---
abstract: 'Charge transport through a small topological superconducting island in contact with a normal and a superconducting electrode occurs through a cycle which involves coherent oscillations of Cooper pairs and tunneling in/out the normal electrode through a Majorana bound state, <span>*the Josephson-Majorana cycle*</span>. We illustrate this mechanism by studying the current-voltage characteristics of two-terminal superconductor – topological superconductor – normal metal single-electron transistors. At low bias and temperature the Josephson-Majorana cycle is the dominant mechanism for transport. The non-local character of the Majorana bound states can be directly probed in a three-terminal setup.'
author:
- Nicolas Didier
- Marco Gibertini
- 'Ali G. Moghaddam'
- Jürgen König
- Rosario Fazio
title: 'Josephson-Majorana cycle in topological single-electron hybrid transistors'
---

<span>*Introduction*</span> — Majorana bound states are zero-energy states that occur at the boundary or in the vortex core of topological superconductors @qi11 [@beenakker11; @alicea12]. Besides the genuine interest in understanding their properties, they play a fundamental role in the realization of a topological quantum computer @kitaev01 [@nayak08]. For these reasons an intense research has started to find physical systems that support Majorana excitations. A quantum wire with spin-orbit coupling in close proximity to a superconductor and in the presence of an external magnetic field has been considered among the most promising proposals @lutchyn10 [@oreg10]. The very recent experiment @leo of the Delft group on this system provides the first evidence of the existence of Majorana bound states in the condensed matter world.

The presence of Majorana bound states in topological superconductors has a number of distinct signatures. It is responsible for a fractional Josephson effect @kitaev01 [@lutchyn10; @fu09; @jiang11], it leads to a resonant Andreev reflection @andreev [@nilsson] and to anomalies in interference experiments @interference, and it can be detected by measuring the local density of states @dos.

In the presence of quantum fluctuations, affecting the phase coherence of the topological superconductor, new effects appear due to a parity constraint linking the dynamics of the superconducting phase and of the Majorana fermions. A paradigmatic situation to explore these phenomena, in the setup of Refs. @kitaev01 [@lutchyn10; @oreg10], is when the superconducting island in contact with the wire is small enough so that charging effects @grabert-devoret come into play @footnote. Fu @fu10 first pointed out that the parity constraint leads to non-local effects in electron transport. The role of charging effects on the fractional Josephson effect and on the Coulomb blockade through a topological superconducting transistor was studied in Refs. @vanheck11 and @zazunov11, respectively.

In this work we introduce and analyze what turns out to be the dominant charge transport mechanism occurring at low voltages in hybrid topological single electron transistors: The Josephson-Majorana cycle. It takes place when a topological superconducting island is coupled to superconducting and to normal leads (see Fig. \[setup\]). Charges can flow through the island due to the combined effect of the coherent oscillations of Cooper pairs in the island and the tunneling between the Majorana state and the normal leads. Although the process bears some similarities to the Josephson-quasiparticle mechanism present in Cooper pair transistors @averin89 [@vandenbrink91; @choi01] there are also important differences. In the concluding part of the paper we will address this issue and show that some features of the Josephson-Majorana cycle are true consequences of the non-local character of the Majorana bound states. Thus the observation of transport mediated by this cycle can provide further support to elucidate the properties of Majorana fermions.

![The system consists of a superconducting grain coupled to a quantum wire connected to normal metals and to a superconductor. Parameters are chosen so that the wire hosts two Majorana bound states at the two ends.<span data-label="setup"></span>](fig1){width="\columnwidth"}

