A Supplemental material for“Single microwave-photon detector using an artificial \Lambda-type three-level system”

Single microwave-photon detector using an artificial -type three-level system

Abstract

Single photon detection is a requisite technique in quantum-optics experiments in both the optical and the microwave domains. However, the energy of microwave quanta are four to five orders of magnitude less than their optical counterpart, making the efficient detection of single microwave photons extremely challenging. Here, we demonstrate the detection of a single microwave photon propagating through a waveguide. The detector is implemented with an “impedance-matched” artificial system comprising the dressed states of a driven superconducting qubit coupled to a microwave resonator. We attain a single-photon detection efficiency of with a reset time of  ns. This detector can be exploited for various applications in quantum sensing, quantum communication and quantum information processing.

pacs:
1

Single-photon detection is essential to many quantum-optics experiments, enabling photon counting and its statistical and correlational analyses Hadfield (2009). It is also an indispensable tool in many protocols for quantum communication and quantum information processing Gisin et al. (2002); Knill et al. (2001); O’Brien (2007); Aaronson (2011). In the optical domain, various kinds of single-photon detectors are commercially available and commonly used Hadfield (2009); Eisaman et al. (2011). However, despite the latest developments in nearly-quantum-limited amplification Bergeal et al. (2010); Macklin et al. (2015) and homodyne measurement for extracting microwave photon statistics Lang et al. (2013), the detection of a single microwave photon in an itinerant mode remains a challenging task due to its correspondingly small energy. Meanwhile, the demand for such detectors is rapidly increasing, driven by applications involving both microwave and hybrid optical-microwave quantum systems.

In this report we demonstrate an efficient and practical single-microwave-photon detector based on the deterministic switching in an artificial -type three-level system implemented using the dressed states of a driven superconducting quantum circuit. The detector operates in a time-gated mode and features a high quantum efficiency , a low dark-count probability , a bandwidth  MHz, and a fast reset time  ns. It can be readily integrated with other components for microwave quantum optics.

Our detection scheme carries several advantages compared with previous proposals. It uses coherent quantum dynamics, which minimizes energy dissipation upon detection and allows for rapid resetting with a resonant drive, in contrast to schemes that involve switching from metastable states of a current-biased Josephson junction into the finite voltage state Romero et al. (2009); Peropadre et al. (2011); Chen et al. (2011). Moreover, our detection scheme does not require any temporal shaping of the input photons, nor precise time-dependent control of system parameters adapted to the temporal mode of the input photons, in contrast to recent photon-capturing experiments Yin et al. (2013); Palomaki et al. (2013); Wenner et al. (2014). It also achieves a high efficiency without cascading many devices Romero et al. (2009); Sathyamoorthy et al. (2014).

The operating principle of the detector fully employs the elegance of waveguide quantum electrodynamics, which has recently attracted significant attention in various contexts surrounding photonic quantum information processing Duan and Kimble (2004); Chang et al. (2007); Witthaut and Sørensen (2010); Zheng et al. (2013). When electromagnetic waves are confined and propagate in an one-dimensional (1D) mode, their interaction with a quantum emitter/scatterer is substantially simplified and enhanced compared with three-dimensional cases. These advantages result from the natural spatial-mode matching of the emitter/scatterer with a 1D mode and its resulting enhancement of quantum interference effects. Remarkable examples are the perfect extinction of microwave transmission for an artificial atom coupled to a 1D transmission line Astafiev et al. (2010); Hoi et al. (2011), the photon-mediated interaction between two remote atoms coupled to a 1D transmission line van Loo et al. (2013), and the perfect absorption — and thus “impedance matching” — of a -type three-level system terminating a 1D transmission line Koshino et al. (2013); Inomata et al. (2014). In the latter system, the incident photon deterministically induces a Raman transition which switches the state of the system Pinotsi and Imamoglu (2008); Koshino et al. (2013). This effect has recently been demonstrated in both the microwave and optical domains Inomata et al. (2014); Shomroni et al. (2014), indicating its potential for photon detection Koshino et al. (2015a) as well as for implementing deterministic entangling gates with photonic qubits Koshino et al. (2010).

