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Abstract
Laser intensity noise is one of the main limiting factors in pulsed vapor cell clocks. To reduce the contribution of the laser intensity noise to detection signal in the pulsed optically pumped atomic clock, a scheme based on the differential Faraday rotation angle is proposed. Theoretically, the Ramsey fringes, the sensitivity of clock frequency to laser intensity fluctuation and the signal to noise ratio for absorption, differential, and Faraday rotation angle methods are calculated and compared. Using a Wollaston prism rotated 45°relative to the incident polarization, and two photodetectors, Ramsey fringes of three detection methods are obtained simultaneously. In the proposed scheme, the long-term Faraday rotation angle fluctuation is 0.66% at 30000s, which is much smaller than fluctuation of traditional absorption signal 3.9% at 30000s. And the contribution of laser intensity noise to clock instability is also reduced. Using optimized photodetector with high common mode rejection ratio, a better performance should be expected. This proposed scheme is attractive for the development of high performance vapor clock based on pulsed optically pumped.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the advantages of simple physical operation, small size, light weight, and high reliability, the rubidium (Rb) vapor cell atomic clock has wide applications in radio navigation, space exploration, satellite position, and telecommunication [1–3].The best frequency stability performance in vapor cell atomic clock has been demonstrated based on pulsed optically pumped (POP) scheme [3–8], in which the pumping, interrogation, and detection phases are separated in time. In this manner, the light shift is strongly reduced, and the smaller light-induced instability contributions have been experimentally demonstrated [4,9].
Although the light shifts in the POP atomic clock are much smaller compared with those in the continuous double resonance atomic clock, they are not completely eliminated. In the POP scheme, the clock transition is commonly obtained by monitoring the transmission intensity of a probe laser. The noises in the amplitude and frequency of the probe laser will degrade the clock stability through laser amplitude modulation to amplitude modulation (AM-AM) transfer and frequency modulation to amplitude modulation (FM-AM) conversion [10,11]. The residual coherence due to non-ideal optical pumping, and the residual light during microwave interrogation due to inadequate light extinction (through the AC Stark shift effects [12]), also enhance the light shifts. Finally, the clock short-term instability contribution of the relative intensity noise (RIN) of the laser is a few parts in 10−13, which is one of the main limiting factors [4]. Meanwhile, the fluctuation of the laser power is the dominant contribution to the medium-to-long term frequency stability of the POP clock, with a contribution of the order of 10−14[5,6].
Several approaches have been investigated to reduce or eliminate the light shifts in vapor cell atomic clocks. A signal theory approach has been proposed and tested in [13,14] to estimate the contribution of the laser intensity fluctuations to the short-term stability of the POP Rb cell clock. With high-pressure buffer gas filled in the vapor cell, broadening of the optical absorption line reduced the efficiency of FM-to-AM conversion and the light-shift coefficient by several orders of magnitude, causing a negative impact on the shot-noise-limited performance of the clock [15,16].To reduce the long-term instability contribution induced by light intensity fluctuations, active laser intensity stabilization is required and can be implemented using external power actuators, such asacousto-optical modulator (AOM) [17], electro-optic amplitude modulator (EOM) [18], polarizer and photo-elastic modulator [19]. The power fluctuations measured by the in-loop detector are not synchronized with the out-of-loop part [20]; therefore, the power stability of probe laser is worse than that of the feedback loop. Moreover, the active stabilization system is considerably complicated.
Motivated by the previous studies based on the detection of the forward-scattering signals from magneto-optical rotation in the pulsed atomic clock [21,22] and the studies employing differential detection to reduce laser noise in coherent population trapping (CPT) atomic clocks [23,24], Faraday rotation measurements [25–27], electromagnetically induced transparency (EIT) atomic clock [28], magnetometer [29], and radio-frequency-dressed states detection [30], we have been conducting theoretical and experimental investigations on the differential Faraday rotation angle detection for the POP atomic clock. Theoretical study shows that the sensitivity of clock frequency to laser intensity fluctuation in the differential Faraday rotation angle detection is 10 orders of magnitude smaller than the traditional absorption method, and with optimized high common mode rejection ratio (CMRR) photodetector the stability can be further improved. Experimentally, based on a combination of a Wollaston prism and two photodetectors, the Ramsey fringes of the differential Faraday rotation angle can be obtained for the POP atomic clock. Employing a differential detection scheme, the fluctuation of detection signal is significantly suppressed caused by laser intensity fluctuation, and the contribution of laser intensity noise to short-term stability is also reduced.
