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Abstract
A new method for the detection of atomic spin precession based on the Mach-Zehnder interferometer (MZI) is proposed and experimentally demonstrated. Different from the conventional polarization detection methods which obtain the atomic spin precession signal by measuring the change of the probe laser power, the proposed method uses the laser modulated by an electro-optic phase modulator (EOM) as the source of the interferometer, and obtains the atomic spin precession signal by measuring the phase difference between the two arms of the MZI. The output of interferometer is independent of the probe laser power, which avoids the system error caused by the fluctuation of the probe laser power, and the long-term stability of the system is effectively improved. At the same time, the method adopts high-frequency electro-optic modulation, which can effectively suppress low-frequency noise, such as 1/f noise, and can significantly improve the detection sensitivity. The rotation sensitivity and long-term stability of the atomic comagnetometer were tested using the MZI detection method and a typical detection method, respectively. The comparison results show that the proposed method has the highest low frequency sensitivity and the potential to improve the long-term stability of the system.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the continuous development of quantum mechanics theory, laser technology and atom manipulation technology, quantum sensors using atoms as sensitive media [1,2] have attracted extensive attention of researchers. The atomic spin comagnetometer is widely used in Lorentz and CPT violation [3–5], EDM measurements [6,7], dark matter detection [8,9], and inertial rotation measurements [10–12] due to their ultra-high sensitivity. In 2005, the Princeton University research team firstly realized the gyroscopic effect by using a $\mathrm {K}-^{3}\mathrm {He}$ atomic comagnetometer [10] working in the spin-exchange relaxation-free(SERF) regime [13,14], and obtained rotation sensitivity of $5 \times 10^{-7} \mathrm {rad} / \mathrm {s} / \sqrt {\mathrm {Hz}}$ and the low frequency angle drift is about $0.04 \mathrm {deg} / \mathrm {h}$. Subsequently, in order to explore the ultimate sensitivity and inertial measurement accuracy of SERF atomic comagnetometer (SERF ACM), Beihang University has carried out technical researches on this system, such as high-efficiency optical pumping [15], low-noise magnetic shield [16], and laser power stabilization [17,18], and has achieved a rotation sensitivity of $2.1 \times 10^{-8} \mathrm {rad} / \mathrm {s} / \sqrt {\mathrm {Hz}}$ and a bias drift of $10^{-2} \mathrm {deg} / \mathrm {h}$ in a $\mathrm {K}-\mathrm {Rb}-^{21}\mathrm {Ne}$ comagnetometer [19].
SERF ACM utilizes atomic spin precession signal to characterize the inertial rotation signal. Therefore, a highly sensitive, stable and non-interference atomic spin precession detection technique is essential for the realization of quantum sensing. The precession and reorientation of atomic spins modifies the optical absorptive and dispersive properties of the atoms, and this modification can be detected by measuring the light transmitted through the atomic medium. There are usually two detection modes of atomic spin precession: optical rotation mode [20,21] and optical absorption mode [22–24]. The optical absorption mode is based on the absorption of the circularly polarized light passing through the alkali metal vapor. The probe laser and the pump laser are the same beam, with a simple structure but low measurement accuracy and poor signal-to-noise ratio [25]. For high-precision spin precession detection, the optical rotation mode is usually chosen [26]. A variety of optical rotation measurement methods have been reported. The most common detection methods include the balanced polarimetry technique (BPT) [27], laser modulation detection [28–30] with external modulators such as Faraday modulators, photoelastic modulators (PEM) and acousto-optic modulators (AOM), and magnetic field modulation detection based on the working characteristics of SERF ACM [31]. These methods are based on Malus’s Law and use the variation of laser power to achieve the detection of atomic spin progression. Therefore, the fluctuation of laser power is easily coupled with the polarization change caused by the optical rotation angle, resulting in laser power errors being included in the output signal. This makes it difficult to achieve high-precision and stability measurement in experiment [32]. Some scholars proposed that the phase instead of amplitude to detect optical rotation could be separated with the laser power noise, and realized the atomic spin precession detection based on Sagnac interferometer [33]. The optical fiber Sagnac interferometer detection method utilizes the reciprocal structure to suppress common mode noise, however, the high coherence of the laser source causes a large backscatter noise, resulting in a low signal measurement sensitivity. The Mach-Zehnder interferometer (MZI) has been widely used in the measurement of organic solution concentration by virtue of its high sensitivity [34,35], but no relevant research has been found in the measurement of atomic spin precession.
