A Simplified Laser System for Atom Interferometry Based on a Free-Space EOM

Abstract

In this paper, a compact laser system for ⁸⁷Rb atom interferometry based on only one free-space electro-optic modulator (EOM) was realized, where repumping and Raman beams were generated with a free-space EOM. In addition, this laser system does not require a laser amplifier compared to fibered EOM since fibered EOM cannot transmit high-power lasers. However, due to the narrow modulation linewidth of free-space EOM, it is impossible to obtain the frequencies of repumping and Raman beams separately, which would lead to some complicated effects. Therefore, a theoretical analysis was carried out to solve this problem, and a new frequency scheme for AI is proposed. For the experiment, the laser system of AI was built up. Moreover, the atomic interference fringes were obtained with a contrast of 20.7% (T = 60 ms) and the fitted phase resolution is approximately 1.25 mrad. The presented laser system could provide a new solution for compact AI systems in the future.

1. Introduction

Atom interferometers (AIs) have a wide range of applications, including measuring gravity [1,2,3], rotation [4,5], gravity gradient [6,7], fundamental constants of physics [8,9], and testing the weak equivalence principle [10,11]. There are many inherent advantages of AIs [12,13,14]: they have incredibly high sensitivities and their intrinsic calibration leads to excellent long-term stability. Therefore, AIs have been the ideal candidates for experiments of precision measurement.

Usually, the laser systems of AIs are complicated since many laser frequencies need to be generated. For instance, more than three lasers are typically used to produce cooling, repumping, detection, Raman laser, etc. Moreover, the optical path system of AIs is subject to drifts or vibrations [15]. To date, many laser systems have been proposed to solve these problems. AIs frequently adopt electro-optic modulators (EOMs) to modulate frequencies, which reduce the number of lasers and the complexity of the optical path system [16,17]. The frequencies of repumping and Raman beams could be yielded by modulation based on EOMs so that the laser system could be simplified and the optical path could be miniaturized. Furthermore, the stability of the laser system can be improved. As an example of AI-based on the ⁸⁷Rb atom, a fibered EOM was adopted in the laser system of AIs. A laser with a wavelength of 1560 nm was modulated by a fibered EOM and then frequency-doubled to 780 nm [18,19]. For the sake of simplicity, the EOM at the wavelength of 780 nm could be an option. However, the fibered EOM could not transmit the high-power laser [20,21]. The free-space EOM could withstand the high-power transmission, however, it has a narrow modulation bandwidth, which means that two free-space EOMs should be implemented to obtain repumping beams (modulation at the frequency of 6.57 GHz) and Raman beams (modulation at the frequency of 6.83 GHz). A laser system with two EOMs would have problems such as greater power consumption and more complex optical paths.

In this paper, a simplified laser system of AI was proposed based on one free-space EOM of 780 nm. The modulation realized the repumping and Raman beams with one free-space EOM. Theoretically, the analysis of this scheme was carried out. The influence of the Raman transition and single-photon transition on the experiments of the AI was analyzed. As for the experiments, cold atom interference fringes were obtained, and the contrast was 20.7%.

2. The Laser System of Atom Interferometer

The principle of AI is the interference of cold atom matter waves. The cold atom is prepared by using the technology of laser cooling and trapping. For a detailed description, please refer to ref. [22]. Usually, the laser system of AI needs four kinds of laser beams, including the cooling beam, repumping beams, Raman beams, and detection beam (blow beam). The design of the frequency scheme and the optical path scheme is required. In our experiments, the frequency of the laser is referred to as the D2 line of the rubidium atom. Figure 1a shows the modulation frequencies of different beams at different stages, which require the frequency modulation of the AOM and frequency shifting of the main laser to complete. The repumping beams contain repumping 1 beam and repumping 2 beam because the repumping beams are generated by modulation with only one EOM at a depth of 33%.

