Phase space representation of density operator: the coherent state and squeezed state Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 21, 2017)
Here we discuss the phase space representation of the density operator, including Wigner function for the coherent state and squeezed state in quantum optics. The density operator of a given system includes classical as well as quantum mechanical properties. ______1. P function representation for the Density operator We start with a density operator defined by
ˆ d 2P() where P() is called the Glauber-Sudarshan P function (representation) and is an coherent state. We note that
Tr[ˆ] Tr d 2P() d 2P()Tr[ ] d 2P() d 2P() 1
or
Tr[ˆ] Tr d 2P() 1 d 2P() d 2 1 2 d 2P() d 2 1 2 2 d 2P() d 2 exp( * * ) 1 2 2 d 2P()exp( ) d 2 exp( 2Re[ * ])) 1 2 d 2P() exp( )exp( 2 ) d 2P() 1 where
1 d 2 1ˆ (formula)
2 exp{ 2 2 2Re[ * ]}
Note that and are complex numbers,
x iy , a ib
Since
Re[ *] Re[(a ib)(x iy)]
ax by we get the integral as
d 2 exp( 2 2Re[ *])) dxexp(x2 2ax)dx dyexp(y2 2by) exp(a2 b2 ) exp( 2 )
We also note that the density operator can be rewritten as
1 ˆ d 2 d 2 ˆ 2
1 d 2 d 2 ˆ 2 using the formula
1 d 2 1ˆ
((Example)) The average number of photons can be written as
nˆ aˆ aˆ Tr[ˆaˆ aˆ] Tr[ d 2P() aˆ aˆ] d 2P() aˆ aˆ d 2P() 2
So P() is normalized as a classical probability distribution.
((Note)) Here we note that
ˆ d 2P() 2 d 2P() d 2P()exp[ 2 ] since
2 exp{ 2 2 2Re[ *]} exp[ 2 ]
Since exp[ 2 ] is not a delta function, the diagonal elements
ˆ P( )
Although ˆ must be positive. However, the integral d 2P()exp[ 2 ] does not require to be positive.
2. Two-dimensional Dirac delta function of the complex variable Two-dimensional Dirac delta function of the complex variable, 'i".
(2) () (') (") 1 dx dy exp[i('x " y)] (2 )2 1 d 2 exp( * * ) 2 1 d 2 exp( * * ) 2 or
1 (2) () d 2 exp( * * ) 2 where
* * 2i Im( * ) i('x " y) and
y x 1 i (y ix) 2 2 2
We also note that the 2D Dirac delta function is given by
1 (2) ( ) d 2 exp[( ) * ( * *) ] 2
3. Expression of P() How can we find the expression of P( ) ? We consider the density operator which is defined by
ˆ d 2P()
Using the formula for the scalar product for the coherent states,
1 2 1 2 exp[ * ] 2 2 we get the matrix element of the density operator as
u ˆ u d 2P() u u
exp[u 2 ] d 2P()exp[ 2 ]exp[ *u u*] or
exp[u 2 ] u ˆ u d 2P()exp[ 2 ]exp[ *u u* ]
Here we use the 2D Dirac delta function.
d 2u exp[u 2 ] u ˆ u exp[ *u u* ] d 2 P( )exp[ 2 ] d 2u exp[ *u u* ]exp[ *u u* ] d 2 P( )exp[ 2 ] d 2u exp[( )u* ( )*u] 2 d 2 P( )exp[ 2 ] ( ) 2 P()exp[ 2 ] or
