New Normally Ordered Four-Mode Squeezing Operator for Standard Squeezing of Four-Mode

Quadratures

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2007 Commun. Theor. Phys. 47 135

(http://iopscience.iop.org/0253-6102/47/1/026)

Download details:

IP Address: 165.190.89.176

The article was downloaded on 17/04/2013 at 14:51

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 135–138c© International Academic Publishers Vol. 47, No. 1, January 15, 2007

New Normally Ordered Four-Mode Squeezing Operator for Standard Squeezing of

Four-Mode Quadratures∗

FAN Hong-Yi1,2 and CAO He-Lin1

1Department of Material Science and Engineering, University of Science and Technology of China, Hefei 230026, China

2Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China

(Received October 2, 2005; Revised June 2, 2006)

Abstract By virtue of the technique of integration within an ordered product of operators a new four-modesqueezing operator that squeezes the four-mode quadrature operators of light field in the standard way is found.This operator differs from the direct product of two two-mode squeezing operators. It is the exponential operatorV ≡ exp [ir(Q1P2 + Q2P3 + Q3P4 + Q4P1)]. The Wigner function of the new four-mode squeezed state is calculated,which quite differs from that of the direct-product state of two usual two-mode squeezed states.

PACS numbers: 42.50.DvKey words: four-mode quadrature operator, squeezing operator, the technique of IWOP, Wigner function

1 IntroductionQuantum entanglement is a weird, remarkable feature

of quantum mechanics. In recent years, various entan-gled states have brought considerable attention and in-terests of physicists because of their potential uses inquantum communication.[1] Among them the two-modesqueezed state exhibits quantum entanglement betweenthe idle-mode and the signal-mode in a frequency do-main manifestly, and is a typical entangled state of con-tinuous variable. Theoretically, the two-mode squeezedstate is constructed by the two-mode squeezing operatorS = exp

[λ(a1a2 − a†1a

†2

)]acting on the vacuum state,

S|00〉 = sechλ exp[−a†1a

†2 tanh(λ)

]|00〉 , (1)

where λ is a squeezing parameter.[2] The two-mode squeez-ing operator can also be recast into the form

S = exp[iλ(Q1P2 + Q2P1)] , (2)

after using the relation Qi = (1/√

2)(ai + a†i ), Pi =(1/i

√2)(ai − a†i ). Remarkably, S squeezes the entangled

state |η〉,

|η〉 = exp(−1

2|η|2 + ηa†1 − η∗a†2 + a†1a

†2

)|00〉 ,

η = η1 + iη2 (3)in the most natural way,[3−6] i.e.

S|η〉 =1µ|η/µ〉 , µ = eλ . (4)

The |η〉 state was constructed in Ref. [3] following theidea of Einstein, Podolsky, and Rosen in their argumentthat quantum mechanics was incomplete.[5] |η〉 obeys theeigenvector equation,

(Q1 −Q2)|η〉 =√

2η1|η〉 ,(P1 + P2) = |η〉 =

√2η2|η〉 , (5)

and the orthonormal-complete relation∫d2η

π|η〉 〈η| = 1 , 〈η′|η〉 = πδ (η − η′) (η∗ − η′∗) . (6)

Thus the two-mode squeezing operator has a neat andnatural representation on the 〈η| basis,

S =1µ

∫d2η

π|η/µ〉 〈η| , µ = eλ . (7)

By introducing the two-mode quadrature operators oflight field as in Ref. [2],

x1 =Q1 + Q2

2, x2 =

P1 + P2

2, (8)

the variances of x1 and x2 in the state S |00〉 are in thestandard form,〈00|S†x2

2S |00〉 = e−2λ , 〈00|S†x21S |00〉 = e2λ , (9)

thus we get the standard squeezing for the two quadra-tures: x1 → e2λx1, x2 → e−2λx2. Then a questionnaturally rises: what is the four-mode squeezing operatorwhich can engender the standard squeezing for the four-mode quadratures? In this work we shall find it. As onecan see shortly later, the new four-mode squeezing opera-tor differs from the direct product of two two-mode squeez-ing operators, even though both of them may produce thesame standard squeezing for four-mode quadratures. Inthe following we try to study the following operator,

V = exp [ir (Q1P2 + Q2P3 + Q3P4 + Q4P1)]

= exp[ir

∑i,j=1

4QiAijPj

], (10)

where

A =

0 1 0 00 0 1 00 0 0 11 0 0 0

, (11)

and its corresponding squeezed state. We arrange thiswork in this way. Firstly we use the technique of inte-gration within an ordered product (IWOP) of operatorsto derive the normally ordered expansion of V ; then weexamine the variances of the four-mode quadrature oper-ators in the state V |0000〉, and find that it just exhibits

∗The project supported by the President Foundation of the Chinese Academy of Sciences and National Natural Science Foundation of

China under Grant No. 10475657

136 FAN Hong-Yi and CAO He-Lin Vol. 47

squeezing in the standard way. The Wigner function ofnew four-mode squeezed state is calculated which signifi-cantly differs from that of the direct-product of two usualtwo-mode squeezed states.

