Quantum Theory of Distributed-Feedback Parametric Amplifiers and Oscillators

TL;DR

The paper develops a fully quantum-mechanical, analytic model for distributed-feedback fiber parametric oscillators based on Bragg gratings and degenerate four-wave mixing. It derives input-output relations, threshold conditions, and complete quantum statistics (squeezing, correlations, and entanglement) for various cavity geometries, including Fabry-Perot and phase-shift configurations, while incorporating intracavity losses. The framework enables calculation of threshold pump powers, squeezing levels, and output entanglement, and it highlights applications in quantum-enhanced fiber sensing. The results are generalizable to integrated platforms and support future deployment of DFB FOPOs as versatile quantum light sources.

Abstract

Optical parametric oscillators are among the best-developed quantum light sources, having already been adopted in precision measurement and underpinning various quantum computing and communication paradigms. Meanwhile, progress in photonic structures such as Bragg gratings has enabled distributed feedback oscillators to become widely established as classical laser sources with desirable properties, as well as enabling a new generation of precision optical sensors. Recent work in fabricating and processing photonic structures in nonlinear media opens the path to combining these two programs to realize distributed feedback parametric oscillators. Such devices have great potential as sources of quantum light, especially for squeezed vacuum, a crucial resource state in emerging quantum technologies. We present an analytic and fully quantum-mechanical model of the dynamics of such devices. This approach yields the key properties of these sources, such as the parametric oscillation threshold, intracavity mode, tunability, and quantum statistics (including entanglement) of the output modes. We also discuss the application of these devices as quantum-enhanced sensors. These results underpin future work on a versatile class of next-generation quantum light sources.

Figures (5)

• Figure 1: Left: Micrograph of UV-photosensitive germanium-doped PCF, with the Ge-doped region visible as a lighter spot in the fibre core. Centre: Seeded near-degenerate FWM spectra through 10m of this fibre with seed wavelength 1278 nm and pumps $\lambda_p= 1080$ nm and $\lambda_q= 1568$ (blue), $1569$ (orange) and $1570$ nm (red). Right: Normalized transmission spectrum of 50mm Bragg grating inscribed in this fiber.
• Figure 2: Top: DFB OPO diagram based on dual-pumped four-wave mixing, with key modes labelled, including the noise modes $\hat{c}'$ and $\hat{d}'$. Bottom: Degenerate pumping scheme. Pumps $p$ and $q$ generate photons pairs (squeezing) at their average frequency $\omega_s$.
• Figure 3: Left: Dependence of sub-threshold squeezing in the $\hat{b}$ mode with pump power and relative grating length for a Fabry-Perot cavity, with $\kappa L_1=3$ and $L_3=4L_1$. Right: Semilogarithmic plot of the oscillation threshold pump power for various combinations of $L_2,~L_3$. Here $\kappa L_1=3$, $L_1=0.1$m and $\gamma_{NL}=0.25$ W$^{-1}$km$^{-1}$.
• Figure 4: LEFT: Expectation value for the photon number per mode volume in the forward propagation direction along the length of a phase-shift grating with $L_1=L_2=L$, $\kappa L=1.5$ and $P_1P_2=0.9P_\theta^2$ with the defect located at $z=0$. The double-exponential mode envelope characteristic of phase-shift gratings is modified by the gain. Note that in the final section of the device the intensity is increasing again. RIGHT: Suppression of the quadrature noise ($10\log_{10}\langle \hat{x}^2\rangle$) of the most squeezed supermode of the two outputs of a phase shift grating with $\kappa L_1=\kappa L_2=3$, for varying levels of loss and pump power.
• Figure 5: Schematic diagram of a balanced interferometric fiber sensor using the round-trip phase of a fiber cavity to measure an applied strain. With a coherent-state probe (top right), phase sensitivity is limited by noise in the phase quadrature of the signal passing through the sensor. When inline squeezing is employed (bottom right) the phase noise can be squeezed around the values of $\theta$ where the interferometer is set, giving a more precise measurement. FBS- fibre beam splitter. PD- photodiode. $I_-$- difference photocurrent.