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Physica Scripta

Quantization of radiation emitted at discontinuities of nonlinearity

Vlasta Peřinová, A. Lukš, Jan Peřina

Published Mar 1, 2013• 4 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/2013/t153/014050

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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.
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.
Starting with the canonical quantization procedure for the electromagnetic field inside an effective (linear or nonlinear) medium, we present a direct-space formulation of the theory of quantum optics. This approach does not use the conventional modal decomposition of the field, but relies on the electromagnetic momentum operator defined in terms of local electric- and magnetic-field operators. The momentum operator contains all the information on the spatial characteristics of the field, and can describe the translation of a short light pulse in a nonlinear medium, without a modal analysis of the pulse. Propagation is described through an operatorial wave equation that relates the temporal evolution of an electromagnetic pulse to its spatial progression. Through this equation, the direct-space approach to quantum optics can treat traveling-wave nonlinear-optical phenomena and, at the same time, account for their quantum statistics. The theory is applied to squeezed-light generation by the parametric down-conversion of a short laser pulse, as an illustration.
The familiar input-output relations for an optical beam splitter are generalized to allow for linear absorption by the medium forming the mirror. Beam-splitter losses generally affect the noise levels detectable in experiments involving nonclassical light. When employed to investigate two-photon interference effects, a lossy beam splitter can lead to apparent nonlinear absorption, which, in the most extreme case, leads to either both or neither of the photons being absorbed. The degree of second-order coherence of antibunched light can be maintained on transmission through the beam splitter but any amplitude squeezing in the incident light is degraded.
Several authors1,2 have studied the lossless beam splitter on a quantum mechanical basis. In this approach the action of the beam splitter is described by unitary transformations that act on the operators for the radiation fields present. In this paper we study quantum mechanically the general case of reflection and transmission from a boundary. We obtain the quantum analog of the classical Fresnel relations. We find that the general operators that describe reflection and transmission at a boundary are a series of unitary operators involving appropriate radiation fields. The generalization for the nonlinear media presents many interesting situations where the classical laws of reflection and refraction are modified. This provides an interesting description of phase conjugation.
Classically, light is an electromagnetic phenomenon, described by Maxwell's equations. However, under certain conditions, such as low intensity or in the presence of certain nonlinear optical materials, light starts to behave differently, and we have to construct a ‘quantum theory of light’. We can exploit this quantum behaviour of light for quantum information processing, which is the subject of this book. In this chapter, we develop the quantum theory of the free electromagnetic quantum field. This means that we do not yet consider the interaction between light and matter; we postpone that to Chapter 7. We start from first principles, using the canonical quantization procedure in the Coulomb gauge: we derive the field equations of motion from the classical Lagrangian density for the vector potential, and promote the field and its canonical momentum to operators and impose the canonical commutation relations. This will lead to the well-known creation and annihilation operators, and ultimately to the concept of the photon. After quantization of the free electromagnetic field we consider transformations of the mode functions of the field. We will demonstrate the intimate relation between these linear mode transformations and the effect of beam splitters, phase shifters, and polarization rotations, and show how they naturally give rise to the concept of squeezing. Finally, we introduce coherent and squeezed states.
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.