Post on 08-Aug-2020
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 2 Pd
where )(P is called the Glauber-Sudarshan P function (representation) and is an coherent
state. We note that
1
)(
)(
][)(
)(]ˆ[
2
2
2
2
Pd
Pd
TrPd
PdTrTr
or
1
)(
)exp()exp()(1
]))Re[2exp()exp()(1
)exp()(1
)(1
)(1
)(]ˆ[
2
222
*2222
**2222
222
22
2
Pd
Pd
dPd
dPd
dPd
dPd
PdTrTr
where
11 2
d (formula)
]}Re[2exp{ *222
Note that and are complex numbers,
iyx , iba
Since
byax
iyxiba
)])(Re[(]Re[ *
we get the integral as
)exp(
)exp(
)2exp()2exp(]))Re[2exp(
2
22
22*22
ba
byydydxaxxdxd
We also note that the density operator can be rewritten as
ˆ1
ˆ1
ˆ
222
222
dd
dd
using the formula
11 2
d
((Example)) The average number of photons can be written as
22
2
2
)(
ˆˆ)(
]ˆˆ)([
]ˆˆˆ[
ˆˆˆ
Pd
aaPd
aaPdTr
aaTr
aan
So )(P is normalized as a classical probability distribution.
((Note)) Here we note that
]exp[)(
)(
)(ˆ
22
22
2
Pd
Pd
Pd
since
]exp[]}Re[2exp{2*222
Since ]exp[2 is not a delta function, the diagonal elements
)(ˆ P
Although ˆ must be positive. However, the integral ]exp[)(22 Pd 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 .
)exp(1
)exp(1
)]"'(exp[)2(
1
)"()'()(
**22
**22
2
)2(
d
d
yxidydx
or
)exp(1
)( **22
)2(
d
where
)"'()Im(2 *** yxii
and
)(2
1
22ixy
xi
y
We also note that the 2D Dirac delta function is given by
])()exp[(1
)( ***22
)2(
d
3. Expression of )(P
How can we find the expression of )(P ? We consider the density operator which is defined
by
)(ˆ 2 Pd
Using the formula for the scalar product for the coherent states,
]2
1
2
1exp[ *22
we get the matrix element of the density operator as
]exp[]exp[)(]exp[
)(ˆ
**222
2
uuPdu
uuPduu
or
]exp[]exp[)(ˆ]exp[ **222uuPduuu
Here we use the 2D Dirac delta function.
]exp[)(
)(]exp[)(
])()exp[(]exp[)(
]exp[]exp[]exp[)( ]exp[ˆ]exp[
22
222
**222
****222**22
P
Pd
uuudPd
uuuuudPduuuuuud
or
udeuueeP uuu 2)(2
**22
ˆ1
)(
4. Another expression of )(P ))
The expression of )(P can be also obtained as follows.
**
*
*
*
)(
)(
])( [
]ˆ[
2
ˆˆ2
ˆˆ2
ˆˆˆˆ
ePd
eePd
eePdTr
eeTree
aa
aa
aaaa
Here we use the expression of the 2D Dirac function as
])()exp[(1
)( ***22
)2(
d
Thus we get
)(
)()(
1)(
))()( (11
)2(2
)()(22
2
222
ˆˆ22
***
*******
P
Pd
edPd
eePddeeed aa
or
]ˆ[1
1)(
)ˆ()ˆ(22
ˆˆ22
**
***
aa
aa
eeTrd
eeedP
Here we note that Tr under the integral is characteristic function of the normally ordered a , a operators,
]ˆ[)( ˆˆˆˆ ** aaaaN eeTree
Therefore we have
**
***
)( 1
1)(
22
ˆˆ22
ed
eeedP
N
aa
This relation represents a mapping from the density operator , which is a function of the two
operators a and a , 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
)exp(
]2
1
2
1exp[]
2
1
2
1exp[
ˆ
**22
*22*22
uuu
uuuu
uuuu
where
]2
1
2
1exp[ *22
uuu
]2
1
2
1exp[ *22 uuu
Then we get
)(
)(
1
1
1
ˆ1
)(
)2(
)2(
2)()((2
2)()(2
2)()(
2
2)(2
22
***22
****22
****2222
**22
e
udee
udeee
udeeee
udeuueeP
uu
uuuu
uuuuuu
uuu
or
)()( )2( P 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
**
*
*
*
ˆˆ
ˆˆ
ˆˆ
][
]ˆ[)(
e
ee
eeTr
eeTr
aa
aa
aaN
**
***
)( 1
1)(
22
ˆˆ22
ed
eeedP
N
aa
)(
1
1)(
)2(
)()(22
22
***
****
ed
eedP
7. P function for the number state n (I)
We consider the density operator for the pure coherent state n ;
nn .
