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Abstract
Magnetic-free nonreciprocal optical devices have attracted great attention in recent years. Here, we investigated the magnetic-free polarization rotation of light in an atom vapor cell. Two mechanisms of magnetic-free nonreciprocity have been realized in ensembles of hot atoms, including electromagnetically induced transparency and optically-induced magnetization. For a linearly polarized input probe light, a rotation angle up to 86.4° has been realized with external control and pump laser powers of 10 mW and is mainly attributed to the optically-induced magnetization effect. Our demonstration offers a new approach to realize nonreciprocal devices, which can be applied to solid-state atom ensembles and may be useful in photonic integrated circuits.
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1. Introduction
Optical nonreciprocal devices play an important role in photonic information processing, and are indispensable in practical optical applications, including the protection of lasers and amplifiers from backward noise and the routing of optical signals in optical networks. Conventional nonreciprocal devices, such as optical isolators, are achieved through the magneto-optical effect of dielectrics in an externally biased magnetic field. However, the application of this method is greatly limited by the available magneto-optics materials, and their application in photonic integrated circuits is challenging [1,2]. Additionally, strong magnetic fields are demanding, preventing the switchable functionality of nonreciprocal devices and also introducing stray magnetic fields for other optical or electronic devices. Therefore, the realization of magnetic-free optical nonreciprocity has attracted great attention in the past decade [3–8]. Great progress has been achieved by researchers by bringing distinct physical mechanisms to break the time-reversal symmetry of light, including coherent nonlinear optics effects [9–15], optomechanical and Brillouin interactions between mechanical vibrations and photons [16–24], the Fizeau effect in rotating microresonators [25,26], the spatial-temporal modulation of dielectrics through microwave fields [27–30], "moving" Bragg mirrors in atom ensembles [31,32], and the Doppler effect of hot atom ensembles [33–36].
Among these different platforms that realized magnetic-free nonreciprocities, the atomic medium provides an excellent test bed with versatile atom-light interactions for realizing functional nonreciprocal devices. On the one hand, the abundant transitions of atoms permit strong and coherent nonlinear optics interactions, with the interaction strength being greatly enhanced through near-resonant couplings and the collective effects of atoms. For instance, the electromagnetic-induced transparency (EIT) effect [37–41] in an atom ensemble allows the transmission of a signal when its propagation direction is the same as that of the external control laser while preventing backward propagating light due to the absorption of atoms [33,42]. On the other hand, the atoms have Zeeman magnetic sublevels, and the selection rule of ground spin states provides the polarization-dependent interaction between light and atoms. Therefore, using an external circularly polarized pump laser, the ground state population of atoms could be biased to a certain Zeeman state due to the effect of optically-induced magnetization (OIM) [43–45], and then provide the selective absorption of circularly polarized signals. However, these previous investigations of nonreciprocity are limited to the isolation of optical signals, i.e., direction-dependent absorption, while the potential direction-dependent refractive index is omitted. The absorption and refractive index are complementary parameters for describing the linear optical properties of optical media, which correspond to the imaginary and real parts of dielectric constants. The demonstration of the direction-dependent refractive index can lead to the nonreciprocal phase shift of the signal, which has the potential for realizing a magnetic-free Faraday rotator and circulator.
In this work, we demonstrated the magnetic-free polarization rotation of light with a hot atom ensemble using electromagnetically induced transparency and optically-induced magnetization. In an atomic vapor cell at a temperature of $40^{\circ }\mathrm {C}$, a control beam with a power of $10\,\mathrm {mW}$ can rotate the polarization plane of the probe beam by about $\pi /2$. This magnetic-free polarization rotation of light can be utilized to realize nonreciprocal devices, such as all-optical switchable isolators and circulators [46–48]. In addition, our demonstration could be generalized to solid-state atom ensembles, such as doped ions and defects in crystals, and may find applications in photonic integrated circuits.
