Time scales for Majorana manipulation using Coulomb blockade in gate-controlled superconducting nanowires

We numerically compute the low-energy spectrum of a gate-controlled superconducting topological nanowire segmented into two islands, each Josephson coupled to a bulk superconductor. This device may host two pairs of Majorana bound states and could provide a platform for testing Majorana fusion rules. We analyze the crossover between (i) a charge-dominated regime utilizable for initialization and readout of Majorana bound states, (ii) a single-island regime for dominating interisland Majorana coupling, (iii) a Josephson-plasmon regime for large coupling to the bulk superconductors, and (iv) a regime of four Majorana bound states allowing for topologically protected Majorana manipulations. From the energy spectrum, we derive conservative estimates for the time scales of a fusion-rule testing protocol proposed recently (D. Aasen et al., arXiv:1511.05153). We also analyze the steps needed for basic Majorana braiding operations in branched nanowire structures.

Figure 1

Segmented nanowire setup for gate-controlled fusion of MBS. The device consists of two tunnel-coupled superconducting islands (orange), formed by a nanowire in proximity to a superconductor (not shown). In addition to that, both islands are connected to nontopological bulk superconductors (blue) with a tunable coupling. In (a), when the outer valves are maximally open (largest coupling) and the central valve is closed, the device hosts four MBS at zero energy (crosses). In (b), when all junctions are closed (minimal coupling), the pairs of MBS on each island are fused (connected circles). In this situation, the device may still possess a subgap state provided $E_{C,\alpha }<\mathrm{\Delta }$ ($\alpha =L,R$), rendering the sketched setup distinct from a conventional nontopological Cooper-pair box.

Figure 2

Sketch of the parameter regimes for the energy-spectra characteristics of the coupled topological superconducting island Hamiltonian (4). In the boxes, we sketch the energy level spectrum and indicate the different energy scales dominating the level structure. The lower part of the boxes shows the couplings (lines) between the MBS, which are fused when denoted as circles and close to zero energy when denoted as crosses. We assume symmetric islands $E_{J,L}=E_{J,R}=E_J, E_{C,L}=E_{C,R}=E_C$, and have set the central Josephson coupling to zero, $E_{J,C}=0$. The white arrows indicate the paths taken in the $(E_J,E_M)$ plane for the plots in Fig. 3.

Figure 3

Low-energy spectrum of coupled topological superconducting islands. Solid lines are the numerically computed energy splittings (blue) between succeeding excited states (energy $E_k$) and the ground state (energy $E_0$) in the even total parity sector. Dashed lines indicate analytic approximations to the energy gaps (see below). The parameters are changed along the paths in the $(E_J,E_M)$ plane as denoted by the white arrows in Fig. 2 and the colors on the horizontal axis correspond to the regimes in Fig. 2. In (a), we show the spectrum as a function of the Majorana coupling $E_M$ for $E_J=0$. We indicate in darker color those states with zero excess charge, $N=0$, while the bright color corresponds to other excess charges, $N\ne 0$. The red dashed lines indicate the analytic approximations from Eq. (35). In (b)–(d), we show the spectra as a function of the bulk Josephson coupling $E_J$ for different values of $E_M$ as indicated. Here, the darker color highlights the 12 lowest eigenstates and the bright color corresponds to higher-lying states. The dashed lines indicate the transition energies $E_k-E_0=k\sqrt{8E_J{E}_C}$ related to Josephson-plasma oscillations (red), the parity splitting $2{\varepsilon }_P$ [Eq. (37)] (yellow), and the Majorana splitting $2E_M$ (green). We assume $E_{C,L}=E_{C,R}=E_C, E_{J,L}=E_{J,R}=E_{J,C}=0, n_{g,L}=n_{g,R}=0, E_{J,C}=0$, and use a number-state cutoff $N_{max}=25$ for the numerical calculations.

