Influence of branch points in the complex plane on the transmission through double quantum dots

We consider single-channel transmission through a double quantum dot system consisting of two single dots that are connected by a wire and coupled each to one lead. The system is described in the framework of the $S$ matrix theory by using the effective Hamiltonian of the open quantum system. It consists of the Hamiltonian of the closed system (without attached leads) and a term that accounts for the coupling of the states via the continuum of propagating modes in the leads. This model allows one to study the physical meaning of branch points in the complex plane. They are points of coalesced eigenvalues and separate the two scenarios with avoided level crossings and without any crossings in the complex plane. They influence strongly the features of transmission through double quantum dots.

Figure 1

The evolution of $\mathrm{Re}\left(z_k\right)$ (a) and $\mathrm{Im}\left(z_k\right)$ (b), $k=1,3$ (solid lines), $k=2$ (dashed line), of the three eigenvalues of the effective Hamiltonian $H_{\text{eff}}$ as a function of $v$ at $E=E_c=0.9847$. The parameters of the double DQ system are chosen as ${\epsilon }_1=1$, $ϵ\left(L\right)=2-L∕5$, $u=1∕4$, $L=3$. At $v=v_c=0.9013$ the two eigenvalues $z_1$ and $z_3$ coalesce. The $\mathrm{Re}\left(z_1\right)$ and $\mathrm{Re}\left(z_3\right)$ approach each other when $v<v_c$, while the corresponding $\mathrm{Im}\left(z_1\right)$ and $\mathrm{Im}\left(z_3\right)$ bifurcate when $v>v_c$. At the branch point in the complex plane $E_c\ne E_k,E_l$.

Figure 2

The transmission probability through the double QD vs $v$ and energy. Each single QD has one level at ${\epsilon }_1=1$. It is $ϵ\left(L\right)=2-L∕5$ and $L=3$. The eigenenergies of the double QD are shown by stars. The case (a) corresponds to Fig. 1. The coupling constant between the single dots and the wire is $u=1∕4$. The point of coalesced eigenvalues is $v_c=0.9013$, $E_c=0.9847$ and the solutions of the fixed point equations (14) give $E_k=E_l\ne E_c$ as can be seen from Fig. 1. In the case (b), the coupling constant $u=u_c=0.1443$ between the single dots and the wire is chosen in correspondence to Eq. (21). Therefore, $E_k=E_l$ coincides with $E_c=7∕5$ at $v=v_c$.

Figure 3

The evolution of $\mathrm{Re}\left(z_k\right)$ (a) and $\mathrm{Im}\left(z_k\right)$ (b), $k=1,3$ (solid lines), $k=2$ (dashed line), of the three eigenvalues of the effective Hamiltonian $H_{\text{eff}}$ as a function of $v$ at $E=E_c=0$. The parameters $u=1∕4$, $L=10$, ${\epsilon }_1=0$, $ϵ\left(L\right)=2-L∕5$ of the double QD system are chosen in such a manner that $ϵ\left(L\right)={\epsilon }_1=0$ at $E=0$. Here, the two eigenvalues coalesce; $v_c=8^{1∕4}{u}^{1∕2}=0.8409$.

Figure 4

(a) The transmission probability through the double QD vs $v$ and energy for the case shown in Fig. 3. (b) The same as (a) but for fixed $v=0.2$ (dashed line), $v=0.53$ (solid line), and $v=0.83$ (dot-dashed line). At $v=0.53$, the double QD is a perfect filter.

Figure 5

The evolution of the real and imaginary parts of the eigenvalues $z_k$, $k=1,3$ (solid lines), $k=2$ (dashed line), as a function of the length $L$ for the same double QD system as in Fig. 1 but $v=1$. $E_c=\pm \sqrt{2}$. The critical values of the length $L$ at the two points of coalescence of eigenvalues are $L_{1c}=1.4645,L_{2c}=8.5355$. (a,b) $E=-\sqrt{2}-0.1$, (c,d) $E=-\sqrt{2}$, (e,f) $E=-\sqrt{2}+0.1$, and (g,h) $E=\sqrt{2}$.

Figure 6

(a) The probability $T$ for transmission through the double QD vs $E$ and $v$ for $L_c=8.5355$. (b) The transmission probability as a function of $E$ for fixed $v=0.85$. It has one narrow peak on the background caused by the two broad resonance states. Parameters: ${\epsilon }_1=1,ϵ\left(L\right)=2-L∕5,u=1∕4$ as in Fig. 1.