In this paper, we propose a differential Faraday rotation angle detection method for the POP atomic clock. Firstly, we calculate the Ramsey fringes for Faraday rotation angle detection under the condition of linearly polarized light using a three-level V system model. Secondly, the sensitivity of clock frequency to laser intensity fluctuation and the signal to noise ratio (SNR) for three detection methods are compared theoretically. Thereafter, we discuss the experimental results obtained based on this differential Faraday rotation angle detection scheme, including the Ramsey fringes and the suppression of laser intensity fluctuations and laser intensity noise.
2. Theory
2.1 Theory of POP Rb atomic clock
The POP atomic clock runs periodically, the timing sequence of which is shown in Fig. 1(a). After optical pumping, 87Rb atoms are pumped to the three Zeeman sublevels of $|{5{S_{1/2}},F = 1} \rangle$. Thereafter, 87Rb atoms interact with two separated microwave pulses in Ramsey scheme for the interrogation of the clock transition. At the end of the second Ramsey pulse, the clock transition is detected by monitoring the transmitted intensity of a probe laser through the cell.
Fig. 1. (a) Timing sequence of POP atomic clock; (b) three-level approximation of 87Rb. Here, |μ'› and |μ› represent the two atomic clock levels |F=2;mF=0› and |F=1;mF=0›; ωL and ω0 are the angular frequencies of laser and microwave, respectively; γ1 and γ2 are the ground state population and coherent relaxation rates; Г* is the decay rate from the excited state to the ground state; Δ0 is the detuning frequency of the laser; tp, tm, and td are the duration of the laser optical pumping, microwave, and optical detection pulses, and T is the Ramsey time.
Using the approximate analytical equations in [9], the Ramsey interaction can be expressed by the following matrix equation:
$R[t]$ is the Bloch vector defined as:2.2 Differential Faraday rotation angle detection
The linearly polarized probe light propagates along the z-axis, with the initial polarization along the x-axis. This incident light can be written as a superposition of two orthogonal circularly polarized lights, σ+ and σ-, as follows:
Schematic diagram of differential Faraday rotation angle detection is shown in Fig. 2. After the Rb cell, a Wollaston prism is placed before the two photodetectors and its axis are defined as ${\hat{n}_1} = \hat{x}\cos \phi - \hat{y}\sin \phi$ and ${\hat{n}_2} = \hat{x}\cos \phi + \hat{y}\sin \phi$, where $\phi$ is the angle between the optical axis of Wollaston prism and x-axis. Differential detection method requires $\phi$ equal to $45^\circ$, in such case the amplitudes of the two orthogonally polarized light intensities would be equal in the absence of Faraday rotation, and the differential output would be zero. When the Faraday rotation angle changes, the difference between the two photodetectors can be expressed as following:
Here, I1 and I2 are the intensities of the two orthogonally polarized light outputs. In the absorption detection method, the intensity of the detection pulse at the end of the vapor cell equal to the sum of the intensity emerging from the two faces of the Wollaston prism is given by
From Eq. (7a) and (7b), the Faraday rotation angle can be evaluated by
For a hot Rb vapor cell and a weak magnetic field, the Faraday rotation angle is at the level of mrad, so the hypothesis that $\Delta \theta = \sin (\Delta \theta )$ can be used in Eq. (7c). The differential Faraday rotation angle method can greatly reduce the light intensity fluctuations and noise compared with the absorption detection method, which is the main reason we propose this novel detection scheme.
