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SCIENCE CHINA Physics, Mechanics & Astronomy • Article • August 2012 Vol.55 No.8: 1345–1350 doi: 10.1007/s11433-012-4749-y Generalized Fresnel operators corresponding to optical Fresnel diffraction and the squeezed states LI HongQi1,2* & REN TingQi1 1 Department of Physics, Qufu Normal University, Qufu 273165, China; 2 Department of Physics, Heze University, Heze 274015, China Received June 16, 2011; accepted March 8, 2012; published online May 2, 2012 Corresponding to optical Fresnel diffraction, we show that the exponential quadratic operator exp{i[P2+Q2+(PQ+QP)]/2} is actually a generalized single-mode Fresnel operator (GFO) in compact form, where [Q,P]=iħ. We also demonstrate that exp{i[(Q1+Q2)2+(P1P2)2]+i[(Q1Q2)2+(P1+P2)2]+i(Q1P2+Q2P1)} is a two-mode GFO. Their disentangling formula and normal ordering form are derived with the use of technique of integration within an ordered product (IWOP) of operators and the coherent state representation. The squeezed states generated by these two GFOs are obtained. Fresnel diffraction, single-mode Fresnel operator, two-mode Fresnel operator, IWOP technique PACS number(s): 03.65.-w, 42.50.Dv, 42.25.Fx Citation: Li H Q, Ren T Q. Generalized Fresnel operators corresponding to optical Fresnel diffraction and the squeezed states. Sci China-Phys Mech Astron, 2012, 55: 13451350, doi: 10.1007/s11433-012-4749-y 1 Introduction Fresnel diffraction is a core of classical optics. Fan and Hu [1] following Dirac’s assertion [2]: “ for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”, suggested the appropriate Fresnel operator (FO) to correspond to the optical Fresnel transform (FT): g q2 K A B C D q2 q1 f q1 dq1 (1) i 2 2 exp Aq2 2 q2 q1 Dq1 2B 2π i B F r s exp r 2 1 1 r a exp a a ln exp a 2 2 2s s 2 s (3) where a a 1 the fundamental bosonic commutator s,r are related to A, D, B and C by K A B C D q2 q1 respectively. The so-called single-mode FO is [1] where the integration kernel is 1 A B and denotes parameters characterizing an optical C D process (or an optical instrument) obeying AD BC 1, f q1 and g q2 are input and output of optical signals, s (2) 1 1 A D i B C r A D i B C (4) 2 2 which results in s 2 r 2 1 due to AD BC 1 Fan and Hu [1] have demonstrated that the FT kernel K in eq. *Corresponding author (email: [email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2012 phys.scichina.com www.springerlink.com 1346 Li H Q, et al. Sci China-Phys Mech Astron (2) is just the transformation matrix element of FO in the coordinate representation, K A B C D q2 q1 q2 F q1 where qi (5) is the Fock representation of coordinate eigen- state 1 a 2 qi π 1 4 exp qi2 2 qi ai i 0 i 1 2 2 2 (6) FT has been widely implemented in Fourier optics: optical imaging, optical propagation, optical engineering and optical instrument design. Thus, investigating FT in the context of quantum optics is really worthwhile. An important question thus arises: what is the compact form of general FO other than the decomposed form in eq. (3)? In this work we want to show that the general exponential quadratic unitary operator (in compact form) i U exp P 2 Q 2 PQ QP 2 2 2 2 (8) is a two-mode GFO in compact form. We shall derive their disentangling formula and normal ordering form with the use of technique of integration within an ordered product (IWOP) of operators [3–5] and the coherent state representation [6,7]. The squeezed states generated by these two operators are thus obtained. The IWOP technique is useful in constructing quantum mechanical representations and unitary transformations, in deriving new operator identities and new integration formulas [8–10]. Recently, researchers have deduced the s-ordered operator expansion formula of density operator [11]. 