Catching Shaped Microwave Photons with 99.4% Absorption Efficiency
J. Wenner, Yi Yin, Yu Chen, R. Barends, B. Chiaro, E. Jeffrey, J. Kelly, A. Megrant, J. Y. Mutus, C. Neill, P. J. J. O'Malley, P. Roushan, D. Sank, A. Vainsencher, T. C. White, Alexander N. Korotkov, A. N. Cleland, John M. Martinis
CCatching Time-Reversed Microwave Photons with 99.4% Absorption Efficiency
J. Wenner , Yi Yin , , Yu Chen , R. Barends , B. Chiaro , E. Jeffrey , J. Kelly , A.Megrant , , J. Y. Mutus , C. Neill , P. J. J. O’Malley , P. Roushan , D. Sank , A.Vainsencher , T. C. White , Alexander N. Korotkov , A. N. Cleland , and John M. Martinis ∗ Department of Physics, University of California, Santa Barbara, California 93106, USA Department of Physics, Zhejiang University, Hangzhou 310027, China Department of Materials, University of California, Santa Barbara, California 93106, USA and Department of Electrical Engineering, University of California, Riverside, California 92521, USA
We demonstrate a high efficiency deterministic quantum receiver to convert flying qubits to logicqubits. We employ a superconducting resonator, which is driven with a shaped pulse throughan adjustable coupler. For the ideal “time reversed” shape, we measure absorption and receiverfidelities at the single microwave photon level of, respectively, 99.41% and 97.4%. These fidelitiesare comparable with gates and measurement and exceed the deterministic quantum communicationand computation fault tolerant thresholds.
PACS numbers: 85.25.Cp, 42.50.Ct, 03.67.Lx
Systems coupling qubits and cavities provide a naturalinterface between fixed logic qubits and flying photons.They have enabled a wide variety of important advancesin circuit quantum optics, such as generating novel pho-ton states [1–3] and developing a toolbox of quantum de-vices [4–7]. Hybrid quantum devices and computers [8, 9]can be implemented between superconducting coplanarwaveguides and micromechanical oscillators or supercon-ducting, spin, and atomic qubits [10–14]. For such imple-mentations, it can be advantageous to employ determin-istic quantum state transfer [15, 16], which needs highlyefficient quantum transmitters and receivers.For deterministic quantum networks [17], it is partic-ularly challenging to convert flying qubits to station-ary qubits, since absorbing naturally shaped emissionhas a maximum fidelity of only 54% [18, 19]. Theoreti-cal protocols reaching 100% efficiency rely upon sculpt-ing the time dependence of photon wavepackets and re-ceiver coupling [20–22]. To accomplish this, recent exper-iments have developed transmitters with adjustable cou-pling [23–25]. However, experimental reception fidelitieshave reached a maximum of only 88% for optical photons[26] and 81% for microwave photons [14]. These are wellbelow fidelities required for fault-tolerant deterministicquantum communication (96%, [27]) and computation(99.4%, [28]).We demonstrate here a quantum receiver that absorbsand stores photons with a surprisingly high fidelity abovethese thresholds. We classically drive a superconduct-ing coplanar waveguide resonator through an adjustablecoupler, which we use as an on/off switch, with a par-ticularly simple “time reversed” photon shape. At thesingle microwave photon level, we measure an absorptionefficiency of 99.4% and a receiver efficiency of 97.4%.These efficiencies are comparable to fidelities of goodlogic gates and measurements [29–31], enabling new de-signs for quantum communication and computation sys-tems.
FIG. 1: (Color online) Experimental design. (a) Naturaldecay of a photon state from a cavity, showing exponentialdecay with energy leakage rate κ . A time-reversed photonwavepacket is absorbed by a cavity with 100% efficiency. (b)Experimental setup showing shaped microwave pulse (drive)sent to the cavity (resonator) through a coupler ( κ ). An on/offtunable coupler allows separation of the capture, storage andrelease processes, as well as calibration. An off-chip circula-tor separates the drive (brown arrows) from the output V ( t )(blue arrows), measured with an amplifier and comprised ofreflected (purple arrow, 1) and retransmitted (solid green, 2)signals that interfere destructively. The resonator is capaci-tively coupled to a superconducting phase qubit (Q) for cal-ibration. (c) Schematic of tunable coupler. Coupler consistsof two interwoven inductors with negative mutual inductance M and a SQUID with inductance L s tuned by coupler biasline to vary κ . The protocol we implement relies on time-reversal sym-metry [20, 32] between resonator energy absorption andemission. A resonator emits a photon wavepacket whichnaturally decays exponentially in amplitude, as shown inFig. 1(a). By time reversal symmetry, the resonator willthus absorb an exponentially increasing wavepacket with a r X i v : . [ qu a n t - ph ] N ov FIG. 2: (Color online) Measurement protocol and data. (Top) Pulse sequence starts by driving the resonator with the coupleron using (a) a natural exponentially decreasing or (b) time-reversed exponentially increasing microwave pulse. Following thedrive pulse, we close the coupler for 30 ns and reopen it to release the resonator energy. The packets have an energy of onephoton and an amplitude time constant τ = 2 /κ = 100 ns and are truncated after length T =100 ns (natural) or 400 ns (timereversed). (Middle) Measured output voltage versus time. For the natural wavepacket (a), we see an initial reflection (1)and then resonator retransmission (2) in the capture period, followed by the release of the stored energy. For time-reversedpacket (b), little microwave power is observed in the capture period, indicating high efficiency. The complex voltages V ( t )are averaged over 3 × runs. (Bottom) Blue ( κ on) curves show measured energy E ( t ), obtained by subtracting the averagenoise power from | V ( t ) | and then integrating over time. Gold ( κ off) curves show calibration signal of the reflected incomingpacket, with the coupler always off. In (b) the red ( κ on[100 E ( t )]) curve is a 100 × expanded scale, showing small reflectedenergy. 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 ± ± κ off), we measure a [95.5 ± ± perfect efficiency [18, 19, 33].More physically, the resonator perfectly absorbs a trav-elling wave packet if destructive interference occurs be-tween signals reflected off the coupler and retransmitted(leaked) out of the resonator. This is readily obtainedfor an exponentially increasing wavepacket, as calculatedboth classically [22, 33, 34] and quantum mechanically[18, 19], since the reflected and retransmitted signals in-crease in time together with opposite phase. For perfectdestructive interference during the entire pulse, one mustmatch the frequencies and set the drive amplitude timeconstant τ to 2 /κ for coupling leakage rate κ . Underthese conditions, the absorption efficiency equals 1 − e − κT for a pulse of duration T [34]. The absorption efficienciesreach 99.4% for κT ≥ . T → ∞ .Using this idea, other protocols also permit perfect can-celation by temporally varying both the wavepacket and κ [22].Imperfect destructive interference between the re-flected and retransmitted waves results in lower absorp-tion efficiencies. The maximal absorption efficiency isonly 81.45% [34] for a drive pulse with rectangular orexponentially decreasing amplitude because initially thereflected signal has greater amplitude than the retrans-mitted signal, while at long times, retransmission domi-nates. Reflections also do not cancel out retransmissionif the resonator couples significantly to modes besidesthe drive; this results in a decreased efficiency as seen inRefs. [14, 35, 36].To experimentally achieve the strong coupling neces-sary for complete destructive interference, we employ atunable inductive coupler, shown in Fig. 1(b,c), throughwhich we drive a 6.55 GHz superconducting coplanarwaveguide resonator. The coupling is given by a mu- FIG. 3: (Color online) Tuning up 99.4% absorption efficiency.As shown in the pulse sequence of Fig. 2(a), the four parame-ters which must be tuned up precisely are (a) the pulse length T , (b) time constant τ , (c) detuning of the drive frequency f d relative to the resonator frequency f r , and (d) time delayof closing coupler after stopping resonator drive. The absorp-tion efficiencies are maximized for an exponentially increasingpulse by τ = 2 /κ , T ≥ . /κ , f d = f r , and 4.5 ns delay. In allcases the experimental absorption efficiencies (points) fall offas expected theoretically (lines [34]). Data are for single pho-ton drives with κ = 1 / (50 ns) and (a) τ = 2 /κ , (b) T = 20 /κ ,(c)-(d) τ = 2 /κ and T = 8 /κ , as per the colored slices inFig. 4(e). Uncertainties are ≤ .
