Squeezing Transformation of Three-Mode Entangled State

Transcription

1 Commun. Theor. Phys. Beijing, Chin pp. 9 9 c Interntionl Acdemic Publishers Vol. 44, No. 5, November 5, 005 Squeezing Trnsformtion of Three-Mode Entngled Stte QIAN Xio-Qing nd SONG Tong-Qing Deprtment of Physics, Ningbo University, Ningbo 5, Chin Received Jnury 4, 005; Revised June 5, 005 Abstrct We introduce the three-mode entngled stte nd set up n experiment to generte it. Then we discuss the three-mode squeezing opertor squeezed p, χ, χ µ / p/µ, χ /µ, χ /µ nd the opticl implement to relize such squeezed stte. We lso revel tht c-number symmetric shrink trnsform in the three-mode entngled stte, i.e. p, χ, χ µ / p/µ, χ, χ, mps onto kind of one-sided three-mode squeezing opertor exp{iλ/ i= Pi i= Qi λ/}. Using the technique of integrtion within n ordered product IWOP of opertors, we derive their normlly ordered forms nd construct the corresponding squeezed sttes. PACS numbers: 4.50.Dv Key words: three-mode entngled stte, c-number symmetric shrink trnsform, one-sided three-mode squeezing opertor Introduction In recent yers much ttention hs been pid to the squeezed sttes of light.,] The most importnt property of the squeezed sttes of light is tht its quntum fluctution in one qudrture is smller thn tht in coherent stte. Squeezed sttes of light my therefore prove to be useful in quntum detection. In ddition, the squeezed vcuum stte hs been extensively studied,4] becuse of its potentil pplictions in the fields of quntum mesurement nd quntum informtion. Recently, Fn et l. 5] hve introduced one-sided two-mode squeezing opertors. In this pper, we firstly introduce the threemode entngled stte nd set up n experiment to generte it. Then we discuss the squeezing trnsformtion p, χ, χ p/µ, χ /µ, χ /µ nd propose the opticl network for producing such stte. We lso revel tht there exists nother kind of three-mode squeezing opertor { S = exp i λ i= P i i= Q i λ }, which plys the role of symmetric shrink trnsform S p, χ, χ p/µ, χ, χ, µ nd the opertor { S = exp i λ Q Q P P + P ] + i λ } Q Q P P + P ] λ, which plys the role of symmetric shrink trnsform S p, χ, χ µ p, χ /µ, χ /µ. 4 Using the technique of integrtion within n ordered product IWOP, 9] we deduce their normlly ordered forms nd construct the corresponding three-mode vcuum squeezed sttes. Besides, we generte one-sided three-mode squeezed stte by bem splitters. Three-Mode Entngled Stte In Ref. 0] the explicit form of three-mode entngled stte ws constructed, p, χ, χ = exp p π /4 + χ + χ + χ i p χ χ χ ] 000, 5 i, i, i =,,, re the three-mode Bose nnihiltion nd cretion opertors for the i-th mode, 000 is the three-mode vcuum stte, nd p, χ, χ obeys the eigenvectors equtions X X p, χ, χ = χ p, χ, χ, X X p, χ, χ = χ p, χ, χ, 7 P + P + P p, χ, χ = p p, χ, χ 8 with X i = i + i /, Pi = i i / i i =,,. Therefore, p, χ, χ is the common eigenvector of the three opertors, which mutully commute with ech other. The Schmidt decomposition of p, χ, χ is The project supported by Open Foundtion of Lbortory of High-Intensity Optics, Shnghi Institute of Optics nd Fine Mechnics, the Chinese Acdemy of Sciences

