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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 18 — Aug. 31, 2009
  • pp: 15468–15480
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Modulating the multi-wave mixing processes via the polarizable dark states

Huaibin Zheng, Yanpeng Zhang, Utsab Khadka, Ruimin Wang, Changbiao Li, Zhiqiang Nie, and Min Xiao  »View Author Affiliations


Optics Express, Vol. 17, Issue 18, pp. 15468-15480 (2009)
http://dx.doi.org/10.1364/OE.17.015468


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Abstract

We have observed multi-wave mixing (MWM) processes in reversed-Y (RY) type system in 87Rb atoms with electromagnetically induced transparency (EIT) windows at different laser polarization configurations. Interesting rules of changing the MWM processes and EIT profiles are obtained. We have found that the degenerate Zeeman sublevels and their dressed-state effects are responsible for these observed phenomena. Polarizable dark states are used to describe the multi-level dressed states. The experimental data are in good agreement with the results from the theoretical calculation that takes into account all the 16 Zeeman sublevels in the RY system.

© 2009 Optical Society of America

1. Introduction

The generated multi-wave mixing (MWM) signals in multi-level systems can transmit through the resonant atomic medium with little absorption under the electromagnetically induced transparency (EIT) conditions [1

1. S. E. Harris, “Electromagnetically Induced Transparency,” Phys. Today. 50, 36–42 (1997). [CrossRef]

2

2. J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically Induced Transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A. 51, 576–584 (1995). [CrossRef] [PubMed]

]. Enhanced MWM processes due to laser-induced atomic coherence have been experimentally demonstrated in several multi-level atomic systems [3

3. A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, “Transporting and time reversing light via atomic coherence,” Phys. Rev. Lett. 88, 103601-1–103601-4 (2002). [CrossRef]

5

5. B. Lu, W. H. Burkett, and M. Xiao, “Nondegenerate Four-Wave Mixing in a double-Λ system under the influence of coherent population trapping,” Opt. Lett. 23, 804–806 (1998). [CrossRef]

]. Interesting effects, such as quantum destructive interference in inelastic two-wave mixing [6

6. K. J Jiang, L. Deng, and M. G. Payne, “Observation of quantum destructive interference in inelastic Two-wave Mixing,” Phys. Rev. Lett. 98, 083604-1–083604-4 (2007). [CrossRef]

], phase-controlled light switching at low light level [7

7. H. Kang, G. Hernandez, J. Zhang, and Y Zhu, “Phase-contrlled light switching at low light levels,” Phys. Rev. A. 73, 011802(R)-1–011802(R)-4 (2006). [CrossRef]

], entangled images in the probe and signal beams in the four-wave mixing (FWM) process [8

8. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from Four-Wave Mixing,” Science. 321, 544–547 (2008). [CrossRef] [PubMed]

], and generation of correlated photon pairs [9

9. S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave Mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98, 053601-1–053601-4 (2007). [CrossRef]

10

10. V. B. Danielle, A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601-1–183601-4 (2005).

], have been experimentally studied in various coherently-prepared multi-level atomic systems. When a strong light field interacts with an atomic transition, the light-atom system can be considered as coupled harmonic oscillators with split normal modes, called dressed states. When more than one light field interact with the same or connected atomic transitions, multi-dressed states can be formed [11

11. C. Cohen-Tannoudji and S. Reynaud, “Dreesed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B. 10, 345–363 (1977). [CrossRef]

]. Recently, interplay among multi-dressed FWM processes [12

12. H. Zheng, Y. Zhang, Z. Nie, C. Li, H. Chong, J. Song, and M. Xiao, “Interplay among multi-dressed Four-wave Mixing processes,” Appl. Phys. Lett. 93, 241101-1–241101-3 (2008). [CrossRef]

], competition via atomic coherence in a four-level atomic system with two co-existing FWM processes [13

13. Y. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two Four-wave Mixing channels via atomic coherence,” Appl. Phys. Lett. 91, 061113-1–061113-3 (2007). [CrossRef]

] and destructive/constructive interferences in a two-level atomic system [9

9. S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave Mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98, 053601-1–053601-4 (2007). [CrossRef]

] were studied. In recent years, many schemes have been developed to enhance higher-order nonlinear wave-mixing processes. More importantly, with induced atomic coherence and interference, the higher-order processes can become comparable or even greater in amplitude than the lower order wave-mixing processes [14

14. Y. Zhang and M. Xiao, “Generalized dressed and doubly-dressed multi-wave mixing,” Opt. Express. 15.7182–7189 (2007). [CrossRef] [PubMed]

].

In this paper, we report combined theoretical and experimental studies on the polarization dependence of the MWM processes and EIT profile, using rubidium atoms as the nonlinear medium. We present the results obtained for the intensities of the MWM signals as a function of the polarization of the incident laser beams. The relations between the high-order (e.g. third- and fifth-order) nonlinear susceptibilities and the polarizations of the incident beams are shown. To explain the observed EIT profiles and different rules for the MWM signal peaks with different incident beam polarizations, we take into account the degenerate Zeeman sublevels and their dressed effects. We have found that the changes in the spectrum of the MWM processes can be attributed to the modulation in the probe field variation. The experimentally measured data are in good agreements with the results from the theoretically calculated results that take into account all the 16 relevant Zeeman sublevels in the reversed-Y (RY) system in rubidium atom.

