To continue with our discussion on the Coherent
States, ah let us see some more properties
ah on the Coherent of States. And let us start
with ah the scale scalar product of two coherent
states. So, let us see what we ah understand
or what we learn from this scalar product.
um Of course, you know that ah the Fock states
which are the number occupation number states
ah denoted by N ah which are the eigenstates
or eigenfunctions of this operator N, capital
N which has a form which is a dagger a ah
this is a Hermitian operator which means that
it gives a real eigenvalues and also that
ah this are orthonormal states.
So, one ah Fock ah basis is or orthonormal
to another Fock basis. And ah so ah however,
ah the state alpha ah which is a coherent
state ah it has complex eigenvalues and ah
it is an eigenfunction of the operator a which
is ah lowering operator ah which is not a
Hermitian operator. So, its eigenstate of
ah non-Hermitian operator let us write it
here.
So, please take a note that we have explained
this or rather we have stated all of these
facts earlier and um this non-Hermitian operator
is a that is a lowering operator and ah it
has complex eigenvalues. Thus it is not clear
it is a not automatically clear, that ah the
um coherent states are orthogonal 
which is ah and of course, they are distinct
than the Fock ah states as we have seen earlier
and so its needs to be ascertained that whether
these are orthogonal states.
And in order to do that let us take ah two
coherent states 
alpha and beta and look at the inner product
between them I am just writing it as a beat
beta alpha which we know that by definition
its D dagger beta D ah alpha ah 0.
So, this can be written as, ah so these 0s
are the vacuum states and this D ah dagger
and D dagger beta and D alpha are basically
the translation operator that we have learnt
earlier. So, this is a exponential minus beta
a dagger ah exponential beta star a exponential
alpha a dagger exponential minus alpha star
a this is the definition that we have studied
for this ah and along with there is a Gaussian
factor which is equal to alpha square minus
beta square. So, this is ah the expectation
and ah our rather the inner product of the
two coherence states. Now, it has to be seen
that whether these two coherent states are
ah orthogonal.
Now, in order to see that let us see the end
states that is ah this one and this one and
they can be expanded in terms of ah the basically
the exponential can be expanded which is exponential
minus beta a plus and all that and then you
have an exponential minus alpha a ah rather
alpha star a which is equal to 1 minus alpha
star a and plus all that. Now, it is is very
clear that ah all these terms with a dagger
and a in both ah at the edges will yield 0
when they act on vacuum.
So, ah the ones that are saved or rather that
are ah survive those survived are the ones
and in which case we have ah we can actually
ah ignore them and we can write this down
as a 0 and now, we do the expansion to the
other ah ah quantities as well which are this
and this. So, this is equal to ah 1 plus beta
star a plus 1 by 2 factorial ah beta star
a square plus of all that and then we have
a a. So, ah and then we have a 1 plus alpha
a dagger ah plus 1 by 2 factorial alpha a
dagger square and so on and then its 0 and
of course, these are there.
And ah of course, we also have a ah the the
exponential this factor which will let us
write them here exponential minus half alpha
square minus beta square. ah So, this ah yields,
ah so then ah this is like 2, 1 by 2 factorial
root over 2 factorial. ah So, this you will
get it if you see it carefully by ah multiplying
term by term ah and a plus um um 1 ah beta
star ah and a plus a 0 which are for different
values of these ah for basically the Fock
bases which would generate by successively
you know operating the a dagger and things
like that a or a dagger and this has to be
taken. So, this is really a bracket not an
an angular bracket.
And then ah this is multiplied by a 0 plus
alpha 1 plus a 1 by ah 2 factorial root over
2 factorial alpha square 2 and all that ah
plus all this and so on. So, this and of course,
this whole thing is multiplied by the exponential
ah half ah just write it neatly. So, this
is alpha square minus beta square and so on.
ah So, that is that is what comes out of this
a inner product of two coherent states.
