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arXiv:1704.08174 · Physical Review A

Structures far below the sub-Planck scale in quantum phase space through superoscillations

Maxime Oliva, Ole Steuernagel

Published May 15, 2017• 5 citations
arXivHTMLPDF

Abstract

In 2001, Zurek derived the generic minimum scale ${a}_{Z}$ for the area of structures of Wigner's quantum phase distribution. Here we show by construction, using superoscillatory functions, that the Wigner distribution can locally show regular spotty structures on scales much below Zurek's scale ${a}_{Z}$. The price to pay for the presence of such structures is their exponential smallness. For the case we construct, there is no increased interferometric sensitivity from the presence of patches with superoscillatory structure in phase space.

DOI
10.1103/physreva.95.052112

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In 2001, Zurek derived the generic minimum scale aZ{a}_{Z}aZ​ for the area of structures of Wigner's quantum phase distribution. Here we show by construction, using superoscillatory functions, that the Wigner distribution can locally show regular spotty structures on scales much below Zurek's scale aZ{a}_{Z}aZ​. The price to pay for the presence of such structures is their exponential smallness. For the case we construct, there is no increased interferometric sensitivity from the presence of patches with superoscillatory structure in phase space.
We find the existence of sub-Planck scale structures in the P\"oschl-Teller potential, which is an exactly solvable potential with both symmetric and asymmetric features. We analyze these structures in both cases by looking at the Wigner distribution of the state evolved from an initial coherent state up to various fractional revival times. We also investigate the sensitivity to perturbations of the P\"oschl-Teller potential and we verify that, similar to the harmonic oscillator, the presence of sub-Planck structure in phase space is responsible for a high sensitivity to phase-space displacements.
Quantum states extended over a large volume in phase space have oscillations from quantum interferences in their Wigner distribution on scales smaller than variant Planck's over 2pi [W. H. Zurek, Nature (London) 412, 712 (2001)]]. We investigate the influence of those sub-Planck-scale structures on the sensitivity to an external perturbation of the state's time evolution. While we do find an accelerated decay of the Loschmidt Echo for an extended state in comparison to a localized wave packet, the acceleration is described entirely by the classical Lyapunov exponent and hence cannot originate from quantum interference.
Superoscillations, namely regions where a band-limited function f (x) varies faster than the fastest of its Fourier components k, generate the illusion that the Fourier content is ?supershifted? so as to lie outside the spectrum of the function. The relation between supershifts and superoscillations, central to the quantum weak measurements scheme, is explored in terms of two different representations of the local Fourier transform in the ?phase space? (x, k). The Wigner function W(x, k), regarded as a function of k for fixed x, inherits the band-limited property of f (x). Neverthless, its local k average can lie outside the spectrum because W, although real, posesses negative values. The local Wigner average of k equals the local wavenumber at x (local weak value of momentum), defined as the phase variation k loc(x)?=?? x arg f (x). By contrast, the Husimi function H(x, k), i.e. the windowed Fourier transform with window width L, corresponding to squeezing of the coherent state associated with (x, k) (and representing the pointer wavefunction after a weak measurement), is positive-definite. But it is not band-limited, and the local Husimi average of k equals k loc if L is small enough. These properties are illustrated numerically with two superoscillatory functions.
Abstract Superoscillatory wave forms, i.e., waves that locally oscillate faster than their highest Fourier component, possess unusual properties that make them of great interest from quantum mechanics to signal processing. However, the more pronounced the desired superoscillatory behavior is to be, the more difficult it becomes to produce, or even only calculate, such highly fine-tuned wave forms in practice. Here, we investigate how this sensitivity to preparation errors scales for a method for constructing superoscillatory functions which is optimal in the sense that it minimizes the energetic expense. We thereby also arrive at very accurate approximations of functions which are so highly superoscillatory that they cannot be calculated numerically. We then investigate to what extent the scaling and sensitivity results for superoscillatory functions on the real line extend to the experimentally important case of superoscillatory functions that are periodic.
The Wigner function of the compass state (a superposition of four coherent states) develops phase-space structures of dimension much less than the Planck scale $\ensuremath{\hbar}$, which are crucial in determining the sensitivity of these states to phase-space displacements. In the present work we introduce compasslike states that may have connection to the contemporary experiments, which are obtained by either adding photons to or subtracting photons from the superposition of two squeezed-vacuum states. We show that when a significant quantity of photons is added (or subtracted), the Wigner functions of these states are shown to have phase-space structures of an area that is substantially smaller than the Planck scale. In addition, these states exhibit sensitivity to displacements that is much higher than the standard quantum limit. Finally, we show that both the size of the sub-Planck structures and the sensitivity of our states are strongly influenced by the average photon number, with the photon-addition case having a higher average photon number leading to the smaller sub-Planck structures and, consequently, being more sensitive to displacement than the photon-subtraction case. Our states offer unprecedented resolution to the external perturbations, making them suitable for quantum sensing applications.
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.