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A new kind of tripartite entangled state representation in Fock space Gang Ren ∗ , Jian-ming Du Department of Physics, Huainan Normal University, Huainan 232001, China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 2 March 2012 Accepted 12 July 2012

We construct a tripartite entangled sate representation in Fock space, which is the eigenstate of the three compatible operators (Q1 − Q3 ), (Q1 + Q2 + Q3 ) and (P1 − 2P2 + P3 ). Using the technique of integration within an ordered product (IWOP) of operators, we show that |p, 1 , 2 are complete and orthogonal, and hereby qualiﬁed for making up a quantum mechanical representation. Its squeezing property is explored and an optical device for producing this three-mode entangled state are proposed. © 2012 Elsevier GmbH. All rights reserved.

PACS: 03.65.Ud 02.30.Cj Keywords: Tripartite entangled sate representation Three-mode squeezing operator IWOP

1. Introduction It is well known that quantum entanglement and entangled states have been paid much attention because of their potential uses in quantum optics and quantum communication [1–3] in recent years. In 1935, Albert Einstein, Boris Podolsky and Nathan Rosen published a paper which was entitled “Can Quantum Mechanical Description of Physical Reality Be Considered Complete?” [4]. Generally referred to as “EPR”, this paper quickly became a centerpiece in the debate over the interpretation of the quantum theory, a debate that continues today. They introduced the common eigen wave function for two particles’ relative position Q1 − Q2 (with the relative distance q0 ) and their total momentum P1 + P2 (with the eigenvalue p0 ), (Q1 , Q2 ) =

1 p0 exp i (Q1 + Q2 ) 2 2

∞

dp exp[ip(Q1 − Q2 − Q0 )].

(1)

−∞

Inspired by this idea, the simultaneous eigenstate | in Fock space of commutative operators (Q1 − Q2 , P1 + P2 ) can be explicitly constructed, we name it the EPR eigenstate, i.e.,

1 † † † † | = exp − ||2 + a1 − ∗ a2 + a1 a2 |00, 2

(2) †

where = 1 + i2 is a complex number, and |00 is the vacuum state in Fock space. ai and ai , (i = 1, 2) are Bose annihilation and creation operators, respectively, and obey the commutative relation 1 † Qi = √ (ai + ai ), 2

1 † Pi = √ (ai − ai ), 2i

† [ai , aj ]

= ıij , i, j = 1, 2, relating to (Qi , Pi ) by

(3)

From operator identities †

[ai , f (ai , ai )] =

∂f , ∂a†i

i = 1, 2

∗ Corresponding author. Tel.: +86 0554 6672520. E-mail address: [email protected] (G. Ren). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.08.024

(4)

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G. Ren, J.-m. Du / Optik 124 (2013) 2475–2481

and ai |0=0, we have

1 † † † † a1 | = a1 , exp − ||2 + a1 − ∗ a2 + a1 a2 2

†

|00 = ( + a2 )|00.

(5)

Thus, we ﬁnd | state obeys the eigenvalue equations †

(a1 − a2 )| = |.

(6)

Using similar methods, we can get †

(a2 − a1 )| = −∗ |.

(7)

With the help of (6), (7) and (3), we write √ √ (Q1 − Q2 )| = 21 |, (P1 + P2 )| = 22 |.

(8)

The entangled state | satisﬁes the completeness relation,

d2 || = 1,

d2 ≡ d1 d2 ,

(9)

and possesses the orthogonal property | = ı( − )ı(∗ − ∗ ),

(10)

so we name | for the entangled state with continuum variable. The Schmidt decomposition of | is [5]

| = e−i1 2

∞

dp|p +

√ √ 22 1 ⊗ | − p2 e−i 21 p ,

(11)

−∞

where |pi is the momentum eigenstate of Pi

†2 √ a † − + 2ipi ai + i 2 2

|pi = −1/4 exp

p2i

|0i .

