Ultrasensitive and real-time detection of chemical reaction rate based on the photonic spin Hall effect

A chemical reaction is usually accompanied by easily observed physical effects, such as the emission of light, the shift of the optical spectrum, or the change in chirality.¹ All known life-forms show specific chiral properties in chemical structures as well as macroscopic anatomy, development, and behavior.² Therefore, the real-time monitoring of chemical reactions has attracted an enormous amount of interest. Generally, determination of the reaction rate by a chemical method may partly cause impact on the reactants, which increases the difficulty and the complexity of the measurement. However, using the optical method to study the properties of a solution is helpful to maintain the original chemical structures. As polarized light passes through a chiral molecule with optical activity, the plane of polarization will be rotated clockwise or anticlockwise depending on the chirality of molecules.^3–5 Optical rotation has become one of the most useful tools to study the characteristics of chiral molecules. A polarimeter is a standard scientific instrument which is commonly used to measure the angle of rotation caused by passing polarized light through chiral molecules. However, due to the quantum noise of a coherent light source and the limited extinction ratio of polarizers, the accuracy is always lower than 10⁻³ degree.⁶ Moreover, the measurement of chemical reaction inevitably involves manual adjustment of optical elements, such as polarizers and wave plates, resulting in long manipulation times. However, ultrasensitive and real-time detection of chemical reaction is urgently needed in some applications of chemical analysis and biopharmaceuticals.

In recent years, because of the sensitivity to changes in physical parameters of the system, the photonic spin Hall effect (SHE) has received considerable attention.^7–12 The photonic SHE is generally believed to be a result of an effective spin-orbital interaction, which describes the mutual influence of the spin (polarization) and the trajectory of the light beam. When a linearly polarized light is reflected or refracted on an optical interface, the photonic SHE manifested itself as spin-dependent shifts which are perpendicular to the incident plane. The photonic spin-Hall shifts provide important information of optical interface, and therefore, it can be employed as the pointer in precision measurements.^13–20 However, photonic SHE is a weak effect whose initial shift is only a fraction of a wavelength and therefore cannot be detected directly by conventional optical measurements. By incorporating with quantum weak measurements,²¹ however, the initial shift can be enhanced by nearly four orders of magnitude due to the weak-value amplification techniques. Therefore, the photonic SHE has become a useful tool for high precision measurement of weak phenomena and tiny changes of system parameters.^13,14

In this paper, the photonic SHE is proposed to realize the ultrasensitive and real-time detection of the reaction rate of sucrose hydrolysis. The concept of weak measurements was inspired in the context of quantum mechanics. However, the technique of weak-value amplification also works for classical light.^22,23 What can be measured precisely, however, is not the spin state of photons but the postselection state. In our scheme, the postselection polarization state is modified by the change in optical rotation in the process of sucrose hydrolysis and the initial spin-dependent shift induced by photonic SHE acts as the measurement pointer. The amplified pointer shifts in the quantum weak measurements show a high sensitivity to the change in the optical rotation angle. Due to the weak-value amplification in quantum weak measurements, the high measurement resolution with 1.25 × 10⁻⁴ degree is achieved without any mechanical adjustment of optical elements once the experimental setup is established. Our scheme may have potential applications in the observation of the chemical reaction phenomenon and the detection of the parameters.

Sucrose is the main form of storage, accumulation, and transportation of sugar in plants, and it also plays an important role in human nutrition and health.²⁴ Moreover, sucrose hydrolysis plays an important role in the development of chemical kinetics. The rate of sucrose hydrolysis has been found to be very slow in pure water, while it will be greatly increased during HCl-catalysis. It was found that the inversion of sucrose should be considered as pure hydrogen ion catalysis by investigating the electrolyte effect and the evidence of acid molecular catalysis. The hydrolysis of sucrose in dilute HCl solution is shown in Fig. 1(a). Sucrose is linked by a carbon atom of glucose and a carbon atom of fructose through glycoside bonds, while one covalent bond in sucrose breaks to form two molecules during hydrolysis and then individually form additional hydrogen bonds and alter the water molecule structure in the solution.²⁵ Optical rotation is an important characteristic of chiral molecules.^26–28 A linearly polarized light can be regarded as the superposition of two circularly polarized components. When a linearly polarized light passes through chiral solution, the refractive index of the two circularly polarized lights is slightly different and eventually causes the rotation of the polarization plane. To identify the optical rotation ability of various chiral substances, the concept of specific rotation is introduced as follows:

where α is the optical rotation angle (in degrees), l is the thickness of the sample cell (in dm), and c is the concentration of sugar solution (in g/ml). The specific rotation is a constant for a certain temperature and wavelength of light, which indicates that the optical rotation angle α is associated with the thickness l of the chiral sample solution and the concentration c of the sample solution. With the continuous change in the optical rotation angle α in the sample, the final beam displacement will also change. Therefore, the beam displacement can be affected by the thickness of the chiral sample solution, but the final reaction rate cannot be affected. Different chiral solutions have different rotation angles for the light polarization plane, as shown in Fig. 1(b). The glucose is a right-handed substance, while the fructose is a left-handed substance.

The photonic SHE is generally believed to be a result of an effective spin-orbit interaction, which describes the mutual influence of the spin (polarization) and the trajectory of the light beam. The origin of this spin-orbit interaction lies in the transverse nature of the photon polarization: The polarizations associated with the plane-wave components experience different rotations in order to satisfy the transversality after reflection.⁹ In the spin basis set, the horizontal polarization state $|H=(|++|−)/2$ or vertical polarization state $|V=i(|−−|+)/2$ can be decomposed into two orthogonal polarization components, where |+⟩ and |−⟩ represent the left- and right-circular polarization components, respectively. The incident wave function can be written as follows if we only consider the packet spatial extent in momentum space:

where |ψ_i⟩ is the preselection state. We assume the wave function with Gaussian distribution which can be specified by the following expression:

where w is the width of the wave function. For |H⟩ input polarization, the reflected wave function in the momentum space can be obtained as

Here, +δ and −δ denote the initial pointer shifts for the wave packets of the two spin states, and the term of exp(+iσk_yδ) indicates the spin-orbit interaction at the air-glass interface. The spin-dependent shifts generated by the photonic SHE is given by^29,30

Here, θ_i is the incident angle; r_p and r_s are the Fresnel reflection coefficients for parallel and perpendicular polarizations, respectively; and λ is the wavelength of light. In general, the spin-dependent shifts are much smaller than the beam width δ ≪ w, therefore resulting in a weak interaction.

To accurately characterize the shifts, it is necessary to have measurement sensitivities at the angstrom level, so a signal enhancement technique known as quantum weak measurements is applied. In the procedure of weak measurements, the weak-value amplification is divided into three processes: system preselection, weak interaction between the system (observable) and the measuring pointer, and postselection of the system.⁹ In our weak measurement scheme, the spin-orbit interaction of light on an air-glass surface provides the weak coupling. The enhancement effect is achieved by preselecting and postselecting the polarization states of the incoming photons in states |ψ_i⟩ and |ψ_f⟩. Here, we choose

where Δ represents the postselected angle. Given the preselected and postselected states |ψ_i⟩ and |ψ_f⟩, for sufficiently weak measurement strengths, the final pointer shift is proportional to the real part of the so-called weak value of the measured observable,^21,22

Here, $σ3^$ is the observable operator between these two states ⟨ψ_i| and |ψ_f⟩. When the postselection state is nearly orthogonal to the preselection state, ⟨ψ_i|ψ_f⟩ → 0 and a large weak value is introduced. After the preselection of state, weak interaction, and the postselection of state, the wave function evolves to the final state,

The initial pointer shift is significantly enhanced by the large weak value, and the amplified pointer shift is obtained as δ_w = A_wδ. Furthermore, the free evolution can take place, followed by a modified weak-value amplification (see the supplementary material for details). The amplified pointer shift can be rewritten as³¹ 

where z_r = k₀w²/2 is the Rayleigh length and k₀ is the wave vector in vacuum. Therefore, the weak value establishes a quantitative relationship between the amplified pointer shifts and the postselection angle. In our scheme, the postselection polarization state is modified by the change in optical rotation in the process of sucrose hydrolysis Δ = α.

