Observation of the four wave mixing photonic band gap signal in electromagnetically induced grating

For the first time, we experimentally and theoretically research about the probe transmission signal (PTS), the reflected four wave mixing band gap signal(FWM BGS) and fluorescence signal (FLS) under the double dressing effect in an inverted Y-type four level system. FWM BGS results from photonic band gap structure. We demonstrate that the characteristics of PTS, FWM BGS and FLS can be controlled by power, phase and the frequency detuning of the dressing beams. It is observed in our experiment that FWM BGS switches from suppression to enhancement, corresponding to the switch from transmission enhancement to absorption enhancement in the PTS with changing the relative phase. We also observe the relation among the three signals, which satisfy the law of conservation of energy. Such scheme could have potential applications in optical diodes, amplifiers and quantum information processing. ©2014 Optical Society of America OCIS codes: (190.4380) Nonlinear optics, four-wave mixing; (190.4180) Multiphoton processes; (300.2570) Four-wave mixing; (270.1670) Coherent optical effects. References and links 1. P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. P. Shahriar, and P. Kumar, “Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium,” Opt. Lett. 20(9), 982–984 (1995). 2. Y. Q. Li and M. Xiao, “Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms,” Opt. Lett. 21(14), 1064–1066 (1996). 3. D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004). 4. Y. V. Rostovtsev, Z. E. Sariyanni, and M. O. Scully, “Electromagnetically induced coherent backscattering,” Phys. Rev. Lett. 97(11), 113001 (2006). 5. C. B. Li, H. B. Zheng, Y. P. Zhang, Z. Q. Nie, J. P. Song, and M. Xiao, “Observation of enhancement and suppression in four-wave mixing processes,” Appl. Phys. Lett. 95(4), 041103 (2009). 6. H. B. Zheng, X. Zhang, C. B. Li, H. Y. Lan, J. L. Che, Y. Q. Zhang, and Y. P. Zhang, “Suppression and enhancement of coexisting super-fluorescence and multi-wave mixing processes in sodium vapor,” J. Chem. Phys. 138(20), 204315 (2013). 7. J. B. Qi, G. Lazarov, X. J. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: Prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83(2), 288–291 (1999). 8. J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88(17), 173003 (2002). 9. J. B. Qi and A. M. Lyyra, “Electromagnetically induced transparency and dark fluorescence in a cascade three-level diatomic lithium system,” Phys. Rev. A 73(4), 043810 (2006). 10. C. Li, H. B. Zheng, Z. Y. Zhang, X. Yao, Y. Z. Zhang, Y. Q. Zhang, and Y. P. Zhang, “Electromagnetically induced transparency and fluorescence in blockaded Rydberg atomic system,” J. Chem. Phys. 139(16), 164316 (2013). #223118 $15.00 USD Received 15 Sep 2014; revised 23 Oct 2014; accepted 23 Oct 2014; published 19 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029544 | OPTICS EXPRESS 29544 11. H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998). 12. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426(6967), 638–641 (2003). 13. Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106(9), 093904 (2011). 14. Y. P. Zhang, C. Z. Yuan, Y. Q. Zhang, H. B. Zheng, H. X. Chen, C. B. Li, Z. G. Wang, and M. Xiao, “Surface solitons of four-wave mixing in anelectromagnetically induced lattice,” Laser Phys. Lett. 10(5), 055406 (2013). 15. A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30(7), 699–701 (2005). 16. M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006). 17. A. Schilke, C. Zimmermann, P. W. Courteille, and W. Guerin, “Photonic Band Gaps in One-Dimensionally Ordered Cold Atomic Vapors,” Phys. Rev. Lett. 106(22), 223903 (2011). 18. D. W. Wang, H. T. Zhou, M. J. Guo, J. X. Zhang, J. Evers, and S. Y. Zhu, “Optical Diode Made From a Moving Photonic Crystal,” Phys. Rev. Lett. 110(9), 093901 (2013). 19. S. A. R. Horsley, J. H. Wu, M. Artoni, and G. C. La Rocca, “Optical Nonreciprocity of Cold Atom Bragg Mirrors in Motion,” Phys. Rev. Lett. 110(22), 223602 (2013). 20. F. E. Zimmer, A. Andre, M. D. Lukin, and M. Fleischhauer, “Coherent control of stationary light pulses,” Opt. Commun. 264(2), 441–453 (2006). 21. S. A. Moiseev and B. S. Ham, “Generation of entangled lights with temporally reversed photon wave packets,” Phys. Rev. A 71(5), 053802 (2005). 22. K. R. Hansen and K. Molmer, “Trapping of light pulses in ensembles of stationary Lambda atoms,” Phys. Rev. A 75(5), 053802 (2007). 23. K. R. Hansen and K. Molmer, “Stationary light pulses in ultracold atomic gases,” Phys. Rev. A 75(6), 065804 (2007). 24. Z. G. Wang, P. Ying, P. Y. Li, D. Zhang, H. Q. Huang, H. Tian, and Y. Zhang, “Switching suppression and enhancement of fluorescence and six-wave mixing by phase modulation,” Sci Rep 3, 3417 (2013).


