A polarization gradient relaxation simulation calculation method based on three-dimensional polarizability distribution

Abstract

A polarization gradient relaxation simulation calculation method based on three-dimensional polarizability distribution. Through preliminary parameter testing and calculation, a three-dimensional polarizability distribution model within the SERF atomic spin inertia measurement system is constructed. The three-dimensional polarizability simulation calculation inside the gas chamber is realized through the finite element simulation software COMSOL, the polarization gradient distribution inside the gas chamber is obtained, and the polarization gradient relaxation inside the gas chamber is calculated. This calculation method can be used to guide the subsequent suppression of the polarizability gradient and the optimal selection of the working parameters of the gas chamber, providing an evaluation means for reducing the relaxation of the SERF atomic spin inertia measurement system, enhancing the magnetic compensation ability, and improving the nuclear spin magnetic field, etc.

Description

Technical Field

[0001] The invention relates to the field of atomic gyroscopes, in particular to a polarization gradient relaxation simulation calculation method based on three-dimensional polarizability distribution. Through preliminary parameter testing and calculation, a three-dimensional distribution model of polarizability in a SERF atomic spin inertial measurement system is constructed. The three-dimensional polarizability simulation calculation inside an air chamber is realized through finite element simulation software COMSOL to obtain the polarization gradient distribution in the air chamber and calculate the polarization gradient relaxation inside the air chamber. The calculation method can be used to guide the subsequent suppression of polarizability gradients and the optimization selection of air chamber working parameters, and provide an evaluation means for reducing the relaxation of the SERF atomic spin inertial measurement system, enhancing magnetic compensation capability, improving nuclear spin magnetic field, etc. Background Art

[0002] The spin-exchange relaxation-free (SERF) atomic spin inertial measurement device based on the interaction between light and magnetism and atoms has an ultra-high theoretical measurement sensitivity. Atomic spin relaxation is an important error source that limits performance improvement in SERF inertial measurement devices. Among them, polarization gradient relaxation is one of the main sources of transverse relaxation of inert gas nuclear spins. The atomic spin polarization gradient is mainly caused by the attenuation of pumping light when the laser interacts with atoms, and is an unavoidable error source in SERF inertial measurement devices. The atomic spin polarization gradient will cause atomic decoherence, leading to atomic relaxation and reducing system sensitivity. At the same time, atomic spin polarization gradient relaxation will affect the polarization efficiency of inert gas nuclear spins. Reducing the nuclear spin polarization rate will weaken the self-compensation ability of the nuclear spin to the external magnetic field, destroy the stability of the system, and limit the improvement of system performance. However, in the K-Rb-Ne hybrid pumping inertial measurement device, the existing polarization gradient relaxation simulation calculation method is to analyze the variation law of the polarizability gradient according to the one-dimensional linear gradient, and estimate the influence of the polarization gradient on the system by simulating and calculating the linear polarizability gradient. This method only considers the effect of pumping light absorption, and the gradient model only relies on the one-dimensional polarizability variation of the center line of the air chamber. However, due to the influence of factors such as spin exchange, diffusion, and air chamber wall relaxation between alkali metal atoms, the polarizability gradient in the air chamber of the atomic spin inertial measurement device changes in three dimensions and belongs to a space vector. Therefore, it is not comprehensive to simulate and calculate the polarization gradient distribution in the air chamber of the K-Rb-Ne hybrid pumping inertial measurement system according to the linear gradient to evaluate the influence of the polarization gradient inside the air chamber, and the variation law is inconsistent with the actual test results. And the influence of the polarizability gradient on the relaxation performance of the system is not analyzed according to the three-dimensional changes inside the air chamber. Therefore, a polarization gradient relaxation simulation calculation method based on three-dimensional polarizability distribution is provided in the present invention. Summary of the invention

[0003] Aiming at the deficiencies existing in the prior art, the present invention provides a simulation calculation method for polarization gradient relaxation based on three-dimensional polarizability distribution. Through preliminary parameter testing and calculation, a three-dimensional polarizability distribution model within a SERF atomic spin inertial measurement system is constructed. The three-dimensional polarizability simulation calculation inside the gas chamber is realized through the finite element simulation software COMSOL, the polarization gradient distribution inside the gas chamber is obtained, and the polarization gradient relaxation inside the gas chamber is calculated. This calculation method can be used to guide the subsequent suppression of the polarizability gradient and the optimized selection of the working parameters of the gas chamber, providing an evaluation means for reducing the relaxation of the SERF atomic spin inertial measurement system, enhancing the magnetic compensation ability, and improving the nuclear spin magnetic field, etc.

[0004] The technical solution of the present invention is as follows:

[0005] A simulation calculation method for polarization gradient relaxation based on three-dimensional polarizability distribution, characterized by comprising the following steps:

[0006] Step 1, determine the system operating point parameters;

[0007] Step 2, construct the steady-state diffusion equation of the SERF atomic spin inertial measurement system;

[0008] Step 3, use the finite element software COMSOL to establish the alkali metal gas chamber structure model and mesh the structure model;

[0009] Step 4, use the finite element software to simulate and calculate the steady-state diffusion equation of the system to obtain the polarizability corresponding to the grid with coordinates (x, y, z) inside the gas chamber, and obtain the three-dimensional distribution of the electronic polarizability inside the gas chamber;

[0010] Step 5, calculate the polarizability gradients of the polarizability along the x, y, and z directions at each coordinate point in space;

[0011] Step 6, use the relational expression between the polarizability gradient and the polarization gradient relaxation to calculate the polarization gradient relaxation inside the gas chamber.

