Non-Hermitian topology in a single driven-dissipative Kerr-Cat qubit

Abstract

The intriguing physical phenomena associated with exceptional points have established non-Hermitian physics as a frontier of modern research. Recent investigations have extended non-Hermitian physics into the fully quantum domain. However, existing studies predominantly concentrate on discrete-variable quantum systems, while non-Hermitian quantum effects in continuous-variable encoded systems remain largely unexplored. In this work, we investigate the exceptional structure for a driven-dissipative Kerr-cat qubit, realized with a Kerr nonlinear resonator. We find that the dissipation leads to a bidirectional jump between the two basis states of the cat qubit, which is in distinct contrast with the unidirectional jump associated with normal two-level systems. The competition between this jump and a single-photon drive gives arise to the emergence of third-order Liouvillian exceptional points (LEP3s), each corresponds to a crossing point of two lines of LEP2s. We further show that the LEP3 can exhibit the topological character of the Hamiltonian EP3s, which cannot be realized with a single qubit. Our work opens the possibility of realizing non-Hermitian phenomena with continuous-variable quantum systems.

Figures (3)

• Figure 1: Kerr-cat qubit representation and effects of single-photon loss. (a) Bloch sphere representation of the protected Kerr-cat qubit in the large-$\alpha$ limit. Colored markers indicate the cardinal points corresponding to the encoded states, alongside their respective Wigner functions. The Z-basis states are defined as $|\pm Z\rangle=|\mathcal{C}_{\alpha}^{\pm}\rangle=\mathcal{N}_{\alpha}^{\pm}(|\alpha\rangle\pm|-\alpha\rangle)$, and the Y-basis states as $|\pm Y\rangle=|\mathcal{C}_{\alpha}^{\mp i}\rangle=(|\mathcal{C}_{\alpha}^{+}\rangle\pm i|\mathcal{C}_{\alpha}^{-}\rangle)/\sqrt{2}$. (b) The dominant source of noise in the resonator is single-photon loss, resulting from coupling to a bath via single-photon exchange. This dissipation induces random bit flips between the states $|\mathcal{C}_{\alpha}^{\pm}\rangle$, as described by $a=\alpha(p|\mathcal{C}_{\alpha}^-\rangle\langle\mathcal{C}_{\alpha}^+|+p^{-1}|\mathcal{C}_{\alpha}^+\rangle\langle\mathcal{C}_{\alpha}^-|)$.
• Figure 2: Topology of the Liouvillian superoperator. (a) Solid curves depict second-order Liouvillian exceptional points (LEP2s), along which pairs of eigenvectors coalesce. Red dots indicate third-order exceptional points (LEP3s). The solid orange circle (enclosing a single LEP3) and dashed green circle (excluding LEP3s) indicate Winding number calculation contours in parameter space. (b)(c) Winding number of the resultant vector. (b) Trajectory of the normalized resultant vector $\mathcal{R}_{\rm N} \equiv (\mathcal{R}_1 + i\mathcal{R}_2)/|\bm{\mathcal{R}}|$ under parameter variation along $\mathcal{C}_\phi$. The $x$, $y$ and $z$ axes represent $\mathcal{R}_1/|\bm{\mathcal{R}}|$, $\mathcal{R}_2/|\bm{\mathcal{R}}|$ and $\phi/2\pi$, respectively. The red trajectory shows the evolution of $\mathcal{R}_1$ and $\mathcal{R}_2$ as $\phi$ varies. Its projection onto the rescaled $\mathcal{R}_1$-$\mathcal{R}_2$ plane forms a closed circle (solid blue line), indicating a topological winding number $|\mathcal{W}_{\bm{\mathcal{R}}}| = 1$. (c) In contrast, when the projection does not form a closed circle, the winding number is $|\mathcal{W}_{\bm{\mathcal{R}}}| = 0$. The starting point and end point are marked by solid orange dots. Solid arrows indicate counterclockwise winding, while hollow arrows indicate clockwise winding.
• Figure 3: Validation of the Liouvillian dynamics. Color represents the fidelity $F(\rho_{\text{H}},\rho_{\text{L}})$ between states evolved under the Lindblad master equation and the Liouvillian approximation. (a) Initial state $|\mathcal{C}_{\alpha}^{+}\rangle$ with $\varepsilon/2\pi = 0.74$ MHz. (b) Initial state $|\mathcal{C}_{\alpha}^{+}\rangle$ without single-photon driving. (c) Initial state $|\alpha\rangle$ with $\varepsilon/2\pi = 0.74$ MHz. (d) Initial state $|\alpha\rangle$ without single-photon driving. Numerical parameters are taken from Ref. 1: $K/2\pi = 6.7$ MHz, $\kappa = 1/15.5$ MHz, $|\alpha|^2 = 2.3$.