In this paper, the MZI-based atomic spin precession detection method proposed in this paper belongs to a homodyne detection technology. The probe laser modulated by an electro-optic phase modulator (EOM) is used as the interferometric light source, which can effectively suppress the 1/f noise and improve the detection sensitivity. The phase difference between the signal light and the reference light is formed by placing the atomic vapor cell in one arm of the MZI. The two outputs of the interferometer are demodulated by the lock-in amplifier to obtain the phase difference after passing through the polarization components respectively, which realizes the detection of the atomic spin precession. A 1/4 wave plate is inserted in front of one of the polarization components to enhance the change of phase difference. When the azimuth angles of these components are chosen properly, the phase difference determined with interferometric technique is greatly enhanced to result in accurate optical rotation angle. The detection principle and optical path of this method are discussed in Section 2. The performance of the system is tested and compared with the BPT. This method can realize the detection of atomic spin precession by using phase information, and this method is not only applicable to SERF gyroscopes, but also to spin-based atomic sensors such as atomic magnetometers and nuclear magnetic resonance (NMR) gyroscopes.
2. Principle and method
The optical rotation $\theta$ due to atomic spin precession in SERF ACM can be written as
Substituting Eq. (2) into Eq. (1),
The schematic diagram of the atomic spin precession detection scheme based on MZI is shown in Fig. 1. This method is mainly divided into two parts. First, MZI is constructed and phase difference caused by optical rotation angle is generated. Then, the interference signal is detected by a lock-in amplifier. The specific working principle of this method is theoretically derived in detail as follows.
Fig. 1. Schematic diagram of the detection of atomic spin precession based on MZI. P: linear polarizer, $\lambda / 2$: 1/2 wave plate, EOM: electro-optic phase modulator, PBS: polarization beam splitter, $\text {M}_\text {p}$ and $\text {M}_\text {s}$: reflection mirror, BS: beam splitter, $\lambda / 4$: 1/4 wave plate, $\text {A}_\text {r}$ and $\text {A}_\text {t}$: analyzer, $\text {PD}_\text {r}$ and $\text {PD}_\text {t}$: photodetector, LIA: lock-in amplifier, DAQ: data acquisition system
The working principle of the proposed detection method is theoretically analyzed by using the Jones matrix. The probe laser passing through the linear polarizer P and the 1/2 wave plate has the Jones vector
where $\alpha$ represents that the fast axis of the 1/2 wave plate is at $\alpha /2$ to the x-axis. The EOM is driven by a sinusoidal signal which can be expressed as $\phi (t)=\phi _{0} \sin (\omega t)$, where $\phi _{0}$ is the modulation depth, and $\omega$ is the modulation angular frequency. The Jones vector after emerging from the EOM then becomesThe phase-modulated probe laser is split by the PBS into two paths to form the MZI: (a) PBS $\rightarrow$ Cell $\rightarrow$ $\text {M}_\text {p}$ $\rightarrow$ BS and (b) PBS $\rightarrow$ $\text {M}_\text {s}$ $\rightarrow$ BS. The transmitted light after passing through PBS is called p-polarized light, and the reflected light is called s-polarized light. The MZI is constructed using these two polarized beams, and the atomic vapor cell is placed in the optical path of the p-polarized light, which is used as the signal light to detect the atomic spin precession. The s-polarized light is used as the reference light. When the SERF ACM is sensitive to the rotation, the circular birefringence of the atomic vapor cell causes the transmitted linearly polarized light to form an optical rotation angle. This optical rotation phenomenon can also cause a phase difference between the signal light and the reference light. After being reflected by two mirrors ($\text {M}_\text {p}$ and $\text {M}_\text {s}$), these two polarized beams finally meet at the beam splitter(BS) and form a MZI. After the interference light is transmitted to the BS and divided again into two paths: the transmitted light and the reflected light. According to the principle of superposition of light waves, the resultant vibration $E_{t}$ after the superposition of the transmitted p-polarized light and the reflected s-polarized light can be expressed as