2. Experimental setup
Figure 1(a) illustrates the concept of polarization rotation by an atomic medium, where a linearly polarized input signal is converted to another polarization direction when passing through the medium. Distinct from the conventional Faraday effect, due to the magneto-optics material in an external magnetic field, our approach realizes all-optical and magnetic-free polarization rotation, which offers complementary studies of nonreciprocal optical media and promises novel magnetic-free nonreciprocal devices. The principle of the all-optical polarization rotation is the controlled circular birefringence and circular dichroism induced by circularly polarized pump fields, based on two types of interactions: electromagnetically-induced transparency (EIT) and optically-induced magnetization (OIM). The principles of the two interactions are illustrated in the energy level diagram of the ${^{87}} Rb$ atom in Fig. 1(b) for the target probe near resonance with the D2 transition ($5^{2}S_{1/2},F_{g}=2$ and $5^{2}P_{3/2},F_{e}=2$). EIT is realized by introducing an external control field that resonates with the target excited state $5^{2}P_{3/2},F_{e}=2$ and an extra auxiliary ground state $5^{2}S_{1/2},F_{g}=1$, which depicts a typical $\Lambda$-type energy diagram for EIT. For the OIM, a pump field that resonates with the D1 transition enables optical pumping of the $5^{2}S_{1/2},F_{g}=2$ Zeeman states. Note that the control field also assists optical pumping by depleting the atom populations on $F_{g}=1$ to $F_{g}=2$.
Fig. 1. Schematic illustration of the experimental setup and its underlying principle. (a) The polarization rotation of the atom medium induces a change in the polarization direction of the probe laser beam. (b) The energy level diagrams for the light-atom interactions, including both the electromagnetically-induced transparency (EIT) and optically-induced magnetization (OIM). The transitions for the left circularly polarized probe beam are represented by the red solid line, while the transitions for the right circularly polarized probe beam are represented by the red dashed line. The yellow dashed line represents transitions for the right circularly polarized control beam, and the blue dashed line represents transitions for the right circularly polarized pump beam. (c) The experimental setup for the characterization of the magnetic-free polarization rotation of the atom medium with either individual EIT or OIM effects, as well as their combination. The key components employed in the setup include the half wave plate (HWP), quarter wave plate (QWP), polarizing beam splitter (PBS), and photo detector (PD).
Figure 1(c) sketches the experimental setup for the magnetic-free polarization rotation, where three individual lasers provide the control, pump and probe light. The $^{87}$Rb medium is provided by a $7.5\,\mathrm {cm}$-long and $2.5\,\mathrm {cm}$ diameter vapor cell. This cell is placed within a pomalloy structure that provides magnetic shielding and temperature stabilization via a heater. All three lasers are collimated and converted to circular polarization and combined into the same path through beamsplitters and couples with atoms in the Rb atom vapor cell. The polarization of the three inputs is controllable by three quater-wave plates (QWPs), and the intensity and polarization of the output light can be detected by adjusting the half-wave plate (HWP) and QWP before the polarized beam splitter (PBS) and photodetector (PD). The direct detection of the probe beam signal is hindered by the challenge in separating it from the other two lasers, and we applied the lock-in amplifier technique to detect the probe signal. During our experiment, the frequencies of the control and pump lasers are locked to corresponding atomic transitions via the saturated absorption spectrum. Meanwhile, the transmission spectrum of the probe beam through the atomic medium is measured by scanning its frequency.
In the following experiments, we fixed the polarizations of the control and pump laser to $\sigma ^{+}$ polarization, as shown in Fig. 1(b), which leads to the effective pumping of atoms to the $F_g=2,\,m_F=2$ state. The corresponding $\sigma ^{-}$-polarized probe participates in the EIT interaction, and the atoms are transparent for the $\sigma ^{+}$-polarized probe due to the OIM effect [43]. This leads to the emergence of a transmission window and circular birefringence, where the atoms allow the transmission of probe light while changing the phases of different circular polarizations. It is worth noting that the two interactions have been separately studied experimentally, demonstrating the different absorption of light for $\sigma ^{+}$ and $\sigma ^{-}$-polarized light. Combining the vapor cell and an additional HWP at the output port, the backward reflected stray light can be almost completely absorbed by the vapor cell, leading to the high-performance isolation of light. Since the nonreciprocity properties of both EIT and OIM interactions have both been validated previously [33,43], here, we only focus on the demonstration of polarization rotation and do not test the propagation of backward light. The potential application of our magnetic-free polarization rotation in isolation or circulator devices is similar to that demonstrated in conventional magneto-optics-materials-based devices.