2.3 Ramsey fringes of Faraday rotation angle
In an optical medium, the complex susceptibility can be expressed as $\chi = {\rho _{2i}}{{nd_{2i}^2} / {({{\varepsilon_0}\hbar {\Omega _{2i}}} )}}$ [32], where $n$ is the atomic density, ${\varepsilon _0}$ the permittivity, $\hbar$ the reduced Planck’s constant, ${d_{2i}}$ the dipole matrix element, and ${\Omega _{2i}}$ the Rabi frequency. The complex refractive index of the Rb atoms is related to its electric susceptibility by the equation [33]${n_c} = n + i\beta = \sqrt {1 + \chi } \approx 1 + \chi ^{\prime}/2 + i\chi ^{\prime\prime}/2$, where the real part $\chi ^{\prime}$ represents dispersion (refractive index) and the imaginary part $\chi ^{\prime\prime}$ represents absorption. The Faraday rotation angle and absorption coefficients are
In the above equation, the positive and negative subscripts correspond to circularly polarized light σ+ and σ-, respectively.
In the detection phase, we calculate the Ramsey fringes of differential Faraday rotation angle scheme, through a density matrix analysis of the V-type three-level model, as shown in Fig. 3. Using the well-known Liouville-type equation $\dot{\boldsymbol{\mathrm{\rho}}} ={-} i/\hbar [{\mathbf H},\boldsymbol{\mathrm{\rho}}] - 1/2\{ {\mathbf \Gamma },\boldsymbol{\mathrm{\rho}}\}$ that describes the time evolution of the atomic density matrix $\boldsymbol{\mathrm{\rho}}$ under the actions of Hamiltonian ${\mathbf H}$ and relaxation ${\mathbf \Gamma }$, under the rotating wave approximation, the evolution of the coherences and populations can be given as
In the absorption detection method, the power of the transmitted light is represented by ${I_{abs}} = {I_1} + {I_2} = 2E_0^2{e^{ - 2\beta L}}$. In the differential Faraday rotation angle scheme, the signal depends on the susceptibility difference between σ- and σ+. For simplicity, we assume that the optical Rabi frequencies and detuning values are equal for σ- and σ+, which is appropriate for an experimental situation with linearly polarized light and a weak magnetic field. Under the initial condition ${\rho _{11}} = {\rho _{33}} = 0$, with the atomic population ${\rho _{22}}$ derived from Eq. (1), the steady-state solutions for ${\tilde{\rho }_{21}}$ and ${\tilde{\rho }_{23}}$ can be obtained from Eq. (9). Substituting ${\tilde{\rho }_{21}}$ and ${\tilde{\rho }_{23}}$ into the complex susceptibility $\chi = {\rho _{2i}}{{nd_{2i}^2} / {({{\varepsilon_0}\hbar {\Omega _{2i}}} )}}$ and complex refractive index ${n_c} = n + i\beta = \sqrt {1 + \chi } \approx 1 + \chi ^{\prime}/2 + i\chi ^{\prime\prime}/2$, the dispersion and the absorption for σ- and σ+ will be achieved. Then the macroscopic Faraday rotation angle and absorption coefficient should be derived from Eq. (8) and the Ramsey fringes for absorption and differential can be obtained from Eq. (9). The typical calculation results are shown in Fig. 4. Figure 4(a) and (b) show the difference and sum of the laser power emerging from the two faces of the Wollaston prism, and Fig. 4(c) shows the Faraday rotation angle computed from Eq. (7c). These three Ramsey fringes have the same full width at half maximum (FWHM) of 151 Hz, the same atomic quality factor of 4.53×107, and the contrast for absorption, difference and Faraday rotation detection methods are 77%, 52%, and 52% respectively. The Ramsey fringes of the Faraday rotation angle caused by microwave interrogation can be used to lock the local oscillator to atomic transition. The bias is the Faraday rotation angle caused by the axial magnetic field that can be eliminated by choosing a suitable angle for the Wollaston prism.
Fig. 4. Computed differential (Pdifference) Ramsey fringes (a), absorption (Psum) Ramsey fringes (b), and Faraday rotation angle Ramsey fringes (Δθ) (c). Relaxation rates: γ1=γ2=300Hz; time sequence: tm=400μs, T=3.3ms; atomic density: n=2.96×1017/m3; length of Rb cell: L=25mm; microwave pulse area: be*tm=π/2, Zeeman shift: Δ=46kHz.
