High-precision rapid in-situ measurement device and method for polarization parameters of atomic common magnetometer

By applying a transverse excitation magnetic field and demodulating the phase signal with a lock-in amplifier in the atomic gas cell, combined with a three-axis magnetic field coil and photoelectric detection, rapid and high-precision in-situ measurement of the atomic polarization parameters of the SERF gyroscope was achieved, solving the problem of insufficient measurement accuracy in the existing technology and improving measurement accuracy and speed.

CN121409201APending Publication Date: 2026-01-27BEIHANG UNIV
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Patent Information

Application Number
CN202511611218.5
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2025-11-05
Publication Date
2026-01-27

AI Technical Summary

Technical Problem

Existing methods for measuring atomic polarization parameters have limited accuracy and are not in-situ measurements, making it difficult to accurately assess the system state parameters of the SERF gyroscope.

Method used

By applying a transverse excitation magnetic field in the atomic gas cell, demodulating the phase signal using a lock-in amplifier, and scanning the longitudinal compensation magnetic field, in-situ measurements of nuclear spin polarization and electron spin polarization are achieved. Combined with a triaxial magnetic field coil and a photoelectric detection device, the longitudinal compensation magnetic field is gradually reduced until the phase frequency response is minimized, and the polarizability and longitudinal relaxation time are calculated.

Benefits of technology

Rapid and high-precision in-situ measurement of the polarization parameters of the SERF gyroscope atomic ensemble was achieved, improving measurement accuracy and speed, accurately evaluating system state parameters, and providing a foundation for atomic ensemble control.

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Abstract

The invention discloses a high-precision rapid in-situ measurement device and method for polarization parameters of an atomic common magnetometer, and belongs to the technical field of quantum precision measurement and ultrahigh-sensitivity inertial measurement. Based on an atom manipulation technology, a transverse modulation magnetic field is applied to an atom ensemble through a three-axis magnetic field coil, a lock-in amplifier is adopted to extract device output phase frequency response, a longitudinal axis magnetic field is gradually scanned until the minimum phase frequency response is obtained, and therefore a nuclear spin polarization equivalent magnetic field and an electron spin polarization equivalent magnetic field are obtained. And further obtaining the electron spin polarizability, the nuclear spin polarizability and the nuclear spin longitudinal relaxation time. According to the invention, the rapid in-situ measurement of the atom ensemble polarization parameters of the SERF gyroscope is realized, and the measurement precision is improved.
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Description

Technical Field

[0001] This invention belongs to the technical field of quantum precision measurement and ultra-sensitive inertial measurement, specifically relating to a high-precision, rapid in-situ measurement device and method for polarization parameters of atomic common magnetometers. Background Technology

[0002] Atomic polarization technology achieved through optical pumping has been widely used in the field of quantum precision measurement and has received widespread attention in recent years. It has been applied to the measurement of parameters such as inertia, magnetic fields, time, and gravity, and has also been applied to cutting-edge physics exploration. Atomic ensembles operating in the spin-exchange relaxation-free (SERF) state can achieve ultra-high inertial measurement sensitivity due to their longer coherence time and have been applied to SERF gyroscopes.

[0003] The sensitive core of a SERF gyroscope is an atomic ensemble of alkali metal and inert gas atoms. The pump laser transfers rotational momentum to the atoms through interaction with the atomic ensemble, achieving a macroscopically unified orientation of the ensemble, i.e., atomic polarization. The polarization of the atomic ensemble consists of two parts: the electronic spin polarization of the alkali metal atoms and the nuclear spin polarization of the inert gas atoms, manifested as equivalent magnetic fields for electronic spin polarization and nuclear spin polarization, respectively, and these two are strongly coupled together. Decoupling and accurately measuring the atomic polarization parameters is fundamental to evaluating system state parameters and controlling the atomic spin-coupled ensemble. Current methods for measuring atomic polarization parameters calculate the equivalent magnetic field through curve fitting, which is a non-in-situ measurement method with limited accuracy. To further accurately evaluate system parameters and control the atomic ensemble, a high-precision in-situ measurement method for atomic polarization parameters is urgently needed. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention proposes a high-precision, rapid, in-situ measurement device and method for polarization parameters of an atomic common magnetometer. A transverse excitation magnetic field is applied to the horizontal axis, and the phase signal obtained from the lock-in amplifier demodulation is observed. The compensation magnetic field on the vertical axis is scanned until the minimum phase signal is obtained, thus achieving in-situ measurement of the nuclear spin polarization equivalent magnetic field and the electron spin polarization equivalent magnetic field. These are further converted into polarizability and transverse nucleon relaxation time, enabling rapid and high-precision in-situ measurement.

