week ending PRL 108, 230509 (2012) PHYSICAL REVIEW LETTERS 8 JUNE 2012
Measurements of Quasiparticle Tunneling Dynamics in a Band-Gap-Engineered Transmon Qubit
L. Sun,1 L. DiCarlo,1,2 M. D. Reed,1 G. Catelani,1 Lev S. Bishop,1,3 D. I. Schuster,1,4 B. R. Johnson,1,5 Ge A. Yang,1,4 L. Frunzio,1 L. Glazman,1 M. H. Devoret,1 and R. J. Schoelkopf1 1Department of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA 2Kavli Institute of Nanoscience, Delft University of Technology, Delft, The Netherlands 3Joint Quantum Institute and Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA 4Department of Physics and James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA 5Raytheon BBN Technologies, Cambridge, Massachusetts 02138, USA (Received 11 December 2011; published 8 June 2012) We have engineered the band gap profile of transmon qubits by combining oxygen-doped Al for tunnel junction electrodes and clean Al as quasiparticle traps to investigate energy relaxation due to quasiparticle tunneling. The relaxation time T1 of the qubits is shown to be insensitive to this band gap engineering. Operating at relatively low-EJ=EC makes the transmon transition frequency distinctly dependent on the charge parity, allowing us to detect the quasiparticles tunneling across the qubit junction. Quasiparticle kinetics have been studied by monitoring the frequency switching due to even-odd parity change in real time. It shows the switching time is faster than 10 s, indicating quasiparticle-induced relaxation has to be reduced to achieve T1 much longer than 100 s.
DOI: 10.1103/PhysRevLett.108.230509 PACS numbers: 03.67.Lx, 42.50.Pq, 74.55.+v, 85.25. j
Quantum information processing based on supercon- qubit relaxation and odd-even parity switching, with rates qp ¼j j2 ð Þ qp ¼j j2 ð Þ ducting qubits has made tremendous progress toward real- 1!0 M01 Sqp !01 and oe;i Moe;i Sqp i respec- izing a practical quantum computer in the last few years tively, where !01=2 is the qubit ground-to-excited tran- [1–3]. However, the coherence times of superconducting sition frequency, i=2 is the odd-to-even transition qubits still need to improve to reach the error correction frequency for state i ¼ 0, 1, and M01 and Moe;i are the threshold. For example, the single-qubit gate error rate is matrix elements describing the interaction between the limited by qubit decoherence [4–6]. Understanding deco- qubit and quasiparticles [12]. herence mechanisms, in particular those responsible for The dynamics of quasiparticle tunneling in single- qubit relaxation, is therefore crucial. Quasiparticles have Cooper-pair transistors (SCPTs) and Cooper-pair box received significant attention recently as one such possible qubits have been studied extensively, although the under- limiting factor [7–9]. At low temperatures, thermal- standing is still incomplete. The measured odd-even parity equilibrium quasiparticles should be irrelevant because switching time falls in a large range from the order of 1 s their density is exponentially suppressed. In practice, how- to the order of 1 ms [10,17–20], or possibly even longer in ever, nonequilibrium quasiparticles are present from un- some cases [21–23] that have no observable quasiparticle known sources [10,11]. The key question is then whether ‘‘poisoning.’’ One way to suppress quasiparticle tunneling these nonequilibrium quasiparticles are currently limiting is to engineer the band gap profile near the tunnel junction qubit relaxation and, if not, what limit they will ultimately [10,17–21]. This band gap engineering technique has been impose. The answer will be relevant to all superconducting shown to be effective in extending the parity switching qubits. time dramatically from 1 s to above 1 ms [20]. Quasiparticles in qubit electrodes do not themselves In this Letter, we employ band gap engineering for the cause significant qubit decoherence. Instead, it is the dis- first time in transmon qubits, in an attempt to make the sipative and incoherent tunneling of quasiparticles across a odd-even parity switching time in the measurable range Josephson junction that leads to decoherence [12]. The >100 s, which would then eliminate quasiparticles as a tunneling process is characterized by the quasiparticle source of decoherence. We measure the qubit relaxation ð Þ current spectral density Sqp ! . The low frequency com- time T1 as a function of temperature of transmon qubits, ponent in Sqpð!