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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Wigner function in neutron interferometry

M Suda

Published Oct 1, 1995• 7 citations
Publisher

Abstract

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.

DOI
10.1088/1355-5111/7/5/011

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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.
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.
Neutron interferometry provides a powerful tool to investigate particle and wave features in quantum physics. Single particle interference phenomena can be observed with neutrons and the entanglement of degrees of freedom, i.e., contextuality can be verified and used in further experiments. Entanglement of two photons, or atoms, is analogous to a double slit diffraction of a single photon, neutron or atom. Neutrons are proper tools for testing quantum mechanics because they are massive, they couple to electromagnetic fields due to their magnetic moment, they are subject to all basic interactions, and they are sensitive to topological effects, as well. The 4π-symmetry of spinor wave functions, the spin-superposition law and many topological phenomena can be made visible, thus showing interesting intrinsic features of quantum physics. Related experiments will be discussed. Deterministic and stochastic partial absorption experiments can be described by Bell-type inequalities. Neutron interferometry experiments based on post-selection methods renewed the discussion about quantum non-locality and the quantum measuring process. It has been shown that interference phenomena can be revived even when the overall interference pattern has lost its contrast. This indicates a persisting coupling in phase space even in cases of spatially separated Schrödinger cat-like situations. These states are extremely fragile and sensitive against any kind of fluctuations and other decoherence processes. More complete quantum experiments also show that a complete retrieval of quantum states behind an interaction volume becomes impossible in principle, but where and when a collapse of the wave-field occurs depends on the level of experiment.
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.
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.
A quantum-statistical description of noise in interferometers is given in terms of the Wigner distribution function. The interferometer may contain an amplifier in one of its arms. The input field can be in a variety of states such as a Fock state, a coherent state or a squeezed coherent state. The Wigner function of the output field at the detector is shown to have a general Gaussian form with non-zero complex field amplitude and with other parameters related to the characteristics of the input field, amplifier and beam splitters, etc. An explicit form of the photon-number distribution is given. Higher-order correlations of the output field can be obtained from the Gaussian property of the Wigner function.
Single-particle interference phenomena can be observed with neutrons, and the "entanglement of degrees of freedom", i.e. the contextuality, can be verified and used in further experiments. Entanglement of two photons or atoms is a complementary situation to the double-slit diffraction of a single photon, neutron, or atom. In this respect, neutrons are proper tools for testing quantum mechanics, because they are massive, they couple to electromagnetic fields due to their magnetic moment, and they are subject to all basic interactions, and they are sensitive to topological effects as well. The4π-symmetry of spinor wave functions, the spin-superposition law, and many topological phenomena can be made visible, which shows interesting intrinsic features of quantum physics. Related experiments will be discussed. Deterministic and stochastic partial absorption experiments can be described by Bell-type inequalities. Recent neutron interferometry experiments based on post-selection methods have renewed the discussion about quantum non-locality and the quantum measuring process. It has been shown that interference phenomena can be revived even when the overall interference pattern has lost its contrast. This indicates a persisting coupling in the phase space even in cases of spatially separated Schrödinger cat-like situations. These states are extremely fragile and sensitive against any kind of fluctuations and other de-coherence processes. More complete quantum experiments also show that a complete retrieval of quantum states behind an interaction volume becomes impossible in principle.