[0005] The technical solution of the present invention is as follows:

[0006] A high-precision, rapid, in-situ measurement device for polarization parameters of an atomic common magnetometer is characterized by comprising a lock-in amplifier connected to a differential photoelectric amplifier, a signal generator, and a signal acquisition unit. The differential photoelectric amplifier converts the differential intensity signal of the detection light passing through the atomic gas cell into a voltage signal, which is then transmitted to the lock-in amplifier. The lock-in amplifier extracts a phase signal from the voltage signal as a phase frequency response signal. The signal generator generates a reference signal input to the lock-in amplifier to provide a frequency reference, and also generates a modulation voltage input to a triaxial magnetic field coil embedded in the atomic gas cell to generate a transverse modulation magnetic field. The signal acquisition unit is used to observe the magnitude of the phase frequency response signal. The signal generator controls the triaxial magnetic field coil to achieve active compensation of the triaxial magnetic field. Pump light enters the atomic gas cell to achieve polarization of the atomic ensemble. Let the longitudinal axis compensation magnetic field be... The equivalent magnetic field of electron spin polarization is The nuclear spin polarization equivalent magnetic field is , yes The longitudinal compensation magnetic field is gradually reduced until the phase frequency response signal reaches its minimum value, through... Obtaining electronic polarizability ,pass nuclear spin polarization ,pass and The longitudinal relaxation time of nuclear spin was obtained. .

[0007] An oven is provided between the atomic gas chamber and the triaxial magnetic field coil, and the triaxial magnetic field coil is located inside a multi-layer magnetic shielding cylinder.

[0008] The detection light comes from a detection laser, which is connected to the differential photoelectric amplifier in sequence through a second polarizer, an atomic gas cell, and a Wollaston prism. The pump light comes from a pump laser, which is connected to a photodetector in sequence through a beam expander, a first polarizer, a quarter-wave plate, and an atomic gas cell.

[0009] A high-precision, rapid, in-situ measurement method for polarization parameters of an atomic common magnetometer is characterized by employing the aforementioned high-precision, rapid, in-situ measurement device for polarization parameters of an atomic common magnetometer.

[0010] Includes the following steps:

[0011] Step 1: Heat the atomic gas chamber using an oven, and use a signal generator to control the triaxial magnetic field coil to achieve active compensation of the triaxial magnetic field, thus obtaining the longitudinal axis compensated magnetic field. ;

[0012] Step 2: Use a signal generator to generate a modulation voltage, which is applied to the triaxial field coil to generate a transverse modulation magnetic field. A reference signal is generated and transmitted to a lock-in amplifier to provide a frequency reference;

[0013] Step 3: The differential photoelectric amplifier converts the acquired differential signal of the detection light intensity into a voltage signal and transmits it to the lock-in amplifier. The lock-in amplifier extracts the phase signal as the phase frequency response signal and gradually reduces the longitudinal axis compensation magnetic field. Until the phase frequency response signal reaches its minimum value, record the vertical axis compensation magnetic field at this point. The nuclear spin polarization equivalent magnetic field was obtained. Equivalent magnetic field of electron spin polarization ;

[0014] Step 4: Using the equivalent magnetic field of electron spin polarization Calculate electronic polarizability Through nuclear spin polarization equivalent magnetic field Calculation of nuclear spin polarization Calculate the longitudinal relaxation time of nuclear spin. .

[0015] Step 2 includes:

[0016]

[0017] in For modulation amplitude, For modulation frequency, To modulate the phase, For time.

[0018] Step 3 includes:

[0019] .

[0020] Step 4 includes:

[0021]

[0022]

[0023]

[0024] in The Fermi contact constant is... The permeability of free space, For Bohr magneton, For alkali metal density, The density of inert gas atoms. For inert gas nuclear magnetic moments, is the spin exchange rate constant between electron spin and nuclear spin.

