Storage and Retrieval of a Weak Optical Signal Improved by Spontaneously Generated Coherence in an Atomic Assemble

Transcription

1 Commun. Theor. Phys. 57 (2012) Vol. 57, No. 3, March 15, 2012 Storage and Retrieval of a Weak Otical Signal Imroved by Sontaneously Generated Coherence in an Atomic Assemble ZHENG Tao (Ü ), 1 QIU Tian-Hui ( ), 2 and YANG Guo-Jian ( Á ) 2, 1 School of Management, Beijing Normal University, Beijing , China 2 Deartment of Physics, Beijing Normal University, Beijing , China (Received November 16, 2011; revised manuscrit received December 26, 2011) Abstract We investigate the reversible storage of a weak single-mode light signal in a Λ-tye three-level atomic system with sontaneously generated coherence (SGC) and umed by an incoherent field. The scheme is hase-deendent. If the hase of the controlling field is set oositely to that of the signal field, the combination of SGC with weak um overcomes the drawback of incomletely retrieval in the conventional scheme, leading to 100% retrieval fidelity or even to the amlification of the retrieved signal with resect to its initial one. PACS numbers: Gy, Qk, Ct, De Key words: reversible storage, electromagnetically induced transarency, sontaneously generated coherence Reversible storage of light (RSL) in an atomic medium [1 3] is of center imortance for the otential alication in long-distance quantum communication [4] and otical quantum comutation. [5] Its essence is to ma the message carried by a signal light to the atomic sin coherence induced by a time-deendent controlling light. [6 9] However, the RSL fidelity is usually less than one because of hoton losses via sontaneous emission, the decay of sin coherence due to collisions and diffusion of atoms out of electromagnetic beams [10] and the narrowing of the EIT window as the controlling field varies, [11] etc. To solve this roblem, various methods have been roosed, including shaing auxiliary control ulses, [12] the feedbackcontrolled ulse shaing [13] and modulating atom sin coherence by alying a weak microwave field, [14] and so on. In a multi-level atomic system, sontaneously generated coherence (SGC) [15 16] may be detected. It stems from the correlation between two different atomic decay athways via couling to common vacuum modes, and occurs as long as the two related atomic diole moments are not erendicular and atomic energy levels of two transitions are very close to each other. [17] In some cases, SGC affects ones otical uroses ositively and can be used to achieve, for examle, gain without inversion [18] and enhanced Kerr nonlinearity, [19] etc. However, SGC is usually very weak. To enhance it, one can change sontaneous emission by adjusting the oulation in the excited state with an incoherence um. In this aer, we follow this route and study the storage and retrieval of a weak signal light in a Λ-tye three-level atomic medium where SGC is active (see Fig. 1). Our main attention is focused on the roblem whether or not SGC can lay a ositive role in the RSL rocess. We use a gas medium for storage, where the effect of atomic recoil due to thermal motion is included. The result shows that the scheme with SGC in active works well if the system arameters are selected aroriately. Fig. 1 Schematic diagram of atom-light interaction of light storage. The light storage scheme under consideration has a Λ- tye atom-field interaction configuration shown by Fig. 1 and can be generalized as follows: an assemble consists of N identical but indeendent atoms distributing homogeneously in sace. Each atom is with an excited state e and two ground states, g and s. A weak signal light of wave number k 1 and roagating toward +Oz, which is to be stored and retrieved, is alied to the atomic transition e g, a coroagating controlling light of wave Suorted by National Natural Science Foundation of China under Grant Nos , , and by the Fundamental Research Funds for the Central Universities yanggj@bnu.edu.cn c 2011 Chinese Physical Society and IOP Publishing Ltd htt:// htt://ct.it.ac.cn