Þ can cause dephasing, but this dephasing which have significantly different band gaps. At low tem- channel can be eliminated by reducing the qubit’s sensi- peratures, the saturation of T1 of both transmons at ap- tivity to charge noise, as in flux [13,14], phase [15], and proximately the same level suggests a mechanism other transmon [16] qubits. The high frequency component will than thermal quasiparticles for qubit relaxation. To inves- cause energy relaxation in all types of superconducting tigate if nonequilibrium quasiparticles are limiting T1,we qubits. A transmon qubit has the energy level structure then study the quasiparticle kinetics in a band gap– shown in Fig. 1(a). Quasiparticle tunneling causes both engineered transmon operated in the low-EJ=EC regime,
0031-9007=12=108(23)=230509(5) 230509-1 Ó 2012 American Physical Society week ending PRL 108, 230509 (2012) PHYSICAL REVIEW LETTERS 8 JUNE 2012 where EJ and EC are the Josephson and the charging tunnel junction. The energy gaps of oxygen-doped layers energy, respectively. Qubit spectroscopy shows two qubit are determined by independent Tc measurements (not transition frequencies associated with the even- and odd- shown) of thin films evaporated under the same nominal charge states, demonstrating the presence and tunneling of conditions as for the tunnel junction. quasiparticles across the qubit junction. This indicates that, However, our band gap engineering does not appear to with our design, nonequilibrium quasiparticles have not affect T1 (see Table I in Supplemental Material [26] for a been removed by band gap engineering, contrary to the list of devices). A comparison of T1 as a function of results of experiments in SCPTs [10,18,20]. We study the temperature for two representative devices is shown in qp qp ground-state parity switching rate oe oe;0 in the time Fig. 2(a): one qubit is fabricated with clean Al only and qp domain finding 1= oe < 10 s. For typical device parame- the other with oxygen-doped Al without the third layer ters, the expected ratio of parity switching rate to quasiparticle trap. The decrease of T1 with temperatures qp qp quasiparticle-induced qubit relaxation rate, oe= 1!0 above 150 mK for the red curve and 250 mK for the blue 10–100 [12]. Therefore, while we cannot establish quasi- curve, respectively, indicates the effect of thermally gen- particle tunneling as the dominant source of energy relaxa- erated quasiparticles [6,12]. The higher corner temperature tion in our devices, our bound indicates that reducing the confirms that oxygen-doped Al indeed has a larger energy quasiparticle-induced decay rate will be necessary to gap. The saturation of T1 at low temperatures for both achieve T1 much longer than 100 s. clean and oxygen-doped Al devices at the same value Our transmon qubits are measured using a coplanar indicates that thermal quasiparticles are not limiting T1. waveguide cavity in a conventional circuit-quantum- Figure 2(b) shows the saturated T1 as a function of fre- electrodynamics architecture [24]. All devices are mea- quency for four qubits, each of which has its own flux bias sured in a cryogen-free dilution refrigerator with a line allowing individual tuning of the qubit frequency [2], 20 mK base temperature. Oxygen-doped Al, which has fabricated with both oxygen-doped Al and quasiparticle an energy gap 280 eV (Tc 1:9K) about 60% traps described earlier. The qubit lifetimes are limited to a 65 000 higher than clean Al ( 180 eV, Tc 1:2K), is quality factor Q ; , similar to previously reported used as the electrodes of Josephson tunnel junctions de- T1 on qubits fabricated with clean Al only [27]. posited by standard double-angle