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Zhang Da-Wei, Xu Zheng-Yi, Zhou Min, Xu Xin-Ye. Parameter analysis for a nuclear magnetic resonance gyroscope based on 133Cs–129Xe/131Xe. Chinese Physics B, 2017, 26(2): 023201  
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Parameter analysis for a nuclear magnetic resonance gyroscope based on 133Cs–129Xe/131Xe
Zhang Da-Wei, Xu Zheng-Yi, Zhou Min, Xu Xin-Ye†
State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China

 

† Corresponding author. E-mail: xyxu@phy.ecnu.edu.cn

Abstract
Abstract

We theoretically investigate several parameters for the nuclear magnetic resonance gyroscope based on 133Cs–129Xe/131Xe. For a cell containing a mixture of 133Cs at saturated pressure, we investigate the optimal quenching gas (N2) pressure and the corresponding pump laser intensity to achieve 30% 133Cs polarization at the center of the cell when the static magnetic field B 0 is with different 129Xe/131Xe pressure. The effective field produced by spin-exchange polarized 129Xe or 131Xe sensed by 133Cs can also be discussed in different 129Xe/131Xe pressure conditions. Furthermore, the relationship between the detected signal and the probe laser frequency is researched. We obtain the optimum probe laser detuning from the D2 ( resonance with different 129Xe/131Xe pressure owing to the pressure broadening.

PACS: 32.10.Dk;34.10.+x;51.60.+a
Keyword:Nuclear magnetic resonance;Gyroscope;Optical pumping;Spin-exchange collision;Optical rotation
1. Introduction

Atomic gyroscopes are high performance devices sensitive to rotation, which have abundant applications, such as for inertial navigation [1], studying the Earth’s rotation and tests of general relativity.[2] Different from cold atomic gyroscopes based on atomic interference, the nuclear magnetic resonance gyroscope (NMRG) utilizes the nuclear magnetic resonance effect to measure the change of the rotational orientation regarding the nuclear spin Larmor precession rate in the static magnetic field as a stable reference.[3] Basically, NMRG has no moving parts and it is insensitive to the acceleration and the vibration of the gyro housing. It has been developed for several decades since the early table-top version was realized in the 1960s.[4] Recently, NMRG has again attracted considerable attention in that it can meet the requirements of both high precision and miniaturization with the advanced development of microelectronic processing techniques.[5]

NMRGs being developed worldwide mainly employ alkali atoms and noble gases, for instance, a mixture of 87Rb, NMR active medium 129Xe /131Xe together with the buffer gas N2.[6] There are many theoretical or experimental works on studying spin-exchange parameters for alkali-metal-noble-gas pairs, [7, 8] absolute alkali-metal polarization, [9] and surface relaxation of laser-polarized noble-gas.[10] However, the rotation sensitivity associated conditions of these NMRGs, especially the 133Cs–129Xe/131Xe type, are lacking elaborated research. In this work, we detail a discussion on parameters for NMRG based on 133Cs–129Xe/131Xe, including the N2 pressure, the optical pumping laser intensity, the effective field produced by spin-exchange polarized 129Xe/131Xe magnetization vector sensed by 133Cs, and the optimal frequency detuning of the probe laser and so on. This work is to provide theoretical support for the next step of the NMRG experiment based on 133Cs–129Xe/131Xe.

2. Basic principle of NMRG

The basic principle of NMRG is illustrated in Fig. 1. The 133Cs vapor is optically pumped by a circularly-polarized laser beam [11, 12] which propagates along the z axis. A static magnetic field is applied along the z axis. A net electron spin polarization accumulates in 133Cs atoms. It is then transferred to Xe nuclei through spin-exchange collisions, resulting in a macroscopic nuclear spin polarization for Xe. By applying an oscillating magnetic field along the x axis, the Xe spin magnetic moment M tilts from the z axis, and its component along the xy plane will precess about with an angular frequency , where approximately equals to and the proportional constant is the gyromagnetic ratio of Xe. The 133Cs spins precess about the total magnetic field, which is the sum of and the Xe spins generated part. Since the alkali-metal atoms are more susceptible to the magnetic field than the noble gas [13], a linearly-polarized probe laser beam whose frequency is near resonant with the 133Cs optical transition may rotate its polarization of the macroscopic field component along the x axis of the polarized Xe atoms precess about field, the photoelectric detector accepted laser intensity will periodically vary after the probe laser passes through the linear polarizer. Assuming the gyroscope is rotating about the z axis at a frequency , the measured Larmor precession frequency will change from the reference to be

(1)

Fig. 1. (color online) Basic principle of NMRG. The circularly polarized pump light along the z axis, the linear polarized probe light and oscillating magnetic field along the x axis. The combination of linear polarizer and detector is used to detect the polarization rotation of the probe light.