1 2 2 * * P() e e u u ˆ u e(u u )d 2u 2
4. Another expression of P() )) The expression of P( ) can be also obtained as follows.
* eaˆ eaˆ Tr[ˆeaˆ e aˆ ]
* Tr[ d 2 P( ) eaˆ e aˆ ] * d 2P( ) eaˆ e aˆ * * d 2P( )e
Here we use the expression of the 2D Dirac function as
1 (2) ( ) d 2 exp[( ) * ( * *) ] 2
Thus we get
1 * * * 1 * * * * d 2 eaˆ e aˆ e d 2 ( d 2 P( )e )(e ) 2 2
1 * * * d 2 P( ) d 2e ( ) ( ) 2 d 2 P( ) (2) ( ) P() or
1 * * * P() d 2 eaˆ e aˆ e 2
1 * * d 2Tr[ˆe (aˆ )e (aˆ ) ] 2
Here we note that Tr under the integral is characteristic function of the normally ordered aˆ , aˆ operators,
aˆ *aˆ aˆ *aˆ N ( ) e e Tr[ˆe e ]
Therefore we have
1 * * * P() d 2 eaˆ e aˆ e 2
1 * * d 2 ( )e 2 N
This relation represents a mapping from the density operator ˆ , which is a function of the two operators aˆ and aˆ , to the scalar function N ( ) , which is function of the complex variables . P() is the Fourier transform of the normally ordered characteristic function.
5. P function for the coherent state (I) We consider the density operator for the pure coherent state ;
ˆ .
We note that
u ˆ u u u
1 2 1 2 1 2 1 2 exp[ u u* ]exp[ u *u] 2 2 2 2 exp(u 2 2 u* *u) where
1 2 1 2 u exp[ u *u] 2 2
1 2 1 2 u exp[ u u* ] 2 2
Then we get
1 2 2 * * P() e e u u ˆ u e(u u )d 2u 2
1 2 2 2 2 * * * * e e u e( u u u)e(u u )d 2u 2
1 2 2 * * * * e e(u u)e(u u )d 2u 2
1 2 2 * * * e e(u ( )u( )d 2u 2 2 2 e (2) ( ) (2) ( ) or
P() (2) ( ) for the density operator (pure state ˆ ).
6. P function for the coherent state (II) We consider the density operator for the pure coherent state ;
ˆ .
Thus we get
aˆ *aˆ N ( ) Tr[ˆe e ] * Tr[ eaˆ e aˆ ]
* eaˆ e aˆ
* * e
1 * * * P() d 2 eaˆ e aˆ e 2
1 * * d 2 ( )e 2 N
1 * * * * P() d 2e e 2
1 * * * d 2 e ( ) ( ) 2 (2) ( )
7. P function for the number state n (I) We consider the density operator for the pure coherent state n ;
ˆ n n .
We note that
* n 2 (u u) u ˆ u u n n u e u n!
Thus we get
1 2 2 * * P() e e u u ˆ u e(u u )d 2u 2 * n 1 2 2 2 (u u) * * e e u e u e(u u )d 2u 2 n! 2 e 1 * * (u*u)n e(u u )d 2u n! 2
Formally, we can write
2 2n e 1 * * P() e(u u )d 2u n * 2 n! 2 e 2n n () n! n *
8. P function for the number state n (II) We consider the density operator for the pure number state n ;
ˆ n n .
Thus we get
aˆ *aˆ N ( ) Tr[ˆe e ] * Tr[ n n eaˆ e aˆ ]
* n eaˆ e aˆ n
1 * * P() d 2 ( )e 2 N
1 * * * d 2 n eaˆ e aˆ n e 2
1 * * * d 2 n eaˆ e aˆ n e 3
1 1 * * * d 2 n n d 2e ( ) ( ) 2 1 d 2 n n (2) ( ) 1 2 n or
2n 1 2 2 P() n e n! which is a Poisson function. where
* * e aˆ n * n eaˆ e n
* * e aˆ n e n
9. Q-representation There are other orderings possible for the two operators aˆ and aˆ . This time we use the antisymmetric ordering and define the antinormally ordered characteristic function
*aˆ aˆ A ( ) Tr[ˆe e ]
Its Fourier transform is called Q representation. This representation is sometimes called the Husimi function.
1 * * Q() d 2 ( )e 2 A and has a simple form
1 Q() ˆ 0
Thus Q( ) is the probability of finding a system in the density operator ˆ in the coherent state . This proof is given below.
