A Fast and Ultrasensitive Electrometer Operating at the Single-Photon Level
B. L. Brock, Juliang Li, S. Kanhirathingal, B. Thyagarajan, M. P. Blencowe, A. J. Rimberg
AA Fast and Ultrasensitive Electrometer Operating at the Single-Photon Level
B. L. Brock, ∗ Juliang Li, † S. Kanhirathingal, B. Thyagarajan, M. P. Blencowe, and A. J. Rimberg ‡ Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA (Dated: March 3, 2021)We demonstrate fast and ultrasensitive charge detection with a cavity-embedded Cooper pairtransistor via dispersive readout of its Josephson inductance. We report a minimum charge sensi-tivity of 14 µe/ √ Hz with a detection bandwidth on the order of 1 MHz using 16 attowatts of power,corresponding to the single-photon level of the cavity. In addition, our measured sensitivities arewithin a factor of 5 of the quantum limit for this device. This is the first ultrasensitive electrometerreported to operate at the single-photon level and its sensitivity is comparable to rf-SETs, whichtypically require picowatts of power. Our results support the feasibility of using this device tomediate an optomechanical interaction that reaches the single-photon strong coupling regime.
Fast and ultrasensitive electrometers have been instru-mental to the advancement of basic science. They havebeen used to detect in real time the tunneling of electronsin a quantum dot [1], determine the tunneling rates ofquasiparticles in superconducting devices [2], and studythe properties of Majorana zero modes in nanowires [3].In addition, the rapid detection of single electrons is cru-cial for the readout of quantum-dot-based qubits [4], forwhich operating at lower photon numbers reduces mea-surement backaction [5]. In this same vein, ultrasensitiveelectrometers are at the heart of many schemes for sens-ing the displacement of charged mechanical resonators[6–8], as well as for coherently coupling mechanical res-onators to microwave cavities [9–11]. To observe and takeadvantage of quantum effects in such hybrid systems it isoften essential that their coupling be strong at the single-photon level, a regime that has been achieved for quan-tum dots [12, 13] but not yet for mechanical resonatorsdespite significant effort [14–17]. Reaching the single-photon strong optomechanical coupling regime, where asingle cavity photon causes sufficient radiation pressureto displace the mechanical resonator by more than itszero-point uncertainty, would enable the generation ofnonclassical states of both light and motion [18, 19], aswell as provide a rich platform for studying the quantum-to-classical transition and other fundamental physics [20].Electrometers based on the single electron transistor(SET) are among the fastest and most sensitive reportedin the literature to date. Radio-frequency single electrontransistors (rf-SETs) are the best known of these devices,having achieved sensitivities below 1 µe/ √ Hz [21] andbandwidths greater than 100 MHz [22]. The rf-SET func-tions by encoding the charge gating the SET in the powerdissipated by the SET, which is embedded in a tank cir-cuit to enable radio-frequency readout of this dissipation.This dissipative detection typically requires picowatts ofpower, corresponding to hundreds of thousands of pho-tons in the tank circuit, which renders the rf-SET un-suitable for some of the aforementioned applications andmakes it impossible to integrate the rf-SET with mod-ern near-quantum-limited parametric amplifiers [23–25](which typically saturate well below the picowatt scale). Dispersive electrometers based on the SET have also beendeveloped, which encode the gate charge in the resonantfrequency of a tank circuit. Such electrometers have beenoperated using femtowatts of power [2, 26], correspondingto tens or hundreds of photons, and have achieved chargesensitivities as low as 30 µe/ √ Hz [27]. More recently, dis-persive gate-based sensors have been developed [28] thathave surpassed the performance of SET-based electrom-eters. These devices have achieved sensitivities as lowas 0 . µe/ √ Hz with detection bandwidths approaching1 MHz while using only 100 attowatts of power, corre-sponding to hundreds of photons [29].In this letter we demonstrate ultrasensitive dispersivecharge detection with a cavity-embedded Cooper pairtransistor (cCPT) [30, 31]. Using 16 attowatts of power,corresponding to the single-photon level of the cavity,we measure a minimum charge sensitivity of 14 µe/ √ Hz.Relative to theory, we find