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Journal of Experimental and Theoretical Physics

Paradox of a nonlinear beam splitter and its resolution

Aleksandr V. Belinsky, D. V. Volkov, A. V. Dmitriev, M. Kh. Shulman

Published Nov 1, 2013• 8 citations
Publisher

Abstract

A nonlinear beam splitter is shown to be an interesting object of investigation for the following reasons. First, the classical and quantum theories of its description give directly opposite behaviors of the phase fluctuations: according to the classical theory, the phase is unchanged; according to the quantum theory, the phase fluctuations increase or decrease, depending on the suppression or growth of amplitude fluctuations. The fundamental cause of these differences has been established. Second, the quantum fluctuations of the input mode can be separated into the amplitude and phase ones, so that the predominantly phase fluctuations are directed into one output mode, say, the reflected one, while the amplitude fluctuations are directed into the other (transmitted) mode.

DOI
10.1134/s106377611312011x

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A nonlinear beam splitter is shown to be an interesting object of investigation for the following reasons. First, the classical and quantum theories of its description give directly opposite behaviors of the phase fluctuations: according to the classical theory, the phase is unchanged; according to the quantum theory, the phase fluctuations increase or decrease, depending on the suppression or growth of amplitude fluctuations. The fundamental cause of these differences has been established. Second, the quantum fluctuations of the input mode can be separated into the amplitude and phase ones, so that the predominantly phase fluctuations are directed into one output mode, say, the reflected one, while the amplitude fluctuations are directed into the other (transmitted) mode.
It has been found that a nonlinear beam splitter is an interesting object for investigation for the following reasons. First, the classical and quantum descriptions of this object give opposite results for the behavior of phase fluctuations: according to the classical theory, the phase remains unchanged, whereas the quantum theory predicts that phase fluctuations increase or decrease in correlation with a decrease or increase in amplitude fluctuations. Second, quantum fluctuations of the input mode can be separated into amplitude and phase fluctuations, so that phase fluctuations are directed predominantly into one output mode, say the reflected mode, whereas amplitude fluctuations are directed predominantly into the other (transmitted) mode.
This is a review of a very interesting (in the authors' view) phenomenon — the operation of a nonlinear light beam splitter. The beam splitter is a flat interface between two transparent dielectrics, at least one of which exhibits Kerr nonlinearity, i.e., its refractive index depends on the transmitted radiation intensity. Interestingly, quantum and classical theories make directly opposite predictions about the phase fluctuations of the output radiation of this device. In classical theory, the phases remain unchanged; in quantum theory, the phases fluctuate in accordance with the amplitude–phase uncertainty relation. The origin of this difference is established at the fundamental level. A further remarkable point about this quantum paradox is that not only is the source beam split in two but one can also create conditions where the two split parts are respectively dominated by amplitude noise and phase noise, thus allowing the selection of photon fluctuations. Results of original studies are summarized and further developed.
We provide two quantum descriptions of a beam splitter with second-order nonlinearity. Models of nonlinear couplers could be the same. For simplicity, a symmetry of the device is assumed, which does not exclude various reflectivities. The models differ in taking into account the co-propagation of beams and replacing it by their counter-propagation. The unitary operators that evolve the input states into the output ones are calculated by using a momentum operator and the effective distance of presumed co-propagation and counter-propagation, respectively.