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arXiv:1207.6888 · Physical Review Letters

Dynamical Formation and Manipulation of Majorana Fermions in Driven Quantum Wires in Contact with a Superconductor

Enrico Perfetto

Published Feb 19, 2013• 41 citations
arXivHTMLPDF

Abstract

Controlling the dynamics of Majorana fermions (MF) subject to time-varying driving fields is of fundamental importance for the practical realization of topological quantum computing. In this work we study how it is possible to dynamically generate and maintain the topological phase in cold-atom nanowires after the temporal variation of the Hamiltonian parameters. Remarkably we show that for a sudden quench the system can never relax toward a state exhibiting fully developed MF, independently of the initial and final Hamiltonians. Only for sufficiently slow protocols the system behaves adiabatically, and the topological phase can be reached. Finally we address the crucial question of how "adiabatic" a protocol must be in order to manipulate the MF inside the topological phase without deteriorating their Majorana character.

DOI
10.1103/physrevlett.110.087001

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Controlling the dynamics of Majorana fermions (MF) subject to time-varying driving fields is of fundamental importance for the practical realization of topological quantum computing. In this work we study how it is possible to dynamically generate and maintain the topological phase in cold-atom nanowires after the temporal variation of the Hamiltonian parameters. Remarkably we show that for a sudden quench the system can never relax toward a state exhibiting fully developed MF, independently of the initial and final Hamiltonians. Only for sufficiently slow protocols the system behaves adiabatically, and the topological phase can be reached. Finally we address the crucial question of how "adiabatic" a protocol must be in order to manipulate the MF inside the topological phase without deteriorating their Majorana character.
We theoretically explore quench dynamics in a finite-sized topological fermionic p -wave superconducting wire with the goal of demonstrating that topological order can have marked effects on such non-equilibrium dynamics. In the case studied here, topological order is reflected in the presence of two (nearly) isolated Majorana fermionic end bound modes together forming an electronic state that can be occupied or not, leading to two (nearly) degenerate ground states characterized by fermion parity. Our study begins with a characterization of the static properties of the finite-sized wire, including the behavior of the Majorana end modes and the form of the tunnel coupling between them; a transfer matrix approach to analytically determine the locations of the zero energy contours where this coupling vanishes; and a Pfaffian approach to map the ground state parity in the associated phase diagram. We next study the quench dynamics resulting from initializing the system in a topological ground state and then dynamically tuning one of the parameters of the Hamiltonian. For this, we develop a dynamic quantum many-body technique that invokes a Wick's theorem for Majorana fermions, vastly reducing the numerical effort given the exponentially large Hilbert space. We investigate the salient and detailed features of two dynamic quantities—the overlap between the time-evolved state and the instantaneous ground state (adiabatic fidelity) and the residual energy. When the parity of the instantaneous ground state flips successively with time, we find that the time-evolved state can dramatically switch back and forth between this state and an excited state even when the quenching is very slow, a phenomenon that we term 'parity blocking'. This parity blocking becomes prominently manifest as non-analytic jumps as a function of time in both dynamic quantities.
We theoretically explore quench dynamics in a finite-sized topological fermionic p-wave superconducting wire with the goal of demonstrating that topological order can have marked effects on such non-equilibrium dynamics. In the case studied here, topological order is reflected in the presence of two (nearly) isolated Majorana fermionic end bound modes together forming an electronic state that can be occupied or not, leading to two (nearly) degenerate ground states characterized by fermion parity. Our study begins with a characterization of the static properties of the finite-sized wire, including the behavior of the Majorana end modes and the form of the tunnel coupling between them; a transfer matrix approach to analytically determine the locations of the zero energy contours where this coupling vanishes; and a Pfaffian approach to map the ground state parity in the associated phase diagram. We next study the quench dynamics resulting from initializing the system in a topological ground state and then dynamically tuning one of the parameters of the Hamiltonian. For this, we develop a dynamic quantum many-body technique that invokes a Wick's theorem for Majorana fermions, vastly reducing the numerical effort given the exponentially large Hilbert space. We investigate the salient and detailed features of two dynamic quantities-the overlap between the time-evolved state and the instantaneous ground state (adiabatic fidelity) and the residual energy. When the parity of the instantaneous ground state flips successively with time, we find that the time-evolved state can dramatically switch back and forth between this state and an excited state even when the quenching is very slow, a phenomenon that we term `parity blocking'. This parity blocking becomes prominently manifest as non-analytic jumps as a function of time in both dynamic quantities.
We develop an approach to realizing a topological phase transition and non-Abelian braiding statistics with dynamically induced Floquet Majorana fermions (FMFs). When the periodic driving potential does not break fermion parity conservation, FMFs can encode quantum information. Quasienergy analysis shows that a stable FMF zero mode and two other satellite modes exist in a wide parameter space with large quasienergy gaps, which prevents transitions to other Floquet states under adiabatic driving. We also show that in the asymptotic limit FMFs preserve non-Abelian braiding statistics and, thus, behave like their equilibrium counterparts.
Topological Quantum Computing has largely evolved towards a paradigm of manipulating edge localized Majorana within ppp-wave topological superconducting nanowires. To bridge the gap between physical qubit systems and quantum algorithms, we perform a dynamical analysis to extend what is known in the adiabatic regime, providing time-dependent gate elements for further qubit and algorithm modeling efforts. Our analysis covers dynamical considerations for two methods of shuttling domain edge bound Majoranas in a single nanowire system which both function by applying spatiotemporally dependent onsite and hopping parameters within the system's Hamiltonian. We then complicate this model by converting it into the T-qubit to calculate the finite-time gate representation of the shuttling techniques used in a more practical setting. These contributions provide insight for realistic experimental setups in the next-generation of qubit implementation and will hopefully facilitate fault tolerant scalable systems and universal gate design.
The fermionic and Majorana edge mode dynamics of various topological systems are compared, after a sudden global quench of the Hamiltonian parameters takes place. Attention is focused on the regimes where the survival probability of an edge state has oscillations either due to critical or off-critical quenches. The nature of the wave functions and the overlaps between the eigenstates of different points in parameter space determine the various types of behaviors, and the distinction due to the Majorana nature of the excitations plays a lesser role. Performing a sequence of quenches, it is shown that the edge states, including Majorana modes, may be switched off and on. Also, the generation of Majoranas due to quenching from a trivial phase is discussed.
The prospects for realizing a topological quantum computer have brightened since the apparent detection of Majorana fermions at the ends of semiconducting nanowires. These Majorana zero-modes persist in the presence of the strong disorder that may be present in such systems, protected by the mobility gap of localized systems. Recent proposals to manipulate quantum information in a Majorana chain involve adiabatically adjusting gate voltages in one-dimensional nanowire networks. However, as we show, in the adiabatic limit, a disordered system with multiple zero-mode Majorana fermions will undergo a non-local response to a local perturbation via a multi-level Landau-Zener transition. This will disperse the quantum information stored in the target zero-modes amongst other zero-modes that are present in the system, complicating the realization of controlled gates. There is however a solution that may work in practice: we show that there is an optimum window of intermediate time-scales during which the quantum information is approximately preserved under manipulation.