[Paper Review] Super-phenomena in arbitrary quantum observables

Andrew N. Jordan, Yakir Aharonov+7 more|arXiv (Cornell University)|Sep 12, 2022

Quantum Mechanics and Applications4 citations

TL;DR

This paper generalizes superoscillations—normally observed in band-limited functions—to arbitrary quantum observables via weak values. By preselecting a superposition of eigenstates with bounded eigenvalues and postselecting on position, the weak value of any operator can exceed its eigenvalue range, leading to 'superbehavior' such as superenergy or superangular momentum. The key result is a mathematical construction of a finite-energy quantum state built entirely from states with asymptotically zero energy, demonstrating 'energy out of nothing' in the limit of large superposition size.

ABSTRACT

Superoscillations occur when a globally band-limited function locally oscillates faster than its highest Fourier coefficient. We generalize this effect to arbitrary quantum mechanical operators as a weak value, where the preselected state is a superposition of eigenstates of the operator with eigenvalues bounded to a range, and the postselection state is a local position. Superbehavior of this operator occurs whenever the operator's weak value exceeds its eigenvalue bound. We give illustrative examples of this effect for total angular momentum and energy. In the later case, we demonstrate a sequence of harmonic oscillator potentials where a finite energy state converges everywhere on the real line, using only bounded superpositions of states whose asymptotic energy vanishes - "energy out of nothing". This limit requires postselecting the particle in a region whose size diverges in the considered limit. We further show that superenergy behavior implies that the state superoscillates in time with a rate given by the superenergy divided by the reduced Planck's constant. This example demonstrates the possibility of mimicking a high-energy state with coherent superpositions of nearly zero-energy states for as wide a spatial region as desired. We provide numerical evidence of these features to further bolster and elucidate our claims.

Motivation & Objective

• To extend the concept of superoscillations beyond momentum to any quantum observable using weak values.
• To address the open question of whether superoscillatory behavior can be generalized to other observables like energy and angular momentum.
• To construct a sequence of quantum states where a finite-energy state emerges in the limit from superpositions of states with vanishing asymptotic energy.
• To demonstrate that superbehavior in space implies superoscillations in time, with a frequency proportional to the superenergy.
• To provide a mathematical and numerical foundation for applying superoscillations to differential operators and spectral theory beyond quantum mechanics.

Proposed method

• Define superbehavior as the weak value of an arbitrary operator exceeding its eigenvalue bounds when postselected on position.
• Use preselected states as superpositions of eigenstates of the observable with eigenvalues confined to a bounded interval.
• Apply the formalism of weak values in quantum mechanics to compute the effective operator behavior at specific positions.
• Construct a sequence of harmonic oscillator potentials where the energy eigenvalues of the superposed states tend to zero in the limit.
• Use spectral decomposition and asymptotic analysis to show convergence of the superposition to a finite-energy wavefunction over the entire real line.
• Perform numerical simulations to verify convergence of the wavefunction and the emergence of superoscillations in time, particularly for the harmonic oscillator case.

Experimental results

Research questions

• RQ1 Can superoscillations be generalized from momentum to arbitrary quantum observables using weak values?
• RQ2 Is it possible to construct a finite-energy quantum state from a superposition of states each with asymptotically zero energy?
• RQ3 Does superbehavior in space imply superoscillations in time, and if so, with what frequency?
• RQ4 What is the relationship between the spatial superenergy and the temporal oscillation rate of the wavefunction?
• RQ5 Can superoscillatory behavior be extended to differential operators and spectral theory beyond quantum mechanics?

Key findings

• A finite-energy state can be constructed in the limit of a superposition of states with asymptotically vanishing energy, demonstrating 'energy out of nothing' in a mathematically rigorous way.
• The weak value of the energy operator exceeds its eigenvalue bounds in a spatial region, defining a point of superenergy behavior.
• The wavefunction converges everywhere on the real line to a finite-energy state, even though each component state has zero asymptotic energy.
• Superenergy behavior implies superoscillations in time with a frequency given by the superenergy divided by ℏ, confirmed numerically for the harmonic oscillator.
• The convergence of the superposition to a plane wave in time is observed numerically, with good agreement for N = 1000, showing stable superoscillations near t = 0.
• The region of time where superoscillations occur expands with increasing N, suggesting a broad temporal range of validity for the superenergy effect.