[0012] The system operating point parameters in the above-mentioned step 1 include the initial pumping rate Rp, spin-exchange relaxation spin collision relaxation and and the diffusion coefficient D i :

[0013]

[0014] where φ is the light flux, σ is the photon absorption cross-section area, P is the pumping laser power, A is the cross-sectional area of the pumping light spot, h is Planck's constant, c is the speed of light, v is the pumping laser frequency, λ is the pumping laser wavelength, v c the central absorption frequency of alkali metal atoms under the action of gas, Γ Lis the pressure broadening caused by the gas in the gas chamber, r e is the electron radius, f is a constant characterizing the vibration intensity. For the D1 line: For the D2 line:

[0015]

[0016] Among them, is the spin-exchange relaxation of the alkali metal electron to the inert gas nuclear spin. The alkali metals are Rb and K, and the inert gas is Ne, n e is the alkali metal atomic density, is the spin-exchange coefficient between Ne atoms and Rb, K atoms;

[0017]

[0018]

[0019] Among them, is the spin-collision relaxation of the alkali metal electron to the inert gas nuclear spin, is the spin-collision relaxation of the inert gas nuclear spin to the alkali metal electron. The alkali metals are Rb and K, and the inert gas is Ne, n e is the alkali metal atomic density, n n is the density of Ne, is the exchange-collision coefficient between Ne atoms and Rb, K atoms, υ ne is the spin-collision velocity between the inert gas Ne atoms and the alkali metal atoms;

[0020]

[0021] Among them, is the standard diffusion coefficient of the atom in the gas chamber, T is the gas chamber temperature in Kelvin, p Ne is the gas chamber pressure. The subscript i of D represents the atomic species K, Rb, and Ne respectively. D i are respectively D K 、D Rb and D Ne .

[0022] The steady-state diffusion equation in step 2 is expressed by the Bloch-Torrey equation as follows:

[0023]

[0024] Among them, t is the time, is the K electron spin polarization rate vector, is the Rb electron spin polarization rate vector, P n is the inert gas nuclear spin polarization rate vector, is the Laplace operator, x, y, and z represent Cartesian coordinates in the x - y - z three - dimensional space, D K , D Rb , D Ne are the diffusion coefficients of K, Rb, and Ne atoms in the buffer gas inside the gas cell. Both the above - mentioned electron spin polarization rate vector and the nuclear spin polarization rate vector can be expressed as components in the three directions of x, y, and z. × represents the cross product. γ e is the gyromagnetic ratio of the electron spin, γ n is the gyromagnetic ratio of the nuclear spin of the noble gas, Q K is the K electron spin relaxation factor, Q Rb is the Rb electron spin relaxation factor, B is the external magnetic field vector, L K and L Rb are the optical displacement vectors;

[0025] λ K and λ Rb are the Fermi contact enhancement factors of K and Ne, λ Rb is the Fermi contact enhancement factor of Rb and Ne, and are the magnetic susceptibilities when the electron spin of the alkali metal and the nuclear spin of the noble gas are completely polarized, is the magnetic field generated by the nuclear spin of the noble gas that the K electron spin senses, is the magnetic field generated by the nuclear spin of the noble gas that the Rb electron spin senses, is the magnetic field generated by the K electron spin that the nuclear spin of the noble gas senses, is the magnetic field generated by the K electron spin that the nuclear spin of the noble gas senses. Ω is the relative rotational angular velocity vector sensed by the atomic spin, is the spin - exchange relaxation between the K electron spin and the nuclear spin of the noble gas that the K electron spin senses, is the spin - exchange relaxation between the nuclear spin of the noble gas and the K electron spin that the nuclear spin of the noble gas senses, is the spin - exchange relaxation between the K electron spin and the Rb electron spin that the K electron spin senses, is the spin - exchange relaxation between the Rb electron spin and the nuclear spin of the noble gas that the Rb electron spin senses, is the spin - exchange relaxation between the nuclear spin of the noble gas and the Rb electron spin that the nuclear spin of the noble gas senses, is the spin - exchange relaxation between the Rb electron spin and the K electron spin that the Rb electron spin senses;

[0026] R p is the rate of pumping - light - polarized atoms, R m is the polarization rate of the detection light, is the total relaxation rate of the K electron spin, is the total relaxation rate of the Rb electron spin, is the collision relaxation rate of the inert gas nuclear spin. S p and S m are the degrees of circular polarization characterizing the circularly polarized components of the pumping light and the detection light.

[0027] The steady-state diffusion equation in step 2 is expressed as follows:

[0028]

[0029] The condition for equation (2) to hold is: After the pumping action of the pumping light on the gas chamber reaches equilibrium with relaxation, diffusion and other effects, the internal polarizability of the gas chamber reaches a steady state, the change rates of the electron spin and the nuclear spin are 0, the directions of the pumping light and the main magnetic field are along the Z axis, the direction of the detection light propagates along the X direction. At steady state, the main direction of the polarizability is on the z axis, and the steady-state distribution of the Z-axis polarizability in the gas chamber is determined by equation (2).

[0030] In step 2, it includes selecting three groups of convective diffusion equations in the finite element simulation software COMSOL to describe the diffusion steady-state distribution of the electron and nuclear spin polarizabilities inside the gas chamber. The change rate of the polarization distribution with time is 0, the initial value is 0, and the boundary absorption term is the collision rate between the polarizability and the boundary, which is related to the diffusion coefficient and pumping rate of the colliding atoms; when the alkali metal contacts the uncoated gas chamber wall, the atomic polarizability rapidly decays to 0 in a short time. The boundary conditions for the electron spin polarizability and the nuclear spin polarizability are set as:

[0031]

[0032] In step 3, it includes setting the alkali metal gas chamber as a spherical glass gas chamber with an inner diameter of 8 mm, using the finite element simulation software COMSOL to construct the structure of the alkali metal gas chamber, selecting a relatively refined refinement degree, and the refinement degree will affect the accuracy of the relaxation rate simulation calculation. Through refinement, the inside of the gas chamber is divided into multiple small volume grid units for subsequent calculations.

[0033] In step 4, it includes considering the interaction of three atoms according to equations (2) and (3) above, as well as the relaxation effect of the external pumping light and the gas chamber wall. After refining the grid, the polarizabilities of the electron spin and the nuclear spin inside the gas chamber are simulated and calculated by the finite element software COMSOL in each small volume grid unit. In each grid, the polarizability corresponding to the three-dimensional coordinate point (x, y, z) at a specific position is obtained That is, the polarizability in each grid inside the gas chamber is obtained. The origin of the three-dimensional coordinate is the center point of the incident end face, that is, the contact point between the center point of the pumping Gaussian light and the top of the gas chamber.