(12)

√ √ 21 2 ei 22 q ,

(13)

The Schmidt decomposition of | can also be

| = e−i1 2

∞

dq|q1 ⊗ |q − −∞

where |qi is the coordinate eigenstate of Qi

|qi = −1/4 exp

−

q2i 2

+

√

†2

ai

†

2qi ai −

|0i .

2

(14)

For the case of two-mode, a transform of coordinates can be considered as

1 −1 1

1

Q1

=

Q2

Q1 − Q2 Q1 + Q2

The transform of momentums is

1 −1 1

1

P1

=

P2

P1 − P2 P1 + Q2

.

(15)

.

(16)

From Eqs. (15) and (16), we can have a canonical transform by a orthogonal transformation 1 R= √ 2

1 −1 1

1

.

(17)

From Eq. (17), one can get the following relations: [Qi , Pj ] = ıij ,

[Qi , Qj ] = [Pi , Pj ] = 0,

(18)

where Qi = Rij Qj , Pi = Rij Pj . Therefore we are easy to ﬁnd that the conjugate of |, which is the eigenstate of (Q1 + Q2 ) and (P1 − P2 ), i.e.,

1 † † † † | = exp − ||2 + a1 + ∗ a2 − a1 a2 |00, 2 which obeys another pair of eigenvector equations √ √ (Q1 + Q2 )| = 21 |, (P1 − P2 )| = 22 |.

(19)

(20)

G. Ren, J.-m. Du / Optik 124 (2013) 2475–2481

2477

A interesting question is naturally put forward: can we extend the method of ﬁnding conjugate of two-mode entangled state to tripartite case in a direct way, so that a new kind of tripartite entangled states (TES) of continuum variables can be constructed? The answer is afﬁrmative. We obtain a matrix as

⎛ 1 2 −√ √ 6 ⎜ 6 ⎜ 1 R3 = ⎜ 0 ⎝ − √2 1

1

1 √ 6 1 √ 2

⎞

⎟ ⎟ ⎟. ⎠

1

Enlighted by Eq. (18), we have

⎛

⎞

⎛

⎜ Q1 ⎜ ⎜ ⎟ ⎜ ⎝ Q2 ⎠ = ⎜ ⎜ ⎝ Q 3

and

⎛

⎞

⎛

⎜ ⎜ ⎜ ⎟ ⎜ ⎝ P2 ⎠ = ⎜ ⎜ ⎝ P P1

3

(21)

⎞

1 √ (Q1 − 2Q2 + Q3 ) ⎟ 6 ⎟ 1 ⎟ − √ (Q1 − Q3 ) ⎟ , ⎟ 2 ⎠ 1 √ (Q1 + Q2 + Q3 ) 3

(22)

⎞

1 √ (P1 − 2P2 + P3 ) ⎟ 6 ⎟ 1 ⎟ − √ (P1 − P3 ) ⎟ . ⎟ 2 ⎠ 1 √ (P1 + P2 + P3 ) 3

(23)

Now we can have the following commutation relations [(Q1 − Q2 ), (P1 + P2 + P3 )] = 0,

[(Q1 − Q3 ), (P1 + P2 + P3 )] = 0,

(24)

the eigenstate of this three operators has been studied in Ref. [5]. The purpose of this work is to construct eigenvector for another set of three-particle compatible observable {(Q1 − Q3 ), (Q1 + Q2 + Q3 ), (P1 − 2P2 + P3 )} . So far as we know, this state has not been reported in the literature before. As [(Q1 − Q3 ), (P1 − 2P2 + P3 )] = 0,

[(Q1 + Q2 + Q3 ), (P1 − 2P2 + P3 )] = 0,

(25)

we make effort to search for the common eigenvector of these three compatible observable operator notated as |p, 1 , 2 . By virtue of the technique of integration within an ordered product (IWOP) of operators [6], we prove that |p, 1 , 2 makes a complete representation and its corresponding squeezing property is also discussed. 2. Three-mode entangled state in Fock space After some attempts, we ﬁnd that the common eigenket of {(Q1 − Q3 ), (Q1 + Q2 + Q3 ), (P1 − 2P2 + P3 )} is given in three-mode Fock space as

|p, 1 , 2 = exp

3 √ √ √ † 2 2 2 1 2 1 2 1 2 † † † † † p − 1 − 2 + ip(a1 − 2a2 + a3 ) + 1 (a1 − a3 ) + 2 − ai 12 4 6 6 2 3