<span>*The model*</span> — The system we consider is illustrated in Fig. \[setup\]. It consists of a topological superconducting island (a nanowire in close proximity to a superconducting island) tunnel-coupled to normal and superconducting electrodes in a three-terminal configuration. The island hosts two Majorana bound states at the ends of the wire associated with the Majorana operators $\gamma_{i}$ ($i=L,R$), $\{\gamma_i,\gamma_j\}=\delta_{i,j}$ and $\gamma_i=\gamma_i^{\dag}$. At low energies (much smaller than the superconducting gap), the island couples to the superconducting lead only through Josephson tunneling. The coupling to the normal leads occurs only via the Majorana bound states. Throughout the paper we set the two normal electrodes at the same voltage, the only net current will be from the superconductor to the normal electrodes. The three-terminal setup will be however crucial to discuss the non-local character of the Josephson-Majorana cycle. The mesoscopic scale of the central island considered here requires to take into account the charging energy of the island. The Hamiltonian of the system reads: $${\cal H} = {\cal H}_M +  {\cal H}_{Ch} + {\cal H}_{J} + {\cal H}_{leads} + {\cal H}_T~,
\label{hamiltonian}$$ where the five terms describe coupling of Majorana states, island charging energy, Josephson coupling to the superconducting electrode, metallic normal electrode, and tunneling to the normal lead, respectively. The coupling between the Majorana fermions is given by ${\cal H}_M = i\lambda \gamma_L\gamma_R$ where the overlap $\lambda$ between them is exponentially small for distances much larger than the superconducting coherence length. Two Majorana states can form a zero-energy fermionic level described by an annihilation operator $d=(\gamma_L+i\gamma_R)/\sqrt{2}$ which can be either occupied or empty. Therefore not only the number of excess Cooper pairs $N$ but also the occupation of the $d$-level enters the charging energy of the island, $${\cal H}_{Ch}= E_C(2N+n_d-n_g)^2~,
\label{charging}$$ where $n_d=d^\dag d$ counts the occupation of the $d$-level, $n_g$ is the gate charge which can be varied continuously by changing the gate voltage, and $E_C = e^2/(2C)$ is the charging energy expressed in terms of the total capacitance $C$ of the island. For later convenience we will label the eigenstates of the charging energy with the notation $|N,n_d \rangle$. To describe the Josephson coupling to the superconducting electrode we use the effective Hamiltonian $${\cal H}_{J}= -E_J\cos\left(\varphi_S-\varphi\right)~,$$ where $E_J$ is the Josephson coupling energy and $\varphi_S$ and $\varphi$ are the phase of the superconducting electrode and the island condensate, respectively. Here, any possible coupling to the Majorana state can be safely ignored @zazunov12. The normal lead Hamiltonian is given by the noninteracting model ${\cal H}_{leads} = \sum_{i=L,R} \sum_k\epsilon_k c_{k,i}^\dag c_{k,i}$ with creation (annihilation) operator $c^{\dagger}_{k,i}\;(c_{k,i})  $ corresponding to a spinless fermion in the free particle state $k$ with energy $\epsilon_k$ inside the left/right (L/R) normal lead. Finally, the tunneling through the Majorana state takes the form @zazunov11, $${\cal H}_T =   \sum_{i=L,R} \sum_k t_{k,i} c_{k,i}^\dag(d+\delta_i e^{-i\varphi}d^\dag)+ {\rm H.c.}~,
\label{tunneling}$$ where $t_{k,i}$ is the hopping amplitude to the $k$-state in the $i$-th lead and $\delta_{L/R} =\pm 1$. The advantage of this formulation introduced by Zazunov <span>*et al.*</span> @zazunov11 is the fact that it includes automatically the constraint on the Hilbert space, linking the occupation of the Majorana bound state to the parity of the superconducting condensate. A non-trivial dynamics in the problem arises since $\varphi$ and $N$ are canonically conjugated variables, $[N, e^{i\varphi}] = e^{i\varphi}$. The two terms in the tunneling Hamiltonian correspond to regular and anomalous tunneling. The first one describes the transfer of an electron from the $d$-level to the normal lead, and the second one the annihilation of a Cooper pair inside the island accompanied by the creation of two electrons, one in the $d$-level and one in a normal electrode.

![Josephson-Majorana cycle for $n_g\sim 1$. The states differing by one Cooper pair, $|0,0 \rangle$ and $|1,0 \rangle$, are coupled via the Josephson term. The coupling between $|0,1 \rangle$ and $|0,0 \rangle$ is given by the normal term in the tunneling Hamiltonian, the connection between by $|0,1 \rangle$ and $|1,0 \rangle$ by the anomalous one. <span data-label="cycle"></span>](fig2){width="0.7\columnwidth"}