3. Results
3.1 Effects due to control and pump lasers
Figure 2 shows the transmission spectra of the $\sigma _{\pm }$-polarized probe laser under different external drive conditions through sweeping the probe laser frequency. To calibrate the laser frequency, we simultaneously measure the saturated absorption spectrum of Rb D2 transitions as a reference, i.e., the pink curves in Fig. 2, and align all spectra with respect to the transition frequency of $5^{2}S_{1/2},F_{g}=2\leftrightarrow 5^{2}P_{3/2},F_{e}=2$. The other curves are the results for four conditions:
Fig. 2. Absorption spectra for $\sigma ^{+}$ (a) and $\sigma ^{-}$ (b) polarized probes under different external drive conditions. $P_{\mathrm {c}}$ and $P_{\mathrm {p}}$ are the power of the control beam and pump beam, respectively. The power of the probe beam was $600\, {\mathrm{\mu} \mathrm{W}}$, and the temperature of the atomic cell was set at $40^{\circ }\mathrm {C}$. The red lines are the saturated absorption spectra (SAS) of the ${^{87}}Rb$ $D_{2}$ lines for transitions between $5^{2}S_{1/2},F_{g}=2$ and $5^{2}P_{3/2}, F_{e}=1,2,3$.
(i) No external drives, with $P_{\mathrm {p}}=0$ and $P_{\mathrm {c}}=0$. The black curves represent the bare absorption spectrum of the Rb cell for two circular polarizations. At a relatively low cell temperature $T=40^{\circ }\mathrm {C}$, both spectra show a Gaussian-like dip with a width of around $500\,\mathrm {MHz}$ due to Doppler broadening.
(ii) EIT interaction, with $P_{\mathrm {p}}=0$ and $P_{\mathrm {c}}=10\,\mathrm {mW}$. Due to the coherent coupling-induced quantum interference between the ground hyperfine states, the absorption of the probe laser is suppressed at the detuning $\delta \approx 0$. Therefore, the atoms become transparent for the probe lasers, giving rise to the well-known EIT phenomena. Additionally, as shown by the red curves, the envelope of the Doppler-broadened absorption spectrum is deeper than that in the case without external drives, which is attributed to the increased atom populations in the $F_{g}=2$ states. In the vapor cell, the atoms at thermal equilibrium are equally populated in the $F_{g}=1$ and $F_{g}=2$ states, while the probe can only couple with the $F_{g}=2$ states. The control laser could efficiently excite the atoms from the $F_{g}=1$ to excited states and transfer the populations of atoms to $F_{g}=2$ Zeeman states. Therefore, although the density of the atoms is not changed, the optical depth of the atomic medium for the probe lasers effectively increases, and the absorption increases.
(iii) OIM interaction, with $P_{\mathrm {p}}=10\,\mathrm {mW}$ and $P_{\mathrm {c}}=0$. As shown by the blue curves, the absorption of both circular polarizations is significantly suppressed when compared with the results without external drive (black curves). However, there are two features worth noting: (1) the absorption of $\sigma _{+}$ polarization is much stronger than that of $\sigma _{-}$ polarization, and (2) there is a transparency window at around $250\,\mathrm {MHz}$, corresponding to the transitions $5^{2}S_{1/2},F_{g}=2\leftrightarrow 5^{2}P_{3/2},F_{e}=3$. The first feature could be explained by the twofold effects induced by the pump field: the circularly polarized pump laser effectively drives the ground populations on $|F_{g}=2,m_f=-2,-1,0,1\rangle$ Zeeman states to the $|F_{g}=2,m_f=2\rangle$ Zeeman state and to the $|F_{g}=1\rangle$ states. Such a pump field effectively reduces the population in the $F_{g}=2$ states, just as a counterexample of the control field, and thus, the absorption is reduced. In addition, according to the Clebsch-Gordan coefficients, it can be inferred that the probability of an atom transitioning from $|5^{2}S_{1/2},F_{g}=2,m_{f}=2\rangle$ to $|5^{2}P_{3/2},F_{e}=3,m_{f}=3\rangle$ is 15 times higher than the probability of transitioning to $|5^{2}P_{3/2},F_{e}=3,m_{f}=1\rangle$. Therefore, the absorption of both polarizations decreases, while $\sigma _{-}$ has a much lower absorption. For the second feature, we attributed it to the mechanism of hole burning in a hot atom ensemble. Since the atoms travel through the pump laser beam with a Doppler frequency shift, the pumping effect is strongest when the longitudinal velocity is around 0 as its transition frequency detuning with respect to the pump laser is lowest. As a result, these low longitudinal velocity atoms have lower populations, and thus lower absorptions for the probe around the transition frequency of $5^{2}S_{1/2},F_{g}=2\leftrightarrow 5^{2}P_{3/2},F_{e}=3$.
