1. Introduction

Under the condition of electromagnetically induced transparency (EIT) [1, 2], laser induced quantum coherence in atomic samples has been extensively studied both experimentally and theoretically, e.g., to enhance resonant optical nonlinearities with little absorption [3, 4, 5, 6], to manipulate the propagation dynamics of weak light pulses [7, 8, 9], to achieve quantum storage and retrieval of photonic states [10, 11, 12, 13, 14], etc. These applications of laser induced quantum coherence are essential and stimulating for the further developments of quantum nonlinear optics and quantum information science as well as other related fields in physics. Originally, laser fields dressing atomic samples are always set in the traveling-wave (TW) pattern to establish spatially homogeneous quantum coherence. Recently, the standing-wave (SW) driving configuration has been well adopted to induce spatially periodic quantum coherence, which is advantageous for both the coherent generation of photonic bandgaps (PBGs) [15, 16, 17, 18, 19] and the dynamic creation of stationary light pulses (SLPs) [20, 21, 22, 23, 24]. All-optically controlled PBGs and SLPs in atomic samples represent, in fact, an important progress for achieving the flexible manipulation of photonic flow and interaction [25, 26] as required in classical and quantum information processing.

So far atomic samples dressed by TW or SW laser fields are always assumed to have spatially homogeneous densities for the simplicity in theoretical treatments, which is not true at least for cold atoms confined in magneto-optical traps (MOT). It is well known that cold atomic clouds in MOT usually have the Gaussian [27] or cigar-shaped [28] profiles in spatial density. In this paper, we investigate by numerical calculations how the spatial inhomogeneity affects both steady and dynamic optical properties of cold atoms driven (by a weak probe and a strong coupling) into the EIT regime. Steady spectra of probe reflectivity and transmissivity are examined by the transfer-matrix method with the probe susceptibility analytically solved from density matrix equations while the propagation dynamics of an incident pulse is explored via the Fourier-transform method starting from the probe reflectivity and transmissivity.

Our numerical results show that, in the TW driving configuration, the probe transmissivity characterized by an EIT window is irrelevant to the spatial inhomogeneity of cold atoms. Therefore the transmitted pulses are not distinguishable for three typical samples (one homogeneous and two inhomogeneous) with the same amounts of cold atoms but very different density functions. In the SW driving configuration, however, both spectra of probe reflectivity and transmissivity are very sensitive to the spatial inhomogeneity of cold atoms. For instance, coherently induced PBGs denoted by a platform of reflectivity near 100% become narrower and narrower when the inhomogeneity of cold atoms is gradually increased. Consequently, an incident pulse perfectly reflected by the homogeneous sample will be split into a partially reflected one and a partially transmitted one in the two inhomogeneous samples with the same driving parameters. The underlying physics is that constructive and destructive interferences between forward (FW) and backward (BW) probe photons (scattered by an atomic grating generated by the SW coupling) depend critically on the spatial inhomogeneity of cold atoms. These results are expected to be instructive for the design of relevant experiments for light information manipulation in cold atomic samples driven by at least one SW field.

2. Model and Equations

We consider here a simplified three-level Λ system as shown in Fig. 1. The transition from level |1〉 to level |3〉 is probed by a weak light beam of frequency ω_p and amplitude E_p while the transition from level |2〉 to level |3〉 is coupled by a strong light beam of frequency ω_c and amplitude E_c, which may be either in the TW pattern or in the SW pattern. Under the electric-dipole and rotating-wave approximations, we can write the interaction Hamiltonian of this coherently driven Λ system as

 

Fig. 1 (Color online) Schematic diagram of a three-level Λ-type atomic system interacting with a weak probe field ω_p and a strong coupling field ω_c. The coupling field may be either in the TW pattern or in the SW pattern.

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Starting from Eq. (1) and Eq. (2), it is straightforward to attain the following (expanded) Liouville equations for density matrix elements in the interaction picture

In the limit of a weak probe, we are allowed to assume ρ₁₁ = 1 and ρ₂₂ = ρ₃₃ = ρ₂₃ = 0 so that the steady-state solution of ρ₃₁ can be found from Eqs. (3) to the first order in Ω_p, from which the probe susceptibility may be written as

Due to joint modulations of the spatially periodic coupling field and the spatially inhomogeneous atomic density, the complex refractive index n_p (Δ, z) in Eq. (5) becomes very complicated so that the derivation of relevant transmission and reflection spectra are rather intractable. That is, we have to first partition the atomic sample into a large number of laminas, then derive the individual transfer matrix of each lamina with the complex refractive index, and finally multiply the individual transfer matrices of all laminas in succession to attain the total transfer matrix of the whole sample [29]. To be specific, the propagation of a probe field through the nth atomic lamina (extending from z = nd − d to z = nd) is described by a 2 × 2 unimodular transfer matrix M_n (Δ, d) via