<span>*Josephson-Majorana cycle*</span> — We start by considering the current-voltage characteristics at the second order in the tunneling amplitudes $t_{k,i}$. At bias voltage and temperature smaller than the superconducting gap, quasiparticle tunneling is suppressed and coherent Josephson (Cooper pair) tunneling is necessary for transport through the transistor. From Eq. (\[charging\]) it follows that the resonance condition for coherent Cooper pair oscillations between two charge states that differ by one Cooper pair is fulfilled at integer values of the gate charge $n_g$. Around $n_g\sim 1$, e.g., the lowest energy states are $|N,n_d \rangle = |0,0 \rangle$, $|0,1 \rangle$ and $|1,0 \rangle$. A Josephson-Majorana cycle involving these three states is illustrated in Fig. \[cycle\]. Starting from $|0,1 \rangle$, a regular tunneling process releases an electron into the (left or right) normal lead. Then, the island is recharged with an extra Cooper pair provided by the Josephson coupling. A second (anomalous) tunneling which annihilates a Cooper pair in order to create an electron inside the normal electrode and another one filling the $d$-level, completes the cycle. (A cycle with the reverse direction can be obtained by the conjugates of each tunneling process.) Since the energy of the state $|0,1 \rangle$ is lower than the ones connected by Josephson tunneling, there will be a threshold voltage for the onset of current. In the following we will support this simple picture sketched above with more detailed calculations.

As already mentioned, the two normal leads are kept at the same voltage. The average of the total current operator, $I=ie \,\sum_{i=L,R}\sum_k[c_{k,i}^\dag c_{k,i},H]  =
-2e\,\mathrm{Im}\, \sum_{i=L,R} \sum_k t_{k,i} c^{\dagger}_{k,i} (d+\delta_i e^{-i\varphi}d^{\dagger})$, can be conveniently expressed in terms of the reduced density matrix $\rho$ of the topological superconducting island which is obtained after tracing out the fermionic degrees of freedom of the normal metals. The steady-state current is $$\begin{gathered}
    \langle I \rangle  =  \frac{e}{2} \sum_{i=L,R}  \sum_{l,n,m} \Gamma_i \left[ D^{(i)\ast}_{nl}D^{(i)}_{ml} (F_{ln}+ F_{lm}^*)
    \right. \\ \left.
    -D^{(i)}_{ln}D^{(i)\ast}_{lm}  (\bar{F}_{nl}^*+\bar{F}_{ml} ) \right] \rho_{nm}\,,    \label{exprcurrent}\end{gathered}$$ where $D^{(i)}_{nm}$ and $\rho_{nm}$ are the matrix elements of the operators $D^{(i)}=d+\delta e^{-i\varphi}d^{\dagger}$ and $\rho$ in the basis defined by the eigenstates $|\psi_n \rangle$ of the Hamiltonian ${\cal H}_{Ch} + {\cal H}_{J}$ with eigenvalues $\mathcal{E}_n$ and $F_{nm}=f(\Delta_{nm}) - (i/\pi) \, \mathrm{Re}\,\Psi[1/2+i(\Delta_{nm}-eV)/(2\pi k_BT)]$, $\bar{F}=1-F$, where $\Delta_{nm}=\mathcal{E}_n-
\mathcal{E}_{m}$, $f(\epsilon)=1/[1+e^{(\epsilon-eV)/k_BT}]$ is the Fermi function of the normal electrode, and $\Psi$ is the digamma function. Finally $\Gamma_i=2\pi|\tau|^2N(\epsilon_F)$ is the tunneling rate with $t_{k,i}\sim \tau$ assumed constant close to the Fermi energy $\epsilon_F$, and $N(\epsilon_F)$ is the density of states in the normal metal. For simplicity we will assume $\Gamma_L \sim \Gamma_R = \Gamma /2$.

![Current-voltage characteristics for $k_BT=E_C/100$, $\Gamma=k_BT/10$, $E_J=E_C/5$.<span data-label="iv-gate3d"></span>](fig3){width="0.9\columnwidth"}

