Elliptic polarization measurement is a technology that accurately measures various physical properties by detecting polarization changes before and after the sample^[1,2], including optical constants, film thicknesses, lattice vibration modes, and local atomic structures^[3]. However, in practical applications, the characteristics of a material could be affected by the strong probe light. For example, photosensitive materials exposed to strong light may produce photochromic, polymerization, fracture, deformation, and other changes in physical properties, and the light intensity reaching the damage threshold of the material may cause irreversible photo-thermal damage and others. Hence, enhancing detection accuracy under low incident light intensity is an essential topic in the field of precision measurement.

The entangled photon source based on the unique quantum feature provides a feasible method for accurately measuring the optical properties of a material using ultralow light intensity illumination. There are several ways to introduce entanglement into the precision measurement. The best known is to use an N-photon entangled state instead of a classical N-photon state to pass through the target sample, with all photons participating in the parameter estimation process. In this case, the standard quantum limit can be improved by a factor of N under equivalent light intensity^[4–7]. Additionally, entanglement is often introduced as an auxiliary measurement technique, where the ancilla state is entangled with the probe but does not itself participate in the estimation^[8–10]. In a non-ideal transmission channel, it has been proven that the entanglement-assisted quantum metrology can achieve higher measurement accuracy for the estimation of channel-correlated parameters, such as the amplitude damping possibility and depolarizing noise^[11,12].

Our situation is somewhat similar but not exactly the same as the analyses in the previous research. On the one hand, the relative phase retardance of the target sample determines the final photon counts we can obtain, i.e., the estimated parameter can be related to the transmission rate of the channels. Therefore, the previous analysis of entanglement-assisted quantum metrology can be applied to the polarization measurement. However, on the other hand, the polarimeter studies the unitary evolution of the polarization state rather than the direct generation and annihilation of the photon numbers caused by the channel noise. Only by projection measurement can we obtain various photon numbers determined by the birefringence of the samples. This measurement is bound to destroy the fixed relative phase between each possible state, turning the previously pure state into a mixed state. Therefore, the application of entanglement-assisted measurement in this scenario needs to be further discussed. Note that there is little existing research analyzing the basic experimental configurations^[13,14], the general theoretical framework^[15], and the fundamental quantum limit^[16] of the quantum polarimeter (or ellipsometer). However, we have not found works that have directly demonstrated the advantages of using an entanglement-assisted measurement system for the transmitted polarization measurement both by theoretical analysis and the experimental comparison.

In this work, we calculate the Fisher information for the coherent state and the maximally entangled photon pair with one photon used as an auxiliary bit in a polarizer-sample-analyzer (PSA) polarimeter. The introduction of the Senarmont method is considered to improve the measurement precision of certain samples that cannot be estimated by the PSA polarimeter. Additionally, in combination with the experimental setup, we give the maximum likelihood estimation for calculating the sample’s phase retardance in a lossy channel. Based on the calculation, we select suitable measurement bases for both the established entanglement-assisted quantum polarimeter and the traditional polarimeter utilizing a coherent laser source and systematically compare the accuracy and precision of each system.

In Fig. 1(a), we show a typical configuration of a classical PSA polarimeter in which the polarization of the input photon will be first determined by the input polarizer, then changed by the anisotropic sample, and finally figured out by a rotating analyzer. By analyzing the polarization changes caused by the sample, we can derive its anisotropic optical properties. Analogous to this classical PSA polarimeter, we use the maximally polarization-entangled photon source to build an entanglement-assisted polarimeter. The basic configuration is shown in Fig. 1(b), where the probe photon measures the sample in the upper path while the auxiliary photon entangled with the probe photon passes through the lower path.

