Download Waxman

Coherent and squeezed
states of the radiation field
By Amir Waxman
1
Introduction
Classically, an electric field consists of waves which are well defined
both in amplitude and phase. That is not the case for the quantum
radiation field. An electromagnetic field in the state |n> got a well
defined amplitude, but completely uncertain phase. We can also
describe the field in terms of 2 conjugates quadrature components
If those components have minimum uncertainty relations, the state is
said to be a coherent state. In general, it is possible to create a state
with different uncertainties that steel has a minimum uncertainty
relations. in that case, the fluctuations in one of the components will
be reduced, on the expanse of greater fluctuations. In the conjugate
component. That is a squeezed state.
We will start with defining a the field operator for a single mode, and
then on to definition of the number state. we will present the
coherent state as a superposition of the number state, and then we
will go to squeezed vacuum and then to squeezed coherent states,
with all their properties.
2
Single mode field operator
when we made the quantization of the radiation field, we defined the field
operator:
ˆ  i  aˆ †ei 
Eˆ (  )  Eˆ  (  )  Eˆ  (  )  ( h  / 2 0V )1/ 2 ae
Where:
  t  kz 

2
Is a phase angle. By a convention we can get rid from the constants:
1 i 1 † i


ˆ
ˆ
ˆ
ˆ
E (  )  E (  )  E (  )  ae
 aˆ e
2
2
And by the transformation to quadrature operators:
3
aˆ  Xˆ  iYˆ ; aˆ †  Xˆ  iYˆ
We get:
Eˆ (  )  Xˆ cos(  )  Yˆ sin(  )
From the commutation realations of the “a” operators we get:
i
 Eˆ ( 1 ), Eˆ (  2 )    sin  1   2 


2
This commutation relation is only for the single mode while the complete
field continue to commute. The uncertainty relations can be derived
from the last equation:
Eˆ ( 1 )Eˆ (  2 )  1/ 4 sin  1   2 
Where the variance of the electric field is defined by:
(Eˆ (  ))  ( Eˆ (  ))  Eˆ (  )
2
4
2
2
By this we can see that if the phase differs in л so we can measure the
field precisely but then uncertainties of other pairs of angles will
increase.for the vacuum state the mean is:
Eˆ (  )  0
And the variance is:
(Eˆ (  )) 2  1/ 4
The coherent signal S is defined as the expectation value of the field:
S  E(  )
From the matrix elements of destruction and creation operators we
know that a coherent signal occurs only for a state that contains
superposition of photons-number states differ in value by 1.
5
The uncertainty of the field can be refered to as noise:
  (Eˆ (  )) 2
And the signal to noise ratio will thus be:
E( )
S
SNR 

 (Eˆ (  )) 2
2
2
SNR can be seen to depend on phase.
The first order degree of coherence of the single mode
state (g) can be defined as:
g 1 ( z1 , t1; z2 , t2 )  g (1) ( 1; 2 )  g (1) ( )  exp{i( 1  2 )}
And in second order:
6
ˆ ˆ / aˆ † aˆ
g (2) ( )  aˆ † aˆ † aa
  1   2
We can define the degree of coherence with the mean
square photon number (by using commutators and
definition of the number operator):
g (2) ( ) 
n(n  1)
n
2

n2  n
n
2
 1
n 2  n
n
2
The photon number must be a an affirmative quantity:
1
1
 g (2) for n  1
n
And than, the coherence from second order must satisfy:
0  n 2  n 2  n
We thus see that in a single mode the second order
coherence is independent in time and space.
7
Number states
Number states are the basic states of the quantum theory of light. They
form a complete set for the state of single mode. They are easy to
manipulate in calculation of quantum optical properties. They are on
the contrary less easy to generate experimentally. The eigen value
equations for n states is :
n̂ n  n n
There for:
(n)2  0
The sec. degree of coherence then follows:
1
g ( )  1  for n  1
n
(2)
For n<1 the limit is zero.
8
The energy eigenvalue relation can be written in some forms:


ˆ n  h   aˆ † aˆ  1/ 2  n  h  Xˆ 2  Yˆ 2 n  h   n  1/ 2  n

Thus the number state has the quadrature-operator eigenvalue property:
 Xˆ
2

 Yˆ 2 n  h   n  1/ 2  n
The quadrature- operator expectation values are:
n | Xˆ | n  n | Yˆ | n  0
And the variance is:
1
1
(X ) 2  (Y ) 2   n  
2
2
We can see that the number of state has got the same property for each
one of the quadrature operators. For the vacuum state (n=0) the
variance have the smallest values. We thus say that the vacuum state
is a minimum uncertainty state.
9
By the expectation value and variance of the electric field
we can now calculate the signal and the noise:


S  n | Eˆ (  ) | n  0
1
1
2
2
ˆ
ˆ
  (E (  ))  n | (E (  )) | n   n  
2
2
10
Coherent states
The most commonly found single mode states correspond not to the
number state, but to a superposition of number states. The coherent
state is an important example for such a state as a single mode laser,
operated well above threshold, generates a coherent state excitation.
The coherent state is defined:
 e
 /2
2


n 0
n
n!
n
When α is a complex number, and the normalization can be easily
verified:
  e
11

2

 n n
n 0
n!

