Optical Multi-wave Mixing Process Based on Electromagnetically Induced Transparency

Transcription

1 Commun. Theor. Phys. (Beijing China 41 (004 pp c International Academic Publishers Vol. 41 No. 1 January Optical Multi-wave Mixing Process Based on Electromagnetically Induced Transparency LI Jia-Hua 1 PENG Ju-Cun and CHEN Ai-Xi 13 1 Department of Physics Huazhong University of cience and Technology Wuhan China Department of Physics Xiaogan Normal University Xiaogan China 3 Wuhan Institute of Physics and Mathematics the Chinese Academy of ciences Wuhan China (Received April 003; Revised June Abstract In this paper we propose and analyze an optical multi-wave mixing scheme for the generation of coherent light in a five-level atomic system in the context of electromagnetically induced transparency. A detailed semiclassical study of the propagation of generated mixing and probe fields is demonstrated. The analytical dependence of the generated mixing field on the probe field and the respective detuning is predicted. uch a nonlinear optical process can be used for generating short-wavelength radiation at low pump intensities. PAC numbers: 4.50.Gy 4.65.Ky 4.50.Hz Key words: optical multi-wave mixing electromagnetically induced transparency Rabi frequency 1 Introduction Electromagnetically induced transparency (EIT is the optical transparency of a three-level medium at a resonant transition induced by application of a coherent electromagnetic field at an adjacent transition. [1 8] The accompanying reduction of the group velocity of light by many orders of magnitude in the EIT transparency window [3] has been both intensively and actively investigated in connection with many potential applications especially lowintensity nonlinear optics (i.e. it can reach a level of few photons quantum information processing and quantum information storage. [9] In essence the transparency may be viewed as the result of a combination of the dynamic ac-tark splitting or Autler Townes splitting and the interference between the two dressed states that are created by the strong coupling laser. Recently chmit and co-workers have investigated a nonlinear double-λ scheme without considering the wave-mixing process. It turns out that based on electromagnetically induced transparency the nonlinear susceptibility of the probe field can greatly increase by many orders of magnitude. [10] Li et al. have demonstrated a clear experimental observation of an enhanced nondegenerate four-wave mixing (NDFWM process making use of the EIT effect in a three-level Λ- type system of Rb atoms with cw diode lasers. [] Later on Deng et al. have proposed an EIT-based double-λ system with the inclusion of the generated field and by means of the time-dependent perturbation treatment they have shown that the possible quantum constructive and destructive interference between different excitation pathways can result in enhancement and suppression of the Kerr nonlinearity. [11] More recently Wu et al. analyzed a four-wave-mixing (FWM scheme in a five-level atomic system and hyper-raman scattering (HR in resonant coherent media by the use of EIT which leads to suppressing both two-photon and three-photon absorptions in both FWM and HR schemes and enabling the four-wave mixing to proceed through real resonant intermediate states without absorption loss. [113] In particular with the advent of Bose Einstein condensation in an atomic gas there has been much interest in studying four-wave mixing with matter wave both experimentally and theoretically in the framework of nonlinear atomic optics. [14 19] In this context in the present work by applying the method of Wu et al. [1] we present and analyze an optical multi-wave mixing scheme for the generation of coherent light in the five-level atomic system by means of electromagnetically induced transparency under the condition that we apply the weak probe pulsed laser. A detailed semiclassical study of the propagation of the generated mixing and probe fields is demonstrated. The influences of the probe field and the respective detuning on the generated mixing field are discussed. uch a nonlinear optical process can be used for producing the coherent shortwavelength radiation. To conclude we give a brief discussion on the experimental realization of the proposed scheme. Model and olutions of Atomic Equations of Motion Let us consider the energy scheme depicted in Fig. 1. In this five-level atomic scheme a weak probe pulsed laser (pulse length τ is tuned near the 0 resonance. The project supported by National Fundamental Research Program of China under Grant No. 001CB and National Natural cience Foundation of China under Grant Nos and