And just to remind you that we are trying
to find out that whether two um coherent states
are orthogonal or they form a complete set
of states. And so this if you use the ah orthogonality
of the Fock bases that is this equal to delta
n m then your ah beta alpha ah becomes equal
to exponential minus half. ah So, there is
a sign problem that I have introduced from
the beginning there is a plus there and there
is a plus, there is a plus there.
So, this is alpha mod square plus a beta mod
square um and a 1 plus alpha beta star plus
a 1 by 2 factorial alpha beta star um and
so on ah plus. So, this is nothing but exponential
ah that is the expansion of the exponential
ah alpha beta star. So, its exponential alpha
beta star, so this is equal to exponential
minus half alpha mod square plus beta mod
square plus alpha beta star. ah Now, if you
look at ah this alpha minus beta mod square.
ah So, that can be written as alpha minus
beta and alpha star minus beta star ah which
is equal to alpha mod square if you just open
up that is multiplied term by term its alpha
square plus beta square minus alpha beta star
plus alpha star beta ah and the.
So, this is equal to, so beta alpha ah mod
square is nothing, but equal to exponential
alpha minus beta square. ah So, the ah so
the main thing is that ah after doing all
this algebraic turns out that the coherent
states are not orthogonal and in fact, they
are only orthogonal when alpha is much greater
than beta or alpha minus beta is much greater
than 1. So, in general coherent states are
not orthogonal. The transition probabilities
only vanish if alpha minus beta much greater
than 1. So, this is the ah conclusion for
this.
And ah, but of course, ah they form a complete
set of states. ah So, this I will not prove
it here, but one can show that which we may
actually ah do it in a tutorial that the coherent
states 
form a complete set of states. So, even if
they are not ah orthonormal to each other
they still form a complete set of states and
which can be shown ah by showing that ah this
um summing over the coherent states this gives
ah 1 and ah so on. So, these are ah some of
the properties of the coherent states.
Let us now ah go to study what are called
as squeezed states, ok. So, ah this is very
well known that ah the uncertainty principle
ah which Heisenberg had proposed ah puts ah
an impediment in the um in the study of ah
various quantum systems or rather positional
measurements of quantum systems. And ah this
of course, is ah sort of dampens you know
the enthusiasm by the quantum engineers ah
where they want to do a lossless transmission
of quantum information by optical means. So,
ah just to summarize that ah uncertainty principle
is an impediment to transmission of information
by optical means.
So, there are of course, quantum noise. ah
So, quantum noise is an essential ingredient
to optical communication. So, ah the ah very
relevant and ah pertinent question in this
regard is that can we beat the uncertainty
principle or can we reduce the effect of the
uncertainty principle, and ah so this is the
question that we want to box it and possibly
the squeeze states um give a clue to this
answer. So, ah it is important to understand
that the uncertainty principle is actually
a statement about the area in the phase space.
So, is about area in the phase space which
what I what we mean by that is that it is
this ah product of a a small you know the
pixel in the phase space. So, ah if we can
ah divides the procedure to deform and squeeze
this area ah which can be effectively used
for reducing the noise. Such squeezing process
is of course, ah it does not de validate the
uncertainty principle, but it is just an engineering
that one can think of doing and ah the possible
means of doing it is via squeezed states we
will see what that is.
So, the question that one asks is that ah
can one squeeze or deform this area in the
phase space, phase space, ok. Again a question
that we want to box because it is a very relevant
question and to ah keep in mind that ah this
procedure 
does not de validate the uncertainty principle
that we all are familiar with. So, let us
see that how it can be squeezed. Before that
let us try to ah see some basics which we
already are quite aware well aware of and
ah ah consider consider two non-commuting
operators a and b um I mean which are which
are conjugate to each other 
each other such that x and p committed, ok.