1
The coherent states are right eigen states of the destruction
operator as can be seen form:
aˆ   e
1 2
  
2
n
 n!
n 0
1/ 2
n
n 1   
1/ 2
The creation operator thus satisfies
 ↠   
Remembering that:
(a† ) n 0  n ! n
We can now write the coherent state using the vacuum
state:
 e
1 2
  
2

n 0
12
 aˆ 
† n
n!
0  exp( aˆ †  1/ 2  ) 0
2
According to Baker-Hausdorf formula, any pair of operators
that keep the relations:
 A, B  , A   A, B  , B   0
Also keep:
e
A B
e
 A, B / 2 A B
ee
In our case if A and B are:
†
ˆ
A   a ; B    aˆ
We get (with using the vacuum condition) :
  exp( aˆ †   aˆ) 0
Which defines the displacement operator:
†

ˆ
ˆ
D( )  exp( a   aˆ )
13
D is an equivalent to a creation operator for the complete
state, analougus to the number state creation operator N
D keeps the condition of unitarity:
Dˆ † ( ) Dˆ ( )  Dˆ ( ) Dˆ † ( )
And the effect on the destruction operator is:
ˆ ˆ ( )  aˆ  
Dˆ † ( )aD
With the hermitian conjugate relations:
Dˆ † ( )aˆ † Dˆ ( )  aˆ †   
14
Some properties of orthogonal states:
Different coherent states are not orthogonal this can be shown from:
  e
1 2
1 2 
 
 
2
2

e

n
n
n!
n 0
 1 2 1 2

0  exp          
2
 2

Thus:
 
2

 exp    
2

The coherent-state expectation values for the number operator are:
n   nˆ    aˆ † aˆ   
2
And for the second moment:
n
15
2
  nˆ       n  n
2
4
2
2
The photon number variance will then be:
 n 
2
  n
2
And the fractional uncertainty in the photon number:
n 1


n

1
n
And this decreases while the coherent state amplitude increases. In a
classical field the variance of intensity can vanish for a single wave.
In quantum optics it cannot because of the particle-like aspects in
quantum theory.
The probability to find n photons in the mode is there for:
P ( n)  n 
2
e

2

2n
n!
Which is a Poisson distribution. For large values of n it approaches a
gaussian distribution.
16
the poisson photon number distribution for various mean photon
numbers
17
The coherent state expectation values of the quardature operators we
defined earlier are the following:
1
1
 Xˆ    aˆ †  aˆ         Re    cos 
2
2
Where we have put:
   ei
In a similar way we can calculate:
1
1
 Yˆ    aˆ †  aˆ         Im    sin 
2i
2i
The expectation values of the squares, can be now calculated, when we
use the normal ordering procedure we used for the value of the
square photon number n
1
Xˆ 2   a † a †  2a † a  aa  1
4
18
1
Yˆ 2   a † a †  2a † a  aa  1
4
The quadrature variances will there for be :
1
ˆ
ˆ
X  Y 
4
From that we can conclude that the coherent state is a
quadrature minimum uncertainties state for all photon
mean number <α> ( unlike the situation in the n state).
Using the expectations values for the quadrature
components, we can easily calculate the coherent signal:
S   Eˆ (  )    cos     
And the variances help us to calculate the noise.
1
2
ˆ
N  (E (  )) 
4
The noise is there for phase independent and it has the
minimum value allowed according to uncertainty
principle. Next to be obtained is the signal to noise ratio:
2
SNR  4  cos 2       4 n cos 2     
With maximum value at χ=θ
19
In the picture we present the coherent states properties. the arrow
point on the mean field value. The length of the arrow is |α| and it is
inclined in the χ-θ angle from the real field. Χ is determined by the
evaluation in position and time of the field averages and there for is
a property of the measurement and can be controlled by the
experimentalist. Θ is a property of the field excitation on which the
measurement is made and in principle is not controlled by the
experimentalist. The black disc in the picture stands for the field
uncertainty, and resolved to amplitude and phase
20
The amplitude result gives us the same uncertainty we got for the
photon number uncertainty:
2
2
1/ 2
 1/ 2 1   1/ 2 1 
n   n     n    n
4 
4

The uncertainty in the phase may also be extracted, by geometric
calculations:
 
