Superoscillations and the quantum potential ^*

Abstract

For quantum or other waves that are band-limited, the quantum potential in the Madelung–Bohm representation vanishes on the boundaries of regions where the waves are superoscillatory (i.e. where they vary faster than any of their Fourier components). This connection is illustrated by calculations of the quantum potential zeros for a superoscillatory superposition of plane waves, and by Aharonov–Bohm waves.

Broadband Asymmetric Light Transport in Compact Lithium Niobate Waveguides

“…To realize the mode transfer in a more compact device, we apply the shortcut to adiabaticity (STA) to accelerate the mode evolution. First proposed by M. Berry, [ 27 ] the transitionless driving is a kind of STA method that can suppress the instantaneous eigenmode coupling induced by fast parameter manipulation via adding counterdiabatic driving terms. [ 27–31,33 ] The transitionless driving method has been adopted in quantum systems to speed up the state evolution, [ 29,30 ] but it requires complex coupling modulation (i.e., both real and imaginary coupling) and induces long‐range couplings, [ 29 ] which are difficult to be implemented in LNOI‐coupled waveguides.…”

“…First proposed by M. Berry, [ 27 ] the transitionless driving is a kind of STA method that can suppress the instantaneous eigenmode coupling induced by fast parameter manipulation via adding counterdiabatic driving terms. [ 27–31,33 ] The transitionless driving method has been adopted in quantum systems to speed up the state evolution, [ 29,30 ] but it requires complex coupling modulation (i.e., both real and imaginary coupling) and induces long‐range couplings, [ 29 ] which are difficult to be implemented in LNOI‐coupled waveguides. Here, we circumvent the difficulty by the following steps: First, we have noticed that the original Hamiltonian H 0 ( z ) satisfies the so‐called one‐photon resonance condition (i.e., the mode constants of waveguides I‐III are equal), which allows one to map the three‐level system H 0 ( z ) to an effective two‐level one $${H_{{\mathrm{eff}}}}( z ) = [ { \def\eqcellsep{&}\begin{array}{cc} {{c_2}(z)/2}&{{c_1}(z)/2}\\ {{c_1}(z)/2}&{ - {c_2}(z)/2} \end{array} } ]$$ (refs.…”

Quantum potential in time-dependent supersymmetric quantum mechanics

“…However Planck's constant is very small, but not zero, and this term can be regarded as an additional self-generated potential that the classical particle experiences to give it its quantum nature. Furthermore, a number of authors have argued that a quantum particle behaves at its most classical in places where the quantum potential is zero [8][9][10]. From the point of view of the current work, the key thing to note from this formalism can be seen in the second of equations ( 2).…”

“…If this is not possible the calculation may be mathematically interesting, but there is very unlikely to be any physical applications of the results. Next we calculate Â0 (x, t) from equation (10) and then Â(x, t) from equation ( 9). Finally we find φ(x, t) from equation ( 8) or (11) and that is the wavefunction corresponding to the potential V 1 (x, t).…”

“…[9] have shown that wavefunctions with fold caustics are the most classical because the zeros of the quantum potential coincide with the caustic and the evolution of the caustic is governed by the Hamilton-Jacobi equation. Berry [10] has shown that, for quantum wavepackets, the Bohm potential vanishes on the boundaries of regions where the oscillations become superoscillatory. In a connection with relativistic quantum theory Salesi et.…”