2 No. 1 Optical Multi-wave Mixing Process Based on Electromagnetically Induced Transparency 107 One of the two lower levels is coupled to the upper level ( 1 by a strongly coherent coupling laser to create an Autler Townes doublet and two sign laser fields are tuned to the transition 1 3 and 3 4 respectively. To avoid significant group velocity mismatch and for mathematical simplicity we will assume that all the involving fields are continuous single-frequency laser (cw laser with the exception of the probe field. It should be noted however that the transition 1 3 is dipoleforbidden in line with the well-known selection rule yet we can choose sufficiently strong laser field to stimulate electric quadrupole transitions which still leads to observed emission and absorption. In the present analysis the semiclassical Hamiltonian describing the atom-field interaction for the system under consideration can be written as in the chrödinger picture (assuming h = 1 Fig. 1 EIT-based five-level atomic diagram for analysis. H = 4 ε j j j + (Ω p e iθp 0 + h.c. + (Ω c e iθc 1 + h.c. + (Ω s1 e iθs h.c. j=0 + (Ω s e iθs h.c. + (Ω m e iθm h.c. (1 where θ n = k n r ω n t corresponds to the positive-frequency part of the respective field Ω n (n = c p s1 s m are one-half Rabi frequencies for the corresponding transitions i.e. Ω p = D 0 E(ω p /( h Ω c = D 1 E(ω c /( h Ω s1 = D 31 E(ω s1 /( h Ω s = D 43 E(ω s /( h Ω m = D 40 E(ω m /( h with D kl denoting the dipole moment for the transition between levels k and l and ε j = hω j is the energy of the atomic state j. For simplicity of analysis in what follows we will take ε 0 = 0 for the ground state 0. Turning to the interaction picture the Hamiltonian can be rewritten as H 0 = (ω p ω c ω p + (ω p ω c + ω s (ω p ω c + ω s1 + ω s 4 4 ( H int = ω c ω p + ω ω (Ω p e ikp r 0 + Ω c e ikc r 1 + Ω s1 e iks1 r Ω s e iks r Ω m e ikm r h.c. (3 where ω p = p = (ω ω 0 ω p is the single-photon detuning ω c = p c = (ω 1 ω 0 (ω p ω c ω 3 = p c + δ 1 = (ω 3 ω 0 (ω p ω c ω s1 and ω 4 = p c + δ 1 + δ m = (ω 4 ω 0 (ω p ω c ω s1 ω s are the separate two-photon three-photon and four-photon detunings respectively. Let us assume that the wave function has the form Ψ = A A 1 e i(kp kc r 1 + A e ikp r + A 3 e i(kp kc+ks1 r 3 + A 4 e ikm r 4 (4 with A j standing for the probability amplitude of the atomic state respectively. Making use of the chrödinger equation in the interaction picture i / t Ψ = H int Ψ the equations of motion for the probability amplitude of the atomic wave functions can be readily obtained as t A 0 = iω pa iω ma 4 t + γ 1 A 1 = iω ca iω s1a 3 t + γ A = iω c A 1 iω p A 0 t + γ 3 A 3 = iω s1 A 1 iω s e iδk r A 4 t + γ 4 A 4 = iω m A 0 iω s e iδk r A 3 (5 where we have introduced the definitions γ 1 = γ 1 + i ω c γ = γ + i ω p γ 3 = γ 3 + i ω 3 γ 4 = γ 4 + i ω 4 and phase match relation δk = k p + k s1 + k s k c k m. γ j (j = 1 4 is the relaxation rate of the state j.

