Dark states in the magnetotransport through triple quantum dots

We consider the transport through a system of three coupled quantum dots in a perpendicular magnetic field. At zero field, destructive interference can trap an electron in a dark state—a coherent superposition of dot states that completely blocks current flow. The magnetic field can disrupt this interference, giving rise to oscillations in the current and its higher-order statistics as the field is increased. These oscillations have a period of either the flux-quantum or half the flux quantum, depending on the dot geometry. We give results for the stationary current, and the shot noise and skewness both at zero and finite frequency.

Figure 1

Three quantum dots are coupled coherently to one another via tunnel couplings $t_{ij}$ and incoherently to source and drain leads with rates ${\Gamma }_i$. Each dot contains a single level and by adjusting the relative positions of these levels, the system can be prepared in a dark state where no current flows despite the applied bias. In a perpendicular magnetic field, the structure encloses a magnetic flux $\Phi$, which causes a phase difference between different paths around the system that can disrupt the dark state and lead to current flow.

Figure 2

At zero field, the stationary current $⟨I⟩$ through the three-dot system shows a pronounced antiresonance with complete current blocking at a detuning of $\Delta ={\Delta }_0$, where the dark-state forms. The three curves show the current for different values of the coupling parameters $(t_{12},t_{23})$, with $t_{13}=\Gamma$ in each case. At zero field, a dark state always exists whenever both $t_{13}$ and $t_{23}$ are finite.

Figure 3

The stationary current through the three-dot system shows pronounced oscillations as a function of the applied flux. Plotted here is $\overline{I}=⟨I⟩∕{⟨I⟩}_{\max }$, the ratio of the current to its maximum value, which always occurs at $\Phi =\frac{1}{4}{\Phi }_0$. The different curves are for different values of $t_{13}$. Other parameters were $t_{12}=t_{23}=\Gamma$, $ϵ=0$, and $\Delta ={\Delta }_0$, such that CPT occurs at zero field. In the symmetric case with $t_{13}=t_{23}$ (solid curve), CPT trapping occurs at $\frac{n}{2}{\Phi }_0$, $n=0,1,2,\dots$, and the flux period of the oscillations is thus $\frac{1}{2}{\Phi }_0$. As $t_{13}$ moves away from symmetry, the dark-state current blocking at $\frac{n}{2}{\Phi }_0$, odd $n$ disappears, and the period of the oscillations doubles to ${\Phi }_0$.

Figure 4

The zero-frequency (shot noise) Fano factor $F^{\left(2\right)}\left(0\right)$ for the three-dot system as a function of magnetic flux also exhibits oscillations. We observe strong super-Poissonian peaks with $F^{\left(2\right)}\left(0\right)=3$ at the values of flux for which the dark-state forms (Ref. 18). Away from these points, the noise is sub-Poissonian. The period of these oscillations is the same as for the current. Same parameters as for Fig. 3, except for the displayed values of $t_{13}$.

Figure 5

(Color online) Contour plot of the finite-frequency Fano factor $F^{\left(2\right)}\left(\omega \right)$ as a function of magnetic flux $\Phi$ and frequency $\omega$. Colors white, red, and blue correspond to Poissonian, super-Poissonian, and sub-Poissonian values, respectively. Large super-Poissonian peaks occur at the zero-freqeuncy limit only. The shot noise also shows a series of inflexion points at a set of frequencies corresponding to the energy differences $\Delta E_{ij}$ of the Hamiltonian (dotted lines). Same parameters as in Fig. 3 with all $t_{ij}=\Gamma$.

Figure 6

The zero frequency skewness Fano factor $F^{\left(3\right)}\left(0\right)$ as a function of flux $\Phi$. The behavior is similar to that of the shot noise, with the highly super-Poissonian maximum value of $F^{\left(3\right)}{\left(0\right)}_{\max }=13$. Parameters as in Fig. 4.

Figure 7

(Color online) Contour plots of the finite-frequency skewness Fano factor $F^{\left(3\right)}\left(\omega \right)$ as a function of its two frequency arguments $\omega$ and ${\omega }^{\prime }$ for values of the magnetic flux $\Phi ∕{\Phi }_0=0,1∕12,1∕6,1∕4$. Parameters and color scheme as in Fig. 5. Strong super-Poissonian behavior only occurs close to $\Phi ∕{\Phi }_0=\frac{n}{2}$; otherwise, the skewness is predominantly sub-Poissonian. The fine structure arises from resonance between the frequencies $\mid \omega \mid$, $\mid {\omega }^{\prime }\mid$, and $\mid \omega -{\omega }^{\prime }\mid$ and the level splittings of the isolated dot system.