Based on the parameters shown in Fig. 1(a), the angles of the polarizer and the analyzer are fixed at 2h1 and 2h2 [h1 and h2 are the angles of the half-wave plates (HWPs) that are used as the polarizers], respectively, and the sample’s phase retardance is marked as δ while the optical axis is set to θ. Then, considering an input single-photon with a polarization fixed at the horizontal direction, that is, |H⟩=(10)T, we can derive the output polarization state by calculating its Jones vector, $$\begin{gathered} Eout=(1000)(cos2h2sin2h2−sin2h2cos2h2)A⁢(cosθ−sinθsinθcosθ)(100eiδ)(cosθsinθ−sinθcosθ)S \\ ×(cos2h1sin2h1−sin2h1cos2h1)P(Ein0)Ein. \end{gathered}$$(1)

The output photon intensity is therefore shown as Iout=|Eout|2=12(1+cosacosb+cosδsinasinb)|Ein|2≡P1Iin.(2)

Equation (2) indicates that a single photon has a probability P1 of successfully transmitting through the given configuration and being collected by the detector in an ideal transmission channel. Conversely, the probability of the photon failing to pass through the transmission channel is P2=1−P1.

Based on the above analysis, after the projection measurement, the state of the single photon can be described as ρ=P1|1⟩⟨1|+P2|0⟩⟨0|.(3)

The Fisher information for a quantum state ρ can be calculated as^[17]: ∑i(∂αλi)2λi+2∑i≠j(λi−λj)2λi+λj|⟨ei|∂αej⟩|2, where λi and |ei⟩ are the eigenvalue and eigenvector of the density matrix, respectively. For the diagonal density matrix in Eq. (3), the Fisher information reduces to Fs|δ=∑m=121Pm(∂Pm∂δ)2=sin2asin2bsin2δ1−(cosacosb+cosδsinasinb)2.(4)

In between, a=4h1−2θ and b=4h2−2θ stand for the relative angle difference between the analyzers and the sample’s optical axes. In real applications, ideal single photons are difficult to prepare and are therefore mainly used for analyzing the principle of the experimental configuration. Instead, the coherent state is the most common situation when we use laser input in the classical polarimeter.

When the coherent state passes through the configuration in Fig. 1(a), the transmission amplitude of a single input photon should still be η(δ)=P1 in an ideal channel with no extra photon loss. Therefore, a coherent state |α⟩ with an average photon number of |α|2=N will become |ηα⟩ after passing through this transmission channel^[18], the possibility that the detector receives m photons is Pm=|⟨m|ηα⟩|2=e−η2Nη2mNmm!.(5)

Then, the Fisher information is shown to be Fc|δ=∑m=0∞1Pm(∂Pm∂δ)2=N=1sin2asin2bsin2δ2(1+cosacosb+cosδsinasinb).(6)

Here, the average photon number of the coherent state is set to 1 since we need to compare it with the entanglement-assisted single photon detection.

Then, let us consider the Fisher information of the entanglement-assisted polarimeter based on Fig. 1(b). The polarization-entangled photon source is chosen to be one of the four Bell states, described as |ϕ+⟩=1/2(|HV⟩+|VH⟩). In Fig. 1(b), we regard JS=S (Jones matrix of the retarder) and JI=I (unit matrix) as the evolution operators of the signal (upper) and idler (lower) paths, respectively, so that the two-photon state after passing through the sample is |ϕsample+⟩=JI⊗JS|ϕ+⟩=12(A|HH⟩+B|HV⟩+C|VH⟩+D|VV⟩).(7)

Here, we have A=D=−sinθcosθ(eiδ−1), B=sin2θ+eiδcos2θ, and C=cos2θ+eiδsin2θ. A set of HWPs and polarizing beam splitters (PBS) is employed in both the signal and idler paths to perform the joint projection measurement [shown as the analyzer in Fig. 1(b)]. Similar to Fig. 1(a), assuming the HWPs have an optical angle of h1 in the signal path and h2 in the idler path, the photon state will collapse to a mixed state in the occupation number representation, i.e.,  ρ=Pa|11⟩⟨11|+Pb|10⟩⟨10|+Pc|01⟩⟨01|+Pd|00⟩⟨00|.(8)

Here, we have Pa=Pd=(1−cosa′cosb+cosδsina′sinb)/4 and Pb=Pc=(1+cosa′cosb−cosδsina′sinb)/4. In between, Pa, Pb, Pc, and Pd stand for the probabilities of obtaining one photon in each detector, obtaining one photon only in the signal/idler path, and no photon detected, respectively. a′=4h1+2θ and b=4h2−2θ. Using the above expressions, the Fisher information for the mixed state is Fq|δ=∑m=a,b,c,d1Pm(∂Pm∂δ)2=sin2a′sin2bsin2δ1−(cosa′cosb+cosδsina′sinb)2.(9)

