1. Introduction

Thanks to its simple atomic structure, Helium atoms play an important role in metrology, e.g. measurement of fine-structure constant [1–3], neutron electric dipole moments [4, 5], and is useful for high sensitivity magnetometer [6–10]. Two isotopes exist, ${ }^3\text{He}$ and ${ }^4\text{He}$, both with two electrons, and can be distinguished by nuclear spin I. ${ }^3\text{He}$ has richer energy level structure with $I=1/2$ (see, Fig. 1), while ${ }^4\text{He}$ having no nuclear spin at all. For both ${ }^3\text{He}$ and ${ }^4$He, since ground state $1^1{\mathrm{S}}_0$ is normally spectroscopically inaccessible, the lowest metastable state $2^3{\mathrm{S}}_1$, which is usually produced by discharge in atomic vapor, is more favorable.

Study on interaction between ${ }^3\text{He}$ ground state $1^1{\mathrm{S}}_0$ and metastable state $2^3{\mathrm{S}}_1$ can be traced back to the 1960s [11, 12]. The original motivation is to indirectly produce populationdifference between Zeeman sublevels of the ground state $1^1{\mathrm{S}}_0$, or “polarization” in the language of multipole moment formalism (see, Chap. 5 of [13]) by the means of optical pumping induced repopulation among metastable state $2^3{\mathrm{S}}_1$ sublevels. This interaction is named “metastability exchange collisions” (MECs), where during a rapid collision process between ${ }^3\text{He}$ atoms, there is a good chance that the nuclei of those in ground state combines with the electron of those in metastable state to get “metastability”. Here “metastability” stands for electron energy level $1\mathrm{s}2\mathrm{s}$ of metastable state $2^3{\mathrm{S}}_1$, in contrary to $1{\mathrm{s}}^2$ of $1^1{\mathrm{S}}_0$. Angular momentum is conserved in MECs, so the aforementioned process can be viewed as transfer of polarization between $1^1{\mathrm{S}}_0$ and $2^3{\mathrm{S}}_1$. A natural extension would be transfer of coherence, i.e., off-diagonal elements of density matrix in angular momentum representation, which enables full capacity of manipulation on ground state $1^1{\mathrm{S}}_0$ by optical pumping of the metastable state $2^3{\mathrm{S}}_1$ [14, 15].

 

Fig. 1 Energy-level diagram for ³He atoms (not in scale). The nuclear spin for ³He is 1/2. There is no electronic spin or orbit angular momentum for the ground state 1¹ S₀. For the metastable state 2³ S₁, the electronic spin is 1 and the orbit angular momentum is 0. For the excited state 2³ P₀ the electronic spin is 1 and angular momentum is 0. The ground-state atoms can be excited to the metastable state by RF discharge, and atomic polarization canbe exchanged between the ground state and the metastable state by metastability-exchange collisions (MECs). The gyromagnetic ratios for the ground and the metastable states are different, i.e., γ_g= -32 Hz/μT, γ_μ= -38 kHz/μT, and ${\gamma }_{{\mu }^{\prime }}=$ -19 kHz/μT for the ground state and the two metastable hyperfine levels F = 1/2 and F = 3/2, respectively [11]. Atoms in the metastable state can be optically pumped or detected by C₈ or C₉ line with a 1083.353 nm or 1083.326 nm wavelength laser beam [14].

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In this work, a quintuplet spectrum is observed in the metastable-state ${ }^3\text{He}$ atoms, which is coupled with the magnetically-driven ground-state ${ }^3\text{He}$ atoms via the metastable state by MECs. The quintuplet spectrum includes Larmor frequency, sidebands of Rabi frequency and twice the Rabi frequency of the ground state, and only can be detected by linearly polarized light in $F=3/2$ hyperfine level in pure ${ }^3$He atomic cell. Furthermore, the quintuplet spectrum can also be observed in metastable ${ }^4$He atoms with the angular momentum quantum number to be J = 1 in a hybrid atomic vapor cell mixed with ${ }^3$He atoms. The novel quintuplet spectrum detected in the metastable-state atoms has not been reported, which is relevant to the rank-two multipole moment (alignment component) coupled with transferred Rabi nutation via MECs.

