An axion rocket is a form of theoretical space drive that would take advantage of primordial axions to provide energy and thrust.
According to axion theory, if axions exist, they should've been produced in enormous quantities during the big bang, and subsequently cooled, forming a Bose-Einstein Condensate permeating all of space. Axions are supposed to transform into photons in the presence of a sufficiently strong electric or magnetic field. An axion rocket would use such a strong magnetic field at the focus of a parabolic reflector to produce a directional beam of light from the conversion of ambient axions for thrust, thus making it a form of photon rocket.
The power output and thrust of an axion rocket would be limited by the flux of axions through the reaction chamber. Thus, power output and thrust would be expected to increase with speed (although relativistic redshift would offset the thrust gains at near-light speeds). Additionally, axion density would be expected to increase near galactic centers, and decrease in the intergalactic voids; density near Earth is predicted to be around 500 billion per cubic centimeter, but their extreme low mass means that the actual energy density is quite small- much less than the enegy density in sunlight hitting Earth.
unperson questions whether or not an axion rocket might violate the Second Law of Thermodynamics:
Interstellar space is essentially a heat bath of photons at a temperature of 2.7 K (i.e. the CMBR). Generally, all of space is permeated with a "gas" of photons of this temperature traveling in every which direction....
The reflector for the axion rocket will have these CMBR photons incident on it from all sides, so it will get no thrust from them, of course. When you turn on your magnet, it creates a new light source by transforming axions to photons. If the transformation preserves energy-momentum (i.e. 4-momentum), then I'd expect that the newly created photons are also at the CMBR temperature, since that's also the temperature of the condensate. Nevertheless, this seems to put more photons on one side of the reflector than the other, leading to net thrust.
Perhaps I am nit-picking, but temperature is an equilibrium phenomenon, thus the photons produced by the device can't be said to have any particular temperature. We would expect that the equilibrium state would have a higher temperature than the microwave background, however, because radiant energy is added to the system from the conversion of axion mass.
There's a catch, though. If the process of conversion between axions and photons is time reversal invariant, then it will be creating photons from axions but at the same time it will be creating axions from photons; it will be sucking up light. What's more, if the temperatures of the axion gas and the photon gas are the same, the light it's sucking up will be exactly equal to the light it's creating. As a result, there would, in fact, be no thrust at all.
There are multiple ways around this. To begin with time-reversal symmetry (which is expected) does not imply time-reversal invariance, or that the probability of interactions is the same in both directions, only that both directions are possible (this fact forms part of the basis for the Second Law of Thermodynamics, in fact).
But, consider the case where the per-particle interaction probability is the same in both directions (i.e., time-reversal invariance holds). If the device produces light of sufficient intensity that outgoing radiation is more intense than incident microwave background radiation, then there will not be sufficient incident photons for back-conversion to occur. This means that the rocket may not work below a certain axion flux rate, and thus may have a minimum velocity, which reinforces the idea that it works best at high speed, but does not preclude it from working at all. Additionally, at sufficiently high speeds, blue-shifted photons will be blocked by the reflector (which does induce a small amount of drag in relativistic starships), while red-shifted photons may not have sufficient energy for back-conversion into axions.
It's possible that the relevant mechanisms of linear momentum conservation during the transformation may also favor the rocket, but I'm not sufficiently versed in the theory to examine that.
In fact, the interaction probability in each direction is not expected to be the same. If I understand the theory correctly, photon-axion conversion should only occur when the energy of light in a particular volume is equal or greater than the axion energy in that volume.) In this case, the majority of incident background radiation would pass through the conversion device with negligible effect. The rate of light-eating would increase as the axions in any particular volume are depleted, but this, again, simply means that the device would work best at high speeds near galactic centers.