Which is closest to the minimum power required by the Carnot limit to pump 1 µW of heat out of a 10 mK stage and reject it at 300 K?

Answer: B. About 30 mW

For a Carnot refrigerator, the required power is W = Q × (T_H − T_C)/T_C = 1 µW × (300 − 0.01)/0.01 ≈ 30 mW. The same 1 µW at a 4 K stage would need only about 74 µW, so removing heat becomes far more expensive at colder stages. The cooling power of the coldest stage of a real dilution refrigerator is roughly ten to several tens of µW near 20 mK, far smaller than that of the 4 K stage (around 1 W), so heat flowing in through the wiring and heat generated by circuits are handled at the warmest stage possible. For the same reason, control circuits placed at cryogenic temperatures are mostly placed at a stage of about 4 K.

• A. This assumes the work equals the heat pumped. Pumping heat out of a low temperature requires much more work, in proportion to the temperature ratio.
• B. Correct. 1 µW × (300 − 0.01)/0.01 ≈ 30 mW.
• C. This is the value if 1 µW is mistaken for 1 mW.