−

1 1 †2 †2 †2 † † † † † † (2a1 − a2 + 2a3 ) − (2a1 a2 − a1 a3 + 2a2 a3 ) |000, 6 3

i=1

(26)

where |000 is the three-mode vacuum state. Below we will prove that |p, 1 , 2 is just the TES we are looking for. Using Eq. (4) and operating ai , (i = 1, 2, 3, ) on |p, 1 , 2 , respectively, one can have √ 2 1 † † † (ip + 31 + 22 ) − (2a1 + 2a2 − a3 ) |p, 1 , 2 , a1 |p, 1 , 2 = (27) 6 3

√

and a3 |p, 1 , 2 =

1 † 2 † † (−ip + 2 ) + (a2 − 2a1 − 2a3 ) |p, 1 , 2 , 3 3

a2 |p, 1 , 2 =

1 3

√

(28)

2 † † † (ip − 31 + 22 ) − (2a3 − a1 + 2a2 ) |p, 1 , 2 . 2

(29)

The sum of Eqs. (27)–(29) leads to (Q1 + Q2 + Q3 )|p, 1 , 2 = 2 |p, 1 , 2 .

(30)

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G. Ren, J.-m. Du / Optik 124 (2013) 2475–2481

Further, {Eq . (27) − Eq . (29)} shows (Q1 − Q3 )|p, 1 , 2 = 1 |p, 1 , 2 ,

(31)

while {Eq . (27) − 2 × Eq . (28) + Eq . (29)} yields (P1 − 2P2 + P3 )|p, 1 , 2 = p|p, 1 , 2 .

(32)

We now examine if |p, 1 , 2 satisﬁes a completeness relation. By virtue of IWOP and the normal ordering form of the three-mode vacuum projector †

†

†

|000000| =: exp(−a1 a1 − a2 a2 − a3 a3 ) :,

(33)

we can easily perform the following integration:

1

dpd1 d2 |p, 1 , 2 p, 1 , 2 |

63/2 =

∞

−∞

1

√ √ 2 2 1 2 1 2 1 2 † † † † † dpd1 d2 : exp − p − 1 − 2 + ip(a1 − 2a2 + a3 − a1 + 2a2 − a3 ) + 1 (a1 − a3 + a1 + a3 ) 6 2 3 6 2 −∞

63/2

∞

3 √ 3 † 2 1 †2 †2 †2 + 2 ai + ai − (2a1 − a2 + 2a3 + 2a21 − a22 + 2a23 ) 3 6 i=1

i=1

1 † † † † † † † † † − (2a1 a2 − a1 a3 + 2a2 a3 + 2a1 a2 − a1 a3 + 2a2 a3 − a1 a1 − a2 a2 − a3 a3 )] := 1, 3

(34)

where in the last step we have used the mathematical integral formula

∞ 2

exp(−˛x + ˇx)dx = −∞

exp ˛

ˇ2 4˛

.

Using Eqs. (30)–(32), we can also calculate p , 1 , 2 |(Q1 − Q3 )|p, 1 , 2 = 1 p , 1 , 2 |p, 1 , 2 = 1 p , 1 , 2 |p, 1 , 2 ,

(35)

p , 1 , 2 |(Q1 + Q2 + Q3 )|p, 1 , 2 = 2 p , 1 , 2 |p, 1 , 2 = 2 p , 1 , 2 |p, 1 , 2 ,

(36)

p , 1 , 2 |(P1 − 2P2 + P3 )|p, 1 , 2 = p p , 1 , 2 |p, 1 , 2 = pp , 1 , 2 |p, 1 , 2 .

