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arXiv:quant-ph/0607098 · Europhysics Letters (EPL)

How to measure the wave function absolute squared of a moving particle by using mirrors

Volker Hannstein, Gerhard C. Hegerfeldt

Published Feb 28, 2007• 0 citations
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Abstract

We consider a slow particle with wave function ψ t ( x ), moving freely in some direction. A mirror is briefly switched on around a time T and its position is scanned. It is shown that the measured reflection probability then allows the determination of |ψ T ( x )| 2 . Experimentally available atomic mirrors should make this method applicable to the center-of-mass wave function of atoms with velocities in the cm/s range.

DOI
10.1209/0295-5075/77/60003

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We consider a slow particle with wave function ψ t ( x ), moving freely in some direction. A mirror is briefly switched on around a time T and its position is scanned. It is shown that the measured reflection probability then allows the determination of |ψ T ( x )| 2 . Experimentally available atomic mirrors should make this method applicable to the center-of-mass wave function of atoms with velocities in the cm/s range.
When a stationary reflecting wall acting as a perfect mirror for an atomic beam with well-defined incident velocity is suddenly removed, the density profile develops during the time evolution an oscillatory pattern known as diffraction in time. The interference fringes are suppressed or their visibility is diminished by several effects such as averaging over a distribution of incident velocities, apodization of the aperture function, atom-atom interactions, imperfect reflection, or environmental noise. However, when the mirror moves with finite velocity along the direction of propagation of the beam, the visibility of the fringes is enhanced. For mirror velocities below beam velocity, as used for slowing down the beam, the matter wave splits into three regions separated by space-time points with classical analogs. For mirror velocities above beam velocity a visibility enhancement occurs without a classical counterpart. When the velocity of the beam approaches that of the mirror the density oscillations rise by a factor 1.8 over the stationary value.
We present a simple spectroscopic method based on Autler-Townes spectroscopy to determine the center-of-mass atomic wave function. The detection of spontaneously emitted photons from a three-level atom, in which two upper levels are driven by a classical standing light, yields information about the position and momentum distribution of the atom [A. M. Herkommer, W. P. Schleich, and M. S. Zubairy, J. Mod. Opt. 44,44,44, 2507 (1997)]. In this paper, we show that both the amplitude and phase information of the center-of-mass atomic wave function can be obtained from these distributions after a series of conditional measurements on the atom and the emitted photon.
Quantum mechanics relates probability of an observable event to the absolute square of the corresponding probability amplitude. It may, therefore, seem that the information about the amplitudes' phases must be irretrievably lost in the experimental data. Yet, there are experiments which report measurements of wave functions, and closely related quantities such as Bohm's velocities and positions of bohmian particles. We invert the question, and ask under which conditions the values of quantum amplitudes can be recovered from observed probability distributions and averages.
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.
Motivated by the technical requirements of quantum information processing and nanotechnology, the control of individual quantum systems such as single atoms, ions or even photons has become a highly desirable aim.The monitoring of quantum systems-a direct test and in many cases a prerequisite of their control-has been investigated so far only for certain observables such as the position or momentum of quantum particles.Here, we describe a method to monitor in real time the complete state of a quantum particle with unknown initial state moving in a known potential.The method is based on successively updating an estimate by the results of a continuous position measurement.We demonstrate by numerical simulations that even in a chaotic potential tracking the wave function of a particle is possible, and we show with an example that the monitoring scheme appears to be robust against sudden random perturbations.