Measuring the angular rotations with high sensitivity has an increasing interest recently, for its growing potential in a wide range of optical science and applications. For example, precise measurement of rotations plays a vital role in atom interferometer gyroscopes , optical tweezers , rotational Doppler effect [3–5], and magnetic field measurements . Traditionally, the basic laser beam with Gaussian profile is incapable of taking angular rotations because it is rotational symmetry . Motivated by the studies of light endowed with orbital angular momentum (OAM) , some related efforts are proposed to increase the sensitivity of angular rotation’s measurement. But it is worth noting that the pure OAM lights like Laguerre–Gaussian (LG) beams are still rotational symmetry, and therefore quantum resources are involved for angular rotations measurement in addition, such as quantum entanglement of high OAM values  and N00N states in the OAM bases [10,11]. Recently, D’Ambrosio
Those previous works concentrated on the precision improvement via higher OAM values, but we find that the ultimate precision on rotation measurement is decided by the variance of OAM distribution instead of OAM value. Therefore, we employ the Hermite–Gaussian (HG) pointer to achieve an ultrahigh sensitivity on the measurement of angular rotations because of its large OAM variances. Previously, the -order HG pointer was employed on the measurement of spatial displacement [18,19], where the corresponding quantum Cramér–Rao (QCR) bound  was improved linearly with mode number . In this work, we employ the -order HG pointer in a rotational-coupling weak measurement scheme. After the post-selection, the information of angular rotation is taken by an HG mode state, which is orthogonal to the initial pointer state. Then projecting the final light beam to the state taking angular rotations, the quantum limit precision of rotation measurement can be achieved, which is enhanced with a factor of .
For demonstration, we set up an optical experiment to implement the precision measurement of angular rotation. Instead of the tomography of OAM distribution in Ref. , we demodulate the angular rotations from the projection intensity directly, and the imaginary weak value is also not necessary. The measurement precision in our experiment achieves 0.89 μrad with the -order HG beam. Our results shed new light on the precise measurement of angular rotation and have potential for optical metrology, remote sensing, biological imaging, and navigation systems [4,21,22].
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2. THEORETICAL MODEL
A. Enhanced Quantum Limit via HG Pointer
To be clear, we first consider the general weak measurement process with post-selection as depicted in Fig. 1. For simplicity, we consider a two-level system with initial state and a pointer with initial state . They couple together during the weak interaction procedure with an impulse Hamiltonian , where is a translation operator on the pointer, and is the corresponding interaction strength. Here is a Pauli operator on the two-level system. In weak measurement scheme, the interaction strength , and then the unitary evolution operator of weak interaction procedure can be approximately calculated as . (Without loss of generality, we adopt units making in this paper.)
Figure 1.Post-selected weak measurement scheme.
To individually read out the measurement information from the pointer, we post-select the system by state and turn the final state in whole to be , where is the pointer’s final state, and is the normalized factor with , . is a weak value calculated by [23,24].
To analyze the estimating precision in our weak measurement scenario, we employ the quantum Fisher information (QFI) as a figure of merit [25,26]. The QFI of final pointer state about interaction strength parameter can be calculated as , where . For classical measured samples, the variance of estimator satisfies the QCR inequality , which leads to an uncertainty relation
In this work, we apply the weak measurement scheme to the measurement of angular rotation. Thus, the interaction strength corresponds to the rotation angle of pointer, and the translation operator is the OAM operator. Traditionally, the laser beam with a Gaussian profile is widely used in weak measurement, and the corresponding spatial wave function is , where is the spatial variance of the Gaussian beam. Obviously, because of the rotational symmetry of the Gaussian beam. Thus, it is necessary to devise an appropriate pointer for rotation measurement. In addition, as revealed from Eq. (1), increasing the variance of the OAM of the pointer is beneficial for higher precision on measuring angular rotations.
