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Physical review. B, Condensed matter

Uncertainty relation in fluctuations in the space and time domains in disordered optical media

Makoto Tomita, Kenshi Shimano, Kazunari Nakata

Published Aug 1, 1996• 6 citations
Publisher

Abstract

We confirmed that the uncertainty relation between time and energy is established in fluctuations in the space and time domains that appear when coherent optical pulses propagate through a disordered static medium. The widths of temporal and spectral correlation functions of these fluctuations satisfy the relation $\ensuremath{\delta}\ensuremath{\tau}\ensuremath{\delta}\ensuremath{\omega}\ensuremath{\sim}1$, that is, $\ensuremath{\delta}\ensuremath{\tau}\ensuremath{\delta}E\ensuremath{\sim}\ensuremath{\hbar}$. The number of fluctuations is discussed on the basis of the uncertainty relation.

DOI
10.1103/physrevb.54.r3687

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We confirmed that the uncertainty relation between time and energy is established in fluctuations in the space and time domains that appear when coherent optical pulses propagate through a disordered static medium. The widths of temporal and spectral correlation functions of these fluctuations satisfy the relation \ensuremathδ\ensuremathτ\ensuremathδ\ensuremathω\ensuremath∼1\ensuremath{\delta}\ensuremath{\tau}\ensuremath{\delta}\ensuremath{\omega}\ensuremath{\sim}1\ensuremathδ\ensuremathτ\ensuremathδ\ensuremathω\ensuremath∼1, that is, \ensuremathδ\ensuremathτ\ensuremathδE\ensuremath∼\ensuremathℏ\ensuremath{\delta}\ensuremath{\tau}\ensuremath{\delta}E\ensuremath{\sim}\ensuremath{\hbar}\ensuremathδ\ensuremathτ\ensuremathδE\ensuremath∼\ensuremathℏ. The number of fluctuations is discussed on the basis of the uncertainty relation.
We investigate the fluctuations of light transmitted through static disordered media in time and frequency domains. When the pulse duration of the incident light is much shorter than the typical time of flight through the medium, fluctuations appear both in time and frequency domains and the uncertainty relation, δτδω∼1 or δτδE∼ℏ\delta \tau \delta E \sim \hbarδτδE∼ℏ, holds good in these fluctuations. We discuss the necessary condition to observe this relation. Precise calculations of the correlation functions also reveal that the exact shape and width of the correlation functions depend on the pulse duration and the observation time. Experimental results with picosecond pulses through thick and thin samples show good agreement with the theoretical calculations.
Light is known to exhibit quantum uncertainty in terms of its amplitude, phase, and polarization. However, quantum uncertainty related to coherence, which is also a fundamental physical property of light, remains to be considered. Here, we formulate and explore the concept of quantum optical coherence uncertainty. We focus on the first-order coherence of the simplest possible light field, a purely monochromatic plane wave, which is classically completely stable. Starting from a scalar treatment, we show that the field displays zero coherence uncertainty only for a number state. We then proceed to the vectorial regime and establish that any state leads to coherence fluctuations, governed by a set of uncertainty relations depending on the polarization state and space-time points. Our work thus provides fundamental insights into the quantum character of optical coherence, with potential benefits in applications using highly sensitive interferometric and polarimetric techniques.
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We analyze some interesting and peculiar phenomena that appear both in the well-known classical domain and in the quantum domain, related to the behavior of the visibility in interference experiments. One can show that, according to its usual definition, the visibility is not a monotonic function of appropriate quantities, such as the optical path difference. In the quantum domain, the visibility can be used to infer the quantum mechanical coherence properties of a quantum particle, like for example a neutron, interacting with a fluctuating environment. In some cases the wave function is more coherent even though it has interacted with a more disordered medium. We will refer to these results as anomalous, because they are against naive expectations.