(37)

and

From Eqs. (36) and (37), one obtain the overlap of |p, 1 , 2 p , 1 , 2 |p, 1 , 2 = ı(p − p)ı(1 − 1 )ı(2 − 2 ),

(38)

which indicated that |p, 1 , 2 is orthogonal. From Eqs. (34) and (38), we see that |p, 1 , 2 is qualiﬁed for making up a representation. However, it must be clariﬁed that |p, 1 , 2 is different from the three-mode representation which is common eigenvector of {(Q1 − Q3 ), (Q1 − 2Q2 + Q3 ), (P1 + P2 + P3 )} [8]. For revealing the entanglement of |p, 1 , 2 , we make the one-fold Fourier transformation of |p, 1 , 2 over dp, and have 1 2

∞

dpe

−iqp

|p, 1 , 2 =

−∞

3 1 1 1 1 1 |q + 1 + 2 1 ⊗ | − 2q + 2 2 ⊗ |q − 1 + 2 3 , 2 3 3 2 3

where its inverse transformation is

|p, 1 , 2 =

3 −ip((1/2)1 +(1/3)2 ) e

(39)

∞

dq|q1 ⊗ | − 2q + 1 + 2 2 ⊗ |q − 1 3 eiqp ,

(40)

−∞

From Eq. (40), we show that for the state once particle 1 is measured in the state |p, 1 , 2 , particle 2 immediately collapses to the coordinate eigenstate | − 2q + 1 + 2 , while particle 3 immediately collapses to |q − 1 3 . So Eq. (40) is the Schmidt decomposition of |p, 1 , 2 , from which we see that |p, 1 , 2 is just TES. On the other hand, we can introduce the state vector |, p1 , p2 , which is the conjugate state of |p, 1 , 2 ,

1

|, p1 , p2 = √ exp 63/4

3 √ √ † 1 2 1 2 1 2 2 2 † † − ai − p1 − p2 + ip1 (a1 − a3 ) + ip2 12 4 6 2 3 i=1

√ 1 1 2 † † † †2 †2 †2 † † † † † † + (a1 − 2a2 + a3 ) + (2a1 − a2 + 2a3 ) + (2a1 a2 − a1 a3 + 2a2 a3 ) |000. 6 6 3

(41)

As the proof in the front process, one can ﬁnd |, p1 , p2 obeys the eigenvector equations (P1 − P3 )|, p1 , p2 = p1 |, p1 , p2 ,

(42)

G. Ren, J.-m. Du / Optik 124 (2013) 2475–2481

2479

(P1 + P2 + P3 )|, p1 , p2 = p2 |, p1 , p2 ,

(43)

(Q1 − 2Q2 + Q3 )|, p1 , p2 = |, p1 , p2 .

(44)

and

The term “conjugate” originates the canonical conjugate commutator of the operators in Eqs. (30)–(32) and those in Eqs. (42)–(44), i.e., [(Q1 − Q3 ), (P1 − P3 )] = 2i,

(45)

[(Q1 + Q2 + Q3 ), (P1 + P2 + P3 )] = 3i,

(46)

[(P1 − 2P2 + P3 ), (Q1 − 2Q2 + Q3 )] = 4i.

(47)

and

The Schmidt decomposition of |, p1 , p2 is |, p1 , p2 =

1 √ ei((1/2)p1 +(1/3)p2 ) 2

∞

|p1 ⊗ | − 2p + p1 + p2 2 ⊗ |p − p1 3 e−ip dp.