3 108 LI Jia-Hua PENG Ju-Cun and CHEN Ai-Xi Vol. 41 In order to correctly describe the propagation of the optical pulse the atomic equations of motion must be simultaneously solved with the Maxwell s equations for both the probe and the generated mixing fields in a selfconsistent manner. In the limit of plane waves and slowly varying amplitude the positive frequency parts of these fields satisfy ( E p = 4πik p P (+ P E m = 4πik m P m (+ (6 where E p (P (+ P and E m(p m (+ are the slowly varying envelopes of the probe and generated mixing field amplitudes (polarizations. In order to treat the propagation of the generated MWM field through a resonant medium the polarization of the medium must be obtained. In the following using the atomic wave function (4 we have P (r 0 = N Ψ ˆD Ψ = P p (+ (ω p e ikp r + P c (+ (ω c e ikc r + P (+ s1 (ω s1 e iks1 r + P (+ s (ω s e iks r + P (+ m (ω m e ikm r + h.c. (7 where N and ˆD are the atomic concentration and the dipole moment operator of the relevant transition respectively. P n (+ (ω n represents the positive frequency part of the polarization at the corresponding field i.e. P p (+ (ω p = ND 0 A A 0 P c (+ (ω c = ND 1 A A 1 P (+ s1 (ω s1 = ND 13 A 3 A 1 P (+ s (ω s = ND 34 A 4 A 3 P m (+ (ω m = ND 04 A 4 A 0. For both the probe at ω p and generated mixing fields combining P p (+ (ω p = ND 0 A A 0 and P m (+ (ω m = ND 04 A 4 A 0 with Ω p = D 0 E(ω p /( h and Ω m = D 40 E(ω m /( h the slowly varying envelope equations for the Rabi frequencies Ω p and Ω m are given by Ω p = ik 0 A A 0 Ω m = ik 04 A 4 A 0 (8 where we have introduced the notations k 0 = πnω p D 0 /( hc k 04 = πnω m D 04 /( hc. Following the standard procedure described before in Refs. [1] [13] [0] and [1] the probe pulsed laser is assumed to be sufficiently weak that the ratio (Ω p /Ω c is much less than unity so that in essence all of the atomic population remains in the ground state 0 i.e. A 0 1. Again we consider the coupling field ω c and sign fields ω s1 and ω s to be continuous single-frequency laser but the probe field ω p and the generated mixing field ω m have a time-dependent beam profile. With these assumptions performing Fourier transformations A j (t = (1/ π α j(ω exp( iωtdω j = 1 4 and Ω j (t = (1/ π W j(ω exp( iωtdω j = p m for both Eqs. (5 and Eqs. (8 we straightforwardly obtain the following equations ω 1 α 1 = Ω cα + Ω s1α 3 ω α = Ω c α 1 + W p ω 3 α 3 = Ω s1 α 1 + Ω s e iδk r α 4 ω 4 α 4 = W m + Ω s e iδk r α 3 (9 where we have introduced the notations ω 1 = ω + i γ 1 ω = ω + i γ ω 3 = ω + i γ 3 and ω 4 = ω + i γ 4. Equations (9 can be solved in terms of W p and W m with the result Likewise combining α = Ω cω s1ω s α 4 = Ω cω s1 Ω s W m e iδk r + ω 1 ω 3 ω 4 ω 1 Ω s ω 4 Ω s1 W p W p e iδk r + ω 1 ω ω 3 ω 3 Ω c ω Ω s1 W m. (10 z i ω c W p = ik 0 α then inserting Eqs. (10 into Eqs. (11 we have W ( p ω z i c + f W p = ig Wm where we have defined the new parameters z i ω c W m = ik 04 α 4 (11 W m z W m = W m e iδkz ω 1 ω 3 ω 4 ω 1 Ω s ω 4 Ω s1 f = k 0 g = k 0 Ω c Ω s1ω s f 4 = k 04 ω 1 ω ω 3 ω 3 Ω c ω Ω s1 i( ω c + f 4 δk Wm = ig 4 W p (1 g = k 04 Ω cω s1 Ω s = ω 1 ω ω 3 ω 4 ω 1 ω Ω s ω ω 4 Ω s1 ω 3 ω 4 Ω c + Ω s Ω c.

4 No. 1 Optical Multi-wave Mixing Process Based on Electromagnetically Induced Transparency 109 After some algebra but with no further assumptions the solutions to Eqs. (1 are W p (z ω = Q exp(iλ + z + R exp(iλ z W m (z ω = [Q 4 exp(iλ + z + R 4 exp(iλ z] exp(iδkz (13 where the constants Q j and R j are determined by the boundary condition at z = 0 and λ ± = ω c + f + f 4 δk ± 1 (f f 4 + δk + 4g g 4. 3 Discussions For given W p (0 ω and the MWM emission process W m (0 ω = 0 the generated MWM field W m (z ω is then given by g 4 W m (z ω = W p (0 ω ( e iλ+z e iλ z e iδkz λ + λ W p (z ω = W p(0 ω [( λ + ω λ + λ c f ( e iλ z λ ω c f e iδkz] (14 which is the main result of the present study. If we use Gaussian pulse shape for the probe pulsed laser at the entrance to the medium Ω p (0 t = Ω p (0 0 e (t/τ. Carrying out Fourier transformation then we find W p (0 η = Ω p (0 0(τ/ e η /4 where we have introduced the dimensionless variable η = ωτ. Under perfect phase match conditions δk = 0 we then arrive at where W m (z η = i Ω p(0 0τ W p (z η = Ω p(0 0τ e η /4 g 4(η Λ(η e id(ηz/ sin e η /4 e id(ηz/ { i f 4(η f (η Λ(η f (η = k 0 D 1D 3 D 4 D 1 τω s D 4 τω s1 (η [ Λ(ηz ] [ Λ(ηz sin ] + cos [ Λ(ηz g (η = k 0 (τω c(τω s1(τω s (η f 4 (η = k 04 D 1D D 3 D 3 τω c D τω s1 g 4 (η = k 04 (τω c(τω s1 (τω s (η (η D(η = η + f (η + f 4 (η [( f (η f 4 (η 1/ Λ(η = + g (η g 4 (η] (η = D 1 D D 3 D 4 D 1 D τω D D 4 τω 1 D 3 D 4 τω c + τω s τω c D 1 = η ( p c τ + iγ 1 τ D = η p τ + iγ τ D 3 = η ( p c + δ 1 τ + iγ 3 τ D 4 = η ( p