2. Theoretical model and analysis

The relevant experimental energy-level diagram is shown in Fig. 1(a). Four energy levels from the 87Rb atoms are involved in the experimental schemes used in this work. The laser beams are spatially aligned as shown in Fig. 1(b). In Fig. 1(a), energy levels |0〉(5S 1/2, F=2), |1〉 (5P 3/2, F′=2), |2〉 (5D 3/2, F″=1) and |3〉 (5S 1/2, F=1) form the RY-type four-level atomic system. Strong coupling laser beam E 2 (ω 2, k 2, and Rabi frequency G 2) together with E2 (ω 2, k2, and Rabi frequency G2) with small angle (0.5°) and the same frequency detuning Δ2 (=ω 21-ω 2), connecting the transition between |1〉 to |2〉, propagate in the opposite direction of the weak probe beam E 1 (ω 1, k 1, and Rabi frequency G 1), which has the frequency detuning Δ1 (=ω 10-ω 1) and connects the transition between |0〉 to |1〉, as shown in the inset of Fig. 1(b). Two additional coupling beams E 3 (ω 3, k 3, and Rabi frequency G3) and E3 (ω 3, k3, and Rabi frequency G3), with the same frequency detuning Δ3 (=ω 13-ω 3) connecting the transition between |3〉 to |1〉, also propagate in the opposite direction of the weak probe beam E 1 [Fig. 1(b)] with small angles. When all five laser beams (E 1, E 2, E2, E 3 and E3) are turned on simultaneously, only one ladder-type EIT subsystem will form due to the two-photon Doppler-free configuration [2

2. J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically Induced Transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A. 51, 576–584 (1995). [CrossRef] [PubMed]

] and an EIT window opens, whose position in frequency domain depends on the frequency detuning of the ω 2 laser beams. Meanwhile, there will be co-existing MWM processes that generate signal beams at frequency ωM (ωM=ω1) in such multi-level system. First, without the coupling beams E 3 and E3, one pure FWM EF process will be generated in the ladder system (|0〉→|1〉→|2〉) satisfying the phase-matching condition of k F=k 1+k 2-k2. The signal emerging from the “P” polarization direction is detected by an avalanche photodiode detector (APD). Second, when all beams (except with E2 blocked) are turned on, the strong coupling beam E2 will dress the energy level |1〉 to create the dressed states |+〉 and |-〉. In such case, there exist a dressed FWM process kF=k 1+k 3-k3 and a coexisting six-wave mixing (SWM) process, in which two photons from E 2 and one photon each from E 1, E 3 and E3 participate in the SWM process to generate ES with phase-matching condition of k S=k 1+k 2-k 2+k 3-k 3′.

Fig. 1. (a) Energy level diagram of relevant 87Rb energy levels. (b) The schematic diagram of the experiment. D denotes the photodiode and APD denotes the avalanche diode detector; PBS denotes the polarized beam splitter and WP denotes the wave plate. Inset: the spatial alignment of the laser beams.

To quantitatively investigate the polarization dependence of the MWM signals, we develop a theoretical model to treat the polarization-dependent nonlinear processes. The observed MWM signal intensities in the experiments are proportional to the square of the polarizations induced in Rb vapor at frequency ωM. First, for the FWM signal EF (by blocking laser beams E 3 and E3), the nonlinear atomic polarization P (3)(ω 1) along the i (i=x,y) direction, from first-order perturbation theory, is given by [15

15. C. J. Zhu, A. A. Senin, Z. H. Lu, J. Gao, Y. Xiao, and J. G. Eden, “Polarization of signal wave radiation generated by parametric four-wave mixing in rubidium vapor: Ultrafast (~150-fs) and nanosecond time scale excitation,” Phys. Rev. A. 72, 023811-1–023811-7 (2005). [CrossRef]

]

Pi(3)(ω1)=ε0jklχijkl(3)E1j(ω1)E2k*(ω2)E2l(ω2),
(1)

Px(3)(ω1)=ε0χxE22E1,
Py(3)(ω1)=ε0χyE22E1,
(2)

where χx and χy are the effective susceptibilities. If there is a half-wave plate placed on the path of the probe beam E 1, then

χx(3)=χxyyxcos2θ,
χy(3)=χyyyysin2θ.
(3)

For a quarter-wave plate in the probe beam,

χx(3)=χxxxxsin4ϕ+cos4ϕ,
χy(3)=χyxxy2sinϕcosϕ2.
(4)

θ, ϕ are the rotated angles of the half-wave plate and quarter-wave plate, respectively, relative to the x-axis.