Figure 1: Experimental setup and pulse sequence. (A) Image of the sample chip containing a flux qubit and a superconducting microwave resonator coupled capacitively and operated in the dispersive regime. For certain proper conditions of the qubit drive, the coupled system functions as an impedance-matched -type three-level system. (B) Schematic of the itinerant microwave-photon detector consisting of the coupled system and connected to a parametric phase-locked oscillator (PPLO) via three circulators in series. The circuit has three input ports: signal, qubit drive, and pump for the PPLO. (C) Energy-level diagram of the coupled system and the pulse sequence for single-photon detection. The system is first prepared in the ground state. During the detection stage, we concurrently apply the drive and signal pulses. The drive is parameterized to fulfill the impedance-matched condition such that a signal photon (blue arrow) induces a deterministic Raman transition. A down-converted photon (green arrow) is emitted in the process and discarded. In the readout stage, we detect the qubit excited state nondestructively by sending a qubit readout pulse. The qubit-state-dependent phase shift in the reflected pulse is discriminated by the PPLO.

Our device consists of a superconducting flux qubit capacitively and dispersively coupled to a microwave resonator (Fig. 1B) Inomata et al. (2012). With a proper choice of the qubit drive frequency and power , the system functions as an impedance-matched system with identical radiative decay rates from its upper state to its two lower states (Fig. 1A) Koshino et al. (2013); Inomata et al. (2014). The qubit-resonator coupled system is connected to a parametric phase-locked oscillator (PPLO), which enables fast and non-destructive qubit readout Lin et al. (2014).

Figure 1C shows the level structure of the qubit-resonator system and the protocol for the single photon detection. We label the energy levels and their eigenfrequencies , where and respectively denote the qubit state and the photon number in the resonator. In the dispersive coupling regime, the qubit-resonator interaction renormalizes the eigenfrequencies to yield and , where and are the renormalized frequencies of the qubit and the resonator, respectively, and is the dispersive frequency shift of the resonator due to its interaction with the qubit. Only the lowest four levels with or are relevant here.

We prepare the system in its ground state (Fig. 1C, Initialization) and apply a drive pulse to the qubit (Fig. 1C, Detection). In a frame rotating at , the level structure becomes nested, i.e., , for in the range . On the plateau of the drive pulse (), the lower-two levels and (higher-two levels and ) hybridize to form dressed states and ( and ). Under a proper choice of , the two radiative decay rates from (or ) to the lowest-two levels become identical. Thus, an impedance-matched system comprising , , and (alternatively, , , and ) is realized. An incident single microwave photon (Gaussian envelope, length ), synchronously applied with the drive pulse through the signal port and in resonance with the transition, deterministically induces a Raman transition, , and is down-converted to a photon at the transition frequency. This process is necessarily accompanied by an excitation of the qubit Koshino et al. (2013); Inomata et al. (2014).

Figure 2: Impedance matching and itinerant microwave-photon detection. (A) Amplitude of the reflection coefficient of the input signal pulse with mean photon number as a function of the qubit drive power and the signal frequency . The PPLO is not activated during this measurement. The impedance-matched point is resolved (dark-blue region), where the input microwave photon is absorbed almost completely. In the inset, we also observe another dip in , corresponding to the Raman transition of . Microwave power levels stated in this article are referred to the value at the corresponding ports on the sample chip. (B) Detection efficiency of an itinerant microwave photon. The efficiency hits the maximum at the impedance-matched point, where the Raman transition of takes place. (C) and (D) Theoretical predictions corresponding to A and B. (E) Cross-sections of B (blue dots) and D (red dashed line) at . The error bars are due to the uncertainty in the input power calibration.

To detect the photon, we adiabatically switch off the qubit drive and dispersively read out the qubit state (Fig. 1C, Readout). We apply a readout pulse with the frequency through the signal port, which, upon reflection at the resonator, acquires a qubit-state-dependent phase shift of or . This phase shift is detected by the PPLO with high fidelity: in the present setup, the readout fidelity of the qubit is , which is primarily limited by qubit relaxation prior to readoutLin et al. (2014).

