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.
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.
The nonlinear evolution of certain quantum systems such as a Gaussian wave packet in an infinite potential well, can lead to fractional revivals of the initial wave packet. Analogously, a coherent state propagating through a Kerr medium can form the so-called Schrödinger cats and cat-like states. Recently, it was shown that a superposition of Schrödinger cats can give rise to sub-Planck structures in the Wigner distribution. The purpose of this review is to introduce and apply, in the above context, a phase space distribution ‘which differs but little from Wigner's and yet, had remained obscure for decades. This is the Kirkwood – Rihaczek (KR) distribution, which has aroused considerable interest in recent years. It is thus appropriate and timely to see how it compares with the Wigner distribution in analysing the above phenomena. Our detailed analysis shows that the KR distribution is just as adequate in explaining these phenomena as the Wigner distribution. Sub-Planck structures are obtained in the KR distribution as well, and despite important differences, the two phase space pictures are comparable.
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) 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-enhanced sensitivity against displacements in all phase-space directions (N 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 N with the total angular momentum.
The position and momentum space information entropies, of the ground state of the P\"oschl-Teller potential, are exactly evaluated and are found to satisfy the bound obtained by Beckner, Bialynicki-Birula, and Mycielski. These entropies for the first excited state, for different strengths of the potential well, are then numerically obtained. Interesting features of the entropy densities, owing their origin to the excited nature of the wave functions, are graphically demonstrated. We then compute the position space entropies of the coherent state of the P\"oschl-Teller potential, which is known to show revival and fractional revival. Time evolution of the coherent state reveals many interesting patterns in the space-time flow of information entropy.
In 2001, Zurek derived the generic minimum scale 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. 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.