2 464 Communications in Theoretical Physics Vol. 57 number k 2 to its another transition e s. An incoherent um field with intensity 2Λ coules to the atomic transition g e. For this system the atomic olarization induced by a light is described by the z-deendent collective atomic density oerator, ρ ge (z, t), for examle, and defined in the slowly varying aroximation as ρ ge (z, t) = 1 g j e j ex[ ik 1 z]. N j In the conventional treatment, the variables describing the system are ρ αβ (z, t) (α, β = e, g, s) and thus any effects related to the hase factors ex[ ik 1,2 z] can not be seen directly. In rincile atomic recoil resulting from absorbing or emitting a hoton is distinguishable, which imlies that the atomic external degree of freedom is quantized. Considering the relation ex[±ikz] = ρ ge (z, t) can be rewritten as where ρ ge (z, t) 1 N = N j=1 ρ ge ( k 1, ) = 1 N ± k, j g j, j k 1 e j, j ρ ge ( k 1, ), (1) N g j, j k 1 e j, j j=1 is introduced as the -deendent collective atomic density oerator. [20] To obtain the last equation in Eq. (1) we have sorted the atoms into grous according to their momenta, thus the sequence of the two summations over atomic number j and momentum j can be exchanged and then subscrit j is omitted. Similarly, we can define the other collective atomic density oerators, for examle, ρ gs (z, t) 1 g j s j ex[ i(k 1 k 2 )z] N = where the middle equation ρ gs ( (k 1 k 2 ), ) = j ρ gs ( k 1, k 2 ), (2) ρ gs ( k 1, k 2 ) has been considered. ρ gs (z, t) illustrates that the atom gets net recoil momentum (k 1 k 2 ) in a Ramam twohoton transition as a result of illumination by both the signal and the controlling fields. Accordingly, we can introduce -deendent collective atomic states in the way α, c (1/ N) N j=1 α j, j (α = g, s, e) and, as being an extension of Ref. [21], a momentum family F() = { e, c, g, k 1 c, s, k 2 c }. It can be easily verified and understood by Fig. 1 that F() is closed with resect to the stimulated emission or excitation induced by the signal and controlling fields. Therefore, we can make for convenience the following abbreviation ρ ge ( k 1, ) ρ ge (), ρ gg ( k 1, k 1 ) ρ gg (), ρ gs ( k 1, k 2 ) ρ gs (), and ρ ee (, ) ρ ee (), etc. In the electric-diole and rotating-wave aroximations, oerators ρ αβ () (α, β = g, s, e) fulfill the following Heisenberg equations, ρ eg ρ es ρ gg ρ gs ρ ss = (i eg +Λ)ρ eg i 2 ( Gâρ gg Ωρ sg +Gâρ ee ) (γ 1 + γ 2 ) ρ eg, 2 = i es σ es i 2 [ Gâρ gs Ω(ρ ss ρ ee )] (γ 1 + γ 2 ) ρ es, 2 = i 2 ( G â ρ g + Gâρ ge ) 2Λρ gg + γ 1 k1 k 1 duw 1 (u)ρ ee ( k 1 + u), = (i 12 + Λ)ρ es i 2 [ G â ρ es + Ωρ ge ] + k γ 1 γ 2 σ cosϕ duw(u)ρ ee ( k+ u), k = i 2 ( Ω ρ es + Ωρ se ) + γ 2 k2 k 2 duw 2 (u)ρ ee ( k 2 + u), (3) and their conjugations ρ αβ / = ( ρ βα /) (α, β = e, g, and s) restricted by ρ ee + ρ gg + ρ ss = 1. Equation (3) describes a system where the atomic internal degree of freedom is couled with its external one. â is the annihilation oerator of a hoton in the signal field detuned g from atomic transition e g and G is the relative couling constant, Ω stands for the Rabi frequency of the controlling field detuned s from the atomic transition e s, γ 1 (γ 2 ) is the decay rate of the atom from e to g ( s ). αβ () is the single-hoton (α = e, β = g or s) or Raman two-hoton (α = g, β = s) atom-field detuning with Doler-shift, and read eg = k 1 /M k 2 1 /2M g, es = k 2 /M k 2 2 /2M s, gs = (k 1 k 2 )/M + (k 2 1 k2 2 )/2M + g s, resectively. The term γ1 γ 2 σ cosϕ k k duw(u)ρ ee ( k + u)

3 No. 3 Communications in Theoretical Physics 465 reresents the thermal motion-deendent SGC effect with ϕ being the angle between the two atomic diole moments. It has been known that the atomic oulation in the excited state feeds a ground state via sontaneous emission in the rocess of atomic coherent oulation traing, which takes lace between two different families F() and F( ) with = u ( u k i, i = 1, 2). [21] Now we see that in the resence of SGC this rocess also contributes to the atomic sin coherence. The function W i (u) given by W i (u) = 3 1 ( ) 1 + u2 8 k i 2 k 2 i describes the momentum distribution robability of a hoton sontaneously emitting