2.4 Sensitivity to laser intensity fluctuation
In this section, the benefit of differential Faraday rotation angle method will be studied theoretically, by comparing the sensitivity of clock frequency to laser intensity. The sensitivity to laser intensity of three methods could be calculated from the Ramsey fringes. The Rabi frequency of laser is ${\Omega _{2i}} = {{|{{\mu_{2i}}} |{E_0}} / \hbar }$, with the help of expression for laser intensity versus field strength ${E_0} \approx \sqrt {{I_0}}$, and substitute the ${\Omega _{2i}}$ in Eq. (8) with ${{|{{\mu_{2i}}} |\sqrt {{I_0}} } / \hbar }$. The coefficient of the amplitude of central Ramsey fringes to laser intensity at microwave detuning equaling to 0.5 FWHM is calculated as $ef{f_{laser}} = {{\partial P} / {\partial {I_0}}}$, and the coefficient of the amplitude of central Ramsey fringes to microwave detuning is $ef{f_\mu }_w = {{\partial P} / {\partial {\Omega _\mu }}}$. Then the sensitivity of clock frequency to laser intensity fluctuation will be obtained from:
where $\nu$ is the clock frequency, and $\Delta I$ is fluctuation of laser intensity. The results of coefficients and sensitivities for three methods are shown in Table 1. In the differential Faraday rotation angle method, the sensitivity of clock frequency to laser intensity fluctuation is about 10 orders of magnitude smaller than the traditional absorption method.
Table 1. Calculated coefficients and sensitivity for three detection methodsa
2.5 Signal to noise ratio
It is difficult to make a direct comparison between Faraday rotation angle and absorption methods on the noises and stabilities using one theoretically model. Because the absorption method and Faraday rotation angle method monitor the absorption coefficient and rotation angle after the cell respectively, which are two physical parameters with different units. However, the theoretical frequency stability in terms of Allan deviation for POP atomic clock can be expressed as [4]: ${\sigma _y} = \frac{1}{{\pi {Q_a}{R_{sn}}}}\sqrt {\frac{{{T_c}}}{\tau }}$, where Qa is the quality factor of the atomic resonance and Rsn is the signal-to-noise ratio. In this paragraph, from the point of signal to noise ratio, the mentioned three methods will be compared theoretically.
Taking the extinction ratio of Wollaston polarizer d into account, the intensity of the ordinary ${I_ \bot }$ and the extraordinary ${I_\parallel }$ output beam behind the Wollaston polarizer is found:
The revised differential intensity, absorption and Faraday rotation angle expressions are:
For practical values of extinction ratio of ${10^{ - 2}} \le d \le {10^{ - 4}}$, Eq. (12) is the same with Eq. (7). The signal amplitude is the difference of Ramsey fringes for microwave detuning at 0 and FWHM, which is $S = |{I(\Omega = FWHM) - I(\Omega = 0)} |$.
There exist different sources of noises in optical detection. One is the photon shot noise, which is related to the Poissonian distribution of photons and represents the minimum possible noise with which an optical measurement can be achieved. Another fundamental source of noise is electronics, or thermal noise that originates from thermal fluctuations of charge carriers in any kind of conductor. The RMS of these noises is most important one is the intensity fluctuation of light, due to temperature vibrations of the laser cavity, mechanical vibrations of the laser or of the optical components.
