1. Introduction

Recently the slow down of optical group velocity in dispersive medium relative to the speed of light in vacuum has attracted great interests, and several approaches of slow light have been reported [1–11]. It is well known that slow light propagation can be achieved using the technique of electromagnetically induced transparency (EIT), and the remarkable experiment of slow light using EIT was made in ultracold atomic gas [1]. As an optional approach to obtain slow light, one can use the spectral hole which is created by a coherent field due to coherent population oscillation (CPO) [5]. Using this approach, Bigelow et al. showed that slow light can be enabled in Ruby at room temperature [6, 7]. Excepting the experiments made in homogeneous broaden medium [1, 6, 7], there are some researches on slow light in inhomogenerously broaden medium [2–4, 8–11]. In the medium of crystal having strong inhomgenerous broadening, Baldit et al. reported group delays of order of 1.1s [8]. In the Doppler-broadened atomic systems, more experiments of slow light propagation were reported using EIT [2–4]. In addition, Agarwal et al. showed that a group index of the order of 10³ can be obtained in the configuration of saturated absorption spectroscopy in Doppler broadened two-level systems [10], and R. M. Camacho et al. reported a modified experimental realization of the proposal of Agarwal [11]. It is noted that the method of CPO in two-level systems requires that the transverse and longitudinal relaxation times of the system are of very different order, and such case is hardly satisfied in atomic systems. In the Doppler broadened atomic systems, these approaches of slow light have some limitations. In the regime of EIT, if the Rabi frequency of the coupling field is much smaller than the Doppler width, one can take the advantage of the Doppler-free scheme to achieve slow light in the thermal gaseous medium. However, the Doppler-free configurations require the wavelengths of probe and coupling waves are very close with each other. Using the method of saturated absorption spectroscopy, one can not obtain large group delay due to the reason that the dispersion is very flat.

In this paper, we theoretically present an approach of slow light based on coherent holeburning (CHB) in a Doppler broadened three-level Λ-type atomic system. The phenomena of CHB has been reported in several Refs.[12]-[14], and here we use the CHB dip at resonance to achieve slow light. Due to the coherent effect of the coupling wave, we can obtain a greater group index than that using the method of saturation absorption spectroscopy in Dopplerbroadened two-level atomic systems, and the wavelength mismatch between the wavelengths of the probe and coupling waves can be very large. Our results should be helpful to realize slow light in diverse Doppler-broadened atomic systems.

2. The model and equations

The atomic system of CHB under consideration is shown in Fig.1(a). It can be described by the three-level Λ-type configuration.A weak probe wave with frequencyω _p and amplitude E _p cou-ples to the atomic transition |3〉-|1〉. A strong coupling wave with frequency ω_c amplitude E_c interacts with another atomic transition |3〉-|2〉. Another saturating wave with frequencyω_s and amplitude E_s, which is much stronger (weaker) than the probe (coupling) wave, also interacts with the transition |3〉-|1〉The transition |3〉-|1〉 and |3〉2〉 are electric dipole allowed, while the transition |2〉-|1〉 is forbidden. Δ_p= ω_p-ω ₃₁, Δ_s= ω_s-ω₃₁, and Δ_c= ω_c-ω ₃₂ are detunings of the probe, the saturating, and coupling waves, respectively. s

 

Fig. 1. (a) Schematic diagram of the three-level Λ-type atomic system driven by the strong coupling wave E_c, the saturating wave E_s and probed by a weak wave E_p. (b) A block diagram of case (i). (c) A block diagram of case (ii).

