>> 
SAROVAR: Let me begin by thanking the organizers,
too, for the invitation to speak here. I think
it'll be an interesting workshop and I'm looking
forward to some stimulating discussion. So
I'm--yeah, I'm going provide the introduction
to quantum mechanics of photosynthetic light
harvesting. You can't quite see my title but
it's quantum mechanics of photosynthetic light
harvesting machinery. So, I'm going to provide
a bit of introduction, then talk about some
of the work in our group in this area, okay?
So, right. And since this is a quantum biology
workshop, I thought I'd motivate a little
bit and give, you know, start at the beginning
and talk about quantum biology and what it
should mean or what it does mean. And the
way I like to think about it, I like to classify
it into three levels of quantum influence
in biological systems. So going down the list
here, I mean, the first one is the trivial.
Since we believe quantum mechanics is the
theory of microscopic systems, then everything
is quantum mechanical at some level, right?
But that's--at this level of reductionism,
it's--you can't really add much to knowledge.
You know, quantum mechanics dictates energy
levels, molecular orbitals, and so on, but
does not really tell us anything new. The
next level up, which is more interesting,
is the level of molecular dynamics and chemical
kinetics. So, for the vast majority of biology--systems
of biological interest, you can describe chemical
kinetics, molecular dynamics by just classical
physics, okay? But in a few cases, there are
quantum corrections which turned out to be
significant. Just some examples here, this
non-adiabatic transitions through conical
intersections, which are a popular research
these days, I mean, you need to have quantum
mechanics in order to describe the rates and
the correct transfer in these cases. Yes,
and then there are chemical reactions involving
tunneling of electrons and maybe protons.
There again, the quantum corrections become
important, okay? And then the next level,
which is maybe the most tantalizing, is where
quantum mechanics becomes a functional necessity,
okay? And, you know, the division between
these two is kind of hazy, where you throw
one and where you throw the other is not clear.
But at least the third class of quantum influence,
you should be able to make the connection
between if the quantum mechanics wasn't there,
if the system was entirely classical, the
organism wouldn't be able to survive, right?
And I've listed some examples of some potential
examples from this class, the way geckos hold
on to things is maybe through van der Waals
forces, magneto-reception in birds, olfaction,
and finally, photosynthetic light harvesting,
which is what I'm going to focus on. I should
say that the last class is really the only
one where we have convincing experimental
evidence for quantum effects. You could still
debate whether it belongs in class three or
class two. I think--I'll talk more about the
role of quantum mechanics in light harvesting,
but I tend to put it in class three. Okay,
so then, I should also go over the skeptic's
view, right? This is--if you say to a, even
the normal scientist, you expect quantum mechanics
to play a role in biology, there are few arguments
for why this should not be the case that most
people bring up. The first of these is a classical
kind of thermal argument. They say that there's
too much thermal energy in biological systems,
right? The typical energy scales that you're
concerned with is just overwhelmed and drowned
up by just the thermal noise and thermal energy
in a biological system, okay? And that's a
fairly weak argument in my point of view because
it's a very equilibrium argument. I mean,
the temperature is an equilibrium concept,
right? And in most biological systems, it's
the non-equilibrium dynamics that matters.
And you can have systems that take a very
long time to actually equilibrate. And you
can be very quantum mechanical in that region
before equilibration or before thermalization
and still have a state that's thermal at the
end, okay? So this is not a convincing argument,
I don't think. A more sophisticated way of
putting this comes in the framework of decoherence
theory where you really consider the timescales
in the system. You say there are two timescales
that matter. One is the timescales that of
interactions within the system. And by system,
I mean the biological thing that we're looking
at, the structure that we're in and that's
of interest, right, that's parameterized by
this J. So this is the rate at which dynamics
occurs within the system, and one on J is
the times--the coherent evolution timescale.
And then there's a rate of interaction with
the environment, and that's parameterized
by this lambda here. And so you can see that
what J does is it likes to create super positions
between states, between classically distinguishable
states, okay? What lambda does, it likes to
decohere these super positions into classical
mixtures, right? And so there's an intrinsic
competition between these two rates and so
a nice way to talk about how quantum the system
is, is to think about this ratio: J on lambda,
okay? And so you can draw the scale down and
say that, well, we know that all the way on
the right, where J on lambda is large, is
where things are very quantum. And here, we
have classic systems from, well, these days,
from quantum computing. Trapped ions, you
know, super inducting systems, maybe protons
and atoms low opticalizes and so on. Here,
J on lambda can reach, you know, tens of hundreds.
On the other end of the scale, and this is
typically the skeptic's view, right, that
this is where biochemical processes like daylight,
all the way on very small J on lambda, okay?
But one should ask the question, "Is that
a priority necessary?" I mean, should biochemical
systems only lie on that side? And what I
will hope to show you is that we know of at
least one biochemical system which is much
closer to the right-hand side of the scale,
okay, and maybe there are more. Okay, so that
brings me to the main topic of the talk: quantum
effects in light harvesting systems. I'm going
to provide a little bit of introduction in
the absence of our line of talk here so bear
with me if you already know all this stuff.
So the structure of photosynthesis really
is that you can break it down into two sections,
what typically we call light reactions and
dark reactions. The dark reactions are the
chemistry essentially; the Calvin cycle, everything
that goes from sugars--everything that goes
from units of biochemical energy to sugars.
And then the light reactions, which is what
I'm going to focus on, is what the things
that are initiated by light, by photons, okay?
If I zoom in on those light reactions, this
is a cartoon picture of what happens. You
get light impinging on absorbing pigments.
There could be a whole variety of them going
over multiple energy scales so you have blue,
orange, red. Eventually, the resulting excitation
migrates its way to something called the reaction
center which is where most of the chemistry
is initiated, okay? So this is where you get
a setting up of electrochemical potential,
which is essentially a battery for the rest
of the reactions. So, I'm going now zoom in
a little bit more and just focus on this first
part, the--what are called the antenna systems.
