## Modulation Transfer Spectroscopy of Coherent Population Trapping based on phase-coherent lasers

Optics Express, Vol. 17, Issue 8, pp. 6741-6746 (2009)

http://dx.doi.org/10.1364/OE.17.006741

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### Abstract

A new source of two diode laser beams, spatially separated but optically phase-locked with each other, is used to study the modulation transfer spectroscopy of coherent population trapping resonance (CPT). The spectrum for the ^{87}Rb D2 line is obtained with narrow linewidth and high signal-to-noise ratio, and analyzed with different experimental parameters. A theoretical analysis of the CPT modulation transfer spectra is deduced from the density matrix equation of motion, and found to be in good agreement with the experimental results.

© 2009 Optical Society of America

## 1. Introduction

1. N. Cyr, M. Têtu, and M. Breton, “All-Optical Microwave Frequency Standard: A Proposal,” IEEE Trans. Instrum. Meas. **42**, 640–649 (1993). [CrossRef]

2. J. Vanier, A. Godone, and F. Levi, “Coherent population trapping in cesium: Dark lines and coherent microwave emission,” Phys. Rev. A **58**, 2345–2358 (1998). [CrossRef]

^{-11}and 8×10

^{-12}for an averaging time of 1s and 100s respectively. On the other hand, some people are drawing their attention to the best performance now, with secondary concerns about the size and power consumption of the clock [3

3. H. S. Moon, S. E. Park, Y. H. Park, L. Lee, and J. B. Kim, “Passive atomic frequency standard based on coherent population trapping in ^{87}Rb using injection-locked lasers,” J. Opt. Soc. Am. B **23**, 2393–2397 (2006). [CrossRef]

4. R. K. Raj et al., “High-Frequency Optically Heterodyned Saturation Spectroscopy Via Resonant Degenerate Four-Wave Mixing,” Phy. Rev. Lett. **44**, 1251–1254 (1980). [CrossRef]

5. L. S. Ma, L. E. Ding, and Z. Y. Bi, “Doppler-Free Two-Photon Modulation Transfer Spectroscopy in Sodium Dimers,” Appl. Phys. B **51**, 233–237 (1990). [CrossRef]

*δ*[6

6. W. Chen, X. Qi, L. Yi, K. Deng, Z. Wang, J. Chen, and X. Chen, “Optical phase locking with a large and tunable frequency difference based on a vertical-cavity surface-emitting laser” Opt. Lett. **33**, 357–359 (2008). [CrossRef] [PubMed]

^{87}Rb D2 atomic system, and find that the experimental results are in good agreement with the theoretical predictions. We also give theoretical simulations which, by changing the modulation frequency and modulation index, gain the largest signal gradient to optimize the parameters which will be used for microwave frequency locking in the atomic clock.

## 2. Theoretical analysis

*ω*and

*ω*+ Ω, named probe and pump, are resonant with energy levels ∣

*c*〈 ↔ ∣

*b*〉 and ∣

*a*〉 ↔ ∣

*b*〉, respectively. The pump beam is frequency modulated with a sinusoidal wave at frequency

*δ*, leaving the sidebands at frequencies

*ω*+ Ω +

*2δ*, where

*n*is the order of the sidebands. The modulated pump and the co-propagating unmodulated probe beams are aligned collinearly through a vapor cell. If the interaction of the pump and probe beams with the atomic vapor are sufficiently nonlinear, the modulation sidebands appear on the unmodulated probe beam. The optical heterodyne beating between the weak new sidebands and the probe beam can be demodulated by the original modulation signal at frequency

*δ*.

*ω*+ Ω and the sidebands separated by the modulation frequency

*δ*:

*β*is the modulation index and

*J*(

_{n}*β*) is the Bessel function of order

*n*. The unmodulated probe beam is expressed as

*E*with the frequencies

_{r}*ω*=

_{r}*ω*

_{1}−

*ω*

_{2}+

*ω*

_{3}=

*ω*±

*δ*, according to the request of four-wave mixing, when up to the

*s*order of the pump beam sidebands are considered. In Table 1,

^{th}*n*is the order of the sidebands (

*n*= −

*s*, −

*s*+ 1,…,

*s*− 1,

*s*). Using perturbation theory and the rotating-wave approximation, the third-order elements of the density matrix are given by Eq. (12) – (17) in [7

7. M. Ducloy and D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened media. II. Doppler-free heterodyne spectroscopy via collinear four-wave mixing in two- and three-level systems,” J. Physique **43**, 57–65 (1982). [CrossRef]

^{87}Rb, the relaxation rates

*γ*,

_{a}*γ*and

_{c}*γ*are of the order of several hundred hertz, while

_{ac}*γ*,

_{b}*γ*and

_{ab}*γ*are approximately several hundred mega-hertz. They are all much less than the hyper-fine splitting,

_{cb}*ω*~ 6.8GHz, of the ground state. The modulation frequency

_{ac}*δ*is also ignorable compared with

*ω*. Using all the approximations above, we obtain the third-order elements of the density matrix in our system:

_{ac}*δ*=

_{ab}*ω*+ Ω −

*ω*,

_{ab}*δ*=

_{cb}*ω*−

*ω*,

_{cb}*δ*= Ω −

_{ac}*ω*,

_{ac}*ν*is the velocity component along

*z*axis, and

*s*is the maximum order of sidebands considered.

