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arXiv:1911.05795 · Physical Review Letters

Unitary Subharmonic Response and Floquet Majorana Modes

Oles Shtanko, Ramis Movassagh

Published Aug 20, 2020• 9 citations
arXivHTMLPDF

Abstract

Detection and manipulation of excitations with non-Abelian statistics, such as Majorana fermions, are essential for creating topological quantum computers. To this end, we show the connection between the existence of such localized particles and the phenomenon of unitary subharmonic response (SR) in periodically driven systems. In particular, starting from highly nonequilibrium initial states, the unpaired Majorana modes exhibit spin oscillations with twice the driving period, are localized, and can have exponentially long lifetimes in clean systems. While the lifetime of SR is limited in translationally invariant systems, we show that disorder can be engineered to stabilize the subharmonic response of Majorana modes. A viable observation of this phenomenon can be achieved using modern multiqubit hardware, such as superconducting circuits and cold atomic systems.

DOI
10.1103/physrevlett.125.086804

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Detection and manipulation of excitations with non-Abelian statistics, such as Majorana fermions, are essential for creating topological quantum computers. To this end, we show the connection between the existence of such localized particles and the phenomenon of unitary subharmonic response (SR) in periodically driven systems. In particular, starting from highly nonequilibrium initial states, the unpaired Majorana modes exhibit spin oscillations with twice the driving period, are localized, and can have exponentially long lifetimes in clean systems. While the lifetime of SR is limited in translationally invariant systems, we show that disorder can be engineered to stabilize the subharmonic response of Majorana modes. A viable observation of this phenomenon can be achieved using modern multiqubit hardware, such as superconducting circuits and cold atomic systems.
Obeying non-Abelian statistics, Majorana fermions holds a promise to implement fault-tolerant quantum computing. It was found that Majorana fermions can be simulated by the zero-energy excitation in a nanowire with strong spin-orbit coupling interacting with an sss-wave superconductor under a magnetic field. However, the signal of Majorana fermion in that system is obscured by the disorder in the nanowire and the confinement potential at the wire end. Thus, more controllable platforms are desired to simulate Majorana fermions. We here propose an alternative scheme to simulate the Majorana fermions in a trapped-ion system. Our dimerized-ion configuration permits us to generate the Majorana modes not only at zero energy but also at the nonzero ones, which enlarge the family of Majorana modes and supply another qubit carrier for quantum computing. We also investigate the controllability of the Majorana modes by Floquet engineering. It is found that a widely tunable number of Majorana modes are created on demand by applying a periodic driving on the trapped-ion system. Enriching the platforms for simulating Majorana fermions, our result would open another avenue for realizing fault-tolerant quantum computing.
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.
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.
We demonstrate longtime coherent subharmonic motion of a many-boson system subjected to an external time-periodic driving force. The underlying mechanism is exemplified numerically through analysis of a periodically driven Bose-Hubbard dimer, and clarified conceptually by semiclassical requantization of invariant tubes pertaining to the system’s mean-field description. In this way, one arrives at pre-Floquet states that relate to the actual many-body Floquet states in a manner similar to the relation of site-localized Wannier states to lattice-extended Bloch states in solid-state physics. It is argued that even high-order subharmonic response can be systematically engineered, and be observed experimentally, with weakly interacting Floquet condensates comprising a sufficiently large number of particles.
Periodically driven quantum systems can exhibit subharmonic response, usually characterized through physical observables and often discussed in interacting settings. Here we show that a sharp subharmonic signature already appears in the entanglement spectrum of a number-conserving free-fermion system. We study a two-step driven Su-Schrieffer-Heeger chain whose Floquet operator supports symmetry-protected edge modes at quasienergies 000 and πππ. When the initial state is a coherent superposition of these two edge sectors, we show that the subsystem correlation matrix alternates between two stroboscopic structures, and the entanglement spectrum is period-doubled as a set, while an overlap-tracked entanglement level shows a robust period-doubling response with Fourier weight concentrated at half the drive frequency. By contrast, diagonal edge densities remain flat by sublattice symmetry, while an off-diagonal edge-bond observable provides the corresponding linear one-body comparator. The effect disappears both when the initial state is replaced by a stroboscopically stationary Floquet eigenstate built from the same topological mode content, and when the system is placed in the topologically trivial phase where no edge modes exist. Altogether, these establish zero-πππ Floquet topology as a necessary condition and coherent nonequilibrium preparation as the additional sufficient ingredient. Our results identify entanglement spectroscopy as a sharp subsystem-resolved probe of Floquet topological coherence.
Periodically driven quantum systems host exotic phenomena that often do not have any analog in undriven systems. Floquet prethermalization and dynamical freezing of certain observables, via the emergence of conservation laws, are realized by controlling the drive frequency. Recent experimental developments in synthetic quantum matter, such as superconducting qubits and cold atoms, have opened avenues for implementing local Floquet engineering which can achieve spatially modulated quantum control of states. Here, we uncover the novel memory effects of local periodic driving in a nonintegrable spin-half staggered Heisenberg chain. For a boundary-driven protocol at the dynamical freezing frequency, we show the formation of an approximate strong zero mode, a prethermal quasilocal operator, due to the emergence of a discrete global Z2 symmetry. This is captured by constructing an accurate effective Floquet Hamiltonian using a higher-order partially resummed Floquet-Magnus expansion. The lifetime of the boundary spin can be exponentially enhanced by enlarging the set of suitably chosen driven sites. We demonstrate that in the asymptotic limit, achieved by increasing the number of driven sites, a strong zero mode emerges, where the lifetime of the boundary spin grows exponentially with system size. The nonlocal processes in the Floquet Hamiltonian play a pivotal role in the total freezing of the boundary spin in the thermodynamic limit. The novel dynamics of the boundary spin is accompanied by a rich structure of entanglement in the Floquet eigenstates where specific bipartitions yield an area-law scaling while the entanglement for random bipartitions scales as a volume-law. Our work addresses the long-standing question of the existence of a strong zero mode in a nonintegrable model and elucidates the complex nature of thermalization in locally driven systems. Published by the American Physical Society 2024