These are the things responsible for absorbing
the light and also eventually transporting
the resulting excitation to the reaction centers,
okay? So there's a huge variety of antenna
centers. This is just a picture from Blankenship's
book. So, if you want to make generalizations,
then something you can say is that they're
all composed of densely packed pigments. So
pigments of things like chlorophyll molecules
which absorb light very efficiently, okay?
And most of these pigment or molecular aggregates
or these pigment aggregates are encased in
these protein cages such as these that you
see, which lend some stability and some energetic
structure to the system, okay? And all light
harvesting complexes are also embedded within
membranes. And I'll come back to this point
at the end, why this could be important, okay?
So another generalization you can make, which
is perhaps the most important at this point,
is that energy transfer is very efficient
in these systems. Almost--well, greater than
95% efficient in most cases, and it happens
at very fast timescales, picosecond time scales,
okay, so these are some of the fastest processes
we know in biology and arguably some of the
most efficient as well. So, let me introduce
to you, well, a specific light harvesting
complex here. Now, I'm going to concentrate
most of the theoretical studies later on this
complex so I'm going through them in a little
more detail. This is the apparatus of green
sulfur bacteria. So these are bacteria, these
are unoxygenic so they don't produce oxygen,
but they live in these sulfur-rich environments,
typically in oceans or swamps. This is a picture
of them. If you zoom in to the light harvesting
apparatus it looks like this. It's essentially,
most of the light harvesting, the antenna
of the system, is this large complex you see
at the top here. This is called a chlorosome.
It's composed of very densely packed pigment--pigments
and that's where most of the light is absorbed.
And eventually the resulting excitation from
the absorption migrates its way down to the
reaction center which you can see here. Can
anyone--can people see the cursor on the screen?
>> Yes.
>> SAROVAR: Is that okay? Okay, all right,
I'll keep using that. But in a--so in this
migration of this excitation energy, right
in the middle is this thing called the FMO
complex, okay? So it's essentially a wire
that channels the energy from the main antenna
to the reaction center, okay? I'm going to
zoom in on that a little bit. It has this
structure, it's a trima of identical units
that looks like this. If I just look at one
of the monomers this is its picture. And if
I strip away the protein cage, all this gray
matter around it, it's essentially just seven
chromophores, seven chloroform molecules,
okay? There's a standard labeling; one to
seven. And if it--the way I've drawn it here
is it's actually what we believe it looks
like in-vivo, so this is the orientation.
The light enters through chromophores one
and six from the base plate and it leaves
through chromophore three, right? So it really
is just a wire that connects the main antenna
to the reaction center, okay? And it is very
efficient. It's amazingly efficient at this
job it's been given of funneling this energy
down. So, as you probably all know, there
were experiments on this but--on precisely
on this complex. So, in the FMO complex in
2007, which showed that not only are the static
features quantum mechanical but the dynamic
features in the system are also quantum mechanical,
meaning that how energy moves about among
these many different pigments is not by just
hopping from pigment to pigment, it rather
moves around in a wave-like manner. It samples
a lot or many of the pigments at once, okay,
as it's moving around. And if you look at--so
I mean, I like to think of this as basically
getting--refuting this classic picture of
energy transfer in these systems, of excitation
jumping from pigment to pigment and rather
moving to this other picture where this is
now an energy space. This is the energy levels
of the system and it's just coherently sampling
the energy levels. It's moving up and down
more like a wave, okay? I know it's a very
cartoony picture but I think it's useful to
think about that the C-change in the picture
in which the dynamics is happening, okay?
Yes, and then as Hartmut mentioned, there
had been several experiments since then, but--so
this initial experiment was done at cryogenic
temperatures and now there's been multiple
experiments at room temperature which confirm
the same effect and in multiple organisms
too, so not just in green sulfur bacteria.
This is not an outlier in the space of light
harvesting complexes, okay, it's been confirmed
in marine algae, in purple bacteria, also
in higher plants. So LHC-II is a light harvesting
complex in--present in spinach, all right,
so--and in most of the green plants that we
see is--contain LHC-II, and we know that there
are quantum coherent effects in LHC-II. Okay,
so now moving on to the theory, there's been
a lot more theory on these systems. And this
is just a small sampling of the theoretical
papers out there. People have looked at efficiency
of these systems and how quantum mechanics
plays into the efficiency. I think Mosuda
will talk a little more about this in the
next talk. People have also looked at a more
exotic quantum properties like quantum entanglement
and speedup and dynamical, in a dynamical
sense. And I'm going to talk more about these
two topics in--later on in the talk. But just
the--just a survey, there are other things
that people have looked at and Mosuda will
again present some more on this. There's optimality
of the system. Is it really tuned to do what
it's meant to do or is it not really optimal
at all? And I'll let Mosuda answer that question,
because since he's done a lot there. And then
on purple bacteria, which are in a sense more
heavily studied, because I guess the quantum
mechanical effects there, at least, the static
quantum mechanical effects were known for
a longer time--a longer period of time. And
people who looked at delocalizations, so how
delocalized, coherently delocalized, is the
energy in the system? How will the transfer
rates in between modules of this purple bacteria
light harvesting complex dictated by quantum
mechanics? And also in higher plants, people
have looked at entanglement in these light
harvesting LHC-II complexes of higher plants.