To determine , precise knowledge and careful control of are needed. A two-NMR-isotope method fundamentally uncouples from , reducing the difficulty in stabilizing . To this end, both 129Xe and 131Xe are contained in the vapor cell which means they have the same . Since γ of 129Xe and 131Xe have different values and opposite signs, they have their respective precession frequencies [14]

(2)
where and are gyromagnetic ratios of 129Xe and 131Xe. Therefore, can be deduced by simultaneously monitoring and , where is seen as a removable common-mode parameter and
(3)
where denotes the cesium atomic density, the normalized polarization is the 133Cs polarization and defined as ) with being the total spin-relaxation rate, here we consider as a fixed value, and is the photon absorption cross-section at the resonance frequency . For the D1 line of 133Cs, the linewidths broadened by the 129Xe/131Xe and N2 (for ) pressures are 6 GHz and 6.8 GHz [20] respectively for mixture II. For comparison, the natural linewidth is 4.6 MHz and the Doppler broadening linewidth is 427 MHz. Thus is dominated by the pressure broadening, approximately equals to , where is the classical electron radius, c is the speed of light, is the D1 transition oscillator strength, and is the transition linewidth.

The principal value of the Lambert W-function (the inverse of the function ) describes the solution of Eq. (3), which is given by

(4)
where is the pump rate at z = 0, i.e., at the front of the cell. The 133Cs polarization , we perform the 133Cs polarization as a function of different conditions at the center of the cell, as shown in Fig. 2. When the pump rate is comparable to the total spin-relation rate, the polarization is small. When the pump rate is much higher than the total spin-relation rate, the polarization rapidly increases first and then slowly increases. In NMRG, obtaining high polarization of 133Cs is difficult and the obtainable highest 133Cs polarization is about 20% at present.[15] Here we will discuss the relevant parameter to achieve 30% 133Cs polarization. In order to obtain 30% 133Cs polarization at the cell center, from Fig. 2 we know should be controlled around 12.48 by adjusting the pump laser intensity.

Fig. 2. (color online) The 133Cs polarization as a function of different conditions at the center of the cell.

A phenomenon known as radiation trapping may degrade the polarization of the alkali vapor. Radiation trapping happens when the unpolarized photons scattered by excited 133Cs atoms through spontaneous emission are reabsorbed by other adjacent atoms. In our case, a quenching gas N2 [11, 12] is added to it to create a pathway for the atoms to decay to the ground state without emitting photons at all. The fraction of atoms that reach the ground state by emitting a photon rather than quenching is given by , where is the N2 pressure, and is a characteristic pressure. For the D1 line of 133Cs at 110 °C, is 473 Pa.[22] The spin-relaxation rate due to radiation trapping is , where K describes the degree of depolarization caused by reabsorbing a photon, N is the average number of times that a photon is emitted before it leaves the vapor cell. Note that is proportion to , so the radiation trapping effect is also location-dependent. In the presence of the quenching gas, the obtainable 133Cs polarization at the cell center is

(5)
where is the pump rate at the center of the cell and is the relaxation rate due to effects other than radiation trapping. K is always below 1 due to uncompleted depolarization. In theory, a reasonable value of K is 0.06 according to Ref. [22] and N is calculated about 75 for mixture II when N2 pressure is .[23] In fact, when is set to be dozens of Torr (1 Torr = 1.33322×102 Pa), will be much larger than , which means the polarization is not mainly limited by the radiation trapping.

Figure 3 shows the 133Cs polarization at the center of the cell as a function of N2 pressure with N around 75 when assuming . The figure indicates that the polarization will keep growing until saturation as the N2 pressure increases. It shows that at the cell center 133Cs polarizability is about 29.5% (less than 30%) when , which indicates that the radiation trapping cannot be ignored. Meanwhile, the similar tendency of three lines suggests that the polarization is insensitive to N when , which means that the 133Cs polarization is stable when various errors are taken into consideration in this regime.

Fig. 3. (color online) The 133Cs polarization at the center of the cell as a function of the N2 pressure with different N.

Apart from the radiation trapping, several other effects contribute to the total relaxation rate, which is given by

(6)
where is the Cs–Cs spin-exchange collisional term, is the spin-destruction collisional term, is the Cs–Xe spin-exchange collisional term, is the term of diffusion to the walls, and is the probe laser pumping term.

As aforementioned, the spin-relaxation rate due to radiation trapping is related to , which is nonuniform throughout the cell, here we assume , and are the mean pump rate and mean polarizability from z = 0 to the cell center respectively, thus . In actual experiments, the probe laser frequency is tuned far from resonance and the laser power is usually relatively low. Therefore, the probe laser pumping term can also be ignored. Spin-exchange collisions between alkali metal atoms are very fast and can be estimated as [24]

(7)
where is the slowing down factor, denotes the spin-exchange rate, is the gyromagnetic ratio of 133Cs, and I is 133Cs nuclear spin. In the magnetic field , is calculated to be . The relaxation term of spin-destruction collisions between 133Cs and other elements in the cell is [7]
(8)
where , , , and are spin-destruction cross-sections of Cs–129Xe, Cs–131Xe, Cs–N2, and Cs–Cs collisions respectively, and represents the mean velocity. The relaxation rate due to spin-exchange collisions between 133Cs atoms and the NMR medium Xe is
(9)
where , are spin-exchange cross-sections of Cs–129Xe, Cs–131Xe. Here we approximate and to calculate the relaxation rates in Eq. (8) and Eq. (9). When an alkali atom encounters the glass surface of the cell wall, it will experience the large local electric and magnetic fields produced by ions and molecules within the glass. The polarized spins undergo a depolarization process with a rate [22–25]
(10)
where χ is a diffusion constant. Equation (10) may be written in terms of the total pressure as . For , . The relaxation rate is . Together with the result of turns out to be .