Similarly, for the generated SWM signal ES (when blocking the coupling laser beam E2), the fifth-order nonlinear polarization P (5)(ω 1) along the i (i=x,y) direction is then given by

Pi(5)(ω1)=ε0jklmnχijklmn(5)E1j(ω1)E2k*(ω2)E2l(ω2)E3m*(ω3)E3n(ω3),
(5)

χxxxxxx=χyyxxxx+χyxyxxxχyxxyxxx+χyxxxyx+χyxxxxy
+χxyyxxx+χxyyyxx+χxyxxyx+χxyxxxy+χxxyyxx
+χxxyyxx+χxxyxyx+χxxyxxy+χxxxyxy+χxxxxyy.
(6)

Under the same condition as mentioned above for the FWM case, the generated SWM polarization will have two components, parallel and perpendicular to the beam’s polarization,

Px(5)(ω1)=ε0χxE22E32E1,
Py(5)(ω1)=ε0χyE22E32E1.
(7)

According to the experimental arrangement, we assume that the probe field can have linear polarization in any direction in the xy plane determined by the wave-plate, while the other coupling fields have fixed polarization along the same y-axis (S polarization). In this case, the polarization of the generated SWM signal beam will have two components, i.e. parallel and perpendicular to the coupling beams’ polarization. For a half-wave plate placed in front of the probe beam, the polarizations of the SWM signal beam become

χx=χxyyyyxcos2θ,
χy=χyyyyyysin2θ.
(8)

In addition, for a quarter-wave plate, these fifth-order nonlinear susceptibility components are

χx=χxyyyyxsin4ϕ+cos4ϕ,
χy=χyyyyyy2sinϕcosϕ2.
(9)

In our experimental [Fig 1(b)], we first use a half-wave plate to horizontally polarize the probe beam (P polarization). Then the beam passes through a half-wave plate (or a quarter-wave plate) that is rotated by an angle θ (or ϕ) before going into the 87 Rb atomic vapor cell. By blocking different incident beams, the probe beam and the generated FWM signal (or SWM signal) [13

13. Y. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two Four-wave Mixing channels via atomic coherence,” Appl. Phys. Lett. 91, 061113-1–061113-3 (2007). [CrossRef]

] both pass through a polarized beam splitter (PBS) before being detected. The detector D receives the P polarization component of the transmitted probe beam, E 1(ω 1), whereas the APD receives the horizontally polarized component of the generated MWM signal EM(ωM).

3. Experimental results

The experiment is done with hot 87 Rb atoms by three external cavity diode lasers (ECDL) and linewidths of less than or equal to 1 MHz. Each output power is as follow: 0.7 mw of probe field E 1; 65 mw of coupling beams E 2, E2 and 15 mw of coupling laser beams E 3, E3. The cell whose length is 5cm is heat up to 60°C and the density is 2.5×1011/cm3.

Figure 2 shows the EIT spectra (lower dark curves) and the SWM spectra (upper red curves) of the 87 Rb atoms, by blocking laser beam E2, for probe field with different polarizations rotated by a half-wave plate. The coupling beams are linearly polarized in the S polarizations direction and the probe beam is linearly polarized in the P polarization direction initially (defined as θ=0°).

As shown in Fig. 2, various polarized configurations of the coupling and probe beams, by rotating the half-wave plate, result in quite different spectra. The absorption curve of the probe beam changes from EIT to a dispersion-like curve, and then back to EIT again while the polarization of the input probe beam changes one period. Meanwhile, the rules of the two SWM signal peaks change quite differently. These observed experimental results reveal that the degenerate Zeeman sublevels [16

16. H. Y. Ling, Y. Q Li, and M. Xiao, “Coherent population trapping and electromagnetically induced transparency in multi-Zeeman-sublevel atoms,” Phys. Rev. A , 53, 1014–1026 (1996). [CrossRef] [PubMed]

] and the dressed-state effects play crucial roles in the EIT and SWM spectra.

Fig. 2. (color online) Polarization dependence of the transmitted probe and SWM signal beams versus the rotating angle of the half-wave plate. (a1–a6) EIT (lower curves) and SWM (upper curves) spectra versus the rotating angle. The experimental parameters are G 1=2π×5MHz, G 2=G2=2π×80MHz, G 3=G3=2π×35MHz, and Δ23=0.

To understand the above experimental results, we show the theoretical calculation of the probe absorption spectrum that considers all the 16 relevant Zeeman sublevels [17

17. S. Li, B. Wang, X. Yang, Y. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A. 74, 033821-1–033821-12 (2006). [CrossRef]

] in the RY system. Our calculations are limited to the cases that the coupling fields are linearly polarized and the magnetic field is absent. Three examples of the excitations of the coupling and the probe beams are illustrated in Fig. 3. Figure 3(a) shows the case with the probe beam polarization to be orthogonal to the coupling beams and Fig.3(c) corresponds to the case with parallel polarization. Figure 3(b) gives the results with probe input polarization in between those two cases. For simplicity, we consider the cascade three-level system (by blocking coupling fields E 3 and E3), which forms an EIT configuration. For the case as given in Fig. 3(a), the density-matrix ρ considering all the Zeeman levels is solved with the following equations:

ρ˙=(1i)[Hatom+Hcoupling+Hprobe,ρ]+{dρdt},
(10)
Hatom=ω10i=610i><i+ω21i=1113i><i,
(11)
Hcoupling=exp[iωct][(Ωc12)(7><11+9><13)
+(Ωc22)8><12]+c.c.,
(12)
Hprobe=exp[iωpt][(Ωp12)(1><7+5><9)
+(Ωp22)(2><8+4><8)
+(Ωp32)(3><9+3><7
+(Ωp42)(2><6+4><10)]+c.c.,
(13)
{dt}=Γ1i=1113ρiii><iΓ2i=610ρiii><i
+i=110(j=1113Γjiρjj)i><i+i=15(j=610Γjiρjj)i><i
12[Γ1i=1113j=110ρiji><j+Γ2i=610j=15ρiji><j+c.c.]
[Γi=1113j=610ρiji><j+Γi=610j=15ρiji><j+Γi=1113j=15ρiji><+c.c.],
(14)

where ω 10 is the F=2→F′=2 transition frequency and ω 21 is the F′=2→F″=1 transition frequency. ωc is the frequency of the coupling field and ωp is frequency of the probe laser. ΩN is the Rabi frequency for the transition indicated by its subscripts. The expression of {/dt} describes all the relaxation processes in the system. Γ1 and Γ2 are the spontaneous decay rates of the 5D 3/2, F′=1 and 5P 3/2, F′=2 excited states, respectively. Γji is the spontaneous emission rate from an excited state |j> to a ground state |i>. Γ′, Γ″ and Γ‴ are the decoherence rates for the relaxation processes other than the spontaneous decay. To obtain linear susceptibility, we need to solve the density-matrix equations (11) under the steady-state condition. Under the weak probe field approximation [2

2. J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically Induced Transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A. 51, 576–584 (1995). [CrossRef] [PubMed]

], the expressions of the first-order matrix elements can be easily calculated.

(1) For the right-hand-circularly (RHC) polarized sub-ladder systems:

ρ1,7+=iΩp1+iΔp1+Γ1,7+Ωc12i(Δpl+Δcl)+Γ11,1ρ1,7(0),
ρ2,8+=iΩp2+iΔp2+Γ2,8+Ωc22i(Δp2+Δc2)+Γ12,2ρ2,8,(0)
ρ3,9+=iΩp3+iΔp3+Γ3,9+Ωcl2i(Δp3+Δcl)+Γ13,3ρ3,9,(0)
ρ4,10+=iΩp4+iΔp4+Γ4,10ρ4,10(0).
(15)

(2) For the left-hand-circularly (LHC) polarized sub-ladder systems:

ρ2,6=iΩp4iΔp4+Γ2,6ρ2,6(0),
ρ3,7=iΩp3iΔp3+Γ3,7+Ωcl2i(Δp3+Δcl)+Γ11,3ρ3,7(0),
ρ4,8=iΩp2iΔp2+Γ4,8+Ωc22i(Δp2+Δc2)+Γ12,4ρ4,8(0),
ρ5,9=iΩp1iΔp1+Γ5,9+Ωc12i(Δp1+Δc1)+Γ13,5ρ5,9(0),
(16)

where frequency detuning parameters Δij are the differences between the laser frequency and the corresponding atomic transition frequency.

Fig. 3. (color online) The excitations of the coupling and probe fields among the Zeeman levels in the RY system. (a), (b) and (c) correspond to the case with a half-wave plate in the probe beam; (d) (e) and (f) correspond to a quarter-wave plate.For the level configuration shown in Fig. 3(c), the probe Hamiltonian is changed to
Hprobe=exp[iωpt][(Ωp12)(1><6+5><10)
+(Ωp22)(2><7+4><9)+c.c.,
(17)

and the remaining calculations are similar. We can then get the expressions of the matrix elements for the linearly polarized sub-ladder systems as:

ρ1,60=iΩp1iΔp1+Γ1,6ρ1,6(0),
ρ5,100=iΩp1iΔp1+Γ5,10ρ5,10(0),
ρ2,70=iΩp2iΔp2+Γ2,7+Ωcl2i(Δp2+Δcl)+Γ11,2ρ2,7(0),
ρ4,90=ρ2iΔp2+Γ4,9+Ωc12i(Δp2+Δc1)+Γ13,4ρ4,9(0).
(18)

The case in Fig. 3(b) is an integration of the results from Figs. 3(a) and 3(c). All the theoretical results of the different polarization configurations are obtained in the same way.