14% [34]. tual inductance modulated by a tunable SQUID (super-conducting quantum interference device) inductance [23]and is calibrated using a superconducting phase qubit[34]. The resulting coupling can be tuned from nega-tive couplings through zero (off) to +1 / (20 ns) in a fewnanoseconds; as the drive pulse is similarly tuned in a fewnanoseconds, the coupling can be adjusted with the drivepulse to ensure reflection cancellation. We choose a cou-pling κ = +1 / (50 ns) for resonator driving and retrans-mission, which dominates over the intrinsic resonator loss κ i ’ / (3 µ s). By turning off the coupler, the absorbedenergy is stored instead of immediately retransmitted.We drive the resonator with a shaped pulse and mea-sure the output voltage V ( t ) versus time t to character-ize the destructive interference. We generate a classicalsingle-photon drive pulse using heterodyne mixing withan arbitrary waveform generator [34] and concurrentlyset the coupling to κ = +1 / (50 ns) to capture the driveenergy. Upon stopping the drive, we idle the coupler at κ off = 0 for 30 ns and then reset the coupling to κ torelease the energy [see pulse sequence in Fig. 2]. Duringthe entire sequence, we measure the complex V ( t ) us-ing two-channel heterodyne detection near the resonator frequency [34], averaging over 3 × repetitions.When the reflection and retransmission interfere de-structively, V ( t ) is comparable to the noise prior to thedrive. This occurs only around 50 ns for an exponentiallydecreasing drive pulse [arrow in Fig. 2(a)]; elsewhere, ei-ther reflection [(1) in the middle panel] or retransmission(2) dominate. In contrast, the destructive interferencelasts the entire drive for a properly shaped exponentiallyincreasing pulse [Fig. 2(b)], implying high absorption ef-ficiency.We quantify this interference by the energy absorp-tion efficiency. The energy measured through time t isproportional to E ( t ) = R t [ | V ( t ) | − N ] dt for averagenoise power N [see bottom panel, Fig. 2]. The absorbedenergy E abs is the difference between the total driveenergy E ( ∞ ) and the near-constant energy E (idle) atthe idle. The absorption efficiency E abs /E ( ∞ ) equals(99.41 ± ± V ( t ) is av-eraged and E ( ∞ ) for E abs = (0 . →
50) photons [34].Note that we calibrate the energy scale by using the qubitto measure E abs [34].To achieve this high 99.4% absorption efficiency, wetune the pulse length T , exponential time constant τ ,drive frequency f d , and the timing of closing the coupler(see Fig. 3). For τ = 2 /κ , longer pulse lengths correspondto higher absorption efficiencies, reaching 99.4% for T ≥ /κ [Fig. 3(a)]. If τ is varied at T = 20 /κ [Fig. 3(b)], theabsorption efficiency reaches the maximal 99.4% within10% of τ = (2 /κ ) but falls to 90% for τ / (2 /κ ) = 2 , / f d about the res-onator frequency f r for a T = 8 /κ , τ = 2 /κ exponen-tially increasing drive pulse [Fig. 3(c)]. The absorptionefficiency is maximized for f d = f r and is at least 90%within ± f r . According to theory, achieving ap-preciable absorption efficiency requires f d = f r to withina linewidth κ/ π =3 MHz [34].To achieve the 99.4% absorption efficiency, we mustdelay closing the coupler relative to turning off the driveeven after calibrating the coupler-resonator timing [34].The absorption efficiency is reduced by 10% when thedelay differs by 3.5 ns from the optimal 4.5 ns [Fig. 3(d)].The efficiency decreases since the entire drive is reflectedif the coupler is closed too early and some of the capturedenergy is retransmitted if the coupler is closed too late.The scaling is linear for delays longer (shorter) than theoptimum due to the sharpness of turning off the drive(closing the coupler). We observe experimental devia-tions from the linear slope for shorter delays, so our cou-pler pulse shaping is nonideal, possibly explaining whythe offset is required.To demonstrate the necessity of appropriate pulseshaping, we measured the absorption efficiencies for FIG. 4: (Color online) Absorption efficiencies vs pulse shape.Theoretical and experimental absorption efficiencies (color,grayscale) are plotted versus pulse length T and exponen-tial time constant τ . Experimental data are for single photondrives with couplings (a,c) κ = 1 / (40 ns) and (e) κ = 1 / (50 ns)and have uncertainties ≤ .
