## Femtowatt-light-level phase measurement of slow light pulses via beat-note interferometer |

Optics Express, Vol. 18, Issue 17, pp. 18498-18505 (2010)

http://dx.doi.org/10.1364/OE.18.018498

Acrobat PDF (949 KB)

### Abstract

We report on an experimental demonstration of applying the beat-note interferometer to simultaneously measure the phase and amplitude variations of light pulses after propagating through an electromagnetically induced transparency medium at femtowatt-light levels. Furthermore, we observe that the measured phase noise approaches the shot-noise level arising from the fluctuations of detected photons.

© 2010 Optical Society of America

## 1. Introduction

6. I. Fushman, D. Englund, A. Faraon, N. Stoltz, P. Petroff, and J. Vučković, “Controlled phase shifts with a single quantum dot,” Science **320**, 769–772 (2008). [CrossRef] [PubMed]

9. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today **50**, 36–42 (1997). [CrossRef]

10. M. Fleischhauer, A. Imamoğlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. **77**, 633–673 (2005). [CrossRef]

11. S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. **81**, 3611–3614 (1998). [CrossRef]

12. H. Schmidt and A. Imamoğlu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. **21**, 1936–1938 (1996). [CrossRef] [PubMed]

13. S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. **82**, 4611–4614 (1999). [CrossRef]

*µ*s [14–16

14. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature **409**, 490–493 (2001). [CrossRef] [PubMed]

*µ*s single-photon pulse is 100 fW, where the laser wavelength is 780 nm. Hence, the ultralow-light-level phase measurement in the EIT-based system is an important issue that allows us to determine the phase properties of light pulses at the single-photon level for realizing practical applications in quantum information processing, such as quantum phase gates [17–21

17. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. **90**, 197902 (2003). [CrossRef] [PubMed]

## 2. Experimental details

^{87}Rb atoms produced by a vapor-cell magneto-optical trap (MOT). Around 3 × 10

^{9}atoms are trapped in the MOT, as measured by the optical-pumping method. Figure 1(a) shows the relevant energy levels of

^{87}Rb

*D*

_{2}transitions that is used for the Λ-scheme EIT experiment. This scheme consists of the two levels ∣

*F*= 1〉 ≡ ∣1〉 and ∣

*F*= 2〉 ≡ ∣2〉 of the 5

*S*

_{1/2}ground states, and the ∣

*F*′ = 2〉 ≡ ∣3〉 level of the 5

*P*

_{3/2}excited state. A strong coupling field with frequency

*ω*and a weak probe field with frequency

_{c}*ω*are employed to drive ∣2〉 ↔ ∣3〉 and ∣1〉 ↔ ∣3〉 transitions, respectively. These two fields, which are circularly polarized with

_{p}*σ*+ polarization, form a three-level Λ-type configuration of EIT system.

*χ*(

*ω*) [10

_{p}10. M. Fleischhauer, A. Imamoğlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. **77**, 633–673 (2005). [CrossRef]

*T*= exp[−Im

*χ*(

*ω*)

_{p}*kL*] and

*k*is the wave number of the probe field and

*L*is the optical path length of the medium. They are given by

*nσ*

_{13}

*L*is the optical density for the probe transition (

*n*is the atomic density, and

*σ*

_{13}is the atomic cross section of the ∣1〉 ↔ ∣3〉 transition), and Ω

_{c}is the Rabi frequency of the coupling transition. The one-photon (probe) detuning is defined as Δ

_{p}=

*ω*−

_{p}*ω*

_{31}, where

*ω*

_{31}is the frequency of the ∣1〉 ↔ ∣3〉 transition, and the two-photon detuning is defined as

*δ*= Δ

_{p}− Δ

_{c}, where Δ

_{c}is the detuning of coupling field (see Fig. 1(a)).

*γ*

_{21}is the decoherence rate between the two ground states, and

*γ*

_{31}= Γ +

*γ*

_{3deph}is the total coherence decay rate out of the excited state ∣3〉, where Γ = 2

*π*× 6 MHz is the spontaneous decay rate of the excited state, and

*γ*

_{3deph}is the dephasing rate of the state ∣3〉 arising from the linewidth of laser fields and elastic collisions between atoms.

