Light shift of Ramsey coherent population trapping resonance using drive current modulation

We investigated the light shift of Ramsey coherent population trapping (CPT) resonance by drive current modulation (DCM), which can be realized with low power consumption in a small system, to improve the performance of a CPT atomic clock. The light shift makes the greatest contribution to the degradation of long-term frequency stability of a CPT atomic clock. We also devised digitizer-based DCM to reduce the uncertainty in the frequency of light shift due to wavelength variation at the time of observation. The results show that light shift with DCM is reduced compared with cw excitation and is consistent with theoretical calculations. The estimated frequency uncertainty of the light shift due to wavelength variation using the proposed method is 2.1 × 10−13, which is approximately 165 times better than that of the conventional method without a digitizer, thus demonstrating the effectiveness of the proposed method for future small atomic clocks.


Introduction
Frequency standards play an important role in supporting critical infrastructure for information and communication systems, such as positioning, network synchronization, satellite constellations and long-term data acquisition. 1,2)he frequency references in these systems necessitate both high accuracy and stability in addition to properties such as small size, low weight and low power consumption (low-SWaP).It is expected that the chip-scale atomic clock (CSAC) could be used as a low-SWaP frequency reference for these systems.The CSAC is an atomic clock whose frequency is stabilized to a microwave transition of an alkali atom using coherent population trapping (CPT) resonance.Because the microwave transition can be easily detected through laser light using the CPT phenomenon it does not require a microwave cavity and is suitable for miniaturization.In 2004, a millimeter-sized physics package was demonstrated at the research and development stage, 3) and it was commercialized in 2011. 4)Zhang et al. achieved a frequency stability of 2.2 × 10 −11 at an averaging time of 10 5 s with a volume of 15 cm 3 and a power consumption of 59.9 mW. 5) Atomic clocks must possess high long-term frequency stability.][11][12][13][14][15][16] However, these methods involve a trade-off between short-term stability and a reduction in the rate of the light shift, and they cannot simultaneously improve both quantities.][19][20][21][22][23][24][25][26][27][28][29] In this method, the CPT resonance is detected by irradiating the atom with two laser pulses that are separated by a time interval.][32] This method requires an external optical modulator, such as an acousto-optic modulator, to generate the laser pulse.However, the large size and power consumption of an acousto-optic modulator limit its use in CSACs, wherein a low-SWaP is required.
With the aim of realizing the Ramsey CPT method with low-SWaP, we previously proposed the drive current modulation (DCM) method, wherein the laser drive current is pulsed in a two-stage waveform. 33)The DCM method possesses the ability to detect Ramsey CPT resonances by the addition of a simple two-stage current generation circuit.Also, because no additional optical elements are required, DCM can be seamlessly integrated in a CSAC without any changes to the optical setup.Our previous study confirmed that Ramsey CPT resonances can be obtained using the DCM method. 33)It was shown that this suppresses the power broadening effect and produces narrow-linewidth Ramsey CPT resonances.However, evaluation of the light shift using DCM was abandoned because the measurement results varied significantly and reproducibility could not be obtained.We assumed that this variation was caused by laser wavelength fluctuations due to injection current modulation during DCM.Moreover, the frequency uncertainty of the light shift caused by wavelength fluctuations at the time of observation is large.Therefore, further investigation regarding the light shift with DCM was required to reveal the correctness of this assumption.
In this study, we first investigated the light shift with respect to wavelength variation to evaluate the relationship between frequency uncertainty and wavelength variation.The wavelength variation was also measured to estimate the frequency uncertainty.The results show that wavelength variation is the dominant factor in DCM.Next, we devised a pulse excitation method based on DCM using a digitizer to suppress the impact of wavelength variation during observation.We investigated the light shift of Ramsey CPT resonance using the proposed method, compared the measured results with theoretical calculations and confirmed the consistency of the results for several free evolution times.Finally, we demonstrated that the proposed method can significantly improve the uncertainty of the light shift compared with the conventional method.The results indicate that the DCM method without external modulators is an effective approach for observing Ramsey CPT resonances suitable for CSACs.

