Fluctuations in an established transmission in the presence of a complex environment

In various situations where wave transport is preeminent, like in wireless communication, a strong established transmission is present in a complex scattering environment. We develop a nonperturbative approach to describe emerging fluctuations which combines a transmitting channel and a chaotic background in a unified effective Hamiltonian. Modeling such a background by random matrix theory, we derive exact results for both transmission and reflection distributions at arbitrary absorption that is typically present in real systems. Remarkably, in such a complex scattering situation, the transport is governed by only two parameters: an absorption rate and the ratio of the so-called spreading width to the natural width of the transmission line. In particular, we find that the established transmission disappears sharply when this ratio exceeds unity. The approach exemplifies the role of the chaotic background in dephasing the deterministic scattering.

Figure 1

Schematic drawing of the model Hamiltonian. The single level ${\varepsilon }_0$ is coupled to scattering channels and a chaotic background $H_{\mathrm{bg}}$, leading to decay rates ${\mathrm{\Gamma }}_0$ and ${\mathrm{\Gamma }}_{\downarrow }$, respectively. Note that although the background states are not connected to the channels directly, they may have uniform broadening ${\mathrm{\Gamma }}_{\mathrm{abs}}$ to account for homogeneous dissipation.

Figure 2

Transmission distribution for a stable chaotic background. The probability density (top) and cumulative distribution function (bottom) are shown at several values of the coupling parameter $\eta =0.2,0.5,1,2,5$. The curves stand for the analytical results (23) and (24). The histograms correspond to numerical simulations with $5\times {10}^4$ realizations of random $200\times 200$ GOE matrices. These simulations are shown only for $\eta =0.2$ and 5; the agreement for the other values is similar. The numerics for the cumulative distribution (bottom, dashed black lines) is indistinguishable from the theory within the linewidth.

Figure 3

Transmission distribution for a chaotic background at finite absorption. The curves in color correspond to Eq. (31) and are shown at the absorption rate $\gamma =0.1$ (solid yellow curve), $\gamma =1$ (dotted blue curve), and $\gamma =5$ (dashed green curve). The coupling to the chaotic background is chosen to be $\eta =0.2$, 1, and 5 from top to bottom. Histograms stand for numerical simulations, as detailed in Fig. 2. Notice the disappearance of the bimodal shape of the distribution at moderate absorption.

Figure 4

Distribution of reflection coefficients $R_1$ (left) and $R_2$ (right), where $R_c={\left|S_{cc}\right|}^2$. The curves in color correspond to Eq. (37) and are shown at the different absorption rates $\gamma =0.1,1$, and 5 (solid yellow, dotted blue, and dashed green lines, respectively) and background coupling $\eta =0.2$, 1, and 5 (from top to bottom). The numerical results of RMT simulations are shown as histograms (the same statistics as in Fig. 3 was used). The chosen deterministic transmission $T_0=0.8$.

Figure 5

Accuracy of the interpolation formula. The exact distributions of $T, R_1$, and $R_2$ are shown as solid blue, yellow, and green lines, respectively. Dashed black lines show the same functions from Eqs. (31) and (37) but using the interpolation formula ${\mathbb{P}}_0^{\left(\mathrm{int}\right)}$ from Eq. (A1). All curves are for $\eta =1, \gamma =1$, and $T_0=0.8$. For other parameter values the correspondence is equally good.