Collision induced magnetic resonance in Cesium atoms

Guoqing Yang; He-bin Zhang; Xu-xing Geng; Shang-qing Liang; Yifu Zhu; Jing-tao Mao; Guang-ming Huang; Gao-xiang Li

doi:10.1364/OE.26.030313

1. Introduction

Atomic magnetometers based on the radiofrequency-optical double magnetic resonance have been shown to be useful for practical applications in space physics [1], geomagnetism [2], medical imaging [3] and ocean science [4]. The properties of these resonances have been well studied, which are the basis for high performances in sensitivity and accuracy [5–8]. In a typical radiofrequency-optical double magnetic resonance configuration, the radiofrequency field is applied to excite magnetic dipole transitions and reduce the population imbalance created by the optical field. Obviously, the population transfer between magnetic sublevels is an important process. The spin-exchange collision is also lead to population transfer. But in the literature, it is mainly considered as a relaxation mechanism in the radiofrequency-optical double magnetic resonances with one atomic species [9–11]. However, the contribution of the spin-exchange collisions induced population transfer between the magnetic sublevels in different hyperfine states to the spin orientation was studied [12].

In this paper, we demonstrate the first observation of a new magnetic resonant signal involving the spin-exchange collisions in the M_Z type magnetic resonance spectrum. The physical mechanism is similar to transferring spin polarization from optically pumped alkali-metal atoms to noble gas atoms in a spin-exchange optical pumping process [13]. The resonant laser renders atoms in one hyperfine state polarized. The spin polarization is then transferred from the polarized atoms to atoms in other hyperfine states. When a resonant radiofrequency field is applied, the population distribution changes and the new resonant signal occurs. The new signal exhibits two important features: 1. the spectral linewidth is not broadened by the high laser intensity; 2. the amplitude of the signal does not change when the laser intensity is above a threshold value. Therefore, it indicates that when the laser intensity is below the threshold, the ratio of the linewidth and the amplitude of the signal decreases as the laser intensity increases; but when the laser intensity is above the threshold, the ratio is constant. This implies that the sensitivity would only depend on the noise of the magnetic resonance spectrum, if the new magnetic resonant signal is used for the magnetic field measurement in a magnetometer. In a closed-loop atomic magnetometer for geomagnetic field, an invariable ratio of the linewidth and the amplitude of the signal means that the feedback gain is a constant, which is very important for a stable system.

2. Theoretical analysis

When cesium atoms are placed in a weak external magnetic field $B_{z}$ along the z direction, the magnetic sublevels split linearly according to $Δ E_{| F, m_{F} 〉} = μ_{B} g_{F} m_{F} B_{Z}^{0}$ , where $μ_{B}$ is Bohr magneton and $m_{F}$ is magnetic quantum number. The hyperfine Landé $g$ -factor is given by [14]:

g_{F} = g_{J} \frac{F (F + 1) - I (I + 1) + J (J + 1)}{2 F (F + 1)} + g_{I} \frac{F (F + 1) + I (I + 1) - J (J + 1)}{2 F (F + 1)}

Since

g_{I}

is much smaller than

g_{J}

, the nuclear term is usually neglected in most experiments. However, the second term can induce a significant correction in the geomagnetic field measurement, which is important for the precision measurements. Importantly, Eq. (1) implies that there are two different resonant frequencies for magnetic resonance with alkali atoms. The

g

-factors of the ground states of cesium atoms are

g_{4} = (1 / 8) g_{1 / 2} + (7 / 8) g_{7 / 2}

and

g_{3} = (- 1 / 8) g_{1 / 2} + (9 / 8) g_{7 / 2}

. When

B_{Z}^{0}

is in the geomagnetic field range, the difference between the Zeeman shifts of the two ground states is several hundred hertz. If a circular polarized laser field coupled to the transition

F_{g} = 4 \to F_{e} = 3

and a radiofrequency field

B_{t} = B_{X}^{1} \cos ω t

are applied to the atoms normal to the static magnetic field, the radiofrequency-optical double magnetic resonance might be excited when the frequency

ω

equals the Zeeman shifts, as shown in Fig. 1. We define z-direction as the quantization axis and consider the spontaneous decays from the excited states to the two ground states. There are 23 Zeeman sublevels to be considered. We first introduce the Liouville equations for the time evolution of the Zeeman sublevels formulated as following [10,15–17]:

\dot{ρ} = \frac{1}{i ℏ} [H, ρ] - \frac{1}{2} {\hat{R}, ρ} + Λ_{γ} + T r (F ρ) + L_{e c o} ρ + Γ_{\sec} (ρ) .

where

H = H_{0} + H_{E} + H_{B} .

