How can one model quantum walks in photosynthesis?

Question

I have been working on quantum biology and found something interesting that I would like to write an equation for.

Scientists have wondered how plants have such a high efficiency in photosynthesis; they always thought that the photons' energy (needed for photosynthesis) reach the "reaction center" by jumping from cell to cell, but this doesn't predict the high efficiency. Then scientists realized that maybe the photons' energy goes into superposition to try all the possible paths (which would be the states) and when the shortest path is found, the photons' wave functions collapse into the states correlating to the shortest path to the "reaction center." According to Discover magazine:

Within chlorophyll, so-called antenna pigments guide energy from light collecting molecules to nearby reaction-center protein along a choice of possibles pathways... To explain the near-perfect performance of plants, biophysicists reasoned, the energy must exist in a quantum superposition state, traveling along all the molecular pathways at the same time...Once the quickest road is found, the idea a goes, the system snaps out of superposition and onto this route, allowing all the energy to take the best path every time

So, how would I write an equation for this? I was thinking about path integrals but couldn't find a way to use it. I know that for quantum superposition $$ \psi(x) = a |S1\rangle + b |S2\rangle + c |S3\rangle + \ldots $$

and the probability is

$$\int|\psi(x)|^2dx = |a|^2 \langle S1|S1 \rangle + |b|^2 \langle S2|S2\rangle \, .$$

I am not sure how to write a wave function collapse equation. Can I combine this knowledge into one equation, or should I still use path integrals?

Extra information on the topic

Answer 1

I actually don't think that this view of light being in a quantum superposition is anything new: what Discover magazine is describing (I believe) is the stock standard picture of how one would describe a system of cells, molecules, chloroplasts, fluorophores, whatever interacting with the quantised electomagnetic field.

My simplified account here (answer to Physics SE question "How does the Ocean polarize light?") addresses a very similar question. The quantised electromagnetic field is always in superposition before the absorption happens and, as light reaches a plant, it becomes a superposition of free photons and excited matter states of many chloroplasts at once.

To learn more about this kind of thing, I would recommend

M. Scully and M. Zubairy, "Quantum Optics"

Read the first chapter and the mathematical technology for what you are trying to describe is to be found in chapters 4, 5 and 6.

The truth is, photons do not bounce from cell to cell like ping pong balls. So that theory happens to be incorrect.

---

Further questions and Edits:

But this is about the energy FROM the photon... Would whatever you are saying still work for that? Plus, I would like to see some math...

Energy is simply a property of photons (or whatever is carrying it): there has to be a carrier to make any interaction happen. All interactions we see are ultimately described by this. See eq (1) and (2) here, this is for the reverse process (emission) but you are ultimately going to write equations like this. To get a handle on this quickly look into this Wikipedia article (Quantization of the electromagnetic field) and then read Chapter 1 from Scully and Zubairy.

Ultimately, you're going to need to write down a one-photon Fock state, and add to the superposition excited atom states. The neater way to do this is with creation operators acting on the universal, unique quantum ground state $\left|\left.0\right>\right.$: we define $a_L^\dagger(\vec{k},\,\omega),\,a_R^\dagger(\vec{k}\,\omega)$ to be the creation operators for the quantum harmonic oscillators corresponding to left and right handed plane waves with wavenumber $\vec{k}$ and frequency $\omega$. Then a one-photon state in the oscillator corresponding to the classical solution of Maxwell's equation with complex amplitudes $A_L(\vec{k},\,\omega), A_R(\vec{k},\,\omega)$ in the left and right handed classical modes is:

$$\left|\left.\psi\right>\right.=\int d^3k\,d\omega\left(A_L(\vec{k},\,\omega)\,a_L^\dagger(\vec{k},\,\omega)+A_R(\vec{k},\,\omega)\,a_R^\dagger(\vec{k}\,\omega)\right)\,\left|\left.0\right>\right.$$

To define an absorption, Scully and Zubairy show that the probability amplitude for an absorption at time $t$ and position $\vec{r}$ is proportional to:

$$\left<\left.0\right.\right| \hat{E}^+(\vec{r},t)\left|\left.\psi\right>\right.$$

where $\hat{E}$ is the electric field observable and $\hat{E}^+$ its positive frequency part (the part with only annihilation operators and all the creation operators thrown away).