 

Fig. 2 Experimental setup. PBS, polarization beam splitter; QWP, quarter wave plate; HWP, half wave plate; BE, beam expander; BT, beam trap and PD, photo detector.

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2. Experimental results

The experimental setup is shown in Fig. 2. The metastable ${ }^3$He atoms are optically pumped with a circular polarized laser beam propagating in the z direction with the $1/e^2$ waist diameter to be ∼ 20 mm and the power to be ∼ 15 W. The probe beam propagating in the x direction is applied to detect the atomic polarization in the metastable state with the $1/e^2$ waist diameter to be ∼ 1 mm and the power to be ∼ 0.3 W. Both the pump and probe beams are from the laser source NKT Photonics Y10 and the pump beam is power-enhanced with a laser amplifier (LEA Photonics MLXX-EYFA-CW-SLM-P-TKS). The home-made pure ${ }^3\text{He}$ cylindrical atomic cell (0.6 Torr, ϕ = 50 mm, $L=70$ mm) is located in the seven-layer magnetic shield and excited with a radio-frequency power source module (50 MHz, 0.8 W) to continuously discharge and generate metastable-state atoms. The static magnetic field B₀ oriented in the z axis is generated by the solenoid and a set of Helmholtz coils is applied to generate an oscillating magnetic field $B_{\mathrm{M}}\cos (2\pi f_{\mathrm{M}}t)$ aligned with the y axis. The digital processing system used to control the Helmholtz coils and acquire the signals consists of the sound and vibration modules (PXI-4461 and PXI-4462, resolution@24-Bit, sampling rate@204.8 kS/s) of National Instruments. The oscillating magnetic field can be switched on or off by the digital processing system and its frequency is fixed at the ground-state Larmor frequency.

 

Fig. 3 Frequency spectrums for C₈- or C₉-line probe. Black dots are the experimental data, and red lines are the connecting lines of adjacent data points. (a) and (b) are probed with C₈ line, and (c) and (d) are probed with C₉ line. The probe beams for (a) and (c) are circularly polarized, and for (b) and (d) are linearly polarized.

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Both the pump and probe beams continuously interact with atoms, and the oscillating magnetic field interacts with atoms impulsively. The Larmor frequency of ground state is $f_g={\gamma }_g{B}_0$, where γ_g is the gyromagnetic ratio. The frequency of the oscillating magnetic field is set as Larmor frequency of the ground state, i.e., $f_{\mathrm{M}}=f_g$, and it induces Rabi nutation whose frequency is $f_{\mathrm{R}}=1/2{\gamma }_g{B}_{\mathrm{M}}$ in ground state [16]. When the probe beam is circularly polarized, and tuned to the C${ }_8$ or C${ }_9$ line, a triplet spectrum is detected, as is shown in Fig. 3(a) and 3(c). The spectrum is similar to Ref. [17], where the center peak is located at Larmor frequency f_g and the two sidebands are with an interval of Rabi frequency f_R to the central peak. It demonstrates that the Rabi nutation (or dressed effect) is transferred from the ground state to the metastable state via MECs, as both the Larmor and the Rabi frequencies bear the ground-state features. Additional two peaks appear in the spectrum when the probe beam is linearly polarized and tuned to the C${ }_9$ line, while almost no signal is detected when the linearly polarized light is tuned to the C${ }_8$ line, as is shown in Fig. 3(d) and 3(b). The interval between the central peak and the two additional peaks is twice the Rabi frequency. We also find that the quintuplet spectrum varies with the polarization direction of the probe beam, as is shown in Fig. 4. When the polarization of the linearly polarized probe beam is rotated in the y − z plane, denoting the angle between the polarization direction and the y axis by θ, one can find that the heights of the peaks in the quintuplet spectrum are almost zero at θ = 0 and 90°, and reach the maximum at $\theta ={45}^{\circ }$ and 135°. We measure the height of the central peak at different angles, and find that it can be well described by the function $\sin 2\theta$, as is shown in Fig. 4(d). The intervals of the peaks are investigated at different strength of the oscillating magnetic field, as is shown in Fig. 5(a). We fix the angle θ at 45° and change the strength of oscillating magnetic field B_M. The interval between the two adjacent peaks is almost linearly dependent on B_M, consistent with the formula to calculate Rabi frequency at resonance, i.e., $f_{\mathrm{R}}=1/2{\gamma }_{\mathrm{g}}{B}_{\mathrm{M}}$. The fact that spectral line is not strictly straightis due to the instability of the oscillating field’s frequency while changing its amplitude. We also measure the spectrum when the oscillating field is detuned from Larmor frequency, as is shown in Fig. 5(b). The effective Rabi frequency at detuning can be calculated by $f_{\mathrm{R},\text{eff}}=\sqrt{{(1/2{\gamma }_{\mathrm{g}}{B}_{\mathrm{M}})}^2+{(f_{\mathrm{M}}-{\gamma }_{\mathrm{g}}{B}_0)}^2}$.

