Atomic transitions and photon absorption

Atoms have various intrinsic energy levels that their electrons can occupy. These form a discrete ladder of levels, and electrons can jump up or down a rung. Atoms in general can absorb a photon of a certain frequency, provided the photon’s frequency matches the energy difference between two of the atomic energy levels. The photon absorption transfers its momentum to the atom. If the photon was traveling in the same direction as the atom, the atom gains momentum and travels faster. Since temperature is - roughly and simplistically speaking - a measure of the speed of an atom, if the photon was traveling in the same direction as the atom it will increase the “temperature” of the atom. The atom will eventually and spontaneously re-emit a photon in a random direction. The emitted photon will provide a kick in the opposite direction of the photon emission (conservation of momentum). So if the photon emits “backward” with respect to the atom, the kick will increase the speed and temperature of the atom further.

The opposite goes for the case where there’s a head-on collision between the photon and the atom. This will happen if the photon and atom are initially traveling in opposite directions. This will decrease the speed and temperature of the atom. So if we want to cool a group of atoms zipping around in space we’ll want those head-on collisions to occur more frequently. How can we make that happen?

The Doppler effect

The Doppler effect as applied to light waves and photons is a phenomenon that occurs from the point of view of an “observer” observing light traveling through space. If the light is traveling toward the observer, the observer will perceive an upward frequency shift as the wavefronts bunch up.

The opposite will occur if the light is moving away - the perceived frequency is lower (redshift). So… how can we exploit this and the above to cool a collection of atoms?

Putting it all together

If we have a laser that is shining light on the atoms from one direction, if we exactly match an atomic transition, we might be equally likely to speed up atoms (moving away from the laser) as slowing them down. Thus, no net cooling effect.

On the other hand, what if we tune the laser’s frequency (photons’ frequencies) to be a bit below the atom’s transition energy (spacing between atomic energy levels)? In that case, it’s far more likely for atoms traveling toward the laser light source to absorb the photon and get slowed down. This will serve to cool the atom. This happens due to the Doppler effect - atoms traveling toward the light source see a blueshifted photon frequency, which shifts the photon’s frequency up to match the atomic transition. The photons traveling in the opposite direction will see a redshifted photon, and therefore the frequency will be even lower than the atomic spacing → no absorption occurs.

Eventually the atoms will be traveling much more slowly and so the laser’s frequency will need to be tuned down even further, as the Doppler effect is less pronounced as the atoms get slower and cooler. If you keep tuning the laser down, you can get those atoms extremely cold, and extremely close to absolute zero temperature. In the next post I’ll walk through the math, which will only require high school-level geometry.

This can be extended to multiple lasers pointing in different, opposite and orthogonal directions. That way atoms moving away from one laser will be likely to absorb a photon from an opposite-pointing laser.

The full source for this series is on GitHub. The derivations, code, and prose are all mine. However, I did consult Claude to proofread and for help setting up the project and rendering equations.

Solution using the Schrödinger equation

The Schrödinger equation is a second-order partial differential equation that describes the time evolution of a quantum system. In classical mechanics, using the Euler-Lagrange equations gives you a deterministic trajectory of a particle or system through space and time. If you know the initial state of the system at a given time, you can calculate the system state at any subsequent time, provided you know the form of the Lagrangian (energy “function” for the environment the system is in).

In quantum mechanics, there is no such determinism. It is the probabilities of the state being in a state at a given time that evolve according to the equations of “motion.” So here’s the famous Schrödinger equation, (technically speaking, the position- or space-representation of the general equation):

where is the wavefunction, which varies in space and time and is the potential energy. The wavefunction describes the probability of finding a particle at a given point in space at a given time.

The setup

Now let’s make this concrete with our experiment. A free electron leaves the origin traveling in toward a screen a distance away. The screen has two slits: the top slit spans and the bottom slit , so the central barrier between them has height and each opening has height .

Code

import matplotlib.pyplot as plt
import dslit

fig, ax = plt.subplots(figsize=(8, 4.2))
dslit.draw_schematic(ax)
plt.show()

The derivation

Here we let the transmitted wave keep going to a detection screen a distance beyond the slits and derive the interference pattern in closed form. The trick is to model each slit as a Gaussian aperture: a Gaussian packet times a Gaussian slit is still Gaussian, and free evolution maps Gaussians to Gaussians, so every step stays analytic and no grid is needed.

Throughout, is the transverse coordinate and the along-axis drift just supplies a clock: the packet reaches the slits at and the screen at , with . The incident state is the standard free Gaussian wave packet of Cohen-Tannoudji (Complement ).

NoteThe complex Gaussian integral

Every evolution step below is a single application of the complex Gaussian integral

which stays valid by analytic continuation when is complex with positive real part.

The incident packet

Start with a Gaussian wave packet along , following the convention found in many graduate texts (see e.g. Cohen-Tannoudji, Sakurai).

Free evolution to the slit plane

Convolve Equation 1 with the free-particle kernel over , writing :

Collecting the terms gives , , and . The integral identity, together with , collapses the exponent to and the prefactor to , with

This is the Cohen-Tannoudji spreading packet: the modulus of Equation 2 gives . Both the analytic track here and the hard-slit numeric track saved for later start from this arrived packet.

