Effects of Fluctuations on Majorana Edge Modes in Atomic Topological Kitaev Wire with Molecular Reservoir

We discuss the effects of quantum and thermal fluctuations on the Majorana edge states in a topological atomic wire coupled to a superfluid molecular gas with gapless excitations. We find that the coupling between the Majorana edge states remains exponentially decaying with the length of the wire, even at finite temperatures smaller than the energy gap for bulk excitations in the wire. This exponential dependence is controlled solely by the localization length of the Majorana states. The fluctuations, on the other hand, provide the dominant contribution to the preexponential factor, which increases with temperature and the length of the wire. More important is that thermal fluctuations give rise to a decay of an initial correlation between Majorana edge states to its stationary value after some thermalization time. This stationary value is sensitive to the temperature and to the length of the wire and, although vanishing in the thermodynamic limit, can still be feasible in a mesoscopic system at sufficiently low temperatures. The thermalization time, on the other hand, is found to be much larger than the typical time scales in the wire and is sufficient for quantum operations with Majorana fermions before the temperature-induced decoherence sets in.We discuss the effects of quantum and thermal fluctuations on the Majorana edge states in a topological atomic wire coupled to a superfluid molecular gas with gapless excitations. We find that the coupling between the Majorana edge states remains exponentially decaying with the length of the wire, even at finite temperatures smaller than the energy gap for bulk excitations in the wire. This exponential dependence is controlled solely by the localization length of the Majorana states. The fluctuations, on the other hand, provide the dominant contribution to the preexponential factor, which increases with temperature and the length of the wire. More important is that thermal fluctuations give rise to a decay of an initial correlation between Majorana edge states to its stationary value after some thermalization time. This stationary value is sensitive to the temperature and to the length of the wire and, although vanishing in the thermodynamic limit, can still be feasible in a mesoscopic system at sufficiently low temperatures. The thermalization time, on the other hand, is found to be much larger than the typical time scales in the wire and is sufficient for quantum operations with Majorana fermions before the temperature-induced decoherence sets in.

arXiv:1412.2547Physical Review ANov 17, 201513 citations

Majorana fermions in noisy Kitaev wires

Ying Hu, Zi Cai, Mikhail A. Baranov +1 more

Robustness of edge states and non-Abelian excitations of topological states of matter promises quantum memory and quantum processing, which are naturally immune to microscopic imperfections such as static disorder. However, topological properties will not in general protect quantum systems from time-dependent disorder or noise. Here we take the example of a network of Kitaev wires with Majorana edge modes storing qubits to investigate the effects of classical noise in the crossover from the quasistatic to the fast fluctuation regime. We present detailed results for the Majorana edge correlations, and fidelity of braiding operations for both global and local noise sources preserving parity symmetry, such as random chemical potentials and phase fluctuations. While in general noise will induce heating and dephasing, we identify examples of long-lived quantum correlations in the presence of fast noise due to motional narrowing, where external noise drives the system rapidly between the topological and nontopological phases.Robustness of edge states and non-Abelian excitations of topological states of matter promises quantum memory and quantum processing, which are naturally immune to microscopic imperfections such as static disorder. However, topological properties will not in general protect quantum systems from time-dependent disorder or noise. Here we take the example of a network of Kitaev wires with Majorana edge modes storing qubits to investigate the effects of classical noise in the crossover from the quasistatic to the fast fluctuation regime. We present detailed results for the Majorana edge correlations, and fidelity of braiding operations for both global and local noise sources preserving parity symmetry, such as random chemical potentials and phase fluctuations. While in general noise will induce heating and dephasing, we identify examples of long-lived quantum correlations in the presence of fast noise due to motional narrowing, where external noise drives the system rapidly between the topological and nontopological phases.

