Landau-Zener-Stückelberg Interferometry for Majorana Qubit

Stimulated by a recent experiment observing successfully two superconducting states with even- and odd-number of electrons in a nanowire topological superconductor as expected from the existence of two end Majorana quasiparticles (MQs) [Albrecht et al., Nature 531, 206 (2016)], we propose a way to manipulate Majorana qubit exploiting quantum tunneling effects. The prototype setup consists of two one-dimensional (1D) topological superconductors coupled by a tunneling junction which can be controlled by gate voltage. We show that the time evolution of superconducting phase difference at the junction under a voltage bias induces an oscillation in energy levels of the Majorana parity states, whereas the level-crossing is avoided by a small coupling energy of MQs in the individual 1D superconductors. This results in a Landau-Zener-Stückelberg (LZS) interference between the Majorana parity states. Adjusting pulses of bias voltage and gate voltage, one can construct a LZS interferometry which provides an arbitrary manipulation of the Majorana qubit.

as given in the main text. Because the Hamiltonian is time periodic, Eq. (S1) can be tackled in terms of the Floquet theory. A previous study 1 shows that, up to the lowest-order perturbation the time evolution of the quantum state is governed by the following effective Floquet Hamiltonian, where z = 2E m /ω and J 0 is the Bessel function. The evolution operator for this effective Floquet Hamiltonian is,Û which gives an oscillation of the form, |ψ 0 (t)| 2 = cos 2 (ω m t), |ψ 1 (t)| 2 = sin 2 (ω m t).
starting from an initial state ψ 1 = 0 with, This is the LZS interference in the Majorana qubit addressed in the main text.
Hamiltonian for Two Qubits. In this section we derive the Hamiltonian for the quantum mechanical dynamics of two Majorana qubits in a matrix form. The system consists of two tunneling junctions between three 1D topological superconductors as illustrated in Fig. 6(a) in the main text. The Hamiltonian is given as, where θ L = ϕ 2 − ϕ 1 and θ R = ϕ 3 − ϕ 2 is the phase difference, E m,L and E m,R is the coupling energy of MQs at the left and right junction respectively, and δ 1,2,3 are the coupling energies due to the wavefunction overlappings of MQs in the three topological superconductors. In terms of the six MQs one can define three complex fermions, Then Hamiltonian (S6) can be transformed into the fermionic form, Because the total parity of the system is conserved, one can concentrate on the even-parity subspace without losing generality. In this subspace, we can take basis states, |00⟩ = |0⟩, , with |0⟩ the vacuum state. Then the Hamiltonian is rewritten in a matrix form with E L = E m,L cos θ L 2 and E R = E m,R cos θ R 2 . The superconducting phase differences of the two junctions θ L and θ R are controlled by the two bias voltages. The coupling energies E m,L and E m,R can be tuned by gate voltages.
Solving the Schrödinger equation with the matrix Hamiltonian presuming the linear timedependence of θ L and θ R and δ 1 = δ 2 = δ 3 = δ we obtain the results displayed in Fig. 6 in the main text.
RCSJ phase dynamics. The LZS interferometry for Majorana qubit can also be achieved under current bias. In order to show this explicitly, we analyze the resistively-andcapacitively-shunted dynamics of superconducting phase difference at the junction 2,3 where C and R are the effective capacitance and resistance of the junction [see Fig. 1 The tunneling junction has both conventional Cooper-pair and MQ channels, and the total supercurrent depends on the superconducting phase difference in a way given by (S11) We observe that, in a static state, the Majorana parity state with |ψ 0 | = 1 and |ψ 1 | = 0 bypasses a supercurrent larger than other cases, presuming I s,m > 0. It is easy to derive the phase difference for the critical supercurrent which gives (S13) In the present scheme, when the current injected through the junction is larger than the critical current I c , a voltage drop is induced at the junction and drives the time evolution of superconducting phase difference according to the ac Josephson effect 6,7 . Conventionally the RCSJ dynamics of the superconducting phase difference mimics a Newtonian particle moving in a tilted washboard potential 2,3 . In the present case, the phase particle acquires an additional pseudo-spin degree of freedom associated with the Majorana parity states as described by Eq. (S10).
The LZS oscillation of the Majorana qubit changes the supercurrent through the last term in Eq. (S10), and then recoils to influence the time evolution of superconducting phase difference. Therefore, the full dynamics of the system obey simultaneously the Schrödinger equation (6) in the main text and the RCSJ dynamics (S10). As shown in Fig. S1, (S15) As an ansatz solution capturing the fundamental modes we presume where A, B, B ′ , ω, ϕ 1 , ϕ 2 and ϕ ′ 2 are parameters to be determined. Plugging it into the RCSJ equation (S15), we encounter double-sine functions such as ) . (S17) With the Jacobi-Anger expansion and checking the static term and those depending on time sin ωt, cos ωt, sin( ω parameters in the ansatz solution 3 , ω = 2eRI ext / (S18) where ω p = √ 2eI s / C is the plasma frequency and Q = √ 2eI s R 2 C/ is the quality factor 2 .
The voltage drop at the junction is given by the time derivative of the phase difference dt . Accordingly, the time averaged voltage is given by V = ω /2e, and the oscillating part of the voltage is characterized by a spectrum function which is the Fourier (S19) The power spectrum of the voltage oscillation evaluated by the Fourier transformation on data in Fig. S2 Fig. S1, one has β C = 100 ≫ 1, corresponding to a highly underdamped junction. In this case, decoherence of the LZS interference cannot be found over sufficiently long time as shown in Fig. S3. In a highly underdamped Josephson junction there is a hysteresis loop in the I-V characteristics, and a phase dynamic state with a finite voltage drop across the junction is possible even though the current injection is below the critical current 3 , which may also be used for achieving the LZS interferometry for Majorana qubit.
In the above discussions, we have presumed a pure Majorana qubit state (even-or oddparity state) as the initial condition of quantum mechanical evolutions. As read from E- q. (S10), the maximal supercurrent in the parallel circuit of Cooper-pair channel and MQ channel is achieved by a pure state. This feature can be exploited to initialize the qubit into a pure state by increasing gradually the current injection up to the maximal supercurrent.
In a very recent experiment, rf radiations were detected from a Josephson junction in the heterostructure of the topological insulator HgTe and an s-wave superconductor, a candidate system for realizing topological superconductivity 8 . A half-frequency component with broad line width was observed in addition to the conventional Josephson one, which may be relevant to the physics discussed in the present work. It is anticipated that experiments with finer frequency resolution based on our protocol will reveal the quantum mechanical dynamics of MQ, and pave the way for building a universal gate for Majorana qubit.
To finish this section we notice that the current injected into the system beyond the critical current triggers a large voltage drop across the junction, which then may induce undesired coupling between the MQs and quasiparticles in the continuum spectrum. In this sense, the voltage bias is a better implementation of the LZS interferometry for the topological superconductors, the smallest energy scale of the present system, is δ = 5 µeV.