Mathematical formalism to show that an atom casts a small shadow in the photon field that illuminate it

Question

This questions regards the relationship between photon absorption and the spatial mode of light. In the question I have some physical intuition which I think I understand and which is born out by experiment which is sprinkled throughout. However, the mathematical formalism I have to tackle the question at hand seems to fall short of being able to describe the physical situation I am concerned with and the formalism also raises causality issues for me. Because of all of this I spend most of the text in this post laying out the mathematical formalism as I understand it in hopes of seeking further understanding of this formalism or to be pointed towards a more sophisticated formalism which can address my concerns.

Background

In quantum optics the electric field can be quantized as

$$ \hat{\boldsymbol{E}}(\boldsymbol{x}, t) = i\sqrt{\frac{\hbar}{2\epsilon_0 V}} \sum_{\boldsymbol{k}, s}\sqrt{\omega_{\boldsymbol{k}}}\left(\boldsymbol{f}_{\boldsymbol{k}, s}(\boldsymbol{x})\hat{a}_{\boldsymbol{k},s}(t) - \boldsymbol{f}_{\boldsymbol{k}, s}^*(\boldsymbol{x})\hat{a}_{\boldsymbol{k},s}^{\dagger}(t)\right) $$

Bold symbols represent vector quantities. The is an equation for the quantum electric field in space and time. We sum over all wavevectors $\boldsymbol{k}$ which have, by the Helmholtz equation, related temporal frequencies $\omega_{\boldsymbol{k}} = c|\boldsymbol{k}|$. $s$ is a polarization index and takes on the values 1 or 2.

$\boldsymbol{f}_{\boldsymbol{k},s}(\boldsymbol{x})$ is a dimensionless vector valued spatial mode function which is determined by the boundary conditions*. For example, commonly, if we consider quantization in box of volume $V$ the mode functions are given by

$$ \boldsymbol{f}_{\boldsymbol{k}, s}(\boldsymbol{x}) = \boldsymbol{\epsilon}_{\boldsymbol{k},s} e^{i \boldsymbol{k}\cdot\boldsymbol{x}} $$

Here $\boldsymbol{\epsilon}_s$ is the polarization vector. Note that this is only one possible choice for the complete set of modes arising from solving the Helmholtz equation. The $\boldsymbol{f}_{\boldsymbol{k},s}(\boldsymbol{x})$ could also be, for example, Hermite-Gaussian or Laguerre-Gaussian modes as may be helpful to consider for this problem.

The mode volume or quantization volume is related to the spatial modes by**

$$ \int d\boldsymbol{x}\boldsymbol{f}_{\boldsymbol{k}, s}(\boldsymbol{x})\cdot\boldsymbol{f}_{\boldsymbol{k}',s'}^*(\boldsymbol{x}) = \delta_{\boldsymbol{k}\boldsymbol{k}'}\delta_{ss'}V $$

The $\hat{a}_{\boldsymbol{k},s}(t)$ and $\hat{a}^{\dagger}_{\boldsymbol{k},s}(t)$ are the bosonic, photonic annihilation and creation operators. These operators are related to the number of photons occupying a single mode. We see that quantum statistical properties of $\hat{\boldsymbol{E}}$ depend on the quantum statistical properties of the $a_{\boldsymbol{k},s}$

If we remove the hats from this expression we can see that the $a_{\boldsymbol{k},s}(t)$ are time-dependent coefficients of the spatial mode decomposition of the electric field. Putting the hats back in we see that these mode coefficients, $\hat{a}_{\boldsymbol{k},s}(t)$ are now quantum random variables rather than fixed amplitudes.

Shining Laser on a Screen

First a thought experiment. Suppose we have a light source which outputs, say a Gaussian beam*** which is focused down to a spot size $w_0$ at a certain location. Suppose we are able to arbitrarily tune the power of this source. Suppose for the sake of argument that it outputs coherent states of light. In one mode (high power) the output can be tuned so that the coherent state flux is composed of many many photons per second (like in a usual laser that we think of) or in another mode (low power)it can be tuned so that the output is less than one photon per second.

