Fluctuation response of a minimal Kitaev chain in nonequilibrium states

Abstract

Minimal Kitaev chains provide a unique platform to engineer Majorana states in quantum dots interacting via normal tunneling and crossed Andreev reflection specified by their amplitudes $|η_{n,a}|$. Here we analyze fluctuations of electric currents in a double quantum dot Kitaev chain using the differential effective charge $q$, that is the ratio of the differential shot noise and conductance. At low bias voltages $V$ we find that $q=e/2$ in a very narrow vicinity of the point $|η_n|=|η_a|$ whereas $q=3e/2$ almost in the whole sweet spot region and marks the range where the poor man's Majorana states largely govern the fluctuations. At high $V$ we show that the sweet spot region is still characterized by $q=3e/2$ uniquely identifying the poor man's Majorana states using the high voltage tails. For $|η_n|=0$ or $|η_a|=0$ we obtain $q=e$ at any $V$. Remarkably, before the asymptotic value $q=e$ is reached for very high $V$, the maximal value $q=2e$ is formed at $|eV|=2\sqrt{|η_n|^2+|η_a|^2}$. The unique nature and potentially rich fluctuation behavior revealed in this work provide a stimulating ground for the next generation experiments on nonequilibrium shot noise in minimal Kitaev chains.

Figures (7)

• Figure 1: A schematic representation of the system composed of two QDs, QD1 and QD2, grounded superconductor inbetween and two normal metallic contacts. Each QD is characterized by one non-degenerate single-particle energy level, $\varepsilon_1$ and $\varepsilon_2$, localized, respectively, in QD1 and QD2. The superconductor governs both the normal tunneling and crossed Andreev reflection between the QDs. The corresponding processes occur with the amplitudes $|\eta_n|$ and $|\eta_a|$. Both contacts are coupled to QD1 via normal tunneling of strength $\Gamma_{L,R}$. The difference in their chemical potentials is the direct source for nonequilibrium states in this system and the degree of nonequilibrium is controlled by the bias voltage $V$. The Hamiltonian of this composite nanostructure is specified in Eqs. (\ref{['Ham']})-(\ref{['Symm_coupling']}).
• Figure 2: The differential effective charge, $q\equiv\partial_VS/\partial_VI$, is shown in universal units of the elementary charge $e$ as a function of the ratio between the amplitudes $|\eta_n|$ and $|\eta_a|$ for low bias voltages, $|eV|\ll\Gamma$. Specifically, $|eV|/\Gamma=10^{-2}$. For the black and red curves $|\eta_n|=|\eta|\cos\alpha$, $|\eta_a|=|\eta|\sin\alpha$ with $|\eta|/\Gamma=10^2$ (black curve) and $|\eta|/\Gamma=10^3$ (red curve). For the green curve $|\eta_n|=|\eta_1|\cos\alpha$, $|\eta_a|=|\eta_2|\sin\alpha$ with $|\eta_1|/\Gamma=10^3$, $|\eta_2|=|\eta_1|\cot(3\pi/16)$. The values of the other parameters: $k_\text{B}T/\Gamma=10^{-7}$, $\varepsilon_1/\Gamma=10$, $\varepsilon_2/\Gamma=10^{-9}$. The inset illustrates in more detail a very narrow vicinity of the point $\alpha=\pi/4$.
• Figure 3: The differential effective charge, $q\equiv\partial_VS/\partial_VI$, is shown in universal units of the elementary charge $e$ as a function of the ratio between the amplitudes $|\eta_n|$ and $|\eta_a|$ for high bias voltages, $|eV|\gg\Gamma$. Specifically, $|eV|/\Gamma=10$ for the black curve and $|eV|/\Gamma=10^2$ for the red and green curves. For the black and red curves $|\eta_n|=|\eta|\cos\alpha$, $|\eta_a|=|\eta|\sin\alpha$ with $|\eta|/\Gamma=10^2$ (black curve) and $|\eta|/\Gamma=10^3$ (red curve). For the green curve $|\eta_n|=|\eta_1|\cos\alpha$, $|\eta_a|=|\eta_2|\sin\alpha$ with $|\eta_1|/\Gamma=10^3$, $|\eta_2|=|\eta_1|\cot(3\pi/16)$. The values of the other parameters are the same as in Fig. \ref{['figure_2']}. The inset zooms in relevant details in a vicinity of the point $\alpha=\pi/4$.
• Figure 4: Panel ( a). $\partial_VS$ and $\partial_VI$ as functions of the bias voltage, $\Gamma\ll|eV|<\sqrt{|\eta_n|^2+|\eta_a|^2}$. Here $|\eta_n|=|\eta|\cos\alpha$, $|\eta_a|=|\eta|\sin\alpha$ with $|\eta|/\Gamma=10^3$ and $\alpha=\pi/4$ (or $|\eta_n|=|\eta_a|$). These high voltage tails do not depend on the temperature: the black ($\partial_VS$) and red ($\partial_VI$) curves remain unchanged when the temperature is drastically increased from $k_\text{B}T/\Gamma=10^{-7}$ to $k_\text{B}T/\Gamma=2\times 10^{-1}$. The values of the other parameters are the same as in Fig. \ref{['figure_2']}. Inset: $q\equiv\partial_VS/\partial_VI$ obtained using $\partial_VS$ and $\partial_VI$ from the main plot. Panel ( b). $\partial_VI$ around $V=0$. As can be seen, at high $T$ the zero bias resonance in $\partial_VI$ is greatly suppressed below its universal maximum.
• Figure 5: The differential effective charge, $q\equiv\partial_VS/\partial_VI$, is shown in universal units of the elementary charge $e$ as a function of the ratio between the amplitudes $|\eta_n|$ and $|\eta_a|$ for $|eV|=2\sqrt{|\eta_n|^2+|\eta_a|^2}$. Here $|\eta_n|=|\eta|\cos\alpha$, $|\eta_a|=|\eta|\sin\alpha$ with $|\eta|/\Gamma=10^2$ (black curve) and $|\eta|/\Gamma=10^3$ (red curve). With this parameterization the bias voltage does not vary along the curves, $|eV|=2|\eta|$. The values of the other parameters are the same as in Fig. \ref{['figure_2']}.