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 87Rb 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

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References

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  1. N. Cyr, M. Tetu, 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]
  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 87Rb 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]
  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]
  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]

2008 (1)

2006 (1)

1998 (1)

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]

1993 (1)

N. Cyr, M. Tetu, and M. Breton, "All-Optical Microwave Frequency Standard: A Proposal," IEEE Trans. Instrum. Meas. 42, 640-649 (1993).
[CrossRef]

1990 (1)

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]

1982 (1)

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]

1980 (1)

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]

Bi, Z. Y.

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]

Bloch, D.

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]

Chen, J.

Chen, W.

Chen, X.

Cyr, N.

N. Cyr, M. Tetu, and M. Breton, "All-Optical Microwave Frequency Standard: A Proposal," IEEE Trans. Instrum. Meas. 42, 640-649 (1993).
[CrossRef]

Deng, K.

Ding, L. E.

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]

Ducloy, M.

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]

Godone, A.

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]

Kim, J. B.

Lee, L.

Levi, F.

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]

Ma, L. S.

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]

Moon, H. S.

Park, S. E.

Park, Y. H.

Qi, X.

Raj, R. K.

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]

Vanier, J.

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]

Wang, Z.

Yi, L.

Appl. Phys. B (1)

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]

IEEE Trans. Instrum. Meas. (1)

N. Cyr, M. Tetu, and M. Breton, "All-Optical Microwave Frequency Standard: A Proposal," IEEE Trans. Instrum. Meas. 42, 640-649 (1993).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Physique (1)

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]

Opt. Lett. (1)

Phy. Rev. Lett. (1)

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]

Phys. Rev. A (1)

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]

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Figures (6)

Fig. 1.
Fig. 1.

Λ-configuration three-level model for calculating the FWM response with optical frequency modulation. The angular frequencies of the two Raman lasers, one of which is modulated at frequency δ, are ω and ω + Ω + , where n is the order of the sidebands. ωab , ωcb , and ωca are the transition frequencies corresponding to the three Raman energy levels ∣a〈, ∣b〉 and ∣c〉.

Fig. 2.
Fig. 2.

The calculated in-phase component, quadrature component and maximum-amplitude signal with normalized modulation frequency of 1 and modulation index of 1. The maximum-amplitude signal is obtained at φ = 142°.

Fig. 3.
Fig. 3.

Experimental setup to obtain the absorption spectra and MTS spectra of CPT resonance. ISO: isolator; λ/2: half-wave plate; PBS: polarizing beam splitter; BS: beam splitter; NDF: neutral density filter; λ/4: quarter-wave plate; PD: PIN photo diode.

Fig. 4.
Fig. 4.

(a) CPT absorption spectrum with pump and probe power of 50 μW each. The linewidth obtained is 400Hz. (b) The MTS spectra with the modulation index of 1 and modulation frequencies of 400 Hz.

Fig. 5.
Fig. 5.

The MTS spectra with the modulation index of 1 and the modulation frequencies (and detector phases) of (a) 400 Hz (ϕ=142°), (b) 1 kHz (ϕ=97°) and (c) 3 kHz (ϕ=83°) respectively. The solid lines show the theoretical predictions. The “o” curves show the experimental results.

Fig. 6.
Fig. 6.

The curves show the maximum signal gradient under different normalized modulation frequencies δ/γ, where γ is the measured linewidth of the CPT resonance. Each curve is shown in different modulation index, which is indicate by numbers.

Tables (1)

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Table 1. Possible combinations for regenerating the field of Er

Equations (15)

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E m = 1 2 E m 0 [ n = 0 J n ( β ) e i ( ω + Ω + n δ ) t + n = 1 ( 1 ) n J n ( β ) e i ( ω + Ω + n δ ) t ] e i k z + c . c .
E u = 1 2 E u 0 e i ( ω t k z ) + c . c .
ρ a b ( 3 ) ( ω ± δ ) 0
ρ c b ( 3 ) ( ω ± δ ) = ± 1 16 ( i h ¯ ) 3 N E m 0 2 E u 0 μ c b μ a b 2 e t [ ( ω ± δ ) t k z ] γ c b + i ( Δ c b ± δ + k υ ) n = 1 s J n 1 ( β ) J n ( β )
× { 1 γ a c i ( Δ a c n δ ) [ 1 γ a b i ( Δ a b n δ )
+ 1 γ c b + i ( Δ c b k υ ) ] 1 γ a c i [ Δ a c ( n 1 ) δ ]
× [ 1 γ a b i [ Δ a b ± ( n 1 ) δ k υ ] + 1 γ c b + i ( Δ c b k υ ) ] }
E r = i k L 2 ε 0 u π + e υ 2 / u 2 [ μ a b ρ a b ( 3 ) + μ c b ρ c b ( 3 ) + c . c . ] d υ
C = + e υ 2 / u 2 γ cb i k υ ( 1 γ a b + i k υ + 1 γ c b i k υ ) d υ
E r ( ω + δ ) E u * ( ω ) + E r * ( ω δ ) E u ( ω ) + c . c .
I ( δ ) C k L μ c b 4 N E u 0 2 64 h ¯ 3 ε 0 u π S ( δ ) e i δ t + c . c .
S ( δ ) = n = 1 s J n 1 ( β ) J n ( β )
× { 1 γ a c i [ Δ a c + ( n 1 ) δ ] 1 γ c b + i [ Δ a c ( n 1 ) δ ]
+ 1 γ a c + i ( Δ a c + n δ ) 1 γ a c i ( Δ a c n δ ) }
[ S ( δ ) ] cos ( δ t + ϕ ) + [ S ( δ ) ] sin ( δ t + ϕ )

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