Pure phase decoherence in a ring geometry

We study the dynamics of pure phase decoherence for a particle hopping around an $N$-site ring, coupled both to a spin bath and to an Aharonov-Bohm flux which threads the ring. Analytic results are found for the dynamics of the influence functional and of the reduced density matrix of the particle, both for initial single wave-packet states, and for states split initially into two separate wave packets moving at different velocities. We also give results for the dynamics of the current as a function of time.

Figure 1

At left, an eight-site ring with nearest-neighbor hopping between sites. At right a potential $U(\mathbf{R})$ with eight potential wells (shown here symmetric under rotations by $\pi /4$), depicted as a contour map (with lower potential shown darker). When truncated to the eight lowest eigenstates, this is equivalent to the eight-site model.

Figure 2

A particular path in a path integral for the particle, shown here for an $N=3$ ring. This path, from site $0$ to site $1$, has winding number $p=1$.

Figure 3

Results for the free particle for $N=3$ and for a particle initially on site $1$. (Left) The probabilities to occupy site $1$ (full line), $2$ (large dashes), and $3$ (small dashes). (Right) The current from site $1$ to site $2$. (Top) $\Phi =0$. (Bottom) $\Phi =\pi /2$.

Figure 4

Plot of $P_{j0}(t)$ for a three-site ring, for a particle initially on site $1$, in the intermediate decoherence limit, with $\lambda =.02$. (Left) The probability to occupy site $0$ (full line), $1$ (large dashes), and $2$ (small dashes). (Right) The current from site $0$ to site $1$. (Top) $\Phi =0$. (Bottom) $\Phi =\pi /2$.

Figure 5

Plot of $P_{j0}(t)$ for a three-site ring, for a particle initially on site $1$, in the strong decoherence limit. (Left) The probability to occupy site $0$ (full line), $1$ (large dashes), and $2$ (small dashes). (Right) The current from site $0$ to site $1$ (compare Fig. 3). The results do not depend on $\mathit{\Phi }$.

Figure 6

Plot for $P_j(t)$ as a function of both $j$ and ${\Delta }_{o}t$ in the strong decoherence limit. $j_o=50$ and $N=100$. The relative velocity is $\frac{\pi }{2}$, in phase units. (Top) Global view. (Bottom) A particular peak.