Whereas, the resultant vibration $E_{r}$ after the superposition of the reflected p-polarized light and the transmitted s-polarized light can be expressed as
The transmission direction and reflection direction of PBS are equivalent to two polarizers, and the corresponding Jones matrix can be written as
The transmitted light of the BS is detected by $\text {PD}_\text {t}$ after passing through a 1/4 wave plate (the fast axis lying at the y-axis) and an analyzer $\text {A}_\text {t}$ (with the transmission axis being at $\beta _{1}$ to the y-axis). The Jones vector $E_{t}^{\prime }$ can be written as
The light intensity detected by $\text {PD}_\text {t}$ is
On the other hand, the reflected light of the BS is detected by $\text {PD}_\text {r}$ after passing through an analyzer $\text {A}_\text {r}$ (with the transmission axis being at $\beta _{2}$ to the y-axis). The Jones vector $E_{r}^{\prime }$ can be written as
The light intensity detected by $\text {PD}_\text {r}$ is
Finally, the signals output of the two photodetectors (${\text {I}_\text {t}}$ and ${\text {I}_\text {r}}$) are used as reference and signal input to a lock-in amplifier for phase demodulation to obtain the phase difference information between the two interference signals. The phase difference demodulated by the lock-in amplifier is
According to Eq. (23), the phase difference between the signal light and the reference light caused by the optical rotation of the atomic vapor cell can be obtained as
According to Eq. (14) and Eq. (20), the relationship between the phase difference $\phi$ and the optical rotation $\theta$ can be calculated as
Then, the optical rotation $\theta$ can be expressed as
whereTherefore, the optical rotation can be obtained by measuring the phase difference information from the output of the lock-in amplifier, thereby realizing the detection of the atomic spin precession signal. The output of this method is phase information, independent of the optical power of the probe laser, and the signal drift caused by the unsatisfactory performance of the laser is avoided from the source.
3. Experimental setup
The experimental apparatus of the SERF ACM based on the MZI detection method is shown in Fig. 2. The atomic vapor cell, as the core sensitive component, is an aluminosilicate glass spherical cell with an outer diameter of 10 mm. The cell contains a small drop of alkali metal potassium (K) and rubidium (Rb) mixture, 1 atm of Neon ($^{21}\mathrm {Ne}$), and 50 Torr of nitrogen ($\text {N}_{2}$). The cell is fixed in a boron-nitride oven whose temperature is stabilized at 185 $^{\circ }$C by a non-magnetic high-frequency AC current heating coil and a PID controller. The atomic number density of alkali metal can reach the order of $10^{14}$ at this temperature. Three layers permalloy and an inner layer of manganese-zinc (Mn-Zn) ferrite form a high-performance and low-noise magnetic shield. The magnetic shield combined with a three-axis high-precision magnetic compensation coil creates a low magnetic environment for the electron spin to ensure that the electron spin is in the SERF regime. The pump laser with a center wavelength of 770nm (D1 resonance line of K) is emitted from a distributed Bragg reflection (DBR) laser. After passing through the saturated absorption frequency stabilization module (SAS), the laser power stabilization system (LPSS) and the lens beam expansion, a linear polarizer (P) and a 1/4 wave plate ($\lambda / 4$) convert the pump laser into a circularly polarized laser. The circularly polarized laser is incident on the atomic vapor cell to spin polarize the atoms along the $z$-axis.