 

Fig. 4 Quintuplet spectrums for linearly polarized probe laser beams with different polarization directions. Black dots in (c) and (d) are the experimental data, red lines in (c) are the connecting lines of adjacent data points, and blue line in (d)is a mathematical curve. (a) the polarization direction of the linearly polarized laser beam; it rotates in the z - y plane and θ is the titled angle with respect to y axis. (b) and (c) show the heights of the quintuplet spectrum at different angles. (d) The heights of the central peaks at differen angles; the black dots is the measured and normalized value and the blue line is a mathematical curve with function sin (2θ). The static magnetic field is ∼41 μT.

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Fig. 5 Quintuplet spectrums at different oscillating magnetic field. (a) the oscillating magnetic field’s frequency is tuned to Larmor frequency while changing its amplitude; (b) the oscillating magnetic field’s amplitude is fixed at 3 μT while changing its detuning. Larmor frequency is equal to ∼1300 Hz.

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3. Discussion

According to the observed phenomena, we give an interpretation that the quintuplet spectrum is induced by three process including Rabi nutation’s generation in ground state $1^1{\mathrm{S}}_0$, transfer to metastable state $2^3{\mathrm{S}}_1$ by MECs,and detection of alignment component among rank-two multipole moment.

3.1. Rabi nutation of ground state $1^1{S}_0$

The ground state of ${ }^3$He atom splits into two sublevels $m_I=1/2$ and $m_I=-1/2$, with the frequency difference of the Larmor frequency $f_g={\gamma }_g{B}_0$. The frequency of oscillating magnetic field is set as $f_{\mathrm{M}}=f_g$ to maintain magnetic resonance. The magnetic moment of ground state is ${\mu }_I={\gamma }_{g}I$, where I is the nuclei spin, since the spin electrons s = 0. This two-level system connected by dipole experiencing external electromagnetic field would have a typical Rabi oscillation, called Rabi nutation in particular, which would result in two symmetric sidebands to the main peak with Lamour frequency f_g, each separated by Rabi frequency $f_{\mathrm{R}}=1/2{\gamma }_g{B}_{\mathrm{M}}$. Please note that γ_g is the gyromagnetic ratio of the ground state.

3.2. Metastability exchange collisions transfer coherence

The MECs of pure ${ }^3$He atoms are modeled that the metastable-state atoms immediately after the collision are constructed by the nuclei of ground-state atoms ρ_g and electrons of metastable-state atoms before MECs, and the model can be expressed as the following equation [14, 15]

Rabi nutation of ground state is a kind of coherence among Zeeman sublevels, which can be transferred to metastable states via MECs according to Eq. (1). The experimental results shown in Fig. 3(a) and 3(c) are detected in both $F=3/2$ and $F=1/2$ metastable states. The central peak and sidebands are corresponding with the Larmor and Rabi frequencies of ground state, respectively. Figures. 3(a) and 3(c) demonstrate that the Rabi nutation of ground state $1^1{\mathrm{S}}_0$ has transferred to metastable state clearly. To be more clear, the evolution equations of ${ }^3$He metastable-state’s orientation component based on Eq. (1), when no coherence between hyperfine structure levels exists, are [21]

3.3. Alignment of $\mathrm{F}=3/2$ metastable-state and quintuplet with twice Rabi frequency

The creation of twice the Rabi frequency can be further interpreted with rank-two multipole moment (alignment) of $F=3/2$ metastable-state, for which the $F=1/2$ hyperfine structure does not have. The metastable-state density matrix ρ_m can be expressed in terms of multipole moments $m_{k,q}$ according to [18]

