Quantum Noise in Fibers with Arbitrary Nonlinear Profiles

In this Letter we introduce a novel equation addressing the effect of quantum noise in optical fibers with arbitrary frequency-dependent nonlinear profiles. To the best of our knowledge, such an endeavor has not been undertaken before despite the growing relevance of fiber optics in the design of new quantum devices. We show that the stochastic generalized nonlinear Schrödinger equation, derived from a quantum theory of optical fibers, leads to unphysical results such as a negative photon number and the appearance of a dominant anti-Stokes sideband when applied to this kind of waveguides. Starting from a recently introduced master-equation approach to propagation in fibers, we derive a novel stochastic photon-conserving nonlinear Schrödinger equation suitable for modeling arbitrary nonlinear profiles, thus greatly enhancing the study of fiber-based quantum devices.

Figure 1

Numerical results of the stoGNLSE and its positive $P$ representation for a 1-cm standard single-mode fiber pumped by a high-power CW laser at 1550 nm. Curves were obtained by averaging 10 000 noise realizations (the pump laser is not shown for the sake of clarity.) The $+P$ representation does not display an unphysical negative photon number as predicted by the stoGNLSE.

Figure 2

(top panel) Frequency-dependent nonlinear profile. (bottom panel) Numerical results of the positive $P$ stoGNLSE and the proposed sto-pcGNLSE. Curves correspond to the average of 10 000 noise realizations (the CW is not shown for the sake of clarity.) The sto-pcGNLSE avoids the unphysical negative photon number as predicted by the $\mathrm{stoGNLSE}(+P)$.

Figure 3

Numerical results of the positive $P$ stoGNLSE and the sto-pcGNLSE for a HNLW with a negative nonlinear coefficient. Curves were obtained as an average of 1000 noise realizations. Unlike the sto-pcGNLSE, the $\mathrm{stoGNLSE}(+P)$ predicts an unphysical growth of the anti-Stokes sideband.