When we take into account all the Zeeman sublevels in the RY system, the unexpected profiles of the EIT spectra and different changing rules of the two SWM signal peaks can then be easily explained. For instance, initially, the polarizations of the probe beam and coupling beams are perpendicular [Fig. 3(a)]. There exist three RHC (blue arrows) and three LHC (green arrows) EIT subsystems with one RHC and one LHC probe transitions without the interference of the coupling fields. The observed transmission spectrum is the combination of the Lorentzian absorption profile and the EIT profile for the probe beam. However, in the Fig. 3(a), the intensity of the Lorentzian absorption profile is much smaller than that of the EIT profile, which is the result of different dipole momentums between the Zeeman sublevels [18

18. For detail in transition probabilities of D1 and D2 line in Rb, see http://steck. Us/alkalidata.

]. In the case of Fig. 3(b), as the half-wave plate rotating, there are two linearly polarized EIT subsystems (red arrows) with the RHC and LHC EIT subsystems decreasing. Also, there are two additional linear probe transitions. The intensity of the Lorentzian absorption profile is comparable to that of the EIT subspectrums. Therefore, a narrower Lorentzian shape peak can be seen clearly on top of the broader EIT profile as shown in Fig. 2(a3). Due to the detuning of the coupling field, the curve appears to be dispersion-like. When all the laser beams have the same polarization [Fig. 3(c)], there are only two linearly-polarized EIT subsystems and two linear probe transitions. Because of the different dipole momentums among different Zeeman sublevels, the Lorentzian absorption profile is much larger than that of the EIT profile, so the probe transmission has an absorption-type shape with subnatural linewidth [19

19. Y. Chen, C. Lin, and I. A. Yu, “Role of degenerate Zeeman levels in electromagnetically induced transparency,” Phys. Rev. A. 61, 053805-1–053805-6 (2000). [CrossRef]

]. This effect doesn’t present in Fig. 2 due to our current experimental configuration, i.e., the signal is detected only in the P polarization.

In addition, if the coupling fields E 3 and E3 are open, there are doubly-dressed effects for the EIT curve. The analytical solutions will be changed as following.

(1) For the RHC-polarized sub-RY systems:

ρ1,7+=iΩρ1+iΔp1+Γ1,7+Ωc12i(Δρ1+Δc1)+Γ11,1+Ωd12i(Δp1Δd1)+Γ14,1ρ1,7(0), ρ2,8+=iΩp2+iΔp2+Γ2,8+Ωc22i(Δp2+Δc2)+Γ12,2+Ωd22i(Δp2Δd2)+Γ15,2ρ2,8(0),,

ρ3,9+=iΩp3+iΔp3+Γ3,9+Ωc12i(Δp3Δc1)+Γ13,3+Ωd12i(Δp3Δd1)+Γ16,3ρ3,9(0).,

(2) For the LHC-polarized sub-RY systems:

ρ3,7=iΩp3iΔp3+Γ3,7+Ωc12i(Δp3+Δc1)+Γ11,3+Ωd12i(Δp3Δd1)+Γ14,3ρ3,7(0), ρ4,8=iΩp2iΔp2+Γ4,8+Ωc22i(Δp2+Δc2)+Γ12,4+Ωd22i(Δp2Δd2)+Γ15,4ρ4,8(0), ρ5,9=iΩp1iΔp1+Γ5,9+Ωc12i(Δp1+Δc1)+Γ13,5+Ωd12i(Δp1Δd1)+Γ16,5ρ5,9(0)..

(3) For the linearly-polarized sub-RY systems:

ρ2,70=iΩp2iΔp2+Γ2,7+Ωc12i(Δp2+Δc1)+Γ11,2+Ωd12i(Δp2Δd1)+Γ14,2ρ2,7(0), ρ4,90=iΩp2iΔp2+Γ4,9+Ωc12i(Δp2+Δc1)+Γ13,4+Ωd12i(Δp2Δd1)+Γ16,4ρ4,9(0).,

These expressions indicate that the additional coupling fields will modulate the EIT profiles. Different frequency detuning configurations of the three stronger coupling fields will affect the EIT spectral shape. By controlling the frequency detuning of the coupling fields, either EIT or electromagnetic induced absorption (EIA) spectrum will appear.

In order to quantitatively compare with the observed changes of the SWM signal peaks as probe polarization changes, we choose to calculate the area of the spectrum under the peak [see Fig. 4(b1)] and the height of each peak [see Figs. 4(b2) and 4 (b3)]. As a comparison, the area curve of the EIT [Fig. 4(a1)] and the absorption height of each peak [Figs. 4(a2) and 4(a3)] are also presented. From Figs. 4(a1) and 4(a1), the changes of the SWM spectrum, as well as for the EIT peaks, are well described by the function cos2 2θ, as calculated above. It is interesting to see the polarization dependence of the SWM emission signal following the polarization of the probe field. The theoretical curves (the solid lines) are in good agreement with the experimental data. The different change rules of the two SWM signal peaks can be attributed to the modulation of the modified EIT spectral profile. The dotted lines of Figs. 4(a2) and 4(a3) represent the absorption peak heights of the probe field at different polarization states. The dotted lines of Figs. 5(a1) and 5(a2) depict the EIT depths which show different coupling processes of the total EIT effect via the different polarization dark states. As can be seen from the figures, the rule of change in the positive part of the EIT dispersion-like curve [Fig 5(a1)] is dominated by the polarization property. However, in the negative part [Fig. 5(a2)], it gradually changes from EIT to absorption in a half period. After the absorption is larger than the EIT dip, the increase of absorption is faster than the polarized attenuation, and then the roles of them are switched. Figures 4(b2) and 4(b3) illustrate the measured dependence of the relative SWM signal intensity on the rotation angle θ (half-wave plate) when the diode laser (probe) is tuned to the 5S 1/2, F=2→5P 3/2, F′=2 one-photon transition. Besides its direct dependence on the polarization of the probe beam, the SWM signal spectrum is also modulated by the EIT effect since it transmits through the medium in the EIT window. Figures 5(b1) and 5(b2) show the corresponding differences of the experiment data and theoretically calculated curves which represent different coupling paths of the total SWM processes via different polarization configurations. The left peak is always in the positive part of the dispersion-like EIT curve which gets dramatically modulated effect due to EIT. This means that the fifth-order susceptibility χ (5) ijklmn must be taken into account due to dressed effect, which affects the evolution of the peak together with the polarization dependence of cos2 2θ. The right peak is in the negative region of the dispersion-like EIT shape, where it is dominated by the polarization property due to the increase in absorption. We can conclude that the evolution of the SWM spectrum is modulated by the modified EIT spectrum.