[0025] The technical effects of this invention are as follows: This invention provides a high-precision, rapid, in-situ measurement device and method for atomic common magnetometer polarization parameters. Based on atomic manipulation technology, it applies a transversely modulated magnetic field to the atomic ensemble through a triaxial magnetic field coil. A lock-in amplifier is used to extract the phase frequency response, gradually scanning the longitudinal magnetic field until the minimum phase frequency response is obtained. This yields the equivalent magnetic field of nuclear spin polarization and the equivalent magnetic field of electron spin polarization, further providing the electron spin polarizability, nuclear spin polarizability, and nuclear spin longitudinal relaxation time. This invention enables rapid in-situ measurement of the atomic ensemble polarization parameters of a SERF gyroscope, improving measurement accuracy.

[0026] This invention enables rapid and high-precision in-situ measurement of atomic ensemble polarization parameters based on the system's minimum phase frequency response. An AC magnetic field excitation is applied along the horizontal axis of the atomic ensemble, and the phase frequency response is extracted using a lock-in amplifier, achieving both speed and high precision. The compensation magnetic field along the vertical axis is gradually reduced until the minimum phase frequency response is obtained; this compensation magnetic field value is the equivalent magnetic field for nuclear spin polarization, thus achieving in-situ measurement. The polarizability and nuclear spin longitudinal relaxation time are further obtained. This method overcomes the shortcomings of previous indirect measurement methods, improves measurement accuracy and speed, and accurately evaluates the state parameters of the atomic ensemble, laying the foundation for further atomic ensemble manipulation. Attached Figure Description

[0027] Figure 1 This is a schematic diagram of the structure of the high-precision and rapid in-situ measurement device for atomic common magnetometer polarization parameters that implements the present invention.

[0028] Figure 2 This is a flowchart of the high-precision and rapid in-situ measurement method for atomic common magnetometer polarization parameters according to the present invention. Figure 2 The process includes: Step 1, actively compensating the triaxial magnetic field to obtain the longitudinal compensated magnetic field Bc (the longitudinal axis is the z-axis); Step 2, applying a transverse magnetic field Bx (Bx is the x-axis magnetic field) through a signal generator; Step 3, extracting the phase signal, gradually reducing the longitudinal compensated magnetic field until the phase frequency response is minimized, and recording the longitudinal compensated magnetic field as... Step 4: Calculate the electronic polarizability, nuclear spin polarizability, and nuclear spin longitudinal relaxation time.

[0029] The reference numerals in the attached diagram are explained as follows: 1-Pump laser; 2-Beam expander assembly; 3-First polarizer; 4-1 / 4 wave plate; 5-Oven; 6-Atomic gas cell; 7-Triaxial magnetic field coil; 8-Photodetector; 9-Multilayer magnetic shielding cylinder; 10-Detection laser; 11-Second polarizer; 12-Wollaston prism; 13-Differential photoelectric amplifier; 14-Lock-in amplifier; 15-Signal generator; 16-Signal acquisition unit. Detailed Implementation

[0030] The following is in conjunction with the attached diagram ( Figures 1-2The present invention will be described in conjunction with the embodiments.

[0031] Figure 1 This is a schematic diagram of the structure of the high-precision and rapid in-situ measurement device for atomic common magnetometer polarization parameters that implements the present invention. Figure 2 This is a flowchart illustrating the high-precision, rapid, in-situ measurement method for atomic common magnetometer polarization parameters according to the present invention. (Reference) Figures 1 to 2 As shown, a high-precision, rapid in-situ measurement device for polarization parameters of an atomic common magnetometer includes a lock-in amplifier 14 connected to a differential photoelectric amplifier 13, a signal generator 15, and a signal acquisition unit 16. The differential photoelectric amplifier 13 converts the differential intensity signal of the detection light formed by the detection light passing through the atomic gas cell 6 into a voltage signal, which is then transmitted to the lock-in amplifier 14. The lock-in amplifier 14 extracts a phase signal from the voltage signal as a phase frequency response signal. The signal generator 15 generates a reference signal input to the lock-in amplifier to provide a frequency reference, and also generates a modulation voltage input to the triaxial magnetic field coil 7 built into the atomic gas cell 6 to generate a transverse modulation magnetic field. The signal acquisition unit 16 is used to observe the magnitude of the phase frequency response signal. The signal generator 15 controls the triaxial magnetic field coil 7 to achieve active compensation of the triaxial magnetic field, and the pump light enters the atomic gas cell to achieve polarization of the atomic ensemble. Let the longitudinal axis compensation magnetic field be... The equivalent magnetic field of electron spin polarization is The nuclear spin polarization equivalent magnetic field is , yes The longitudinal compensation magnetic field is gradually reduced until the phase frequency response signal reaches its minimum value, through... Obtaining electronic polarizability ,pass nuclear spin polarization ,pass and The longitudinal relaxation time of nuclear spin was obtained. .