We note that
!
)(ˆ
*2
n
uueunnuuu
nu
Thus we get
udeuun
e
uden
uueee
udeuueeP
uun
uun
uu
uuu
2)(*2
2)(*
2
2)(2
**
2
**222
**22
)(1
!
!
)(1
ˆ1
)(
Formally, we can write
)(!
1
!)(
*
2
2)(2*
2
2
**
2
nn
n
uun
n
n
e
uden
eP
8. P function for the number state n (II)
We consider the density operator for the pure number state n ;
nn .
Thus we get
neen
eennTr
eeTr
aa
aa
aaN
ˆˆ
ˆˆ
ˆˆ
*
*
*
][
]ˆ[)(
2
)2(2
)()(22
2
ˆˆ23
ˆˆ22
22
1
)(1
11
1
1
)( 1
)(
***
***
***
**
n
nnd
ednnd
eneend
eneend
edP
aa
aa
N
or
2
!
1)(
22
en
nPn
which is a Poisson function. where
neenne aa ** ˆ*ˆ
nene a ** ˆ
9. Q-representation
There are other orderings possible for the two operators a and a . This time we use the antisymmetric ordering and define the antinormally ordered characteristic function
]ˆ[)( ˆˆ* aaA eeTr
Its Fourier transform is called Q representation. This representation is sometimes called the Husimi function.
**
)( 1
)( 22
edQ A
and has a simple form
0ˆ1
)(
Q
Thus )(Q is the probability of finding a system in the density operator in the coherent state
. This proof is given below.
aa
aaA
eeTrd
eeTr
ˆˆ2
ˆˆ
*
*
ˆ[ 1
]ˆ[)(
ˆ1
)(ˆ[ 1
1
]ˆ[ 1
]ˆ[ 1
1
]ˆ[ 1
1
)(
)2(2
)()(22
2
ˆ222
ˆˆ222
***
****
***
d
edTrd
eeTrded
eeTrdedQ
a
aa
((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
]exp[11
)( **222
Q
since
]2
1
2
1exp[ *22
11. Q representation for the pure number state
For the density operator
nn
we have
)exp(!
1)(
22
2
n
nQn
which is the Poisson function 12. Wigner representation
For symmetric order of two operators a and a . The characteristic function is defined as
)(ˆ)](ˆˆ[)]ˆˆexp(ˆ[)( * DDTraaTrS
Its Fourier transform is called Wigner representation.
)](ˆˆ[ 1
)( 1
)(
**
**
22
22
DTred
edW S
13. Another expression of Wigner function
)ˆˆ(2
1)ˆˆ(
2
1
2ˆ
2ˆ aaaaxX
,
)ˆˆ(2
1)ˆˆ(
22
1ˆ
2
1ˆ 0 aai
aai
mpY
or
YiXa ˆˆˆ , YiXa ˆˆˆ . Note that
12
1]ˆ,ˆ[
2
1]ˆ,ˆ[
2
1
2]ˆ
2
1,ˆ
2[]ˆ,ˆ[ ipxpxpxYX
,
)ˆ'2exp()ˆ"2exp()"'exp(
]ˆ'2ˆ"2exp[
]ˆ)(ˆ)exp[(
)]ˆˆ()ˆˆ(exp[
)ˆˆexp()(ˆ
**
*
*
YiXii
YiXi
YiX
YiXYiX
aaD
This operator can be rewritten as
)ˆ'2
exp()ˆ"2exp()"'exp(
)ˆ2
1'2exp()ˆ
2"2exp()"'exp()(ˆ
pixii
pixiiD
Then we get
'2')'"2exp()"'exp(
'2')
'2'("2exp()"'exp(
'2')ˆ"2exp()"'exp(
')ˆ'2
exp()ˆ"2exp()"'exp(')(ˆ
xxii
xxii
xxii
xp
ixiixD
Wigner function
'2'ˆ')'"2"'exp('
1
')(ˆˆ'' 1
)](ˆˆ[ 1
)( 1
)(
**
**
**
**
22
22
22
22
xxxiidxed
xDxdxed
DTred
edW S
Note that
'"2"'2** ii
")'2''2("'2
'"2"''"2"'2'"2"'**