Here the Hamiltonian

{\hat{H}}_{0}

describes the unperturbed atom:

H_{0} = \sum_{F m_{F}} ℏ ω_{F m_{F}} | F m_{F} 〉 〈 F m_{F} | + \sum_{F' m_{F'}} ℏ ω_{F' m_{F'}} | F' m'_{F} 〉 〈 F' m_{F'} | .

where

F

and

m_{F}

are the total quantum number and the angular quantum number of the ground states, respectively.

F'

and

m_{F'}

are the total quantum number and the angular quantum number of the excited states, respectively. The Hamiltonian

H_{E}

and

H_{B}

describe the interactions with laser light and magnetic field:

H_{E} = - d \cdot E = \sum_{F, F', m_{F}, m_{F'}} 〈 F' m_{F'} | d | F m_{F} 〉 \cdot E | F' m_{F'} 〉 〈 F m_{F} | + h . c .

H_{B} = - μ \cdot B = \frac{- g_{F} μ_{B}}{ℏ} \sum_{F, m_{F}, F', m'_{F}} 〈 F m_{F} | F | F m_{F'} 〉 \cdot B | F m_{F} 〉 〈 F m'_{F} | .

where magnetic field

B = B_{Z}^{0} + B_{t}

.

Fig. 1 Energy structure of the cesium atoms in the external magnetic field $B_{Z}^{0}$ . The red solid arrows represent the $σ^{+}$ polarized laser field that couples the atomic transitions $F_{g} = 4, m_{F} \to F_{e} = 3, m_{F} + 1$ (D₁ lines at 895 nm). The radiofrequency field RF is scanned and the range of the frequency covers $ω_{1}$ and $ω_{2}$ which are the Zeeman shifts of the two hyperfine states. SEC represents the spin-exchange collisions.

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In Eq. (2), $\hat{R}$ describes the relaxation induced by the spontaneous emission and limited interaction time of cesium atom with the laser field:

\hat{R} = \sum_{F m_{F}} γ | F m_{F} 〉 〈 F m_{F} | + \sum_{F' m_{F'}} (γ + Γ) | F' m_{F'} 〉 〈 F' m_{F'} | .

where

γ

is the relaxation rate of the transit time effect and

Γ

is the rate of the spontaneous emission.

The matrix $Λ_{γ}$ describes the repopulation of the ground-state levels due to the transit time effect:

Λ_{γ} = \sum_{F m_{F}} γ | F m_{F} 〉 〈 F m_{F} | ρ_{m_{F} m_{F}}^{e q} .

where

ρ_{m_{F} m_{F}}^{e q}

is the equilibrium population of state

| F m_{F} 〉

.

In the Eq. (2), $T r (F ρ)$ describes the repopulation of the ground-state levels due to the spontaneous emission, here $\hat{F}$ is the spontaneous emission operator [15]. The matrix elements are formulated as:

T r {(F ρ)}_{m n} = \sum_{r s} F_{m n}^{s r} ρ_{r s} .

where

F_{m n}^{s r} = \frac{4 ω_{r m}^{3}}{3 ℏ c^{3}} d_{m r} \cdot d_{s n} .

The dephasing rate of the excited-state coherences are given by

{\hat{L}}_{e c o} \hat{ρ} = \sum_{m_{F'} m'_{F}} Γ^{*} D [σ_{m_{F'} m_{F'}}, σ_{m'_{F} m'_{F}}] .