4% [34]. Theory has no fit param-eters. (a) Rectangular drive pulse (theory and experiment).Absorption effiencies are maximized at 79% (81.5% theory)for T ≈ . /κ . (b,c) For truncated exponentially decreasingdrive pulse, theoretical (b) and experimental (c) absorptionefficiencies are maximized as τ → ∞ , the rectangular pulselimit, with efficiencies similar to (a). (d,e) For truncated ex-ponentially increasing drive pulse, theoretical (d) and exper-imental (e) absorption efficiencies are maximized for τ = 2 /κ and T ≥ /κ at 99.4% (100% theory). rectangular [Fig. 4(a)], truncated exponentially decreas-ing [Fig. 4(c)], and exponentially increasing drive pulses[Fig. 4(e)] versus pulse length T and exponential timeconstant τ . All three pulse shapes have similar shapesand hence absorption efficiencies for τ > T . Forrectangular pulses, the maximal absorption efficiency is ∼
79% compared to the predicted 81.5%; this limit alsoprovides the maximal absorption efficiencies for exponen-tially decreasing pulses. The efficiency is reduced be-cause no initial excitation exists for which retransmissioncan cancel the initial reflections [see (1) in middle panel,Fig. 2(a)]. In addition, these experimental efficienciesagree with the theoretical efficiencies [Fig. 4(a),(b),(d)],to within ∼ τ = 2 /κ and T ≥ /κ .However, the absorption efficiencies neglect intrinsicresonator losses κ i . To measure this effect, we drive theresonator with the coupler off ( κ off) and measure the to- tal reflected energy E off. We compare this to the totalmeasured energy E on when we drive the resonator at κ . The fraction of energy not lost is E on /E off=96.1%[Fig. 2(b), bottom panel]. The storage efficiency forthe entire process, (95.5 ± E on /E off times theabsorption efficiency. However, this efficiency includeslosses from κ i during both the capture and release phases.During just the drive, we expect to keep κ/ ( κ + κ i ) =98 .
4% of the energy [34] and actually keep approximately p E on /E off = 98 .
1% assuming near-perfect absorption[14]. This fraction times the absorption efficiency is thereceiver efficiency, which equals (97.4 ± /κ i = 45 µ s [37], we should be able toincrease this receiver efficiency to above 99%. Hence, de-terministic quantum state transfer between macroscopi-cally separated cavities is possible with high fidelity. ACKNOWLEDGMENTS
Devices were made at the UC Santa Barbara Nanofab-rication Facility, part of the NSF-funded National Nan-otechnology Infrastructure Network. This research wasfunded by the Office of the Director of National Intelli-gence (ODNI), Intelligence Advanced Research ProjectsActivity (IARPA), through Army Research Office grantW911NF-10-1-0334. All statements of fact, opinion orconclusions contained herein are those of the authors andshould not be construed as representing the official viewsor policies of IARPA, the ODNI, or the U.S. Government. ∗ Electronic address: [email protected][1] G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H.Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M.Girvin, and R. J. Schoelkopf, Nature (London) , 205(2013).[2] H. Wang et al. , Phys. Rev. Lett. , 060401 (2011).[3] A. F. van Loo, A. Fedorov, K. Lalumi`ere, B.C. Sanders, A. Blais, and A. Wallraff, ScienceDOI:10.1126/science.1244324 (2013).[4] I.-C. Hoi, C. M. Wilson, G. Johansson, J. Lindkvist, B.Peropadre, T. Palomaki, and P. Delsing, New J. Phys. , 025011 (2013).[5] A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, andF. Nori, Phys. Rev. A , 023809 (2013). [6] B. Peropadre, G. Romero, G. Johansson, C. M. Wilson,E. Solano, and J. J. Garc´ıa-Ripoll, Phys. Rev. A ,063834 (2011).[7] A. Reiserer, S. Ritter, and G. Rempe, ScienceDOI:10.1126/science.1246164 (2013).[8] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev.Mod. Phys. , 623 (2013).[9] M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P.Zoller, Phys. Scr. , 014001 (2009).[10] J. Majer et al. , Nature (London) , 443 (2007).[11] S. D. Hogan, J. A. Agner, F. Merkt, T. Thiele, S. Filipp,and A. Wallraff, Phys. Rev. Lett. , 063004 (2012).[12] D. I. Schuster et al. , Phys. Rev. Lett. , 140501 (2010).[13] Y. Kubo et al. , Phys. Rev. Lett. , 140502 (2010).[14] T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Sim-monds, and K. W. Lehnert, Nature (London) , 210-214 (2013).[15] S. Ritter, C. N¨olleke, C. Hahn, A. Reiserer, A. Neuzner,M. Uphoff, M. M¨ucke, E. Figueroa, J. Bochmann, andG. Rempe, Nature (London) , 195 (2012).[16] L. Steffen, Y. Salathe, M. Oppliger, P. Kurpiers, M. Baur,C. Lang, C. Eichler, G. Puebla-Hellmann, A. Fedorov,and A. Wallraff, Nature (London) , 319 (2013).[17] H. J. Kimble, Nature (London) , 1023 (2008).[18] M. Stobi´nska, G. Alber, and G. Leuchs, Euro. Phys. Lett. , 14007 (2009).[19] Y. Wang, J. Min´aˇr, L. Sheridan, and V. Scarani, Phys.Rev. A , 063842 (2011).[20] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi,Phys. Rev. Lett. , 3221 (1997).[21] K. Jahne, B. Yurke, and U. Gavish, Phys. Rev. A ,010301(R) (2007).[22] A. N. Korotkov, Phys. Rev. B , 014510 (2011).[23] Y. Yin et al. , Phys. Rev. Lett. , 107001 (2013). [24] S. J. Srinivasan, N. M. Sundaresan, D. Sadri, Y. Liu, J.M. Gambetta, T. Yu, S. M. Girvin, and A. A. Houck,arXiv:1308.3471.[25] M. Pechal, C. Eichler, S. Zeytinoglu, S. Berger, A. Wall-raff, and S. Filipp, arXiv:1308.4094.[26] M. Bader, S. Heugel, A. L. Chekhov, M. Sondermann,and G. Leuchs, arXiv:1309.6167.[27] A. G. Fowler, D. S. Wang, C. D. Hill, T. D. Ladd, R.Van Meter, and L. C. L. Hollenberg, Phys. Rev. Lett. , 180503 (2010).[28] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N.Cleland, Phys. Rev. A , 032324 (2012).[29] J. M. Chow et al. , Phys. Rev. Lett. , 060501 (2012)[30] J. E. Johnson, C. Macklin, D. H. Slichter, R. Vijay, E. B.Weingarten, J. Clarke, and I. Siddiqi, Phys. Rev. Lett. , 050506 (2012).[31] K. Geerlings, Z. Leghtas, I. M. Pop, S. Shankar, L. Frun-zio, R. J. Schoelkopf, M. Mirrahimi, and M. H. Devoret,Phys. Rev. Lett. , 120501 (2013).[32] Jackson, J. D. Classical Electrodynamics (Wiley, 1999).[33] S. Heugel, A. S. Villar, M. Sondermann, U. Peschel, andG. Leuchs, Laser Phys. , 100 (2010).