Introduction
Four-wave mixing (FWM) is a nonlinear optical effect that generates light with different frequencies and different quantum properties.Experimental and theoretical studies show that atomic coherence of the electromagnetically induced transparency (EIT) plays a critical role in the nonlinear wave-mixing process [1][2][3][4][5][6].Under EIT conditions FWM signals can be allowed to transmit through the atomic medium and also the fluorescence induced by spontaneous emission can be generated [7][8][9][10].Due to the two counter propagating coupling fields, the EIT-based nonlinear schemes can be driven by traveling wave beams as well as by a standing wave (SW).The large nonlinearity was obtained in an atomic system driven by two counter propagating coupling fields which form a SW if the two counter propagating coupling fields have the same frequency [11,12].Interaction of the SW with the atomic coherent medium results into an electromagnetically induced grating (EIG) [13,14], which possesses photonic band gap (PBG) structure as shown in Fig. 1(c).Such EIG has a potential use in all optical switching [15], manipulation of light propagation to create a tunable photonic band gap [16,17].This research can also be used to make optical diodes.The idea proposed here for optical diode implementation is a photonic crystal generated from a periodic modulation of the optical properties of a medium, which leads to the formation of a band gap.Light is transmitted through the crystal if its frequency is outside the gap, but it is not transmitted if the frequency is inside the gap.Due to the Doppler Effect, the counter propagating light are blue shifted and the co-propagating light is red shifted in the reference frame of the moving photonic crystal [18].If only one of the shifted frequencies is inside the gap optical diode is formed.Optical non reciprocity of cold atom Bragg mirrors in motion may also be used to make optical diode [19].
In this paper, we investigate the optical response of hot rubidium ( 85 Rb) atoms driven by a stationary SW coupling field and probe field, from which the PBG structure and four wave mixing band gap signal (FWM BGS) and probe transmission signal (PTS) can be obtained.For the first time we observe optically controllable PBG structure based on an EIT medium driven by a SW field.By optically controllable PBG structure not only we experimentally observe PTS, FWM BGS and Fluorescence signal (FLS), but also observe the enhancement and suppression of these signals under the double dressing effect in an inverted Y-type four level system.Furthermore by measuring the above signals we observed that they satisfy the law of conservation of energy, this information can be used to enhance any one of the signals at the cost of suppressing the others.By finding a strong relation among the signals we are able to find the optimal condition of any one of these signal which can be of interest for particular applications.One example of this is the optimized enhancement of FWM BGS may be used to make optical amplifiers.We also demonstrate that the characteristics of PTS, FWM BGS and FLS can be controlled by frequencies, powers, and relative phase of dressing fields; this information may be useful for the enhancement and modulation of the signals which highlights the signification of this research work.Optical response of medium is also examined by resorting to a set of nonlinear coupled wave equations, which is a powerful tool [20][21][22][23] for describing the nonlinear interaction of light fields in such a dressed medium and are here used to test the validity of the experiment result.The experiment was implemented in a cell with rubidium ( 85 Rb) atomic vapors at temperature of 53 °C with four energy levels consisting of 5S

