Quantum Theory of Light and Noise Polarization in Nonlinear Optics

Abstract

We present a consistent quantum theory of the electromagnetic field in nonlinearly responding causal media, with special emphasis on χ (2) media. Starting from QED in linearly responding causal media, we develop a method to construct the nonlinear Hamiltonian expressed in terms of the complex nonlinear susceptibility in a quantum mechanically consistent way. In particular we show that the method yields the nonlinear noise polarization, which together with the linear one is responsible for intrinsic quantum decoherence.PACS numbers: 42.50. Nn, 42.65.Lm, 42.50.Ct, 42.50.Lc Recent advances in quantum information technologies have been the main driving forces behind the desire to build parametric down-conversion sources of entangled photon pairs (in the low-intensity limit) [1,2,3,4,5] or two-mode squeezed states (in the high-intensity limit) [6] with high fidelity. It is known that single-photon states of non-unit efficiency as produced by heralded single-photon sources using parametric down-conversion, cannot be purified by using linear optical elements and photo detection to yield states with higher efficiency [7,8]. That in turn means that post-processing of single-photon sources is impossible and the sources themselves have to be improved. In order to achieve the maximally possible purity of heralded single-photon states or correlated (entangled) twin-beam photons it is therefore necessary to investigate the theoretical limits nature imposes on us.An important step in this direction is to provide a quantum theory of light that takes into account nonlinear processes such as parametric down-conversion, and at the same time decoherence mechanisms due to unavoidable absorption losses of the nonlinear material the light interacts with. The theory of quantized electromagnetic fields in linearly and causally responding materials (with the linear response function satisfying the Kramers-Kronig relations) is well established (see, e.g., Refs. [9, 10, 11]). It has been known for some time that analogous Kramers-Kronig relations do also hold for nonlinear susceptibilities [12]. Hence, it will be interesting to see how these causal relations appear in a nonlinear quantum theory.Previous work on electromagnetic field quantization in nonlinear materials have focused on strictly lossless materials where Lagrangian methods and mode decompositions apply [13,14,15,16]. A first attempt to include in the field quantization both linear and nonlinear losses was made in Ref. [17] for Kerr media, by extending the linear harmonic-oscillator model used in the Huttner-Barnett quantization scheme [9] to a nonlinear one. A consistent approach that includes-for given nonlinear susceptibility-absorption and dispersion has not yet been formulated within the frame of (macroscopic) QED.In this article we will exemplify, on focusing on χ (2) media, how to consistently quantize the electromagnetic field in the presence of nonlinearly responding causal materials. This theory provides the starting point for further investigations of theoretical ...

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