(48)

−∞

Using the overlap of the coordinate eigenstate |qi and the momentum eigenstate |pi i q|pi

= (2)−1/2 eipi qi ,

i = 1, 2, 3,

(49)

and with the help of Eqs. (40) and (48), we obtain the overlap of |p, 1 , 2 and |, p1 , p2 p, 1 , 2 |, p1 , p2 =

1 1 (2)−3/2 exp − i(p − 31 p1 − 22 p2 ) . 12 6

(50)

3. Three-mode squeezed operator in TES Inspired by the two-mode squeezing operator has a natural representation in the | basis [9,10], we construct the three-mode squeezed operator in the TES by building a ket–bra integral as S3 =

1

∞

dpd1 d2 |

3/2

−∞

p 1 2 , , p, 1 , 2 |,

(51)

where is compressed parameters. Using Eq. (26) and the IWOP technique, we perform the above integration and ﬁnally get

√ S3 = 2 2

2 + 1

(3/2)

: exp

1 2 − 1 †2 †2 †2 † † † † † † (2a1 − a2 + 2a3 + 4a1 a2 − 2a1 a3 + 4a2 a3 ) − − 6 2 + 1

2 1− 2 +1

†

†

†

(a1 a1 + a2 a2 + a3 a3 )

1 2 − 1 + (2a21 − a22 + 2a23 + 4a1 a2 − 2a1 a3 + 4a2 a3 ) : . 6 2 + 1

(52)

Let = e , tanh = ((2 − 1)/(2 + 1)), sech = ((2)/(2 + 1)), we have

1

S3 = (sech )3/2 : exp − +

6

†2

†2

†2

† †

† †

† †

†

†

†

tanh (2a1 − a2 + 2a3 + 4a1 a2 − 2a1 a3 + 4a2 a3 ) − (1 − sech )(a1 a1 + a2 a2 + a3 a3 )

1 tanh (2a21 − a22 + 2a23 + 4a1 a2 − 2a1 a3 + 4a2 a3 ) : . 6

(53)

Using the operator formula eka

†

a

=: exp[(ek − 1)a† a] :,

(54)

we can express Eq. (53) as S3 = exp(A† tanh ) exp(B ln sech ) exp(−A tanh ),

(55)

where 1 †2 †2 †2 † † † † † † A† ≡ − (2a1 − a2 + 2a3 + 4a1 a2 − 2a1 a3 + 4a2 a3 ), 6

†

†

†

B ≡ a1 a1 + a2 a2 + a3 a3 ,

(56)

composed a closed SU(1, 1) Lie algebra, [A† , B] = −2A† ,

[B, A] = −2A,

[A† , A] = −B.

(57)

From Eqs. (34) and (51), it is easy to obtain S3 |p, 1 , 2 = −3/2 |

p 1 2 , , .

(58)

1 1 [A, [A, B]] + [A, [A, [A, B]]] + · · ·, 2! 3!

(59)

Using the operator identity eA Be−A = B + [A, B] +

2480

G. Ren, J.-m. Du / Optik 124 (2013) 2475–2481

we see that S3 a1 S3−1 = a1 cosh +

1 † † † (2a1 + 2a2 − a3 ) sinh , 3

(60)

S3 a2 S3−1 = a2 cosh +

1 † † † (2a1 + 2a3 − a2 ) sinh , 3

(61)

S3 a3 S3−1 = a3 cosh +

1 † † † (2a2 + 2a3 − a1 ) sinh . 3

(62)

and

By introducing the two quadratures as 1 X0 = √ Xi , 6

1 P0 = √ Pi , 6

3

3

i=1

[X0 , P0 ] =

i , 2

(63)

i=1

and using the S3 transformation in Eqs. (60)–(62), we calculate the following equations, S3 X0 S3−1 = e X0 ,

S3 P0 S3−1 = e− P0 .

(64)

Operating Eq. (55) on the three-mode vacuum state |000, we have S3 |000 = (sech )3/2 exp(A+ tanh )|000.

(65)

Calculating the quantum ﬂuctuation of the two photon-ﬁeld quadratures in the state S3 |000 via Eq. (64), we obtain ( X0 )2 = 000|S3−1 X02 S3 |000 = 000|e2 X02 |000 = ( P0 )2 = 000|S3−1 P02 S3 |000 = 000|e−2 P02 |000 =

1 2 e , 4 1 −2 . e 4

(66)

(67)

and X0 = 000|S3−1 X0 S3 |000 = 0,

P0 = 000|S3−1 P0 S3 |000 = 0.

