Chin. Phys. B
Vol. 19, No. 9 (2010) 090307
Nonlinear two-mode squeezing obtained by analysing two-mode exponential quadrature operators in entangled state representation∗ Liu Tang-Kun(刘堂昆)a)† , Shan Chuan-Jia(单传家)a) , Liu Ji-Bing(刘继兵)a) , and Fan Hong-Yi(范洪义)b) a) College of Physics and Electronic Science, Hubei Normal University, Huangshi 435002, China b) Department of Material Science and Engineering, University of Science and Technology of China, Hefei 230026, China (Received 7 December 2009; revised manuscript received 3 March 2010) By analysing the properties of two-mode quadratures in an entangled state representation (ESR) we derive from ESR some complicated exponential quadrature operators for nonlinear two-mode squeezing, which directly leads to wave function of the nonlinear squeezed state in ESR.
Keywords: nonlinear two-mode squeezing, two-mode exponential quadrature operators, entangled state representation PACC: 0365, 0290
1. Introduction
parametric down conversion is an entangled state in
Because the quantum fluctuation of one quadrature of the light field in the squeezed state is less than in a coherent state, squeezed states is a very important topic in quantum optics[1−7] and has potential applications in quantum communication.[8−10] Besides, it has been recognised that the two-mode squeezed state (idle mode and signal mode) produced from a
a1 + a2 + a†1 + a†2 = (Q1 + Q2 ) /2, 23/2
the frequency domain.[11] The conception of entangled state, has received more and more attention as it can be employed in quantum teleportation, quantum cryptography, and quantum superdense coding.[12] For analysing two-mode light field, the two-mode quadratures that are characteristic of the quantum fluctuation of various states are
a1 + a2 − a†1 − a†2 = (P1 + P2 ) /2, 23/2 i
(1)
with [(Q1 + Q2 ) /2, (P1 + P2 ) /2] = i. It has been known that the exponential quadrature operator e iλ(P2 Q1 +Q2 P1 ) = e iλ[(P1 +P2 )(Q1 +Q2 )−(Q1 −Q2 )(P1 −P2 )]/2−λ is a standard two-mode squeezing operator, an interesting question 2 thus challenges us, i.e. how a complicated exponential quadrature operator, say, e iλ(P1 +P2 ) (Q1 +Q2 )/2 , affects squeezing. This problem seems tough. In this work we show that by analysing the properties of two-mode quadratures in an entangled state representation[13,14] we can derive some generalised two-mode squeezing operators in the entangled state representation (ESR), these complicated operators generate nonlinear two-mode squeezing. The merit of the ESR of these operators lies in that they can directly lead to the wave function of the nonlinear squeezed state in ESR.
2. Brief review of two bipartite entangled states In Refs. [13]–[15] it has been shown that operators P1 + P2 and Q1 − Q2 possess a common eigenvector, that is, [ ] 1 2 |η⟩ = exp − |η| + ηa†1 − η ∗ a†2 + a†1 a†2 |00⟩ , η = η1 + iη2 , (2) 2 ∗ Project
supported by the National Natural Science Foundation of China (Grant No. 10904033), and the Natural Science Foundation of Hubei Province, China (Grant No. 2009CDA145). † Corresponding author. E-mail:
[email protected] c 2010 Chinese Physical Society and IOP Publishing Ltd ⃝ http://www.iop.org/journals/cpb http://cpb.iphy.ac.cn
090307-1
Chin. Phys. B
Vol. 19, No. 9 (2010) 090307
with the eigenvector equations (Q1 − Q2 ) |η⟩ =
√
2η1 |η⟩ , (P1 + P2 ) |η⟩ =
where a†i[ (i = ] 1, 2) are the bosonic operators † obeying ai , aj = δij , and |00⟩ is the vacuum state in Fock space. It is Einstein–Podolsky–Rosen (EPR)[16] who first used the commutative relation [Q1 − Q2 , P1 + P2 ] = 0 to propose the concept of quantum entanglement and argued the incompleteness of quantum mechanics. With using the technique of integration within an ordered product (IWOP) of operators,[17−19] |η⟩ state spans a complete and orthonormal space ∫ 2 d η |η⟩ ⟨η| = 1, (4) π ⟨ξ|ξ ′ ⟩ = πδ(η − η ′ )δ(η ∗ − η ′∗ ). (5) Thus we name |η⟩ the EPR entangled state. The state conjugate to |η⟩ is |ξ⟩ , which is the common eigenvector of another pair of EPR commutative operators Q1 + Q2 and P1 − P2 , the explicit form of |ξ⟩ is ] [ 1 2 † † † ∗ † |ξ⟩ = exp − |ξ| + ξa1 + ξ a2 − a1 a2 |00⟩ , (6) 2 which satisfies the eigenvalue equations √ (Q1 + Q2 ) |ξ⟩ = 2ξ1 |ξ⟩ , √ (P1 − P2 ) |ξ⟩ = 2ξ2 |ξ⟩ , and the completeness relation ∫ 2 d ξ |ξ⟩ ⟨ξ| = 1, ξ = ξ1 + iξ2 . π ⟨ξ|ξ ′ ⟩ = πδ(ξ − ξ ′ )δ(ξ ∗ − ξ ′∗ ).