[0034] In step 5, it includes:

[0035]

[0036] In (4), is the gradient of the ternary function f(x, y, z), are the unit vectors along the x, y, and z directions, is the polarizability corresponding to the coordinate point (x, y, z);

[0037]

[0038] Equation (5) is the expression of the polarizability gradient of the Z-direction polarizability along three directions. The subscripts i, j, k represent the grid numbers along the x, y, and z directions, and d i is the distance between the centers of the i-th (i can be replaced by j, k) grid and the (i + 1)-th grid along the X direction, is the Z-direction polarizability at the coordinate point (x i , y, k), the gradient along the x direction, is the Z-direction polarizability at the coordinate point (x, y j , z), the gradient along the y direction, is the Z-direction polarizability at the coordinate point (x, y, z k ), the gradient along the z direction. Through the above formula, the transverse polarization gradient relaxation inside the gas chamber can be calculated;

[0039]

[0040] Equation (7) is the expression of the polarizability gradient of the X-direction polarizability along three directions. Among them, is the X-direction polarizability at the coordinate point (x i , y, k), the gradient along the x direction, is the X-direction polarizability at the coordinate point (x, y j , z), the gradient along the y direction, is the X-direction polarizability at the coordinate point (x, y, z k ), the gradient along the z direction;

[0041]

[0042] In Equation (8), is the Y-direction polarizability at the coordinate point (x i , y, k), the gradient along the x direction, is the Y-direction polarizability at the coordinate point (x, y j , z), Gradient along the y direction is the Y-direction polarizability at the coordinate point (x, y, z k ). Gradient along the z direction. The polarization gradient relaxation inside the gas chamber can be calculated through the above formula. The longitudinal polarization gradient relaxation inside the gas chamber can be calculated through the above formula.

[0043] Step 6 includes:

[0044]

[0045] In formula (6), the subscripts i, j, k represent the grid numbers along the x, y, z directions, m, n, b are the total number of grids along the x, y, z directions, V is the volume of the gas chamber, λ is the Fermi contact enhancement factor between the alkali metal electron spin and the noble gas nuclear spin, and are the magnetic susceptibilities when the alkali metal electron spin and the noble gas nuclear spin are completely polarized respectively, D Ne is the diffusion coefficient of Ne atoms under the buffer gas inside the gas chamber, γ n is the gyromagnetic ratio of the noble gas nuclear spin;

[0046]

[0047] Formula (9) is for the longitudinal polarization gradient relaxation and the X-direction polarizability gradient Y-direction polarizability gradient relation formula, where the subscripts i, j, k represent the grid numbers along the x, y, z directions, m, n, b are the total number of grids along the x, y, z directions, V is the volume of the gas chamber, Bz is the magnetic field in the Z direction. λ is the Fermi contact enhancement factor between the alkali metal electron spin and the noble gas nuclear spin, and are the magnetic susceptibilities when the alkali metal electron spin and the noble gas nuclear spin are completely polarized respectively, D Ne is the diffusion coefficient of Ne atoms under the buffer gas inside the gas chamber. γ n is the gyromagnetic ratio of the noble gas nuclear spin.

[0048] Use formulas (5), (7), and (8) to calculate the polarizability gradients of the polarizabilities along the x, y, and z directions at each coordinate point in space, and use formulas (6) and (9) to calculate the polarization gradient relaxation inside the gas chamber.

[0049] The technical effects of the present invention are as follows: In the present invention, various parameters required for the simulation calculation of the measurement system are first measured, and a three-dimensional polarizability distribution model inside the K-Rb-Ne gas cell of the hybrid pumping inertial measurement system is established through the steady-state diffusion equation and boundary conditions. Then, the finite element software COMSOL is used to mesh the inside of the gas cell, and the polarizabilities of each grid point inside the gas cell are calculated. According to the polarizabilities of each grid point and the distances between the grids, the polarizability gradient distribution inside it is obtained. Finally, according to the polarizability gradient, the relaxation caused by the polarization gradient is calculated by summation and averaging. Through this simulation calculation method, the variation law of the polarization gradient inside the gas cell with working parameters such as the gas cell temperature, pumping laser power, gas cell pressure, and density ratio can be quickly obtained, providing a simulation evaluation means for the optimization of pumping parameters and gas cell parameters, which is of great significance for suppressing the polarization gradient in the K-Rb-Ne hybrid pumping inertial measurement system, reducing nuclear spin relaxation, enhancing the magnetic field suppression ability of the system, and the nuclear spin polarizability. Description of the Drawings

[0050] Figure 1 It is a schematic flow chart of a polarization gradient relaxation simulation calculation method based on three-dimensional polarizability distribution for implementing the present invention. Figure 1 It includes the following steps: Step 1, determine the working point parameters such as the initial pumping rate, spin-exchange relaxation, spin-collision relaxation, diffusion coefficient, etc. of the system according to the gas cell temperature, pumping laser power, and gas pressure; Step 2, construct the steady-state diffusion equation of the system through each parameter of the working point and the SERF atomic spin inertial measurement equation; Step 3, establish the structural model of the alkali metal gas cell, and use the finite element software COMSOL to mesh the inside of the structural model; Step 4, use the finite element software to simulate and calculate the steady-state diffusion equation of the system to obtain the polarizability corresponding to the grid with coordinates (x, y, z) inside the gas cell, and obtain the three-dimensional electronic polarizability distribution inside the gas cell; Step 5, use Equation (5), Equation (7), and Equation (8) to calculate the polarization gradients of the polarizabilities along the x, y, and z directions at each coordinate point in space; Step 6, use the relational expressions (6) and (9) between the polarization gradient and the polarization gradient relaxation to calculate the polarization gradient relaxation inside the gas cell.

[0051] Figure 2 It is a distribution diagram of the Z-direction polarizability on the (0, y, z) plane. Figure 2 The y-axis scale in it includes -4, -2, 0, 2, 4 mm, and the z-axis scale includes 0, 2, 4, 6, 8 mm. Figure 2 The circle in it is the cross-section of the gas cell, the Z-axis is the propagation direction of the pumping light, the right side is the color legend of the polarization distribution inside the gas cell and its marked values (0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5). From bottom to top, the polarizability value increases from 0 to 0.5, and the corresponding color gradually changes from dark blue - green - yellow - dark red. Figure 2The in - circle polarization rate shows an equivalent - water - droplet - shaped change. The color changes from dark blue - green - yellow - dark red from the outside to the inside, that is, the polarization rate value gradually increases in a water - droplet shape. The Z - axis is the propagation attenuation direction of the pump light. Therefore, the polarization rate gradually attenuates along the longitudinal axis, and the distribution along the horizontal axis is mainly caused by the Gaussian distribution of the original pump light.

