this week as we have been looking at the quantum implementation part we will explore another
approach of quantum computing which has been very popular which is known as the ion
trap quantum computing now in this ion trap method the principles that will be using
will introduce you at the beginning and then we will look into the aspects that are necessary
for ion trap quantum computing so in this case we are doing the ion trap implementation
of quantum computing i have set up a small outline so that its easier to look at how
we are going to do the principles of quantum computers as we have been discussing is the
process that we are looking at here the overview of the ion trap quantum information processor
is the first thing that we are going to look at and in that respect we have to understand
the concept of the ion trapping that we do once we have seen that then we are interested
to get into the initialization and detection part of the problem one or two small single
qubit gates internal will be looking at so there are two aspects one is the internal
aspects and the other one would be the consolidated together approach so we will
discuss those two aspects of the of this particular ion trap case and the coupling of internal
external qubits as we discuss that this will become clear in nutshell we are basically
talking about in one case the principle of polarization and in the other case the
principle of the the motion so in one case the polarization is one particular aspect
and the other one could be the state of the system whether its in ground or excited and
the third one would could be the motional so all these the three things can be looked
at while we look at ion trap case so this could be momentum aspect which is the motional
thing and the other one could be polarization aspect and finally the third one and the other
very important one could be the state of the system that we are looking at whether it is
the ground state or the excited state now in terms of building the quantum computer
as we have been looking at from the very beginning we need the qubits to work and
in most cases this two level quantum systems the way we look at it
there are super position aspects which are isolated from the outside outside world and
confined characterizable and scalable these are the issues that we are really looking
at and let me tell you while we go ahead with this area that trap time quantum computing
has been quite successful in the sense that we have managed to and make up to i
mean the groups which work on this have managed to make ten qubits available in terms of ion
trap cases so in case of ion trap as we know ten qubits have been achieved we looked
into the n m r case where we showed that eight qubit was the maximum which would achieved
so in that sense the number of qubits in case of ion traps have been slightly higher but
all of them have their problems thats why will be discussing in detail as to how to
look at them in case of the preparation the preparation of the computer is in is the standard
start state that we have to put it up and then comes the read outs and the logical gates
which are associated with this would be the once where the controllable interactions can
occur with the outside world and and that could be through single or two qubit gates
sometimes most of these cases that will be discussing here single and two qubit gates
are sufficient in the set of problem that we have first looking at it is not yet necessary
to get very high number of qubits to start with thats why its perhaps not very important
to say all this but generally speaking at this point of time will be looking at the
single and two qubit gates and for the preliminary operations to prove the point so why atomic
qubits atoms and ions are used in conjunction here thats because atomic qubits have certain
advantages they are unparalleled persistence of quantum superposition that is one of the
best features of having atomic qubits because we know that we have the most precise and
accurate clocks which are available in terms of the atomic clocks this also provides control
over quantum states both internal as we mentioned as well as in external and there are many
versions available bose einstein condensate and it allows fermi degeneracy which is controllable
and then there is this mott insulator transition that we can look at quantum squeezing quantum
states engineering so there are many other properties to work on and get them to
the explode as we look into this particular concept of atomic qubits atomic ions demonstration
of building blocks of scalable quantum computer thats what the effort has been in terms
of get into scalability of our quantum architecture
so here is an example of the simplest possible atoms that we talk about so atoms in a
trap on more precisely atomic ions or atoms in traps can be looked at as if the state
of the system can be changed where the the ground state and the excited state can be
considered as the state zero and state one so this is roughly what we are we are
talking about the internal state of the system for instance and the laser interaction
can takes state zero to state one and this is an example of how we can understand
the concept of quantum and this i think i had opened with this kind of demonstration
at the very beginning of this course where we have shown this particular example where
we had showed how we can take a light pulse to produce coherence position of states one
and zero by shinning a light so that a superposition occurs between the states one and zero instead
of being only in one or in two state classically you can only be in state zero or state one
whereas in case of quantum mechanical aspects you can produce this particular state which
is coherent super position of states one and zero so that was the basic idea behind bringing
up this concept of quantum gates now making it go forward what we are looking at is not
just a single atom but several of them which can be put together in a space which is controllable
under the conditions of the environment provided which will you is what we will be
discussing today and under that point there are two different cases one is the say the
internal qubits state which could be when the spin of the these individuals ions are
in the anti parallel case or when they are in the parallel case with respect to each
other whereas the motional qubits states could be while they are having at rest which one
we treated as n equal to zero state or there is a collective motion which can be treated
as n equal to one state so its almost like a simple harmonic motion of this collective
states which can be then looked at as a qubit condition for this case so thats how
this is working
so a little bit detail now as we go into this basic principle of this so we are talking
in terms of a collection or string of of trapped atomic ions and which forms our qubit in terms
of two conditions as we discussed one is the internal atomic levels and so that has to
do with the principle that the de coherence has to be long enough compared to the gate
operation that we want to do with these things so that comes to our principle of quantum
memory which is raise to the requirements that you need that the face delay has to be
long enough so that the face of the system does not decay or the coherence is long live