*µ*s, the 50-

*µ*s probe square pulse is switched on for measurement.

## 3. Beat-note interferometer

*ω*, of the probe laser is 2

_{a}*π*× 80 MHz, which is sufficiently large that the interaction between the zeroth-order beam and the EIT medium is negligible. Both reference and probe beat notes, which carries the beat frequency of

*ω*, display the waveforms on the oscilloscope (OSC, Agilent MSO6034A).

_{a}*I*

_{0r},

*I*

_{1r},

*I*

_{0p}and

*I*

_{1p}are the intensities of zeroth- and first-order beams along the path of reference and probe beat notes, respectively. Δ

*φ*is the phase shift induced by the medium.

*φ*and

_{r}*φ*are phases that result from optical paths, components, or other factors. The difference between

_{p}*φ*and

_{r}*φ*is always fixed, thus we can directly measure the phase shift of a probe pulse by comparing the two beat signals using OSC. At the same time, the probe transmission (

_{p}*I*

_{1p}) can be obtained from the amplitude of probe beat notes according to Eq. (4). Throughout the experiment, only the phase shift within 200 ns of the end of the probe pulse is considered in order to obtain the steady-state results of EIT, and the power of the zeroth-order beam for the probe beat note is fixed at 2 nW. In addition, the measurement uncertainty is evaluated using 10 samples, with each sample averaged 4096 times using OSC.

22. Y.-F. Chen, Y.-C. Liu, Z.-H. Tsai, S.-H. Wang, and I. A. Yu, “Beat-note interferometer for direct phase measurement of photonic information,” Phys. Rev. A **72**, 033812 (2005). [CrossRef]

## 4. Experimental results and discussion

^{7}V/W in the experiment. We note that the measured beat-note amplitudes are only 80% of that estimated from the gain of the PMT module due to the finite frequency response of the PMT module.

*q*= 0.006 rad. Here,

*q*represents an unknown phase noise, which is not decreased to zero with increasing the light power. The fluctuation of optical paths from BS2 to two photodetectors, which are caused by mechanical vibrations, would result in phase noise for the measurement [22

22. Y.-F. Chen, Y.-C. Liu, Z.-H. Tsai, S.-H. Wang, and I. A. Yu, “Beat-note interferometer for direct phase measurement of photonic information,” Phys. Rev. A **72**, 033812 (2005). [CrossRef]

*ω*/

_{a}*c*or about 1.7 × 10

^{−3}rad/mm in our experiment, where

*c*is the speed of light. The unknown phase noise is much larger than the mechanical noise, hence it does not arise from the mechanical vibrations. The first term in function g exhibits the measured phase noise approaches the shot-noise level. The shot noise caused by the fluctuations of detected photons is predicted by the uncertainty principle Δ

*ϕ*Δ

*n*≥ 1, where Δ

*ϕ*and Δ

*n*are the uncertainty in the optical phase and the photon number, respectively. For example, a probe pulse with a peak power of 100 pW contains 80 photons within the temporal length of 200 ns. Δ

*n*for the coherent-state light is equal to the square root of the mean photon number per light pulse. Therefore, the phase noise, Δ

*ϕ*, is estimated to be about 0.001 rad among 10 samples, with each sample averaged 4096 times. Under the same conditions, the measured phase noise is about 0.002 rad, which is larger than the shot noise due to additional noise sources or other factors.

_{c}is 0.31Γ, as obtained from the separation of two absorption peaks in the EIT spectrum.

*γ*

_{21}is 0.002Γ, as estimated from the degree of transparency in the EIT spectrum.

*γ*

_{31}= 1.25Γ and Δ

_{c}= −0.7 MHz are determined from the spectrum of one-photon absorption for the probe transition. Of note,

*nσ*

_{13}

*L*is 16, as derived from the group delay time of slow light pulses [10

10. M. Fleischhauer, A. Imamoğlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. **77**, 633–673 (2005). [CrossRef]

*e*

^{−16}. Figure 3(b) presents the synchronous observation of EIT phase shift of a probe pulse as a function of probe detuning. The red dashed line is the theoretical curve predicted by Eq. (2) with the same parameters in Fig. 3(a). Figures 3(a) and 3(b) show the good agreement between experimental data and theoretical predictions using the same parameters.