[34] See Supplemental Material at [URL will be inserted bypublisher] for derivations of theoretical absorption effi-ciencies, details of the experimental setup (including cal-ibrations, independence on drive power, and effect of av-eraging), and derivation of the error analysis.[35] S. A. Aljunid, G. Maslennikov, Y. Wang, H. L. Dao, V.Scarani, and C. Kurtsiefer, Phys. Rev. Lett. , 103001(2013).[36] S. Zhang, C. Liu, S. Zhou, C.-S. Chuu, M. M. T. Loy,and S. Du, Phys. Rev. Lett. , 263601 (2012).[37] A. Megrant et al. , Appl. Phys. Lett. , 113510 (2012). upplement to “Catching Shaped Microwave Photons with 99.4% AbsorptionEfficiency” J. Wenner , Yi Yin , , Yu Chen , R. Barends , B. Chiaro , E. Jeffrey , J. Kelly , A.Megrant , , J. Mutus , C. Neill , P. J. J. O’Malley , P. Roushan , D. Sank , A.Vainsencher , T. C. White , Alexander N. Korotkov , A. N. Cleland , and John M. Martinis Department of Physics, University of California, Santa Barbara, California 93106, USA Department of Physics, Zhejiang University, Hangzhou 310027, China Department of Materials, University of California, Santa Barbara, California 93106, USA and Department of Electrical Engineering, University of California, Riverside, California 92521, USA
THEORETICAL CAPTURE EFFICIENCIES
Here, we calculate the reflected signal and receiver ef-ficiency for tunably-coupled resonator driven with sev-eral different drive pulse shapes. For a time constantsnear 2 /κ , exponentially increasing drive pulses are moreefficient than rectangular and exponentially decreasingpulses. We then show that optimal efficiencies requirethat the resonator have zero intrinsic loss and be drivenat the resonance frequency. Transmission Coefficients
Consider a resonator driven through a tunable couplerwith transmission and reflection coefficients defined inFig. S1. These coefficients are related by [1] t = R R t , | t | R R + | r | = 1 , r = − r ∗ t t ∗ , (S1)where R ’
50 Ω ( R ) is the drive (resonator) impedance, t ( t ) is the transmission coefficient into (from) the res-onator, and r ( r ) is the reflection coefficient on thedrive (resonator) side, where | r | = | r | = | r | .The transmission coefficients are related to the cou-pling quality factor Q c by Q c = ωτ rt | t | R R , where τ rt is the ratio of the resonator energy to the trav-elling wave power and ω/ π = 6 .
55 GHz is the resonatorfrequency. For a λ/ τ rt ≈ π/ω. (S2)With our device, κτ rt (cid:28) κ is within the range [1 / (3 µ s) , / (50 ns)]. Accordingto [2], κ is given by Q c = ω/κ , so | t | τ rt = κ R R . (S3)Note that | t | (cid:28) FIG. S1: Transmission and reflection coefficient notation.An R ’
50 Ω drive transmission line is coupled to an R ’
80 Ω resonator via a tunable coupler (barrier). A isthe incoming signal from the drive line, whereas B is the sig-nal reaching the coupler from the resonator. The coupler hasreflection coefficient r on the drive side, reflection coefficient r on the resonator side, transmission coefficient from thedrive line t , and transmission coefficient from the resonator t . Absorption Efficiencies
Suppose that two signals reach the coupler at time t :an incoming drive A ( t ) and a signal B ( t ) from the res-onator (see Fig. S1). Then, B ( t ) is described by [1]˙ B = r e iφ − τ rt B + t e iφ τ rt A − T B, (S4)where T is the intrinsic resonator energy decay time and φ = τ rt δω − arg( r ) arises from a detuning δω betweenthe drive and resonator frequencies. Note that e iφ = r ∗ | r | e iτ rt δω , so Eq. (S4) simplifies to˙ B = (cid:18) − κ B + t τ rt r ∗ | r | A (cid:19) − T B + δω (cid:18) iB + t r ∗ | r | A (cid:19) (S5)with Eq. (S3) and κτ rt , τ rt δω, | t | (cid:28)
1. Here, the time-dependence is contained in A , B , and potentially x .We initially consider a simple case where the couplingis time-independent, the drive is on resonance, and theresonator intrinsic quality factor is infinite. In this case, a r X i v : . [ qu a n t - ph ] N ov FIG. S2: Receiver efficiency for infinite exponentially de-creasing drive. If the drive is infinitely long ( T → ∞ ) butthe coupler closes at T , the receiver efficiency is maximizedwhen − τ = T = 2 /κ and is then 54.1%. solving for B ( t ) gives B ( t ) = B (0) e − κt/ + e − κt/ t τ rt r ∗ | r | Z t A ( t ) e κt / dt . (S6)The power in the resonator is | B ( t ) | / R . The energy E res in the resonator is thus | B ( t ) | τ rt / R from the def-inition of τ rt . If the resonator is initially unpopulated, E res = 12 R κe − κT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T A ( t ) e κt/ dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (S7)after a time T . The total energy E tot equals the integral E tot = 12 R Z T | A ( t ) | dt (S8)of the drive power | A ( t ) | / R applied for t ∈ (0 , T ).The ratio E res /E tot is defined to be the receiver effi-ciency; note the factors 1 / R cancel. Rectangular Pulses
Suppose the resonator is driven by a rectangular pulse, A ( t ) = A for 0 ≤ t ≤ T = T . Then, the receiverefficiency is E res E tot = 4 κT (cid:16) − e − κT/ (cid:17) , (S9)which is plotted in Fig. 4(a) of the main text. Thisefficiency reaches a maximal value of 81.4529% when T = 2 . /κ . Exponential Pulses
Suppose that the resonator is driven by an exponentialpulse, A ( t ) = A e t/τ for 0 ≤ t ≤ T and the coupler FIG. S3: Receiver efficiency for exponentially increasingdrive. The efficiency is shown versus (a) pulse length T ( τ = 2 /κ ) and (b) time constant τ ( T → ∞ ). The efficiencyis maximized for τ = 2 /κ , T → ∞ . The regime for efficiencies >
90% is shown in Fig. 3(a),(b) of the main text. closed at t = T . Then, the receiver efficiency is E res E tot = 4 (cid:0) κτ (cid:1) (cid:0) e T/τ − e − κT/ (cid:1) (cid:0) κτ (cid:1) (cid:0) e T /τ − (cid:1) . (S10)This is true for both exponentially decreasing ( τ < τ >