2 The normal ordered expansion of U In order to identify U as a GFO, let us recall the FO’s coherent state representation [1] s 2 d z sz rz z π d 2 z s r z π r s (10) is the coherent state [6,7]. Eq. (9) indicates that the c-number transform z sz rz in coherent state basis maps into the operator F. In fact, using the normal ordering of the vacuum projector 0 0 exp a† a and the IWOP technique we can perform the integral in eq. (9) and obtain d2 z 2 2 : exp s z sza z a ra π r s 2 rs 2 z z a a : 2 2 F r s s exp r 2 1 1 r a exp a a ln exp a 2 . 2 s s 2s 2 (11) q z exp i pQ qP 0 z q i p 2 (12) p F r s A D i( B C ) 2 dqd p 2π z z z , (9) A B q) C D p q p F A B C , (13) and eq. (11) becomes F A B C 2 A D iB C A D i B C a 2 :exp A D B C 2 i 2 1 a a A D i B C F r s s 1 2 z z za† 0 z 2 and eq. (4) we can rewrite eq. (11) as: 2 i Q1 Q2 P1 P2 i Q1 P2 Q2 P1 , z exp Alternately, using the canonical coherent state form [6] state. Then we shall show that where (7) is actually a generalized single-mode Fresnel operator (GFO), where , and are real, and U 0 is a squeezed U 2 exp i Q1 Q2 P1 P2 August (2012) Vol. 55 No. 8 A D iB C a2 : 2 A D i B C (14) From eq. (11) we see that F engenders the transformation Q A B Q F 1 F P C D P a a a a P Q 2 2i (15) Li H Q, et al. Sci China-Phys Mech Astron Let us now examine U in eq. (7), using the Baker-Hausdorff formula e A Be A B A B 1 A A B 2! 1 A A A B . 3! the Fresnel transformation under the GFO, for example, from eqs. (21) and (22) we know that GFT of the vacuum state is (16) 2 i sinh a 2 0 , 4 cosh i 2 sinh exp i 1 1 U QU Q P 2 PQ QP Q 2 2! 3! Q P Q 1 1 i 1 P 2 Q 2 PQ QP P Q 2! 2 3! Q cosh 2 2 cosh i sinh U 0 Then we can derive 1347 August (2012) Vol. 55 No. 8 P Q sinh (17) which is just a special squeezed state. 3 1 P Q sinh (18) Disentangling of U Due to the decomposition A B 1 0 A 0 1 1 C D C A 1 0 A 0 and U 1 PU P cosh U A where (19) Q A B Q U 1 U P C D P sinh cosh A B C D sinh (20) sinh sinh cosh B D U U A 1 0 C A 1 0 U1 0 A1 0 B A 1 , (24) i C 2 i B 2 i Q exp PQ QP ln A exp P 2A 2 2A i sinh Q 2 2 cosh sinh exp i sinh exp PQ QP ln cosh 2 i sinh P 2 . 2 cosh sinh exp with its determinant being 1 and comparing eq. (20) with eq. (11) we can identify U as a generalized FO, so * r 1 1 r U exp * a 2 exp a a ln * exp * a 2 2 s s 2s 2 (21) where 1 A D i B C 2 cosh i sinh , 2 1 r A D i B C 2 i sinh . 2 (23) which means the disentangling U exp Eqs. (17) and (18) in a compact form B A , 1 so C 2 (25) We immediately know the Fresnel transformation kernel q2 U q1 1 2π i B i 2 2 (26) Aq2 2 q2 q1 Dq1 2B exp 4 The two-mode generalized FO Similar in spirit to the single-mode case eq. (9), we introduce the two-mode FO F2 r s through the following s 2-mode coherent state representation [12] (22) The advantage of eq. (21) lies in that we can directly know F2 r s s d 2 z1 d 2 z2 sz1 rz2 rz1 sz2 z1 z2 (27) π2 which indicates that F2 r s is a mapping of classical symplectic transform phase space, where z1 z2 sz1 rz2 rz1 sz2 in 1348 Li H Q, et al. z1 z2 exp Sci China-Phys Mech Astron August (2012) Vol. 55 No. 8 so 1 2 1 2 z1 z2 z1 a1† z2 a2† 00 2 2 F2 Q1 F21 is a usual two-mode coherent state. Concretely, the ket in eq. (27) is sz1 rz2 rz1 sz2 sz1 rz2 rz1 sz2 1 and s 2 still r satisfy the unimodularity A D Q2 B C P2 (28) 2 F2 Q2 F21 condition s r 1. 