A convenient way to represent and compute the reduced density matrix, in particular when higher-order tunneling processes are taken into account, is to use a real-time diagrammatic technique that has been developed to describe transport through a metallic single-electron transistor @schoeller94 [@koenig98]. The time evolution of the reduced density matrix $\rho(t)$ can be cast in the form @footnote-rules $$\label{rhodot}
\nonumber
\dot{\rho}_{nm}+i\Delta_{nm}\rho_{nm}=\sum_{n',m'}\int_{-\infty}^{t} \mathrm{d}t'W_{nm,n'm'}(t,t'){\rho}_{n'm'}(t')
\, .$$ Expanding to second order in the tunneling, this yields for the steady-state limit ($\dot \rho=0$) @footnote-MEA, $$\begin{aligned}
\nonumber
        &  i\Delta_{nm}{\rho}_{nm} = \frac{\Gamma}{4} \sum_{i,n',m'} \left[
        D^{(i)\ast}_{n'n} D^{(i)}_{m'm} (F_{mn'}+ F_{nm'}^{\ast} ) \right. \\
\nonumber & \left. \hspace{18.5mm}+D^{(i)}_{nn'}D^{(i)\ast}_{mm'} (\bar{F}_{n'm}^*+\bar{F}_{m'n}  ) \right] {\rho}_{n'm'}\\
\nonumber &  -\frac{\Gamma}{4}  \sum_{i,l,n'}  \left[    D^{(i)}_{nl}D^{(i)\ast}_{n'l} F_{lm}^{\ast}  +
        D^{(i)\ast}_{ln}D^{(i)}_{ln'} \bar{F}_{ml}   \right]  {\rho}_{n'm} \\
               &   -\frac{\Gamma}{4} \sum_{i,l,m'} \left[    D^{(i)}_{m'l}D^{(i)\ast}_{ml} F_{ln}   +
               D^{(i)\ast}_{lm'}D^{(i)}_{lm} \bar{F}_{nl}^{\ast}   \right]    {\rho}_{nm'} \, .
        \label{rate}\end{aligned}$$

![Current-voltage characteristics and its dependence on the bias charge $n_V=CV/e$ and the gate charge $n_g=CV_g/e$. Parameters are: $k_BT=E_C/100$, $\Gamma=k_BT/10$, $E_J=\Gamma/\sqrt{3}$. The threshold voltage to observe a current is $n_V\simeq\pm0.5$. The slight displacement of the resonance condition from $n_g=1$ is due to the Lamb shift from the normal lead.<span data-label="iv-gate"></span>](fig4){width="0.9\columnwidth"}

In Fig. \[iv-gate3d\] we show the current as a function of the gate charge $n_g=CV_g/e$ and the bias charge $n_V=CV/e$. The current increases stepwise (smeared by temperature) with the bias voltage and shows resonance peaks at integer values of $n_g$. In the range of the gate charge where only the three charge states $|0,0 \rangle$, $|0,1 \rangle$ and $|1,0 \rangle$ are involved, a simple analytical solution can be obtained for zero temperature when neglecting the imaginary parts of $F_{nm}$, which are associated with energy renormalization induced by the coupling to the bath. We find the two threshold voltages $V^{\pm}$ $$eV^{\pm}=E_C \left[1\pm2\sqrt{(1-n_g)^2+(E_J/4E_C)^2} \right]~.
\label{eqthreshold}$$ For positive voltage $V>0$ and $1/2<n_g<3/2$, the steady-state current is $$\langle I \rangle=e\Gamma E_J^2\left\{
\begin{aligned}
&2/[\Gamma^2+ 4\mathcal{E}^2- E_J^2] &V>V^+ \\
&1/[\Gamma^2g_c(n_g) + 2 \mathcal{E}^2] &|V^-|<V< V^+ \\
&0 &V<|V^-|
\end{aligned}\right.
\label{eqIVC}$$ where $\mathcal{E} = \sqrt{[4E_C(1-n_g)]^2+E_J^2}$, $g_c(n_g) = (21+13~x-x^2-x^3)/16$, and $x = 4E_C(1-n_g)/\mathcal{E} $. We immediately see that the current vanishes when the Josephson coupling goes to zero. The width of the resonance peaks is $\delta n_g=E_J/4E_C$ and the peak value for $V>V^{+}$ becomes maximal for $\Gamma=\sqrt{4 \mathcal{E}^2-E_J^2}$.

The effect of energy renormalization terms (Lamb shifts) that were neglected in the analytical formulas presented above, is highlighted in Fig. \[iv-gate\] where the resonance condition is slightly displaced from the value $n_g=1$. We remark that the current is antisymmetric under the transformation $(V,n_g)\to(-V,2-n_g)$ @footnote-symmetry.