that the cCPT operates withina factor of 5 of its quantum-limited sensitivity, this dis-crepancy being due to the presence of frequency fluctu-ations comparable to the cavity linewidth, the inherentnonlinearity of the device, and the noise of our amplifierchain. Another major limitation of the present deviceis quasiparticle poisoning [32], which prevents us fromstudying the cCPT at its theoretically-optimal operatingpoint. Based on these results we expect an optimizedsample could realistically achieve a sensitivity as low as0 . µe/ √ Hz, rivaling that of the best gate-based sensor[29]. This is the first ultrasensitive electrometer reportedto operate at the single-photon level; for this reason, thecCPT has been proposed as a platform for reaching thesingle-photon strong coupling regime of optomechanics[9]. Our results support the feasibility of this proposaland represent an important step toward its realization.We study the same device characterized experimen-tally in Ref. [30]. The cCPT, depicted schematically inFig. 1(a), has two components: a quarter-wavelength( λ/
4) coplanar waveguide cavity and a Cooper pair tran-sistor (CPT). The CPT consists of two Josephson junc-tions (JJs) with an island between them that can begated via the capacitance C g . The CPT is connected be-tween the voltage antinode of the cavity and the ground a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r FIG. 1. (a) Schematic of the cCPT. (b) Schematic of themeasurement circuitry. The cavity behaves as a parallel RLCcircuit when driven near its fundamental frequency [33], andthe CPT behaves as an inductance L J in parallel with thecavity. (c) System noise referred to the sample plane (solidblack line). Shaded areas show the contribution of each noisesource. The dashed white line is the quantum limit. plane, such that the two form a SQUID loop. Embeddedin this way, the CPT behaves as a nonlinear Josephson in-ductance L J in parallel with the cavity that can be tunedby both the number of electrons n g gating the island andthe flux Φ ext threading the SQUID loop. The gate charge n g is thus encoded in the resonant frequency ω of thecavity, which can then be detected via microwave reflec-tometry. The theoretical charge sensitivity of the cCPTin this mode of operation is derived from first principlesin Ref. [31]. This device can be operated at much lowerpowers than comparable SET-based dispersive electrom-eters [2, 26, 27] for two key reasons. First, we use a dis-tributed superconducting microwave cavity rather thana lumped-element LC circuit, yielding much lower dissi-pation. Second, we can tune the CPT band structure viathe external flux Φ ext , which provides us greater flexibil-ity in biasing the device to an optimally-sensitive point.To measure the charge sensitivity of the cCPT we drive the cavity with a resonant carrier signal while mod-ulating the gate about a dc bias point n g such that n g ( t ) = n g + √ q rms /e ) cos( ω g t ), which in turn mod-ulates the resonant frequency such that ω ( t ) = ω + √ ∂ω /∂n g )( q rms /e ) cos( ω g t ). As a result, the reflectedcarrier signal is phase-modulated leading to output power P out proportional to q at the sideband frequencies ω ± ω g . Since the charge sensitivity δq is defined asthe rms charge amplitude per √ Hz that yields a signalto noise ratio of unity, it can be expressed as δq = q rms (cid:115) S noise P out ( ω + ω g ) + P out ( ω − ω g ) . (1)where S noise is the power spectral density of the noisefloor near the sideband frequencies. Here we considerthe total power at the two sidebands to be the signalof interest, since it is possible to combine them via ho-modyne mixing [21]. Thus, given the rms charge mod-ulation amplitude q rms , we can use a spectrum analyzerwith resolution bandwidth B to measure both the outputsideband power and the noise floor P noise = B × S noise ,from which we can extract the charge sensitivity δq . As-suming symmetric sidebands, Eq. (1) can be rewrittenin the more familiar form δq = q rms / √ B × SNR / ,where SNR is the single-sideband signal to noise ratioexpressed in decibels [21, 34]. In practice, however, weobserve asymmetric sidebands and therefore use Eq. (1)to extract the charge sensitivity from our measurements.We do not yet understand the origin of this asymmetry,which can be several dB, but we find it is correlated withthe detuning of the carrier from resonance such that theright sideband tends to be larger than the left for nega-tive