So, let us ah do this commutator and let us
write this as i C, where i C is another. ah
So, these are ah Hermitian operators ah just
like our p and x are. ah So, C is another
Hermitian operator. So, in general it is a
Hermitian operator, but it could be just a
C number in some cases that we will just see
in a moment. So, this is the commutation relation
and it does not commute, ok, um the commutation
of these two ah Hermitian operators ah give
me another Hermitian operators which is i
C and ah the uncertainty principle satisfies
delta A ah multiplied by delta B that should
be greater than equal to half of C, where
C is the as I told it is an Hermitian operator
and we have taken an expectation value of
that operator and ah this expectation value
is taken with respect to some state psi. And
as usual our definition of delta A is nothing
but ah A square average minus A average square
and similarly for ah B and C, ah etcetera.
So, as I told that ah the expectation value
of ah A, B and C are calculated within some
ah given state psi and this state psi ah the
expectation values are 
are computed within a state psi, and ah this
psi will earn a name as a minimum uncertainty
state if ah the one of the observables say
ah delta A ah will satisfy certain relation
we will just write that. So, this psi will
be called as or rather ah will be called as
a better word, will be called as um a minimum
uncertainty state, state provided um. So,
if if a delta A and ah delta B they follow
an equality rather than an inequality which
appears here. So, that is the minimum ah answer
or rather that is a minimum uncertainty and
ah they would follow this ah relation where
an equality exists.
Now, ah these states will be called as squeezed
um if if one of the observables say a ah satisfies
a delta A square should be less than half
C square of course, along with that the um
minimum uncertainty condition has to be satisfied.
So, a state can actually satisfy a minimum
uncertainty condition, but may not satisfy
the squeezed condition as its written here.
But for a squeezed state to occur this condition
has to satisfy it has to be satisfied along
with um along with what we have just said.
So, delta A delta B should be equal to half
of a C and ah this ah will be called as, so
this is called as ideal squeezed states.
So, the whole idea is that the quantum fluctuations
of a squeezed state ah in one observable ah
say A are reduced below these half C, ah if
at the cost of the fluctuations in the other
observable. So, let us write this because
it is an important statement ah thus quantum
fluctuations of a squeezed state in one of
the variables or observables say A are reduced
are reduced to values lower than half of mod
C um at the expense of fluctuations 
in the other observable. So, let us give some
examples in order to validate ah our claims
or at least to check the definition of the
squeeze state.
So, ah you remember that we have ah introduced
ah the canonical variables or observables
x and p in terms of a and a dagger and we
are going to take those ah say say um A equal
to half of a plus a dagger and B equal to
1 by 2 i ah a minus a dagger. So, if you simply
ah so these are our ah x and p as we have
said that these applies to two canonically
conjugate variables which are which could
be x and p. So, if I take A and B commutator
and want to see that how it behaves this commutator
has a value which is i over 2. So, ah it is
very clear that C is equal to 1 here which
is a constant.
So, the Heisenberg uncertainty principle takes
ah delta A into delta B should be greater
than half, ok. So, that is the ah uncertainty
principle. ah Now, let us consider 
consider the expectation values 
values of A and B and the variances delta
A 
and delta B within the coherent states, ok.
ah So, define so coherent state is given by
alpha which is equal to a D alpha and 0, um
just to reiterate the notation 0 is the vacuum
D alpha is the translation operator which
produces an a state coherent state alpha that
is written with a ket on the left hand side.
So, ah my alpha a alpha ah that is equal to
A alpha plus i B alpha. So, this is equal
to 0 ah D alpha ah a D alpha 
and this is nothing but equal to alpha. So,
it is very clear that A alpha is equal to
real value of alpha and B alpha is equal to
imaginary alpha, alpha.
So, one of the ah canonical variables ah has
an expectation with respect to the coherent
state which is real part of alpha. And the
other one the imaginary one has got an overlap
in the coherent state as imaginary ah beta
sorry I mean some imaginary alpha where alpha
is of course, a coherent state. So, now, ah
if we want to calculate the variances, so
that is equal to alpha delta A square alpha
which is equal to 0 or delta A square 0 equal
to one-fourth ah alpha delta B ah 0 ah alpha
which is equal to 0 delta B 0 which is equal
to one-fourth itself.