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It is worthwhile to point out that the system of CHB is different from the two-level systems of CPO [5–9] and saturation absorption spectroscopy [10, 17–19]. In these two-level systems, the saturating wave and the probe wave interact with the same transition, and only the pump effect of the saturating wave has been considered. Here for the system of CHB as shown in Fig.1(a), we not only need to consider the pump effect of the saturating wave, but also investigate the coherent effect of the strong coupling wave which interacts with another transition. In the framework of the semiclassical theory and under the rotating wave approximation, the density matrix equations of motion for the closed system of CHB are given by

where T ₁ and T ₂ present longitudinal and transverse relaxation times of the transition |3〉-|1〉, respectively. The interaction Hamiltonian are given by V ₃₁ = -μ₃₁(E_pe ^-iω_pt+E_se ^-iω_st) and V ₃₂ = -μ₃₂ E_ce ^-iω_ct , where μ₃₁ and μ₃₂ are the dipole matrix elements. The density-matrix equation of motion Eqs.(1) can not be solved in closed form with above interactions.We obtain a solution to the density matrix equation that is correct to all orders in the amplitude of the coupling wave and saturating wave and is correct to the first order in the amplitude of the probe wave. With this approximation, we represent the terms in Eqs.(1) as

which were deduced by Mollow in order to solve the condition where the probe and saturating waves interact with the same transition [15], and here δ = Δ_p -Δ_s. With inserting of Eqs.(2) into Eqs.(1), we obtain the first-order steady state solution of the σ⁺ ₃₁ which gives rise to the response of the system to the probe wave as

where g=μ₃₁ E_p/2h¯, Ω_s =μ₃₁ E_s/2h¯ and Ω_c =μ₃₂ E_c/2h¯ are the Rabi frequencies of the probe, saturating and coupling waves, respectively. For simplicity, we take these Rabi frequencies as real.

To consider the Doppler frequency shift, we substitute Δ_p, Δ_s, and Δ_c with Δ_p +ω_pν/c, Δ_s +α₁ ωsν/c, and Δ_c +α₂ ω_sν/c, where α_i = 1 (α_i = -1) presents that the saturating or coupling wave is co-propagating (counter-propagating) with the probe wave. From Eqs.(3), we obtain the linear susceptibility χ (ω_p) which is to be averaged over the Doppler distribution of velocities as

here N = N ₀(λ_p/2π)³ is the scaled average atomic density, and N ₀ presents atomic density. f(ν) =exp(-ν²/ν² _p)/ν_p √π is the Maxwllian distribution and ν_p = √2kT/M = √2RT/M represents the most probable atomic velocity. In Eq.(4), the real and imaginary parts of χ correspond to the dispersion and absorption, respectively.We introduce the group index n_g = c/ν_g where c is the speed of light in vacuum. The group velocity ν_g is given by [16]

3. Numerical results and discussion

Based on the above formulas in the previous section, we consider the following two cases of CHB:

It should be noted that the wavelengths according to the two transitions are mismatched (ω ₃₂ ≠=ω ₃₁), the Doppler-free configuration can not be fulfilled in case (i). In case (ii), there are no Doppler-free schemes even in the condition that the wavelengths are matched (ω ₃₂ =ω ₃₁)

We assume γ/2π = 1.5 MHz, T ₁ = T ₂/2 = 1/2γ , and choose the modest values for the frequency of the probe wave as ω _p = 2.5×10⁸γ . At first, we display the imaginary and real parts of susceptibility χ (ω_p) of the probe wave in Fig.2. With only the saturating wave copropagating with the probe wave, the imaginary part of χ (ω_p) shows the typical Lamb dip [17–19] at resonance, which is used for slow light by Agarwal et al. [10]. Here we are interested in the burning-hole at resonance in the condition of CHB. Fig.2(a) shows that the hole is narrower than that of the Lamb dip no matter for what cases (case (i) or case (ii)) of CHB. The reason of this phenomenon can be understood by the quantum interference arising from the coupling wave, which is clear in Eqs.(1) that the Rabi frequency of coupling wave (Ω_c) induces the coherence between level |2〉 and |1〉 (σ₂₁). We study the effect of quantum interference of the coupling wave on the dispersion of the weak probe wave in the condition of CHB in Fig.2(b). The results of Fig.2(b) show that the slope of dispersion of CHB is positive and much pronounced around resonance than the slope of Lamb dip.

 

Fig. 2. (Color online) (a) and (b) The imaginary and real parts of susceptibility χ(ω_p). The inset shows a magnified part of the same. Other common parameters of above curses are Δ_c=Δ_s=0, ν_p = 250 ms^-1, ω ₃₂ = 0.9ω ₃₁, atomic density N ₀ = 2×10¹¹ cm^-3, and γ/2π =1.5 MHz.

