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Physics Letters A 295 (2002) 65–73 www.elsevier.com/locate/pla

Nonlinear entangled state representation in quantum mechanics ✩ Hongyi Fan a,b,∗ , Hailing Cheng b a CCAST (World Laboratory), P.O. Box 8730, 100080, Beijing, China b Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui, 230026, PR China 1

Received 21 November 2001; received in revised form 7 February 2002; accepted 13 February 2002 Communicated by P.R. Holland

Abstract We develop Dirac’s representation theory in quantum mechanics by constructing the nonlinear entangled state |ηnl and its non-Hermite conjugate state nlη| with continuum variable. By virtue of the technique of integration within an ordered product of operators we show that |ηnl and nlη| make up an orthonormal and complete representation. From |ηnl we also deduce another kind of entangled states. Application of |ηnl in studying two-mode squeezed state is demonstrated. 2002 Elsevier Science B.V. All rights reserved. PACS: 03.67.Hk; 42.50.-p; 03.65.Bz

Dirac’s book The Principles of Quantum Mechanics used symbolic method to depict quantum mechanics. As Dirac himself wrote in the foreword of the first edition of his book: “There is the symbolic method, which deals directly in an abstract way with the quantities of fundamental importance. . .”. “The symbolic method, however, seems to go more deeply into the nature of things. It enables one to express the physical law in a neat and concise way, and will probably be increasingly used in the future as it becomes better understood and its own special mathematics gets developed. For this reason I have chosen the symbolic method. . .”. In this work, in response to Dirac’s expectation, we shall establish nonlinear entangled state representation which is generalization of Dirac’s coordinate and momentum representation. The conception of entanglement has played a key role in quantum optics, quantum teleportation, quantum computation and quantum measurement [1–7]. In an entangled quantum state, measurement performed on one part of the system provides information on the remaining part, as first pointed out by Einstein, Podolsky and Rosen (EPR) [8] in their famous paper arguing the incompleteness of quantum mechanics. EPR revealed the quantum entanglement involved in the common eigenfunction of two particles’ relative position X1 − X2 (with center of mass coordinate x0 ) and their total momentum P1 + P2 (with

✩

Work supported by National Natural Science Foundation of China under grant 10175057.

* Corresponding author.

E-mail address: [email protected] (H. Fan). 1 Mailing address.

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 1 6 4 - 0

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eigenvalue p0 = 0). The corresponding bipartite Wigner function is ψ(x1 , p1 ; x2, p2 ) = δ(x1 − x2 + x0 )δ(p1 + p2 ),

(1)

Remarkably, the simultaneous eigenstate |η of the commutative operators (X1 − X2 , P1 + P2 ) in two-mode Fock space can be explicitly constructed [9], which is 1 2 † ∗ † † † |η = exp − |η| + ηa − η b + a b |00, (2) 2 where η = η1 + iη2 is a complex number, |00 ≡ |01 |02 is the two-mode vacuum state; a † , b† are two-mode creation operators related to (Xi , Pi ), i = 1, 2, by 1 1 a − a† , P1 = √ X1 = √ a + a † , 2 2i The |η state obeys the eigenvector equations a − b† |η = η|η, b − a † |η = η∗ |η.

1 X2 = √ b + b† , 2

1 b − b† . P2 = √ 2i

It then follows from (3) and (4) that √ √ (X1 − X2 )|η = 2 η1 |η, (P1 + P2 )|η = 2 η2 |η.

(3)

(4)

(5)

From (4) we can immediately see η |η = πδ(η − η )δ(η∗ − η ∗ ).

(6)

|η span a completeness relation which can be directly demonstrated by virtue of the technique of integration within an normally ordered product (IWOP) of operators [10–12], i.e., 2 d η |ηη| π 2 d η = exp −|η|2 + ηa † − η∗ b† + a † b† exp −a †a − b† b exp η∗ a − ηb + ab π 2 d η = (7) exp −|η|2 + η a † − b + η∗ a − b† − a † − b a − b† := 1, π where d 2 η ≡ dη1 dη2 and we have used the normal ordering form of two-mode vacuum projector |0000| = : exp −a † a − b† b : , and integration formula 2 d η λ|η|2 +µη+η∗ ν −1 −µν/λ e e = , π λ

(8)

Re λ < 0.