We first determine the operating point where the system deterministically absorbs a signal photon. We simultaneously apply a drive pulse of length  ns and a signal pulse of length  ns, and proceed to measure the reflection coefficient of the signal pulse as a function of the drive power and the signal frequency (Fig. 2A). The signal pulse is in a weak coherent state with mean photon number . A pronounced dip with a depth of  dB is observed in at , in close agreement with theory (Fig. 2C). The dip indicates a near-perfect absorption condition, i.e., impedance matching, where the reflection of the input microwave photon vanishes due to destructive self-interference. Correspondingly, a deterministic Raman transition of is induced, and the qubit state is flipped.

Figure 3: Optimization of the efficiency. (A) Single photon detection efficiency as a function of the signal pulse length . The mean photon number for the weak-coherent signal pulse is . (B)  as a function of . Dashed lines indicate theoretical predictions.
Figure 4: Demonstration of the fast reset protocol. (A) Pulse sequence used to evaluate the reset efficiency. The initial -pulse mimics a single-photon detection and excites the qubit. During the reset stage, a drive pulse and a reset pulse with the mean photon number of are concurrently applied, inducing an inverse Raman transition: . The remaining population in the state is then detected. (B) Population of the qubit excited state after the reset operation , as a function of the reset-pulse frequency and the drive-pulse power . (C) Theoretical prediction for B with no free parameters. (D) Cross sections of B (blue dots) and C (red dashed line) at  GHz.

To obtain a ‘click’ corresponding to single-photon detection, we read out the qubit state by using the PPLO immediately after the Raman transition. Before initiating readout, the drive pulse is turned off to suppress unwanted Raman transitions induced by the readout pulse, e.g., . We repeatedly apply the pulse sequence in Fig. 1C times and evaluate the single-photon detection efficiency , where and are the probabilities for the qubit being in the excited state and the signal pulse being in the vacuum state, respectively. Figure 2B depicts as a function of and . The dark count probability of the detector — mainly caused by the nonadiabatic qubit excitation due to the drive pulse and the imperfect initialization — is subtracted when evaluating  Sup (). We observe that is maximized at the dip position in Fig. 2A in accordance with the impedance-matching condition. We also confirm that the result agrees with numerical calculations based on the parameters determined independently (Fig. 2D). The maximum value, , is obtained at (Fig. 2E) Sup (). The efficiency exceeds 0.5 over a signal-frequency range of  MHz, which is comparable to the bandwidth of the detector,  MHz Sup ().

In the Fig. 3A, we plot efficiency as a function of the signal pulse length . Here, we fix and at the values which maximize in Fig. 2E. The drive pulse duration is set to be  ns, which empirically maximizes at each . We observe that is a non-monotonic function of and attains a maximum at  ns. The initial increase of at short is due to the narrowing of the signal bandwidth resulting in an improved overlap with the detection bandwidth. For longer , the qubit relaxation limits  Koshino et al. (2015a). Next, we examine how the photon detector behaves when in the signal pulse is varied. Figure 3B shows as a function of for fixed signal pulse lengths at , , and  ns. The detection efficiencies stay constant for regardless of the pulse lengths. This validates the determination of in our measurements using signal pulses in the weak coherent states. For , slightly depends on because of the possibility to drive multiple Raman transitions.

After a single-photon detection event, the qubit remains in the excited state until it spontaneously relaxes to the ground state, which leads to a relatively long dead time of the detector. However, our coherent approach allows us to implement a fast reset protocol (Fig. 4A): in conjunction with the drive pulse that forms the system, we apply a relatively strong reset pulse through the signal port which induces an inverse Raman transition, . We optimize the drive-pulse power and the reset-pulse frequency such that the resulting qubit excitation probability is minimized (Fig. 4B). At the optimal reset point , attains a minimum value , equivalent to the value obtained in the absence of the initial -pulse used to mimic a photon absorption event. Without a reset pulse, we obtain . A comparison of the two results indicates that the reset pulse is highly efficient. Moreover, we confirm that the reset protocol does not affect the succeeding detection efficiency and that the time-gated operation can be repeated in a rate exceeding 1 MHz Sup ().

For the moment, the detection efficiency of this detector is limited by the relatively short qubit relaxation time,  s. Nonetheless, our theoretical work indicates that efficiencies reaching are readily achievable with only a modest improvement of the qubit lifetime Koshino et al. (2015a). An extension from the time-gated-mode to the continuous-mode operation is also possible Koshino et al. (2015b).

