Superconducting islands with topological Josephson junctions based on semiconductor nanowires

We theoretically study superconducting islands based on semiconductor-nanowire Josephson junctions and take into account the presence of subgap quasiparticle excitations in the spectrum of the junction. Our method extends the standard model Hamiltonian for a superconducting charge qubit and replaces the Josephson potential by the Bogoliubov–de Gennes Hamiltonian of the nanowire junction, projected onto the relevant low-energy subgap subspace. This allows us to fully incorporate the coherent dynamics of subgap levels in the junction. The combined effect of spin-orbit coupling and Zeeman energy in the nanowires forming the junction triggers a topological transition, where the subgap levels evolve from finite-energy Andreev bound states into near-zero-energy Majorana bound states. The interplay between the microscopic energy scales governing the nanowire junction (the Josephson energy, the Majorana coupling, and the Majorana energy splitting), with the charging energy of the superconducting island, gives rise to a great variety of physical regimes. Based on this interplay of different energy scales, we fully characterize the microwave response of the junction, from the Cooper pair box to the transmon regimes, and show how the presence of Majoranas can be detected through distinct spectroscopic features. In split-junction geometries, the plasma mode couples to the phase-dispersing subgap levels resulting from Majorana hybridization via a Jaynes–Cummings-like interaction. As a consequence of this interaction, higher order plasma excitations in the junction inherit Majorana properties, including the $4\pi$ effect.

Figure 1

Sketch and spectrum of a NW-based superconducting island. (a) Sketch of the NW-based superconducting island simplified circuit. All the microscopic details of the NW junction (bowtie shape) are encoded in the Josephson potential $V_J\left(\phi \right)$, while the combination of a shunting capacitor $C_J$ and the gate capacitance $C_g$ define the charging energy $E_C=e^2/2(C_J+C_g)$. The dashed region is a blowup showing the schematics of the NW junction with all the relevant energies involved in the problem (the Josephson coupling $E_J$, the Majorana coupling $E_M$, and the Majorana splitting $E_0$). All these energy scales are calculated microscopically from the BdG Hamiltonian of the junction, modeled as two Lutchyn-Oreg segments coupled through a weak link. Orange circles with ${\gamma }_i$ represent MBSs. (b) Spectrum of the island in the charging regime ($E_J\ll E_C$) and for $B\sim B_c$ (where $B_c$ is the field after which the two segments of the NW become topological), showing all the competing energy scales in the problem. Blue/orange dashed curves denote even/odd parity Coulomb parabolas in the absence of tunneling coupling across the junction. A finite coupling generates both standard Josephson coupling (avoided crossings $\sim E_J$ between same-color parabolas with minima differing by two electron charges $2e$ in gate space) and Majorana coupling (avoided crossings $\sim E_M$ between different-color parabolas with minima differing by one electron charge $e$ in gate space). The energy difference between the odd and even parity sectors $\delta$ (which is in turn given by the Majorana splitting on each segment, $\delta =2E_0$) changes with $B$ field. When it becomes smaller than $E_C$ (as in the case shown here), the ground state of the island around $n_g=0.5$ becomes odd.

Figure 2

BdG spectrum as a function of Zeeman field $B$ and Majorana oscillations. BdG spectrum of NWs of increasing lengths $L_S=2.2\mu m$ (a), $L_S=3\mu m$ (b) and $L_S=5\mu m$ (c), as a function of the ratio $B/B_c$. For $B>B_c$ the lowest mode (orange line), corresponding to weakly overlapping MBSs, shows clear oscillations of amplitude $E_0$ around zero energy in (a) and (b). These oscillations become progressively reduced as $L_S$ increases (c). For $B\gg B_c$, the Majorana mode around zero energy is separated from the quasi-continuum formed by the rest of BdG excitations (grey lines) by a so-called topological gap ${\mathrm{\Delta }}_T$, which varies with the $B$ field and depends on microscopic parameters of the wire ($\mu$ and ${\alpha }_{\mathrm{SO}}$). Near $B_c$ the relevant gap is the one that closes and reopens at the topological transition (the zero momentum gap ${\mathrm{\Delta }}_0$).

Figure 3

Low-energy Andreev spectrum as a function of $\phi$. Different phase-dependent low-energy spectra of a junction where each of the two segments is a NW like the one shown in Fig. 1. Each panel shows decreasing $\tau =1,0.8,0.4,0.2$, from (a)–(d). The Zeeman field is fixed at $B/B_c\approx 1.84$, corresponding to one of the minima of the Majorana oscillations. The two lowest modes (orange lines) are originated from the overlap of four MBSs, as discussed in the main text. For $\tau =1$, they touch the quasicontinuum formed by the rest of levels (grey lines) at the so-called topological gap ${\mathrm{\Delta }}^T$, which provides an upper bound for $E_M$ (see the main text). For $\tau \ne 1$, the Majorana modes detach from this quasicontinuum of levels.