D [σ_{m_{F'} m_{F'}}, σ_{m'_{F} m'_{F}}] \equiv σ_{m_{F'} m_{F'}} \cdot ρ \cdot σ_{m'_{F} m'_{F}} - \frac{1}{2} ρ \cdot σ_{m'_{F} m'_{F}} \cdot σ_{m_{F'} m_{F'}} - \frac{1}{2} σ_{m'_{F} m'_{F}} \cdot σ_{m_{F'} m_{F'}} \cdot ρ

where

σ_{m'_{F} m'_{F}}

is the transition operator,

Γ^{*}

is the rate of phase-changing collisions [18].

At last, the relaxation induced by the spin-exchange collision is considered in the Eq. (2) [10]:

\frac{{\hat{Γ}}_{\sec} (ρ)}{R_{\sec}} = - \frac{3}{4} ρ + \hat{S} \cdot ρ \cdot \hat{S} + 〈 \hat{S} 〉 ({\hat{S}, ρ} - 2 i \hat{S} \times ρ \cdot \hat{S}) .

In Eq. (13), the third term represents nonlinear effect, which is neglected in our system since it is much weaker than the effect induce by the radiofrequency field.

3. Experimental setup and results discussion

The experimental setup is shown in Fig. 2, which is a conventional configuration of the M_Z magnetometer [7]. To detect the M_Z signal, we place a 25 mm diameter, 30 mm long paraffin-coated cell filled with cesium atoms in a solenoid coil. In a paraffin-coated cell the depolarization atom-wall collisions are suppressed, which leads to a reduced linewidth of several tens hertz. The static magnetic field $B_{Z}^{0}$ produced by the coil can be tuned from zero to $5 \times 10^{4}$ nT (geomagnetic field) with a precision current source (Keysight B2912A). The coils and the cell are magnetically shielded with four layers of μ-metal cylinders and the magnetic field gradient in the middle zone of the coil is below 2 nT/cm even when the magnetic field is $5 \times 10^{4}$ nT. We use an 895 nm DBR laser as the light source which is locked to the transition $F_{g} = 4 \to F_{e} = 3$ with the modulation-free polarization spectroscopy. The power of the laser light sent to the experiment can be tuned with a half wave plate and a polarized beam splitter. The diameter of the light is 3 mm and the polarization is transformed with a quarter wave plate. The radiofrequency field $B_{t} = B_{X}^{1} \cos ω t$ is produced with a Helmholtz coil (not drawn in Fig. 2). When the frequency of $B_{t}$ is on resonance, the magnetic resonance occurs and the laser is strongly absorbed. A photodetector (Newport optical receiver 2031) is used to collect the transmitted laser light and a digital oscilloscope (Tektronix TDS2014C) is used to record the spectrum.

Fig. 2 Experimental setup. DBR: distributed Bragg reflector laser; PS: polarization spectroscopy; PID: proportional-integral-derivative controller; HWP: half-wave plate; QWP: quarter-wave plate; PBS: polarization beam splitter; cell: paraffin coated vapor cell filled with cesium atoms; RF: radiofrequency field; Det.: optical receiver; OSC: oscillator.

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The static magnetic field $B_{Z}^{0}$ is about 27806 nT in the experiment. The frequency of the radiofrequency field is scanned from 96.8 kHz to 98 kHz, which covers $ω_{1}$ and $ω_{2}$ . The measured spectrum (solid line) is plotted in Fig. 3 and agree very well with the theoretical calculations (dashed line). The left absorption peak is the normal M_Z signal corresponding to $F_{g} = 4$ , while the right peak, whose resonant frequency equals exactly to $ω_{2}$ , represents the collision induced magnetic resonance and has not been observed before. To understand this newly observed peak, we calculate the population distributions of all the Zeeman sublevels in two ground states versus the frequency of the radiofrequency field with and without considering spin-exchange collisions, and the results are shown in Fig. 4.

Fig. 3 The magnetic resonance spectrum vs. the frequency of the oscillating magnetic field $B_{t}$ . The red solid line represents the experimental data. The blue short dash line is the theoretical calculation.