0) pulses.The case τ < , T → ∞ corresponds to a source exci-tation which naturally decays via static coupling. Here,the receiver efficiency reduces to E res E tot = 4 (cid:16) κ ( − τ ) (cid:17) (cid:0) e − T/ ( − τ ) − e − κT/ (cid:1) (cid:16) − κ ( − τ ) (cid:17) (S11)and is plotted in Fig. S2. It is maximized when − τ = T = 2 /κ with an efficiency of 4 /e = 54 . τ < , T = T case, a truncated natural drive,the efficiency is plotted in Fig. 4(b) of the main text. Theefficiency is continuous even at τ = − /κ where2 R E res = A T κe − κT . For a constant pulse length, the efficiency is maximizedin the limit τ → −∞ . This limit corresponds to a rect-angular pulse, and E res /E tot reduces to the rectangularvalue. For a constant time constant, the efficiency is max-imized at pulse lengths which increase asymptotically as τ → −∞ to the rectangular maximum of 2 . /κ . Inaddition, the efficiency always approaches zero for infi-nite pulse lengths as the excitation has infinite time todecay.An exponentially increasing ( τ > , T = T ) drivepulse is ideal for static coupling; the receiver efficiencyEq. (S10) is plotted in Fig. 4(d) of the main text. Aspredicted [1], the efficiency for constant pulse lengths ismaximized when τ = 2 /κ , where it reduces to 1 − e − κT and so asymptotically approaches unity as T → ∞ (seeFig. S3(a)). For a fixed time constant, the efficiency is3 FIG. S4: Receiver efficiency versus offset between closingcoupler ( T ) and stopping drive ( T ). The efficiency is shownfor an exponentially increasing pulse with τ = 2 /κ , T = 8 /κ , κ = 1 / (50 ns). The regime for efficiencies >
90% is shown inFig. 3(d) of the main text. The exponential decay for
T < T is due to the particular drive pulse shape, while that for T
5, as shownin Fig. S3(b).
Coupler Closing Delay
Now suppose that the coupler is closed and the drivestopped at slightly different times T and T , respectively.If the drive is stopped first ( T < T ), the receiver effi-ciency is decreased by a factor of e − κ ( T − T ) according toEq. (S6), regardless of the pulse shape. If the coupler isclosed first ( T < T ), the receiver efficiency is changed bya factor which depends on the particular drive pulse. Foran exponential pulse, the receiver efficiency is reduced bya factor of (cid:18) e T/τ − e − κT/ e T /τ − e − κT / (cid:19) , as shown in Fig. S4. Reflections - Destructive Interference
The basis for these high absorption efficiencies is de-structive interference between the reflection r A and re-transmission t B signals. This requires opposite phasesfor these signals. Since the phase of A adds to the phaseof B , we assume without loss of generality that A is pos-itive. The phase of the reflection signal is then given by r , which by Eq. (S1) is the phase of − r ∗ t . The phaseof the retransmission signal is given by the phase of t ,which by Eq. (S1) is the phase of t , and the phase of B , which by Eq. (S6) is the phase of t r ∗ . Thus, the re-flection and retransmission signals are always opposite inphase. Drive Frequency
We now start to consider the effects of relaxing our sim-plifying assumings. First, we detune the drive frequencyby a small δω from the resonator frequency ω while westill assume that the resonator is lossless and the couplingis static. The capture efficiencies can then be solved withEq. (S5). The resonator frequency is included here notonly in δω but also τ rt according to Eq. (S2).For the exponentially increasing pulse, a nonzero de-tuning reduces the maximum receiver efficiency. As thedetuning increases, not only is the maximum receiver effi-ciency reduced but the values of τ and T which maximizethe efficiency are reduced as shown in Fig. S5(a)-(c). Forthe ratio of the efficiency to the efficiency with δω = 0,the frequency width of the peak is independent of T for τ = 2 /κ and T ≥ /κ [Fig. S5(d)] but is proportionalto the coupling κ [Fig. S5(e)]. In addition, the receiverefficiency is maximized in the case of δω = 0.We have also verified numerically that the receiver ef-ficiency is maximized when δω = 0 for rectangular, expo-nentially increasing, and exponentially decreasing drivepulses, even when the pulse lengths and time constantsare varied. The efficiency is also an even function of thedetuning. In addition, this is even true when the intrinsicresonator loss is nonzero. Intrinsic Loss
Suppose the resonator has intrinsic loss characterizedby a decay time T but the detuning is zero and thecoupling is static. Then, Eq. (S5) reduces to˙ B = (cid:18) − κ − T (cid:19) B + t τ rt r ∗ | r | A, where the receiver efficiency can be found as before. Nowsuppose that the time-dependence of A ( t ) can be ex-pressed as a function of t/τ for some time constant τ ;this is valid for both rectangular and exponential pulses.Then, this efficiency incorporating T equals that cal-culated from Eqs. (S7)-(S8) with the following modifica-tions:1. T → κ +1 /T κ T T → κ +1 /T κ T τ → κ +1 /T κ τ FIG. S5: Effect of detuned drive frequency on receiver efficiency. Unless otherwise stated, theory is for the experimental systemwith frequency ω/ π = 6 .
55 GHz, coupling κ = 1 / (50 ns), pulse length T = 8 /κ , and exponential time constant τ = 2 /κ . (a)-(c)The receiver efficiency is plotted vs T and τ for three fractional detunings δω/ω : (a) 0, (b) 0.0002 [1.3 MHz], and (c) 0.0005[3.3 MHz]. (d) The receiver efficiency is plotted vs T and δω/ω . The efficiency is maximized when the drive and resonatorfrequencies are equal with the frequency width effectively independent of T for κT >
3. (e) The receiver efficiency is plottedvs 1 /κ and δω/ω for fixed κT , κτ . The frequency width is proportional to κ .
4. The receiver efficiencies are reduced by κκ + T .For example, the receiver efficiency for an exponentialpulse is E res E tot = 4 (cid:18) (cid:16) κ + T (cid:17) τ (cid:19) (cid:18) e T/τ − e − (cid:16) κ + T (cid:17) T/ (cid:19) (cid:18) (cid:16) κ + T (cid:17) τ (cid:19) (cid:0) e T /τ − (cid:1) κκ + T , which is just Eq. (S10) with these modifications. Notethat (cid:16) κ + T (cid:17) is just the measured decay constant whenperforming a typical T measurement with a given κ .5 EXPERIMENTAL METHODS
Here we describe our experimental setup along withhow we extract absorption efficiencies from the raw data.We then describe how to use the superconducting phasequbit to experimentally calibrate the couplings betweenthe resonator and the transmission line, delay times be-tween the various control lines, and the resonator driveenergy. We demonstrate the independence of the absorp-tion efficiency on the drive power and number of averages.