This work was partially supported by JSPS KAKENHI (Grant Number 25400417, 26220601, 15K17731), ImPACT Program of Council for Science, and the NICT Commissioned Research.

Appendix A Supplemental material for “Single microwave-photon detector using an artificial -type three-level system”

a.1 Device

Our device is composed of a superconducting coplanar waveguide (CPW) resonator and a superconducting flux qubit (Fig. 1A). The CPW resonator is made of a 50-nm-thick Nb film sputtered on a 300-m-thick undoped silicon wafer with a 300-nm-thick thermal oxide on the surface. It is patterned by electron-beam (EB) lithography using the ZEP520A-7 resist and reactive ion etching. The flux qubit with three Josephson junctions, where one is made smaller than the other two by a factor of , is fabricated by EB lithography and double-angle evaporation of Al using PMMA ( nm)/Ge ( nm)/MMA ( nm) trilayer resist (Fig. 5B). The thicknesses of the bottom and the top Al layers separated by an layer are 20 and 30 nm, respectively. In order to realize a superconducting contact between Nb and Al, the surface of Nb is cleaned by Ar ion milling before the evaporation of Al. The qubit is located at one end of the resonator and is coupled to the resonator dispersively through a capacitance of 4 fF, while it is coupled to the drive port inductively (Fig. 5A).

The flux qubit is always biased with a half flux quantum where the transition frequency of the qubit from the ground state to the excited state is  GHz ( ns during photon-detection experiments), while the resonator frequency is  GHz ( factor ) when the qubit is in the state. It is shifted by a dispersive interaction with the qubit of  MHz, which is enhanced by the straddling effect and the capacitive coupling Inomata et al. (2012) when the qubit is in the state. Note that and denote not their bare frequencies but the renormalized ones including the dispersive shifts Inomata et al. (2014).

A parametric phase-locked oscillator (PPLO) Lin et al. (2014), which is previously operated as a flux-driven Josephson parametric amplifier (JPA) Yamamoto et al. (2008) consists of a superconducting CPW resonator terminated by a dc-SQUID (superconducting quantum interference device). A pump port is coupled to the SQUID loop inductively. The device was fabricated by the planarized Nb trilayer process at MIT Lincoln Laboratory. The resonator and the pump port are made out of a 150-nm-thick Nb film sputtered on a Si substrate covered by a 500-nm-thick SiO layer. A static resonant frequency of the PPLO is  GHz. The PPLO chip is the same as the one used in Ref. Lin et al. (2014).

Figure 5: Image of the qubit-resonator coupled system. (A) False-colored optical image of the device magnified at the qubit part. The qubit (white) is coupled to the center conductor of the coplanar waveguide (CPW) resonator (green) through a capacitance of  fF. (B) Scanning electron micrograph of the three-junction flux qubit. The areas shaded by yellow indicate the Josephson junctions.

a.2 Experimental setup

A schematic of the measurement setup including the wiring in a cryogen-free He/He dilution refrigerator, circuit components, and instruments used in the experiment is shown in Fig. 6.

The qubitresonator and the PPLO circuits are fabricated on separate chips and are separately mounted in microwave-tight packages equipped with an independent coil for the flux bias. They are protected independently by the Cryoperm magnetic shield from an external flux noise such as the geomagnetic field.

Microwave pulses for the drive, signal, and pump ports are generated by mixing the continuous microwaves with pulses which have independent IF frequencies generated by DACs (digital to analog converter) developed by Martinis group at UCSB DAC (). The pulses are applied through the input microwave semi-rigid cables, each with attenuators of dB in total, and DC-blocks for the drive and pump ports. For the signal port, the microwave pulses are further attenuated by 20 dB, and are input to the resonator through a circulator to separate the input and reflected waves. The reflected waves are routed to PPLO via three circulators (- GHz) and are reflected there again, and are propagated through a low-pass ( GHz) and band-pass filters (- GHz), two isolators (- GHz), and the circulator (- GHz) with a termination. Finally, the signals are amplified by a cryogenic HEMT amplifier and a room-temperature amplifier with a total gain of  dB, and mixed with a local oscillator at an I/Q mixer down to the IF frequency. The I component of the reflected signals are sampled at 1 GHz/s by an ADC (analog to digital converter).

