Modulation Transfer Spectroscopy of Coherent ... - OSA Publishing

Modulation Transfer Spectroscopy of Coherent Population Trapping based on phase-coherent lasers Xiang Hui Qi, Wen Lan Chen, Lin Yi, Xiao Ji Zhou and Xu Zong Chen Institute of Quantum Electronics, School of Electronics Engineering and Computer Science, Peking University, Beijing, 100871, China [email protected]

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 OCIS codes: (300.6210) Spectroscopy, atomic; (300.6290) Spectroscopy, four-wave mixing; (300.6310) Spectroscopy, heterodyne; (020.1670) Coherent optical effects.

References and links 1. N. Cyr, M. Tˆetu, and M. Breton, “All-Optical Microwave Frequency Standard: A Proposal,” IEEE Trans. Instrum. Meas. 42, 640-649 (1993). 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). 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). 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). 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). 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). 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).

1.

Introduction

Atomic clocks based on coherent population trapping (CPT) effect have been of great concern for several years [1, 2]. Since this is a promising scheme for micro atomic clocks, people’s interests are focused on the compactness and low power consumption. And due to all the efforts, the first commercial CPT atomic clock (SA3Xm, Symmetricom Inc.) came out recently with the size as small as 51mm×51mm×18mm, while providing a stability of 3 × 10−11 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]. #108413 - $15.00 USD

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Received 5 Mar 2009; revised 4 Apr 2009; accepted 4 Apr 2009; published 8 Apr 2009

13 April 2009 / Vol. 17, No. 8 / OPTICS EXPRESS 6741

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 + nδ , 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.

The remarkable advantages of conventional MTS come from the four-wave mixing (FWM) process [4]. Due to the nonlinear process of FWM, the detected signal is insensitive to the linear absorption of the background. Hence the MTS technique can readily generate dispersion-like lineshapes which sit on a flat, zero background and is especially suitable for detection of weak transitions and could obtain Doppler-free and high signal-to-noise-ratio (SNR) spectra [5]. In this paper, we deduce a simple line-shape formula for the modulation transfer spectra of the Λ-type three-level atomic system from the matrix equation of motion, in the case that the frequency difference between two ground states is much higher than Doppler linewidth of the upper energy level, and verify the correctness of the formula experimentally. To observe the MTS signal in Λ configuration of the three-level atomic system, two spatially separated and optically phase-locked laser beams are necessary. We have constructed an optical phase-locking system to generate these coherent Raman beams with one of them modulated at angular frequency δ [6]. Using this source of coherent beams, we obtained the MTS signal at different modulation frequencies and different demodulating phases. We calculated the lineshape formula for Λ configuration of the three-level 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

Consider a Λ-configuration of the three-level quantum systems showed in Fig. 1. Two coherent Raman beams with angular frequency of 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 + nδ , 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. The fields of the frequency-modulated pump beam are represented in terms of the carrier frequency ω + Ω and the sidebands separated by the modulation frequency δ : ∞ ∞ 1 n i(ω +Ω+nδ )t i(ω +Ω−nδ )t Em = Em0 ∑ Jn (β ) e e−ikz + c.c. + ∑ (−1) Jn (β ) e (1) 2 n=0 n=1 #108413 - $15.00 USD

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where β is the modulation index and Jn (β ) is the Bessel function of order n. The unmodulated probe beam is expressed as 1 Eu = Eu0 ei(ω t−kz) + c.c. (2) 2 All the possible combinations we are concerned about are listed in Table 1. These combinations regenerate the fields of Er with the frequencies ωr = ω1 − ω2 + ω3 = ω ± δ , according to the request of four-wave mixing, when up to the sth order of the pump beam sidebands are considered. In Table 1, n is the order of the sidebands (n = −s, −s + 1, ..., s − 1, s). Using perturbation Table 1. Possible combinations for regenerating the field of Er