Fig 4. (a) Measured (dots) and calculated (solid curves) areas and heights of the two transmitted EIT peaks. (b) Measured (dots) and calculated (solid curves) areas and heights of the SWM signal peaks.
Fig 5. (a) Peak depths for the two EIT transmission peaks. (b) Differences between the theoretical results and experimental data.

Next, we concentrate on the experimentally measured and theoretically calculated probe transmission and FWM spectra by rotating the quarter-wave plate (in front of the input probe beam) whose period is 180 degree as can be seen in Fig. 6. Here, the MWM process is the FWM by blocking laser beams E 3 and E3 while other beams are turned on. The theoretical results can be obtained from the same procedure as above by eliminating the absent items, that is, LHC subsystems in the RY system. The EIT spectrum also shows the same profile, including the dispersion-like curve. However, the positive peak changes from a single peak into two peaks. Although the way of polarization with a quarter-wave plate is different from with a half-wave plate, its rules of evolution can be explained by the same method as for the half-wave plate. Schemes (d), (e) and (f) in Fig. 3 show the ways of coupling in different polarization configurations. Differ from using a half-wave plate whose period is 90 degrees, the right-hand elliptically polarized beam is present during 0~45 and 45~90 degrees [Fig. 3(e)], and a pure RHC-polarized beam at 45 degrees [Fig. 3(f)]. However, within 0~45 degrees, the RHC component increases gradually while the linear component decreases. In the region of 45~90 degrees the opposite process is true. Taking the case of 0~45 degrees as an example, we can take the right-hand elliptically-polarized beam being composed of a vertical, linearly-polarized beam and a RHC-polarized beam. Therefore, in our experiment, the original symmetric EIT configurations [Fig. 3(d)] are replaced by two linear EIT and three RHC EIT subsystems [Fig. 3(e)] that are asymmetric due to the difference in the dipole moments among different Zeeman sublevels. The destruction of this symmetry results in different polarizable dark states leading to the modified EIT spectrum. It is different from the case with half-wave plate, which does not destroy the symmetry in the EIT spectral shape. The situation in the rest of the period (45~90 degree) can be discussed in the same way. So, the spectrum by rotating a quarter-wave plate is different from the case of using a half-wave plate.

Fig. 6. (color online) Polarization dependence of the transmitted probe and FWM signal beams versus the rotation angle of the quarter-wave plate. (a1–a7) The EIT (lower curves) and FWM (upper curves) spectra versus the rotation angle. The experimental parameters are G 1=2π×5MHz, G 2=G′2=2π×80MHz, and Δ2=0.

The FWM spectra are modulated by the EIT spectra as shown in Fig. 6. As the quarter-wave plate rotates, the FWM spectrum changes from a two-peak structure to a three-peak one. The rule of evolution for each peak includes not only the polarized property, but also effects of dark-state modulation due to the asymmetric coupling structure in EIT spectrum. Similar to the analysis used for the half-wave plate, the rule of evolution in the height of each absorption peak in the EIT profile follows the polarization effect of the probe beam. Although each peak [Figs. 7(a1)–(a3)] is significantly modulated, the general trend in its evolution is not destroyed. Moreover, in studying the detail dark-state coupling configuration for each peak, we have investigated the peak depths of the EIT curves [Figs. 8(a1)–(a3)]. All of them undergo the combined interactions of the selectively-polarized probe beam and the polarized dark-state modulation. The two peaks in the positive part of the dispersion-like curve can be attributed to the asymmetric EIT configuration, RHC polarization and vertical linearly-polarized EIT subsystems. Due to the increase in absorption [Fig. 3(e)], the right peak in the probe transmission spectrum changes from EIT to EIA [Fig. 8(a3)]. Therefore, based on the above analysis of the EIT spectrum, we can understand the rule of change in FWM spectrum easily. The left FWM peak [Figs. 7(b1) and 8(b1)], corresponding to the left EIT peak [Fig. 8(a1)] dominated by the polarized dark-state component, is dramatically modulated and the contribution from polarization change of the probe beam is suppressed. As mentioned above, there could be combined contributions from the third-order nonlinear susceptibility χ (3) ijkl(ωF;ω 1,-ω 2,ω 2 and the polarization effect of the probe field in the form of sin4ϕ+cos4ϕ. Although it is affected by the polarized dark-state modulation, the trend of change for the middle peak [Fig. 8(a2)] follows the polarization change of the probe beam. As a result, the polarization property of the middle FWM peak survives [Figs. 7(b2) and 8(b2)]. The right FWM peak is modulated and its height increases gradually due to the increased absorption of the right peak in the probe transmission spectrum. Meanwhile, the middle FWM peak decreases gradually. These two peaks have a competitive relationship, which can explain their opposite phase evolutions [Figs. 7(b3) and 8(b3)].