(Note that Bose operators commute within : : and can be considered as parameters during the integration.) Thus |η is qualified to make up a new quantum mechanical representation, we name it the entangled state representation of continuous variables. The |η representation has applications in solving two-body dynamic problem in the presence of some kinetic coupling [13], in studying path integral and the Wigner function of two-mode correlated system [14,15] as well as the two-mode nonlinear phase operator in quantum optics theory [16]. Thus the |η state enriches Dirac’s representation theory originally proposed in his famous book The Principles of Quantum Mechanics. As the interests in nonlinear coherent states (NCS) grows rapidly (because they exhibit nonclassical features and many

H. Fan, H. Cheng / Physics Letters A 295 (2002) 65–73

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quantum optical states can be viewed as a sort of nonlinear coherent states [17–20], for a tutorial review, see [17]), a question naturally arises: can we generalize the conception of entangled state to some nonlinear cases by constructing a kind of nonlinear entangled states (NES) |ηnl ? The answer is affirmative. By generalizing the IWOP technique to nonlinear Bose operators we shall show that |ηnl make up a new quantum mechanical representation too. Its applications in deriving another kind of entangled states and in obtaining two-mode squeezed state’s natural representation will be discussed. Let Q ≡ a † a − b† b be two-mode number-difference operator, it is easily seen Q, a − b† = − a − b† , (9) Q, a † − b = a † − b . Consequently,

and

f (Q)a = af (Q − 1),

f (Q)a † = a † f (Q + 1),

f (Q)b† = b† f (Q − 1), f (Q) a − b† = a − b† f (Q − 1),

f (Q)b = bf (Q + 1), f (Q) a † − b = a † − b f (Q + 1)

f (Q) a − b† ,

† 1 a − b = 0. f (Q − 1)

(10)

(11)

Thus f (Q)(a − b† ) and [1/f (Q − 1)](a † − b) can have common eigenvector, denoted as |ηnl . We search for such |ηnl as to obey the eigenvector equations f (Q) a − b† |ηnl = η|ηnl , (12) 1 a † − b |ηnl = η∗ |ηnl . (13) f (Q − 1) In comparison with the definition of nonlinear coherent state |zf , which comes from the eigenvector equation f (a † a)a|zf = z|zf , we have reason to name Eqs. (12) and (13) as the definition equations of NES, but the properties of NES is totally different from NCS. By considering the following relations † † 1 1 † † a + b = 2, a − b = 2, f (Q) a + b , f (Q) a − b , f (Q − 1) f (Q − 1) † † 1 1 † † f (Q) a − b , f (Q) a + b = 0, a −b , a + b = 0, f (Q − 1) f (Q − 1) † † † 1 † † † (14) = 0, a + b , f (Q) a + b f (Q) a − b , a b = f (Q)b , f (Q − 1) we construct the eigenstate |ηnl as |ηnl = exp g(η) exp a † b† |00,

(15)

where † η∗ η a + b − f (Q) a + b† . 2f (Q − 1) 2 To see that Eq. (15) really fits Eq. (12), we notice a − b† exp a † b† |00 = 0, and operate f (Q) a − b† on |ηnl , with the aid of Eq. (14) and operator identity, g(η) ≡

eA Be−A = B + [A, B] +

1 [A, [A, B]] + · · · , 2!

(16)

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we have f (Q) a − b† |ηnl = exp g(η) exp −g(η) f (Q) a − b † exp g(η) exp a † b† |00 = η|ηnl .

(17)

Similarly, using (14) and (16) we can prove Eq. (13). Due to [a †b† , Q] = 0 and e−a

† b†

† † † a + b ea b = 2a † + b,

e−a

† b†

† † b † + a ea b = a + 2b†,

we can recast Eq. (15) as |ηnl = e

a † b† −a † b†

e

a † b† a † b† exp g(η) e |00 = e exp

Further, by observing 1 1 † f (Q−1) a , f (Q−1) b = 0, 1 † , f (Q)b † = 0, a f (Q−1)

f (Q)b† = 1, 1 a † = 0, a † b† , f (Q−1) 1 f (Q−1) b,

† η∗ η † 2a + b − f (Q) a + 2b |00. (18) 2f (Q − 1) 2

f (Q)a = −1, 1 b, f (Q)a = 0, f (Q−1) 1 † f (Q−1) a ,

f (Q)a, f (Q)b† = 0, a †b† , f (Q)b† = 0,

we can finally rewrite Eq. (18) in a more compact form |η|2 η † ∗ † † † |ηnl = exp − + a − η f (Q)b + a b |00. 2 f (Q − 1)