Now we come to estimate the pumping laser intensity. According to the pumping scheme shown in Fig. 1, the optical pumping rate can be expressed as

(11)
where P is the pump laser power, h is the Planck constant, and A is the pump laser facula area, is the broadening linewidth of the D1 line, and is the D1 transition oscillator strength of 133Cs. To meet the requirement of , the pumping laser intensity P/A is calculated to be 1171 mW/cm2. In this way, we discuss the pump laser intensity of the three kinds of gas mixture with different N2 pressures to achieve 30% 133Cs polarization at the center of the cell, which is shown in Fig. 4. The pump intensity drops sharply at first then slowly rises. The minimum intensity of mixture I, mixture II and mixture III are 1020 mW/cm2, 1169 mW/cm2, and 1371 mW/cm2 when the N2 pressure is about , higher 129Xe/131Xe pressure corresponding to higher minimum pump laser intensity to achieve the same 133Cs polarization. At present we have developed a Distributed Bragg Reflector (DBR) diode-laser [26] at 895 nm. The laser optical fiber coupling power output exceeds 60 mW. For this purpose, the laser beam diameter could be expanded to 3.87 mm, 3.62 mm, and 3.34 mm for mixture I, mixture II, and mixture III respectively.

Fig. 4. (color online) The pump laser intensity of the three kinds of gas mixture as a function of the N2 pressure. The result is obtained to achieve 30% 133Cs polarization at the center of the cell.

The net polarization of 133Cs atoms is transferred to 129Xe/131Xe atoms through spin-exchange collisions. To obtain the same macroscopic nuclear magnetic moment, the pressure ratio of 129Xe and 131Xe has been set to 1:4 [16] throughout our discussions. The polarization of 129Xe can be estimated as

(12)
where is the 133Cs–129Xe spin-exchange rate, is the 129Xe atomic density, and is the longitudinal spin relaxation time of 129Xe. can reach about 20 s even without anti-relaxation coating of the cell wall. By applying a specified oscillating magnetic field perpendicular to , the Xe spins will precess about the direction, generating a precessing macroscopic magnetic field. The magnitude of the macroscopic magnetic field has an important influence on the detected signal and can be expressed as [27]
(13)
where is the vacuum permeability, is the enhancement factor of the Cs–129Xe interaction, and is the 129Xe nuclear magnetic moment. The effective field sensed by 133Cs due to polarized 129Xe or 131Xe magnetization is about 181 nT, 302 nT, and 484 nT for mixture I, mixture II, and mixture III respectively.

Since alkali metal atoms are very sensitive to the magnetic field, the Larmor precession frequencies of 129Xe and 131Xe can be measured by using the polarized 133Cs atoms. An off-resonance linearly-polarized probe laser beam will rotate its polarization when it interacts with the polarized 133Cs, which is known as the Faraday rotation effect. As shown in Fig. 1, the probe beam propagates along the x-axis, which is orthogonal to the pump beam. The detected rotation signal is given by [22]

(14)
where is the D2 transition oscillator strength of 133Cs, and v are the 133Cs D2 hyperfine transitions resonance frequency and the probe laser frequency respectively, is the broadening linewidth of the D2 line, is the relative strength of the hyperfine transition , and is the probe laser intensity. From Eq. (14), we know that the probe laser intensity is proportional to the detect signal and it does not affect the optimal probe laser frequency detuning. The photon absorption cross section of cesium is written by
(15)
which contains the overall contributions of the D2 line hyperfine transitions. The dependences of the optical rotation signal on the probe laser detuning of the three types of 129Xe/131Xe mixtures are shown in Fig. 5. The signals are somewhat like the dispersion shaped type. The differences are that there are little optical rotations around the resonant frequency. The maximum rotation signals are obtained when the probe laser frequencies are 92 GHz, 106 GHz, and 124 GHz for mixture I, mixture II, and mixture III detuning from the D2 resonance.

Fig. 5. (color online) The dependence of optical rotation signal on the probe laser detuning of the three kinds of gas mixture.
4. Conclusions

In summary, we have theoretically investigated the interaction between light and atoms, and the interaction between atoms and atoms of the NMRG based on 133Cs–129Xe/131Xe. We discussed the relationship between quenching gas (N2) pressure and the corresponding optimal pump laser intensity in different 129Xe/131Xe pressure under given parameter conditions. Meanwhile, we estimated the polarized noble gas macroscopic magnetic field, the higher polarized magnetic field corresponding to the higher pump laser intensity. We also studied the influence of probe laser frequency detuning on the probe signal. The theoretical results in this paper will provide guidance for our subsequent experiments of NMRG based on 133Cs–129Xe/131Xe.

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