Figure 4

Low-energy Andreev spectrum as a function of $\phi$. Same as Fig. 3 but with the Zeeman field fixed at $B/B_c=2$, corresponding to one of the maxima of the Majorana oscillations.

Figure 5

Dependence of the superconducting energy scales on $B$. Apart from $E_C$, all the energy scales entering the superconducting-island Hamiltonian in Eq. (16) are sensitive to various NW properties (including its topological transition/gap closing and reopening, the emergence of MBSs and their characteristic oscillatory pattern with $B$ due to finite overlap). The overall qualitative dependence of $E_J$ and $E_M$ on $B$ varies significantly with junction transparency factor $\tau$ but does not change significantly with $L_S$ and hence it is not shown (the opposite holds for $\delta$ which is $\tau$ independent). Parameters (if not specified): $L_S=2.2\mu \mathrm{m}$, $\tau =0.8$.

Figure 6

Spectrum of a SIS junction with no magnetic field. For $B=0$ and $\tau \to 0$, we recover the standard SIS tunnel junction. These four cases correspond to increasing $E_J/E_C$ values in the transmon regime, $E_J/E_C=1,5,10,50$. ${\omega }_{pl}=\sqrt{8E_J{E}_C}/\hslash$ defines the plasma frequency. Parameters for each NW segment like in Fig. 2.

Figure 7

Spectrum of a NW-based superconducting island with magnetic field (basic phenomenology). The parity content of energy levels is calculated by projecting each eigenstate onto the parity axis defined by ${\widehat{\tau }}_z\equiv |00\rangle \langle 00|-|11\rangle \langle 11|$ (see main text). The even sector is represented by blue parabolas with minima at $n_g=m\in \mathbb{Z}$, while the odd sector is represented by orange parabolas with minima at $n_g=m+n_g^0$, $n_g^0=1/2$. (a)–(d) Coulomb island: Charging regime with $E_C=0.5\mathrm{\Delta }$ and $E_J/E_C={10}^{-4}$. Transparency factor $\tau =0.01$. For zero magnetic field, odd parabolas are shifted in the energy axis by exactly $\delta =2\mathrm{\Delta }$, panel (a). The spectrum is $2e$ periodic. This energy shift $\delta$ decreases for increasing $B$ fields, see panel (b) for $B=0.7B_c$, and vanishes exactly at the topological transition at $B=B_c$, panel (c), where the spectrum becomes $e$ periodic. Panel (d) corresponds to $B=1.5B_c$. (e)–(h) Finite Josephson and Majorana coupling: Same as top panels but with $\tau =0.8$. The finite Josephson coupling (here $E_J=0.8E_C$) results in avoided crossings between parabolas of the same parity. (h) For $B>B_c$ there is also a finite Majorana coupling that induces avoided crossings around $n_g=0.25$ and 0.75 between parabolas of opposite parity. Note that the Majorana-induced avoided crossing ($\sim E_M$) is non-negligible with respect to the maximum at $n_g=0.5$ ($\sim E_C$). Parameters for each NW segment like in Fig. 2.

Figure 8

Deviation of SNS junctions from standard Josephson behavior. Josephson inductance $L_J$ of SNS junctions (solid lines) provides an useful tool to measure deviation from idealized conditions ($L_J^{-1}\sim cos\phi$, dashed lines). Panels describe phase dependence of $L_J^{-1}$ for increasing magnetic fields. The insets on each panel show the corresponding $I_J\left(\phi \right)$. The spikes in panel b) correspond to a Zeeman-induced $0-\pi$ transition in the junction. Parameters: $L_S=2.2\mu \mathrm{m}$, $\tau =0.8$, $B/B_c=0,0.8,1,1.2$ from (a)–(d).

Figure 9

Anharmonicity $\alpha$ of a NW-based superconducting qubit at $n_g=0.5$. Solid (dashed) curves show transmon dependence on $B$ for short (long) wires. Both CPB (a) and transmon limits (b) are displayed. Anharmonicity provides a precise smoking gun to detect topological transitions and Majorana oscillations, especially for the transmon limit. Parameters: $L_S=2.2\mu \mathrm{m}$ (a), $5\mu \mathrm{m}$ (b), $\tau =0.8$, $E_J/E_C=0.5,25$.