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Fig. 4 Theoretical results of the population distributions of all the Zeeman sublevels in two ground states vs. the frequency of the radiofrequency field with (red lines) and without (blue lines) considering spin-exchange collisions. (a) $F_{g} = 4, m_{F} = - 4$ ; (b) $F_{g} = 4, m_{F} = - 3$ ; (c) $F_{g} = 4, m_{F} = - 2$ ; (d) $F_{g} = 4, m_{F} = - 1$ ; (e) $F_{g} = 4, m_{F} = 0$ ; (f) $F_{g} = 4, m_{F} = + 1$ ; (g) $F_{g} = 4, m_{F} = + 2$ ; (h) $F_{g} = 4, m_{F} = + 3$ ; (i) $F_{g} = 4, m_{F} = + 4$ ; (j) $F_{g} = 3, m_{F} = - 3$ ; (k) $F_{g} = 3, m_{F} = - 2$ ; (l) $F_{g} = 3, m_{F} = - 1$ ; (m) $F_{g} = 3, m_{F} = 0$ ; (n) $F_{g} = 3, m_{F} = + 1$ ; (o) $F_{g} = 3, m_{F} = + 2$ ;(p) $F_{g} = 3, m_{F} = + 3$ .

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The M_Z magnetic resonance spectrum shown in Fig. 3 can be understood from the calculated population distribution at different M_Z sublevels plotted in Fig. 4. When a σ + polarized laser drives the transition $F_{g} = 4 \to F_{e} = 3$ of cesium atoms, the laser light is absorbed by the atoms in the states $F_{g} = 4, m_{F} = - 4, \dots, + 2$ . If a far-off-resonance radiofrequency field is applied or the radiofrequency field is turned off, the $σ^{+}$ polarized laser light and the relaxation processes make the atoms reach a dynamic balance under the steady-state condition. The atomic population is mostly concentrated in the $F_{g} = 4, m_{F} = + 2, + 3, + 4$ states due to the optical pumping effect and the spin-exchange collisions. There are only a small amount of atoms in other magnetic sublevels of the $F_{g} = 4$ states. Therefore, the atoms are polarized in the steady-state.

When the frequency of the radiofrequency field is tuned to match the Zeeman shift of the ground state $F_{g} = 4$ , the populations of the states $F_{g} = 4, m_{F} = + 3, + 4$ will be transferred to the states $F_{g} = 4, m_{F} = - 4, \dots, + 2$ . In the steady state, the atoms will accumulate in the states $F_{g} = 4, m_{F} = - 4, \dots, + 2$ (the left peaks in Figs. 4 (a)-4(g)). Then the $σ^{+}$ polarized laser light will be absorbed which gives rise to the observed left peak of the magnetic resonance in Fig. 3. At the same time, the population of the states $F_{g} = 3, m_{F} = + 3$ is transferred to the states $F_{g} = 4, m_{F} = + 3, + 4$ due to the effect of the spin-exchange collision terms $| 4, m + 1 〉〈 3, m |$ and $| 4, m 〉〈 3, m |$ , which also contributes to the magnetic resonance signal represented by the left peak in Fig. 3.

Similarly, when the frequency of the radiofrequency field is tuned to match the Zeeman shift of the ground state $F_{g} = 3$ , the populations of the states $F_{g} = 3, m_{F} = + 2, + 3$ will be transferred to the states $F_{g} = 3, m_{F} = - 3, \dots, + 1$ . In the steady state, the atoms will accumulate in the states $F_{g} = 3, m_{F} = - 3, \dots, + 1$ (the right peaks in Figs. 4 (j)-4(n)). Then the populations in the states $F_{g} = 3, m_{F} = - 3, \dots, + 1$ are transferred to $F_{g} = 4, m_{F} = - 4, \dots, + 2$ by the spin-exchange collision, which gives rise to the magnetic resonance represented by the right peak in Fig. 3. Similarly, the populations of the states $F_{g} = 4, m_{F} = + 4, + 3$ are transferred to the states $F_{g} = 3, m_{F} = + 2, + 3$ due to the effect of spin-exchange collision terms $| 3, m - 1 〉〈 4, m |$ and $| 3, m 〉〈 4, m |$ , which further contributes to the magnetic resonance signal represented by the right peak in Fig. 3 as well.

