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In classical physics we know of coherence of waves and in quantum physics we identify coherent states. While those are clearly defined concepts/terms, in literature we regularly encouter also that a $N$-body quantum system evolves coherently. What does coherent evolution mean in this context? (and is there a trivial example?)

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Schrödinger equation for N-body system
A system evolves coherently, if there are no dephasing/decoherence effects, i.e., its evolution can be simply described by its Hamiltonian and the Schrödinger equation (with obvious modifications for relativistic case): $$ i\frac{\partial \Psi}{\partial t}=H_0\Psi.$$ This equally applies to a many-body system, except the wave-function now is an N-particle wave function - this is where the quantum mechanics enters, in having to describe $N$ particles by a single wave function (whereas in classical mechanics the trajectory of each could be calculated separately.) Note that this also applies in second quantization representation, even though it is rarely explicitly written.

Decoherence and dephasing
In practice the full Hamiltonian is usually more than just $H_0$, but also includes environment and coupling to this environment: $$H=H_0 + H_{env}+H_{int}.$$ The joint evolution of the system AND environment is still coherent, but as we are usually interested only in the evolution in the phase space corresponding to $H_0$, we trace out the environmental degrees of freedom. There are exist different techniques to do so, but in any case we end up with something different from the usual Schrödinger equation, and we usually cannot describe the system by a wave function, but need to resort to using a density matrix.

Another case where the evolution stops being coherent is when we deal with disorder, i.e., we average over many systems with similar but not identical Hamiltonians.

Remark: While there is a good reason why coherent states are called this way, the meaning of the term is a lot mroe specific than what is meant by coherence.

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