Fig. 7. (a) Measured (dots) and calculated (solid curves) heights of the transmitted EIT peaks. (b) Measured (dots) and calculated (solid curves) heights of the FWM signal peaks.
Fig. 8. (a) Peak depths for the EIT transmission peaks. (b) Differences between the theoretical results and experimental data.

4. Conclusion

In summary, we have experimentally and theoretically demonstrated that the modified EIT spectra resulting from degenerate Zeeman sublevels can significantly modulate the MWM processes. The different couple ways of permitted transitions of degenerate Zeeman sublevels due to polarization configuration of incident beam which destroy the symmetric can modulate the spectra [Figs. 2 and 6]. Each peak of the MWM process will be controlled by polarization property [Figs. 4(a3) and 4(b3)], dressed effect [Figs. 4(a2) and 4(b2)], or both of them [Figs. 8(a2) and 8(b2)]. The experimental data are in good agreement with the results from theoretical calculation involving all relevant Zeeman sublevels. Our study indicates that Zeeman sublevels should be taken into account in the analysis of the MWM, such as FWM and SWM, processes involving laser-induced atomic coherence in multi-level atomic systems.

Acknowledgments

This work was supported by NSFC (No. 60678005), FANEDD (No. 200339), RFDP (No. 20050698017), FYTEFYTIHEC (No. 101061), NCET-08-0431.

References and links

1.

S. E. Harris, “Electromagnetically Induced Transparency,” Phys. Today. 50, 36–42 (1997). [CrossRef]

2.

J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically Induced Transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A. 51, 576–584 (1995). [CrossRef] [PubMed]

3.

A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, “Transporting and time reversing light via atomic coherence,” Phys. Rev. Lett. 88, 103601-1–103601-4 (2002). [CrossRef]

4.

P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, “Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium,” Opt. Lett. 20, 982–984 (1995). [CrossRef] [PubMed]

5.

B. Lu, W. H. Burkett, and M. Xiao, “Nondegenerate Four-Wave Mixing in a double-Λ system under the influence of coherent population trapping,” Opt. Lett. 23, 804–806 (1998). [CrossRef]

6.

K. J Jiang, L. Deng, and M. G. Payne, “Observation of quantum destructive interference in inelastic Two-wave Mixing,” Phys. Rev. Lett. 98, 083604-1–083604-4 (2007). [CrossRef]

7.

H. Kang, G. Hernandez, J. Zhang, and Y Zhu, “Phase-contrlled light switching at low light levels,” Phys. Rev. A. 73, 011802(R)-1–011802(R)-4 (2006). [CrossRef]

8.

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from Four-Wave Mixing,” Science. 321, 544–547 (2008). [CrossRef] [PubMed]

9.

S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave Mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98, 053601-1–053601-4 (2007). [CrossRef]

10.

V. B. Danielle, A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601-1–183601-4 (2005).

11.

C. Cohen-Tannoudji and S. Reynaud, “Dreesed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B. 10, 345–363 (1977). [CrossRef]

12.

H. Zheng, Y. Zhang, Z. Nie, C. Li, H. Chong, J. Song, and M. Xiao, “Interplay among multi-dressed Four-wave Mixing processes,” Appl. Phys. Lett. 93, 241101-1–241101-3 (2008). [CrossRef]

13.

Y. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two Four-wave Mixing channels via atomic coherence,” Appl. Phys. Lett. 91, 061113-1–061113-3 (2007). [CrossRef]

14.

Y. Zhang and M. Xiao, “Generalized dressed and doubly-dressed multi-wave mixing,” Opt. Express. 15.7182–7189 (2007). [CrossRef] [PubMed]

15.

C. J. Zhu, A. A. Senin, Z. H. Lu, J. Gao, Y. Xiao, and J. G. Eden, “Polarization of signal wave radiation generated by parametric four-wave mixing in rubidium vapor: Ultrafast (~150-fs) and nanosecond time scale excitation,” Phys. Rev. A. 72, 023811-1–023811-7 (2005). [CrossRef]

16.