Figure 7

The evolution of the real (left column) and imaginary (right column) parts of the eigenvalues $z_k$, $k=1,3$ (solid lines), $k=2$ (dashed line), as a function of energy for transmission through the same double QD system as in Fig. 1, but $v=1$ as in Fig. 5. The point of coalesced eigenvalues is $E_c=\sqrt{2}$. The critical length is $L_c=1.4645$. (a,b) $L=L_c-0.1$, (c,d) $L=L_c$, and (e,f) $L=L_c+0.1$.

Figure 8

Left column: the evolution of the real parts of the eigenvalues of Eq. (23) as a function of $w$ for $v=0.1$, $E=1.0$ (a), $v=0.06$, $E=0.92$ (c), and $v=0.1$, $E=1.26$ (e). The parameters of the closed double QD system are $L=4$, $u=0.15$, ${\epsilon }_1=1$, $ϵ\left(L\right)=2-L∕5$. The circles at the $x$ axes denote the energies $E$. Right column: the transmission probability through the double QD vs coupling $v$ with the left reservoir and $w$ with the right reservoir. The energies $E$ are the same as in the corresponding figures of the left column.

Figure 9

The transmission through a double QD vs $E$ and $L$ for $v=0.25$ (a), 0.5 (b), 0.75 (c), and 1.0 (d). The solid lines represent the five real eigenvalues $E_k^B$ of the Hamiltonian $H_B$ as a function of $L$. The dashed lines show the eigenenergy of the wire $ϵ=3∕2-L∕7$. The eigenenergies of the two single QDs are equal: ${\epsilon }_1=1∕2$, ${\epsilon }_2=1$, and $u=0.25$. The transmission zero at $E_0=3∕4$ is independent of $L$ and $v$.

Figure 10

The evolution of real (left) and imaginary (right) parts of the five eigenvalues of the Hamiltonian $H_{\text{eff}}$ as a function of the length $L$ for a double QD system. The coupling of the system to the continuum is $v=0.35$ (a,b), 0.8 (c,d), and 1.1 (e,f). The parameters of the system are $u=0.25$, $E=0.25$, $ϵ=3∕2-L∕7$. The energies of the two single QDs are the same: ${\epsilon }_1=1∕2$, ${\epsilon }_2=1.$ The transmission of this double QD is shown in Fig. 9

Figure 11

The transmission through a double QD vs $v$ and $E$ with the length $L=2$ (a) and 5 (b). The parameters are $u=0.25$, $ϵ\left(L\right)=2-L∕4$, ${\epsilon }_1=1∕2$, ${\epsilon }_2=1$. The transmission zero at $E_0=3∕4$ is independent of $v$ and $L$.

Figure 12

The evolution of real (left) and imaginary (right) parts of the five eigenvalues of the Hamiltonian $H_{\text{eff}}$ as a function of the coupling strength $v$ for a double QD. The length of the wire is $L=0.7$ (a,b), 2 (c,d), and 3.03 (e,f). Parameters: $u=0.25$, $E=0.75$, $ϵ\left(L\right)=2-L∕4$, ${\epsilon }_1=1∕2$, ${\epsilon }_2=1$. The transmission of this double QD is shown in Fig. 11.

Figure 13

The transmission through a double QD vs $v$ and $E$ with the parameters $L=1.5$ and $u=0.2$. Each single QD has five levels at ${\epsilon }_i=1∕4$, $1∕3$, $1∕2$, $3∕4$, and 1. The energy in the wire is $ϵ=1-L∕8$. The four transmission zeros are independent of $v$ and $L$. The eigenenergies of the closed double QD are shown by stars.

Figure 14

The evolution of the 11 eigenvalues $z_k$ of the effective Hamiltonian $H_{\text{eff}}$ as a function of $v$ at $E=0$. (a) $\mathrm{Re}\left(z_k\right)$, (b) $\text{Im}\left(z_k\right)$. $L=1.5$, $u=0.2$. Each single QD has five levels at ${\epsilon }_i=1∕4$, $1∕3$, $1∕2$, $3∕4$ and 1. The eigenenergy of the wire is $ϵ=1-L∕8$. The transmission of this double QD is shown in Fig. 13.