Okay, so now, I'm going to move on to some
of the work that we've done specifically and
I'm going to begin with looking at entanglement
in the FMO complex itself. So, let me define
entanglement, give a brief introduction. So
entanglement mathematically is this statement,
that if I'm going to write down a quantum
state, if I can write it and it's a quantum
state of many parties, so I've divided my
system into a bunch of subsystems and there's
some natural division, if I can write it down,
if I write down my quantum state as a state
of party A, tens are product in, but a state
of party B tens a particle that would save
party C. If you're not familiar with the tens
of product, I mean, it's just a mathematical
term for saying "this independently of this,
independently of this." So if you can write
down the state of the combined system, independently
of--in a way that each system has it's own
independent state, then we call that a separable
state and then any state that's not written
this way is an entangled state, okay? And
so, I've written an example here of an entangled
state. So this is--you could--these are spins,
I'm spinning these, pointing up or down. So
you can see that you can--you can't factor
out the up component of the spin state, right?
If you factored it out of this state you can't
factor it out of here. So this is party, the
left-hand side is party A and the right-hand
side is party B. Okay, so that's curious and
you might think that's nice math, but why
does it the matter? And really to understand
the fundamental importance of quantum mechanics,
you have to think about what it means for
what the state is telling us about knowledge,
okay, I think this is the clearest way to
imprint up with the system. So the entropy
of a state, we think of how much uncertainty
there--about the state of the system there
is, so it's a measure of uncertainty. Just
like classical entropy is measure of uncertainty,
you can define it in quantum entropy, the
one normal entropy as a measure of uncertainty.
So it's this entropy function here, there's
S of rho. If I calculate the entropy of this
entangled state here as zero, okay, so in
a sense, I know as much as I can know about
this state, okay, it's a pure state and I
know as much as I can know about this state.
But now if I ask the question, what is the
state of either party, of party A or of party
B? The natural way to think about that is
to average over the state of the others--other
party, okay? It's called a partial trace operation
and this is what you get for the part--for
the state of either individual party. It's
a completely mixed state. It's a classical
mixture of up and down. And if you look at
the entropy of that, it's one, which is the
maximum entropy such a system can take. So
what is this saying? And this was first noticed
by Schrödinger way back in '35, and he says,
what this means is that, "the best possible
knowledge of the whole does not imply the
best possible knowledge all--knowledge of
its parts," okay? And this is the reason people
get excited about entanglement, people are
perplexed by it, because this is very counterintuitive.
But if I tell you I know everything about
everything in this room, I should be able
to answer the question about what state is
that chair, what color is this chair, right?
But that's not the case for entanglement.
If--even if I know everything about the global
state, I can have as little knowledge as possible
about individual states, okay? And so, this
is--this was recognized a long time ago and
this is precisely what read on--what lead
Einstein, Podolsky, and Rosen to maybe posit
that the quantum mechanics is not a complete
description of reality. We've progressed a
lot since then and we think that it's a complete
description, but maybe one that's not intuitive.
Okay, and then more recent developments, some
manifestations of entanglement. These days
entanglement is not--it's still weird but
it's not a--it's not necessarily that--something
we don't get our hands work with. Experimental
is great, it's all the time. There are experiments
of some conducting systems, ion traps, protons,
that routine you create entangled states.
And furthermore, we know that it's useful
for certain tasks: information processing,
metrology, communication, and so on. But I
should note that these are low temperature,
low noise experiments. To create an entangled
state with a high fidelity requires a lot
of effort from the experimentalist. And even
small deviations from ideal conditions can
destroy entanglement, so. But there's a more
recent developments that show that even bulk
systems, if I just take a magnetic salt in
this case, if I talk about it's--the state
of its constituents, the nuclei of this magnetic
salt, below a certain temperature, I think
in this case it's about a Kelvin or five Kelvin.
We know that the state of the nuclei are entangled,
okay? We can do bulk measurements which tell
us that the most consistent state is, is an
entangled state. So in a sense, this is a
natural system, I mean it's just a magnetic
salt, that is entangled below a certain temperature,
okay? So it kind of led us to ask the question,
can you actually--can it exist in biological
systems, which is in a sense of a more natural
system and at physiological temperatures,
right? So this must be below a certain temperature.
Can we see it in biological systems at natural
temp--at physiological temperatures? And that's
a question I want to answer today. But before
I do that, let me clarify what I mean by entanglement
in these light harvesting complexes. So I've
seen--I've shown you that they're mostly composed
of chlorophyll or pigment molecules that are
spatially separated. By entanglement, what
I mean is that non-classical correlations
between the electronic state of spatially
separated chromophores, okay? So chromophores
are spatially separated and they have some
electronic degrees of freedom which are moving
around, and if you have non-classical correlations
between them, that's what I'm going to call
entanglement. This is a fairly sensible definitions
of the concept. The only thing is how you
divide up your whole system. And here, I've
chosen a--chosen to divide up the degrees
of freedom of this--the degrees of freedom
of the chromophores themselves. And this is
sensible because they are actually spatially
separated individual components. So how do
you quantify entanglement in such a system?
And it turns out to be very difficult. Quantifying
entanglement is a tricky concept by itself,
and in this case, what we have is something
called multipartite mixed state entanglement.
I'll explain what those things mean. Multipartite,
just because you don't have just two parties
here, right? You have many chromophores, you
have at least seven in this FMO complex, so
you can't just talk about a bipartite system.
And it's mixed-state entanglement because
the system is never in a pure quantum mechanical
state, meaning that it's always interacting
with its environment. It's a biochemical system
after all. It's interacting with its environment
and the state of the system is never in a
pure quantum mechanical state. So we have
to talk about mixed state, my multipartite
entanglement, and there is some--there is
a simplification which we can make which makes
us able, you know, it allows us to actually
quantify the entanglement in the system, and
that's that we're always in the single excitation
subspace in the system. What I mean by that
is that this--particularly for this FMO complex
and more generally for many light harvesting
complexes, the amount of photons that they
receive per unit of time is very small. So
you can be very sure, to a very high level
approximation, you can say that there is at
most a single excitation in the system at
once, right? It transports excitations in
femtoseconds of picosecond timescales and
it receives a photon once every seconds, so
you know it's going to dump that photon much
before it gets--a long time before it gets
the next one. So we're going to use this approximation
that there's at most a single excitation in
the system and what that--and that's very
accurate for at least this FMO system and
more generally for other light harvesting
complexes are and that allows us to form measures
of entanglement in this system. I'm going
to use two measures. One is a global measurement
time, so it did kind of captures the total
amount of entanglement in the system, okay?
