Universal Braess paradox in open quantum dots

We present analytical and numerical results that demonstrate the presence of the Braess paradox in chaotic quantum dots. The paradox that we identify, originally perceived in classical networks, shows that the addition of more capacity to the network can suppress the current flow in the universal regime. We investigate the weak localization term, showing that it presents the paradox encoded in a saturation minimum of the conductance, under the presence of hyperflow in the external leads. In addition, we demonstrate that the weak localization suffers a transition signal depending on the overcapacity lead and presents an echo on the magnetic crossover before going to zero due to the full time-reversal symmetry breaking. We also show that the quantum interference contribution can dominate the Ohm term in the presence of constrictions and that the corresponding Fano factor engenders an anomalous behavior.

Figure 1

The mesoscopic setup consists of two QDs, each one coupled to ideal electronic leads with an independent number of open propagating channels. The two QDs are coupled together with two leads, with the inclusion of a third, overcapacity lead.

Figure 2

Weak localization contribution (top) and Fano factor (bottom, dimensionless), depicted in terms of the number of channels $N_3$ in the overcapacity lead $G_3$, for several values of $N_1,N_2$, and $\Gamma$. The curves represent the analytical results of Eqs. (5) and (9) and the symbols correspond to the numerical simulation.

Figure 3

In the left panel we depict $\Delta G_{wl}$ using Eq. (6) in terms of the overcapacity open channels $\gamma /W$. The analytical result indicates the presence of a minimum for $\eta /W<1$, associated with the constraint for critical congestion $N_{3c}=W-N_1-N_2$. We depict the same quantity in the right panel, but in terms of $x/W$. The curves clearly show the appearance of an echo, that is, the conductance increases and decreases abruptly before going to zero with increasing magnetic field.

Figure 4

Study of the competition between the Ohm law and the weak localization, in the opaque limit. In the top panels we depict $〈G〉$ [Eq. (5)] in terms of $\gamma =N_3{\Gamma }_3$ for the overcapacity lead. In the left and right panels we note that, for certain small values of $W$ and $\eta$, the weak localization overcomes the Ohm term. In the bottom panels we depict the difference $\Delta G=〈G\left(\gamma \right)〉-〈G(\gamma =0)〉$. In these panels we identify analytically a universal signal, shown in the minimum of the same magnitude ${10}^{-3}2e^2/h$. This engenders the Braess paradox of the experiment related in Ref. [2] to a single measure. In particular, in the right panel we see that the magnetic crossover suppress the Braess paradox due to the breaking of the time-reversal symmetry.

Figure 5

The conductance is numerically depicted in the top panel, in the opaque regime, in terms of the tunneling probability ${\Gamma }_3$ of the overcapacity lead. Each curve represents a typical realization of two aleatory and independent QDs and shows a clear nonmonotonic behavior. The curve depicted in the bottom panel refers to the analytical result of Eq. (5) and the circles represent means over ${10}^5$ realizations, like the ones depicted in the top panel. The analytical result is compatible with the simulation for ${\Gamma }_3>0.1$.