Experimental setup
.The probe field E 1 propagates in the same direction of E´3 through the 85 Rb vapors with approximately 0.3° angle between them.The dressing field E 2 propagates in the opposite direction of E´3 with approximately 0.3° angle between them as shown in Fig. 1(c).Generated FWM BGS and PTS are detected by a photodiode (PD1) and avalanche photodiode (PD2) detectors respectively.In addition, two fluorescence signals caused by spontaneous decay are measured.The second order FLS R 0 and fourth order FLS R 1 signals are generated due to the spontaneous emission from 1 to 0 and 2 to 1 , respectively.The fluorescence signals are captured by another photodiode(PD3).

Basic theory
According to the energy system and Liouville pathways, the first-order and third-order density matrix elements are given as follows where | | / G d by means of altering the incident angle of E 2 [24] in which way the relative phase ϕ Δ related with the orientations of induced dipole moments 1 μ and 2 μ could be manipulated and Eqs. ( 1)-( 2) can be modified as follows: (3) According to the relation ε 0 χE = Nμρ, in which μ is the transition electric dipole moment and N is the atoms density, we have formulas for the first and third order susceptibility as follows.
In order to estimate the reflection efficiency, we start from the nonlinear coupled wave equations, Where E 1 (x) and E F (x) represents the probe and FWM BGS fields, respectively, α = (ω 1 /c)Imχ´( 1) /2 is the attenuation of the field due to the absorption of the medium and k = i(ω 1 /c)χ´( 3) /2 is the gain due to the nonlinear susceptibility.χ´( 1) and χ´( 3) are the zero order coefficients from Fourier expansion of χ (1)  and χ (3) , respectively.
is the phase mismatch magnitude, in which θ is the angle between probe 1 E and 3 E′ .If length of the sample in x direction is d x , then by solving above equations, the reflected FWM BGS (R) and PTS (T) are given as Where (2)   11 ρ related to second order FLS R 0 and (4)   22 ρ related to fourth order FLS R 1 can be obtained by solving the density-matrix equations.The fluorescence signal R 0 is described by solving the coupled density-matrix equations, the expression of the singly dressed density-matrix element (2)   11SD ρ (where SD stand for singly dressed) can be obtain the amplitude squared of which is proportional to the intensity of R 0 .When the beams E 2 is turned on, the fluorescence process R 0 doubly dressed, described by the Liouville pathway Therefore the expression of (2)   11SD ρ can be modified as d i = Γ + Δ , the amplitude squared of (4)   22 ρ is proportional to the intensity of R 1 .By considering the dressing effect of E 2 , the dressed fluorescence process R 1 is given as The system obeys law of conservation of energy, according to which where I in is intensity of the incident probe, I R0 is intensity of fluorescence signal R 0 , R is reflected four wave mixing band gap signal and T is the transmission of probe signal.The condition of generating PBG structure is that the medium should have a periodic refractive index.According the relation of the refractive index with the susceptibility, i.e., 1 Re( ) , in order to get the periodic refractive index, the susceptibility should also be periodic.Further we should generate the periodic energy level structure for getting the periodic susceptibility.Hence, by introducing periodic standing wave field E 31 , we can obtain the periodic energy levels as shown in Fig. 2. In Figs.When the probe reaches two-photon resonance Δ 1 -Δ 3 = 0, absorption will be suppressed, i.e. the PTS becomes strong.At the same time, the FWM BGS will be suppressed correspondingly.Thus, we define Δ 1 -Δ 3 = 0 as the suppression condition.