Uploaded by GoogleTechTalks on 28.10.2010

Transcript:

>>

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.

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.