along Oz direction. The system with the active SGC, where σ = 1 in Eqs. (3), otherwise σ = 0 for the out-active SGC case, becomes quite sensitive to the hase difference Φ = φ φ c between the signal field hase φ and the controlling field hase φ c. To see this we searate the hases from the atom-field coulings and the atomic olarization in the way G =, Ω = Ωe Ḡe iφ iφc, ρ eg = ρ eg e iφ, ρ es = ρ es e iφc, ρ sg = ρ sg e iφ. It is found that the equations for ρ αβ (α, β = e, g and s) are identical to Eqs. (3) excet that coefficient σ is relaced by σ e iφ, G by Ḡ, and Ω by Ω. The equation of motion for the signal field to be stored can be written as â = i 2 G ρ ge (). (4) Because the resent RSL analysis is on the semiclassical level, we relace oerator â with its average value ā in the dynamic equations Eqs. (3) and (4). Furthermore, we assume that the signal is very weak ā 1, where ā t=0 = 0.05 is taken, to meet the requirement that the number of signal hotons must be much less than the number of atoms. Fig. 2 Couling field via time. Followings are the numerical solutions of Eqs. (3) and (4). As has been known that it is not ossible to sto a light field comletely through stationary EIT, [22] we choose a time-deendent couling field varying adiabatically in the way: [1] Ω(t) = 30.0{1 0.5 tanh[β (t t1 )]+ 0.5 tanh[β (t t 2 )]}, where β, t 1, and t 2 are adjustable, taken as β = 0.1, t 1 = 50, t 2 = 150, and the related shae of Ω(t) is shown by the Fig. 2. The incoherent um is active in time interval 50 t 150 when the coherent controlling field is turned down. The atoms are initially reared in the ground state g and their thermal motion fulfills the Gaussian momentum distribution with standard half-width. For simlicity, we focus on throughout this aer the degenerated and resonant case where 1 = 2 = 0.0 and k 1 = k 2 = k, and take Ḡ = 10, = 50, and γ 1 = γ 2 = 50. We note here that all arameters used have been scaled: tω r t, 2 /2M ω r 2, k 2 /2Mω r k 2, Ḡ/ω r Ḡ, Ω/ωr Ω, Λ/ω r = Λ, i /ω r i, γ i /ω r γ i (i = 1, 2), where ω r = k 2 /2M is the recoil frequency related to the wave number k. It has been known that SGC could be ositive for one s otical urose in aroriate circumstances. The same thing haens to our RSL scheme. Before discussing this, we note that, as our numerical simulations have shown, if only SGC or the incoherent um is considered, the system behaves no much difference from that when both of them are not considered. [23] Therefore, the ossible imrovement of the RSL oeration comes from their combination action. Figures 3(a), 3(b) stand out the role of the incoherent um Λ laying in the RSL rocess for a given SGC effect described by cosϕ (= 0.9) and given hase difference Φ (= 0.0) between two otical fields. The signal field evolutes similarly as the controlling field does, where the signal mas into the atomic sin coherence in the time interval t 1 < t < t 2. But the retrieved signal changes dramatically with the um rate (Fig. 3(a)). The Λ-deendence of the retrieved signal can be seen much clearly at a fixed time (t = 200) when the retrieval rocess has finished. It has a V -tye feature (Fig. 3(b)), where the level of the dotted (dashed) horizontal line indicates the intensity of the retrieval signal without considering both SGC and incoherent um (the intensity of the signal field before it enters the medium). (This definition also holds for the other figures of Fig. 3) Thus, the segment of the V -tye curve above the dotted horizontal line means that SGC under the corresonding um condition affects the RSL oeration ositively. One can obtain the RSL oeration with 100% retrieval efficiency if the um rate is taken as Λ = , the horizontal ordinate of the intersection between the dashed line and the V -curve, and even an amlified retrieved signal if Λ > Figures 3(c), 3(d) focus the attention mainly on the role of SGC (cos ϕ) for given Λ (= ) and Φ (= 0.0). The resonse of the system to cosϕ is now same as its to Λ when cosϕ and Φ are fixed. Obviously, choosing of

