There is great interest in designing photonic devices capable of disorder-resistant transport and information processing. In this work we propose to exploit three-dimensional integrated photonic circuits in order to realize two-dimensional discrete-time quantum walks in a background synthetic gauge field. The gauge fields are generated by introducing the appropriate phase shifts between waveguides. Polarization-independent phase shifts lead to an Abelian or magnetic field, a case we describe in detail. We find that, in the disordered case, the magnetic field enhances transport due to the presence of topologically protected chiral edge states that do not localize. Polarization-dependent phase shifts lead to effective non-Abelian gauge fields, which could be adopted to realize Rashba-like quantum walks with spin-orbit coupling. Our work introduces a flexible platform for the experimental study of multiparticle quantum walks in the presence of synthetic gauge fields, which paves the way towards topologically robust transport of many-body states of photons.There is great interest in designing photonic devices capable of disorder-resistant transport and information processing. In this work we propose to exploit three-dimensional integrated photonic circuits in order to realize two-dimensional discrete-time quantum walks in a background synthetic gauge field. The gauge fields are generated by introducing the appropriate phase shifts between waveguides. Polarization-independent phase shifts lead to an Abelian or magnetic field, a case we describe in detail. We find that, in the disordered case, the magnetic field enhances transport due to the presence of topologically protected chiral edge states that do not localize. Polarization-dependent phase shifts lead to effective non-Abelian gauge fields, which could be adopted to realize Rashba-like quantum walks with spin-orbit coupling. Our work introduces a flexible platform for the experimental study of multiparticle quantum walks in the presence of synthetic gauge fields, which paves the way towards topologically robust transport of many-body states of photons.

arXiv:1503.07172Physical Review AJan 23, 201724 citations

One-Dimensional Continuous-Time Quantum Walks

Daniel ben‐Avraham, Eric M. Bollt, Christino Tamon

Springer eBooksApr 2, 20070 citations

Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk

Hideaki Obuse, János K. Asbóth, Yuki Nishimura +1 more

Quantum walks, whose dynamics is prescribed by alternating unitary coin and shift operators, possess topological phases akin to those of Floquet topological insulators, driven by a time-periodic field. While there is ample theoretical work on topological phases of quantum walks where the coin operators are spin rotations, in experiments a different coin, the Hadamard operator, is often used instead. This was the case in a recent photonic quantum walk experiment, where protected edge states were observed between two bulks whose topological invariants, as calculated by the standard theory, were the same. This hints at a hidden topological invariant in the Hadamard quantum walk. We establish a relation between the Hadamard and the spin rotation operator, which allows us to apply the recently developed theory of topological phases of quantum walks to the one-dimensional Hadamard quantum walk. The topological invariants we derive account for the edge state observed in the experiment; we thus reveal the hidden topological invariant of the one-dimensional Hadamard quantum walk.Quantum walks, whose dynamics is prescribed by alternating unitary coin and shift operators, possess topological phases akin to those of Floquet topological insulators, driven by a time-periodic field. While there is ample theoretical work on topological phases of quantum walks where the coin operators are spin rotations, in experiments a different coin, the Hadamard operator, is often used instead. This was the case in a recent photonic quantum walk experiment, where protected edge states were observed between two bulks whose topological invariants, as calculated by the standard theory, were the same. This hints at a hidden topological invariant in the Hadamard quantum walk. We establish a relation between the Hadamard and the spin rotation operator, which allows us to apply the recently developed theory of topological phases of quantum walks to the one-dimensional Hadamard quantum walk. The topological invariants we derive account for the edge state observed in the experiment; we thus reveal the hidden topological invariant of the one-dimensional Hadamard quantum walk.

