Majorana qubit decoherence by quasiparticle poisoning

We consider the problem of quasiparticle poisoning in a nanowire-based realization of a Majorana qubit, where a spin-orbit-coupled semiconducting wire is placed on top of a (bulk) superconductor. By making use of recent experimental data exhibiting evidence of a low-temperature residual nonequilibrium quasiparticle population in superconductors, we show by means of analytical and numerical calculations that the dephasing time due to the tunneling of quasiparticles into the nanowire may be problematically short to allow for qubit manipulation.

Figure 1

Schematic circuit representation of the superconductor/topological-nanowire system. The interface that separates the two subystems acts as a tunnel junction, with tunnel resistance $R_{\mathrm{T}}$ and capacitance $C$. The internal impedance $Z_{\mathrm{S}}$ and $Z_{\mathrm{NW}}$ of the superconductor and the nanowire are combined in the text in a single global environmental impedance $Z\left(\omega \right)$. An external voltage bias between the two sides of the junction can be present.

Figure 2

Equivalent circuit for the SC/TSC system in the case of a single-mode environment. The environment is modeled with a single inductance $L$ with impedance $Z\left(\omega \right)=i\omega L$, corresponding to a total impedance $\Re \mathrm{e}\left[Z_{\mathrm{t}}\left(\omega \right)\right]\sim [\delta (\omega +{\omega }_{\mathrm{R}})+\delta (\omega -{\omega }_{\mathrm{R}})]$ (plotted in the inset).

Figure 3

Same as in Fig. 2, but with a different environmental impedance. Here, the environment is modeled by $Z=R$, and the resulting Lorentzian total impedance is shown in the inset, where ${\omega }_{\mathrm{C}}=1/\left(RC\right)=g{E}_{\mathrm{C}}/\pi \hslash$.

Figure 4

Behavior of the probability function $P\left(E\right)$ as a function of the energy exchange measured in units of $E_{\mathrm{C}}$. For large environment resistances (small $g$) the junction releases a typical energy amount of the order of the charging energy. For small resistances (large $g$), the energy that is exchanged shrinks to zero, and one recovers a situation with independent quasiparticles and junction degrees of freedom. The curves have been obtained through numerical integration.

Figure 5

Dimensionless quasiparticle tunneling rate as a function of environmental dimensionless conductance $g$. The red curve refers to the case ${\Delta }_{\mathrm{S}}=10E_{\mathrm{C}}$, ${\Delta }_{\mathrm{T}}=5E_{\mathrm{C}}$, and no observable dependence on $g$ is noticed at this scale. The cyan curve has been obtained for ${\Delta }_{\mathrm{S}}=100E_{\mathrm{C}}$, ${\Delta }_{\mathrm{T}}=50E_{\mathrm{C}}$, and the corresponding values of ${\Gamma }_{\mathrm{qp}}$ are slightly lower in this case.

Figure 6

Same as in Fig. 5, but with different parameter values. The cyan curve corresponds to ${\Delta }_{\mathrm{S}}=2E_{\mathrm{C}}$, ${\Delta }_{\mathrm{T}}=E_{\mathrm{C}}$ and the red curve refers to the case ${\Delta }_{\mathrm{S}}=E_{\mathrm{C}}$, ${\Delta }_{\mathrm{T}}=0.5E_{\mathrm{C}}$. In the first case, $({\Delta }_{\mathrm{S}}-{\Delta }_{\mathrm{T}})$ equals $E_{\mathrm{C}}$ and leads to an unbounded increase in ${\Gamma }_{\mathrm{qp}}$ for $g\to 0$. In the second case (and, in general, for $E_{\mathrm{C}}>\delta \Delta$), one observes ${\Gamma }_{\mathrm{qp}}(g\to 0)\to 0$ because the energy exchange $E_{\mathrm{C}}$ provided by the environment is too large to be absorbed by $\delta \Delta$.

Figure 7

Dispersion relation in a one-dimensional wire in the presence of Rashba spin-orbit and Zeeman interaction. The gap at $k=0$ is entirely due to the Zeeman energy $V_{\mathrm{Z}}$. For $\alpha k_{\mathrm{F}}\gg V_{\mathrm{Z}}$, the position of the two minima $\varepsilon ={\varepsilon }_0$ is approximately given by $\pm k_{\mathrm{so}}\equiv \alpha m/{\hslash }^2$. The topological regime requires having the chemical potential lying inside the gap, as shown here; $k_0$ denotes the point at which the dispersion crosses the representative midgap level $\varepsilon =0$.