Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities

Transcription

1 PHYSICAL REVIEW A 8, 8 (11) Multitability of electromagnetically induced tranparency in atom-aited optomechanical cavitie Yue Chang, 1 T. Shi, 1 Yu-xi Liu,,,4 C. P. Sun, 1 Franco Nori,,5 1 Intitute of Theoretical Phyic, Chinee Academy of Science, Beijing 119, China Advanced Science Intitute, The Intitute of Phyical Chemical Reearch (RIKEN), Wako-hi , Japan CREST, Japan Science Technology Agency (JST), Kawaguchi, Saitama -1, Japan 4 Intitute of Microelectronic, Tinghua National Laboratory for Information Science Technology (TNLit), Tinghua Univerity, Beijing 184, China 5 Center for Theoretical Phyic, Phyic Department, Center for e Study of Complex Sytem, The Univerity of Michigan, Ann Arbor, Michigan , USA (Received 5 May 9; publihed June 11) We tudy how an ocillating mirror affect e electromagnetically induced tranparency (EIT) of an atomic enemble confined in a ga cell, which i placed inide a microcavity wi an ocillating mirror in one end. The cavity field act a a probe of e EIT ytem alo produce a light preure on e mirror. The back-action from e mirror to e cavity field reult in everal (from one to five) teady-tate for i atom-aited optomechanical cavity, producing a complex tructure in it EIT. The multiequilibrium poition of e ocillating mirror have been calculated, en e uceptibility i tudied wi repect to e different equilibrium poition. We find at e EIT of e atomic enemble can be ignificantly changed by e ocillating mirror, alo e variou teady tate of e mirror have different effect on e EIT. DOI: 1.11/PhyRevA.8.8 PACS number(): 4.5.Gy, 4.5.Pq,.8.Qk I. INTRODUCTION Fat development are now occurring in tudie at e interface between different kind of phyical ytem. Example of ee include: coupling between light nano- or micromechanical ytem (e.g., in Ref. [1 1] ), called optomechanical ytem; interaction between uperconducting artificial atom (e.g., charge or flux qubit); tranmiion line reonator (e.g., in Ref. [11 1]); o on. Thee tudie are partly motivated by poible phyical implementation of quantum information proceing, to explore potential future device, to tudy e intereting phyic in ee hybrid tructure. In optomechanical ytem (e.g., in Ref. [1 1]), e radiation preure act on e ocillating mirror induce e interaction between e mechanical ytem e optical field. The back-action from e mirror to e cavity field [17] can reult in everal teady-tate olution for e equilibrium poition of e mirror, e.g, it ha been proved [1] at i ytem experience bitability in ome parameter region. It ha alo been hown at light preure can be ued to realize entanglement between e cavity field a microcopic object (e.g., a movable mirror) [1 4], can alo cool down e mirror [5 8]. Furermore, Ref. [18,19] tudied optomechanical cavitie containing a two-level atomic enemble. The atomic enemble effectively enhance e radiation preure of e cavity field on e ocillating mirror, producing a cavity-atom-mirror entanglement [18,19]. In i paper we conider i atom-optical ytem wi a ree-level atomic enemble, which can enable quantum interference, e.g., electromagnetically induced tranparency. Electromagnetically induced tranparency (EIT) i a remarkable quantum interference phenomenon, which enable e active control of light propagation in a coherent medium [ ]. Uually, e baic population tranfer configuration for e atom in EIT i of type, where e two tranition from a common upper energy level to two lower energy level are induced by two different optical field (e.g., claical control field quantized probe field) [4 ], repectively. One i a trong field, e oer i a weak one. The tronger field can effectively modify e uceptibility of e medium o at e weak one (a a probe ignal) can pa rough e medium tranparently at e two-photon reonance [7 1]. Recently, it ha been hown at i effect can alo work well at e ingle photon level for e probe light [], u e weak field mut be treated quantum mechanically. In e quantum approach, a dark tate wi dreed photon can be invoked to tore quantum information of photon on e atomic enemble a quantum memory [,4]. Thee quantum manipulation at e ingle photon level require frequency matching wi extremely high preciion for one- or two-photon reonance. When e quantum field i provided by a microcavity wi a one-end ocillating mirror, e ocillation of e mirror might affect uch a precie frequency-matching condition u affect e EIT. Motivated by ee work, here we tudy how e ocillating