Figure 4

Protocol for testing the Majorana fusion rules. The steps of the protocol are sketched in (a) and the corresponding path taken in the $(E_J,E_M)$ parameter space in (b) (compare with Fig. 2). White arrows indicate that these processes have to be adiabatic, while the orange arrow indicates a diabatic step, in which the system should not follow the ground-state evolution. The yellow arrow indicates the resetting step, which should be done on the time scale of the charge relaxation to the ground state. The color scale gives the ground-state expectation value of $〈-i{\widehat{\gamma }}_1{\widehat{\gamma }}_2〉$ [see Eq. (46)], indicating the preferred parity combination of the ground state. In the charge-dominated regime ($E_J,E_M\ll E_C$), the Majorana pairs (${\gamma }_1,{\gamma }_2$) and (${\gamma }_3,{\gamma }_4$) are fused $[〈-i{\widehat{\gamma }}_1{\widehat{\gamma }}_2〉\to 1]$, while in the tunneling-dominated regime ($E_M\gg {\varepsilon }_P$), the Majoranas (${\gamma }_2,{\gamma }_3$) are fused $[〈-i{\widehat{\gamma }}_1{\widehat{\gamma }}_2〉\to 0]$. To bring all the parity states of the two islands as close to zero energy as possible, the path should cover the Majorana regime ($E_M,E_J\gg E_C$ and $E_M,E_J$ should be, in principle, as large as possible). We use $E_{J,C}=5E_M^2/\mathrm{\Delta }, E_{J,L}=E_{J,R}=E_J, \mathrm{\Delta }=100E_C, n_{g,R}=-n_{g,L}=0.3$, and $N_{max}=21$.

Figure 5

Energy gap between the ground state and first excited state for even total parity (in some intervals the gap to other excited states is shown as well). The parameters $E_M$ (and accordingly $E_{J,C}=5E_M^2/\mathrm{\Delta }$) as well as $E_J$ on the horizontal axis are changed logarithmically along the path $\text{A}\to \text{B}\to \text{C}\to \text{D}\to \text{A}$ marked in Fig. 4. The solid lines correspond to the numerically computed splittings while the dashed lines are the approximation formulas for the splittings (see text). Gray arrows indicate allowed transitions and we mark the lowest accessible excited state from the ground state in blue. In several steps, transitions into the lowest excited states (red, green) are prohibited by parity or charge conservation as explained next: In step 2, the lowest excited state $|1_{23},1_{14}\rangle$ is decoupled from the ground state $|0_{23},0_{14}\rangle$ for a large energy range because the tunneling Hamiltonian $H_T$ conserves the nonlocal parities $p_{23}, p_{14}$. However, when ${\varepsilon }_P$ becomes dominant, the parity character of the ground state changes to $|0_{12}{0}_{34}\rangle$. Thus, when $E_M$ approaches ${\varepsilon }_P$, transitions to the lowest excited state become possible. By tuning fast the system stays in the desired state $|0_{23},0_{14}\rangle =\left(\right|0_{12}{0}_{34}\rangle +|1_{12}{1}_{34}\rangle )/\sqrt{2}$, i.e., the system can be both in the ground and first excited states after step 2. Since the islands are decoupled then, the local parities $p_{12},p_{34}$ are conserved in step 3 and transitions from even-even to odd-odd parity or vice versa are prohibited. From the ground state, one can therefore only reach the third-lowest excited state [blue, parity $(0_{12},0_{34})$]. In addition, transitions from the first to the second excited state are possible [red, parity $(1_{12},1_{34})$], which determines the adiabaticity condition. Finally, in step 4, the islands host a different total charge $N=\pm 2$ in the lowest excited state, while transitions are only possible into the lowest $N=0$ state ($N=n_L+n_R$ is the total number of electrons). The parameters are $n_{g,R}=-n_{g,L}=0.3, \mathrm{\Delta }=100E_C, E_{C,0}=2E_C(1+n_{g,L}-n_{g,R})$ (see text), and the number-state cutoff is $N_{max}=25$. Note that we chose a rather large value for $\mathrm{\Delta }$ to be consistent with the assumption $E_C\ll E_M^{max}\ll \mathrm{\Delta }$ for our numerical approach. We expect that the protocol should also work for smaller values of $\mathrm{\Delta }$ as we further explain in Appendix pp2.

