Download Generalized Fresnel operators corresponding to optical Fresnel

SCIENCE CHINA
Physics, Mechanics & Astronomy
• Article •
August 2012 Vol.55 No.8: 1345–1350
doi: 10.1007/s11433-012-4749-y
Generalized Fresnel operators corresponding to optical Fresnel
diffraction and the squeezed states
LI HongQi1,2* & REN TingQi1
1
Department of Physics, Qufu Normal University, Qufu 273165, China;
2
Department of Physics, Heze University, Heze 274015, China
Received June 16, 2011; accepted March 8, 2012; published online May 2, 2012
Corresponding to optical Fresnel diffraction, we show that the exponential quadratic operator exp{i[P2+Q2+(PQ+QP)]/2}
is actually a generalized single-mode Fresnel operator (GFO) in compact form, where [Q,P]=iħ. We also demonstrate that
exp{i[(Q1+Q2)2+(P1P2)2]+i[(Q1Q2)2+(P1+P2)2]+i(Q1P2+Q2P1)} is a two-mode GFO. Their disentangling formula and
normal ordering form are derived with the use of technique of integration within an ordered product (IWOP) of operators and
the coherent state representation. The squeezed states generated by these two GFOs are obtained.
Fresnel diffraction, single-mode Fresnel operator, two-mode Fresnel operator, IWOP technique
PACS number(s): 03.65.-w, 42.50.Dv, 42.25.Fx
Citation:
Li H Q, Ren T Q. Generalized Fresnel operators corresponding to optical Fresnel diffraction and the squeezed states. Sci China-Phys Mech Astron,
2012, 55: 13451350, doi: 10.1007/s11433-012-4749-y
1 Introduction
Fresnel diffraction is a core of classical optics. Fan and Hu
[1] following Dirac’s assertion [2]: “  for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact
transformation in the classical theory”, suggested the appropriate Fresnel operator (FO) to correspond to the optical
Fresnel transform (FT):

g  q2    K  A B C  D q2  q1  f  q1  dq1 

(1)

i  2
2 

exp
 Aq2  2 q2 q1  Dq1   


2B
2π i B

F  r s 

 exp  




r 2 
1 1 
 r

a  exp  a  a   ln   exp   a 2  




2
2s
s
2
s






(3)
where  a a    1 the fundamental bosonic commutator 
s,r are related to A, D, B and C by
K  A B C  D q2  q1 





respectively. The so-called single-mode FO is [1]


where the integration kernel is
1
A B
and 
 denotes parameters characterizing an optical
C D
process (or an optical instrument) obeying AD  BC  1,
f  q1  and g  q2  are input and output of optical signals,
s
(2)
1
1
 A  D  i  B  C    r    A  D  i  B  C    (4)
2
2
which results in  s 2   r 2  1 due to AD  BC  1 Fan
and Hu [1] have demonstrated that the FT kernel K in eq.
*Corresponding author (email: [email protected])
© Science China Press and Springer-Verlag Berlin Heidelberg 2012
phys.scichina.com
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1346
Li H Q, et al.
Sci China-Phys Mech Astron
(2) is just the transformation matrix element of FO in the
coordinate representation,
K  A B C  D q2  q1   q2 F q1 
where qi
(5)
is the Fock representation of coordinate eigen-
state
 1
a 2 
qi  π 1 4 exp   qi2  2 qi ai  i  0  i  1 2
2 
 2
(6)
FT has been widely implemented in Fourier optics: optical
imaging, optical propagation, optical engineering and optical instrument design. Thus, investigating FT in the context
of quantum optics is really worthwhile. An important question thus arises: what is the compact form of general FO
other than the decomposed form in eq. (3)? In this work we
want to show that the general exponential quadratic unitary
operator (in compact form)
 i

U  exp   P 2   Q 2    PQ  QP   
 2


2


2
2



(8)
is a two-mode GFO in compact form. We shall derive their
disentangling formula and normal ordering form with the
use of technique of integration within an ordered product
(IWOP) of operators [3–5] and the coherent state representation [6,7]. The squeezed states generated by these two
operators are thus obtained.
The IWOP technique is useful in constructing quantum
mechanical representations and unitary transformations, in
deriving new operator identities and new integration formulas [8–10]. Recently, researchers have deduced the
s-ordered operator expansion formula of density operator
[11].
2 The normal ordered expansion of U
In order to identify U as a GFO, let us recall the FO’s coherent state representation [1]
 s
2
d z
sz  rz  z
π
d 2 z  s r   z
π



r


  

s
(10)
is the coherent state [6,7]. Eq. (9) indicates that the
c-number transform z  sz  rz  in coherent state basis
maps into the operator F. In fact, using the normal ordering
of the vacuum projector 0 0  exp a† a  and the