mirror change e propertie of e EIT. We will tudy e atom-aited optomechanical microcavity, rough which we explore e poibility to interface oer ytem, via ome phyical mechanim, uch a EIT. Here, e ree-level atomic enemble for each atom wi -type tranition i placed inide a cavity wi a one-end ocillating mirror. Due to e mirror ocillation, e uceptibility of i atomic medium diplay a multitability correponding to e multiequilibrium poition of e mirror. Anoer prediction of our tudy i at e mirror ocillation ignificantly alter e propertie of bo e real imaginary part of e uceptibility, u make e EIT phenomena change. We alo invetigate in detail how e different teady tate of e mirror influence e diperion relation aborption propertie of e light. The paper i organized a follow. In Sec. II, we introduce e model for e optomechanical ytem wi e EIT. In Sec. III, we preent e Heienberg-Langevin equation for /11/8()/8(1) American Phyical Society

2 CHANG, SHI, LIU, SUN, AND NORI PHYSICAL REVIEW A 8, 8 (11) Fixed mirror p Probe light b a Λ -type atomic enemble Driving field c Claical light c Ocillating mirror FIG. 1. (Color online) Schematic diagram for e atom-aited optomechanical ytem conidered here. There are ree main component: (i) an optical cavity wi one fixed mirror; (ii) anoer ocillating mirror, which i modeled a a quantum-mechanical harmonic ocillator (denoted by a black pring); (iii) an enemble of identical ree-level atom, which are confined inide e cavity. Each atom i aumed to have -type tranition. Here, p = ω a ω + ω x /l c = ω a ω c ν. i ytem obtain everal (from one to five, depending on e ytem parameter) teady-tate olution for e poition of e ocillating mirror. In Sec. IV, we tudy e EIT wi uceptibilitie for different (from one to two) equilibrium poition of e mirror. In Sec. V, we ummarize our reult. II. ATOM-ASSISTED OPTOMECHANICAL SYSTEM A hown in Fig. 1, we conider an enemble of N identical ree-level atom, which are confined inide an optical cavity wi a one-end ocillating mirror. The excited, metatable, ground tate of e i atom are denoted by a i, c i, b i. Each atom i aumed to have -type tranition. That i, for e i atom, a claical light field wi e frequency ν induce a tranition between a i c i, which i often ued a a control field. The quantized cavity field, wi e frequency ω when e ocillating mirror i fixed, induce e tranition between a i b i. We aume at i cavity field act a a probe field. Here, e tranition between b i c i i aumed to be forbidden. The ocillating mirror i modeled a a quantum-mechanical harmonic ocillator wi e frequency ω M e ma M. Thi harmonic ocillator can alo be conidered a a pring wi an elatic coefficient MωM. Baed on e above conideration, uing h = 1, e Hamiltonian of e total ytem H = H C + H M + H A + H M-C + H A-L (1) ha five term correponding to (i) e cavity (H C ), (ii) e ocillating mirror (H M )ofemam, (iii) e atom ga (H A ), (iv) e mirror cavity (H M-C ), (v) e atom-light term (H A-L ). Explicitly, ee are decribed below: (i) e free Hamiltonian H C = ω a a, () of e ingle-mode cavity field wi e annihilation creation operator a a ; i term ( ω ) refer to a cavity wi two fixed mirror; (ii) e free Hamiltonian H M = p M + 1 Mω M x, () of e ocillating mirror, where p i e momentum of e ocillating mirror wi a mall diplacement x; (iii) e free Hamiltonian H A = ( ωa σ aa (i) + ω cσ cc (i) ), (4) of e N ree-level atom wi e operator σ αα (i) = α ii α α = a,c; here, ω a (ω c ) i e energy-level pacing between a i b i ( c i b i )forei atom, we have aumed e ground tate b i a an energy reference point; (iv) e interaction Hamiltonian H M-C = ω xa a, (5) l between e cavity field e ocillating mirror [1], preent a radiation preure on e mirror due to e mall change x of e cavity leng when e mirror ocillate [5], where l i e cavity leng when e mirror i fixed; (v) e interaction Hamiltonian H A-L = ( e iνt ac + gaσ(i) ab + H.c.), () between e ree-level atom e claical a well a e quantized field. In Eq. (), i e Rabi frequency aociated wi e coupling between e claical field e ree-level atom. The frequency ν i aumed to atify e condition ν = ω a ω c c. (7) Here, c i e detuning between e frequency of e claical control field e tranition frequency from e energy level a i to e energy level c i for e i atom. The parameter g = µ ω /Vɛ (8) in () decribe e coupling between e quantized cavity field e ree-level atom, where µ i e electric-dipole tranition matrix element between level a i b i, V decribe e volume of e cavity, ɛ i e permittivity of e vacuum. We note at e effect of e ocillating mirror on e coupling between e atom e quantized cavity field [5] ha been neglected when e Hamiltonian in Eq. (1) wa derived, becaue we do not conider e trong coupling between e quantized field e atom. The dynamic of e ytem governed by e Hamiltonian H in Eq. (1) i depicted by e Heienberg-Langevin equation t x = p M, (9a) t p = γ M M p + ω l a a Mω M x γ M ɛ in (t), (9b) 8-