I'm not going to go through the technical
details of what these--how you come up with
these measures of or how to interpret them,
but I'm happy to talk about that at the end
if there's time. And then, the other measure
of entanglement I'm going to use is the standard
measure called the concurrence. It's just
the measure of bipartite entanglement so I
single out two particular chromophores and
I ask, "How entangled are they," okay, and
it's--I'm going to use a measure of concurrence
for that purpose. And, okay, so then now I've
talked about how to quantify entanglement,
I'm going to talk about how to actually theoretically
simulate these systems so that you have some
way of actually calculating how much entanglement
there is. And these are maybe too technical
for people who aren't interested in the details,
but I'm just going to go through it very quickly
for people who are. A very good approximation
of the dynamics here is what's called the
Frenkel Hamiltonian. What it says is that
each chromophores, so this notation here of
N, you should think of that as the state of
a single chromophore, okay, as the state of
a single pigment. But it says that each pigment
has a certain characteristic energy to it.
This is the energy of its excited state and
there's a matrix on it that couples pigments.
And this is the physical cool on with couplings.
So if there's an excitation on here and there's
a nearby chromophore right next to it, it
will eventually move over to here, okay? And
that's--and how quickly it moves over to here
is characterized by this, is characterized
by the J, okay? So, it's technically called
the transition dipole couplings. So that describes
the movement of excitation in these systems.
But that's not all, right, because I've mentioned
a couple of times that it's actually embedded
within this protein cage. It's embedded within
a lot of other dynamics happening around it
so you need to actually talk the talk about
the open system dynamics and that's the Hamiltonian
there. What describes is it the dynamic there
is not only this part but also something that
describes the coupling between the chromophores
to the environment and also the dynamics of
the environment. That ends up being a lot
of things to keep track of so what--the usual
way to do this is to average out over the
dynamics of the environment. What you say
is that, "I'm going to ignore what the environment
is doing. I'm just going to look at the system
of interest," and that's--you compose things
like the master equation which describes the
dynamics of the system. Now the tricky thing
about this particular system and about light
harvesting complexes in general, and we can
talk about why this is, is that there are
no perturbative timescales in the system.
The standard way of going from this complete
description of the system in the environment,
everything around it and just focusing on
the environment is to perturb around some
timescale. You say that the environment is
not moving very quickly, so I'm just going
to say it's a constant--it's constant or moving
very slowly, right? And this is a standard
mathematical technique. You separate timescales
and you just talk about the timescale of interest.
You ignore things that are happening too fast
and you ignore things that are happening too
slow. But the thing about these systems is
there are no perturbative timescales. It's
really difficult to do this, and so you need
a theory that can take this into account.
And very recently, just last year, such a
theory was developed in the Fleming Lab at
Berkeley, and Akihito Ishizaki, was the first
who thought of that, and we're going to use
that model for the dynamics. So we showed
that we're capturing the dynamics of the system
in the regimes that are interesting. Precisely
in this regime where there are no perturbative
timescales, okay? Right, so here are the results.
So what--these come out of drawing the simulations
according to this very accurate physical model
and measuring the entanglement using the measures
that I mentioned. So if we just looked at
global entanglement, right, so this is the
total amount of entanglement in the system,
the two curves here, the blue curve here is
called 77K, which is where the first experiments
were done, and the red curve is for the room
temperature or 300 Kelvin. What we see is
that there's an initial spike of entanglement.
So the initial state here is that the excitation
is just on one of the chromophores. So here,
I mentioned that the FMO receives its excitation
from--through 1 and 6, right? So I'm just
choosing the excitation to be on 1 in this
case. And then its--as it's spreading out
over the chromophores these entanglement build
up over all the chromophores that aren't involved
in the transport. And you see that the temperature
dependence is not drastic. So, you know, there's
a viewpoint that increasing temperature will
kill off quantum effects very quickly, right?
But this is not what you see here. You've
increased temperature almost four-fold from
77K to 300K and the entanglement has decreased
by maybe three quarters, right, so there's
no kind of linear or super linear scaling
of--I should say there's no super linear scaling
of the suppression of quantum effects, in
this case, the temperature. And this--and
furthermore, the significant entanglement
at long times, it's the steady state of the
system which is way over here. So, sorry,
so the main graph shows time up to 100--1,000
femtoseconds and the sub-graph is showing
times up to 5,000 femtoseconds. So there's
entanglement even--up to 5 picoseconds. Now
that's on absolute timescales or at least
on human timescales that's very short. But
you have to remember that these systems dump--these
things do their job in femtoseconds timescales,
right? So these are certainly functioning
relevant timescales and that's the important
thing to keep in mind. Okay, so as I mentioned,
this is a measure of just the total entanglement
in the system. It doesn't really tell us about
what things are entangled. You might say,
"Well, this is maybe not so surprising because,
you know, you have chromophores that are very
close to each other, they're talking, you
know, they're strongly coupled to each other
and therefore they will have entanglement
with each other so maybe this is not so surprising.
And to look at more closely of where exactly
the entanglement is spatially, what you have
to do is result to this kind of bipartite
measure. So I'm going to look at--I'm going
to choose two chromophores and then just look
at how entangled they are, okay? And so, let's
just focus on the left-hand graph here, the
77K. I think you can see this. This is showing
which two chromophores each line corresponds
to. So the blue is 1, chromophores 1 and 2,
the red is chromophores 1 and 3 and so on.
And so you see the intuition was right there.
