# How do we know which states are decohered by the environment?

If I take an atom in a momentum eigenstate, or just a very narrow gaussian in momentum space (with a very large spread in position space), and then I throw it into a gas, it will quickly decohere into a much narrower gaussian in position space, and it will spread in momentum space. Is there a way to calculate the new size of the wave packet? More generally, how do I know in what way the environment will decohere my quantum state? I.e. what is the basis that my environment "measures" in?

• Answer to part of your question: the environment measures the subsystem in whatever basis diagonalizes the interaction Hamiltonian. – DanielSank Apr 1 '19 at 6:39

(Tegmark's paper is actually critiquing a modification of quantum theory, but calculations using ordinary quantum theory are included for comparison.) I don't have a copy of Joos and Zeh in front of me, but Tegmark at least provides a quantitative estimates of the size of a particle's wavepacket after decoherence is well-developed, as well as the time required for decoherence to become well-developed. The width is $$\Delta x \sim\frac{\hbar\tau}{m\lambda_\text{eff}}$$ where $$m$$ is the mass of the atom, $$\lambda_\text{eff}$$ is an effective wavelength of the gas particles, and $$\tau$$ is the coherence time, which in turn is given by $$\tau=1/(\sigma\Phi)$$ where $$\sigma$$ is the total scattering cross section and $$\Phi$$ is the average flux of gas-particles per unit area per unit time (regarded as a flux of incident particles being scattered by the atom of interest).