[0032] An oven 5 is provided between the atomic gas chamber 6 and the triaxial magnetic field coil 7, and the triaxial magnetic field coil 7 is located inside the multi-layer magnetic shielding cylinder 9.

[0033] The detection light comes from the detection laser 10, which is connected to the differential photoelectric amplifier 13 in sequence through the second polarizer 11, the atomic gas cell 6 and the Wollaston prism 12. The pump light comes from the pump laser 1, which is connected to the photodetector 8 in sequence through the beam expander 2, the first polarizer 3, the quarter-wave plate 4 and the atomic gas cell 6.

[0034] A high-precision, rapid, in-situ measurement method for polarization parameters of an atomic common magnetometer includes the use of the aforementioned high-precision, rapid, in-situ measurement device for polarization parameters of an atomic common magnetometer.

[0035] Includes the following steps:

[0036] Step 1: Heat the atomic gas chamber using an oven, and use a signal generator to control the triaxial magnetic field coil to achieve active compensation of the triaxial magnetic field, thus obtaining the longitudinal axis compensated magnetic field. ;

[0037] Step 2: Use a signal generator to generate a modulation voltage, which is applied to the triaxial field coil to generate a transverse modulation magnetic field. A reference signal is generated and transmitted to a lock-in amplifier to provide a frequency reference;

[0038] Step 3: The differential photoelectric amplifier converts the acquired differential signal of the detection light intensity into a voltage signal and transmits it to the lock-in amplifier. The lock-in amplifier extracts the phase signal as the phase frequency response signal and gradually reduces the longitudinal axis compensation magnetic field. Until the phase frequency response signal reaches its minimum value, record the vertical axis compensation magnetic field at this point. The nuclear spin polarization equivalent magnetic field was obtained. Equivalent magnetic field of electron spin polarization ;

[0039] Step 4: Using the equivalent magnetic field of electron spin polarization Calculate electronic polarizability Through nuclear spin polarization equivalent magnetic field Calculation of nuclear spin polarization Calculate the longitudinal relaxation time of nuclear spin. .

[0040] Step 2 includes:

[0041]

[0042] in For modulation amplitude, For modulation frequency, To modulate the phase, For time.

[0043] Step 3 includes:

[0044] .

[0045] Step 4 includes:

[0046]

[0047]

[0048]

[0049] in The Fermi contact constant is... The permeability of free space, For Bohr magneton, For alkali metal density, The density of inert gas atoms. For inert gas nuclear magnetic moments, is the spin exchange rate constant between electron spin and nuclear spin.

[0050] like Figure 1 As shown, the high-precision, rapid in-situ measurement device for polarization parameters of an atomic common magnetometer of the present invention includes a pump laser 1, a beam expander 2, a first polarizer 3, a quarter-wave plate 4, an oven 5, an atomic gas cell 6, a triaxial magnetic field coil 7, a photodetector 8, a multi-layer magnetic shielding cylinder 9, a detection laser 10, a second polarizer 11, a Wollaston prism 12, a differential photoelectric amplifier 13, a lock-in amplifier 14, a signal generator 15, and a signal acquisition unit 16, the positional relationship of which is as follows:

[0051] The beam emitted from pump laser 1 passes sequentially through beam expander 2, first polarizer 3, quarter-wave plate 4, atomic gas cell 6, and photodetector 8. The beam emitted from detection laser 10 passes sequentially through second polarizer 11, atomic gas cell 6, Wollaston prism 12, and differential photoamplifier 13. The output of differential photoamplifier 13 is connected to lock-in amplifier 14, which is used to extract phase frequency response signal. Signal generator 15 generates reference signal input to lock-in amplifier 14 to provide frequency reference and generates modulation voltage input to triaxial magnetic field coil 7 to generate transverse modulation magnetic field. The output of lock-in amplifier 14 is connected to signal acquisition device 16, which is used to observe phase frequency response magnitude. Atomic gas cell 6 is fixed inside oven 5. Oven 5, atomic gas cell 6, and triaxial magnetic field coil 7 are fixed inside multi-layer magnetic shielding cylinder 9.