arXiv:1506.06977Physical Review BOct 16, 201541 citations

Thermally enhanced Majorana-mediated spin transport in the Kitaev model

Hirokazu Taguchi, Yuta Murakami, Akihisa Koga

We study how stable the Majorana-mediated spin transport in a quantum spin Kitaev model is against thermal fluctuations. Using the time-dependent thermal pure quantum state method, we examine finite-temperature spin dynamics in the Kitaev model. The model exhibits two characteristic temperatures, ${T}_{L}$ and ${T}_{H}$ , which correspond to energy scales of the local flux and the itinerant Majorana fermion, respectively. At low temperatures $(T\ensuremath{\ll}{T}_{L})$, an almost flux-free state is realized, and the spin excitation propagates in a similar way to that for the ground state. Namely, after the magnetic pulse is introduced at one of the edges, the itinerant Majorana fermions propagate the spin excitations even through the quantum spin liquid state region, and oscillations in the spin moment appear in the other edge with a tiny magnetic field. When $T\ensuremath{\sim}{T}_{L}$, larger oscillations in the spin moments are induced in the other edge, compared with the results at the ground state. At higher temperatures, excited ${Z}_{2}$ fluxes disturb the coherent motion of the itinerant Majorana fermions, which suppresses the spin propagation. Our results demonstrate a crucial role of thermal fluctuations in the Majorana-mediated spin transport.We study how stable the Majorana-mediated spin transport in a quantum spin Kitaev model is against thermal fluctuations. Using the time-dependent thermal pure quantum state method, we examine finite-temperature spin dynamics in the Kitaev model. The model exhibits two characteristic temperatures, ${T}_{L}$ and ${T}_{H}$, which correspond to energy scales of the local flux and the itinerant Majorana fermion, respectively. At low temperatures $(T\ensuremath{\ll}{T}_{L})$, an almost flux-free state is realized, and the spin excitation propagates in a similar way to that for the ground state. Namely, after the magnetic pulse is introduced at one of the edges, the itinerant Majorana fermions propagate the spin excitations even through the quantum spin liquid state region, and oscillations in the spin moment appear in the other edge with a tiny magnetic field. When $T\ensuremath{\sim}{T}_{L}$, larger oscillations in the spin moments are induced in the other edge, compared with the results at the ground state. At higher temperatures, excited ${Z}_{2}$ fluxes disturb the coherent motion of the itinerant Majorana fermions, which suppresses the spin propagation. Our results demonstrate a crucial role of thermal fluctuations in the Majorana-mediated spin transport.

arXiv:2201.12517Physical review. B./Physical review. BMar 28, 20228 citations

Probability Distribution of Majorana End-State Energies in Disordered Wires

Piet W. Brouwer, Mathias Duckheim, Alessandro Romito +1 more

One-dimensional topological superconductors harbor Majorana bound states at their ends. For superconducting wires of finite length L, these Majorana states combine into fermionic excitations with an energy ε(0) that is exponentially small in L. Weak disorder leaves the energy splitting exponentially small, but affects its typical value and causes large sample-to-sample fluctuations. We show that the probability distribution of ε(0) is log normal in the limit of large L, whereas the distribution of the lowest-lying bulk energy level ε(1) has an algebraic tail at small ε(1). Our findings have implications for the speed at which a topological quantum computer can be operated.One-dimensional topological superconductors harbor Majorana bound states at their ends. For superconducting wires of finite length L, these Majorana states combine into fermionic excitations with an energy ε(0) that is exponentially small in L. Weak disorder leaves the energy splitting exponentially small, but affects its typical value and causes large sample-to-sample fluctuations. We show that the probability distribution of ε(0) is log normal in the limit of large L, whereas the distribution of the lowest-lying bulk energy level ε(1) has an algebraic tail at small ε(1). Our findings have implications for the speed at which a topological quantum computer can be operated.

arXiv:1104.1531Physical Review LettersNov 1, 2011187 citations

Topological protection of Majorana qubits

Meng Cheng, Roman M. Lutchyn, S. Das Sarma

We study the stability of the topological quantum computation proposals involving Majorana fermions against thermal fluctuations. We use a minimal realistic model of a spinless ${p}_{x}+i{p}_{y}$ superconductor and consider the effect of excited midgap states localized in the vortex core as well as of transitions above the bulk superconducting gap on the quasiparticle braiding, interferometry-based qubit readout schemes, and quantum coherence of the topological qubits. We find that thermal occupation of the midgap states does not affect adiabatic braiding operations but leads to a reduction in the visibility of the interferometry measurements. We also consider quantum decoherence of topological qubits at finite temperatures and calculate their decay rate which is associated with the change of the fermion parity and, as such, is exponentially suppressed at temperatures well below the bulk excitation gap. Our conclusion is that the Majorana-based topological quantum computing schemes are indeed protected by virtue of the quantum nonlocality of the stored information and the presence of the bulk superconducting gap.We study the stability of the topological quantum computation proposals involving Majorana fermions against thermal fluctuations. We use a minimal realistic model of a spinless ${p}_{x}+i{p}_{y}$ superconductor and consider the effect of excited midgap states localized in the vortex core as well as of transitions above the bulk superconducting gap on the quasiparticle braiding, interferometry-based qubit readout schemes, and quantum coherence of the topological qubits. We find that thermal occupation of the midgap states does not affect adiabatic braiding operations but leads to a reduction in the visibility of the interferometry measurements. We also consider quantum decoherence of topological qubits at finite temperatures and calculate their decay rate which is associated with the change of the fermion parity and, as such, is exponentially suppressed at temperatures well below the bulk excitation gap. Our conclusion is that the Majorana-based topological quantum computing schemes are indeed protected by virtue of the quantum nonlocality of the stored information and the presence of the bulk superconducting gap.