So, ah the coherent states 
are indeed minimum uncertainty states , but
whether they are squeezed states or not that
we are not sure, but they are definitely minimum
uncertainty states. ah So, ah these are so.
So, now, the squeezed states as we said earlier
offer possibilities of 
of beating the quantum uncertainty, the quantum
uncertainty limit in measurements. So, let
me um box this this is an important statement.
So however, ah a neither Fock which is n nor
coherent which is alpha are ah squeezed states.
So, in fact, ah one can actually calculate
that ah that delta A equal to delta B equal
to half for coherent states whereas, delta
A equal to delta B equal to half 2 n plus
1 ah for the Fock. ah So, this is for ah the
Fock states. So, it is clear that the Fock
states are actually the ah states in the occupation
number basis which was quite largely used
um in dealing with quantum harmonic oscillator.
So, a squeezed state can be obtained from
a coherent state 
by applying 
a squeezing operator 
operator which is S ah by xi, which is xi
star ah a square by 2 exponential minus xi
a dagger square by 2. So, ah S xi, S xi acting
on a vacuum which is 0, ah now it is important
to understand that the coherent states are
actually ah built from the the vacuum states
ah by the translation operator. So, if the
translation operator yields one then coherent
state yields a vacuum and there is a special
vacuum state, and so if S xi is made to act
on 0 that gives me a state ah coherent state
ah xi. So, this acting on 0 is gives me xi
and S of xi acting on alpha gives me another
state which is alpha xi. So, these are the.
ah
So, they show that ah S xi operating on the
trivial coherent state gives a xi which is
a a coherent state, and a S xi acting on a
coherent state gives ah another coherent state
which is alpha xi. ah So, the ah in fact,
the squeezed states um are of 
of 4 types, um one is the ah number squeeze
states, ah two is the face squeeze 
state state, and of course, in addition to
that we have a space squeeze state and a momentum
squeeze states. So, let us ah give some demonstrations
or pictorial ah representations of these states.
So, let us look at. ah So, this is my A axis
which is real alpha and this is my B axis
which is imaginary alpha, and so there is
a vacuum state 
and this vacuum state can be translated to
get a coherent state ah pardon this ah shapes
that are not coming very well. So, these are
ah intended to be circular. So, these are
squeezed. ah So, there is a vacuum vacuum
state and this is a coherent state ah and
ah coherent state which is all we know that.
So, there is a D alpha operation that gives
a a state, so a vacuum state becomes a coherent
state and similarly ah for the squeezing part.
So, there is a ah this, ah and then this,
this. ah So, this is A, this is B again, this
is real part of alpha, and this is imaginary
part of alpha, and ah so this is of course,
a squeezed vacuum. So, we have simply written
it as xi and again ah this ah is operated
upon and one gets coherent squeeze state.
And this is given as alpha xi according to
the definitions that we have given.
So, this ah is more or less the introduction
about the coherent states and squeezed states
ah they are ah taken up as applications of
the quantum mechanics on quantum harmonic
oscillators that we have learnt. And of course,
there are many many examples that are used
I mean in optical communications, in teleportation
and so on and various branches of optomechanics.
ah We will not go into very specific details
of these applications, but rather ah referred
to you ah or, so references ah on squeezed
states are ah one is a Y B ah Band um. So,
it is a Light and Matter ah um and Electromagnetism,
Optics, ah Spectroscopy 
and Lasers, and it is a Wiley publications.
Number two is ah Scully and M S Zubairy, and
so this is a quantum optics book we will give
you this ah let us let me just type out this,
and this is from the Cambridge University
Press, ok. So, these are some of the references
on the ah squeezed and ah the coherent states
which me you may want to look at. And we will
move on to another topic.
