Ultrawide-range photon number calibration using a hybrid system combining nano-electromechanics and superconducting circuit quantum electrodynamics
Philip Schmidt, Daniel Schwienbacher, Matthias Pernpeintner, Friedrich Wulschner, Frank Deppe, Achim Marx, Rudolf Gross, Hans Huebl
UUltrawide-range photon number calibration using a hybrid system combiningnano-electromechanics and superconducting circuit quantum electrodynamics
Philip Schmidt,
1, 2, 3, a) Daniel Schwienbacher,
1, 2, 3
Matthias Pernpeintner,
1, 2, 3
Friedrich Wulschner,
1, 2
FrankDeppe,
1, 2
Achim Marx, Rudolf Gross,
1, 2, 3 and Hans Huebl
1, 2, 3, b) Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Walther-Meißner-Str. 8, 85748 Garching,Germany Physik-Departement, Technische Universit¨at M¨unchen, James-Franck-Str. 1, 85748 Garching,Germany Nanosystems Initiative Munich, Schellingstraße 4, 80799 M¨unchen, Germany (Dated: 17 August 2018)
We present a hybrid system consisting of a superconducting coplanar waveguide resonator coupled to ananomechanical string and a transmon qubit acting as nonlinear circuit element. We perform spectroscopyfor both the transmon qubit and the nanomechanical string. Measuring the ac-Stark shift on the transmonqubit as well as the electromechanically induced absorption on the string allows us to determine the aver-age photon number in the microwave resonator in both the low and high power regimes. In this way, wemeasure photon numbers that are up to nine orders of magnitude apart. We find a quantitative agreementbetween the calibration of photon numbers in the microwave resonator using the two methods. Our ex-periments demonstrate the successful combination of superconducting circuit quantum electrodynamics andnano-electromechanics on a single chip.The field of optomechanics allows to investigate theinteraction of light with mechanical degrees of free-dom. It enables the optical readout of the mechanicaldisplacement as well as the control of the mechanicalstate. Over the past decade, optomechanics has beensuccessfully used to study the interplay between mechan-ical modes and quantized electromagnetic waves in a res-onator on the quantum level . One successful experi-mental implementation is based on superconducting cir-cuits as they, straightforwardly, can be operated in theresolved sideband limit .In addition, superconducting nano-electromechanicalcircuits are compatible with the field of circuit quan-tum electrodynamics (cQED) regarding fabricationtechnology, operation temperature and frequency range.In cQED, the strong and ultra-strong couplingregimes have been achieved and the generationof non-classical states of microwave light is wellestablished . Therefore, the combination of nano-electromechanics with cQED is an ideal approach to delveinto the quantum nature of mechanical motion.Recent experiments show that the combination ofa superconducting qubit, a microwave resonator and ananomechanical element can enhance the phonon-photoninteraction, allow for the controlled preparation of non-classical phonon states, and enable entanglement gener-ation. One envisaged state preparation protocol is thegeneration of a non-classical microwave state in a mi-crowave resonator coupled to both a qubit and a me-chanical resonator. It makes use of a well-defined qubitstate and its transfer to the mechanical system via a red-sideband drive pulse. For this, one critical parameter is a) Electronic mail: [email protected] b) Electronic mail: [email protected] the average photon number in the microwave resonator.Here, we present an experimental study of a hybridquantum system consisting of a transmon qubit, a dou-bly clamped high-Q nanomechanical string resonator,and a superconducting microwave resonator. We showthat the average photon number determined from the ac-Stark shift and electromechanically induced absorption(EMIA) measurements are in good quantitative agree-ment.The average photon number inside a λ/ κ ext is given by ¯ n c = 2 P appl (cid:126) ω p (cid:0) κ + 4∆ (cid:1) Λ κ ext (cid:124) (cid:123)(cid:122) (cid:125) ≡ x . (1)Here, (cid:126) is the reduced Planck constant, ∆ p ≡ ω p − ω c de-notes the detuning between the probe tone frequency ω p and the resonant frequency ω c . Additionally, κ ≡ κ int + κ ext describes the total loss rate of the microwave res-onator given by the sum of the internal ( κ int ) and exter-nal ( κ ext ) losses. The applied power P appl describes thetotal calibrated output power of the microwave sourcesbefore sending it to the dilution refrigerator. Since theattenuation Λ of the microwave lines and κ ext can onlybe estimated, we introduce the product Λ · κ ext as a cal-ibration factor x in Eq. (1). We demonstrate that x canbe quantified and corroborate its value via two indepen-dent approaches: (i) We measure the ac-Stark shift ofthe qubit transition frequency as a function of P appl inthe dispersive regime which we will call x qb and (ii)we measure EMIA resulting from the electromechanicalinterference effect between the anti-Stokes field of thecoupled electromechanical resonator system and a probefield determining x EMIA .Taking the linewidth κ of the microwave resonator intoaccount, the ac-Stark shift for a transmon qubit is given a r X i v : . [ qu a n t - ph ] A ug Ω m ω d κ Ω m anti-StokesStokes ω c ω p ∆ mc ωω q (n c )ω q,0 δω∆ qc s Ω m Γ eff Γ m ∆ p FIG. 1.
