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arXiv:1007.3983

Diffraction limit of the sub-Planck structures

Raman Kumar Sharma, Prasanta Kumar Panigrahi

Published Jul 22, 2010• 0 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 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.

DOI
10.48550/arxiv.1007.3983

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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 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 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 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.
Coherent-state superpositions are of great importance for many quantum subjects, ranging from foundational to technological, e.g., from tests of collapse models to quantum metrology. Here we explore various aspects of these states, related to the connection between sub-Planck structures present in their Wigner function and their sensitivity to displacements (ultimately determining their metrological potential). We review this for the usual Heisenberg-Weyl algebra associated with a harmonic oscillator and extend it to find analogous results for the su(2)\mathrm{su}(2)su(2) algebra, typically associated with angular momentum. In particular, in the Heisenberg-Weyl case, we identify phase-space structures with support smaller than the Planck action in both Schr\"odinger-cat-state mixtures and superpositions, the latter known as compass states. However, as compared to coherent states, compass states are shown to have N\sqrt{N}N​-enhanced sensitivity against displacements in all phase-space directions (NNN is the average number of quanta), whereas cat states and cat mixtures show such enhanced sensitivity only for displacements in specific directions. We then show that these same properties apply for analogous SU(2) states provided (i) coherent states are restricted to the equator of the sphere that plays the role of phase space for this group, (ii) we associate the role of the Planck action with the size of SU(2) coherent states in such a sphere, and (iii) we associate the role of NNN with the total angular momentum.
We show that entanglement of two superposed coherent states produces sub-Planck structures in phase space. The origin of these structures depends entirely on entanglement, unlike those in Zurek [Nature (London) 412, 712 (2001)], where the effect of entanglement was not considered. The constituent states are sensitive enough to carry out the Heisenberg sensitive measurements. However, their usefulness is limited in providing high sensitivity in either the position (x)(x)(x) direction or in the momentum (p)(p)(p) direction of the phase-space. Interestingly, the entanglement causes the proposed states to be sensitive in both xxx and ppp directions. This state offers a better sensitivity compared to the single partite compass state in Zurek [Nature (London) 412, 712 (2001)] in carrying out precision measurements. It is also argued that such states are easy to create and are better suited for carrying out precision measurements.