1. Introduction

Techniques to manipulate propagation dynamics of light signals have been attracting considerable interests for their importance in both fundamental science and practical applications. One example is electromagnetically induced transparency (EIT) [1, 2] which can be used to eliminate the resonant absorption of a laser beam incident upon a coherently dressed medium with appropriate energy levels. Using EIT techniques, one may greatly slow down the moving optical pulses [3–5] and even stop them to attain reversible storage and retrieval of light signals [6–9]. One EIT prototype is a three-level Lambda system driven by a weak probe field and a strong coupling field both in the traveling-wave (TW) configuration. Yet when a standing-wave (SW) coupling field [10, 11] is applied instead, the dressed medium may work as an one-dimension photonic crystal because its probe refractive index is periodically modulated in space. This realizes in fact energy bands separated by a region in which light signals cannot propagate, i.e. the so-called photonic bandgap (PBG) [12–14].

Several studies have been done to generate a tunable PBG in cold atomic samples [15–18] by controlling the parameters of a SW coupling in the EIT regime. Similarly works have also been implemented in impurity doped solid materials such as Pr³⁺: Y₂SiO₅ and diamond containing nitrogen-vacancy (N-V) color centers [19, 20] where the dynamically induced PBG can be attained even in the presence of inhomogeneous broadening. The active control of PBG structures by pure optical approaches, instead of growing crystals with predetermined PBGs once and for all, has important applications in quantum nonlinear optics and quantum information processing [21, 22]. One example is to realize a micro cavity sandwiched between two dynamically induced Bragg mirrors to confine, manipulate, and release a slow-light pulse on demand [23]. Another example is to serve as an efficient all-optical two-port switching and routing scheme for weak light signals even at the single-photon level [24, 25]. To the best of our knowledge, however, no studies have been done in the SW-EIT regime to simultaneously generate two or more PBGs in atomic or solid media.

 

Fig. 1. (color online) Schematic diagram of a four-level tripod-type atomic system dressed by a standing-wave coupling laser ω_c and a standing-wave driving laser ω_d. Level ∣ 0⟩ is the only populated one in the limit of a weak probe as denoted by the filled circles.

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In this paper we extend the dynamically induced PBG study to a four-level tripod system [26–29] of cold atoms dressed by a SW coupling field and a SW driving field. The refractive index experienced by a probe field is space-dependent in a rather complicated way, i.e. not periodic even on the cm scale, when the two SW fields have different periodicities. Thus, in numerical calculations, we have to first partition the sample into a large number of laminas and then derive the total transfer matrix of the whole medium by successively multiplying the transfer matrices of each lamina. We find that it is viable to obtain a pair of tunable PBGs inside two space-dependent EIT windows on the probe resonance, whose locations are determined by the coupling and driving detunings, respectively. In particular each bandgap is simultaneously governed by both SW fields if they have similar frequency detunings. In this case well developed double PBGs can only be attained when spatial periodicities of the two SW fields are close enough, and likewise for their initial phases. If the coupling detuning is greatly different from the driving detuning, however, it is possible to control one bandgap by manipulating a corresponding SW field without large influence on the other bandgap, and both bandgaps become less sensitive to the field parameters such as spatial periodicities and initial phases.

2. Theoretical Model

We consider here a four-level tripod system [26–29] referring to the D2 line of cold ⁸⁷Rb atoms as shown in Fig. 1 where levels ∣0⟩, ∣1⟩, ∣2⟩, and ∣3⟩ may correspond to the hyperfine states ∣5S _1/2,F = 2,m_F = 1⟩, ∣5S _1/2,F = 1,m_F = − 1⟩, ∣5S1/2,F = 1,m_F = 1⟩, and ∣5P _3/2,F = 2,m_F = 0⟩, respectively. Thus transitions ∣0⟩ ↔ ∣3⟩, ∣1⟩ ↔ ∣3⟩, and ∣2⟩ ↔ ∣3⟩ are electric-dipole allowed and interact with a probe field of frequency ω_p and polarization σ ^-, a coupling field of frequency ω_c and polarization σ ⁺, and a driving field of frequency ω_d and polarization σ ^-, respectively. Moreover we assume that level ∣0⟩ is the only populated atomic state at the initial time, which holds true at any time if the probe is very weak compared with the other two fields.

In the limit of a weak probe, we can analytically solve the Liouville equations in the steady state to have the off-diagonal density matrix element

where γ′₁₀ = γ ₁₀ − i(Δ_p − Δ_c), γ′₂₀ = γ ₂₀ − i(Δ_p − Δ_d), and γ′₃₀ = γ ₃₀ − iΔp are complex dephasing rates of atomic coherences ρ ₁₀, ρ ₂₀, and ρ ₃₀, respectively. Δ_p = ω_p − ω ₃₀ is the detuning of the probe field from transition ∣3⟩ ↔ ∣0⟩, Δ_c = ω_c − ω ₃₁ is the detuning of the coupling field from transition ∣3⟩ ↔ ∣0⟩, while Δ_d = ω_d − ω ₃₂ is the detuning of the driving field from transition ∣3⟩ ↔ ∣2⟩. Ω_p = E_pd ₃₀/2h̄, Ω_c = E_cd ₃₁/2h̄, and Ω_d = E_dd ₃₂/2h̄ are Rabi frequencies of the probe, coupling, and driving fields, respectively.