Figure 6: Experimental setup diagram.

For the impedance-matching measurement (Fig. 2A), the PPLO is kept off. Namely, pump pulse is off (the output from the DAC in the pump port is turned off) and is far detuned from so that the PPLO acts as a perfect mirror. In other measurements, the PPLO is kept on.

a.3 Input-power calibration

To estimate the photon detection efficiency precisely, calibration of the signal microwave power level on the sample chip is required. For the calibration, we measure the reflection coefficient as a function of the signal microwave frequency and the drive power and determine the impedance-matching points (Fig. 7). Here, we use continuous microwaves for both the signal and the qubit drive, and set the drive frequency at  MHz. We observe two dips representing the impedance matching, similarly to the inset of Fig. 2A. In the limit of weak signal power and no intrinsic loss of the resonator, these dips are expected to appear at the same , where the two radiative decay rates of the system are balanced, and  Koshino et al. (2013), where is the radiative decay rate for the transition in the impedance-matched system. In the actual system, however, the finite population in the level as well as the intrinsic loss of the resonator weakens the elastic photon scattering from the system, and the impedance matching occurs when the radiative decay rates are not balanced, and  Inomata et al. (2014). This yields a difference in the drive power, , between the two dips. is sensitive to the input signal power: As we increase the signal power, the level is more populated and gets larger. Note that the small observed in the inset of Fig. 2A is attributed to the intrinsic loss of the resonator, since the pulsed signal field is sufficiently weak in this measurement.

We use to calibrate the signal power level. We determine the signal power level which reproduces  dB (Fig. 7B) by the numerical simulation, following Ref. Koshino et al. (2013). In the numerical simulation, we employ the following parameters which are estimated by independent measurements: the qubit decay rate  MHz (during this measurement shows  ns) and the ratio of the external and total decay rates of the resonator photon (for other parameters, see “Device” section of this supplementary material). As a result, the signal power is estimated to be  dBm at

Figure 7: (A) Amplitude of the reflection coefficient of a continuous input signal as a function of its frequency and the drive power . Two dips corresponding to absorptions of the input microwave due to the impedance matching are observed. (B) Cross-sections of A at  GHz (blue curve) and  GHz (red curve). Difference in between two dips is  dB.

maximum ( MHz and ) and  dBm at minimum ( MHz and ). Therefore,  dBm. This agrees well with an independent estimation of  dBm by taking into account the total losses in the input port.

a.4 Protocol for single photon detection

In the main text (Fig. 1C), we show the protocol for single photon detection. Here, we present the detailed parameters of the pulses in the protocol.

The drive frequency is set at , where  MHz () is the detuning from the qubit energy, and is fixed through all the experiments described in the main text. The drive pulse is synchronized with the signal pulse with a Gaussian envelope with a length corresponding to its full width at half maximum (FWHM) in the voltage amplitude. The duration of the drive pulse is optimized as  ns so that the signal pulse is completely covered by the drive pulse and is efficiently absorbed by the system. In order to suppress unwanted nonadiabatic qubit excitations, the rising and falling edges of the drive-pulse envelope are smoothed by Gaussian function with FWHM of  ns in the voltage amplitude.

The readout pulse (the frequency  GHz, the length  ns, and the mean photon number ) is applied after the delay of from the center of the drive and signal pulses. The reflected readout pulse works as a locking signal for the PPLO output phase, and the pump pulse (the frequency , the length  ns, and the power  dBm) is applied after  ns. The parametric oscillation signal with either or phase is output from the PPLO during the application of the pump pulse, and the data acquisition time of  ns is required to extract the phase.

a.5 Photon detection efficiency

In Figs. 2B, 2E, and 3A, a mean photon number in a signal pulse is kept to be , which implies that of the weak-coherent signal pulses contain multiple photons. Our detector responds to the multi-photon pulses but cannot discriminate them from single-photon pulses. The efficiency includes those counts.

a.6 Dark count in the detector

Figure 8 shows the dark count probability in the detector, which is the click probability without applying the signal pulse in the pulse sequence of Fig. 1C. The dark count is mainly caused by the nonadiabatic qubit excitation due to the drive pulse and the imperfect initialization. The probability induced by the latter factor is constant and is measured to be , while the probability induced by the former factor depends on the power and the length of the drive pulse and remains finite even with the Gaussian envelope. We determine the dark count probability before and after each measurement of Figs. 2B, 2E, and Fig. 3 and subtracted the averaged value from the measurement result.