xii
xiiiixii
We use the Dirac delta function as
)'2''2(2 ]")'2''2(exp["d xxi
Then we have
)''22
ˆ')]'2
'("4exp['2
)'2'2(2
'ˆ')]'2'2("2exp['2
'2'ˆ')'2''2()"'2exp(''
2)(
xxxidx
xxxxidx
xxxiddxW
Here we define
2'
uxx ,
2''
22 uxx
x'2
, p"2
dudx2
1'
pu
u
xu
x
x
xx
2
"2
2
)2
("2
4
)'2
'("2
4
)'2
'("4)'2
'("4
2ˆ
2
1),()( 2 u
xu
xedupxWW ipu
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
),( 2
ˆ2
2 pxWedpu
xu
x ipu
13. Displacement operator
With help of the 2D Dirac delta function, we introduce a new notation for the quantum states in phase space,
)(ˆ)exp(1
)ˆˆexp()exp(1
])ˆ()ˆexp[(1
)ˆ(
**22
***22
**22
)2(
Dd
aad
aada
where )(ˆ D is the displacement operator,
)ˆexp()ˆexp()2
1exp(
)ˆexp()ˆexp()2
1exp(
)ˆˆexp()(ˆ
*2
*2
*
aa
aa
aaD
We note that
)(
)](ˆˆ[)exp(1
)]ˆ(ˆ[)ˆ(
**22
)2()2(
W
DTrd
aTra
15. Wigner representation for a pure coherent state
For ˆ
)exp()2
1exp(
)(ˆ
)](ˆ[
)(ˆ)](ˆˆ[
**2
D
DTr
DDTr
where
)exp()2
1exp(
)ˆexp()ˆexp()2
1exp()(ˆ
**2
*2
aaD
Thus we get the Wigner representation
]})()[(2exp{2
]2exp[2
)]()(exp[)2
1exp(
1
)exp()2
1exp()exp(
1
)](ˆˆ[)exp(1
)(
20
20
2
***22
2
**2**22
**22
yyxx
d
d
DTrdW
or
]})()[(2exp{2
)( 20
20 yyxxW
We use '' iyx , ''2 dydxd , )( 00 yyixx (x, y, p, p0, q and q0 are real). Using
the Mathematica, we get
)])(2)(2exp[2
]')(2')(2)''(2
1exp['')]()(exp[)
2
1exp(
20
20
0022***22
yyxx
yxxixyyiyxdydxd
16. Wigner representation for a pure squeezed state
For ˆ
0ˆ)(ˆˆ0
)(ˆ
)](ˆ[
)(ˆ)](ˆˆ[
SDS
D
DTr
DDTr
Then the Wigner function is obtained as
0ˆ)(ˆˆ0)exp(1
)](ˆˆ[)exp(1
)(
**22
**22
SDSd
DTrdW
where
)ˆˆexp()(ˆ *aaD
We use the Bogoliubov transformation
aabSaS ˆˆˆˆˆˆ , aabSaS ˆˆˆˆˆˆ **
where
])ˆ(2
1ˆ
2
1exp[ˆ 22* aaS ,
SaaS ˆ])ˆ(2
1ˆ
2
1exp[ˆ 2*2
122 , ise
scosh , sei sinh
00ˆ b
Thus we have
aa
aaaa
bbSaaS
ˆ)(ˆ)(
)ˆˆ()ˆˆ(
ˆˆˆ)ˆˆ(ˆ
****
***
**
Here we switch the variable as , leading to the change of parameters ( and
SS ˆˆ ).
aaSaaS ˆ)(ˆ)(ˆ)ˆˆ(ˆ *****
or
aa
aaSaaS
ˆˆ
ˆ)(ˆ)(ˆ)ˆˆ(ˆ
*11
*****
where **1
By using the relation 1ˆˆ SS we get
2*11
**2*
)ˆˆ(
ˆ)ˆˆ(ˆˆ)ˆˆ(ˆˆ)ˆˆ(ˆ
aa
SaaSSaaSSaaS
which leads to the relation
)(ˆ)(ˆˆ)(ˆˆ **1 DDSDS
Using the relation
)2
1exp(
0)ˆexp()ˆexp(0)2
1exp(
0)ˆexp()ˆexp()2
1exp(00)(ˆ0
2
1
*11
2
1
*11
2
11
aa
aaD
we get
)2
1exp(
1
)2
1exp()exp(
1
)2
1exp()exp(
1
0)(ˆ0)exp(1
)(
2****22
2****22
2
1**2
2
1**2
2
d
d
d
DdW
where 122 . This integral can be calculated by using the Mathematica with
''2 dydxd and '' iyx .