Figure 1b presents the designed frequency scheme. Two lasers are utilized for the laser system of AI. The reference laser is locked to the transitions of ⁸⁵Rb $F_g=3\to F_e=4$ via polarization spectroscopy, which has a relatively large signal in the spectrum. The frequency of the main laser is locked via the beat note with the reference laser, and the frequency difference between the two lasers can be controlled by means of a frequency–voltage–convert (FVC) (frequency difference adjustment range 1.1–2 GHz) [16]. The main laser generates the frequencies required for the AI system. Firstly, scanning the voltage applied to the piezo actuator of the reference laser could linearly change the frequency of the laser so that the spectroscopy [23] could be obtained. Then, the frequency of the reference laser is locked. The specific frequency modulation is shown in Figure 1b. The main laser is locked at 92 MHz below the ⁸⁷Rb $F_g=2\to F_e=3$ transition line. The frequency shift of the main laser is achieved by controlling the frequency difference between the main and reference lasers, and the adjustment of the frequency difference can be achieved using the FVC module. Therefore, we adjust the output frequency of the main laser.

Figure 1c shows the diagram of our optical path. The generation of different beams in the optical path is accompanied by the modulation of AOMs. Firstly, the cooling beam is generated through two AOMs; however, it passes through the AOM2 (the modulation frequency is −87.3 MHz) as a zero-order diffraction. There is no effect on the cooling beam, and then it is modulated by AOM3 (the modulation frequency is +80 MHz). Secondly, Raman beams are generated by two AOMs (AOM2, AOM4) and one EOM modulating, where the modulation frequency of two AOMs is −87.3 MHz, and the modulation frequency of the EOM is 6.834 GHz (the bandwidth is approximately 5 MHz). Thirdly, the frequencies of the detection and blow beams were generated by modulation with AOM1 (the modulation frequency is +92 MHz). Finally, repumping beams are special in the optical path, because they are separated from Raman beams and then coupled into the cooling beam. The photograph of the optical path is shown in Figure 1d.

3. Theoretical Analysis of the Cooling Caused by the Laser System

In universal AIs, the frequencies of repumping beams are in resonance with the $F_g=1\to F_e=1$ or the $F_g=2\to F_e=1$ transitions of the ⁸⁷Rb D2 line. However, in our scheme, the frequencies of repumping beams are resonate with the $F_g=1\to F_e=2$ and $F_g=2\to F_e=2$ transition of the ⁸⁷Rb D2 line, with an intensity ratio of approximately 1:2 between two beams. The frequencies of the repumping beams change with the frequency of the main laser (${\delta }_c$), which will cause the frequencies of repumping beams to change as well during the polarization-gradient cooling (PGC) stage. Therefore, the action of repumping beams on atoms causes atomic Raman transition, which can complicate the cooling stage of atoms. The theoretical analysis of the impact was carried out. Basically, the detuning and intensity of the repumping beams may affect the process of laser cooling. We started with the atomic transition rate and damping force, and finally obtained the optimal scheme based on the experimental results.

In our experiments, the power of the cooling beam is 60 mW, the power of the repumping beams is 4 mW in the cooling process, and the beams diameter of both is 0.9 cm. Subsequently, we calculated the photon scattering rates and damping coefficients for different atomic states, so that the effect of the beams on the atoms process and the effect on their motion can be clarified during the cooling. The photon scattering rate of could be written as [24,25]:

$$W_{ai}=\frac{1}{2}\frac{s}{1+s+{(2\delta /\mathsf{\Gamma })}^2},$$

(1)

where $\delta ={\omega }_{laser}-{\omega }_0+{\delta }_c$ are the detuning of the laser, ${\omega }_{laser}$ is the laser frequency (the frequency of the cooling beam is 12 MHz below the $F_g=2\to F_e=3$ transition of ⁸⁷Rb; the repumping beams are approximately resonant with the $F_g=1\to F_e=2$ and $F_g=2\to F_e=2$ transition of ⁸⁷Rb), and ${\omega }_0$ is the atomic energy levels, ${\delta }_c$ is the frequency shift of the main laser. $\mathsf{\Gamma }=2\pi \mathsf{\Gamma }{}_v$ is the natural linewidth. $s=I/I_{\mathrm{Sat}}$ is the saturation parameter of each of the two beams that compose the standing wave. $I=(1/2)c{\epsilon }_0{\left|E\right|}^2$ is the laser intensity (where c is the speed of light, ${\epsilon }_0$ is permittivity of vacuum, and E is the electric intensity). $I_{\mathrm{Sat}}=1.67\mathrm{mW}/\mathrm{c}{\mathrm{m}}^2$ represents the saturation intensity of the two-level transition.

