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Physica Scripta

A beam splitter with second-order nonlinearity modelled as a nonlinear coupler

Vlasta Peřinová, A. Lukš, J. Křepelka

Published Feb 1, 2011• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/2011/t143/014020

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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.
The second-order coherence of optical fields at output ports of a beam splitter is investigated. The properties of the second-order coherence are not only strongly dependent on the input states but also quite involved with the nature of the beam splitter. When parameters of both the input states and the beam splitter are chosen appropriately, the fields at the output ports will exhibit quantum behaviour. Between the two output modes, there stands mode–mode correlation which can even be anticorrelation. Some nonclassical properties, such as the photon antibunching effect of the input fields, may be destroyed by the beam splitter.
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.
We investigate both codirectional and contradirectional nonlinear couplers composed of two nonlinear waveguides operating by second-harmonic generation. We take into account possible mismatches inside the waveguides and between them. We calculate the photon number distribution, its factorial moments, quadrature and integrated intensity variances and quadrature uncertainty product taking into account the effects of transmission of light between waveguides. Incident beams are assumed to be coherent, squeezed and mixed with external noise. Second-harmonic modes are assumed to be pumped with strong coherent beams. We show that non-classical behaviour of beams generated by the nonlinear waveguides can be transferred between them and controlled by linear and nonlinear mismatches. In the contradirectional regime of propagation, asymptotic non-classical states can be generated.
We consider a substituting scheme for nonlinear optical couplers operating by means of degenerate parametric down-conversion with strong coherent pumping. The proposed scheme, which provides the same unitary input-output transformation as the original coupler, consists of simple linear and nonlinear optical devices: beam splitter and optical parametric amplifiers. Using a group-theoretical approach, we find analytical formulas for parameters of these optical elements. The scheme allows us to get a better insight into the coupler behavior, because the complex dynamics of the coupler is transformed into a sequence of simpler evolutions governed by the beam splitter and parametric amplifiers, whose properties are well known and understood. We demonstrate that various dynamical regimes of the coupler are clearly reflected by the substituting device.
It is appropriate to base the quantum description of light propagation and down-conversion in a nonlinear medium on a Hamiltonian operator. In pumping with femtosecond pulses, which is not yet treated here, the description is so complicated that the discontinuities of the linear and nonlinear susceptibilities present a specific problem. In contrast, the analogy with a quantum nonlinear beam splitter can be utilized for the quantization of classical input–output relations. On numerical simulation, nonvanishing commutators between output field operators of different modes are revealed and a method of adaptation of these results to quantum input–output relations is also proposed.