 

Fig. 2. (color online) Photonic bandgap structure (left) and reflection and transmission spectra (right) for an cold atomic sample of density N ₀ = 1.0 × 10¹² cm^-3 and length L = 2.0 cm. Other parameters are λ_p = 780.792 nm, λ_c = λ_d = 780.778 nm, γ ₁₀ = γ ₂₀ = 1.0 kHz, γ ₃₀ = 6.0 MHz, Ω_c0 = Ω_d0 = 60.0 MHz, Δ_c = 0, Δ_d = 6.0 MHz, R_m = 0.82, α = β = 0, and δ = 0.

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In the following, we just consider the case where both coupling and driving fields are in the SW pattern as generated from retro-reflecting upon impinging on a mirror of reflectivity R_m in the x direction. Then the squared coupling and driving Rabi frequencies vary periodically along x, respectively, as

with initial phases Φ_C and Φ_d at the sample entrance and spatial periodicities a_c = λ_c/2 and a_d = λ_d/2 inside the sample. The forward (FD) and backward (BD) beams of the coupling and driving fields may be misaligned through two small angles α and β so that the spatial periodicities change, respectively, into

The linear susceptibility determining the probe response is proportional to ρ ₃₀ in the form of

with N ₀ being the atomic density. Then we can attain the space-dependent refractive index n_p experienced by the probe field through $n_p=\sqrt{1+{\chi }_p}.$We note, however, that the probe refractive index n_p may not be periodic along the x direction, even on the cm scale, due to the mutual influence of the two SW fields of different periodicities. This is unlike the simpler case for a Lambda EIT system in the presence of a single SW field [16].

 

Fig. 3. (color online) Probe transmission (left) and reflection (right) spectra for the same atomic sample as in Fig. 2 Black-solid curves are obtained when the two SW fields have different periodicities (upper:α = 0 and β = 1 mrad; middle: α = 0 and β = 3 mrad; lower: α = 0 and β = 10 mrad), while red-dashed curves correspond to α = β = 0 in all panels. Other parameters are the same as in Fig. 2.

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To examine the potential double PBGs on the probe resonance due to nonlinear Bragg scattering, we first partition the atomic sample of length L into N laminas of thickness d, which should be much smaller than the spatial periodicities a_c and a_d. Then with the space-dependent refractive index n_p, we can evaluate a 2×2 unimodular transfer matrix M_n [30] describing the propagation of a monochromatic probe field through the nth atomic lamina via

where E ⁺ and E ^- denote the FD and BD probe electric fields, respectively. In a periodic medium with a_c = a_d = a, Eq. (5) is further restricted by the following Bloch condition

which then allows us to check the expected double PBGs via the complex Bloch wave vector κ = κ′ + iκ”. In a quasi-periodic medium with a_c ≠ a_d, however, it is unsuited to describe the PBG structure with κ and only the reflection and transmission spectra can be used to verify the existence of two dynamically induced PBGs. Multiplying transfer matrices of all partitioned laminas, we can attain further the total transfer matrix M = M ₁ ⋯ M_n⋯ M_n and finally the probe reflectivity and transmissivity

with M_ij being one matrix element of M. Note that Eqs. (6), (7) are the basis of all numerical calculations done in the next section.

 

Fig. 4. (color online) Probe transmission (left) and reflection (right) spectra for the same atomic sample as in Fig. 2. Black-solid curves correspond to δ = π/100, π/20, π/2 in the upper, middle, and lower panels, respectively, while red-dashed curves correspond to δ = 0 in all panels. Other parameters are the same as in Fig. 2.

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3. Numerical Results

We report in this section our numerical results about the steady optical response of a cold atomic sample dressed by a SW coupling and a SW driving in the tripod configuration (see Fig. 1). A probe field with frequency ω_p ≈ πc/a_c ≈ πc/a_d, when incident upon such a sample, will experience photonic Bragg scattering through a quasi-periodic structure described by the refractive index n_p, i.e. a mixture of two periodic structures with similar periodicities a_c ≈ a_d. Then a pair of PBGs are expected to appear within two space-dependent EIT windows on the probe resonance [31] contributed by the two SW fields.

We first consider the ideal case of a_c = a_d to let the quasi-periodic structure degenerate into a periodic one. As we can see from Fig. 2, the probe field experiences a pair of rather good PBGs within two adjacent EIT windows centered at Δ_p = Δ_c = 0 and Δ_p = Δ_d = 6 MHz, respectively. The corresponding reflectivities are found to be homogeneously over 95% inside the two PBGs. Both bandgaps can be easily tuned in widths (positions) by changing Rabi frequencies (frequency detunings) of the two SW fields as in Ref. [16]. In addition, the left bandgap is much narrower than the right one because its development toward right is severely restricted by the absorption peak (centered at Δ_p = 3 MHz) separating the two EIT windows.