(iv) The combination of EIT and OIM interactions, with $P_{\mathrm {p}}=10\,\mathrm {mW}$ and $P_{\mathrm {c}}=10\,\mathrm {mW}$. When both the control beam and the pump beam are present, the combined effect of the lasers on the atoms will redistribute the populations to the target Zeeman state $|5^{2}S_{1/2},F_{g}=2,m_{f}=2\rangle$ , which shows the strongest magnetization of the system. Consequently, the hole burning effect is prohibited, as shown by the green curves. The absorption of the $\sigma _{+}$ polarized probe is enhanced, similar to the pure EIT interaction case because it is mainly determined by the transition to $|F_{e}=3,m_f=3\rangle$ state. The EIT phenomena are only profound in the spectra of $\sigma _{-}$ polarization, as expected from the energy level diagram in Fig. 1(b), and the absorption at around $250\,\mathrm {MHz}$ is much weaker due to the weak transition strength.
3.2 Comparison between two interactions
The experimental results provide a direct comparison between the two mechanisms from the aspect of nonreciprocal absorption. In Fig. 3, we further characterized the effect of the control and pump fields by varying the power of the two fields. For a $\sigma ^{+}$ probe, the absorption is mainly determined by the populations of atoms in the $|F_{g}=2,m_f=2\rangle$ state. Therefore, when the pump field is fixed, the increase of control field would repump the leakage of the atom population in the $F_{g}=1$ states to the target state, and thus, the absorption decreases with increasing $P_{\mathrm {c}}$, as shown in Fig. 3(a). In contrast, in Fig. 3(c), when $P_{\mathrm {c}}$ is fixed, increasing the pump field $P_{\mathrm {p}}$ induces more atom population transfer to $F_{g}=1$ states, and thus the absorption increases with $P_{\mathrm {p}}$. We can also observe a slight increase in transmission when the probe detunes near zero, indicating the effect of EIT for the $\sigma ^{+}$ probe. This is because the atoms fly around in the cell, and their populations are redistributed to thermal equilibrium when they collide on the vapor cell boundaries. Thus, the atoms cannot be perfectly prepared to the target states. Instead, weak populations on $|F_{g}=2,m_f=-1,0,1\rangle$ could coherently couple with $|F_{g}=1,m_f=-1,0,1\rangle$ through the $\Lambda$-type EIT interaction path [Fig. 1(b)], and the corresponding absorption due to these energy levels is weakened when increasing $P_{\mathrm {c}}$.
Fig. 3. Experimental spectra measured with different control beam and pump beam powers. (a), (b) Experimental spectra measured with probe beams of ${\sigma }^{-}$ polarized and ${\sigma }^{+}$ polarized. The pump beam power is fixed at $10\,\mathrm {mW}$, while the control beams have powers ranging from $0\,\mathrm {mW}$ to $10\,\mathrm {mW}$. (c), (d) Experimental spectra measured with probe beams of ${\sigma }^{-}$ polarized and ${\sigma }^{+}$ polarized. The control beam power is fixed at $10\,\mathrm {mW}$, while the pump beams have powers ranging from $0\,\mathrm {mW}$ to $10\,\mathrm {mW}$.
For a $\sigma ^{-}$ probe, the effect of the control laser is more profound due to the strong EIT interactions. Similar to the case with the $\sigma ^{-}$ probe, when fixing $P_{\mathrm {p}}$ and increasing $P_{\mathrm {c}}$, the overall absorption increases, while the transparency window due to EIT is more profound as the contrast and width of the window increase. Comparing the results at different $P_{\mathrm {c}}$, we found that the changes are weaker when $P_{\mathrm {c}}$ approaches $10\,\mathrm {mW}$. However, when $P_{\mathrm {c}}=10\,\mathrm {mW}$ is fixed, the spectra are almost unchanged for $P_{\mathrm {p}}>2\,\mathrm {mW}$, which indicates saturation of the OIM effect with a relatively weak pump field intensity.