detuning and vice versa for positive detuning.Theoretically, P out ( ω ± ω g ) can be expressed as [30] P out ( ω ± ω g ) = 2 κ κ ( ω g + κ / (cid:12)(cid:12)(cid:12)(cid:12) q rms e ∂ω ∂n g (cid:12)(cid:12)(cid:12)(cid:12) P in , (2)where κ ext and κ tot are the external and total dampingrates of the cavity, respectively, and P in is the input car-rier power at the plane of the sample. The theoreticalcharge sensitivity can therefore be expressed as [31] δq = κ tot κ ext (cid:115) S noise P in (cid:18) ω g + κ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ∂ω ∂n g (cid:12)(cid:12)(cid:12)(cid:12) − e. (3)To evaluate this expression we use the sample-referred S noise and P in (discussed below), as well as the values of κ ext , κ tot , and ω ( n g , Φ ext ) determined from a detailedcharacterization of the device [30]. The damping ratesare approximately κ ext / π ≈ . κ tot / π ≈ . ω ( n g , Φ ext ) and canvary by 10% − S QLnoise = (cid:126) ω , as discussed below.It is important to note that both Eqs. (1) and (3) areonly valid when q rms /e (cid:28) ω g / ( ∂ω /∂n g ), which ensuresthat the amplitude of the resulting frequency modulationis small compared to ω g and that P out ( ω ± ω g ) is propor-tional to q . In all of our measurements we use suffi-ciently small q rms to satisfy this constraint. Furthermore,Eq. (3) is most accurate in the linear response regime forwhich n (cid:28) κ tot / | K | , where n = 4 κ ext P in / (cid:126) ω κ is theaverage number of intracavity photons and K is the Kerrnonlinearity of the cCPT [30]. Experimentally, we findthat for n (cid:28) κ tot / | K | the output sideband power growslinearly with P in as expected from Eq. (2), but as n ap-proaches κ tot / | K | this trend becomes sub-linear. Nearthis threshold, P out ( ω ± ω g ) begins to decrease with in-creasing P in . We therefore use sufficiently small inputpowers such that n (cid:46) κ tot / | K | > . n g and Φ ext [30].The detection bandwidth of the present device, whichdetermines the maximum rate at which the cavity canrespond to changes in the gate charge n g , is set by κ tot and is on the order of 1 MHz. This can be improved byusing a larger coupling capacitance C c , thereby increas-ing κ ext , but this improved bandwidth also affects thecharge sensitivity. Restricting ourselves to single-photon-level operation and assuming negligible internal loss suchthat κ tot ≈ κ ext , Eq. (3) becomes δq = 12 (cid:114) κ tot S noise (cid:126) ω (cid:12)(cid:12)(cid:12)(cid:12) ∂ω ∂n g (cid:12)(cid:12)(cid:12)(cid:12) − e (4)for ω g (cid:28) κ tot . Thus, the bandwidth can be improved atthe expense of sensitivity, and vice versa.The cCPT is housed in a dilution refrigerator with abase temperature of T (cid:46)
30 mK and measured usingthe circuitry depicted schematically in Fig. 1(b), whichis nearly identical to that used in Ref. [30]. The onedifference here is that we use a near quantum-limitedTWPA [24] as a first-stage amplifier, followed by a cryo-genic HEMT amplifier and room temperature amplifier.We use the techniques described in Ref. [30] to referall input and output powers, as well as the system noise S noise ( ω ), to the plane of the sample. The measured sys-tem noise, shown in Fig. 1(c), is due to the half-photonof vacuum noise S vac = (cid:126) ω/ S amp , such that S noise = S vac + S amp . For all ofthe charge sensitivity measurements we report, the noisefloor near the sideband frequencies is dominated by thissystem noise, which is why we use the same notation forthese two quantities. At sufficiently low gate modulationfrequencies, however, the noise floor is dominated by 1 /f charge noise [36]. In our case this regime is below about1 kHz [30]. We determine the noise added by the TWPAand HEMT independently by measuring the gain of theamplifier chain and total system noise twice: once withthe TWPA pump on and once with it off. Over the work-ing range of resonant frequencies of the cCPT (between Φ ext / Φ − . − . − . . . . . n g (a) Experiment Φ ext / Φ Theory δ q (cid:16) µ e / √ H z (cid:17) ω − ω g − − − P o u t ( d B m ) (b) ω ω + ω g FIG. 2. (a) Measured and theoretical charge sensitivities, ob-tained using Eqs. (1) and (3) respectively, as a function ofgate and flux. Data is omitted where the sidebands could notbe resolved from the noise floor. (b) Sample-referred spec-trum analyzer trace of the optimal charge sensitivity mea-surement, corresponding to δq = 14 µe/ √ Hz. The carrierfrequency is ω / π = 5 .
