图 1  Mach-Zehnder干涉仪. a, b为入射端; e, g为输出端; $ {\mathrm{U}}_{1} $为分束镜; $ {\mathrm{U}}_{2} $为合束镜. 入射光a和b经过第一个分束镜$ {\mathrm{U}}_{1} $后, 产生的光场线性映射到c和d两个路径上, 然后进行相位积累后再合束在一起, 实现相干态叠加, 最后输出探测场是e和g

Figure 1.  Schematic diagram of Mach-Zehnder interferometer. a, b are incident end; e, g are output end;$ {\mathrm{U}}_{1} $ is the beam-splitting mirrors;$ {\mathrm{U}}_{2} $ is the beam-closing mirrors. After the incident light a and b pass through the first beam division mirror $ {\mathrm{U}}_{1} $, the resulting light field is linearly mapping to the two paths c and d, then phase accumulation and then beam together to achieve coherent state superposition, and the final output detection fields are e and g.

图 2  在Bloch球上的Ramsey干涉仪的序列　(a)第一个光束对应于一个量子态围绕$ {J}_{x} $轴旋转$ \mathrm{\pi }/2 $, 输入态为指向北极的自旋相干态; (b)自由演化得到相位差$ \theta $, 即围绕$ {J}_{z} $旋转$\theta $; (c)第二个分束器再次围绕$ {J}_{x} $轴旋转$ \mathrm{\pi }/2 $; (d) Ramsey序列的整体效应是初始状态沿$ {J}_{y} $轴旋转一个角度$ \theta $. 通过测量$ {J}_{z} $即可读出$ \theta $^[22]

Figure 2.  Sequences of the Ramsey interferometer are illustrated with the Bloch sphere: (a) First beam corresponds to a $ \mathrm{\pi }/2 $rotation of the quantumstate around the $ {J}_{x} $axis, and he input state is a CSS pointing toward the north pole; (b) free evolution picks up a phase difference θ, which corresponds to a rotation around the $ {J}_{z} $ by an angle of $ \theta $; (c) second beam splitter again revolve around $ {J}_{x} $ axis of $ \mathrm{\pi }/2 $; (d) overall effect for the Ramsey sequence is a rotation of the initial state by an angle θ along the $ {J}_{y} $ axis. By measuring the $ {J}_{z} $ can read $ \theta $^[22].

图 3  LIGO探测结果^[25], 红线有压缩光的观测噪声, 黑线为无压缩光的观测噪声

Figure 3.  Detection results of LIGO detector, red line compression light observation noise, black line for no compression light observation noise.

图 4  (a)一个沿正x轴的自旋相干态在Bloch球上的旋转; (b)对应的沿y方向的的自旋概率为${P}_{y}\left(m\right)={\left|\langle{m}_{y}|\varPsi\rangle\right|}^{2}$(右图)^[27]

Figure 4.  (a)A coherent spin state along the x axis in the rotation of the Bloch sphere; (b) spin probability ${P}_{y}\left(m\right)= $$ {\left|\langle{m}_{y}|\varPsi\rangle\right|}^{2}$ along the $ y $ direction^[27].

图 5  非线性干涉仪. 将分束镜和合束镜用非线性的哈密顿量压缩到哈密顿量来产生, $ {U}_{1} $为分束镜, $ {U}_{2} $为合束镜. 将相同的压缩机制进行反压缩成真空态后, 信号便得到了放大

Figure 5.  Schematic diagram of the nonlinear interferometer. The beam-splitting mirrors and beam-closing mirrors are produced by squeezing to the Hamiltonian with a nonlinear Hamiltonian, $ {U}_{1} $ is the beam-splitting mirrors, $ {U}_{2} $ is the beam-closing mirrors. After the same compression mechanism is inversely compressed into a vacuum state, the signal is amplified.

图 6  不同压缩角时的量子噪声谱　(a)7°; (b) 24°; (c)46°. 每个数据集都采用经典噪声减法绘制, 并绘制了有注入压缩态时相应的量子噪声模型曲线(铜线)和没有注入压缩态的模型(蓝线)以进行比较. 结果表明随着注入状态的变化, 量子辐射压力噪声的贡献可以增加和减少^[48]

Figure 6.  Quantum noise spectra at different additional squeezing angles: (a) 7°; (b)24°; (c) 46°. Each dataset is plotted with the classical noise subtraction, The corresponding quantum noise model curve (copper line) with the injected compressed state and the uninjected compressed state model (blue line) are also plotted for comparison. The results show that, the quantum radiation pressure noise contribution can be increased and decreased as the injected state is varied^[48].

