The abbreviation here has a double meaning-- both "Open Access" and "Operator Algebra." In my Quantum Optics class yesterday, I was talking about how to describe "coherent states" in the photon number state formalism. Coherent states are the best quantum description of a classical light field-- something like a laser, which behaves very much as if it were a smoothly oscillating electromagnetic field with a well-defined frequency and phase.
Mathematically, one of the important features of a coherent state is that it is unchanged by the photon annihilation operator (in formal terms, it's an "eigenstate" of that operator). That is, if you have a coherent state with an average of five photons in it, and you subtract one photon from it (say, by detecting that photon), you end up with... a coherent state with an average of five photons in it. It's not that the photon subtraction makes a negligible change in the state of the light, it makes absolutely no change at all.
This is kind of odd, and at least 16.7% of the class was bothered by this. So, even though I hadn't planned for it as part of the class, I remembered a recent experimental paper that did just this measurement (I wrote it up for ResearchBlogging earlier this year). And because it's an Open Access journal (New Journal of Physics), it was easy to Google up the paper and show it to the class. so, rather than talking about Hanbury Brown and Twiss type experiments in the photon number formalism, we spent the last tfifteen or so mintues of the class talking about the experimental demonstration.
So, hooray for OA. Both types.
The relevant lecture notes are below the fold, for those who care:
- Lecture12-13.pdf: Coherent states and squeezed states in the phasor picture, experimental generation and detection of squeezed states.
- Lecture14.pdf: The simple harmonic oscillator using ladder operators; photon creation and annihilation operators.
- Lecture15-16.pdf: Coherent states as eigenstates of the annihilation operator, Hanbury Brown and Twiss experiments in the number-state formalism.
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Ah yes, all those a's and a-daggers remind me of gamma-5's from field theory, which in turn reminds me of all the physics I have forgotten since I passed the qualifier...
I like your description of what it means to be in an eigenstate of the annihilation operator and the framing as a spooky result, but I wonder if the terminology in your description isn't slightly misleading.
First, I would guess that most things one would think of as "subtracting a single photon" are not described by applying the annihilation operator alone.
Second - now referring to the experimental paper - while I agree it's true that the outgoing state of beamsplitter+single-photon-detector has a mean photon number that is independent of whether the detector clicks, it is not true that the outgoing state photon number is independent of whether or not you encountered a beamsplitter+single-photon-detector.
One can only have the case where you have a mean photon number of 5 before you encounter the beamsplitter+single-photon-detector and a mean photon number of 5 afterwards in the limit that the beamsplitter reflectivity goes to zero. But in that limit you'll never get a click on the detector.
Which perhaps makes things less spooky?
I just got a chance to read the paper and your notes. It's interesting to put together lectures 15 & 16. Since subtracting a photon from a coherent state yields a coherent state, you can't get single photon states by putting a bunch of filters in front of a laser. Thats' why something like spontaneous downconversion is necessary.
Some introductory books use a laser and filters as an example of a single photon source (Six Ideas That Shaped Physics, Vol. Q, for example). TeachSpin also sells a device described as "Two Slit Interference, One Photon at a Time" which uses a laser or light bulb.