Sunday, October 10, 2021

The Purcell effect - still mind-blowing.

The Purcell effect is named after E. M. Purcell, a Nobel-winning physicist who also was a tremendous communicator, author of one of the great undergraduate textbooks and a famous lecture about the physical world from the point of view of, e.g., a bacterium.  I've written about this before here, and in a comment I include the complete (otherwise paywalled) text of the remarkable original "paper" that describes the effect.

When we calculate things like the Planck black-body spectrum, we use the "density of states" for photons - for a volume $V$, we are able to count up how many electromagnetic modes are available with frequency between $\nu$ and $\nu + \mathrm{d}\nu$, keeping in mind that for each frequency, the electric field can be polarized in two orthogonal directions.  The result is $(8\pi/c^3)\nu^2 \mathrm{d}\nu$ states per unit volume of "free space".

In a cavity, though, the situation is different - instead, there is, roughly speaking, one electromagnetic mode per the bandwidth of the cavity per the volume of the cavity.  In other words, the effective density of states for photons in the cavity is different than that in free space.  That has enormous ramifications:  The rates of radiative processes, even those that we like to consider as fundamental, like the rate at which electrically excited atoms radiatively decay to lower states state, can be altered in a cavity.  This is the basis for a lot of quantum optics work, as in cavity quantum electrodynamics.  Similarly, the presence of an altered (from free space) photon density of states also modifies the spectrum of thermal radiation from that cavity away from the Planck black-body spectrum.

Consider an excited atom in the middle of such a cavity.  When it is going to emit a photon, how does it "know" that it's in a cavity rather than in free space, especially if the cavity is much larger than an atom?  The answer is, somehow through the electromagnetic couplings to the atoms that make up the cavity.  This is remarkable, at least to me.   (It's rather analogous to how we picture the Casimir effect, where you can think about the same physics either, e.g., as due to altering local vacuum fluctuations of the EM field in the space between conducting plates, or as due to fluctuating dipolar forces because of fluctuating polarizations on the plates.)

Any description of a cavity (or plasmonic structure) altering the local photon density of states is therefore really short-hand.  In that approximation, any radiative process in question is tacitly assuming that an emitter or absorber in there is being influenced by the surrounding material.  We just are fortunate that we can lump such complicated, relativistically retarded interactions into an effective photon density of states that differs from that in free space.

Steve said...

I haven't read this in depth, but I've been meaning to ..

https://www.nature.com/articles/srep12956#Abs1

Mayer Landau said...

Hi Professor Natelson
You state, "somehow through the electromagnetic couplings to the atoms that make up the cavity ... we can lump such complicated, relativistically retarded interactions into an effective photon density of states that differs from that in free space."
Is a good way then, to think about this, is that the atom emits a photon, but if the photon determines that it does not obey the cavity boundary conditions then it is virtual and is reabsorbed by the atom? Following that reasoning, because only some photons make the transition from virtual to real, the free space radiation rate is altered by the cavity.
Does that reasoning make any sense or is it off the mark?

Anonymous said...

An interesting aspect of the Purcell effect is that when you go through to the ultra-strong and deep-strong coupling regimes of a two-level system coupled to light you actually get a *reversal* of the Purcell effect. Odd as it sounds, light and matter decouple at strong enough coupling!

Check out this paper:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.016401

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