图 7  非线性干涉仪原理图. OPA1-2, 光学参量放大器; HD1-2, 平衡零拍探测器, ${\hat{a}}_{0}$为信号光场, ${\hat{b}}_{0}$为闲频光场; G₁, g₁, G₂, g₂为参量放大器振幅增益系数^[49]. 将相干态$|\alpha \rangle$光场和真空态$|0\rangle$光场同时输入OPA1中进行非相敏放大, 利用产生的具有纠缠特性的闲频光场与信号光场作为探测光场, 将被测信号加载到纠缠光上, 然后将携带被测信号的两束光同时输入OPA2中进行相敏放大

Figure 7.  Principle diagram of the nonlinear interferometer. OPA1-2, optical parametric amplifier; HD1-2, setup of balanced homodyne detection, ${\hat{a}}_{0}$ is signal light field, ${\hat{b}}_{0}$ is idle frequency light field; G₁, g₁, G₂, g₂ are parametric amplifier amplitude gain coefficient. The coherent state$|\alpha \rangle$light field and vacuum state $|0\rangle$ light field simultaneously input in OPA1 for nonphase sensitive amplification, using the entanglement of idle frequency light field and signal light field as detection light field, the measured signal on the entangled light, and then carry the two light of the measured signal simultaneously input OPA2 for phase sensitive amplification^[49].

图 8  平衡零拍探测装置^[59]. 将输入光场$ a $与待测光场$ b $通过50∶50分束器进行合束, 合束后, 用光电探测器分别对输出的两束光场$ c $与$ d $进行光强探测, 然后将探测信号相减, 得到的即为待测光场的正交分量值

Figure 8.  Setup of balanced homodyne detection ^[59]. The input light fields $ a $ and $ b $ light fields to be measured are combined through a 50∶50 beam splitter. After the beam combined, the two output light fields $ c $and $ d $ are detected by photodetector, and then let the detection signal subtraction. The result is the orthogonal component value of the light field to be measured.

图 9  左图为不同的量子态的旋转, 右图为对应不同量子态沿y方向的自旋概率为${P}_{y}\left(m\right)={\left|\langle{m}_{y}|\varPsi\rangle\right|}^{2}$　(a)自旋压缩态; (b)双Fock态, 由不可区分的玻色粒子组成, $|\uparrow \rangle$和$|\downarrow \rangle$的原子数相等, 在Bloch球中, 沿纬度方向的量子投影噪声为零即不确定度为零, 但沿经度方向的分布则完全不确定; (c) NOON态^[27]

Figure 9.  Rotation of different quantum states (left), spin probability ${P}_{y}\left(m\right)={\left|\langle{m}_{y}|\varPsi \rangle\right|}^{2}$ along the $ y $ direction of different quantum states (right): (a) Spin-squeezed state. (b) twin-Fock state. Comconsists of indistinguishable Boseonic particles, with equal atomic numbers for $|\uparrow\rangle$ and $|\downarrow\rangle.$ The quantum projection noise along the latitude direction is zero and the uncertainty is zero, but the distribution along the longitude direction is completely uncertain. (c) NOON state ^[27].

图 10  弱值放大原理图, $ g $为耦合强度, $ A $为量子系统的可观测量, $ P $为测量仪器态的可观测量, $|{\psi }_{\mathrm{i}}\rangle$为测量仪器与量子系统的联合态, $ \delta \left(t-{t}_{0}\right) $为该哈密顿量在$ {t}_{0} $时刻的瞬时相互作用, $|\phi \rangle$为仪器态, $|{\phi }_{\mathrm{f}}\rangle$为后选择成功的测量仪器态^[74]

Figure 10.  Schematic for the weak-value amplification, the g is the coupling strength, the A is the observable of the quantum system, and the P is the observance of the measured instrumental state, $|{\psi }_{\mathrm{i}}\rangle$ is the joint state of the measurement instrument and the quantum system, $ \delta \left(t-{t}_{0}\right) $ indicates the instantaneous interaction of this Hamiltonian at time $ {t}_{0} $, $|\phi \rangle$ is Instrument state, $|{\phi }_{\mathrm{f}}\rangle$ is the successfully post selective measurement instrument state^[74].