H. Y. Ling, Y. Q Li, and M. Xiao, “Coherent population trapping and electromagnetically induced transparency in multi-Zeeman-sublevel atoms,” Phys. Rev. A , 53, 1014–1026 (1996). [CrossRef] [PubMed]

17.

S. Li, B. Wang, X. Yang, Y. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A. 74, 033821-1–033821-12 (2006). [CrossRef]

18.

For detail in transition probabilities of D1 and D2 line in Rb, see http://steck. Us/alkalidata.

19.

Y. Chen, C. Lin, and I. A. Yu, “Role of degenerate Zeeman levels in electromagnetically induced transparency,” Phys. Rev. A. 61, 053805-1–053805-6 (2000). [CrossRef]

OCIS Codes
(020.1670) Atomic and molecular physics : Coherent optical effects
(190.4180) Nonlinear optics : Multiphoton processes
(270.1670) Quantum optics : Coherent optical effects
(190.4223) Nonlinear optics : Nonlinear wave mixing

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: June 22, 2009
Revised Manuscript: July 19, 2009
Manuscript Accepted: July 20, 2009
Published: August 17, 2009

Citation
Huaibin Zheng, Yanpeng Zhang, Utsab Khadka, Ruimin Wang, Changbiao Li, Zhiqiang Nie, and Min Xiao, "Modulating the multi-wave mixing processes via the polarizable dark states," Opt. Express 17, 15468-15480 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-15468


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References

  1. S. E. Harris, "Electromagnetically Induced Transparency," Phys. Today. 50, 36-42 (1997). [CrossRef]
  2. J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, "Electromagnetically Induced Transparency in ladder-type inhomogeneously broadened media: Theory and experiment," Phys. Rev. A. 51, 576- 584 (1995). [CrossRef] [PubMed]
  3. A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, "Transporting and time reversing light via atomic coherence," Phys. Rev. Lett. 88, 103601 (2002). [CrossRef]
  4. P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, "Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium," Opt. Lett. 20, 982-984 (1995). [CrossRef] [PubMed]
  5. B. Lu, W. H. Burkett, and M. Xiao, "Nondegenerate Four-Wave Mixing in a double-? system under the influence of coherent population trapping," Opt. Lett. 23, 804-806 (1998). [CrossRef]
  6. K. J Jiang, L. Deng, and M. G. Payne, "Observation of quantum destructive interference in inelastic Two-wave Mixing," Phys. Rev. Lett. 98, 083604 (2007). [CrossRef]
  7. H. Kang, G. Hernandez, J. Zhang, and Y. Zhu, "Phase-contrlled light switching at low light levels," Phys. Rev. A. 73, 011802 (2006). [CrossRef]
  8. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, "Entangled images from Four-Wave Mixing," Science. 321, 544-547 (2008). [CrossRef] [PubMed]
  9. S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, "Four-wave Mixing and biphoton generation in a two-level system," Phys. Rev. Lett. 98, 0536014 (2007). [CrossRef]
  10. V. B. Danielle, A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, "Generation of paired photons with controllable waveforms," Phys. Rev. Lett. 94, 183601 (2005).
  11. C. Cohen-Tannoudji and S. Reynaud, "Dreesed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam," J. Phys. B. 10, 345-363 (1977). [CrossRef]
  12. H. Zheng, Y. Zhang, Z. Nie, C. Li, H. Chong, J. Song, and M. Xiao, "Interplay among multi-dressed Four-wave Mixing processes," Appl. Phys. Lett. 93, 241101 (2008). [CrossRef]
  13. Y. Zhang, B. Anderson, A. W. Brown, and M. Xiao, "Competition between two Four-wave Mixing channels via atomic coherence," Appl. Phys. Lett. 91, 061113 (2007). [CrossRef]
  14. Y. Zhang and M. Xiao, "Generalized dressed and doubly-dressed multi-wave mixing," Opt. Express. 15. 7182-7189 (2007). [CrossRef] [PubMed]
  15. C. J. Zhu, A. A. Senin, Z. H. Lu, J. Gao, Y. Xiao, and J. G. Eden, "Polarization of signal wave radiation generated by parametric four-wave mixing in rubidium vapor: Ultrafast (~150-fs) and nanosecond time scale excitation," Phys. Rev. A. 72, 023811 (2005). [CrossRef]
  16. H. Y. Ling, Y. Q Li, and M. Xiao, "Coherent population trapping and electromagnetically induced transparency in multi-Zeeman-sublevel atoms," Phys. Rev. A,  53, 1014-1026 (1996). [CrossRef] [PubMed]
  17. S. Li, B. Wang, X. Yang, Y. Han, H. Wang, M. Xiao, and K. C. Peng, "Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms," Phys. Rev. A. 74, 033821 (2006). [CrossRef]
  18. For detail in transition probabilities of D1 and D2 line in Rb, see http:// steck. Us/ alkalidata.
  19. Y. Chen, C. Lin, and I. A. Yu, "Role of degenerate Zeeman levels in electromagnetically induced transparency," Phys. Rev. A. 61, 053805 (2000). [CrossRef]

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