Dynamical topology of chiral and nonreciprocal state transfers in a non-Hermitian quantum system

Abstract

The fundamental concept underlying topological phenomena posits the geometric phase associated with eigenstates. In contrast to this prevailing notion, theoretical studies on time-varying Hamiltonians allow for a new type of topological phenomenon, known as topological dynamics, where the evolution process allows a hidden topological invariant associated with continuous flows. To validate this conjecture, we study topological chiral and nonreciprocal dynamics by encircling the exceptional points (EPs) of non-Hermitian Hamiltonians in a trapped ion system. These dynamics are topologically robust against external perturbations even in the presence dissipation-induced nonadiabatic processes. Our findings indicate that they are protected by dynamical vorticity -- an emerging topological invariant associated with the energy dispersion of non-Hermitian band structures in a parallel transported eigenbasis. The symmetry breaking and other key features of topological dynamics are directly observed through quantum state tomography. Our results mark a significant step towards exploring topological properties of open quantum systems.

Figures (16)

• Figure 1: Generation of chirality and nonreciprocity in a dissipative trapped-ion qubit. (A) Schematic diagram of the energy levels of a $^{171}$Yb$^{+}$ ion. The five involved levels include $|F=0, m_F =0\rangle$ and $|F=1, m_F =0, \pm1\rangle$ in the electronic ground state $^2 S_{1/2}$, and $|F=0, m_F =0\rangle$ in the electronic excited state $^2 P_{1/2}$. (B) Encircling paths in the parameter space (coupling strength $J$ and detuning $\Delta$) and the state evolution trajectories projected onto the eigenvalues' Riemann sheets, starting in the $\mathcal{PT}$-symmetric ($\mathcal{PTS}$) regime of the $\mathcal{PT}$ Hamiltonian. The solid (dashed) trajectory in (B) denotes the clockwise (counterclockwise) evolution of $|\alpha_A(0)\rangle$. (C) Chiral symmetry (depicted by red and green curves in the left panel) and broken chiral symmetry (depicted by red and blue curves in the left panel) serve as mathematical criteria for determining chiral dynamics. Time-reversal symmetry (illustrated by red and green curves in the right panel) and broken time-reversal symmetry (illustrated by red and blue curves in the right panel) serve as mathematical criteria for determining nonreciprocal dynamics.
• Figure 2: The time-varying evolutionary state $|\psi(t)\rangle$ starts from either the $\mathcal{PTS}$ or $\mathcal{PTB}$ regime with the $\mathcal{PT}$ Hamiltonian. The solid (dashed) box depict the state evolution trajectories in the visual four-dimensional picture of the eigenspectrum as a function of the $\Delta$ and $J$, with the colors on the Riemann sheet corresponding to the respective imaginary (real) values. Clockwise and counterclockwise encircling the EP starting from $|\alpha_{A}(0)\rangle$ ($|\alpha_{B}(0)\rangle$ ) and $|\beta_{A}(0)\rangle$ ($|\beta_{B}(0)\rangle$) are represented by trajectory 1-4 in (A), (C), (E) and (G) (trajectory 5-8 in (I), (K), (M) and (O)). Figs. (B), (D), (F) and (F) ((J), (L), (N) and (P)) represent the overlap between the two instantaneous eigenstates and the evolutionary state $\langle \alpha_{A}(t)|\psi(t)\rangle$, $\langle \beta_{A}(t)|\psi(t)\rangle$ ($\langle \alpha_{B}(t)|\psi(t)\rangle$, $\langle \beta_{B}(t)|\psi(t)\rangle$ ) from $\mathcal{PTS}$ ($\mathcal{PTB}$) regime, where the nonadiabatic dynamics emerge in the cyan shaded regions. Circles and lines represent experimental measurements and numerical simulation, respectively.
• Figure 3: Investigation of the nonadiabatic effects with clockwise encircling the EP starting from the initial eigenstate $|\alpha_A(0)\rangle$ (trajectory 1). (A-B) The dependence of the ratio $\tau_{\textrm{crit}}/T$ on the encircling period and radius. (C-D) The change in the fidelity of state transfer with varying periods and radii. The contour map and solid lines denote the simulation results, while the circles represent the experimental data. (E-F) The overlap $\langle\alpha_{A}(t)|\psi(t)\rangle$ (solid line) and $\langle\beta_{A}(t)|\psi(t)\rangle$ (dash line) with varying period for the initial state (E) $|\alpha_A(0)\rangle$ and (F) $|\beta_A(0)\rangle$, respectively. The blue, red, and black circles correspond to $T=16.67~\mu s$, $100~\mu s$, and $250~\mu s$, respectively, which match the vertical lines in (A). (G-H) The overlap with varying radii for the initial state (E) $|\alpha_A(0)\rangle$ and (F) $|\beta_A(0)\rangle$, respectively. The blue, red, and black squares correspond to $r =$0.003 MHz, 0.008 MHz, and 0.03 MHz, respectively, which correspond to the vertical lines in (B).
• Figure 4: Investigation of the robustness of dynamically encircling an EP. (A, C, E and G) The color map shows the overlap between the instantaneous eigenstate and the evolutionary state under varying noise intensity. (B, D, F, and H). The experimental results corresponding to the three dashed lines in (A, C, E, and G) demonstrate robustness against noise, as all data match the predicted values (solid lines). The insets in (B) and (F) display clockwise encircling trajectories in the two-dimensional parameter space $\{\Delta, J\}$ with random noise intensities $ir =$ 0 (black), 0.3 (red), and 0.5 (blue), respectively. In Fig. A-D, the encircling starts from point A in the $\mathcal{PTS}$ regime, while in Fig. E-H, it starts from point B in the $\mathcal{PTB}$ regime.
• Figure 5: Experimental verification of the piecewise strategy with the initial eigenstate $|\alpha_A(0)\rangle$ in the $\mathcal{PT}$-symmetric regime. (A) The overlap between the evolutionary state $|\psi(t)\rangle$ and instantaneous eigenstate $|\alpha_A(t)\rangle$. (B) The overlap between the evolutionary state $|\psi(t)\rangle$ and instantaneous eigenstate $|\beta_A(t)\rangle$. The experimental results with $N=$10, 20, 30, 50, 80 and 100 are represented by black, red, green, blue, cyan, and magenta squares, respectively. The yellow shaded region represents the numerical calculation with $5\%$ uncertainty in the density matrix of the measured state $|\psi(t)\rangle$, accounting for fluctuations in the dissipation strength and the coupling strength.