*N*, at Δ

_{p}= −0.31 MHz [see Fig. 3(b)] corresponding to the transmitted probe power of 10 fW under EIT conditions. It is worth noting that a 50-

*µ*s probe pulse with a power of 10 fW contains only ~ 2 photons. The measured phase noise is decreased from 0.45 to 0.06 rad when the number of average is increased or, equivalently, SNR is enhanced. The experimental data are fitted by the function of

*h*=

*α*/√

*N*+

*β*, where

*β*= 0.009 rad that is slightly larger the unknown phase noise of 0.006 rad (see Fig. 2) due to the instability of EIT system. It shows a well-known result that increasing N improves the SNR by √

*N*.

## 5. Conclusion

23. M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. **84**, 1419–1422 (2000). [CrossRef] [PubMed]

24. M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the electromagnetically-induced-transparency-enhanced refractive Kerr nonlinearity: Theory,” Phys. Rev. A **74**, 013812 (2006). [CrossRef]

25. G. F. Sinclair, “Time-dependent cross-phase-modulation in ^{87}Rb,” Phys. Rev. A **79**, 023815 (2009). [CrossRef]

## Acknowledgments

## References and links

1. | D. Bouwmeester, A. K. Ekert, and A. Zeilinger, |

2. | M. A. Nielsen and I. L. Chuang, |

3. | Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. |

4. | M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature |

5. | Y.-F. Chen, C.-Y. Wang, S.-H. Wang, and I. A. Yu, “Low-light-level cross-phase-modulation based on stored Light pulses,” Phys. Rev. Lett. |

6. | I. Fushman, D. Englund, A. Faraon, N. Stoltz, P. Petroff, and J. Vučković, “Controlled phase shifts with a single quantum dot,” Science |

7. | N. Matsuda, R. Shimizu, Y. Mitsumori, H. Kosaka, and K. Edamatsu, “Observation of optical-fibre Kerr nonlinearity at the single-photon level,” Nat. Photonics |

8. | H. P. Specht, J. Bochmann, M. Mücke, B. Weber, E. Figueroa, D. L. Moehring, and G. Rempe, “Phase shaping of single-photon wave packets,” Nature Photon. |

9. | S. E. Harris, “Electromagnetically induced transparency,” Phys. Today |

10. | M. Fleischhauer, A. Imamoğlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. |

11. | S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. |

12. | H. Schmidt and A. Imamoğlu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. |

13. | S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. |

14. | C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature |

15. | D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. |

16. | Y.-F. Chen, Z.-H. Tsai, Y.-C. Liu, and I. A. Yu, “Low-light-level photon switching by quantum interference,” Opt. Lett. |

17. | C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. |

18. | S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A |

19. | A. Joshi and M. Xiao, “Phase gate with a four-level inverted-Y system,” Phys. Rev. A |

20. | B. He, Y.-H. Ren, and J. A Bergou, “Universal entangler with photon pairs in arbitrary states,” J. Phys. At. Mol. Opt. Phys. |

21. | A. V. Gorshkov, J. Otterbach, E. Demler, M. Fleischhauer, and M. D. Lukin, “Photonic phase gate via an exchange of fermionic spin waves in a spin chain,” arXiv:1001.0968v2 [quant-ph] (2010). |

22. | Y.-F. Chen, Y.-C. Liu, Z.-H. Tsai, S.-H. Wang, and I. A. Yu, “Beat-note interferometer for direct phase measurement of photonic information,” Phys. Rev. A |

23. | M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. |

24. | M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the electromagnetically-induced-transparency-enhanced refractive Kerr nonlinearity: Theory,” Phys. Rev. A |

25. | G. F. Sinclair, “Time-dependent cross-phase-modulation in |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(020.3320) Atomic and molecular physics : Laser cooling

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: June 17, 2010

Revised Manuscript: August 5, 2010

Manuscript Accepted: August 11, 2010

Published: August 13, 2010

**Citation**

Hsiang-Yu Lo, Po-Ching Su, Yu-Wei Cheng, Pin-I Wu, and Yong-Fan Chen, "Femtowatt-light-level phase measurement of slow light pulses via beat-note interferometer," Opt. Express **18**, 18498-18505 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-18498