)2exp[2
)(2**
W
When iyx
)],(2exp{2
),( yxFyxW
where
)]2sinh(]sin2cos))([()2cosh()(),( 22 sxyyxyxsyxyxF
When 0
ss eyex
syxyxsyxyxF2222
22 )]2sinh()])(()2cosh()(),(
((Note))
yexe
yexe
y
x
e
e
y
x
ss
ss
s
s
2cos
2sin
2sin
2cos
2cos
2sin
2sin
2cos
0
0
'
'
under the rotation and squeezed processes. Thus we have
xys
yssxssyx
sin)2sinh(2
]cos)2sinh()2[cosh(]cos)2sinh()2[cosh('' 2222
17. Wigner representation for a pure squeezed coherent state
For )(ˆˆ00ˆ)(ˆ,,ˆ DSSD
0ˆ)(ˆ)(ˆ)(ˆˆ0
,)(ˆ,
)](ˆ,,[
)(ˆ)](ˆˆ[
SDDDS
D
DTr
DDTr
Then the Wigner function is obtained as
0ˆ)(ˆ)(ˆ)(ˆˆ0)exp(1
)](ˆˆ[)exp(1
)(
**22
**22
SDDDSd
DTrdW
where
)ˆˆexp()(ˆ *aaD
Using the formula
DDD ˆ)2
exp(ˆˆ**
We calculate
D
D
DDDDD
ˆ)exp(
ˆ)2
exp()2
exp(
ˆˆ)2
exp(ˆˆˆ
**
****
**
0ˆ)(ˆˆ0)]()(exp[1
)( ***22
SDSdW
where
)(ˆ)(ˆˆ)(ˆˆ **1 DDSDS
with
**1
Using the relation
)2
1exp(
0)ˆexp()ˆexp(0)2
1exp(
0)ˆexp()ˆexp()2
1exp(00)(ˆ0
2
1
*11
2
1
*11
2
11
aa
aaD
Thus we have
]2
1)()(exp[
1)(
2*****22
dW
or
))()(2exp[2
)(2**
W
When iyx and 00 iyx
)],(2exp{2
),( yxFyxW
where
)]2sinh(]sin2cos))([()2cosh()(),( 2200 syxyxyxsyxyyyxxxF
18. Wigner function for the pure number state
We have the Wigner function for the density operator nn as
nDnd
DnnTrd
DTrd
aTrW
)(ˆ)exp(1
)](ˆ[)exp(1
)](ˆˆ[)exp(1
)]ˆ(ˆ[)(
**22
**22
**22
)2(
Here we note that
)()2
1exp(
!
)()
2
1exp(
)ˆ)ˆ()!(
)()
2
1exp(
)ˆexp()ˆexp()2
1exp()(ˆ
22
0
22
02
22
*2
n
n
m
m
n
m
mmm
L
m
n
m
naanm
naannDn
where )(xLn is a Laguerre polynomial. Thus we have
)4()2exp()1(2
)()2
1exp()exp(
1)(
22
22**22
nn
n
L
LdW
where
222yx
n
k
k
n k
n
k
xxL
0 !
)()(
19. Relations between the Wigner, Q and P representations (a) Relation between W and P
Using the expression for
)(ˆ 2 Pd
we have
]2exp[)(2
])]()(exp[)2
1(e
1)(
])(ˆ 1
)(
)](ˆ[ 1
)()(
22
***222
2
22
2
22
2
**
**
Pd
xpdPd
DedPd
DTredPdW
where
)exp()2
1exp(
)ˆexp()ˆexp()2
1exp()(ˆ
**2
*2
aaD
(b) Relation between Q and P
]exp[)( 1
)(1
ˆ1
)(
22
22
Pd
Pd
Q
where
)2
exp( *
22
]exp[
])(exp[
)*2
exp()2
exp(
2
**22
22
*
222
(c) Fourier transform
)exp()exp()( 1
)exp(
)exp()exp()( 1
]exp[)( 1
)(
**222
**222
22
Pd
Pd
PdQ
or
)exp()exp()( 1
)()exp( **222
PdQ
Thus )()exp(2 Q is the Fourier transform of the function )exp()(
2 P . On taking the
Inverse Fourier transform, we have
)exp()()exp( 1
)exp()( **222
QdP
or
)exp()]()exp([ )exp(1
)( **222
QdP
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, …).
)4()2exp()1(2
)(22
n
n LW
where 222yx , ),()( yxWW n
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
]})()[(2exp{2
)( 20
20 yyxxW
,
where iyx and 00 iyx .
