How can one model quantum walks in photosynthesis?
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.
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.
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