Figure 6

Basic operations for gate-controlled manipulations of MBS in nanowire networks. Panels (a) and (b) show the positions of the four and six MBS in the segmented and trijunction nanowire geometry, respectively. In (c), we show the steps of transferring a MBS from one end of the left wire segment to the opposite end of the right segment. During the entire transfer process, a MBS ${\gamma }_0$ remains at zero energy [see Eq. (61) and the explanation below], which is a linear combination with changing weights of ${\gamma }_2$ and ${\gamma }_4$ as indicated in the figure. We put the last step in parentheses since opening the valve to the right superconductor is not necessary, provided the coupling of the central junction can be made large enough (as specified in Sec. 6a). In (d), we show the steps to transfer a MBS from the right side of the trijunction to the left side. (e) Elementary braid operation, composed of operations (c) and (d). Analogous to (c), the vertical island needs not be connected to a bulk superconductor for large couplings across the trijunction. We emphasize that all operations in (c) [(d) and (e)] must be performed such that the ground-state degeneracy is not changed during the operation, meaning that during the braid operation the system should never be in the configurations shown in (a) [(b)] with 4 (6) uncoupled MBS.

Figure 7

Transfer of MBS between two nanowire segments. Numerically computed energy gap (solid blue) between the ground state and the first excited state. The two curves are for two different values of $E_{J,C}$. We change $E_M$ in the left part and $E_{J,R}$ in the right part as illustrated in the sketches below the plots. We also show analytic approximations for the energy splitting (dashed lines) as denoted and further discussed in the text. Due to the gating of the right island, the relevant charging energy is given by $E_{C,0}=E_C(1-2n_{g,R})$. The other parameters are given by $E_{C,L}=E_{C,R}=E_C, n_{g,L}=0, n_{g,R}=0.25$, and $N_{max}=20$.

Figure 8

Gate-controlled transfer of a MBS between two nanowire segments. We show the numerically computed coefficient $a$ in Eq. (61) as a function of the Majorana coupling $E_M$. The calculation of the coefficient $a$ is explained in Appendix pp7. The parameters used here are $E_{J,L}=100E_C, E_{J,R}=E_{J,C}=0, n_{g,L}=0, n_{g,R}=0.25$, and $N_{max}=10$.

Figure 9

Comparison of the convergence of the numerical approach in number space and phase space. (a), (b) Energy difference of the four lowest eigenenergies $\mathrm{\Delta }{E}_k\left(N_{max}\right)=E_k(N=N_{max})-E_k(N=N_{max}-2)$ as a function of $N$ included basis states. The states are labeled in ascending order ($k=0,1,2,3$). We show the result both in number space (a) and in discretized phase space (b). The eigenenergies are obtained from a single-island Hamiltonian, i.e., $H=E_C{\left(\widehat{n}-n_g\right)}^2+E_J\left(1-cos\left(\widehat{\phi }\right)\right)$ [see Eq. (5)]. In (c) and (d), we show the corresponding probability densities in number and discretized phase space for the case $N_{max}=25$. Note that we only include even electron numbers [i.e., the points in (c) differ by $\mathrm{\Delta }n=2$]. To compute the eigenenergies and eigenfunctions in phase space, we express the corresponding Schrödinger equation in phase space with the replacement ${\widehat{n}}^2\to -{\partial }^2/\partial {\phi }^2$ and discretize the phase variable $\psi \left(\phi \right)\to \psi \left({\phi }_j\right), j=1,...,N, {\phi }_j=j2\pi /N_{max}$ and the second derivative ${\partial }^2\psi \left(\phi \right)/\partial {\phi }^2=\psi [\left({\phi }_{j+1}\right)+\psi \left({\phi }_{j-1}\right)-2\psi \left({\phi }_j\right)]/{(2\pi /N_{max})}^2$. For the even parity sector, we impose periodic boundary conditions, i.e., $\psi \left({\phi }_{N_{max}+1}\right)=\psi \left({\phi }_1\right)$. The results are shown for $E_J/E_C=100$ and $n_g=0$.

