Ground-state quantum geometry in superconductor–quantum dot chains

Multiterminal Josephson junctions constitute engineered topological systems in arbitrary synthetic dimensions defined by the superconducting phases. Microwave spectroscopy enables the measurement of the quantum geometric tensor, a fundamental quantity describing both the quantum geometry and the topology of the emergent Andreev bound states in a unified manner. In this work we propose an experimentally feasible and scalable multiterminal setup of $N$ quantum dots connected to $N+1$ superconducting leads which allows us to deterministically study nontrivial topology in terms of the Chern number of the noninteracting ground state. An important result is that the nontrivial topology in a linear chain appears beyond a threshold value of the nonlocal proximity-induced pairing potential which represents the novel theoretical key ingredient of our proposal. Moreover, we generalize the microwave spectroscopy scheme to the multiband case and show that the elements of the quantum geometric tensor of the noninteracting ground state can be experimentally accessed from the measurable oscillator strengths at low temperature.

Figure 1

Model of the superconductor–quantum dot chain. The chain consists of $N$ quantum dots (each with on-site energy ${\varepsilon }_j, j=1,...,N$) which are connected to $N+1 s$-wave superconducting leads (each with a pairing phase ${\phi }_k, k=0,...,N$). The couplings between the quantum dots are $w_{l,l+1}$ ($l=1,...,N-1$), while the couplings between the quantum dots and the superconducting leads are $t_{j,j-1}$ and $t_{jj}$.

Figure 2

(a) ABS energies ${\varepsilon }_{\mathrm{A},-3},...,{\varepsilon }_{\mathrm{A},3}$ (from bottom to top) for $N=3$ dots. Energy bands with the same color represent PH-symmetric ABS with energies ${\varepsilon }_{\mathrm{A},\pm j}$. Parameters: ${\varepsilon }_1=-{\varepsilon }_2={\varepsilon }_3=-{\mathrm{\Gamma }}_{\mathrm{L}}/2, {\phi }_0=0, {\phi }_3=\pi , w=0.2{\mathrm{\Gamma }}_{\mathrm{L}}, {\mathrm{\Gamma }}_{\mathrm{NL}}=0.8{\mathrm{\Gamma }}_{\mathrm{L}}$. (b) Locations of the four Weyl points ${\mathit{\phi }}_{\mathrm{W}}\in {\mathbb{T}}^3$ for which ${\varepsilon }_{\mathrm{A},\pm 1}\left({\mathit{\phi }}_{\mathrm{W}}\right)=0$ for the same parameters as in (a). The color indicates the topological charge of the Weyl points. The two blue Weyl points in the ${\phi }_3=\pi$ plane are visible as zero-energy band crossings in (a). (c) Topological phase diagrams of the ground-state Chern number $C_{12}^{\left(\mathrm{GS}\right)}$ for $N=3$ dots for different values of the direct tunneling $w$ between the dots (columns) as a function of the phase ${\phi }_3$. The upper row shows the phase diagram as a function of the nonlocal coupling ${\mathrm{\Gamma }}_{\mathrm{NL}}$ for fixed ${\varepsilon }_2={\mathrm{\Gamma }}_{\mathrm{L}}/2$, while the lower row shows it as a function of the central dot level ${\varepsilon }_2$ for fixed ${\mathrm{\Gamma }}_{\mathrm{NL}}=0.8{\mathrm{\Gamma }}_{\mathrm{L}}$. All other parameters are the same as in (a). The dashed line with blue and red dots in the phase diagrams for $w=0.2{\mathrm{\Gamma }}_{\mathrm{L}}$ mark the locations of the planes with fixed ${\phi }_3$ in which the Weyl points of the same color appear in (b).

Figure 3

Different contributions of the elements $q_{\alpha \beta }^{\left(1m\right)}$ from other ABS with energies ${\varepsilon }_{\mathrm{A},m}(m\ne 0,1)$ to the QGT $q_{\alpha \beta }^{\left(1\right)}$ of the ABS with energy ${\varepsilon }_{\mathrm{A},1}$ according to Eq. (8).

Figure 4

Ground-state metric tensor $G_{12}^{\left(\mathrm{GS}\right)}$ and Berry curvature $F_{12}^{\left(\mathrm{GS}\right)}$ for $N=3$ dots at ${\phi }_3=0.4\pi$ in the topological phase with $C_{12}^{\left(\mathrm{GS}\right)}=-1$ [cf. Fig. 2]. The elements of the QGT are strongly peaked for values of $\mathit{\phi }$ close to a Weyl point. The coordinates of the Weyl point indicated in this plot are ${\mathit{\phi }}_{\mathrm{W}}=({\phi }_{\mathrm{W},1},{\phi }_{\mathrm{W},2},{\phi }_{\mathrm{W},3})\approx (1.603,0.781,0.384)\pi$ [cf. lower red dot in Fig. 2]. The visible range of values of the elements of the QGT has been limited from $-5$ to $+5$ to make the overall structure for small values visible. In fact, the values of the elements of the QGT at the point $({\phi }_{\mathrm{W},1},{\phi }_{\mathrm{W},2})$ are exceeding $\pm 800$. Parameters: ${\mathrm{\Gamma }}_{\mathrm{NL}}=0.8{\mathrm{\Gamma }}_{\mathrm{L}}, w=0.2{\mathrm{\Gamma }}_{\mathrm{L}}, {\varepsilon }_1=-{\varepsilon }_2={\varepsilon }_3=-{\mathrm{\Gamma }}_{\mathrm{L}}/2, {\phi }_0=0$.