3 MULTISTABILITY OF ELECTROMAGNETICALLY INDUCED... PHYSICAL REVIEW A 8, 8 (11) t a = γ ( a iω 1 x ) a ig l t t N N ba = (γ 1 + iω a ) ig a bc = γ +iga j=1 j=1 [ σ (j) bb bc iω c ba i e iνt ba + γ a in (t), bc (9c) σ aa (j) ] + Nf1 (t), (9d) bc i eiνt ba σ (j) ac + Nf (t). (9e) Here, for convenience, we have neglected detailed derivation a in Ref. [4], jut phenomenologically introduce e damping γ M of e ocillating mirror, e decay rate γ for e cavity field, e decay rate γ 1 (γ )from a to b ( c to b ). The decay from c to b i an equivalent reult alough ere i no direct electric dipole tranition between e energy level c b. Phyically, i decay i from e electric dipole tranition between a b a well a a c, which can be referred to Ref. [4]. We alo aume at e quantum fluctuation of e cavity field, mirror, e atom atify e condition ɛ in (t) = f 1 (t) = f (t) = (1) a in (t) = α in (t). (11) Here, α in (t) can be undertood a an input driving field. That i, a in (t) can be rewritten a a in (t) = α in (t) + δa in (t), (1) where e quantum fluctuation of e input field δa in (t) atifie δa in (t) =. III. HEISENBERG-LANGEVIN EQUATIONS AND MULTISTABILITY A. Steady-tate poition of e movable mirror: Analytical reult We are intereted in e teady-tate olution. In e derivation of e Hamiltonian in Eq. (1), we have aumed at e linear ize of e atomic enemble i much maller an e waveleng of e light field. In i cae, e coupling between e atom e light field are homogeneou, we can define e collective operator of e atomic enemble a in Ref. [5,]: S = T + = (T ) = aa, A = 1 N N ac, C = 1 N ab, N bc. (1a) (1b) Togeer wi Eq. (1a) (1b), e interaction Hamiltonian H A-C in Eq. () can be rewritten a H A-L = e iνt T + + g NaA + H.c. (14) by uing e collective operator. Let u now aume: (i) e number N of atom i large enough; (ii) mot of e atom are in e ground tate, i.e., e atomic ytem i in very low excitation. Baed on ee two aumption, we can find at e collective operator in Eq. (1a) (1b) atify e commutation relation a in Ref. [19,5,]: [C,S] =, [A,S] = A, (15a) [T,C ] =, [T +,C ] = A, (15b) [A,C] =, [T +,A] = C, (15c) [T,C] = A, (15d) [A,A ] = 1 N [C,C ] = 1 N [A,C ] = 1 N T, j=1 j=1 [ σ (j) bb [ σ (j) bb (15e) σ aa (j) ] 1, (15f) σ cc (j) ] 1. (15g) Here, it i clear at σ (j) bb, σ aa (j) σ cc (j) are e population operator at e i atom i in different tate. In e following calculation, we are intereted in e average value in e teady tate. Baed on e aumption at mot of e atom are in e ground tate, e average σ bb 1, but oer average σ cc σ aa. Thu e average of collective operator N j=1 σ aa (j) / N N j=1 σ cc (j) / N are much maller an one due to e low excitation in e excited tate a c, however, N j=1 σ (j) bb / N, A C are large quantitie due to e large number of atom in e ground tate b. Moreover our calculation below are not related to e correlation pectrum, u e mall contribution of e average of e operator N j=1 σ ac (j) / N, which characterize e tranition between two upper level, i neglected below. Equation (15a) (15b) preent a dynamical ymmetry in our ytem decribed by e emidirect-product algebra containing e algebra SU() wi it generator T ±.Iti eay to prove, a in Ref. [19], at e collective operator e commutation relation can alo be given in a imilar way a in Eq. (), (15a) (15g) for e cae when e coupling between different atom e light field are inhomogeneou. Therefore, our tudy here can be generalized to e cae of inhomogeneou coupling. Uing e commutation relation in Eq. (15a) (15g) ine low excitation of e atomic ytem, e Heienberg-Langevin equation (9a) (9e) are now implified to t x = p M, t p = γ M M p + ω l a a Mω M x γ M ɛ in (t), (1a) (1b) 8-