3.3 Magnetic-free polarization rotation
The magnetic-free polarization rotation of the probe light is measured by sending a linearly polarized probe to the system and measuring the transmission spectra of different output polarizations. With the experimental setup shown in Fig. 1(c), the input probe is set to horizontal polarization, and the measurement bases of the output probe are horizontal (H), vertical (V), left-circular (L, ${\sigma ^{-}}$), and right-circular (R, ${\sigma ^{+}}$) polarizations. The bases are changed by adjusting the HWP and QWP before the detector, while the other parts of the experimental setup are fixed.
The experimental results are summarized in Figs. 4(a) and (b), with vapor cell temperatures of $T=30\,^\circ \mathrm {C}$ and $40\,^\circ \mathrm {C}$, respectively. Here, the combined OIM and EIT are presented with the pump and control fields fixed as $P_{\mathrm {p}}=10\,\mathrm {mW}$ and $P_{\mathrm {c}}=10\,\mathrm {mW}$. As a comparison, the transmissions without any external drives are recorded, as shown by the dashed lines. The H-polarized input can be treated as the superposition of $\sigma ^{+}$ and $\sigma ^{-}$, and with the assumption that the weak probe cannot change the populations of the atom ensemble, the outputs of L and R are independent of each other and should be half the transmission of the two circular polarizations. As expected, we found that the results are consistent with the transmission spectra in Fig. 2. Compared with the dashed lines, it confirms that the two circular polarizations have almost the same transmissions when there are no external drives.
Fig. 4. Measured transmittance and derived optical rotation angle for a horizontally polarized probe beam with a power of $1\,\mathrm {mW}$. The solid lines are the results when EIT and OIM are turned on with both the control beam and pump beam power $10\,\mathrm {mW}$, and the dashed lines are the results when both the control and pump beams are absent. (a) and (c) are the results for cell temperature $T=30^\circ \mathrm {C}$, and (b) and (d) are the results for $T=40^\circ \mathrm {C}$.
Due to the difference in the absorption spectra of the ${\sigma }^{-}$ and ${\sigma }^{+}$ polarized probes, the two polarizations experience different refractive indices within the rubidium medium. This discrepancy in refractive indices leads to circular birefringence and circular dichroism, which are phenomena that involve the differential propagation of light with different circular polarizations through a medium. As a result, the H-polarized input could be converted to V-polarized output, manifesting the polarization rotation effect.
To evaluate the polarization rotation by the atomic medium, we processed the spectra and derived the rotation angle. Fig. 4(c) and (d) display the absolute values of the optical rotation angles calculated at $30~^\circ \mathrm {C}$ and $40~^\circ \mathrm {C}$, respectively. With respect to the probe beam propagating direction along the $z$-axis, the H-polarization corresponds to an electric field direction in the $x-y$ plane aligned along the x-axis. Thus, the Jones vector of the input probe light can be expressed as:
Then, the rotation angle can be derived as
where $I_V$ and $I_H$ are the intensities of transmitted H-polarized and V-polarized probe. Since there is ambiguity of the phase that we cannot distinguish the rotation phases $\theta$ and $-\theta$, we plot $|\theta |$ instead. We also note that during the experimental measurements, the control beam also contributes to the detected signal, as the modulated probe light can slightly change the absorption of the control and pump light by atoms. Although such back-action of probe light on the drive fields is very weak in most cases, it is not negligible when the transmitted probe field is near zero. As a result, we observed negative values for the obtained probe beam intensities, and we excluded these cases when calculating the rotation angle, as represented by shaded areas in Figs. 4(c) and (d). It is found that the rotation angle is maximum at detunings of around $200\,\mathrm {MHz}$ and $400\,\mathrm {MHz}$, corresponding to the half-height of the Doppler-broadened absorption dip. This can be explained by the fact that the maximum refractive index change due to the atoms should be obtained when the derivative of absorption against the frequency is largest. This is also in general agreement with our numerical simulation results (Figs. 5(b)). The largest achieved rotation angles are $28.9$ and $86.5$ degrees for $T=30^{\circ }\mathrm {C}$ and $T=40^{\circ }\mathrm {C}$, respectively. The larger rotation angle in Fig. 4(d) is attributed to the higher atom density at higher temperature. According to the vapor pressure model provided in Ref. [49] where $P_{\mathrm {Rb}}$ is the pressure in Torr and $T$ is the temperature in Kelvin. We estimated that the atomic density is enhanced by a factor of about $2.7$ times in Fig. 4(d). This factor agrees with the enhancement of the rotation angle of $2.92$.Fig. 5. (a) Energy level structure of atoms for theoretical analysis. (b) The polarisation rotations of the probe beam obtained by numerical simulation without (black line) and with (red line) the Doppler shift.
