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arXiv:1301.0077 · Physical Review A

Quantum decoherence scaling with bath size: Importance of dynamics, connectivity, and randomness

Fengping Jin, Kristel F. L. Michielsen, M. A. Novotny, Seiji Miyashita, Shengjun Yuan, Hans A. De Raedt

Published Feb 20, 2013• 20 citations
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Abstract

We consider the decoherence of a quantum system $S$ coupled to a quantum environment $E$. For states chosen uniformly at random from the unit hypersphere in the Hilbert space of the closed system $S+E$ we derive a scaling relationship for the sum of the off-diagonal elements of the reduced density matrix of $S$ as a function of the size ${D}_{E}$ of the Hilbert space of $E$. This sum decreases as $1/\sqrt{{D}_{E}}$ as long as ${D}_{E}\ensuremath{\gg}1$. We test this scaling prediction by performing large-scale simulations which solve the time-dependent Schr\"odinger equation for a ring of spin-$1/2$ particles, four of them belonging to $S$ and the others to $E$, and for this ring with small world bonds added in $E$ and/or between $S$ and $E$. The spin-1/2 particles experience nearest-neighbor interactions that are identical for the interactions within $S$ and random for the interactions within $E$ and between $S$ and $E$, or that are all identical. Provided that the time evolution drives the whole system from the initial state toward a scaling state, a state which has similar properties as states belonging to the class of quantum states for which we derived the scaling relationship, the scaling prediction holds. We examine various interaction parameters and initial states for our model system to find whether or not the time evolution reaches the class of states that have the scaling property. For the homogeneous ring we find that the evolution for select initial states does not reach these scaling states. This conclusion is not modified if we add some homogeneous random connections. For a ring we find that some randomness in the interaction parameters is required so that most initial configurations are driven toward the scaling state. Furthermore, if the amount of randomness is small the time required to reach the scaling states may be very large. For the case of all random interactions in $E$ the ring is driven toward the scaling state. Adding small world bonds between $S$ and $E$ with random interaction strengths may decrease the time required to reach the scaling state or may prevent the scaling state from being reached. For the latter case we show that increasing the complexity of the environment by adding extra connections within the environment suffices to observe the predicted scaling behavior.

DOI
10.1103/physreva.87.022117

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