Allen D. Parks

Exact pointer theories for von Neumann projector measurements of pre- and postselected and preselected-only quantum systems: statistical mixtures and weak value persistence

Exact expressions for the mean pointer position, the mean pointer momentum and their variances are obtained for projection operator measurements performed upon ensembles of pre- and postselected (PPS) and preselected-only (PSO) quantum systems. These expressions are valid for any interaction strength which couples a measurement pointer to a quantum system, and consequently should be of general interest to both experimentalists and theoreticians. To account for the ‘collapse’ of PPS states to PSO states that occurs as interaction strength increases and to introduce the concept of ‘weak value persistence’, the exact PPS and PSO pointer theories are combined to provide a pointer theory for statistical mixtures. For the purpose of illustrating ‘weak value persistence’, mixture weights defined in terms of the Bhattacharyya coefficient are used and the statistical mixture theory is applied to mean pointer position data associated with weak value projector measurements obtained from a recent dynamical quantum non-locality detection experiment.

Methods used to observe a dynamical quantum nonlocality effect in a twin Mach–Zehnder interferometer

Straightforward novel methods for stabilizing, tuning, and controlling a twin Mach-Zehnder interferometer for the purpose of observing a subtle dynamical quantum nonlocality effect in a recent optical experiment are presented and discussed. Weak measurements were required for observing a subtle quantum dynamical nonlocality effect that reveals itself in a change of a weak value. Consequently, emphasis is placed upon describing the approaches to apparatus stabilization and interaction strength control between photons and the apparatus. The details discussed in this paper should be of general interest to experimentalists engaging in weak measurement and weak value research.

Weak value amplification of an off-resonance Goos-Hänchen shift in a Kretschmann-Raether surface plasmon resonance device.

During the past two decades there has been increased interest in the optical excitation of surface plasmon resonance (SPR) at a metal-dielectric interface. This is due in large part to its potential applications in such areas as medical diagnostics and pharmaceutical research. Also occurring during this time has been a growing recognition by the quantum physics community that weak value amplification (WVA) can serve as a valuable metrological research resource. Recently WVA has been used to amplify very small optical Goos-Hänchen (GH) shifts in glass and it has also been shown that SPR can greatly enhance optical GH shifts at the metal/air interface in Kretschmann-Raether (KR) devices. This paper demonstrates experimentally the WVA of an off-resonance GH shift in a KR device and explains why WVA of sufficiently SPR enhanced optical GH shifts cannot be achieved.

Weak value amplification of an optical Faraday differential refraction effect

In the presence of a longitudinal magnetic field B, a beam of linearly polarized light incident from a Faraday medium of Verdet constant V refracts at its interface with a medium of negligible Verdet constant and emerges as two opposite circularly polarized beams that are separated by a small divergence angle δ that is proportional to the product BV. Judicious postselection of the polarization state of the emergent light can be used to amplify the measured value of δ by several orders of magnitude. This technique makes it possible to optically measure either very small V values when B is known or small magnetic fields when V is known.

Weak Energy: Form and Function

The equation of motion for a time-dependent weak value of a quantum mechanical observable contains a complex valued energy factor—the weak energy of evolution. This quantity is defined by the dynamics of the pre-selected and post-selected states which specify the observable’s weak value. It is shown that this energy: (i) is manifested as dynamical and geometric phases that govern the evolution of the weak value during the measurement process; (ii) satisfies the Euler-Lagrange equations when expressed in terms of Pancharatnam (P) phase and Fubini-Study (FS) metric distance; (iii) provides for a PFS stationary action principle for quantum state evolution; (iv) time translates correlation amplitudes; (v) generalizes the temporal persistence of state normalization; and (vi) obeys a time-energy uncertainty relation. A similar complex valued quantity—the pointed weak energy of an evolving quantum state—is also defined and several of its properties in PFS coordinates are discussed. It is shown that the imaginary part of the pointed weak energy governs the state’s survival probability and its real part is—to within a sign—the Mukunda-Simon geometric phase for arbitrary evolutions or the Aharonov-Anandan (AA) geometric phase for cyclic evolutions. Pointed weak energy gauge transformations and the PFS 1-form are defined and discussed and the relationship between the PFS 1-form and the AA connection 1-form is established. [Editors note: for a video of the talk given by Prof. Parks at the Aharonov-80 conference in 2012 at Chapman University, see quantum.chapman.edu/talk-25.]

A note concerning the modular valued von Neumann interaction operator

The modular valued operator

Persistent homology in graph power filtrations

{\hat{V}}_m

Method and apparatus for clock synchronization using quantum mechanical non-locality effects

V^m of the von Neumann interaction operator for a projector is defined. The properties of