The chromophores that get very entangled quickly,
and there's a lot entanglement, are the ones
that are right next to each other, 1 and 2,
okay? But then--and then as the excitation
moves around, other things get entangled and
the really surprising thing for us here was
this 1 and 3, the red curve here. You see
this entanglement build up over chromophores
1 and 3 and it stays around for long timescales,
right least, til a picosecond at 77K and further.
So that's this red curve. And why this is
surprising, is that 1 and 3 are almost as
far apart as they could be in this complex.
They're about 3 nanometers separated and they
have negligible direct coupling, right, so
if you just look at the direct coupling between
these things they're not very coupled at all.
So this entanglement, this non-local correlation
between these two is mediated by the interaction
with the rest of the system, okay? And so--and
if you look at--I've just plotted the curves
which have--which showed the most significant
amount of entanglement, but if you look at
other bipartite cuts as well they also show
entanglement. And what it means is that there's
true multipartite entanglement. So if you
look at entanglement there, people make the
distinction between things that are just bipartite
entangled, and for our three systems, only
these two entangled are, or are they all entangled
with each other, which is a higher, more complex
description of the entanglement system. And
turns out that it's multipartite entanglement
in this case. Okay, so that brings you to
the end of the entanglement discussion, and
now I'm going to talk about quantum speedup.
So what do I mean by quantum speedup? We were
motivated to look at this by statements like
this that came out right after the experiments
were done in 2007. So people made the analogy
to quantum computation in these systems, saying
that, well, plants are actually performing
quantum computation in the process of absorbing
energy. And for anyone who works in quantum
computing, and that's my background, this
is a very striking statement. If you're talking
to experimentalists, they've spent years and
years, you know, building up a single qubit
to do virtually nothing. And so, you know,
it's very kind of--I mean, almost insulting
if you talk to a quantum computing experimentalist,
to say that such a messy system is performing
quantum computation so we wanted to really
examine this question closely. And the way
we do it is--so the most direct analogy to
a quantum computation is through this thing
called the quantum walk. So, quantum walk,
if you--most of you probably know about random
walk. A random walk is just where you move
randomly between spots and then you eventually
diffuse out in a system, right? So if I have
a line and I move to the right with probability
P and I move to the left with probability
P, after a certain amount of time, I'll have
a Gaussian distribution for my position, right,
this is a classical diffusion. If now instead,
I say that I'm a quantum walker, I can move
in superposition to the right and left. If
I look at my distribution after a certain
amount of time, it will be this black curve.
So the blue curve is the classical distribution,
this is Gaussian. It will be this black curve,
and it's really strange. Not only is it a
solitary, it's also bimodal, right? So you
have maximum probability of being on the end,
not in the middle anymore. So, if you just
look at the variance of that walker, it scales
lineally in time as opposed to--sorry, the
variant scales lineally in time, right, or
quadratically? Yes, the variance scale quadratically
in time, I think. But it scales polynomially
faster than the classical walker, okay, the
variance of this system. And you can embed
a quantum algorithm, like a quantum search,
within a quantum walk. So you can say that
the walker is searching for a particular node
in the graph and you can embed the whole idea
of a growth research algorithm within a quantum
walk. And this is the most natural way in
which we think about photosynthetic systems
as performing quantum computation. They're
actually randomly walking around this chromophore
complex and maybe they're doing some computation
in this process, right? But one thing to keep
in mind is that--think this is not an ideal
quantum walk at all. For an ideal quantum
walk, you assume that all the nodes are of
the same energy, there's no decoherent. And
in these systems, there's a certain energy
landscape, and you need an energy landscape
because you need energy to be funneled to
reaction centers, right? So there's a certain
energy landscape in the system. There's disorder
in the system. There's also decoherence in
the system, which we'll talked about extensively.
So can you really have speedup in this environment?
Can you really have this kind of quadratic
speedup over propagation in this environment?
And to look at this, what we did was actually
look at the FMO complex and map it to a 1-D
quantum walk since we know the properties
of the 1-D quantum walk very well. If you
look at the Hamiltonian, so what does this
Hamiltonian mean? The diagonal elements of
the Hamiltonian are the energies of each chromophore,
and the off-diagonal elements are how strongly
they're coupled, okay? So this element here
will tell you how strongly chromophore one
is coupled to chromophore two, okay? Now,
if you just get rid of the smallest elements
and you just look at the largest elements
here, what you'll notice is that it looks
very much like a 1-D quantum walk. You have
two source nodes, one and six at the end,
and you have a sink node in the middle, three,
okay? There's a slight ambiguity here that
this five and seven are equally or very strongly
coupled to six and also four. But you can
collapse those two into a single node in a
quantum walk, you know, so you have a six-node
quantum walk and you have things coming from
the ends and moving to the middle, okay? And
now, a good way to see if you have a quantum
speedup is just to say, "Look at the scaling
of the variance," and say, "If it's classical
it's going to scale lineally, and if it's
quantum it's going to scale quadratically,"
right? So it's direct analogy to this picture
here, okay? And so, what do you see? So in
the left-hand side is a graph of the actual--the
variance with respect to time, and the right-hand
side is its power law. I think the right-hand
side is probably more--it's easy to interpret.
But we can look at both. Let's see. So in
the--what you see is that after about 70 femtoseconds,
you have ballistic motion so you truly have
a quadratic scaling of variance. It moves
very quickly, right, so it's easy to see in
the power law. The power--if you just look
at t to the b, b is two, right, so you have
quadratic scaling of how quickly it's moving.
But then very quickly, it goes down. It goes
actually sub-diffusive. It goes to below what
you expect for a classical random walk, and
then it eventually gets trapped in the trapping
sensor. So this includes trapping, that's
why it's going down continuously. But what
this tells you is that there is no quantum
speedup in the sense that you need for quantum
computation beyond 70 femtoseconds, okay?
But the coherence is still there up to 700
femtoseconds, an order of magnitude greater.
