The Optimization of Cold Rubidium Atom Two-Photon Transition Excitation with an Erbium-Fiber Optical Frequency Comb

: We demonstrated the observation of cold rubidium atom two-photon transition excitation by a ﬁber optical frequency comb. In addition to this, we optimized the repetition rate of optical frequency comb to enhance two-photon intensity by controlling cavity length and pump source of optical comb. This technique can ﬁne tune repetition rate to corresponding stepwise two-photon transition resonance frequency and improve the transition intensity by three times. This method is useful in Doppler laser cooling and detection of macromolecules.

natural advantages of Erbium (Er) optical frequency make the whole scheme more simplified and robust. In order to implement accuracy and stability of the system, we eliminate the Dopplerbroadening, which is the main line broadening. The laser cooling technologies, such as magneto optical trap [4] and molasses, can eliminate most of the Doppler-broadening background. However, the number of atoms in the magneto-optical trap is much lower than the number of atoms in a normal gas cell.. Cold Rb TPT transition intensity is weak for establishing frequency standard or Doppler laser cooling. Therefore, following the previous research [5,6], we try to optimize repetition frequency of optical frequency comb to achieve higher transition intensity with the same average probe laser power. As the result of measurement, transition intensity is improved threefold.

Principle
Specific theoretical analysis is as follows. Each mode of optical frequency comb can be expressed as 0 n r e p F f nf = + (1) where f0 is the carrier-envelop offset frequency, frep is the pulse repetition rate, and n is an integer on the order of 10 6 . The sum frequency of two comb modes is given by Whenever the sum frequency of two modes coincides transition frequency, the optical comb excites the two-photon transition. A hundred thousand pair modes can satisfy the resonant condition at the same time. When two comb modes are near resonance with the 5S-5P and 5P-5D transitions, the intermediate 5P state enhances stepwise two-photon transitions (S-TPT). We utilize this phenomenon to enhance the two-photon transition signal intensity.
Due to the negligible Doppler broadening in cold atom experiments, the S-TPT excitation conditions are more critical. S-TPT can be achieved only when the 5S-5P and 5P-5D transition frequency difference is exactly an integral multiple of the repetition frequency,. If frep deviates (f776nm-f780nm)/n, no S-TPT can be excited, because no other comb excites another single-photon transition when one comb frequency is aligned with a single-photon transition. To validate this feature, we consider all 14 S-TPT transition pathways for 5S1/2-5P3/2-5D5/2 of 87 Rb. 14 S-TPT pathways and the TPT intensity of each TPT pathway are shown in Table 1. Table 1. 14 two-photon transition (TPT) pathways and transition intensity of these pathways.

Rb 87 TPT Pathways
Intensity 5S1/2(F=1)→5P3/2(F=0)→5D5/2(F=1) 0.1666 5S1/2(F=1)→5P3/2(F=1)→5D5/2(F=1) 0.125 5S1/2(F=1)→5P3/2(F=1)→5D5/2(F=2) 0.2916 In order to implement accuracy and stability of the system, we eliminate the Doppler-broadening, which is the main line broadening. The laser cooling technologies, such as magneto optical trap [4] and molasses, can eliminate most of the Doppler-broadening background. However, the number of atoms in the magneto-optical trap is much lower than the number of atoms in a normal gas cell.. Cold Rb TPT transition intensity is weak for establishing frequency standard or Doppler laser cooling. Therefore, following the previous research [5,6], we try to optimize repetition frequency of optical frequency comb to achieve higher transition intensity with the same average probe laser power. As the result of measurement, transition intensity is improved threefold.

