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Physical Review A

Wigner-function calculations for the coherent superposition of matter waves

P. B. Lerner, H. Rauch, M. Suda

Published May 1, 1995• 9 citations
Publisher

Abstract

Spectral filtering of the time-separated neutrons passing through an interferometer reconstitutes the interference pattern through the formation of a macroscopic superposition out of the neutron beams and produces squeezed states. In this paper we use matter-waves squeezed states produced in this way to manipulate the rate of expansion of the neutron wave packet. We demonstrate that these quantum states also have nontrivial correlations 〈\ensuremath{\Delta}k\ensuremath{\Delta}x〉. The Heisenberg relation for the spread of the wave packet holds even in this case and the reversal of the wave packet spreading is impossible. However, one can manipulate the rate of the wave packet expansion by changing the phase shift in one of the arms of a neutron interferometer.

DOI
10.1103/physreva.51.3889

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Spectral filtering of the time-separated neutrons passing through an interferometer reconstitutes the interference pattern through the formation of a macroscopic superposition out of the neutron beams and produces squeezed states. In this paper we use matter-waves squeezed states produced in this way to manipulate the rate of expansion of the neutron wave packet. We demonstrate that these quantum states also have nontrivial correlations 〈\ensuremath{\Delta}k\ensuremath{\Delta}x〉. The Heisenberg relation for the spread of the wave packet holds even in this case and the reversal of the wave packet spreading is impossible. However, one can manipulate the rate of the wave packet expansion by changing the phase shift in one of the arms of a neutron interferometer.
The quantum-mechanical superposition of the two coherent states behind the Laue neutron interferometer is described in terms of the Wigner function which depends on the action of a phaseshifter in one arm of the interferometer. The wavepacket formalism is used and the coherent superposition of simple Gaussian wavepackets is investigated in detail with regard to the components of the momentum vector parallel and normal to the reflecting planes of the crystal. The characteristic features of interference are already apparent without closer consideration of the dynamical theory of diffraction in single crystals. These entangled states of superposition exhibit characteristic squeezing phenomena if one applies the method of mean square deviation of momentum. On the other hand the method of Shannon entropy does not show squeezing. That is what is expected, because the Shannon entropy is an integral quantity which is not so sensitive to details of the momentum spectrum. The Q function is discussed briefly in an appendix.
A striking spectral-modulation effect has been observed by means of a proper postselection procedure under conditions where the spatial shift of the wave trains greatly exceeds the coherence length of the neutron beams traversing an interferometer. It is shown that Schr\"odinger-cat-like states are created by the superposition of two coherent states generated in the interferometer. These entangled states exhibit, under certain circumstances, characteristic squeezing phenomena, indicating a highly nonclassical behavior. Analogies with optical experiments are discussed.
The neutron interferometer is a unique instrument that allows one to construct a neutron wave packet of macroscopic size, divide it into two components separated by centimeters, and then coherently recombine them. A number of experiments clearly showing the difference between quantum and classical theory have been performed with it, which are suitable for presentation in elementary quantum courses. This article presents a simple mathematical model of the interferometer, which can be used to illustrate clearly many of the surprising features of quantum systems. For example, one can describe an experiment to determine which component beam the neutron takes (an analog of the two-slit electron experiment). One can then trace in detail the loss of coherence of the wave function, rather than merely invoke the usual "handwaving" uncertainty arguments. The author discusses the effect of gravity on the neutron beam [the classic COW (Colella, Overhauser, and Werner) experiment], including a simple analysis in an accelerated reference frame, and its relation to the equivalence principle, the red shift, and the twin paradox. Also described are the effect of rotation of the neutron by 360\ifmmode^\circ\else\textdegree\fi{} to change its phase, the effect on the wave function of measuring the absence of the particle from a beam ("Dicke's paradox"), and a realizable version of Wheeler's "delayed-choice" experiments, as well as their relation to the problem of "Schr\"odinger's cat." The treatment is suitable for bright undergraduates and first-year graduate students.
The coherence length of a neutron wave packet depends on the spectral width of the distribution of wave vectors that make up the packet. The coherent overlap of the wave packets traversing the two beam paths in a perfect-silicon-crystal neutron interferometer (NI) is altered by placing a material with a neutron-nuclear optical potential in one of the beam paths in the NI. If the optical potential is positive, it causes a delay of the wave packet and a loss of fringe visibility. By use of an analyzer crystal, we narrow the spectral distribution after the mixing and interference has occurred in the last crystal slab of the NI. This increases the coherence length and restores some of the fringe visibility.
Fluctuations of the magnetic field lead to a dephasing of polarized neutron wave packets and to an attenuation of the interference term of the Wigner distribution. It turns out that the Schrödinger equation can be solved exactly by using a constant space-dependent magnetic potential plus an arbitrary number of time-dependent cosine- and sine-shaped potentials with different frequencies and amplitudes. If the time oscillations are not resolved or if frequency fluctuations are taken into consideration, one can work out a procedure to take the damping characteristics of the interference term into account. An arbitrary statistical noise function can be constructed by Fourier's theorem. This potential leads to depolarization due to a loss of phase correlation. One can simulate the transition from coherence to decoherence which can be used in ongoing spin-echo and quantum state reconstruction experiments.
We directly observe the quantum interference between two well-separated trapped-ion mechanical oscillator wave packets. The superposed state is created from a spin-motion entangled state using a heralded measurement. Wave packet interference is observed through the energy eigenstate populations. We reconstruct the Wigner function of these states by introducing probe Hamiltonians which measure Fock state populations in displaced and squeezed bases. Squeezed-basis measurements with 8 dB squeezing allow the measurement of interference for Δα=15.6, corresponding to a distance of 240 nm between the two superposed wave packets.