4 CHANG, SHI, LIU, SUN, AND NORI PHYSICAL REVIEW A 8, 8 (11) t a = γ ( a iω 1 x ) a ig NA+ γ a in (t), l (1c) t A = γ 1 A iω a A i e iνt C ig Na+ f 1 (t), (1d) t C = γ C iω c C i e iνt A + f (t). (1e) We note: (i) e equation of motion for e operator T i decoupled from e above equation; (ii) we can alo derive e equation of motion wiout conidering e low excitation approximation firt, en we can obtain e ame reult a e above by taking i approximation when we olve e teady-tate equation. To obtain e teady-tate olution, let u firt remove e fat varying factor by e following rotating tranformation: a = ã exp( iω L t), (17a) A = à exp( iω L t), (17b) C = C exp[i( p c ω c )t], (17c) a in (t) = ã in (t) exp( iω L t) (17d) = [ α in (t) + δã in (t)]exp( iω L t), (17e) where e detuning p between e tranition frequency ω a of e atom, from e energy level a to b, e effective frequency ω L of e cavity field, i given by p = ω a ω L, (18) wi e effective cavity frequency ω L = ω ω x. (19) l Here, x denote e mean value of x. Equation (19) how at e effective frequency ω L of e cavity can be changed by e ocillating mirror. In e rotating frame, e Heienberg-Langevin equation in Eq. (1b) (1e) become t p = γ M M p + ω ã ã MωM l x γ M ɛ in (t), (a) t à = (γ 1 + i p )à i C ig N ã + f 1 (t), (b) t C = [γ + i( p c )] C i à + f (t), (c) [ γ t ã = i ω ] l ( x x) a ig N à + γ ã in (t), (d) wi e fluctuation ( f 1 (t) = f 1 (t)exp [iω 1 + x ) ] t (1a) l f (t) = f (t)exp[ i( p c ω c )t]. (1b) We are intereted in e teady-tate regime. Let u now aume at all e time derivative of e mean value for e operator in Eq. (a) (d) are equal to zero, en we obtain e teadytate equation ω l ã ã MωM x =, (a) γ ã ig N à + γ α in =, (b) (γ 1 + i p, ) à i C ig N ã =, (c) [γ + i( p, c )] C i à =. (d) Here, O (O repreent e operator in e above teady-tate equation) i e mean value of e operator O under e teady tate. The parameter p, i e detuning decribed in Eq. (18) when e ytem reache teady-tate. In Eq. (a) (b), e correlation ã ã xã have been approximately replaced by ã ã x ã, repectively. Thee approximation indicate at e correlation between e fluctuation near e teady tate are very mall compared to e correponding mean value in e teady tate. Thi can be quantitatively hown a [] (δã )(δã) ã ã 1, (δx)(δã) x ã 1. () From Eq. (b) (d), when e ytem reache e teadytate, e mean value à ã can be eaily derived a à = ig Nγ α in ( p, ) G( p, ) ( p, ) + γ (4) ] ã = α in γ [1 g N ( p, ) G( p, ) ( p, ) + γ, (5) wi e function ( p, ) = γ + i( p, c ) () G( p, ) = g N + 1 γ (γ 1 + i p, ). (7) Uing Eq. (a) (5), we can elf-conitently derive a nonlinear implicit equation for p, a below 1 g N ( p, ) G( p, ) ( p, ) + γ = γ κ ( 4 α in p, ), ω (8) wi e parameter κ = Mω M l, (9) = ω a ω. () For e detuning between ω a (e highet frequency of e atom) ω [e cavity frequency in Eq. () when e two mirror are fixed]. Equation (8) i a fif-power implicit equation for p,. Thu e ytem may have everal olution (i.e., multitability in ome parameter region, correponding to everal teady-tate poition of e mirror). Thee mirror poition are determined by x = ω ã ã Mω M l. (1) 8-4

5 MULTISTABILITY OF ELECTROMAGNETICALLY INDUCED... PHYSICAL REVIEW A 8, 8 (11) B. Steady-tate poition of e movable mirror: Numerical reult Let u firt analyze e atom-cavity detuning p, e equilibrium poition x of e mirror. In principle, we can obtain p, by olving e fif-order equation in Eq. (8). However, intead of i, we obtain p, by plotting e left Y L ( p, ) right Y R ( p, ) h ide of Eq. (8) a function of p,, repectively. The real root of Eq. (8) can be repreented by e croing point of two curve correponding to Y L ( p, ) Y R ( p, ). Here, for clarity, ee are hown a Y L ( p, ) = 1 g N ( p, ) G( p, ) ( p, ) + γ Y R ( p, ) = γ κ 4 α in ω, () ( p, ). () The curve e line, correponding to Y L ( p, ) Y R ( p, ), are chematically hown in Fig., which i ued to analyze e olution of e elf-conitent equation in Eq. (8). In Fig. (a), e double-well-like curve e traight line how how Y L ( p, ) Y R ( p, ) change wi p, for a given κ but for larger = ω a ω, repectively. Figure (b) how e olution of e elf-conitent equation in Eq. (8) (a) (b) p () = p () when i in e region inide Fig. (a) marked by e red dahed box. The lope of e line in Fig. (b) are proportional to κ in Eq. (9). The different traight line in Fig. (b) correpond to different value of. Recall at i e atom-cavity detuning when e two mirror are fixed. Each parameter region for can have at mot five teady-tate olution ree table olution. Thi ytem ha eight parameter. However, during mot of i tudy, we will be varying e atom-cavity detuning (for fixed mirror) e parameter κ a hown in Fig. (b). Phyically, it i important to tudy e olution near e atom-cavity detuning, e untable olution hown by e yellow dot in Fig. (a) can be neglected. In e following dicuion on e teady-tate olution, we only conider e croing point between two curve