Berry phase in superconducting multiterminal quantum dots

We report on a study of the nontrivial Berry phase in superconducting multiterminal quantum dots biased at commensurate voltages. Starting with the time-periodic Bogoliubov–de Gennes equations, we obtain a tight-binding model in Floquet space, and we solve these equations in the semiclassical limit. We observe that the parameter space defined by the contact transparencies and quartet phase splits into two components with a nontrivial Berry phase. We use the Bohr-Sommerfeld quantization to calculate the Berry phase. We find that if the quantum dot level sits at zero energy, then the Berry phase takes the values ${\phi }_B=0$ or ${\phi }_B=\pi$. We demonstrate that this nontrivial Berry phase can be observed by tunneling spectroscopy in the Floquet spectra. Consequently, the Floquet-Wannier-Stark ladder spectra of superconducting multiterminal quantum dots are shifted by half-a-period if ${\phi }_B=\pi$. Our numerical calculations based on the Keldysh Green's functions show that this Berry phase spectral shift can be observed from the quantum dot tunneling density of states.

Figure 1

(a) A superconducting three-terminal QD biased on the quartet line at voltages $V_{a,b}=\pm V$ and $V_c=0$, with in addition a tunnel-contacted normal lead to probe the quantum dot density of states. (b) The values of $({\phi }_a\left(t\right),{\phi }_b\left(t\right))$ during one period of Josephson oscillations. Commensurate bias voltage implies that $({\phi }_a,{\phi }_b)$ encloses a cycle on a two-dimensional torus.

Figure 2

Ternary diagram for the gap closing condition: The nodal lines, displayed in magenta and calculated for ${\phi }_q/2\pi =0.2$, represent the values of the parameters for which the gap between the two Andreev bound-state bands vanishes [see Eq. (31)]; below these two lines, the Berry phase takes the value ${\phi }_B=\pi$. The smaller shaded inner triangle shows all the possible values of the nodal lines when $0<{\phi }_q/2\pi <1$.

Figure 3

Tunnel spectroscopy of the Berry phase: The figure features the logarithm of the local density of states on the quantum dot (in color scale) as a function of inverse voltage $\mathrm{\Delta }/eV(x$-axis) and tunnel probe bias voltage $e{V}_{\mathrm{tun}}/\mathrm{\Delta }(y$-axis). The Berry phase is ${\phi }_B=0$ in panel (a) and ${\phi }_B=\pi$ in panel (b). The tunnel spectra reveal the Floquet-Wannier-Stark ladders, and they are compared to the tilted white lines, which correspond to $w=0$ in Eq. (39). The half-a-period shift appearing in panel (b) is a signature of the nontrivial Berry phase ${\phi }_B=\pi$, while ${\phi }_B=0$ for the nonshifted tunnel spectrum in panel (a). The three-terminal superconducting QD has ${\mathrm{\Gamma }}_a/\mathrm{\Delta }=0.4,{\mathrm{\Gamma }}_b/\mathrm{\Delta }=0.2$, and ${\phi }_q=0$, with (a) ${\mathrm{\Gamma }}_c/\mathrm{\Delta }=1.0$ and (b) ${\mathrm{\Gamma }}_c/\mathrm{\Delta }=0.3$.

Figure 4

Classical trajectories: The figure shows the classical trajectory ${\mathcal{T}}_{E,\pm }$, together with the tunneling paths, i.e., the four complex solutions $k_{\alpha }$ of Eq. (A11), each with a given value of $E+\xi$. In these plots, we use the complex variable ${\lambda }_{\alpha }=exp\left(i{k}_{\alpha }\right)$. Panels (a), (c) and (b), (d) show, respectively, ${log}_{10}\left|{\lambda }_{\alpha }\right|$ and $arg\left({\lambda }_{\alpha }\right)/2\pi$ on the $x$-axis. The $y$-axis in each panel features $E+\xi$ normalized to the gap $\mathrm{\Delta }$. The dispersion relation $E_A\left(k\right)$ (in magenta) has two local minima and two local maxima ${\mathcal{N}}^{(a,b)}=2$ in panel (b), as $k$ varies in the interval $-\pi <k<\pi$. Panel (d) corresponds to a single local minimum and maximum ${\mathcal{N}}^{(c,d)}=1$. The color code is explained in the text.

Figure 5

Ternary diagrams for the number of minima in $E_A\left(k\right)$: the domain in parameter space in which the dispersion relation for $\mathcal{N}=1$ and 2 minima is shown in blue and red, respectively, for ${\phi }_q/2\pi =0.2$.

Figure 6

Semiclassical wave function for $\mathcal{N}=1$: The semiclassical trajectory ${\mathcal{T}}_{E,-}\left({\mathcal{T}}_{E,+}\right)$ exhibits turning points at ${\xi }_1$ and ${\xi }_2({\xi }_3$ and ${\xi }_4)$. The wave vector $k\left(\xi \right)$ jumps by $2\pi$ at ${\xi }_5$ and ${\xi }_6$. Classical trajectories ${\mathcal{T}}_{E,-}$ and ${\mathcal{T}}_{E,+}$ are depicted by full lines, and they are connected by a pair of tunneling paths depicted by dashed lines. Arrows near turning points indicate either an increasing phase function $\theta \left(\xi \right)$ along classical trajectories, or an increasing modulus along tunneling paths.

Figure 7

Semiclassical wave function for $\mathcal{N}=2$: The semiclassical trajectory ${\mathcal{T}}_{E,-}\left({\mathcal{T}}_{E,+}\right)$ exhibits now four turning points. Classical trajectories ${\mathcal{T}}_{E,-}$ and ${\mathcal{T}}_{E,+}$ are depicted by full lines, and they are connected by a pair of tunneling loops depicted by dashed lines. These tunneling loops are characterized by tunneling amplitudes ${\eta }_1,{\eta }_2$ and by tunneling phases ${\theta }_1,{\theta }_2$. Arrows near turning points indicate an increasing phase function $\theta \left(\xi \right)$ along classical trajectories.