Comparison and analysis of methods for measuring the spin transverse relaxation time of rubidium atomic vapor

The precise measurement of magnetic fields has very important application potential in basic physics research [1], [2], geophysics [3], clinical medicine [4], and cosmic dark matter measurement [5]. The transverse relaxation time (${T_{2}}$) of atoms is an important indicator to characterize the performance of magnetic field precision measurements. Especially, in atomic magnetometers [6],[7],[8], the sensitivity improves with ${T_{2}}$ of atoms. Because the ${T_{2}}$ of atoms contains a large amount of information, it is necessary to accurately measure the ${T_{2}}$ of atoms to help select the ideal type of atomic vapor cell.

Currently, the traditional free induction decay(FID) method is a commonly used method for measuring the ${T_{2}}$ of atoms. In 2017, Jiang et al [9] proposed the fitting-ratio method and the magnetic resonance broadening method to measure the ${T_{2}}$ of atoms, and compared with the traditional FID method to explore the advantages of the new methods. In 2020, the group proposed a perturbation-free method to measure the ${T_{2}}$ [10] of atoms in nuclear magnetic resonance oscillator. In 2023, Wei et al [11] measured the ${T_{2}}$ of cesium using three methods.

In addition, the spin noise spectrum signal in the spin noise spectroscopy technique also contains a significant information, and is an important method for the measurement of the ${T_{2}}$. In 2007, Katsoprinakis et al [12] analyzed the spin noise spectrum properties of atoms in detail from theoretical and experimental perspectives, revealing the relationship between the spin spectrum and the ${T_{2}}$ of atoms.

In this paper, we select spin noise spectrum fitting, improved FID signal fitting, $\omega_{m}$-broadening fitting and magnetic resonance broadening fitting, which are the four methods for measuring the ${T_{2}}$ of atoms. In addition, five types of rubidium (Rb) vapor cells are selected to measure their ${T_{2}}$, to visually observe the merits of various methods and the effects of different types of buffer gases on the ${T_{2}}$ of Rb through the comparative analysis of the experimental results.

The relaxation time of an atom is divided into longitudinal relaxation time (${T_{1}}$) and transverse relaxation time. Longitudinal relaxation is the relaxation of the atomic state population to a certain equilibrium value, which is related to the spin of the atom, that is, the lifetime of the atomic state. Transverse relaxation is phase-dependent and refers to the decoherence time of phase.

For alkali metal atomic ensembles, under the action of the static magnetic field ${B_{z}}$ along the z-axis, the components of macroscopic magnetization of atoms in thermal equilibrium are:

Here ${M_{0}}$ is a constant value. If a pump light is simultaneously applied along the z-axis to polarize the alkali metal atoms, an oscillating magnetic field with amplitude ${B_{1}}$, frequency $\Omega$ is introduced, whose direction is perpendicular to the static magnetic field ${B_{z}}$, to deflect the spin polarization of the alkali metal atoms away from the z-axis. Then the pump light and the oscillating magnetic field are withdrawn, the alkali metal atoms are in a non-equilibrium state, and some relaxation mechanism restores it to a thermal equilibrium state. In a rotating coordinate system, the evolution of the macroscopic magnetization of alkali metal atoms can be expressed by the Bloch equation, which is described as follows [13]:

Here, ${\Delta\omega=\omega_{L}-\Omega}$ is the mismatch between the Larmor frequency ${\omega_{L}}$ and the oscillating magnetic field frequency ${\Omega}$, and $\gamma$ is the ground-state gyromagnetic ratio. In this process, the macroscopic magnetization of alkali metal atoms is decomposed into a component $M_{z}$ (parallel to the static magnetic field) and components $M_{x}$, $M_{y}$ (perpendicular to the static magnetic field). The characteristic time from $M_{z}$ to $M_{0}$ is referred to as the ${T_{1}}$. The characteristic time for $M_{x}$ and $M_{y}$ to gradually return to 0 is called the ${T_{2}}$, in which the spin precession phase of the alkali metal atoms is redistributed until it is disordered.

