Deterministic Remote Entanglement of Superconducting Circuits through Microwave Two-Photon Transitions

Large-scale quantum information processing networks will most probably require the entanglement of distant systems that do not interact directly. This can be done by performing entangling gates between standing information carriers, used as memories or local computational resources, and flying ones, acting as quantum buses. We report the deterministic entanglement of two remote transmon qubits by Raman stimulated emission and absorption of a traveling photon wave packet. We achieve a Bell state fidelity of 73%, well explained by losses in the transmission line and decoherence of each qubit.

Figure 1

(a) Minimal logical circuit for remote entanglement. Alice is entangled with the ancillary system C by a Hadamard and a cnot gate. The information propagates to ${\mathrm{C}}^{\prime }$ (green wave) where it is swapped to Bob. (b) Setup schematics and (c) energy level diagram. Two transmon qubits Alice (in dark blue, dressed frequency ${\tilde{\omega }}_{qA}$, see text for details) and Bob (in red, dressed frequency ${\tilde{\omega }}_{qB}$) are dispersively coupled to two resonant cavities (in green, dispersive couplings ${\chi }_{A,B}$). The cavities’ lowest energy modes are frequency matched (${\tilde{\omega }}_{cA}-{\chi }_A\simeq {\tilde{\omega }}_{cB}$) and are strongly coupled to a directional transmission line routing photons from Alice to Bob. By simultaneously driving Alice (Bob) with the detuned purple microwave at ${\omega }_{1A}$ (orange, at ${\omega }_{1B}$) and her cavity with the detuned light blue microwave at ${\omega }_{2A}$ (light pink, at ${\omega }_{2B}$), we drive a Raman-type two-photon transition. For Alice, we choose ${\omega }_{1A}+{\omega }_{2A}={\tilde{\omega }}_{qA}+{\tilde{\omega }}_{cA}-{\chi }_A$ to resonantly drive $|g0⟩\leftrightarrow |e1⟩$ (see (c), left-hand diagram). A photon can eventually be emitted in the line (green wave). The wave packet is shaped by modulating the pump amplitude. This photon is absorbed by Bob by driving $|g1⟩\leftrightarrow |e0⟩$ with ${\omega }_{2B}-{\omega }_{1B}={\tilde{\omega }}_{cB}-{\tilde{\omega }}_{qB}$ (right-hand diagram). After a full photon pitch and catch, the system is in $|e0{⟩}_A|e0{⟩}_B$ (in magenta). After a “half” pitch, the qubits are entangled. The output field from Bob is directed to a high efficiency detection line (JPC) used for calibration and qubit measurements (see text).

Figure 2

Top: Rabi oscillations when driving a two-photon transition for a varying duration $t_{\text{pulse}}$ are recorded in the qubit excited state populations (dots). Alice is initialized in $|g⟩$ and Bob in $|e⟩$. The pump amplitude values ${\xi }_1$ and ${\xi }_2$ are calibrated through Stark-shift measurements (see text and Ref. [36]). As for all population measurements presented in this Letter, statistical error bars are smaller than the dot size. Lines are fits for the two-photon drive strengths $g_s$ and $g_c$. Inset: Pulse sequence schematics. Pump pulse edges are smoothed to 128 ns and the pump 1 pulse is 100 ns longer for accurate control of the drive ramp-up and -down. Bottom: The extracted drive strengths are plotted when varying ${\xi }_2$ (dots, the green stars are from the top panel fits). For each point, the cavity pump frequency is tuned to match the resonance condition Eq. (3). Lines are linear fits of the nonsaturated regions and their slopes are used as a calibration for the release and capture of a shaped photon. Dashed black lines are the drive strengths $|g_{c,s}|=\chi |{\xi }_1{\xi }_2|$ predicted from Stark shift calibration of ${\xi }_{1,2}$ [36].

Figure 3

Release and capture of a shaped photon. (a) Calculated complex amplitude of the two-photon drive strength for Alice (blue) and Bob (red) to transfer a photon in a Gaussian traveling mode centered at Bob’s cavity resonance frequency with deviation $\sigma =800\mathrm{ns}$. These controls are realized by holding ${\xi }_1$ constant and varying ${\xi }_2$ as represented on the right-hand axis. (b) Excited state populations of Alice and Bob during the transfer (dots), measured by interrupting the transfer control pulses after a duration $t$ and subsequent dispersive readout of the qubits. Lines are predictions from cascaded quantum system simulation including all imperfections.

Figure 4

With Alice and Bob initially in $|g⟩$, a pump control signal is applied on Alice to release half a photon (see text) while the capture sequence of Fig. 3 is played for Bob. (a) Measured excited state populations and correlator (with $Z=2|e⟩⟨e|-1$) when interrupting the control pulses after a duration $t$ and then performing simultaneous dispersive readout on both qubits. Plain lines are simulations including all imperfections. Dashed lines are the same simulations assuming perfect final readouts. (b) Real part of the density matrix of the final entangled state measured by tomography of the two-qubit state (colored bars) and reconstituted by simulation (black contours). Fidelity to the Bell state $(|gg⟩+|ee⟩)/\sqrt{2}$ is 73%.