2 9 QIAN Xio-Qing nd SONG Tong-Qing Vol. 44 p, χ, χ = π e ipχ+χ/ + dx x x χ x χ e ixp, 9 the three single-mode sttes ll belong to the set of coordinte eigenvectors x i = π /4 exp x + x i ] i 0 i. 0 p, χ, χ cn lso be Schmidt-decomposed s p, χ, χ = + p π u υ p + u p + υ e iuχ+νχ dudν, the three single-mode sttes ll belong to the set of momentum eigenvectors p i = π /4 exp p + i p i + ] i 0 i. Using the IWOP technique 7 0] nd the norml product form =: e :, we cn prove tht p, χ, χ stisfies the completeness reltion ] + dpdχ dχ p, χ, χ p, χ, χ =, 4 nd the orthogonl property When p = χ = χ = 0, eqution 5 becomes p, χ, χ p=χ=χ =0 = p, χ, χ p, χ, χ = δp p δχ χ δχ χ. 5 π /4 exp = π ] 000 dx x x x. According to Vn Loock nd Brunstein s method, ] we cn generte this stte from squeezed modes of the field emitted by opticl prmetric oscilltors bellow threshold nd ppropritely blnced bem splitters, ˆB ij θ : i i cos θ + j sin θ, 7 j i sin θ j cos θ, θ 0, π], ˆN,, = ˆB π ˆB cos. 8 4 Applying the bem splitter opertor ˆN,, to zeromomentum eigenstte p = 0 nd zero-position eigenstte x = 0 x = 0, we cn obtin the three-mode entngled stte in Eq.. Opticl Implement We now mke clssicl trnsformtion, p p/µ, χ χ /µ, χ χ /µ, in the ste p, χ, χ to build ket-br integrl, + S = dpdχ dχ µ / p/µ, χ /µ, χ /µ p, χ, χ = exp tnhλa ] exp ln sech λb ] exp tnhλa ], 9 e λ = µ, tnhλ = µ + µ, sech λ = µ + µ, 0 A = A = B = i= i i + i= i j, i j, i i +. i= These obey n SU, Lie lgebr, 4] A, A] = B, A, B] = A, B, A] = A. 4 Thus we cn further compct the three-mode squeezing opertor 9 s S = exp tnhλ A A ]. 5 Operting S in Eq. 9 on the three-mode vcuum stte yields new squeezed stte S 000 = sech / λ exp A tnhλ] 000, which is lso n entngled stte. We pply opticl implement to generte such n squeezed stte. It 5,] ws pointed out tht the bsic opertions of opticl devices bem splitters, mirrors, opticl fibre, nd phse shifter

3 No. 5 Squeezing Trnsformtion of Three-Mode Entngled Stte 9 bsed on quntum optics components re the unitry trnsformtion of set of incoming sttes into nother set. Such trnsformtion cn be relized by using opticl networks. The liner networks re relized by use of pssive opticl elements. It is well known tht two-mode entngled stte cn be produced by operting the symmetric 50 : 50 bem-splitter on pir of input modes: 7] one is zero-momentum eigenstte p = 0, nd the other is the zero-position eigenstte x = 0. They re usully considered s two light fields mximlly squeezed in P nd X direction, respectively. In this pper, we wnt to design n opticl network such tht the light bem entering the three input ports of this network will be chnged into triprtite entngled stte. The light bem is mde up of zero-position eigenstte x = 0 nd two modes of zero-momentum eigenstte p = 0 p = 0. We set unitry opertor R, which stisfies the following requirement: R x = 0 p = 0 p = 0 sech / λ exp i + i= i i j 000, 7 in Fock spce p = 0 i exp 0 i, 8 Letting x = 0 i exp i 0 i E = 0 0, R should engender the trnsformtion Letting R R = Rã E R = i= i + 4 i j = ã A. Rã R = ã G, Ri R = G ij j = i, nd combining Eq. with Eq., we cn derive A = GEG = A, nd / / / G = / / + / / + / /. 4 To obtin the opticl trnsfer evolution in Eqs. nd, we need to know the Hmiltonins chrcterizing the interction between the modes of light entering n pproprite network relized by the use of pssive opticl elements. In Refs. 8] nd 9], it ws pointed out tht by mpping the clssicl c-number trnsformtion in coherent stte bsis onto quntum-mechnicl opertors of Fock spce nd using IWOP technique, we cn find the Hmiltonin. Here we construct the ket-br integrl opertor in the coherent stte representtion. R = i= d z i π G ijz j z i = i= d z i π : exp zi + i j Let ln G = itk nd K = K, then we cn get the time-evolution opertor, nd the corresponding Hermitin Hmiltonin i G ijz j + z i i i i ] : = exp ã ln G ]. 5 Rt = exp itã K ], H = ã K. 7 4 Squeezing Trnsformtion of p, χ, χ From the Schmidt decomposition of p, χ, χ in the coordinte representtion, we hve Q + Q + Q p, χ, χ = + e ipχ+χ/ x χ χ x x χ x χ e ixp dx π which mens tht in the p, χ, χ representtion, Combining with Eq. 8 we hve = i p p, χ, χ, 8 Q + Q + Q i p. 9 p, χ, χ P + P + P Q + Q + Q = ip p p, χ, χ. 40

