The three-level-amplifier is the template of a quantum device. It operates by employing a hot and cold bath
to maintain population inversion between two energy levels which is used to amplify light by stimulated emission
The ground state level (1-g) and the excited level (3-h) are coupled to a hot bath of temperature .
The energy gap is  . When the population on the levels equilibrate 
where  is the Planck constant and  is the Boltzmann constant.
The cold bath of temperature  couples the ground (1-g) to an intermediate level (2-c) 
with energy gap .
When levels 2-c and 1-g equilibrate then
The device operates as an amplifier when  levels (3-h) and (2-c) are coupled to an external field of frequency .
For optimal resonance conditions  . 
The efficiency of the amplifier in converting heat to power is the ratio of work output to heat input:
Amplification of the field is possible only for positive gain (population inversion)
This is equivalent to .
Inserting this expression into the efficiency formula  leads to:
where is the Carnot cycle efficiency. 
Equality is obtained under a zero gain condition .
The relation between the quantum amplifier and the Carnot efficiency was first pointed out by Scovil and Schultz-DuBois:
Reversing the operation driving heat from the cold bath to the hot bath by consuming power constitutes a refrigerator.
The efficiency of the refrigerator defined as the coefficient of performance (COP) for the reversed device is: