Pointer supershifts and superoscillations in weak measurements

Michael Berry, Pragya Shukla

The association between large shifts of a pointer in a weak measurement and fast oscillations in an associated function involving the pre- and post-selected states has been clarified in a recent paper (Aharonov et al 2011 J. Phys. A: Math. Theor . 44 365304 ). Here we explore the association further for the case of an observable with N discrete eigenvalues, by calculating and illustrating how the supershift emerges, even for N = 2, as the uncertainty in the pointer position increases. This happens if the initial pointer wavefunction is Gaussian or Lorentzian but not if it is exponential.

Andrew N. Jordan, John C. Howell, A. Nick Vamivakas +1 more

Michael Berry, Nimrod Moiseyev

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.

Allen Lobo, Carlos Alberto de Lima Ribeiro

Yakir Aharonov, Jussi Behrndt, Fabrizio Colombo +1 more

Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time-dependent Schrödinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper, we give a unified approach to determine the supershift property for the solution of the time-dependent one-dimensional Schrödinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Green's function, but not its explicit form. With this efficient general technique, we are able to treat various potentials.

Erik Sjöqvist