we have been doing a lot of introspection
of the entire course in this particular week
as this is the final week of our course on
quantum information and quantum computing
and mostly the implementation part of it in
doing so we went through all the basics we
went through the implementation aspects that
we have looked into in this course now as
the final part of this course is coming to
an end let me try to give an overview of some
of the aspects of the problem that we are
in particular dealing with so in our case
we are basically starting with the same principle
that we have discussed throughout this course
the ideal two level system which essentially
forms the qubit for most of the systems that
we have discussed in this particular case
the main point of our work have been focusing
on the idea of using light as one of the three
as one of the key ingredients for doing the
quantum computing so that's why we distinguish
ourselves from the rest of the work which
is going around so in the simplest possible
sense some of the earlier maths that you have
looked in is summarized in this particular
slide where the interactions and the way the
system evolves is being looked at in terms
of an ensemble of the system that is been
looked at and that has been and that is interacting
with the applied field in most cases for our
particular case it's optical
it is also important to mention that as we
have developed during this entire course any
system that can interact resonantly with assists
with an applied field whether it is radio
frequency or it is electromagnetic or it is
magnetic they all have similar interaction
features and they end up producing hamiltonian
interactions which are sort of what we go
ahead with describing the system the most
important part which we discussed in this
course was also the fact that whenever we
are looking at a practical aspect of the problem
we are not looking at a single quantum system
and so instead of solving the schrodinger
equation we always end up solving the liouville
equation which is how this entire process
evolves with time and under the condition
where the two states are being interacted
with not just a resonance single photon excitation
but with the resonant several photons adding
together to give that excitation a picture
similar to the one shown here is often used
and the simple picture for a multi photon
process as long as it doesn't create any other
complicacy works quite well in terms of describing
what is going on with these basis many of
the discussions which have been also made
in this course are relevant to the kind of
work that we are doing and have been developing
over the years
you have also seen these kinds of features
as we discussed during the course where the
ground and the excited states keep on oscillating
with the typical rabi frequency based on the
interaction with applied field and as the
field is not in resonance but goes away from
the exact resonance point to other frequencies
which are non resonant with the two states
that we are looking at the fall off of the
population goes in this particular fashion
and in this theoretical limit the idea has
been taken that there is nothing but a pure
two state system which is being excited and
therefore beyond a certain range of detuning
there is no resonance available and so it
falls off in terms of excitation of the system
going from ground to the excited
however it is important to note that there
is a range over which the effect of the pulse
is still there because we are not really talking
in terms of delta functions but we are talking
about actual width of these lasers or pulses
that we are talking about and these fourier
definitions often always help us in defining
and finding out as to how far the excitations
can exist even when they are not exactly on
resonance so all these are important concepts
to understand as we address and work with
quantum systems and qubits in particular there
are effects of the actual profile of the laser
which is often not very importantly looked
at but the consequence of that is the idea
of solitons which i think i might have mentioned
during propagation of laser pulses through
optical fibers and others while we were doing
the laser concepts but there are certain shapes
of which are more resistant or which maintain
their shapes as they interact with systems
and so hyperbolic seek and for example is
a better shape in that respect with
as compared to let say the gaussian one and
so there are different principles which keep
on working at every respect of the application
of interaction in these kinds of areas of
research so in general the ground state excitation
of a laser pulse also has to be thought of
an envelope profile which is having a carrier
frequency and in the carrier frequency included
laser pulse for instance looks like this and
when we often for convenience right only the
amplitude profile we take a look and write
it this way which automatically removes the
carrier frequency of the laser which goes
into the resonant excitation of the field
and most often we go into the rotating frame
where this carrier frequency of the laser
is considered to be the frame in which the
system is rotating or revolving and therefore
we can only work simply by using the amplitude
profile of the laser beam instead of wondering
instead of worrying too much about the frequency
of the laser for instance so the populations
and how they work are important aspects to
understand because that's how laser matter
interaction plays which play a major role
in some forms of quantum information and computing
can be relevant in these kinds of understanding
and studies
so the model therefore is built on this principle
that if you have a simple single photon case
where the resonance is when the exact gap
of the energy between zero and one state is
provided otherwise given the fact that there
is a bandwidth does it with the laser even
if the laser does not exactly match the bandwidth
match the la[ser]- match the energy gap between
zero and one the two states can be connected
through dipole interactions and excitation
can occur and because of that a detuning factor
