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Acta Physica Sinica

Quantum description of interference effect with two holes

Zhixin Yao, Zhong Jian-Wei, Mao Bang-Ning, Gang Chen, Bailiang Pan, 浙江大学物理学系，杭州 310027

Published Jan 1, 2007• 4 citations
Publisher

Abstract

By using one-dimensional photon state-vector function obtained previously, a general analytic expression is deduced for the optical two-hole interference experiment, which not only greatly enlarges the range of the observable, but also completely represents the probability distribution for a photon in the whole space. On the basis of the basic rules of quantum optics, we quantitatively analyze and discuss the domain of definition of the experiment, and the quantum behavior of the interference item on the average of space.

DOI
10.7498/aps.56.3185

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By using one-dimensional photon state-vector function obtained previously, a general analytic expression is deduced for the optical two-hole interference experiment, which not only greatly enlarges the range of the observable, but also completely represents the probability distribution for a photon in the whole space. On the basis of the basic rules of quantum optics, we quantitatively analyze and discuss the domain of definition of the experiment, and the quantum behavior of the interference item on the average of space.
The conditions are examined under which interference effects should be observable in the superposition of two light beams derived from independent sources. The quantum-mechanical description of these effects differs from the classical description principally in that it is necessarily based on expectation values of the light intensity at one or more space-time points. It is shown that pure states of the radiation field which are not energy states give rise to interference effects, but in that case the two beams cannot meaningfully be described as statistically independent. For a realistic description mixed states have to be introduced, when the expectation value of the intensity gives no indication of interference effects. On the other hand, the intensity correlation at two space-time points is a periodic function of the separation of the points and indicates the presence of transient interference effects. The effects become readily observable only when the average photon occupation number of each cell of phase space is appreciably greater than 1, and this explains why laser beams were needed for the experimental observations. It is pointed out that, even in these experiments, each photon may be regarded as interfering only with itself.
Based on Born's statistical interpretation for the normalized wave function, the twin two-photon thought experiment is reexamined from point of view of quantum mechanics. The emphasis is no longer the pair of twin photons itself, but rather a group of photons with common attributes. All the photons have the characteristics of identical energy and angular momentum, but possess same probability in the opposite direction of momentum, that is to say, this is a group of photon collections, which have the same number in the opposite direction of momentum, while all other quantum characteristics are identical. Following the example of the previous procedure dealing with optical two-hole effect by use of quantum mechanics, the total normalized photonic state-vector function that described the twin two-photon thought experiment has been specifically constructed, which does not describe the entangled behavior of the two photons, but the probability of a single photon, and thus deduces the probability distribution of a photon in the space. When the total number of photons satisfies the statistical requirements to form a complete picture, some fringes of light and dark interval will be displayed. On the basis of retrospective historical origins of entanglement, the point of view of quantum mechanics about the twin two-photon thought experiment was reverted. The so-called EPR paradox, and the entangled two-photon state have been re-interpreted, and a new understanding of quantum information has been put forward.
We investigate interference of optical fields by examining the probability distribution of photon detection. The usual description of interference patterns in terms of superposition of classical mean fields with definite phases is elucidated in quantum fashion. Especially, for interference of two independent mixtures of number states with Poissonian or sub-Poissonian statistics, despite lack of intrinsic phases, it is found that the joint probability has a distinct peak manifold in the multi-dimensional space of the detector outcomes, which is along the trajectory of the mean-field values as the relative phase varies on the unit circle. Then, an interference pattern should mostly appear in each shot of measurement as a point in the peak manifold with a randomly chosen relative phase. On the other hand, for super-Poissonian sources the mean-field description is likely invalidated with rather broad probability distributions.
Using the formula of Wigner function in coherent representation, we have obtained the analytical expression for Wigner function of N00N state. Based on phase space method, we study the quantum interference with N00N state as input. We derive the analytical expression of conditional probability related with the input parameter N and phase parameter φ and analyze it numerically. It is shown that, when φ is 0 or π, the output is just N00N state. It is also shown that, for 2002 state as input, the output must be 2002 state, which is independent of phase parameters. Moreover, as the number of input photon N increases, the phase probability distributions remain to have one, two, three and four peaks and get narrower. All these results can offer theoretical reference for experiments.
Optical interference is regarded as a classical phenomenon and is usually completely described in classical terms, in which optical fields are represented by classical waves. Does it mean that optical interference, which is fundamentally ascribable to the phenomenon of a superposition of wave amplitudes, cannot be applied to test quantum phenomena? The superposition principle is at the heart of quantum physics and one could expect that it should distinguish the quantum nature of light from thewave nature. Our experience based on extensive theoretical analysis and experimental observations shows that classical and quantum theories of optical interference readily explain the presence of an interference pattern resulting from the first-order coherence. However, there are interference effects involving the higher order (second-order) coherence that distinguish the quantum (photon) nature of light from the wave nature. Quantum interference has recently returned to prominence because of its utility in manipulating spontaneous emission and other radiative properties of atomic systems. In this chapter, we present elementary concepts and definitions of both the classical and quantum theory of optical interference and discuss similarities and differences between these two approaches [80].