806 GHz, the gate modulation fre-quency is ω g / π = 350 kHz, the span of each segment is 1kHz, and the resolution bandwidth is B = 10 Hz. The noisefloor near the carrier is due to 1 /f charge noise [30, 36]. .
68 GHz and 5 .
82 GHz), the TWPA contributes 1 . . S noise . Thequantum limit of noise in this system is one photon, suchthat S QLnoise = (cid:126) ω , since phase-insensitive amplifiers mustadd at least a half-photon of noise [37]. Thus, our aver-age system noise is only a factor of 2 . √ . δq as a function of both the gate charge n g andexternal flux Φ ext . Although we can access a full pe-riod of Φ ext (from 0 to the magnetic flux quantum Φ ),we can only access the gate range − . < n g < . P in = −
141 dBm ≈ q rms = 10 − e .Ideally we would set ω g to be significantly less than κ tot / ≈ π ×
700 kHz to minimize Eq. (3), but in ourexperiments we observe cross-talk between our gate andflux lines at frequencies below about 650 kHz. We there-fore use ω g / π = 800 kHz, such that the gate modulationdoes not also induce a flux modulation. To measure thereflected power and noise floor at ω ± ω g we use a reso-lution bandwidth B = 1 Hz. Electrometer δq ( µe/ √ Hz) P in (aW) .
25 100 190Best rf-SET[21] 0 . × × Andresen et al.[39] 2 . × × L-SET[27]* 30 1 × × × × rf-QPC[41] 200 1 × × TABLE I. Comparison of the cCPT with a representative setof fast and ultrasensitive electrometers. Asterisks indicatedispersive electrometers.
The results of this measurement are shown in Fig. 2(a).We find that the variation of δq with n g and Φ ext is ingood agreement with our theory, but our measured sen-sitivities are about 3 times worse than theory. We at-tribute this discrepancy to two factors. First and fore-most, the resonant frequency fluctuates due to 1 /f chargeand flux noise [30, 38] over the course of each measure-ment, which means our carrier is not always on reso-nance. On average, this reduces the output sidebandpower yielding worse charge sensitivity than expected.Second, we used a sufficiently high input power that P out ( ω ± ω g ) scales sublinearly with P in due to the Kerrnonlinearity. Although this improves the sensitivity over-all and was necessary to resolve the sidebands over a largearea of the gate/flux parameter space, it causes the mea-sured sensitivity to diverge from theory since the latterassumes proportionality between P in and P out ( ω ± ω g ).Finally, accounting for the fact that S noise /S QLnoise ≈ . µe/ √ Hz at( n g , Φ ext ) = (0 . , . µe/ √ Hz and 6 µe/ √ Hz, respectively.In order to optimize δq we narrow our search to thegate range 0 . ≤ | n g | ≤ .
65 and the flux points Φ ext =0 , Φ /
2. At these flux points the resonant frequency ofthe cCPT is insensitive to flux, so we can reduce ourgate modulation frequency to ω g / π = 350 kHz withoutthe gate/flux cross-talk interfering with our results. Tomaintain a small frequency modulation amplitude rela-tive to ω g , we also reduce q rms to 5 × − e . For thismeasurement we use a resolution bandwidth B = 10 Hz.We find a minimum charge sensitivity of 14 µe/ √ Hzat ( n g , Φ ext ) = (0 . , .
0) using an input power P in = −
138 dBm ≈
16 aW. Under these conditions our pre-dicted theoretical and quantum-limited sensitivities are5 µe/ √ Hz and 3 µe/ √ Hz, respectively. The spectrumanalyzer trace of this optimal measurement is shown in Fig. 2(b). At this bias point the resonant fre-quency is ω / π = 5 .
