Optimal and Robust Energy Transfer in Light Harvesting Complexes (Google Wkshop on Quantum Biology)

Uploaded by GoogleTechTalks on 28.10.2010

MOHSENI: Good morning, everyone. I'm going to continue essentially on talk by--an excellent
talk by Mohan on exploring quantum effects in photosynthetic complexes. In particular,
I'm going to, in this talk, discuss how an interplay between quantum coherence and environmental
fluctuations can lead to optimal and robust and actually transfer within light-harvesting
complexes. This work is done in collaboration with Aleriza Shabani, Seth Lloyd and Hersch
Rabitz. So, one of the major challenges that they are facing today in designing efficient
solar cells and also sensors is to be able to manipulate and control excitation energy
transport in these solar materials. And this is especially a major obstacle to design materials
with long diffusion lengths. Typically, the diffusion lengths in these solar materials
are about 10 nanometer and so this has led to serious complications in designing photovoltaic
cells to create this kind of a mix blend of donor and acceptors and to have like an average
distance of about 10 nanometer. So that the exciton, which is the electron hole created
after light has been absorbed by a molecule to transfer to here like an electron acceptor,
and charge separation happens. So, this is a major problem because collecting this electron
through these percolating networks are very inefficient. So, if you have methods to enhance
exciton diffusion lengths we could actually go back to simpler designer structure for
photovoltaic cells like a by layer structure and consider an electron donor with enough
volume to absorb light and being able to transfer all this excitation to an electron acceptor
and so we are now dealing with this complication of electron collection. But this has been
done in nature for a long time, you know? So, you can consider this photosynthesis,
which is the source of like energy on Earth, as a major R&D, nanotechnology R&D operations,
for four billion years. And so, it's assumed usually that the output of this operation
should be pretty efficient. But the question is that, is that really the case? And in order
to explore this hypothesis we should probably test one prototype. So in this talk, I'm going
to concentrate on finding materials on protein complex in green-sulphur bacteria which Mohan
talked in detail about it so I'm going to skip lots of the more introductory concepts.
I'm going to a more technical details of how the energy transport happens, what are the
environmental interactions, and addressing the question, is this protein complex is so
optimal and robust with respect to variation in these parameters, both internal and external
parameters. And also, one more important is that, how likely to end up in this particular
geometry? Is there anything particular about the--this chromophoric complexes that make
them so efficient or they are just that trivial, you know, random structure that are operating
because of certain time or skills operation? I'm going to discuss this in detail. So, green-sulphur
bacteria lives at the bottom of ponds and ocean. And in the conditions that are virtually
no lights, they can capture one photon every hour so and there's--so, you assume that they
should be pretty efficient. And this Fenna-Matthews-Olson protein is connecting antenna of these bacteria
to reaction center. Essentially, it's like a channel gliding the energy, gliding the
energy from this antenna complex to reaction center. And--but you have to know that, as
Mohan talked earlier, there are many different variety of this light-harvesting complexes
and they have different geometries. For example, this is the light-harvesting complex of purple
bacteria and you see that these systems have a certain symmetry, the light absorbed by
this light-harvesting complexes 2 and being transferred to LH-1 and then the reaction
center is here that the energy is stored in a biochemical form of energy. And as you probably
all know, in last few years, there have been a variety of experimental demonstration of
existence of quantum coherence in this light-harvesting complexes. This is started by Graham Fleming
group on FMO protein and purple bacteria, conjugated polymer by Greg Scholes group,
Ian Mercer, using a different technique that--to the electronic spectroscopy, and more recently,
also by Greg Scholes at room temperature for marine algaes, and again, on FMO by Greg Engel
group. At room temperature, they observed oscillating beating of cross-peaks in a 2D
electronic spectroscopy that demonstrates quantum coherence in excitonic basis for these
light-harvesting complexes. So, one might ask why quantum coherence could even exist
at this kind of warm and wet environment? This was typically considered, you know, most
of physicists dismissed any potential role of coherence because of simple argument like
that, that this is just too hot, too wet, just cannot happen. And beyond explaining
that, we have to explain it--or address this question of, "Is there any role for quantum
dynamical effects?" Is there, this is the focus of my talk, is that these are like any
interplay with the environment that actually is important here not that merely coherent
effects. And this is the--I'm going to discuss and tell you that we believe that there is
actually a very helpful collaboration between environment and quantum coherence that leads