so that is how embedding the t two structure of the story this is something which comes
which has appeared earlier in n m r if you remember and in terms of the other applications
it has been very highly applied in the as i mentioned before in terms of clocks because
it has high accuracy and stability as high as ten to the power has to be quite
high and that is what is being provided s w t has to be greater than one over ten to
the power fifteen so once you apply an electric field in this particular case the state can
either be in the round case which is the zero state co instance or once the field is applied
as we have discussed it could also go to the state one and so we have the two different
possibilities so this is the internal atomic state the other one which is sought of like
a entire data set which has been once loaded can undergo common mode motion and therefore
have the applications of properties or discussions or it can actually undergo changes
or it can be actually utilized say for doing operations then would like to have the common
mode motion in that case
so for example we can have all of them under the same condition once again in this condition
it has to also satisfy that the coherences to be greater than the gate timing and which
is in this order and here looking at the condition where they are moving which is
our state one lets say verses the case where it is in the initial condition so the total
system going between this initial condition where all of them are in one kind verses the
other case where they have been moved closer together so once again when they are separated
out is our state zero and then once we composite a little bit then it becomes our state
one but this is a as we have been trying to value more and more times is that is the
entire bunch so this is like the data bus the entire bunch which is undergoing a motion
and thats why this is a transitory condition which is what we are looking at so these are
the two different conditions of the trapped ions that will be utilizing when we discuss
the application of ions for doing quantum processing informations and all that so the
basic in some sense the basic operation principle of this quantum system which we are going
to utilize for an quantum information processing or computing with their with therefore involved
these kinds of principles say quantum logic gate between two different ions lets say say
i and j or our two ions that we are looking at and we are going to look at their basic
operations here so the lets look at how we can get them to operate so first is to
prepare the qubits using single qubit gates so as we have mentioned before preparing single
qubit gate means addressing individual elements so this is
for example addressing the individual element i and individual element j so this is our
preparation where we have producing the qubits using the single qubit gate principle so we
can take the view that we have essentially taken these two particular cases under the
applied field condition to make them go to state one
for example where is anybody else in this entire set still remains a state zero next
when this is been prepared by using single qubit gates you can map the qubit i state
to the motion with lasers so now that this has been achieved in terms of labeling or
preparing the preparation has been happened then we can actually utilize the principle
of collectiveness to map qubit i state to the two motion with lasers and for that we
can actually provide a laser which will done gives some sort of a movement as we
have just now shown here in this case with respect to the state i so in this case
for example when the the laser is on the operative on i specifically to make sure that thats
the one which undergoes a change in this particular case due to the motional aspect
whereas the j state remains the same so this one therefore has a different case
because of the selective application of the mapping that we just date so which this means
that we have a two qubit gate between the motion and the ion j now because the ion j
has been prepared from the principle of the internal condition where as the the
state i has been put back to this other motional condition and so the two qubit gate between
the motion and the ion j is therefore created because the rest of the gates are all the
same
so finally what we have done is that now created a gate which is specific created a set
which is now two qubit set because of the condition of j so we can put the information
from motion back into ion i by putting the laser back on i state and so we can essentially
have the interaction go between that different condition so its like saying that we can
prepare by using internal qubit states for both for single qubit conditions then we can
map one of them with the motion with lasers and then we can get back to the case were
we now have the two different conditions available for j where is the i has gone back to the
original case and these two qubit gate between motion and ion of the j can then do its
job and then put the information from the motion because it has now come back to the
original case back to back into ion i so this is basically the set procedure that can be
applied to get multi qubit operations going in terms of both the motional or the collective
mode verses the individual ions that we are talking about
so now for studying these kinds of ions states and to be able to do something with how we
are going to apply them eventually so we would like to have dynamical trapping of this
entire process and that is typically attain with r f trapping and the applied field
of the laser is used for doing the addressing so the dynamical r f trapping it helps us
because we want to confine the charged atoms in the field and we can use the eherenfest
gauss principles and and we know that we cannot simply use static field because they are not
going to help us in this particular condition when we have this charged atoms which were
going to have their own fields and we are going to also apply lasers from outsiders
individual fields so we have to use oscillatory fields because if you want to confine
them in three dimensions its just to use the oscillatory field and in three d we have a
form of the potential which has to be of a certain kind we are not going into the details
of this but these essentially provides us the relative motional condition or the
the exhilaration and the associated force which are provided and to balance this entire
process and we can undergo the mathematical transformation sphere two kind of c how the
reduced mass motion of the system is finally going to be coupled to the applied frequency
of the fields that we are going to provide for this to operate so this dynamical r f
trapping therefore can be see how they map with respect to the positional variance
as well as the frequency as a result of this and that would give raise to the oscillatory
motion that we are expecting from this trap system the quantum motion of this is equivalent
to the quantum harmonic oscillator motion and which gives raise to the wave packet breathing
at about the same frequency as these particular motion that we have just now discussed
and so this particular dynamical trapping concept helps us in providing the oscillatory
quantum like features that we are able to utilize so a coupling of this particular approach
along with the in the linear ion trap allows us to make use of the principle and so this
