Postselected amplification and photon recycling applied to optical sensing of magnetic fields

TL;DR

This work addresses enhancing the precision of optical magnetic-field sensing by integrating postselected amplification with photon recycling in a Mach-Zehnder interferometer. The authors derive analytic expressions for two pulsed recycling strategies, showing that recycling preserves the amplified signal per postselected event while increasing the total postselected photon count, thereby boosting the overall signal-to-noise ratio (SNR) without sacrificing measurement integrity. In Scheme I, external recycling increases the effective dark-port postselection probability and yields an SNR enhancement factor scaling with the recycled dark-port yield; Scheme II achieves further SNR gains via internal recycling with a tunable amplification factor that depends on the recycling round. Numerical results demonstrate convergence to the infinite-recycling limits for large numbers of cycles and quantify improvements over conventional and non-recycled postselected measurements, including practical considerations for pulse timing and component switching. Overall, the work shows that combining postselection with recycling can yield substantial SNR gains for static magnetic-field sensing and highlights directions for continuous-wave recycling and applications beyond this specific setup.

Abstract

We apply the combined technique of postselected amplification and photon-recycling to an optical setup of magnetic field precision measurement. We propose two recycling schemes and carry out analytic expressions for the amplified signal and measurement sensitivity. The results show significant improvement of performance over conventional measurement. The underlying reason is twofold. On one aspect, introducing the technique of recycling eliminates the shortcoming of data discarding in postselection, thus maintains similar noise level of conventional measurement (without postselection). On the other aspect, performing intentional postselection within the recycling framework, which was originally proposed in the context of gravitational wave detection, can amplify the signal. Thus, the measurement signal-to-noise ratio is enhanced.

Figures (6)

• Figure 1: (a) Schematic of an optical magnetometer, in which a Faraday crystal (FC) is placed between two electric coils, which can introduce a relative phase between the left circularly polarized light and the right one, equivalently, cause a Faraday rotation (FR) of a linearly polarized light, with the FR angle proportional to the magnetic field to be measured. (b) Following Ref. Niu24, the magnetometer (characterized by the FR angle $\theta$) is proposed to embed into an MZI. However, beyond Ref. Niu24, an external optical circuit is introduced to recycle the discarding light of postselection in the bright port back to the input port, and re-inject it into the MZI. Optical elements in this set-up: BS (beam splitter); PBS (polarizing beam splitter); PC (Pockels cell); HP (horizontal polarizer); HWP (half wave plate); D1 and D2 (two photo-detectors). Cooperative action of these elements can fulfill the requirement of the pulse-recycling scheme. In particular, in order to properly control the polarization of light, while the PC2 in the external circuit is keeping on "off" state, the PC1 in the input port should be switched from the initial "off" state (to inject the initial pulse with "H" polarization into the interferometer), to the later "on" state (to ensure the successive recycling of light re-injected with "H" polarization). In the final stage of recycling, the PC2 is switched on to convert the polarization of light from "V" to "H", releasing the residual light in the interferometer via transmitting through the PBS1, and allowing next pulse's input and recycling.
• Figure 2: Internal recycling scheme, without introducing the external circuit as shown in Fig. 1(b). The role of HP1 and HP2 is filtering the "V" component of the light, allowing only the "H" component of light transmitted through them. The role of the PC in the input port, first set in "off" state to allow the transmission of the initial pulse with "H" polarization, and later switched to "on" state, is converting the polarization from "H" to "V" and from "V" to "H", in order to resend the outgoing light back into the interferometer, with "H" polarization.
• Figure 3: (a) Scaled SNR of the rPSM scheme (I), associated with the schematic plot of Fig. 1. It shows that stronger postselection (smaller $\beta$) can result in a better enhancement of the SNR, especially for measuring weaker magnetic field, which corresponds to the smaller FR angle $\theta$. This result indicates the remarkable role of postselection in the recycling measurements. (b) Total postselection probability of photons into the dark exit port "d", after accounting for all the infinite number of times of recycling. The larger loss of $P_d$, for larger $\theta$, is because of the stronger loss of photons caused by the polarization filtering element HP, in the bright exit port "c".
• Figure 4: (a) Scaled SNR of the rPSM scheme (II), associated with Fig. 2; and (b), the total postselection probability of photons into the dark exit port "d", after accounting for all the infinite number of times of recycling. The qualitative behaviors of both are similar to the results shown in Fig. 3, for the rPSM scheme (I).
• Figure 5: (a) Scaled SNR $\widetilde{R}^{(n)}_{S/N}$ and (b) the total postselection probability $P^{(n)}_d$, for finite ($n$) times of recycling based on the rPSM scheme (I). Results of two postselection parameters are displayed and compared with the results of infinite number of times of recycling (the solid and dashed lines), showing that both can become the same for sufficiently large $n$. In the main text, the number of recycling times $n=10^5$ is estimated based on certain practical parameters.