[0052] The oven 5 is a hollow cylinder with a spherical cavity inside for fixing the atomic gas chamber 6. A non-magnetic electric heating film is symmetrically pasted on the outer surface of the oven 5 for high-temperature heating of the atomic gas chamber 6. The atomic gas chamber 5 is a spherical glass gas chamber filled with potassium, rubidium, neon and nitrogen atoms.

[0053] The multi-layer magnetic shielding cylinder 9 is used to shield the magnetic field of the Earth's environment, and the triaxial magnetic field coil 7 is used to compensate for the remaining magnetic field. The multi-layer magnetic shielding cylinder 9 and the triaxial magnetic field coil 7 together improve the weak magnetic environment required for atomic polarization.

[0054] The modulation voltage signal generated by the signal generator 15 acts on the triaxial magnetic field coil 7 to generate a transverse modulation magnetic field, and the magnetic field strength is controlled by the signal generator 15.

[0055] like Figure 2 As shown, this invention also provides a high-precision, rapid in-situ measurement method for polarization parameters of an atomic common magnetometer, comprising the following steps:

[0056] Step 1: Heat the atomic gas chamber 6 using the oven 5; control the triaxial magnetic field coil 7 using the signal generator 15 to achieve active compensation of the triaxial magnetic field, obtaining a longitudinal axis compensated magnetic field. ;

[0057] Step 2: Use the signal generator 15 to generate a modulation voltage, which is applied to the triaxial magnetic field coil 7 to generate a transverse modulation magnetic field. A reference signal is generated and transmitted to lock-in amplifier 14 to provide a frequency reference;

[0058] Step 3: The differential photoelectric amplifier 13 converts the acquired differential signal of the detection light intensity into a voltage signal and transmits it to the lock-in amplifier 14. The lock-in amplifier 14 extracts the phase signal as the phase frequency response signal; the longitudinal axis compensation magnetic field is gradually reduced. Until the phase frequency response signal reaches its minimum value, record the vertical axis compensation magnetic field at this point. The nuclear spin polarization equivalent magnetic field was obtained. Equivalent magnetic field of electron spin polarization ;

[0059] Step 4: Using the equivalent magnetic field of electron spin polarization Calculate electronic polarizability Through nuclear spin polarization equivalent magnetic field Calculation of nuclear spin polarization ; Calculate the longitudinal relaxation time of nuclear spin .

[0060] in The Fermi contact constant is... The permeability of free space, For Bohr magneton, For alkali metal density, The density of inert gas atoms. For inert gas nuclear magnetic moments, denoted as the spin exchange rate constant between electron spin and nuclear spin, both of which are known values.

[0061] The working principle of this invention is as follows: When an excitation magnetic field is applied to the horizontal axis of the SERF gyroscope, the steady-state response model of the atomic spin inertial measurement system to the high-frequency modulated magnetic field on the horizontal axis is obtained, and the phase frequency response is further obtained, which is related to the total magnetic field on the vertical axis, i.e. .in It is the phase frequency response. To compensate for the residual magnetic field along the vertical axis after the magnetic field and polarization magnetic field cancel each other out, the phase frequency response is compared with... Taking the derivative, we find that when hour, This will reach a minimum value. Therefore, the polarization magnetic field can be compensated to the operating point first, at which point the vertical axis compensation magnetic field... Then, an excitation magnetic field is applied to the horizontal axis, and the compensation magnetic field is scanned along the vertical axis until a minimum phase response is obtained. The magnetic field applied along the vertical axis at this point can be used to obtain the nuclear spin polarization equivalent magnetic field. Then the equivalent magnetic field of electron spin polarization .

[0062] Contents not described in detail in this specification are prior art known to those skilled in the art. It is hereby indicated that the above description is intended to help those skilled in the art understand this invention, but does not limit the scope of protection of this invention. Any equivalent substitutions, modifications, improvements, and / or simplifications of the above descriptions that do not depart from the essential content of this invention fall within the scope of protection of this invention.

