Due to the excellent detection performance, levitated optomechanical systems have become intriguing in the field of precision measurement. The detection schemes of such systems mainly contain the balanced photodetector (BPD) and quadrant photodetector (QPD). The BPD has a current differential structure that can eliminate common-mode DC signals, which results in extra low noise. However, the BPD scheme has the disadvantage of complicated detection systems because one BPD usually measures the particle displacement in one direction. In contrast, a QPD can simultaneously detect displacements in three directions, which effectively simplifies the detection system. Usually, QPDs amplify not only the AC photocurrent generated by the fluctuations of the incident optical power, which mainly arise from the displacements to be measured, but also the DC photocurrent corresponding to the average incident optical power. As a consequence, QPDs always show worse noise performance than BPDs. Therefore, this work proposes a QPD scheme, which reduces the electrical noise of a QPD through electrical filtering and achieves a common-mode rejection ratio (CMRR) as high as possible, and the response coefficients of the four quadrants are calibrated. The QPD is a promising alternative scheme to the BPD, which has high detection sensitivity and is beneficial to the miniaturization of levitated optomechanical systems.

This work proposes a QPD scheme by building the noise model in levitated optomechanical systems, according to which the proportions of electrical noise, optical shot noise, and relative intensity noise (RIN) can be obtained. A current filter circuit is built with an operational amplifier, and the DC and AC components of the photocurrent are separated without any influence on the junction capacitance of the QPD sensor. As a result, the AC transimpedance gain is free from the limitation of the DC component, and meanwhile, the electrical noise is reduced. The converted voltage signal from the AC component is used to demodulate the three-axis displacement information of the particle, and that converted from the DC component is used to indicate the position of the incident light on the QPD. Moreover, a QPD sensor with high saturated optical power is used to suppress the shot noise to the maximum extent. In addition, the bias voltage of the QPD sensor is adjusted to reduce the junction capacitance differences between the four quadrants, and the feedback resistance of each quadrant is calibrated through the connection of an adjustable potentiometer in series with a high-precision fixed resistance resistor. Consequently, the response coefficients of four quadrants are unified, and the CMRR of the detector is improved to a large degree.

Attributed to the calibration of the response coefficients, the junction capacitance difference between QPD quadrants is less than 1%, the circuit gain difference is less than 0.04%, and the phase difference is less than 0.02° (Fig. 4). The CMRR and noise performance of the QPD are tested in a levitated optomechanical system and are compared with those of a commercial BPD which meets the requirements of such a system with superior noise performance. The results show that the power spectral density (PSD) of the electrical noise of the QPD is -129.4 dBV²/Hz with a circuit gain of 10⁵ and reaches the same level as that of the commercial BPD (Tables 1 and 2). The CMRR of QPD is greater than 45 dB (50-250 kHz), which is better than that of the commercial BPD [Fig. 5(b)]. When the incident optical power is in the range of 1-40 mW, the noise performance of the QPD is only limited by shot noise (Fig. 6, Table 2). Overall, compared with the commercial BPD, the QPD proposed in this work can better suppress RIN and has higher saturated optical power to effectively limit the influence of shot noise. Therefore, the detection performance of the proposed QPD can meet the requirements of levitated optomechanical systems.

An active current filtering scheme with operational amplifiers is designed in this work to improve the detection performance of the QPD-based levitated optomechanical system. A 10⁵ single-stage transimpedance gain is achieved so that the electrical noise is optimized. The response coefficients of four quadrants are calibrated so that the CMRR is improved, which is greater than 45 dB in the frequency range of 50-250 kHz. When the incident optical power is in the range of 1-40 mW, the noise performance of the QPD is only limited by shot noise. The QPD scheme has been successfully applied for extremely weak force sensing in levitated optomechanical systems, which lays the foundation for the high-performance, miniaturized detection system. In the future, the delay introduced by the acoustic-optic modulator can be compensated with a calibration module, so as to further improve the common-mode rejection performance of the QPD scheme.