Experimental setup
Figure 1 depicts the experimental setup of the proposed method.Pulse excitation using DCM has the same optical setup as cw excitation without any dark time, which is used in CSACs.The gas cell is a glass cell containing 133 Cs and 4 kPa of Ne, and its temperature was stabilized at 42 °C.The gas cell is cylindrical and has a diameter and optical path length of 20.0 mm and 22.5 mm, respectively.A static magnetic field was applied with a Helmholtz coil to obtain a clock transition that is relatively less sensitive to magnetic fields by lifting Zeeman degeneracy.The gas cell and Helmholtz coil were covered with a magnetic shield to prevent the undesired effect of the external magnetic field.The laser diameter was set to 2 mm and was linearly polarized in front of the gas cell parallel to the static magnetic field.An 894.6 nm single-mode verticalcavity surface-emitting laser (VCSEL) was used as the laser source, and the temperature of the VCSEL was stabilized at 38 °C using a thermo-electric cooler and its controller.The bichromatic laser light for observing CPT resonances was generated by injecting a RF signal of 4.6 GHz, which was the half-frequency of the microwave transition between two ground states of the Cs atom, into the VCSEL through a bias tee.The RF power was set to maximize the amplitude of the CPT resonance with cw excitation.To obtain large CPT amplitudes, the laser wavelength was stabilized to the wavelength that excites F′ = 3 of the Cs atom.The light transmitted through the gas cell was converted to an electrical signal by a photodiode.The signal was connected to a lock-in amplifier and digitizer.To stabilize the wavelength of the laser, the drive current of the VCSEL was modulated using a sinusoidal waveform.The error signal, derived from the absorption line and demodulated from the signal of the photodiode via the lock-in amplifier, was used to control the output wavelength of the VCSEL, ensuring it aligned with the center of the absorption line.A digitizer was used to acquire the Ramsey CPT resonance signal, and its sampling rate was 1 gigasample per second (Gsps).
Figure 2(a) shows the two-stage drive current waveform of DCM.By applying a two-stage drive current, the output wavelength of the VCSEL rapidly approaches the absorption wavelength at which a target absorption line is obtained.The two-stage drive current waveform sets five parameters: two current values I 1 and I , 2 their periods t 1 and t 2 and the period T set during which no current is applied.Figure 2(b) shows the output laser intensity waveform driven by the two-stage current to the VCSEL.The laser intensity has a first-order lag response because of an inductor included in a bias tee.Further, at the rise of the output light dead times occur because considerable time is required from the start of laser current injection to exceed the threshold current.Thus, in DCM there is a difference between the timing of injection current into the VCSEL and the light intensity irradiated to the atoms.Figure 2(b) illustrates the relationship of the free evolution time T off during which the laser pulse does not irradiate atoms, the observation time t m and the time duration for turn-off of the laser light t .
fall Although the free evolution time T off and the observation time t m are consistent with the definitions of the conventional method, 34) to consider the effect of the first-order delay response when the current is turned off we define t fall as the time from the end of t 2 until   the laser light intensity becomes less than 10%.Here, T off indicates the time when the laser light is not irradiated and is defined as the time interval between the end of t fall and the beginning of t .
m Further, t m is defined as the time interval between the laser light incidence and Ramsey CPT signal acquisition, starting from the point where the maximum light intensity is 10%.
The two-stage drive current was generated by a digital to analog (D/A) converter that was controlled by a field programmable gate array (FPGA).The FPGA can set the five parameters (I , 1 I , 2 t , 1 t 2 , and T set ).To obtain the Ramsey CPT resonance, it is necessary to detect the transmitted light intensity immediately after the pulse rise and to set t m to about several tens of microseconds.In this study, I 1 was set to 4.00 mA and t 1 to 112 μs; as a result, the wavelength coincided with the target absorption peak at t m = 85 μs.These parameter values were determined due to the limitation of the response speed of the photodetector.Further, I 2 was set to 3.5 mA, and this value yields a wavelength that continuously excited the target absorption line.The period t 2 for outputting I 2 was set to 6.9 ms, which is sufficient, considering the transition rate of the CPT state, to prepare an adequate number of atoms in the CPT state at the falling edge of the pulse.By repeating this sequence periodically, the signal of Ramsey CPT resonance was acquired.The output laser intensities corresponding to the current values of I 1 and I 2 under this condition were 3.85 and 3.03 mW cm −2 , respectively.

