Detection of a weak magnetic field is vital in many areas such as fundamental physics, biomagnetism, and geophysical exploration.^{[1– 3]} An optically pumped quantum magnetometer (OPQM) is excellent in these particular applications. The OPQM approach was first proposed by Dehmelt in 1957.^[4] Later, it was implemented by Bloom.^[5] After about five decades’ development, great progress has been achieved.^{[6– 8]} The sensitivity of OPQM was demonstrated to be in a spherical cell with a diameter of 7  cm and could reach in the limit of shot noise.^[6]

Based on the form of a magnetic resonance signal which is associated with the projection of the magnetic moment, general OPQMs are divided into M_z magnetometer and M_x magnetometer. The M_z magnetometer detects the absorption signal of the longitudinal component of the magnetic moment (M_z signal).^[9] It possesses a relatively high sensitivity which is limited by the linewidth and signal-to-noise ratio but a slow response speed in the limit of longitudinal relaxation time T₁. The M_x magnetometer detects the phase of the oscillating transverse component of the magnetic moment (M_x signal).^[9] Its response time is not limited by the relaxation time, so it possesses a high operational speed. However, it is so sensitive to the phase error that its sensitivity is relatively low. In many situations, a magnetometer with a high operational speed and sensitivity is necessary.^[10] An idea of an M_x– M_z tandem which combines the advantages of M_z and M_x magnetometers was proposed and further developed.^{[11– 13]} It detects both the amplitude (M_R signal) and the phase of the oscillating transverse component of the magnetic moment. The M_R signal is similar to the M_z signal and also possesses high sensitivity but low operational speed which can be compensated for by the M_x signal.

As for the M_R magnetometer and M_z magnetometer, the sensitivity δ B is given by

where, Δ ω is the resonance linewidth, γ is the gyromagnetic ratio of sensory atom (alkali-metal atom), and S_NR is the signal-to-noise ratio. So the sensitivity can be improved by reducing Δ ω and improving S_NR. In a traditional measuring method, the Rabi frequency ω ₁ satisfies

where, B₁ is the amplitude of oscillating magnetic field, and T₂ is the transverse relaxation time. Under this condition, the resonance linewidth is limited to 2/T₂.

In this paper, a dip of the M_R signal as a function of resonance frequency-detuning will emerge when ω ₁ is larger than (T₁T₂^{)− 1/2}, which is demonstrated theoretically and experimentally. The M_x– M_R magnetometer which operates under this condition is analyzed. Compared with the M_x– M_R magnetometer which operates when ω ₁ < (T₁T₂)^{− 1/2} in Ref.  [12], this magnetometer has a linewidth that can exceed the limit of 2/T₂, and more important is that the M_x– M_R magnetometer can possess a much higher signal-to-noise ratio. When B₁ is properly chosen, the sensitivity can be improved by an order of magnitude. The M_x– M_R magnetometer with higher sensitivity can be achieved without losing the operational speed.

As for a vapor cell filled with alkali-metal atoms, considering a constant field B₀ directed along the Z axis in the laboratory coordinate system XYZ as shown in Fig.  1(a), the energy levels of alkali-metal atoms will split along the Z axis. In the equilibrium state, the numbers of atoms in different Zeeman energy levels obey the Boltzmann distribution, and it can be described by a macroscopic magnetization M₀ along the Z axis. As the difference in the number of atoms between different Zeeman energy levels is tiny, M₀ is quite small. If circularly polarized light is used to pump alkali-metal atoms along the Z axis, ^[14]M₀ will be improved by several orders of magnitude. In Fig.  1(a), an oscillating magnetic field B₁ = 2B₁x̂ cos(ω _at) exciting the atomic transition is applied along the X axis.

Fig.  1.

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	Fig.  1. Magnetization vectors in (a) a laboratory coordinate system XYZ and (b) a rotating coordinate system X′ Y′ Z.

A rotating coordinate system X′ Y′ Z which rotates clockwise around the Z axis with a frequency of ω _α is introduced and shown in Fig.  1(b). The magnetization vector M due to spin polarization is represented as (M_x(t), M_y(t), M_z(t))^T in the XYZ coordinate system. In the X′ Y′ Z coordinate system, the magnetization vector M′ is represented as (M_x′ (t), M_y′ (t), M_z(t))^T. The following relationship can be easily obtained:

in the above equation, , and ϕ (t) = arctan(M_x′ (t)/M_y′ (t)).

For simplifying computation, we divide B₁ into the clockwise components and counterclockwise components. The clockwise components are B_x = B₁cos(ω _α t) and B_y = − B₁sin(ω _α t), and the counterclockwise components are B_x = B₁cos(ω _α t) and B_y = B₁sin(ω _α t). The counterclockwise components have little influence on the evolution of M unless B₁ is quite large.^[15] When we ignore the counterclockwise components with the rotating-wave approximation, the evolution of the magnetization vector M′ can be described by

where, ω ₂ = γ B₀− ω _α is the detuning of resonance frequency. By setting the above equations to be equal to zero, the stationary solution can be obtained as follows:

Substituting Eq.  (5) into Eq.  (3), we obtain

By setting dM_R/dω ₂ = 0, the following solution can be easily obtained:

If ω ₁ is larger than (T₁T₂)^{− 1/2}, a dip of the M_R signal as a function of the resonance frequency-detuning emerges. The width of two peaks Δ ω _{p − p} is

Therefore, the linewidth (full-width at half-maximum) of the dip can exceed the limit of 2/T₂.

