Memory effects induced by initial switching conditions

Initial switching refers to the way in which the decay of an initially confined state begins, as the barrier isolating it from the exterior is relaxed. We study these effects in the context of Longhi’s version of the Fano-Anderson model. Most authors assume the sudden approximation where the coupling is turned on instantaneously. We consider a finite rise time $T$ both numerically and analytically. When the coupling is ramped up linearly over a switching time $T$, we show that the asymptotic survival amplitude acquires a phase $T$ and is modulated by a factor $\left(\text{sin}T\right)/T$. Several other results relating to the solution of the model are obtained. All site amplitudes have the same decay constant during the exponential decay regime. In the asymptotic regime, the amplitude and phase of decay oscillations depend on the initial-switching profile, but the period does not.

Figure 1

Semi-infinite array of identical 1D wells coupled to a defect well separated by a thicker barrier.

Figure 2

A segment of a semi-infinite array of tunnel-coupled optical waveguides made by etching. The $z$ axis is parallel to the guides and $z_T$ at the top of the drawing. A logarithmic dependence on $z$ of the separation leads to linear dependence of the coupling.

Figure 3

Survival probability $P\left(t\right)$ for constant $\Delta =0.3$ in the region of changeover from exponential to asymptotic decay. Here and in the subsequent figures, time $t$ (dimensionless) is in units $\hslash /g$.

Figure 4

Diagrams which begin at site 1, with $m=1$ and 2 lines. The meaning of the various symbols is explained in text.

Figure 5

Three line diagrams which begin at site 1.

Figure 6

Four line diagrams which begin and end at site 1.

Figure 7

(Color online) Functions $S_o\left(T\right)/{\Delta }^2$ and $\text{Im}\left[S_e\left(T\right)/{\Delta }^2\right]$ for the linear rise model with site 1 initially $\left(t=0\right)$ occupied and the rest are empty. Various dashed lines: exact numerical calculations for $\Delta =0.3$, 0.5, and 0.9 according to key. Adjacent continuous lines: expressions of Eqs. (64, 65).

Figure 8

(Color online) Survival probability in the linear rise model for $T=1$ and $\Delta =0.3$. The continuous (red) line is the exact result. Practically hidden by it, the dashed (green) line is the value from the asymptotic expressions in the text. For comparison, the sudden approximation is also shown as a (blue) dotted line.

Figure 9

(Color online) Linear rise model, $\Delta =0.3$: survival probabilities for $T=0.5$, 1, and 2 from left to right. Continuous (red) lines: exact. Dotted (blue) lines: the exponential decay component [Eq. (40)].

Figure 10

(Color online) ${\left|c_1\left(t\right)\right|}^2$ for, from top to bottom, rise times $T=0.5$, 1, and 2. Linear rise model with $\Delta =0.3$, but now site 2 is initially occupied and the others empty. Continuous (red) lines: exact. Dotted (blue) straight lines: the exponential decay component [Eq. (40)].