图 11  弱值实现实验装置图　(a)单模光纤射出的平行光符合高斯强度分布, 由1/4波片和半波片实现初始态, 偏振片形成后选择态, CCD 用来检测偏振相关的位移变化情况; (b)弱值实部的实验装置图; (c)弱值虚部的实验装置图^[76]

Figure 11.  Experimental setup for showing how weak values can be obtained: (a) Light emitted from a single mode fiber is Gaussian distribution, which is collimated by a len and is preselected by an initial polarization state constructed by a quarter-wave plate (QWP) and a half-wave plate(HWP). A polarizer plays an role of postselection and the following CCD then measure the intensity dependent position information. (b) real weak value realization setup. (c) imaginary weak value realization setup^[76].

图 12  弱测量结果示意图^[76]　(a)“扰动”(实线)和”无扰动“(虚线)在不同$\varepsilon$ 情况下探测概率比较; (b)完全阶数情况下的(实线)和一阶弱值(虚线)在不同 $\epsilon $ 情况下的比较, 在 $\varepsilon $极小时, 二者近似

Figure 12.  (a) Comparisons between perturbed situation (solid) and unperturbed situation (dashed) with different $\varepsilon $; (b) diagram between exact ratio of $ {P}_{\varepsilon}/p $ (solid curve) and the first order approximation in weak measurement linear regime(dashed curve), both of them are approaching when the $\epsilon $ is ultra-small^[76].

图 13  具有海森伯精度的实验测量结果, 绿色实线表示对点的拟合, 红色实线表示混合态测量精度的一个界限^[77]

Figure 13.  Experimentally obtained precision showing Heisenberg scaling. shown as a green line, obtained by fitting these points. The red line is a bound on the precision for mixed state^[77].

图 14  (a)无测量的幺正演化; (b)弱测量下的连续演化^[80]. 结果表明, 随着时间的演化, 若在没有进行测量或跃迁时为幺正演化, 会形成一个相对光滑的图形; 若投影到子空间或进行弱测量时, 会形成具有明显跳跃幅度的图形, 即为非厄米哈密顿量最终表达的效果

Figure 14.  (a) Without measurement: unitary evolution; (b) continuous evolution under the weak measurements^[80]. The results show that with the evolution of time, a unitary evolution occurs when no measurements or transitions are made, forming a relatively smooth figure, if projected to the subspace or weak measurement, forming a figure with significant jump amplitude, that is, the final expression of the non-Hermitian Hamiltonian.

图 15  (a)PT对称系统构建逻辑图实验装置^[81](流程见图). (b)图表和设置都分为6个模块: (1)系统准备和扩展; (2)预选状态准备; (3)指针状态准备; (4)指针与嵌入了PT对称系统的膨胀系统的耦合; (5)后选; (6) 弱值读出. HWP, 半波片; QWP, 1/4波片; BS, 分束器; PBS, 偏振分束器; PP, 相位板; BD, 光束置换器; ppKTP, 周期性极化的磷酸氧钛钾; BF, 带通滤波器; SPAD, 单光子雪崩二极管

Figure 15.  (a) Logic diagram of the construction of PT-symmetric system. (b) Experimental setup. Both the diagram and setup are divided into 6 modules: (1) system preparation and dilation; (2) preselection state preparation; (3) pointer state preparation; (4) coupling of the pointer and dilated system in which the PT-symmetric system is embedded; (5) postselection; (6) weak value readout. HWP, half-wave plate; QWP, quarter-waveplate; BS, beam splitter; PBS, polarizing beam splitter; PP, phase plate; BD, beam displacer; ppKTP, periodically poled potassium titanyl phosphate; BF, band-pass filter; SPAD, single-photon avalanche diode ^[81]

图 16  PT对称系统的能谱$ {E}_{+}\left(s\right) $直接由弱值${\langle{\tilde{\boldsymbol{H}}\left(s\right)}\rangle}_{\mathrm{W}}^{+}$测量^[81]. 带有误差条的黑色圆圈是弱值的实部和虚部, 红线是直接从PT对称哈密顿方程计算的相应本征能量. 即${E}_{+}\left(s\right){|}_{r=\sqrt{2}, \theta = {\mathrm{\pi }}/{4}}=1+\sqrt{{s}^{2}-1}$