linear ion trap which which is been the main principle here utilizes the r f field as we
just discuss in this radio frequency field this r f ions would interact by exchanging
vibrational excitations and that is because of the principle that we just discussed were
the oscillatory motions almost may make the simple harmonic principles shown by the
harmonic oscillator mode of a quantum system almost each of the ion can be controlled by
polarized properly focused laser beam in addition to what we just now discussed in terms
of the r f so that we can have the coupling that we just discussed before that the
picture have shows how the electrode structure typically looks for a particular linear
ion trap experimentally and the electrodes are roughly about millimeter thick which is
what is used for making this systems work in in practice or in schematic case this is
how the picture looks like we are applying the oscillatory field as we discuss
before so because we know that static work for it so this actual confinement
parties static and the radial confinement part is dynamic so there are many different
places were this particular approach of linear traps has been used one of the pictures
that i showed you as from was which has been used mostly in oxford then there are these
other places were similar traps have been very successful in max planck and as well
as this hand book as well as in many places next and in us many places include in
maryland and others
so this micro motion which we show in terms of the r f frequency is the one which is on
top of the oscillatory motion that is there for these things now these these are
very small at an different frequency so they do not really do not really effect one
or the other but they can be actually utilize in the way these confinements have been done
we have to very careful in terms of making sure that our applied radio frequencies always
much much larger than the internal confinement the dynamical principles that exists because
this but as you can see in the diagram because of the because of the large difference
in the two frequencies we applied r f always is consistent version that can be utilize
with respect to the other micro motions that are that are already existing in the
system so there are various different ion traps set ups has been discussed for typically
the they applied fields are of the range where we have about ten volts of dc
volts and then the r f is about seven fifty volts as you can as i have mentioned before
the ratio of the two are quite different and the frequency oscillatory part is about
two hundred and thirty megahertz which is the very high frequency so what we are talking
about here as you can see are the three different frequencies which are extremely different
from each other and they are not going to actually create any problem with respect
to the different applications so we have the megahertz coming in one hand the
radio frequency coming in the other and the applied field coming in the other hand so
we have these three different kinds oscillatory motions and this is our harmonic oscillator
which is essentially working at about ten megahertz and conditions so here is a micro
expanded blow up picture i dont know how where you can see here but these are the few trapped
ions which have been depicted inside the actual experimental set up which is what we
have essentially earlier used and blown up in the cartoons to show you exactly how they
look and if you the the single ion life time is can be pretty high and under conditions
it can almost always staying there so given all this backgrounds ion traps have some very
useful advantages as now has we have been talking about that their properties and
conditions can be maintained with very high accuracy and precision over a very large number
of days
so in terms of the motion inside the trap a single ion is almost like a mass
on a spring has we were discussing and it can be seen like this and there are multiple
cold ions which are simultaneously used the normal modes of the string moves as one
there are n ions so there are n modes of the direction so that depends on the center of
mass and that gives c o m and that gives raise to the overall frequency omega x which
we have seen here oscillatory frequency so this is how this whole picture looks like
where if you look at the single ion its like a mass on a spring where it has all these
probabilities of being finding an overview certain range which is your frequency of the
oscillator in the multiple cold ion conditions the normal modes the string moves as one and
the whole picture is essentially looking like this where you can take the center of mass
of the entire process and provide the frequency associated with this and so there are n modes
per direction that this can be associated with so depending on the applied laser
and the properties as we go along we can actually make this and do it this way
the other important aspects is that we could also have different properties such as
such as the stretch mode such as the stretch mode so one one as we discussed was a combined
motion which is center of mass then the other one would be like a stretch mode which will
be like the harmonic oscillator conditions and so there are different kinds of modes
and oscillation that we can discuss in this particular case there are however important
aspects to remember that in terms of the technical details there is motional heating which can
happen and for extract of particles which are like this been set to a state in cold
conditions it might not be possible for it remain under the same condition have although
the life time can be high but after cooling to the ground state of motion the ion can
heat back up and so the timescales of motion manipulations are roughly in the range of
ten micro seconds and this has been this has been consistently becoming better with
the number we also have experiments
so for example they have been cases which this has been improved upon the number
of b e r so that these difficulties can be made better and also the motion can be
made to be sensitive to the noise spectrum which are not going to hurt it or do something
about it however there are issues which has still to be addressed and these are the few
points that i thought i will point it out and these have all been presented in the
previous work and in the most recent developments in the current times we have this few wonderful
developments which have been shown were h this bolder i has has been running the best
possible condition with um less than one over four milliseconds timescale so they have the
longest level life time where as the others are also getting closer but they have
their own difficulties and they have been trying to do their work quite well and so
with this starting point of our discussion on ion traps with some connection to how these
become useful to quantum computing let me close this lecture because this is a a
new area that we have been en blocking on for this particular application and so
i thought i will give you some of the background material which goes into understanding the
ion trap principles which we have just discussed here in the next class we will be detailing
further on the application of the ion traps into making of quantum information processing
and further more thank you see you in the next class