Session Q29: Focus Session: Topologically Protected Qubits II

Topological superconducting states and protected qubit manipulations

Topological superconducting states supporting Majorana fermion excitations have been recently proposed as platforms for topological quantum computation. Of particular importance are semiconductor-superconductor and topological insulator-superconductor heterostructures, which have been shown to support Majorana fermions at order parameter defects under appropriate external conditions. Here I will focus on topologically non-trivial properties of two-dimensional semiconductors and one-dimensional quantum wires placed adjacent to superconductors, and discuss the possible protected qubit manipulations that may eventually lead to topological quantum computation.   

Majorana Zero Modes in 1D Quantum Wires Without Long-Ranged Superconducting Order

We show that long-ranged superconducting order is not necessary to guarantee the existence of Majorana fermion zero modes at the ends of a quantum wire. We formulate a concrete model which applies, for instance, to a semiconducting quantum wire with strong spin-orbit coupling and Zeeman splitting coupled to a wire with algebraically-decaying superconducting fluctuations. We solve this model by bosonization and show that it supports Majorana fermion zero modes. We argue that a large class of models will also show the same phenomenon. We discuss the implications for experiments on spin-orbit coupled nanowires coated with superconducting film and for LaAlO3/SrTiO3 interfaces.   

Coulomb stability of the $\mathbf{4\pi}$-periodic Josephson effect of Majorana fermions

The Josephson energy of two superconducting islands containing Majorana fermions is a $4\pi$-periodic function of the superconducting phase difference. If the islands have a small capacitance, their ground state energy is governed by the competition of Josephson and charging energies. We calculate this ground state energy in a ring geometry, as a function of the flux $\Phi$ enclosed by the ring, and show that the dependence on the Aharonov-Bohm phase $2e\Phi/\hbar$ remains $4\pi$-periodic regardless of the ratio of charging and Josephson energies---provided that the entire ring is in a topologically nontrivial state. If part of the ring is topologically trivial, then the charging energy induces quantum phase slips that restore the usual $2\pi$-periodicity [B.\ van Heck, F.\ Hassler, A.\ R. Akhmerov, and C.\ W.\ J. Beenakker, Phys. Rev. B {\bf 84}, 180502(R) (2011)].   

Unconventional Josephson signatures of Majorana bound states

A junction between two topological superconductors containing a pair of Majorana fermions exhibits a `fractional' Josephson effect, $4\pi$ periodic in the superconductors' phase difference. An additional fractional Josephson effect, however, arises when the Majoranas are spatially separated by a superconducting barrier. This new term gives rise to a set of Shapiro steps which are essentially absent without Majorana modes and therefore provides a unique signature for these exotic states. Other new signatures associated with Majorana fermions will also be discussed.   

Signature of Majorana Fermions in Charge Transport in Semiconductor Nanowires

We investigate the charge transport in a semiconductor nanowire that is subject to a perpendicular magnetic field and in partial contact with an \textit{s}-wave superconductor. We find that Majorana fermions, existing at the interface between superconducting and normal sections of the nanowire within certain parameter region, can induce resonant Andreev reflection of electrons at the interface, which yields a zero energy peak in the electrical conductance of the nanowire. The width of the zero energy conductance peak for different experimental parameters is characterized. While the zero energy peak provides a signature for Majorana fermions in one dimensional nanowires, it disappears in a two-dimensional semiconductor thin film with the same experimental setup because of the existence of other edge states in two dimensions. The proposed charge transport experiment may provide a simple and experimentally feasible method for the detection of Majorana fermions in semiconductor nanowires.   

Searching for Majorana Zero-Energy Modes in Semiconductor Nanowires

Majorana fermions are proposed elementary particles with the unique property of being their own antiparticle, whose discovery remains elusive. Because of their non-Abelian statistics, Majorana fermions provide a promising opportunity to realize fault-tolerant topological quantum computation. Recently, semiconductor nanowires have been proposed as a possible platform for realizing Majorana physics in solid state systems. An s-wave superconductor inducing the proximity effect in a one-dimensional semiconductor nanowire would create chiral p-wave superconductivity in the nanowire; this superconductor would have Majorana modes as zero-energy excitations. We use nanofabrication techniques to create such nanowires out of MBE-grown two-dimensional electron gas formed in III-V semiconductor heterostructures. We present measurements which show the promise of this approach to creating and controlling Majorana excitations.   

Majorana fermions in superconducting helical magnets

In a variety of rare-earth based compounds singlet superconductivity coexists with helical magnetism. Here we demonstrate that surfaces of these system are expected to generically host a finite density of zero-energy Majorana modes. When confined to a wire geometry, a discrete number of Majorana modes can be isolated, in close analogy with the Rashba superconductors proposed recently as a framework for topological quantum computing. In contrast to the latter systems, however, the larger characteristic energy scales for superconductivity and magnetism, as well as the lack of need for fine-tuning, make helical magnetic superconducting compounds favorable for the observation and experimental investigation of Majorana fermions.   