to optimization of this photosynthetic complexes. But other more interesting question is that--are
that--that like, this quantum effects could be there and help the energy transfer efficiency
of the FMO protein, for example, but do they have any evolutionary role? Do they, you know,
have like a biological impact at the higher level? We don't know that. And can we make
new predictions by including this quantum effects that are not possible using just simple
classical dynamics for excitation transport and also applications for designing artificial
excitonic devices for a variety of different purposes of absorption and transport, storage
and sensing? So, as Mohan mentioned, one of the first ideas that came out of the Greg
Engel paper in 2007 was that there are certain form of quantum computation are happening
in these final materials from protein. It was speculated that there is a connection
with Grover search algorithm or quantum walks. We explored this possibilities of--we immediately
noticed that there is no--there's not--nothing, anything too close to have Grover search algorithm
in these systems. Essentially, if you would evolve the pre-Hamiltonian of this Fenna-Matthews-Olson
protein, the energy, the overlap of excitation energy state, with the trapping side is never
exceed like 40%. But if this was supposed to be as something close to like a continuous
time evolution of the form of a Grover, you should expect that at a certain time you have
a major overlap with the target side, which is the--essentially the state you are searching
for. But that's not the case here, at least not in the context of a unitary evolution
of it. And also, the connection with quantum walks, as Mohan mentioned, and this is the
work done by Stepfan Weigert and Brierley's group, that there's not any speed up in the
context of quantum walks. So it's a--there is no--that's not the right measure to look
at the potential contribution of quantum interference effects. So it doesn't matter for this system
how fast they get, like the excitation gets to reaction center. Its like--it's a matter--that's
not really leads to efficience of these devices as I discussed. But I think, still the quantum
walk picture a good picture because you can think about--just look at this dendrimers
that these are artificial system that you can have multi-branching of polymers and that
they have like a similarity with these kind of binary tree structures well-studied in
quantum information science. In this binary tree structure you can write a diffusion equation
describing a classical random walk of hopping a--for example, an excitation can hop toward
the root of this binary tree and this transition matrix is described by the connectivity of
this graph. You can, in analogy, define like a continuous-time quantum walk just--this
is--you can write this for any quantum system in which the Hilbert space have a spatial
structure so you can talk about continuous time quantum walk. Essentially, what it means
that the elements of Hamiltonian in the Hilber space that they have a spatial structure denotes
your transition matrix and this is like a quantum walk of a probability amplitude. But
these systems are really interacting with the warmer so their continuous-time quantum
walks are not good descriptions. And essentially, it's not clear, you have to have a good measure
of what part to actually quantify if there is a quantum walk and how does it contribute
to the dynamics. In order to do that, let's just go up on a step back, just look at the
FMO complex. As you see in earlier in Mohan's talk, that these are like connecting this
antenna chromosome to reaction center; that this is like a trimer consist of three monomer
and there's a protein scaffold and a--in the--there are seven bacterial chlorophyll that are like
actually doing this energy transfer, they're operating as an energy transfer channel. There
are two different bases that is discussed in this community. There's a side base which
denotes the spatial coordinate of bacterial chlorophyll and there's an exciton base, which
is the bases that diagonalize the three Hamiltonian of this system. Essentially, what it means
that, these states are delocalized or are extended spatial structure over multi-chromophores,
at least two or more. And the, using 2D electronic spectroscopy, these pathways have been studied.
There are too many pathways of excitation starting at site 1 or 6, these are close to
antenna and so--and they end up at site 3 and 4 which are close to reaction center.
And--but in order to really explore this, you need to start from a like a more formal
mathematical formalism. And this is a Frenkel exciton Hamiltonian. This is--that become
right for any multichromophoric system. This denotes that site energies at site M, and
this creation annihilation operator of an excitation at that particular site. This "Vmn"
denote a coupling between two chromophores and this system interacts with the thermal
phonon bath and radiation field in general. To a good accuracy you can ignore off diagonal
couplings and consider only site fluctuations due to interaction with a phonon bath. And
radiation field can later transition between different excite--multiple--different excitation
manifold. And in order to study these systems, you have to write essentially the unitary
evolution of the entire system to take and trace over environmental degrees of freedom.