correponding to Y L ( p, ) Y R ( p, ), in e red quare in Fig. (a). Inide i red quare, e number of real root of Eq. (8) can be characterized by ree critical value of e detuning : (cd), (bc), (ab), when κ L <κ<. The pace of root are divided into four region preented by e four letter a, b, c, d at e bottom of Fig. (b): (a) when e atom-cavity detuning > (cd), ere i no real root; (b) when (bc) < < (cd), ere alway exit two root; (c) when (ab) < < (bc), ere are four real root; (d) when < (ab), ere are two real root. Alo at e rehold point for = (cd), (bc), (ab), e number of teady-tate root i one, ree, ree, repectively. However, in e cae when κ<κ L, e rehold value (ab) doe not exit, ere are only two rehold value, (cd) (bc), which divide e -parameter pace into ree region for e root of Eq. (8). In i cae, e number of root will be explained below for given et of parameter. We now imulate e four teady-tate olution (i) olution label) p, (particular probe, teady-tate, (i = 1,,, 4) a ( ab) b ( bc) ω c ( cd ) c d p () FIG.. (Color online) Schematic diagram for e function of e left Y L ( p, ) e right Y R ( p, ) h ide of Eq. (8)veru p, in (a) for a large value of (b) for in e region of (a) indicated by e red dahed box. In (b), e brown triangle, red croe, green dot blue quare denote four different teady tate olution of e elf-conitent equation (8). The brown triangle (far, e four blue quare (top left) denote e olution () p,, e red dot on e curve denote e olution () p,, e green dot denote e olution (4) p,. The olution (1) p, i not hown in e region c d. Here, (i) (i = ab,bc,cd) denote e rehold value of at eparate region wi different number of olution. Namely, (ab) i e boundary point of e region a b, (bc) i e boundary point of e region b c, (cd) i e boundary point of e region c d. right) denote e olution (1) p, for given parameter a in Ref. [5], e.g., c =, ω a = 1, γ = 1, γ = 1 4, g N = 1, =, a in = 1. Hereafter, all quantitie are meaured in unit of γ 1. Wi e above parameter, we can find at e lower bound κ L i about 4. We firt tudy e cae for κ<κ L = 4, e.g., κ = 1. In i cae, ere are two critical value (a) = 5 =.95. Figure how e teady-tate atom-cavity (b) detuning (i) p, (when e mirror move) veru (when e mirror i fixed). We find at when e difference between e atomic frequency ω a e cavity frequency ω i larger an 5, ere i no teady-tate near e detuning c. When.95 5, ere are two teady-tate olution, i.e., (1) p, () p, hown in Fig.. In i region, we find at (1) p, decreae linearly, but () p, increae wi increaing. When.95, a hown in Fig., ere are four teady-tate olution, i.e., (i) p, (i = 1,,, 4). Figure how at when e detuning pae rough.95, from e right to e left, two additional olution ( () p, (4) p, ) appear. We alo find at two olution ( () p, (4) p, ) gradually approach c = torealizee 8-5

6 CHANG, SHI, LIU, SUN, AND NORI PHYSICAL REVIEW A 8, 8 (11) p p p 1 p p 9 atom cavity detuning FIG.. (Color online) The teady-tate olution (i) p, (i = 1,,, 4) veru e atom-cavity detuning (when e two mirror are fixed) for given parameter, e.g., κ = 1, c =, ω a = 1, γ = 1, γ = 1 4, g = 1, =, a in = 1. Hereafter, all e quantitie are meaured in unit of γ 1. Note at here eight parameter determine e ytem. Recall at p, i e teady-tate atom-cavity detuning when one mirror i movable. The inet how e teady-tate olution (1) p,, which correpond to a very large diplacement of e mirror. A chematically hown in Fig. (b), e dotted blue, dahed red, continuou green line denote e olution () p,, () p,, (4) p,, repectively. For example, e green cro in Fig. (b) correpond to a ingle value of. Here, i wept, e green cro in Fig. (b) become a continuou curve. Same for one red dot one blue quare in Fig. (b). two-photon-reonance condition. Moreover, note at () p, increae almot linearly wi increaing. Wi e ame parameter a oe in Fig., wehavealo plotted in Fig. 4 e -dependent location x (i) of e mirror correponding to (i) p,. We find at ere i no olution for e teady-tate value of x (i) when 5. When.95 5, e mirror poition x (1), correponding to e olution (1) p,, exhibit a very large (compared wi x (), 9 1 x x 4 x x 9 atom cavity detuning FIG. 4. (Color online) Steady-tate poition x (i) (i = 1,,, 4) for e movable mirror veru e atom-cavity detuning, for e ame parameter lited in Fig.. Here, e continuou black, dotted blue, dahed red, continuou green line denote x (1), x (), x () x (4), when e atom-cavity detuning p, take e following teady-tate value (1) p,, () p,, () in Fig.. x 1, p,, (4) p,, repectively, a