New approach for solving the Wigner function from the Husimi function in the two-mode entangled state representation

Optik 125 (2014) 5303–5308

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New approach for solving the Wigner function from the Husimi function in the two-mode entangled state representation夽 Qin Guo a,∗ , Wen Yuan a , Hong-Yi Fan b a b

College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, China Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China

a r t i c l e

i n f o

Article history: Received 5 January 2013 Accepted 22 May 2014 PACS: 03.65.Db 02.30.-f 03.65.-w

a b s t r a c t We introduce a new method to calculate the Wigner function when its corresponding Husimi function is given. A new formula is derived for calculating conveniently the Wigner function in two-mode entangled state representation. As application, we derive Wigner functions of some quantum states, such as twomode entangled state, the electron’s two-mode squeezed canonical coherent state, and the electron’s coordinate eigenstate. © 2014 Elsevier GmbH. All rights reserved.

Keywords: Wigner function Husimi function The two-mode entangled state representation Two-variable Hermit polynomial

1. Introduction Phase space technique is very effective in various branches of physics and it is a popular tool to study semiclassical physics since the phase-space distribution function allows one to describe the quantum aspects of a system with as much classical language allowed. It becomes a useful tool to study quantum computer recently [1]. Among various phase space distributions, the Wigner function [2–6], Husimi distribution [7], and the Q function are the most popularly used distribution functions. Studying the Wigner function has been a major topic in quantum statistical physics and quantum optics [8–10]. Wigner functions of some cavity fields can be measured in the whole phase space by a scheme based on interaction between cavity fields and atoms [11]. The Wigner function can take negative values and exhibits complex patterns due to fast oscillations. To overcome this inconvenience, the Husimi distribution function is introduced. A smoothing of the Wigner function with a squeezed Gaussian gives the Husimi distribution [7], which has a much simpler structure, is easier to interpret and, therefore, more useful for the study of quantum classical correspondence. The Husimi representation is a quasi-probability distribution commonly used in quantum mechanics and also to represent the quantum state of light [12]. It is also applied to study the quantum effects in superconductors [13] and the quantum phase-space picture of Bose–Einstein condensates in a double well [14]. In addition, in the preceding work of Fan Hong-yi and Yang Yan-li [15], the Husimi operator h (q, p ; ) corresponding to the Husimi distribution function Fh (q, p, ) in phase space was proposed for the first time through the technique of integration within an ordered product (IWOP) of operators [16]. In their work, the single-particle Husimi operator h (q, p ; ) is just the single-mode squeezed coherent state projector, i.e. the pure state density matrix, h (q, p ; ) = |p, q p, q|, where |p, q is

夽 This project was supported by the National Natural Science Foundation of China (Nos. 11264016 and 11364022), the Natural Science Foundation of Jiangxi Province of China (Nos. 2009GZ W0006 and 20142BAB202004) and the Research Foundation of the Education Department of Jiangxi Province of China (Nos. GJJ12171 and GJJ12172). ∗ Corresponding author. Tel.: +86 791 88120370; fax: +86 791 88120370. E-mail address: [email protected] (Q. Guo). http://dx.doi.org/10.1016/j.ijleo.2014.05.043 0030-4026/© 2014 Elsevier GmbH. All rights reserved.

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a new kind of squeezed coherent state. This result provides us with a convenient approach for deriving Husimi distribution functions of various density matrices (), i.e. Fh (q, p, ) = Tr[h (q, p; )]= p, q||p, q ,

(1)

which is a pure state expectation value. The Husimi function of the excited squeezed vacuum state was calculated through this method [17]. More application of this formula see reference [18]. Shortly afterwards, this generalization from single-mode case to entangled two-mode case was discussed in reference [19] by Fan Hongyi and Guo Qin who presented the entangled two-mode Husimi operator h (, , ) corresponding to the entangled two-mode Husimi function, and found that it is also a pure state density matrix h (, , ) = |,  , |, where |,  is the two-mode squeezed coherent state. This result also provides us with a convenient approach for deriving Husimi distribution functions of various density matrices () in two-mode case, i.e. Fh (, , ) = Tr[h (, , )]= , ||,  .