Now we briefly introduce the Fourier transform method [19] for studying the propagation dynamics of an incident light pulse with r(Δ, L) and t(Δ, L) given in Eqs. (7). In the weak probe limit, the linear response functions r(Δ, L) and t(Δ, L) contain all relevant information on the optical responses of the atomic system under consideration to a monochromatic probe field. The basic procedure for the Fourier transform method is: first to write the incident pulse in the time domain E_It (t) and decompose it into the Fourier components in the frequency domain E_If (Δ); then to multiply the incident Fourier components E_If (Δ) with r(Δ, L) and t(Δ, L) to attain the reflected and transmitted Fourier components E_Rf (Δ) = E_If (Δ) ·r(Δ, L) and E_Tf (Δ) = E_If (Δ) ·t(Δ, L); finally to perform the inverse Fourier transform so that the reflected pulse E_Rt (t) at the sample entrance (z = 0) and the transmitted pulse E_Tt (t) at the sample exit (z = L) can be reconstructed as

3. Results and Discussions

With the formulas developed in the last section, we now perform numerical calculations to investigate the probe spectra of reflectivity and transmissivity as well as the propagation dynamics of an incident pulse in one homogeneous sample and two inhomogeneous samples of cold atoms. The three samples under consideration have the same medium length L = 1.5 mm, the same average atomic density N₀ = 2.0 × 10¹⁰ mm⁻³, but very different density functions N_ai(z), which is constrained by ${\int }_0^L{N}_{ai}\left(z\right)dz=N_{0}L$ (see Fig. 2). In particular, the two inhomogeneous samples have either the sinusoid profile or the Gaussian profile as denoted by the caption of Fig. 2 and the latter is quite typical for cold atomic clouds in MOT [27].

 

Fig. 2 (Color online) Atomic density functions N_ai(z) versus the spatial variable z. Black-solid: N_a1(z) = N₀; Red-dashed: N_a2(z) = N′ sin(πz/L) with N′ = N₀π/2; Blue-dotted: N_a3(z) = N″ exp[−20(z − L/2)²/L²] with $N^{″}=N_0\sqrt{20/\pi }$. The three samples have the same medium length L = 1.5 mm and the same average atomic density N₀ = 2.0 × 10¹⁰ mm⁻³.

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First we consider the simple case where a TW coupling field is switched on, which can be examined by setting Ω_c− = 0 in Eq. (2). We see from Fig. 3(a) that the transmission spectra with a characteristic EIT window are clearly independent of atomic density functions as long as all three samples contain the same amounts of cold atoms. Fig. 3(b) further shows that the three transmitted pulses with a remarkable time delay (relative to the incident pulse) are almost indistinguishable for different atomic density functions constrained by ${\int }_0^L{N}_{ai}\left(z\right)dz=N_{0}L$. Thus we may conclude that, in the TW driving configuration, it is a good choice to simplify relevant calculations and analyses by ignoring the spatial inhomogeneity of real atomic samples. Note, however, that the pulse distribution in the homogeneous sample is distinct from those in the inhomogeneous samples because the pulse intensity and the group velocity at a given spatial position are inversely proportional to the local atomic density.

 

Fig. 3 (Color online) Transmissivity versus probe detuning Δ_p (a) and scaled intensities versus time t (b) with d₁₃ = 1.0375 × 10⁻²⁹ C·m, λ_p = 794.983 nm, λ_c = 794.969 nm, γ₂₁ = 0.2 kHz, γ₃₁ = 5.75 MHz, Δ_c = 0, Ω_c+ = 10 MHz, and Ω_c− = 0. The black-solid, red-dashed, and blue-dotted curves are respectively obtained with N_a1(z), N_a2(z), and N_a3(z). The thin grey curve in (b) denotes the incident pulse with δt = 20 μs, t₀ = 35 μs, and Δ₀ = 0.

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Then we consider the complicated case where a SW coupling field is switched on, i.e. both Ω_c+ and Ω_c− are nonzero in Eq. (2). We see from Fig. 4(a) that three PBGs, characterized by a platform of reflectivity over 95%, are well developed around the probe resonance with suitably chosen parameters. But these dynamically induced PBGs have quite different widths indicating that they are very sensitive to atomic density functions. Fig. 4(b) further shows that the three transmission spectra depend critically on atomic density functions due to their correlation with the three reflection spectra in Fig. 4(a). In Fig. 4(c) and 4(d), we examine instead the propagation dynamics of a probe pulse with its most carrier frequencies fallen into the widest PBG in Fig. 4(a). It is clear that the incident pulse is perfectly reflected with a short time delay and experiences little energy loss and profile deformation in the homogeneous sample. For the inhomogeneous samples, however, the reflected pulses become depleted and distorted more or less due to the loss of some carrier frequencies. Among the lost carrier frequencies, most make up the transmitted pulses with a long time delay at the sample exits while little are indeed absorbed because we always work in the regime of EIT. It is clear that, in the SW driving configuration, the spatial inhomogeneity of real atomic samples have to be duly addressed, otherwise theoretical predictions may deviate largely from relevant experimental results.