*aˆ aˆ A ( ) Tr[ˆe e ]
1 * d 2 Tr[ˆe aˆ eaˆ
1 * * 1 * Q() d 2 e d 2 Tr[ˆe aˆ eaˆ ] 2
1 * * 1 * * d 2 e d 2 Tr[ˆ eaˆ ]e 2
1 1 * * * d 2 Tr[ˆ ] d 2 e ( ) ( ) 2 1 d 2 [ ˆ (2) ( ) 1 ˆ
((Husimi function))
Kôdi Husimi (June 29, 1909 – May 8, 2008, Japanese:) was a Japanese theoretical physicist who served as the president of the Science Council of Japan. Husimi trees in graph theory, and the Husimi Q representation in quantum mechanics, are named after him. https://en.wikipedia.org/wiki/K%C3%B4di_Husimi
Fig. Photo of Prof. Dodi Husimi. http://www.47news.jp/PN/200805/PN2008050901000355.-.-.CI0003.jpg
10. Q representation for a pure coherent state For the density operator
ˆ we have
1 2 1 2 2 Q() exp[ * * ] since
1 2 1 2 exp[ * ] 2 2
11. Q representation for the pure number state For the density operator
ˆ n n we have
2n 1 2 2 Q() n exp( ) n! which is the Poisson function
12. Wigner representation For symmetric order of two operators aˆ and aˆ . The characteristic function is defined as
* ˆ ˆ S ( ) Tr[ˆ exp(aˆ aˆ)] Tr[ˆD( )] D( )
Its Fourier transform is called Wigner representation.
1 * * W () d 2 ( )e 2 S
1 * * d 2 e Tr[ˆDˆ ( )] 2
13. Another expression of Wigner function
1 1 Xˆ xˆ (aˆ aˆ ) (aˆ aˆ ) , 2 2 2 2
1 1 m 1 Yˆ pˆ 0 (aˆ aˆ ) (aˆ aˆ ) 2 2 2i 2i or
aˆ Xˆ iYˆ , aˆ Xˆ iYˆ .
Note that
1 1 1 1 [Xˆ ,Yˆ] [ xˆ, pˆ] [xˆ, pˆ] [xˆ, pˆ] i1ˆ , 2 2 2 2 2 2
Dˆ ( ) exp(aˆ *aˆ) exp[ (Xˆ iYˆ) * (Xˆ iYˆ)] exp[( * )Xˆ i( * )Yˆ] exp[2i " Xˆ 2i 'Yˆ] exp(i ' ")exp(2i " Xˆ )exp(2i 'Yˆ)
This operator can be rewritten as
1 Dˆ ( ) exp(i ' ")exp(2i " xˆ)exp(2i ' pˆ) 2 2 2 ' pˆ exp(i ' ")exp(i 2 "xˆ)exp( i )
Then we get
2 ' pˆ Dˆ ( ) x' exp(i ' ")exp(i 2 "xˆ)exp( i ) x' 2 ' exp(i ' ")exp(i 2 "xˆ) x'
2 ' 2 ' exp(i ' ")exp(i 2 "(x' ) x' 2 ' exp(i ' ")exp(i 2 "x') x'
Wigner function
1 * * W () d 2 ( )e 2 S
1 * * d 2 e Tr[ˆDˆ ( )] 2
1 * * d 2 e dx' x' ˆDˆ ( ) x' 2
1 * * 2 ' d 2 e dx'exp(i ' "i 2 "x') x' ˆ x' 2
Note that
* * 2i '"2i "'
* * i ' "i 2 "x' 2i '"2i "'i ' "i 2 "x'
2i '"i(2' ' 2x') "
We use the Dirac delta function as
d "exp[i(2' ' 2x') "] 2 (2' ' 2x')
Then we have