delta may come in and that's how all these
discussions appear if multiple pulses are
possible to interact simultaneously then we
can get two photon or multi photon cases and
at every point there will be some aspects
related to the line width and the exact energy
gap differences that are allowed because of
the way these interactions go so the resultant
is that we were able to see the effect of
these interactions in our last lecture where
we looked at how the energy used to flow from
the excite state to many other state and for
simplistic real molecules that are real we
actually saw how i v r can really be modeled
and that can that information and their ideas
can in fact go ahead to form the principle
of adiabatic interactions to get to cases
where they can be utilized for computing one
of the simplest applications we discussed
was the idea of using taylor series expansion
of the instantaneous phase of the electric
field
where the phase was actually in expanded by
using the taylor series and then his derivative
gave rise to the frequency sweep which resulted
in shape pulses and the application of those
and their interactions with this is model
systems can be understood in terms of an ensemble
of states by using the liouville equation
and that's typically the result of all these
experiment all these theoretical models which
have been described and discussed in many
of the lectures previously and the convenient
unit for many of the interactions are often
in terms of rabi frequency which
essentially has the connection of how well
the two states are connected as well as the
intensity of the applied field both of them
together defines the rabi frequencies mu dot
e over h cross mu is the coupling constant
e is the applied field and h cross is the
plancks constant so that's how typically these
parameters are placed so that they can be
related to how the actual interactions as
well as the theory can be looked at between
each other
once again the sweeping which results in adiabaticity
can be looked at as a result of different
shapes of the pulses and the complex shapes
of certain pulses are much more are complex
laser pulses have very specific interactions
and one of the interesting ones is hyperbolic
secant with hyperbolic tangent frequency sweep
because it essentially results in a rectangular
inversion profile so this principle where
the population is being looked at with respect
to the applied energy and the detuning applied
field and the detuning is often called as
the inversion profile and the shape of the
inversion profiles often enables to understand
how the interaction goes and it is important
when these interactions are well defined rather
than they are
oscillating or changing because once they
are well defined they can be utilized for
certain applications
so for instance a a perfect way of taking
one state to the other precisely all the time
is for instance applying a not gate but if
a simple pulse is applied to a gaussian pulse
without any other property a changing of the
laser is applied to a two level system it
will
start flopping so the rabi flopping which
means the population will go back and forth
between ground and excited state so we will
it will not really undergo a clean not gate
but an oscillation and so for generating simple
gates which are abuse it is important to have
interactions which can be predefined in a
certain way and can be utilized in terms of
computational principles that's the basic
idea behind the start of these kinds of applications
in this format
so under these kinds of implications and the
pulses interaction with the system can be
looked at a little bit more and completely
and even under the condition where there is
an ideal two level system the effect of the
pi pulses can be understood in terms of
realizing how the coherence terms now that
we know that in a just density matrix the
diagonal elements represent the population
and the off diagonal elements which are rho
one two and rho two one essentially represent
the coherence of the system ah those are not
observables however for simulation these are
important parameters which can be looked at
and to in order to understand what is going
on with the system
so by looking at these it is possible to understand
how the different pulses affect the interaction
of the system which can which is to be used
as a qubit so here is the model of looking
at the off diagonal density matrix elements
that is in other words looking at the coherence
of the system under the impact of an applied
field the population may oscillate as in this
particular case it's only a simple pulse which
is been provided without any modulating field
associated with it so there will be a rabi
flopping and that depends on the pulse area
of the interacting pulse with the system and
the interesting part here which cannot be
an observable now this part population part
is an observable the part which is not an
observable is the coherence part and it can
be modeled and understood in these terms and
by using the principle of adiabatic passage
where the frequency is being changed slowly
from below resonance to resonance and to beyond
resonance the population can be made to follow
the applied field and so these ideas can be
demonstrated and understood in the same way
as the principles are built up and it is possible
to see that these principles are really well
followed as the applications are being made
to these kinds of understandings and the model
is being checked at every point through simulations
and applications
so based on these understandings and checking
it through many different intensity patterns
and the conditions we have managed to distill
a few of the important aspects which is related
to probing the coherence and coming from the
off diagonal elements of the density matrix
and as expected all absorptions are associated
with dispersion and this is known from spectroscopy
as it is known as the famous kramer kronig
relationship all absorptions are composed
of the real part and the imaginary part where
the real part corresponds to the dispersive
and interaction and the imaginary part is