Experimental Setup and Measurement
The device is the same as was used in Ref. [3], asshown in Fig. S6. This multi-layer device was patternedusing standard photolithography [4]. All metal filmswere sputter-deposited Al, with in situ
Ar ion millingprior to deposition; they were etched with a BCl /Cl inductively-coupled plasma [5]. The phase qubit andSQUID (superconducting quantum interference device)are comprised of Al/AlO x /Al junctions. The phase qubitcapacitor and crossovers were created using a hydro-genated amporphous silicon dielectric.The chip with the resonator is mounted on the 30 mKstage of a dilution refrigerator. The chip is located inan Al sample mount [6] placed in a high-permeabilitymagnetic shield. This protects against magnetic vortexlosses and prevents magnetic fields from other compo-nents, such as circulators and switches, from changingthe device calibration.We generate the resonator drive pulse with a digital-to-analog converter (DAC) [3, 4, 7, 8]. Each board con-tains two channels with a one-gigasample-per-second 14-bit DAC chip; the outputs correspond to the I (cosine)and Q (sine) quadratures. Both outputs are mixed by an IQ -mixer with a continuous ∼ f sb =165 MHz from the resonatorfrequency to prevent spurious residual signal at the LOfrequency from exciting the resonator; we compensatefor this with the I and Q signals. We calibrate out mixerimperfections as explained in [8].To reach the single-photon level, we attenuate thedrive signal and amplify the reflected signal, as shownin Fig. S7. In particular, we attenuate themal noise with20 dB of attenuation at 4 K and 40 dB of attenuation at30 mK. We separate out the reflections on the drive linewith a circulator. The reflections then pass through twocirculators to isolate the resonator from thermal noiseand are then amplified by ∼
35 dB at 4 K with a lownoise HEMT (high-electron mobility transistor) ampli-fier; there is additional amplification at room tempera-ture.We measure the amplified reflection signal with a roomtemperature analog-to-digital converter (ADC) [3, 7]. We
FIG. S6: Photomicrograph of experimental device, show-ing chip (a), superconducting phase qubit (b), and tunablecoupler (c). Qubit, resonator, coupler, coupler bias line, anddrive/measure line are respectively in black, green, dark pur-ple, light purple, and red. Qubit is used for calibrating cou-pler.FIG. S7: Experimental schematic. The resonator is driventhrough a tunable coupler (consisting of a mutual inductance M modulated by a SQUID with inductance L s . The drivesignal is generated by a digital-to-analog converter (DAC),mixed with local oscillator (LO) at 6.4 GHz, and attenuated.Signals reflected from the coupler and leaking from the res-onator are separated from the drive path using a circulator.These output signals then pass through two circulators forthermal noise isolation, are amplified at 4 K by a HEMT am-plifier and then further at room temperature, are heterodynemixed with an IQ mixer using the drive LO, and then aremeasured using an analog-to-digital converter (ADC). FIG. S8: Calibrating coupling. (a) Pulse sequence. We firstreset the coupler while resetting the qubit into the groundstate. We then excite the qubit and swap the excitation intothe resonator. After a time t with the excitation in the res-onator, any remaining excitation is swapped back into thequbit and then measured. (b) The decay time T is plottedversus the coupler bias current, expressed as a flux Φ throughthe SQUID divided by the flux quantum Φ = h/ e . Decaytimes are shown for two coupler reset biases, +6 . (yel-low) and − . (blue). With the − . bias, we definethe coupling κ on to be on when T = 1 /
50 ns; this is wherewe drive the resonator and allow retransmission. When T ismaximized, the coupling κ off is zero and so is turned off. first down-convert the signal with an IQ -mixer using theLO signal. The resulting I and Q quadrature voltages arethen measured versus time using two 500-megasample-per-second 8-bit ADC chips and are then averaged ∼ , ,
000 times.We then filter the raw V ( t ) = I + iQ data. We firstrescale the Q data by an experimentally measured factorto accomodate differneces in electronics between the twoquadratures. We then subtract the average value of V ( t )as measured prior to the drive to remove DC components.We then multiply by e i πf sb to determine the signal at thedrive frequency. To remove crosstalk signals, we digitallyapply a sharp low-pass filter at 150 MHz.In calculating the energy R | V ( t ) | dt in each portionof the pulse sequence, any noise appears to be additionalenergy. To subtract out this spurious contribution, ourprocedure (as rigorously derived in the “Error Analysis”section) involves: • Calculate the energy prior to driving the resonator,where there is no signal
FIG. S9: Correcting for inter-line delays. (a) To measurethe coupler-qubit delay t cq , we excite the qubit, swap theexcitation into the resonator detuned by opening the coupler,and measure the qubit. (b) To measure the resonator-couplerdelay t rc , we drive the resonator and open the coupler for thesame duration but offset by t rc . We then swap any resonatorexcitation to the qubit, which is then measured. • Rescale the noise energy to determine the noise en-ergy in the desired integration region • Subtract this noise energy from the total measuredenergy in a desired region.
