Steady-state entanglement of two superconducting qubits engineered by dissipation

We present a scheme for the dissipative preparation of an entangled steady state of two superconducting qubits in a circuit quantum electrodynamics (QED) setup. Combining resonator photon loss—a dissipative process already present in the setup—with an effective two-photon microwave drive, we engineer an effective decay mechanism which prepares a maximally entangled state of the two qubits. This state is then maintained as the steady state of the driven, dissipative evolution. The performance of the dissipative state preparation protocol is studied analytically and verified numerically. In view of the experimental implementation of the presented scheme we investigate the effects of potential experimental imperfections and show that our scheme is robust to small deviations in the parameters. We find that high fidelities with the target state can be achieved both with state-of-the-art three-dimensional, as well as with the more commonly used two-dimensional transmons. The promising results of our study thus open a route for the demonstration of a highly entangled steady state in circuit QED.

Figure 1

Setup. The internal levels of two transmons (a) are coupled by coherent interactions (b) to mimic the $\Lambda$ system in (c). Two microwave fields ${\Omega }_{1/2}$ provide virtual couplings of the transitions $|0\rangle \leftrightarrow |1\rangle$ and $|1\rangle \leftrightarrow |2\rangle$ (b) which combine to an effective two-photon drive ${\Omega }_{\mathrm{eff}}$ of the transition $|0\rangle \leftrightarrow |2\rangle$. The transmon-resonator coupling ($g$) is resonant with the upper transition and detuned by ${\delta }_1-{\delta }_{\mathrm{c}}$ from the lower transition. Spontaneous emission ($\gamma$) and resonator photon loss ($\kappa$) are present as decoherence processes. The detunings are defined in the text.

Figure 2

(a),(b) Dissipative state preparation mechanisms, (c) loss mechanisms, and (d) effective lower-level decay processes. (a) Effective resonator decay from $|00\rangle$ into $|S\rangle$ involves coherent coupling to $|S_0\rangle$. $|S_0\rangle$ and $|S\rangle |1\rangle$ are strongly coupled ($\sqrt{2}g$) so that these states hybridize and form dressed states $|S_{\pm }\rangle$ (shown here for a choice of ${\delta }_{\mathrm{c}}={\delta }_2-{\delta }_1$, by which the resonator is resonant with the upper transition). By setting ${\delta }_2=\sqrt{2}g$ the driving from $|00\rangle$ is resonant with the lower dressed state $|S_{-}\rangle$. Population from $|00\rangle$ is thus rapidly excited and decays into $|S\rangle$ via the effective engineered resonator decay ${\kappa }_{+}$. (b) The population of the bright states $|11\rangle$ and $|T\rangle$ is shuffled to $|00\rangle |0\rangle$ by the resonator coupling $g$ and successive resonator decay at an effective rate of ${\kappa }_{\mathrm{eff}}$. (c) The two-photon drive also causes an undesired coupling of the otherwise dark target state $|S\rangle$ to an excited state $|T_1\rangle$. $|T_1\rangle$, in turn, couples to a number of (resonator-) excited states which form dressed states at different energies (indicated) and eventually decay to other states. These can generally be made off-resonant with the drive from $|S\rangle$ by an appropriate choice of the resonator and microwave detunings so that the effective resonator decay ${\kappa }_{-}$ from $|S\rangle$ is suppressed. In addition, since $|S\rangle$ is a dark state of the cavity interaction, the only direct decay mechanism is through the weak qubit decay $\gamma$ to $|00\rangle$. The effective decay processes of the lower levels are summarized in (d).

Figure 3

Dressed state energy vs anharmonicity. An effective two-photon drive ${\Omega }_{\mathrm{eff}}$ from $|00\rangle$ (solid black line) to $|S_{-}\rangle$ (green dashed) is implemented as two consecutive single-photon excitations by two microwave fields, ${\Omega }_1$ and ${\Omega }_2$. The individual drives are mediated by $|T_{+}\rangle$ (short-dashed red), which is a dressed state of $|T\rangle$, and made virtual through a detuning of $\epsilon$ (not shown). The two virtual excitations combine to an effective drive ${\Omega }_{\mathrm{eff}}$ resonant with the transition $|00\rangle \leftrightarrow |S_{-}\rangle$; $|S_{-}\rangle$ then decays into $|S\rangle$ (indicated). The same field couples to the transition from $|S_{-}\rangle$ to the dressed states of $|T_1\rangle$ (dashed-dotted). By an appropriate choice of the oscillator detuning ${\delta }_{\mathrm{c}}$ (here plotted for ${\delta }_{\mathrm{c}}={\delta }_2-{\delta }_1$ with $\omega =20g$), this coupling to $|T_1\rangle$ is made off-resonant (left set of arrows). In the case that $|T_1\rangle$ is hit by the drive (right set of arrows), ${\delta }_{\mathrm{c}}$ needs to be chosen differently to make the coupling off-resonant.

Figure 4

Evolution of the system towards an entangled steady state. Initially prepared in an equal mixture of the lower states ($|00\rangle$: green, dotted line; $|11\rangle$: red, dashed-dotted line; $|T\rangle$: blue, dashed line; $|S\rangle$: purple, solid line) the system evolves towards its steady state which is close to the maximally entangled singlet state of the two transmons. Parts (a) and (b) show the result for an anharmonicity of $A=g$ and $A=4.75g$, respectively. The remaining parameter values are ${\Omega }_{1/2}=g/3$, $\kappa =3g/10$, and $\gamma =g/5400$ for all plots. The values of $\overline{\omega }$, ${\Delta }_{1/2}$, and ${\delta }_c$ are obtained through numerical optimization. The inset in (a) shows the region in the ${\Delta }_A-{\Delta }_g$ plane where the singlet fidelity is high, $F_S>90\%$, for $A=g$. The number on each contour line indicates the preparation time in units of $1/g$. The inset in (b) shows the singlet state fidelity at $t=1000/g$ as a function of anharmonicity.

Figure 5

The fidelity as a function of the difference in resonance frequency $\Delta \omega$ between the two transmons. The parameters are as in Fig. 4 with $t=400/g$ and $A=g$. The inset shows the fidelity when varying the amplitude and phase of the microwave signals.