corresponding to the absorption of this applied
field and which results in there are be flopping
and this is coupling through absorption in
case of the adiabatic process coupling is
through that dispersive part so there is no
absorption process and so there is no population
flopping so this is the key concept of the
adiabatic process which means that the adiabatic
process essentially works through the interactions
through the dispersive part and so no absorption
process occurs and no population flopping
is seen
whereas in case of the resonant interaction
absorption rabi flopping occurs because the
coupling is through absorption so the benefits
of such understanding of theoretical models
and study is the fact that we can quantify
the two level character in a multi level system
which is very important it's almost like saying
that we are tracing out the density matrix
to get the information about the rest of the
system off diagonal l symmetric elements switch
from real to imaginary and the excited process
changes from being resonant to completely
adiabatic and this is one of the most one
of the interesting observations that was observed
as a result of such a study so some of these
fundamental ideas that we have been discussing
has profound implications as expected and
these were the models to test and find that
they do in fact confirm to these understandings
these are also very important as we would
like to make molecules as or qubits because
molecules are richer in terms of handling
as well as interacting as well as the number
of states concerned and so they would become
good quantum computing qubits
however there are many challenges one of the
biggest challenge is the process of i v r
which we discussed in earlier classes and
there is also there is an other important
point that since molecules are not really
two level systems to know it is important
to realize that in order to use the molecules
as qubits we would need to demonstrate a true
two level nature of the molecules so for instance
as i mentioned before all real molecules are
going to be multi level and we have to find
a way to see how we can understand their two
level nature increasing decreasing time of
the states have to be also very important
as we would like to use them as our qubits
so this is a coming from the temporal side
the other important part is the isolation
or controlling the molecules in such a way
so that they can be made to interact under
experimenters discretion so that has to do
with the spatial processing so that they can
be under control as we interact or to work
with them and this has been achieved through
principles of molecular beams or in the gas
phase which is equivalent to the idea of atomic
or ionic traps of environments were such possibilities
exist
optical tweezers are the other interesting
ways of doing it in terms of liquids and so
they are also very attractive which we look
at it and we have developed the pulse optical
tweezers in this context to make this possible
so the ideas of control essentially exists
both temporarily and spatially and they can
be coupled at some point by using pulsed optical
tweezers for instance where both temporal
and spatial aspects can be put into action
so in this context is perhaps useful to also
discuss about the half passage instead of
the full passage that we discussed earlier
where the property of the laser is changed
from below resonance to resonance and is held
there so that the property of the pulse is
changed from below resonance to resonance
and it is either kept like that or after a
while it is switched off so that the states
follow to the point where they are coherent
between the ground and excise state
however the states which are going to create
i v r or the dark states we take the energy
away do not interact with the ground and excited
states when they are coupled this is one of
the principles which we discussed before in
terms of i v r and their control once the
pulse is turned off they all go back into
their original conditions so this is the work
that i have already presented in one of the
lectures in this week so i am not going to
explain it once more it suffice to say that
it is possible to use these same principles
and this is our work in particular which shows
that it is possible to show molecules can
be tuned to be working as qubits and in terms
of molecules realistic molecules a lot more
work can also be done to show that they all
would be interacting in one way or the other
depending on the other kinds of shapes that
can be applied not just the ones where the
pulse is just going to counter resonance and
turn off and go through resonance in a very
slow manner and so on and so forth
so there has been some work in terms of looking
at all the components both real and imaginary
components as it undergoes these changes so
as to be able to understand how the coherence
flows in these kinds of interactions and how
the energy localization may be looked at as
a function of the applied field in this regard
it is possible to bring the states to coherence
by using the population to follow a pattern
where the amplitude profile of the pulse is
following the frequency profile or may be
undergoing a certain kind of a phase modulation
where this kind of coherence between the two
states exists and the other states are not
possible to interact because they are not
able to get any interactions with them in
these kinds of cases it is important to find
that the real part is important as we can
see the off diagonal elements to understand
how the coherence of this entire process is
going and working
this particular idea of how molecules are
interacting because of their many many states
is also possible to understand in terms of
another model of intra molecular vibrational
relaxation where it is not just a few state
interacting with the original two states which
get coupled due to applied
field but there are many other states which
sort of interact in a different way and that
particular model is shown here which is known
as a tier model of intramolecular vibrational
relaxation where the energy of the states
are sort of star interaction and the interactions