Compare Eq. (9) with Eq. (4), the entanglement-assisted polarimeter has the same precision as the single-photon input in an ideal transmission channel, which is quite reasonable since entanglement-assisted detection involves only one photon of the photon pair entering the sample, while the other only serves as an auxiliary detection bit. Additionally, compare Eq. (9) with Eq. (6), the maximum value of the entanglement-assisted polarimeter is fixed to 1 when we use the joint measurement bases a′=b=π/2 or a′=π/2,b=3π/2, while the maximum value for coherent state is sin2(δ/2) (a=b=π/2) or cos2(δ/2) (a=π/2,b=3π/2). This result shows that although the entanglement-assisted system has no ability to enhance the upper limit of the measurement precision, but it helps to achieve higher measurement precision for samples with various relative phase retardance.

Equations (6) and (9) also tell another truth that the PSA polarimeter is not able to precisely measure the samples with the phase retardance of δ=kπ(k∈Z) because both formulas remain zero for arbitrary measurement bases. To solve this problem, we can simply introduce a 0° quarter-wave plate (QWP) after the target sample in the signal path. This approach is called the Senarmont compensation method, where the compensated QWP is often fixed at an angle parallel to the polarizer^[19].

With the introduction of the compensator, the output polarization state is shown to be |ϕsample+′⟩=I⊗(UQWPS)|ϕ+⟩, and the probabilities of the four cases after making joint projective measurements are $$\begin{gathered} Pa=Pd=14{1−cos4h2[cos4h1cos2δ2+cos(4h1+4θ)sin2δ2] \\ −sin4h2sin(4h1+2θ)sinδ}, \\ Pb=Pc=14{1+cos4h2[cos4h1cos2δ2+cos(4h1+4θ)sin2δ2] \\ +sin4h2sin(4h1+2θ)sinδ}. \end{gathered}$$(10)

Then, the Fisher information in this case is calculated to be Fq|δ′=sin2a′(cosδsin4h2−cos4h2sinδsin2θ)21−[cos4h2(cosa′cos2θ+sina′sin2θcosδ)+sin4h2sina′sinδ]2.(11)

Different from the PSA polarimeter, Eq. (11) shows that we can obtain the maximum value of 1 for any unknown phase retardance δ by flexibly selecting the measurement basis h1 and h2. In the following description, we will mark the PSA configuration with the Senarmont compensation method as the PSCA configuration, where “C” is the compensator for short.

Generally speaking, transmission and detection losses are inevitable in a quantum system. In this section, we discuss the parameter estimation under lossy transmission and detection processes based on the system configuration. Given that we are dealing with a short free-space propagation, it is postulated that the main sources of channel loss are the transmittance of the optical components, the coupling efficiency of the fiber, and the detection efficiency of the photodetectors. These factors are collectively represented by a constant, denoted as γ, which is independent of the length of the channel. The amplitude damping channel, which stands for the energy dissipation with a possibility of γ, is appropriate for describing this process.

Following the Kraus operators of the amplitude damping channels^[20], and with γ1 and γ2 representing the loss rates of the signal and idler paths, respectively, the Kraus operators for the evolution of the photon pair are given by $$\begin{gathered} E00=(100001−γ200001−γ100001−γ11−γ2),E01=(0γ2000000000γ2(1−γ1)0000), \\ E10=(0γ2000000000γ2(1−γ1)0000),E11=(000γ1γ2000000000000). \end{gathered}$$(12)

The output density matrix considering the channel loss is $$\begin{gathered} ρ2=∑i,j=0,1EijρEij†=P0|00⟩⟨00|+Pi|01⟩⟨01|+Ps|10⟩⟨10|+Pc|11⟩⟨11| \\ =[γ1+γ22+(1−γ1)(1−γ2)Pa]|00⟩⟨00|+{1−γ22[1−2(1−γ1)Pa]}|01⟩⟨01| \\ +{1−γ12[1−2(1−γ2)Pa]}|10⟩⟨10|+[(1−γ1)(1−γ2)Pa]|11⟩⟨11|. \end{gathered}$$(13)