Calibrating Coupling & Delays
The tunable coupler employs a tunable mutual induc-tance between the drive and resonator [3]. The mutualinductance M consists of two interwound coils whichare galvanically connected, as shown in Fig. S6(c) andFig. S7. From this connection, a SQUID (superconduct-ing quantum interference device) is attached with tuninginductance L s . This tuning arises from a flux inducedby an on-chip coupler bias current which is externallygenerated.We calibrate the coupling κ ∝ ( M − L s ) [3] in termsof this current with a superconducting phase qubit ca-pacitively coupled to the resonator [3, 9, 10]. Using thequbit, we generate a single photon and swap it to theresonator [Fig. S8(a)]. We then apply the desired couplerbias for varying times, after which we swap to the qubitand measure the residual excitation. From the decay ofthe excitation probability P e , we extract the resonator7 FIG. S10: Calibrating drive energy. (a) With the coupler open, we drive the resonator with various drive amplitudes. We thenswap any excitation to the qubit for varying times t swap and measure the qubit. (b) The qubit excited state ( | e i ) probability isshown versus the drive amplitude and t swap . For these pulses, we used a 1 µ s exponentially increasing pulse with τ = 2 /κ . (c)Probability for the resonator to be in the n th Fock state, fit to the data in (b), versus drive amplitude. States with n > decay time T , shown in Fig. S8(b). At the bias maximiz-ing T , the coupling is zero and hence closed, as verifiedby trying to drive the resonator; here, T is the intrinsicresonator decay time 1 /κ i ’ µ s since 1 /T = κ + κ i .We define the coupler to be open and drive the resonatorwhen T ’
50 ns.However, the open coupling can range from 1/(50 ns)to 1/(30 ns), since the SQUID has multiple potentialwells, each with a different coupling (see Fig. S8(b)). Wereproducibly select a particular well by adding a couplerreset pulse prior to all pulse sequences.Adjusting the SQUID to tune the coupler modulatesthe resonator inductance to ground [3]. This adjusts theresonator frequency by ∼
20 MHz between the openedand closed biases. Hence, opening the coupler to detunethe resonator blocks swaps between the qubit and res-onator as tuned with the coupler closed.We use this to tune the temporal delay t cq between thequbit and coupler [Fig. S9(a)]. We first excite the qubit,swap the excitation to the resonator while opening thecoupler, and measure the qubit. As we vary t cq , thetime for which the qubit excited state probability P e ismaximized is the actual delay.To calibrate the delay t rc between the resonator driveand the coupler, we employ the sequence in Fig. S9(b).Here, we drive the resonator with a many-photon pulse to ensure resonator excitation. For this time but offsetby t rc , we open the coupler. Any induced resonator ex-citation is then swapped into the qubit and measured.Hence, the actual delay is when P e is maximized. As ex-plained in the main text, we verify this timing by varyingwhen the coupler is closed relative to stopping the driveand maximizing the absorption efficiency. Drive Energy
We use the qubit to calibrate the resonator drive en-ergy in terms of the drive amplitude [3, 11]. We varythe drive amplitude for a particular drive pulse shape.For each amplitude, we drive the resonator, swap be-tween the qubit and resonator for varying swap times t swap, and measure the qubit excited state probability P e [Fig. S10(a),(b)]. We simulate this probability versus t swap using the Linblad master equation for n -photonFock states. We least-squares fit the experimental andtheoretical probabilities to determine for each drive am-plitude the experimental Fock state distribution, shownin Fig. S10(c). We fit this measured distribution to aPoisson distribution, P Poisson n = h n i n e −h n i n ! , FIG. S11: Drive energy independence of absorption effi-ciency. The absoprtion efficiency is independent of the res-onator drive energy for energies of 0.5-50 photons. Data areshown for exponentially increasing drive pulses with time con-stant τ = 2 /κ and length 20 /κ , and error bars indicate sta-tistical errors from Eq. (S16). as the resonator is in a classically-driven coherent state.From this fit, we extract the mean number of photons h n i captured in the resonator [Fig. S10(d)] and find alinear fit between drive amplitude and p h n i . We thenrescale this according to the measured absorption effi-ciency to determine the drive amplitude necessary for asingle photon drive ( h n i = 1).To determine if the exact calibration matters, we mea-sured the absorption efficiency versus resonator drive am-plitude. As shown in Fig. S11, the capture efficiency isindependent of the drive energy between 0.5 and 50 pho-tons. This demonstrates that, although the theoreticalcapture efficiencies were calculated in the classical limit,they are still valid in the quantum regime. We note thateven if the energy calibration is mistuned, the absorptionefficiencies quoted in the main text are still valid. Averaging
We also checked whether the number of averages affectsthe absorption efficiencies. As the number of averagesis increased, the signal amplitude would tend to zero ifthe signal lacked phase coherence. However, as shown inFig. S12(a), the signal energy instead is constant once thesignal dominates over the noise. Similarly, the absorptionefficiencies [Fig. S12(b)] are nearly constant in this regimebut are frequently unphysical when the noise is dominant.The noise energy scales inversely with the number ofaverages. This makes sense as the energy is proportionalto the square of the measured voltage. For large num-bers of averages, the noise energy becomes constant; wemeasure the absorption efficiencies near the start of thisregime to ensure maximal signal-to-noise ratio. The un-certainties in the absorption efficiency, as calculated withEq. (S16), are approximately 6% the ratio of the releasephase noise energy to the total signal energy.
FIG. S12: Varying number of averages. (a) The noise (blue)and release signal (yellow) energies are shown for varyingnumbers of averages. The noise energy has been rescaled bythe duration of the energy release phase of the pulse sequence(See Fig. 2 of the main text). These are measured for expo-nentially increasing drive pulses with time constant τ = 2 /κ and length 20 /κ . The release energy flattens out where itno longer equals the noise energy. (b) Absorption efficienciesversus number of averages. Data are only shown where thenoise energy is substantially less than the release energy sothat absorption efficiencies make sense. Error bars indicatestatistical errors from Eq. (S16) and are approximately 6%the noise-to-signal ratio. ERROR ANALYSIS
Here we explain how to subtract noise to get unbi-ased estimates of energies. We then calculate the randomerrors in terms of experimental quantities and considerpossible sources of systematic errors.
