Lifetime of Majorana qubits in Rashba nanowires with nonuniform chemical potential

We study the lifetime of topological qubits based on Majorana bound states hosted in a one-dimensional Rashba nanowire (NW) with proximity-induced superconductivity and nonuniform chemical potential needed for manipulation and readout. If nearby gates tune the chemical potential locally so that part of the NW is in the trivial phase, Andreev bound states (ABSs) can emerge which are localized at the interface between topological and trivial phases with energies significantly less than the gap. The emergence of such subgap states decreases the Majorana qubit lifetime at finite temperatures due to local perturbations that can excite the system into these ABSs. Using the Keldysh formalism, we study such excitations caused by fluctuating charges in capacitively coupled gates and calculate the corresponding Majorana lifetimes due to thermal noise, which are shown to be reduced in comparison with those in NWs with uniform chemical potential.

Figure 1

(a) Sketch of proximitized Rashba NW consisting of two sections. (b) The nonuniform chemical potential $\mu \left(x\right)$ is controlled by a nearby gate, so that the right section (gray) of length $L_{+}$ is in the trivial ($|{\mu }_{+}|>{\mu }^{*}$) or topological ($|{\mu }_{+}|<{\mu }^{*}$) phase, with ${\mu }^{*}$ being the critical value (see text), while the left section (blue) of length $L_{-}$ with $\mu =0$ stays always topological. At the interface, $\left|x\right|<l_0$, $\mu \left(x\right)$ grows linearly in $x$. The proximity gap ${\mathrm{\Delta }}_{sc}$ and Zeeman energy ${\mathrm{\Delta }}_Z$ are uniform.

Figure 2

Excitations spectrum of the NW Hamiltonian (1) as function of ${\mu }_{+}$. The topological phase transition in the right NW section occurs at ${\mu }_{+}={\mu }^{*}$. In addition to the bulk modes above the gap ${\mathrm{\Delta }}_{-}$ (blue crosses), there are bulk modes in the right section (green squares) with energies above the gap ${\mathrm{\Delta }}_{+}$ (green dashed line). There also emerge RABSs (black empty squares) and IABSs (red circles). The IABSs occur if $l_0\gtrsim min\{{\xi }_{+},{\xi }_{-}\}$ and have energies far below the gap ${\mathrm{\Delta }}_{-}$ as ${\mu }_{+}$ approaches ${\mu }^{*}$. The parameters chosen are $(E_{so},{\mathrm{\Delta }}_{sc},{\mathrm{\Delta }}_Z)=(0.05,0.5,1)\mathrm{meV}$, corresponding to $(t,{\mathrm{\Delta }}_{-},{\mu }^{*})=(10,0.21,0.87)\mathrm{meV}$ and ${\xi }_{-}\approx 30a$. For these parameters, $l_0=60a\approx 2{\xi }_{-}$ and $L_{-}=L_{+}=300a\approx 10{\xi }_{-}$. The inset shows the IABS energy as function of $l_0$ for different ${\mu }_{+}$: from bottom to top ${\mu }_{+}/{\mu }^{*}=(1.3,2,3,4)$. The IABS energy decreases with increasing $l_0$ as $\varepsilon \propto l_0^{-1/2}$ (see Appendix pp1).

Figure 3

(a) Feynman diagram corresponding to the correlators ${\mathcal{D}}^{R,A,K}$ (thick wavy line) for fluctuations of the gate potential. The electrons in the NW interact with electrons in the gates with Green functions ${\mathcal{G}}_g^{R,K,A}$ (blue lines) via the Coulomb potential $U(\mathbf{R},x)$ (thin wavy lines). (b) Diagram for the self-energy $\mathrm{\Sigma }$ corresponding to the process where gate fluctuations [thin wavy line; see (a)] promote the system from its GS to an excited state $m$ (straight red line).

Figure 4

Lifetime $\tau$ as function of temperature $T$ and chemical potential ${\mu }_{+}$ for (a) long ($L_{+}=300a\approx 10{\xi }_{-}$) and (b) short ($L_{+}=60a\approx 2{\xi }_{-}$) right NW section. The blue lines hold for $\tau ({\mu }_{+},T_{\mu s})=1\mu \mathrm{s}$, and ${\mu }_{+}=0$ corresponds to uniform $\mu \left(x\right)$. As ${\mu }_{+}\to {\mu }^{*}$, $\tau$ decreases, being much shorter for case (a). If the right section is tuned to the trivial phase, ${\mu }_{+}\gg {\mu }^{*}$, the temperature required for maintaining the same $\tau$ becomes approximately twice smaller than for uniform $\mu \left(x\right)={\mu }_{+}=0$. For ${\mu }_{+}<{\mu }^{*}$, the required $T$ is similar for (a) and (b); $\tau$ and $T$ oscillate for ${\mu }_{+}>{\mu }^{*}$ for (b) but not for (a) (see Appendix pp2). The parameters chosen are $L_{-}=300a=10{\xi }_{-}$, ${\mathrm{\Delta }}_{-}=2.4\mathrm{K}$, ${\varepsilon }_F^{\left(g\right)}=10\mathrm{eV}$, $v_F^{\left(g\right)}=2\times {10}^6\mathrm{m}/\mathrm{s}$, $d_0=100\mathrm{nm}$, $\lambda =300\mathrm{nm}$, $\epsilon =15$, and the rest as in Fig. 2. The value for the effective coupling constant [see Eq.(8)] is then given by $g=1.05$.