The photon scattering rate increases in the direction opposite to the atomic velocity, which produces a damping force over a large number of photon recoils [22]. Theoretically, this dissipative force should slow the atoms to zero velocity. The force can be expressed as $F=-\beta v$, where v is the velocity of atoms, and $\beta$ is the damping coefficient which can be given by (a cooling beam is a standing wave):

$$\beta =-\hslash {\left|\overrightarrow{k}\right|}^2\frac{8s(\delta /\mathsf{\Gamma })}{{(1+{(2\delta /\mathsf{\Gamma })}^2)}^2},$$

(2)

where $\overrightarrow{k}$ is the wave vector of a laser.

Furthermore, we calculated the Raman transition rates at different frequency shifts during the cooling process to illustrate the Raman transition of atoms caused by repumping beams. The Raman transition rate was calculated by the following formula:

$$W_{Raman}\approx {\mathsf{\Omega }}_{eff}=\frac{{\mathsf{\Omega }}_{a0i}{\mathsf{\Omega }}_{b0i}}{2\delta +i\mathsf{\Gamma }},$$

(3)

where ${\mathsf{\Omega }}_{a0i}=2〈J=1/2∥er∥J^{\prime }=3/2〉·E{\gamma }_{ai}/\hslash$ is the Rabi frequency corresponding to the magnon-energy level of the ground state. ℏ is the reduced Planck constant, and ${\gamma }_{ai}$ is the dipole matrix elements for the ${\sigma }^{+}$ transition (please refer to the ref. [26]).

The photon scattering rate of atoms under different $\delta$ is shown in Figure 2a. Normally, the spontaneous emission rate of atoms is proportional to the scattering rate, so the big peaks should be avoided when choosing frequency shifts. The spontaneous emission rate will affect the atomic temperature and the experimental results [27,28]. In addition, the Raman transition rate is also very high at the peak of the scattering rate in Figure 2b. A high Raman transition rate will also heat up the atoms. However, when the Raman transition rate is proper, it can make atoms continuously be pumped from the state of $F_g=2$ to $F_g=1$. Figure 2c indicates the relationship between the damping coefficient and the frequency shift. When the damping force is negative, the damping force can reduce the velocity of atoms and then cool the atom down. Therefore, we set the frequency shift ($0<{\delta }_c<5\mathrm{MHz}$) in the stage of Doppler cooling, where the frequency of the cooling beam is approximately red detuned ($2{\mathsf{\Gamma }}_v$) to the atomic resonant transition of $F_g=2\to F_e=3$. When the frequency shift is close to 154.8 MHz, the scattering damping coefficient reaches the minimum value, and the damping force will accelerate atoms.

Finally, we studied the effect of different intensity ratios of the repumping beams and cooling beam in the atomic cooling process. The results are shown in Figure 2d. Different frequency shifts are set up. It was found that there is a negative damping coefficient with the increase in intensity of repumping beams. This will cause no damping effect, so the intensity of repumping beams cannot be too large. When the frequency shift is small, the damping coefficient is positive. A low intensity of repumping beams can increase the number of $F=2$ atoms, which results in a large amplitude of the detection signal. However, the Raman transition rate will increase with the power of repumping beams, and it will directly bring about a very complicated heating mechanism. This is because all six cooling beams are combined with repumping beams in the cooling stage. Due to the fixed modulation frequency of 6.834 GHz of the EOM, the two repumping beams would cause the process of Raman transition. The imparted recoil momentums in the three perpendicular directions would produce the extension heating and increase the atom temperature. In addition, it is theoretically difficult to calculate an optimal intensity ratio, so the subsequent experiments were carried out to select the optimal intensity ratio of repumping beams and cooling beam.

4. The Experiment and Results

4.1. The Optimization of Laser System Configurations in the PGC Process

The experiment was carried out in a three-dimensional magneto-optical trap (MOT). The number of loaded atoms is approximately $2.58\times {10}^7$ during 798 ms in the MOT. Then, the process of PGC is applied to further cool the atoms, and it takes 2 ms for the process of PGC. Then, the cold atoms free fall and the Raman pulses were applied. In the end, the atomic fluorescence was collected by the method of time of flight (TOF).

Figure 3a shows that frequency shift affects the atomic temperature in the PGC process. When the frequency shift is 154.67 MHz, the temperature of the atoms is close to the maximum. The temperature of the atoms is proportional to the diffusivity. At this frequency shift, the reason for the high atomic velocity is that the damping force of beams on the atom causes the atom to accelerate rather than decelerate. Additionally, the Raman transition rate is close to the maximum at this time (as shown in Figure 2), which will bring about a great heating mechanism to the atoms. As the frequency shift is close to 154.8 MHz, the detection signals gradually become smaller and almost disappear.