And you could ask, so why is this happening,
right? What is limiting the quantum speedup?
And it turns out it's a combination of two
things. One is, it's localization. And this
is well-known from the [INDISTINCT] and it's
usually called Anderson localization of very
large systems. It's where if you have any
disorder in your system, so if it's not a
plane, if you're not just moving along an
energetic plane, if you have like disorder
in your system, you're going to get localized.
You're going to get trapped in local minima.
You're going to--you're not going to propagate
very far in the system. But at the same time,
there is a dephasing mechanism which is the
effect of the environment. And what the dephasing
does is it actually--it moves energy levels
around, okay? So if you imagine a static picture
where the energy levels are just fixed you're
going to get trapped. But then the dephasing
actually helps you because it moves energy
levels around and you can get pushed over
boundaries. And so the dephasing moves you
beyond being trapped. So if you didn't have
the dephasing, this would go from two and
rapidly drop down to zero, right? So the effect
of the dephasing is it pushed it back up,
but it still doesn't push it back up to one
so it's actually sub-ballistic all the way
through up to 70 femtoseconds. So we think
this is a pretty convincing argument that
there is no quantum speedup in the system.
The combined effects of the disorder, the
localization and the dephasing preclude any
kind of quantum mechanical speedup that's
required for quantum computing in the system,
okay? Okay, so how much time do I have left
now?
>> As of now, you have 15 more minutes.
>> SAROVAR: Fifteen, great. Okay, so I'm going
to talk a little bit more about--well, maybe
this question is naturally motivated at this
point. What's so special about light harvesting
complexes, right? You know, I've presented
you some results and you'll see some more
about their efficiency and so on, but I want
to address what is special outside light harvesting.
These are the only, arguably, the only biological
systems where we know quantum effects play
such an important role and so what makes these
things so special? And this is not just to
build up a question, I think this is a very
practical question, because if we want to
find more systems in biology which might benefit
from quantum mechanical effects then we should
really isolate what is special about these
systems. We should really find out what makes
these different from, you know, any other
biological system. And so, it's useful to
go back to this picture that I introduced
about the system and environment coupling.
So J is the rate that which dynamics happens
within the system, and lambda is the rate
of dynamics between the system and the environment.
And it turns out, for light harvesting systems,
they increase J by doing a very nifty thing
of just having very dense packing of molecules,
right? So these chlorophyll molecules are
packed very close together, of separation
of lengths of a nanometer or sub-nanometer
length scale. This leads to a very strong
interaction between them. So J is kind of
pushed up naturally by this, by this engineering
in a way. And furthermore, lambda is pushed
down. And there's a couple of tricks that
it uses to do this. So, the main trick is
that the light harvesting complexes are always
embedded within membranes, okay? So you have
some membrane and the light harvesting complex
lives within the membrane. And there's a very
good reason for this. It's really to set up
an electrochemical potential. If you want
to separate electrons from protons, positive
charges from negative charges, you need to
find a way to keep them separated, otherwise,
they'll recombine again. So what you do is
you send the positive charges to one side
of the membrane and the negative charges to
the other, okay? So this is the membrane,
in a sense, is a battery here or the terminals
of the battery. But maybe that it's doing
something more. If you put things inside the
membrane, they're in a way in a more controlled
environment than just in some bare solvent,
okay? And it turns out, if you look at how
strong the electron-proton coupling, how strongly
the chromophores are coupled to the vibrations
in the system, its orders of magnitude is
smaller than what you'd expect in a typical
solvent. So if you look at what you'd expect,
this coupling is called the reorganization
energy, if you look at what the reorganization
energy that you calculate in a typical solvent,
it's about 200 to 2,000 wave numbers in this
unit. And if you look at--for the FMO, what
it is, it's about 35 wave numbers. So it's
drastically smaller than what you see in a
typical solvent system. And we believe this
has to do with the membrane embedding, it's
a much more controlled environment. And another
aspect that comes out of this is that maybe
the fluctuations that it sees are much more
controlled. Let's say, they're not just random
fluctuations, there's some correlation time
and there's some spatial correlation to the
fluctuations as well. And people, including
Massoud, have shown that this has--this can
actually help in the transport. If you have
correlated fluctuations as opposed to a completely
random environment, you can preserve coherence
for a longer period of time. Okay, so this
could be--so what does this mean? It means
that for these systems, J and lambda is about
1. They don't lie all the way on this left-hand
side of this--this graph that I drew. And
maybe this is an engineering choice, right?
You push J up and you push lamda down in order
to achieve this kind of coincidence of timescales.
Okay, and so this leads me to structure and
this is some work that we recently started.
I'm going to talk about some of the work that
we're doing now and some of the things we're
looking at. So, these concentrations lead
you to think about structure in these systems
and how structure plays a role. And so in
biology, I guess, these are normally called
structure function questions. You want to
know, you know, why the structure is so that
it performs its own function. And you can
really kind of form this picture of light
harvesting complexes going from a monomer
or a single pigment. You aggregate them, you
form them together in there both, and then
you put some protein cage around it in order
to give it some stability, okay? So how does
each aspect of this construction affect the
quantum mechanical effects and also the functional--the
function of this device? So you can ask, you
know, which structural features actually determine
the quantum behavior? What's the influence
of the pigment density? What's the influence
of the pigment orientations that you choose,
of the actual choice of pigments that you
use? You can ask whether you can engineer
the protein backbone. So I mentioned that
the membrane maybe a way that you have some
shielding from environmental risk. Can you
engineer the protein backbone so you have
this similar kind of shielding? And how the--how,
you know, can you enhance quantum behavior
by its suitable design? And so, how does also--how
does the aggregation, how do the protein determine
both the transport properties and also the
optical properties? So, I haven't mentioned
much about the optical properties but really,
these things are antenna, right? So not only
do they need to be good at transporting excitations
but they also need to be very good at absorbing
the light that they receive. It's a whole
another story. They tend--it turns out that
most light harvesting complexes are really
finely tuned to the actual spectrum that they
received. So to investigate these questions,
what we're doing is actually building test
systems. We're basing this on the tobacco
mosaic virus. So this is a common virus of--this
is just a monomer of the protein. And what
you can do is you can mutate certain sites
of this protein and you can add chromophores,
you can add pigments, right? So the pigments
will be attached to these sites and then they
self-assemble into these beautiful structures.