Then, you end up with a like a so-called master equations. This is very well-known. As a form
of the study in this system, is to express the time variation of the density operator
which contains all knowledge about the system. As a different operation, these are--this
denotes the unitary evolution of system under the influence of free Hamiltonian. This is
Lamb shift, which is essentially related to reorganization energy of this system and these
are due to non-unitary evolution due to interaction with the phonon bath and radiation field.
And this is related to how much decoherence you have, this is essentially the coherence
rate, these coefficients, and these operators are just nothing but other product of a projection.
Essentially, you can consider two exciton bases or a jump between different excitation
bases and this decoherence rate is just for you to transfer phonon bath correlation function
and can be expressed linearly as a spectral density. And in this relation which N is a
busonic distribution function at temperature T, it's--you can consider Ohmic spectral density
and which will be the cut off frequency and the spectral density becomes linearly related
to reorganization energy. So, we have studied this system using a quantum trajectory picture
and you can arrive, if you do the math, you arrive at this equation. What it means, this
is a non-unitary damping evolution because this Hamiltonian is not--have emission so
this is like a damping evolution. And these are the jump in a fixed excitation manifold
between different sites. And this is very important. I want to emphasize this that this
is essentially what I'm talking about in the Born-Markov approximation as interplay between
quantum coherence and environment because as you might remember, in the Hamiltonian
itself, these jumps are not really--the thermal phonon bath does not lead to any jumps between
different sites, just fluctuating it's site energy, so it's not contributing to the transport
itself. But if you look at these terms, these are relate--this actually creates jump between
different sites. And they happen to be there because of environmental interaction and free
Hamiltonian so if you do not have the coherent couple in between this thing their phonon
bath interaction doesn't lead to that jump. These terms create jumps between different
excitation manifold. These are separated by more than a thousand wave number and the timescale
of these jumps are on the order of one nanosecond compared to energy transfer time of one picosecond.
So you can ignore these jumps in this description and just simulate these first two term. And
this is more a detail expression for this non-Hamiltonian term of decoherence and you
can understand this as a quantum walk in a little space and writing this transition matrix
in this form as a function of this non-Hamiltonian and these quantum jumps. So it's possible
to understand this in a context of quantum work but it doesn't really matter, that's
just the interpretation of the dynamics. You need to have, like, a measure to quantify
the performance of this photosynthetic complexes. So what we used was to look at the success
probability of the excitation being trapped at the target site with different rate "KJ".
And this is--we, you know, define this as an energy transfer efficiency, which is generally
less than one, because due to interaction with the environment we can also have the
loss of excitation due to electron-hole recombination. So what we did was to explore this energy
transfer efficiency as we looked at the different numbers for reorganization energy and so this
just denotes the interaction of the system with the environment. The more--the larger
reorganization energy, you have a much stronger environment and you see that energy transfer
efficiency enhances by 15% or about 30% from, like, in this particular initialization, from
70% to 99%, if you enhance the reorganization energy. So here, environment is actually helping
you in this case to have a larger efficiency. Of course, because this is a perturbative
method you cannot really go and explore reorganization energy larger than about 30 wave number because
the site energies are about 300 wave number. So this is like about one tenths to a perturbative
approximation if it's still valid. But there is no way to explore beyond that. And we want
you to see if this is really an optimal point but in this model we couldn't do it because
it's just--the model would collapse. And you see that the transfer time of the excitation
to the reaction center reduces by one order of magnitude, which is related to energy transfer
efficiency definition and it doesn't contain any more information. So in order to see all
this energy transfer, what are its mechanisms for environment-assisted transport, it's better
to look at these two different scenario. It's either you have funneling, which the interaction
with the environment in this exciton base is helping you to relax to the ground state
which is close to the energy at site three. But also, it helps you--this is a phenomenon,
like, a tunneling in quantum mechanical effects that helps you to overcome a potential barrier
where in essentially, in increasing the spectral density overlap between two chromophores.