For this reason, we employ the -order HG beam as initial pointer for the measurement of angular rotations, where and are the transverse mode numbers of the component and component, respectively. Though the -order HG beam takes zero-mean OAM , its variance of OAM distribution increases quadratically with the mode numbers
B. Operator Algebra of HG Beams
In detail, we can relate the HG beams to the harmonic oscillators (HOs) here. The wave function of the HG beam is 
From the view of quantum mechanics, wave function is the time-independent solution for the Schrödinger equation of 2D harmonic oscillators:
Here, we denote the -order HG beam state as
HG beam states are dependent, and their wave functions are the solutions of the paraxial wave equation
Moreover, it is easy to determine that the momentum operators and for 2D HO states, which have the same expression as those of HG beam states. In other words, the result of inflicting displacement on the -order HG beam state is the same as that of the -order HO state. Then we can derive the OAM operator by Eq. (17):
C. Saturating Quantum Limit via Projective Measurement
Taking the initial pointer state as , then the final pointer state can be calculated as
In a complete metrological process, a classical measurement strategy is necessary for the final state to read out the unknown parameters . In this case, the final estimating precision of angular rotation is evaluated by the classical Fisher information (CFI):
Figure 2.Lower bound of estimation variances
3. EXPERIMENTAL SCHEME
A. Experimental Materials and Setup
To experimentally verify that the enhancement on rotation measurement with an HG pointer, we setup a practical optical system to implement it as shown in Fig. 3. A light beam from a laser working at 780 nm is expanded and then converted to -order HG mode via a spatial light modulator (SLM) and a spatial filter system . Here the beam’s polarization states and are set as the basis of the two-level system. We employ a Dove prism to introduce a pair of inverse weak rotations for the and components in a polarizing Sagnac interferometer. The Pauli operator is denoted as . In the post-selected weak measurement scheme, pre-selection and post-selection states are nearly orthogonal to amplify the estimated parameter [17,31–35]. Thus, we choose and , where . Thus, the weak value . Considering measurement samples (effective measured number of photons in the experiment), the minimum detectable rotation given by QCR bound is
Figure 3.Diagram of the experimental setup. (a) The
The laser employed in this experiment is a distributed Bragg reflector (DBR) single-frequency laser of Thorlabs Inc. (part number: DBR780PN), which works at 780 nm with 1 MHz typical linewidth. To generate the high-order HG beams, we used an SLM of Hamamatsu Photonics (part number: X13138-02), which has pixels with 12.5 μm pixel pitch. The focal length of the Fourier lens in the system is 5 cm. A 200 μm square pinhole is used as the spatial filter.
In this work, we set up a polarized Sagnac interferometer to introduce a pair of inverse rotation signals for horizontal and vertical polarization states. However, the extinction ratio of the reflection port of the polarizing beam splitter (PBS) cube (part number CCM1-PBS25-780/M of Thorlabs Inc.) is from 20:1 to 100:1 in practice, which deteriorates the degree of polarization of the output beam. Hence, we added a polarizer behind the reflection port of the PBS to improve the degree of polarization. A half-wave plate (HWP), whose optic axis is at 45° angle with respect to the horizontal plane, was employed to exchange the polarization states in the clockwise loop and counterclockwise loop.
In our experimental scheme, the Dove prism is driven by piezoelectric transducer (PZT) chips, and we exert an cosine driving signal on the PZT to generate the tiny rotation signal. Here, we pasted four PZT chips on the reflection side of the Dove prism as illustrated in Fig. 3(b). The four PZT chips are arranged as a array, and the vertical distance of this array is 10 mm. Here we used the NAC2013 PZT chip of the Core Tomorrow company, which shifts 22 nm with 1 V driving voltage. We exert cosine signals (with 1/2 amplitude DC bias) on the PZT chips, where a -phase difference is introduced between the top-row PZT chips and bottom-row PZT chips. Therefore, a cosine driving signal with 1 V peak-to-peak voltage corresponds to a 2.2 μrad maximum rotation of the Dove prism, which leads to a 4.4 μrad transverse rotation of the input light beam. Besides this cosine driving signal, the initial rotation bias of the Dove prism, which is denoted as and on the milliradian (mrad) scale, is non-negligible. Thus, the total rotation is , and it is easy to determine that .
After the post-selection, another SLM is employed to project the final pointer to carrying state with a Fourier transfer lens and a spatial filter from single-mode fiber (SMF) coupling detected photons to an avalanche photodiode (APD, part number: APD440A of Thorlabs Inc.), which has maximum conversion gain of and 100 kHz bandwidth. Then the detected voltage signal is analyzed by the spectrum analyzer module of Moku:Lab, which is a reconfigurable hardware platform produced by Liquid Instruments. The resolution bandwidth (RWB) of spectrum analyzer was 9.168 Hz in our experiment, which leads to the detecting time of .
B. Experimental Results
In practice, before exerting the driving signal, we project the final pointer to state to fix the measured photons number for different HG pointers. In the experiment, the detected power of the APD is fixed as at the beforehand projection step. Theoretically, the detected optical power is given as , where is the energy of a single photon at and the detecting time duration in our experiment. Thus, the effective measured photons number is fixed as in this experiment.
