General two-mode squeezed states

Condensed Z. Phys. B - Condensed Matter 71, 527-529 (1988) Ze,tso.r,. M a t t e r for PhysikB 9 Springer-Verlag 1988 General two-mode squeezed states R.F. Bishop ~ and A. Vourdas 2. Department of Mathematics, University of Manchester, Institute of Science and Technology, Manchester, United Kingdom 2 Fachbereich Physik, Universit~it Marburg, Federal Republic of Germany Received October 16, 1987 The states OIA1A2) are considered, where the operators 0 are associated with a unitary representation of the group Sp(4, R), and the two-mode Glauber coherent states IA~ A2 ) are joint eigenstates of the destruction operators a I and a 2 for the two independent oscillator modes. We show that they are ordinary coherent states with respect to new operators bl and b2, which are themselves general linear (Bogoliubov) transformations of the original operators al, az and their hermitian conjugates a~, a*2. We further show how they may be regarded as the most general two-mode squeezed states. Most previous work on two-mode squeezed states appears to be based on more restrictive definitions than our own, and thereby reduces to special cases which are unified within our treatment. We first consider the annihilation and creation operators a and a t associated with a single oscillator mode, and which obey the usual canonical commutation relation, [a,.a*] = 1. (1) The one-mode squeezed states associated with these operators are by now well-known (see e.g. [1-6]). In terms of the vacuum state 10), defined as usual by a l0) = 0, they are defined as, ]A;p02) -~ U2(pO2)]a ) = U2(pO2)UI(A)IO ) (2) AcC (3) and the squeezing operator; U2(p02), given by U2 (p 0)~)---exp ( - 88 p,O, 2 ~ , p>0, U2atUtz --- b t = 5*a+7*a t 7 -- e-iZcosh( 89 (5) 6 = e-i~z+~ which are linear (Bogoliubov) transformations of the original operators a and a*. The transformation is easily seen from (1) and (5) to be a canonical one, (6) Two-mode squeezed states were first considered as trivial products of two one-mode squeezed states, IA1A2; PtO1)q, P20222) - ioa t 2 + 88 w a 2) exp (i2a* a) 0 < 0 < 2re, U2aUt2 ~ b = 7a+Sa t, [b, b*] = 1. where the displacement operator, UI(A), given by Ut(A ) - e x p ( A a t - A*a), are unitary representations of respectively the Weyl group and the group SU(1, 1). These squeezed coherent states are just ordinary coherent states with respect to the operators b and b*, U(2t)(pl O121) UpI(PzO222)IA1A 2 ), (4) * Present address: Department of Electrical Engineeringand Electronics, University of Liverpool, Brownlow Hill, P.O. Box 147, Liverpool L69 3BX, United Kingdom (7) where IA1A2) = I A 1 ) I A z ) are two-mode Glauber coherent states associated with two independent modes described respectively in terms of operators a t and a 2 (and their hermitian conjugates), and where U~)(piO~2i), i = 1, 2 are the squeezing operators given in (4) in terms of annihilation and creation operators at 528 and a~. In fact, Milburn [7] proved that a certain particular definition of two-mode squeezing, leads precisely to the states defined in (7) as the most trivial generalisation of the one-mode squeezed states. Caves and Schumaker [8] later adopted a different definition of two-mode squeezing and studied the associated squeezed states. More recently we have ourselves shown [9] that these latter squeezed states are based on the so-called discrete series representation of the group SU(1, 1). Both this type of squeezing and also a separate type of squeezing based on the