Principle
Specific theoretical analysis is as follows. Each mode of optical frequency comb can be expressed as where f 0 is the carrier-envelop offset frequency, f rep is the pulse repetition rate, and n is an integer on the order of 10 6 . The sum frequency of two comb modes is given by Whenever the sum frequency of two modes coincides transition frequency, the optical comb excites the two-photon transition. A hundred thousand pair modes can satisfy the resonant condition at the same time. When two comb modes are near resonance with the 5S-5P and 5P-5D transitions, the intermediate 5P state enhances stepwise two-photon transitions (S-TPT). We utilize this phenomenon to enhance the two-photon transition signal intensity.
Due to the negligible Doppler broadening in cold atom experiments, the S-TPT excitation conditions are more critical. S-TPT can be achieved only when the 5S-5P and 5P-5D transition frequency difference is exactly an integral multiple of the repetition frequency. If f rep deviates (f 776nm − f 780nm )/n, no S-TPT can be excited, because no other comb excites another single-photon transition when one comb frequency is aligned with a single-photon transition. To validate this feature, we consider all 14 S-TPT transition pathways for 5S 1/2 -5P 3/2 -5D 5/2 of 87 Rb. 14 S-TPT pathways and the TPT intensity of each TPT pathway are shown in Table 1.
Each transition path has a different single-photon transition frequency, as well as a different two-step transition frequency difference, which has a different repetition frequency resonance value. The results in Figure 2 show the simulation curves of the total TPT intensities produced by these 14 S-TPT pathways with repetition rates. It can be seen that the S-TPT intensity exhibits a periodicity of about 8 kHz with f rep , which corresponds to the range of variation of the repetition frequency from one resonance value to the next resonance value (∆f/n→∆f /(n + 1)). Take the transition path  Table 1. 14 two-photon transition (TPT) pathways and transition intensity of these pathways.

Rb 87 TPT Pathways
Intensity Each transition path has a different single-photon transition frequency, as well as a different two-step transition frequency difference, which has a different repetition frequency resonance value. The results in Figure 2 show the simulation curves of the total TPT intensities produced by these 14 S-TPT pathways with repetition rates. It can be seen that the S-TPT intensity exhibits a periodicity of about 8 kHz with frep, which corresponds to the range of variation of the repetition frequency from one resonance value to the next resonance value (Δf/n→Δf/(n+1)). Take the transition path 5S1/2(   Based on the above calculation and analysis, in order to further enhance the signal intensity of two-photon transition signal in cold atoms, we need to finely control f rep . The standard technique to change repetition rate of an Er fiber optical frequency comb is to change the laser cavity length by a piezoelectric transducer (PZT) or a translation stage. The relationship between f rep and cavity length can be given as where L is the original length, l(t) is length of the PZT or the translation stage at time t, c is the light velocity, and k is the average refractive index of the cavity. The resolution of the laser's phase shift is given as in Reference [22]: where ∆θ o (t) is the laser's phase shift, l res (most larger than 0.5 nm) is the length-tuning resolution of the PZT or the translation stage, and t res (usually in sub-microsecond scale) is the response time. However, limited by length-tuning resolution and long response time, it is hard for the laser's phase shift to be smaller than 40 fs [22]. The cavity-length-controlling technique can demonstrate a wide range adjustment of f rep , but is not suitable for fine adjustment.
The other way to change the repetition rate of the fiber optical frequency comb is using pump power modulation. This is different with Ti as with the sapphire frequency comb, gain fiber is added into the fiber optical comb cavity. Changes in pump power affect the interaction between atoms in gain fiber that result in changes of f rep [23,24]. Based on the experiment and analysis in Reference [22], we know changing the pump source affects f rep in a linear way. This method of adjustment is not limited by PZT length-tuning resolution and response time, and the laser's phase resolution can increase by two orders of magnitude. By controlling the pump source, fine f rep adjustment can be achieved.
In summary, combined with cavity length controlling and pump source controlling, we can find the f rep corresponding S-TPT resonance to increase the two-photon transition intensity.