Figure 10

Majorana versus standard Josephson coupling. Both panels show the evolution of the ratio $E_M/E_J$ as the transparency factor $\tau \in [0,1]$ is increased for different $B$ fields in the topological regime. For clarity, both linear (a) and logarithmic (b) plot scales are provided. For small transparencies below $\tau \approx 0.2$, the Majorana coupling $E_M$ becomes larger than $E_J$. Parameters for each NW segment like in Fig. 2.

Figure 11

Majorana versus standard Josephson coupling. $E_M/E_J$ against $\sqrt{E_{\mathrm{SO}}/B}$. Panel (a), calculated with the NW parameters of Fig. 2 ($\mu =2\mathrm{\Delta }=0.5\text{meV}$), illustrates how by decreasing $\tau$ one gets progressively larger values $E_M/E_J$ for all $E_{\mathrm{SO}}/B$. Panel (b) shows the behavior at fixed $\tau =0.2$ for decreasing chemical potentials. The estimation $E_M/E_J\propto \sqrt{E_{\mathrm{SO}}/B}$ of Eq. (10) is recovered for small chemical potentials in the $E_{\mathrm{SO}}/B\to 0$ limit.

Figure 12

MW spectroscopy of a NW-based superconducting island in the $E_M/E_C\ll 1$ regime (with $E_J/E_C\approx 5$). Here, starting from a single channel calculation we artificially increase $E_J$ (simulating many channels) while keeping $E_M$ and $E_C$ constant. The ratios used in these plots are $E_M/E_C\approx 0.17$ and $E_M/E_J\approx 0.036$. The superconducting island is formed by two coupled NW segments, each of them like in Fig. 2, and $\tau =0.8$. (a) Contour plot of MW absorption spectrum $S_N\left(\omega \right)$ versus $\omega$ and $B/B_c$ at $n_g=0$. Bright lines signal allowed transitions in the MW response. (b) Transition frequencies and spectral weights (shadowed widths). (c) Spectrum versus $B/B_c$ at $n_g=0$. (d), (f) Gate dependence of $S_N\left(\omega \right)$ at three magnetic fields (color bars) marked in (a). (g)–(i) Spectra corresponding to panels (d)–(f).

Figure 13

Spectral weights and spectra in the $E_M/E_C\ll 1$ regime for increasing $E_J/E_C$ ratios. The $E_J/E_C$ ratio is tuned by artificially increasing the $E_J$ amplitude, keeping $E_C$ constant. This regime imposes $E_C>E_M$, similarly to the cases considered in previous references [28, 29, 35]. (a), (b) Spectral weights of the first transitions as a function of $E_J/E_C$ and fixed $B/B_c=1.2$ at different $n_g=0$ (a) and $n_g=0.5$ (b). (c)–(f) Spectra for increasing ratios $E_J/E_C$ from the CPB to the transmon regime [$E_J/E_C=2,5,10,25$ from (c)–(f), corresponding to the colored bars at the top]. For transmon regimes $E_J/E_C\gg 1$, we recover the almost doubly degenerate transmon spectrum that is expected for $E_M\ll E_C$. Note that only the first transmon transitions ${\omega }_{02}$, ${\omega }_{03}$ are allowed. Parameters for each NW segment like in Fig. 2 and $\tau =0.8$.

Figure 14

MW spectroscopy of a NW-based superconducting island in the $E_M\approx E_C$ regime (with $E_J/E_C\approx 2$). The ratios used in these plots are $E_M/E_C\approx 0.56$ and $E_M/E_J\approx 0.28$. (a) Contour plot of MW absorption spectrum $S_N\left(\omega \right)$ versus $\omega$ and $B/B_c$ at $n_g=0$. (b) Transition frequencies and spectral weights (shadowed widths). (c), (d) Same as (a), (b) but at $n_g=0.5$. (e), (f) Same as (a), (b) but at $n_g=0.25$. (g)–(i) Gate dependence of $S_N\left(\omega \right)$ at the three magnetic fields (color bars) marked in (a), (c), and (d). (j)–(l) Spectra corresponding to panels (g)–(i). Parameters for each NW segment as in Fig. 2 and $\tau =0.8$.