Claims

1. A high-precision, rapid, in-situ measurement device for polarization parameters of an atomic common magnetometer, characterized in that, The system includes a lock-in amplifier connected to a differential photoelectric amplifier, a signal generator, and a signal acquisition unit. The differential photoelectric amplifier converts the differential intensity signal of the detection light formed by the detection light passing through the atomic gas cell into a voltage signal, which is then transmitted to the lock-in amplifier. The lock-in amplifier extracts a phase signal from the voltage signal as a phase frequency response signal. The signal generator generates a reference signal input to the lock-in amplifier to provide a frequency reference, and also generates a modulation voltage input to a triaxial magnetic field coil built into the atomic gas cell to generate a transverse modulation magnetic field. The signal acquisition unit is used to observe the magnitude of the phase frequency response signal. The signal generator controls the triaxial magnetic field coil to achieve active compensation of the triaxial magnetic field. Pump light enters the atomic gas cell to achieve polarization of the atomic ensemble. Let the longitudinal axis compensation magnetic field be... The equivalent magnetic field of electron spin polarization is The nuclear spin polarization equivalent magnetic field is , yes The longitudinal compensation magnetic field is gradually reduced until the phase frequency response signal reaches its minimum value, through... Obtaining electronic polarizability ,pass nuclear spin polarization ,pass and The longitudinal relaxation time of nuclear spin was obtained. .

2. The high-precision, rapid in-situ measurement device for polarization parameters of an atomic common magnetometer according to claim 1, characterized in that, An oven is provided between the atomic gas chamber and the triaxial magnetic field coil, and the triaxial magnetic field coil is located inside a multi-layer magnetic shielding cylinder.

3. The high-precision, rapid in-situ measurement device for polarization parameters of an atomic common magnetometer according to claim 1, characterized in that, The detection light comes from a detection laser, which is connected to the differential photoelectric amplifier in sequence through a second polarizer, an atomic gas cell, and a Wollaston prism. The pump light comes from a pump laser, which is connected to a photodetector in sequence through a beam expander, a first polarizer, a quarter-wave plate, and an atomic gas cell.

4. A high-precision, rapid, in-situ measurement method for polarization parameters of an atomic common magnetometer, characterized in that: This includes the in-situ measurement device for atomic polarization parameters based on phase frequency response as described in any one of claims 1-3.

5. The high-precision, rapid in-situ measurement method for polarization parameters of an atomic common magnetometer according to claim 4, characterized in that, Includes the following steps: Step 1: Heat the atomic gas chamber using an oven, and use a signal generator to control the triaxial magnetic field coil to achieve active compensation of the triaxial magnetic field, thus obtaining the longitudinal axis compensated magnetic field. ; Step 2: Use a signal generator to generate a modulation voltage, which is applied to the triaxial field coil to generate a transverse modulation magnetic field. A reference signal is generated and transmitted to a lock-in amplifier to provide a frequency reference; Step 3: The differential photoelectric amplifier converts the acquired differential signal of the detection light intensity into a voltage signal and transmits it to the lock-in amplifier. The lock-in amplifier extracts the phase signal as the phase frequency response signal and gradually reduces the longitudinal axis compensation magnetic field. Until the phase frequency response signal reaches its minimum value, record the vertical axis compensation magnetic field at this point. The nuclear spin polarization equivalent magnetic field was obtained. Equivalent magnetic field of electron spin polarization ; Step 4: Using the equivalent magnetic field of electron spin polarization Calculate electronic polarizability Through nuclear spin polarization equivalent magnetic field Calculation of nuclear spin polarization Calculate the longitudinal relaxation time of nuclear spin. .

6. The high-precision, rapid in-situ measurement method for polarization parameters of an atomic common magnetometer according to claim 5, characterized in that, Step 2 includes: in For modulation amplitude, For modulation frequency, To modulate the phase, For time.

7. The high-precision, rapid in-situ measurement method for polarization parameters of an atomic common magnetometer according to claim 5, characterized in that, Step 3 includes: 。 8. The high-precision, rapid in-situ measurement method for polarization parameters of an atomic common magnetometer according to claim 5, characterized in that, Step 4 includes: in The Fermi contact constant is... The permeability of free space, For Bohr magneton, For alkali metal density, The density of inert gas atoms. For inert gas nuclear magnetic moments, is the spin exchange rate constant between electron spin and nuclear spin.