so-called Schwinger representation of the S U(2) group, have been used recently both in the study of interferometers [10] and in other applications in quantum optics I-1 1]. It is one of our purposes in the present paper to show that all of these different types of twomode squeezing which have been considered separately in the literature, may be unified within our treatment which is based on the larger group Sp(4, ~). In terms of the annihilation and creation operators for the two independent modes, which obey the usual canonical commutation relations, [a s, a[] = 1 = [a2, at2] [as, a2] = [a,, a'z] = Jail, a2] = Earl, a~] = 0, [1] that with the particular realisation of these operators considered here, this is simply the " 883 representation of SU(1, 1)." The operators K(+2), K~ > and K(o2) similarly close under commutation into the same subalgebra. The three operators L+, L_ and Lo - K(o1) + K(o2) also close into the same SU(1, 1) subalgebra, but we explained elsewhere [9] that in this case the particular realisation of (8) and (9) now leads to the "discrete series representation of SU(1, i)." Finally, 7.1(i) k - ( 2 ) the three operators J+, J _ and Jo - -~o - - ~ o also close under commutation, in this case into (the Schwinger representation of) the angular momentum subalgebra of SU(2). We now consider a unitary representation of the group Sp(4, ~) which is realised by the ten-parameter family of operators 0 - 0 ( o ~ , r49, p l 0 i 2 i , p 2 0 2 / ~ 2 ) , defined by, b - W2(o, 0)V2(r, 49)U(21)(psOl"~'l)U(22)(p202"~2) where, W2(a), 0) - e x p ( - 89 co, 0 e ~ V2(r, 49) =- exp(-- 89 K(2) s ~ - =- ~a2, K(o2) ~ ~a2a2 1 * + 88 a~ a*2 + 89176as a2); r, 49e R (12) U~O(pjO/y) _ e x p ( - zpje s -~o~aj,z s ~.,2 , K ~ ) =- ~,.2 at2); a2 + 89 (8) we consider each of the ten possible pairing operators, s.t2 s t s + 88 K ( + I) ~ ~t~ 1 , K(1) - ~ ~,a l2, K(o I) ~ )-asa (11) +~pje (9) pj, O j , 2 j E ~ , i0j aj)expO2ja jt aj), 2 " 9 j=1,2. ~ L _ =- asa2, J+ - at~a2, J_ = ala*2, L+ - a*a 1 2, which satisfy the Sp(4, ~) algebra. From (8) and (9) it is a simple business to verify the commutation relations, K~), [K~), K~)] = 2K(os) [Kto:), K ~ )] = + K ~ ), [K~ ), K(+2)] = 2K~o2' ld'(1) ) - Id ~ at~ 0 , xI (x( 1_+ ..}- [L_, L+ ] = 2(K(ou + K(o2)), [d+, J_ ] = ")iV(l) -~Vt x O - - J t~-"(2)~ xO ! [K~ ~,L_] = - 8 9 [K~), L + ] = 89 [K~ ) , d _ ] = - 8 9 " [K~), J + ] = 8 9 [ K ~ ) , L _ ] = -- 89 [K~), J + ] = -- 89 [K~ ) , L + ] = 89 [K~ ) , J _ ] = 89 [K~o~),L_+] = +_ 89 [K~o'),d+] = +_ 89 [ K ~ # , L + ] = +}L+_, [K~o~),J+_] = -7- 89 (lO) As we have already explained, the following special cases of (11) and (12) apply: (i) for either P2 = )-z = r = e ) = 0 or P x = 2 1 = r = o ) = 0 , we have the ~, representation of SU(1, 1); (it) for Pl = P2 = 21 - 22 = co = 0, we have the discrete series representations of SU(1, 1); and (iii) for Pl = P2 = 2s + 2z = r = O, we have the Schwinger representation of SU(2). We note next that it is not difficult to show that the mode of action of the unitary operators 0 on the basic operators a i and a~ (i = 1, 2) is as follows, OasO t = bx = ala~ + zta~ + # l a z + vsatz OatO t = b~ =C{a 1 + a * a ~ + v * a 2 + # ~ a t 2 (13) Oa2 O* -- b 2 = a 2al + 272a~ + #2 a2 + v2 atE 0atz0t=b~2 = 2729a s + O"2* a ts "-l- V 29a 2 + # 2* a 2t [L+, J+ ] = - 2K~ ), [L_, J+ ] = 2Kt_2) where the constants o-i, % #i and vi (i = 1, 2) are defined by, [L +, J _ ] = -- 2K(+2), [L_, J _ ] = 2 K 9 ' , a 1 _-- e-iZleosh( 89189 and that all other commutators are