The Gaussian slits

For the sake of reaching a closed form solution: model the two openings as Gaussian apertures centered at , with and width (chosen so a uniformly illuminated Gaussian slit transmits the same flux as a hard slit of height ):

Expanding the square in the slit- term of Equation 3 writes its exponent as a single quadratic with

Evolution to the screen

Convolve with the free kernel over , writing :

The -integrand is again with

Since is pure imaginary, and the integral converges. Applying the identity,

Summing the slits: envelope times fringes

The screen amplitude is . Expand . The piece is the same for both slits (it depends on ), and only the middle term carries the sign of . Defining the single complex constant

the two slits sum to a hyperbolic cosine, , where gathers the common (non-) factors. Taking the modulus squared and writing , the identity gives

The term is the interference; the term and together form the single-slit envelope.

The pattern

Evaluating Equation 7 on a grid gives the closed-form pattern below. The fringe spacing measured from the peaks matches ?@eq-spacing.

Code

import numpy as np
import matplotlib.pyplot as plt
import dslit

yD = np.linspace(-60e-6, 60e-6, 4000)
psi_p, psi_m = dslit.per_slit_analytic(yD)
I = dslit.screen_intensity(psi_p, psi_m, mu=1.0)
I = I / I.max()
fr, env = dslit.fringe_spacing()

fig, ax = plt.subplots(figsize=(8, 4))
ax.plot(yD * 1e6, I, color="#185FA5", lw=1.5)
ax.set_xlabel(r"$y_D$ on detection screen [$\mu$m]")
ax.set_ylabel("normalized intensity")
ax.set_title(r"Gaussian-slit fringes, $\Delta = \lambda L_2/(a+b)$"
             f" = {fr*1e6:.2f} µm")
plt.show()

Code

from scipy.signal import find_peaks

peaks, _ = find_peaks(I, height=0.05)
measured = np.median(np.diff(np.sort(yD[peaks])))
print(f"predicted  lambda*L2/(a+b) = {fr*1e6:.3f} um")
print(f"measured   peak spacing    = {measured*1e6:.3f} um")

predicted  lambda*L2/(a+b) = 6.464 um
measured   peak spacing    = 6.452 um

Classical picture: light

Many of you may know that light has both particle- and wave-like properties. A photon is an “atom” of light, an indivisible and smallest packet of light that exhibits particle-like behavior. However, light at a macroscopic level - and that which we encounter in our daily life - behaves as a wave. And waves exhibit two phenomena that are critical to understanding the double-slit experiment: diffraction and interference.

Quantum picture: also with light

Let’s now focus on the particle-like properties of light. The particle behavior of light, by the way, is inherently “quantum.” No classical theory can explain phenomena like the photoelectric effect or other experiments where photons’ particle-like behaviors dominate. So just remember this: light’s “quantumness” is roughly indicated by how much it’s behaving like a particle. For matter - including the electrons we’ll be dealing with - the opposite is true: the wave behavior only comes on when quantum effects dominate.

Aside: this second statement is crucial to understanding the profundity of the results we’re going to see. Because it turns out that the quantumness disappears whenever you try to figure out the particle’s state (“observe” it).

Let’s make our detector screen a theoretical, 2-dimensional screen that can detect where a single photon lands. When the light is on full blast like in the above experiment, the detector will detect a large number of photons over a given (short) timespan. The resulting pattern will exhibit wave interference, as we described before: you’ll see fringes with bright areas and dark areas. This is fully classical so far. Waves diffract when they hit the slits, and interfere when they recombine.

However, now dial down the light intensity to be sufficiently low that single photons arrive sequentially at the detector screen, passing through either the top or bottom slit (or… both or neither, as we’ll see). Okay, so admittedly this is really a semi-classical picture since we’re dealing with individual photons. But hear me out. Classically, we would expect the screen to have two blotches where the photons hit the screen, each directly, line-of-sight behind one of the slits. We wouldn’t expect interference, since each photon passes through sequentially with no overlap.

BUT, we in fact do see the interference fringes on the screen, provided we don’t set up a way to detect which slit each photon went through. I’ll get to the mathematics and maybe even philosophy of what happens when the observer (us) detects which slit the photon passed through. But for now, take it as a given that this is the case.

Let’s stop for a moment and appreciate how crazy this is: individual photons shot at the screen, one by one and, result in that interference pattern. If the photon went through a slit, there’s no second photon to overlap or recombine with. How is it possible for there to be interference? As we’ll eventually see, the photon essentially went through both slits simultaneously, and interfered with itself before hitting the detector. But wait, there’s more! If you somehow determine whether each photon went through the top or bottom slit, the pattern disappears, and we get that classical two-smudge pattern!

Note: if we’re being pedantic, we might say that the path the light took is indeterminate if you don’t measure it, and that it’s the trajectories or possible paths that interfere. I, for one, don’t think this makes things any less magical. But I also think my previous description is better for our layperson understanding of this.

Quantum picture: electrons

Let’s now turn to the actual experiment of interest: a double-slit setup with an electron beam instead of a light beam. Now prior to quantum theory, there were two disparate theories to describe particles and waves. These are (somewhat loosely) the fields of classical mechanics and wave mechanics. Whether you’re dealing with sound wave propagation, light, or a spreading circular wave in water, you use a different set of equations than if you’re looking at particles with mass. The equations of motion in both cases give you the state of the system at some time in the future, provided you know the initial state of your particle(s) or wave(s).

This is adequate for almost everything we interact with in our daily lives. Single electrons, however, are so tiny and interact so differently with each other and light or matter that we need quantum theory to calculate their trajectories and interactions.

And so when we fire electrons at the screen, we might know what to expect: fringes on the detector if quantumness was preserved, or smudges with no fringes if the quantumness disappeared. This is indeed the case! If we don’t observe which slit the electron went through, we see fringes. If we somehow figure out the trajectory and which slit the electron passed through, the quantumness and fringes disappear!