Eigenfrequencies and excitation tones of the hybrid system consisting of a superconducting microwave resonator, atransmon qubit, and a nanomechanical string resonator.
The mechanical element (blue) has an eigenfrequency Ω m . For theso-called red-sideband configuration, the frequency ω d of the microwave drive tone is chosen to position the anti-Stokes field ator close to the resonance frequency of the microwave resonator ω c . When the frequency ω p of the probe tone is resonant withthe anti-Stokes field, electromechanically induced absorption is observed. The average photon number ¯ n c causes an ac-Starkshift δω (¯ n c ) of the bare qubit transition frequency ω q, . The photon number dependent qubit frequency, ω q (¯ n c ) = ω q,0 − δω ,is determined using a spectroscopy tone with frequency ω s . by δω = 2 g ∆ qc αα + ∆ qc ¯ n c ( κ ) , (2)when probing the microwave resonator on resonance(∆ p = 0). Here, the coupling between the transmonand the microwave resonator is g q and ∆ qc = ω q − ω c (cf. Fig. 1). The non-linearity of the transmon qubit isdefined by α , quantifying the deviation in energy of thesecond mode from twice the ground mode. Evidently, wecan determine ¯ n c and, in turn, the calibration factor x qb by measuring δω if we know the qubit parameters.Next, we turn to EMIA which is known to result in anincrease of the linewidth Γ m of the mechanical oscillator,leading to an effective linewidth Γ eff given by Γ eff = Γ m (cid:18) g κ Γ m P appl x EMIA (cid:126) ω d ( κ + 4∆ ) (cid:19) . (3)Thus, measuring Γ eff as a function of P appl allows us todetermine the calibration factor x EMIA , if we know therelevant parameters of the mechanical oscillator and theelectromechanical vacuum coupling constant g m0 . Forthis experiment, we chose ω d = ω c − Ω m , i.e., ∆ mc = − Ω m (cf. Fig. 1).To quantitatively compare the resonator photon num-bers determined with the two methods described above,we fabricate a hybrid device consisting of a superconduct-ing microwave resonator, a transmon qubit and a doublyclamped nanomechanical string resonator, as depicted inFig. 2. All these parts consist of superconducting alu-minum thin films deposited by electron beam evaporationon a single crystalline silicon substrate. Patterning of themicrowave and nanomechanical resonator is achieved byelectron beam lithography and a lift-off process. After Aldeposition, the sample is annealed at 300 ◦ C for 30 min-utes to generate a high tensile stress in the aluminum thinfilm. Then, the transmon qubit is defined again by elec-tron beam lithography and fabricated using a two-angleshadow evaporation (see Ref. 37 and Ref. 38). In the last step, we release the nanostring resonator by reactive ionetching and critical point drying.The 60 µ m long, 120 nm thick, and 230 nm wide nano-string resonator has a mass of about 2 pg. It is sepa-rated by a 160 nm gap from the ground plane resultingin g m0 / π = 0 .
31 Hz. At the experimental temperatureof T cryo (cid:39)
50 mK we observe a mechanical resonance fre-quency of Ω m / π = 3 . m / π = 12 . µ s.Tuning the transmon qubit to its minimum frequency,far away from the resonator frequency, we find the baremicrowave resonator frequency ω c / π = 5 .
875 GHz. Itslinewidth depends on the setpoint of the transmon qubitand the photon occupation inside, see supplementary ma-terial for details.As further detailed in the supplementary material,we find for the transmon qubit an eigenfrequency of ω q / π = 7 .