[0052] Figure 3 It is the distribution diagram of the Z - direction polarization rate on the (x, y, 4) plane. Figure 3 In the lower left corner of is the coordinate axis, showing the XOY section. Figure 3 In the circle is the section of the gas chamber. The horizontal axis is the X - axis coordinate with the unit of mm. The diameter of the gas chamber is 8 mm, and the horizontal axis range is from - 4 mm to 4 mm. The vertical axis is the Y - axis coordinate with the unit of mm, and the vertical axis range is from - 4 mm to 4 mm. XOY is the incident plane of the pump light. Figure 3 On the right is the color legend of the polarization distribution in the gas chamber and its marked values (0.05, 0.1, 0.15, 0.2, 0.25, 0.3). From bottom to top, the polarization rate value increases from 0 to 0.3, and the corresponding color gradually changes from dark blue - green - yellow - dark red. Figure 3 The polarization rate inside the circle in shows an equivalent - circle - shaped change. The color changes from dark blue - green - yellow - dark red from the outside to the inside, that is, the polarization rate gradually increases in a circular shape. The XY plane is the incident plane of the pump light, and the polarization rate distribution on this plane is mainly caused by the Gaussian distribution of the incident pump light. Specific implementation mode

[0053] The following combines the attached drawings ( Figures 1 - 3 ) and embodiments to illustrate the present invention.

[0054] Figure 1 It is a schematic flow diagram of a polarization - gradient relaxation simulation calculation method based on three - dimensional polarization rate distribution for implementing the present invention. Figure 2 It is the distribution diagram of the Z - direction polarization rate on the (0, y, z) plane. Figure 3 It is the distribution diagram of the Z - direction polarization rate on the (x, y, 4) plane. Refer to Figures 1 to 3As shown in the figure, a polarization gradient relaxation simulation calculation method based on three-dimensional polarizability distribution calculates system parameters through working point conditions, constructs a polarizability diffusion distribution equation in a SERF atomic spin inertia measurement system using the system parameters, establishes an alkali metal gas cell structure model, performs mesh division on the structure through COMSOL software, performs polarization rate distribution simulation calculation on the diffusion distribution equation and boundary conditions, obtains the polarizability magnitudes corresponding to each grid point in the gas cell space, calculates the polarization gradients of each grid point in the gas cell in the x, y, and z coordinate axes directions using the polarizabilities of each grid point, gives the expressions of polarization gradient relaxation and the polarizability gradients at each point in space, and superimposes and calculates the polarization gradient relaxation in the SERF atomic spin inertia measurement system. Through this simulation calculation method, it is possible to guide the subsequent suppression of polarizability gradients and the optimization of gas cell working parameters. It provides an evaluation method for reducing the relaxation of SERF atomic spin gyroscopes, enhancing magnetic compensation capabilities, and improving nuclear spin magnetic fields.

[0055] In the present invention, first, various parameters required for system simulation calculation are measured, a three-dimensional polarizability distribution model inside a K-Rb-Ne gas cell in a hybrid pumping inertia measurement system is established through a steady-state diffusion equation and boundary conditions, then the interior of the gas cell is meshed using the finite element software COMSOL, the polarizabilities of each grid point inside the gas cell are calculated, and the polarizability gradient distribution inside it is obtained based on the polarizabilities of each grid point and the distance between grids. Finally, according to the polarizability gradient, the relaxation caused by the polarization gradient is calculated through summation and averaging. Through this simulation calculation method, the variation law of the polarization gradient inside the gas cell with working parameters such as gas cell temperature, pumping laser power, gas cell pressure, and density ratio can be quickly obtained, providing a simulation evaluation method for the optimization of pumping parameters and gas cell parameters, and is of great significance for suppressing the polarization gradient in the K-Rb-Ne hybrid pumping inertia measurement system, reducing nuclear spin relaxation, and enhancing the system's magnetic field suppression ability and nuclear spin polarizability.

[0056] (1) Determine the system working point parameters

[0057] When the working conditions of the K-Rb-Ne hybrid pumping inertia measurement system and the gas cell parameters are known, parameters such as pumping rate, spin exchange relaxation, collision relaxation, slowing factor, and diffusion coefficient can be calculated. Substitute the parameters into the steady-state diffusion equation for three-dimensional polarizability distribution simulation.

[0058] A. Initial pumping rate R p

[0059] According to the pumping laser power, frequency, gas cell pressure, and pumping spot area, the initial pumping rate R can be calculated p As follows:

[0060]

[0061] where φ is the luminous flux, σ is the photon absorption cross-section, P is the pumping laser power, A is the cross-sectional area of the pumping light spot, h is Planck's constant, c is the speed of light, v is the pumping laser frequency, λ is the pumping laser wavelength, v c is the central absorption frequency of alkali metal atoms under the action of gas, Γ L is the pressure broadening caused by the gas in the gas chamber. r e is the electron radius. f is a constant characterizing the vibration intensity (for D1 line: D2 line: ).

[0062] B. Spin-exchange relaxation

[0063]

[0064] Among them, is the spin-exchange relaxation of alkali metal electrons to the nuclear spin of inert gas. The alkali metal electron spin e in the subscript represents Rb and K atoms, and the inert gas nuclear spin n represents Ne atoms, n e is the alkali metal atom density, is the spin-exchange coefficient between Ne atoms and Rb and K atoms.

[0065] C. Spin-collision relaxation

[0066]

[0067]

[0068] is the spin-collision relaxation of alkali metal electrons to the nuclear spin of inert gas, is the spin-collision relaxation of the inert gas nuclear spin to alkali metal electrons. The alkali metal electron spin e in the superscript and subscript represents Rb and K atoms, and the inert gas nuclear spin n represents Ne atoms. is the exchange collision coefficient between Ne atoms and Rb and K atoms, n e 、n n are the densities of alkali metal electrons e (Rb, K) and inert gas nucleons n (Ne) respectively. υ ne is the spin-collision velocity between inert gas atoms and alkali metal atoms.

[0069] D. Diffusion coefficient D i

[0070]

[0071] Among them, is the standard diffusion coefficient of inert gas in the gas chamber, T is the gas chamber temperature, and the unit is Kelvin. pNe is the air chamber air pressure. Table i below shows the atomic species, D i includes D K , D Rb , D Ne .

[0072] (2) Construct the steady-state diffusion equation of the SERF atomic spin inertia measurement system to obtain a three-dimensional distribution model.

