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arXiv:quant-ph/0402105 · Journal of Optics B Quantum and Semiclassical Optics

Nonclassicality of thermal radiation

Lars M. Johansen

Published Oct 16, 2004• 23 citations
arXivHTMLPDF

Abstract

It is demonstrated that stochastic c-number theories fail as a model of weak measurements on thermal radiation of low occupation numbers. This occurs, for example, at the wavelength of maximum irradiation in blackbody radiation. The classical model is approximately capable of describing weak measurements at high occupation numbers. The nonclassical effect is only present for quantum states with a negative Terletsky–Margenau–Hill quasi-probability distribution.

DOI
10.1088/1464-4266/6/12/l01

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It is demonstrated that stochastic c-number theories fail as a model of weak measurements on thermal radiation of low occupation numbers. This occurs, for example, at the wavelength of maximum irradiation in blackbody radiation. The classical model is approximately capable of describing weak measurements at high occupation numbers. The nonclassical effect is only present for quantum states with a negative Terletsky–Margenau–Hill quasi-probability distribution.
We examine weak measurements of arbitrary observables where the object is prepared in a mixed state and on which measurements with imperfect detectors are made. The weak value of an observable can be expressed as a conditional expectation value over an infinite class of different generalized Kirkwood quasiprobability distributions. ``Strange'' weak values for which the real part exceeds the eigenvalue spectrum of the observable can only be found if the Terletsky-Margenau-Hill distribution is negative or, equivalently, if the real part of the weak value of the density operator is negative. We find that a classical model of a weak measurement exists whenever the Terletsky-Margenau-Hill representation of the observable equals the classical representation of the observable and the Terletsky-Margenau-Hill distribution is non-negative. Strange weak values alone are not sufficient to obtain a contradiction with classical models. We propose feasible weak measurements of photon number of the radiation field. Negative weak values of energy contradict all classical stochastic models, whereas negative weak values of photon number contradict all classical stochastic models where the energy is bounded from below by the zero-point energy. We examine coherent states in particular and find negative weak values with probabilities of 16% for kinetic energy (or squared field quadrature), 8% for harmonic oscillator energy, and 50% for photon number. These experiments are robust against detector inefficiency and thermal noise.
The photon-number distribution for the gray-body radiation given by Bekenstein and Schiffer [Phys. Rev. Lett. 72, 2512 (1994)] is used to study its nonclassical effects. It is shown that as the absorptivity varies from 0 to 1, the gray-body state changes from an extremely nonclassical state to an extremely classical one. The critical point of this transition is located. It is discovered that the negative binomial state is a special case of the gray-body radiation at the transition point; hence, it is an example of borderline states such as the coherent state. The explicit expressions for the factorial moments, the Q parameter, the P function, and the nonclassical depth of the gray-body state as functions of absorptivity and temperature are derived.
According to Born's rule quantum probabilities are given by the overlap between the system state and measurement states in a quite symmetrical way. This means that both contribute to any observed nonclassical effect that is usually attributed just to the observed light state. This is relevant since typical measurement are highly nonclassical by themselves, such as number states and quadrature eigenstates. We show that nonclassical effects only arise provided that the measurement is itself nonclassical. Otherwise there is a classical-like model accounting for the observed statistics.
As we have seen in the previous chapters, there are quantum fluctuations associated with the states corresponding to classically welldefined electromagnetic fields. The general description of fluctuation phenomena requires the density operator. However, it is possible to give an alternative but equivalent description in terms of distribution functions. In the present chapter, we extend our treatment of quantum statistical phenomena by developing the theory of quasi-classical distributions. This is of interest for several reasons. First of all, the extension of the quantum theory of radiation to involve nonquantum stochastic effects such as thermal fluctuations is needed. This is an important ingredient in the theory of partial coherence. Furthermore, the interface between classical and quantum physics is elucidated by the use of such distributions. The arch type example being the Wigner distribution. In this chapter, we introduce various distribution functions. These include the coherent state representation or the Glauber–Sudarshan Prepresentation. The P -representation is used to evaluate the normally ordered correlation functions of the field operators. As we shall see in the next chapter, the P -representation forms a correspondence between the quantum and the classical coherence theory. This distribution function does not have all of the properties of the classical distribution functions for certain states of the field, e.g., it can be negative. We also discuss the so-called Q -representation associated with the antinormally ordered correlation functions. Other distribution functions and their properties are also presented.
Quasiprobability distributions offer a useful way of describing nonclassical features of quantum systems through their departure from classical probability theory. In this work, we investigate the quantification of nonclassicality associated with the Margenau--Hill quasiprobability (MHQ) distribution using its moments, without requiring reconstruction of the full distribution. First we introduce the logarithmic MHQ negativity as a quantifier of nonclassicality, and then derive a hierarchy of rigorous lower bounds in terms of low-order moments. Within the resulting hierarchy, the fourth-moment bound is the strongest among the bounds based on even moments. For qubits, we further derive a tight upper bound on the exact MHQ negativity and identify the corresponding extremal state. We also obtain an optimal positivity threshold based on the fourth moment for arbitrary pairs of qubit observables. Finally, we show that the relevant moments admit exact multicopy representations, providing a route to their estimation using interferometry or classical shadow techniques. Our results establish a moment-based framework for extracting quantitative information about MHQ negativity from a finite set of low-order observables.