806 GHz, the external damp-ing is κ ext / π = 1 .
24 MHz, and the total damping is κ tot / π = 1 .
62 MHz, such that the number of intracav-ity photons is n = 4 κ ext P in / (cid:126) ω κ ≈
1. This is thefirst ultrasensitive electrometer reported to operate atthe single-photon level. Furthermore, its sensitivity is ri-valed only by gate-based sensors [29], rf-SETs [21], andcarbon nanotube-based rf-SETs [39]. In Table I we com-pare the performance of the cCPT to a representative setof fast (detection bandwidth (cid:38) δq < − e/ √ Hz) electrometers. Clearly, the cCPTis unparalleled in its ability to operate at low powers andphoton numbers. As discussed earlier, this makes it idealfor mediating an optomechanical interaction that reachesthe single-photon strong coupling regime [9].There remains significant room for improving the sensi-tivity of the cCPT, with two distinct approaches for doingso. The most promising approach is to reduce quasipar-ticle poisoning (QP) [32], which prevents us from operat-ing at gate biases above | n g | ≈ .
65 [30]. If we were ableto operate the present device at ( n g , Φ ext ) = (0 . , Φ / δq ≈ . µe/ √ Hz, assuming the same factor of 3 discrepancywith theory as we observe experimentally. The presentdevice was designed with a 9 nm thick CPT island [30] tosuppress QP [42], but other fabrication techniques couldbe employed to reduce it further. These include oxygen-doping the CPT island [32] and embedding quasiparticletraps near the CPT [43].The other approach is to mitigate the discrepancybetween our measured sensitivities and the quantumlimit of sensitivity for the cCPT. One such improve-ment would be to use a truly quantum-limited ampli-fier, which would improve our sensitivities by a factor of (cid:112) S noise / (cid:126) ω ≈ √ .
4. Another such improvement wouldbe to stabilize the resonant frequency using a Pound-locking loop [44], thereby reducing the scale of frequencyfluctuations induced by 1 /f noise, which we expect wouldsubstantially improve the sensitivity of the cCPT.Finally, it may be possible to improve the sensitivity ofthe cCPT by employing more sophisticated charge detec-tion schemes. For example, the cCPT can be driven toits bifurcation point, set by the Kerr nonlinearity, wherethe slope of the cavity response with respect to detun-ing diverges. At this point the charge modulation wouldinduce a stronger modulation of the reflected signal andthus improved sensitivity [45, 46]. It is difficult to oper-ate at this point, however, since frequency fluctuationsblur the sharp cavity response. Another possibility isto incorporate parametric pumping of the flux at 2 ω toimprove sensitivity. If biased near the threshold of para-metric oscillation, small changes in the gate charge wouldyield huge changes in the oscillation amplitude. Such ascheme has previously been used to achieve single-shotreadout of a superconducting qubit [47].We thank W. F. Braasch for helpful discussions and W.Oliver for providing the TWPA used in these measure-ments. The sample was fabricated at Dartmouth Collegeand the Harvard Center for Nanoscale Systems. B.L.B.,S.K., and A.J.R. were supported by the National Sci-ence Foundation under Grant No. DMR-1807785. J.L.was supported by the Army Research Office under GrantNo. W911NF-13-1-0377. M.P.B. was supported by theNational Science Foundation under Grant No. DMR-1507383. ∗ [email protected] † Present address: High Energy Physics Divison, ArgonneNational Laboratory, 9700 South Cass Avenue, Argonne,IL 60439, USA ‡ [email protected][1] W. Lu, Z. Ji, L. Pfeiffer, K. W. West, and A. J. Rimberg,Real-time detection of electron tunnelling in a quantumdot, Nature , 422 (2003).[2] O. Naaman and J. Aumentado, Time-domain measure-ments of quasiparticle tunneling rates in a single-cooper-pair transistor, Phys. Rev. B , 172504 (2006).