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We continue with our discussion on quantum states of electromagnetic field. We started
to discuss coherent states in the last lecture, towards the end. You have any questions from
the last lecture?
So, let us continue with our discussion on coherent states. Let me recall, we are looking
at single mode states that means, states in which the electromagnetic field corresponding
to one mode is exited. This particular coherent state is a state, which is defined as the
eigen state of the annihilation operator a. Now, because a is not annihilation operator,
alpha can be complex, the eigen values alpha can be complex. So, this equation implies
alpha a dagger is equal to alpha star. In the last class, we wrote this alpha as a superposition
of the n ket states. Because, n ket states from a complete set of function, complete
set of states, you can expand any state in terms of these n ket states and c n of the
expansion coefficients. We had calculated the values of c n in the
last class. We have shown that this is a exponential minus mod alpha square by 2 sigma alpha raise
power n by square root of n factorial get n. Alpha satisfies of course the normalization
condition, alpha alpha is equal to 1. So, this is a state, which is a superposition
of the various n ket state. n ket state is a state, which is an eigen state of the Hamiltonian
operator. To measure the energy of a n ket state, you will always precisely get the same
value every time. This is a superposition of n ket state, let me try to calculate what
is the probability of observing n photons in the state.
p n is equal to mod n alpha; if you use this expression for alpha, you can show that this
is minus mod alpha square; mod alpha square this per n by n factorial.
Now, I can relate alpha to the expectation value of the number of photons. So, let me
calculate the expectation value of n operator, this is alpha a dagger a alpha. a alpha is
alpha alpha, alpha a dagger is alpha star alpha, because alpha is normalized, so this
is mod alpha. This is the expectation value, which I just wrote as the average number of
photons n bar. So, the probability observing and photons is exponential minus n bar n bar
raise power by n factorial, this is the Poisson distribution.
The probability observing n photons is given by this equation. For example, the probability
observing 0 photon is exponential minus n bar, n is equal to 0. So, probability observing
10 photons is exponential minus n bar, n bar raise power 10 by 10 factorial. So, this is
a state of superposition of the n ket states. What you have calculated if the probability
observing a certain number of photons in that state?
Now, let me calculate what is the variance in the number of photons in that state? So,
remember, variance is given by n square expectation value minus n expectation valve whole square.
So, for this, I need to calculate n square expectation value, so n square expectation
value alpha a dagger a a dagger a alpha. Now, I have this relation a a dagger is equal
to 1. So, this implies a a dagger minus a dagger a is equal to 1. So, I replace this
a a dagger by 1 plus a dagger a and I get n square expectation value is alpha a dagger
into a dagger a plus 1 into a alpha. So, this is equal to alpha a dagger square
a square alpha plus alpha a dagger a alpha. a operating twice on alpha, gives me alpha
square, a dagger operating twice on alpha bra, gets me alpha star square. So, this is
mod alpha raise power 4 plus, this is mod alpha square that expectation value of n square;
expectation value of n is mod alpha square.
So, I can substitute these two expressions in this equation, to find out what is the
variance in the number of photons in this coherent state. That comes out to be, so delta
n square is equal to mod alpha 4 plus mod alpha square minus mod alpha 4, which is equal
to mod alpha square, which is equal to n bar. So, the variance in the number of photons
in this state is equal to the average number of photons in that state. The uncertainty
in the number of photons is square root of n bar. So, this means that the fractional
uncertainty in the number of photons is given by 1 by square root of n bar.
So, if you have an electromagnetic field generated in the single mod coherent state, is fractional,
uncertainty in the number of photons in that state decreases as n bar increases. That means,
as alpha increases, as mod alpha increases, the fractional uncertainty in the number of
photons in that state decreases, is given by, decreases as 1 by square root of n bar;
this is a characteristic feature of Poisson distribution.
Now, to relate this coherent sate to a classical electromagnetic field, let us try to calculate
what are the expectation values of electric field in this state?