In one experiment we put a screen at the location of the focus and shine the laser beam in high power onto the screen. We will of course see a spot on the screen with a Gaussian shape.

In another experiment we put the screen at the same location of the focus but we now turn the laser down to low power. Now if we look at the screen we will not see a brightly illuminated spot. What we will see, is as time goes on we will see little**** spots appear on the screen one at a time (the temporal spacing between the appearance of spots will be statistical but related to the photon flux). If we keep track of all of the spots that we see then over time the distribution of the spots will look exactly like the gaussian spot we had for high power.

This sort of story is familiar to those who know about Young's double slit experiment.

Now imagine we put a little disk in front of the screen, say a few optical wavelengths in front of the screen. In the high power case we will just see a shadow of the disk. In the low power case we will see the shadow of the disk when we look at the distribution of bright spots.

Shadow of a Single Atom

Now imagine instead of a disk in front of the screen we place a single atom which has a transition resonant with the frequency of the laser beam. The atom can absorb a little bit of light and thus cast a shadow. The question sort of goes like this:

1) What does the shadow look like? Actually I know the answer to this question thanks to Absorption Imaging of Single Atom. The answer is that a small shadow of size $\approx \lambda \approx 1\text{ $\mu$m}$ will appear on the screen. Note that $w_0\gg \lambda$.

2) My question is how do describe in the formalism laid out in the background section?

We can consider the (dipole)***** coupling between an atom light of the form $H = -\boldsymbol{E}\cdot \boldsymbol{d}$ and we will see something like

\begin{align} \hat{H}_{AF} = \sum_{\boldsymbol{k},s} \hbar g_{\boldsymbol{k},s} \hat{\sigma}^{\dagger}\hat{a}_{\boldsymbol{k},s} + \hbar g_{\boldsymbol{k},s} ^*\hat{\sigma} \hat{a}_{\boldsymbol{k},s}^{\dagger} \end{align}

Here $\hat{\sigma} = |G\rangle\langle E|$ is the atomic lowering operator which takes the atom from the excited to the ground state. The coupling operator for each mode is given by

\begin{align} g_{\boldsymbol{k},s} = \sqrt{\frac{\omega}{2\hbar \epsilon_0 V}}d^{GE}_{\boldsymbol{k},s} \end{align}

Here

\begin{align} d^{GE}_{\boldsymbol{k},s} = \langle G|e\boldsymbol{x}\cdot \boldsymbol{\epsilon}_{\boldsymbol{k},s}|E\rangle \end{align}

$e$ is the electron charge. Note that if we consider, for example, an $s\rightarrow p$ atomic tranisition there are actually multiple excited states which makes the coupling of the atom to the different optical modes isotropic. That is the total coupling is the same for light coming from all directions.

My thinking would be that the answer to how the shadow is formed is that the atom preferentially absorbs modes with certain wavevectors but not others. As a result, the mode decomposition for light "after" the atom is different than the decomposition "before" the atom. This means the optical field will look different, i.e. it can have a shadow in it. however, The fact that the coupling is isotropic seems to put a wrench in this hope..

The question itself

A) If the coupling of light to all spatial modes is the same then wouldn't the affect of the atom on the field be to suppress the transmitted amplitude of the ENTIRE optical pattern by the same amount? Thus dimming the whole pattern rather than creating a shadow?

B) Of course, if the proposition in A is correct (I don't think it is, especially given the cited reference above) then there seem to be some serious locality issues. How can the presence of the atom in the center of the gaussian beam affect the transmitted intensity near the edge of the beam when they are separated by many many wavelengths?