√
2η2 |η⟩ ,
(3)
3. Exponential quadrature operators for standard linear squeezing We begin with elucidating the properties of quadrature operators in |η⟩ state in more detail. From Eq. (2) we see ( ) ( ) a1 − a†2 |η⟩ = η |η⟩ , a2 − a†1 |η⟩ = −η ∗ |η⟩ , (13) and using ) ) 1 ( 1 ( Pi = √ ai − a†i , Qi = √ ai + a†i , 2i 2
(14)
we have the quadrature operators behaviour in ⟨η| representation √ ∂ ⟨η| , ⟨η| (P1 − P2 ) = −i 2 ∂η1 √ ∂ ⟨η| (Q1 + Q2 ) = i 2 ⟨η| . ∂η2
(15) (16)
One can check Eq. (15) by using Eq. (12), (7) = (8)
=
(9)
Then in Refs. [20] and [21] it has been )] [ (shown that the two-mode squeezing operator exp λ a†1 a†2 − a1 a2 is neatly expressed as [ ( )] ∫ d 2 η † † exp λ a1 a2 − a1 a2 = |η/µ⟩ ⟨η| , µ = e λ , µπ (10) or ∫ µd 2 ξ |µξ⟩ ⟨ξ| . (11) S2 = π The overlap between these two mutually conjugate states is 1 ⟨η| ξ⟩ = exp [(η ∗ ξ − ηξ ∗ ) /2] . (12) 2 Thus an interesting question that naturally arises is whether we can derive the standard squeezing operator directly from the properties of two-mode quadratures in the entangled state representation (|η⟩ or |ξ⟩). The answer is affirmative.
= =
⟨η| (P1 − P2 ) |ξ⟩ √ 2ξ2 ⟨η| ξ⟩ 1 √ ξ2 exp [i (η1 ξ2 − ξ1 η2 )] , 2 ⟨η| (Q1 + Q2 ) |ξ⟩ √ 2ξ1 ⟨η| ξ⟩ 1 √ ξ1 exp [i (η1 ξ2 − ξ1 η2 )] . 2
(17)
(18)
So ⟨η|
1 ∂ (P1 + P2 ) (Q1 + Q2 ) = iη2 ⟨η| . 2 ∂η2
(19)
Letting η2 ≡ e y yields 1 (P1 + P2 ) (Q1 + Q2 ) 2 ∂y ∂ = i ey ⟨η1 , η2 = e y | ∂η2 ∂y ⟨η|
= i
∂ ⟨η1 , η2 = e y | . ∂y
(20)
From e −λ ∂y f (y) = f (y − λ) and Eq. (20) we see
090307-2
∂
Chin. Phys. B
Vol. 19, No. 9 (2010) 090307
⟨ ⟨ ∂ ⟨η1 , η2 | e iλ(P1 +P2 )(Q1 +Q2 )/2 = e −λ ∂y ⟨η1 , η2 = e y | = η1 , e y−λ = η1 , e −λ η2 ,
(21)
which indicates that unitary operator e [iλ(P1 +P2 )(Q1 +Q2 )−λ]/2 plays a role of one-sided squeezing, it squeezes ⟨ ⟨η| e [iλ(P1 +P2 )(Q1 +Q2 )−λ]/2 = e −λ/2 η1 , e −λ η2 . Using complete-orthonormal relation (4), we have ∫ 2 ∫ 2 ⟨ d η d η iλ(P1 +P2 )(Q1 +Q2 )/2 iλ(P1 +P2 )(Q1 +Q2 )/2 e = |η⟩ ⟨η| e = |η⟩ η1 , e −λ η2 , π π
(22)
(23)
which is the natural representation of e iλ(P1 +P2 )(Q1 +Q2 )/2 in |η⟩. Similarly, from Eq. (15) we can prove ⟨ ⟨η| e −iλ(Q1 −Q2 )(P1 −P2 )/2 = e −λ η1 , η2 .