Then, exerting the driving signal on the PZT chips and projecting the final pointer to , the detected photons number is
In practice, we detected the peak level from spectrum analyzer at 1 kHz to demodulate the amplitude of rotation signal. Generally, the peak level consists of three parts: signal level, shot-noise floor, and electrical-noise floor, which is denoted as . Here, is the signal level, is the shot-noise level, and they vary with the different HG modes. In our experiment, the electrical noise level is a constant value in the experiment, which was detected without inputting light on the APD. The detected level of the total noise floor with a -order HG beam is . Thus, the detected signal-to-noise ratio with a -order HG beam in our scheme is obtained as
Figure 4.Detected electrical spectrum of HG11 to HG66 modes at 500 Hz to 5 kHz. The driving voltage of the PZT is 5 V, which corresponds to 22 μrad rotation signal. The first line is the spectrum of the electrical-noise floor of the APD detector, which is detected without input light on the APD.
As is shown in Fig. 4, the electrical noise is , and the detected shot-noise level of the -order HG beam can be calculated by . We list the results in Table 1. Experimental Results of Detected Noise Levels Total detected noise levels in the APD.
HG mode HG11 HG22 HG33 HG44 HG55 HG66 Noise level 41.43 μV 44.43 μV 49.39 μV 54.08 μV 57.27 μV 61.70 μV Shot-noise level 5.68 μV 8.68 μV 13.64 μV 18.33 μV 21.52 μV 25.95 μV
Experimental Results of Detected Noise Levels
Total detected noise levels in the APD.
Figure 5.Experimental results. (a) Detected peak signal level of the HG11, HG33, and HG55 modes at 1 kHz. (b) Detected signal-to-noise ratio of the HG11, HG33, and HG55 modes. The driving voltage of the PZT increases from 0 to 2 V, which corresponds to 0–8.8 μrad rotation signal.
A. Technical Advantages of a Weak Value
In the theoretical frame, the post-selection is employed for individually reading out the measurement parameters from the pointer, and the precision enhancement comes from the mode entanglement of the HG pointer, but not the weak values. Thus, our main conclusion still holds in the post-selection-free scheme. In the experimental scheme, we still employed the weak value amplification technology, though the weak value takes no enhancement for the theoretical minimum detectable rotation in Eq. (28) because the detected number of photons is attenuated by the post-selection, where is the number of photons before post-selection. However, the weak value amplification technology has been proved efficient for suppressing technical noises, such as reflection of optical elements  and detector saturation [36,37]. The detector saturation is non-negligible in our experiment since the saturation power of our APD detector is only 1.54 nW. Considering the projection demodulation of SLM, only 10% of photons can be modulated on the first-order diffraction, so the maximum efficient received power of our detector is about 154 pW, which is easily saturated without post-selection. For example, the efficient detected light power in our experiment is , and the post-selection angle . Therefore, for the post-selection-free scheme, an detected light power is needed to achieve the same precision of the post-selected scheme, which is far larger than the saturation power of the APD detector.
B. Rotation-Coupling Weak Measurement for Hamiltonian Estimation
Though we only investigate the enhanced measurement on angular rotation, the precision enhancement of employing HG pointers can be applied in various missions in quantum physics. The most obvious application of our scheme in quantum physics is the estimation of Hamiltonian [38,39]. In this case, we do not only concentrate on the interaction strength parameter but we are also interested in the information of operator . For a two-level system, the unknown operator can be represented as , where is the direction vector of the measurement operator, and , where , , and are Pauli matrices. Thus, there are two unknown parameters and to be estimated for identifying the operator . As we calculated in the theoretical model, the final pointer’s state in our post-selected scheme is . Then the QFI of parameter can be calculated as . Combined with the QCR inequality , the estimation precision of parameter satisfies the uncertainty relation
C. Rotation-Coupling Weak Measurement for Monitoring the Quantum Bit
Besides Hamiltonian estimation, our precision-enhanced method also has the potential for monitoring the quantum bit, which is a vital mission in quantum metrology . Generally, the state of an arbitrary quantum bit (qubit) can be represented as
As is illustrated in Fig. 6, an ancillary device (pointer) is employed to monitor the quantum system via a weak interaction procedure, which is described by the von Neumann measurement theory  via an impulse Hamiltonian (here we take the monitoring of qubits on the basis of , for example):
Figure 6.Schematic of monitoring quantum bits.