(2)

More application of this formula see references [20–22]. Now the Husimi distribution function of a density matrix  in single-mode case or in two-mode case can be calculated directly through the single-particle Husimi operator or the entangled two-mode Husimi operator, while need not through the Wigner function. An interesting question thus naturally arises: if a Husimi distribution function is known then what is the corresponding Wigner function? To put it in another word, can we deduce the Wigner function when the Husimi function is known? This question, to our knowledge, has not been reported in the literature before. And exploring this problem may enrich the transform relation between the Wigner function and the Husimi function. In the following we shall solve an integration equation which relate the Husimi function and the Wigner function, and use the orthonormal and complete properties of two-variable Hermite polynomials [23] to derive the calculating formula of Wigner function. The work is arranged as follows: In Section 2 we derive a new formula (see Eq. (16)) showing how to obtain the Wigner function W(  ,   ) from the Husimi distribution function H(, , ) in two-particles case. In Sections 3–5 we derive the two-mode Wigner function in entangled state representation |  ,    , derive the Wigner function of the electron’s two-mode squeezed canonical coherent states |ε ,    and derive the Wigner function of the electron’s coordinate eigenstate |, by using the formula (16), respectively. Thus several examples are discussed by using this formula (16). In doing so, the Husimi distribution function theory and Wigner function theory for quantum optics can be enriched. 2. A new formula for deriving the Wigner function when the Husimi distribution function is known How to obtain the Wigner function through the Husimi distribution function? To solve this question we start from the following relation [19]:





d2   d2   W (  ,   ) × exp

H(, , ) = 4

−|  − |2 −

|  − |2 



(3)

where W(  ,   ) is the two-mode Wigner function and H(, , ) is the corresponding two-mode Husimi function, which is defined in a manner that guarantees it to be non-negative. Its definition is smoothing out the Wigner function W(  ,   ) by averaging over a “coarse graining” function exp(− |  − |2 − ((|  − |2 )/)). This integration equation is a Fredholm equation of the first-kind with the kernel  2  2 e−| −| −((| −| /) [24]. According to the following formula [25] 

∞  t m t n

 ∗

e−tt +tz+t z =

m,n=0

m!n!

Hm,n (z, z ∗ ),

(4)

where Hm,n ( , * ) is the two-variable Hermite polynomial,

 m!n!(−1)l zm−l z∗n−l

min(m,n)

Hm,n (z, z ∗ ) =

l=0

l!(m − l)!(n − l)!

(5)

,

we can obtain the following relations: 

∞ √ m √ n  (  ∗ ) ( )

2

e−| −| =

m,n=0

m!n!

√ √  2 × Hm,n (   ,  ∗ )e−| | ,

(6)

and e−((|

 −|2 )/)

=

∞ √ n √ m  ( ∗ / ) (/ )

m !n !

m ,n =0

× Hm ,n

    ∗  √ ,√  

e(−|

 |2 /)

.

(7)

Substituting (6) and (7) into Eq. (3), then H(, , ) can be expressed as



H(, , ) = 4

 2

d2   d2   e−| | W (  ,   ) ×

∞ √ √ √  m √ n (  ∗ ) ( ) Hm,n (   ,  ∗ )

m!n! m,n=0

×

∞ √ m √ n √ √  2  ( ∗ / ) (/ ) e−(| | /) Hm ,n ((  / ), ( ∗ / ))

m !n ! m ,n =0

.

(8)

Q. Guo et al. / Optik 125 (2014) 5303–5308

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Assuming that W(  ,   ) can be expanded as W (  ,   ) =

∞ 

√ √ ∗ Clk,ij Hlk (   ,  ∗ )Hij∗

lk,ij=0

    ∗  √ ,√  

,

(9)

then H(, , ) can be further expressed as the following form,

 d2   d2  

H(, , ) = 4

∞ 

√ √ ∗ Clk,ij Hlk (   ,  ∗ )Hij∗

    ∗ 

lk,ij=0

×

∞ √ m √ n  ( ∗ / ) (/ )

m !n !

m ,n =0

Hm ,n

√ ,√  

    ∗  √ ,√  

×

√ 2 ∞ √ m √ n  (  ∗ ) ( ) e−|  |

m!n!

m,n=0

√ √ Hm,n (   ,  ∗ ) (10)

e

√ −|(  / )|2

.

