A highly readable book I picked up a few years ago is Principles of Quantum Electronics by Dietrich Marcuse. One of the fun parts about this book is that it beigins discussion of quantizing electronmagnetism by starting with the quantization of simple LC circuits. Of course, Marcuse, writes

It is true that the quantum theory of the LC circuit must be regarded as more correct than the classical theory, but the difference between the results of classical and quantum theory are unobservable by experiments with LC circuits.

This was written in 1970. What is interesting, of course, in our modern day of cool quantum experiments, is whether this is true today. In many ways, it does remain true, but there is a notable exception and that is in superconducing circuits. Very interesting, and readable works on this (with a focus on decoherence) are cond-mat/0308025 and cond-math/0408588 (works by Guido Burkard, Roger Koch, and David DiVincenzo.) So if a quantized theory of quantum circuits can be used to describe some superconducting quantum systems can we also do things like couple these to electromagnetic fields which are themselves quantized?

Well the answer is yes! And a group at Yale has been doing some excellent experiments in this direction. Here is a Nature paper just published about coupling superconducting circuits to microwaves transmission lines (Nature 445, 515-518, “Resolving photon number states in a superconducting circuit”, D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin and R. J. Schoelkopf) Previos work had shown how to get this coupling into the “strong coupling regime” where a single photon can be absorbed and emitted multiple times, whereas in this work they show how to get into a “strongly disapative regime” where the a single photon can have a large effect on the superconducting circuit without ever being absorbed. This allows the authors to perform experiments where they measure the photon number of the microwave field. Pretty cool! They call experiments like this circuit quantum electrodynamics. A new field is born.

So what consequences are there for quantum computerists? Well certainly this gives a possible method for a bus in some of the superconducting quantum systems. It should also allow for the preparation on nonclassical states of the microwave field, something which might have potential impact on quantum communication like protocols for quantum computing. But this is all in the future. Right now it’s just cool to sit back and read about the experiment!

Wait, I have a bone to pick with the wording. He says the quantum theory should “be regarded as more correct than the classical theory” but closes by saying “the results of classical and quantum theory are unobservable.” Now, maybe that isn’t true anymore and, as a quantum physicist I’m inclined to think the quantum theory is

alwaysmore accurate in some sense of the term, but – and remember that I’m a theorist here – isn’t science based on evidence? That’s precisely my problem with string theory. One can dream up all the nice theories and equations one wants, but how well does it describe reality? That’s my measure of how good a theory is. As such, his comment seems to be a bit of a non-sequitor taken in this light.Actually, Ian, I think the “unobservable” caveat was technically wrong even in the ’70s. There

isan observable consequence: thermodynamics. If the classical theory was literally correct, then the entire apparatus should vanish in a blinding flash of ultraviolet catastrophe!So we have two theories, both of which predict (in 1970)

completelyindistinguishable results for direct measurements, but one of which predicts a UV catastrophe that doesn’t happen. Ergo, the classical theory is false. Since the quantum theory hasn’t been falsified yet, it’s “more correct,” although this phrasing is technically just as meaningless as “very unique”.I could go on about effective theories and realms of applicability and such, but I’ll shut up, after noting that with string theory we’ve got a different choice between: (a) a theory that is clearly wrong, as it contains contradictions, but is still very effective in normal regimes; or (b) a theory that is not yet known to be wrong, but hasn’t predicted a single thing (even things predicted by the old theory).

Aside from Robin’s nice point about thermodynamics I read Marcuse as saying that while in general we know from experiment that conductors (e.g. copper wire) and electromagnetic fields must be understood by quantum mechanics, in a particular class of LC circuit experiments the difference between quantum and classical is not observable.

OK, JK’s point is well taken, though it points to either a potential editing snafu (or maybe just an editorial poor choice) or a logical contradiction in the wording. However, Robin, you can’t deny the wording – at least in the way it is included here – appears inherently contradictory regardless of the state of quantum and classical physics in 1970. It is a poor choice of words. In any case, I really don’t know much about the state of LC circuits circa 1970, but I will say that I would agree that the extrapolation of the inherent correctness of quantum theory to a variety of systems not yet proven (circa 1970 – and by his own contradictory and perhaps incorrect admission) would appear valid, but it gets at something that eats at me constantly: injecting certitude where there is high probability. They are

notthe same thing.Measuring the (complex) excitation amplitude on an LC oscillator definitely does exhibit quantum effects, even if the oscillator is in a coherent state. Because, after we measure such an coherent amplitude, we can “clone” it with any gain we want. Hey, this means that quantum limits on measurement, and quantum limits on amplification must be the same limits! The quantum limits on amplification were derived by Caves in the 80s, they boil down to “the noise figure of a linear stationary quantum amplifier must be at least 3 dB”. In the world of LC circuits, this is readily shown to be equivalent to “S_q S_V > hbar^2”, i.e., you can measure the charge on the capacitor, or the voltage on the coil, but not both at the same time to arbitrary accuracy. In the world of test mass measurement, this same limit is “S_f S_x > hbar^2”, where f is the force noise and x is the position of the test mass. LIGO hits this limit, but (needless to say) does not surpass it. Defined in this way, the “standard quantum limit” is strict and exact.