Moreover, we express the HG beams via harmonic oscillator model, which has been widely used in quantum computation and metrology topics, such as superconducting qubits [43,44] and optomechanics [45–47]. Thus, our theoretical model can be applied in such scenarios naturally and can provide a significant method in quantum metrology.
In summary, we have implemented a practical scheme for measuring the tiny rotation by employing an -order HG pointer in a post-selected weak measurement scheme, where the precision limit is theoretically improved by a factor of . Experimentally, we demodulate the angular rotation parameter via a single projective measurement, and precision up to 0.89 μrad is achieved with -order HG beams. Moreover, we have found that the precision enhancement of the rotation-coupling method with an HG pointer still holds in a wide range of applications in quantum physics, such as Hamiltonian estimation and monitoring qubits. Thus, our results constitute valuable resources not only for measurement and control of light’s angular rotation in optical metrology but also for sensitive estimating and control of evolution procedure in quantum physics.
Acknowledgment. We thank Lijian Zhang and Kui Liu for the helpful discussions.
APPENDIX A: DERIVATION OF QUANTUM LIMITS IN A POST-SELECTED WEAK MEASUREMENT SCHEME
In this section, we give the calculation details about the quantum limits of weak interaction parameters , , and in post-selection weak measurement, where is the weak interaction strength and , and are the Hamiltonian parameters of the two-level system. In this case, the operator is given as the generalized formalism , , where , , and are the Pauli matrices. The weak interaction procedure described by an impulse Hamiltonian is , and then the evolution operator of the weak interaction procedure can be calculated as
The initial state of the whole system before weak interaction can be denoted as . Then the final state of the whole system after weak interaction and post-selection can be calculated as
For a parameterized state , its corresponding QFI for a single parameter can be given by [
Taking , we can calculate
In this work, we investigate two types of pointer in the weak measurement scheme, a Gaussian pointer and an HG pointer. For the Gaussian pointer, the measurement parameters are coupled to the pointer’s spatial displacement, which leads to a constant QCR bound. For the HG pointer, its mode numbers in the direction and direction are respectively and , and the QCR bound is improved with pointer’s mode numbers and . Moreover, the 2D HG pointer theoretically equals a 2D harmonic oscillator (HO), which has been proved in the main text. Therefore, our results can be extended to any HO-formalism pointer here. Coupling the measurement parameters to the pointer’s displacement ( direction), the QCR bound is improved by factor , which is enhanced linearly. Coupling the measurement parameters to the pointer’s rotation, the QCR bound is improved by factor , which is enhanced quadratically. Here we calculate these three QCR bounds for parameters , , and at , , and , respectively. Without loss of generality, we normalize the value of pointer’s spatial uncertainty to and set the measured samples number . In Fig.
Figure 7.QCR bounds of the measurement parameters. (a) QCR bounds of parameter
Figure 8.QCR bound of
APPENDIX B: DERIVATION OF QFI FOR MONITORING QUBITS WITH A WEAK MEASUREMENT SCHEME
As is elucidated in the main text, the ancillary pointer is adopted to monitor the qubit via a weak interaction procedure . Then the measurement information of the Pauli operator on the qubit is transferred to the pointer shift of the ancillas via a translation operator , and the final state of the whole system is
To calculate the QFI in Eq. (
The expression of QFI in Eq. (
APPENDIX C: GENERATION METHOD OF HG BEAMS AND EXPERIMENTAL RESULTS
Traditionally, a mode cleaner cavity is necessary for generating high-order HG beams [
In this scheme, the light beam from a 780 nm DBR laser was expanded to a 8.6-mm-width Gaussian beam by a fiber coupler. The complex amplitude of expanded Gaussian beam is denoted as . Then, inputting this light into a SLM, where the phase map is displayed, the output amplitude of the SLM can be denoted as
By employing a 4 spatial filter system with an aperture at the first-order diffraction point, we can generate any target beam amplitude with the displayed phase map on the SLM. Here we illustrate the experimentally generated results of HG00 to HG66 beams and calculate the corresponding purity in the above table (Table Experimental Results of Generated HG Beams
HG Mode HG00 HG11 HG22 HG33 HG44 HG55 HG66 Phase map Simulation Experiment Purity 97.87% 94.91% 93.76% 92.54% 90.17% 88.24% 86.75%
Experimental Results of Generated HG Beams
APPENDIX D: IMPLEMENTATION OF PROJECTIVE MEASUREMENT
In our experimental scheme, the rotation signal was finally detected by projective measurement [
From Eq. (
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