evaporation [25]. The Nonequilibrium quasiparticles have been observed in oxygen dopants are introduced with a continuous O2 flow the band gap–engineered devices, as will be shown later, during the Al deposition. The same technique has also been so here we will only focus on the effect on T1 from those used in SCPTs to realize a large band gap [10,18]. To nonequilibrium quasiparticles. The lack of qualitative create quasiparticle traps, a third layer of clean Al is 5 5 deposited to cover the whole oxygen-doped Al layers to (a) (b) Qubit 1 100 nm Qubit 2 within from the junctions [Fig. 1(b)]. Figure 1(c) Qubit 3 shows the expected band gap profile near the junction. This Qubit 4 s) s) μ μ ( profile is expected to trap quasiparticles away from the ( 1 1 1 1 T T
clean Al oxygen-doped Al oxygen-doped Al + quasiparticle traps 0.2 0.2 0.10 0.20 0.30 4 6 8 10 temperature (K) frequency (GHz)
FIG. 2 (color online). (a) Measured qubit relaxation time T1 as a function of temperature for two transmon qubits: one qubit is fabricated with clean Al only with a transition frequency f01 ¼ 4:25 GHz and the other with oxygen-doped Al, but without the third layer of quasiparticle traps, with f01 ¼ 5:16 GHz. The device layout is identical to Fig. 1(a) in Ref. [1]. Dashed black and magenta lines are theory for relaxation due to thermal FIG. 1. (a) Low energy–level structure of a transmon qubit. quasiparticles, assuming 180 and 280 eV, respectively The parity state with larger separation between j0i and j1i qubit [8]. The solid green and cyan lines represent the sum of the states is assigned to odd without loss of generality. 0 and 1: theoretical expectations for thermal-equilibrium quasiparticles Energy difference between odd-even parities for the ground and and the best-fit saturated T1. (b) T1 vs frequency on qubits odd even ¼ first excited state, respectively, 1 0. !01 !01 !oe fabricated with oxygen-doped Al and quasiparticle traps. The qp 1 is the charge dispersion. oe;i is the switching rate between device layout is identical to Fig. 1(a) in Ref. [2]. Dashed red line: qp qp ¼ qp qubit states of different parities. 1o!0e 1e!0o 1!0 is the Purcell-induced relaxation time; solid blue line: Q ¼ 65; 000; odd even qubit relaxation rate because !01 !01 ¼ !01 1. solid black line: a combination of the Purcell effect and a (b) SEM image of a typical device fabricated by three-angle constant Q. Neither oxygen-doped Al nor quasiparticle traps evaporation to engineer the band gap profile. (c) Expected band improve T1 qualitatively. The black arrow indicates the cavity gap profile near the tunnel junction. frequency.
230509-2 week ending PRL 108, 230509 (2012) PHYSICAL REVIEW LETTERS 8 JUNE 2012 improvements of T1 suggests three possible scenarios: T1 is both parity switching times o (odd-to-even) and e not limited by nonequilibrium quasiparticles, the quasipar- (even-to-odd) are shorter than the 50 ms averaging time ticle contribution to T1 has not been affected by our band at each pixel. gap engineering, or both. In any case, we wish to find the To study the dynamics of quasiparticle tunneling, a qp quasiparticle-induced qubit relaxation rate 1!0. However, selective pulse is applied repeatedly every 10 s at this rate cannot be measured directly in the presence of one of two parity frequencies (for simplicity, we choose other sources of relaxation, so we instead measure the the odd parity frequency for all data presented here), qp dynamics of quasiparticle tunneling oe in the time do- immediately followed by a measurement of the qubit state qp main, from which we can estimate 1!0 [12]. In order to [Fig. 4(a)]. This selective pulse will excite the qubit only qp measure oe, we operate the transmon qubit in the when the parity is odd. We note that, due to the smallness low-EJ=EC regime where the qubit spectrum will have of 0 [Fig. 1(a)], the selective pulse will also cover the two distinct parities caused by quasiparticle tunneling transition j0; eveni to j1; oddi, but for nondegenerate qua- qp across the junction. Thus, oe can be studied by