2 2 ' W () dx' d 'exp(2i '") (2' ' 2x') x' ˆ x' 2 2 dx'exp[2i"(2' 2x')] x' ˆ x' (2' 2x') 2 2 2 dx'exp[4i"(' x')] x' ˆ 'x') 2
Here we define
u 2 2 u x' x , 'x' x 2 2
2 ' x , " p 2
1 dx' du 2
4"(' x') 4"(' x') 2 2 2 4 "(x' ') 2 u 4 "(x x) 2 2 2 "u 2 2 pu
1 u u W () W (x, p) du e2ipu x ˆ x 2 2 which is the Fourier transform of non-diagonal density matrix element in the position eigenstates basis. Using the inverse Fourier transform, the non-diagonal matrix element of the demsity operator is
u u x ˆ x dp e2ipuW (x, p) 2 2
13. Displacement operator With help of the 2D Dirac delta function, we introduce a new notation for the quantum states in phase space,
1 (2) (aˆ ) d 2 exp[(aˆ *) (aˆ ) *] 2 1 d 2 exp( * *)exp(aˆ *aˆ) 2 1 d 2 exp( * *)Dˆ ( ) 2 where Dˆ ( ) is the displacement operator,
Dˆ ( ) exp(aˆ *aˆ)
1 2 exp( )exp(aˆ )exp( *aˆ) 2 1 2 exp( )exp( *aˆ)exp(aˆ ) 2
We note that
(2) (aˆ ) Tr[ˆ (2) (aˆ )] 1 d 2 exp( * * )Tr[ˆDˆ ( )] 2 W ()
15. Wigner representation for a pure coherent state
For ˆ
Tr[ˆDˆ ( )] Dˆ ( )
Tr[ Dˆ ( )]
Dˆ ( )
1 2 exp( )exp( * * ) 2 where
1 2 Dˆ ( ) exp( ) exp(aˆ )exp( *aˆ) 2 1 2 exp( )exp( * * ) 2
Thus we get the Wigner representation
1 W () d 2 exp( * * )Tr[ˆDˆ ( )] 2 1 1 2 d 2 exp( * * )exp( )exp( * * ) 2 2 1 1 2 exp( )d 2 exp[ * ( ) ( * * )] 2 2 2 2 exp[2 ] 2 exp{2[(x x )2 (y y )2 ]} 0 0 or
2 W () exp{2[(x x )2 (y y )2 ]} 0 0
2 We use x'iy' , d dx'dy ' , x x0 i(y y0 ) (x, y, p, p0, q and q0 are real). Using the Mathematica, we get
1 2 1 exp( )d 2 exp[ * ( ) ( * *)] dx'dy'exp[ (x'2 y'2 ) 2i(y y )x'2i(x x )y'] 2 2 0 0 2 2 2 exp[2(x x0 ) 2(y y0 ) )]
16. Wigner representation for a pure squeezed state For ˆ
Tr[ˆDˆ ( )] Dˆ ( )
Tr[ Dˆ ( )]
Dˆ ( ) ˆ ˆ ˆ 0 S D( )S 0
Then the Wigner function is obtained as
1 W () d 2 exp( * * )Tr[ˆDˆ ( )] 2
1 d 2 exp( * * ) 0 Sˆ Dˆ ( )Sˆ 0 2 where
Dˆ ( ) exp(aˆ *aˆ)
We use the Bogoliubov transformation
ˆ ˆ ˆ ˆ ˆ ˆ * * S aˆS b aˆ aˆ , S aˆ S b aˆ aˆ where
1 1 1 2 1 Sˆ exp[ *aˆ 2 (aˆ )2 ], Sˆ exp[ aˆ * (aˆ)2 ] Sˆ 2 2 2 2
2 2 1, sei
cosh s , ei sinh s
bˆ 0 0
Thus we have
ˆ * ˆ ˆ * ˆ S (aˆ aˆ)S b b (*aˆ *aˆ) * (aˆ aˆ ) (* *)aˆ ( * * )aˆ
Here we switch the variable as , leading to the change of parameters ( and ˆ ˆ S S ).
