Optimal phase control of a measurement-assisted quantum refrigerator

A tilted nonselective measurement can close the upper transition of a three-level quantum refrigerator, but it also generates coherence that feeds back on the populations and suppresses the cooling current. We formulate randomized open-loop control of the measurement azimuthal phase $\alpha$, sampled independently at each event from a probability distribution $P(\alpha )$, while keeping the event rate, polar angle, and single-event population-transfer probability fixed. Direct population transfer is phase independent, whereas the coherence source carries the factor $e^{-i\alpha }$. Exact elimination of the stationary coherence gives $0\le k_{\mathrm{e}\mathrm{f}\mathrm{f}}[P]\le A_m$, where $k_{\mathrm{e}\mathrm{f}\mathrm{f}}[P]$ is the effective population-transfer rate between the upper levels and $A_m$ is the bare direct mixing rate. When the fixed measurement couples populations and coherence, the upper bound is attained if and only if $⟨e^{i\alpha }{⟩}_P=0$, where $⟨\cdot {⟩}_P$ denotes an average over the phase distribution. The minimal realization uses two equally weighted phases separated by $\pi$: the population-to-coherence source and its return to the populations then vanish at the generator level, while the full direct mixing remains. Because the cold current increases strictly with $k_{\mathrm{e}\mathrm{f}\mathrm{f}}$ in the refrigeration window, the same condition maximizes the stationary cooling power over the independent random-phase distributions considered here. For one representative parameter set, phase cycling increases the fixed-gap cooling power by about $32\mathrm{\%}$ relative to a fixed-phase measurement, while leaving the ratio of extracted cold heat to supplied measurement energy unchanged.