*N*is the total number of atoms, and we assume that

*n*=

_{ab}*n*=

_{cb}*N*/2. The reemitted field is given by [7

7. M. Ducloy and D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened media. II. Doppler-free heterodyne spectroscopy via collinear four-wave mixing in two- and three-level systems,” J. Physique **43**, 57–65 (1982). [CrossRef]

*C*is real. Since the beating current at the photodiode is proportional to

^{87}Rb quantum system is expressed by

*γ*is the linewidth of the CPT resonance, and ∆

_{ac}*is the detuning from the resonance center. The real part*

_{ac}*ℜ*[

*S*(

*δ*)] represents the quadrature component of the signal and the imaginary part

*ℑ*[

*S*(

*δ*)] represents the in-phase component of the signal. The real demodulated signal is a combination of these two parts with a form of

*ϕ*is the detector phase with respect to the modulation field applied to the pump laser. The maximum signal amplitude is obtained at

## 3. Experiment

6. W. Chen, X. Qi, L. Yi, K. Deng, Z. Wang, J. Chen, and X. Chen, “Optical phase locking with a large and tunable frequency difference based on a vertical-cavity surface-emitting laser” Opt. Lett. **33**, 357–359 (2008). [CrossRef] [PubMed]

*μ*W of each beam. Two quarter plates at the entrance and the exit of the cell change the polarization of the beams to be circular to ensure the maximum CPT signal. The photodiode (PD) detects the probe beam and the reemitted sidebands by placing a PBS in front of it, filtering the pump beam out.

*μ*W/cm

^{2}, and is mainly broadened by saturation broadening and collision broadening.

## 4. Conclusion

*S*

_{1/2}↔ 5

*P*

_{3/2}three-level system of the

^{87}Rb atom. We recorded the MTS signal of CPT resonances in this system and studied the dependence of the MTS signal gradient on the modulation frequency and modulation index. The high-phase-coherence laser beams allows us to obtain the MTS signal of narrow linewidth and high signal-to-noise ratio. A theoretical analysis of the CPT modulation transfer spectra is deduced from the density matrix equation of motion, and found to be in good agreement with the experimental results.

## References and links

1. | N. Cyr, M. Têtu, and M. Breton, “All-Optical Microwave Frequency Standard: A Proposal,” IEEE Trans. Instrum. Meas. |

2. | J. Vanier, A. Godone, and F. Levi, “Coherent population trapping in cesium: Dark lines and coherent microwave emission,” Phys. Rev. A |

3. | H. S. Moon, S. E. Park, Y. H. Park, L. Lee, and J. B. Kim, “Passive atomic frequency standard based on coherent population trapping in |

4. | R. K. Raj et al., “High-Frequency Optically Heterodyned Saturation Spectroscopy Via Resonant Degenerate Four-Wave Mixing,” Phy. Rev. Lett. |

5. | L. S. Ma, L. E. Ding, and Z. Y. Bi, “Doppler-Free Two-Photon Modulation Transfer Spectroscopy in Sodium Dimers,” Appl. Phys. B |

6. | W. Chen, X. Qi, L. Yi, K. Deng, Z. Wang, J. Chen, and X. Chen, “Optical phase locking with a large and tunable frequency difference based on a vertical-cavity surface-emitting laser” Opt. Lett. |

7. | M. Ducloy and D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened media. II. Doppler-free heterodyne spectroscopy via collinear four-wave mixing in two- and three-level systems,” J. Physique |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(300.6210) Spectroscopy : Spectroscopy, atomic

(300.6290) Spectroscopy : Spectroscopy, four-wave mixing

(300.6310) Spectroscopy : Spectroscopy, heterodyne

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: March 5, 2009

Revised Manuscript: April 4, 2009

Manuscript Accepted: April 4, 2009

Published: April 8, 2009

**Citation**

Xiang H. Qi, Wen L. Chen, Lin Yi, Xiao J. Zhou, and Xu Z. Chen, "Modulation Transfer Spectroscopy of Coherent Population Trapping based on phase-coherent lasers," Opt. Express **17**, 6741-6746 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-8-6741

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### References

- N. Cyr, M. Tetu, and M. Breton, "All-Optical Microwave Frequency Standard: A Proposal," IEEE Trans. Instrum. Meas. 42, 640-649 (1993). [CrossRef]
- J. Vanier, A. Godone, and F. Levi, "Coherent population trapping in cesium: Dark lines and coherent microwave emission," Phys. Rev. A 58, 2345-2358 (1998). [CrossRef]
- H. S. Moon, S. E. Park, Y. H. Park, L. Lee, and J. B. Kim, "Passive atomic frequency standard based on coherent population trapping in 87Rb using injection-locked lasers," J. Opt. Soc. Am. B 23, 2393-2397 (2006). [CrossRef]
- R. K. Raj et al., "High-Frequency Optically Heterodyned Saturation Spectroscopy Via Resonant Degenerate Four-Wave Mixing," Phy. Rev. Lett. 44, 1251-1254 (1980). [CrossRef]
- L. S. Ma, L. E. Ding, and Z. Y. Bi, "Doppler-Free Two-Photon Modulation Transfer Spectroscopy in Sodium Dimers," Appl. Phys. B 51, 233-237 (1990). [CrossRef]
- W. Chen, X. Qi, L. Yi, K. Deng, Z. Wang, J. Chen, and X. Chen, "Optical phase locking with a large and tunable frequency difference based on a vertical-cavity surface-emitting laser" Opt. Lett. 33, 357-359 (2008). [CrossRef] [PubMed]
- M. Ducloy and D. Bloch, "Theory of degenerate four-wave mixing in resonant Doppler-broadened media. II. Doppler-free heterodyne spectroscopy via collinear four-wave mixing in two- and three-level systems," J. Physique 43, 57-65 (1982). [CrossRef]

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