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Physical Review A

Reconstruction of Bohm trajectories and wave functions from interferometric measurements

Wolfgang P. Schleich, Matthias Freyberger, Muhammad Suhail Zubairy

Published Jan 16, 2013• 29 citations
Publisher

Abstract

We propose a generalization of the technique of quantum state reconstruction to obtain the space derivative of the phase of a wave function, which is central to the hydrodynamic formulation of quantum mechanics in the manner of Madelung and Bohm. In this way we can construct Bohm trajectories and make contact with a recent experiment [Kocsis et al., Science 332, 1170 (2011)] extracting these trajectories after a double slit.

DOI
10.1103/physreva.87.014102

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We propose a generalization of the technique of quantum state reconstruction to obtain the space derivative of the phase of a wave function, which is central to the hydrodynamic formulation of quantum mechanics in the manner of Madelung and Bohm. In this way we can construct Bohm trajectories and make contact with a recent experiment [Kocsis et al., Science 332, 1170 (2011)] extracting these trajectories after a double slit.
Abstract In 1952 David Bohm proposed an interpretation of quantum mechanics, in which the evolution of states results from trajectories governed by classical equations of motion but with an additional potential determined by the wave function. There exist only a few experiments that test this concept and they employed weak measurement of non-classical light. In contrast, we reconstruct the Bohm trajectories in a classical hydrodynamic system of surface gravity water waves, by a direct measurement of the wave packet. Our system is governed by a wave equation that is analogous to the Schrödinger equation which enables us to transfer the Bohm formalism to classical waves. In contrast to a quantum system, we can measure simultaneously their amplitude and phase. In our experiments, we employ three characteristic types of surface gravity water wave packets: two and three Gaussian temporal slits and temporal Airy wave packets. The Bohm trajectories and their energy flows follow the valleys and bounce off the hills in the corresponding quantum potential landscapes.
Bohmian mechanics, developed by Bohm in 1952, provides an alternative interpretation to nonrelativistic quantum mechanics [1-3]. In the hydrodynamic formulation of quantum mechanics, the continuity equation and the quantum Hamilton-Jacobi equation (QHJE) are obtained by substituting the wave function expressed in terms of the real amplitude and the real action function into the time-dependent Schrödinger equation. Bohm’s analytical approach has been used to compute and interpret realvalued quantum trajectories from a precomputed wave function for a diverse range of physical processes [3-8]. In the synthetic approach, the quantum trajectory method has been developed as a computational tool to generate the wave function by evolving ensembles of real-valued quantum trajectories through the integration of the hydrodynamic equations on the fly [9]. Remarkable progress has been made in the use of T&F Cat # K11224, Chapter 18, Page 284, 15-9-2010 real-valued quantum trajectories for solving a wide range of quantum mechanical problems [10].
A method is presented that reconstructs the amplitude and phase of an unknown time-dependent pure-state wave function entirely from experimental position data. No assumptions about the wave function are needed. At a series of times, a large collection of position data is taken, but no measurements of momentum or energy are necessary. The momenta, however, are required for the reconstruction of the wave function and are supplied by analysis of the position data via quantum particle trajectories. Simulations of the method are provided for helium atoms in single and double slit experiments.