ˆ * ˆ * * * * S (aˆ aˆ)S ( )aˆ ( )aˆ or
Sˆ (aˆ *aˆ)Sˆ (* *)aˆ ( * *)aˆ * 1aˆ 1 aˆ
* * where 1
ˆ ˆ ˆ By using the relation S S 1 we get
Sˆ (aˆ *aˆ)2 Sˆ Sˆ (aˆ *aˆ)Sˆ Sˆ (aˆ *aˆ)Sˆ * 2 (1aˆ 1 aˆ) which leads to the relation
ˆ ˆ ˆ ˆ ˆ * * S D( )S D(1) D( )
Using the relation
1 2 0 Dˆ ( ) 0 0 exp( )exp( aˆ )exp( *aˆ) 0 1 2 1 1 1 1 2 exp( ) 0 exp( aˆ )exp( *aˆ) 0 2 1 1 1 1 2 exp( ) 2 1 we get
1 W () d 2 exp( * * ) 0 Dˆ ( ) 0 2 1 1 1 2 d 2 exp( * * )exp( ) 2 2 1
1 1 2 d 2 exp( * * )exp( * * ) 2 2 1 1 2 d 2 exp( * * * * ) 2 2 where 2 2 1 . This integral can be calculated by using the Mathematica with d 2 dx'dy' and x'iy' .
2 2 W () exp[2* * )
When x iy
2 W (x, y) exp{2F(x, y)] where
F(x, y) (x2 y2 )cosh(2s) [(x y)(x y)cos 2xysin]sinh(2s)]
When 0
F(x, y) (x2 y2 )cosh(2s) (x y)(x y)]sinh(2s)]
x2e2s y2e2s
((Note))
x' es 0 cos sin x 2 2 y' 0 es y sin cos 2 2
es cos x es sin y 2 2 es sin x es cos y 2 2 under the rotation and squeezed processes. Thus we have
x'2 y'2 [cosh(2s) sinh(2s)cos ]x2 [cosh(2s) sinh(2s)cos ]y2
2sinh(2s)sin xy
17. Wigner representation for a pure squeezed coherent state
ˆ ˆ ˆ ˆ For ˆ , , D( )S 0 0 S D( )
Tr[ˆDˆ ( )] Dˆ ( )
Tr[ , , Dˆ ( )]
, Dˆ ( ) , ˆ ˆ ˆ ˆ ˆ 0 S D ( )D( )D( )S 0
Then the Wigner function is obtained as
1 W () d 2 exp( * * )Tr[ˆDˆ ( )] 2
1 d 2 exp( * * ) 0 Sˆ Dˆ ( )Dˆ ( )Dˆ ( )Sˆ 0 2 where
Dˆ ( ) exp(aˆ *aˆ)
Using the formula
* * Dˆ Dˆ exp( )Dˆ 2
We calculate
* * Dˆ Dˆ Dˆ exp( )Dˆ Dˆ 2 * * * * exp( )exp( )Dˆ 2 2 * * ˆ exp( )D
1 W () d 2 exp[ * ( ) ( * * )] 0 Sˆ Dˆ ( )Sˆ 0 2 where
ˆ ˆ ˆ ˆ ˆ * * S D( )S D(1) D( ) with
* * 1
Using the relation
1 2 0 Dˆ ( ) 0 0 exp( )exp( aˆ )exp( *aˆ) 0 1 2 1 1 1 1 2 exp( ) 0 exp( aˆ )exp( *aˆ) 0 2 1 1 1 1 2 exp( ) 2 1
Thus we have
1 1 2 W () d 2 exp[ * ( ) ( * * ) * * ] 2 2 or
2 2 W () exp[2( )* ( )* )
When x iy and x0 iy0
2 W (x, y) exp{2F(x, y)] where
2 2 F(x x x0 ,y y y0 ) (x y )cosh(2s) [(x y)(x y)cos 2xysin]sinh(2s)]
18. Wigner function for the pure number state We have the Wigner function for the density operator ˆ n n as
W () Tr[ˆ (2) (aˆ )] 1 d 2 exp( * * )Tr[ˆDˆ ( )] 2 1 d 2 exp( * * )Tr[ n n Dˆ ( )] 2 1 d 2 exp( * * ) n Dˆ ( ) n 2
Here we note that
1 2 n Dˆ ( ) n exp( ) n exp(aˆ )exp( *aˆ) n 2 2 n m 1 2 ( ) exp( ) n (aˆ )m aˆ m ) n 2 (m!)2 m0 2 n m 1 2 ( ) n exp( ) 2 m0 m! m
1 2 2 exp( )L ( ) 2 n
where Ln (x ) is a Laguerre polynomial. Thus we have
1 1 2 2 W () d 2 exp( * * )exp( )L ( ) 2 2 n
2 2 2 (1)n exp(2 )L (4 ) n where
2 x2 y2
n (x)k n Ln (x) k 0 k! k
19. Relations between the Wigner, Q and P representations (a) Relation between W and P Using the expression for ˆ
ˆ d 2P() we have
1 * * W () d 2P( ) d 2 e Tr[ Dˆ ( )] 2