Figure 10

Effect of the conventional Josephson coupling of the central junction $\left(E_{J,C}\right)$ on the low-energy spectrum. Solid lines show the numerically computed energy splitting between the first four excited states with zero excess charge $N=0$ (at energies $E_k$) and the ground state (at energy $E_0$) as a function of the Majorana-Josephson coupling $E_M$. The center Josephson coupling is given by $E_{J,C}=5E_M^2/\mathrm{\Delta }$, which we exclude $(E_{J,C}=0)$ and include $(E_{J,C}\ne 0)$ from the computation as indicated. The red dashed lines indicate the approximations for the splittings from Eq. (D2) for $E_{J,C}=0$ and from Eq. (D5) for $E_{J,C}\ne 0$. All the states are continuously connected to charge states $(n_L,n_R)$ for $E_M=0$ as indicated at the curves on the left. The other parameters are $E_{C,L}=E_{C,R}=E_C, E_{J,L}=E_{J,R}=0, \mathrm{\Delta }=100E_C$, and $n_{g,L}=n_{g,R}=0$.

Figure 11

Low-energy spectrum of two hybridized topological superconducting islands asymmetrically coupled to bulk superconductors. The figure shows the numerically computed energy splittings between succeeding excited states (at energy $E_k$) and the ground state (at energy $E_0$) as a function of the average Josephson coupling ${\overline{E}}_J=(E_{J,L}+E_{J,R})/2$ to the bulk superconductors. We highlight the 12 lowest states in darker color for comparison with Fig. 3 and show all higher-lying states in brighter color. The Majorana coupling $E_M$ is changed from one row to the other while the asymmetry $E_{J,R}/E_{J,L}$ of the Josephson couplings is changed from one column to the other as indicated. We use $n_{g,L}=n_{g,R}=0, E_{J,C}=0$, and $N_{max}=20$.

Figure 12

Parity-dependent charge pumping. (a) Sketch of the steps of the pumping process. (b) Sketch of the energy spectrum for the left island as a function of the gating $n_{g,L}$. Sweeping the gating from $n_{g,L}^{\left(1\right)}\to n_{g,L}^{\left(2\right)}$ changes the charge on the island by 0 if the parity is even (blue) and by $2e$ if the parity is odd (red). This works oppositely for the right island with $n_{g,R}^{\left(1\right)}=n_{g,L}^{\left(2\right)}$ and $n_{g,R}^{\left(2\right)}=n_{g,L}^{\left(1\right)}$. Note that the labeling of the charge states is not valid near the charge-degeneracy points, where states that differ by one Cooper pair mix with each other.

Figure 13

Numerically computed energy gap ${\varepsilon }_{P,R}$ [Eq. (G6)] (solid blue lines) between odd and even parity states on the right island as a function of the central Majorana coupling $E_M$. We also indicate the parity energy splitting ${\varepsilon }_{P,R}(E_M\to 0)=E_{C,R}/2$, corresponding to the two lowest charge states in the limit of vanishing Majorana coupling (green dashed). For the central Josephson coupling, we use $E_{J,C}=5E_M^2/\mathrm{\Delta }$ if nonzero as indicated and the other parameters are given by $E_{C,R}=E_{C,L}=E_C, n_{g,L}=n_{g,R}=0,E_{J,L}=100E_C,E_{J,R}=0, \mathrm{\Delta }=100E_C$.