Recall, the electrical field operator is given by i times square root of h cross by h cross
omega by 2 epsilon 0 V into a exponential minus i omega t minus k z minus a dagger exponential
i omega t minus k z. We are assuming the snood, the coherent state to be in a mod, which is
propagating along this z direction. It is linearly polarized, so I am not writing any
vector here, so it is a scalar electric field. Now, let me define two quantities E naught,
which is 2 h cross omega by epsilon 0 E V and chi, which is omega t minus k z.
So, in terms of these two expressions, the electric field operator becomes I times E
naught by 2 a exponential minus i chi minus a dagger exponential i chi. Please note that
this electric field is a time dependent operator, we are using the Heisenberg picture in this
electric field operator, is a time dependent operator. The state is time independent, given
by this expression for this.
You can do the same calculations in the Schrodinger picture, all expectations values that we are
calculating will exactly have the same values in the Schrodinger picture. So, this is an
electric field operator, chi contents time, z depends here. So, let me calculate what
is the expectation value of the electric field in the coherent state?
Expectation value of E is equal to alpha E alpha, which is equal to i E naught by 2 alpha
a alpha exponential minus i chi minus alpha a dagger alpha exponential i chi, which is
equal to i E naught by 2. What is this? This is alpha into alpha alpha, which is 1 minus
i chi minus alpha a dagger is alpha star alpha, so this is alpha star exponential i chi.
So, alpha is complex, so let me write alpha is equal to mod alpha exponential i theta.
So, expectation value of E becomes i E naught by 2mod alpha e to the power minus i i minus
theta minus mod alpha e the power i i minus theta, which is equal to i E naught by 2 mod
alpha. This reaming factor within this bracket is minus 2 i sin chi minus theta.
So, this two cancels off, i into minus i is 1, so E naught mod alpha sin omega t minus
kz minus theta. So, notice is that the expectation value in the electric field varies a sin omega
t minus kz minus theta, exactly like a classical electromagnetic field.
The amplitude of this is given by E naught times mod alpha. If you recall, when we looked
at the n ket state, the electric field expectation value of n ket state was equal to 0. So, the
n ket state is an eigen state of the Hamiltonian operator, it as a fixed number of photons
exactly a well-defined number of photons. The expectation value electric field was 0
there. That is not a representation of the classical
electromagnetic field. This state alpha, which we are looking at the coherent state, which
is an eigen state of the annihilation operator a. If you prepare a state in this, if you
prepare a electromagnetic field in this state, expectation value of the electric field in
the state is E naught mod alpha sin omega t minus kz minus theta, varies in exactly
the same passion as a cluster at the electromagnetic field.
Of course, this is a quantum field; it is not a classical field, because as you can
see now, as you will see, the electric field as uncertainties, this is the expectation
values of electric field to find out what is the uncertainty in the electric field.
We need to calculate expectation value in E square. So, for example, let me calculate,
the expectation value of E square. This is alpha, E square alpha, what is E square?
Now E, remember, we had written as - E was written in terms of a and a dagger like this.
So, e square is minus e naught square by 4 a exponential minus i chi minus a dagger exponential
i chi into a exponential minus i chi minus a dagger exponential i chi. That is equal
to minus E naught square by 4 a square exponential minus two i chi minus a a dagger minus a dagger
a plus a dagger square exponential 2 i chi. So, expectation value of E square is minus
E naught square by 4 alpha a square alpha exponential minus 2 i chi that we bring it
in the front, so plus, alpha a dagger square alpha exponential 2 i chi minus alpha a dagger
alpha minus alpha a dagger a. So, this is equal to minus E naught square by 4, so this
is nothing but alpha square. So, alpha square exponential minus 2 i chi, this is alpha star
square exponential 2 i chi.