Figure 15

Spectral weights and spectra for non-negligible $E_M/E_C\gtrsim 1$ ratios and increasing $E_J/E_C$ ratios. (a), (b) Spectral weights of the first transitions as a function of $E_J/E_C$ and fixed $B/B_c=1.2$ (before a parity crossing) at different $n_g=0$ (a) and $n_g=0.5$ (b). (c)–(f) Different spectra at this magnetic field, $B/B_c=1.2$, for increasing ratios $E_J/E_C$ from the CPB to the transmon regime ($E_J/E_C=2,5,10,25$ from (c)–(f), corresponding to the colored bars at the top). (g), (h) and (i)–(l): Same as before but for $B/B_c=1.3$ (after a parity crossing). In the CPB regime the ${\omega }_{01}$ transition has some weight, but deep in the transmon regime the only allowed transition is a transmon line ${\omega }_{03}$. Note how the matrix elements are fully exchanged between $n_g=0$ and $n_g=0.5$ after a parity crossing [i.e., compare panel (a) with (h) and (b) with (g)]. The exact cancellation at $E_J/E_C=5$ of the ${\omega }_{01}$ transition in (b) results from parity degeneracy at $n_g=0.5$ [panel (d)]. After a parity crossing, the full spectrum is shifted by one $e$ unit, while flipping parity, and the parity degeneracy occurs now at $n_g=0$ and $n_g=1$ [panel (j)]. Consequently, the exact cancellation at $E_J/E_C=5$ of ${\omega }_{01}$ in (g) occurs now at $n_g=0$. Parameters for each NW segment like in Fig. 2 and $\tau =0.8$.

Figure 16

MW spectroscopy of a NW-based superconducting island in the $E_J\to 0$ regime for long wires. In contrast to previous cases, here we consider a tunnel junction for the wire, $\tau \ll 1$, so that the Josephson term $E_J$ almost vanishes. This in turn makes the Majorana coupling $E_M$ much larger that $E_J$ (see Fig. 10 for $E_M/E_J$ vs $\tau$ dependence). We use, in particular, $\tau \simeq 0.01$, which translates into $E_M/E_J\sim 20$. This results in $E_C$ being the dominant energy scale of the island, as can be seen from ratios $E_M/E_C\approx 0.2$ and $E_J/E_C\approx 0.01$ (this regime is relevant for the experiments reported in Ref. [50]). Contour plots of $S_N(\omega ,B)$ and transition frequencies are alternatively shown for different gates: $n_g=0$ (a), (b); $n_g=0.5$ (c). (d); and $n_g=0.25$ (e), (f). Transition frequency lines are shadowed according to their spectral weight. Panels (g)–(l) render gate dependence of $S_N\left(\omega \right)$ (g)–(i) and spectra (j)–(l) before, at, and after a parity crossing as functions of $n_g$. Parameters for each NW segment like in Fig. 2 (with $L_S=5\mu \mathrm{m}$).

Figure 17

MW spectroscopy of a NW-based superconducting island in the $E_J\to 0$ regime, for short wires. Similar to Fig. 16, a strong charging regime is considered owing to a tunnel junction factor $\tau \ll 1$. Energy scales are similar to those of Fig. 16, $E_M/E_C\approx 0.25$, $E_J/E_C\approx 0.01$. Now we focus on a shorter wire with $L_S=2.2\mu \mathrm{m}$. Majorana splitting increases significantly over that of Fig. 16 such that $\delta$ can be greater than $E_C$ in large ranges of $B$ field. As a consequence, transitions are a little more involved regarding parity mixing, except in quite narrow $B$ windows around parity switches of the ground state. Almost always, both sets of even-odd parabolas are strongly shifted from each other, making it impossible to observe parity events at lower transitions. (a)–(f) MW response and transition frequency of the lowest transition as a function of $B$, at three gates $n_g=0,0.5,0.25$. Gate dependence of MW response $S_N\left(\omega \right)$ (g)–(i) and of the corresponding spectra (j)–(l). Parameters for each NW segment as in Fig. 2.

Figure 18

Sketch of a NW-based superconducting island in a split-junction geometry. The total phase across the split junction $\varphi ={\phi }_1+{\phi }_2$ is fixed by an external applied flux $\varphi \equiv 2\pi \frac{\mathrm{\Phi }}{{\mathrm{\Phi }}_0}$, where ${\mathrm{\Phi }}_0=h/2e$ is the flux quantum. In strongly asymmetric junctions with $E_L\gg E_C$, quantum fluctuations of phase are small and the external phase mostly drops on the NW JJ with Josephson potential $V_J$.

Figure 19

MW spectroscopy in a split-junction geometry with $E_L/E_J=60$. Parameters for each NW segment as in Fig. 2. (a) Phase dispersion of the transition frequencies and their spectral weights (widths of the lines) in the trivial regime $B=0.33B_c$. (b) Blowup of the corresponding MW spectrum showing the avoided crossing between the plasma mode and the Andreev level in the NW junction. (c), (d) Same in the topological regime, $B=1.25B_c$, showing plasma replicas of the underlying $4\pi$-periodic Josephson effect in the NW junction. (e), (f) Magnetic field dependence near $\varphi =\pi$. (g), (h) Blowup near the first minimum at $B\approx 1.25B_c$.