(24)
[(P1 + P2 ) (Q1 + Q2 ) , (Q1 − Q2 ) (P1 − P2 )] = 0,
(25)
Due to we see ⟨ ⟨η1 , η2 | e iλ[(P1 +P2 )(Q1 +Q2 )−(Q1 −Q2 )(P1 −P2 )]/2−λ = ⟨η1 , η2 | e iλ(P2 Q1 +Q2 P1 ) = e −λ e −λ η1 , e −λ η2 or
⟨ † † ⟨η1 , η2 | e iλ(a1 a2 −a1 a2 ) = e −λ e −λ η1 , e −λ η2 .
From completeness (4) we indeed have usual two-mode squeezing operator ∫ 2 ⟨ † † d η |η⟩ e −λ η1 , e −λ η2 = e λ(a1 a2 −a1 a2 ) = e iλ(P2 Q1 +Q2 P1 ) . e −λ π
(26)
(27)
(28)
We can make a similar discussion on the squeezing unitary operator in ⟨ξ| representation.
4. Entangled state representation of some complicated exponential quadrature operators for nonlinear squeezing Now we turn to the nonlinear squeezing, in Eq. (2) letting η2 ≡ 1/w, so ∂w/∂η2 = −1/η22 = −w2 , we consider the following product of quadrature operators in ⟨η| representation: ⟨ ⟨ √ 2 ∂w ∂ √ ∂ 1 1 1 2 ⟨η| (P1 + P2 ) (Q1 + Q2 ) = i 2η2 η1 , = −i 2 η1 , , (29) 2 ∂η2 ∂w w ∂w w ∂
then using e λ ∂y f (y) = f (y + λ) we have ⟨η1 , η2 | e iλ(P1 +P2 ) Noticing
2
(Q1 +Q2 )/2
√
= e
∂ 2λ ∂w
⟨ ⟨ ⟨ 1 1 η . √ = η1 , √2 η1 , = η1 , w w + 2λ 1 + 2λη2
[ ] 2 2 (P1 + P2 ) (Q1 + Q2 ) , (P1 + P2 ) (Q1 + Q2 ) = −2i (P1 + P2 ) (Q1 + Q2 ) ,
(30)
(31)
and using the operator identity e A+B = e A( e
λ
−1)/λ
eB ,
[A, B] = −λA,
(32)
we have 2
e iλ(P1 +P2 ) (Q1 +Q2 )/2−if (P1 +P2 )(Q1 +Q2 )/2 [ ] ef − 1 2 = exp iλ (P1 + P2 ) (Q1 + Q2 ) /2 e −if (P1 +P2 )(Q1 +Q2 )/2 . f 090307-3
(33)
Chin. Phys. B
Vol. 19, No. 9 (2010) 090307
From Eqs. (33) and (30) we can deduce 2
⟨η1 , η2 | e iλ(P1 +P2 ) (Q1 +Q2 )/2−if (P1 +P2 )(Q1 +Q2 )/2 [ ] ef − 1 2 = ⟨η1 , η2 | exp iλ (P1 + P2 ) (Q1 + Q2 ) /2 e −if (P1 +P2 )(Q1 +Q2 )/2 f ⟨ = ⟨η1 , η˜2 | e −if (P1 +P2 )(Q1 +Q2 )/2 = η1 , e f η˜2 , where η˜2 = 1+
√
η2 . ef − 1 2λ η2 f
(34)
(35)
Using completeness relation (4) yields the following entangled state representation of this exponential quadrature operator for nonlinear squeezing: ∫ 2 2 d η e iλ(P1 +P2 ) (Q1 +Q2 )/2−if (P1 +P2 )(Q1 +Q2 )/2 = |η⟩ ⟨η1 , η˜2 | . (36) π
Similarly, let τj ≡ 1/ηjn−1 , (j = 1, 2) , and ∂τj /∂ηj = −(n − 1)/ηjn , (where n ≥ 2), we see 1 n (P1 + P2 ) (Q1 + Q2 ) 2 √ ∂ ⟨η1 , η2 | = i 2n−1 ⟨η| η2n ∂η2 ⟨ √ ∂ 1 = −i (n − 1) 2n−1 η1 , n−1 √ , ∂τ2 τ2 ⟨η|
(37)
and 1 n (Q1 − Q2 ) (P1 − P2 ) 2 ⟨ √ ∂ 1 , = i (n − 1) 2n−1 , η √ 2 ∂τ1 n−1 τ1
5. Wave function of nonlinear squeezed state in ESR The advantage of Eqs. (42) and (36) lies in the fact that it directly leads us to derive the wave function of the nonlinear squeezed state in ESR, i.e., it is very convenient for us to calculate the wave function (WF) by using the above results. For example, for two-mode nonlinear squeezing 2
⟨η|
e iλ(P1 +P2 )
f (y) = f (y − λ) we have ⟨ n η2 ⟨η1 , η2 | e iλ(P1 +P2 ) (Q1 +Q2 )/2 = η1 , , (39) K2 ⟨ n η1 , η2 , (40) ⟨η1 , η2 | e −iλ(Q1 −Q2 ) (P1 −P2 )/2 = K1
then using e
where
√ Kj =
n−1
√ 1 + λ (n − 1) 2n−1 ηjn−1 ,
(j = 1, 2) .