Fig. 2. Schematic diagram of SERF atomic co-magnetometer based on MZI detection method. SAS: the saturated absorption frequency stabilization module, LPSS: laser power stabilization system, P: linear polarizer, $\lambda / 4$: 1/4 wave plate, EOM: electro-optic phase modulator, PBS: polarization beam splitter, $\text {M}_\text {p}$ and $\text {M}_\text {s}$: reflection mirror, BS: beam splitter, $\text {A}_\text {r}$ and $\text {A}_\text {t}$: analyzer, $\text {PD}_\text {r}$ and $\text {PD}_\text {t}$: photodetector, LIA: lock-in amplifier, DAQ: data acquisition system
The probe laser is emitted from a distributed feedback (DFB) laser, and the laser wavelength is detuned by 100GHz (0.2nm) from 795 nm (D1 resonance line of Rb). The optical rotation of the cell is detected by the MZI detection method shown in Fig. 1. The EOM (iXblue, NIR-MPX800-LN-0.1) is used to generate phase modulation with a frequency of 500 kHz. The PBS divides the probe laser into signal light and reference light. The signal light propagates along the $x$-axis and passes through the atomic vapor cell. After being reflected by a mirror, the signal light meets and interferes with the reference light at the BS. The interference signals are detected by the photodetectors ($\text {PD}_\text {r}$ and $\text {PD}_\text {t}$) after passing through the polarization components. These two signals are used as reference input and signal input respectively and are sent to the lock-in amplifier for phase demodulation. The lock-in amplifier (LIA, Zurich Instruments, HF2LI) is adapted to pick up the phase difference between $\text {PD}_\text {r}$ and $\text {PD}_\text {t}$. Finally, the phase difference was collected by the data acquisition system (DAQ). A National Instruments USB-4431 acquisition system is used for data collection, and the sampling rate is 200Hz. The phase difference output from LIA contains the optical rotation information of the atomic vapor cell due to sensitive inertial rotation . Thus, the phase measurement method of inertial rotation is realized.
4. Experimental results and discussion
According to Eq. (26), the optical rotation $\theta$ produced by the precession of atomic spins can be measured using the phase difference $\phi$. However, the phase difference $\phi$ in the expression is not only related to the $\theta$, but also related to the angle ($\phi _{B S}$, $\beta _{1}$ and $\beta _{2}$) selection of the polarization component. Properly adjusting the angle of the polarization component can improve the measurement sensitivity of the optical rotation. In order to obtain a high sensitivity of optical rotation measurement, we simulated the relationship between optical rotation and the phase difference under different angles of polarization components. First, under the condition of $\beta _{2}=84^{\circ }, \phi _{B S}=20^{\circ }$, the relational expression between the phase difference $\phi$, the optical rotation $\theta$ and the angle $\beta _{1}$ of the analyzer $\text {A}_\text {t}$ is simulated and the result is shown in Fig. 3. The relationship curve between $\phi$ and $\theta$ shows that $\phi$ increases monotonously with $\theta$, and the increasing trend shows a sharp increase followed by a flattening out. This indicates that the measurement sensitivity of the MZI detection method is higher when the optical rotation angle is smaller. The relationship curve between $\phi$ and $\beta _{1}$ shows that $\phi$ also increases monotonously with $\beta _{1}$, and the rate of increase in $\phi$ becomes more intense as $\beta _{1}$ approaches $90^{\circ }$. That is, the resolution of $\phi$ increases as $\beta _{1}$ approaches $90^{\circ }$.
Fig. 3. The relationship curve between phase difference $\phi$, the optical rotation $\theta$ and the angle $\beta _{1}$ of the analyzer $\text {A}_\text {t}$ when $\beta _{2}=84^{\circ }, \phi _{B S}=20^{\circ }$.
Subsequently, under the condition of $\beta _{1}=89^{\circ }$, the simulation analysis of the relationship between the phase difference $\phi$, the optical rotation $\theta$ and the angle $\beta _{2}$ of the analyzer $\text {A}_\text {r}$ is carried out under the conditions that $\phi _{B S}$ is $20^{\circ }$ and $50^{\circ }$, respectively. The results are shown in Fig. 4 and Fig. 5. It can be seen that the magnitude of $\phi _{B S}$ affects the detection range of the optical rotation $\theta$ and the selection of the analyzer angle $\beta _{2}$. And the value of $\phi _{B S}$ is smaller, the continuous test range of the optical rotation is larger. $\phi _{B S}$ is the phase difference between the two interfering beams after passing through the BS. According to the Fresnel formula, the reflected light changes in phase depending on the angle of incidence laser. Therefore, the value of $\phi _{B S}$ can be optimized by adjusting the incident angle of laser.
Fig. 4. The relationship curve between phase difference $\phi$, the optical rotation $\theta$ and the angle $\beta _{2}$ of the analyzer $\text {A}_\text {r}$ when $\beta _{1}=89^{\circ }, \phi _{B S}=20^{\circ }$.
