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My understanding is that microwaves use the dipole moments of water molecules to heat things. The microwaves resonate with the dipole moment, and add energy to the system.

How exactly does the microwave-molecule interaction work, and why can't that same resonance be used to reduce molecular movement?

For analogy: If there was a bell with a resonant frequency, you could add energy to the bell (and ring it) by hitting it with sound at that frequency. If the bell was already ringing, you could slow it down by hitting it with the same frequency, but 90-degrees phase shifted. Why can't the same be done with water molecules?

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Your question is based on a misapprehension. The heating in microwave ovens is not a resonant process. See the answers to Does a domestic microwave work by emitting an electromagnetic wave at the same frequency as a OH bond in water? for a discussion of this.

But let's leave this aside, because there is some interesting physics in your question. Suppose there was something that created an energy difference between orientations of the water dipole. For example, suppose we apply an external electric field so that the dipole aligned with the field has a lower energy than the dipole aligned against it. We'll tweak the field so that the energy difference between the two alignments is equal to the energy $h\nu$ of the microwave radiation.

A dipole aligned with the field can absorb a photon and flip into anti-alignment. This is like hitting the bell i.e. it adds energy to the bell. However the radiation can also cause stimulated emission of the excited state so it decays to the ground state by emitting a photon - you'd put one photon in and get two out. This is like hitting the bell in antiphase - hitting the bell in antiphase means your hammer rebounds with more energy than you put in.

The problem is you don't have a single bell. You have $6.023 \times 10^{23}$ bells for every 18g of water and those bells have no mechanism to keep themselves in phase with each other. So when you turn on the microwaves you end up with an equilibrium between absorption and stimulated (and spontaneous) emission. Typically you get roughly equal numbers of dipoles in the ground and excited state.

But actually this still isn't really heating in the sense we normally use the word. Heating occurs when the excited dipoles can relax by transferring their energy to lattice vibrations rather than by emitting a photon. So the overall process is photon $\rightarrow$ excited dipole $\rightarrow$ lattice vibrations = heat. This is more like what happens in a microwave oven. The reverse, i.e. cooling, process is when lattice vibrations cause transient dipoles and these radiate photons. This is just black body radiation.

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