1 * * d 2P( ) d 2 e Dˆ ( ) ] 2
1 1 2 d 2P( ) d 2 exp( )exp[ ( * *) * ( )]] 2 2 2 2 d 2P( )exp[2 ] where
1 2 Dˆ ( ) exp( ) exp(aˆ )exp( *aˆ) 2 1 2 exp( )exp( * * ) 2
(b) Relation between Q and P
1 Q() ˆ 1 2 d 2P( ) 1 2 d 2 P( )exp[ ] where
2 2 exp( * ) 2
2 2 2 2 2 exp( * )exp( * ) 2 2 exp[( 2 2 ) * * ] exp[ 2 ]
(c) Fourier transform
1 2 Q( ) d 2 P()exp[ ] 1 2 2 d 2 P()exp( )exp( * * ) 2 1 2 exp( ) d 2 P()exp( )exp( * * ) or
2 1 2 exp( )Q( ) d 2 P()exp( )exp( * * )
Thus exp( 2 )Q( ) is the Fourier transform of the function P()exp( 2 ) . On taking the Inverse Fourier transform, we have
2 1 2 P()exp( ) d 2 exp( )Q( )exp( * * ) or
1 2 2 P() exp( ) d 2 [exp( )Q( )]exp( * * )
20. Example of Wigner representation for the density operator with a pure number state. Wigner function for the pure number state n (n = 1, 2, 3, …).
2 2 2 W () (1)n exp(2 )L (4 ) n
2 2 2 where x y , W () Wn (x, y )
Fig. Wigner function for the density operator with the pure state n (n = 0, 1, 2, 3, 4).
21. Example of Wigner representation for the density operator with a coherent state The Wigner function for the density operator with a coherent state is given by
2 W ( ) exp{2[(x x )2 (y y )2 ]}, 0 0
where x iy and x0 iy0 .
Fig. Plot of W (x, y ) with the density operator (a pure coherent state) with a fixed point
(x0 , y0 ) .
Fig. Plot of W (x, y) with x r cos and y r sin , where n (n = 0, 1, 2, 3, 4, 5). 0 0 3
22. Example of Wigner representation for the density operator with a squeezed coherent state
The wigner function for the density operator with a squeezed coherent state is given by
2 2 W () exp[2( )* ( )* )
where x iy and x0 iy0 .
2 W (x x x ,y y y ) exp{2F(x,y)] 0 0 where
2 2 2 2 F(x x x0 ,y y y0 ) (x y )cosh(2s) [(x y )cos 2xysin]sinh(2s)] where cosh(s ) and ei sinh(s ) .
Fig. Plot of W (x, y ) for the density operator (a pure squeezed coherent state) with fixed values of s and .
REFERENCES M. Suda, Quantum Interferometry in Phase Space, Theory and Applications (Springer, 2006). C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge, 2006) W.P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001). Z. Ficek and M.R. Wahiddin (Pan Stanford Publishing, 2014). U. Leonhardt, Essential Quantum Optics (Cambridge, 2010). S. Haroche and J.M. Raimond, Exploring the Quantum Atoms, Cavities, and Photons (Oxford, 2006).
______APPENDIX I Formula The formula which are used in this chapter are listed below.
(a)
1 d 2 1
(b)
1 2 1 exp[ exp[ ( * * )] 2 2 1 2 1 2 exp[ *] 2 2
(c)
1 2 2 d 2 exp( * * )exp( ) 2 exp(2 ) 2
APPENDIX II 2D Dirac delta function
1 (2) () d 2 exp( * * ) 2
1 (2) (aˆ ) d 2 exp( * *)Dˆ ( ) 2 where Dˆ ( ) is the displacement operator.