Dependence on observation timing of light shift uncertainty of Ramsey CPT resonance acquisition and improvement by the proposed method
In this section we discuss the light shift and its uncertainty arising from variations in laser wavelength due to DCM, and propose a method to reduce the uncertainty of the light shift.In conventional Ramsey CPT resonance acquisition the observation time t m is fixed [Fig.][35] This is the same as in Ramsey CPT resonance acquisition using a sample-and-hold circuit.To acquire Ramsey CPT resonances periodically under the same conditions, signals at the same wavelength must be acquired in each sequence.For example, during the acquisition of Ramsey CPT resonance using DCM, the Ramsey CPT signal is detected when the laser wavelength matches the absorption wavelength.However, variations in laser wavelength are slightly affected in the direction of the time axis by electrical noise and temperature variation.is acquired, and this affects the light shift of the Ramsey CPT resonance for each sequence.Thus, the frequency uncertainty is expected to increase.
To determine the frequency uncertainty of the light shift when using the sample-and-hold circuit, we measured the light shift versus t offset and the variation of t .
p Figure 4 shows a graph of the light shift versus t .offset The light shift has a maximum value at t = 0 offset and a decreasing trend away from t = 0.
offset This feature may be related to the laser sidebands, observation time t m and free evolution time.However, this trend is not entirely clear, and further investigation is necessary.Further, Fig. 5 shows the variation of t .
p Here, I 1 was controlled to ensure that the t p was within 60 ns.This fluctuation range is limited by the minimum resolution of the D/A converter used in this experiment.Because the standard deviation of t p is 29 ns, and  p Upon deriving the frequency uncertainty when measuring with constant observation time t m from the uncertainty of t offset from Figs. 4 and 5, we obtained a frequency uncertainty of 0.30 Hz, which is 3.3 × 10 −11 of the fractional frequency.Thus, the frequency uncertainty is limited to the 10 −11 level via the conventional method with DCM because of the variation in t .p Figure 6 illustrates the Ramsey CPT observation method that is not affected by the variation of t ; p it uses a digitizer that can sample the signal with high speed at the specified acquisition window aimed at overcoming the τ p issue.In the proposed method, the digitizer is used to acquire the transmitted light near t , p and the signal at t p is selected from the data set.Because the proposed method always obtains a signal with t @ 0, offset it can reduce the frequency uncertainty in t offset caused by the variation of t .
p In the proposed method, the uncertainty of t offset becomes the uncertainty of the sampling rate of the digitizer.Because the sampling rate of the digitizer used in this experiment is 1 Gsps, the time resolution is 1 ns.As shown in Fig. 4, the frequency uncertainty corresponding to a t offset uncertainty of 1 ns is 0.940 mHz, and this fractional frequency uncertainty is ´-1.02 10 . 13Regarding the frequency uncertainty caused by uncertainty of the laser intensity, the sensitivity of the frequency to fluctuations in the laser intensity when using the auto-balanced Ramsey method was reported to be −2.9 × 10 −15 μW −1 . 36)This value is smaller than that of the frequency uncertainty by more than one order of magnitude, and this may be attributed to the previous variation in t .p Thus, the uncertainty of the light shift with conventional DCM can be considered to be dominated by the uncertainty of t .offset Therefore, the proposed method can significantly reduce the uncertainty in the variation of t .p Figure 7 shows Ramsey CPT fringes for both sample-andhold-based and digitizer-based DCM methods.In the sampleand-hold-based approach, the fringe shape deteriorates as one moves outward from the center frequency.By contrast, in the digitizer-based method, the fringe shape remains consistent even when deviating from the center frequency.The asymmetry of the Ramsey CPT fringe is attributed to the imbalance of the laser sidebands caused by RF modulation of the VCSEL.