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### References

- D. Bouwmeester, A. K. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, New York, 2000).
- M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
- Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, "Measurement of conditional phase shifts for quantum logic," Phys. Rev. Lett. 75, 4710-4713 (1995). [CrossRef] [PubMed]
- M. D. Lukin, and A. Imamoǧlu, "Controlling photons using electromagnetically induced transparency," Nature 413, 273-276 (2001). [CrossRef] [PubMed]
- Y.-F. Chen, C.-Y. Wang, S.-H. Wang, and I. A. Yu, "Low-light-level cross-phase-modulation based on stored Light pulses," Phys. Rev. Lett. 96, 043603 (2006). [CrossRef] [PubMed]
- I. Fushman, D. Englund, A. Faraon, N. Stoltz, P. Petroff, and J. Vučković, "Controlled phase shifts with a single quantum dot," Science 320, 769-772 (2008). [CrossRef] [PubMed]
- N. Matsuda, R. Shimizu, Y. Mitsumori, H. Kosaka, and K. Edamatsu, "Observation of optical-fibre Kerr nonlinearity at the single-photon level," Nat. Photonics 3, 95-98 (2009). [CrossRef]
- H. P. Specht, J. Bochmann, M. Mücke, B. Weber, E. Figueroa, D. L. Moehring, and G. Rempe, "Phase shaping of single-photon wave packets," Nat. Photonics 3, 469-472 (2009). [CrossRef]
- S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997). [CrossRef]
- M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005). [CrossRef]
- S. E. Harris, and Y. Yamamoto, "Photon switching by quantum interference," Phys. Rev. Lett. 81, 3611-3614 (1998). [CrossRef]
- H. Schmidt, and A. Imamoǧlu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21, 1936-1938 (1996). [CrossRef] [PubMed]
- S. E. Harris, and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999). [CrossRef]
- C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature 409, 490-493 (2001). [CrossRef] [PubMed]
- D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, "Storage of light in atomic vapor," Phys. Rev. Lett. 86, 783-786 (2001). [CrossRef] [PubMed]
- Y.-F. Chen, Z.-H. Tsai, Y.-C. Liu, and I. A. Yu, "Low-light-level photon switching by quantum interference," Opt. Lett. 30, 3207-3209 (2005). [CrossRef] [PubMed]
- C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, "Polarization qubit phase gate in driven atomic media," Phys. Rev. Lett. 90, 197902 (2003). [CrossRef] [PubMed]
- S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, "Polarization phase gate with a tripod atomic system," Phys. Rev. A 70, 032317 (2004). [CrossRef]
- A. Joshi, and M. Xiao, "Phase gate with a four-level inverted-Y system," Phys. Rev. A 72, 062319 (2005). [CrossRef]
- B. He, Y.-H. Ren, and J. A. Bergou, "Universal entangler with photon pairs in arbitrary states," J. Phys. At. Mol. Opt. Phys. 43, 025502 (2010). [CrossRef]
- A. V. Gorshkov, J. Otterbach, E. Demler, M. Fleischhauer, and M. D. Lukin, "Photonic phase gate via an exchange of fermionic spin waves in a spin chain," arXiv:1001.0968v2 [quant-ph] (2010).
- Y.-F. Chen, Y.-C. Liu, Z.-H. Tsai, S.-H. Wang, and I. A. Yu, "Beat-note interferometer for direct phase measurement of photonic information," Phys. Rev. A 72, 033812 (2005). [CrossRef]
- M. D. Lukin, and A. Imamoǧlu, "Nonlinear optics and quantum entanglement of ultraslow single photons," Phys. Rev. Lett. 84, 1419-1422 (2000). [CrossRef] [PubMed]
- M. V. Pack, R. M. Camacho, and J. C. Howell, "Transients of the electromagnetically-induced-transparency enhanced refractive Kerr nonlinearity: Theory," Phys. Rev. A 74, 013812 (2006). [CrossRef]
- G. F. Sinclair, "Time-dependent cross-phase-modulation in 87Rb," Phys. Rev. A 79, 023815 (2009). [CrossRef]

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