APPENDIX III 2D Fourier Transform The 2D Fourier transform is defined as
1 * * F( ) d 2 f ()e (Fourier Transform)
The Inverse Fourier transform is given by
1 * * f () d 2 F( )e( ) (Inverse Fourier transform)
((Proof))
1 * * 1 * * 1 * * d 2 F( )e( ) d 2 e( ) d 2 f ( )e
1 * * * d 2 f ( ) d 2 e ( ) ( ) 2 d 2 f ( ) (2) ( ) f ()
APPENDIX IV Formula related to displacement operator ˆ D is called the displacement operator,
ˆ * D exp(aˆ aˆ) .
ˆ ˆ * D D exp(aˆ aˆ)
ˆ ˆ ˆ D aˆD aˆ 1
ˆ ˆ ˆ D aˆD aˆ 1
ˆ ˆ *ˆ D aˆ D aˆ 1
ˆ ˆ ˆ D D 1. (Unitary operator).
1 2 exp(aˆ )exp( *aˆ) exp(aˆ *aˆ ) 2 1 2 exp( )exp(aˆ *aˆ) 2
1 2 Dˆ exp(aˆ *aˆ) exp( )exp(aˆ )exp( *aˆ) 2
ˆ ˆ 2 ˆ ˆ 2 D aˆ aˆ D aˆ aˆ 1 2aˆ (aˆ 1)
ˆ ˆ * 2 ˆ * * ˆ 2 D aˆ aˆ D aˆ aˆ ( ) 1 2 aˆ (aˆ 1)
ˆ ˆ ˆ * ˆ D f (aˆ ,aˆ )D f (aˆ 1, aˆ 1)
1 2 Dˆ exp(aˆ *aˆ) exp( )exp(aˆ )exp( *aˆ) 2
1 2 Dˆ exp(aˆ *aˆ) exp( )exp(aˆ )exp( *aˆ) 2
ˆ * D exp[( )aˆ ( ) aˆ)
1 2 exp( )exp(aˆ )exp(aˆ )exp( *aˆ)exp( *aˆ) 2
1 2 1 2 Dˆ Dˆ exp( )exp( )exp(aˆ )exp( *aˆ)exp(aˆ )exp( *aˆ) . 2 2
ˆ ˆ * * ˆ ˆ D D exp( )D D
* * Dˆ Dˆ exp( )Dˆ 2
1 exp( *aˆ)exp(aˆ ) exp( * )exp( *aˆ aˆ ) 2
1 exp(aˆ )exp( *aˆ) exp(aˆ *aˆ)exp( *[aˆ,aˆ ]) 2 1 exp( * )exp( *aˆ aˆ ) 2
exp( *aˆ)exp(aˆ ) exp( * )exp(aˆ )exp( *aˆ) .
i(xˆ pˆ) APPENDIX V Equivalence between exp[ ] and displacement operator Using the expressions of aˆ and aˆ , we have
i(xˆ pˆ) exp[ ] exp(aˆ *aˆ) with
m 0
ipˆ aˆ (xˆ ) , 2 m0
ipˆ aˆ (xˆ ) 2 m0
1 xˆ aˆ aˆ aˆ aˆ 2 2m0
1 m 1 m pˆ 0 aˆ aˆ 0 aˆ aˆ 2 i i 2
i(xˆ pˆ) The operator can be rewritten in terms of { aˆ , aˆ },
i(xˆ pˆ) i m aˆ aˆ 0 aˆ aˆ 2m0 2 i m i m [ 0 ]aˆ [ 0 ]aˆ 2m0 2 2m0 2 aˆ *aˆ where is a complex number,
i m 1 i 0 ( ) 2m0 2 2
i m 1 i * 0 ( ) 2m0 2 2
Thus we have
i(xˆ pˆ) exp[ ] is the displacement operator,
1 2 Dˆ exp(aˆ *aˆ) exp( )exp(aˆ )exp( *aˆ) 2