 

Fig. 4 (Color online) Reflectivity and transmissivity versus probe detuning Δ_p (a, b) and scaled intensities versus time t (c, d). Black-solid, red-dashed, and blue-dotted curves are respectively obtained with N_a1(z), N_a2(z), and N_a3(z). Thin grey curves in (c, d) denote the incident pulse with δt = 10 μs, t₀ = 15 μs, and Δ₀ = −0.72 MHz. Other parameters are the same as in Fig. 3 except Ω_c+ = 30 MHz and Ω_c− = 20 MHz.

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It is straightforward to understand why the inhomogeneity of cold atomic samples is trivial in the TW driving configuration: the probe susceptibility χ_p depends on the spatial variable z only through the atomic density N_a(z) so that the probe transmissivity is proportional to ∫N_a(z)dz (i.e. the total amounts of cold atoms) while the probe reflectivity is always negligible. In the SW driving configuration, however, it is not so simple to explain the importance of the inhomogeneity of cold atomic samples. In this case, the probe susceptibility χ_p depends on z not only through N_a(z) but also through Ω_c(z), which makes both transmissivity and reflectivity significant around the probe resonance. In particular, dynamically induced PBGs may be observed as a platform of high reflectivity when perfect constructive (destructive) interference occurs between the BW (FW) probe photons scattered by an atomic grating generated by the SW coupling. In a homogeneous sample, different lattices of the atomic grating are exactly the same in optical responses and thus have identical scattering abilities, which is favorable for achieving perfect interference between the probe photons in a large spectral region. When the sample becomes inhomogeneous, however, each lattice of the atomic grating may be very different from others in optical responses, especially in scattering abilities, so that perfect interference can only be achieved between the probe photons in a small spectral region. This is why distinct steady optical spectra and light propagation dynamics have been observed in Fig. 4 for three samples with the same amounts of cold atoms but very different density functions.

One may wonder whether can we compensate the undesirable effect of the spatial inhomogeneity of cold atomic samples to attain wide enough PBGs? Fig. 5 shows that, for the two inhomogeneous samples with fixed medium lengths and density functions, it is viable to widen the dynamically induced PBGs while maintaining their high reflectivities by suitably increasing Rabi frequencies Ω_c+ and Ω_c− of the FW and BW coupling beams. It is also clear that the amended PBGs (red-dashed) in Fig. 5(a) and Fig. 5(b) well match the widest PBG (black-solid) in Fig. 4(a) as far as widths and heights are concerned.

 

Fig. 5 (Color online) Reflectivity versus probe detuning Δ_p for N_a2(z) in (a) and N_a3(z) in (b). The black-solid curves are attained with Ω_c+ = 30 MHz and Ω_c− = 20 MHz. The red-dashed curve in (a) is attained with Ω_c+ = 35 MHz and Ω_c− = 24 MHz while that in (b) is attained with Ω_c+ = 41 MHz and Ω_c− = 30 MHz. Other parameters are the same as in Fig. 3.

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4. Conclusions

In summary, we have calculated steady optical spectra and light propagation dynamics in three samples of cold atoms with either homogeneous or inhomogeneous density functions. These samples are assumed to be driven into the EIT regime by a weak probe and a strong coupling in the Λ configuration. When the coupling is a TW field, the transmissivity spectra characterized by typical EIT windows are exactly identical for the three samples with the same amounts of cold atoms but very different spatial inhomogeneity while the reflectivity spectra are not necessary to be taken into account. Consequently, an incident pulse with its most carrier frequencies fallen into the overlapped EIT windows will go through the three samples with the same time delay while suffering little energy loss and profile deformation. The spatial inhomogeneity of cold atomic samples becomes rather important when the coupling is a SW field instead. That is, both transmissivity and reflectivity spectra may change dramatically if the spatial inhomogeneity of a sample is increased or decreased. In particular, the dynamically induced PBG for the homogeneous sample looks much wider than those for the two inhomogeneous samples. As a result, an incident pulse perfectly reflected by the homogeneous sample is seen to split into a partially reflected one and a partially transmitted one when impinging upon the two inhomogeneous samples. Qualitative analyses have been given in terms of constructive (destructive) interference between the BW (FW) probe photons scattered by an atomic grating generated by the SW coupling. These findings should be helpful to amend relevant theoretical results attained without considering the spatial inhomogeneity of real atomic samples coherently driven by one or more SW fields.