Fig. 5. The relationship curve between phase difference $\phi$, the optical rotation $\theta$ and the angle $\beta _{2}$ of the analyzer $\text {A}_\text {r}$ when $\beta _{1}=89^{\circ }, \phi _{B S}=50^{\circ }$.
After determining the optimal angle of optical components, we simulated the output response of this detection method. For the high-precision SERF ACM, the value of optical rotation is relatively small ($\theta \ll 1$) and Eq. (25) can be simplified to
According to the above equation, the simulation results of using the phase difference to measure the optical rotation $\theta$ at different $\beta _{1}$ angles are shown in the Fig. 6. The curve clearly shows that the phase difference $\phi$ is approximately linear with the optical rotation $\theta$ and the slop increases with $\beta _{1}$ approaches $90^{\circ }$. The slope is called the scale factor in the SERF ACM. For the measurement of the scale factor, the subsequent content of this paper is described in detail.
Fig. 6. The relationship curve between the optical rotation $\theta$ and the phase difference $\phi$ at different $\beta _{1}$ angles.
In order to verify the effectiveness of the proposed method in measuring the atomic spin precession signal, the scale factor was first calibrated on the experimental platform, that is, the output response of SERF ACM at different input angular rates was measured. Usually, a turntable is used to calibrate the scale factor. For the SERF ACM, it is inconvenient to use a turntable due to the large size of the experimental setup. An equivalent magnetic field can be used instead of the rotational angular rate according to $B=\Omega / \gamma ^{n}$ [19]. The method of calibrating the scale factor using an equivalent magnetic field is shown in Fig. 7. A square wave modulation magnetic field with an amplitude of 0.3 nT and a frequency of 10 mHz was applied in the $y-$axis. Different output responses were obtained by scanning the magnetic field near the $z-$axis compensation point. After leaving the compensation point, the steady-state response difference changed in both sign and amplitude under the excitation of different square wave modulated magnetic fields. Then, the calibration relationship between the magnetic field and the output response can be obtained by fitting the steady-state response difference measured under different $z-$axis magnetic fields.
Fig. 7. Using the equivalent magnetic field to measure the scale factor: (a) At the compensation point, the steady-state response difference is zero. (b) Scanning the $B_{z}$ magnetic field near the compensation point and defining that the difference is recorded as a positive value if the output signal varies from high to low as the modulated magnetic field also varies from high to low. (c) When the change of the output signal is opposite to the modulated magnetic field, the difference is recorded as a negative value.
According to the different resolution of the optical rotation measurement at different $\beta _{1}$ angles in Fig. 6, we measured the scale factors at different $\beta _{1}$ angles. The scale factor was measured as described above, resulting in a standard dispersion shape response curve as shown in the Fig. 8. The horizontal coordinate represents the magnitude of the magnetic field $\delta B_{z}$ deviating from the self-compensation point of the nuclear spin magnetic field. And the vertical coordinate represents the steady-state response difference $\Delta S$ of the SERF ACM output signal. The slope at the zero-crossing point of the dispersion curve is a function of the scale factor [19]. The scale factors obtained after fitting the formula to the experimental data are $K_{\beta _{1}=89^{\circ }}=5.70 \mathrm {~V} /(^\circ / \mathrm {s})$, $K_{\beta _{1}=88^{\circ }}=3.42 \mathrm {~V} /(^\circ / \mathrm {s})$ and $K_{\beta _{1}=85^{\circ }}=2.79 \mathrm {~V} /(^\circ / \mathrm {s})$, respectively, which verifies the simulation results in Fig. 6.
Fig. 8. Scale factor measured using equivalent magnetic field with different angles of $\beta _{1}$. The test values are obtained by measuring the steady-state response difference when scanning the $B_{z}$ magnetic field near the magnetic compensation point. The fitting curves can be obtained using the formula $y=A \frac {x-B_{c}}{\left (x-B_{c}+L_{z}\right )^{2}+B^{2}}$. According to the parameters obtained when fitting the curve, the scale factor can be calculated using $K=\frac {A}{L_{z}^{2}+B^{2}} \frac {B_{c}}{\Delta B_{y}} \frac {1}{\gamma _{n}}$, and this value is related to the magnitude of the slope of the zero-crossing point of the dispersion curve [19].
