every week we are looking at certain basics and the implementation of the aspects of quantum
computing so last week we looked at the nmr implementation of quantum computing and we
introduced you to some of the basics of quantum mechanics in the beginning and then went ahead
to look in to the basics of nmr and the implementation via nmr
in this week we will look at the optical aspects of quantum computing and in that regard as
we know very well lasers play a very vital role so we will introduce you to lasers in
this week so that you are up to date with the concepts of laser that will be necessary
for quantum computing purposes so we start off with the basics on lasers and then we
will looking to the various implementation aspects involving lasers in this week it may
over to the following week also but this is how we start
so laser as we know is in acronym light amplification by stimulated emission of radiation but it
is much more than that it is a device that produces coherent beam of optical radiation
by stimulating electronic ironic or molecular transition to higher energy levels it is mainly
used in single mode systems light emission ranges from five to ten degrees which means
that it is highly directional and it does require a higher complex driving circuitry
when we are looking at certain aspects compare to lets say light emitting diodes
these are action occurs from three main processes photon absorption spontaneous emission and
stimulated emission so we look in to all these aspects in more detail just remember that
whatever we say here none of them are to be avoided by in full detail as we go along and
will explain as how that happens
so this principles of light amplification by stimulated emission of radiation or laser
can transform energy from other forms of energy into coherent and highly directional electromagnetic
radiation and this other form of energy can be anything as diverse as chemical energy
electron beam electric current electromagnetic radiation and so on and so forth and this
is based mostly on the work of the postulates provided by einstein who first gave the aspects
of photons and stimulated emission by his famous a and b con coefficients which we will
go through here and the first microwave laser was demonstrated by townes shawlow and prokhorov
in nineteen fifty four and it was called maser and in nineteen sixty the first optical
maser as that time it was called optical maser its which is eventually became laser was
demonstrated though the novel prize was given to townes prockhorov basov for the demonstration
of the principle of this entire process that is maser in nineteen sixty four maiman
eventually got the patent after a long battle on the laser and the laser patent is held
by maiman
so we know that there have been several theories of light starting from classical to quantum
and we have already looked into the aspects in quantum mechanics in terms of the plan
constant and einstein whereas the wave and the classical nature of light is better understood
and propagation is better understood in terms of these either wave or particle later on
this corpusculler replace by particle and the the propagation of electromagnetic radiation
is understood in terms of maxwells equations
so all these aspects are quite well preserved in the concept of laser the interesting
part is the quantum aspects also which is utilize as we get into the principle of the
laser so the properties of laser which are important are the concept of monochromaticity
or monochromatic which means that the lights is concentrated on a narrow range of wavelengths
ideally one specific color but in real sense that's not exactly how it is we will
come to it and the other important part is the coherence where all the emitted photons
bear a constant phase relationship with each other in both time and phase its highly directional
which means a very tight beam which is very strong and concentrated let us look at it
a little bit more technically monochromaticity is looked at in terms of the wavelength spread
function which essentially tells how single wavelength or how much of a particular wavelength
is been generated
coherence is the range over which the phase has constant relationship it can be looked
at the region of the length over which the coherence is maintained which means that the
constant phase relationship will be maintain is given in terms of coherence length lc
lambda squared over two delta lambda which is the monochromaticity divergence is similarly
another aspect which measures how much of the laser is spreading out with distance and
there are both longitudinal and transfers modes which are there in the laser which will
discuss so the monochromaticity as we mentioned is essentially the single possible wavelength
that can be generated which in theory can be as simple as a stick or a single line however
in reality its always a spread which is delta lambda as we have discussed before and in
fact this has a fourier relationship to time so in case you are going to have a laser which
is going to measure events which are in time then we have to give up more and more on the
single color feature or the monochromaticity of the laser in wave length now will discuss
into that more as we get along this area
the other important part as we mentioned is the divergence or directionality which can
be quantified in terms of lets say the angles spread as the distance keeps on increasing
from the source point so roughly it can be looked at as if it is a function which
is the spread function which can be somewhere of this dimension when you measure its somewhere
at this point radiation coming out of the laser has a directionality and spread with
a define divergence angle theta as we mentioned here and this angular spreading of a laser
beam is very small compare to other sources of electromagnetic radiation and is describe