We also studied a binary tree structure in here like, a pure dephasing model. This is
like having considering just white nodes which is killing the octagonal element of density
operator. And you see that in presence of disorder, without the noise, the energy transfer
efficiency drops significantly. But with noise, it has enhanced efficiency for this structure
as well. We use this pure dephasing model for FMO. The reason we did that is that in
this thing you can explore all variety of the environmental strengths of--in this pure
dephasing model and so we finally observe this optimal energy transport in this model.
Although, this is not an adequate model to describe the environment in this system but
this was good enough for us to demonstrate that as--there's a--just enough--there's a
certain amount of environmental interaction that could be just the right one. So if it's
not there, the quantum, there is a destructive quantum interference effects leading to localization,
which is like Enders localization for large system. But at a very high, the coherence
limit, that you have this quantum Zeno effect which is the efficiency drop to essentially
to zero. But there's an optimal regime here; and the estimated value for FMO is just sitting
in that optimal regime. But it's still, you know, this wasn't convincing because the model
is not describing the actual non-perturbative, non-Markovian baths of this photosynthetic
complexes. The reason is that, this is a very interesting feature of this light-harvesting
complex, is that the magnitude of the diagonal, off-diagonal, and system bath interaction
is essentially they're trapping everything sits close to 100 wave number. So the time
and scale of these things are very similar and that makes it very hard to study this
system. So this is the major challenge to simulate this system efficiently because most
of the techniques that develop to study this complex open quantum systems are--do not work
here. But there are more fundamental question, is this--like, you can ask why. Why there
should be this kind of--why this photosynthetic complexes operate at this regime that there's
this convergence of time-scale? Is this--have anything to do with the optimality of the
system or the robustness of these complexes and what is the likelihood of just being at
that regime? And so, in order to address this question, you really have to develop more
advanced techniques to simulate. This is really tough area to work with. It's like essentially
sailing in really stormy condition. Environment has lots of unusual characteristic. It has
a memory and it--so it can interact with the system and put, like, essentially giving back
some of the coherences that already, you know, absorbed. And so, informally, you can write
the evolution of the system as this propagator. This is an interaction picture. And this is
like a super operator due to system bath interaction. And this denotes the time ordering. So because
of non-Markovian effects, you have to keep track of the time ordering. And there is a
famous theorem that says that this multi-order of correlation function, if this is expanding
this exponential function, you have--you deal with this higher order correlation, bath correlation
function, and you can write this as a different combination of two order correlation function.
This is true only for busonic baths out there, with Gaussian properties, which is good because
the--most photonic environment for this system have that feature. And so, if you, like, schematically,
if you do the math, like you end up with the master equation, this is--there are many different
methods. Actually I should mention here that to a study they said it's known as hierarchy
equation approach to study this system in this realistic environment. And this is developed
originally by KuboTanimura and recently completed by Ishizaki Fleming. And there is also other
technique known by polaron transformation, time-local master equation, by Jang. But here,
I'm going to explore interesting feature of the pioneering work of Jang Seogjoo at about
13 years ago on essentially, at approximate techniques to stimulate this system. We have
a different revision of this master equation which leads--allows you to estimate how much
error you have by using a simpler model. Essentially, you end up with a pair of couple master equation
which is the contributing terms due to coherent evolution, recombination and trapping and
this is like a schematically demonstrating bath response operator. And mathematically,
you can express it in this form. This is a non-unitary evolution due to loss and trapping,
and this is like a response of the bath with the system. This is like a, in context of
quantum computing, we know this as like, error operator interacting on the system. I have
that kind of background so this is the way I look at it. So considering--although we
know that this is nothing like quantum computation in that sense, but there is a quantum interference
effects that happens which is leading to an optimal regime due to interaction with the
environment. So it's not helping in the transferring, like, your first passage time, as stuff on