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Fig. 3. Group index n_g at resonance versus the Rabi frequency of the saturating wave, (a) for case (i) and (b) for case (ii). Other parameters are the same as Fig.2.

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Secondly, we supply the calculated group index n_g at resonance of the probe wave for two cases of CHB as a function of ω_s in Fig.3. In Fig.3(a) of case (i), we can get a large group index when the wavelengths of two transitions become matched. And under the case of ω₃₂ =0.9ω₃₁, we can obtain a nearly ten times larger group index than that of Lamb dip. As a contrary result, in Fig.3(b), it is found that when the frequencies of ω₃₁ and ω₃₂ become much mismatched, the larger group index we obtain. However, the results of Fig.3(a) and Fig.3(b) are consistent, they can be understood that the group index of CHB depends on the quantum interference effect induced by the coupling wave, and this effect is destroyed by the Doppler shifts due to the motions of atoms (k_pν - α₂ k_cν). In case (i), as the two wavelengths of the two transitions become matched, the Doppler shifts as k_pν -k_cν become small, and the effect of quantum interference is enhanced, as a result, the group index becomes large. When the wavelengths are complete matched (ω ₃₂ =ω ₃₁), the Doppler shift is zero, the effect of quantum interference by the coupling wave becomes dominant, and this phenomenon is EIT [20]. The Doppler shifts of case (ii) is k_pν+k_cν, this value can only be smaller as the of k_c becomes smaller, this is agree with results of Fig.3(b) that the smaller the frequency of ω ₃₂ is, the larger group index we can have. And it should be noted that due to the quantum interference of the coupling wave, we can get a lager group index in the two cases of CHB than that of saturated absorption spectroscopy.

In the following, we also consider the propagation of a Gaussian pulse through the sample having a length L=1cm in order to confirm the above results. The solution of the field profile for the probe pulse is [21]

where $\epsilon \left(0,{\omega }_p\right)=\frac{\tau }{{\left(2\pi \right)}^{\frac{1}{2}}}\exp \left[-\frac{1}{2}{\left({\omega }_p-{\omega }_{31}\right)}^{2_{\tau }2}\right]$ is the Fourier transform of the probe field at the entrance of the cell, z is the propagation distance, and n(ω_p)=√1+χ(ω_p) is the refractive index. For probe pulses with τ =2μs, we show the delays due to the medium in Fig.4. From the relative delay between the reference pulse and the output pulse, we calculate the group velocity of the pulse, and find it is consistent with the results as shown in Fig.3. The spectral width of the Gaussian pulses are well contained within the region of the burning-hole at resonance of CHB in the medium, so there are no distortions for the propagation of the pulse. And the transmission of Gaussian pulse in the condition of CHB is the same as that of saturated absorption spectroscopy. If the spectral width of the pulse becomes too broad relative to the width of the dip of CHB at resonance, the results of Eq.(5) does not hold, and Eq.(6) has to be used to calculate the output pulse numerically.

 

Fig. 4. The solid curve presents the Gaussian pulse propagating at speed c through 1 cm of vacuum. The dash-dot curse shows the same pulse in case (i) of CHB through a atomic medium of length 1cm having the energy levels as ω₃₂ = 0.9ω₃₁. The dash curse shows the pulse in case (ii) of CHB through a atomic medium of length 1cm having the energy levels as ω₃₂ = 0.01ω₃₁. Δ_s = Δ_c = 0 and other parameters are the same as Fig.2. The inset shows a magnified part of the same.

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4. Conclusion

In summary, we theoretically present a calculation of slow light based on CHB in a Dopplerbroadened Δ-type atomic medium without the Doppler-free configurations. We consider two cases of the laser arrangements of CHB, and find that in the both cases, due to the quantum interference effect arising from the coupling wave, we can obtain larger group index than that of saturated absorption spectroscopy in two-level atomic systems. Our results should be helpful to realize slow light propagation in various Doppler-broadened atomic systems.