by a small divergence angle of the order of milli radians
so for example a he ne laser which we will discuss here with the moderate is a power
of one milli watts has a very low divergence so which is about two mm in diameter will
grow to may be about two point one mm after quite a long distance and which means that
you are intensity of the laser remains quite high irrespective of the space at which you
are intercepting the laser so the photon flux availability of a laser
is extremely high so in a little bit more technical term divergence is the measure
of this directionality and it has certain important terms associated with it the rayleigh
range over which no diameter spread happens about thar laser beam coming out and so here
for example its shown that the range is the region over which the laser just comes
out of the output point and remains study over the region and it is technically defined
in terms of the diameter squared over the wavelength so it is depending on the wavelength
and so on so of the laser
the far field divergence can be defined into two l sin of theta in similar to the way i
defined the divergence in the earlier slide where i talked about the time tan theta
as a ratio of two numbers of length versus the spread
similarly divergence at the far field is going to be defined in terms of wavelength over
the waste or the minimum width of the beam inside the laser cavity which is never going
to be zero and that is roughly the divergence which is stated for a laser which is very
low essentially showing that this directionalities preserved very high and the intensity availability
is very high over a large distance coherence on the other hand is the fact that the phase
relationship within the electromagnetic waves emitted are going to be very high and so for
a laser for example another waves with same phase start almost at the point in time
and therefore as it goes long it is always bearing a constant relationship with the other
waves as a result you get and the phase to remains study over distance however if
you have waves with different phases they interfere and they eventually die out in terms
of the aptitude so there laser radiation is composed of waves of the same wavelength which
start at the same time and keep the relative phase as the advance giving its uniqueness
in terms of his capability of retaining its photon flux as well as this properties
over a very long distance
so the basic concepts of a laser in terms of its operation depends on certain principles
which are the principle of absorption spontaneous emission stimulated emission and a concept
known as population inversion we will go through them now so let us look at the different principles
one after the other and the stimulated versus the spontaneous emission so stimulated emission
requires the presence of a photon and incoming photon stimulates a molecule in an excited
state to decay to the ground state by emitting a photon now this is under the principles
that when we have light coming in the process of taking the system from the ground state
to the excited state is absorption as will discuss but in this particular case the
photon is essentially not going to do that part but it is looking for case the state
already exist in the excited state and the photon is going to bring it down so that is
the stimulated emission case
the stimulated photons travel in the same direction as the incoming photon spontaneous
emission does not require the presence of a photon on the other hand so instead of a
molecule in excited state being hit by a photons so that it come down in this case the molecule
can just come down on its own which is a relaxation process to the ground state by spontaneous
emitting a photon so that's the reason why spontaneously emitting photons are emitted
in all directions whereas in the other case it follows a directions of the photons coming
in
so the stimulated photons have unique properties which is in phase with the incident photon
same wavelength as the incident photon and travel in the same direction as the incident
photon essentially giving rise to the principle of the laser when light travels through an
absorbing medium the medium absorbs the light the amount of light absorbed is determined
by beers law which we know now which means that the moment we have photon being absorbs
the light availability as it goes through the system keeps on going down so for a medium
to operate as lasing medium the transmitted light intensity should be greater than the
intensity of the light incident on the material when an atom or molecule in the lasing medium
absorbs light it is excited so that is the absorbing process the excited molecule then
decays to a lower level either through emission of a photon which can be either stimulated
or spontaneous or via a non radiative loss of the energy now in this particular case
of non radiative loss of energy this is not doing anything to us is just a loss factor
for lasing action stimulated emission must dominate as determined by the boltzmann factor
the population of the ground state is always greater than the population of the excited
state so typically absorption would dominate that's what is the natural way of thinking
of things for stimulated emission to be the dominant process the excited state population
must be larger than the lower state population now this is one of the place where its important
in other words for a medium to produce laser light there must be population inversion so
that was the fourth term that was there when i mentioned what are the essential requirements
for a laser so this population inversion essentially ensures that the upper state population is
greater than the lower state population
so how can a population inversion we created when the population in the ground state is