In Eq. (13), γ1 and γ2 are constants that can be obtained before the sample measurement. Supposing that we set the sample axis at θ=0°, while two measurement bases are fixed at h1=0° and h2=45°, then we have the possibility Pa0=1/2. In this situation, the number of photons obtained from two single detectors and the joint measurement are marked as N10, N20, and Nc0, respectively. Then, we can obtain the loss rate from Nc0N10=PcPs+Pc=θ=0,h1=0,h2=π/41−γ2andNc0N20=PcPi+Pc=θ=0,h1=0,h2=π/41−γ1.(14)

In our system, the optical paths at the signal and idler sides are basically the same. To simplify the calculation, we mark γ1≈γ2≈γ. Then, the Fisher information for a quantum polarimeter with a lossy channel is Fq|δγ=(∂Pa∂δ)2=(1−γ)2[γ+2Pa(1−γ)]Pa[1−2Pa(1−γ)][γ+Pa(1−γ)2].(15)

Using the different expressions of Pa mentioned in Eqs. (8) and (10), we can obtain the Fisher information distribution under different measurements based on combinations in the PSA and PSCA configurations, respectively.

Additionally, the parameter estimation is obtained by the log-likelihood function. If we detect N1 photons in the signal path, N2 photon in the idler path, and Nc coincidence photons, then the likelihood function can be established as L(δ,θ,γ1,γ2,h1,h2|N0,Ns,Ni,Nc)=P0N0PsNsPiNiPcNc.(16)

Here, Ns(i)=N1(2)−Nc describes the number of events in which only signal (idler) photons reach the corresponding detectors. N0 is the number of events that no photon reaches either of the two detectors. Using N to stand for the total number of events, then we have $$\begin{gathered} N=Ns+NcPs+Pc=2N11−γ1=2N21−γ2, \\ N0=N−Ns−Ni−Nc=γ1N11−γ1+γ2N21−γ2+Nc. \end{gathered}$$(17)

To get the maximum likelihood estimation, ∂δ(logL) should be equal to 0. When γ1≈γ2≈γ is satisfied, the estimation of the phase retardance is shown below, Pc=(1−γ)Nc2N1(2)=(1−γ)2Pa.(18)

For a PSA quantum polarimeter, the estimator yields δ˜=arccos[2NcN1(2)(1−γ)−1+cos(4h1+2θ)cos(4h2−2θ)sin(4h1+2θ)sin(4h2−2θ)].(19)

For the PSCA quantum polarimeter, the estimator can be derived from the data fitting of the following equation: $$\begin{gathered} 1−2NcN1(2)(1−γ)−cos(4h1+4θ)cos4h2cos2θ \\ =sin(4h1+2θ)(cos4h2sin2θcosδ˜+sin4h2sinδ˜). \end{gathered}$$(20)

Based on the two equations above, using different combinations of the measurement bases (h1,h2), together with the selected angle θ of the sample axis, we can derive the phase retardance δ by the photon numbers detected in the single path N1(2) and dual paths Nc.

The experimental setup of the quantum polarimeter is sketched in Fig. 2(a). In the left triangular area, a high-quality polarization-entangled photon source is generated via the spontaneous parametric down conversion (SPDC) process in a type II phase-matched PPKTP crystal (Raicol Crystals Ltd.) embedded in a Sagnac interferometer^[21–23]. A continuous-wave (CW) laser (Kunteng Quantum Technology Co. Ltd) with a 2 mW pump power at 405 nm is focused on the center of the PPKTP crystal by two symmetrical parabolic silver mirrors with a focal length of 101.6 mm. The PPKTP crystal has dimensions of 1mm×2mm×20mm, with a polling period of 10.02 µm and is set at 23.6°C with a temperature stability of ±0.002°C. The generated photon pairs then pass through separate polarization analysis parts and are finally detected by single-photon avalanche diodes (Excelitas, SPCM-AQRH) with a detection efficiency of 60% at a wavelength of 810 nm.