Figure 5

Temperature $T_{\mu s}$ for which $\tau \gtrsim 1\mu \mathrm{s}$ as function of ${\mu }_{+}$ plotted for different values of $l_0$. While the dip at ${\mu }_{+}\approx {\mu }^{*}$ does not depend on $l_0$, for ${\mu }_{+}>{\mu }^{*}$, $T_{\mu s}$ is determined by the IABS energy, which decreases with increasing $l_0$. For an abrupt transition ($l_0=0$, black top line), $T_{\mu s}$ for a trivial right section (${\mu }_{+}\gg {\mu }^{*}$) remains practically the same as for a uniform chemical potential, $\mu \left(x\right)={\mu }_{+}=0$. The parameters are the same as for Fig. 4.

Figure 6

(a) Excitation spectrum of the NW [obtained numerically by diagonalizing Eq. (1)] as function of ${\mu }_{+}$ for a short right section $L_{+}=l_0=2{\xi }_{-}$ and a uniform ${\mathrm{\Delta }}_{sc}$ over the entire NW length. If ${\mu }_{+}$ is larger than the critical value, ${\mu }_{+}>{\mu }^{*}$, in addition to the bulk modes above the gap ${\mathrm{\Delta }}_{-}$ (blue crosses), there emerge bulk modes in the right section (green squares), whose energies show oscillatory dependence on ${\mu }_{+}$ approaching the gap ${\mathrm{\Delta }}_{-}$ for ${\mu }_{+}\gg {\mu }^{*}$. The parameters used are the same as for Fig. 2. (b) The same as (a) but for a normal right section (${\mathrm{\Delta }}_{sc}=0$). The spectrum of a normal section can be only partially gapped by the magnetic field. If ${\mu }_{+}<{\mathrm{\Delta }}_Z$, the spectrum of the NW Hamiltonian, see Eq. (1), consists of states above the gap ${\mathrm{\Delta }}_{-}$ localized in the left superconducting section (blue crosses) and states localized in the right normal section (green squares) originating from the exterior branches of the spectrum with quantized energies, whereas the interior branches of the spectrum in the right section are gapped. If ${\mu }_{+}$ becomes larger than ${\mathrm{\Delta }}_Z$ (shaded region) the number of states with energies below ${\mathrm{\Delta }}_{-}$ increases since both exterior and interior branches of the spectrum are ungapped at these energies. The energy levels exhibit characteristic oscillations as ${\mu }_{+}$ is changed.

Figure 7

Spatial dependence of the parameters in an SN junction. The nonuniform chemical potential $\mu \left(x\right)$ is controlled by a nearby gate similar to the setup shown in Fig. 1. The proximity-induced superconducting gap is nonzero, ${\mathrm{\Delta }}_{sc}\ne 0$, in the left (S) section, and vanishes in the right (N) section. The Zeeman energy ${\mathrm{\Delta }}_Z$ is uniform.

Figure 8

(a) Lifetime $\tau$ as function of temperature $T$ and chemical potential ${\mu }_{+}$ for an SN junction with short right (N) section, $L_{+}=30a={\xi }_{-}$; see Fig. 7. The blue line holds for $\tau ({\mu }_{+},T_{\mu s})=1\mu \mathrm{s}$. As ${\mu }_{+}$ becomes larger than ${\mathrm{\Delta }}_Z$ (shown with a green dashed line), the lifetime $\tau$ and $T_{\mu s}$ oscillate with increasing ${\mu }_{+}$ with a period of order $h{v}_F^{\left(i\right)}/L_{+}$. The parameters chosen are the same as for Fig. 4. (b) Temperature $T_{\mu s}$ as function of ${\mu }_{+}$ plotted for different lengths of the right section $L_{+}$. For $\mu <{\mathrm{\Delta }}_Z$ the temperature $T_{\mu s}$ decreases as $L_{+}$ increases, $T_{\mu s}\propto L_{+}^{-1}$. For larger values of the chemical potential ${\mu }_{+}>{\mathrm{\Delta }}_Z$, the oscillations of $T_{\mu s}$ become more rapid when the right section increases.