[0073] The evolution process of the SERF atomic spin inertia measurement system considering the diffusion effect over time t can be represented by the following Bloch-Torrey equation:

[0074]

[0075] Among them, is the K electron spin polarization rate vector, is the Rb electron spin polarization rate vector, P n is the noble gas nuclear spin polarization rate vector, is the Laplace operator, x, y, z represent the Cartesian coordinates in the x - y - z three-dimensional space, D K , D Rb , D Ne are the diffusion coefficients of K, Rb, and Ne atoms in the buffer gas inside the air chamber. The above electron spin polarization rate vectors and nuclear spin polarization rate vectors can all be expressed as components in the three directions of x, y, and z, and × represents the cross product. γ e is the gyromagnetic ratio of the electron spin, γ n is the gyromagnetic ratio of the noble gas nuclear spin, Q K is the K electron spin relaxation factor, Q Rb is the Rb electron spin relaxation factor, B is the external magnetic field vector, L K and L Rb are the optical displacement vectors.

[0076] λ K and λ Rb are the Fermi contact enhancement factors of K and Ne, λ Rb is the Fermi contact enhancement factor of Rb and Ne, and are the magnetic susceptibilities when the electron spin of the alkali metal and the nuclear spin of the noble gas are completely polarized, is the magnetic field generated by the noble gas nuclear spin felt by the K electron spin, is the magnetic field generated by the noble gas nuclear spin felt by the Rb electron spin, is the magnetic field generated by the K electron spin felt by the noble gas nuclear spin, is the magnetic field generated by the K electron spin felt by the noble gas nuclear spin. Ω is the relative rotational angular velocity vector felt by the atomic spin, is the spin-exchange relaxation between the K electron spin and the noble gas nuclear spin is the spin-exchange relaxation between the noble gas nuclear spin and the K electron spin is the spin-exchange relaxation between the K electron spin and the Rb electron spin is the spin-exchange relaxation between the Rb electron spin and the noble gas nuclear spin is the spin-exchange relaxation between the noble gas nuclear spin and the Rb electron spin is the spin-exchange relaxation between the Rb electron spin and the K electron spin

[0077] R p is the rate of pumping light to polarize atoms, R m is the polarization rate of the detection light is the total relaxation rate of the K electron spin is the total relaxation rate of the Rb electron spin is the collision relaxation rate of the noble gas nuclear spin. S p and S m are the degrees of circular polarization characterizing the circularly polarized components of the pumping light and the detection light (for linearly polarized light, |s| = 0; for circularly polarized light, |s| = 1).

[0078] When the gas cell reaches equilibrium through the pumping, relaxation, diffusion, etc. of the pumping light, the internal polarizability of the gas cell reaches a steady state, and the change rates of the electron spin and the nuclear spin are 0. The pumping light and the main magnetic field direction are along the Z axis, and the detection light direction propagates along the X direction. At steady state, the main direction of the polarizability is on the z axis, and the change rate of the polarizability on the left side of the above formula with respect to time is 0. The polarization gradient relaxation simulation calculation method proposed in this paper is applicable to the calculation of the polarization gradient relaxation in the x, y, and z axes. For the sake of explanation, we mainly analyze along the main axis direction Z axis. The steady-state distribution of the Z-axis polarizability in the gas cell can be expressed in the following form:

[0079]

[0080] In the finite element simulation software COMSOL, three groups of convection-diffusion equations are selected to describe the diffusion steady-state distribution of the electron and nuclear spin polarizabilities inside the gas cell. The change rate of the polarization distribution with respect to time is 0, the initial value is 0, and the boundary absorption term is the collision rate between the polarizability and the boundary, which is related to the diffusion coefficient and pumping rate of the colliding atoms. When the alkali metal contacts the uncoated gas cell wall, the atomic polarizability rapidly decays to 0 in a short time. The boundary conditions for the electron spin polarizability and the nuclear spin polarizability are set as:

[0081]

[0082] (3) Establish an alkali metal gas chamber structure model and perform mesh division.

[0083] The alkali metal gas chamber is a spherical glass gas chamber with an inner diameter of 8 mm. The finite element simulation software COMSOL is used to construct the alkali metal gas chamber structure, and the refinement level is selected as a relatively fine one. The refinement level will affect the accuracy of the relaxation rate simulation calculation. Through refinement, the interior of the gas chamber is divided into multiple small volume grid units for subsequent calculations.

[0084] (4) The steady-state diffusion equation is calculated using finite element software to obtain the three-dimensional polarizability distribution inside the air chamber.

[0085] According to the above equations (2) and (3), considering the interaction between the three atoms and the relaxation effect between the external pumping light and the gas chamber wall, the polarizability of the electron spin and nuclear spin in each small volume grid unit (coordinate x, y, z) inside the gas chamber can be calculated by finite element software COMSOL simulation. In each grid, the polarizability corresponding to the three-dimensional coordinate point (x, y, z) at a specific position can be obtained. That is, the polarizability in each grid inside the air chamber is obtained. The origin of the three-dimensional coordinate is the center point of the incident end face, that is, the contact point between the center point of the pumping Gaussian light and the top of the air chamber. According to the above simulation steps, the z-direction polarizability in the air chamber is obtained in space. The YOZ plane and XOY plane are shown in the following figure, where Figure (a) shows the interior of the air chamber. The cross-sectional distribution diagram of the center of the air cell. The lower left corner is the coordinate axis, which shows the YOZ cross-section. The circle in the figure is the cross-section of the air cell. The horizontal axis is the Y-axis coordinate, the unit is mm, the diameter of the air cell is 8mm, the horizontal axis range is -4mm to 4mm, the vertical axis is the Z-axis coordinate, the unit is mm, the vertical axis range is 0mm to 8mm, and the Z axis is the propagation direction of the pumping light. On the right is the color legend of the polarization distribution in the air cell. From bottom to top, the polarization value increases from 0 to 0.5, and the corresponding color gradually changes from dark blue → green → yellow → dark red. The polarization value in the circle in the figure shows an equal-valued water droplet-like change, and the color changes from outside to inside are dark blue → green → yellow → dark red, that is, the polarization value gradually increases in a water droplet-like shape. The Z axis is the propagation attenuation direction of the pumping light. Therefore, the polarization value gradually decays along the vertical axis, and the distribution along the horizontal axis is mainly caused by the Gaussian distribution of the original pumping light. Figure (b) is the inside of the air cell. The distribution diagram of the central section of the gas chamber, with the coordinate axes at the lower left corner, showing the XOY section. The circle in the figure is the section of the gas chamber. The horizontal axis is the X-axis coordinate, with the unit of mm. The diameter of the gas chamber is 8 mm, and the range of the horizontal axis is from -4 mm to 4 mm. The vertical axis is the Y-axis coordinate, with the unit of mm, and the range of the vertical axis is from -4 mm to 4 mm. XOY is the incident plane of the pumping light. On the right is the color legend of the polarization distribution in the gas chamber. From bottom to top, the polarization rate value increases from 0 to 0.3, and the corresponding color changes gradually from dark blue → green → yellow → dark red. The polarization rate inside the circle in the figure shows an equipotential circular change, and the color changes from dark blue → green → yellow → dark red from outside to inside, that is, the polarization rate increases gradually in a circular shape. The XY plane is the incident plane of the pumping light, and the polarization rate distribution on this plane is mainly caused by the Gaussian distribution of the incident pumping light.

