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Physical Review A

Classical and nonclassical interference

Krzysztof Wódkiewicz, G. H. Herling

Published Feb 1, 1998• 21 citations
Publisher

Abstract

The relation between quantum interference and classical interference is discussed in terms of the Wigner function and an analogous classical expression. For two displaced coherent states and two displaced electric-field Gaussian pulses, both the quantum-mechanical and classical Wigner functions exhibit oscillatory behavior. Classical analogs of squeezing, photon-number oscillation, and the $Q$ function are presented.

DOI
10.1103/physreva.57.815

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The relation between quantum interference and classical interference is discussed in terms of the Wigner function and an analogous classical expression. For two displaced coherent states and two displaced electric-field Gaussian pulses, both the quantum-mechanical and classical Wigner functions exhibit oscillatory behavior. Classical analogs of squeezing, photon-number oscillation, and the QQQ function are presented.
In this paper we discuss the nonclassical properties of quantum superpositions of coherent states of light. Using general expressions for the Wigner functions of superposition states we analyze the consequences of quantum interference between coherent states. We describe in detail nonclassical properties of a superposition of two coherent states. In particular, we study the oscillatory behavior of the photon number distribution of the even and odd coherent states. We show under which conditions a superposition of two coherent states can exhibit second- and fourth-order squeezing or sub-Poissonian photon statistics. We examine the sensitivity of nonclassical effects such as oscillations in the photon number distribution or second-order squeezing to dissipation. We demonstrate that quantities such as the photon number distribution and interferences in phase space are highly sensitive to even a quite small dissipative coupling, because they depend on all moments of the field observables, and higher moments decay more rapidly than lower moments. Quantities such as quadrature squeezing, on the other hand, are more robust against dissipation because they involve only lower moments. Finally, we find a remarkable effect whereby fourth-order squeezing is generated by damping.
We discuss a phase-space description of the photon number distribution of non-classical states which is based on Husimi's Q (α) function and does not rely on the WKB approximation. We illustrate this approach using the examples of displaced number states and two photon coherent states and show it to provide an efficient method for computing and interpreting the photon number distribution. This result is interesting in particular for the two photon coherent states which, for high squeezing, have the probabilities of even and odd photon numbers oscillating independently.
It is shown that a particular combination of so-called even coherent state dubbed the orthogonal-even coherent state, while exhibiting no ``ordinary'' or second-order squeezing, can furnish near-optimal simultaneous-quadrature fourth-order squeezing [where the latter term is defined in the sense of Hong and Mandel, Phys. Rev. Lett. 54, 323 (1985)]. The Wigner function of the particular quantum-mechanical superposition of states is calculated and various plots compared to the Wigner function of a simple statistical mixture of the same states. Rather uninteresting behavior is found in the latter case, which illustrates that the higher-order squeezing effect is due to nonclassical, quantum-mechanical ``interference in phase space.''
Approximating a continuous superposition by a discrete one leads to an understanding of the role of quantum interference in generating squeezed coherent states, and in its quantum properties. It was shown that discrete superposition with a large enough number of coherent states with appropriate coefficients can have practically the same features as the squeezed coherent state. Dependent on the parameters W, r and 8, different numbers of coherent states are required for the interference fringe to form a Wigner function resembling that of the squeezed vacuum state and to reduce the deviation below a certain value. For the case of rotating the direction of squeezing, about 20 states were required to lower the deviation to 0.05%, while 640 states would be necessary when a unit displacement along the imaginary axis is included. When only a few coherent states are superimposed, the resulting state changes chaotically. In the case when the approximated squeezed coherent state was displaced along the imaginary axis an interesting Schrodinger cat-like state was found with about 20 coherent states on the real axis. At this point a significant peak occurs in the variances of the quadratures. It turned out that non-classical states can be effectively produced by a discrete superposition of coherent states with the coefficients derived from the one-dimensional representation of the given state.
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].
The phase space representation is used in the classical mechanics of particles but is not appropriate for describing interference phenomena that occur in classical field theory such as classical optics. The methods of classical optics and quantum mechanics are amazingly similar. That is principally because both use Hilbert spaces to represent states. The only difference is in the use of non-commuting operators to represent observables in quantum mechanics. The quantum theory of light predicts perfect anti-coincidence for single detections on the two sides of a beam splitter, whereas classical light, however weak, would be divided on a beam splitter and produce a minimum degree of correlated or coincidence counts. In mathematics the Hermite polynomials are classical orthogonal polynomial sequences that arise in probability and numerical analysis as Gaussian quadrature. They are also used in systems theory in connection with non-linear operations on Gaussian noise and other sciences.