Figure 8: Dark count probability in the detector. The data was taken ten times each and averaged before and after the measurement in Fig. 2B. The dark count probability including the imperfect initialization shows at  dBm where the single-photon-detection efficiency hits the maximum. The error bars represent the standard deviation in twenty identical measurements.

a.7 Time constant of an impedance-matched system

We denote the overall decay rate of the resonator by and the radiative decay rate for the transition in the system by . Figure 9 shows as a function of the drive power , calculated based on the experimental parameters. In the experiment, we choose  dBm where the photon detection efficiency reaches the maximum. At this point, . The time constant of the impedance-matched system for the voltage amplitude decay, , is estimated to be  ns, where  MHz. The shortest signal pulse length is  ns in Fig. 3B, which is comparable with .

Figure 9: Radiative decay rates of the impedance-matched system, which are calculated based on the experimental parameters, as a function of the drive power. The two relevant decay rates, and or and , become identical at  dBm, where the impedance matching takes place.

a.8 Protocol for reset

In the main text (Fig. 4A), we show the reset protocol for the single photon detection. Here, we describe how to optimize the parameters of pulses in the protocol.

We first apply a pulse with the length of  ns to directly excite the qubit from the to the state. Then, we apply the drive and reset pulses to induce the transition. To find the operating point to maximize the resetting efficiency, we swept the frequency of the reset pulse and the drive power . After fixing and , we optimize the drive pulse length , and the mean photon number in the reset pulse to minimize . Finally, we measure as a function of and using the reset protocol with optimized parameters. Parameters for the readout and pump pulses are the same as the ones in Fig. 1C.

At the optimal reset point, shows the minimum value of , which results in twice larger occupation of the qubit excited state compared to obtained in the initialization in the equilibrium condition. This indicates the small probability of unwanted nonadiabatic excitations by the drive pulse in the reset protocol.

We demonstrate the microwave photon detection combined with the fast reset protocol. We apply the drive and the signal pulses (the same conditions as in the measurement in Fig. 2B) after the reset protocol and readout the qubit. We accomplish of which is consistent with the maximum obtained in Fig. 2E. This indicates that the photon detection efficiency is unaffected by the reset protocol.

It takes  ns to reset the system and  ns to detect the single photon for  ns. Both of the durations are determined by the drive pulse widths including  ns. The qubit readout is completed by accumulating data for  ns after  ns. The period of the single photon detection including the reset protocol is  ns, which allows the photon counting rate of  MHz.

Footnotes

  1. thanks: These authors contributed equally to this work.