arXiv:1505.03264Physical Review BJul 23, 201579 citations

Complementarity in quantum walks

Andrzej Grudka, Paweł Kurzyński, T. P. Polak +3 more

Abstract The eigenbases of two quantum observables, <?CDATA $\{|a_i\rangle\}_{i = 1}^D$ ?> { | a i ⟩ } i = 1 D and <?CDATA $\{|b_j\rangle\}_{j = 1}^D$ ?> { | b j ⟩ } j = 1 D , form mutually unbiased bases (MUB) if <?CDATA $|\langle a_i |b_j \rangle| = 1/\sqrt{D}$ ?> | ⟨ a i | b j ⟩ | = 1 / D for all i and j. In realistic situations MUB are hard to obtain and one looks for approximate MUB (AMUB), in which case the corresponding eigenbases obey <?CDATA $|\langle a_i |b_j \rangle| \leqslant c/\sqrt{D}$ ?> | ⟨ a i | b j ⟩ | ⩽ c / D , where c is some positive constant independent of D. In majority of cases observables corresponding to MUB and AMUB do not have clear physical interpretation. Here we study discrete-time quantum walks (QWs) on d-cycles with a position and coin-dependent phase-shift. Such a model simulates a dynamics of a quantum particle moving on a ring with an artificial gauge field. In our case the amplitude of the phase-shift is governed by a single discrete parameter q. We solve the model analytically and observe that for prime d the eigenvectors of two QW evolution operators form AMUB. Namely, if d is prime the corresponding eigenvectors of the evolution operators, that act in the D-dimensional Hilbert space ( <?CDATA $D = 2d$ ?> D = 2 d ), obey <?CDATA $|\langle v_q|v^{^{\prime}}_{q^{^{\prime}}} \rangle| \leqslant \sqrt{2}/\sqrt{D}$ ?> | ⟨ v q | v q ′ ′ ⟩ | ⩽ 2 / D for <?CDATA $q\neAbstract The eigenbases of two quantum observables, <?CDATA $\{|a_i\rangle\}_{i = 1}^D$?> { | a i ⟩ } i = 1 D and <?CDATA $\{|b_j\rangle\}_{j = 1}^D$?> { | b j ⟩ } j = 1 D , form mutually unbiased bases (MUB) if <?CDATA $|\langle a_i |b_j \rangle| = 1/\sqrt{D}$?> | ⟨ a i | b j ⟩ | = 1 / D for all i and j. In realistic situations MUB are hard to obtain and one looks for approximate MUB (AMUB), in which case the corresponding eigenbases obey <?CDATA $|\langle a_i |b_j \rangle| \leqslant c/\sqrt{D}$?> | ⟨ a i | b j ⟩ | ⩽ c / D , where c is some positive constant independent of D. In majority of cases observables corresponding to MUB and AMUB do not have clear physical interpretation. Here we study discrete-time quantum walks (QWs) on d-cycles with a position and coin-dependent phase-shift. Such a model simulates a dynamics of a quantum particle moving on a ring with an artificial gauge field. In our case the amplitude of the phase-shift is governed by a single discrete parameter q. We solve the model analytically and observe that for prime d the eigenvectors of two QW evolution operators form AMUB. Namely, if d is prime the corresponding eigenvectors of the evolution operators, that act in the D-dimensional Hilbert space ( <?CDATA $D = 2d$?> D = 2 d ), obey <?CDATA $|\langle v_q|v^{^{\prime}}_{q^{^{\prime}}} \rangle| \leqslant \sqrt{2}/\sqrt{D}$?> | ⟨ v q | v q ′ ′ ⟩ | ⩽ 2 / D for <?CDATA $q\ne

arXiv:2205.05445Journal of Physics A Mathematical and TheoreticalJun 8, 20231 citation

Quantum Walks with Phase Parameters

Marcos Villagra

Mar 24, 20110 citations

Abstract The eigenbases of two quantum observables, <?CDATA

\{|a_i\rangle\}_{i = 1}^D

?> { | a i ⟩ } i = 1 D and <?CDATA

\{|b_j\rangle\}_{j = 1}^D

?> { | b j ⟩ } j = 1 D , form mutually unbiased bases (MUB) if <?CDATA

|\langle a_i |b_j \rangle| = 1/\sqrt{D}

?> | ⟨ a i | b j ⟩ | = 1 / D for all i and j. In realistic situations MUB are hard to obtain and one looks for approximate MUB (AMUB), in which case the corresponding eigenbases obey <?CDATA

|\langle a_i |b_j \rangle| \leqslant c/\sqrt{D}

?> | ⟨ a i | b j ⟩ | ⩽ c / D , where c is some positive constant independent of D. In majority of cases observables corresponding to MUB and AMUB do not have clear physical interpretation. Here we study discrete-time quantum walks (QWs) on d-cycles with a position and coin-dependent phase-shift. Such a model simulates a dynamics of a quantum particle moving on a ring with an artificial gauge field. In our case the amplitude of the phase-shift is governed by a single discrete parameter q. We solve the model analytically and observe that for prime d the eigenvectors of two QW evolution operators form AMUB. Namely, if d is prime the corresponding eigenvectors of the evolution operators, that act in the D-dimensional Hilbert space ( <?CDATA

D = 2d

?> D = 2 d ), obey <?CDATA

|\langle v_q|v^{^{\prime}}_{q^{^{\prime}}} \rangle| \leqslant \sqrt{2}/\sqrt{D}

?> | ⟨ v q | v q ′ ′ ⟩ | ⩽ 2 / D for <?CDATA $q\ne