# Photo-Carnot engine

A photo-Carnot engine is a Carnot cycle engine in which the working medium is a photon inside a cavity with perfectly reflecting walls. Radiation is the working fluid, and the piston is driven by radiation pressure.

A quantum Carnot engine is one in which the atoms in the heat bath are given a small bit of quantum coherence. The phase of the atomic coherence provides a new control parameter.[1]

The deep physics behind the second law of thermodynamics is not violated; nevertheless, the quantum Carnot engine has certain features that are not possible in a classical engine.

## Derivation

The internal energy of the photo-Carnot engine is proportional to the volume (unlike the ideal-gas equivalent) as well as the 4th power of the temperature (see Stefan–Boltzmann law):

$U = \varepsilon\sigma T^{4} \,.$

The radiation pressure is only proportional to this 4th power of temperature but no other variables, meaning that for this photo-Carnot engine an isotherm is equivalent to an isobar:

$P = \frac{U}{3 V} = \frac{\varepsilon \sigma T^{4}}{3 V} \,.$

Using the first law of thermodynamics ($dU = dW + dQ$) we can determine the work done through an adiabatic ($dQ = 0$) expansion by using the chain rule ($dU = \varepsilon \sigma dV T^{4} + 4 \varepsilon \sigma V T^{3} dT$) and setting it equal to $dW = -P dV = -\frac{1}{3} \varepsilon \sigma T^{4} dV \,.$

Combining these gives us $\frac{2}{3} T^{4} dV = 4 V T^{3} dT$ which we can solve to find $\frac{V^{1/6}}{T} = const \,.$

....

The efficiency of this reversible engine must be the Carnot efficiency, regardless of the mechanism and so $\eta = \frac{T_H - T_C}{T_H} \,.$