# .The same way 31 -G is further dressed into two second level dressed states 31 2 -G G ± as shown in Fig. 2(c4)-2(c5), the Eigenvalues of which are .In Fig. 2(c3), because of three photon resonance with appear.Thus we also obtain the double dressed periodic energy levels as shown in Fig. 2 (d1)-2(d5).First, we observed the PTS, FWM BGS and FLS when we scan the probe frequency detuning Δ 1 .When E 2 is blocked in Fig. 3(a1), there is a Doppler absorption background at the single-photon resonant condition Δ 1 = 0 due to the term d 1 = Γ 10 + iΔ 1 in (1)   10 ρ and small PTS peak caused by the dressing effect of E 3 (E´3) according to the term |G 31 | 2 /d 3 in Eq. (1).When E 2 beam is on, due to the dressing term | | / G d in (1)   10 ρ of Eq. ( 1), the energy level 1 can be dressed to influence PBG structure so that the PTS increases as shown in Fig. 3(a2).The PTS reaches maximum at Δ 1 = Δ 3 = -Δ 2 due to double dressing effect of E 2 and E 3 (E´3) according to (1)   10 ρ .In Fig. 3(b1) the emission peak is FWM BGS give by R in Eq. ( 9) which is from the reflection of the PBG structure.With E 2 on, the FWM BGS will be suppressed due to the dressing effect of E 2 according to emission peak appears in the profile curve shown in Fig. 3(c2), which is the fourth order FLS R 1 related to (4)   22 ρ .Compared to Fig. 3(c1) the dip becomes deeper because of the double dressing effect of E 2 and E 3 (E´3) according to (2)   11DD ρ .Compared Figs.3(a1)-3(c1) with Figs.3(a2)-3(c2), due to the dressing effect of E 2 , the intensity of PTS increases and the intensities of FWM BGS and FLS R 0 decrease, and the whole energy is conserved according to Eq. ( 17).| | / G d of (3)   10 ρ in Eq. ( 2).The deepest dip appears at Δ 2 = -Δ 1 = -Δ 3 corresponding to the total strongest PTS.In Fig. 4(c1), the profile(dashed curve) of the baselines is the second order FLS R 0 signal suppressed by E 3 and E′ 3 and it reaches minimum at Δ 1 -Δ 3 = 0 due to the term |G 31 | 2 /d 3 in Eq. (12).In each sub curve, the peak is the fourth order FLS R 1 related to (4)   22 ρ , and it reaches the smallest one at Δ 1 = 0 because of the strongest dressing effect of E 2 according to | | / G d in (3)   10 ρ of Eq. ( 2).The dip is shallow at small values of power and become deeper with increasing P 2 due to the enhanced dressing effect of E 2 .In Fig. 5(c), the baselines represent the second order FLS R 0 suppressed by E 3 .By changing P 2 from small to large values the dip change from shallow to a deeper one because of the enhanced dressing effect of E 2 .Variation of the dips shows that R 0 are further suppressed by

Results and discussions
11DD ρ of Eq. ( 12).R 0 signal with the deepest dip correspond to the FWM BGS with deepest dip and PTS with higher peak.Peaks in the baseline are R 1 related to (4)   22DD ρ , which become higher with increasing the power of the dressing field E 2 .By observing these three signals we conclude, with the PTS increasing, the FWM BGS and FLS R 0 decrease to ensure the conservation of energy according to Eq. ( 17).15)-( 16).The controllable enhancement of FWM BGS may be used to make optical amplifiers.

Conclusion
In summary, the single-dressed and double-dressed PTS, FWM BGS and FLS are compared for the first time to deeply comprehend the double-dressing effect on the PBG.We experimentally and theoretically demonstrated that, PTS and FWM BGS and FLS can be manipulated by multiple parameters like, changing power, detuning and relative phase of incident beams.We also observed, the three types of signals satisfy law of conservation of energy.Such research could find its applications in optical amplifiers and quantum information processing.