Now, please note that a dagger alpha is not alpha star alpha. a dagger operates on bra
alpha, I need to replace a a dagger in terms of a dagger a by using the commutation relation.
a a dagger is a dagger a plus 1, so I get minus alpha a dagger a alpha which is small
alpha square and its minus 1 minus mod alpha square.
This a dagger a expectation value is small alpha square, a a dagger expectation value
is mod alpha square plus 1. Expectation if E square becomes, expectation value E square
is minus E naught square by 4. This is mod alpha square exponential minus 2 i chi minus
theta, because alpha square is mod alpha square exponential 2 i theta, so I take it is here,
plus mod alpha square exponential 2 i chi minus theta minus 2 mod alpha square minus
1, which is equal to minus E naught square by 4.
So, this is mod alpha square into some of these two exponential, is 2 cosin of 2 chi
minus theta minus 2 mod alpha square minus 1. Let me take the minus sign inside, so I
get E naught square by 4 2 mod alpha square into 1 plus cos 2 chi minus theta plus 1.
So, there is a minus sign here. This I take the minus sign inside, so this becomes 2 of
mod alpha square plus 1 and then, I minus 2 mod alpha square cos 2 chi minus theta.
So, what is this value? cos 2 theta is 1 minus 2 sin square theta. So, I can replace this
and I get expectation value of E square, is equal to E naught square by 4 into 1 plus
4 mod alpha square sin square chi minus theta. Remember that expectation value of E, we obtained
earlier, was equal to E naught mod alpha sin chi minus theta. So, the variance in the electric
field is equal to expectation value of E square minus expectation value of E Whole Square,
which is equal to - so expectation value of E square is this, so the second term is E
naught square mod alpha square sin square chi minus theta that cancels with this, then
I get E naught square by 4. E naught square if E replace, so this is 1 by 4 2 h cross
omega by epsilon 0 V; so this is H cross omega. So, delta E the uncertainty in the electric
field is square root of h cross omega by 2 epsilon 0, independent of mod alpha; independent of mod alpha, independent
of time, it is a constant. The expectation value of electric field increases as mod alpha
increases, the uncertainty in the electric field remains constant.
So, I can actually picturize the electric field variation in this form. So, the electric
field expectation value - let me draw a curve representing the expectation value in the
electric field first.
So, this is E naught mod alpha sin omega t minus kz minus 0, so let me plot at z is equal
to 0. So, this will go as E naught mod alpha sin omega t minus theta, so it has some variation
likes this.
So, let me plot- let me plot for example, if theta is to 0, it goes like this, is the
expectation value going like this. There is an uncertainty of square root of h cross omega
by 2 epsilon 0 V of the electric field, at heavy instant of time.
So, let me plot the upper bound, the lower bound here and the lower region. So, this will represent
- so this is uncertainty in the electric field at any time, this remains constant, independent
of time. So, this is a representation of the coherent state.
Remember, this uncertainty is independent of mod alpha, so if you take a coherent state
with a large value of mod alpha, this expectation, this height is quite large, amplitude is quite
large, because amplitude is e naught mod alpha and the uncertainty is always square root
of h cross omega by 2 epsilon 0V.
So, this field resembles more and more of a classical field. For very large values of
mod alpha, this noise that is represented in the electric field become negligibly small
compare to the amplitude itself.
The field tends more and more towards a classical field is always noisy, there is always noise
in the electric field. Please remember of this amount, in this coherent state and this
state. This noise is actually the same noise that you will find in a state, which is the
vacuum state. Because, a vacuum state, if you go back and look at this equation, for
the definition of coherent state, a vacuum state correspondence to a alpha is a 0, is
equal to 0. So, vacuum state correspondence to alpha is equal to 0, so the expectation
value of electric field in the vacuum state becomes 0, because mod alpha is 0, but there
is uncertainty electric field, which is the vacuum fluctuation.