4 94 QIAN Xio-Qing nd SONG Tong-Qing Vol. 44 Eqution 40 cn be rewritten s p, χ, χ P p p + P + P Q + Q + Q = i e p p p = ep, χ, χ = i p p = ep, χ, χ. 4 It then follows from Eq. 4 nd e λ / y fy = fy λ tht p, χ, χ e iλ P +P +P Q +Q +Q / = e λ / p p = e p, χ, χ = e λ p, χ, χ, 4 which mens tht unitry opertor S = exp { iλ P + P + P Q + Q + Q / λ / } squeezes p, χ, χ s π / p, χ, χ S = µ p µ, χ, χ. 4 Now using the completeness reltion 4, we hve S = e λ / dpdχ dχ p, χ, χ p e λ, χ, χ = e λ / dpdχ dχ p e λ, χ, χ p, χ, χ. 44 Eqution 44 revels tht the quntum mechnicl imge of clssicl diltion p e λ p in the stte p, χ, χ is just the one-sided squeezing opertor. We now present the dynmic Hmiltonin for generting such squeezing evolution. Let the squeezing prmeter e λ = µ be time-dependent e λ t = µt, we seek for the interction Hmiltonin which cn generte the continuous squeezing trnsform p, χ, χ p e λ, χ, χ. For this purpose we differentite S with respect to t nd obtin i S µt = i λt S µt. 45 t t Compring Eq. 45 with the stndrd form of the Schrödinger eqution i S µt = HtS µt t in n interction picture, we know the Hmiltonin Ht = λt P + P + P Q + Q + Q + ] t i. 4 From the Schmidt decomposition of p, χ, χ in the momentum representtion, we cn lso hve { iλ p, χ, χ exp Q Q P P + P ]} = p, χ e λ, χ. 47 Similrly { iλ p, χ, χ exp Q Q P P + P ]} = p, χ, χ e λ. 48 Letting λ = λ = λ nd using Eqs. 47 nd 48, we hve π / S = e iλ {Q Q P P +P ]/+Q Q P P +P ]/} λ = e λ dpdχ dχ π / p, χ, χ p, e λ χ, e λ χ. 49 Using Eqs. 5 nd, s well s the IWOP technique, we cn perform the integrtion in Eq. 44 nd obtin its normlly ordered expnsion, S = e λ dpdχ dχ p, χ π /, χ p e λ, χ, χ = : e λ/ e λ tnhλ e λ + exp tnhλ sech λ ] : = exp tnhλ k ] exp ln sech λ ] ] k 0 exp tnhλ k, 50 These obey n SU, Lie lgebr, k = + +, 5 k = + +, 5 k 0 = k, k] = k 0, k, k 0 ] = k, k 0, k] = k. 54 Thus we cn further compct the three-mode squeezing opertor 50 s S = exp tnhλ k k ]. 55