Stability of Majorana fermions in proximity-coupled topological insulator nanowires

It has been shown previously [1] that a finite-length topological insulator nanowire, proximity-coupled to an ordinary bulk s-wave superconductor and subject to a longitudinal applied magnetic field, realizes a one-dimensional topological superconductor with an unpaired Majorana fermion localized at each end of the nanowire. Here we show that the unpaired Majorana fermions persist in this system for any value of the chemical potential inside the bulk band gap of order 300 meV in Bi$_2$Se$_3$, and, remarkably, also outside this gap in smaller domains, by computing the Majorana number. From this calculation, we also show that the unpaired Majorana fermions persist when the magnetic flux through the nanowire cross-section deviates significantly from half flux quantum. Lastly, we demonstrate that the unpaired Majorana fermions persist in strongly disordered wires with fluctuations in the on-site potential ranging in magnitude up to the size of the bulk band gap. These results suggest this solid-state system should exhibit the elusive Majorana particles under conditions accessible enough for their long sought-after experimental realization.\\[4pt] [1] A. Cook and M. Franz, Phys.\ Rev.\ B (in press, arXiv:1105.1787)   

Josephson current in finite-lenght nanowire SNS junctions with Majorana fermions

The dc Josephson effect (JE) through infinite-lenght junctions of one-dimensional topological superconductors exhibits an anomalous $4\pi$ periodic phase ($\phi$) dependence which originates from a parity-protected level crossing of zero-energy Majorana bound states (MBS) at $\phi=\pi$. This ``fractional'' JE provides an important experimental detection tool for MBS. In this talk, I will discuss the JE in more realistic SNS junctions of arbitrary transparency and when both the normal and the nanowire regions are of finite length, namely beyond the short-junction and infinite topological superconductor limits. In general, the spectrum of Andreev bound states can become rather intricate and dense as opposed to the infinite-lenght case. Moreover, the low-energy spectrum around $\phi=\pi$ shows always anticrossings, originated from hybridization of four MBS, which may preclude the experimental observation of the fractional JE. At finite bias voltages, Landau-Zener dynamics involving the MBS and quasi-continuum Andreev levels gives rise to a nontrivial ac Josephson current. Interestingly, the ac current phase diagram as a function of the Josephson frequency/normal transmission shows fractional JE regions which are tunable through bias/gate voltages.   

Detecting a Majorana-Fermion Zero Mode Using a Quantum Dot

We propose an setup for detecting a Majorana zero mode consisting of a spinless quantum dot coupled to the end of a p-wave superconducting nanowire [1]. The conductance through the dot is monitored by adding two external leads. We find that the Majorana bound state at the end of the wire strongly influences the conductance through the quantum dot: driving the wire through the topological phase transition causes a sharp jump in the conductance by a factor of 1/2. In the topological phase, the zero temperature peak value of the dot conductance (i.e. on resonance and symmetric coupling) is e$^2$/2h. In contrast, if the wire is in its trivial phase, the peak is e$^2$/h, or if a regular fermionic zero mode occurs, the conductance is 0. We also consider coupling the dot to both ends of the wire (two MBS), with a magnetic flux f through the loop. The conductance as a function of phase shows peaks at f/f0 = (2n+1)*pi which can be used to tune Flensberg's qubit system [PRL (2011)] to the energy degeneracy point. \\[4pt] [1] D. E. Liu and H. U. Baranger, PRB in press (2011); arXiv/1107.4338.   

Majorana chain coupled to a microwave cavity

We study the Majorana end-states in a one-dimensional Kitaev model in the presence of a microwave cavity. We analyze both the resonant and off-resonant coupling to the cavity for different cavity states. In the resonant regime, we find that the topology of the system can be modified depending on the number of photons in the cavity, while in the off-resonant regime (large detuning), the cavity could be used to detect the topological transition from a non-trivial to trivial state, being thus an optical alternative to the transport-based detection of the transition point. We also analyze the effects of the coupling to the cavity on the braiding of the emerging Majoranas.   

Majorana modes in time-reversal invariant s-wave topological superconductors

We present a time-reversal invariant $s$-wave superconductor supporting Majorana edge modes. The multi-band character of the model together with spin-orbit coupling are key to realizing such a topological superconductor. We characterize the topological phase diagram by using a partial Chern number sum, and show that the latter is physically related to the parity of the fermion number of the time-reversal invariant modes. By taking the self-consistency constraint on the $s$-wave pairing gap into account, we also establish the possibility of a direct topological superconductor-to-topological insulator quantum phase transition.   

Majorana Fermions in Disordered Quasi-One-Dimensional Topological Superconductors

Majorana fermions have long been predicted to emerge in certain quantum Hall states and other naturally occurring p-wave superconductors. However, these materials are quite delicate and consequently the experimental realization of Majorana fermions remains elusive. The possibility of engineering 1D networks of topological superconducting wires from conventional materials offers a promising alternative route to realize Majorana fermions and probe their predicted non-Abelian statistics. In practice, it is impossible to fabricate perfectly clean and strictly one-dimensional structures; how do these non-idealities affect the proposed Majorana states? This talk will show that Majorana end states are robust away from the strict 1D limit, so long as the sample width is not much larger than the superconducting coherence length. The effects of disorder are potentially more severe, as impurity scattering is generally pair-breaking and tends to suppress the gap protecting the Majorana modes. Finally, we propose new candidate materials and geometries that greatly simplify the experimental setup and mitigate the harmful effects of disorder.