Catching Time-Reversed Microwave Coherent State Photons with 99.4% Absorption Efficiency

We demonstrate a high-efficiency deterministic quantum receiver to convert flying qubits to stationary qubits. We employ a superconducting resonator, which is driven with a shaped pulse through an adjustable coupler. For the ideal “time-reversed” shape, we measure absorption and receiver fidelities at the single microwave photon level of, respectively, 99.41% and 97.4%. These fidelities are comparable with gates and measurement and exceed the deterministic quantum communication and computation fault-tolerant thresholds, enabling new designs of deterministic qubit interconnects and hybrid quantum computers.

Figure 1

Experimental design. (a) Natural decay of a photon state from a cavity, showing exponential decay with energy leakage rate $\kappa$. A time-reversed photon wave packet is thus absorbed by a cavity with 100% efficiency, as verified here (box). (b) Experimental setup showing shaped microwave pulse (drive) sent to the cavity (resonator) through a coupler ($\kappa$). An on/off tunable coupler allows separation of the capture, storage, and release processes, as well as calibration. An off-chip circulator separates the drive (brown arrows) coherent-state pulse generated by an arbitrary waveform generator (AWG) from the output $V(t)$ (blue arrows), measured with an amplifier and comprised of reflected (purple arrow, 1) and re-emitted (solid green, 2) signals that interfere destructively. The resonator is capacitively coupled to a superconducting phase qubit (Q) for calibration. (c) Schematic of tunable coupler. Coupler consists of two interwoven inductors with negative mutual inductance $M$ and a SQUID with inductance $L_s$ tuned by coupler bias line to vary $\kappa$.

Figure 2

Measurement protocol and data. Top: pulse sequence starts by driving the resonator with the coupler on using (a) a natural exponentially decreasing or (b) time-reversed exponentially increasing microwave pulse. Following the drive pulse, we close the coupler for 30 ns and reopen it to release the resonator energy. The packets have an energy of one photon and an amplitude time constant $\tau =2/\kappa =100\mathrm{ns}$ and are truncated after length $T=100\mathrm{ns}$ (natural) or 400 ns (time reversed). Middle: measured output voltage versus time. The complex voltages $V(t)$ are averaged over $3\times {10}^6$ runs. For the natural wave packet (a), we see an initial reflection (1) and then resonator re-emission (2) in the capture period, followed by the release of the stored energy. At the arrow, the reflection and re-emission signals cancel and the phase of $V(t)$ changes by $\pi$ (see the Supplemental Material [39]). For the time-reversed packet (b), little microwave power is observed in the capture period, indicating high efficiency. Bottom: blue (${\kappa }_{\mathrm{on}}$) curves show measured energy $E(t)$, obtained by subtracting the average noise power from $|V(t){|}^2$ and then integrating over time. Gold (${\kappa }_{\mathrm{off}}$) curves show the calibration signal of the reflected incoming packet, with the coupler always off. In panel (b) the red (${\kappa }_{\mathrm{on}}[100E(t)]$) curve is a $100\times$ expanded scale, showing small reflected energy. The energy at the end of the capture period, normalized to the total energy at long times, gives the absorption error; we find absorption efficiencies of $(61.0\pm 0.3)\%$ for the natural case and $(99.41\pm 0.06)\%$ for the time-reversed case. Normalizing to the total incident energy (gold, ${\kappa }_{\mathrm{off}}$), we measure a $[95.5\pm 1.2]\%$ ($[97.4\pm 0.6]\%$) storage (receiver) efficiency for the time-reversed wave packet.

Figure 3

Tuning up 99.4% absorption efficiency. As shown in the pulse sequence of Fig. 2, the four parameters which must be tuned up precisely are (a) the pulse length $T$, (b) time constant $\tau$, (c) the detuning of the drive frequency $f_d$ relative to the resonator frequency $f_r$, and (d) the time delay of closing the coupler after stopping the resonator drive. The absorption efficiencies are maximized for an exponentially increasing pulse by $\tau =2/\kappa$, $T\ge 5.3/\kappa$, $f_d=f_r$, and a 4.5 ns delay. In all cases the experimental absorption efficiencies (points) fall off as expected theoretically (lines: see the Supplemental Material [39]). Data are for single-photon drives with $\kappa =1/(50\mathrm{ns})$ and (a) $\tau =2/\kappa$, (b) $T=20/\kappa$, and (c)–(d) $\tau =2/\kappa$ and $T=8/\kappa$, as per the colored slices in Fig. 4. Uncertainties are $\le 0.14\%$ [39].

Figure 4

Absorption efficiencies versus pulse shape. Theoretical and experimental absorption efficiencies (color, grayscale) are plotted versus pulse length $T$ and exponential time constant $\tau$. Experimental data are for single-photon drives with couplings (a), (c) $\kappa =1/(40\mathrm{ns})$, and (e) $\kappa =1/(50\mathrm{ns})$ and have uncertainties $\le 0.4\%$ (see the Supplemental Material [39]). Theory has no fit parameters. (a) Rectangular drive pulse (theory and experiment). Absorption effiencies are maximized at 79% (81.5% theory) for $T\approx 2.5/\kappa$. (b)–(c) For a truncated exponentially decreasing drive pulse, theoretical (b) and experimental (c) absorption efficiencies are maximized as $\tau \to \infty$, the rectangular pulse limit, with efficiencies similar to (a). (d)–(e) For a truncated exponentially increasing drive pulse, theoretical (d) and experimental (e) absorption efficiencies are maximized for $\tau =2/\kappa$ and $T\ge 6/\kappa$ at 99.4% (100% theory).