According to the above test results of the scale factor, we set $\beta _{1}=89^{\circ }$ to detect the atomic spin precession. In order to demonstrate that the proposed detection method can improve the sensitivity and long-term stability of the atomic spin precession. We compared the performance of the proposed MZI detection method and the classical balanced polarimetry technique (BPT) method in detecting the atomic spin precession signal. When the BPT method was used for the detection of the atomic spin precession, comparative tests were carried out under the conditions of stable and unstable laser power respectively. First, the scale factor was measured as described above. Then, the stability of the static SERF ACM was tested for two hours without input angular rate. The rotation sensitivity curve can be obtained by selecting 2 minutes of the test data for FFT analysis with scale factor. Allan deviation analysis of the test data can be used to evaluate the long-term stability of the system. The results are shown in Fig. 9.
Fig. 9. The rotation sensitivity (a) and Allan deviation curves (b) of the SERF ACM tested using two detection methods (BPT and MZI). When using the BPT method for the atomic spin precession detection, the tests were carried out under the conditions of stable (black curve) and unstable (blue curve) laser power, respectively.
For inertial measurement, we pay more attention to the low frequency sensitivity of the system. As shown by the comparison between the blue curve and the black curve in Fig. 9 (a), the stability of the probe laser power directly affects the rotation sensitivity. The MZI detection method used EOM for high frequency modulation, so the red curve showed a clear advantage in the frequency band lower than 1Hz. The rotation sensitivity measured by the MZI detection method was $2.28 \times 10^{-5} {~}^\circ / \mathrm {s} / \mathrm {Hz}^{1 / 2} @ 0.1 \mathrm {~Hz}$, which was increased by 3.5 times than $7.89 \times 10^{-5} {~}^\circ / \mathrm {s} / \mathrm {Hz}^{1 / 2} @ 0.1 \mathrm {~Hz}$ of the BPT method, indicating that the low frequency noise was effectively suppressed. The bias instability is used to evaluate the long-term stability of the comagnetometer. In the Allan deviation curve, the low-frequency noise corresponds to a horizontal curve with zero slope, and its ordinate refers to the bias instability of the system. Fig. 9 (b) shows the Allan deviation curves. After the laser power was stabilized, the bias instability of the BPT method was reduced from 0.057 $^\circ / \mathrm {h}$ at an averaging time of 4s to 0.035 $^\circ / \mathrm {h}$ at 21s. The bias instability of the MZI method (red curve) was 0.064$^{\circ }$/h at the average time of 73s. Compared with the BPT method, the bias instability of the MZI method did not show its advantage. The possible reason is that the complex optical structure of the MZI method introduced a lot of noise. However, comparing the Allan deviation value of the two detection methods at the 100s position, it was reduced from 0.256 $^\circ / \mathrm {h}$ to 0.067 $^\circ / \mathrm {h}$, which is about 4 times smaller. In addition, the trend term of the BPT method after 100s shows an upward state, which is caused by the drift of the probe laser power. The MZI method detects the optical rotation through the phase difference. And the high-frequency modulation suppresses the low-frequency drift caused by the slow variation of temperature, electronic detection and other factors. Therefore, the corresponding average time of the trend item of the MZI method has been extended, indicating that the use of the MZI method has great potential to improve the long-term stability of the system.
5. Conclusion
In order to improve the detection sensitivity and long-term stability of the atomic spin precession signal in the SERF ACM, an MZI-based atomic spin precession detection method is proposed, and the theoretical analysis and experimental verification were carried out. Different from other detection methods that obtain the optical rotation by measuring the variation of the probe laser power, this method obtains the optical rotation by extracting the phase information. The effects of different detection methods are compared in the experiment, the balanced polarimetry technique and the MZI detection method were used to measure the optical rotation. From the comparison results of the rotation sensitivity and the Allan deviation, the method proposed in this paper is more sensitive at the low frequency band, the rotation sensitivity at 0.1Hz has been increased by 3.5 times. The Allan deviation at the 100s position has been reduced by about 4 times, from 0.256 $^\circ / \mathrm {h}$ to 0.067 $^\circ / \mathrm {h}$. The MZI based detection method has great potential to improve the rotation sensitivity and long-term stability of the SERF ACM.
Funding
National Natural Science Foundation of China (62003024, 62103026).
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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