1
2
1

2 n
1/ 2
The product will there for be:
1
n 
2
This equation represents the trade off relation between amplitude and
phase of the field in a coherent state.
We can see that both the change in phase and fractional uncertainty
are dependent in 1/α, which means that if we have a large number
of photons the field is better defined both in amplitude and phase.
21
In the following graph we can see the phase dependence
of the electric field ( <n>=4). The dashed lines stand for
the noise band.
22
We use the formula for phase distribution :
P( ) 
1
2

 c(n)e
2
 in
n 0
And:
 e
 /2
2


n 0
n
n!
n
To get the phase distribution of coherent states:
P ( ) 
1
2

e
n 0
1 2
 
2

2
n
n!
ein (  )
P is an even function of θ-φ and it follows that the mean phase equals:
 0
23
The variance of the phase is more difficult to determine,
and the curve which is shown here is a result of a
numeric simulation. The dashed curve shows the large n
approximatiom.
24
Squeezed vacuum
A field excitation said to be quadrature squeezed, when its field uncerainty keeps:
1
0  (Eˆ (  )) 2 
4
For some values of the measurement phase angle χ. Of course this “squeeze”
has to be compensated by a variance larger than ¼ at the phase angle
perpendicular to χ.
We will start with the squeezed vacuum state which is defined:
  Sˆ ( ) 0
With the squeeze unitary operator:
1
1

Sˆ ( )  exp    aˆ   aˆ †ei  ; Sˆ † ( ) Sˆ ( )  Sˆ ( ) Sˆ † ( )  1
2
2

That has amplitude and phase defined as following:
25
  sei
When those definition are similar to those were made for the coherent
states ( with operator D)
From the exponent in the squeezed state definition it is clear the
squeezed state consists only from even n states, but in order to
show it is necessary to use general operator ordering methods. The
number of state expansion will there for be brought directly:

(2n)!1/ 2
  (sec hs) 
n!
n 0
1/ 2
n
 1 i


e
tanh
s
 2
 2n
We will now turn for calculating some of the expectation values. First,
for the mean number of photons we can write:
ˆ ˆ †   0 Sˆ † ( )aˆ † Sˆ ( ) Sˆ † ( )aS
ˆ ˆ ( ) 0
n   aa
Where we use unitarity.
By operator relations, we can get to:
26
ˆ ˆ ( )  aˆ cosh s  aˆ †ei sinh s
Sˆ † ( )aS
ˆ  i sinh s
Sˆ † ( )aˆ † Sˆ ( )  aˆ † cosh s  ae
And now we can calculate the mean photon number:
n  sinh 2 s
It is shown that if we don’t have any squeezing (s=0), the mean
number of photons vanishes, and we return to ordinary vacuum
state.
Using the same methods of algebra we can derive the second moment:
n 2  3sinh 4 s  2sin h 2 s  3 n 2  2 n
And the variance is accordingly:
n 2  2 n

n 1

The degree of second order coherence will be
1
2
g   ( )  3 
n
The photon number flactuations of the squeezed vacuum are thus
superpoissonian .
27
The interest of the squeezed vacuum lies in the quadrature operators
properties. For destruction and creation operators :
 aˆ    aˆ †   0
Which again shows that squeezed vacuum states are superpositions of
n states with even n only. Now it is easy to show that:
 Xˆ    Yˆ   0
The variance of quadrature operator can be calculated with the help of:
ˆ ˆ   ei sinh s cosh s  aˆ †aˆ †   ei sinh s cosh s
 aa
And thus be:
28
 X 2  
1  2 s 2  1  2 s

21
e
sin


e
cos






4
2 
 2 
 Y 2  
1  2s
 2 s 2  1  
21
e
cos



  e sin    
4
2 
 2 
A presentation of the quadrature expectation values is
shown in the picture. S is given by exp(s)=2.
The lengths of the ellipse axis and the inclination angle
are written. The black disc is the uncertainty area. if s=0
this is a coherent state but otherwise the values of the
variances are either greater or smaller than ¼.
29
The expectation values of the field can be also calculated in a similar
way. The mean field can be then shown to vanish:
S   Eˆ (  )   0
And the noise is:
1
1 
1 


N  (Eˆ (  )) 2  e2 s sin 2       e2 s cos 2      
4
2 
2 


Which means completely phase dependent, unlike coherent states. That
can be shown from the graph:
When s equals 0 we get constant noise, like in coherent states.
30
It can be shown from the noise equation that:
1
 1
Emin  E    m   e  s
2
 2
And this is a quadrature squeezed field. The max. value of the field
variance is:
1
 1
Emax  E       m   e s
2
 2
And the product satisfy the uncertainty relations:
Emin Emax
1