IWOP technique we can perform the integral in eq. (9) and
obtain
d2 z
2
2
: exp   s z  sza   z   a  ra  

π

r  s 2 rs 2
z 
z  a a :

2
2

F  r s   s 

 exp  






r 2 
1 1 
 r

a  exp  a  a   ln   exp   a 2  .




2
s
s
2s
2






(11)
q
z     exp  i  pQ  qP   0  z   q  i p   2  (12)
 p
F  r s  
A  D  i( B  C )
2
dqd p

2π
z

















z
z
,
(9)
 A B   q) 

 
C D p 
q
 
 p
 F  A B C  ,
(13)
and eq. (11) becomes
F  A B C  
2
A  D  iB  C
 A  D  i  B  C 
a 2
:exp 


A
D
B
C



2
i



 


2

 1 a  a
 A  D  i  B  C  

F  r s   s 



1 2
 z 
z  za†  0    

z 
2



and eq. (4) we can rewrite eq. (11) as:
2


 i   Q1  Q2    P1  P2    i   Q1 P2  Q2 P1  ,




z  exp  
Alternately, using the canonical coherent state form [6]
state. Then we shall show that

where
(7)
is actually a generalized single-mode Fresnel operator
(GFO), where ,  and  are real, and U 0 is a squeezed
U 2  exp i    Q1  Q2    P1  P2 
August (2012) Vol. 55 No. 8
A  D  iB  C

a2  : 
2  A  D  i  B  C   
(14)
From eq. (11) we see that F engenders the transformation
Q
 A B Q
F 1   F  
 
P
 
C D P 
a  a
a  a
P

Q
2
2i
(15)
Li H Q, et al.
Sci China-Phys Mech Astron
Let us now examine U in eq. (7), using the Baker-Hausdorff
formula
e A Be  A  B   A B  
1
 A  A B 
2! 
1
  A  A  A B    .
3!
the Fresnel transformation under the GFO, for example,
from eqs. (21) and (22) we know that GFT of the vacuum
state is
(16)

 2  i      sinh 

a 2  0 ,
 4 cosh   i 2     sinh 





 exp 
i
1
1
U QU  Q   P 2    PQ  QP   Q     
2
2!
3!
 Q  P   Q
1
1 i
 1
 P 2   Q 2    PQ  QP     P   Q   
2!  2 
 3!
 Q cosh  
2
2 cosh   i     sinh 
U 0 
Then we can derive

1347
August (2012) Vol. 55 No. 8

 
P  Q  sinh  


 
(17)

which is just a special squeezed state.
3
1

 P   Q  sinh  
(18)
Disentangling of U
Due to the decomposition
 A B   1 0   A 0   1



 
1  
 C D   C A 1   0 A   0
and
U 1 PU  P cosh  
U A
where
(19)

Q
 A B Q
U 1   U  
 
P
C D P 
 sinh 

cosh  
A B 



C
D


sinh

  





(20)
 sinh 




 sinh  
cosh  

 
B

D
 U
U A
1 0


 C A 1 


0

U1



0
A1 

0
B A 

1 
,
(24)

i C 2 
i B 2 
 i

Q  exp    PQ  QP  ln A exp  
P 



2A 
 2

 2A





 i  sinh 
Q 2 
 2   cosh    sinh  



 exp 
 i
 sinh   

 exp    PQ  QP  ln  cosh  
  
2






 i  sinh 
P 2  .
 2   cosh    sinh  



 exp 
with its determinant being 1 and comparing eq. (20) with
eq. (11) we can identify U as a generalized FO, so
*




r
1 1 
 r

U  exp  * a 2  exp  a  a   ln *  exp  * a 2  





2
s
s
2s
2











(21)
where
1
 A  D  i  B  C  
2
 
 cosh   i
sinh  ,
2
1
r     A  D   i  B  C  
2

    
  i
 sinh  .
2 
 
(23)
which means the disentangling
U  exp 
Eqs. (17) and (18) in a compact form
B A 
,
1 
so

