Impact of Boundary Conditions on the Double-Kicked Quantum Rotor

TL;DR

The paper addresses how finite momentum-space boundaries influence the Floquet spectrum and the Mean Chiral Displacement (MCD) in the on-resonance Spin-1/2 Double-Kicked Quantum Rotor (DKQR). It combines Floquet analysis with the MCD as a bulk topological probe to compare Open Boundaries, Periodic Boundaries, and the ideal infinite system, linking edge states to winding numbers $W_0$ and $W_\pi$ via bulk-edge correspondence. It finds that Open Boundary Conditions produce edge-localized quasienergy states at $\epsilon=0$ and $\epsilon=\pm \pi$, while Periodic Boundaries alter momentum-space distribution and reduce the MCD plateau; after accounting for mean-momentum shifts, the MCD remains a reliable bulk indicator of the underlying topology in both BCs. The results quantify boundary-induced effects relevant for cold-atom experiments and suggest practical strategies to mitigate them, thereby enhancing the use of MCD as a topological diagnostic in Floquet systems.

Abstract

We study the on-resonance Spin-1/2 Double Kicked Rotor, a periodically driven quantum system that hosts topological phases. Motivated by experimental constraints, we analyze the effects of open and periodic boundary conditions in contrast to the idealized case of infinite momentum space. As a bulk probe for topological invariants, we focus on the Mean Chiral Displacement (MCD) and show that it exhibits a pronounced sensitivity to boundary conditions, which can be traced to the dynamics in momentum space. Under open boundaries, states that would otherwise extend freely become localized at the edges of the finite momentum space, forming quasienergy edge states. While the bulk response measured by the MCD is strongly affected once the evolving wave packet reaches the boundaries, the persistence of these edge states still reflects the bulk-edge correspondence and provides reliable signatures of topological transitions.

Figures (4)

• Figure 1: (Left panel) Quasienergy spectrum of the antiresonant QKR under open (red) and periodic (blue) BCs. The eigenenergies are plotted against their eigenstate index, which is chosen such that the quasienergies are sorted in ascending order. While OBCs yield flat bands at $\epsilon = 0, \pm 0.5\pi$, PBCs introduce discrete quasienergies that deviate from this flat structure. For the sake of legibility, only every fifth quasienergy is plotted, as well as all deviating quasienergies. (Right panel) Occupation probability of the eigenvectors corresponding to the deviating quasienergies under PBCs. Only the eigenvectors with index 27 (dark blue) and 28 (light blue) are shown; the remaining two exhibit the same edge-localized distribution.
• Figure 2: (Left panel) Quasienergy spectrum of the on resonant DKQR as a function of $k_2$ under open (red) and periodic (blue) BCs. Edge states appear at $\epsilon = 0, \pm \pi$ only for OBCs. The pairs of numbers in the areas separated by the black dotted lines are the winding numbers and uniquely define the topological phases. (Middle panel) Localization of the eigenvectors corresponding to the edge states of the quasienergy spectrum at $\epsilon = 0$ and $\epsilon = \pm \pi$. (Right panel) Edge states appear only under OBCs, and localize at the boundaries of the momentum basis, consistent with the bulk-edge correspondence. The gray bars show the occupation probabilities of all six edge states, illustrating their boundary localization; overlaps occur where several edge states occupy the same momentum class $n$. To resolve these overlaps and analyze the symmetry structure of the edge states more clearly, the insets display the occupation probabilities of each edge state individually in distinct colors, with bars at the same $n$ plotted side-by-side. The insets reveal that the edge states form antisymmetric pairs given by the indices (0,1), (60,61), and (120,121).
• Figure 3: MCD under open (red solid line) and periodic (blue dotted line) boundary conditions, as well as for the ideal case (black dashed line). The inset shows the MCD for the Spin-$\uparrow$- and -$\downarrow$-component separately. We see that the $\downarrow$-component dominates the behavior of the MCD after $k_2 \approx 1.7\pi$. The initial state is chosen as $\ket{\psi_0} = \ket{\delta_{n,0}} \otimes \ket{\uparrow}$.
• Figure 4: Comparison of the probability distribution of the $\ket{\downarrow}$-component under PBCs minus that of the ideally infinite system. We see a shift of the occupation probability towards positive momenta for PBCs compared to the ideal case. The mean momentum is plotted in black. Again, the initial state is chosen as $\ket{\psi_0} = \ket{\delta_{n,0}} \otimes \ket{\uparrow}$. As shown in the inset, after $t \approx 8$ kicks, the mean momentum shifts towards positive values as well. Notice that the spin-down component contributes negatively to the MCD, see its definition in Eq. \ref{['eq: MCD definition']}, explaining the drop at $k_2 \gtrsim 1.7\pi$ in Fig. \ref{['fig:3']}.