We also found that the EIT effect could only slightly modify the rotation angle around the zero detunings. EIT requires a control laser power higher than $10\,\mathrm {mW}$ and can only create a small polarization rotation angle, with a bandwidth of around $10\,\mathrm {MHz}$. In contrast, as noted in Fig. 3, a pump power of $2\,\mathrm {mW}$ is sufficient for $\pi /2$ polarization rotation, with a bandwidth of about $100\,\mathrm {MHz}$, and the bandwidth could potentially be enhanced to exceeding $1\,\mathrm {GHz}$ by introducing buffer gas into the vapor cell, as the atomic transitions are broadened due to collisions [45]. These results indicate that the OIM effect is more energy efficient in realizing nonreciprocal absorption of light. Therefore, we suggest that OIM could find more applications in future nonreciprocal devices.
Lastly, we theoretically studied the polarization rotation, with a simplified model for the Rubidium atom’s multi-energy level system as shown in the Fig. 5(a). The pump laser in the experiment acts as a polarization to align the atoms to the target state, and the theoretical simulation only needs to consider the action of the control beam versus the probe beam. Taking $|2\rangle$ as the zero energy surface, $\omega _{12}$ and $\omega _{13}$ are the eigenfrequencies corresponding to $|1\rangle$ and $|3\rangle$. Energy levels $|1\rangle$ and $|2\rangle$ are $|F_{g}=2,m_{f}=2\rangle$ and $|F_{g}=1,m_{f}=0\rangle$ for the ground state $5^{2}S_{1/2}$, and energy levels $|3\rangle$, $|4\rangle$,and $|5\rangle$ are $|F_{e}=2,m_{f}=1\rangle$, $|F_{e}=3,m_{f}=1\rangle$ and $|F_{e}=3,m_{f}=3\rangle$ for the excited state $5^{2}P_{3/2}$.
The probe beam (frequency $\omega _{p}$) drives the transitions $|1\rangle \leftrightarrow |3\rangle$, $|1\rangle \leftrightarrow |4\rangle$, $|1\rangle \leftrightarrow |5\rangle$ at the Rabi frequencies $\Omega _{1}$, $\Omega _{2}$ and $\Omega _{3}$, while the control beam (frequency $\omega _{c}$) drives the transition $|2\rangle \leftrightarrow |3\rangle$ at the Rabi frequencies $\Omega _{c}$. Consider the Doppler shift due to atomic motion, the detuning of the probe beam and the corresponding atomic leaps is $\Delta _{p} = \omega _{13}+\vec {k} _{p}\cdot \vec {v}-\omega _{p}$, and the two-photon detuning is $\Delta _{c} = \omega _{12}+(\vec {k} _{p}-\vec {k} _{v})\cdot \vec {v}-(\omega _{p}-\omega _{c})$, the difference between $|3\rangle$ and $|4\rangle$ is $\Delta _{34}=266.65\,\mathrm {MHz}$ . The Hamiltonian of the system can be expressed as
The evolution of the atomic variables in the system is governed by the master equation
4. Conclusion
The magnetic-free polarization rotation is experimentally demonstrated in an atom vapor cell. Two potential approaches, electromagnetically-induced transparency (EIT) and optically-induced magnetization (OIM) in a multilevel atomic system, for realizing magnetic-free nonreciprocal have been studied systematically with the same experimental setup. We realized a nearly $\pi /2$ rotation angle in a $40^\circ \mathrm {C}$ vapor cell under $10\,\mathrm {mW}$ pump and $10\,\mathrm {mW}$ control lasers, manifesting an efficient all-optical polarization rotator. We found that OIM could provide more profound magnetic-free polarization rotation with a higher bandwidth and lower laser power, and suggest the OIM-effect for realizing novel and practical nonreciprocal devices in future applications.
Funding
National Key Research and Development Program of China (2021YFF0603701); National Natural Science Foundation of China (12074067, 12274079, 92265108, U21A20433); Fundamental Research Funds for the Central Universities; USTC Research Funds of the Double First-Class Initiative.
Acknowledgments
We thank Qin-Xuan Jie, Dong-Qi Ma, and Yi-Chen Zhang for valuable discussions. This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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