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Scientific Reports

Phase-space studies of backscattering diffraction of defective Schrödinger cat states

Damian Kołaczek, Bartłomiej J. Spisak, M. Wołoszyn

Published Jun 2, 2021• 3 citations
Publisher

Abstract

The coherent superposition of two well separated Gaussian wavepackets, with defects caused by their imperfect preparation, is considered within the phase-space approach based on the Wigner distribution function. This generic state is called the defective Schrödinger cat state due to this imperfection which significantly modifies the interference term. Propagation of this state in the phase space is described by the Moyal equation which is solved for the case of a dispersive medium with a Gaussian barrier in the above-barrier reflection regime. Formally, this regime constitutes conditions for backscattering diffraction phenomena. Dynamical quantumness and the degree of localization in the phase space of the considered state as a function of its imperfection are the subject of the performed analysis. The obtained results allow concluding that backscattering communication based on the defective Schrödinger cat states appears to be feasible with existing experimental capabilities.

DOI
10.1038/s41598-021-90738-x

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The coherent superposition of two well separated Gaussian wavepackets, with defects caused by their imperfect preparation, is considered within the phase-space approach based on the Wigner distribution function. This generic state is called the defective Schrödinger cat state due to this imperfection which significantly modifies the interference term. Propagation of this state in the phase space is described by the Moyal equation which is solved for the case of a dispersive medium with a Gaussian barrier in the above-barrier reflection regime. Formally, this regime constitutes conditions for backscattering diffraction phenomena. Dynamical quantumness and the degree of localization in the phase space of the considered state as a function of its imperfection are the subject of the performed analysis. The obtained results allow concluding that backscattering communication based on the defective Schrödinger cat states appears to be feasible with existing experimental capabilities.
Much of the discussion of decoherence has been in terms of a particle moving in one dimension that is placed in an initial superposition state (a Schrödinger "cat" state) corresponding to two widely separated wave packets. Decoherence refers to the destruction of the interference term in the quantum probability function. Here, we stress that a quantitative measure of decoherence depends not only on the specific system being studied but also on whether one is considering coordinate, momentum or phase space. We show that this is best illustrated by considering Wigner phase space where the measure is again different. Analytic results for the time development of the Wigner distribution function for a two-Gaussian Schrodinger "cat" state have been obtained in the high-temperature limit (where decoherence can occur even for negligible dissipation) which facilitates a simple demonstration of our remarks.
We analyze some features of the class of discrete Wigner functions that was recently introduced by Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] to represent quantum states of systems with power-of-prime dimensional Hilbert spaces. We consider ``cat'' states obtained as coherent superpositions of states with positive Wigner function; for such states we show that the oscillations of the discrete Wigner function typically spread over the entire discrete phase space (including the regions where the two interfering states are localized). This is a generic property, which is in sharp contrast with the usual attributes of Wigner functions that make them useful candidates to display the existence of quantum coherence through oscillations. However, it is possible to find subsets of cat states with a natural phase-space representation, in which the oscillatory regions remain localized. We show that this can be done for interesting families of stabilizer states used in quantum error-correcting codes, and illustrate this by analyzing the phase-space representation of the five-qubit error-correcting code.
Non-Gaussian quantum states whose Wigner functions can take negative values are important both for fundamental tests of quantum physics and for the quantum information technologies that have been under active development recently.A typical example of a non-Gaussian state is the so-called Schr odinger's cat state.Its very interesting feature is that its `classical' part (two Gaussian maxima) is geometrically separated from the `nonclassical' part (interference fringes).In this paper, several methodological issues related to these fringes are considered.
The Schrödinger-cat state, the quantum-mechanical superposition of two coherent states, has been investigated extensively due to its many nonclassical features. The properties of Schrödinger-cat state highly depend on the overlap between the two coherent states, i.e. the interference of the two coherent states. It has been shown that the interference is determined by the phase. The schrödinger-cat state can also be consider as the superposition of two coherent states with the same amplitudes, but different phase. In contrast, the superposition of two coherent states with the same phases but different amplitudes will produce the phase-cat states.
The orthogonality of cat and displaced cat states, underlying Heisenberg limited measurement in quantum metrology, is studied in the limit of large number of states. The mesoscopic superposition of the generalized cat state is correlated with the corresponding state overlap function, controlled by the sub-Planck structures arising from phase space interference. The asymptotic expression of this overlap function is evaluated and the validity of large phase space support and distinguishability of the constituent states, in which context the asymptotic limit is achieved, are discussed in detail. For large number of coherent states, uniformly located on a circle, the overlap function significantly matches with the diffraction pattern for a circular ring source with uniform angular strength. This is in accordance with the van Cittert-Zernike theorem, where the overlap function, similar to the mutual coherence function, matches with a diffraction pattern. The physical situation under consideration is delineated in phase space by utilizing Husimi-Q function.
The orthogonality of cat and displaced cat states, underlying Heisenberg limited measurement in quantum metrology, is studied in the limit of a large number of states. The mesoscopic superposition of the generalized cat state is correlated with the corresponding state overlap function, controlled by the sub-Planck structures arising from phase-space interference. The asymptotic expression of this overlap function is evaluated, and the validity of large phase-space support and distinguishability of the constituent states, in which context the asymptotic limit is achieved, are discussed in detail. For a large number of coherent states, uniformly located on a circle, the overlap function significantly matches the diffraction pattern for a circular ring source with uniform angular strength. This is in accordance with the van Cittert--Zernike theorem, where the overlap function, similar to the mutual coherence function, matches a diffraction pattern. The physical situation under consideration is delineated in phase space by utilizing the Husimi QQQ function.