hown x (), x (4) ) diplacement of e mirror, x (1) increae a increae. The mirror diplacement x (), correponding to e olution () p,, i nearly zero. When.95, e four teady-tate olution for e diplacement x exit imultaneouly. Two of em, x () x (4), how at e pring i compreed, eir diplacement linearly increae when decreae. One of em, x (), nearly vanihe. From Eq. (19), we know at e ocillating mirror can affect e two-photon reonance by changing e effective frequency ω L of e cavity field. Becaue when e mirror i fixed, e two-photon reonant condition become However, i condition i modified to = ω a ω = c. (4) p, = ω a ω L = c, (5) when e mirror i ocillating. We note at e two-photon reonant condition in Eq. (5) i furer modified to (i) p, = ω a ω L = c, () (bc) when e ytem reache it teady tate. We aume at e two-photon reonant condition in Eq. (4) i atified when e mirror i fixed. Recall at c = infig. 4. Thi mean at e two-photon reonant condition i (i) p, = in i cae. We find at x i nearly zero a hown in Fig. 4 when > c =. In i cae, ere i no value of p, approaching zero, ahowninfig. ; o e two-photon reonance cannot happen. From Fig. 4, we find at e two-photon reonance (i) p, = c = might happen when.95. Becaue in i region, () p, (4) p, can approach zero a in Fig., which correpond to e teady-tate value of e mirror poition x () x (4), repectively, a in Fig. 4. We now turn to tudy e teady-tate value of p, x for e cae when 4 κ<, e.g., κ = 1 4. In i cae, a chematically hown in Fig. (b), ere are ree critical value (i) (i = ab, bc, cd): (cd) =.5 1, =. (ab) = The number of olution for p, ha e ame decription a oe in Fig. (b). Similarly to Fig. 4, weplotfig.5 for p, (i) eir correponding x (i), repectively. The two-photon reonance condition in Eq. () for c = might alo be atified in i cae. Becaue when., two teady-tate value ( () p, (4) p, )of p, are near zero, a hown in Fig. 5, which correpond to e teady-tate olution of e mirror poition x () x (4),ahown in Fig.. It i alo found at one, x (), of e teady-tate olution of x nearly vanihe, a hown in Fig., when > c =. Finally, we note at ere i only one teady tate olution p, = for Eq. (8) in e limit κ. Thi implie when e elatic coefficient MωM i very large for given cavity leng l, it i difficult for e photon preure to make e mirror to even have a tiny diplacement, e ocillating mirror doe not affect e optomechanical ytem []. 8-

7 MULTISTABILITY OF ELECTROMAGNETICALLY INDUCED... PHYSICAL REVIEW A 8, 8 (11) p p p 1 p p atom cavity detuning FIG. 5. (Color online) Steady-tate olution (i) p() (i = 1,,, 4) chematically hown in Fig. (b) veru wi κ = 1 4. The oer parameter here are e ame a in Fig.. The inet how e teady tate olution (1) p() at correpond to a very large diplacement of e mirror. A hown in Fig. (b), e dotted blue, dahed red, continuou green line denote e olution () p(), () p(), (4) p(), repectively. IV. ELECTROMAGNETICALLY INDUCED TRANSPARENCY IN THE OPTOMECHANICAL SYSTEM To explore how e mirror ocillation affect e EIT, let u now tudy e uceptibility of e atomic medium. Becaue we can write e probe cavity field a E(t) = ωl Vɛ ae iω Lt + H.c. = εe iω Lt + H.c., (7) en e linear repone of e atomic enemble to e weak probe field E(t) can be decribed by e uceptibility 1 x x 4 x χ = p ε ɛ. (8) x x atom cavity detuning FIG.. (Color online) Steady-poition x (i) (i = 1,,, 4) of e movable mirror veru e atom-cavity detuning,wie ame parameter a in Fig. 5. Here, e continuou black, dotted blue, dahed red, continuou green line denote x (1), x (), x (), x (4), when e atom-cavity detuning p() (for a moving mirror) take e following teady-tate value: (1) p,, () p(), () p,, (4) p,, repectively, a hown in Fig. 5. Here, e average polarization p of e atomic enemble i p = µ ba V. (9) Uing Eq. (4), (7), (8), we obtain e uceptibility χ χ = F γ ( p, c ) + wi +if γ + ( p, c ) +, (4) F = µ N ɛ V, (41) = γ 1 ( p, c ) + γ p,, (4) (where F i proportional to e denity N/V) = p, ( p, c ) + γ 1 γ. (4) It i well known at e real part, Re(χ), e imaginary part, Im(χ), of e uceptibility χ decribe e diperion aborption of light, repectively. A dicued in e lat ection, one of e olution correponding to x (1) (1) p, i untable, anoer olution, correponding to x (4) (4) p,, i imilar to e olution x () () p, in e region ( (b), (c) ). Therefore, we only conider two olution below. In Fig. 7, Re(χ i ) Im(χ i )(i = 1, ) veru are plotted for e two olution () p, () p, tudied in Fig.. Here, χ 1 χ denote e uceptibilitie correponding to () p, () p,, repectively. All parameter in Fig. 7 are e ame a oe in Fig.. It i found at when > c =, e curve for Re(χ 1 ) Im(χ 1 ) are imilar to oe in e uual EIT phenomenon []. Thi mean at e teady-tate value x () of e mirror electrical uceptibility..1 Imχ Imχ 1 ReΧ ReΧ atom cavity detuning FIG. 7. (Color online) The uceptibilitie χ 1 χ for e detuning between e atom e control field c = κ = 1. We plot only two uceptibilitie becaue ere are ree table olution in i parameter region but two of em are quite imilar. Here, e dotted olid curve correpond to e imaginary real part, repectively. The blue e green color correpond to χ 1 e dark red brown color correpond to χ, repectively. The EIT-like region i located between e two peak of Imχ 1 Imχ. 8-7

8 CHANG, SHI, LIU, SUN, AND NORI PHYSICAL REVIEW A 8, 8 (11) electrical uceptibility..1.1 ReΧ Imχ Imχ 1 ReΧ 1 atom cavity detuning FIG. 8. (Color online) The uceptibilitie χ 1 χ for c = κ = 1. Here, e dotted olid curve correpond to e imaginary real part, repectively. The blue e green color correpond to χ 1 e dark red brown color correpond to χ, repectively. The EIT-like region i located between e two peak of Imχ 1 Imχ. diplacement i near zero in i region of parameter, e ocillating mirror ha no effect on e EIT. However, when < c =, e mirror diplacement x () make e two-photon-reonance condition () p, = ω a ω L c = approximately atified. A a reult, in i region, each of e curve Re(χ 1 ) Im(χ 1 ) i almot cloe to zero, like an infinite tail. When.95 a hown in Fig., anoer electrical uceptibility..1.1 electrical uceptibility 4 Imχ Imχ 1 Reχ Reχ 1 atom-cavity detuning 1 4 Re χ Im χ Re χ 1 Im χ 1 () a (b) atom-cavity detuning FIG. 9. (Color online) (a) The uceptibilitie χ 1 χ for 7 c = κ = 1. (b) The magnified figure of (a) for mall. Here, e dotted olid curve correpond to e imaginary real part, repectively. The blue e green color correpond to χ 1 e dark red brown color correpond to χ, repectively. It i hown in ee two figure at e ocillating mirror influence on EIT can alo be oberved if e detector reolving power i high enough. olution () p, emerge, u we alo have a () p, -dependent uceptibility χ. A hown in Fig. 7, e curve correponding to e real part Re(χ ) e imaginary part Im(χ ) of e uceptibility χ are imilar to oe in e uual EIT [], ince e mirror diplacement x () i nearly zero. From Fig. 7, we find at e right part of e curve Im(χ ) almot merge wi e left part of e curve Imχ 1, o at a tranparent window i formed. To know how c affect e EIT, we can alo tudy e uceptibility χ for e detuning c = when oer parameter are aumed to be e ame a oe in Fig. 7. A hown in Fig. 8, e teady-tate olution of p, x are imilar to oe for c =. The curve correponding to ee olution are jut rightward hift for e curve in Fig. 4, but e hape of e curve are almot e ame. Similar to Fig. 7, we chooe two teady-tate olution plot Re(χ i ) Im(χ i )(i = 1, ) veru. Obviouly, e eential concluion remain unchanged a oe in Fig. 7, but all curve have a rightward hift. Baed on e analyi in Sec. III, (bc) (ab) approach zero from e left ide when κ i increaed. When κ>κ L, e.g., κ 4 in Fig. 5, e larger κ correpond to horter tail of e curve correponding to Re(χ 1 ) Im when.95. A hown in Fig. 9(a) 9(b), fore parameter in e current experiment [7,8], uch a κ 1 dicued a above, e tail phenomenon can be detected if e reolution of e detector i high enough. Actually, a mall elatic contant can be obtained in optomechanical ytem formed by a quantum-degenerate Fermi ga [7], where e mechanical motion of e fermionic particle-hole electrical uceptibility..1.1 ReΧ Imχ ReΧ 1 Imχ 1 atom cavity detuning FIG. 1. (Color online) The uceptibilitie for c = κ = Note at i value of e elatic contant κ i huge, correponding to an almot fixed mirror. In i cae, e Reχ 1 Imχ 1 occur for motly poitive value of e atom-cavity detuning. When e pring contant become ofter, a in Fig. 7, e Reχ 1 Imχ 1 have a repone at extend over a huge range of value of e atom-cavity detuning,evenfor c. When κ tend to infinity, e uceptibility become e ame a at in e uual EIT phenomenon. Thi conitency check i reauring for our calculation. Here, e dotted olid curve correpond to e imaginary real part, repectively. The blue e green color correpond to χ 1 e dark red brown color correpond to χ, repectively. The EIT-like region occur near zero detuning. 8-8