To determine the hydrolysis rate of sucrose, the weak measurement system is established, as shown in Fig. 2(a). In our experiment, the z axis of the laboratory Cartesian frame (x, y, z) is normal to the air-prism interface. First, the Gaussian beam generated by the He-Ne laser (λ = 632.8 nm) passes through the half-wave plate (HWP) for the use of adjusting the intensity of the light beam. The control of preselection and postselection is realized by using GLP1 and GLP2, respectively. Lens1 and Lens2 are used to focus and collimate the transverse distribution of the input beam. After the preselection of GLP1, the wave function remains Gaussian distributed, as shown in Fig. 2(b). Then, the beam impinges onto the prism interface and photonic SHE arises. The spin-dependent shifts generated by the photonic SHE are much smaller than the width of the transverse distribution, which makes the wave function with different spin eigenstates overlap to a large extent. The optical activity of the chiral solution makes the polarization plane rotate a small angle which can be regarded as the postselection state. These two spin components will interfere with each other after the postselection, and the centroid of the output beam will be amplified significantly. Finally, the intensity distribution and amplified pointer shift of the output beam are recorded by the CCD.

Before using this device to determine the hydrolysis rate of sucrose, without putting a chiral solution sample into this system, we calibrate the system by changing the postselection angle of GLP2 to measure the amplified beam displacement in order to make the subsequent measurement more accurate and sensitive. At the same time, the calibration graph also provides a reference for the subsequent data processing. Figure 3 shows the amplified beam displacements varying with the postselected angles at an incident angle of θ_i = 30°. It can be seen from the changing trend of the curve that the absolute value of the amplified beam displacements will first increase and then decrease and the slope will gradually slow down with the increase in the postselected angle. It is proved that the amplified beam displacement is sensitive to the tiny postselected angle (see the supplementary material for details). The experimental results coincide well with the theoretical prediction. The insets of Fig. 3 represent the output intensity of the light corresponding to different postselection angles in theory and in experiment, respectively. The intensity of the output beam taken by CCD clearly shows the shifts in the centroid of the beam. The experimental results coincide well with the theoretical prediction.

In our scheme, the optical rotation in the process of sucrose hydrolysis acts as the postselection state. GLP2 is rotated to a vertical polarization state so that it is orthogonal to the preselection state. To investigate the effect of catalyst concentration on the hydrolysis rate of sucrose, we introduce a 1 cm thickness quartz cuvette into the beam path with different catalyst concentrations of the sucrose solution which are fixed at 0.04 g/ml. The hydrolysis of sucrose is a first-order reaction, and the initial concentration of the reactant does not affect the reaction rate constant, so there is no special requirement for the concentration of the sucrose solution in the experiment. The beam propagation is perpendicular to the quartz cuvette to avoid other unnecessary beam shifts. The first pointer shift was read from CCD when the reaction time reached 5 min, and then read every 5 min until 90 min.

Figure 4(a) shows the relationship between amplified pointer shifts and the time when different concentrations of hydrochloric acid are added. With the increase in hydrochloric acid concentration, the amplified pointer shifts very faster. Based on the calibration data of weak measurement without the sucrose solution, the change in the optical rotation angle of the reaction solution varying with time is obtained. From Fig. 4(b), it is shown that the initial sucrose solution has the right-handed property. With the hydrolysis reaction going on, the right-handed angle of the system will be reduced and the left-handed angle reaches the maximum value until the sucrose is completely converted. This is because the left-handed properties of fructose in the product are larger than the right-handed properties of glucose in the product.

The hydrolysis of sucrose catalyzed by hydrochloric acid is a first-order reaction. According to chemical kinetics, the equation of the first order reaction can be written as (see the supplementary material for details)

where k is the hydrolysis rate constant which is determined by the concentration of hydrochloric acid in our scheme. α₀ is the initial optical rotation of the system when sucrose has not converted. α_t represents the optical rotation at which the reaction lasts for t minutes. α_∞ is the final optical rotation obtained by water bath heating. As shown in Fig. 5, there is a positive correlation between the hydrochloric acid concentration and the reaction rate constant k without changing the temperature, and the hydrolysis rate of sucrose in pure water is very slow, which indicates that the concentration of catalyst hydrochloric acid has a great influence on the hydrolysis rate of sucrose. The measuring process does not involve any mechanical adjustment of optical elements once the experimental setup is established and thereby realizes a real-time detection of the dynamic chemical reaction.