Alternatively you can in principle model absorption by writing down the Hamiltonian which is going to look something like:

$$\int d^3k\,d \omega\left(a_L^\dagger(\vec{k},\,\omega)\,a_L^\dagger(\vec{k},\,\omega)+a_R^\dagger(\vec{k}\,\omega)\,a_R(\vec{k},\,\omega) \right)+\sum\limits_{\text{all chloroplasts }j} \int d^3k\,d\omega\,\sigma^\dagger_j\left(\kappa_{j,L}(\vec{k},\,\omega)\,a_L(\vec{k},\,\omega)+\kappa_{j,L}(\vec{k},\,\omega\,a_R(\vec{k},\,\omega) \right)+\\\sum\limits_{\text{all chloroplasts }j} \int d^3k\,d\omega\,\left(\kappa_{j,L}(\vec{k},\,\omega)\,a^\dagger_L(\vec{k},\,\omega)+\kappa_{j,L}(\vec{k},\,\omega)\,a^\dagger_R(\vec{k},\,\omega) \right)\sigma_j$$

where $\sigma_j^\dagger$ is the creation operator for a raised chlorophore at site $j$ and the $\kappa$s measure the strength of coupling.

This is complicated stuff and takes more than a simple tutorial to write down.

Answer 2

The role of coherence in biological electron transport, e.g. within chromophores, is an open and actively researched problem in quantum optics/quantum chemistry. The two classic theoretical treatments which kick-started the field are by Plenio & Huelga and Mohseni et al.. Since then an enormous literature has emerged on the topic.

A basic, generic model which contains the relevant physics is to consider a quantum network of sites, each of which can either have one or zero excitations present, and is thus equivalent to a spin-1/2 particle. The network could be governed by the following generic Hamiltonian: $$ H = \sum_i \epsilon_i \sigma^+_i\sigma^-_i + \sum_{i\neq j} V_{ij} \sigma^+_i \sigma^-_j, $$ where the operator $\sigma^+_i$ creates an excitation on site $i$ (that is, $\sigma^{\pm}_i = 1/2(\sigma^x_i \pm \mathrm{i}\sigma^y_i)$. This Hamiltonian describes excitations with energies $\epsilon_i$ which hop around the network according to the couplings $V_{ij}$. If you calculate the quantum dynamics under this Hamiltonian then you may (depending on the parameters, see Caruso et al.) find the kind of delocalised transport behaviour alluded to in your pop-sci article. However, this is not even close to touching the main current issues relevant for quantum biology.

In a biological setting one also has a strongly-coupled vibrational environment due to the surrounding water and protein structures. Traditionally one would expect that environmental fluctuations would destroy any quantum coherent effects, and that transport would occur due to incoherent transitions between energy eigenstates. The interesting feature of many natural chromophores is that the environment produces highly structured noise, which tends to promote long-lived coherences (compared to the time scales relevant for electronic transport).

How to model the complicated environment to successfully account for the spectroscopic data is one of the main open problems. See, for example, Chin et al. for some recent theoretical efforts in this direction. Since barely any in vivo experimental data is available, the actual biological relevance of this phenomenon is moot. However, some have conjectured that it has been naturally selected to provide a transport enhancement, which, for example, would be advantageous in low-light environments.

Answer 3

This is not an answer for the obvious reason that this question cannot be answered easily, hence why it is an open area of research. What I will provide though is links to how something like this is done.

The idea resides in the dynamics of open quantum systems, which are systems that are constantly interacting with the environment and hence tend to become entangled. These entangled states of the system and the environment will need to be described in a density matrix formalism.

These stuff are not generally found in standard quantum textbooks. One that includes a discussion on open quantum systems is

• Quantum Physics by Michel Le Bellac

Since unitary operators preserve purity, then it is impossible to create a mixed state from a pure state via unitary evolution. This means that there must be additional types of quantum evolution, which can change the purity of the state.