monitoring siparticles the probability of this unwanted transition is the dynamics of one particular parity in real time. small compared to that of the direct photon transition We have fabricated a single transmon qubit with both j0; oddi to j1; oddi. Thus, by interrogating the qubit state oxygen-doped Al and quasiparticle traps, and we operate it after the pulse, the qubit parity can be inferred. To ensure at EJ=EC 12:5 by tuning the qubit frequency to f01 that the qubit begins in the ground state and that the qubit 4:5 GHz, where T1 ¼ 2 s. To protect the qubit from frequency has stabilized after the rapid tuning from 7 to spontaneous emission [27] through the low-Q cavity 4.5 GHz at the beginning of each experimental cycle, the ¼ 500 (Q ), a Purcell filter is also integrated on the chip. repetition rate is set to ts ¼ 10 s ¼ 5T1. As shown in The device layout is identical to Fig. 1(a) in Ref. [28]. Ref. [20], if slow quasiparticle tunneling dynamics of ¼ At this large detuning from the cavity frequency fc the order of 1 ms could be achieved in our band gap– 8:072 GHz, a direct readout of the qubit state is difficult engineered transmon qubits, the presented method would because of the weak dispersive interaction between the easily resolve these dynamics, thus completely ruling out qubit and the cavity. Instead, to enhance readout fidelity, quasiparticles as a source of qubit relaxation. the qubit is pulsed to f01 7 GHz prior to measurement. Because quasiparticles tunnel across the qubit junction By making use of the high-power Jaynes-Cummings read- randomly, a random telegraph signal (RTS) is expected. To out [29], we achieve a single-shot fidelity F ¼ 42%. overcome low single-shot readout fidelity, we perform a The presence and tunneling of nonequilibrium quasipar- Fourier transform of the time domain data to study the ticles is demonstrated by the clearly observed ‘‘eye’’ pat- power spectral density (PSD) to better extract the RTS tern in the qubit spectroscopy as a function of gate-induced information. Background charge motion limits the experi- charge ng [Fig. 3(a)][11]. The two peaks in Fig. 3(b) have ment duration, since the qubit frequency shifts noticeably almost equal height, implying no preferred parity, and every few minutes. In total, 12 million measurements are recorded and Fourier-transformed. The stability of the readout is particularly important 4.56 (a) 19.0 (b) even odd for this measurement. Thus, to minimize the drift during 4.52 18.5 the experiment, the readout result is digitized by thresh- 4.48 18.0 olding the measurement results [Fig. 4(b)]. From each 4.44 17.5 readout (mV) frequency (GHz) 17.0 0.3 0.4 0.5 0.6 0.7 4.44 4.48 4.52 4.56 ng (2e) frequency (GHz)
FIG. 3. (a) Spectroscopy of a qubit with engineered band gap as a function of gate-induced charge ng. The gate voltage is applied to the qubit through a bias tee at the cavity input port. For each pixel, a Gaussian pulse ( ¼ 20 ns, corresponding to a pulse on resonance) is applied at the indicated frequency and the qubit is immediately measured. Each pixel is an average of 5000 repetitions (50 ms). Darker pixels correspond to higher homo- FIG. 4 (color online). (a) Schematic of the measurement. A dyne readout voltages that are proportional to the probability of selective pulse is applied to one of the two parity peaks in Fig. 3, the qubit in the excited state. An ‘‘eye’’-shaped pattern indicates followed by an immediate readout of the qubit state at about charge-e jumps associated with the tunneling of nonequilibrium 7 GHz, where readout fidelity is improved. Lines: The flux pulse quasiparticles. (b) Cross section of (a) at ng ¼ 0:5 as indicated sequence. The process is repeated every 10 s. (b) Histogram of by the black arrow in (a). The charge dispersion is 60 MHz. We the readout at the odd parity peak in Fig. 3(b). A threshold Vth ¼ refer to the lower (upper) frequency branch as the even (odd) 19 mV is chosen to digitize the readout. Inset: Schematic of an parity peak. imperfect readout with false positives (negatives) 0 ( 1).