4 466 Communications in Theoretical Physics Vol. 57 ϕ 0 is good for the RSL urose, but at the same time the couling efficiency between the otical field and the corresonding atomic transition decreases. The last factor we should discuss is the hase difference between two otical fields Φ. Figures 3(e), 3(f) are lotted for given Λ (= ) and cosϕ (= 0.8). The Φ-range desirable for us is centered at Φ = π. Further numerical solutions confirm that in this range, the retrieval signal intensity is roortional inversely to cosϕ, and the requirement for the um rate can be greatly reduced. Fig. 3 Time evolution of signal for (a) cos ϕ = 0.9 and Φ = 0.0 with Λ = 0.0, , , and (related to curves 1 4), (c) Λ = and Φ = 0.0 with cos ϕ = 0.0, 0.2, 0.7 and 0.9 (related to curves 1 4), (e) Λ = and cos ϕ = 0.8 with Φ = 0.0, 0.1, 0.2, and 0.27 (related to curves 1 4). Signal is retrieved at t = 200 via (b) Λ for cos ϕ = 0.9 and Φ = 0.0, (d) cos ϕ for Λ = and Φ = 0.0, and (f) Φ for Λ = and cos ϕ = 0.8. The other arameters are γ 1 = γ 2 = 50.0, Ḡ = In hysics, dynamically induced coherence (DIC) resulting from the time-deendent couling between otical fields and atomic transitions is the main reason for RSL. If SGC is active, the quantum interference between DIC and SGC takes lace. Usually the SGC effect is very weak so that no detectable effects is obtained. As the incoherent um is alied, however, the DIC-SGC interference, no matter what it is destructive or constructive, is greatly enhanced. Furthermore, the RSL action originates actually from the atomic internal motion. The influence of the atomic external motion on RSL realizes through the couling between the atomic internal and external degrees of freedoms. That is the exlanation for what we have observed. We have investigated the reversible storage of a weak otical signal in a Λ-tye atomic assemble with the SGC effect and umed by a weak incoherence field. We have shown that by setting the hase of the controlling field oosite to that of the signal field, SGC combined with a um action can be used to overcome the drawback

5 No. 3 Communications in Theoretical Physics 467 of incomletely retrieval in the conventional RSL scheme, leading to the retrieval fidelity aroaching 100% or even with amlification of the signal with resect to its initial one. The resent work is another examle that SGC lays a ositive role in achieving one s goals in coherent otical rocesses. The analysis we have made may be helful for one to see the hysics of quantum coherence and interference that haen in this light storage scheme. References [1] M. Fleischhauer and M.D. Lukin, Phys. Rev. Lett. 84 (2000) [2] C. Liu, Z. Dutton, C.H. Behroozi, and L.V. Hau, Nature (London) 409 (2001) 490. [3] D.F. Phillis, A. Fleischhauer, A. Mair, R.L. Walsworth, and M.D. Lukin, Phys. Rev. Lett. 86 (2001) 783. [4] L.M. Duan, et al., Nature (London) 414 (2001) 423. [5] P. Kok, et al., Rev. Mod. Phys. 79 (2007) 135. [6] M.D. Lukin, Rev. Mod. Phys 75 (2003) 457. [7] B. Julsgaard, et al., Nature (London) 432 (2004) 482. [8] B. Kraus, W. Tittel, N. Gisin, M. Nilsson, S. Kröll, and J.I. Cirac, Phys. Rev. A 73 (2006) [9] Rong Yu, Liugang Si, and Xiaoxue Yang, Ot. Commun. 282 (2009) [10] Y. Xiao, I. Novikova, D.F. Phillis, and R. Walsworth, Phys. Rev. Lett. 96 (2006) [11] I. Novikova, D.F. Phillis, and R.L. Walsworth, Phys. Rev. Lett. 99 (2007) [12] I. Novikova, N.B. Phillis, and A.V. Gorshkov, Phys. Rev. A 78 (2008) [13] F. Beil, M. Buschbeck, G. Heinze, and T. Halfmann, Phys. Rev. A 81 (2010) [14] A. Eilam, A.D. Wilson-Gordon, and H. Friedmann, Ot. Lett. 34 (2009) [15] J. Javanainen, Eurohys. Lett. 17 (1992) 407. [16] P. Zhou and S. Swain, Phys. Rev. Lett. 77 (1996) [17] R.M. Whitley and C.R. Stroud. Jr, Phys. Rev. A 14 (1976) [18] S.E. Harris, Phys. Rev. Lett. 62 (1989) [19] Y.P. Niu and S.Q. Gong, Phys. Rev. A 73 (2006) [20] G.J. Yang, T.H. Qiu, K.G. Wang, and M. Xie, Phys. Rev. A 81 (2010) [21] A. Asect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61 (1988) 826. [22] M. Fleischhauer, A. Imamoglu, and J.P. Marangos, Rev. Mod. Phys. 77 (2005) 633. [23] Xie Min, Qiu Tianhui, Meng Jianyu, and Yang Guojian, J. Ot. Soc. Am. B 26 (2009) 2261.