There are three splittings appearing in the population distribution peaks in Figs. 4(g) and 4(n). Both of the radiofrequency field and the spin-exchange collisions make contribution to the splittings. When the radiofrequency field is far off the resonance, the atoms are mainly in the states $F_{g} = 4, m_{F} = + 4, + 3$ and $F_{g} = 3, m_{F} = + 2, + 3$ (Figs. 4(h), 4(i), 4(o) and 4(p)) with the resonant laser and the role of spin-exchange collisions. There are also some atoms in the states $F_{g} = 4, m_{F} = + 2$ and $F_{g} = 3, m_{F} = + 1$ . When the radiofrequency field is near the resonance, the atoms are transferred from the states $F_{g} = 4, m_{F} = + 4, + 3$ and $F_{g} = 3, m_{F} = + 2, + 3$ to other magnetic sublevels, which leads to the population increase in the states $F_{g} = 4, m_{F} = + 2$ and $F_{g} = 3, m_{F} = + 1$ . When the radiofrequency field is on the resonance, there are still obvious population imbalance between states $F_{g} = 4, m_{F} = + 2$ and $F_{g} = 4, m_{F} = + 1$ ( $F_{g} = 3, m_{F} = + 1$ and $F_{g} = 3, m_{F} = 0$ ). Then the radiofrequency field will transfer the atoms from $F_{g} = 4, m_{F} = + 2$ to $F_{g} = 4, m_{F} = + 1$ ( $F_{g} = 3, m_{F} = + 1$ to $F_{g} = 3, m_{F} = 0$ ). This is the reason for the splitting of the left peak in Fig. 4(g) and the splitting of the right peak in Fig. 4(n).

As mentioned above, there is strong coupling between the states $F_{g} = 4, m_{F} = + 2$ and $F_{g} = 3, m_{F} = + 2$ by the spin-exchange collisions. When the frequency of the radiofrequency field equals the Zeeman shift of the ground state $F_{g} = 3$ , the population of the state $F_{g} = 3, m_{F} = + 2$ is transferred to $F_{g} = 3, m_{F} = + 1$ by the radiofrequency field (Fig. 4(o)). Then, the population of the state $F_{g} = 4, m_{F} = + 2$ is transferred to $F_{g} = 3, m_{F} = + 2$ by the spin-exchange collisions, which leads to the reduction of the population in the state $F_{g} = 4, m_{F} = + 2$ (Fig. 4(g)). This is the reason for the splitting of the right peak in Fig. 4(g).

For further verification, we change the scanning time of the RF field and see if it reduces the amplitude of collisional induced magnetic resonance signal. For reference, we define the amplitudes of two signal peaks as 1 when the scanning time is 10s. The experimental results are shown in Fig. 5. The right peak amplitude of the collisional induced resonance is limited by the collision time and decreases rapidly as the scanning time is decreased. The signal disappears when the scanning time is 0.1s or less. By contrast, the signal amplitude of the left peak form the normal resonance decreases slowly and is still observed when the scanning time is 0.1 s or less.

Fig. 5 The amplitudes of the two magnetic resonances vs. scanning time: 0.1s, 0.5s, 1s, 5s and 10s. The upper blue line is for the left peak (normal resonance) and the lower red one is for the right peak, the collision induced resonance.