zero. We see immediately that the three operators K~ ), K(_~> and K~o~) close under commutation into a subalgebra of Sp(4, g~). We explained in detail elsewhere _ el(C- zl - 0, + 0)sinh( 89 s ) sinh( 89 sin tim) 271 = e-i(z' +~189 _ )cosh( 89189 e-i(~, +r +O)cosh( 89189189 ~ 529 ]21 ~ ei(~P--z, -Ol)sinh( 89189189 ) + e -i~1 +*)cosh( 89189189 V1 = e - i ( . h +4')cosh(Xpl)sinh(Xr)cos( 89 + ei(O-x,-~189189189 a 2 - ei~g'-z2-~189 (14) (i) r = c~ = 0, which corresponds to squeezing of the two modes independently; (ii) Pa = P2 = 2 1 - 2 2 = ~ = 0, which corresponds precisely to the squeezing considered by Caves and Schumaker [8]; and (iii) Pl = P2 = 21 +-~2 = r = 0, which corresponds to SU(2) squeezing. sinh( 89189 -ei~O- Z])cosh( 89189189 z 2 - e-i~z]+0)cosh( 89189189 ) -e-i(o+ z2+~189189189 #2 -= e - i Z 2 c ~ 1 8 9 1 7 6 1 8 9 1 7 6 1 8 9 co) +ei(O-z~-0~ - q,)sinh ( 892) sinh ( 89 v2 - e-i(z2 + 0~)sinh( 89189189 in conclusion, it is clear that we have completely generalised and unified the various different types of two-mode squeezing that have been studied in the literature. ( 89 ) + e i(q'- ~ -O)cosh( 89189 Equations (13) and (14) provide us with the most general homogeneous linear transformations among the four operators a 1, at, a2, at2 that preserve the commutation relations (8), [bl, b t l ] = 1 = [b 2, It is clear from (18) and (19) that the states IA1Az)sq are ordinary (Gtauber) coherent states with respect to the transformed operators bx and be. As special cases of our general two-mode squeezed states, we mention the following: bt]] RFB gratefully acknowledges support for this work in the form of a research grant from the Science and Engineering Research Council of Great Britain. AV gratefully acknowledges support from GSI (Darmstadt). (15) Ebb, b2] = [b~, b'z] = [bI, b2] = [bl, b'z] = 0. The operators 0 thus most completely generalise to two modes, the Bogoliubov transformation (5) for a single mode, which is in turn generated by the operator U2 9 From (13) and the fact that 0 is unitary, we may trivially prove that for an arbitrary function f(ax, a], a2, a'z) we have, ~?f(a 1, a~, a 2, atz)Ot=f (bl, btl, b2, b~).~ Of(a1, al, a], a~) = f (b 1, bl, b], b~)~. (16) If we now define the general two-mode squeezed states as IAiAa)sq - [gIA1A2) , (18) (19) A. Vourdas Department of Electrical Engineering and Electronics University of Liverpool Brownlow Hill P.O. Box 147 Liverpool L69 3BX United Kingdom We note also the relation, O exp(A l a ~ - A*~al )exp(A 2a*2 -A~a2)lO, O) = exp(A~ bl - A*bl)exp(A2b~2 A*b2)[0, 0)~q. Stoler, D.: Phys. Rev. Lett. 33, 1397 (1974) Yuen, H.P.: Phys. Rev. A 13, 2226 (1976) Caves, C.M.: Phys. Rev. D 26, 1817 (1982) Walls, D.F.: Nature 306, 141 (1983) Bishop, R.F., Vourdas, A.: J. Phys. A 19, 2525 (1986) Schumaker, B.L.: Phys. Rep. 135, 317 (1986) Milburn, G.J.: J. Phys. A 17, 737 (1984) Caves, C.M., Schumaker, B.L.: Phys. Rev. A 31, 3068, 3093 (1985) 9. Bishop, R.F., Vourdas, A.: J. Phys. A 20, 3727 (1987) 10. Yurke, B., McCall, S.L., Klauder, J.R.: Phys. Rev. A 33, 4033 (1986) 11. Wodkiewicz, K., Eberly, J.H.: J. Opt. Soc. Am. B2, 458 (1985) R.F. Bishop Department of Mathematics University of Manchester Institute of Science and Technology P.O. Box 88 Manchester M60 1QD United Kingdom bllA1A2)sq = AI[A1A2)sq [AIA2)sq : 1. 2. 3. 4. 5. 6. 7. 8. (17) then (16) implies that 0ax = blO and [9a2 = b20, and hence that the states IA ~A2 )~q are simultaneous eigenstates of the transformed operators bz and b2, b]IA1A2)~q= A2IA1A])~q. References