In our experiment, the same photodetectors and probe laser beam are used during the three detection methods, so the shot noise and the thermal noise are equal and much smaller than the intensity fluctuation of light. Here, we focus on the intensity fluctuations of laser and compare the SNR for three detection methods. The fluctuation of laser can be expressed as $\Delta I = \gamma {I_0}$, and the noise induced by this fluctuations for absorption, differential and Faraday rotation angle methods are:
The SNR for different detection methods can be expressed as $SNR = S/N$. The calculated ratios of $R1 = SN{R_{Faraday}}/SN{R_{abs}}$ and $R2 = SN{R_{diff}}/SN{R_{abs}}$ are shown in Fig. 5. With $\gamma = 0.001,\;{I_0} = 0.00025$, the SNR for absorption is 1602, and the corresponding short-term stability is 3.2E-13, which agrees with the stability induced by laser noise in Ref. [1,4]. When the rest of intensity is g=0.28d=0.00028, the SNR of Faraday rotation angle method is equal to that of absorption method, and the smaller g, the bigger the SNRFaraday. This result suggests that by optimizing photodetector with a smaller g value, POP atomic clock with a better performance will be obtained by differential Faraday rotation angle detection method.
Fig. 5. The calculated ratios of R1 and R2, Relaxation rates: γ1=γ2=300Hz; time sequence: tm=400μs, T=3.3ms; atomic density: n=2.96×1017/m3; length of Rb cell: L=25mm; microwave pulse area: be*tm=π/2, Zeeman shift: Δ=46kHz, γ=0.001, intensity of laser I0=0.25mW, d=0.001.
3. Experimental setup
Figure 6 depicts the experimental setup for the POP atomic clock using the differential Faraday rotation angle detection method. The laser source was a 795-nm DFB laser (eagleyard, EYP-DFB-0795) and it was locked to the transition $F = 2 \to F^{\prime} = 2$ on the 87Rb D1 line via the saturated absorption spectrum (SAS). The reason for choosing the transition is that the D1 hyperfine dipole matrix elements for σ+ and σ- transitions $|{F = 2;{m_F} = 0} \rangle \to |{F^{\prime} = 2;{m_F} ={\pm} 1} \rangle$ are three times larger than those in the transitions $|{F = 2;{m_F} = 0} \rangle \to |{F^{\prime} = 1;{m_F} ={\pm} 1} \rangle$, therefore more effective atoms are populated to the level $|{F = 1;{m_F} = 0} \rangle$ related to the clock transition. The main part of the laser beam double-passed through an AOM (A.A Sa, MT80-B30A1.5-IR). On the one hand, the AOM acts as an optical switch in the generation of optical pulses for optical pumping and detection; on the other hand, it also blue-shifts (160 MHz) the laser frequency, to match the red shift induced by buffer gases. After the light passed a polarizer (105 extinction ratio),it was sent into the physical package of the atomic clock. In the physical package, a quartz Rb cell filled with 87Rb atoms and a mixture of Ar and N2 (${P_{Ar}}:{P_{{N_2}}} = 1.6$) with a total pressure of 25Torrwas placed in a TE011-mode microwave cavity. The quartz cell has a diameter of 30 mm and a length of 25 mm. The cavity is made of copper, with a loaded quality factor of 2000, its temperature was stabilized at 65.5 °C (for minimum temperature frequency shift) and its temperature sensitivity is 120 kHz/°C. A solenoid around the cavity produced a homogenous longitudinal static magnetic field of 3 μT to eliminate the degeneracy of the Rb ground states. A three-layer μ-metal magnetic shield was used to prevent perturbations from the ambient magnetic field. To detect the Faraday rotation angle, a Wollaston prism with its axis oriented at 45° with respect to the incident polarization was placed behind the physical package. The powers of the two output beams behind the Wollaston prism were detected by two photodetectors (Thorlabs PDA 100A), and the voltage were recorded and computed using a computer based on a high-resolution data acquisition card (NI 6281). As illustrated in Fig. 1(a), the durations of the three interrogation phases are set as follows: optical pumping time tp=1ms; microwave pulse time tm=0.4ms; Ramsey time T=3.3ms; optical detection time td=0.25ms, and the cycle time is 5.35ms. The laser powers at the input of the cell are 14mW and 0.25mW for the optical pumping and optical detection respectively.
Fig. 6. Schematic setup of the POP Rb clock with differential Faraday rotation angle detection. DFB: distributed feedback laser, OI: optical isolator, BS: beam splitter, SAS: saturated absorption spectrum, PBS: polarization beam splitter, AOM: acoustic optical modulator, QWP: quarter-wave plate, M1: 0° reflection mirror, M2: 45° reflection mirror, WP: Wollaston prism, PD: photodetector, PID: proportional-integrate-differential, LO: local oscillator.