that Mohan showed, but it helps to have a higher energy transfer efficiency. But here,
we want to study this in this kind of mode--appropriate model to describe the system in the actual
setting they're operating in the biological environment, so. And here, the action of this
environment, the memory effects, it can be represented by this. This is just nothing
but the--just action of these errors. You can consider as errors acting on the density
operator in interaction picture rated by bath correlation function. And this is a bit more
technical. I apologize for the general audience. But this is--I want to talk in detail to say
what are the complications of really simulating this system and so this bath correlation function
have this imagined and real part. And there are two different spectral density that you
can use that are related to this reorganization energy through a Lorentzian function or an
exponential decay. So, in order to study this system--so the whole point was that, if we
wanted to study this optimality of this system, we have to simulate this system for over a
wide range of parameters. And using the hierarchy equation approach, which is the general benchmark
for simulating this system, you have to solve 50,000 differential equations to--for FMO
complex at a reasonable temperature to simulate this. This is what is done by Ishizaki Flemming
at UC Berkeley and published last year, showing coherent oscillation of excitation of the
timescale of closer 400, 600 femtosecond which is relevant. The timescale of trapping is
one picosecond so this is the timescale of this system. So, using only this pair of couple
differential equation we actually can reproduce these results within good accuracy seeing
these oscillations. But this is much faster simulation and this allows us to explore a
whole variety of the parameter range for this system. This shows energy transfer efficiency
as a function of bath cutoff frequency and reorganization energy. So this axis is the
strength of environment. And here is the non-Markovian effect so this is the inverse of this bath
cutoff frequencies, the coherence of the--coherence timescale of the environment. So here, this
can be measured--used as a quantifier for non-Markovianity of this system. And so, the
closer you are here, it's more non-Markov. You see that these are the estimated value
for Fenna-Matthews-Olson protein that are sitting at the really optimal point, which
is pretty robust with variation in either of these two parameters. And you see that
for a very large reorganization energy, energy transfer efficiency dropped significantly
when it's non-Markovian. But in the--reasonable to reorganization energy as close to natural
setting for Fenna-Matthews-Olson protein, it's--this non-Markovian effect is essentially
helping a bit. But the system is robust with this kind of non-Markovian effect. You can
explore the robustness by mentioning the second order derivative and seeing that it has really
a flat structure. And this shows this. This is like a lateral point of this plot, showing
this optimal environmental transport at this reorganization energy of the--close to 20
wave number. And so, we studied this system as--in various temperatures as well and you
see that this is actually very intuitive. At the very large reorganization energy, so
a very strong environment and at very high temperature you expect the energy transfer
efficiency drops. And that's what happens. But at the values of environment for FMO and
room temperature, it's sitting at the optimal robust point again. And so, this is--at significant
point, you know, that--it seems that the parameter of this environmental reorganization and memory
and temperature and all these relevant parameters to be just at the right regime for the FMO.
But one caveat in using energy transfer efficiency to quantify this system is the fact that this
hasn't been measured experimentally, and we estimated about one picosecond timescale based
on that special separation and so, we consider this as a free parameter to explore energy
transfer efficiency. This is very interesting because you see at this regime, if the trapping
is much faster, this is like one picosecond, and so if the trapping is very slow the excitation
just sitting there until it dissipate to environment; and that makes sense. But here, you see that
even if the trapping is very fast also, you expect that the, you know, energy transfer
efficiency becomes more efficient, but it's not the case. It becomes less efficient. So
there's an optimal regime here. That doesn't make sense at the beginning but, you know,
I can give you a classical example. Suppose there is a gas chamber and someone is sentenced
to death, like sitting in there, like, you know, this is--and there's like a--the gas
act within a timescale of a minute and there's a revolving door that rotates in a reasonable
timescale. If this timescale is very slow, like, on the order of hours for one rotation,
it doesn't really matter if the door is actually rotating it's very unlikely that the person
can escape on the gas chamber. But if the door is rotating at the right speed, you know,
on the order of like a 30-second or something, it's very likely that the person can escape.