[3] D. M. T. van Zanten, D. Sabonis, J. Suter, J. I.V¨ayrynen, T. Karzig, D. I. Pikulin, E. C. T. O’Farrell,D. Razmadze, K. D. Petersson, P. Krogstrup, and C. M.Marcus, Photon-assisted tunnelling of zero modes in amajorana wire, Nature Physics , 663 (2020).[4] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,and A. C. Gossard, Coherent manipulation of coupledelectron spins in semiconductor quantum dots, Science , 2180 (2005).[5] B. D ' Anjou and G. Burkard, Optimal dispersive read-out of a spin qubit with a microwave resonator, PhysicalReview B , 245427 (2019).[6] R. G. Knobel and A. N. Cleland, Nanometre-scale dis-placement sensing using a single electron transistor, Na-ture , 291 (2003).[7] M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab,Approaching the quantum limit of a nanomechanical res-onator, Science , 74 (2004).[8] A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A.Clerk, M. P. Blencowe, and K. C. Schwab, Coolinga nanomechanical resonator with quantum back-action,Nature , 193 (2006).[9] A. J. Rimberg, M. P. Blencowe, A. D. Armour, and P. D.Nation, A cavity-cooper pair transistor scheme for inves-tigating quantum optomechanics in the ultra-strong cou-pling regime, New Journal of Physics , 055008 (2014).[10] T. T. Heikkil¨a, F. Massel, J. Tuorila, R. Khan, and M. A.Sillanp¨a¨a, Enhancing optomechanical coupling via thejosephson effect, Phys. Rev. Lett. , 203603 (2014).[11] J.-M. Pirkkalainen, S. Cho, F. Massel, J. Tuorila,T. Heikkil¨a, P. Hakonen, and M. Sillanp¨a¨a, Cavity op-tomechanics mediated by a quantum two-level system,Nature Communications , 6981 (2015).[12] X. Mi, J. V. Cady, D. M. Zajac, P. W. Deelman, andJ. R. Petta, Strong coupling of a single electron in siliconto a microwave photon, Science , 156 (2016). [13] X. Mi, M. Benito, S. Putz, D. M. Zajac, J. M. Taylor,G. Burkard, and J. R. Petta, A coherent spin–photoninterface in silicon, Nature , 599 (2018).[14] D. Zoepfl, M. Juan, C. Schneider, and G. Kirchmair,Single-photon cooling in microwave magnetomechanics,Physical Review Letters , 023601 (2020).[15] P. Schmidt, M. T. Amawi, S. Pogorzalek, F. Deppe,A. Marx, R. Gross, and H. Huebl, Sideband-resolved res-onator electromechanics based on a nonlinear josephsoninductance probed on the single-photon level, Communi-cations Physics , 233 (2020).[16] M. Kounalakis, Y. M. Blanter, and G. A. Steele, Flux-mediated optomechanics with a transmon qubit in thesingle-photon ultrastrong-coupling regime, Physical Re-view Research , 023335 (2020).[17] T. Bera, S. Majumder, S. K. Sahu, and V. Singh, Largeflux-mediated coupling in hybrid electromechanical sys-tem with a transmon qubit, Communications Physics ,12 (2021).[18] A. Nunnenkamp, K. Børkje, and S. M. Girvin, Single-photon optomechanics, Physical Review Letters ,063602 (2011).[19] P. Rabl, Photon blockade effect in optomechanical sys-tems, Physical Review Letters , 063601 (2011).[20] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt,Cavity optomechanics, Reviews of Modern Physics ,1391 (2014).[21] H. Brenning, S. Kafanov, T. Duty, S. Kubatkin, andP. Delsing, An ultrasensitive radio-frequency single-electron transistor working up to 4.2 k, Journal of Ap-plied Physics , 114321 (2006).[22] R. J. Schoelkopf, The radio-frequency single-electrontransistor (RF-SET): A fast and ultrasensitive electrom-eter, Science , 1238 (1998).[23] M. A. Castellanos-Beltran and K. W. Lehnert, Widelytunable parametric amplifier based on a superconduct-ing quantum interference device array resonator, AppliedPhysics Letters , 083509 (2007).[24] C. Macklin, K. O’Brien, D. Hover, M. E. Schwartz,V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi,A near–quantum-limited josephson traveling-wave para-metric amplifier, Science , 307 (2015).[25] V. Sivak, N. Frattini, V. Joshi, A. Lingenfelter,S. Shankar, and M. Devoret, Kerr-free three-wave mix-ing in superconducting quantum circuits, Physical Re-view Applied , 054060 (2019).[26] M. A. Sillanp¨a¨a, L. Roschier, and P. J. Hakonen, Induc-tive single-electron transistor, Physical Review Letters , 066805 (2004).