4. Results and discussion
4.1 Ramsey fringes
Figure 7 shows the experimental Ramsey fringes of the POP clock for the absorption, differential, and Faraday rotation angle methods that are proportional to the sum (I1+I2), difference (I1-I2), and division (0.5(I1-I2)/(I1+I2)) of the photodetector signals. The central fringe contrast is 37%, 21%, and 20% for the absorption, differential, and Faraday rotation angle method, respectively, and the full-width at half-maximum (FWHM) is the same 151 Hz. An appropriate qualitative agreement is observed between the experimental results in Fig. 7 and the theoretical predictions in Fig. 4.
Fig. 7. Experimental Ramsey fringes for POP clock. (a) absorption method (Pabs); (b) differential method (Pdifference); (c) Faraday rotation angle (Δθ). tp=1ms, tm=0.4ms, T=3.3ms, td=0.25ms, and microwave pulse area θ=π/2.
There is a DC component in the Faraday rotation angle Ramsey fringes, mainly induced by the axial magnetic field. The theoretical calculation and experimental setup are based on the condition that the Wollaston prism is rotated 45° with respect to the incident polarization in the absence of axial magnetic field. In this manner, the laser noise can suppressed, nonetheless a DC component caused by the steady magnetic field reduces the contrast of the Ramsey fringes of the Faraday rotation angle.
4.2 Fluctuation of detection signals
To verify the suppression of laser intensity fluctuation in the differential Faraday rotation angle scheme, the powers of the two laser beams after the Wollaston prism were measured. Without the axial magnetic field, the Wollaston prism was rotated 45° with respect to the incident polarization and the powers of the two laser beams were equal. Then the magnetic field was turned on and the two beams were detected separately by two photodetectors (Thorlabs PDA 100A, PD1 and PD2), whose outputs were recorded by a high-resolution data acquisition card (NI 6281) with a sampling rate of 1 kHz. The Faraday rotation angle and the absorption signal can be calculated with Eq. (7a), Eq. (7b) and Eq. (7c). The probe laser before the physical package was continuous with a power of 250μW. The laser power was not actively stabilized, and its fluctuations mainly originated from the influence of environment on the characters of the laser electronics, optical elements, AOM, and physical package. The absorption signal fluctuates almost synchronously with the fluctuation of ambient temperature in the laboratory, as shown in Fig. 8 (a), and has a period of approximately1.5 hours. Besides periodic fluctuations, the absorption signal has a linear drift of -1.52%/1000s, and a fluctuation of 3.9% at 30000s. The measured linear drift and fluctuation for Faraday rotation angle signal (Fig. 8 (b) red line) are 6.8E-6/1000s and 0.66% at 30000s respectively, which are much smaller than that of absorption method. In the differential Faraday rotation angle method, the linear drift of laser intensity is almost eliminated and the intensity fluctuation is strongly suppressed, that is, the differential Faraday rotation angle signal is insensitive to laser fluctuation, which agrees with the theoretical prediction in section 2.4.
Fig. 8. (a) Laser intensity fluctuations (black line) and temperature of cavity (blue line); (b) Fluctuations of the detection signals in the Faraday rotation angle scheme (red line) and the absorption scheme (black line).
4.3 Suppression of laser intensity noise
In the traditional optical detection mode, amplitude fluctuation of the laser probe is another effect limiting the short-term stability of the POP clock. To estimate its contribution to the clock Allan deviation, the power spectral density of the fractional intensity fluctuations of the probe signal reaching the photodetector ${S_{AM}}(k{f_c})$ [4] should be measured. It contains both the laser relative intensity noise (RIN) transferred at the output of the cell (AM-AM) and the laser frequency noise converted into amplitude fluctuations (PM-AM). While in Faraday rotation angle method, the angular sensitivity is given by $\frac{{\Delta \theta }}{{\sqrt {\Delta f} \times SNR}}$ [25], SNR is the ratio of root-mean-square (RMS) value divided by the standard deviation, $\Delta f$ the measurement bandwidth, and $\Delta \theta$ the Faraday rotation angle. Therefore the laser intensity noise in absorption and Faraday rotation angle methods are hardly comparable.