(19)

(20)

Especially, when f (Q) = f (Q − 1) = 1, |ηnl reduces to |η. By straightforward calculation we find that nlη |ηnl is not a delta-function, instead, we introduce another nonlinear entangled state (ket) |η|2 † ∗ 1 † † † b +a b − |ηnl = exp ηf (Q − 1)a − η (21) |00. f (Q) 2 We can prove that |ηnl satisfies the equations † † a − b f (Q)|ηnl = η∗ |ηnl , nlη|f (Q) a − b = η nlη|, † 1 1 |ηnl = η|ηnl , a − b = η∗ nlη|. a − b† nlη| f (Q − 1) f (Q − 1) Using Eqs. (12), (13), (22) and (23), we have † nlη |f (Q) a − b |ηnl = ηη |ηnl = η nlη |ηnl , † 1 a − b |ηnl = η∗ nlη |ηnl = η ∗ nlη |ηnl . nlη | f (Q − 1) It then follows the orthonormal property ∗ ∗ nlη |ηnl = πδ(η − η)δ η − η ,

(22) (23)

(24)

(25)

H. Fan, H. Cheng / Physics Letters A 295 (2002) 65–73

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which is in sharply contrast to NCS. We now show that a new completeness relation can be composed by |ηnl and nlη|. For this purpose we need to know generalized normal ordering form of |0000|. Because of 1 1 a † = a, a † = 1, b, f (Q)b† = b, b† = 1, f (Q)a, f (Q − 1) f (Q − 1) 1 b = f (Q)a, f (Q)b† = 0, f (Q)a, f (Q − 1) 1 1 1 a†, b = a † , f (Q)b† = 0, f (Q − 1) f (Q − 1) f (Q − 1) 1 1 a † = aa †, bf (Q)b† = bb†, f (Q)a (26) f (Q − 1) f (Q − 1) 1 b) behave just like a † (b† ) and a(b), so we can introduce we see that [1/f (Q − 1)]a †(f (Q)b† ) and f (Q)a( f (Q−1) 1 b. Correspondingly, the the generalized normal ordering for [1/f (Q − 1)]a † and f (Q)a, f (Q)b† and f (Q−1) generalized IWOP technique for them can be introduced. Now we turn Eq. (8) (the usual normal ordering) to generalized normal ordering, by expanding

|0000| = :

∞ (−)n (a † a + b† b)n n=0

n!

:,

(27)

and using the property (10) we have n

n n a †l b†n−l a l bn−l : a † a + b† b : = l l=0 n−l l n l 1 1 n † † n−l a b = f (Q)b f (Q)a l f (Q − 1) f (Q − 1) l=0 n 1 1 a † f (Q)a + f (Q)b† b ◦◦ , = ◦◦ f (Q − 1) f (Q − 1) where

◦ ◦ ◦ ◦

1 f (Q−1) b,

denotes generalized normal ordering with respect to the operators

1 † f (Q−1) a

(28)

and f (Q)a, f (Q)b† and

in this way

|0000| = ◦◦ exp −

1 1 a † f (Q)a − f (Q)b† b f (Q − 1) f (Q − 1)

◦ ◦

.

Based on Eqs. (20), (21), (29) and 1 a † f (Q)b† = a † b† , f (Q − 1)

f (Q)a

1 b = ab, f (Q − 1)

as well as the generalized IWOP technique we can directly prove the completeness relation of NES 2 2 d η d η η 2 † ∗ † † † |ηnl nlη|= exp −|η| + a − η f (Q)b + a b π π f (Q − 1)

1 1 † † ◦ a f (Q)a − f (Q)b b ◦◦ × ◦ exp − f (Q − 1) f (Q − 1) 1 ∗ + ab × exp η af (Q − 1) − ηb f (Q)

(29)

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H. Fan, H. Cheng / Physics Letters A 295 (2002) 65–73

