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Physica Scripta

Quantum states with minimum phase uncertainty for the Süssmann measure

B. Daeubler, C. A. Miller, H. Risken, L. Schoendorff

Published Jan 1, 1993• 19 citations
Publisher

Abstract

We investigate those quantum states, which lead to minimum phase uncertainty for a fixed mean intensity. In contrast to the work of Bandilla, Paul, and Ritze (BPR) we employ instead of the usual variance measure for the phase the Süssmann measure. It turns out that our minimal states are of the form |ψ⟩ ∝ ∑ γ n | n ⟩. Because of this simple expression the phase distribution can be given analytically. It is shown that this distribution is about twice as peaked than the BPR phase minimum states. The Q-function is also obtained. It shows an asymmetry whereas the BPR states lead to a Q-function with a nearly symmetric ellipsoidal shape.

DOI
10.1088/0031-8949/1993/t48/018

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We investigate those quantum states, which lead to minimum phase uncertainty for a fixed mean intensity. In contrast to the work of Bandilla, Paul, and Ritze (BPR) we employ instead of the usual variance measure for the phase the Süssmann measure. It turns out that our minimal states are of the form |ψ⟩ ∝ ∑ γ n | n ⟩. Because of this simple expression the phase distribution can be given analytically. It is shown that this distribution is about twice as peaked than the BPR phase minimum states. The Q-function is also obtained. It shows an asymmetry whereas the BPR states lead to a Q-function with a nearly symmetric ellipsoidal shape.
The minimization problem of finding the number-phase minimum uncertainty states (MUS) is considered and its solutions are found either numerically or, under some special conditions, analytically. The phase uncertainty measure is based on the Bandilla-Paul dispersion. The problem is treated (i) in a finite-dimensional Hilbert space and (ii) for a countably infinite-dimensional Hilbert space (i.e. the standard quantum harmonic oscillator), with the constraint of a given mean photon number. The MUS relations between the photon number uncertainty and phase uncertainty are presented. Connections to some other minimization problems are discussed.
Starting from a convenient measure for the phase uncertainty delta phi , a rigorous solution to the variational problem in which ( delta phi )2 is minimized under the constraint that the mean photon number N is held fixed, was found. In an approximation valid for large values of N, closed-form expressions for both ( delta phi )2 and the variance of the photon number ( Delta n)2 were obtained. In the final result, the approximation is very similar to an approximation scheme recently discussed by Summy and Pegg Opt. Commun., vol.77, p.75, 1990. The phase uncertainty for the phase optimized states is found to decrease as (N+1)-1 with growing N.
Recent theoretical studies have considered various possibilities for experimental measurements of phase using quantum states of a finite mean number of photons. Currently, the most experimentally accessible of these possibilities is an interferometric phase measurement using a squeezed state that has minimum variance for the phase quadrature; this possibility can yield, in principle, a phase measurement with uncertainty of order l/(mean number of photons). Shapiro, Shepard, and Wong1 (SSW) have proposed another possibility. They maximize the peak likelihood in an abstract measurement of the Susskind-Glogower phase operator, and they claim a phase sensitivity that goes as l/(mean number of photons).2 The SSW phase probability distribution consists of an extremely narrow peak sitting on a broad base that covers the entire 2 radians of phase; as a result, a single SSW phase measurement has poor sensitivity. Shapiro, Shepard, and Wong argue that their claimed sensitivity can be achieved in a sequence of measurements. We study the statistics of the maximum-likelihood estimator in a sequence of SSW phase measurements, and we report how the uncertainty in the estimator scales with the mean number of photons used in the entire sequence.
Based on the statistical concept of the median, we propose a quantum uncertainty relation between semi-interquartile ranges of the position and momentum distributions of arbitrary quantum states. The relation is universal, unlike that based on the mean and standard deviation, as the latter may become non-existent or ineffective in certain cases. We show that the median-based one is not saturated for Gaussian distributions in position. Instead, the Cauchy-Lorentz distributions in position turn out to be the one with the minimal uncertainty, among the states inspected, implying that the minimum-uncertainty state is not unique but depends on the measure of spread used. Even the ordering of the states with respect to the distance from the minimum uncertainty state is altered by a change in the measure. We invoke the completeness of Hermite polynomials in the space of all quantum states to probe the median-based relation. The results have potential applications in a variety of studies including those on the quantum-to-classical boundary and on quantum cryptography.