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High-finesse optical cavity assisted quantum nondemolition (QND) measurement is an important method of generating high-gain spin or momentum squeezed states, which can enhance the sensitivity of atom interferometers beyond the standard quantum limit. Conventional two-mirror Fabry-Perot cavities have the drawback of a standing wave pattern, leading to inhomogeneous atom-light coupling and subsequent degradation of squeezing enhancement. In this study, we present a novel method of achieving homogeneous quantum nondemolition measurement by using an optical ring cavity to generate momentum squeezed states in atom interferometers. We design and develop a high-finesse ($ {\cal{F}} = 2.4(1) \times 10^{4} $), high-vacuum compatible ($ 1\times 10^{-10} $ mbar) optical ring cavity. It utilizes the properties of traveling wave fields to address the issue of inhomogeneous atom-light interaction. A strontium cold atomic ensemble is prepared and coupled into the cavity mode; the nondemolition measurement of atom number is achieved by extracting the dispersive cavity phase shift caused by the passage of atoms through differential Pound-Drever-Hall measurement. Experimental results indicate that under a probe laser power value of $ 20\; $μW, the dispersive phase shift of the ring cavity is measured to be $ 40\; $mrad. The effective number of atoms coupled into the cavity mode is around $ 1 \times 10^{6} $. The consistency between the ring cavity dispersive phase shift and QND measurement theory is verified by adjusting parameters such as matching the atomic position with the cavity mode and tuning the frequency of the probe laser. The optical ring cavity developed in this work provides an important method for generating spin or momentum squeezed states in atom interferometers. Therefore it holds promise for enhancing their sensitivity, and it is expected to be widely applied to cavity-enhanced quantum precision measurements.
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Keywords:
- Optical ring cavity /
- quantum nondemolition measurement /
- spin squeezing /
- atom interferometer
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图 1 (a)原子与光学环形腔耦合模型, 一对Bragg激光($ \text{B}_{1} $和$ \text{B}_{2} $)用于制备基态的动量叠加态, 探测光和参考光沿相反方向耦合至腔内, 且与环形腔同时共振; (b)$ ^{88} {\mathrm{S}}$r原子能级跃迁示意图, 其中$ \Delta_{1} $和$ \Delta_{2} $分别为参考光和探测光相对于$ ^{1} {\mathrm{S}} _{0} $-$ ^{3} {\mathrm{P}} _{1} $ 跃迁的失谐量; (c)光学环形腔内光束腰半径沿切平面(tangential plane, T-plane)和矢状面(sagittal plane, S-plane)的演化规律
Figure 1. (a) Atom-optical ring cavity coupling model. A pair of Bragg beams ($ \text{B}_{1} $ and $ \text{B}_{2} $) are used to induce the momentum state superposition on the ground state, the probe and reference beams are coupled into the cavity with counter-propagating directions and are resonant with the cavity simultaneously. (b) $ ^{88} {\mathrm{S}}$r atomic transition energy diagram, where $ \Delta_{1} $ and $ \Delta_{2} $ are detunings of the reference and probe beam with respect to the $ ^{1} {\mathrm{S}} _{0} $-$ ^{3} {\mathrm{P}} _{1} $ transition. (c) Optical ring cavity beam waist evolution at the tangential plane (T-plane) and the sagittal plane (S-plane).
图 2 自制腔镜支架原理图, 右插图为用于俯仰角度调节的分立零件
Figure 2. Schematic for the homemade cavity mirror holder, right inset shows the parts for adjusting the rolling direction.
图 3 (a)光纤EOM输出的探测光束腰半径演化规律, 实线为拟合曲线; (b)探测光与光学环形腔模式匹配光路示意图. EOM: 电光调制器; PBS: 偏振分束器; CCD: 电荷耦合器件; PC: 计算机; PD: 光电探测器; Scope: 示波器
Figure 3. (a) Probe beam waist propagation at the output of the fiber-EOM, solid traces are fitting results; (b) mode-matching schematic for the probe beam and the optical ring cavity. EOM: Electro-optic modulator; PBS: polarization beam splitter; CCD: charge coupled device; PC: personal computer; PD: photo detector; Scope: oscilloscope.
图 4 (a)利用翻转光学面板的方式实现环形腔腔镜整体粘连; (b)光学环形腔粘连至真空法兰示意图; (c)真空装配光学环形腔实物图
Figure 4. (a) Schematic for epoxying the cavity mirrors by flipping the optical breadboard; (b) optical ring cavity schematic assembled on a vacuum flange; (c) optical ring cavity assembled into a vacuum chamber.
图 5 实验测试装置, 详见正文阐述. ECDL: 外腔半导体激光器; TA: 锥形放大器; PID: PID控制器; AOM: 声光调制器; OI: 光学隔离器; MOD: 调制信号; LO: 本振信号; MX: 混频器; ADC: 模数转换器; PZT: 压电传感器
Figure 5. Schematic for the experimental setup, see main text for more details. ECDL: external cavity diode laser; TA: tapered amplifier; PID: proportional-integral-derivative controller; AOM: acousto-optic modulator; OI: optical isolator; MOD: modulation; LO: local oscillator; MX: mixer; ADC: analog to digital converter; PZT: piezoelectric transducer.
图 6 (a) 光学环形腔自由光谱区测量, 蓝色实线为拟合曲线, 插图为电光调制产生的载波和边带; (b) 环形腔真空装配前后腔内光子数衰减振荡时间测量, 实线为指数衰减拟合曲线
Figure 6. (a) Measurement of the FSR of the optical ring cavity, the blue solid trace is a fit of the data. Inset shows the carrier and sidebands as a result of electro-optic modulation; (b) cavity photon decay time constant measurement before and after the cavity assembly, the solid traces are exponential decay fits of the data.
图 7 (a)锶原子冷却和环形腔内QND测量时序. (b)环形腔内QND测量结果. 蓝色点线为磁场梯度变化, 提供时序参考; 灰色虚线为环形腔透射信号, 监测PDH锁定状态. 红色实线为滤波处理后的差分PDH误差信号, 粉红色区域内显示原子对腔造成的相移
Figure 7. (a) Experimental sequence for Sr atom cooling and QND measurement in the optical ring cavity. (b) QND measurement results in the optical ring cavity. The blue dotted trace shows the recording of the magnetic gradient, which provides a reference for time sequence; the gray dashed trace shows the cavity transmission signal, which monitors the PDH locking state; the red solid trace is the filtered differential PDH error signal, where the atom-induced cavity phase shift is shown in the pink shaded region.
图 8 (a)耦合进腔内原子数目随x方向补偿磁场电流大小变化, 蓝色实线为高斯拟合; (b)腔色散相移随探测光失谐量变化, 蓝色实线为基于函数$ y=a+b/x $的拟合
Figure 8. (a) Effective atom number coupled into the cavity mode as a function of the current for the x-compensation coils. The blue trace is the Gaussian fit. (b) Cavity dispersive phase shift as a function of the frequency detuning of the probe beam. The blue trace is a fit with function $ y=a+b/x $.
表 1 光学环形腔真空制备前后主要参数测试结果
Table 1. Test results of the relevant optical ring cavity parameters before and after vacuum assembly.
Parameter Symbol Value (before) Value (after) Units Free spectral range FSR $ 1.4475(5) $ $ 1.4475(5) $ GHz Linewidth $ \delta\nu $ $ 56.9(4) $ $ 60.4(1) $ kHz Finesse $ {\cal{F}} $ $ 2.5(4)\times 10^{4} $ $ 2.4(1)\times 10^{4} $ $ - $ 
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