2 New Four-Mode Squeezing Operator andIts Corresponding StateClearly, Q1P2, Q2P3, Q3P4, and Q4P1 terms do not

make up a closed Lie algebra, so it is hard to use Lie alge-bra to analyze V . Thus we turn to appealing to the IWOPtechnique to disentangle V . Using the Baker–Hausdorffformula, we have

V −1QkV = Qk − rQiAik +12!

ir2 [QiAijPj , QlAlk] + · · ·

= Qi( e−rA)ik = ( e−rA)kiQi ,

V −1PkV = Pk + rAkiPi +12!

ir2 [AkiPj , QlAlmPm] + · · ·

= ( erA)kiPi , (12)

where the repeated indices represent the Einstein sum-mation notation. From Eq. (12) we see that when Vacts on the four-mode coordinate eigenstate |~q 〉, where~q = (q1, q2, q3, q4), it squeezes |~q 〉 in the way of

V |~q 〉 = |Λ|1/2 |Λ~q 〉 , Λ = e−rA , |Λ| ≡ detΛ . (13)Thus V has the representation on the 〈~q | basis,

V =∫

d4qV |~q 〉 〈~q | = |Λ|1/2∫

d4q |Λ~q 〉 〈~q | ,

V † = V −1 . (14)Using the expression of eigenstate |~q 〉 in Fock space,

|~q 〉 = π−1 exp[−12~q~q +

√2~qa† − 1

2a†a†]|~0〉 ,

|~0〉 ≡ |0000〉 , a† = (a†1, a†2, a

†3, a

†4) , (15)

and |~0 〉〈~0 | = : exp[−a†a†] :, we can put V into normalordering form by virtue of the technique of integrationwithin an ordered product (IWOP) of operators,

V = π−2|Λ|1/2

∫d4q : exp

[−1

2~q (1 + ΛΛ)~q +

√2~q (Λa† + a)− 1

2(aa + a†a†)− a†a

]: . (16)

To compute the integration in Eq. (16), we use the mathematical formula∫dnx exp

(−xFx + xv

)= πn/2(detF )1/2 exp

(14vF−1v

), (17)

to obtain

V = (det Λ)1/2(detN)−1/2 exp[12a†

(ΛN−1Λ− I

)a†

]: exp

[a†(ΛN−1 − I)a

]: exp

[12a(N−1 − I)a

], (18)

where N = (1 + ΛΛ)/2. Noticing A4 = I, I is the 4×4 unit matrix, from the Cayley–Hamilton theorem we know thatthe expanding form of exp(−rA) must be

Λ = exp(−rA) = a(r)I + b(r)A + c(r)A2 + d(r)A3 . (19)

To determine a(r), b(r), c(r), and d(r), we take A being 1, e i(1/2)π, e iπ, e i(3/2)π, respectively, then we haveexp(−r) = a(r) + b(r) + c(r) + d(r) ,

exp(−r e i(1/2)π) = a(r) + b(r) e i(1/2)π + c(r) e iπ + d(r) e i(3/2)π ,

exp(−r e iπ) = a(r) + b(r) e iπ + c(r) e i2π + d(r) e i3π ,

exp(−r e i(3/2)π) = a(r) + b(r) e i(3/2)π + c(r) e i(6/2)π + d(r) e i(9/2)π . (20)Its solution is

a(r) =12(cosh r + cos r) , b(r) =

12(− sinh r − sin r) , c(r) =

12(cosh r − cos r) , d(r) =

12(− sinh r + sin r) . (21)

It follows thatΛΛ =

[a(r)I + b(r)A3 + c(r)A2 + d(r)A

] [a(r)I + b(r)A + c(r)A2 + d(r)A3

]=

12

cosh(2r) + 1 − sinh(2r) cosh(2r)− 1 − sinh(2r)− sinh(2r) cosh(2r) + 1 − sinh(2r) cosh(2r)− 1

cosh(2r)− 1 − sinh(2r) cosh(2r) + 1 − sinh(2r)− sinh(2r) cosh(2r)− 1 − sinh(2r) cosh(2r) + 1