Experiment & Results
The experiment setup for our system is shown in Figure 3. An Erbium-fiber-based frequency comb with the 1556 nm center wavelength is used as the source to excite atoms. The Er frequency comb emits 100 fs pulses with 100 mW average power and a repletion rate of 129 MHz. The cavity length can be adjusted via a PZT and a translation stage in the cavity. A home-built two-stage erbium-doped fiber amplification module is used to amplify the output of 1556 nm mode-locked laser to 200 mw. In order to improve pulse peak power, the comb is compensated the pulse broadening in optical fiber with a pair of prisms made of silica. The properties of the optical pulse are adjusted to focus the power of the optical frequency comb spectrum to the vicinity of 1556 nm. Then we focus the beam into a periodically poled lithium niobate crystal (PPLN) by a lens to double the frequency of the comb. A 20 mW frequency comb with center wavelength of 778 nm is obtained in the end.
Secondly, the 778 nm frequency comb is directed into magneto optical trap (MOT) cell to probe the cold 87 Rb atom cloud which is cooled and trapped in MOT. To eliminate the influence of magnetic fields in MOT, we design a timing control cycle. We switched off the magnetic fields in MOT when the 778 nm frequency comb is directed into MOT cell to excite the Rb atom for 2 ms. Then the magnetic fields are switched on and the atom cloud is captured and cooled in MOT for 8 ms, and the 778 nm frequency comb is switched off by Acousto-optical Modulators (AOM) at the same time. Repeating this 10 ms cycle, we demonstrate the excitation of the two-photon transition.
To verify the relationship between f rep and TPT intensity, we scan f rep and detect 420 nm fluorescence signal from cascade decay via the 6P-5S state with a photomultiplier (PMT), used in photon counting mode. First, in order to verify the simulation results in Figure 2, we fix the pump current at 92.2 mA in experiments and adjust the repetition frequency to observe the fluorescence signal at the same time. When we scan f rep 50 Hz, the comb teeth at 780 nm will scan 129 MHz. So, we can get the direct frequency comb spectroscopy which illustrates all 5S→5D transition lines by scanning f rep 50 Hz with controlling voltage on PZT. The intensity of each line depends on the center scanning frequency of f rep . The peak intensity of the DFCS is recorded. Then, we change the center f rep scanning frequency by adjusting the translation stage in cavity and repeat the above steps. The obtained result is shown in Figure 4. Blue dots show the peak intensity of the DFCS with different center f rep scanning frequency. Red lines show simulation results. It can be seen that the experiment data and the simulation results are almost perfectly matched, which fully illustrates the influence of the repetition frequency on the two-photon intensity and verifies our theoretical analysis. Then, we verify the small change of repetition frequency caused by the pump power change. In ensuring the laser cavity length and average power fixed, we change the pump power and measure TPT intensity, and the results are shown in Figure 5. We can observe the small shift of the TPT peaks in the spectrum because the frequency of the two photon transitions is fixed. It can be seen that different pump powers affect the repetition frequency of the frequency comb. Secondly, the 778 nm frequency comb is directed into magneto optical trap (MOT) cell to probe the cold 87 Rb atom cloud which is cooled and trapped in MOT. To eliminate the influence of magnetic fields in MOT, we design a timing control cycle. We switched off the magnetic fields in MOT when the 778 nm frequency comb is directed into MOT cell to excite the Rb atom for 2 ms. Then the magnetic fields are switched on and the atom cloud is captured and cooled in MOT for 8 ms, and the 778 nm frequency comb is switched off by Acousto-optical Modulators (AOM) at the same time. Repeating this 10 ms cycle, we demonstrate the excitation of the two-photon transition.
To verify the relationship between frep and TPT intensity, we scan frep and detect 420 nm fluorescence signal from cascade decay via the 6P-5S state with a photomultiplier (PMT), used in photon counting mode. First, in order to verify the simulation results in Figure 2, we fix the pump current at 92.2 mA in experiments and adjust the repetition frequency to observe the fluorescence signal at the same time. When we scan frep 50 Hz, the comb teeth at 780 nm will scan 129 MHz. So, we can get the direct frequency comb spectroscopy which illustrates all 5S→5D transition lines by scanning frep 50 Hz with controlling voltage on PZT. The intensity of each line depends on the center scanning frequency of frep. The peak intensity of the DFCS is recorded. Then, we change the center frep scanning frequency by adjusting the translation stage in cavity and repeat the above steps. The obtained result is