always greater than the population in the excited state this is what we know from our
boltzmann distribution so what kind of materials will allow for an
inversion of population in its electronic state so we can ask the question how a population
inversion can be created one way of looking at it is by exciting the laser atom or molecules
and this process is called pumping if the pumps source is very intense the number of
atoms or molecules excited can be large however (Refer time:17:00) once excited the atoms
and molecules must stay in the excited state long enough to create an excited population
which is greater than the ground state population so essentially we are looking at an equation
of motion here in some sense you are creating some population in the excited state and that
has to stay therefore certain time so that an inversion process can happen that is what
we are asking
so the absorption and emission process and the spontaneous and stimulated process as
we have looked at essentially means that they are equally likely at thermal equilibrium
probability of stimulated emission however depends on the cross section because you need
another photon to be present or absorbed during this process the concept of stimulated emission
was first proposed by einstein as i mentioned so stimulated emission therefore has this
curious property of being in phase in contrast to that of spontaneous which can be in any
direction so the absorption and spontaneous emission can now be looked at in many different
ways let us consider the simplest of them which is a two level system so in a two level
system we have say the exact energy in necessary to to the system so that it get excited so
its a absorption process which would be followed by an emission process which is spontaneous
so light bulb work in this process so light that i am standing in front you the scatter
that you see from regular light is based on this process of spontaneous emission on the
other hand if after absorption the photon are available in plenty so that it can still
push the population from the excited state so basically you make an absorption with the
photon and then you have more of that available so that you can now stimulate the excited
state to come down then there will be a phase relationship as we mentioned with the input
photon which will result in stimulated emission and that is the process through which the
laser happens so it has to be made to happen so in summary we have this overall process
that we been looking at that it's a absorption process simply taking it to level picture
we have the exact energy gap which is been provided by the applied field then it can
either undergo in spontaneous emission the excite state can just come down committing
the same energy photon or if the same energy photon is available it can also stimulate
the excited state to come down and then (Refer time: 20:00) two photons of energy having
the same phase as the incident photon would be coming out so this is the unique case of
stimulated emission
pictorially this is what we are looking at we have unexcited molecule which is being
excited by the radiation we get the excited molecule if it under goes just simple emission
of photon then its spontaneous emission however if you another photon available for it to
bring it down to the ground state then we get stimulated emission which is the one with
the phase now the other very important thing in this
process is to have this process go on and have a gain in the process as we continue
so here is the principle here we have the incident and the output intensity going in
and the output going out as long as there is an amplification factor a then we can say
that there is a gain associate with this process and the length of the absorber that we are
looking at or the length of the gain medium is our l so as long as there is an amplification
process we will have this expression valid and so we can have a gain in the system so
it can look something like that the stimulated emission leads to a chain reaction and laser
emission so this is the cartoon of such a situation if a medium has many excited molecules
one photon can become many and here is a cartoon showing that in availability of a inverted
system would allow for multiple photon to be generated from itself so much so that
you finally get an output and this is the essence of the laser the factor by which an
input beam is amplified in a medium is called a gain as we just discussed and its represent
by g and its will follow the expression that we will discuss that the they will be a
gain which will be giving rise to the amplification factor for a given length of the gain medium
so the laser therefore is a medium that stores energy surrounded by two mirrors now why do
you need this two mirrors that's because we want have a situation where we want to keep
on having the process over and over again so its the repeatability of the principle
which is what we are looking at in the simplest possible case the laser medium with gain g
is placed within a cavity which is essentially these two mirror in which the system can continuously
keep going back and forth and the laser will lase if the beam increases in intensity during
a round trip that is if my ith is going to be increasing with respect to my input usually
there are additional losses in intensity such as absorption catering and reflections however
those are taken in terms of loss parameter in general the laser will lase if in a round
trip the gain is greater than the loss and this is known as threshold because you need
a certain amount of gain process to occur so that the increase of the entire process
happens and you can look at it only a very small amount of this entire process of the
gain circulating in the system is taken out through a partially reflecting mirror that
what is the simplest design of the laser
