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