shown in Figure 4. Blue dots show the peak intensity of the DFCS with different center frep scanning frequency. Red lines show simulation results. It can be seen that the experiment data and the simulation results are almost perfectly matched, which fully illustrates the influence of the repetition frequency on the two-photon intensity and verifies our theoretical analysis. Then, we verify the small change of repetition frequency caused by the pump power change. In ensuring the laser cavity length and average power fixed, we change the pump power and measure TPT intensity, and the results are shown in Figure 5. We can observe the small shift of the TPT peaks in the spectrum because the frequency of the two photon transitions is fixed. It can be seen that different pump powers affect the repetition frequency of the frequency comb.    Finally, we improve the two-photon transition by adjusting the repetition frequency to corresponding S-TPT resonance frequency. The PMT signal is delivered into a lock-in amplifier to generate an error signal. The error signal is fed back to tune the PZT and pump source to lock center scanning frequency of frep to corresponding 5S1/2(F=2)-5P3/2(F=3)-5D5/2(F=4) S-TPT resonance frequency. Next, we fine scan frep to get Rb TPT signals. Direct frequency comb spectroscopy of cold Rb TPTs is shown in Figure 6. The peak fluorescence intensity detected through PMT at the optimization spectrum in Figure 6 is about 1189 photons per millisecond when we tune frep with PZT and pump power. If we only optimize DFCS with controlling PZT, the peak intensity is 903 per Finally, we improve the two-photon transition by adjusting the repetition frequency to corresponding S-TPT resonance frequency. The PMT signal is delivered into a lock-in amplifier to generate an error signal. The error signal is fed back to tune the PZT and pump source to lock center scanning frequency of f rep to corresponding 5S 1/2 (F = 2)-5P 3/2 (F = 3)-5D 5/2 (F = 4) S-TPT resonance frequency. Next, we fine scan f rep to get Rb TPT signals. Direct frequency comb spectroscopy of cold Rb TPTs is shown in Figure 6. The peak fluorescence intensity detected through PMT at the optimization spectrum in Figure 6 is about 1189 photons per millisecond when we tune frep with PZT and pump power. If we only optimize DFCS with controlling PZT, the peak intensity is 903 per millisecond. As a comparison, the peak intensity of most DFCS (about 75% DFCS) without frep optimization is around 400 photons per millisecond. The final TPT intensity can increase by up to three times before optimization. The experimental results are consistent with the simulation in Figure 4 which shows that the peak intensity (transition intensity after optimization) is 850 photons per millisecond and the bottom intensity (transition intensity without optimization) is 320 photons per millisecond. Due to weak Doppler Effect in MOT, the linewidth of the cooled Rb TPT resonance is improved. We achieve better linewidth in cooled atoms (about 900 kHz) compared with linewidth in thermal atoms in our previous work [6] (about 2 MHz). The residual linewidth is mainly due to the 600-kHz natural linewidth and the 300-kHz transit time broadening. The reduced linewidth is approximately equal to the Doppler broadening of the thermal atoms (about 1.2 MHz). This shows that we can ignore Doppler broadening in cold atoms.
millisecond. Due to weak Doppler Effect in MOT, the linewidth of the cooled Rb TPT resonance is improved. We achieve better linewidth in cooled atoms (about 900 kHz) compared with linewidth in thermal atoms in our previous work [6] (about 2 MHz). The residual linewidth is mainly due to the 600-kHz natural linewidth and the 300-kHz transit time broadening. The reduced linewidth is approximately equal to the Doppler broadening of the thermal atoms (about 1.2 MHz). This shows that we can ignore Doppler broadening in cold atoms. Figure 6. Spectrum of cold Rb TPTs directly excited by the Erbium-fiber optical frequency comb. frep,peak is the corresponding repetition frequency to intensity peak of each DFCS.

Conclusions
In conclusion, we have demonstrated cold Rb atom two-photon transition excitation with an Erfiber optical frequency comb. In addition, we optimized the repetition rate of optical frequency comb to enhance TPT intensity by controlling cavity length and pump source. The optimization method not only excites rubidium atoms to establish frequency standard, but also has applications for utilizing fiber optical comb to excite other atom TPTs. For example, there have been studies of Doppler cooling using fiber optical combs [25]. However, most of these studies are limited by the fiber optical combs power and have failed to expand in depth. The optimization method can greatly reduce the power required for Doppler cooling to solve power problem. It has many potential applications in Doppler laser cooling and detection of macromolecules [16].