[0086] Figure 2 It shows the distribution of the Z-direction polarization rate on the (0, y, z) plane. Figure 3 It shows the distribution of the Z-direction polarization rate on the (x, y, 4) plane.

[0087] (5) Calculate the polarization gradients of the polarization rates of each grid in the gas chamber space along the x, y, and z directions in the finite element software COMSOL.

[0088] The gradient of the ternary function f(x, y, z) can be expressed as:

[0089]

[0090] are the unit vectors along the x, y, and z directions. is the gradient of the function u = f(x, y, z) at point P. The objective function f(x, y, z) is the corresponding value of the polarization rate at different position points in space, that is Divide the gas chamber into several small grids by dividing the mesh, and obtain the polarization rate corresponding to each grid (spatial coordinates are (x, y, z)) from step (4) Express the polarization rate in the form of discrete points. The ratio of the difference between the current grid and the adjacent grids along the x, y, and z directions to the distance d between the two grids can be expressed as the polarization rate gradient at this point The gradients of the Z-direction polarization rate in the x, y, and z directions can be respectively expressed as:

[0091]

[0092] The above are the expressions of the polarization rate gradients of the Z-direction polarization rate along the three directions. The subscripts i, j, k represent the grid numbers along the x, y, and z directions, d i is the distance between the centers of the i-th (i can be replaced by j, k) grid and the (i + 1)-th grid along the X direction. is the coordinate point (x i, y, k) the Z - direction polarizability at The gradient along the x - direction,

[0093] is the coordinate point (x, y j , z) the Z - direction polarizability at The gradient along the y - direction, is the coordinate point (x, y, z k ) the Z - direction polarizability at The gradient along the z - direction. Through the above formula, the transverse polarization gradient relaxation inside the gas chamber can be calculated.

[0094] (6) Superpose the polarization gradient relaxations in space according to the polarization gradient to obtain the total polarization gradient relaxation.

[0095] According to the analysis of the classical gradient relaxation expression, the transverse polarizability gradient will cause longitudinal relaxation, and the longitudinal polarizability will cause transverse relaxation. The transverse polarization gradient relaxation and the polarizability gradient The relationship is:

[0096]

[0097] In the above formula, the subscripts i, j, k represent the grid numbers along the x, y, z directions, m, n, b are the total number of grids along the x, y, z directions, V is the volume of the gas chamber, λ is the Fermi contact enhancement factor between the alkali - metal electron spin and the inert - gas nuclear spin, and are the magnetic susceptibilities when the alkali - metal electron spin and the inert - gas nuclear spin are fully polarized respectively, D Ne is the diffusion coefficient of Ne atoms under the buffer gas inside the gas chamber. γ n is the gyromagnetic ratio of the inert - gas nuclear spin.

[0098] When the incident pumping light is not aligned with the coil direction, the polarizability has components in the X and Y directions The steady - state distributions of the X - and Y - axis polarizabilities inside the gas chamber can be given by the corresponding expressions according to Equation (2). The distributions of the X - direction polarizability and the Y - direction polarizability inside the gas chamber are obtained through the finite - element simulation software COMSOL. The gradients of the X - direction polarizability in the x, y, z directions can be respectively expressed as:

[0099]

[0100] Among them, is the X - direction polarizability at the coordinate point (x i , y, k) The gradient along the x - direction, is the X - direction polarizability at the coordinate point (x, y j , z), the gradient along the y - direction, is the X - direction polarizability at the coordinate point (x, y, z k ). The gradient along the z - direction.

[0101] The gradients of the Y - direction polarizability in the x, y, z directions can be respectively expressed as:

[0102]

[0103] Among them, is the Y - direction polarizability at the coordinate point (x i , y, k), the gradient along the x - direction, is the Y - direction polarizability at the coordinate point (x, y j , z), the gradient along the y - direction, is the Y - direction polarizability at the coordinate point (x, y, z k ). The gradient along the z - direction. Through the above formula, the polarization gradient relaxation inside the gas chamber can be calculated. Through the above formula, the longitudinal polarization gradient relaxation inside the gas chamber can be calculated.

[0104] Longitudinal polarization gradient relaxation and the X - direction polarizability gradient Y - direction polarizability gradient The relationship is:

[0105]

[0106] In the above formula, the subscripts i, j, k represent the grid numbers along the x, y, z directions, m, n, b are the total number of grids along the x, y, z directions, V is the volume of the gas chamber, Bz is the magnetic field in the Z - direction. λ is the Fermi contact enhancement factor between the alkali - metal electron spin and the inert - gas nuclear spin, and are the magnetic susceptibilities when the alkali - metal electron spin and the inert - gas nuclear spin are completely polarized respectively, D Ne is the diffusion coefficient of Ne atoms under the buffer gas inside the gas chamber. γ n is the gyromagnetic ratio of the inert - gas nuclear spin.