arXiv:1112.3662Physical Review BApr 16, 201287 citations

Distribution of Majorana modes in the extended-range Kitaev chain

Pedro Barbisan Widniczck, Gerardo Martinez

The topological properties of the Kitaev chain model with extended-range interactions are investigated, focusing on cases where the topological winding number is preserved. We assume that the pairing and hopping terms decay algebraically in space with exponents $α$ and $β$ , respectively. We show that in the truncated-range scenario, there are as many distinct topological phases as the number of coupled neighboring sites. In addition, an explicit analytical formulation is provided to evaluate the topological invariant and the phase transitions that emerge in these systems. Besides the analytical description, we introduce a new physical insight into the topological excitations of the ground-state by measuring the spatial distribution of the edge modes with the Majorana average position. Taking the next-nearest neighbor Kitaev chains as a probe, various numerical calculations of Majorana edge states were performed in finite-size clusters to determine the sensitivity of the topological zero energy modes to the parameters of interest. The occupation of edge-to-edge non-local fermion states is computed and defined as an effective parity. Such an effective parity exhibits new interesting features beyond the energetic exchange from the ground-state fermion parity switches, which are related to the distribution of the respective edge modes. Our calculations show a direct correlation between the ground-state fermion parity and the edge occupation numbers, which are translated into localization and delocalization of the Majorana average position.The topological properties of the Kitaev chain model with extended-range interactions are investigated, focusing on cases where the topological winding number is preserved. We assume that the pairing and hopping terms decay algebraically in space with exponents $α$ and $β$, respectively. We show that in the truncated-range scenario, there are as many distinct topological phases as the number of coupled neighboring sites. In addition, an explicit analytical formulation is provided to evaluate the topological invariant and the phase transitions that emerge in these systems. Besides the analytical description, we introduce a new physical insight into the topological excitations of the ground-state by measuring the spatial distribution of the edge modes with the Majorana average position. Taking the next-nearest neighbor Kitaev chains as a probe, various numerical calculations of Majorana edge states were performed in finite-size clusters to determine the sensitivity of the topological zero energy modes to the parameters of interest. The occupation of edge-to-edge non-local fermion states is computed and defined as an effective parity. Such an effective parity exhibits new interesting features beyond the energetic exchange from the ground-state fermion parity switches, which are related to the distribution of the respective edge modes. Our calculations show a direct correlation between the ground-state fermion parity and the edge occupation numbers, which are translated into localization and delocalization of the Majorana average position.

arXiv:2606.10043Jun 8, 20260 citations

{T}_{L}

and

{T}_{H}

, which correspond to energy scales of the local flux and the itinerant Majorana fermion, respectively. At low temperatures $(T\ensuremath{\ll}{T}_{L})$, an almost flux-free state is realized, and the spin excitation propagates in a similar way to that for the ground state. Namely, after the magnetic pulse is introduced at one of the edges, the itinerant Majorana fermions propagate the spin excitations even through the quantum spin liquid state region, and oscillations in the spin moment appear in the other edge with a tiny magnetic field. When $T\ensuremath{\sim}{T}_{L}$, larger oscillations in the spin moments are induced in the other edge, compared with the results at the ground state. At higher temperatures, excited

{Z}_{2}

fluxes disturb the coherent motion of the itinerant Majorana fermions, which suppresses the spin propagation. Our results demonstrate a crucial role of thermal fluctuations in the Majorana-mediated spin transport.

We study the stability of the topological quantum computation proposals involving Majorana fermions against thermal fluctuations. We use a minimal realistic model of a spinless

{p}_{x}+i{p}_{y}

superconductor and consider the effect of excited midgap states localized in the vortex core as well as of transitions above the bulk superconducting gap on the quasiparticle braiding, interferometry-based qubit readout schemes, and quantum coherence of the topological qubits. We find that thermal occupation of the midgap states does not affect adiabatic braiding operations but leads to a reduction in the visibility of the interferometry measurements. We also consider quantum decoherence of topological qubits at finite temperatures and calculate their decay rate which is associated with the change of the fermion parity and, as such, is exponentially suppressed at temperatures well below the bulk excitation gap. Our conclusion is that the Majorana-based topological quantum computing schemes are indeed protected by virtue of the quantum nonlocality of the stored information and the presence of the bulk superconducting gap.

α

and

β

, respectively. We show that in the truncated-range scenario, there are as many distinct topological phases as the number of coupled neighboring sites. In addition, an explicit analytical formulation is provided to evaluate the topological invariant and the phase transitions that emerge in these systems. Besides the analytical description, we introduce a new physical insight into the topological excitations of the ground-state by measuring the spatial distribution of the edge modes with the Majorana average position. Taking the next-nearest neighbor Kitaev chains as a probe, various numerical calculations of Majorana edge states were performed in finite-size clusters to determine the sensitivity of the topological zero energy modes to the parameters of interest. The occupation of edge-to-edge non-local fermion states is computed and defined as an effective parity. Such an effective parity exhibits new interesting features beyond the energetic exchange from the ground-state fermion parity switches, which are related to the distribution of the respective edge modes. Our calculations show a direct correlation between the ground-state fermion parity and the edge occupation numbers, which are translated into localization and delocalization of the Majorana average position.