C) This sort of raises a general question for me about the locality of atom-light interactions. Viewed in this way $\hat{a}_{\boldsymbol{k},s}$ is the quantum amplitude of an entire extended, non-local spatial mode with spatial pattern $\boldsymbol{f}_{\boldsymbol{k},s}(\boldsymbol{x})$. If one photon is emitted or absorbed into this field by the atom then it seems like the atom is doing something highly non-local in this mathematical description. That is, the atom occupies a very very small subwavelength volume of the field but in this mathematical description it can affect the amplitude of the field millions of wavelengths away instantaneously by absorbing or emitting a photon. Is there a more sophisticated mathematical formalism for treating this physical situation that would clarify these issues.

Footnotes

*Boundary conditions are assumed to be finite, like a large but finite box. I don't know exactly how to treat what I am asking in the case of infinite space and I think that this might be implicated in the answer to my question.

**Note other normalizations for mode volume are possible but this is the one that I take. Note that in this setup all modes have the same mode volume.

***For what follows, even though the light is a Gaussian mode I will consider $\boldsymbol{f}_{\boldsymbol{k},s}(\boldsymbol{x})$ to be plane waves. This means that the optical field coming out of the laser is actually composed of many plane wave modes with different wavevectors. That is, the field is in a (quantum) superposition of occupying many different modes.

****How little actually? I guess in principle as little as whatever is absorbing or scattering the light on the screen so perhaps atomic scale meaning, because of the diffraction limit the spots would appear upon imaging to be about the size of an optical wavelength, $\lambda$.

*****I wonder if part of the answer to my question has to do with high order multipole coupling terms? I don't think so. We can suppose there are not nearby transitions with the appropriate selection rules so that these higher order couplings play no role.

Answer 1

Upon reading the question carefully, I believe that the problems of the OP have nothing to do with the quantum nature of the interaction, but simply with the understanding of how modes work. To see this, let us simply write the interaction term in a different form which is in fact also mentioned in the question. Putting in the relevant functional dependences

$$ \hat{H}_{AF} = -\hat{\mathbf{E}}(\mathbf{r_a}, t) \cdot \hat{\mathbf{d}}, $$

where $\mathbf{r}_a$ is the position of the atom. This interaction the starting point for deriving the modal picture which is given by the OP. It comes from the minimal coupling prescription and involves for example the dipole approximation and fixing the gauge appropriately.

So let us look at this problem on a conceptual level. What we have is an electric field operator (an operator valued function of space and time) which is coupled to the atom.

• The field operator is governed by the operator version of Maxwell's equations.
• The atomic operators are governed by the standard Hamiltonian for whatever level structure you have in the atom.
• The Hamiltonian makes these two operator evolution equations coupled.

Your task is not to start with a certain initial condition for the electric field operator (or density matrix) and solve these evolution equations. By this we can at least answer C)

Answer to C): There is nothing non-local here, the coupling to the electric field is only at the position of the atom (this assumes the dipole approximation of course).

Solving these operator equations is of course difficult. But as far as I understand the question is about conceptual issues, not about how to solve this problem in a certain context.

This makes it clear that the only problem is the mode decomposition. Let's work backwards and first look at question B):

B) How can the presence of the atom in the center of the gaussian beam affect the transmitted intensity near the edge of the beam when they are separated by many many wavelengths?

The answer is simple: light couples to the atom, which causes a local change of the quantum field, which then propagates according to the propagation equations. Nothing difficult here either.

Here we can already see why this changes in the modal picture. The modes themselves are a non-local basis in some sense. That is you do not work in position space. If you want to describe how a localized field behaves, you therefore have to look at superpositions and cannot consider the modes individually.

This prepares us for A):

A) If the coupling of light to all spatial modes is the same then wouldn't the affect of the atom on the field be to suppress the transmitted amplitude of the ENTIRE optical pattern by the same amount? Thus dimming the whole pattern rather than creating a shadow?

Well, the coupling constant may be the same, but the population of each of the modes is not. If you are looking at linear scattering, you can simply imagine to replace the atom by a little refractive sphere, which is entirely equivalent for linear scattering. What would happen then is exactly the classical intuition which the OP described by the examples in the question, just that the atom is refractive instead of a fully absorbing material.