The experimental setup is shown in Fig. 1, in which a cubic purified $\rm{{}^{87}Rb}$ atomic vapor cell, contains 100 Torr $N_{2}$, with a length of 15 mm was used for the experiment. The AC-driven non-magnetic heating films and Servo loop act as a heating system to control the temperature of the atomic vapor cell at 75 °C. The cell is housed within four layers of $\mu$-metal magnetic shielding to shield the ambient magnetic field. The three-axis Helmholtz coil is placed inside to generate the magnetic field, where the direction of the static magnetic field $B_{0}$ is along the z-axis, and the static magnetic field $B_{0}$ is 6.32 $\mu$T.

The magnetometer configuration is shown in Fig. 1(a). The pump light from the 795 nm external cavity semiconductor laser (ECDL) passes through the acousto-optic modulator(AOM), beam expander, Glen-Taylor prism, and quarter-wave plate into a circularly polarized pumped light with a beam diameter of $\sim$ 10 mm, and then enters the rubidium atomic vapor cell along the z-axis. In the experiment, depending on the type of atomic vapor cell, the frequency of the pump light is locked at $\rm{{}^{87}Rb}$ D1 line F=2-F’=1 transition line. The pump light intensity entering the cell is 12.74 $mW/cm^{2}$. The probe light from the 780 nm distributed Bragg reflective (DBR) laser also passes through the AOM and the polarizer to become linearly polarized light with a beam diameter of $\sim$ 2 mm. The frequency of the probe light is blue detuned 18 GHz from $\rm{{}^{87}Rb}$ D2 line F=2-F’=2 transition line, and the light intensity entering the cell is 3.18 $mW/cm^{2}$. After traversing the cell along the x-axis, the probe light enters a balanced polarimeter where the optical signal is converted into an electrical signal and fed into the LIA.

For the spin noise spectrum configuration, as shown in Fig. 1(b), the direction of the static magnetic field is along the z-axis. Only a 780 nm linearly polarized probe light is used to traverse the vapor cell along the x-axis. The rotation of the probe light polarization is received by the balanced polarimeter, subsequently into the fast Fourier transform (FFT) to convert the time-domain signal into a frequency-domain signal.

In this paper, four methods were employed to measure the $T_{2}$ of Rb based on the Rb atomic vapor cell filled with different types and pressures buffer gas. We demonstrate the dependence between different atomic number densities and the $T_{2}$ of Rb. The effect of the relaxation mechanism on the $T_{2}$ of Rb was suppressed by filling with buffer gas. Spin-exchange collisions can be further suppressed by optical narrowing effects[24],[25],[26] or by heating the cell to extremely high temperatures to keep atoms in the spin-exchange relaxation-free state[27],[28].

Among these four methods, the $T_{2}$ of Rb obtained by the spin noise spectrum fitting method is less than the actual value owing to the limitation of the measurement technology, and a high-resolution spin noise spectrometer is required for experimental acquisition. However, this method provides a roundabout method for comparing the $T_{2}$ in different types of atomic vapor cells. The experimental setup for the magnetic resonance broadening fitting method is relatively simple. It is easier to implement experimentally, and even the $T_{2}$ of atoms can be measured only by measuring the pump light [9]. The $\omega_{m}$ - broadening fitting method allows real-time monitoring of the drift of the static magnetic field, which can be compensated by a servo-loop system. Although the experimental setup for the improved FID signal fitting method is relatively complex, it ensures that the negative effects of the RF magnetic field and pump light are not introduced during the measurement by applying a timing sequence control system. In addition, the $T_{2}$ of atoms measured by this method is relatively accurate.

These four methods can be used in different measuring environments. The $T_{2}$ of atoms obtained by different measuring methods are slightly different, mainly because the magnetic field environment is different during the measurement. However, all of them intuitively show the relationship between the buffer gases of different pressures and the $T_{2}$. This provides a variety of feasible methods for future precision measurement fields, such as the improvement of the sensitivity of atomic magnetometers and the selection of atomic vapor cells. Meanwhile, by comparing the $T_{2}$ of Rb obtained by five different types of Rb atomic vapor cells under the same conditions, we can obtain an optimal type of atomic vapor cell. This study has important application reference value in the fields of the fabrication of atomic vapor cells, and the Selection of buffer gas types and pressures.

This work was supported by the National Key R & D Program of China (Grant No. 2021YFA1402002), the National Natural Science Foundation of China (Grant Nos. 11974226), the Shanxi Provincial Graduate Education Innovation Project (Grant No.2022Y022) and the 1331 Key Subjects Construction of Shanxi Province, China.

The authors have no conflicts to disclose.