4
The state can’t be squeezed for all values of s. the condition for
squeezing is:
1
2
    e s
The range of angles for which squeezing is possible thus decreases
when s increases.
31
Squeezed coherent states
Those states are defined by:
 ,   Dˆ ( )Sˆ ( ) 0
D is the displacement operator:
Dˆ ( )  exp( aˆ †   aˆ )
We will see that squeezed coherent states keep our possibility of
reducing the noise in the system, but can also acquire a non-zero
signal, unlike the squeezed vacuum.
32
In order to find the mean photon number we have to calculate some
important relations, similar to those we calculate for the vacuum. With
the help of the properties of D operator, we get:
ˆ ˆ ( ) Sˆ ( )  aˆ cosh s  aˆ †ei sinh s  
Sˆ † ( ) Dˆ † ( )aD
ˆ i sinh s   
Sˆ † ( ) Dˆ † ( )aˆ † Dˆ ( ) Sˆ ( )  aˆ † cosh s  ae
These transformations provide eigen value relations:
(aˆ cosh s  aˆ †ei sinh s)  ,   ( cosh s   ei sinh s)  , 
And they reduce to squeezed vacuum relations where α=0. the mean
photons number is then calculated:
n    sinh 2 s
2
This is actually a sum between the coherent and squeezed vacuum
mean numbers.
33
The photon number variance is then obtained:
1 
1 
2


(n)2   e2 s sin 2       e2 s cos 2        2sinh 2 s  sinh 2 s  1
2 
2 



We can see that taking s to zero will give us a coherent state, and taking
α to zero will give us a squeezed vacuum.
The main interest lies in the expectation values of the quadrature
operators. We start with:
 ,  | Xˆ |  ,   Re    cos 
 ,  | Yˆ |  ,   Im    sin 
Which are identify to the coherent states result. The quadrature
variances will thus be:
 X   14 e
2
2s
1 
 1 
sin 2     e2 s cos2    
2 
 2 

 Y 2   14 e2s cos2  12    e2s sin 2  12  


Identical to the results for the squeezed vacuum state.
34
In the figure below we see a comparison between ordinary
vacuum squeezed vacuum state and the coherent
squeezed state.
The figure well shows how the mean quadrature values are
solely depended on the coherent parameter α and while
the variance values solely determined by ζ the vacuum
parameter.
35
From the former results, we easily obtain the expectation values of the
electric field operator:
S   Eˆ (  )    cos(    )
Which is identical to the coherent state signal. The field variance, or
noise is:
1  2s 2 
1  2 s
1 
2
2
N  (Eˆ (  )) 
e sin       e
4
2 

cos      
2 

Which is identical to the phase dependent squeezed vacuum noise. The
signal to noise ratio reads:
4  cos 2 (    )
SNR 
1 
1 


e 2 s sin 2       e 2 s cos 2     
2 
2 


The SNR is dependent by distinct phase angles whose relative values
depend of the method of generation of the squeezed coherent state.
The source of light can be adjusted to optimize the SNR :
SNRmax  4e 2 s 
2
1
for     
2
The coherent squeezed state there for enjoys both world. His coherent
36 signal noise is such of a coherent state, and his SNR can be
improved, matching the noise of the squeezed vacuum state.
In the figure we can see a representation for a single mode
amplitude squeezed coherent state with <n>=4 θ=v/2
and exp(s)=2, showing the mean and uncertainty of the
electric field. The maximum SNR is achieved. This is an
example for an amplitude coherent squeezed state.
37
The uncertainties occur in the limit when the coherent contribution is
much larger:

n 
es
2
Then the uncertainty in photons number is:

n   n

2
1/ 2
1
 
 e s    n
4  
2
1/ 2
1

 e s   n
4 
1/ 2
e s
The uncertainty in phase is:
es
es
 

2  2 n 1/ 2
And the product of photon number and phase uncertainties is again:
1
n 
2
The amplitude squeezed coherent state has reduced photon number
uncertainty and an enhanced phase uncertainty.
38
In the figure bellow we can see the mean field
representation of the amplitude squeezed state in the
former figure. The dashed lines represent the noise
band. The increased phase uncertainty and the
decreased amplitude uncertainty make this noise band
deferent from the one in the coherent state graph where
the noise band was constant.
39
In this figure we present the case of       / 2 . The light in this case is
said to be phase squeezed coherent state. The effects of the
squeezing on the amplitude and phase uncertainties are now
1/ 2 s
interchanged:
n  n
 
e
es
2 n
1/ 2
The major axis of the noise ellipse is enhanced to show an increased
amplitude uncertainty, and the minor axis is reduced to show a
decreased amplitude uncertainty.
40
This figure shows experimental results for the
noise.
The measurements are made by homodyne
detection, with a local oscillator phase
angle that varies linearly with the time.
(a) is a coherent state (b) is a squeezed
vacuum state
(c) Is amplitude squeezed state (d) phase
squeezed state
(e) A squeezed state with 48 degrees
between the coherent vector, and the axis
of the noise ellipse.
41
42