916 GHz at the sweet spot, corresponding toa detuning of ∆ qc / π = 2 .
056 GHz, a transmon nonlin-earity of α/ π = −
188 MHz, and a transmon-resonatorcoupling of g q / π = 134 MHz.For the measurement of the qubit transition frequencyas a function of P appl in the few photon regime, we tunethe qubit to its sweet spot by an applied magnetic field.We then perform two-tone spectroscopy by driving thetransmon qubit via its antenna while probing the res-onator transmission with a microwave tone of varyingpower P appl . In this way, we obtain the qubit frequencyas a function of P appl . In addition, we determine the mi-crowave resonator linewidth as ¯ n c is influenced by thisparameter. The product δω · κ is shown in Fig. 3. Asexpected [c.f. Eq. (2)], this product linearly depends on P appl . We find a slope of − (2 π ) · (1 . ± . · / s nW.Combining this slope, as well as Eq. (1), and the systemparameters, we obtain x qb = (5 . ± .
23) s − . Note thatwe used probe powers corresponding to ¯ n c ≤
28, well be-low the critical photon number of ¯ n crit = ∆ / (2 g q ) ≈
60, set by the assumptions of the dispersive limit . VNA in out SAin ϕ ω s - B c r y o s t a t - B cpw resonator stringtransmon (c) (b) μm μm (a)
80 µm (d) ω d ω lo ω p FIG. 2.
Sample layout, images, and spectroscopy setup.
Panel(a) shows the chip layout including the λ/ µ m long nanostring including an enlarged view of the re-gion close to the right clamp. The enlarged image is tiltedto view the successful release. An additional antenna struc-ture is placed close to the qubit (green) in panel (c). In panel(d) multiple microwave sources act as red-sideband drive tone ω d , qubit spectroscopy tone ω s , and local oscillator ω lo . Thevector network analyzer (VNA) supplies the probe tone andallows direct analysis of the transmitted signal. In addition,a spectrum analyzer is used to analyze the sideband fluctu-ations of the mechanical element. The output signal of themicrowave resonator is preamplified with a cryogenic HEMTamplifier at 4 K, followed by post-amplification at room tem-perature (not shown). For the determination of the resonator photon numbersat higher occupations, we turn to the two-tone EMIAspectroscopy scheme (cf. Fig. 1). We set the red-sidebanddrive tone to ω d = ω c − Ω m and probe the anti-Stokesfield with the probe tone ω p , close to ω c . Dependingon the red-sideband drive amplitude, i.e. ¯ n c , we ob-tain the spectra depicted in Fig. 4(a). This figure showsthe EMIA, which manifests as an additional absorptionaround ω p = ω d + Ω m . By fitting a Lorentzian lineshapeto the EMIA data, we extract the effective interferencelinewidth Γ eff . A quantitative analysis of the transmis-sion data shows an EMIA signature with a minimum cor- dw k / (2 p )3 (MHz)3 P a p p l ( n W )0 1 0 2 0n c FIG. 3.
Ac-Stark shift of the transmon.
The product δωκ (green dots) is plotted as a function of the applied microwavepower P appl . A linear model [see Eq. (2)] is fitted to the data(black solid line). The determined photon numbers are shownon the top axis. - 1 0 0 0 1 0 00 . 0 00 . 2 50 . 5 00 . 7 51 . 0 00 . 0 0 0 . 0 5 0 . 1 01 02 03 0 P a p p l ( W ) 0 . 0 9 7 0 . 0 2 8 0 . 0 0 3 |S21,n|2 (w p - w d - W m ) / 2 p ( H z ) G e f f / 2 p G eff / 2 p (Hz) P a p p l ( W )0 . 0 0 . 4 0 . 8 1 . 2( b ) n c ( 1 0 )( a ) FIG. 4.
EMIA spectroscopy on the electromechanical hybridsystem.
Panel (a) shows the normalized EMIA signature (dot-ted) for three drive powers P appl . The EMIA dip deepens andwidens with increasing drive power. Lorentzian models arefitted to the data (black solid lines) determining Γ eff . Panel(b) displays the extracted effective linewidth Γ eff (blue dots)as a function of the red-sideband drive amplitude at a fridgetemperature of about 50 mK. Using the model described inthe main text (black solid line), we find the calibration factor x EMIA and the photon numbers in the microwave resonator(top axis). responding to 25 % of the unperturbed microwave trans-mission parameter | S | .Figure 4 (b) displays the extracted EMIA linewidth Γ eff as a function of the applied red-sideband drive power,confirming the linear increase predicted by Eq. (3). Thecalibration factor x EMIA is determined by fitting Eq. (3)to the data. Having characterized the sample parameters,the calibration factor x EMIA and the intrinsic linewidthΓ m are the remaining free parameters in the model. Weobtain x EMIA = (5 . ± .