Before evaluating the performance of the quantum polarimeter, we tested the quality of the polarization-entangled photon source. Notably, high fidelity is fairly important in this measurement since the theoretical model is based on a preconfigured, maximally entangled state. By removing the sample from the idler path, we measure the two-photon polarization interference curve and perform quantum state tomography to characterize the quality of the generated polarization-entangled state in Figs. 2(b) and 2(c). The raw interference visibilities are calculated as 99.75%±0.01% and 99.12%±0.01% for the H-base (fixed at 0°) and the D-base (fixed at 22.5°), respectively, far beyond the 71% shown in the Bell inequality. Moreover, the S-parameter is calculated to be 2.754±0.006 in 10 s, violating the CHSH inequality by 122 standard deviations^[24]. The fidelity between the experimental state and the ideal Bell state is estimated to be 99.79%±0.013%. Additionally, the coincidence count is 2×104 counts/s (cps) with a 1.6 ns coincidence window. Total collective efficiencies are measured to be 14.23% and 13.68% for the control and measurement arms, respectively. The above series of properties show that the current photon source is fairly close to the maximum entangled state, providing a solid basis for making an accurate birefringence measurement of target samples.

Using the high-quality entangled source, we can move one step further to show the process and result of the quantum polarimeter. Based on Eq. (19), the main constraint of the joint measurement basis is sin(4h1+2θ)sin(4h2−2θ)≠0 for a PSA polarimeter. Note that the parameters {h1,h2,θ} can be selected in the measurement process, while N1(2)(1−γ) is a constant photon number for a maximally entangled photon state in a fixed system. Therefore, we can change either the measurement basis h1(2) or the sample axis θ to obtain a series of the output coincidence photon numbers Nc. Then, the estimated relative phase retardance δ˜ can be derived from the curve-fitting of the Nc. Similar fitting curves can be derived from Eq. (20) for a PSCA system.

A unique feature for the quantum polarimeter is the ability to non-locally control the polarization of the light incident on the sample. Here, we choose two commercial wave plates, a multi-order QWP (808 nm, MFOPT Ltd.), and a true zero-order HWP (808 nm, Thorlabs) as the target samples. In Fig. 3, we fixed the measurement basis in the signal arm while changing the basis in the idler arm. The fixed bases are chosen to be H-base (h1=0°, purple line) and D-base (h1=22.5°, blue line), respectively. Especially in the right four sub-figures, the left y-axis shows the estimated phase retardation and marked by the solid lines in the upper part of each sub-figure, while the right y-axis shows the relative errors defined as Δδ=|δexp−δstd|/δstd and described by dotted lines with star marks. Additionally, the black dotted lines are regarded as the reference values, which are measured from a commercial polarimeter with strong light input. For each sample, we record the measurement results of a single test, results with varying sample axes, and results with long time durations. All the data are summarized in Table 1.

In Table 1, the upper values in the second and third columns show the average and standard deviations of the estimated retardance, and the lower numbers in the brackets are the average relative errors. The single test is equivalent to one point in the varying sample axis measurement with the axis of the sample fixed at 22.5°. The measurement results in the table exhibit the feasibility of measuring the phase retardance using a quantum polarimeter. Two things that may be noticed are as follows: 1) For either the HWP or the QWP, the rotating sample axis induces unstable results [Fig. 3(b) and 3(e)] compared to the long-time measurements with the fixed sample axis [Figs. 3(c) and 3(f)]. This is primarily due to the fact that if the sample is not perfectly aligned with the incident light during rotation, it can lead to variations in the angle of incidence as well as changes in the measurement point; 2) The measurement precision of the QWP is apparently higher than the HWP measurement. This can be seen in the standard deviation values for the long duration test (the bold text in Table 1) and the long error bars in Fig. 3(b) compared with Fig. 3(e). The explanation can be found in Eq. (3). Since the PSA configuration cannot measure samples with a phase retardance of δ=kπ(k∈Z), we need to employ the Senarmont method to solve this problem in the next section.

