Phonon-mediated decoherence in triple quantum dot interferometers

We investigate decoherence in a triple quantum dot in a ring configuration, in which one dot is coupled to a damped phonon mode while the other two dots are connected to a source and a drain, respectively. In the absence of decoherence, single-electron transport may get blocked by an electron falling into a superposition decoupled from the drain; this is known as a dark state. Phonon-mediated decoherence affects this superposition and leads to a finite current. We study the current and its shot noise numerically within a master equation approach for the electrons and the dissipative phonon mode. A polaron transformation allows us to obtain a reduced equation for only the dot electrons, which provides analytical results in agreement with numerical ones.

Figure 1

Triple quantum dot in a ring configuration with mutual tunnel couplings $\tau$. Dots 1 and 3 possess on-site energies ${\epsilon }_{1,3}=0$ and are tunnel-coupled to the source and the drain, respectively. Dot 2 interacts with a damped vibrational mode with frequency ${\omega }_0$, while its on-site energy can be tuned by a gate voltage such that ${\epsilon }_2=V_{\mathrm{gate}}/e$. Dot 2 has a vibrational degree of freedom, while dots 1 and 3 are rigidly attached to the contacts.

Figure 2

Current at zero detuning, ${\epsilon }_2=0$, as a function of the scaled magnetic flux $\varphi$ for two values of the phonon damping strengths $\gamma$ compared to the current in the absence of the phonon ($\lambda =0$). The dot-lead tunneling rate is $\Gamma =0.1{\omega }_0$. In the noninteracting case, the current drops to zero at semi-integer values of the quantum flux. The Aharonov-Bohm amplitude is reduced by the phonon-mediated decoherence of the dark state.

Figure 3

Current as a function of the detuning ${\epsilon }_2$ for various values of the electron-phonon coupling $\lambda$. The interdot tunneling and the dot-lead tunneling rates are $\tau =0.01\hslash {\omega }_0$ and $\Gamma =0.1{\omega }_0$, respectively, while the dissipation strength is $\gamma =0.05{\omega }_0$. The temperatures are (a) $T=0$ and (b) $T=1.5\hslash {\omega }_0/k_B$. Insets: enlargement of the region near ${\epsilon }_2=0$ demonstrating a shift of the current minimum with increasing electron-phonon coupling.

Figure 4

Current and Fano factor as a function of the electron-phonon coupling for the dark state, i.e., for the detuning $\epsilon ={\epsilon }_2-{\lambda }^2/\hslash {\omega }_0=0$, at zero temperature and two values of the dissipation strength $\gamma$. All other parameters are as in Fig. 3a. The dotted lines mark the results obtained with the reduced master equation (24). Numerical convergence was reached when considering $N=25$ Fock states of the phonon mode.

Figure 5

Current as a function of the temperature for electron-phonon coupling $\lambda =0.15\hslash {\omega }_0$ and detuning $\epsilon ={\epsilon }_2-{\lambda }^2/\hslash {\omega }_0=0$, corresponding to the dark state. All other parameters are as in Fig. 3. The dotted lines are obtained with the reduced master equation (24) for the dot electrons.

Figure 6

Comparison of the results with the full quantum master equation (10) and those of the effective master equation (24) for $\lambda =0.1\hslash {\omega }_0$, $\gamma =0.05{\omega }_0$, $\tau =0.01\hslash {\omega }_0$, and $\Gamma =0.1{\omega }_0$. The temperature is (a) $T=0$ and (b) $T=1.5\hslash {\omega }_0/k_B$.