Quantum Decoherence with Bath Size: Dynamics, Randomness, and Connectivity

The decoherence of a quantum system $S$ coupled to a quantum environment $E$ is considered, where $S+E$ is a closed quantum system. For typical states $X$ of the Hilbert space, i.e. for states chosen randomly from the Hilbert space unit hypersphere, we derive a scaling relation for the sum of the off-diagonal elements of the reduced density matrix $\rho_S$ of $S$. This sum is a measure of the decoherence of $S$, and decreases as $D_E^{-\frac{1}{2}}$ as the dimension of the environment Hilbert space $D_E$ increases. We present long-time calculations of the time dependent Schr\"odinger equation (TDSE) of spin $\frac{1}{2}$ particles comprising $S+E$ in order to test this scaling. The Hamiltonian has uniform or random Heisenberg couplings of a spin chain for $S+E$. Factors that affect the approach to the predicted scaling relation for the Heisenberg $d=1$ ring include how quickly and successfully the dynamics drives an initial configuration to an $X$ state, and this depends on the randomness of the coupling strengths in the Hamiltonian and the addition of other connections either within $E$ or between $S$ and $E$.