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arXiv:1008.0630 · Physical Review A

Mesoscopic quantum superposition of the generalized cat state: A diffraction limit

Suranjana Ghosh, Raman Sharma, Utpal Roy, Prasanta Kumar Panigrahi

Published Nov 6, 2015• 13 citations
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Abstract

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 $Q$ function.

DOI
10.1103/physreva.92.053819

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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.
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 large number of states. The asymptotic expression for the corresponding state overlap function, controlled by the sub-Planck structures arising from phase space interference, is obtained exactly. For large number of coherent states, uniformly located on a circle, it identically matches with the diffraction pattern for a circular ring with uniform angular source strength. This is in accordence with the van Cittert-Zernike theorem, where the overlap function, similar to the mutual coherence function matches with a diffraction pattern. Interestingly, the convergence to asymptotic limit is quite rapid, reminiscent of the convergence of superposed incoherent sources.
The superposition of coherent states has attracted extensive and intensive attention, exemplified by intriguing instances such as Schr\"odinger cat states and compass states. In this study, we derive analytical expressions of the Wigner functions and phase-space displacement sensitivity of circular states, which are superpositions of coherent states distributed equidistantly on a circle. The visualization of these phase-space functions exhibits fascinating quantum kaleidoscopic patterns, and intuitively reveals the internal coherence and symmetric structure in these states. Moreover, we unveil the inherent sub-Planck structure of circular states in terms of the zeros of sensitivity functions.
We propose the cavity quantum electrodynamics method, using dispersive interaction between atoms and a high-quality cavity to realize the mesoscopic superposition of coherent states that would exhibit sub-Planck structures in phase space, i.e., the structures at a scale smaller than the Plank's constant $(\ensuremath{\hbar})$. These structures are direct signatures of quantum coherence and are formed as a result of interference between the two superposed cat states. In particular we focus on a superposition involving four coherent states. We show interferences in the conditional measurements involving two atoms.
Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.
Weyl functions are shown to be an important tool in quantum phase-space studies. Their properties are studied and relations with other quantities are derived. The use of Weyl functions for the understanding of quantum interference phenomena is discussed. The general theory is applied to superpositions of mmm coherent states uniformly distributed on a circle (generalized Schr\"odinger cats). The properties of these states are explored and their interference behavior is discussed, using Weyl functions.