keep on increasing as the number of states
across keep on getting coupled more and more
such a model hamiltonian can also be built
and many molecules have such interacting hamiltonians
which can be further generalized to make a
good model of the molecule using such models
it is possible to have simple hadamard gate
in molecules which was also mentioned earlier
in one of the lectures and this is the same
slide that was used there to essentially show
if that more than one state is coupled to
the ground state then they can all be simultaneously
put to an equals position whereas the dark
states which are the ones which are not coupled
through dipoles but are going to interact
with the excited states to take the energy
away are kept at bay and not allowed to interact
and so it automatically creates an equilibration
of bright states
so these have their interesting implications
and perhaps can be utilized as qubits has
as been suggested by earlier work so one of
the key features in all these interactions
is the idea of having control knobs where
the spatial modulation is being used to get
the individual molecular control in condensed
phase and in case of gas phase one can use
molecular beam conditions as we discussed
before the other kind of control that can
be utilized in any kind any in many of these
applications in terms of control knobs is
the laser polarization so one is the laser
spatial modulation the other one is the laser
polarization and in terms of temporal modulation
the simplest of all that we have been discussed
is the frequency chirping so a natural question
under these conditions to ask whenever we
are looking at control knobs is how important
are these parameters or control knobs in the
in concept of molecular control because the
idea of the molecular control is intimately
connected to the concept of quantum computing
is they are the ones which end up producing
the gates
so here is a an example of how the simple
idea of frequency chirping which is basically
linearly changing the frequency across the
pulse can be put to a use this is connected
to the idea that chirp is a term which has
been around for a very long time it has been
adapted from the concept that birds make the
frequency modulating sound and that's how
the chirping of birds have been associated
with this principle of chirp pulses so they
have these principle ideas where the chirp
parameters can be defined in terms of the
change of the frequency as it undergoes within
the pulse so they can be interacting and their
interaction with the molecular system can
be understood in as a function of their parameter
so one of the places where this has become
very interesting is the idea of using such
pulses in the gas phase for controlling the
fragmentation process which may have different
applications but in terms of our particular
goal we will present how this is done however
this particular technology itself as a lot
of bear in to the the controlled environment
principles that we have always been talking
about in terms of quantum computing ideas
and applications where in this case the the
beam chamber is generated through a supersonic
jet expansion in a very low pressure environment
which is effectively generated by use of turbo
pumps and diffusion pumps and any other pumping
techniques so that the their zone over which
the molecules expand are under the condition
that they they have a laminar flow and the
laser can in fact interact across that in
under that condition with a well known mark
number and and the proper temperature associated
with these molecules these interaction may
lead to the fragmentation of these molecules
and those fragments are then measured through
a time of light mass spectrometer and measured
through a multi channel plate
the signal which gives rise to the fragments
that we are finally looking at in this particular
case a laser operating at a thousand hertz
was used with ample intensity to be able to
undergo break down of the molecules that were
being looked at now what was notice very interestingly
is the fragmentation pathway for a simple
enough molecule n propyl benzene dependent
on the way the frequency of frequency content
of the laser that was used for this fragmentation
was and that resulted in different fragmentation
ratios that was dependent on the
frequency content of the laser pulse
so if there was no chirp associated with a
laser as it has been shown here transform
limited with each of the chirp parameter is
zero this is the transform limit or the regular
gaussian beam this is the distribution of
the ions which has seen however the mass spectra
of the n propyl benzene changes as the frequency
chirp is changed between positive and negative
frequencies and as they are mapped in a more
systematic manner what is able to be seen
is there is a laser induced control of the
fragmentation of the n propyl benzene because
of the applied chirp parameter and this is
interesting because that would mean that these
fragments can essentially relate to the applied
field conditioning and that can in turn act
as the understanding parameter for some bit
of information this is similar to the ideas
which was originally propounded by bergsman
and others where they had used the fragmentation
principle to finally read off how certain
states were more populated than the others
in terms of the interaction of the cesium
grade berg state with respect to the applied
field which had some information on it
so given the fact that the information content
on this particular cases can be related to
the frequency content of these pulses they
can also interact to show how these information
is being translated into the fragment ratios
which have been seen here so in particular
for example c six h five plus has and as a
very strong positioning at and a positive
value whereas the c five h five plus as a
very strong positioning at a negative value
and so these two for instance can serve as
markers for that whereas the beta zero which
is essentially now as the has the maximum
coming for the c seven eight seven at the
beta zero position so they all have different