For simplicity without loss of generality, we have set λ_c = λ_d in Fig. 2 so that the difference between periodicities ac and ad are just determined by misalignments α and β [see Eqs. (3)]. In practice, α and β are difficult to be kept exactly equal, so it is reasonable to set a small difference between a_c and a_d. We define here a dimensionless ratio

to denote the periodicity difference, which is scaled to have values between 0 and 1. When Eqs. (3) are inserted into, Eq. (8) turns out to be

which implies that we can control the scaled periodicity difference g just by arranging angles α and β.

 

Fig. 5. Probe reflection (black-solid) and transmission (red-dashed) spectra for the same atomic sample as in Fig. 2. We have set Δ_d = −200 MHz, Δ_c = 10 MHz, and α = 3 mrad (upper); α = 6 mrad (middle); α = 10 mrad (lower). Other parameters are the same as in Fig. 2.

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The PBG structure shown in Fig. 2 changes little when g is very small, namely when α and β are close enough. For example, the curves for g = 6.25 × 10^-8 (α = 0 and β = 1 mrad) and those for g = 0 (α = β = 0) completely overlap in the two upper panels of Fig. 3. As we gradually increase g, however, the resulting spectra of reflection and transmission deviate more and more from those for g =0 (see the middle and lower panels with g = 5.63 × 10^-7and g = 6.25 × 10^-6, respectively). This implies that the double PBGs will experience more and more breakage when a periodic structure gradually evolves into a quasi-periodic one. In particular, the left narrower bandgap is found to suffer more breakage than the right wider one, which is also true if we change α but fix β (not shown). Thus we may conclude that the periodicity difference is critical for both dynamically induced PBGs and its increasing influences more the narrower bandgap restricted by the absorption peak at Δ_p = 3 MHz.

In the following we further demonstrate the importance of initial phases Φ_c and Φ_d of the coupling and driving fields at the sample entrance. In general we may set Φ_d = Φ_c + δ with 0 ≤ 5 < 2π. As we can see from the reflection and transmission spectra in Fig. 4, both PBGs are kept well developed when the phase difference δ is very small but becomes more and more malformed when δ is gradually increased. This situation is quite similar to that in Fig. 3 where g is increased instead. The underlying physics is that a larger δ or g will result in a stronger mutual influence between the coupling and driving fields due to the spatial separation of their nodes and antinodes. In addition, the left narrower bandgap experiences, once again more breakage than the right wider one.

 

Fig. 6. Probe reflection (black-solid) and transmission (red-dashed) spectra for the same atomic sample as in Fig. 2. We have set Δ_d = −200 MHz, Δ_c = 10 MHz, and δ = 0 (upper); δ = π/6 (middle); δ = π/2 (lower). Other parameters are the same as in Fig. 2.

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Hereinbefore we have verified that, when ∣Δ_c − Δ_d∣ is very small, the coupling and driving fields interact so strongly that steady optical properties of the tripod system are very sensitive to their parameters variation. Now we show that, if ∣Δ_c − Δ_d∣ is large enough, the reflection and transmission spectra in a small frequency region of interest may become rather dull to modulations of the periodicity difference g and the phase difference δ. In Fig. 5, we observe once again a pair of fully developed PBGs, which are generated within two well-separated EIT windows centered at Δ_p = Δ_d = −200 MHz and Δ_p = Δ_c = 10 MHz, respectively. It is clear that the two bandgaps have quite similar widths because both of them experience little restriction from the absorption peak centered at Δ_p ≈ −95 MHz. When we modulate the periodicity difference g by changing α, only the right bandgap is largely modified while the left bandgap seems unchanged. In stead if we change β but fix α, it is the left bandgap that is largely modified with the right bandgap unchanged (not shown). This means that the bandgap near Δ_p = Δ_c (Δ_p = Δ_d) is essentially contributed and controlled by the coupling (driving) field and the mutual influence between the two SW fields is very weak in the case of a large ∣Δ_c − Δ_d∣. This is further confirmed by Fig. 6 where the modulation of δ results in little change inside both PBGs though more oscillations are found at the right wing of the right bandgap when δ becomes larger.

4. Conclusions

In summary, we have investigated a four-level tripod-type atomic system driven by two SW fields propagating in the same direction. It is found that two PBGs located at different positions may be simultaneously induced and well developed on the probe resonance. Specifically, we have examined two different cases where the absolute detuning difference ∣Δ_c − Δ_d∣ is either very small or large enough. For the former, each bandgap is controlled by both SW fields and seems quite sensitive to the periodicity difference g (misalignments α and β) and the phase difference δ (initial phases Φ_c and Φ_d). That is, both g and δ should be kept very stable to have fully developed double PBGs and, when we tune the coupling or driving field to manipulate one bandgap, it is inevitable to significantly alter the other bandgap. For the latter, each SW field can effectively control a single bandgap, which then allows us to manipulate the two bandgaps, respectively. In addition, both bandgaps seem more stable in this case with respect to parameter fluctuations of the two SW fields. We expect that these new findings be instructive to devise novel photonic devices, e.g. all-optical switching and routing, for simultaneous information processing of two weak light signals.