. (22)

Then

N =12(ΛΛ + I

)=

14

cosh(2r) + 3 − sinh(2r) cosh(2r)− 1 − sinh(2r)− sinh(2r) cosh(2r) + 3 − sinh(2r) cosh(2r)− 1

cosh(2r)− 1 − sinh(2r) cosh(2r) + 3 − sinh(2r)− sinh(2r) cosh(2r)− 1 − sinh(2r) cosh(2r) + 3

, (23)

which is a circulant matrix (see Appendix) and has a mathematical notation,

N = Cire

[14(cosh(2r) + 3)− 1

4sinh(2r),

14(cosh(2r)− 1),−1

4sinh(2r)

]. (24)

No. 1 New Normally Ordered Four-Mode Squeezing Operator for Standard Squeezing of Four-Mode Quadratures 137

Its inverse can be calculated the way Eq. (A1) demonstrates,

N−1 =

1 1

2 tanh r 0 12 tanh r

12 tanh r 1 1

2 tanh r 00 1

2 tanh r 1 12 tanh r

12 tanh r 0 1

2 tanh r 1

. (25)

It then follows that

ΛN−1 − I =12

sech r + cos r − 2 − sin r sech r − cos r sin r

sin r sech r + cos r − 2 − sin r sech r − cos r

sech r − cos r sin r sech r + cos r − 2 − sin r

− sin r sech r − cos r sin r sech r + cos r − 2

, (26)

and

ΛN−1Λ− I = −12

tanh r

0 1 0 11 0 1 00 1 0 11 0 1 0

= −(N−1 − I) . (27)

The relation between ΛN−1Λ− I and N−1− I in Eq. (27), which are separately involved in the first and the last termof Eq. (18), implies some algebra symmetry. Substituting Eqs. (26) and (27) into Eq. (18) yields

V = sech r exp[−1

2tanh(r)

(a†1a

†2 + a†2a

†3 + a†3a

†4 + a†4a

†1

)]× : exp

[12(sec r + cos r − 2)

4∑i=1

a†iai −12

sin r(a†1a2 + a†2a3 + a†3a4 + a†4a1

)+

12(sech r − cos r)

(a†1a3 + a†3a1 + a†4a2 + a†2a4

)+

12

sin r(a†1a4 + a†4a3 + a†3a2 + a†2a1

)]:

× exp[12

tanh r(a1a2 + a2a3 + a3a4 + a4a1)], (28)

which is not the direct product of two two-mode squeezing operators. Operating V in Eq. (28) on the four-modevacuum state leads to the new four-mode squeezed vacuum state,

V |~0 〉 = sech r exp[−1

2tanh r

(a†1a

†2 + a†2a

†3 + a†3a

†4 + a†4a

†1

)]|~0 〉 . (29)

Now we see the new four-mode squeezed state in Eq. (29) is not the same as the direct product of the direct productof two two-mode squeezed state S12 |00〉12 ⊗ S34 |00〉34.

3 Standard Squeezing for Four-Mode FieldThen, let us check the variances of the four-mode quadratures. The quadratures in the four-mode case should be

defined as

X1 =Q1 + Q2 + Q3 + Q4

2√

2, X2 =

P1 + P2 + P3 + P4

2√

2, (30)

whose commutation relation is [X1, X2] = i/2. The expectation values of X1 and X2 in the state V |~0 〉 are 〈X1〉 =〈X2〉 = 0, and their variances are

(4X1)2 = 〈|X21 |〉 =

18〈~0 |V −1X2

1V |~0 〉 =116

∑i,j

(ΛΛ)ij =e−2r

4,

(4X2)2s = 〈|X22 |〉 =

18〈~0 |V −1X2

2V |~0 〉 =116

∑i,j

(ΛΛ)−1ij =

e2r

4, (31)

and

4X1 · 4X2 =14

, (32)

which shows the standard squeezing similar to Eq. (9). Next we should point out that though the direct-product of twousual two-mode squeezing operator can generate the same quadrature fluctuation for four-mode quadratures as shownin Eqs. (31) and (32), it significantly differs from the new state (29). To explain this more explicitly, we calculate theWigner function of new four-mode squeezed state by using the coherent state representation[7] of the Wigner operator[8]

∆i(αi) =∫

d2zi

π2|αi + zi〉 〈αi − zi| exp (αiz

∗i − ziα

∗i ) , αi =

1√2

(xi + ipi) , (33)