So, this noise that is present in a coherent state is exactly the same amount of noise
that is present in the vacuum state. As the amplitude of the coherent state increases,
a fractional uncertainty in the electric field decreases, because delta E remains constant,
while the amplitude keeps on increasing and finally, you get close and closen to the classical
state of electromagnetic field. So, in this state, expectation value of electric
field is E naught mod alpha sin omega t. This is by assuming theta is equal to 0 and delta
E is equal to square root of h cross omega by 2 epsilon 0.
Now, there is another pictorial representation that is used to represent these states, which
we discussed earlier. That is in terms of the uncertainties and the expectation values
of the quadrature operators x and y. please remember, we defined a quadrature operators
x, is a plus a dagger by 2, another quadrature operator y by a minus a dagger by 2 i.
So, let us calculate what the expectation value is and what are the uncertainties in
the two quadrature operators when the field is in coherent state? So, expectation value
of x is equal to alpha a plus a dagger by 2 alpha, which is equal to half of alpha plus
alpha star, which is equal to half of mod alpha e to the power of i theta plus mod alpha
e to the power minus i theta, which is equal to mod alpha cos theta.
That is the expectation value of x, what is the expectation value of x square? We need
to calculate the variance of the uncertainty in x, so for that I need to calculate expectation
value of x square. Which is equal to alpha a plus a dagger whole square by 4 alpha, which
is 1 by 4 alpha a square plus a dagger square plus a a dagger plus a dagger a alpha, which
is equal to 1 by 4. Now, I can replace a a dagger by a dagger
a plus 1, I get alpha a square plus a dagger square plus 2 a dagger a plus 1 mod alpha,
which is equal to 1 by 4 a square expectation value is alpha square, a dagger square expectation
value is alpha star square, a dagger a expectation value is mod alpha square plus 1. Which is
equal to 1 by 4 mod alpha square e to the power 2 i theta plus mod alpha square e to
the power minus 2 i theta plus 2 mod alpha square plus 1. This is equal to 1 by 4 mod
alpha square into 2 cosine 2 theta plus 2 mod alpha square plus 1.
So, this is 2 mod alpha square into 1 plus cos 2 theta, which it can be written as in
terms of cos theta, so you get 2 mod alpha square into 1 plus cos 2 theta plus 1.
So, cos 2 theta is 2 cos plus theta minus 1, so this becomes expectation value of x
square, becomes 1 by 4 2 mod alpha square into 2 cos square theta plus 1, which is equal
to mod alpha square cos square theta plus 1 by 4. That is the expectation value of x
square, so the variation in x operator is equal to x square average minus x average
square. x average we have just now calculated, is mod alpha cos theta, so this becomes 1
by 4.
So, the uncertainty in the x coordinator is half, again independent of the denominator
alpha. I can similarly calculate the expectation value of y, the expectation value of y square;
from the two I can calculate delta y. The expression, I leave it you to calculate, this
comes out to be again 1 by 2, delta y is also 1 by 2, delta x is also 1 by 2. Please remember,
x and y satisfy commutation relation, which we have discussed earlier, x comma y is equal
to i by 2. Now, when you have a commutation relation
between two operators, they have to satisfy a uncertainty relationship between them. In
fact, if you have two operators satisfy in this relation A B is equal to i c, then you
can show that delta A into delta B the product of the uncertainty is greater than equal to
half of mod of expectation value of c. If you have two operators a and b satisfying
the commutation relation a b is equal to i c, then delta a can delta b, must be greater
than or equal to half of the modulus of the expectation value of c. Because, x and y satisfy
this equation, delta x times delta y, because c is just 1 by 2, this must be greater than
or equal to 1 by 4. So, the product of the uncertainties in the x quadrature and the
y quadrature must be greater than equal to 1 by 4. Or the coherent state we just now
calculated, delta x is half and delta y is half, so for the coherent state, delta x delta
y is equal to 1 by 4 and this is called a minimum uncertainty state. This satisfies
the minimum value of the product of the uncertainties in x and y, hence it is called the minimum
uncertainty state. This is smallest value of the product of the
uncertainties in x and y, you may have states for which this product is greater than 1 by
4. Do you have minimum uncertainty states? But, these states are special states, for
which the product of the uncertainties in x and y is actually 1 by 4. Coherent state,
which we are discussing, is one of the minimum uncertainty states.