Using the orthonormal relation of two-variable Hermit polynomial Hm,n ( , * ) [25],



√ d2 z −|z|2 ∗ ∗ e Hm,n (z, z ∗ )Hm m!n!m !n !ım,m ın,n ,  ,n (z, z ) =

we have √ m!n!l!k!ım,l ın,k = 



√ √ √ √ d2   −|√  |2 ∗ × Hm,n (   ,  ∗ )Hl,k (   ,  ∗ ), e

and



m !n !i!j!ım ,i ın ,j =

(11)



1 

d2   −|(  /√)|2 e × Hm ,n

    ∗  √ ,√  

∗ Hi,j

    ∗  √ ,√  

(12)

(13)

.

Substituting the above Eqs. (12) and (13) into Eq. (10), we obtain H(, , ) = 4 2

∞ 

 Clk,ij

(l+k−i−j)  ∗l  k  ∗i  j ,

(14)

lk,ij=0

then Clk,ij can be calculated from Eq. (14), Clk,ij =

4 2



∂

l+k+i+j

H(, , )

(l+k−i−j) l!k!i!j!∂ ∗l ∂ k ∂ ∗i ∂ j

|==0 .

(15)

Substituting Eq. (15) into Eq. (9), we have the expression of W(  ,   ). √ √ √ √ ∞ ∗ (   ,  ∗ )H ∗ ((  / ), ( ∗ / )) l+k+i+j  Hlk ∂ ij   H(, , )|==0 , W ( ,  ) = ×  ∂ ∗l ∂ k ∂ ∗i ∂ j 4 2 (l+k−i−j) l!k!i!j! lk,ij=0

(16)

which is a new formula for deriving the Wigner function when the Husimi function is known. This formula can be used to calculate the two-mode Wigner function in entangled state representation. 3. Wigner function of two-mode entangled state representation derived from formula (16) In the preceding work [19], the two-mode Husimi function in entangled state representation |  ,    is derived through the Husimi operator h (, , ), we rewrite it as follow:



H(, , )=   ,   |h (, , )|  ,    where |,  =

 |  − |2 = exp − |  − |2 − 2 2



.

√ 2  −(1/1+)(((||2 )/2)+((||2 )/2)) −(1/1+)(−(+)a† −( ∗ − ∗ )a† −(−1)a† a† ) 1 2 1 2 |00, e e 1+

satisfying the completeness relation 1 4 2

=   ,   |,  , |  ,   





2

d 

1 d |,  , | = 4 2 2



(17)

(18)

 2

d 

d2 h (, , ) = 1.

Substituting Eq. (17) into Eq. (16), we obtain the corresponding Wigner function: √ √ √ √ ∞ ∗ (   ,  ∗ )H ∗ ((  / ), ( ∗ / )) l+k+i+j  Hlk ∂  2  2 ij   W ( ,  ) = e−(/2)| −| −((| −| )/2) |==0 . ×  ∗l ∂ k ∂ ∗i ∂ j ∂  2 (l+k−i−j) 4

 l!k!i!j! lk,ij=0

(19)

(20)

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Now let us do the partial derivation calculus. According to the formula [25],

 ∂ exp −tt  + tz + t  z ∗ |t=t  =0 = Hm,n (z, z ∗ ), ∂t m ∂t n m+n

we have

∂

l+k

∂

i+j



 l  k



 2

 exp − |  − |2 |=0 = 2 ∂ ∗l ∂ k and



∂ ∗i ∂ j

−

exp

|  − |2 2

 |=0 =

(21)

 2

 1 i  1 j √ 2

Then,

√ 2

e−(/2)|

e−(1/2)|

√ l+k−i−j

 |2

 |2

 Hl,k

Hi,j

   , 2

  

 ∗ √ ,√ 2 2

1

 |  |2 ∂  2 − H(, , )|==0 = √ e−(/2)| | e 2 × Hl,k l+k+i+j ∂ ∗l ∂ k ∂ ∗i ∂ j ( 2) l+k+i+j



 ∗  2

 (22)

,

 (23)

.