Fig. Plot of ),( yxW with the density operator (a pure coherent state) with a fixed point
),( 00 yx .
Fig. Plot of ),( yxW with cos0 rx and sin0 ry , where n3
(n = 0, 1, 2, 3, 4, 5).
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
))()(2exp[2
)(2**
W
where iyx and 00 iyx .
)],(2exp{2
),( 00 yxFyyyxxxW
where
)]2sinh(]sin2cos)[()2cosh()(),( 222200 syxyxsyxyyyxxxF
where )cosh(s and )sinh(sei .
Fig. Plot of ),( yxW 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)
11 2
d
(b)
]2
1
2
1exp[
)](2
1exp[
2
1exp[
*22
**2
(c)
)2exp(2)2
1exp()exp(
22**2 d
APPENDIX II 2D Dirac delta function
)exp(1
)( **22
)2(
d
)(ˆ)exp(1
)ˆ( **22
)2(
Dda
where )(ˆ D is the displacement operator.
APPENDIX III 2D Fourier Transform
The 2D Fourier transform is defined as
**
)( 1
)( 2 efdF (Fourier Transform)
The Inverse Fourier transform is given by
)(2 **
)( 1
)(
eFdf (Inverse Fourier transform)
((Proof))
)(
)()(
)( 1
)( 1
1
)( 1
)2(2
)()(222
2)(2)(2
***
******
f
fd
edfd
efdedeFd
APPENDIX IV Formula related to displacement operator
D is called the displacement operator,
)ˆˆexp(ˆ *aaD .
)ˆˆexp(ˆˆ *aaDD
1ˆˆˆˆ aDaD
1ˆˆˆˆ aDaD
1ˆˆˆˆ * aDaD
1ˆˆ DD . (Unitary operator).
)ˆˆexp()2
1exp(
)2
1ˆˆexp()ˆexp()ˆexp(
*2
2**
aa
aaaa
)ˆexp()ˆexp()2
1exp()ˆˆexp(ˆ *2* aaaaD
ˆ D
ˆ a a ˆ D ˆ a a 2 ˆ 1 2 ˆ a ( ˆ a ˆ 1 )2
ˆ D
ˆ a ˆ a ˆ D ˆ a ˆ a ( *)2 ˆ 1 2 * ˆ a ( ˆ a * ˆ 1 )2
ˆ D
f ( ˆ a , ˆ a ) ˆ D f ( ˆ a ˆ 1 , ˆ a * ˆ 1 )
)ˆexp()ˆexp()2
1exp()ˆˆexp(ˆ *2* aaaaD
)ˆexp()ˆexp()2
1exp()ˆˆexp(ˆ *2* aaaaD
)ˆexp()ˆexp()ˆexp()ˆexp()2
1exp(
)ˆ)(ˆ)exp[(ˆ
**2
*
aaaa
aaD
)ˆexp()ˆexp()ˆexp()ˆexp()2
1exp()
2
1exp(ˆˆ **22
aaaaDD .
DDDD ˆˆ)exp(ˆˆ **
DDD ˆ)2
exp(ˆˆ**
)ˆˆexp()2
1exp()ˆexp()ˆexp( *** aaaa
)ˆˆexp()2
1exp(
])ˆ,ˆ[2
1exp()ˆˆexp()ˆexp()ˆexp(
**
***
aa
aaaaaa
)ˆexp()ˆexp()exp()ˆexp()ˆexp( *** aaaa .
APPENDIX V Equivalence between ])ˆˆ(
exp[
pxi and displacement operator
Using the expressions of a and a , we have
)ˆˆexp(])ˆˆ(
exp[ *aapxi
with
0 m
)ˆ
ˆ(2
ˆ0
m
pixa ,
)ˆ
ˆ(2
ˆ0
m
pixa
aam
aax ˆˆ2
ˆˆ2
1ˆ
0
aam
iaa
i
mp ˆˆ
2
1ˆˆ
2
1ˆ 00
The operator
)ˆˆ( pxi can be rewritten in terms of { a , a },
aa
am
m
ia
m
m
i
aam
aam
ipxi
ˆˆ
ˆ]22
[ˆ]22
[
ˆˆ2
ˆˆ2
)ˆˆ(
*
0
0
0
0
0
0
where is a complex number,
)(2
1
220
0
im
m
i
)(2
1
220
0
*
im
m
i
Thus we have
])ˆˆ(
exp[
pxi is the displacement operator,
)ˆexp()ˆexp()2
1exp()ˆˆexp(ˆ *2* aaaaD