Stabilizing Entanglement via Symmetry-Selective Bath Engineering in Superconducting Qubits

Bath engineering, which utilizes coupling to lossy modes in a quantum system to generate nontrivial steady states, is a tantalizing alternative to gate- and measurement-based quantum science. Here, we demonstrate dissipative stabilization of entanglement between two superconducting transmon qubits in a symmetry-selective manner. We utilize the engineered symmetries of the dissipative environment to stabilize a target Bell state; we further demonstrate suppression of the Bell state of opposite symmetry due to parity selection rules. This implementation is resource efficient, achieves a steady-state fidelity $\mathcal{F}=0.70$, and is scalable to multiple qubits.

Figure 1

Cavity-mediated qubit coupling. (a) To-scale schematic of aperture-coupled cavities, with weakly coupled input ports ${\kappa }_i^{\mathrm{in}}$, strongly coupled output port ${\kappa }^{\text{out}}$, and intercavity coupling $J$. (b) Transmission spectrum of the coupled cavity modes, showing the symmetric (blue) and antisymmetric (red) peaks. (c) Pump-probe spectroscopy of the coupled qubit modes, exhibiting an avoided crossing. Cavity $B$ is driven at the symmetric cavity resonance conditioned on the qubit state $|gg⟩$, and cavity $A$ is driven at a swept frequency ${\omega }^R$. A dip in transmission (blue) indicates that ${\omega }^R$ is resonant with a qubit mode. The dashed line is a fit of the spectral data, from which we extract $\delta$.

Figure 2

Protocol for cooling to $|0,0,T_0⟩$ via ${\omega }_{-}^c$ (left) and ${\omega }_{+}^c$ (right). Each set of levels outlined in grey represents a rung on the Jaynes-Cummings ladder; the states $|{\psi }_q⟩$ are the coupled qubit states. The illustrated drives (arrows) represent ${\omega }_{|T_0⟩}^d(\pm )$ from Eq. (4). Parity conservation requires that if cooling via ${\omega }_{+}^c$, the drive must be overall symmetric (indicated by blue lines), with $\varphi =\{0,\pi \}$; if cooling via ${\omega }_{-}^c$, the drive must comprise one antisymmetric (red) photon for each symmetric photon. If this condition is met, leakage of cavity photons (purple, $\kappa$) brings the system to the entangled state $|0,0,T_0⟩$. Leakage from the entangled state is dominated by qubit decay (green, ${\mathrm{\Gamma }}_1$).

Figure 3

Symmetry- and frequency-selective bath engineering. (a) The sequence of drive, qubit, and cavity pulses used in the experiment. We apply the bath drive for time $\tau$, then perform one of a set of tomographic rotations followed by a projective readout. (b) Symmetry and frequency dependence of the cooling drive. We plot ${\mathcal{F}}_{|S⟩}-{\mathcal{F}}_{|T_0⟩}$ such that $|S⟩$ is red and $|T_0⟩$ is blue. At the symmetry points $\varphi =0$, $\varphi =\pi$, and ${\overline{n}}_{+}={\overline{n}}_{-}$ [21], the drive is both frequency and symmetry selective. The $|{\psi }_1⟩\leftrightarrow |{\psi }_2⟩$ notation indicates the transition with which the drive is resonant for the labeled band. Transitions between higher cavity occupation states are red-detuned by ${\chi }_{+}=2\pi \times 2.5\mathrm{MHz}$ for the lower-frequency bands, and ${\chi }_{-}=2\pi \times 1.4\mathrm{MHz}$ for the higher-frequency bands.

Figure 4

Cooling dynamics. (a) We prepare $|S⟩$ using $\varphi =\pi$ by cooling via the antisymmetric cavity mode (inset). (b) Similarly, we prepare $|T_0⟩$ using $\varphi =\pi$ via the symmetric cavity mode. In both cases, we fix $\varphi$ and ${\omega }^d$; apply the drive for time $\tau$; and then tomographically reconstruct the resultant joint qubit state. The experimental data are represented as dots; solid lines are fits to a coupled rate equation with rates as noted. The preparation of $|S⟩$ reaches maximum entanglement in approximately $3.5\mu \mathrm{s}$; $|T_0⟩$ reaches maximum entanglement in $3.8\mu \mathrm{s}$.