Nonlinear quantum evolution of a dissipative superconducting qubit

TL;DR

This work tests the linearity of quantum evolution under an effective non-Hermitian Hamiltonian realized via postselection on no-jump trajectories in a dissipative superconducting transmon. By engineering a decay hierarchy and conditioning on no quantum jumps, the excited subspace $\{|e\rangle,|f\rangle\}$ evolves under $H_{\mathrm{eff}}$, exhibiting an anti-Hermitian nonlinear component and a $\mathcal{PT}$-dimer structure with an exceptional point at $\Delta=0$ and $J=\Gamma_e/4$. Through quantum state tomography and careful comparison between a superposition trajectory and the trajectory formed from the superposition of basis trajectories, the authors demonstrate a breakdown of linearity for pure states in the two-level subspace, while density-matrix analysis shows linearity restoration when considering the full three-level system. The results reveal a genuine quantum nonlinear feature of postselected evolution, with dynamics accelerated near the EP and implications for quantum control and information processing despite the overhead of postselection.

Abstract

Unitary and dissipative models of quantum dynamics are linear maps on the space of states or density matrices. This linearity encodes the superposition principle, a key feature of quantum theory. However, this principle can break down in effective non-Hermitian dynamics arising from postselected quantum evolution. We theoretically characterize and experimentally investigate this breakdown in a dissipative superconducting transmon circuit. Within the circuit's three-level manifold, no-jump postselection generates an effective non-Hermitian Hamiltonian governing the excited two-level subspace and an anti-Hermitian nonlinearity. We prepare different initial states and use quantum state tomography to track their evolution under this effective, nonlinear Hamiltonian. By comparing the evolution of a superposition-state to a superposition of individually-evolved basis states, we test linearity and observe clear violations which we quantify across the exceptional-point (EP) degeneracy of the non-Hermitian Hamiltonian. We extend the analysis to density matrices, revealing a breakdown in linearity for the two-level subspace while demonstrating that linearity is preserved in the full three-level system. These results provide direct evidence of nonlinearity in non-Hermitian quantum evolution, highlighting unique features that are absent in classical non-Hermitian systems.

Figures (5)

• Figure 1: Effective non-Hermitian evolution via postselection on no quantum jumps. (a) Three-level transmon manifold with states ${|g\rangle, |e\rangle, |f\rangle}$. The $|e\rangle$ state decays to $|g\rangle$ at rate $\Gamma_e$, while coherent coupling $J$ hybridizes $|e\rangle$ and $|f\rangle$ with detuning $\Delta$. (b) Illustration of non-Hermitian dynamics. Left: initial population distribution. Middle: coherent coupling and decay during time evolution. Right: postselection yielding the effective dynamics within the ${|e\rangle, |f\rangle}$ subspace.
• Figure 2: Setup. (a) Schematic of the superconducting transmon processor including qubits $Q_1$ and $Q_2$, coupler $C_1$ and readout resonators $R_1$ and $R_2$. $Q_1$ is coupled to a dissipation channel leading to energy decay of the state $\ket{e}$. (b) Parametrically activated SWAP readout: population in the $|e\rangle$ state on $Q_1$ is transferred to $Q_2$ via a parametric modulation of the coupler $C_1$. (c) Readout signal histograms display how joint readout of $R_1$ and $R_2$ yields high fidelity state assignment.
• Figure 3: Characterization of the system dynamics before and after postselection. (a) $P(+z)$ versus $J$ and time. (b) The normalized $P^{\mathrm{(n)}}_{+z}$ after postselection. (c) Measurement probabilities from tomographic projections along different axes in the $\{\ket{e}, \ket{f}\}$ manifold along with the ground state population. (d) Postselected data from (c). (e) First passage time (FPT) versus $J$ (blue squares) compared to the expected FPT for Hermitian evolution ($\pi/(2J)$, dashed line).
• Figure 4: Testing linearity of non-Hermitian quantum evolution. (a) The pure state projection of the tomographically reconstructed density matrix $\rho$ is given by its eigenvector with largest eigenvalue $\ket{\psi_1}$ (inset). The main panel displays three trajectories within the Bloch ball (transparent red, blue, green) reconstructed over time $t\in(0,6)\ \mu\mathrm{s}$ from different initial states (respectively $\ket{e},\ \ket{f},\ \ket{+x}$). The pure state projections are shown in solid colors. (b) We compare the trajectory originating in $\ket{+x}$, denoted $\ket{\psi_{+x}(t)}$ to the trajectory reconstructed from a superposition of $\ket{\psi_{e}(t)}$ and $\ket{\psi_{f}(t)}$ trajectories, denoted $\ket{\psi_s(t)}$ and displayed in magenta. (c) OFS for versus time for different values of $J$ and initial state $\ket{+x}$. (d) OFS versus time for initial state $\ket{+y}$. (e) The postselection ratios as a function of time showing oscillations for values above the exceptional point and decay for those below. (f) The time averaged value of the postselection ratio.
• Figure 5: Linearity of classical mixtures. (a) We show the postselected probability data versus time for different initial states; $\rho_e$, $\rho_f$ and $\rho_m$, evolved with $J = 0.5 \ \mathrm{rad.}/\mu\mathrm{s}$. (b) We compare $P_{\rho_\mathrm{m}}^{(\mathrm{n})}(e)$ to the classical superposition $\frac{1}{2} P_{\rho_f}^{(\mathrm{n})}(e)+\frac{1}{2}P_{\rho_e}^{(\mathrm{n})}(e)$. The curves are in disagreement arising from the breakdown of linearity for classical mixtures. (c) We display the probabilities in the un-postselected three state system for the same initial preparations. (d) Comparing the evolution from the $\rho_\mathrm{m}$ preparation to the classical superposition shows excellent agreement, verifying linearity for the three-state dissipative qubit.