(41)
In particular, when n = 2, Eq. (39) reduces n to Eq. (30). Noting that [(P1 + P2 ) (Q1 + Q2 ), n (Q1 − Q2 ) (P1 − P2 )] = 0, we can further derive e iλ[(P1 +P2 ) (Q1 +Q2 )−(Q1 −Q2 ) ⟨ ∫ 2 η1 η2 d η |η⟩ , = . π K1 K2 n
n
≡ S |ψ⟩ ,
(38)
∂ −λ ∂y
(P1 −P2 )]/2
(42)
Thus we obtain a new approach to deriving two-mode nonlinear squeezing operators.
(Q1 +Q2 )/2−if (P1 +P2 )(Q1 +Q2 )/2
|ψ⟩ (43)
whose wave function in ⟨η| representation is given by Eqs. (36) and (5) as ∫ 2 ′ d η ′ ⟨η| S |ψ⟩ = ⟨η| |η ⟩ ⟨η1′ , η˜2′ | ψ⟩ π ⟨ = η1 , η˜2 e f |ψ⟩ , (44) the wave function of S |ψ⟩ can be caluculated directly by ⟨η| ψ⟩ . In particular, when |ψ⟩ = |00⟩, then from ] [ ) 1( 2 2 (45) ⟨η| 00⟩ = exp − η1 + η2 , 2 we know that the wave function is ] [ ) 1( 2 2 2f , ⟨η| S |00⟩ = exp − η1 + η˜2 e 2 η2 . η˜2 = √ ef − 1 η2 1 + 2λ f
(46)
Thus we provide a new approach to calculating wave functions of some two-mode nonlinear squeezed states.
090307-4
Chin. Phys. B
Vol. 19, No. 9 (2010) 090307
The resolution of unity in differential form for EPR entangled state representation and its application in nonlinear operators normal ordering expansion are shown in Ref. [22]. In summary, after knowing the properties of
References
two-mode quadratures in the entangled state representation, we derive some complicated exponential quadrature operators for nonlinear two-mode squeezing, which directly leads to the wave function of the nonlinear squeezed state.
[12] Preskill J 1998 Quantum Information and Computation (Singapore: World Scientific)
[1] Walls D F and Milburn G J 1994 Quantum Optics (Berlin: Springer-Verlag) [2] Orszag M 2000 Quantum Optics (Berlin: Springer-Verlag) [3] Scully M O and Zubairy M S 1997 Quantum Optics (London: Cambridge University Press) [4] Klauder J R and Sudarshan E C G 1968 Fundamentals of Quantum Optics (New York: Benjamin) [5] Wu Y and Cte R 2002 Phys. Rev. A 66 025801 [6] Yang X X and Wu Y 2003 Commun. Theor. Phys. 40 585 [7] Yang X X and Wu Y 2002 Chin. Phys. Lett. 19 1625 [8] Buˇ zek V 1990 J. Mod. Opt. 37 303 [9] Loudon R and Knight P L 1987 J. Mod. Opt. 34 709 [10] Dodonov V V 2002 J. Opt. B: Quantum Semiclass. Opt. 4 R1–R33 [11] Mandel L and Wolf E 1995 Optical Coherence and Quantum Optics (London: Cambridge University Press)
[13] Fan H Y and Klauder J R 1994 Phys. Rev. A 49 704 [14] Fan H Y and Ye X 1995 Phys. Rev. A 51 3343 [15] Fan H Y and W¨ unsche A 2000 J. Opt. B Quantum Semiclass. Opt. 2 464 [16] Einstein A, Podolsky B and Rosen N 1935 Phys. Rev. 47 777 [17] Fan H Y, Zaidi H R and Klauder J R 1987 Phys. Rev. D 35 1831 [18] W¨ unsche A 1999 J. Opt. B Quantum Semiclass. Opt. 1 R11–21 [19] Hu L Y and Fan H Y 2009 Europhys. Lett. 85 60001 [20] Fan H Y and Fan Yue 2000 Mod. Phys. Lett. B 14 967 [21] Fan H Y and Fan Yue 1996 Phys. Rev. A 54 958 [22] Fan H Y and Yu G C 2001 Mod. Phys. Lett. A 16 2067
090307-5