So, what we have got is the following. We calculated the expectation value of x, so
what you got is expectation value of x is equal to mod alpha cos theta. I left it as
a problem to you to calculate the expectation value of y operator; you can show that this
is mod alpha sin theta. We have also seen that delta x is equal to half and delta y
is equal to half, so using these quantities I can draw the phase of diagram correspondent
to coherent state. This is x, this is y, so I put a point corresponding
to the expectation value of x and y. So, if I join this line, this angle is theta, expectation
value of this distance is mod alpha, so expectation value of x is mod alpha cos theta, expectation
value of y is mod alpha sin theta. There is an uncertainty around this point and this
radius is half. So, the coherent state is represented in the
phase of diagram by a small circle of radius half. The center of the circle is situated
at a distance mod alpha from the origin and this line the radial line makes an angle theta
with the x axis. So that the expectation value of x is mod alpha cos theta, the expectation
value of y is mod alpha sin theta.
So, if you look at a state which happens to be the vacuum state, please note that in the
coherent state, we have these equations. A vacuum state is defined by, so vacuum state
is corresponds to alpha is equal to 0.
So, the same phase I saw, I can actually use these expression that we have got to obtain
the parameters of vacuum state. I get for a vacuum state x average is equal to 0, y
average is equal to 0, delta x is equal to half and delta y is equal to half. So, in
the phase of diagram vacuum state, will be represented by a circle of radius half. On the origin, because the
expectation value of x and y are 0, there is uncertainty of half in both x and y quadratures.
So, for a coherent state, this is mod alpha. Remember, mod alpha is related to the average
number of photons; average number photons is equal to mod alpha square.
So, this distance square is this expectation value of the number of photons in that state,
which is n bar. So, if you increase the value of alpha in a coherent state, this distance
will increase of course, the radius of this circle remains constant, because that is independent
of mod alpha. If you look at a vacuum state, this correspondence to a state in which expectation
values of x and y are 0 and the noise uncertainty in x and y is still half.
So, coherent states are states which are close to the classical electromagnetic field, because
as we have seen here, the expectation value of the electric field in this state, in the
coherent state, varies as exactly like in a classical electromagnetic field. Expectation
value of electric field goes as E naught mod alpha sin omega t minus kz minus theta.
A laser, which is a very well stabilized laser, the light coming out of that kind of laser
is a coherent state. It does not have precise number of photons, there is an uncertainty
in the number of photons and there is an expectation value you can calculate. You can calculate
the variance in the number photons; you can calculate the expectation value of the electric
field, the variance in the electric field and all these you can calculate for the coherent
state; this is a state, which is closest to a classical electromagnetic field.
So, do you have any questions in the discussion that we have had in coherent states? Please
note that we are still discussing single mod states of electromagnetic field, in which,
the excitation is only in one of the modes of the electromagnetic wave.
We will consider another kind of states, later on squeeze states, before we move on to multi-mode
states. So, in this states, in the coherent state, the uncertainty in the x and y quadratures
are equal - equal to the minimum values. There can be states in which this product
is still the product of delta x delta y, which as the minimum value is 1 by 4. The product
is still 1 by 4, but delta x or delta y, one of them could be less than half, at the expense
of the other one being more than half. Do you have reduced uncertainties in one quadrature,
at the expense of increase uncertainties in other quadrature, which leads to state, which
are called squeeze states? Which are very interesting, they are completely non classical
states and we are finding a lot of the applications. So that is the discussion on coherent states,
from next class, we will look at squeeze states and continue with of states of single modes
of electromagnetic field, before we move on to multi-mode states; thank you very much.