=

d 2η ◦ 1 1 2 a † − η∗ f (Q)b† + a † f (Q)b† ◦ exp − |η| + η π f (Q − 1) f (Q − 1) 1 1 a † f (Q)a − f (Q)b† b − f (Q − 1) f (Q − 1) 1 1 ∗ + η f (Q)a − η b + f (Q)a b ◦◦ = 1. f (Q − 1) f (Q − 1)

It must be emphasized that within the symbol

◦ ◦ ◦ ◦

the operator

1 † f (Q−1) a

(30)

commutes with f (Q)a, f (Q)b†

commutes with which means within ◦◦ ◦◦ they can be considered as C-number while the integration over 2 d η is performed. In the same reason we have 2 d η |ηnl nlη| = 1. (31) π It is remarkable that in (30), (31) the bra and ket are not mutual Hermite conjugate which differs from the usual representation theory. As the first application of |ηnl we use (20), (21), and the generalized IWOP technique to calculate the following ket-bra projection operator in an integral form,

2 2 d η◦ |η|2 η∗ η d η 1 |η/unl nlη| = a † − f (Q)b† exp − + 1 + ◦ 2 uπ uπ 2 u uf (Q − 1) u 1 1 1 a † f (Q)b† − a † f (Q)a − f (Q)b† b + f (Q − 1) f (Q − 1) f (Q − 1) 1 η + η∗ af (Q − 1) − b + f (Q)a b ◦◦ f (Q − 1) f (Q − 1) 1 ◦ a † f (Q)b† tanh λ = sec hλ ◦ exp f (Q − 1) 1 1 † † a f (Q)a + f (Q)b b (sec hλ − 1) + f (Q − 1) f (Q − 1) 1 − f (Q)a b tanh λ ◦◦ f (Q − 1) 1 1 † † ◦ W ◦ a f (Q)b tanh λ ◦ e ◦ exp −f (Q)a b tanh λ = sec hλ exp f (Q − 1) f (Q − 1) † † † = exp a b tanh λ exp a a + b† b + 1 ln sec hλ exp(−ab tanh λ) ≡ S, (32) 1 f (Q−1) b,

where

1 1 † † a f (Q)a + f (Q)b b , u=e , W ≡ (sec hλ − 1) f (Q − 1) f (Q − 1) and we have used the operator identity

k † k 1 ka † a † ◦ e (33) a f (Q)a ◦◦ , = : exp e − 1 a a : = ◦ exp e − 1 f (Q − 1)

1 † ekb b = : exp ek − 1 b† b : = ◦◦ exp ek − 1 f (Q)b† (34) b ◦◦ . f (Q − 1) S is just the normally ordered two-mode squeezing operator [21], thus we know that S has a natural representation in NES representation. Using the same technique we have 2 1 d η |ηnl nlη/u|. S= (35) u π

λ

H. Fan, H. Cheng / Physics Letters A 295 (2002) 65–73

71

It then follows from (25) and (32) that 1 |η/unl , (36) u which indicates that the usual two-mode squeezing operator also squeezes the nonlinear entangled states. This is not surprising, because the two-mode squeezed state itself is an entangled state which entangles the idle-mode and signal-mode as an outcome of a parametric-down conversion process [22]. The wave function of two-mode squeezed state in nlη| representation is 2 2 1 1 d η d η |η nl nlη /u|00 = πδ(η − η)δ η ∗ − η∗ nlη /u|00 nlη|S|00 = nlη| u π u π

|η|2 1 1 = nlη/u|00 = exp − (37) . u u 2|u|2 S|ηnl =

As the of nlη| representation we try to find the common eigenvector, denoted as |q, A, of Q application second and a − b† a † − b , Q|q, A = q|q, A, a − b† a † − b |q, A = A|q, A, (38) because [Q, (a − b† )(a † − b)] = 0. For this purpose, we calculate

η 1 1 |η|2 |00 a † f (Q)a − f (Q)b† b exp a † − η∗ f (Q)b† + a † b† − Q|ηnl = f (Q − 1) f (Q − 1) f (Q − 1) 2 1 ∂ a † eiϕ + f (Q)b† e−iϕ |ηnl = −i |ηnl , = |η| (39) f (Q − 1) ∂ϕ it then follows nlη|Q|q, A = i