C
   2   

(25)
We immediately know the Fresnel transformation kernel
 q2 U q1 
1
2π i B


i  2
2 
(26)
 Aq2  2 q2 q1  Dq1   

 2B 


exp 
4 The two-mode generalized FO
Similar in spirit to the single-mode case eq. (9), we introduce the two-mode FO F2  r  s  through the following
s 
2-mode coherent state representation [12]
(22)
The advantage of eq. (21) lies in that we can directly know
F2  r  s   s 
d 2 z1 d 2 z2
sz1  rz2  rz1  sz2 z1  z2  (27)
π2
which indicates that F2  r  s  is a mapping of classical
symplectic
transform
phase space, where
 z1  z2    sz1  rz2  rz1  sz2 
in
1348
Li H Q, et al.

z1  z2  exp  


Sci China-Phys Mech Astron
August (2012) Vol. 55 No. 8
so

1 2 1 2
z1  z2  z1 a1†  z2 a2†  00

2
2

F2 Q1 F21 
is a usual two-mode coherent state. Concretely, the ket in eq.
(27) is
sz1  rz2  rz1  sz2  sz1  rz2  rz1  sz2 
1
and
s
2
still
r
satisfy
the
unimodularity
  A  D  Q2   B  C  P2  
(28)
2
F2 Q2 F21 
condition
s  r 1.
1
2
F2 P1 F
Using the IWOP technique and the normal ordering of
the vacuum projector 00 00 : exp  a1 a1  a2 a2  : we
perform the integral in eq. (28) and obtain


2 2

1 1

2 2









2 2  



1
r
1

exp   a1 a2   : exp    1   a1 a1  a a



s
s

s



 exp  







2 




:
F

r


a
a
 1 2

s




 exp  




r

a a 
 1 2

s

Thus F2  r  s  induces the transform
F2  r  s  a1 F21  r  s   s a1  ra2  F2  r  s  a2 F21  r  s 
(30)
It follows
F2  Q1  Q2  F21  D  Q1  Q2   B  P1  P2  
F2  P1  P2  F21  B  Q1  Q2   D  P1  P2  
F2  Q1  Q2  F21  A  Q1  Q2   C  P1  P2  
F2  P1  P2  F
1
2
(31)
 A  P1  P2   C  Q1  Q2  
a j  a j
2
 Pj 
a j  a j
2i
 j  1 2
F
Q1  Q2 


2 


2 


2 
P1  P
Q1 Q
P1  P
F21
 D

C

 0

 0
B
A
0
0
0   Q1  Q2 


0 0   P1  P2 


A C   Q1 Q2 


B D   P1  P2 
P
Q
F21
P
 A D C  B

B C A D

 A D B C

 BC D A
A D
B  C   Q1 
 
B  C D  A   P1 

  (35)
A  D C  B   Q2 
 
B  C A  D   P2 
A  D  i( B  C )
2
 A  D C  B A  D B  C   q1 

 
2
d qi d pi  B  C A  D B  C D  A   p1 


 
 A  D B  C A  D C  B   q2 
2π
i 1

 
 B  C D  A B  C A  D   p2 











q1 

p1 
  F2  A, B, C  .
q2 

p2 
(36)
Similar to eqs. (24) and (25), F2  A B C  has its canonical
F2  A B C   F2 1 0 C A F2  A 0 0  F2 1 B A 0 
(32)
2
2
 i C 
 exp 
Q1  Q2    P1  P2   

A
4


 exp  i  Q1 P2  Q2 P1  ln A
or





2 






1 


2


2 
operator  Q j  Pj  representation
where
Qj 
Q1 
F2  r  s  
(29)
 s a2  ra1 
1
 A  D  P2   B  C  Q2
2
  A  D  P1   B  C  Q1  ,
It then follows the canonical coherent state form

1
r
 exp  a1 a2  exp  a1 a1  a2 a2  1 ln  s   




s






(34)
and can be re-written in a compact form as:
 z a  z a  a a  a a  :

1 1
1
  A  D  P1   B  C  Q1
2
  A  D  P2   B  C  Q2  
F2 P2 F21 
2
d z1 d z2
:exp    s 2   z1 2   z2 2 
π2
 r  sz1 z2  rs  z1 z2   sz1  rz2  a1   rz1  sz2  a2
F2  r  s   s 
1
 A  D  Q2   B  C  P2
2
  A  D  Q1   B  C  P1  
2
2
1
 A  D  Q1   B  C  P1
2
2
2
 i B 
 exp 
Q1  Q2    P1  P2     (37)