References

  1. R. H. Hadfield, Nature Photon. 3, 696 (2009).
  2. N. Gisin, G. Ribordy, W. Tittel,  and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).
  3. E. Knill, R. Laflamme,  and G. J. Milburn, Nature (London) 409, 46 (2001).
  4. J. L. O’Brien, Science 318, 1567 (2007).
  5. S. Aaronson, Proc. R. Soc. A 467, 3393 (2011).
  6. M. D. Eisaman, J. Fan, A. Migdall,  and S. V. Polyakov, Rev. Sci. Instrum. 82, 071101 (2011).
  7. N. Bergeal, R. Vijay, V. E. Manucharyan, I. Siddiqi, R. J. Schoelkopf, S. M. Girvin,  and M. H. Devoret, Nat. Phys. 6, 296 (2010).
  8. C. Macklin, K. O’Brien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver,  and I. Siddiqi, Science 350, 307 (2015).
  9. C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais,  and A. Wallraff, Nat. Phys. 9, 345 (2013).
  10. G. Romero, J. J. Garca-Ripoll,  and E. Solano, Phys. Rev. Lett. 102, 173602 (2009).
  11. B. Peropadre, G. Romero, G. Johansson, C. M. Wilson, E. Solano,  and J. J. García-Ripoll, Phys. Rev. A 84, 063834 (2011).
  12. Y.-F. Chen, D. Hover, S. Sendelbach, L. Maurer, S. T. Merkel, E. J. Pritchett, F. K. Wilhelm,  and R. McDermott, Phys. Rev. Lett. 107, 217401 (2011).
  13. Y. Yin, Y. Chen, D. Sank, P. J. J. O’Malley, T. C. White, R. Barends, J. Kelly, E. Lucero, M. Mariantoni, A. Megrant, C. Neill, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland,  and J. M. Martinis, Phys. Rev. Lett. 110, 107001 (2013).
  14. T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds,  and K. W. Lehnert, Nature (London) 495, 210 (2013).
  15. J. Wenner, Y. Yin, Y. Chen, R. Barends, B. Chiaro, E. J. J. Kelly, A. Megrant, J. Y. Mutus, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, T. C. White, A. N. Korotkov, A. N. Cleland,  and J. M. Martinis, Phys. Rev. Lett. 112, 210501 (2014).
  16. S. R. Sathyamoorthy, L. Tornberg, A. F. Kockum, B. Q. Baragiola, J. Combes, C. M. Wilson, T. M. Stace,  and G. Johansson, Phys. Rev. Lett. 112, 093601 (2014).
  17. L.-M. Duan and H. J. Kimble, Phys. Rev. Lett. 92, 127902 (2004).
  18. D. E. Chang, A. S. Sørensen, E. A. Demler,  and M. D. Lukin, Nat. Phys. 3, 807 (2007).
  19. D. Witthaut and A. S. Sørensen, New J. Phys. 12, 043052 (2010).
  20. H. Zheng, D. J. Gauthier,  and H. U. Baranger, Phys. Rev. Lett. 111, 090502 (2013).
  21. O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Jr., Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura,  and J. S. Tsai, Science 327, 840 (2010).
  22. I.-C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B. Peropadre,  and P. Delsing, Phys. Rev. Lett. 107, 073601 (2011).
  23. A. F. van Loo, A. Fedorov, K. Lalumire, B. C. Sanders, A. Blais,  and A. Wallraff, Science 342, 1494 (2013).
  24. K. Koshino, K. Inomata, T. Yamamoto,  and Y. Nakamura, Phys. Rev. Lett. 111, 153601 (2013).
  25. K. Inomata, K. Koshino, Z. R. Lin, W. D. Oliver, J. S. Tsai, Y. Nakamura,  and T. Yamamoto, Phys. Rev. Lett. 113, 063604 (2014).
  26. D. Pinotsi and A. Imamoglu, Phys. Rev. Lett. 100, 093603 (2008).
  27. I. Shomroni, S. Rosenblum, Y. Lovsky, O. Bechler, G. Guendelman,  and B. Dayan, Science 345, 903 (2014).
  28. K. Koshino, K. Inomata, Z. Lin, Y. Nakamura,  and T. Yamamoto, Phys. Rev. A 91, 043805 (2015a).
  29. K. Koshino, S. Ishizaka,  and Y. Nakamura, Phys. Rev. A 82, 010301(R) (2010).
  30. K. Inomata, T. Yamamoto, P.-M. Billangeon, Y. Nakamura,  and J. S. Tsai, Phys. Rev. B 86, 140508(R) (2012).
  31. Z. R. Lin, K. Inomata, W. D. Oliver, K. Koshino, Y. Nakamura, J. S. Tsai,  and T. Yamamoto, Nat. Commun. 5, 4480 (2014).
  32. See Supplemental Material for details on the materials and methods, which includes Refs. [34, 35].
  33. K. Koshino, Z. Lin, K. Inomata, T. Yamamoto,  and Y. Nakamura, arXiv:1509.05858  (2015b).
  34. T. Yamamoto, K. Inomata, M. Watanabe, K. Matsuba, T. Miyazaki, W. D. Oliver, Y. Nakamura,  and J. S. Tsai, Appl. Phys. Lett. 93, 042510 (2008).
  35. URL: http://web.physics.ucsb.edu/martinisgroup
    /electronics.shtml.
103579
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
Edit
-  
Unpublish
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel
Comments 0
Request answer
""
The feedback must be of minumum 40 characters
Add comment
Cancel
Loading ...