1 2 3 4

ω1 ω + Ω + nδ ω ω + Ω + (n − 1) δ ω

ω2 ω + Ω + (n − 1) δ ω + Ω + (n − 1) δ ω + Ω + nδ ω + Ω + nδ

ω3 ω ω + Ω + nδ ω ω + Ω + (n − 1) δ

ωr ω +δ ω +δ ω −δ ω −δ

theory and the rotating-wave approximation, the third-order elements of the density matrix are given by Eq. (12) − (17) in [7]. Considering the experimental quantum system of gas-buffered 87 Rb, the relaxation rates γ , γ and γ are of the order of several hundred hertz, while γ , γ a c ac b ab and γcb are approximately several hundred mega-hertz. They are all much less than the hyperfine splitting, ωac ∼6.8GHz, of the ground state. The modulation frequency δ is also ignorable compared with ωac . Using all the approximations above, we obtain the third-order elements of the density matrix in our system: (3) ρab (ω ± δ ) ≈ 0 (3) 2 E μ μ 2 ei[(ω ±δ )t−kz] s 1 i 3 NEm0 u0 cb ab (3) ρcb (ω ± δ ) = ± ∑ Jn−1 (β ) Jn (β ) 16 h¯ γcb + i (Δcb ± δ − kυ ) n=1 1 1 × γac − i (Δac ∓ nδ ) γab − i (Δab ∓ nδ − kυ ) (4) 1 1 + − γcb + i (Δcb − kυ ) γac − i [Δac ± (n − 1) δ ] 1 1 + × γab − i [Δab ± (n − 1) δ − kυ ] γcb + i (Δcb − kυ ) where δab = ω + Ω − ωab , δcb = ω − ωcb , δac = Ω − ωac , υ 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 nab = ncb = N/2. The reemitted field is given by [7] Er = −i

kL √ 2ε0 u π

+∞ −∞

e−υ

2 /u2

(3) (3) μab ρab + μcb ρcb + c.c. d υ

(5)

where an optically thin absorption cell is assumed. The integration is somewhat different from that discussed in [5] because the Doppler limitation ku γab , γcb is unavailable here. When the light field is in near resonance with the atomic system, the detuning Δab , Δcb and the modulation frequency δ are far less than the relaxation γab and γcb . Therefore, the integrations over velocity υ become a constant factor

+∞ −υ 2 /u2 1 e 1 + (6) dυ C= γab + ikυ γcb − ikυ −∞ γcb − ikυ #108413 - $15.00 USD

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Amplitude arb.

inphase 0.5

maximum amplitude quardrature

0.0 0.5 4 2 0 2 4 Normalized frequency detuning ac Γac

Fig. 2. The calculated in-phase component, quadrature component and maximumamplitude signal with normalized modulation frequency of 1 and modulation index of 1. The maximum-amplitude signal is obtained at φ = 142◦ .

Numerical calculation indicates that C is real. Since the beating current at the photodiode is proportional to (7) Er (ω + δ ) Eu∗ (ω ) + Er∗ (ω − δ ) Eu (ω ) + c.c. The modulation transfer signal in case of the Λ configuration of the three-level 87 Rb quantum system is expressed by 4 NE 2 CkLμcb I (δ ) ∝ (8) √ u0 S (δ ) eiδ t + c.c. 64¯h3 ε0 u π s

S (δ ) = ∑ Jn−1 (β ) Jn (β ) n=1

×

1 1 − γac − i [Δac + (n − 1) δ ] γac + i [Δac − (n − 1) δ ] 1 1 − + γac + i (Δac + nδ ) γac − i (Δac − nδ )

(9)

where γac is the linewidth of the CPT resonance, and Δac is the detuning from the resonance center. The real part 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 ℜ [S (δ )] cos (δ t + φ ) + ℑ [S (δ )] sin (δ t + φ )

(10)

where φ is the detector phase with respect to the modulation field applied to the pump laser. ℑ[S(δ )] The maximum signal amplitude is obtained at tan (φ ) = ℜ[S( δ )] . The quadrature component, in-phase component and maximum-amplitude signal are shown in Fig. 2. 3.