The alignment component of ${ }^3$He metastable state $Q_{ij}{ }_{\mathrm{m}}$ can consider $m_{2,q}$ for $F=3/2$ metastable state, where $i,j=x,y,z$. The evolution equations of alignment component of $F=3/2$ sublevel ${ }^{3/2}{Q}_{ij}{ }_{\mathrm{m}}$, when no coherence between hyperfine structure levels exist, are [21]

Equations (2) and (3) reveal that the orientation terms of both the $F=3/2$ and $F=1/2$ metastable states are influenced by the ground state, which demonstrates the Larmor and Rabi frequencies of the ground state transfer to metastablestates. Equation (11) further reveals that the alignment of $F=3/2$ hyperfine structure levels is influenced by orientation terms of $F=3/2$ and $F=1/2$ levels. The products of orientation terms includes twice the Larmor frequency and twice the Rabi frequency of ground state. The longitudinal orientation like ${ }^{1/2}{F}_z$ includes the Rabi frequency, while the transverse orientation like ${ }^{1/2}{F}_x$ includes the Larmor frequency and Rabi frequency. The terms multiplied by longitudinal and transverse orientation (like ${ }^{1/2}{F}_x{ }^{1/2}{F}_z$) give the quintuplet spectrum includes f_g, $f_g\pm f_{\mathrm{R}}$, and $f_g\pm 2f_{\mathrm{R}}$.

The alignment of metastable state can be detected with linearly polarized light, and hyperfine level $F=3/2$ ($F=1/2$) observed quintuplet spectrum (no signal) shown in Fig. 3(d) (3(b)), which demonstrates that alignment of metastable state coupled Rabi nutation of ground state via MECs can create quintuplet spectrum. In order to confirm the interpretation, the similar experiment is achieved in hybrid atomic cell of ${ }^4$He and ${ }^3$He atoms [22]. The central frequency of pump and probe laser is tuned to D${ }_0$ line of ${ }^4$He atoms. The density matrix of ${ }^4$He metastable-state atoms also has rank-two multipole moment (alignment) due to J = 1. The similar quintuplet spectrum with hyperfine level $F=3/2$ of pure ${ }^3$He atomic cell is shown in Fig. 6(b). The experimental results in hybrid cell give more evidence for alignment inducing quintuplet spectrum.

In addition, the angle-dependence of the quintuplet spectrum detected in $F=3/2$ level with linearly polarized light in x axis at the Larmor frequency of ground state shown in Fig. 4 agrees with that of $m_{2,\pm 1}$ component in Eq. (7),

Notice that operators $T_{\pm 1}^{\left(2\right)}{(}^{3/2}F)$ for $F=3/2$ metastable state corresponding to $m_{2,\pm 1}$ can be expressed as [18]

 

Fig. 6 Frequency spectrums for ⁴He atoms in a hybrid atomic vapor cell mixed with ³He atoms. All the experimental conditions are similar to that of the pure ³He atomic vapor, except for that, the metastable ⁴He atoms are optically pumped and probed rather than the ³He. The ³He atoms are polarized via MECs with the ⁴He atoms. Black dots are the experimental data, and red lines are the connecting lines of adjacent data points. (a) triplet spectrum probed with circularly polarized laser beam; (b) quintuplet spectrum probed with linearly polarized laser beam. Both the pump and probe laser beams are tuned to D₀ line and the pump beam is circular polarized.

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In summary, the experimental results provide three major evidences that the observed quintuplet spectrum is induced by rank-two multipole moment (alignment component) of metastable state coupled with transferred Rabi nutation from ground state via MECs:

4. Conclusion

In this work, we present a quintuplet spectrum that is optically observed in the $F=3/2$ metastable ${ }^3$He hyperfine level. The phenomenon is induced by the rank-two multipole moment (alignment component) of metastable state coupled with transferred Rabi nutation from ground state via MECs, and the experimental results support our the preliminary interpretation including comparison of detection via different polarized light in different metastable states and angle-dependence of quintuplet spectrum. We areworking on more details about the quintuplet spectrum like the spatial distribution of metastable-state atoms changed by cell shape and discharge. A comprehensive and in-depth analysis will be presented in our subsequent work.