16 research outputs found
Bulk--Boundary Correspondence for Chiral Symmetric Quantum Walks
Discrete-time quantum walks (DTQW) have topological phases that are richer
than those of time-independent lattice Hamiltonians. Even the basic symmetries,
on which the standard classification of topological insulators hinges, have not
yet been properly defined for quantum walks. We introduce the key tool of
timeframes, i.e., we describe a DTQW by the ensemble of time-shifted unitary
timestep operators belonging to the walk. This gives us a way to consistently
define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using
an "inversion symmetric" pulse sequence. For one-dimensional DTQW's with CS, we
identify the bulk ZxZ topological invariant that controls the number of
topologically protected 0 and pi energy edge states at the interfaces between
different domains, and give simple formulas for these invariants. We illustrate
this bulk--boundary correspondence for DTQW's on the example of the "4-step
quantum walk", where tuning CS and particle-hole symmetry realizes edge states
in various symmetry classes
Topological delocalization in the completely disordered two-dimensional quantum walk
We investigate numerically and theoretically the effect of spatial disorder
on two-dimensional split-step discrete-time quantum walks with two internal
"coin" states. Spatial disorder can lead to Anderson localization, inhibiting
the spread of quantum walks, putting them at a disadvantage against their
diffusively spreading classical counterparts. We find that spatial disorder of
the most general type, i.e., position-dependent Haar random coin operators,
does not lead to Anderson localization but to a diffusive spread instead. This
is a delocalization, which happens because disorder places the quantum walk to
a critical point between different anomalous Floquet-Anderson insulating
topological phases. We base this explanation on the relationship of this
general quantum walk to a simpler case more studied in the literature and for
which disorder-induced delocalization of a topological origin has been
observed. We review topological delocalization for the simpler quantum walk,
using time evolution of the wave functions and level spacing statistics. We
apply scattering theory to two-dimensional quantum walks and thus calculate the
topological invariants of disordered quantum walks, substantiating the
topological interpretation of the delocalization and finding signatures of the
delocalization in the finite-size scaling of transmission. We show criticality
of the Haar random quantum walk by calculating the critical exponent in
three different ways and find 0.52 as in the integer quantum
Hall effect. Our results showcase how theoretical ideas and numerical tools
from solid-state physics can help us understand spatially random quantum walks.Comment: 18 pages, 18 figures. Similar to the published version. Comments are
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Anomalous levitation and annihilation in Floquet topological insulators
Anderson localization in two-dimensional topological insulators takes place via the so-called levitation and pair annihilation process. As disorder is increased, extended bulk states carrying opposite topological invariants move towards each other in energy, reducing the size of the topological gap, eventually meeting and localizing. This results in a topologically trivial Anderson insulator. Here, we introduce the anomalous levitation and pair annihilation, a process unique to periodically driven, or Floquet, systems. Due to the periodicity of the quasienergy spectrum, we find it is possible for the topological gap to increase as a function of disorder strength. Thus, after all bulk states have localized, the system remains topologically nontrivial, forming an anomalous Floquet-Anderson insulator (AFAI) phase. We show a concrete example for this process, adding disorder via on-site potential “kicks” to a Chern insulator model. By changing the period between kicks, we can tune which type of (conventional or anomalous) levitation and annihilation occurs in the system. We expect our results to be applicable to generic Floquet topological systems and to provide an accessible way to realize AFAIs experimentally, without the need for multistep driving schemes
Coherent errors and readout errors in the surface code
We consider the combined effect of readout errors and coherent errors, i.e., deterministic phase rotations, on the surface code. We use a recently developed numerical approach, via a mapping of the physical qubits to Majorana fermions. We show how to use this approach in the presence of readout errors, treated on the phenomenological level: perfect projective measurements with potentially incorrectly recorded outcomes, and multiple repeated measurement rounds. We find a threshold for this combination of errors, with an error rate close to the threshold of the corresponding incoherent error channel (random Pauli-Z and readout errors). The value of the threshold error rate, using the worst case fidelity as the measure of logical errors, is 2.6%. Below the threshold, scaling up the code leads to the rapid loss of coherence in the logical-level errors, but error rates that are greater than those of the corresponding incoherent error channel. We also vary the coherent and readout error rates independently, and find that the surface code is more sensitive to coherent errors than to readout errors. Our work extends the recent results on coherent errors with perfect readout to the experimentally more realistic situation where readout errors also occur
Correlated motion of two atoms trapped in a single mode cavity field
We study the motion of two atoms trapped at distant positions in the field of
a driven standing wave high-Q optical resonator. Even without any direct
atom-atom interaction the atoms are coupled through their position dependent
influence on the intracavity field. For sufficiently good trapping and low
cavity losses the atomic motion becomes significantly correlated and the two
particles oscillate in their wells preferentially with a 90 degrees relative
phase shift. The onset of correlations seriously limits cavity cooling
efficiency, raising the achievable temperature to the Doppler limit. The
physical origin of the correlation can be traced back to a cavity mediated
cross-friction, i.e. a friction force on one particle depending on the velocity
of the second particle. Choosing appropriate operating conditions allows for
engineering these long range correlations. In addition this cross-friction
effect can provide a basis for sympathetic cooling of distant trapped clouds.Comment: 10 pages, 9 figures, accepted for publication in Phys. Rev. A. Minor
grammatical changes to previous versio
Floquet-Anderson localization in the Thouless pump and how to avoid it
We investigate numerically how onsite disorder affects conduction in the
periodically driven Rice-Mele model, a prototypical realization of the Thouless
pump. Although the pump is robust against disorder in the fully adiabatic
limit, much less is known about the case of finite period time , which is
relevant also in light of recent experimental realizations. We find that at any
fixed period time and nonzero disorder, increasing the system size
always leads to a breakdown of the pump, indicating Anderson localization of
the Floquet states. Our numerics indicate, however, that in a properly defined
thermodynamic limit, where is kept constant, Anderson localization
can be avoided, and the charge pumped per cycle has a well-defined value -- as
long as the disorder is not too strong. The critical exponent is not
universal, rather, its value depends on the disorder strength. Our findings are
relevant for practical, experimental realizations of the Thouless pump, for
studies investigating the nature of its current-carrying Floquet eigenstates,
as well as the mechanism of the full breakdown of the pump, expected if the
disorder exceeds a critical value.Comment: 5+5 pages, 4+6 figure
Topological bound states of a quantum walk with cold atoms
We suggest a method for engineering a quantum walk, with cold atoms as
walkers, which presents topologically non-trivial properties. We derive the
phase diagram, and show that we are able to produce a boundary between
topologically distinct phases using the finite beam width of the applied
lasers. A topologically protected bound state can then be observed, which is
pinned to the interface and is robust to perturbations. We show that it is
possible to identify this bound state by averaging over spin sensitive measures
of the atom's position, based on the spin distribution that these states
display. Interestingly, there exists a parameter regime in which our system
maps on to the Creutz ladder.Comment: 17 pages, 16 figure