## Abstract

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

## 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 Pr^{3+}: Y_{2}SiO_{5} 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.

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 ^{87}Rb atoms as shown in Fig. 1 where levels ∣0⟩, ∣1⟩, ∣2⟩, and ∣3⟩ may correspond to the hyperfine states ∣5*S*
_{1/2},*F* = 2,*m _{F}* = 1⟩, ∣5

*S*

_{1/2},

*F*= 1,

*m*= − 1⟩, ∣5

_{F}*S*1/2,

*F*= 1,

*m*= 1⟩, and ∣5

_{F}*P*

_{3/2},

*F*= 2,

*m*= 0⟩, respectively. Thus transitions ∣0⟩ ↔ ∣3⟩, ∣1⟩ ↔ ∣3⟩, and ∣2⟩ ↔ ∣3⟩ are electric-dipole allowed and interact with a probe field of frequency

_{F}*ω*and polarization

_{p}*σ*

^{-}, a coupling field of frequency

*ω*and polarization

_{c}*σ*

^{+}, and a driving field of frequency

*ω*and polarization

_{d}*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

*− Δ*

_{p}*), and γ′*

_{d}_{30}=

*γ*

_{30}−

*i*Δ

*p*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*

_{p}d_{30}/2

*h*̄, Ω

*=*

_{c}*E*

_{c}d_{31}/2

*h*̄, and Ω

*=*

_{d}*E*

_{d}d_{32}/2

*h*̄ 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 *R _{m}* in the

*x*direction. Then the squared coupling and driving Rabi frequencies vary periodically along

*x*, respectively, as

$${\phantom{\rule{.2em}{0ex}}\Omega}_{d}^{2}={\Omega}_{d0}^{2}\left[{\left(1+\sqrt{{R}_{m}}\right)}^{2}{\mathrm{cos}}^{2}\left({k}_{d}x+{\Phi}_{d}\right)+{\left(1-\sqrt{{R}_{m}}\right)}^{2}{\mathrm{sin}}^{2}\left({k}_{d}x+{\Phi}_{d}\right)\right]\phantom{\rule{.2em}{0ex}}$$

with initial phases Φ* _{C}* and Φ

*at the sample entrance and spatial periodicities*

_{d}*a*=

_{c}*λ*/2 and

_{c}*a*=

_{d}*λ*/2 inside the sample. The forward (FD) and backward (BD) beams of the coupling and driving fields may be misaligned through two small angles

_{d}*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*may not be periodic along the

_{p}*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].

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*. Then with the space-dependent refractive index

_{d}*n*, we can evaluate a 2×2 unimodular transfer matrix

_{p}*M*[30] describing the propagation of a monochromatic probe field through the

_{n}*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*, however, it is unsuited to describe the PBG structure with

_{d}*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*and finally the probe reflectivity and transmissivity

_{n}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.

## 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*, when incident upon such a sample, will experience photonic Bragg scattering through a quasi-periodic structure described by the refractive index

_{d}*n*, i.e. a mixture of two periodic structures with similar periodicities

_{p}*a*≈

_{c}*a*. 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.

_{d}We first consider the ideal case of *a _{c}* =

*a*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

_{d}*= Δ*

_{p}*= 0 and Δ*

_{c}*= Δ*

_{p}*= 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*

_{d}*= 3 MHz) separating the two EIT windows.*

_{p}For simplicity without loss of generality, we have set *λ _{c}* =

*λ*in Fig. 2 so that the difference between periodicities ac and ad are just determined by misalignments

_{d}*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*and

_{c}*a*. We define here a dimensionless ratio

_{d}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*.

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^{-7}and *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

*of the coupling and driving fields at the sample entrance. In general we may set Φ*

_{d}*= Φ*

_{d}*+*

_{c}*δ*with 0 ≤ 5 < 2

*π*. 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 ∣Δ* _{c}* − Δ

*∣ 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*

_{d}*− Δ*

_{c}*∣ 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*

_{d}*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

*= Δ*

_{p}*= −200 MHz and Δ*

_{d}*= Δ*

_{p}*= 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*

_{c}*≈ −95 MHz. When we modulate the periodicity difference*

_{p}*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

*= Δ*

_{p}*(Δ*

_{c}*= Δ*

_{p}*) 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*

_{d}*− Δ*

_{c}*∣. This is further confirmed by Fig. 6 where the modulation of*

_{d}*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

*∣ 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*

_{d}*g*(misalignments

*α*and

*β*) and the phase difference

*δ*(initial phases Φ

*and Φ*

_{c}*). That is, both*

_{d}*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.