5 No. 5 Squeezing Trnsformtion of Three-Mode Entngled Stte 95 Moreover, using the opertor identity we derive e A B e A = B + A, B] + A, A, B]]! + A, A, A, B]]] +, 5! S P S = e λ P, S XS = e λ X, 57 P = P + P + P, P i = i i, 58 i X = X + X + X, X i = i + i, 59 X, P ] = i. 0 Operting S in Eq. 50 on the three-mode vcuum stte yields new squeezed stte S 000 = sech / λ exp tnhλ + + ] 000. The quntum fluctution of the opertor qudrtures in the stte S 000 is X = 4µ, P = µ 4. Thus the minimum uncertinty reltion still remins X P = X P = 4, which is the sme s in the ordinry three-mode squeezed vcuum stte. Similrly, using the IWOP technique, we perform the integrtion in Eq. 49 nd obtin S = exp tnhλ k ] expln sech λ k 0] k = exptnhλ k ], 4 + +, 5 k = + +, k 0 = These lso obey n SU, Lie lgebr. simplify Eq. 4 s Thus we cn S = exp tnhλ k k ]. 8 Moreover, using Eq. 8 we derive S P S = P, S XS = X. 9 Operting S in Eq. 4 on the vcuum stte yields new squeezed stte, tnhλ S 000 = sech λ exp + + ] The quntum fluctution of the opertor qudrtures in the stte S 000 is X = 4, P = 4. 7 Thus the minimum uncertinty reltion still remins, X P = X P = Generting One-sided Three-mode Squeezed Stte by Bem Splitters Here we consider tht the following experiment to produce the one-sided squeezed stte S 000 : three single-mode squeezed stte nd the vcuum stte overlpping on bem splitter my produce the stte t the output. According to Refs. 0] ], we set the bem splitter opertor s θ ] N = B θb θ = exp θ ] exp. 7 It opertes on the input stte nd yields tnhλ exp + 4 tnhλ + tnhλ 000, 74 tnhλ N exp + 4 tnhλ + tnhλ { tnhλ exp 000 cos θ sin θ + cos θ + sin θ + cos θ sin θ + cos θ + sin θ ]} When θ/ = π/4 nd θ / = π/4, eqution 75 becomes tnhλ exp + + ] 000, 7 which is just the one-sided squeezed stte in Eq. 70 if we ignore the prmeter sech λ.

6 9 QIAN Xio-Qing nd SONG Tong-Qing Vol. 44 Summry In summry, we hve set up n experiment to generte the three-mode entngled stte. We constructed the opticl network to produce the idel squeezed stte p/µ, χ /µ, χ /µ. We lso reveled tht the c-number symmetric shrink trnsform in the three-mode entngled stte, i.e. p, χ, χ µ / p/µ, χ, χ or p, χ, χ µ p, χ /µ, χ /µ, mps onto kind of one-sided three-mode squeezing opertor in the Hilbert spce. From Eqs. 44 nd 49, we cn see the new reltions between three-mode squeezing nd the trnsforms of entngled sttes. The one-sided squeezing opertors exhibit squeezing effect in the directions p nd χ, χ respectively. The p, χ, χ representtion lso mkes the IWOP technique prcticl in deriving the normlly ordered forms of S, S nd in contrcting the corresponding squeezed sttes. References ] For review, see R. London nd P.L. Knight, J. Mod. Opt ] D.F. Wlls, Nture London ; D.F. Wlls nd G.J. Miburn, Quntum Optics, Springer, Berlin 994. ] A. Furusw, et l., Science ] H.A. Hus, Electromgnetic Noise nd Quntum Opticl Mesurements, Springer, Berlin ] H. Fn nd Y. Fn, J. Phys. A ] H. Fn, H.R. Zidi, nd J.R. Kluder, Phys. Rev. D ] H. Fn nd H.R. Zidi, Phys. Rev. A ] H. Fn nd H. Zou, Phys. Lett. A ] H. Fn nd M. Xio, Phys. Lett. A ] H. Fn, N. Jing, nd H. Lu, Mod. Phys. Lett. B ] P. vn Loock nd S.L. Brunstein, Phys. Rev. Lett ] B.G. Wyboune, Clssicl Groups for Physicists, Wiley, New York 994. ] G. Dttoli, J.C. Gllrdo, nd A. Torre, Riv, Nuovo Cimento 988 No.. 4] A.M. Perelomov, Generlized Coherent Stte nd Its Applictions, Springer-Verlg, New York 98. 5] I. Jex, S. Stenholm, nd A. Zeilinger, Opt. Commun ] S. Stenholm, Quntum Semiclssic. Opt ] For review, see e.g., Quntum Computtion nd Quntum Informtion Theory, eds. C. Mccivello, G.M. Plm, nd A. Zeilinger, World Scientific, Singpore 00. 8] Hong-Yi Fn nd Min Xio, Quntum Semiclssic. Opt ] Hong-Yi Fn, Commun. Theor. Phys. Beijing, Chin ] Z.Y. Ou, S.T. Perin, H.J. Kimble, nd K.C. Peng, Phys. Rev. Lett ] C. Silberhorn, et l., Phys. Rev. Lett ] M.S. Kim, V. Buzek, nd P.L. Knight, Phys. Rev. A