c + δ 1 + δ m τ + iγ 4 τ. ]} (15 In order to show that the MWM scheme is capable of generating the MWM field at low pump intensities we choose the Rabi frequencies of the probe Ω p and one sign field Ω s well below the saturation levels yet another sign field Ω s1 well above the saturation levels and present numerical calculations as shown in Fig.. The results show that W m is maximized at δ 1 τ = 0 and δ m τ = 0 under the usual EIT conditions demonstrating the enhancement of the MWM via the resonant intermediate state manifested by the EIT-induced suppression of the three-photon and four-photon absorptions. Now we give a brief discussion on the experimental realization of our scheme. An experimental candidate for the proposed system can be found in 87 Rb cold atoms with the designated states chosen as follows: 5 1/ F = 1 as 0 5 1/ F = as 1 5P 1/ as 5D 3/ as 3 and np 3/ as 4 (n > 10. The associated transitions are 0 and 1 at 795 nm (γ 5.6 MHz 1 3 at 76 nm (γ 3 = 0.76 MHz and 3 4 at µm (γ MHz all accessible with diode lasers. For the concerning parameters we can choose atomic density N 10 1 cm 3 length of Rb cell L 6 mm the probe pulse length τ s Rabi frequencies Ω c τ = 100 Ω p τ = 10 Ω s1 τ = 100 and Ω s τ = 5. The wavelength of the generated coherent light is of the order of 100 nm. By use of electromagnetically induced transparency strong-coupling field Ω c is applied to the 1 transition first thereby setting up a quantum interference and allowing the weak-probepulsed 0 field to propagate in a refractively thick

5 110 LI Jia-Hua PENG Ju-Cun and CHEN Ai-Xi Vol. 41 Rb atomic medium. Then the 0 field is increased slowly as compared to the Rabi frequency of the 1 field which contributes significantly to the ground-state atoms to evolve smoothly into a population-trapped superposition state with approximately equal and oppositely phased probability amplitudes between states 0 and 1. Fig. Plot of the generated MWM field W m as a function η for a five-level atomic system. The parameters are Ω p τ = 10 Ω c τ = 100 Ω s1 τ = 100 Ω s τ = 5 γ 1τ = 0.01 γ τ = 500 γ 3τ = 5 γ 4τ = 0.1 k 0 = 100 k 04 = 10 z/ =. (a pτ = = δ 1τ = δ mτ = 0; (b pτ = = δ 1τ = 0 δ mτ = 0; (c pτ = = 0 δ 1τ = δ mτ = 0; and (d pτ = 10 = 5 δ 1τ = δ mτ = 0. 4 Conclusions In summary in the present work we present and analyze an optical multi-wave mixing scheme for the generation of coherent light in the five-level atomic system by means of electromagnetically induced transparency under the condition that we apply the weak probe pulsed laser. A detailed semiclassical study of the propagation of the generated mixing and probe fields is demonstrated. The influences of the probe field and the respective detuning on the generated mixing field are discussed. We further show that such a nonlinear optical process can be used for producing the short-wavelength radiation. In addition we give a brief discussion on the experimental realization for our proposed scheme. Acknowledgments The authors would like to thank Y. Wu for many stimulating discussions. References [1] E. Arimondo in Progress in Optics ed. E. Wolf Elsevier cience Amsterdam (1996 pp [] Y. Li and M. Xiao Opt. Lett. 1 ( [3] L.V. Hau.E. Harris Z. Dutton and C.H. Behroozi Nature 397 ( [4] M. Yan E. Rickey and Y. Zhu Phys. Rev. A64 ( [5] H. Wang D. Goorskey and M. Xiao Phys. Rev. Lett. 87 ( [6].E. Harris J.E. Field and A. Kasapi Phys. Rev. A46 (199 R9. [7] M. Yan E. Rickey and Y. Zhu Opt. Lett. 6 ( [8] L. Deng M. Kozuma E. W. Hagley and M.G. Payne Phys. Rev. Lett. 88 ( [9] C. Liu Z. Dutton C.H. Behroozi and L.V. Hau Nature 409 ( [10] H. chmidt and A. Imamoğlu Opt. Lett. 1 ( [11] L. Deng M.G. Payne and W.R. Garrett Phys. Rev. A64 ( [1] Y. Wu J. aldana and Y. Zhu Phys. Rev. A67 ( [13] Y. Wu L. Wen and Y. Zhu Opt. Lett. 8 ( [14] J. Heurich et al. Phys. Rev. A63 ( [15] C.K. Law et al. Phys. Rev. A63 ( [16] L. Deng et al. Nature 398 ( [17] Y. Wu et al. Phys. Rev. A61 ( [18] Q. Yang et al. Phys. Rev. A67 ( [19] P. Villain et al. Phys. Rev. A64 ( [0].E. Harris and Y. Yamamoto Phys. Rev. Lett. 81 ( [1].E. Harris and L.V. Hau Phys. Rev. Lett. 8 (