See, they self-assemble into these--you just
throw a bunch--so, you mutate a bunch of them,
you attach chromophores and you throw them
into a solution with a certain PH and temperature
and they self-assemble into either these disk-like
structures, these stacks of disks, or they
can even assemble into these helix structures,
okay? And these look very similar to natural
light harvesting systems that we have. Both--there's
a natural light harvesting analogy or counterpart
to this helical structures and also to these
disk-like structures. And so the person doing
this synthetic chemistry is Matthew "Matt"
Francis at UC-Berkeley. He's a professor in
the Chemistry Department. And so, he has worked
with his team, this has been for a long time.
He's really an expert in designing this--the
structure and also assembling it in a way.
And so what you can do is mutate different
components and attach chromophores at different
sites to achieve variable separation distance
between chromophores. So what does this mean?
If he attaches it at site 103, right, that's
separated by only about 0.8 nanometers. And
if you make a disk out of them, they'll light
right in the middle here, okay? So, but if
he attaches them at site 123, the green side,
I guess the colors don't come out so well
here, but the point is that where he attaches
them determines what the density of pigments
is and what their orientations is, okay? So
we can kind of make light harvesting complexes
on demand here and that's the idea. So this
is what's called as a bottom-up approach.
We're really trying to understand the structure
function relationships, right? If I make a
light harvesting complex like so, how does
it--how do the quantum properties emerge out
of that and how do they influence function,
like optical and transport properties? Right,
so the goals here, I guess, to mimic the fantastic
efficiency of natural light harvesting complexes,
but by design, by really microscopic design,
and also give it tunable properties. So, one
of the things about natural light harvesting
systems is they don't use a huge variety of
pigments, but by just engineering the pigment
so they can absorb either all the way on the
blue or all the way on the red. And so, can
we have that kind of variability? Can we tune
the absorption of these light harvesting systems
by just changing the structure? By engineering
it's like this so, but not only changing the
pigments. And also, we want structures that
are suitable to biological environments. And,
obviously, there are applications to this,
maybe light senses, also photovoltaics for
the applications, okay. Right, so here's the
summary of what I presented. So, the first
part I presented the entanglement. Hopefully,
I demonstrated that entanglement is a natural
feature of these systems. It turns up as a
result of the excitation transport. What I
didn't really address is the role of the entanglement.
So we haven't been able to pinpoint a precise
role for this entanglement. It could be just
the side effect of the coherence or the fact
that they're strongly coupled. So we're still
working on that. And the class of entangled
states, for people who know about entanglement,
the class of entanglement states of these
things exhibit are very small class of the
total entanglement that they could exhibit.
It's only the so-called single-exciton manifold.
And you could ask whether there's more general
class of entanglement that they could exhibit
as well. The second part I demonstrated that
there really is no quantum speedup in the
quantum information sense in these systems.
It's not--these things aren't built for quantum
computation and the effects, the natural effects,
that they have don't allow for the kind of
quantum speedup that--that's required. But
that's not to say that there's no quantum
advantage. You should make a distinction here
that quantum computing is not the only role
here. As Hartmut mentioned that, "Really overcoming
local energy minima is a possibility. People
have postulated that efficiency is a role.
Robustness--coherence could provide more robustness
in these systems. Unidirectionality." So you
want these things to take energy in one direction
towards the reaction set and you don't want
it to go backwards, right? So maybe that's
a role for the coherence as well. And then
I've briefly presented the new approach that
we're taking, the bottom-up approach of actually
constructing artificial light harvesting systems
and playing around with them to see what the
effects of the coherence are. Okay, thank
you. Let me--all right. This is a--so this
work was done in collaboration with a lot
of people. The speedup work was done in collaboration
with Stephan Hoyer, he's actually here today.
You can talk to him about, if you're interested.
The entanglement work was done in collaboration
with Akihito Ishizaki and Graham Fleming.
And the synthesis work that I've talked about
at the end, the molecular synthesis is done
in collaboration with Dan Finley and Matt
Francis. And Birgitta Whaley is--was involved
in all the work as well. Okay, thank you.
>> You have the entanglement scaling as--with
a power level with X.3 quarters, correct?
>> SAROVAR: The speedup, the variance.
>> Yeah, yeah.
>> SAROVAR: Yes, yes, the variances.
>> I'm interested in how does that exponent
itself scale with the difference between the
energies that you're dealing with which is
several electron volts, and to change to see
at room temperature. I guess, what I'm saying
is if you--suppose if the photosynthesis happened...
>> SAROVAR: Yeah.
>> ...at 2,000 wave numbers instead of whatever
it is, 15,000...
>> SAROVAR: Yes.
>> ...with--would a different exponent apply?
>> SAROVAR: So this is actually transport
within the smaller wave so this is transport
within the excited state manifold only, right?
So the ground to excite it is about 15,000
wave numbers.
>> Right.
>> SAROVAR: But within the excited state manifold,
it's only about a thousand wave numbers or
so separations. So, it's the transport within
that gradation. If the energy difference is
even larger, it would be even more localization.
You wouldn't have dephasing mechanisms which
could bridge the gap of a 15,000 wave numbers.
I think that's...
>> I understand. I guess what I'm saying is
that, one of the reasons why a photosynthetic
apparatus had been--the first place with this
kinds of effects were shown...
>> SAROVAR: Right.