138 FAN Hong-Yi and CAO He-Lin Vol. 47

where |z〉 is the coherent state. The result is

W = 〈~0 |V −14∏

i=1

∆i(αi)V |~0〉

= sech 2 r〈~0 | exp[− tanh r

2(a2 + a4)(a1 + a3)

] 4∏i=1

∫ [ d2zi

π2

]|αi + zi〉〈αi − zi| eαiz

∗i −ziα

∗i

× exp[− tanh r

2(a†2 + a†4)(a

†1 + a†3)

]|~0〉

= sech 2 r4∏

i=1

∫ [ d2zi

π2

]eαiz

∗i −ziα

∗i −|αi|2−|zi|2 exp

[− tanh r

2(α1 + α3 + z1 + z3)(α2 + α4 + z2 + z4)

]× exp

[− tanh r

2(α∗

1 + α∗3 − z∗1 − z∗3)(α∗

2 + α∗4 − z∗2 − z∗4)

]= π−4 exp

{−2 cosh2 r

[ 4∑i=1

|αi|2 − (N + N∗) tanh r + (M + M∗) tanh2 r]}

, (34)

whereM = α1α

∗3 + α2α

∗4 ,

N = α1α2 + α2α3 + α3α4 + α4α1 , (35)and we have used the integration formula,∫

d2z

πexp

(ζ |z|2 + ξz + ηz∗

)= −1

ζexp

(−ξη

ζ

), (36)

whose convergent condition is Re(ζ) < 0. This form dif-fers from the Wigner function of the direct-product ofusual two two-mode squeezed states’ Wigner functionvery much, i.e., the Wigner function of usual two-modesqueezed state is

〈00|S−112

2∏i=1

∆i (αi)S12 |00〉

= π−2 exp(−µ2

∣∣ρ2∣∣− |γ2|

µ2

), (37)

where µ = eλ, ρ = α1−α∗2, γ = α1+α∗

2. Using the Wignerfunction one can calculate particle numbers of each modein the new squeezed state by using Weyl–Wigner corre-spondence theory.

4 SummaryWe have shown that the operator V = exp[ir(Q1P2

+Q2P3 +Q3P4 +Q4P1)] is a new four-mode squeezing op-erator responsible for the four-mode quadratures exhibit-ing the standard squeezing. The corresponding squeezedvacuum state in four-mode Fock space is derived by virtueof the IWOP technique. The new four-mode squeezedstate is quite different from the direct product of usual

two two-mode squeezed state, as their Wigner functionsare quite different.

AppendixA circulant matrix is an arbitrary n × n matrix that

has the form,

a0 a1 a2 · · · an−1

an−1 a0 a1 · · · an−2

an−2 an−1 a0 · · · an−3

······

···. . . ···

a1 a2 a3 · · · a0

. (A1)

We usually mark the above matrix as Cire(a0, a1, a2, · · ·,an−1). By introducing Tn×n =

(0 In−1×n−1

1 0

), and notic-

ing Tn = I (I is the unit n × n matrix), then we canrewrite Eq. (A1) as

Cire(a0, a1, a2, · · · , an−1)

= a0 + a1T + a2T2 + · · ·+ an−1T

n−1

= f(T ) . (A2)The inverse of the circulant matrix Cire(a0, a1, a2, · · ·,an−1) is C−1

ire (b0, b1, b2, · · · , bn−1), where

bj =1n

n−1∑k=0

λkn−j [f(λk)]−1

, j = 1, 2, . . . , n− 1 ,

b0 =1n

n−1∑k=0

λk0 [f(λk)]−1

, (A3)

and the λk = exp(2kπj/n), k = 0, 1, 2, . . . , n− 1.

References

[1] M.A. Horne, H.J. Bernstein, and A. Zeilinger, NASAConf. Publ. 3135 (1992) 33.

[2] R. Loudon and P.L. Knight, J. Mod. Opt. 34 (1987) 709.

[3] H.Y. Fan and J.R. Klauder, Phys. Rev. A 49 (1994) 704;H.Y. Fan and B.Z. Chen, ibid. 53 (1996) 2948; H.Y. Fan,Phys. Lett. A 286 (2001) 81.

[4] H.Y. Fan and Y. Fan, Phys. Rev. A 54 (1996) 958.

[5] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47(1935) 777.

[6] H.Y. Fan, Int. J. Mod. Phys. 18 (2004) 1387; H.Y. Fan,H.L. Lu, and Y. Fan, Ann. Phys. 321 (2006) 480.

[7] H.Y. Fan and H.R. Zaidi, Phys. Lett. A 124 (1987) 303.

[8] E. Wigner, Phys. Rev. 40 (1932) 749.