But if the door is rotating really fast, like, you know, 30 rotations in a second, there's
no way, that it's very unlikely that the person can escape. So it's, like, this is the case,
you know, it's like Zeno effect. It's just absorbing it too much, you know, as the excitation
cannot move, essentially localize, and just dissipates to environment. But there are other
things that are not very well understood so we consider that a bit as a free parameter,
is that the location of trapping is also--it's assumed to be the reaction center close to
site three and four but that's not really known for sure. We consider this like a--to
a--similar to this system based on different trapping site, just rotating essentially this
FMO within that geometry. And observe--first of all, we observe this environment as a transport
no matter where the reaction is but it's interesting that it happens that the site three and four,
which are believed to be close reaction center, are actually provides the most optimal energy
transfer efficiency. And also, there is like a whole discussion about the role of initial
states. That if there is a coherence initial states or incoherent because the solar light
is not coherent, the experiments on with coherent state of light, of lasers and things like
that, so we just consider--to consider the effect of initial states just randomly sampling
over 10,000 different initial states, considering all different coherence fully classical mixture
of statistical measure of states. And considering worst-case and best-case scenario, you see
that at really large reorganization energy, when the environment is very strong, there
is a huge dependence on the initial state so the system is not robust. But it happens
that exactly at the value of the reorganization energy of 35 wave number for FMO, this is
very small dependence to initial stage--initial states about, like, a few percent changes
in the efficiency; which was surprising. Also, the effect of correlation in the bath. So,
there have been a lot of discussion, is that the bath essentially this protein scaffold
is like oscillating in a fashion that creates correlated fluctuation and that helps the--this
complexity to operate efficiently. You see here that the regime of large reorganization
energy, this is the case. And the bath correlation of--in the bath, defined here in this bath
correlation function by this exponential function, this is the distance between two sites, N
an M, and this is a correlation lengths. And showing an exponential decay based on the
separation between two chromophores you see that if you have, like, a higher correlation,
you have a better efficiency. But at the regime that the FMO is actually operating doesn't
really much matter. So this is also robust to correlation and environmental fluctuations.
But one of the things--after studying the system in this all parameter regime, there
was one thing that bothered me in particular and my colleagues, in that maybe this is not
a big deal. Maybe this is just everything we see is that there's a convergence of timescale
about one picosecond for, like, the strength of Hamiltonian, system Hamiltonian, free Hamilto--and
system bath Hamiltonian trapping. But there is a major three order of magnitude timescale
separation between everything that we know with the loss dissipation, which is one nanosecond.
And so, it doesn't matter how actually you go there, you just--you are--excitation has
so long a lifetime that moves around enough to be trapped. And so, it's not a big deal.
Any structure--so this, based on this argument, you expect that any structure pretty much
gives you a very good efficiency within that parameter regime. And so, we consider this
like a--explore this over seven chromophores in random orientation interacted through dipole-dipole
interaction with the distances being bounded between 5 to 50 angstroms that the dipole
approximation is valid and sampling over 100,000 different random configurations. What we observed
was that, and using these known parameters for FMO and environment which is 35 wave number,
50 wave number for bath correlation timescale and trapping one picosecond and loss one nanosecond,
you see that essentially 60% of this random configuration's done, like, less than 10%
efficient. And the one that are over 50% efficient are like 10% of them and just only 1% of efficient
as, like, more than 95%. And close to FMO, it's like 100,000 configurations can be that
efficient. So, this is shows that this geometry is kind of rare but it's not that rare that
it's--so it's kind of, you know, 1% is actually not that bad as well. It shows that there
is certain robustness to these variations. So it's not like, you know--because if it
was one in a million, you expect small changes in a structure of FMO significantly, catastrophically
reduce the energy transfer efficiency so then that's not good for like a system that operates
in a wide way essentially, the environment is completely uncontrolled. So, after this
study, we showed using a variety of different techniques that FMO dynamics we could actually
simulate this in an intermediate and non-Markovian and non-perturbative regime. Yes, I should
mention that the model this approximated model are to be developed based on Jang Seogjoo
is that, the--quantified the errors. And the errors have really blow up in the regime of
really highest point of reorganization and a really low bath frequency cutoff. So, these
are really only appropriate models for intermediate regime. But it happens that we are lucky here
and the biologically-relevant regime is the intermediate. So, we showed environment-assisted
quantum transport in a variety of setting. And always there are parameters rely the--was
in that area that was pretty robust. And also, we explored that a structure and rule of geometry
of Fenna-Matthews-Olson protein and see that this is a rare geometry. Now, that the questions
that everybody's interested now are how can we actually use this quantum coherence effects
to engineered novel materials that outperform classical operating devices for sensing and
light-harvesting, artificial light-harvesting complexes, for and in context of photovoltaic
cells and other potential devices that, you know, could emerge that we cannot even think
about today. And so, there are--this is the--I would to acknowledge the sponsors, financial
funding from NSERC and DARPA and Eni. And yes, that's it. Thank you for your attention.