>> ...it's because your energies are very
high, the excitation energies are quite...
>> SAROVAR: I see, right.
>> ...very high above the [INDISTINCT] that
you have.
>> SAROVAR: Yeah.
>> So, and, of course, this is unique to line
absorption.
>> SAROVAR: Right.
>> It's because everything else in biology
runs at about...
>> SAROVAR: Much of it all, yeah.
>> [INDISTINCT].
>> SAROVAR: Yeah.
>> So I'm kind of interested in how the scales
that, you know...
>> SAROVAR: Right.
>> ...down to half an electron volt from two.
Do you guys--can you generalize...
>> SAROVAR: So I should say the--that energy
scale you're talking about, the optical energy
scale, right, is not relevant here.
>> Okay.
>> SAROVAR: Because it is really in this half
of--it's really in the thousands of wave numbers
when we're talking about the disorder, because
it's transport in this exit--in the excited
manifold, right?
>> Okay.
>> SAROVAR: It's really--I mean, the converse
question of the how would the scale with very
large energy differences, I don't think it
would scale well, I mean you would get localized
very quickly because there isn't, at least
to my knowledge, there aren't de-phasing mechanisms
that could bridge this huge energy gap.
>> Okay, so...
>> SAROVAR: I think that's...
>> ...let's make it a thought experiment.
>> SAROVAR: Yes.
>> Suppose that your photosynthetic pigments
been infrared absorbed.
>> SAROVAR: Yeah, yeah.
>> So if you arrange your energies...
>> SAROVAR: Yeah.
>> ...your orbitals, energy levels to absorb
at let's say, half an electron volt.
>> SAROVAR: Right.
>> ...what would it look like? I mean, you
can actually do that in simulation, I assume.
What would the entanglement look like, if
everything took place at the energies roughly
of ATP hydrolysis systems?
>> SAROVAR: Yeah. We haven't done simulations
so I don't know what it looks like, but it's
an interesting--I mean the thing you have
to be concerned about is actually relaxation
mechanisms in that case because the energy
you have to relax is much smaller, right?
So one of things that photosynthesis is working
for, you get this initial kick of energy,
you go up a huge amount, and then you just
play around in this like large energy scale,
right? Whereas if you push down with an excitation
and it goes all the way here, you have a very--you
have a larger chance of losing the energy
that you initially got. Yeah. So then, you
would--I mean, I think you would lose all
quantum--maybe I shouldn't be so general because
I haven't done the simulations, but I think...
>> You just have to. You don't actually wouldn't,
you know, do it.
>> SAROVAR: Well, I believe it about these
systems, but I don't believe about an artificial
or an infrared absorbing system, I guess.
Yeah.
>> Okay.
>> SAROVAR: Yeah?
>> I don't know if you're familiar with this,
this is a very old story...
>> SAROVAR: Yes.
>> ...going back to the 1920s, the work of
chemists in Germany and England...
>> SAROVAR: Yes.
>> ...added the names tri aggregates and J
aggregates.
>> SAROVAR: Yeah, right.
>> Because their last...
>> SAROVAR: Yeah, because...
>> ...foray into engineering of these systems
reminds me of what they did after they turned
light harvesting artificial bodies, you know,
light quadrant that failed.
>> SAROVAR: Right, right.
>> So perhaps, you can get inspiration from
that because...
>> SAROVAR: Yeah.
>> ...that was also a case where actually
temperature promoted the efficiency of how
the light harvesting...
>> SAROVAR: Right. Yeah, I mean...
>> ...as opposed to what you expect from kind
of quantum mechanics and the equation was
about mechanical...
>> SAROVAR: Okay.
>> ...from [INDISTINCT] states.
>> SAROVAR: Right. Yeah, so we are very much
inspired by that work. And so the artificial
systems, we think of that as actually as J
aggregates, they are encased within a protein.
So the protein degree of freedom is an additional
degree of freedom from these other older aggregates
that were engineered. Yeah.
>> Tobacco mosaic virus is structurally similar
to a microtubule.
>> SAROVAR: Okay.
>> And in fact--so two questions. Do you consider
using that, with microtubules instead of the
tobacco mosaic virus and be--is there a voice
that tell you--tell us--of what your going
to tell us about how microtubules, utilizing
their own geometry, periodic lines of geometry
not with chromophores but with the [INDISTINCT]
amino acids?
>> SAROVAR: Yeah. Okay, so first, we haven't
thought about using microtubules because--so
there are certain advantage to the tobacco
mosaic virus because we know the synthesis
pathways, we know exactly how it aggregates
and so on. Or how it self-assembles. I'm not
sure--it's so--I'm not a synthesis expert
so I didn't know if that same thing is known
for microtubules. The other thing is that
this not a natural thing to make this viruses
to do, right? They don't actually have chromophores
in them. What goes up that helical chain is
actually the RNA so what you've done is you've
stripped away the RNA where normally, the
RNA would bind and actually bound these chromophores.
So, the question about what--in natural--in
nature, they wouldn't have chromophores attached
to them is what I'm saying. And also, we're
more interested in the quantum properties
of the chromophores themselves rather than
the protein, the tobacco mosaic virus protein.
The protein is really the environment for
the chromophores, right, so we're interested
in not necessarily engineering the quantum
properties of the protein, which is a very
difficult task because it's such a large structure,
but rather the quantum properties of the chromophores.
Does that answer the question and...
>> All right. But you can also use the microtubules
as scaffolding [INDISTINCT] associate purposes
for a selection reaction...
>> SAROVAR: Okay.
>> [INDISTINCT]
>> SAROVAR: Okay. I--so, my expertise isn't
synthesis. I mean, this is the--this the expertise
that the lab that we work with has, but I'd
be happy to hear about that. Yeah.
>> [INDISTINCT]
>> SAROVAR: Okay.
>> But first we should thank Mohan for [INDISTINCT].