Noise Subtraction
We calculate the energies for the absorption efficiencyby integrating the magnitude-squared of the signal; how-ever, noise contributes to these calculated energies. As-sume that, at the k th time step, the actual signal is( I k , Q k ), and the noise is ( x k , y k ) in the I - and Q -quadratures, respectively. Then, if the duration of eachtime step is t and there are N time steps, the calculatedenergy is E sig = t N X k =1 [( I k + x k ) + ( Q k + y k ) ] . The energy in the absence of noise is t N X k =1 [ I k + Q k ] . (S12)With no signal, the energy is the noise energy E N = t N X k =1 [ x k + y k ] , so noise energy is measured even without reflections.Hence, the directly measured capture efficiencies arelower than the true values.To remove this noise contribution, we assume the noisein each quadrature is Gaussian and uncorrelated withnoise in the other quadrature, noise at other times, andthe signal amplitude. These assumptions are consistentwith wide bandwidth white noise in our experiment. Wefurther assume the noise has standard deviation σ andzero mean. With these assumptions, the noise for timestep k can be treated as random values g k which areindependent with identical Gaussian distributions. Av-eraging over all such random values gives moments h g pk i = 1 + ( − p p − σ p . Particularly useful moments are h g k i = h g k i = 0, h g k i = σ , and h g k i = 3 σ .Before considering the statistical properties of theseenergies, we first consider the first and second moments oftwo useful sums. The first is a weighted sum of Gaussiandistributed noise signals given by P Nk =1 w k g k , where w k is the weight of the k th time step. By the linearity ofexpectation values, * N X k =1 w k g k + = 0 , while the second moment is * N X k =1 w k g k ! + = N X k =1 N X l =1 w k w l h g k g l i = N X k =1 w k h g k i + X k = l w k w l h g k ih g l i = σ N X k =1 w k , where we use the Gaussian moments and the indepen-dence of noise signals at different times. The other use-ful sum is a weighted sum of the square of Gaussian dis-tributed noise signals, P Nk =1 w k g k for weights w k . By thelinearity of expectation values, * N X k =1 w k g k + = σ N X k =1 w k . The second moment of this sum is given by * N X k =1 w k g k ! + = N X k =1 N X l =1 w k w l h g k g l i = N X k =1 w k h g k i + X k = l w k w l h g k ih g l i = 3 σ N X k =1 w k + σ X k = l w k w l = 2 σ N X k =1 w k + σ N X k =1 w k ! . We can now calculate the mean of E sig and derive howto subtract the noise contribution. The means of E sig and E N equal h E sig i = t " N S X k =1 ( I k + Q k ) + N S σ x + N S σ y (S13) h E N i = t (cid:2) N N σ x + N N σ y (cid:3) (S14)where the signal (noise) is measured with N S ( N N ) timesteps and the noise in the I ( Q ) quadrature has standarddeviation σ x ( σ y ). With E N rescaled by N S /N N , thesemeans differ by the noiseless energy, so we remove thecontribution of noise in a region by: • Calculate E N prior to driving the resonator, so I k = Q k = 0.10 • Rescale E N by the duration of the desired region. • Subtract this noise energy from the total measuredenergy.This gives an energy E NSsig = E sig − N S N N E N , which is an unbiased estimate of the noiseless energy[Eq. (S12)]. Error Analysis
To determine the random uncertainty from this noise,we calculate the variance of both E sig and E NSsig . Weassume that the following pairs are uncorrelated: noiseat different times, noise in the two quadratures, and noisewith the signal amplitude. Using the Gaussian momentscalculated earlier, we find the following variances: h (∆ E sig ) i = 4 t N S X k =1 ( I k σ x + Q k σ y ) + 2 N S t ( σ x + σ y ) h (∆ E N ) i = 2 N N t ( σ x + σ y ) . Since E sig and E N are measured at different times andnoise at different times is uncorrelated, E sig and E N areuncorrelated and so the variance in E NS equals h (∆ E NSsig ) i = h (∆ E sig ) i + (cid:18) N S N N (cid:19) h (∆ E N ) i . The additional uncertainty scales as σ while the uncer-tainty in the raw signal energy scales as σ . With a largesignal-to-noise ratio, the additional noise from this pro-cedure can be neglected.Further, these variances can be reexpressed in terms ofthe measured energies h E NSsig i and h E N i using Eqs. (S13)-(S14). In particular, if σ x = σ y , h (∆ E sig ) i = 2 N N h E NSsig ih E N i + N S N N h E N i h (∆ E NSsig ) i = 2 N N h E NSsig ih E N i + N S ( N S + N N ) N N h E N i h (∆ E N ) i = 1 N N h E N i . (S15)We then use these expressions to determine the un-certainty in the absorption efficiency, the ratio of theenergy E NSabs absorbed and then released by the res-onator to the total measured energy E NStot . However, E NStot = E NSabs + E NSref for reflected energy E NSref , so the un-certainties in E NSabs and E NStot are not independent. Simi-larly, all noise-subtracted energies contain the single term h E N i and so have correlated uncertainties. Since E abs , E ref , and E N are measured at different times, these en-ergies are independent, so their uncertainties can be usedto calculate the overall absorption efficiency uncertainty δ (cid:18) E NSabs E NStot (cid:19) = vuut h (∆ E abs ) i E NSref (cid:0) E NStot (cid:1) ! + h (∆ E ref ) i (cid:0) E NStot (cid:1) + h (∆ E N ) i N ref E NSabs − N abs E NSref N N (cid:0) E NStot (cid:1) ! , (S16)where N ref and N abs are the number of data pointsused to measure E ref and E abs , respectively. To ver-ify this uncertainty, we repeated the measurement ofthe minimal-reflection absorption efficiency 60 times andmeasured a standard deviation in the absorption effi-ciency of 0.0552%, within 1% of the 0.0548% expectedaccording to Eq. (S16), validating this error analysis.The storage efficiency is the ratio of E NSabs to the totalpulse energy E NSoff measured with the coupler off. Sincethese signals and the noise contributions are measured indifferent experiments, the uncertainties in these energiesare independent, so standard error propagation applies.These uncertainties only cover random variations butnot systematic errors. One major source of systematicerrors is poor signal or drive path calibration; this is a multiplicative effect and so changes the raw energiesbut not ratios such as the absorption and receiver ef-ficiencies. These efficiencies can, however, be reducedby imperfections in the pulse calibration. We scan overcoupler closing delay and drive frequency, measure theresulting absorption efficiencies for an exponentially in-creasing pulse, and choose parameters which maximizethe absorption efficiency. However, this does not includeimperfections in the pulse shape, which are likely reduc-ing our measured efficiency as explained in the discussionof the coupler delay in the main text.11 [1] Korotkov, A. N. Flying microwave qubits with nearlyperfect transfer efficiency.
Phys. Rev. B , 014510(2011).[2] Wang, H. et al. Improving the coherence time of su-perconducting coplanar resonators.
Appl. Phys. Lett. ,233508 (2009).[3] Yin, Y. et al. Catch and release of microwave photonstates.
Phys. Rev. Lett. , 107001 (2013).[4] Martinis, J. M. Superconducting phase qubits.
QuantumInf. Process. , 81 (2009).[5] Megrant, A. et al. Planar superconducting resonatorswith internal quality factors above one million.
Appl.Phys. Lett. , 113510 (2012).[6] Wenner, J. et al.
Wirebond crosstalk and cavity modes inlarge chip mounts for superconducting qubits.
Supercond. Sci. Tech. , 065001 (2011).[7] Chen, Y. et al. Multiplexed dispersive readout of super-conducting phase qubits.
Appl. Phys. Lett. , 182601(2012).[8] Hofheinz, M. et al.
Synthesizing arbitrary quantum statesin a superconducting resonator.
Nature , 546-549(2009).[9] Neeley, M. et al.
Process tomography of quantum mem-ory in a Josephson-phase qubit coupled to a two-levelstate.
Nat. Phys. , 523-526 (2008).[10] Wang, H. et al. Measurement of the decay of Fock statesin a superconducting quantum circuit.
Phys. Rev. Lett. , 240401 (2008).[11] Hofheinz, M. et al.
Generation of Fock states in a super-conducting quantum circuit.
Nature454