∂ nlη|q, A = q nlη|q, A. ∂ϕ

(40)

The solution to this equation is (up to a normalization constant regarding to |η|), nlη|q, A = e

−iqϕ

,

(41)

where q must be integers because of the uniqueness of wave function e−iqϕ |ϕ=0 = e−iqϕ |ϕ=2π . On the other hand, from Eqs. (12), (13) and the second equation of (38), we have nlη|

a − b† a † − b |q, A = nlη| a − b†

† 1 a − b f (Q)|q, A f (Q − 1)

= |η|2 nlη|q, A = A nlη|q, A. Combining (42) with (38) yields 2 −iqϕ , nlη|q, A = δ A − |η| e

A 0.

As a result of (31) and (43), 2π 2π √ d|η|2 1 d 2η 2 −iqϕ |ηnl nlη|q, A = dϕ δ A − |η| e |ηnl = dϕ e−iqϕ | A eiϕ nl |q, A = π 2π 2π 0 0 √ 1 A 1 1 exp a † b† − b† |00 = dz q+1 exp A zf (Q − 1)a † − 2πi 2 zf (Q) z

c

(42)

(43)

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H. Fan, H. Cheng / Physics Letters A 295 (2002) 65–73

√ ∞ n n+q 1 A (−)n | A|2n+q f (Q − 1)a † = exp a † b† − b † |00 2 (n + q)!n! f (Q) n=0 √ ∞ (−)n | A|2n+q A f (q − 1)f (q − 2) · · · f (0) = exp a † b† − |n + q ⊗ |n. √ 2 (n + q)!n! n=0

(44)

Considering this result as a Schmidt decomposition [23], we see that |q, A is also an entangled state. Now we make a comparison between the nonlinear entangled state and the nonlinear coherent state (NCS) |zf , the latter is defined as an eigenket of f (N)a, N = a † a, f (N)a|zf = z|zf , and is constructed by operating a 1 a † − z∗ f (N)a on the vacuum state |01 , special displacement operator Df (z) = exp z f (N−1)

1 |z|2 +z a † |01 . |zf = Df (z)|0 = exp − 2 f (N − 1)

(45)

† † By observing the correspondence [N, a] = −a, [N, a ] = a and Eq. (9), one can see that it is natural to between † introduce the eigenket of f (Q) a − b , this is just |ηnl . In another word, the introduction of NES |ηnl actually accompanies the outcome of NCS. To further explain the usefulness of |ηnl , we take the Paul phase operator for example to examine the relationship between NES and NCS. The usual Paul phase operator is defined on the basis of usual coherent state |z through the following expression [24] 2 d z iϕ (46) e |zz| = eiΦ , z = |z|eiφ . π

In Ref. [25] we have proved the overcompleteness relation for NCS 2 d z |zf f z| = 1, π

(47)

where |z|2 + z∗ f (N)a , − 2

f z| = 10| exp

so we can also introduce the generalized Paul phase operator on the nonlinear coherent state basis, 2 d z iϕ e |zf f z| ≡ efiΦ . π From Eqs. (20) and (21) we calculate

1 |η|2 † +η a |01 = |z = ηf , 2 0|ηnl = exp − 2 f (N − 1)

(48)

(49)

(50)

which means that the projection of |ηnl onto the second mode vacuum state is just the NCS |zf . Moreover, from (21) we have

|η|2 ∗ (51) + η af (N − 1) = f z = η|. nlη|02 = 10| exp − 2 It then follows from (50) and (51) that 2 d η iϕ iΦ e 2 0|ηnl nlη|02 , ef = π

(52)

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2 which means that the generalized Paul phase operator is the “expectation value” of the operator dπη eiϕ |ηnl nlη|, this operator, when f (Q) = f (Q − 1) = 1, reduces to the Noh, Fougères and Mandel operational phase operator [26]. In conclusion, we have established the NES representation which corresponds to the nonlinear coherent states, so it is of conceptual importance and physical appealing. We have seen that the two-mode squeezing operator also squeezes |ηnl , since f (Q) is squeezing trans invariant, Sf (Q)S −1 = f (Q). We will further consider the application of NES in quantum teleportation and entanglement swapping in future papers.

Acknowledgement The work is supported by the President Foundation of Chinese Academy of Science.

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