Quantum transport in coupled Majorana box systems

Abstract

We present a theoretical analysis of low-energy quantum transport in coupled Majorana box devices. A single Majorana box represents a Coulomb-blockaded mesoscopic superconductor proximitizing two or more long topological nanowires. The box thus harbors at least four Majorana zero modes (MZMs). Setups with several Majorana boxes, where MZMs on different boxes are tunnel-coupled via short nanowire segments, are key ingredients to recent Majorana qubit and code network proposals. We construct and study the low-energy theory for multi-terminal junctions with normal leads connected to the coupled box device by lead-MZM tunnel contacts. Transport experiments in such setups can test the nonlocality of Majorana-based systems and the integrity of the underlying Majorana qubits. For a single box, we recover the previously described topological Kondo effect which can be captured by a purely bosonic theory. For several coupled boxes, however, non-conserved local fermion parities require the inclusion of additional local sets of Pauli operators. We present a renormalization group analysis and develop a nonperturbative strong-coupling approach to quantum transport in such systems. Our findings are illustrated for several examples, including a loop qubit device and different two-box setups. arXiv:1803.02997v2 [cond-mat.mes-hall]

Quantum Computing with Majorana Kramers Pairs

“…We now proceed by presenting two examples of sequences of intermediate states leading to contributions ∝ J 6 . The first example is given by P ab (λγ b,L,↑ c L,k↑ e iφ2/2 )(λγ a,R,↑ c R,q↑ e iφ1/2 )(λc † R,q↑ γ b,R,↑ e −iφ2/2 )(λc † L,k↑ γ a,L,↑ e −iφ1/2 )P ab = P ab λ 4 (γ b,L,↑ c L,k↑ γ a,R,↑ c R,q↑ c † R,q↑ γ b,R,↑ c † L,k↑ γ a,L,↑ )P ab = P ab λ 4 (γ a,R,↑ γ a,L,↑ γ b,R,↑ γ b,L,↑ )(c L,k↑ c R,q↑ c † R,q↑ c † L,k↑ )P ab = −P ab λ 4 ẑa ẑb (c L,k↑ c R,q↑ c † R,q↑ c † L,k↑ )P ab = −P ab λ 4 ẑa ẑb (u q u k ) 2 (γ L,q↑ γ R,q↑ γ † R,q↑ γ † L,k↑ )P ab = −P ab λ 4 ẑa ẑb (u q u k ) 2 P ab (39) leading to a term ∝ ẑa ẑb and the second example is given P ab (λγ b,L,↑ c L,k↑ e iφ2/2 )(λγ a,R,↓ c R,q↓ e iφ1/2 )(λc † R,q↓ γ b,R,↓ e −iφ2/2 )(λc † L,k↑ γ a,L,↑ e −iφ1/2 )P ab = P ab λ 4 (γ b,L,↑ c L,k↑ γ a,R,↓ c R,q↓ c † R,q↓ γ b,R,↓ c † L,k↑ γ a,L,↑ )P ab = P ab λ 4 (γ a,L,↑ γ a,R,↓ γ b,L,↑ γ b,R,↓ )(c L,k↑ c R,q↓ c † R,q↓ c † L,k↑ )P ab = −P ab λ 4 xa xb (c L,k↑ c R,q↓ c † R,q↓ c † L,k↑ )P ab = −P ab λ 4 xa xb (u k u q ) 2 (γ L,k↑ γ L,q↓ γ † L,q↓ γ † L,k↑ )P ab = −P ab λ 4 xa xb (u k u q ) 2 P ab (40) producing a term ∝ xa xb . The energy denominator for both examples is given by −1/[(E k + U ) 2 (E k + E q + 2U )].…”

“…Finally, we remark that examples for contributions ∝ J 8 and ∝ J 12 can be obtained by appropriately commuting the terms in the round brackets in the first line of Eqs. (39) and (40). The only difference occurs in the energy denominator.…”

“…(2) Quasiparticle poisoning due to thermal excitations within the TRI TSC island is strongly suppressed the SC gap of the island itself. Critically, the energy gap of a TRI TSC island is conceivably larger than the energy gap of TRS-breaking Majorana islands [30][31][32][33][34][35][36][37][38][39][40] since there is no magnetic field that would reduce the SC gap size. As a consequence, the MKQ should benefit from improved coherence times and may be a viable route towards a robust quantum computer.…”

Transport properties of coupled Majorana bound states in the Coulomb blockade regime

“…As MBSs owe their topological protection to the conservation of fermion parity, a single MBS-based qubit typically involves at least four MBSs. Potential structures for such nanowire-based qubits are the T junction [5,[15][16][17] and the Majorana box [18][19][20][21][22][23]. Both systems consist of nanowires placed on top of a mesoscopic superconductor and can be controllably connected to metallic leads.…”

“…When there is no overlap between the MBSs the system can be mapped to a degenerate spin-1/2 system, which leads to a "topological Kondo effect" at temperatures below a Kondo temperature [18,19,21,[46][47][48][49]. Gau et al have recently presented a very detailed analysis on the transport properties in systems with multiple Majorana boxes [22].…”