 15 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.56.2549

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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.
The problem of reconstructing a pure quantum state \ensuremath{\Vert}\ensuremath{\psi}〉 from measurable quantities is considered for a particle moving in a one-dimensional potential V(x). Suppose that the position probability distribution \ensuremath{\Vert}\ensuremath{\psi}(x,t)${\mathrm{\ensuremath{\Vert}}}^{2}$ has been measured at time t, and let it have M nodes. It is shown that after measuring the time evolved distribution at a short-time interval \ensuremath{\Delta}t later, \ensuremath{\Vert}\ensuremath{\psi}(x,t+\ensuremath{\Delta}t)${\mathrm{\ensuremath{\Vert}}}^{2}$, the set of wave functions compatible with these distributions is given by a smooth manifold M in Hilbert space. The manifold M is isomorphic to an M-dimensional torus, TM{\mathcal{T}}^{\mathit{M}}TM. Finally, M additional expectation values of appropriately chosen nonlocal operators fix the quantum state uniquely. The method used here is the analog of an approach that has been applied successfully to the corresponding problem for a spin system. \textcopyright{} 1996 The American Physical Society.
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 quantum state of a one-dimensional molecular vibration can be reconstructed via molecular emission tomography [T. J. Dunn, I. A. Walmsley, and S. Mukamel, Phys. Rev. Lett. 74, 884 (1995)] even in the case of anharmonic potentials. We assume a Morse-oscillator model for the vibration and calculate analytically the irregular wave functions. These mathematical objects serve as ingredients of the reconstruction procedure [U. Leonhardt and M. G. Raymer, Phys. Rev. Lett. 76, 1989 (1996)]. Additionally, we show why the observation time until the revival of order is sufficient for an accurate state determination.
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.