(68)

thus the minimum uncertainty relation is X0 P0 ≡

( X0 )2 ( P0 )2 =

1 . 4

(69)

From the above analysis, S3 is a general three-mode squeezing operator. The classical dilation transformation of variables in such the three-mode EPR state induces a general three-mode squeezing operator which involves an SU(1, 1) Lie algebra. 4. Optical device for producing TES We shall give a experiment scheme that such an ideal TES |p, 1 , 2 can be implemented by using an optical device. For a twomode entangled state can be produced by a symmetric beam-splitter. We operate a pair of input modes on the beam-splitter: one is the vacuum state in momentum representation |p = 01 and the other is the zero-coordinate state |q = 02 [7]. Similarly, we can design such an optical device, characterized by a unitary operator R. Let the three light beams which is composed of by one zero-momentum state †2 |p = 01 → exp((1/2)a1 )|01 and two zero-coordinate states |q = 02 ⊗ |q = 03 , to enter the three input ports of this device. Finally, we get exp(A+ )|000 as an output, i.e., R plays a part in R|p = 01 ⊗ |q = 02 ⊗ |q = 03 → exp(A+ )|000∼|p = 0, 1 = 0, 2 = 0.

(70)

This means that R should cause the transformation 1 2 † † †2 †2 †2 †2 †2 †2 † † † † † † R(a1 − a2 − a3 )R−1 = R a Ea† R−1 = − (2a1 − a2 + 2a3 ) − (2a1 a2 − a1 a3 + 2a2 a3 ) = a Ba† , 3 3 †

†

†

†

a = (a1 , a2 , a3 ), and where

⎡

1

E = ⎣0 0

(71)

0

0

−1

0

0

−1

⎤

⎡

−2

−2

1

−2

⎦ , B = 1 ⎣ −2 3

1

1

⎤

−2 ⎦ .

(72)

−2

To get the concrete Form of R, we set

, R a R−1 = aG †

†

Rai R−1 = Gij aj = ai ,

(73)

then one can have

EG = B. G

(74)

G. Ren, J.-m. Du / Optik 124 (2013) 2475–2481

2481

The solution of Eq. (74) is an orthogonal matrix

⎡

1 √ ⎢ 6 ⎢ ⎢ 1 G = ⎢−√ ⎢ 2 ⎣ 1 √ 3

2 −√ 6

1 √ 6 1 √ 2 1 √ 3

0 1 √ 3

⎤

⎥ ⎥ ⎥ ⎥. ⎥ ⎦

(75)

Using the matrix G in Eq. (75), we show that R is a rotation operator in Hilbert space, which can be constructed by the coherent representation R=

3 d2 zi i=1

|Gij zj zi | =

3 d2 zi i=1

⎡ ⎤ † † † † (−|zi |2 + a Gij zj ) + zi∗ ai − a ai ⎦ :=: exp[ a (G − I)a] := exp[ a (ln G)a], : exp ⎣ i

i

i

(76)

j

where the operator relation †

†

exp[ a a] =: exp[ a ( − I)a] :

(77)

is used. Let ln G = itK, with K† = K, the time-evolution operator of R is †

R(t) = exp(it a RKa),

(78)

the corresponding Hermitian Hamiltonian is †

H = − a Ka.

(79)

5. Conclusion In brief, we have provided the distinct form of new TES representation |p, 1 , 2 for describing a system with three particles. With the help of IWOP, the completeness relation and orthogonal property of TES has been proved. Its Schmidt decomposition shows that |p, 1 , 2 is an entangled state indeed. As the application of this new TES, we have also derived the squeezing operator in the |p, 1 , 2 state using the squeezed transformation of the variables. Finally a design scheme of generating TES is also designed. It is further shown the most squeezed states belong to some entangled states. References [1] [2] [3] [4] [5]

[6]

[7] [8] [9] [10]

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