These data verify the results of the theoretical calculations as well. Furthermore, it is discovered that the detection signals are relatively strong, and the atomic temperature is 25 μK at the frequency shift of 142 MHz. Because the damping coefficient is small in the PGC process, the scattering force has an effect on the high-speed atoms. Moreover, the intensity of Raman transition becomes larger, and the number of $F_g=2$ atoms increases with the process of the Raman transition since the atoms are repumped in the state of $F_g=2$ from $F_g=1$. Therefore, the laser frequency shift of 142 MHz is selected, which can improve the signal-to-noise ratio (SNR) of the subsequent atomic interference fringes. However, the existence of the Raman transition causes the atom to heat up, which is the reason for which the atomic temperature cannot be lowered.

In order to obtain the optimal detection signal, the intensity ratio of the repumping beams and the cooling beam is changed, and the experimental results are shown in Figure 3b. It was found that the amplitude of detection signal is high, and that the atomic temperature is 21.23 μK when the beam intensity ratio is 0.016. Because the intensity of the repumping beams is less at this point, the Raman transition rate of the atoms is lower and does not give the atoms a strong heating mechanism. However, when the intensity of the repumping beam is excessively small, the atom detection signal almost disappears, indicating that the number of atoms in the $F_g=2$ state decreases. Therefore, a certain intensity of repumping beams is required for cooling. Finally, the power of the cooling beam and repumping beams is, respectively, optimized to be 60 mW and 1 mW.

4.2. The Atomic Interference Fringes

Figure 4a shows a schematic diagram of the experimental apparatus. In the experiment, the design of the AI is based on the Ramsey-bordé atom interferometry where the sequence of two co-propagating Raman pulses ($\mathsf{\pi }/2-\mathsf{\pi }/2$) is adopted. In addition, the duration of the Raman pulse is 10 μs and the time interval of T between two pulses is 60 ms. The laser frequency shift (δ_c) is 727.6 MHz when the Raman beams are applied. Subsequently, the atomic interference fringes are obtained by scanning the frequency of Raman beams, which can be shown in Figure 4b. By fitting the fringes with a sine function, the contrast is estimated to be 20.7%, and the phase resolution is approximately 1.25 mrad. The SNR of the interference fringes is approximately 25.

4.3. The Improvement of This Laser System

In the future, the laser system could be further simplified to show its advantages and feasibility. The improved scheme is shown in Figure 5. Only one single laser could be used to build up the laser system of AI. Because a free-space EOM is used to produce laser beams with different frequencies, only the frequency locking of the laser needs to be considered. The optical power required by the frequency-locking optical path is lower, and most of the laser power can be distributed to different beams. The laser power is divided into two parts. One beam could be approximately 30 mW and modulated by the FEOM (NIR-MOX800-LN-10) to generate a first-order sideband. Then, this sideband is locked to the cross peak of (D2 line) ⁸⁵Rb $F_g=3\to F_e=4$.

In addition, the laser frequency could be tuned by changing the modulated frequency of the FEOM. The main laser power of 1 W is used to generate the four different beams through AOM and a free-space EOM. This solution uses only one laser to generate the beams required for AI, which greatly simplifies the optical path system. In addition, there is no need to introduce a frequency doubling and an erbium-doped optical fiber amplifier (EDFA), which could improve the stability of the laser system. The feasibility of this laser system is currently the subject of our ongoing work.

5. Conclusions

First of all, we present a simplified laser system for AIs, which may solve the problem of separated modulation for repumping beams and Raman beams based on a free-space EOM. Secondly, we theoretically calculate the alteration of frequency shifts in the Raman transition rate and single-photon transition rate in this laser system. Then, the frequency shifts are optimized to be 142 MHz in the PGC process, and the intensity ratio of the repumping beams and cooling beam is set to 0.016 to obtain the optimal detection signals. Additionally, the contrast of 20.7% is obtained for the atomic interference fringes ($T=60$ ms). The fitted phase resolution is 1.25 mrad. The SNR of the interference fringes is approximately 25. This gives experimental and theoretical support for subsequent applications based on this scheme. In addition, the scheme simplifies the optical path system, which will play an essential role in the miniaturization of future systems.