[0107] When the air chamber in the inertial measurement system is filled, the air pressure inside the air chamber is fixed. When the system is in the working state, with the known air chamber temperature and pumping light power, the parameters in step (1) are calculated and substituted into the diffusion equation to construct a three-dimensional polarizability distribution equation. By calculating the diffusion equation and boundary conditions of the SERF atomic spin inertial measurement system through finite element software, the polarizability distribution inside the air chamber can be obtained. Substituting the polarizability gradients at each point in the space obtained from the simulation calculation into the above formula and superimposing them can calculate the relaxation of the internal polarization gradient of the air chamber. Through this simulation calculation method, the relaxation caused by the polarization gradient can be quickly obtained from the internal polarization distribution of the air chamber, which can guide the subsequent suppression of the polarizability gradient and the optimal selection of the working parameters of the air chamber. It provides an evaluation method for reducing the relaxation of the SERF atomic spin gyroscope, enhancing the magnetic compensation ability, and enhancing the nuclear spin magnetic field.

[0108] The content not described in detail in the specification of the present invention belongs to the prior art well-known to those skilled in the art. It is hereby pointed out that the above description helps those skilled in the art to understand the present invention, but does not limit the protection scope of the present invention. Any implementation that is an equivalent replacement, modification, improvement, and / or simplification of the above description without departing from the essential content of the present invention falls within the protection scope of the present invention.

Claims

1. A simulation calculation method for polarization gradient relaxation based on three-dimensional polarizability distribution, characterized in that, it includes the following steps: Step 1, determine the system operating point parameters; Step 2, construct the steady-state diffusion equation of the SERF atomic spin inertia measurement system; Step 3, use the finite element software COMSOL to establish the alkali metal gas cell structure model and mesh the structure model; Step 4, use the finite element software to simulate and calculate the steady-state diffusion equation of the system to obtain the polarizability corresponding to the grid with coordinates (x, y, z) in the gas cell, and obtain the three-dimensional distribution of the electronic polarizability in the gas cell; Step 5, calculate the polarizability gradients of the polarizability along the x, y, and z directions at each coordinate point in space; Step 6, use the relationship between the polarizability gradient and the polarization gradient relaxation to calculate the polarization gradient relaxation in the gas cell; The steady-state diffusion equation in Step 2 is represented by the Bloch-Torrey equation as follows: where t is time, is the K electron spin polarization rate vector, is the Rb electron spin polarization rate vector, P n is the noble gas nuclear spin polarization rate vector, is the Laplace operator, x, y, and z represent the Cartesian coordinates in the x - y - z three - dimensional space, D K , D Rb , D Ne are the diffusion coefficients of K, Rb, and Ne atoms in the buffer gas inside the gas chamber. The above - mentioned electron spin polarization rate vectors and nuclear spin polarization rate vectors can all be expressed as components in the x, y, and z directions. × represents the cross product, γ e is the gyromagnetic ratio of the electron spin, γ n is the gyromagnetic ratio of the noble gas nuclear spin, Q K is the K electron spin relaxation factor, Q Rb is the Rb electron spin relaxation factor, B is the external magnetic field vector, L K and L Rb are the optical shift vectors; λ K and λ Rb is the Fermi contact enhancement factor of K and Rb, and λ n is the Fermi contact enhancement factor of Ne, and are the magnetic susceptibilities when the electron spins of alkali metals and the nuclear spins of noble gases are completely polarized, respectively, is or is the magnetic field generated by the nuclear spin of the noble gas felt by the electron spin of K, is the magnetic field generated by the nuclear spin of the noble gas felt by the electron spin of Rb, is the magnetic field generated by the electron spin of K felt by the nuclear spin of the noble gas, is the magnetic field generated by the electron spin of Rb felt by the nuclear spin of the noble gas. Ω is the relative rotational angular velocity vector felt by the atomic spin, is the spin-exchange relaxation between the electron spin of K and the nuclear spin of the noble gas felt by the electron spin of K, is the spin-exchange relaxation between the electron spin of K and the nuclear spin of the noble gas felt by the nuclear spin of the noble gas, is the spin-exchange relaxation between the electron spin of K and the electron spin of Rb felt by the electron spin of K, is the spin-exchange relaxation between the electron spin of Rb and the nuclear spin of the noble gas felt by the electron spin of Rb, is the spin-exchange relaxation between the electron spin of Rb and the nuclear spin of the noble gas felt by the nuclear spin of the noble gas, is the spin-exchange relaxation between the electron spin of Rb and the electron spin of K felt by the electron spin of Rb; R p is the rate of pumping light to polarize atoms, R m is the polarization rate of the detection light, is the total relaxation rate of the K electron spin, is the total relaxation rate of the Rb electron spin, is the collision relaxation rate of the inert gas nuclear spin, S P and S m are the degrees of circular polarization characterizing the circularly polarized components of the pumping light and the detection light.

2. The simulation calculation method for polarization gradient relaxation based on three-dimensional polarizability distribution according to claim 1, characterized in that, The system operating point parameters in step 1 include the initial pumping rate Rp, spin-exchange relaxation spin collision relaxation and as well as the diffusion coefficient D i : Where φ is the luminous flux, σ is the photon absorption cross-section, P is the pumping laser power, A is the cross-sectional area of the pumping light spot, h is Planck's constant, c is the speed of light, v is the pumping laser frequency, λ is the pumping laser wavelength, v c The central absorption frequency of alkali metal atoms under the action of gas, Γ L Is the pressure broadening caused by the gas in the gas chamber, r e Is the electron radius, f is a constant characterizing the vibration intensity. For the D1 line: For the D2 line: Among them, is the spin-exchange relaxation of alkali metal electron pairs on the nuclear spin of inert gas. The alkali metals are Rb and K, the inert gas is Ne, and n e is the alkali metal atomic density, is the spin-exchange coefficient between Ne atoms and Rb and K atoms; Among them, is the spin collision relaxation of the alkali metal electron pair to the inert gas nuclear spin, is the spin collision relaxation of the inert gas nuclear spin to the alkali metal electron. The alkali metals are Rb and K, and the inert gas is Ne. n e is the alkali metal atomic density, n n is the density of Ne, is the exchange collision coefficient between Ne atoms and Rb and K atoms, υ ne is the spin collision velocity between the inert gas Ne atoms and the alkali metal atoms; Among them, is the standard diffusion coefficient of the atom in the gas chamber, T is the temperature of the gas chamber, in Kelvin, p Ne is the gas chamber pressure, and the subscript i of D represents the atomic species K, Rb, and Ne respectively. D i are respectively D K , D Rb and D Ne .

