A tripod atomic system driven by two standing-wave fields (a coupling and a driving) is explored to generate tunable double photonic bandgaps in the regime of electromagnetically induced transparency. Both photonic bandgaps depend critically on frequency detunings, spatial periodicities, and initial phases of the two standing-wave fields. When the coupling and driving detunings are very close, a small fluctuation of one standing-wave field may demolish both photonic bandgaps. If the two detunings are greatly different, however, each standing-wave field determines only one photonic bandgap in a less sensitive way. Dynamic generation and elimination of a pair of photonic bandgaps shown here may be exploited toward the end of simultaneous manipulation of two weak light signals even at the single-photon level.
© 2010 Optical Society of America
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 . 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.
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,mF = 1⟩, ∣5S 1/2,F = 1,mF = − 1⟩, ∣5S1/2,F = 1,mF = 1⟩, and ∣5P 3/2,F = 2,mF = 0⟩, respectively. Thus transitions ∣0⟩ ↔ ∣3⟩, ∣1⟩ ↔ ∣3⟩, and ∣2⟩ ↔ ∣3⟩ are electric-dipole allowed and interact with a probe field of frequency ωp and polarization σ -, a coupling field of frequency ωc and polarization σ +, and a driving field of frequency ωd 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(Δp − Δc), γ′20 = γ 20 − i(Δp − Δd), and γ′30 = γ 30 − iΔp are complex dephasing rates of atomic coherences ρ 10, ρ 20, and ρ 30, respectively. Δp = ωp − ω 30 is the detuning of the probe field from transition ∣3⟩ ↔ ∣0⟩, Δc = ωc − ω 31 is the detuning of the coupling field from transition ∣3⟩ ↔ ∣0⟩, while Δd = ωd − ω 32 is the detuning of the driving field from transition ∣3⟩ ↔ ∣2⟩. Ωp = Epd 30/2h̄, Ωc = Ecd 31/2h̄, and Ωd = Edd 32/2h̄ are Rabi frequencies of the probe, coupling, and driving fields, respectively.
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 Rm in the x direction. Then the squared coupling and driving Rabi frequencies vary periodically along x, respectively, as
with initial phases ΦC and Φd at the sample entrance and spatial periodicities ac = λc/2 and ad = λd/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 np experienced by the probe field through We note, however, that the probe refractive index np 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 .
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 ac and ad. Then with the space-dependent refractive index np, we can evaluate a 2×2 unimodular transfer matrix Mn  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 ac = ad = 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 ac ≠ ad, 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 ⋯ Mn⋯ Mn and finally the probe reflectivity and transmissivity
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 ωp ≈ πc/ac ≈ πc/ad, when incident upon such a sample, will experience photonic Bragg scattering through a quasi-periodic structure described by the refractive index np, i.e. a mixture of two periodic structures with similar periodicities ac ≈ ad. Then a pair of PBGs are expected to appear within two space-dependent EIT windows on the probe resonance  contributed by the two SW fields.
We first consider the ideal case of ac = ad 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 Δp = Δc = 0 and Δp = Δd = 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. . 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 Δp = 3 MHz) separating the two EIT windows.
For simplicity without loss of generality, we have set λc = λd 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 ac and ad. We define here a dimensionless ratio
which implies that we can control the scaled periodicity difference g just by arranging angles α and β.
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 Δp = 3 MHz.
In the following we further demonstrate the importance of initial phases Φc and Φd of the coupling and driving fields at the sample entrance. In general we may set Φd = Φc + 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.
Hereinbefore we have verified that, when ∣Δc − Δd∣ 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 ∣Δc − Δd∣ 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 Δp = Δd = −200 MHz and Δp = Δc = 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 Δp ≈ −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 Δp = Δc (Δp = Δd) 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 ∣Δc − Δd∣. 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.
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 ∣Δc − Δd∣ 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 Φc and Φd). 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.
The authors would like to thank the financial supports from NSFC (10874057), NCET (06-0309), NBRP (2006CB921103), and DYSJ (20070121) of P. R. China.
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