Figure 16.  Energy spectrum $ {E}_{+}\left(s\right) $ of the PT-symmetric systemdirectly measured by weak value ${\langle{\tilde{\boldsymbol{H}}\left(s\right)}\rangle}_{\mathrm{W}}^{+}$. The black circles with error bars are real and imaginary parts of the weak value, and the red lines are the corresponding eigenenergy directly calculated from the PT-symmetric Hamiltonian Eq. (1); i.e., ${E}_{+}\left(s\right){|}_{r=\sqrt{2}, \theta = {\mathrm{\pi }}/{4}}=1+\sqrt{{s}^{2}-1}$^[81].

图 17  3个参数的bloch向量. 量子控制使得3个参数的信息bloch方向上同时相干相长, 最终达到海森伯极限的测量精度^[82]

Figure 17.  Bloch vectors of the three generators. Quantum control enables simultaneous growth in the informative bloch direction of the three parameters, eventually reaching the measurement accuracy in the Heisenberg limit^[82].

图 18  (a)经典独立测量; (b)纠缠独立测量; (c)纠缠并行测量在(a)和(b)这两种独立可分的测量方案中, 演化酉算符的７Ｖ个拷贝被均匀地分为3组, 并且使用一组资源来估计其中一个参数. (a)和(b)之间的区别是(a)仅使用可分探针态和可分测量, 而(b)允许在每组资源中使用纠缠探针态和集体测量. (c)中纠缠同时测量未将N拷贝酉算符分为3组, 而是将它们一起使用以同时估计全部3个参数^[82]

Figure 18.  (a) Classical independent measurement; (b) entangled independent measurement; (c) entangled simultaneous measurement are divided into three groups and use a set of resources to estimate one of the parameters. The difference between (a) and (b) is that (a) only uses separable probe states and separable measurements, while (b) allows the use of entangled probe states and collective measurements in each set of resources. The simultaneous measurement of entanglement in (c) does not divide the N-copy unitary operators into three groups, but uses them together to estimate all three parameters simultaneously^[82].

图 19  控制增强次序方案下的精度实验结果^[82]　(a) $ \alpha $; (b) $ \theta $; (c) $ \varphi $. 将这3个参数在$\alpha =\dfrac{\mathrm{\pi }}{4}, \theta =\phi =\dfrac{\mathrm{\pi }}{6}$时的实验结果与经典态独立测量、纠缠态独立测量、纠缠态并行测量下可以达到的理论极限进行比较, 每个参数的控制增强次序方案的理论实线也代表了在单参数估计中所能达到的最佳精度

Figure 19.  Experimental results of the precision under the control-enhanced sequential scheme: (a) $ \alpha $; (b) $ \theta $; (c) $ \varphi $. The experimental results of these three parameters at $\alpha =\dfrac{\mathrm{\pi }}{4}, \theta =\phi =\dfrac{\mathrm{\pi }}{6}$ are compared with the theoretical limits that can be achieved under the classical individual scheme, the entangled individual scheme, and entangled simultaneous estimation, The theoretical solid line of the control-enhanced sequential scheme for each parameter also represents the best precision that can be achieved in the single-parameter estimation^[82].

图 20  根据不确定性原理, 曲线左下区域禁止对复数$ \alpha $的实部和虚部的使用相干态进行估计的均方误差. 黑色实线为限制关系, 红色虚线为基于右对数导数的几何平均量子Cramér-Rao界, 蓝色虚线为基于右对数导数的算术平均量子Cramér-Rao界, 绿色虚线为基于对称对数导数的谐波平均量子Cramér-Rao界^[83]

Figure 20.  Mean-square errors of estimating the real and imaginary parts of a complex number $ \alpha $ encoded in a coherent state. The regions below the curves are forbidden by the corresponding inequalities. The black solid curve stands for the regret trade-off relation, the red dashed curve stands for the right logarithmic derivative-based geometric-mean quantum Cramér-Rao bound, the blue dash-dotted curve stands for the right logarithmic derivative -based arithmetic mean quantum Cramér-Rao bound, and the green dotted curve stands for the symmetric logarithmic derivative-based harmonic-mean quantum Cramér-Rao bound^[83].