When the influences of the counterclockwise components of B₁ are considered, reference  [15] shows that in the lowest order correction the shape of the resonance curve is unchanged but it is shifted. The resonance frequency is

Here, ω ₀ = γ B₀, and [B₁/(2B₀)]² is the percentage of frequency-shift. Usually, T₁ ≈ 2T₂.^[16] By assuming T₁ = 2T₂, when , ω ₀[B₁/(2B₀)]² = 2/(8ω ₀T₂ × T₂). As for the usually used OPQMs, 8ω ₀T₂ is much larger than 1. So we can ignore the influence of the counterclockwise components. For achieving higher accuracy, we can also correct the resonance frequency by adding ω ₀[B₁/(2B₀)]².

The experimental setup is shown in Fig.  2. A spherical Pyrex cell with a diameter of 2  cm contains ⁸⁷Rb atoms and 100  Torr (1.33322× 10²  Pa) of buffer gas (N₂ and ⁴He) for slowing atomic diffusion and quenching. The cell is placed inside a five-layer magnetic shield and heated with a hot air oven. A constant magnetic field B₀ and an oscillating magnetic field B₁ are generated by the surrounding two pairs of Helmholtz coils. The Helmholtz coils along the Z axis are driven by steady current circuit and the Helmholtz coils along the X axis are driven by a lock-in amplifier. The pump beam is generated by a distributed feedback diode laser which is locked to ⁸⁷Rb D1 transition. It becomes circularly polarized beam after passing through a polaroid and λ /4 plate. Then, it polarizes the ⁸⁷Rb atoms along the Z axis after expansion and collimation. The probe beam is generated by a distributed feedback diode laser which is turned to near ⁸⁷Rb D1 transition. It illuminates the cell after passing through the polaroid and apparatus of expansion and collimation. The M_x(t) signal is measured with a polarized beam splitter (PBS) and a balanced photodetector by utilizing optical rotation of the linearly polarized probe beam.^[17] Finally, the detected signal is acquired and processed by the signal processing system.

Fig.  2.

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	Fig.  2. Schematic diagram of the experimental setup.

The cell is heated and its temperature is stabilized at 110  ° C. The frequency of B₁ is slowly linearly scanned with different amplitudes. The M_R signals are extracted from M_x(t) signals and shown in Fig.  3 when ω ₁ < (T₁T₂)^{− 1/2} and ω ₁ > (T₁T₂)^{− 1/2}. Compared with Fig.  3(a) when ω ₁ < (T₁T₂)^{− 1/2}, figure  3(b) shows an obvious dip when ω ₁ > (T₁T₂)^{− 1/2}.

In order to compare the characteristics of the M_x– M_R magnetometer when ω ₁ < (T₁T₂)^{− 1/2} with those when ω ₁ > (T₁T₂)^{− 1/2}, we measure the sensitivities of the M_x– M_R magnetometer with different values of ω ₁ under the same condition. When ω ₁ is smaller than (T₁T₂)^{− 1/2}, we define the resonance linewidth as the full-width at half-maximum of M_R signal, which is a function of detuning (ω ₂), and the signal intensity as half of the amplitude of M_R at ω ₂ = 0. Similarly, when ω ₁ is larger than (T₁T₂)^{− 1/2}, the resonance linewidth is defined as the full-width at half-maximum of the dip and the signal intensity is defined as half of the depth of the dip. The experimental results of the linewidth and signal intensity as a function of ω ₁ are shown in Figs.  4(a) and 4(b) by using asterisks. In Figs.  4 and 5, T₂₀ is the transverse relaxation time which is obtained by measuring ϕ (t) as a function of ω ₂ when ω ₁ is much smaller than (T₁T₂)^{− 1/2}, ^[18] and T₁₀ is the longitudinal relaxation time which is assumed to be a constant. The computational results of the linewidth and signal intensity can be obtained from Eqs. (6) and (7) by assuming both T₂ and T₁ to be constants, which are shown in Figs.  4(a) and 4(b) by using solid lines. From Figs.  4(a) and 4(b), we can find that the results of experiment and computation are almost identical when ω ₁ is small. As ω ₁ becomes larger, the difference between experimental results and computational results increases. It is mainly because T₂ and T₁ are assumed to be constants in our computation. In fact, T₂ is related to the intensity of oscillating magnetic field. The experimental results of T₂ are obtained at different values of ω ₁ by measuring ϕ (t) as a function of ω ₂, and the results are shown in Fig.  5. From Fig.  5, we can find that the influence of ω ₁ on T₂ becomes much larger as ω ₁ increases. Though T₂ is related to ω ₁, the change in T₂ is less than forty percent. In addition, the operational speed of the M_x magnetometer is independent of relaxation time. Therefore, it has no influence on the operational speed of the M_x– M_R magnetometer.

