Optimal performance of reciprocating demagnetization quantum refrigerators.

A reciprocating quantum refrigerator is studied with the purpose of determining the limitations of cooling to absolute zero. The cycle is based on demagnetization and magnetization of a working medium. We find that if the energy spectrum of the working medium possesses an uncontrollable gap, and in addition there is noise on the controls, then there is a minimum achievable temperature above zero. The reason is that even a negligible amount of noise prevents adiabatic following during the demagnetization stage. This results with a minimum temperature, Tc(min)>0, which scales with the energy gap. The refrigerator is based on an Otto cycle where the working medium is an interacting spin system with an energy gap. For this system the external control Hamiltonian does not commute with the internal interaction. As a result during the demagnetization and magnetization segments of the operating cycle the system cannot follow adiabatically the temporal change in the energy levels. We connect the nonadiabatic dynamics to quantum friction. An adiabatic measure is defined characterizing the rate of change of the Hamiltonian. Closed-form solutions are found for a constant adiabatic measure for all the cycle segments. We have identified a family of quantized frictionless cycles with increasing cycle times. These cycles minimize the entropy production. Such frictionless cycles are able to cool to Tc=0. External noise on the controls eliminates these frictionless cycles. The influence of phase and amplitude noise on the demagnetization and magnetization segments is explicitly derived. An extensive numerical study of optimal cooling cycles was carried out which showed that at sufficiently low temperature the noise always dominated restricting the minimum temperature.