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With the discussion above, we conclude that the observed right peak represents the magnetic resonance induced by the spin-exchange collisions. It has been shown that that the collisional induced resonance peak will be immune to the power broadening of the light [12]. This is indeed the case for the observed right peak in Fig. 3. We measured the linewidth for the left peak and right peak in Fig. 3 versus the laser input power and the results are plotted in Fig. 6. It clearly shows that the linewidth of the left peak from the normal resonance is power broadened while the linewidth of the right peak due to the collision induced resonance exhibits no power broadening and the light narrowing effect is observed in the high intensity regime which was also observed and studied in Refs [12,19,20]. There are three processes that make contribution to the linewidth of the magnetic resonances: collision with walls, spin-exchange relaxation and power broadening effect [12]. The laser light field drives the transition $F_{g} = 4 \to F_{e} = 3$ , so the left normal magnetic resonance exhibits obvious laser power broadening effect while the right collision induced magnetic resonance is not affected. In our experiment, the linewidth of the right collision induced magnetic resonance is dominated by the spin-exchange relaxation which scales linearly with atomic density and depends on the state composition of the atoms [12]. Magnetic sublevels with different populations will have different spin-exchange relaxation rates, which makes the linewidths of the different magnetic resonance transition signals different. When the experiment is operated in the low intensity regime, the atoms distribute in the states $F_{g} = 3, m_{F}$ and have different populations, so the right collision induced magnetic resonance signal consists of multiple magnetic resonance transitions. Therefore, the linewidth of the collision induced magnetic resonance is the superposition of different values. When the experiment is operated in the high intensity regime, the atoms are mostly in the states $F_{g} = 3, m_{F} = + 2, + 3$ , so the right collision induced magnetic resonance signal is mainly from the magnetic resonance transition $F_{g} = 3, m_{F} = 2 \to m_{F} = 1$ . Then the linewidth is dominated by one value, which leads to the narrowing effect.

Fig. 6 Line widths of the two magnetic resonance signals vs. light power. (a) left normal resonance; (b) right collision induced resonance.

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Furthermore, we also measured the signal amplitude of the right peak (from the collision induced resonance) and left peak (the normal magnetic resonance) versus the laser power and the results are plotted in Fig. 7. It shows that as the laser intensity increases, the signal amplitude of the normal magnetic resonance increases monotonically, but the amplitude of the collision induced resonance signal increases first and then is nearly constant when the laser power is increased to be above a threshold value. The phenomenon can be explained as following: First, the atoms will be mostly optically pumped to the $F_{g} = 3$ states in the high laser power regime and the number of atoms in the $F_{g} = 4$ states decreases when the laser power increases, which is the reason why the amplitude of the normal magnetic resonance is very small. Second, the amplitude of the collision induced resonance depends on both the spin-exchange collision rate and the optical pumping rate. When the laser power is low, the optical pumping rate is smaller than the spin-exchange collision rate. Then the amplitude is largely determined by the optical pumping rate and increases when the laser power increases. When the laser power is high, the optical pumping rate is much larger than the spin-exchange collision rate. Then the amplitude is largely determined by the spin-exchange collision rate and shows saturation effect.

Fig. 7 Amplitudes of the two magnetic resonance signals vs. light power. (a) left normal resonance; (b) right collision induced resonance.

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Combination of Fig. 6 and Fig. 7 indicates that at high laser intensities, the ratio of the signal amplitude to the linewidth is nearly constant. Since the sensitivity of a magnetometer is given by $δ B = (Δ B / S) \cdot n o i s e$ : With a constant ratio of the line width $Δ B$ and the signal amplitude S, the sensitivity only depends on the noise, which would be the characteristic feature of the magnetometer based on the collision induced magnetic resonance observed here and provides the advantage for precision measurements of the geomagnetic field.

4. Summary

We experimentally and theoretically investigate the collision induced magnetic resonance in Cs atoms. We show that the population transfer among magnetic sublevels induced by the spin-exchange collisions can lead to a new type of the magnetic resonance. Specifically in the collisional process, the polarization in state $F_{g} = 4$ created by the optical pumping is transferred to the state $F_{g} = 3$ . With the radiofrequency field and the spin-exchange collisions, the imbalanced population in the state $F_{g} = 3$ can be transferred to the magnetic sublevels in the state $F_{g} = 4$ , which are coupled to the resonant laser field. It is a kind of radiofrequency-optical double magnetic resonance involving collisions. We observed that the collision induced magnetic resonance exhibits saturation effect for the signal amplitude with the increase of laser power, but the resonance linewidth is immune to the power broadening. This phenomenon will be useful for precision measurement of the geomagnetic field.

Funding

National Natural Science Foundation of China (Grant No. 11774118) and Fundamental Research Funds for the Central Universities (Grant No. CCNU18CXTD01).

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Collision induced magnetic resonance in Cesium atoms

Abstract

1. Introduction

2. Theoretical analysis

3. Experimental setup and results discussion

4. Summary

Funding

References

Cited By

Figures (7)

Equations (13)

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