To describe this kind of non-unitary evolution we need a new mathematical object called a superoperator, $$S[] := \sum_jK_j\rho K^{\dagger}_j$$

This is called the “operator-sum” representation of the super-operator, or, more commonly, its Kraus representation. A superoperator will transform a density matrix $\rho$ into another density matrix, $\rho' = S[\rho] = \sum_jK_j\rho K^{\dagger}_j = \sum_j p_jU_j\rho U^{\dagger}_j$ which will now describe non-unitary evolution due to the introduction of classical probability int he equation.

Returning back to the open quantum system discussion, what people generally tend to do is to attempt to derive an evolution equation for the reduced state of the system $\rho$ alone:

$$\dot{\rho}(t) = \frac{\partial\rho(t)}{\partial t} = S[\rho(t)]$$

Such an equation has been derived which describe Markovian system-environment interactions, the so called Markovian Master Equation. A simplified form of it called the Lindblad equation for an N dimensional system:

$$ {\dot \rho }=-{i \over \hbar }[H,\rho ]+\sum _{{n,m=1}}^{{N^{2}-1}}h_{{n,m}}\left(L_{n}\rho L_{m}^{\dagger }-{\frac {1}{2}}\left(\rho L_{m}^{\dagger }L_{n}+L_{m}^{\dagger }L_{n}\rho \right)\right) $$

How is all these related to your question? Well, as I currently understand, researchers in the AMOPP(Atomic, Molecular, Optical and Positron Physics) Group in UCL are focussed on the quantum interaction of photosynthetic biomolecules with light to produce photosynthesis, even at room temperature. It turns out these kind of systems are described to some degree by such open quantum systems interacting with a Markovian environment as described above. I had also heard the claim that in these quantum processes the efficiency is as high as $95\%$, so I thought it would be fun to introduce you to this area of research.

I should also state that I am by no means an expert on this and might not be able to answer any questions you might have but I would encourage you to take a look at this link

Answer 4

There are approaches to quantum systems, and photosynthesis from the viewpoint of quantum information systems too. The approach is to basically see photosynthesis as a process in which electrons involved in photosynthesis reactions "sample" different energy-level routes in much the same way quantum-computer algorithms can. To understand the process in such jumps, the search is done like some unsorted database search. The hopping of electrons, is some excited state electrons along some discrete state energy levels. This wavelike characteristic of the energy transfer within the photosynthetic complex can explain its extreme efficiency, in that it allows the complexes to sample vast areas of phase space to find the most efficient path.

In chlorophyll molecules, which are arranged such that neighbouring molecules have different energy levels. When light shines on one of these molecules, an electron is momentarily excited before passing its energy over to a nearby molecule with a slightly lower energy level. In this way energy can flow "downhill" from energy level to energy level until it reaches the crucial "reaction centre" where the actual photosynthesis occur.

Contrast this with the idea of random walk for energy going downhill in a chlorophyll reaction, the random walk notion does not fit the parameter how the energy reaches the reaction core to actuate the photosynthesis reaction. The Quantum Information approach explains this without an inefficient random walk, and uses a quantum "random hopping". so the parallel with a Grover search is that the energy searches the path it has to follow to get to. The temperatures, at which these searching effects are really observed, and effected are still disputed.

Answer 5

It is almost never useful to try to analyze these things with photons. The clorophyll in a plant cell is innundated by light which is properly analyzed as a classical oscillating electric field. All the clorophyll molecules in a cell are driven by the same field, so they are all excited to the same extent. There is no particular constraint whereby the amount of excitation has to be restricted to units of hf. The entire ensemble of molecules is simply in a superposition of the ground state and the exicted states.

Once this state is reached, the transition to a metastable state where energy is stored in one of the molecules does not depend on absorption of a particular "photon". The energy is already present in the cell. The mechanism whereby this happens probably does not depend on any "collapse of the wave function".

The key to understanding how this happens is to realize that molecules in a superposition of two states behave as classical antennas: since the two states have different electron distributions and different frequencies, the result is an oscillating charge distribution at the difference frequency. Depending on the geometry of the ensemble, there can be highly efficient energy transfer between these antennas, resulting in the concentration of energy at a single location.

I have posted an explanation of how this happens through normal time-evolution of the wave function in this blogpost: http://marty-green.blogspot.ca/2014/12/wave-function-collapse-explained-by.html I have analyzed the case of a photographic plate exposed to weak light, but the same principles ought to apply to a plant cell exposed to strong light.