Evaluation of the light shift of Ramsey CPT by DCM
In this section we measure the light shift due to laser intensity of the pulsed excitation by DCM using the observation method proposed in Sect. 3 and verify the effect of the proposed method on light shift reduction.Figure 8(a) shows the results of light shifts by cw excitation and DCM with different free evolution times T off of 200 ms, 600 ms and 1000 ms.The free evolution time T off at which the figure of merit (FoM) of short-term frequency stability reached its maximum was 600 ms. 33)The horizontal axis indicates the laser intensity at t .
m The light shift of the proposed method is significantly smaller than that of cw excitation, and becomes smaller as T off increases.The light shift coefficient at m = T 1000 s off is 6.842 Hz (mW cm −2 ) −1 , which is 53 times smaller than the cw excitation light shift coefficient of 350.3 Hz (mW cm −2 ) −1 .
Next, we compare the experimental results with the theoretical analysis.The light shift S pl of pulsed excitation is expressed as follows: 35) t where S cw is the light shift of the cw excitation, T is the free evolution time and t pump is the pumping rate, which is proportional to the power broadening of the cw excitation.
The free evolution time is the period of time during which  Fig. 6.Relationship between data acquisition timing and wavelength variation for the method of selecting absorption line peaks using a digitizer.This new method tracks the absorption peak, and thus t m becomes variable.light is not irradiated to the atom (i.e.light does not interact with the atom).Although the light is irradiated to the atom for t fall and t m in the DCM method, the impact of laser light is extremely small because the laser wavelength of the laser light is far from the absorption wavelength.Therefore, t fall and t m could be included in T. We calculated the effective free evolution time T eff of the DCM as follows: ´-[ ( )] P 359 s mW cm .pump 2 opt 1 Here, P opt represents the laser intensity.The calculated = T T eff is in better agreement with the experimental results than the calculated T = T off curve, the difference being within 10%.
Figure 9 shows the light shift coefficients as a function of free evolution time T. The markers represent experiment results, and the dashed line shows the theoretical curve derived from Eq. eff Since t t + m fall is constant, the ratio of T off to T eff decreases as T off increases; as a result, the blue circles approach the theoretical curve.Finally, we consider the effect of the free evolution time on the light shift with respect to variation of t .
p Because t t @ m p in the proposed method, t m always varies in sequence.The variation in t m causes a subsequent variation in the free evolution time T , eff assuming t t @ .m p Because the atoms are irradiated by the laser at different free evolution times, the light shift varies accordingly.This variation in the light shift can be calculated using Eq. ( 1) as = T T .eff At m = T 600 s, off when the FoM is maximum, the uncertainty in t p is calculated as 29 ns. Figure 5 indicates that its frequency uncertainty is about 2.08 mHz, and this fractional frequency uncertainty is ´-2.1 10 . 13Therefore, because this frequency uncertainty is approximately twice as large as the frequency uncertainty due to the uncertainty in t offset shown in Sect.3, the frequency uncertainty caused by the variation in t p is the dominant factor in this experiment.The frequency uncertainty due to the fluctuation of the free evolution time could be improved by changing the measurement conditions.For example, the frequency uncertainty could be reduced by adjusting and optimizing the RF power injected to the VCSEL to reduce the light shift coefficient.Further, if we use a D/A converter with high resolution, the variation in t p may be decreased, and it may only cause low-frequency fluctuations.The variation in T eff also induces AM of the Ramsey CPT fringes.This AM effect is expected to deteriorate short-term stability due to the amplification of phase noise.On the other hand, since the integral of AM becomes zero, this effect is considered to be negligible over the long term.
Comparing the light shift of the proposed method with that of the conventional method, it can be inferred that the frequency uncertainty is reduced by a factor of 165 compared with the conventional method.Although the light shift as a function of t offset requires further investigation because the characteristics are affected by laser sidebands, the observation time t , m free evolution time T and so on, it is clear that the proposed method is more effective than the conventional method in reducing the frequency uncertainty due to wavelength variation.

