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Cambridge University Press eBooks

Uncertainty

Patrick Flandrin

Published Aug 22, 2018• 0 citations
Publisher

Abstract

Gaussian functions happen also to be central with respect to the intrinsic conflict that exists between time and frequency and which is referred to as “uncertainty.” This can take various forms depending upon the chosen measure of spread or occupancy in both domains. The striking feature is that most of the intuitive measures such as dispersion or entropy end up with Gaussians as minimizers, making them “quanta” of information (“logons” in Gabor's terminology).

DOI
10.1017/9781108363181.007

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Gaussian functions happen also to be central with respect to the intrinsic conflict that exists between time and frequency and which is referred to as “uncertainty.” This can take various forms depending upon the chosen measure of spread or occupancy in both domains. The striking feature is that most of the intuitive measures such as dispersion or entropy end up with Gaussians as minimizers, making them “quanta” of information (“logons” in Gabor's terminology).
The spreading of the stationary states of the multidimensional single-particle systems with a central potential is quantified by means of Heisenberg-like measures (radial and logarithmic expectation values) and entropy-like quantities (Fisher, Shannon, Rényi) of position and momentum probability densities. Since the potential is assumed to be analytically unknown, these dispersion and information-theoretical measures are given by means of inequality-type relations which are explicitly shown to depend on dimensionality and state's angular hyperquantum numbers. The spherical-symmetry and spin effects on these spreading properties are obtained by use of various integral inequalities (Daubechies-Thakkar, Lieb-Thirring, Redheffer-Weyl, ...) and a variational approach based on the extremization of entropy-like measures. Emphasis is placed on the uncertainty relations, upon which the essential reason of the probabilistic theory of quantum systems relies.
We study quantum gravimetry when the interrogation time carries intrinsic uncertainty, motivated by a fundamental limit on temporal resolution associated with the energy--time uncertainty relation. For linearly gravity-coupled gravimeters, we obtain the effective gravity information by profiling the interrogation time from the two-parameter quantum Fisher information (QFI) matrix. In this class, the time-information block is quadratic in the gravitational parameter, and for quadratic background dynamics, the gravity--time cross term becomes affine in ggg. These properties yield a normalized expression for the fraction of standard single-parameter gravity QFI that remains once interrogation time is treated as a nuisance parameter, with an affine numerator and a Lorentzian denominator. We work out these results in three benchmark models: a freely falling Gaussian wavepacket, the Kasevich--Chu light-pulse atom interferometer, and an idealized closed-unitary optomechanical model. The Gaussian free-fall benchmark yields an exact closed-form expression for the effective gravity information and shows explicitly how nuisance-time profiling suppresses the momentum-spread-dependent part of the standard single-parameter gravity QFI. In the Kasevich--Chu interferometer, internal state population readout gives a rank-deficient measured two-parameter geometry unless independent timing information is supplied, whereas full access to the final motional and internal states restores a full-rank geometry with retention controlled by the competition between initial velocity spread and gravitationally accumulated motion. In atom-interferometric benchmarks, the framework yields explicit conditions for minimizing nuisance-time information loss, together with corresponding constraints on momentum spread, spatial localization, and long-interrogation-time operation.