How long does it take for Schrodingers cat to live or die?

For a quantum system interacting with an environment (containing many degrees of freedom) what causes decoherence? What sets the time scale on which it occurs?

It turns out the decoherence and dissipation are intimately connected. Decoherence arises from fluctuations in the phase of the system quantum state due to its interaction with the environment.

An important calculation was performed by Zurek in the 1980's and elegantly summarised in an expanded and updated version of his famous Physics Today article from 1991). Consider a free particle which is in a superposition state consisting of two Gaussian wave packets a distance Delta x apart.

Let gamma be the damping rate associated with friction from interaction with the environment. Then the decoherence time (i.e., the rate at which the off-diagonal parts of the density matrix decay) is given by

Thus we see that the same physics that causes friction (energy flow from the system to the environment) causes decoherence. But, they occur on completely different timescales.

Note that once Delta x is larger than the deBroglie wavelength [which it must beif we want to think about superpositions of semiclassical states] then the decoherence time is much less than the energy dissipation rate.

This connection between decoherence (fluctuations) and dissipation is another manifestation of the fluctuation-dissipation theorem.

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