Topological Spin Texture of Chiral Edge States in Photonic Two-Dimensional Quantum Walks

When a walker goes through a closed path along the edge anticlockwise, its spin polarization winds a loop along a great circle of the Bloch sphere, with the area subtended close to a hemisphere and resultant solid angle 2π(0.992±0.017) sr.

“When a walker goes through a closed path along the edge anticlockwise, its spin polarization also winds a loop along a great circle of the Bloch sphere.The area subtended by the spin evolution loop in Fig. 4(c) is close to a hemisphere [covering the point (0,1,0)] with resultant solid angle of 2π(0.992 ± 0.017) sr.”

The experiment demonstrated control of the topology of edge-spin textures by varying parameters.

“Further, we demonstrated control of topology of the edge-spin textures by varying parameters.”

Antisymmetric spin texture is observed experimentally: spin is approximately polarized in the opposite direction for the walker going through two positions (x,y) and (-x,-y) on the edge.

“As depicted in Fig. 4(a), antisymmetric spin texture is observed experimentally, namely that spin is approximately polarized in the opposite direction for the walker going through two positions (x, y) and (-x, -y) on the edge.”

Opposite spin winding of chiral edge states can be controlled by tuning experimental parameters.

“Opposite spin winding of chiral edge states controlled by tuning experimental parameters is confirmed by our experiment.”

In the photonic 2D quantum walk, dissipationless chiral edge states in a spin-orbit coupled anomalous Floquet topological phase exhibit topological spin texture on boundaries.

“We experimentally observe that dissipationless chiral edge states in a spin-orbit coupled anomalous Floquet topological phase exhibit topological spin texture on boundaries, realized via a two-dimensional quantum walk.”

When a walker travels over the whole closed boundary, its spin evolves and realizes a spin winding in the σx-σz plane of the Bloch sphere.

“We successfully observe that, when a walker travels over the whole closed boundary, its spin evolves and realizes a spin winding in the σ x -σ z plane of the Bloch sphere, which is linked to chiral-like symmetry of our low-energy Hamiltonian.”

The measured probability distributions have similarities to theoretically simulated distributions of 0.943±0.003 and 0.938±0.003.

“For the measured P exp (x, y) shown in Figs.3(a) and (b), their similarities are 0.943 ± 0.003 and 0.938 ± 0.003, respectively.”

For a walker traveling around a closed loop along the boundary in real space, its spin evolves and winds through a great circle on the Bloch sphere.

“Our experiment shows that, for a walker traveling around a closed loop along the boundary in real space, its spin evolves and winds through a great circle on the Bloch sphere, which implies that edge-spin texture has nontrivial winding.”

The topological spin winding is protected by a chiral-like symmetry emerging for the low-energy Hamiltonian.

“This topological spin winding is protected by a chiral-like symmetry emerging for the low-energy Hamiltonian.”

• AssumptionThe authors state this as an assumption: The low-energy Hamiltonian approximation around the gap-closing points accurately describes the edge states.

For the setting in Fig. 1(e), chiral-like symmetry with Cx=cos(5π/6)σx+sin(5π/6)σz and Cy=cos(π/3)σx+sin(π/3)σz holds, with the sign of the coefficient before σx flipped.

“Similarly, for the setting in Fig. 1(e), we have C x = cos 5π 6 σ x + sin 5π 6 σ z and C y = cos π 3 σ x + sin π 3 σ z , where the sign of the coefficient before term σ x gets flipped [56].”

• cites [56]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The low-energy Hamiltonian approximation around the gap-closing points accurately describes the edge states.

The spin vector along the edge depends on θ1 because the chiral-like symmetry operator S is θ1-dependent.

“Notice that here S is θ 1 -dependent; thus, the spin vector along the edge also depends on θ 1 , which is different from the earlier case [36].”

• cites [36]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The low-energy Hamiltonian approximation around the gap-closing points accurately describes the edge states.

The edge states exhibit richer exotic features of topological spin texture that correspond to the symmetry of the low-energy Hamiltonian, aside from the known bulk-edge correspondence.

“Our observation shows that, aside from known bulk-edge correspondence connecting bulk topology and the number of edge states, the edge states exhibit richer exotic features of topological spin texture that correspond to the symmetry of the low-energy Hamiltonian.”

