## A quantum accelerometer using relativity

I haven’t posted in quite awhile since I’ve been focused on Twitter, but the occasion of sending my latest co-authored paper to the arXiv seems a good opportunity to post something a bit longer, particularly since it relates to a series of blog posts I wrote about five years ago concerning a neutral pion in a large spherical cavity. The more general idea behind those posts was the concept of frustrated spontaneous emission (a rather unfortunate name for anyone with a sixth-grade sense of humor). Imagine a simple two-level quantum system that we’ll assume, for simplicity, emits a photon if it transitions from the excited state to the ground state. As most every first- or second-year undergraduate physics student should know, adding spatial constraints to a quantum system then constrains the energy levels of the system. This is the famous “particle-in-a-box” model. It’s related to the concept behind “Planck’s oven” which is the idea that only certain wavelengths of light can exist inside a finite-sized oven. When a constrained quantum system transitions from an one energy level to a lower one, conservation of energy requires that the energy lost by the system be taken up by something else. In atomic transitions, for instance, that energy is released as a photon. In other words, the amount of energy lost by the system, which is the difference between its initial energy and final energy, $E_i - E_f = \Delta E$, must equal the energy of the photon, $hc/\lambda$. But, since the wavelength $\lambda$ of the photon is proportional to the size of the constraining object (i.e. the “box”) not all transitions will be allowed. For example, suppose a simple quantum system has just two energy levels that we’ll call $|e\rangle$ for the excited state and $|g\rangle$ for the ground state. Suppose that the transition $|e\rangle\to|g\rangle$ produces 4 eV of energy in the form of a photon. That means the photon must have a wavelength of $\lambda=hc/E=(1240\textrm{ eV}\cdot\textrm{nm})/(4\textrm{ eV})=310\textrm{ nm}$. But suppose that the two-level system is in a box that only allows wavelengths of 100 nm, 200 nm, 300 nm, etc. (I’m just making these numbers up—they don’t necessarily correspond to any realistic system). Then the quantum system will be stuck in its excited state since the photon it would have to emit in order to transition to the lower state is not allowed to exist inside the box.

So now suppose that we have one of these quantum systems that is in its excited state but is in a box that won’t allow it to decay. In other words, it is in a state of frustrated spontaneous emission (pause for sophomoric giggling). According to relativity, if we accelerate the box and the quantum system together, the box will Lorentz contract. At some point, if it continues to accelerate and thus contract, it will reach a length that is compatible with the wavelength of the photon. At this point, the system can now transition between the excited state and the ground state. Such a system, if it could be realized, would then be useful as an accelerometer.

That, of course, is the simplified description of our idea. To be rigorous, we approached this problem from the standpoint of having a two-level system weakly coupled to a 1+1-dimensional cavity and used the Klein-Gordon equation to model the transition probabilities for the resonant and off-resonant modes. The proposal utilizes the Purcell effect. I would encourage those of you who are well-versed in the physics to read the paper and offer your comments and suggestions. We’re working on submitting to a journal soon. The paper was done in collaboration with Andrzej Dragan at the University of Warsaw and his undergraduate student Kacper Kożdoń. (That’s the same Andrzej Dragan as the Cannes Lion Award finalist and creator of the Dragan photo effect.)

Regarding actual implementation of this idea, we have several ideas in mind including a solid-state analogue. I’m hoping to get one of my own students working on it soon. There are also some intriguing aspects of the Purcell effect in relativistic settings that are worth exploring.

Finally, it should be noted that, due to the equivalence principle, such a device would also be able to measure motion in gravitational fields and might offer a means by which gravitational field anomalies could be mapped to high precision.