25) s − and Γ m / π = (12 . ± .
3) Hz. Using these results in combination with Eq. (1),we can determine the average photon numbers in themicrowave resonator in a range from about 10 to 10 photons.In conclusion, we have successfully implemented a su-perconducting coplanar microwave resonator coupled toa transmon qubit and a nanomechanical string. Boththe coupled transmon-MW resonator system and thenano-electromechanical system were investigated usingmicrowave spectroscopy. These experiments allowed usto calibrate the MW resonator photon numbers by mea-suring the ac-Stark shift of the transmon qubit in therange from 0 . x qb = (5 . ± .
23) s − . For higher photonnumbers, we used EMIA spectroscopy of the nanome-chanical string to probe the resonator in a populationrange from 1 . × to 1 . × photons. By analyzingthe linewidth of the transmission signature, we found acalibration factor x EMIA = (5 . ± .
25) s − . Both at-tenuation coefficients x qb and x EMIA agree within 5%.Please also note that these two methods determine aver-age resonator photon numbers that are up to nine ordersof magnitude apart.The implementation of a transmon qubit, a high-quality nanostring resonator, and a microwave resonatoron a single chip represents an important step towards therealization of quantum information storage in the vibra-tional degree of freedom of a mechanical element. Dou-bly clamped string resonators are interesting candidatesin this context, due to their high mechanical quality fac-tors, above 10 , corresponding to a thermal coherencetime from micro- to milliseconds at a moderate dilutionfridge temperature of 50 mK. ACKNOWLEDGMENTS
This project has received funding from the EuropeanUnion’s Horizon 2020 research and innovation programunder grant agreement No 736943. We gratefully ac-knowledge valuable scientific discussions with J. Goetz,S. Weichselbaumer, and E. Xie.
Appendix A: Transmon qubit
The transmon qubit is positioned at the electric fieldanti-node of the coplanar waveguide resonator and is ca- w q w g f / a / w q Df (deg) w s / 2 p ( G H z ) a / 2 w g f / F e x t / F w p / 2 p (GHz)
01 02 0| S | ( a r b . u . )( a ) A- 0 . 1 0 . 0 0 . 17 . 77 . 87 . 9 F e x t / F w s / 2 p (GHz)
01 02 03 0 Df ( d e g ) FIG. 5.
Spectroscopy of the coupled resonator-transmon qubitsystem (a) Microwave transmission | S | as a function of theapplied magnetic flux Φ ext . A periodic behavior is found forinteger flux ratios. When the transmon qubit is in resonancewith the microwave resonator an avoided crossing is observed.(b) Transmon transition frequency ω q as a function of the ap-plied flux, measured by two-tone spectroscopy (phase changeof the probe signal, right axis). The maximum frequency isobserved at zero field. (c) High power spectroscopy data of thetransmon qubit at the sweet spot Φ ext / Φ = 0 (green dots)show the single photon | g (cid:105) ↔ | e (cid:105) transition at ω q as well asthe two photon | g (cid:105) ↔ | f (cid:105) transition at ω gf , as schematicallydepicted on the right. pacitively coupled to it. The qubit transition frequencycan be varied by applying a magnetic flux Φ to the dc-SQUID forming the tunable Josephson junction of thetransmon qubit.Figure 5(a) shows the transmission of a weak probetone ω p through the microwave resonator as a func-tion of the applied magnetic flux for an average photonnumber of 1.7. At Φ ext / Φ = ± .
28, we find a pro-nounced anti-crossing demonstrating the strong couplingbetween the transmon qubit and the resonator. Fromthe peak separation at the avoided crossing, for occupa-tions below one photon on average, we find a couplingstrength of g q / π = (134 . ± .
3) MHz. Additionally,we find the expected periodic flux dependence of thequbit transition frequency. When the transmon qubit istuned to its minimum frequency, e.g. at Φ ext / Φ = 0 . ω c / π = 5 .
875 GHz, with a linewidth of κ/ π = (1 . ± . . Thus, driving the qubit with ω s , allowsto perform a two-tone spectroscopy of the qubit, as shownin Fig. 5(b) and (c). From panel (b) we determine a qubitfrequency ( | g (cid:105) ↔ | e (cid:105) transition) of ω q / π = 7 .