 4A

0
(33)
Now we try to derive the disentangling expansion of the
exponential operator U 2 in eq. (8). Noting
Li H Q, et al.
Sci China-Phys Mech Astron
U 2  Q1  Q2  U 21   Q1  Q2  cosh 
  Q1  Q2  2   P1  P2  2  Q1  Q2  2   P1  P2  2 



4
4


  i  PQ
1 2  Q1 P2  






Q1  Q2    P1  P2 
2
2
4






 Q1  Q2    P1  P2 
2
2
4
2
2







i C 
2
2 
Q  Q    P  P   
1
2
1
2
 4 A 
 



U 2  exp 
 exp i  Q1 P2  Q2 P1  ln A



i B 
2
2 
 Q  Q    P  P    
 exp 
1
2
1
2 



4 A



Q1 P2  Q2 P1   Q1  Q2    i  Q1  Q2  



 P1  P2   Q1  Q2    4 i  P1  P2  


Q1 P2  Q2 P1    P1  P2    i  P2  P1  



Q1  Q2    P1  P 
 Q1  Q2  


 P1  P2   sinh  
4

(43)
4 sinh 



  (44)
 sinh  
cosh  




(39)

i  sinh 

2
2
 Q  Q    P  P  
U 2  exp 
1
2
1
2 



  cosh    sinh 


 sinh   

 exp  i  Q1 P2  Q2 P1  ln  cosh  






i  sinh 

2
2
Q  Q    P  P   ,
 exp 
1
2
1
2 



  cosh    sinh 

(45)


2 

or normally ordered expansion
r    
a1 a2  exp[ a1 a1  a2 a2  1
 s *



(40)
 4 i  Q1  Q2  

Q1 P2  Q2 P1    P1  P2    i  P1  P2  

Q1 P2  Q2 P1   Q1  Q2     i Q1  Q2  ,

1



r *

a a  00 ,
* 1 2

s

(46)
1

 A  D   i  B   C    
2
1
r     A  D  i  B   C    
2
s 
  i   Q1 P2  Q2 P1   i   P1  P2    Q1  Q2    
2

  Q1  Q2   4  P1  P2     Q1  Q2   


  Q1  Q2  cosh 
(47)
Then the two-mode squeezed state generated by U 2 is


Q1  Q2   sinh  
4

(41)
and
U 2  P1  P2  U 21   P1  P2  cosh 


where
U 2  Q1  Q2  U 21   Q1  Q2 
 P1  P2  


 ln  s *  ]exp  
and using the Baker-Hausdorff formula we can derive


U 2 00  exp 










2





4
 sinh 

cosh  
A B   

 
C  D   
4  sinh 



which can be proved by using the bipartite entangled state
representation. To identify A B  C  we notice
2
(42)
According to eq. (40) we have the disentangling of U 2 ,







4


 P1  P2   sinh  
4

Then comparing eqs. (42) and (43) with eq. (34) we can
determine
we have

 P1  P2  
   2  16 


i

  PQ
1 2  Q1 P2  

2

4






where

Q1  Q2    P1  P2 


4
  P1  P2  cosh 
(38)
2
4
Q1  Q2    P1  P2 
U 2  P1  P2  U
1
2


2




i

  PQ
1 2  Q1 P2  

2

1349
August (2012) Vol. 55 No. 8
4






 Q1  Q2  

 P1  P  sinh  
4


2 



  2 i      sinh 
U 2 00  exp  
a1 a2  00 .
 cosh   2 i     sinh 


(48)
In summary, for the first time we pointed out that the
general exponential quadratic unitary operator
 i

exp   P 2   Q 2    PQ  QP   
2


1350
Li H Q, et al.
Sci China-Phys Mech Astron
August (2012) Vol. 55 No. 8
is actually a GFO. We also demonstrate that

exp i    Q1  Q2    P1  P2 

2

2


4

 i    Q1  Q2    P1  P2    i   Q1 P2  Q2 P1 


2
2

is a two-mode GFO. Their disentangling formula and normally ordered expansion are obtained which are useful for
knowing Fresnel transformation of quantum states performed by these two unitary operators.
This work was supported by the Natural Science Foundation of Shandong
Province of China (Grant No. Y2008A16) and the Specialized Research
Fund for the Doctoral Program of Higher Education China (Grant No.
20103705110001).
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