230509-3 week ending PRL 108, 230509 (2012) PHYSICAL REVIEW LETTERS 8 JUNE 2012 measurement, one bit of information is extracted. False positives 0 and negatives 1 of the readout [Fig. 4(b) inset] will reduce the RTS amplitude. Note that, due to the switching of the qubit between the two parities, 0 and 1 cannot be extracted directly from Fig. 4(b), but they can easily be obtained by combining histograms of the readout at the even parity peak in Fig. 3(b) (see Supplemental Material [26] for details). We find 0 1 ¼ 0:29,as well as the probability of the qubit at the odd parity Podd ¼ 56%. Thus the readout fidelity is F ¼ 1 0 1 42%. We first test the sensitivity of the experiment to fluctua- tions of parity by applying a ‘‘ mask’’ to the measurement system to imitate the success and failure of a pulse on the qubit. The mask, generated by a field-programmable gate array (FPGA), is a pseudorandom control sequence that enables or disables a pulse applied to the qubit immediately prior to readout. It generates an RTS with a specified time constant ¼ 0 þ 1 and a 50% duty cycle ( 0 ¼ 1). If the parity switches fast compared to FIG. 5 (color online). (a) Power spectral densities of control (as will be confirmed later), the RTS amplitude in the experiments using a mask with different time constants ¼ readout is reduced to Acon ¼ PoddF due to the parity duty 0 þ 1. 0 ¼ 1 are the time constants associated with the cycle and the finite readout fidelity [Fig. 5(a) inset; also see mask. The color curves are data and black curves are theo- Supplemental Material [26]]. Figure 5(a) shows the PSDs retical predictions. Inset: Schematic of the control experiment of the control experiments with a mask with different with a mask, an RTS with specified time constants and time constants ¼ 2ms, 600 s, 200 s, 108 s, generated by an FPGA. The expected RTS amplitude is reduced ¼ ¼ and 72 s, respectively. All measured PSDs agree very to Acon A A0 PoddF, where A and A0 are the statistical well with theory (see Supplemental Material [26] for averages of the readout during the mask assuming parity details). switches fast compared to , Podd is the probability of the qubit We now turn to the measurement of the parity switching to be in the odd parity state, and F is the readout fidelity. qp (b) Measured PSDs and theoretical predictions for the direct rate oe of the qubit, showing in Fig. 5(b) the PSD obtained qp experiment on oe without the mask. Blue curve is the without the mask. The measured spectrum is almost flat measured Fourier spectrum and the color curves are theoretical and white noise–like, suggesting fast qubit parity switch- predictions for different qubit parity switching times ¼ þ qp oe o ing. An upper bound can be placed on oe by comparing e, where o and e are the odd-to-even and even-to-odd switch- with the theoretical predictions for the observed duty cycle ing times, respectively. The red dashed line is the white-noise Podd ¼ o=ð o þ eÞ¼56% and different time constants, spectrum in the limit o, e
230509-4 week ending PRL 108, 230509 (2012) PHYSICAL REVIEW LETTERS 8 JUNE 2012 to quasiparticle traps. On the other hand, the nondegrada- L. Frunzio, L. Glazman, et al., Phys. Rev. Lett. 107, tion of the qubit relaxation time suggests no measurable 240501 (2011). change of the junction quality from oxygen-doped Al. [7] J. M. Martinis, M. Ansmann, and J. Aumentado, Phys. Nonequilibrium quasiparticles leading to the charge-parity Rev. Lett. 103, 097002 (2009). switching have been observed in all devices. Moreover, the [8] G. Catelani, J. Koch, L. Frunzio, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman, Phys. Rev. Lett. 106, 077002 quasiparticle-induced parity switching is shown to be 10 s (2011). faster than , an upper bound limited by the detection [9] M. Lenander, H. Wang, R. C. Bialczak, E. Lucero, M. bandwidth. This fast parity switching rate, different from Mariantoni, M. Neeley, A. D. O’Connell, D. Sank, M. the results of other experiments in SCPTs, could be due to Weides, J. Wenner, et al., Phys. Rev. B 84, 024501 three things. First, the size of the transmon electrodes is (2011). much larger and the quasiparticle density may depend on [10] J. Aumentado, M. W. Keller, J. M. Martinis, and M. H. the electrode size. Second, the qubit readout might gener- Devoret, Phys. Rev. Lett. 92, 066802 (2004). ate quasiparticles or