In the quantum weak measurements, the amplified pointer shifts are determined by the initial pointer shift and the weak-value amplification. Therefore, there are two different ways to realize the high precision metrology. One is to measure the strength of weak interaction which is related to initial pointer shift. For example, in the precision measurement of graphene conductivity, the initial pointer shift or the strength of spin-orbit interaction is sensitive to the conductivity.^32,33 The other is to measure the preselection or postselection angle which determines the weak-value amplification. This scheme can be applied to estimate the optical rotation of chiral molecules where the Faraday rotation angle acts as the preselection or postselection angle. In our present scheme, the Faraday rotation angle acts as the postselection angle to avoid additional effects on weak interactions.

It should be noted that the quantum weak measurements have been successfully applied to biosensing in recent years. Based on the optical rotation of molecular chirality, the weak-value amplification techniques have been applied to chiroptical spectroscopy and optical enantioselectivity.³⁴ By modifying the preselected polarization state with the optical rotation, the molecular chirality can also been precisely detected by weak measurements.^35,36 The proposed weak measurement system not only enriches the types of polarimeters but also exhibits a great potential for chiral molecule detection. Similar to quantum weak measurement, the technique based on the spin-orbit beams can also be applied to detect the optical activity.^37,38 These existing schemes, however, do not involve any chemical reaction process, and therefore, no real-time detection is required. In our present scheme, the chemical reaction rate is monitored by the amplified measurement pointer, thereby realizing a real-time detection of the dynamic chemical reaction. More importantly, catalysts are responsible for engineering the rate of chemical reactions throughout living systems. In our work, we have measured precisely the reaction rate under the catalyst of different concentrations. Bringing together methods of quantum physics and biology, it is possible to constitute an important step toward full development of quantum sensors for biological systems.^39,40

The photonic SHE has been proposed to realize the ultrasensitive and real-time detection of the reaction rate of sucrose hydrolysis. The resolution in our weak measurement system is determined by the factor of weak-value amplification and the measurement resolution of CCD. When the postelection angle ranges from 0° to 0.1°, the amplified pointer shift increases approximately linearly to 800 μm. In the experiment, the resolution of the beam displacement observed by the detector CCD is always lower than 1 μm. Therefore, the measurement resolution of optical rotation detection can reach 1.25 × 10⁻⁴ degree in our weak measurement system. For a standard polarimeter, however, the accuracy is always lower than 10⁻³ degree due to the quantum noise of the coherent light source and the limited extinction ratio of polarizers. In our scheme, only a small amount of sample (2 ml) is needed in the measurements due to the high sensitivity. It is believed that our scheme can be used in lab-on-chip devices in future biochemical sensing applications.

In our scheme, the postselected polarization state is modified by the change in optical rotation in the process of sucrose hydrolysis. The spin-Hall shift acts as the measurement pointer, which enables us to obtain the rate of sucrose hydrolysis by data processing. Compared with the traditional method of polarimeter, the weak measurement method optimizes the sensitivity and resolution of chiral molecular detection in the dynamic chemical reaction. The ultrasensitive and real-time detection of the chemical reaction rate can effectively control the reaction process, which extends the applications of photonic SHE and weak measurement technology. It is believed that our results may have potential applications in high precision chemical and biological sensing. Nevertheless, for the detection of chemical reaction without involving the chirality, the scheme of weak-value amplification still holds with the presence of the external magnetic field. The magnetic-optical rotation angle is sensitive to the change in ion concentration in chemical reaction and thereby plays the role of the postselection angle in quantum weak measurements. Future investigations of the detection of the chemical reaction rate without involving the chirality are critically needed.

See the supplementary material for supporting content.

This work was supported by the National Natural Science Foundation of China (Grant No. 61835004).