3. The simulation calculation method for polarization gradient relaxation based on three-dimensional polarizability distribution according to claim 1, characterized in that, The steady-state diffusion equation in Step 2 is represented as follows: (2) The condition for the equation to hold is as follows: After the pumping action of the pumping light on the gas chamber reaches equilibrium with the relaxation and diffusion actions, the internal polarizability of the gas chamber reaches a steady state, the change rates of the electron spin and the nuclear spin are 0, the directions of the pumping light and the main magnetic field are along the Z axis, and the direction of the detection light propagates along the X direction. At the steady state, the main direction of the polarizability is on the z axis. The steady-state distribution of the Z-axis polarizability in the gas chamber is determined by equation (2), where R rel is the relaxation rate, and R tot is the total relaxation rate.

4. The simulation calculation method for polarization gradient relaxation based on three-dimensional polarizability distribution according to claim 3, characterized in that, In Step 2, three sets of convection-diffusion equations are selected in the finite element simulation software COMSOL to describe the diffusion steady-state distribution of the electronic and nuclear spin polarizabilities inside the gas cell. The rate of change of the polarization distribution with time is 0, the initial value is 0, and the boundary absorption term is the collision rate between the polarizability and the boundary, which is related to the diffusion coefficient and pumping rate of the colliding atoms; when the alkali metal contacts the uncoated gas cell wall, the atomic polarizability rapidly decays to 0 in a short time. The boundary conditions for the electronic spin polarizability and the nuclear spin polarizability are set as:

5. The simulation calculation method for polarization gradient relaxation based on three-dimensional polarizability distribution according to claim 1, characterized in that, In Step 3, the alkali metal gas cell is set as a spherical glass gas cell with an inner diameter of 8 mm. Use the finite element simulation software COMSOL to construct the alkali metal gas cell structure, and select a relatively refined refinement degree. The refinement degree will affect the accuracy of the relaxation rate simulation calculation. Through refinement, the inside of the gas cell is divided into multiple small volume grid units for subsequent calculations.

6. The simulation calculation method for polarization gradient relaxation based on three-dimensional polarizability distribution according to claim 4, characterized in that, In step 4, considering the interaction of three kinds of atoms and the relaxation effect of the external pumping light and the gas cell wall according to the above equations (2) and (3), after refining the grid, the polarization rate of the electron spin and nuclear spin in each small volume grid unit inside the gas cell is simulated and calculated by the finite element software COMSOL. In each grid, the polarization rate corresponding to the three-dimensional coordinate point (x, y, z) at a specific position is obtained. That is, the polarization rate in each grid inside the gas cell is obtained. The origin of the three-dimensional coordinate is the center point of the incident end face, that is, the contact point between the center point of the pumping Gaussian light and the top of the gas cell.

7. The simulation calculation method for polarization gradient relaxation based on three-dimensional polarizability distribution according to claim 1, characterized in that, In Step 5, it includes: In equation (4) is the gradient of the ternary function f(x, y, z), is the unit vector along the x, y, z directions, is the polarizability corresponding to the coordinate point (x, y, z); Equation (5) is the expression for the polarization rate gradients of the Z-direction polarization rate along three directions. The subscripts i, j, k represent the grid numbers along the x, y, z directions, and d i is the distance between the centers of the i-th grid and the (i + 1)-th grid along the X direction, and d j is the distance between the centers of the j-th grid and the (j + 1)-th grid along the Y direction, and d k is the distance between the centers of the k-th grid and the (k + 1)-th grid along the Z direction. is the Z-direction polarization rate at the coordinate point (x i , y, k), is the gradient along the x direction, is the Z-direction polarization rate at the coordinate point (x, y j , z), is the gradient along the y direction, is the Z-direction polarization rate at the coordinate point (x, y, z k ). is the gradient along the z direction. Through the above formula, the transverse polarization gradient relaxation inside the gas chamber can be calculated; Equation (7) is the expression for the polarization rate gradients of the X - direction polarization rate along three directions, where is the X - direction polarization rate at the coordinate point (x i , y, k), is the gradient of the X - direction polarization rate along the x - direction, is the X - direction polarization rate at the coordinate point (x, y j , z), is the gradient of the X - direction polarization rate along the y - direction, is the X - direction polarization rate at the coordinate point (x, y, z k ), is the gradient of the X - direction polarization rate along the z - direction; In formula (8), is the Y-direction polarization rate at the coordinate point (x i , y, k), is the gradient along the x direction, is the Y-direction polarization rate at the coordinate point (x, y j , z), is the gradient along the y direction, is the Y-direction polarization rate at the coordinate point (x, y, z k ). is the gradient along the z direction. The polarization gradient relaxation inside the gas chamber can be calculated through the above formula, and the longitudinal polarization gradient relaxation inside the gas chamber can be calculated through the above formula.

8. The simulation calculation method for polarization gradient relaxation based on three-dimensional polarizability distribution according to claim 1, characterized in that, In Step 6, it includes: In equation (6), the subscripts i, j, k represent the grid numbers along the x, y, z directions, m, n, b are the total numbers of grids along the x, y, z directions, V is the volume of the gas chamber, λ is the Fermi contact enhancement factor between the electron spin of the alkali metal and the nuclear spin of the noble gas, M e is the magnetic susceptibility when the electron spin of the alkali metal is completely polarized, D Ne is the diffusion coefficient of Ne atoms under the buffer gas inside the gas chamber, γ n is the gyromagnetic ratio of the nuclear spin of the noble gas, and R is the relaxation rate; Equation (9) represents the longitudinal polarization gradient relaxation The relationship with the X-direction polarization rate gradient The Y-direction polarization rate gradient where the subscripts i, j, k represent the grid numbers along the x, y, z directions, m, n, b are the total numbers of grids along the x, y, z directions, V is the volume of the gas chamber, Bz is the magnetic field in the Z direction, λ is the Fermi contact enhancement factor between the alkali metal electron spin and the noble gas nuclear spin, M e is the magnetic susceptibility when the alkali metal electron spin is completely polarized, D Ne-Ne is the diffusion coefficient of Ne atoms under the internal buffer gas in the gas chamber, γ n is the gyromagnetic ratio of the noble gas nuclear spin.

9. The simulation calculation method for polarization gradient relaxation based on three-dimensional polarizability distribution according to claim 7 or 8, characterized in that, Calculate the polarizability gradients of the polarizability along the x, y, and z directions at each coordinate point in space using equations (5), (7), and (8), and calculate the polarization gradient relaxation in the gas chamber using equations (6) and (9).

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