Yes? Can you come to the... >> Yes, please. When you ask questions, you
have to come to the microphones because, as we are know, the whole set's recorded.
>> I have a technical question. So, for the non-Markovian case, you used this method by
Cowell and you compare it for--with the Ishizaki, the full hierarchical resolution. You got
good agreement. Is this across a temperature range or is it just at higher temperature?
Because at lower temperature, you start having so many equations it's become almost impossible
to solve. >> MOHSENI: So, actually, yes. It's not easy
to simulate the low temperature using hierarchy approach.
>> Right. >> MOHSENI: Because the--so, I did not mention
this explicitly, so that the problem with hierarchy equation approach is that the complexity
of simulation with the--it grows factorial and that it's grows exponential with respect
to high reorganization energy, low--essentially, passed frequency cutoff when it's very highly
non-Markov. And at the regime of low temperature and also with the size of the system, it's
just--this just grows exponentially at best and so it's just not possible to explore this
at very low temperature. >> So you compare it at high temperature as
in fact and it seems to it? >> MOHSENI: Yes, high temperature which was
like, you know, the most relevant temperature for this system as well.
>> Sure. >> MOHSENI: They're not really operating at
low temperature, anyway. >> Thank you.
>> MOHSENI: Thanks. Any other question? Yes? >> On the temperature...
>> MOHSENI: Yes. >> On the temperature scale. I mean, how sensitive
is it to temperature? >> MOHSENI: So, you mean, how sensitive is--so
this is the plot with respect to temperature. So, it depends on the reorganization energy.
That's the reason we used 3D plots because if you fix the reorganization you observe
something for that particular value, and it doesn't mean much about the different reorganization
energy. So, first of all, I should tell you that really this model beyond this kind of
reorganization energy is collapsing so you have to do like a brute-force hierarchy equation
approach. So, we cannot say much about really large reorganization energy but this is the
border that we can explore this, using this efficient simulation method. And it shows
that it's in--so, in--at--if you increase the temperature, the energy transfer efficiency
drops significantly at the--for large reorganization energy. But the whole point is that for this
kind of--around, you know, between zero to--up to 100 reorganization energy, it shows that
it's--that it's--the system is kind of robust with respect to temperature. And so, this
is what we observe using the [INDISTINCT] master equation or like more Markov technique.
But--so, it's interesting that for this reorganization energy, it looks like that--it's robust with
respect to variation in temperature. So it's not--you see there, based on this definition
of energy transfer efficiency. But, you know, we have to be careful about that too. It depends
what you really look for, you know? What is the actual, you know, function that you are
considering as the measure of the efficiency. But I think this is--what we use is actually
widely being used by other groups. That was originally was used by Klaus Schulten in the
kind of context of energy transfer efficiency of this complex as in--using a diffusion master
equation though. But--and so--and also being used for other system--people in quantum information
science to quantify the energy transfer, like a transport efficiency in binary tree structures
by [INDISTINCT] group. So it's not like they are--that it's just more--our definitions.
Other people are using it and it looks that--it just so that the big, you know, if I want
to summarize the talk in one line that's saying, "Well, what is the significant here," is that
it appears--so there was a conjecture by Engel-Fleming in their 2007 paper that there is a constructive
quantum interference happens to be important to have this in the efficiency of the system.
But although, they speculated about or other things that, you know, might not be relevant
in a context of quantum computing but I think that conjecture was right in the sense that
quantum interference affects all relevant but in a sense that when you consider environmental
interactions, there is a really optimal regime that both quantum coherence and environmental
interaction have to essentially collaborating to have these high efficiencies. And it's
not correct to look at that in the context of first-passage time as Mohan mentioned.
And that's not the right picture. It does not provide any speed up, but it looks that
within that definition, it's very robust with temperature, too. Okay.
>> Okay. Thank you.