[27] M. A. Sillanp¨a¨a, L. Roschier, and P. J. Hakonen, Chargesensitivity of the inductive single-electron transistor, Ap-plied Physics Letters , 092502 (2005).[28] M. F. Gonzalez-Zalba, S. Barraud, A. J. Ferguson, andA. C. Betz, Probing the limits of gate-based charge sens-ing, Nature Communications , 6084 (2015).[29] S. Schaal, I. Ahmed, J. Haigh, L. Hutin, B. Bertrand,S. Barraud, M. Vinet, C.-M. Lee, N. Stelmashenko,J. Robinson, J. Qiu, S. Hacohen-Gourgy, I. Siddiqi,M. Gonzalez-Zalba, and J. Morton, Fast gate-based read-out of silicon quantum dots using josephson paramet-ric amplification, Physical Review Letters , 067701(2020).[30] B. L. Brock, J. Li, S. Kanhirathingal, B. Thyagara-jan, W. F. B. Jr., M. P. Blencowe, and A. J. Rim- berg, A nonlinear charge- and flux-tunable cavity derivedfrom an embedded cooper pair transistor, arXiv preprintarXiv:2011.06298 (2021).[31] S. Kanhirathingal, B. L. Brock, A. J. Rimberg, and M. P.Blencowe, Quantum-limited linear charge detection witha cavity-embedded cooper pair transistor, arXiv preprintarXiv:2012.12313 (2020).[32] J. Aumentado, M. W. Keller, J. M. Martinis, and M. H.Devoret, Nonequilibrium quasiparticles and 2 e periodic-ity in single-cooper-pair transistors, Physical Review Let-ters , 066802 (2004).[33] D. Pozar, Microwave Engineering (Wiley, 2004).[34] A. Aassime, D. Gunnarsson, K. Bladh, P. Delsing, andR. Schoelkopf, Radio-frequency single-electron transis-tor: Toward the shot-noise limit, Applied Physics Letters , 4031 (2001).[35] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Mar-quardt, and R. J. Schoelkopf, Introduction to quantumnoise, measurement, and amplification, Reviews of Mod-ern Physics , 1155 (2010).[36] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Alt-shuler, 1/f noise: Implications for solid-state quantuminformation, Reviews of Modern Physics , 361 (2014).[37] C. M. Caves, Quantum limits on noise in linear ampli-fiers, Phys. Rev. D , 1817 (1982).[38] B. L. Brock, M. P. Blencowe, and A. J. Rimberg, Fre-quency fluctuations in tunable and nonlinear microwavecavities, Phys. Rev. Applied , 054026 (2020).[39] S. E. S. Andresen, F. Wu, R. Danneau, D. Gunnarsson,and P. J. Hakonen, Highly sensitive and broadband car-bon nanotube radio-frequency single-electron transistor,Journal of Applied Physics , 033715 (2008).[40] M. T. Bell, L. B. Ioffe, and M. E. Gershenson, Microwave spectroscopy of a cooper-pair transistor coupled to alumped-element resonator, Physical Review B , 144512(2012).[41] M. C. Cassidy, A. S. Dzurak, R. G. Clark, K. D. Peters-son, I. Farrer, D. A. Ritchie, and C. G. Smith, Single shotcharge detection using a radio-frequency quantum pointcontact, Applied Physics Letters , 222104 (2007).[42] T. Yamamoto, Y. Nakamura, Y. A. Pashkin, O. Astafiev,and J. S. Tsai, Parity effect in superconducting aluminumsingle electron transistors with spatial gap profile con-trolled by film thickness, Applied Physics Letters ,212509 (2006).[43] S. Rajauria, L. M. A. Pascal, P. Gandit, F. W. J.Hekking, B. Pannetier, and H. Courtois, Efficiencyof quasiparticle evacuation in superconducting devices,Physical Review B , 020505 (2012).[44] T. Lindstr¨om, J. Burnett, M. Oxborrow, and A. Y.Tzalenchuk, Pound-locking for characterization of super-conducting microresonators, Review of Scientific Instru-ments , 104706 (2011).[45] C. Laflamme and A. A. Clerk, Quantum-limited amplifi-cation with a nonlinear cavity detector, Physical ReviewA , 033803 (2011).[46] L. Tosi, D. Vion, and H. le Sueur, Design of a cooper-pairbox electrometer for application to solid-state and as-troparticle physics, Physical Review Applied , 054072(2019).[47] P. Krantz, A. Bengtsson, M. Simoen, S. Gustavsson,V. Shumeiko, W. D. Oliver, C. M. Wilson, P. Delsing,and J. Bylander, Single-shot read-out of a superconduct-ing qubit using a josephson parametric oscillator, NatureCommunications7