916 GHz atthe sweet spot of Φ ext / Φ = 0.To access the transmon anharmonicity α we increasethe amplitude of the drive tone at Φ ext / Φ = 0. For highdrive powers two- and multi-photon processes becomevisible . In particular, we observe the two-photontransition | g (cid:105) ↔ | f (cid:105) at ω gf / π = (7 . ± . α/ (cid:126) = 2 ω s − ω gf = − π · (188 ± · s − . For transmonqubits the negative anharmonicity is equivalent to thecharging energy ( − α = E C ) . Via ω c = √ E C E J / (cid:126) we find an E J /E C ratio of 222 for the transmon qubit. Appendix B: Calibration of electromechanical couplingstrengths
Next, we turn to the characterization of the nano-electromechanical system. We use the thermal fluctu-ations of the nanostring, similar to Refs. [32, 33, 44, and45], to determine the electromechanical vacuum couplingconstant g m0 / π = (0 . ± . ω d = ω c to probe the frequency fluctuations of the microwave res-onator, caused by the thermal motion of the nanostringresonator. The transmitted signal is down-converted us-ing a homodyne setup and analyzed with a spectrumanalyzer. The inset of Fig. 6 shows the microwave res-onator sideband noise spectroscopy data representing thethermal motion of the nanostring at 365 mK. We find amechanical resonance frequency Ω m / π = 3 . m / π = (33 . ± .
1) Hz, correspond-ing to a mechanical quality of about Q m (cid:39)
94 000. Thesharp peak on the left side originates from the frequencymodulation of the probe tone (with a modulation fre-quency of Ω mod / π = 3 . φ / π = 80 Hz) used for the calibration of thephase response of the microwave resonator (for details, < dw
2> / 4 p T ( m K ) SP (pW/Hz) W / ( 2 p ) ( M H z )- 3 0 0 0 3 0 0 (W - W m ) / 2 p ( H z ) FIG. 6.
Thermal fluctuations of the nanostring plotted versustemperature.
The inset displays the noise power spectral den-sity of the nanostring at 365 mK. The mechanical resonatorhas an eigenfrequency of Ω m / π = 3 . . The average fluctuations of the resonator arecalculated by a reference peak (left). In the full frame thesefluctuations are plotted versus the sample temperature (bluedots). The slope s , which yields the electromechanical vac-uum coupling according to Eq. (B2), is determined from thelinear dependency. see Refs. 32, 33, 44, and 45). A quantitative comparisonof this amplitude calibration peak S mod with the ampli-tude of the thermal motion peak S pp yields the integrateddisplacement noise (cid:104) δω (cid:105) and hence the vacuum couplingvia : (cid:104) δω (cid:105) = (cid:90) ∞−∞ S ωω ( ω ) dω π = φ Ω Γ m · S pp S mod = 2 g ¯ n m . (B1)Here, we use the measurement bandwidth ENBW = 1 Hz.By repeating this measurement for various tempera-tures, we can calibrate the mechanical coupling with afinite back-action temperature (¯ n m → T ba + k B T / (cid:126) Ω m ).Figure 6 shows the integrated displacement noise as afunction of the temperature T . We find a slope of s/ (2 π ) = (1 . ± . / K and hence a vacuumcoupling of g m0 / π = (cid:114) s (cid:126) Ω m k B = (0 . ± . . (B2) Appendix C: Linewidth of the microwave resonator
As the linewidth (loss rate) of the microwave resonatorinfluences the average photon number in the resonator,see Eq. (1) in the manuscript, we analyze it for each ofthe two calibration methods. We note that for the twomethods different working points of the transmon qubitare used, indicated by A and B in Fig. 5(a), resulting in k / 2 p (MHz) P a p p l ( n W ) w c / 2 p = 5 . 8 6 2 G H z k / 2 p (MHz) P a p p l ( W )( a ) w c / 2 p = 5 . 8 7 5 G H z FIG. 7.
Dependence of the microwave resonator linewidth onthe applied power.
Panel (a) displays the extracted linewidthin the qubit regime (green dots) for a resonator frequency of ω/ (2 π ) = 5 .
862 MHz [point A in Fig. 5(a)]. The applied powerrange is 0 . − . . −
97 mW applied power,where a non-trivial behavior is found. Here, the operationpoint of the transmon qubit is set to point B in Fig. 5(a). different behaviors. The observed dependencies are de-picted versus applied power in Fig. 7 for the qubit (elec-tromechanical) regime in panels (a) and (b), correspond-ing to the working points A and B, respectively.The linewidth for the coupled qubit-resonator systemis measured from 22 pW up to the critical photon num-ber. We find a linear dependence with an offset of(1 . ± .
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