Phase-Space Topology in a Single-Atom Synthetic Dimension

Abstract

We investigate topological features in the synthetic Fock-state lattice (FSL) of a single-atom system described by the quantum Rabi model. By diagonalizing the Hamiltonian, we identify a zero-energy defect state localized at a domain wall of the FSL, whose spin polarization is topologically protected. To address the challenge of applying band topology to the FSL, we introduce a physically motivated and directly measurable topological invariant based on phase-space geometry-the phase-space winding number. We show that the Zak phase, computed using a phase-space parameter, is related to the invariant. This quantized geometric phase reflects the spin polarization of the defect state, demonstrating a bulk-boundary correspondence. The resulting phase-space topology reveals the emergence of single-atom dressed states with contrasting properties-topologically protected spin states and driving-tunable bosonic states. Our results establish phase-space topology as a novel framework for exploring topological physics in single-atom synthetic dimensions, uncovering quantum-unique topological protection distinct from classical analogs.

Figures (8)

• Figure 1: Illustration of (a) the setup under consideration and (b) a domain wall formed for $w \neq 0$. (b) The upper and lower lattices correspond to $v_r=0$ and $v_{cr}=0$, respectively. Each site is labeled by a Fock state $|n\rangle$, with black and gray circles representing the spin states $|e\rangle$ and $|g\rangle$. Solid and dashed lines indicate the stronger and weaker of the intra-cell ($w$) and inter-cell ($v_r$ or $v_{cr}$) couplings, respectively. Red dotted boxes mark the dimerized unit cells. A domain wall appears at $|n\rangle$ when $|v|\sqrt{n}<w<|v|\sqrt{n+1}$, highlighted by a green circle.
• Figure 2: Phase-space trajectories for $\Delta |v| \equiv \bigl||v_r|-|v_{cr}|\bigr|$ at $w=2.5$. Red solid (blue dashed) curves correspond to the regime $|v_r|>|v_{cr}|$ ($|v_r|<|v_{cr}|$). As $\Delta |v|\to 0$, both the center and the radius of the trajectories diverge, while the intersection of the two regimes converges to the location marked by a cross. The phase-space winding number indicates whether the trajectory encloses the origin.
• Figure 3: Noise simulation results averaged over 20 Monte Carlo trials at $w=v_r=1$ and $v_{cr}=0.25$. (a) Fidelity between the exact eigenstates and states obtained by adiabatic preparation. States are adiabatically prepared into $\left|\psi_0\right\rangle$ and $\left|\psi^+_{1,2,3}\right\rangle$ under noise on $v_r$ and $v_{cr}$ with amplitude $\delta v$. Solid and dashed lines show results without ($\Gamma=0$) and with ($\Gamma=0.25$) dissipation, respectively, for the pumping rate $\Gamma$. (b) Spin-flip operation via parameter quenching under fixed noise amplitude $\delta v=0.1$. The parameters $v_r$ and $v_{cr}$ are set to 0 to induce a spin-flip operation due to $w\hat{\sigma}_x$. After the flip, $v_r$ and $v_{cr}$ are restored with their magnitudes also flipped.
• Figure 4: Phase-space trajectories of $\alpha(\phi)$ at $w=2.5$ in the frame rotated by $\phi_{cr}$. Red solid (blue dashed) trajectory corresponds to $|v_r| = 1$, $|v_{cr}| = 0.25$ ($|v_r| = 0.25$, $|v_{cr}| = 1$). The trajectories intersect at $\phi=0$, and the Wigner functions of the defect states at $\phi=\pi/4$ are shown for each regime. Gray dotted lines indicate the squeezing axes, perpendicular to the black dot–dashed lines connecting the phase-space centers $\alpha(\phi)$ between the two regimes.
• Figure 5: Energy spectrum $E^{\pm}_n$ of the Hamiltonian in Eq. (\ref{['eqn:eqn1']}) as a function of $|v_r| - |v_{cr}|$, with $w = 5$. The spectrum is obtained via numerical diagonalization with a Fock-space cutoff $N_{\max} = 3000$, for $\phi = 0$ and $\phi = \pi/2$. The singular region $-0.01 \le |v_r| - |v_{cr}| \le 0.01$, which is excluded from the calculation, is masked out. Each line corresponds to an energy level $E_n$, and the color indicates its sign. Cross markers represent theoretical predictions from Eq. (\ref{['eqn:eqn2']}) for $1 \le n \le 10$, showing good agreement with the numerical results, except near the singular region. When $\phi=\pi/2$, the spectrum displays a distinct feature near the point $|v_r| = |v_{cr}|$, as detailed in the main text.