Fig.  3.

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	Fig.  3. The MR signals extracted from Mx(t) signals when (a) ω 1 < (T1T2)− 1/2 and (b) ω 1 > (T1T2)− 1/2.

Fig.  4.

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Fig.  4. Raw data of the resonance linewidth, the signal intensity, the noise intensity, and the sensitivity each as a function of ω 1. T20 is the transverse relaxation time which is obtained by measuring ϕ (t) as a function of ω 2 when ω 1 is much smaller than (T1T2)− 1/2, and T10 is the longitudinal relaxation time which is assumed to be a constant. (a) Experimental (asterisks) and computational (solid line) results of the resonance linewidth; (b) experimental (asterisks) and computational (solid line) results of the signal intensity; (c) experimental results of the noise intensity; (d) experimental results of the sensitivity.

Although the technical noises can be completely eliminated in theory, they dominate rather than the quantum noises in practice. As for the atomic magnetometer, there are many fluctuations which cannot be avoided, such as the fluctuations in frequency and intensity of pump laser and probe laser, temperature, etc. These fluctuations will induce the fluctuations (δ M₀ and δ T₂) in M₀ and T₂. When the noise intensity is defined by the fluctuation of the M_R signal at ω ₂ = 0, the intensities of these noises (N_M0 and N_T2) induced by δ M₀ and δ T₂ can be obtained from Eq.  (6) as follows:

From the first expression of Eq.  (10), it is easy to see that the noise intensity increases as the M_R signal at ω ₂ = 0 rises. From the second expression of Eq.  (10), the noise intensity increases when the ratio of the square of the M_R signal at ω ₂ = 0 to ω ₁ rises. According to the above definition of the signal intensity, when ω ₁ is smaller than (T₁T₂)^{− 1/2}, the signal intensity is positively associated with the M_R signal at ω ₂ = 0. However, when ω ₁ is larger than (T₁T₂)^{− 1/2}, the signal intensity is negatively associated with the M_R signal at ω ₂ = 0. From Eq.  (6), the M_R signal at ω ₂ = 0 reaches a maximum at ω ₁ = (T₁T₂)^{− 1/2} and will decrease when ω ₁ > (T₁T₂)^{− 1/2}. Considering these conditions, the noise intensity as a function of ω ₁ first increases, and then decreases. The noise intensity is large when the signal intensity is large under the condition of ω ₁ < (T₁T₂)^{− 1/2}. On the contrary, the noise intensity is small when the signal intensity is large under the condition of ω ₁ > (T₁T₂)^{− 1/2}. For obtaining the noise intensity experimentally, the M_R signal at ω ₂ = 0 passes through a low-pass filter with a cut-off frequency of 50  Hz, and then the filtered signals are continually acquired. By computing the standard deviation of 1000 continual data of the acquired M_R signal intensity at ω ₂ = 0, the noise intensity with a bandwidth of 50  Hz is obtained and shown in Fig.  4(c). From Figs.  4(b) and 4(c), we can find that the M_x– M_R magnetometer can possess a strong anti-noise ability and its signal-to-noise ratio can be much larger when ω ₁ > (T₁T₂)^{− 1/2} than when ω ₁ < (T₁T₂)^{− 1/2}.

The sensitivity can be computed by utilizing Eq.  (1), and the result is shown in Fig.  4(d). Under our experimental condition, the sensitivity of the M_x– M_R magnetometer can be improved by an order of magnitude when ω ₁ > (T₁T₂)^{− 1/2} compared with when ω ₁ < (T₁T₂)^{− 1/2}.

Fig.  5.

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	Fig.  5. Raw data of the transverse relaxation time as a function of ω 1.

In this study, a dip of the M_R signal will emerge when the Rabi frequency is larger than (T₁T₂)^{− 1/2}, which has been analyzed. The resonance linewidth of the dip can exceed the limit of 2/T₂. The experimental result shows that the oscillating magnetic field has an influence on transverse relaxation time and the influence becomes much larger as the intensity of oscillating magnetic field increases. However, the operational speed of the M_x magnetometer is independent of relaxation time. So the intensity of oscillating magnetic field has no influence on the operational speed of the M_x– M_R magnetometer. It is demonstrated theoretically and experimentally that the noise intensity is large when the signal intensity is large under the condition of ω ₁ < (T₁T₂)^{− 1/2}. On the contrary, the noise intensity is small when the signal intensity is large under the condition of ω ₁ > (T₁T₂)^{− 1/2}. Therefore, the M_x– M_R magnetometer can possess a strong anti-noise ability and its signal-to-noise ratio can be much larger when ω ₁ > (T₁T₂)^{− 1/2}. Its sensitivity can be improved by an order of magnitude without losing the operational speed under our experimental condition. Further research is needed to further improve the characteristics and solve some open problems.