   , 2

 ∗  2

 Hi,j

  

 ∗ √ ,√ 2 2

 .

(24)

Substituting Eq. (24) into Eq. (20), we have √ √ √ √ ∞ ∗ (   ,  ∗ )H ∗ ((  / ), ( ∗ / ))  2  2 l+k+i+j  Hlk e−(/2)| | e−(1/2)| | ∂ × W (  ,   ) = H (, , ) |==0 =  ij 2 4

∂ ∗l ∂ k ∂ ∗i ∂ j 4 2 (l+k−i−j) l!k!i!j! lk,ij=0 ×

  √ √ ∞ ∗ (   ,  ∗ )H (  Hlk (/2)  , (/2) ∗ ) l,k √ l+k+i+j ( 2) l!k!i!j!

lk,ij=0

    ∗  ∗

× Hij

√ ,√  

Hi,j

  

 ∗ √ ,√ 2 2



(25)

By using the following formula [25,26], ∞  t n sm m,n=0

n!m!

Hm,n (i , i )Hm,n (i , i) =

1 exp 1 − ts

 1



1 − ts

ts − t − s + ts 



,

(26)

we can do the following calculation further   √ √ ∞ ∗ (   ,  ∗ )H (  Hlk (/2)  , (/2) ∗ ) l,k = 2 exp[−|  |2 +    ∗ +  ∗   − (/2)|  |2 ], √ l+k ( 2) l!k! lk and

√

√

√

√

∞  Hij∗ ((  / ), ( ∗ / ))Hi,j ((  / 2), ( ∗ / 2))

√ i+j ( 2) i!j!

ij

(27)

= 2 exp[−(|  |2 /) + (   ∗ /) + ( ∗   /) − (|  |2 /2)].

(28)

Substituting the above Eqs. (27) and (28) into (25), we can obtain the final form of W(  ,   ),



1 |  −   |2 W ( ,  ) = 2 exp −|  −   |2 − 







.

(29)

Thus the two mode entangled Wigner function W(  ,   ) is obtained. The result can be verified through substituting it to Eq. (3) and obtaining the expression of H(, , ), which is just the given condition. 4. Wigner function of the electron’s two-mode squeezed canonical coherent state In Ref. [27], we have introduced the Husimi operator h (, ε; ) for studying Husimi distribution in phase space (, ε) for electron’s states in uniform magnetic field, where  is the Gaussian spatial width parameter. Using the Wigner operator in the entangled state | representation [28] we have found that h (, ε ; ) is just a pure squeezed coherent state density operator |, ε , ε|, which brings convenience for studying and calculating the Husimi distribution Wh (, ε ; k) =  |h (, ε, )| . And we have obtained the Husimi function of |ε ,    ,



Wh (ε, , )= ε ,   |h (ε, , )|ε ,   

=

| ε, |ε ,    |2

 |  − |2 = exp − |ε − ε|2 − 2 2



,

where |ε,  is a two-mode squeezed canonical coherent state, it has been defined as follows [27]: √ 2  −(1/(1+))((|ε|2 /2)+(||2 /2)) ∗ ∗ |ε,  = × e−(1/1+)(−(ε+)K+ +i(ε − )+ −i(−1)+ K+ ) |00. e 1+

(30)

(31)

The relation between the Wigner operator B (ε, ) and the corresponding Husimi operator h (ε, , ) is also given,



h (ε, ; k) = 4



| −   |2 d  d ε B (ε ,  ) × exp −|ε − ε | −  2

 2 





 2



,

(32)