To test the robustness of the Faraday rotation angle method to laser noise, a proof of principle experiment was conducted. The RIN was artificially degraded by adding a white noise with amplitude of 400 mV to the RF signal, which drives the AOM in the detection phase. In this way, the instability is mainly caused by the laser amplitude noise, and all other contributions to the short-term stability are completely negligible. In this particular condition, we measured the atomic clock frequency stability for absorption and differential Faraday rotation angle scheme, shown in Fig. 9. The short-term stability of the POP atomic clock with the Faraday rotation angle is improved by half an order of magnitude.
Fig. 9. Fractional frequency stability of POP atomic clock with absorption and Faraday rotation angle method. Pp = 14mW, Pd = 0.25mW, tp=1ms, tm=0.4ms, T=3.3ms, td=0.2ms.
The result in Fig. 9 shows that the laser amplitude noise is suppressed but not eliminated in differential Faraday rotation angle method. The main reason is that the CMRR of the two individual commercial photodetectors in our experiment is not sufficiently high for the detection of very small Faraday rotation angle with a high SNR. The measured CMRR of the two individual commercial photodetectors is only 20dB, corresponding to g=0.1. If a photodetector with CMRR of 80dB was developed, according to theoretically predicted SNRFaraday, we estimate the short-term stability of POP atomic clock with differential Faraday rotation angle method to $1.13 \times {10^{ - 13}}/\sqrt \tau $.
However, the above short-term stability is an estimation of the ultimate possible stability according to the SNR. Actually, the clock’s short-term stability is also affected by the phase noise of the local oscillator due to the Dick effect. Moreover, in differential Faraday rotation angle scheme, the shape and amplitude of Ramsey fringes are more sensitive to the static magnetic field B than traditional absorption method. The detuning ${\Delta _ - } = ({\omega _ - } - {\omega _0})$ and ${\Delta _ + } = ({\omega _ + } - {\omega _0})$ of σ- and σ+ in Fig. 3 are equal with opposite signs and increases with B. So, in order to reach the theoretical estimated stability, one has to deal with the noise of local oscillator and the fluctuation of magnetic field.
5. Conclusion
We have presented a differential Faraday rotation angle scheme for the POP frequency standard. The Faraday rotation angle induced by the interrogation microwave is a Ramsey fringe versus microwave detuning that can be used as a frequency reference in an atomic clock. The Ramsey fringes, the sensitivity of clock frequency to laser intensity fluctuation and the signal to noise ratio for absorption, differential and Faraday rotation angle methods are calculated and compared. Theoretical study shows that the sensitivity of clock frequency to laser intensity fluctuation in the differential Faraday rotation angle detection is 10 orders of magnitude smaller than that of the traditional absorption method, and with optimized high CMRR photodetector the clock stability can be further improved. Using a Wollaston prism, rotated 45° with respect to the incident polarization, and two photodetectors, the Ramsey fringes of absorption, differential and the Faraday rotation angle were obtained simultaneously. In the proposed scheme, the long-term Faraday rotation angle fluctuation is 0.66% at 30000s, which is much smaller than fluctuation of traditional absorption signal 3.9% at 30000s. A proof of principle experiment demonstrates that the contribution of laser intensity noise to atomic clock instability was reduced by the differential nature of the Faraday rotation angle scheme. According to the theoretically result, the SNR of Faraday rotation angle scheme depends on the CMRR of photodetector, with CMRR of 80dB, a stability of $1.13 \times {10^{ - 13}}/\sqrt \tau $ can be expected. Further, we intend to develop a high CMRR photodetector and improve the frequency stability in future implementations.
Funding
National Key Research and Development Program of China (2017YFB0503200, 2017YFB0503203).
Acknowledgments
The authors thank S.J. Li and J. He for fruitful discussions regarding the theory model.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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