stimulate the tunneling of quasipar- [11] J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. ticles between electrodes. A third possibility is that the Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frunzio, interface between the oxygen-doped and clean Al layers M. H. Devoret, et al., Phys. Rev. B 77, 180502 (2008). might not be transparent enough. Although our results [12] G. Catelani, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman, Phys. Rev. B 84, 064517 (2011). neither prove nor disprove that nonequilibrium quasipar- [13] J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, and J. E. ticles limit T1 in transmon qubits, it does indicate that Lukens, Nature (London) 406, 43 (2000). quasiparticle-induced energy relaxation must be reduced [14] C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. in the future to achieve T1 much longer than 100 s. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and Future experiments could be able to resolve quasiparticle J. E. Mooij, Science 290, 773 (2000). tunneling dynamics by increasing the experimental repeti- [15] J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina, tion rate through either the use of a fast reset [28]orby Phys. Rev. Lett. 89, 117901 (2002). artificially limiting the qubit T1 through other means, such [16] J. Koch, T. M. Yu, J. M. Gambetta, A. A. Houck, D. I. as the Purcell effect. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. We thank E. Ginossar, H. Paik, A. Sears, G. Kirchmair, Girvin, and R. J. Schoelkopf, Phys. Rev. A 76, 042319 S. Shankar, M. Hatridge, and S. Girvin for valuable (2007). [17] A. J. Ferguson, N. A. Court, F. E. Hudson, and R. G. Clark, discussions. L. F. acknowledges partial support from Phys. Rev. Lett. 97, 106603 (2006). CNR-Istituto di Cibernetica. L. D. C. acknowledges partial [18] O. Naaman and J. Aumentado, Phys. Rev. B 73, 172504 support from NWO. This research was funded by the (2006). Office of the Director of National Intelligence (ODNI), [19] M. D. Shaw, R. M. Lutchyn, P. Delsing, and P.M. Intelligence Advanced Research Projects Activity Echternach, Phys. Rev. B 78, 024503 (2008). (IARPA), through the Army Research Office. [20] N. A. Court, A. J. Ferguson, R. Lutchyn, and R. G. Clark, Phys. Rev. B 77, 100501 (2008). [21] P. Joyez, P. Lafarge, A. Filipe, D. Esteve, and M. H. Devoret, Phys. Rev. Lett. 72, 2458 (1994). [1] L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, [22] A. Amar, D. Song, C. J. Lobb, and F. C. Wellstood, Phys. B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Rev. Lett. 72, 3234 (1994). Frunzio, S. M. Girvin, et al., Nature (London) 460, 240 [23] V. Bouchiat, D. Vion, P. Joyez, D. Esteve, and M. H. (2009). Devoret, Phys. Scr. T76, 165 (1998). [2] L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. [24] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Devoret, and R. J. Schoelkopf, Nature (London) 467, 574 Schoelkopf, Nature (London) 431, 162 (2004). (2010). [25] G. J. Dolan, Appl. Phys. Lett. 31, 337 (1977). [3] M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. [26] See Supplemental Material at http://link.aps.org/ Mariantoni, A. D. O’Connell, D. Sank, H. Wang, M. supplemental/10.1103/PhysRevLett.108.230509 for mea- Weides, J. Wenner, et al., Nature (London) 467, 570 sured devices, extraction of parameters from histograms, (2010). and spectrum of an RTS with finite readout fidelity. [4] E. Lucero, M. Hofheinz, M. Ansmann, R. C. Bialczak, N. [27] A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, J. Katz, M. Neeley, A. D. O’Connell, H. Wang, A. N. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Cleland, and J. M. Martinis, Phys. Rev. Lett. 100, Devoret, and S. M. Girvin et al., Phys. Rev. Lett. 101, 247001 (2008). 080502 (2008). [5] J. M. Chow, J. M. Gambetta, L. Tornberg, J. Koch, L. S. [28] M. D. Reed, B. R. Johnson, A. A. Houck, L. DiCarlo, J. M. Bishop, A. A. Houck, B. R. Johnson, L. Frunzio, S. M. Chow, D. I. Schuster, L. Frunzio, and R. J. Schoelkopf, Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 102, 090502 Appl. Phys. Lett. 96, 203110 (2010). (2009). [29] M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun, D. I. [6] H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, Schuster, L. Frunzio, and R. J. Schoelkopf, Phys. Rev. G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, Lett. 105, 173601 (2010).
230509-5