characteristics associated with the fragmentation
property based on the character of the frequency
which has been used this is because the fragmentation
path keeps on changing as a result of the
applied chirp and so they undergo different
directions of getting into the different fragments
as we have been showing here
the same is true for the polarization dependence
of the various fragments of the fragment ions
seals of these n propyl benzene however one
thing is to be noted in case of the polarization
dependence is the fact that the variation
of the different fragment ions with respect
to the polarization angle essentially peak
at given particular values and that is independent
of the fragment character so all the fragments
for instance peak at specific values of the
polarization which sort of provides the polarization
parameter as an intensity parameter where
as in terms of the frequency component the
chirp of the pulse we found that certain fragments
are specific in specifically inclined towards
certain frequency components so the polarization
component can be more like a intensity in
this particular kind of a control knob whereas
the other frequency sweep parameter in this
case is very subject to the character of the
fragment that we are looking at
so having multiple control knobs like this
are very important for controlling quantum
objects because that's the way how these gates
and knobs can be connected and be utilized
for information processing of quantum objects
so this is the idea behind adding the polarization
and the chirp simultaneously on a particular
experiment as we have discussed here and it
is possible to therefore use both the control
knobs which are independent of each other
one acting as an intensity switch and the
other one acting more like a frequency switch
for looking at these different fragments so
this multi parameter control with laser polarization
and pulse chirp essentially are useful because
they are mutually exclusive one acting only
on the
intensity part and the other acting on the
character of the fragment that are being looked
at
another important aspect we looked at in term
in this connection was also to look at how
the chemical dynamics of a system may be affected
as a result of laser frequency changing instantaneously
within the laser as we have been talking about
in terms of frequency sweeper changes and
what we found at here also the parameter of
frequency sweep did create a major change
between the resulting products depending on
how it is interacting so in order to establish
this we essentially used a very symmetric
experiment we used a very rudimentary process
where a dimer of a molecule was being broken
up into the monomers and so in terms of measurements
it was very clear that there was a peak due
to the dimer and there was a peak due to the
monomer and the distinction between the two
could be easily understood as the experiment
was being done so under the same conditions
of the gas phase experiments as we have defined
discussed
in the earlier case these systems could be
looked at by using the frequency modulation
of the laser pulse as it is undergoing interaction
with this system and what we find is that
given our pulse width of the laser the fragments
are very specifically distributed to the dimer
and the monomer which let's our experiment
to work well if the intensity of the laser
goes too high and it becomes complicated in
terms of interpreting the results because
other fragments start appearing but if it
is done in such a way only the two fragments
the monomer and the dimer appear and it is
easy to do these experiments
so we basically set up the experimental condition
in such a way that we were able to look at
only the two major components that are of
interest in this particular kind of work and
what we found is that the result of what we
found was that the affect of the chirp on
the parent ion yield was very symmetric as
expected with respect to the second harmonic
generation of the system essentially showing
that the process was only intensity dependent
however when the relative yield of the monomer
was being looked at it was found that the
negative chirp had a huge impact on the formation
of the enhanced formation of the monomer as
compared to the case when it was positively
chopped pulses were used so that's how one
of the very important parameters were applied
in this particular case to make sure that
we understand that the chirp parameters are
being applied properly to be able to study
this process with care
so overall it was possible to understand that
the relative ion yield of the monomers was
favored when negative chirp was used versus
the positive chirp but the yield of the dimer
essentially remained consistent with the application
of this particular process so the spatial
control with pulse laser opens up possibility
of spatial temporal control polarization can
also play an important role in spatial control
control knobs which were used by a spatial
modulation temporal reputation exploiting
temporal shaping and polarization and for
the traditional molecular control the control
knobs that we explode were frequency chirp
there is a polarization and the control of
dimerization versus is breakdown was also
being looked down so these sort of set the
space for how quantum computing and use of
quantum objects for doing computation can
be utilized
another area of development of quantum computing
in our particular approach has been the idea
of spatial modulation in these particular
cases we were mostly using temporal modulation
the spatial modulation part works extremely
well if one can use a tweezer or a way to
hold on to a particle as we have shown in
terms of ion traps as possible to relatively
move objects and so this is the same principle
that we have used in the case of liquids buy[ers]-
in optical traps and and to discuss that part
i will be going into the next lecture so we
will end this lecture with this principle
that we have managed to show the different
kinds of qubits and optical interactions that
we have done until now in terms of quantum
computing in this respect so we will see you
in the next lecture