9 MULTISTABILITY OF ELECTROMAGNETICALLY INDUCED... PHYSICAL REVIEW A 8, 8 (11) ytem behave a a moving mirror. The frequency of e ocillating mirror i about.1 MHz. We hope at e above dicued phyical effect for e mall elatic contant can be firt oberved in uch ytem [7] or oer condened matter ytem. For e real optomechanical ytem, e frequency of e mirror can reach.5 MHz[8,9] wi e cavity leng (1 nm) [4]. In e limit κ, e tail (for ) diappear e right part of e curve Im(χ ) jut meet e left part of e curve Im(χ 1 ) to form a tranparent window. In i limit, all e phyical propertie return to at in e uual EIT phenomenon. Namely, we recover e tard EIT when κ. Thi aymptotic reult i hown in Fig. 1 wi a large κ (e.g., κ = ), but oer parameter are e ame a in Fig.. InFig.1, around e point > c =, e left part of e curve Re(χ 1 ) Im(χ 1 ) nearly merge wi e right part of e curve Re(χ ) Im(χ ), u tranparent window are formed a in e uual EIT []. V. CONCLUSION AND REMARKS We have tudied e effect of e end mirror ocillation on e EIT phenomenon for an atomic enemble confined in a ga cell placed in a microcavity. Thi tudy can help u to quantitatively conider e quantum interface between an optomechanical ytem an atomic ga diplaying EIT. The reult obtained could be ued to improve e meaurement preciion baed on e EIT effect, when e medium i placed inide a microcavity wi a one-end ocillating mirror e cavity act a e probe light. We have hown at e whole ytem exhibit multitability due to e mirror vibration, we have alo numerically obtained e rehold value of e parameter, which can help determine how many teady tate exit in e correponding parameter region. Thi multitability can be explicitly diplayed rough a modified EIT phenomenon. Conequently, we invetigate e effect of e multi-teady-tate olution on e EIT phenomenon find at in ome parameter region ere are two olution at approximately atify e two-photon reonance condition. Therefore, e propertie of bo e real imaginary part of e uceptibility are ignificantly altered. When e pring elatic contant increae, e mirror become le movable, our tudy how at in i cae all propertie of e ytem gradually revert to oe of e uual EIT phenomenon. ACKNOWLEDGMENTS C.P.S. acknowledge upport by e NSFC wi Grant No , No. 45, No. 174, NFRPC No. CB915 5CB7458. Y.X.L. acknowledge upport by e NSFC wi Grant No F.N. wa upported in part by e National Security Agency (NSA), e Laboratory for Phyical Science (LPS), e Army Reearch Office (ARO), e National Science Foundation (NSF) Grant No. EIA-18, e JSPS- RFBR [1] S. Mancini P. Tombei, Phy.Rev.A49, 455 (1994). [] D. Meier P. Meytre, Phy.Rev.A7, 417 (). [] D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombei, A. Guerreiro, V. Vedral, A. Zeilinger, M. Apelmeyer, Phy. Rev. Lett. 98, 45 (7). [4] W. Marhall, C. Simon, R. Penroe, D. Bouwmeeter, Phy. Rev. Lett. 91, 141 (). [5] S. Mancini, D. Vitali, P. Tombei, Phy.Rev.Lett.8, 88 (1998). [] C. Gene, D. Vitali, P. Tombei, S. Gigan, M. Apelmeyer, Phy. Rev. A 77, 84 (8). [7] M. Bhattacharya P. Meytre, Phy. Rev. Lett. 99, 71 (7). [8] M. Bhattacharya, H. Uy, P. Meytre, Phy.Rev.A77, 819 (8). [9] M. Bhattacharya, P.-L. Gicard, P. Meytre, Phy. Rev. A 77, (8). [1] M. Bhattacharya P. Meytre, Phy. Rev. A 78, 4181 (8). [11] J. Q. You F. Nori, Phy.Rev.B8, 459 (). [1] A. Wallraff, D. I. Schuter, A. Blai, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, R. J. Schoelkopf, Nature (London) 41, 1 (4). [1] J. Q. You F. Nori, Phy. Today 58(11), 4 (5). [14] F. Nori, Nat. Phy. 4, 589 (8). [15] Y. X. Liu, L. F. Wei, J. R. Johanon, J. S. Tai, F. Nori, Phy. Rev. B 7, (7). [1] S. Kleff, S. Kehrein, J. von Delft, Phy.Rev.B7, 1451 (4). [17] P. R. Berman, Cavity Quantum Electrodynamic (Academic Pre, New York, 1994). [18] C. Gene, D. Vitali, P. Tombei, Phy.Rev.A77, 57 (8). [19] H. Ian, Z. R. Gong, Y. X. Liu, C. P. Sun, F. Nori, Phy. Rev. A 78, 184 (8). [] S. E. Harri, Phy. Today 5(7), (1997). [1] M. Fleichhauer A. S. Manka, Phy. Rev. A 54, 794 (199). [] L. V. Hau, S. E. Harri, Z. Dutton, C. H. Behroozi, Nature (London) 97, 594 (1999). [] M. Fleichhauer M. D. Lukin, Phy. Rev. Lett. 84, 594 (). [4] M. O. Scully M. S. Zubairy, Quantum Optic (Cambridge Univerity Pre, Cambridge, 1997). [5] C. P. Sun, Y. Li, X. F. Liu, Phy. Rev. Lett. 91, 1479 (). [] M. D. Lukin, Rev. Mod. Phy. 75, 457 (). [7] O. Kocharovkaya, Y. Rotovtev, M. O. Scully, Phy. Rev. Lett. 8, 8 (1). [8] L. Deng, E. W. Hagley, M. Kozuma, M. G. Payne, Phy. Rev. A 5, 5185(R) (). [9] M. Kozuma, D. Akamatu, L. Deng, E. W. Hagley, M. G. Payne, Phy. Rev. A, 181(R) (). [] Y. Li C. P. Sun, Phy.Rev.A9, 518(R) (4). 8-9

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