Conclusions
In this study, we proposed a Ramsey CPT observation method using a digitizer in pulsed excitation with DCM to reduce the frequency uncertainty caused by wavelength variation during observation.We further investigated the light shift of the Ramsey CPT resonance.We estimated the frequency uncertainty due to wavelength variation of the conventional and proposed methods, and showed that the frequency uncertainty for the proposed method is 2.0 × 10 −13 , which is 165 times better than for the conventional method.We measured the light shift of the proposed method and demonstrated the light shift characteristics of pulsed excitation using the DCM.The results indicate that the light shift coefficient is reduced by 53 times compared with cw excitation.These experimental results were compared with theoretical calculations and confirmed to be consistent with theory.Although the light shift with respect to laser wavelength requires further investigation, the proposed method is suitable for resonance acquisition of pulsed excitation using DCM.Because DCM can only be realized with electronic circuits, Ramsey CPT can easily be applied to small atomic clocks such as miniature atomic clocks and CSACs without a large increase in volume or power consumption, and could potentially improve the frequency stability of small atomic clocks in the future.In future work, we plan to advance studies on improving frequency stability using the proposed method.

Fig. 1 .
Fig. 1.Experimental setup to observe Ramsey CPT resonance using DCM.AM, amplitude modulation; FPGA, field programmable gate array; VCSEL, vertical-cavity surface-emitting laser; PC, personal computer; RF, radio frequency; D/A converter, digital to analog converter; ND Filter, neutral density filter.

Fig. 2 .
Fig. 2. (a) Two-stage drive current waveform using DCM.(b) Output intensity waveform from a VCSEL driven by a two-stage drive current.

Figure 3 (
b) shows the temporal variation of the absorption lines.In repetitive Ramsey pulse sequences, acquisition at the same time will lead to the detection of Ramsey CPT resonance signals at different wavelengths.Hence, the time t p at which the absorption peak is observed in each sequence is different.It is known that the light shift depends on the laser wavelength.If observations are performed at fixed t , m the Ramsey CPT resonance at the difference between t m and t ,

Fig. 3 .Fig. 4 .
Fig. 3. (a) Relationship between data acquisition timing and wavelength variation when the method that acquires Ramsey CPT signals from absorption lines uses a fixed t .m (b) Temporal variation of absorption lines over 20 min.

Fig. 5 .
Fig. 5. Time fluctuation of t .p t p is stabilized at 85 ms controlled by I .

Fig. 7 .
Fig. 7. Comparison of Ramsey CPT fringes acquired by the sample-andhold method and the proposed method.

Figure 8 (
Figure 8(b) shows the experimental results and the curves calculated using Eq.(1) with = T T off and = T T .eff In this calculation, / t m =´-[ ( )] P 359 s mW cm .

2
(1).The circles and triangles indicate the measured light shift coefficient assuming = T T off and = T T , eff respectively.The black square at = T 0 indicates the light shift coefficient for cw excitation, and the dashed line indicates the first derivative of Eq. (1From the comparison of T off and T , eff the triangular marker indicates that T eff is close to the theoretical curve.The relative errors from the theoretical value at m m m = T 200 s, 600 s, and 1000 s off are 53%, 21% and 15%, respectively, and the circles indicating T off approach the theoretical curve as T off increases.These values correspond to the ratio of T off to T .

Fig. 8 .Fig. 9 . 2 5 ©
Fig. 8.Light shift versus laser intensity.(a) Comparison of the light shift between cw excitation and DCM.Markers indicate experimental results and dash-dot lines indicate a linear approximation.(b) Comparison of experimental results and theoretical curves.Markers indicate experimental results, and dashed lines indicate curves obtained from Eq. (1) as = T T off and solid lines as = T T .eff Experimental parameters are t m = 85 s,m