Finally, we comment on the abrupt suppression of the current for large bias voltage $|n_V| > 3/2$ that is visible in Fig. \[iv-gate\]. Beyond the threshold $n_V = 3/2$, the new charge state $|1,1\rangle$ becomes available from $|1,0\rangle$ by tunneling. The island is then trapped in this new state since the Josephson coupling to the state $|0,1\rangle$ is suppressed because of the large energy difference of $|1,1\rangle$ and $|0,1\rangle$. As a consequence, transport is blocked.

As already anticipated in the introduction we now discuss on the differences between the Josephson-Majorana cycle and the Josephson-quasiparticle cycle @averin89 of Cooper pair transistors. A first, obvious but important for experimental detection, difference is the voltage scale at which the Josephson-Majorana cycle takes place. Here there are no excitations above the gap and therefore the threshold voltage is set by the charging energy solely. Second, we note that the Majorana-Josephson cycle shows a $e$-periodicity with respect to $n_g$. If the Majorana level were replaced by a regular spin-degenerate fermionic level, the current would be $2e$-periodic due to an even-odd asymmetry of the current-carrying states and the corresponding cycles. Furthermore, a regular non-degenerate fermionic level could not sustain any Josephson jeremynicholson@example.com.

The strongest proof for the presence of the Majorana states, however, is to detect their non-local character. For this, we consider higher-order current in the fork configuration of Fig. \[setup\]. Below the threshold voltage $V^-$, second-order transport considered so far is suppressed by Coulomb blockade. Nevertheless fourth-order (two-particle) processes sustain a finite current. Let's consider for illustration the case $n_g\sim 1$ where the Coulomb treshold is largest. Here the system remains in its ground state $|0,1\rangle$ and the states $|0,0\rangle$ and $|1,0\rangle$ needed to complete the Josephson-Majorana cycle are only virtually occupied. To evaluate the current, we determine the cotunneling rates for transitions in which two electrons are transferred from the superconducting to the normal leads (or vice versa) while leaving the island in state $|0,1\rangle$. As intermediate (virtual) states, we take the two linear combinations of $|0,0\rangle$ and $|1,0\rangle$ that form eigenstates of ${\cal H}_{Ch}+{\cal H}_J$. All processes which differ only in their intermediate but not initial and final state need to be added coherently. This includes pairs of processes in which the order of the tunneling events of the two normal lead electrons is interchanged. Thereby, it is important to distinguish local from non-local cotunneling. For local cotunneling, the amplitudes of those pairs carry opposite sign. For non-local cotunneling, however, this sign is compensated by the relative sign of $\delta_L=-\delta_R$ in Eq. (\[tunneling\]). As a result, local cotunneling is strongly suppressed, and non-local cotunneling dominates. For zero temperature and to linear order in $eV$, we find that the local contribution vanishes and the non-local one is given by (restoring $\hbar$) $$\langle I \rangle =  \frac{4e^2}{h}   \frac{\Gamma_L \Gamma_R E_J^2}{(E_C^2-E_J^2/4)^2} V\, ,$$ valid for any integer value of $n_g$. The non-local character of the Majorana states can be detected by the finite linear cotunneling conductance. Furthermore the noise is maximally correlated, $\langle \delta I_R^2 \rangle = \langle \delta I_L^2 \rangle = \langle \delta I_R \delta I_L \rangle$, independently of the length of the wire (as in Ref. @nilsson). The difference between local and non-local transport does not appear to second order since, there, tunneling of different lead electrons occurs incoherently.

<span>*Conclusions*</span> — In topological hybrid transistors charge transport is dominated at low voltages by a Josephson-Majorana cycle. This is a process in which the Rabi oscillations of Cooper pairs are accompanied by the tunneling of electrons from/to Majorana bound states. We showed the basic principles of this cycle in a superconductor – topological superconductor – normal metal transistor and discussed how the features of the current-voltage characteristics are linked to the presence of Majorana bound states. The Josephson-Majorana cycle in the three-terminal setup of Fig. \[setup\] allows to reveal the existence and the non-local character of the Majorana bound states.

<span>*Acknowledgments*</span> — We acknowledge stimulating discussions with M. Polini and F. Taddei. We are grateful to C.W.J. Beenakker and R. Egger for useful comments. The work was supported by EU through projects IP-SOLID, QNEMS, GEOMDISS and NANOCTM and by DFG through project KO 1987/5.

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{'source': 'arXiv_out/1202'}