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Journal of Optics B Quantum and Semiclassical Optics

Correlations, visibilities and Wigner functions in a double-loop neutron interferometer

Martin Suda

Published Jun 1, 2003• 0 citations
Publisher

Abstract

In a double-loop neutron interferometer two and three beams are superposed behind the first and second loop respectively. The wavefunctions in different arms of the interferometer are discussed using wavepackets. Various correlations can be defined. Both the momentum and the space distributions which are influenced by phase shifters are calculated. Those distributions are marginal distributions of the corresponding Wigner functions. While the momentum distribution can be measured by means of post-selection, the space distribution is directly accessible via the visibility which is expressed by an appropriate correlation function. It is shown that the space distribution behind the first loop can be represented by the visibility which can be determined by measuring neutron intensities behind the second loop.

DOI
10.1088/1464-4266/5/3/375

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In a double-loop neutron interferometer two and three beams are superposed behind the first and second loop respectively. The wavefunctions in different arms of the interferometer are discussed using wavepackets. Various correlations can be defined. Both the momentum and the space distributions which are influenced by phase shifters are calculated. Those distributions are marginal distributions of the corresponding Wigner functions. While the momentum distribution can be measured by means of post-selection, the space distribution is directly accessible via the visibility which is expressed by an appropriate correlation function. It is shown that the space distribution behind the first loop can be represented by the visibility which can be determined by measuring neutron intensities behind the second loop.
Various phase shifters and absorbers can be put into the arms of a double loop neutron interferometer. The mean intensity levels of the forward and diffracted beams behind an empty four plate interferometer of this type have been calculated. It is shown that the intensities in the forward and diffracted direction can be made equal using certain absorbers. In this case the interferometer can be regarded as a 50/50 beam splitter. Furthermore the visibilities of single and double loop interferometers are compared to each other by varying the transmission in the first loop using different absorbers. It can be shown that the visibility becomes exactly 1 using a phase shifter in the second loop. In this case the phase shifter in the second loop must be strongly correlated to the transmission coefficient of the absorber in the first loop. Using such a device homodyne-like measurements of very weak signals should become possible.
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.
The neutron interferometer set-up S18 at the ILL reactor has been used for the first neutron quantum state reconstruction experiment. By a simultaneous measurement of the coherence function and of the modulated momentum distribution behind the interferometer, the Wigner function of various quantum states can be reconstructed. This is equivalent to knowledge of the particle wavefunction. Its spatial dimension follows from the measurement of the coherence function, which is the autocorrelation function of the wavefunction and the shape in momentum space follows from momentum post-selection experiments. Non-orthogonal traces through the ( x , k )-phase space have to be measured for an unambiguous reconstruction. Nearly classical (coherent) and non-classical states in the form of Schrödinger cat-like states have been identified. Comparisons of calculated and measured Wigner functions will be shown.
We analyse the coherence properties of neutron wavepackets, after they have interacted with a phase shifter undergoing different kinds of statistical fluctuation. We give a quantitative (and operational) definition of decoherence and compare it to the standard deviation of the distribution of the phase shifts. We find that in some cases the neutron ensemble is more coherent, even though it has interacted with a wider (i.e. more disordered) distribution of shifts. This feature is independent of the particular definition of decoherence: this is shown by proposing and discussing an alternative definition, based on the Wigner function, that displays a similar behaviour. We briefly discuss the notion of entropy of the shifts and find that, in general, it does not correspond to that of decoherence of the neutron.
The spatial coherence function in all three orthogonal directions has been measured by means of perfect crystal neutron interferometry. It has been demonstrated that the coherence function is given by the Fourier transform of the related momentum distribution, which in turn is determined by the collimation and monochromatization of the beam incident upon and traversing the interferometer. Thus, a measurement of the coherence function can replace a measurement of the momentum distribution in certain cases. Retrieval of the coherence can be accomplished by phase echo and/or postselection methods. A complete retrieval of the coherence is impossible in principle due to unavoidable loss factors.
A two-loop perfect-crystal neutron interferometer was used for the quantitative determination of which-way (WW) information and the verification of the duality relation between predictability and visibility. Our interferometer has two remarkable features, it allows the absorption-free tuning of WW information and it enables the simultaneous measurement of the duality relation without changing the interferometer setup. In addition, the duality relation of three interfering beams is a good demonstration that WW information should not be misused as particle property because one interferometer beam remains in a coherent superposition of two path alternatives, which is not consistent with the particle picture. The coherent WW tuning was performed without changing the neutron's momentum and without dephasing.