now how can you calculate the gain so in this case it is going back to what einsten had
done in nineteen sixty where he considered two parameter a and b and these where his
einsteins a and b coefficient they are called so he gave this as the various transition
rates between the molecular states say one and two involving the light of irradiance
i the absorption rate will be equal to the b coefficient times the number of absorbers
or in case of a molecule the electron which are there in state one which is going to be
excited so that is the absorption rate
a spontaneous emission rate would them be simply proportional to coefficient a times
the number of electrons or the whatever is been excited into the excite state which
is being pulled down stimulated emission process would however depend on availability of photons
so that will be b coefficient again which will be then the number of excited electrons
present times the intensity of the available photon so that's how it will be equated
and the laser gain can be then be written in terms of the rate equation neglecting spontaneous
emission its a simple equation where the stimulated emission minus the absorption gives
rise to the gain or the change in the flux for their applied intensity gain in the process
the solution therefore is can be written for this differential equation in terms of
an exponential form where the proportionality constant is the absorption or the gain cross
section sigma there can be exponentially gain or loss in irradiance normally since the excited
population going to be less than the ground state population there is loss due to absorption
but if the excited population is greater than the ground state population there is gain
and we define that gain as g so this is again the reason of where you need population inversion
so in terms of laser the idea of population inversion is critical so how to achieve this
laser threshold which means that we would like to have inversion in order to achieve
the threshold where my gain is greater than one that is stimulated emission must exceed
absorption we have to have this satisfied so this is the simplest case ignoring spontaneous
emission which means that i will have to have an inversion in terms of my population
profile which does not occur naturally so in order achieve inversion we must hit the
laser medium very hard in some way and choose our medium correctly so that is the basic
criteria for having the laser to work
so the first question to ask them is that what happens when we consider the simple case
that we have been treating right now the two level system it turns out that inversion
is actually impossible in a two level system so how do you look at it so here is the going
back to the rate equations we can write how the population would change with respect
to time based on these expressions that we have just devised so these coefficient
in terms of these intensity of the available photons and the number of ground state versus
the excited state situation so we will have absorption stimulated emission and spontaneous
emission if the total number of molecules is n this is what we get number of molecules
is equal to n one of absorbing and n two of the excited state and so we can write the
two different equations which will relate to the ground and the excited versus the ground
state dynamics and by looking at with respect to the change of the total number which is
the difference between these two we can write an expression in terms of the total number
of molecules versus the change that we are looking at if we rewrite this expression in
this form we are able to finally get to an expression where we have this form and
at study state what will happen is the rate of change would be equal to zero because whatever
goes to the excited and ground ability becoming zero so in that case we get a situation where
the we have the change that we are looking for delta n its going to be total numbers
over one plus i over i of saturation
now i saturation is defined as the i saturation which we have replaced it here which means
that i saturation is equal to the ratio of a is to two half of the ratio of a by two
b now in this case what we notice is delta n is always going to be positive no matter
how high the value of i is going to be so it is therefore impossible to achieve an inversion
of a two level system so that is the basic reason as to why we cannot actually create
an inventory system for at two level by looking at it in this simple possible manner
because no matter how hard we try the best we can achieve is an equilibrium between the
ground and excited state
so in order to look at in to something known as this very important term of population
inversion a state in which a substance has been energized or excited to a specific energy
levels more atoms or molecules are in a higher excited state than in the ground state the
process of producing a population inversion is called pumping these are all the details
that we have done the examples of such process can be by a by the use of lamps of
appropriate intensity or by using electrical discharge now this forms one of the very critical
forms of a principles that we are looking at for lasing action in this particular
first lecture i think it is important that i conclude it at this point where we are giving
you the basic principles on what we are going to look for in terms of a lasing action and
the critical parameters of all the things that we have looked at are stimulated emission
and finally in order for a laser to happen is the population inversion
so since we reach at this point let me stop here in the next class we will be going in
to the more details of laser process how are these different laser defined and also
i will try to see if i get to your time result or lasers with possibility of measuring time
which will also look in to see you next time