Experiment

The experimental setup is shown in Fig. 3. To obtain the spatially separated and phase coherent Raman beams, we use a microwave frequency-modulated vertical-cavity surface-emitted laser (VCSEL) to phase connect two diode lasers by a two-step injection locking [6]. As a result, the master laser, a narrow-linewidth external-cavity diode laser (ECDL), and the slave laser, a Fabry-Perot diode laser, are phase locked with the frequency difference equal to the modulation frequency, 6.8GHz, of the VCSEL, which is controlled by the frequency synthesizer (E8257D, Agilent). The slave laser could be further modulated at a low frequency, e.g., 400 Hz, by modulating the high modulation frequency of VCSEL. The modulated slave laser playing #108413 - $15.00 USD

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Amplitude V

3.60 a 3.58 400Hz 3.56 3.54 3.52 3.50 20001000 0 1000 2000 Frequency detuning Hz

Amplitude V

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. 3 2 b 1 0 1 2 3 2000 1000 0 1000 2000 Frequency detuning Hz

Fig. 4. (a) CPT absorption spectrum with pump and probe power of 50 μ W each. The linewidth obtained is 400Hz. (b) The MTS spectra with the modulation index of 1 and modulation frequencies of 400 Hz.

the role of the pump beam, and the unmodulated master laser, as the probe beam, are combined with orthogonal polarizations by a polarization beam splitter (PBS). The combined beams are expanded to 12.7 mm in diameter and then incidented into the cell with the light power 50 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. With the 400-Hz modulation turned off, a CPT resonant absorption signal is achieved with direct observation of the PD output signal (Fig.4(a)). With the modulation turned on, the modulation transfer signal can be obtained by demodulating the PD output signal using a lock-in amplifier (Fig.4(b)). The linewidth of the CPT absorption spectrum is measured to be as narrow as 400Hz with the intensity of the superposed Raman lasers less than 100μ W/cm2 , and is mainly broadened by saturation broadening and collision broadening. Fig. 5 shows the MTS spectra with the modulation index of 1 and modulation frequencies of 400 Hz, 1 kHz and 3 kHz respectively. The solid curves are the theoretical predictions with the experimental parameters. And the ”o” curves are the experimental results. It is clear that the experimental results are in good agreement with the theoretical predictions. The most concerning thing in using such dispersion-like spectrum in frequency stabilization is to maximize the gradient of the MTS signal at the center point which determines the frequency discrimination sensitivity. To optimize the gradient of the MTS signal, we theoretically predicted the dependence of signal gradient on the modulation frequency and the modulation index in Fig.6. It indicates that signal gradients can be optimized by using the proper param#108413 - $15.00 USD

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Amplitude V Amplitude V Amplitude V

3 Δ 400 Hz 2 a 1 0 1 2 3 3 Δ 1 kHz 2 b 1 0 1 2 3 3 Δ 3 kHz 2 c 1 0 1 2 3 6000 4000 2000 0 2000 4000 6000

Frequency detuning Hz

Gradient arb.

Fig. 5. The MTS spectra with the modulation index of 1 and the modulation frequencies (and detector phases) of (a) 400 Hz (φ =142º), (b) 1 kHz (φ =97º) and (c) 3 kHz (φ =83º) respectively. The solid lines show the theoretical predictions. The ”o” curves show the experimental results.

1.5

1.25 1.0 1.7 0.75

1.0 0.5

0.5

2.2 3.0 5.0

0.0 0.0

0.5 1.0 1.5 2.0 2.5 Normalized modulation frequency ΔΓac

3.0

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.

eter of the modulation frequency and modulation index. For the rubidium atom used in our experiment, the optimized modulation frequency is about 320 Hz. However, heterodyne detection with this low modulation frequency will be contaminated by acoustic noise. Since the shot noise limitation occurs in detection with high modulation frequency, it is necessary to carefully weigh the benefits of both factors. 4.

Conclusion

We have demonstrated a method for obtaining the MTS spectroscopy of CPT resonance by the example of the 5S1/2 ↔ 5P3/2 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. This work is partially supported by the state Key Development Program for Basic Research of China (No. 2005CB724503, 2006CB921401 and 2006CB921402), and NSFC (No. 10874008). #108413 - $15.00 USD

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