Active thermal control: how I hold a stage where I want it, and prove it.

Active thermal control is a feedback loop: a sensor reads the stage, a controller compares it with the setpoint, a heater corrects it. In practice the loop is the easy part. It only works if three other things are right first.

• The architecture. Heat paths are designed so the loop has little to fight: weak links, low thermal mass, heat-sunk wires, vacuum and radiation shields.
• The sensing. The sensor is calibrated, trusted in the magnetic field, and placed where the sample really is.
• The timing. Nothing is measured until the loop has settled, judged by data, not by a fixed wait.

Getting to about 1 K. A pulse-tube cryocooler, a closed-cycle refrigerator with no liquid helium to refill, cools the first two plates to roughly 50 K and 3 K. The incoming helium is then condensed and expanded through a flow restriction, which takes it below about 1 K.

The key trick. Below about 0.87 K, a mixture of helium-3 and helium-4 separates into two liquid phases, like oil and water: a lighter phase that is almost pure helium-3 floating on a heavier phase that is mostly helium-4 with a few percent helium-3 dissolved in it. When a helium-3 atom crosses from the rich phase into the dilute phase, it absorbs heat, just as a molecule does when a liquid evaporates. That crossing happens in the mixing chamber, the coldest point of the fridge.

Keeping it going. The dilute phase connects to the still, held near 0.7 K. There, helium-3 evaporates far more readily than helium-4, so pumping on the still removes almost only helium-3. That keeps drawing helium-3 across the phase boundary in the mixing chamber. The pumped gas returns down the condensing line, is pre-cooled by heat exchangers against the outgoing dilute stream, and rejoins the rich phase: a continuous cycle.

Cooling power. The heat a dilution fridge can absorb falls roughly as the square of temperature (Q̇ ∝ ṅ₃T², where ṅ₃ is the helium-3 flow). At 10 mK the budget is microwatts, so every wire, support and radiation path matters far more than at 2 K.

• Staged cooling. Each plate, from 50 K to the mixing chamber, intercepts heat before it reaches the next. Wires and supports are anchored at every plate on the way down, so the coldest stage only sees what the stage above lets through.
• Radiation shields on each stage, because a warm surface in view of the mixing chamber can exceed its entire cooling budget.
• Thermal links. In cold-finger mode an experiment bolts directly to the mixing chamber. In top-loader mode it rides on a long probe and is cooled through a gold-plated OFHC copper link to the mixing chamber, which avoids warming the whole fridge to change samples.
• Temperature control with heaters on the still and mixing chamber and resistance thermometers, typically ruthenium oxide, read with very low excitation so the thermometer does not heat itself.
• Cooldown. Operating the fridge meant running the pulse-tube pre-cool, condensing the mixture and starting circulation, while monitoring each stage's temperature as it came down.

To prepare an experiment for it, I designed the tunnel-diode-oscillator cell to fit the top-loading probe, with three thermally separated zones: an electronics stage in its own radiation can, a coil zone isolated on G10 rods, and a sample stage with its own sensor and heater, holding the sample on a sapphire rod inside the coil without touching it. The oscillator was first tested at 77 K in liquid nitrogen; mounting the cell in the fridge was the next step when I finished.