Emergence of Fermi’s Golden Rule

Fermi’s golden rule applies in the limit where an initial quantum state is weakly coupled to a continuum of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final states is discrete, with a nonzero mean level spacing; this question arises in a number of recently investigated many-body systems. For different symmetry classes, we analytically and/or numerically calculate the universal crossovers in the average decay of the initial state as the level spacing is varied, with the golden rule emerging in the limit of a continuum. Among the corrections to the exponential decay of the initial state given by Fermi’s golden rule is the appearance of the spectral form factor in the longtime regime for small but nonzero level spacing.

Figure 1

Diagrammatic representation of the short-time ($\tau <1$) correction to the FGR exponential decay of the initial state. Left-hand diagram (one ladder): classical probability that a particle, escaped into the dot, returns to the weakly coupled level. Right-hand diagram (two ladders): quantum interference correction to classical return probability. Dashed and solid lines here represent the retarded (black) or advanced (red) level and dot Green’s functions $G_i^{\pm }(\epsilon )\sim (\epsilon -{\epsilon }_i\pm i{\mathrm{\Sigma }}_i{)}^{-1}$, dressed by self-consistently calculated self-energies ${\mathrm{\Sigma }}_0\sim g\lambda$ and ${\mathrm{\Sigma }}_{\mu }\sim \lambda$, as indicated in the box. The ladder defines the ergodic quantum dot mode $D(\omega )\sim i{\lambda }^2/(N{\omega }^{+})$, and is recursively defined in the third equation in the box. The solid vertical line represents the coupling between $|0⟩$ and quantum dot states. See also the Supplemental Material for further details and a nonperturbative calculation addressing all times, including $\tau >1$ [26].

Figure 2

The probability $P(t)$ to be in the initial state after time $t$. The total Hilbert space dimension is $N+1={10}^3$ and numerical data are averaged over ${10}^4$ samples. (a)–(c) The time dependence of $P(t)$ for $\gamma \in \{46,0.46,0.022\}$, respectively. The black curves are data from numerical simulations of the model Eq. (1). The red curves are theoretical predictions obtained by evaluating the integral in Eq. (5) numerically. (d) The $t\to \infty$ limit $P_{\mathrm{res}}$. Blue dots are numerical data. The gray curve is the theoretical prediction of Eq. (4). The purple curve is the FGR result $1/\gamma$ [29]. The small black arrows point to data that correspond to the late-time limits of the three upper panels. (e) The spectral form factor arises in the time dependence of $P(t)$ for the three versions of our model derived from the three Wigner-Dyson classes (GOE, GUE, GSE). The black curves are data from numerics, and the colored curves are Eq. (6) with parameters $a_X$ and $b_X$ as given in the main text. In panels (a)–(d) $H$ is GUE.