The 2D discrete-time quantum walk model is topologically nontrivial when a winding number W defined in time-and-momentum space is nonzero, even though the Chern number of the effective Hamiltonian vanishes.

“Though the Chern number of the effective Hamiltonian vanishes, our 2D discrete-time quantum walk (2DQW) model is topolog-ically nontrivial when a winding number W defined in time-and-momentum space is nonzero [50][51][52][53][54][55].”

• cites [50]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [51]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [52]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

When the low-energy Hamiltonian has chiral-like symmetry, the corresponding chiral edge mode exhibits topological spin texture over the one-dimensional closed boundary.

“Furthermore, when the low-energy Hamiltonian has chiral-like symmetry, the corresponding chiral edge mode exhibits exotic topological spin texture over the one-dimensional (1D) closed boundary [36], which is observed experimentally here for the first time.”

• cites [36]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Having chiral-like symmetry, the low-energy edge states on two opposite edges are mutually orthogonal eigenstates of Cx.

“Having the chiral-like symmetry, the edge states with low energies on two opposite edges are the mutually orthogonal eigenstates of the C x , akin to the end states of a 1D chiral TI [36].”

• cites [36]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The low-energy Hamiltonian approximation around the gap-closing points accurately describes the edge states.

The edge-spin texture has nontrivial winding.

“Our experiment shows that, for a walker traveling around a closed loop along the boundary in real space, its spin evolves and winds through a great circle on the Bloch sphere, which implies that edge-spin texture has nontrivial winding.”

The spin winding number ws is ±1 when the edge-spin texture corresponds to ±45° phase boundaries, respectively.

“w s is ±1 when the edge-spin texture corresponding to ±45 • phase boundaries, respectively (see Supplemental Material [56]).”

• cites [56]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The two experimental settings have opposite spin winding ws because the anticlockwise-moving edge states at the interface between phases 1 and 2 (3 and 4) correspond to 45° (-45°) phase boundaries.

“Because the anticlockwise-moving edge states at the interface between phases 1 ○ and 2 ○ ( 3 ○ and 4 ○) correspond to 45 • (-45 • ) phase boundaries, the two settings have opposite w s as shown in Figs.1(d) and 1(f).”

• AssumptionThe authors state this as an assumption: The low-energy Hamiltonian approximation around the gap-closing points accurately describes the edge states.

In the experiments, with θ2=π/2, the low-energy Hamiltonian satisfies chiral-like symmetry (SH=-HS).

“In our experiments, we set θ 2 = π /2.In this case, our low-energy Hamiltonian satisfies chiral-like symmetry (S H = -HS), the central ingredient for the emergence of topological spin texture [36,56].”

• cites [36]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [56]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The low-energy Hamiltonian approximation around the gap-closing points accurately describes the edge states.

For the setting in Fig. 1(c), chiral-like symmetry with Cx=cos(π/6)σx+sin(π/6)σz (Cy=cos(2π/3)σx+sin(2π/3)σz) holds for the low-energy Hamiltonian with edges along x (y) direction.

“For the setting in Fig. 1(c), chiral-like symmetry with C x = cos π 6 σ x + sin π 6 σ z (C y = cos 2π 3 σ x + sin 2π 3 σ z ) holds for low-energy Hamiltonian with edges along x (y) direction.”

• AssumptionThe authors state this as an assumption: The low-energy Hamiltonian approximation around the gap-closing points accurately describes the edge states.

Chiral edge modes exist on the 1D closed boundary when the topological phases of the inner and outer regions are distinct.

“Chiral edge modes exist on the 1D closed boundary when the topological phases of the inner and outer regions are distinct.”

Two-dimensional anomalous Floquet topological systems exhibit topological spin texture on the boundary.

“Our experiment confirms that two-dimensional anomalous Floquet topological systems exhibit topological spin texture on the boundary, which could inspire novel topology-based spintronic phenomena and devices.”

The edge-spin texture in real space reflects spin-orbit coupling in that the spin changes whenever the walker changes local conducting direction, like turning a corner.

“The edge-spin texture in real space also reflects spin-orbit coupling in that the spin changes whenever the walker changes local conducting direction, like turning a corner.”