## Acknowledgments

The authors would like to thank the financial supports from NSFC (10874057), NCET (06-0309), NBRP (2006CB921103), and DYSJ (20070121) of P. R. China.

## References and links

**1. **S. E. Harris, “Electromagnetically induced transparency,” Phys. Today **50** (7), 36 (1997). [CrossRef]

**2. **M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. **77**, 633 (2005). [CrossRef]

**3. **L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London) **397**, 594 (1999). [CrossRef]

**4. **M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. **82**, 5229 (1999). [CrossRef]

**5. **S.-M. Ma, H. Xu, and B. S. Ham, “Electromagnetically-induced transparency and slow light in GaAs/AlGaAs multiple quantum wells in a transient regime,” Opt. Express **17**, 14902 (2009). [CrossRef] [PubMed]

**6. **M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. **84**, 5094 (2000). [CrossRef] [PubMed]

**7. **C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulsed,” Nature (London) **409**, 490 (2001). [CrossRef] [PubMed]

**8. **J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, “Stopped light with storage time greater than one second using electromagnetically induced transparency in a solid,” Phys. Rev. Lett. **95**, 063601 (2005). [CrossRef] [PubMed]

**9. **H.-H. Wang, X.-G. Wei, L. Wang, Y.-J. Li, D.-M. Du, J.-H. Wu, Z.-H. Kang, Y. Jiang, and J.-Y. Gao, “Optical information transfer between two light channels in a Pr^{3+}:Y_{2}SiO_{5} crystal,” Opt. Express **15**, 16044 (2007). [CrossRef] [PubMed]

**10. **R. Corbalan, A. N. Pisarchik, V. N. Chizhevsky, and R. Vilaseca, “Experimental study of bi-directional pumping of a far-infrared laser,” Opt. Commun. **133**, 225 (1997). [CrossRef]

**11. **H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A **57**, 1338 (1998). [CrossRef]

**12. **A. Andre and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. **89**, 143602 (2002). [CrossRef] [PubMed]

**13. **M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) **426**, 638 (2003). [CrossRef] [PubMed]

**14. **X.-M. Su and B. S. Ham, “Dynamic control of the photonic band gap using quantum coherence,” Phys. Rev. A **71**, 013821 (2005). [CrossRef]

**15. **H. Kang, G. Hernandez, and Y. Zhu, “Slow-light six-wave mixing at low light intensities,” Phys. Rev. Lett. **93**, 073601 (2004). [CrossRef] [PubMed]

**16. **M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. **96**, 073905 (2006). [CrossRef] [PubMed]

**17. **D. Petrosyan, “Tunable photonic band gaps with coherently driven atoms in optical lattices,” Phys. Rev. A **76**, 053823 (2007). [CrossRef]

**18. **J.-H. Wu, M. Artoni, and G. C. La Rocca, “Controlling the photonic band structure of optically driven cold atoms,” J. Opt. Soc. Am. B **25**, 1840 (2008). [CrossRef]

**19. **Q.-Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J.-H. Xu, and J.-Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B **73**, 195124 (2006). [CrossRef]

**20. **J.-H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B **77**, 113106 (2008). [CrossRef]

**21. **I. Friedler, G. Kurizki, and D. Petrosyan, “Deterministic quantum logic with photons via optically induced photonic bandgaps,” Phys. Rev. A **71**, 023803 (2005). [CrossRef]

**22. **A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. **94**, 063902 (2005). [CrossRef] [PubMed]

**23. **J.-H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. **103**, 133601 (2009). [CrossRef] [PubMed]

**24. **A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. **30**, 699 (2005). [CrossRef] [PubMed]

**25. **J.-W. Gao, J.-H. Wu, N. Ba, C.-L. Cui, and X.-X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A **81**, 013804 (2010).
[CrossRef]

**26. **D. Petrosyan and Y. P. Malakyan, “Magneto-optical rotation and cross-phase modulation via coherently driven four-level atosm in a tripod configuration,” Phys. Rev. A **70**, 023822 (2004). [CrossRef]

**27. **S. Rebic, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A **70**, 032317 (2004). [CrossRef]

**28. **A. MacRae, G. Campbell, and A. I. Lvovsky, “Matched slow pulses using double electromagnetically induced transparency,” Opt. Lett. **33**, 2659 (2008). [CrossRef] [PubMed]

**29. **L. Karpa, F. Vewinger, and M. Weitz, “Resonance beating of light stored using atomic spinor polaritons,” Phys. Rev. Lett. **101**, 170406 (2008). [CrossRef] [PubMed]

**30. **M. Born and E. Wolf, *Principles of Optics* (Cambridge University Press, Cambridge, 1980), 6th ed.

**31. **E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A **66**, 015802 (2002). [CrossRef]