Lecture 2
. Elements of quantum optics.
Solutions of problems formulated in Lecture 1.
1.
.
. (a,b)
I
=2, (c)
I
=0.
2.
=1/2,
I
y
=1,
, to obtain the circu
lar polarization degree one can
decompose:
and obtain resolving this system the amplitudes:
,
.
.
Or one can argue that
, wh
ich means that the light is composed by linearly polarized wave
and circularly polarized wave. The intensity of the circular polarized component is twice
higher than the intensity of linearly polarized component. Hence, the linear polarization
degree is
, the circular polarization degree is
.
3. The intensity of light
,
, to obtain
I
2
it is
convenient to represent the Jones vector as
. The
in
tensity of light in the diagonal polarization is
. Using the
coefficient A obtained above we express also
. The Stokes
components are
,
,
,
.
4. One can find the angle of rotation of the polarization plane of a linearly polarized
light by representing it as a linear combination of two circularly

polarized waves and
ca
lculating their transmission independently. From Eq. (1.1.11) one can represent the
electric field vector of light as
.
For the circularly polarized light
. For
light
, for
light
.
Let us assume that at the front surface of the slab
, than the electric
field of

polarized light is
, the electric field
of

polarized light is
. At the
back surface of the slab
,
. Consider propagation
of the x

polarized light of the amplitude
A
. At the front surface of the slab its electric
field is
. At the back surface it will become
. Representing
,
we obtain
finally
. Thus the polarization plane is rotated by the
angle
during transmission, which yields the Verdet constant
.
Quantum optics
Quantum optics operates with photons, elementary quanta of the electromagnetic field. It
describes all the effects of classical optics and predicts a
number of peculiar quantum
effects which cannot be described with solely use of Maxwell equations. However,
observation of quantum effects in optics is not a trivial task, it requires special light
sources and
sophisticated
photon counting measurements.
Q
uantum states of the electromagnetic fields.
The quantized electromagnetic fields are mathematically equivalent to an
ensemble of harmonic oscillators. The electric field operator
(with a polarisation in
x

direction) and magnetic f
ield operator
(with a polarisation in the
y

direction) are
expressed in terms of the photon annihilation and creation operators
and
:
,
(
2
.1)
,
(2
.2)
where
E
is the electric field created by a single photon.
The creation and annihilation
operators
can be represented in the matrix form as
,
Using this representation, one can easily obtain some important relations for the creation
and annihilation operators. First,
the commutation relation:
,
(2
.3)
a
nd have the following properties
:
,
(2
.4
)
,
(2
.5
)
.
(2
.6
)
where
is the vacuum state containing 0 photons,
,
, etc.
The
photon number operator
(2
.7
)
has
th
e following property
.
(2
.8
)
is the quantum state of
a
light
mode
which has exactly
n
photons. It is referred to as
the
number state
or
Fock
state.
From (2
.5
)
and
(2
.6
)
one can see that
,
. (2
.9
)
The Fock states are orthogonal with each other:
.
(2
.10
)
That is why the electric f
ield expectation value for the Fock state is always zero:
, (2
.11
)
which means that the average projection of the electric field vector to any axis is zero for
the Fock state. On the other hand, the intensity of light is not
zero:
, (2
.12
)
Note that the intensity of the vacuum state of light having
n
=0 photons is not zero! This
is
a consequence of
so

called
vacuum
field
oscillations
.
The Fock states form a complete orthonormal set of basis vectors th
at describe an
arbitrary single

mode quantized electromagnetic field.
A
coherent state
of light
is the eigen

state of the annihilation operator
:
,
(2
.13
)
where
is a complex number.
Properties of the coherent state:
(2
.14
)
This follows from the fact that for any Fock state
,
. From
(3.14
) directly follows:
.
(2
.15
)
This yields the expansion of the coherent state in the basis of the Fock states.
Using the
normalisation condition
one can show that
, thus
.
(2
.16
)
The probability to find
n
photons in the coherent state
.
(3.17
)
P
(
n
)
for the coherent light
is a non

monotonous function having a maximum near
being
the average number of photons,
(see Figure 3.1).
The coherent light is
produced by lasers.
The natural light sources (e.g.
S
un
) produce so

called
thermal light
, where the
probability to find
n
photons monotonically decreases with increase of
n
:
,
(2
.18
)
where
is the average number of photons
detected at once by our photon counter.
is proportional to the intensity of light.
N
ote that the probabilities (2.17),
(2
.18
) are normalized:
.
(2
.19
)
Figure 3
.1 shows the difference between the photon distribution functions
for the coherent and thermal light. Experimentally,
can
hardly
be measured directly.
Instead, light is conveniently characterized by the
secon
d order coherence
parameter
.
(2
.20
)
Here we
used the commutation relation (2
.3) and relations
,
.
For the coherent distribution
,
, so that
, while for the thermal distribution
.
Figure 2
.1 Coherent
(solid) and thermal (dashed)
distribution
s of photons.
Note also that the second order coherence of the Fock stat
e is
,
(2
.21)
In particular, for
n
=1,
. Observation of the vanishing second order coherence is
considered as an evidence for single

photon emission. Quantum light sources based
on
individual atoms, molecules or semiconductor quantum dots are able to emit photons one
by one, which is proved by observation of
for these emitters.
The second order coherence
can be measured using the Ha
nbury Brown

Twiss
photon counting set

up (Figure 2
.2).
Full information about the photon distribution function of light
and its temporally
evolution is contained in an ensemble of the correlation functions introduced by
R.
Glaub
er:
. (3.
22
)
For
the coherent distribution (3.17
) all
.
The first order coherence
for any monochromatic light.
This is the classical cohe
rence
measured with use of the Michelson or Mach

Zehnder interferometer.
Finally, let us discuss the phase of photons.
The average phase of their ensemble
is given by
if
.
(2
.23)
For the coherent light
. For the thermal light or light in the Fock state the phase
is not defined. The uncertainty principle does not allow us knowing simultaneously the
number of photons and their phase. Those sta
tes of light for which
are referred
to as
number squeezed
or
amplitude squeezed
. For them the number of photons is known
with much better accuracy than their phase (example: a Fock state). On the other hand,
the states having
are referred to as the
phase squeezed
as the uncertainty in the
phase is lower than the uncertainty in number of photons for them.
Figur
e 2
.2 Scheme of the Hanbury

Brown and Twiss set up. The initial light beam is
separated into two beams by a half

silvered mirror, each of these beams goes to the
photon counting detector.
Measuring the
correlations between photon detectio
ns for two
beams
one can detect
the second order coherence of light.
Supplementary reading: Y. Yamamoto, A. Imamoglu,
Mesoscopic Quantum Optics
,
Wiley, New York, 1999. In this book the basic concepts of quantum optics are present
ed
in a simple and compact
way. Experimental methods of quantum optics are well
explained. It also addresses interaction between atoms and fields.
PROBLEMS
1 Prove relations:
a)
,
b)
,
c)
.
2 For th
e coherent
and thermal
distributions
demonstrate that
.
3
Prove that for the coherent distribution
,
.
4 Prove that for the thermal di
stribution
=2.
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