Fig. 1 .
Fig. 1.(a) Four-level energy system.(b) Schematic of an EIG formed by two coupling beams E 3 and E′ 3 .Together with the dressing field E 2 and probe field E 1 , a dressed FWM BGS E F will be generated according to the phase-matching condition K F = K 1 −K 3 + K′ 3 .(c) The setup of our experiment.

Fig. 4 . 10 ρ
Fig. 4. Measured (a1) PTS, (b1) FWM BGS and (c1) FLS versus Δ 2 , when we select five different discrete values of Δ 1 as black(−47 MHz), red(−23 MHz), blue(0 MHz), pink(28 MHz) and green(47 MHz) and Δ 3 = 0 MHz.(a2), (b2) and (c2) are the theoretical calculations of (a1), (b1) and (c1), respectively.Furthermore, we observe second level dressing effect on PTS, FWM BGS and FLS by scanning Δ 2 , at different discrete values of Δ 1 .Figure 4(a1) represents the second level dressed PTS.The profile (dashed curve) of these PTS baselines shows Doppler absorption background with a peak on it.The profile peak is the intensity of single-dressed PTS induced by E 3 and E 3 ', which appear at Δ 1 = Δ 3 according to 2 31 3 | | / G d of (1) 10 ρ in Eq. (1).In each sub curve, the peak on the baseline stands for the enhanced PTS induced by the second level dressing effect of E 2 , which meets Δ 2 = -Δ 1 according to the dressing term 2 2 2 | | / G d in Eq. (1).The total PTS reaches maximum at Δ 2 = -Δ 1 = -Δ 3 .In Fig. 4(b1), profile (dashed curve) of the baselines shows the FWM BGS related to R in Eq. (9) from reflection of the PBG structure.Dip in each sub curve shows that FWM BGS is suppressed due to the dressing effect of E 2 at Δ 2 = -Δ 1 according to 2 2 2 (14) as shown in Fig.4(c3).Next, we observe the power dependences of the PTS, FWM BGS and FLS versus Δ 2 .Variations in the three types of signals are shown from bottom to top with increasing power of E 2 (P 2 ) as shown in Figs.5(a)-5(c).In Fig. 5(a), the baselines of the signals represent the #223118 -$15.00USD Received 15 Sep 2014; revised 23 Oct 2014; accepted 23 Oct 2014; published 19 Nov 2014 (C) 2014 OSAintensity of the PTS induced by E 3 (E´3).Peaks in the baselines at Δ 1 = Δ 3 shows the enhancement of the PTS induced by the second level dressing effect of E 2 .Changing P 2 from small to large values, the peaks becomes higher due to the dressing term .The FWM BGS signal intensity is shown by the baselines in Fig.5(b).Dip in the baseline shows suppression of reflected FWM BGS signal because of the second level dressing effect of E 2 according to the dressing term 2 2 2

Fig. 5 . 3 π 3 π to 2 / 3 π 6 π−
Fig. 5. Measured (a) PTS, (b) FWM BGS and (c) FLS versus Δ 2 from −120 MHz to 120 MHz with Δ 3 = Δ 1 = 0, when we set the power of E 2 (P 2 ) from bottom to top as (1) 9.2 mW, (2) 13.0 mW, (3) 17.1 mW, (4) 21.6 mW, (5) 25.7 mW, respectively.Finally, we regulate the PTS, FWM BGS and FLS with the relative phase of E 2 (Δφ) by changing its incident angle.The experimental results can be obtained by scanning Δ 2 with Δ 1 = 0 as shown in Figs.6(a)-6(c).With the relative phase Δφ changing from -/3 π to 2 /3 π , the PTS in Fig. 6(a) can be switched from a dip to a peak due to the change of dressing effect of E 2 according to (1) 10 ρ′ .The peaks stand for the transmission enhancement of probe signal and the dips stand for the absorption enhancement of the PTS.During this process, the deepest dip and the highest peak separately appear at Δφ = 2 /3 π and Δφ =