In this section, we show the measurement results for the commercial HWP and the QWP with different experimental configurations and compare the performance between each other. The configurations contain the PSA quantum polarimeter, the PSCA quantum polarimeter, and the PSCA classical polarimeter with a continuous laser source. All polarimeters are exposed to the same incident photon level. In Fig. 4, they are shown by the purple, green, and red lines, respectively. Also, to clearly show our measurement results, we conclude all the averages and standard deviations of the measured retardance values in Table 2.

PSCA vs PSA First, let us compare the performance between the two entanglement-assisted polarimeters. In Fig. 4, the PSCA configuration (green line) exhibits apparently short error bars, thus exhibiting higher precision compared to the PSA system (purple line) in the HWP measurement. We also mark the standard deviation of the long-time multiple measurements with bold text in Table 2 to show this comparison. This problem has been already raised in the last section, and here, we intuitively demonstrate that the introduction of the QWP compensator indeed has the ability to enhance the measurement precision for the samples with kπ retardance. Our experimental result shows about five times the precision enhancement with the introduction of a compensator in the HWP measurement and also shows higher precision in the QWP measurement.

Quantum vs Classical Second, let us compare the performance between the entanglement-assisted quantum polarimeter and the classical polarimeter. The comparison is shown by the green and red lines in Fig. 4, as well as the first and second columns in Table 2. It can be seen that, under the same illumination level and PSCA configuration, the quantum system shows slightly higher accuracy and precision than the classical system for both the HWP and the QWP measurement. The relative errors in the table are italicized to directly show the comparison. Based on the theoretical analysis, both systems should have the same measurement precision. However, due to the noise-resistant property of the simultaneously generated photon pair in the quantum polarimeter, we are able to obtain less background noise and defend against the additional perturbation in the measurement process, which contributes to higher measurement performance. As we can see in Fig. 4(a), with a 1.6 ns coincidence window and 10 s acquisition time, the green line shows better visibility than the red line under the same sample axis angle and measurement basis, which is direct evidence that the quantum polarimeter reduces background noise.

Based on the above comparison, the PSCA-typed quantum polarimeter has the best performance within the three configurations. The system accuracy can be exhibited from the average relative errors, which are below 2% for all the tests with a varying sample axis and are within 0.5% for the long duration test, demonstrating the nanometer accuracy in the phase retardance measurement. The system stability comes from the standard deviation of the long duration test, which is all within 0.4% variations compared with their respective average values.

In summary, we give the theoretical analysis and the experimental demonstration for the advantages of the entanglement-assisted quantum polarimeter.

In the theoretical part, we show that the Fisher information of the maximally entangled state in an entanglement-assisted PSA polarimeter, although it has no enhancement on the upper limit, has the advantage of obtaining higher precision than the coherent state for samples with various retardances. Moreover, to solve the confinement of the PSA configuration, we introduce the Senarmont compensation method into the quantum polarimeter and demonstrate that the PSCA configuration can measure arbitrary unknown retarders with the highest precision. Furthermore, in order to accurately estimate the phase retardance in real experiments, we calculate the maximum likelihood estimation in a lossy channel, giving the expression of the estimator that was later used in the experimental analysis process.

In the experimental part, we use a high-quality polarization-entangled photon source to build an entanglement-assisted quantum polarimeter. Taking the commercially available HWP and QWP as samples, we demonstrate that the quantum PSCA polarimeter possesses the highest accuracy and precision in comparison to the classical PSCA and the quantum PSA polarimeter, owing to its noise-resistant characteristics and the introduction of the Senarmont compensation method. Furthermore, leveraging the non-local properties of the entangled photons, we have shown that the quantum polarimeter has the capability to non-locally control the polarization of the incident light.

To conclude, in this work, we describe a detailed theoretical framework for parameter estimation in a two-way entanglement-assisted quantum polarimeter and further established an experimental system with nanometer-level accuracy in retardance measurement. The polarization entanglement and the photon correlation in the two-way system bring unique advantages to our system. It leverages non-locality through basis transformations, enabling remote manipulation of polarization states in interference-sensitive or inaccessible environments. It provides intensity monitoring and feedback using single-photon counting rates as references for light source stability. Also, it demonstrates high measurement accuracy and resistance to background noise through coincidence measurement at extremely low-illumination regimes. These advantages suggest the possible applications in the photosensitive material detection, fragile bio-process observation, and remote control.