Q. Guo et al. / Optik 125 (2014) 5303–5308

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which is similar as formula (3). So we can immediately obtain the corresponding Wigner function of a two-mode squeezed canonical coherent state for describing the electron states in uniform magnetic field. According to the formula (16), we can obtain the Wigner function of |ε ,    : W (ε ,   ) =

√ √ √ √ ∞   ∗  l+k+i+j Hlk ( ε , ε∗ )Hij∗ ((  / ), ( ∗ / )) |  −   |2 1 ∂ Wh (ε, , )|ε==0 = 2 exp −|ε − ε |2 − , ×  

∂ ∗l ∂ k ∂ ∗i ∂ j 4 2

lk,ij=0

(33)

(l+k−i−j) l!k!i!j!

where we omit the similar steps of the calculation as the third section. 5. Wigner function of the electron’s coordinate eigenstate | In Ref. [27], we also calculate the Husimi function of the electron’s coordinate eigenstate | [28], √ Wh (ε, , ) = |h (, ε; )| = | |, ε|2 =  exp{−| − ε∗ |2 }. According to formula (32) we can have the relation between the Husimi function Wh (ε, , ) and the Wigner function





Wh (ε, , ) = 4

2  2







d ε d  W (ε ,  ) × exp

| −   |2 −|ε − ε | − 



 2



(34) W(ε ,



=  exp −| − ε∗ |2 .

Using (16), the Wigner function W(ε ,   ) of the electron’s coordinate eigenstate is given by √  √ ∗ l+k √  √ ∗ √ √ ∗ −||2 ∞ ∞ ∗ ∗   Hlk Hlk ε , ε ε , ε Hk,l ,  e ∂ Wh (ε, , ) W (ε ) = |=ε=0 =   2 l!k! ∗l ∂εk 4

∂ ε 2 (l+k)  l!k! lk=0 4

lk=0 =

−||2

e 4 2

2 /2)

e(||

 2 /2)

e(|ε |

where we use Eq. (21) to calculate ∂

∂

l+k

∂ε∗l ∂εk





exp −| − ε∗ |2 =

l+k

  ), (35)

(36)

1 (2) 1 (2) 

ı ( − ε∗ ) = ı  − ε∗  4





/∂ε∗l ∂εk exp −| − ε∗ |2 as

√ l+k 

√ √ exp (−∗ ) Hk,l ( , ∗ ),

(37)

and we use the following formula [26]: ∞  m,n=0

to calculate

1 2 ∗ Hm,n (,  ∗ )[Hm,n (  ,  ∗ )] e−|| = ı(2) ( −   ), m!n!

∞

√ √ √ √ 1 H ∗ ( ε , ε∗ )Hk,l ( , ∗ ) lk=0 l!k! l,k

√ √ √ √ ∞ ∗ ( ε , ε∗ )H ( , ∗ )  Hl,k k,l lk=0

l!k!

2 /2)

= e(||

(38)

as  2 /2)

e(|ε |

√ √

2  2  | ε∗  = e(|| /2) e(|ε | /2) ı(2) ( − ε∗ ). 

(39)

From the result of formula (36), we can see that it is just the Wigner function of the electron’s coordinate eigenstate |, W (  , ε ) =

1 (2)  ı  − ε∗ , 4

(40)

which is calculated by using another method in Refs. [27,29]. 6. Conclusions In this paper, we have introduced a new method to calculate the Wigner function of some given states when their corresponding Husimi functions are given. A new formula (16) is derived by using the properties of the two-variable Hermite polynomial, which is suitable for calculate the Wigner function in two-mode entangled state. As its applications, we use it to calculate the Wigner function in two-mode entangled state |  ,    , the Wigner function of the electron’s two-mode squeezed canonical coherent state |ε ,    , and the Wigner function of the electron’s coordinate eigenstate |. The results show that the Wigner function of the electron’s coordinate eigenstate obtained by formula (16) is same as the Wigner function obtained by other methods. The new formula for calculating the Wigner function provides us a new method and enriches the phase space theory. References [1] [2] [3] [4] [5] [6]

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