1 2 F2 P1 F Using the IWOP technique and the normal ordering of the vacuum projector 00 00 : exp a1 a1 a2 a2 : we perform the integral in eq. (28) and obtain 2 2 1 1 2 2 2 2 1 r 1 exp a1 a2 : exp 1 a1 a1 a a s s s exp 2 : F r a a 1 2 s exp r a a 1 2 s Thus F2 r s induces the transform F2 r s a1 F21 r s s a1 ra2 F2 r s a2 F21 r s (30) It follows F2 Q1 Q2 F21 D Q1 Q2 B P1 P2 F2 P1 P2 F21 B Q1 Q2 D P1 P2 F2 Q1 Q2 F21 A Q1 Q2 C P1 P2 F2 P1 P2 F 1 2 (31) A P1 P2 C Q1 Q2 a j a j 2 Pj a j a j 2i j 1 2 F Q1 Q2 2 2 2 P1 P Q1 Q P1 P F21 D C 0 0 B A 0 0 0 Q1 Q2 0 0 P1 P2 A C Q1 Q2 B D P1 P2 P Q F21 P A D C B B C A D A D B C BC D A A D B C Q1 B C D A P1 (35) A D C B Q2 B C A D P2 A D i( B C ) 2 A D C B A D B C q1 2 d qi d pi B C A D B C D A p1 A D B C A D C B q2 2π i 1 B C D A B C A D p2 q1 p1 F2 A, B, C . q2 p2 (36) Similar to eqs. (24) and (25), F2 A B C has its canonical F2 A B C F2 1 0 C A F2 A 0 0 F2 1 B A 0 (32) 2 2 i C exp Q1 Q2 P1 P2 A 4 exp i Q1 P2 Q2 P1 ln A or 2 1 2 2 operator Q j Pj representation where Qj Q1 F2 r s (29) s a2 ra1 1 A D P2 B C Q2 2 A D P1 B C Q1 , It then follows the canonical coherent state form 1 r exp a1 a2 exp a1 a1 a2 a2 1 ln s s (34) and can be re-written in a compact form as: z a z a a a a a : 1 1 1 A D P1 B C Q1 2 A D P2 B C Q2 F2 P2 F21 2 d z1 d z2 :exp s 2 z1 2 z2 2 π2 r sz1 z2 rs z1 z2 sz1 rz2 a1 rz1 sz2 a2 F2 r s s 1 A D Q2 B C P2 2 A D Q1 B C P1 2 2 1 A D Q1 B C P1 2 2 2 i B exp Q1 Q2 P1 P2 (37) 4A 0 (33) Now we try to derive the disentangling expansion of the exponential operator U 2 in eq. (8). Noting Li H Q, et al. Sci China-Phys Mech Astron U 2 Q1 Q2 U 21 Q1 Q2 cosh Q1 Q2 2 P1 P2 2 Q1 Q2 2 P1 P2 2 4 4 i PQ 1 2 Q1 P2 Q1 Q2 P1 P2 2 2 4 Q1 Q2 P1 P2 2 2 4 2 2 i C 2 2 Q Q P P 1 2 1 2 4 A U 2 exp exp i Q1 P2 Q2 P1 ln A i B 2 2 Q Q P P exp 1 2 1 2 4 A Q1 P2 Q2 P1 Q1 Q2 i Q1 Q2 P1 P2 Q1 Q2 4 i P1 P2 Q1 P2 Q2 P1 P1 P2 i P2 P1 Q1 Q2 P1 P Q1 Q2 P1 P2 sinh 4 (43) 4 sinh (44) sinh cosh (39) i sinh 2 2 Q Q P P U 2 exp 1 2 1 2 cosh sinh sinh exp i Q1 P2 Q2 P1 ln cosh i sinh 2 2 Q Q P P , exp 1 2 1 2 cosh sinh (45) 2 or normally ordered expansion r a1 a2 exp[ a1 a1 a2 a2 1 s * (40) 4 i Q1 Q2 Q1 P2 Q2 P1 P1 P2 i P1 P2 Q1 P2 Q2 P1 Q1 Q2 i Q1 Q2 , 1 r * a a 00 , * 1 2 s (46) 1 A D i B C 2 1 r A D i B C 2 s i Q1 P2 Q2 P1 i P1 P2 Q1 Q2 2 Q1 Q2 4 P1 P2 Q1 Q2 Q1 Q2 cosh (47) Then the two-mode squeezed state generated by U 2 is Q1 Q2 sinh 4 (41) and U 2 P1 P2 U 21 P1 P2 cosh where U 2 Q1 Q2 U 21 Q1 Q2 P1 P2 ln s * ]exp and using the Baker-Hausdorff formula we can derive U 2 00 exp 2 4 sinh cosh A B C D 4 sinh which can be proved by using the bipartite entangled state representation. To identify A B C we notice 2 (42) According to eq. (40) we have the disentangling of U 2 , 4 P1 P2 sinh 4 Then comparing eqs. (42) and (43) with eq. (34) we can determine we have P1 P2 2 16 i PQ 1 2 Q1 P2 2 4 where Q1 Q2 P1 P2 4 P1 P2 cosh (38) 2 4 Q1 Q2 P1 P2 U 2 P1 P2 U 1 2 2 i PQ 1 2 Q1 P2 2 1349 August (2012) Vol. 55 No. 8 4 Q1 Q2 P1 P sinh 4 2 2 i sinh U 2 00 exp a1 a2 00 . cosh 2 i sinh (48) In summary, for the first time we pointed out that the general exponential quadratic unitary operator i exp P 2 Q 2 PQ QP 2 1350 Li H Q, et al. Sci China-Phys Mech Astron August (2012) Vol. 55 No. 8 is actually a GFO. We also demonstrate that exp i Q1 Q2 P1 P2 2 2 4 i Q1 Q2 P1 P2 i Q1 P2 Q2 P1 2 2 is a two-mode GFO. Their disentangling formula and normally ordered expansion are obtained which are useful for knowing Fresnel transformation of quantum states performed by these two unitary operators. 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