DC measurement of dressed states in a coupled 100~GHz resonator system using a single quasiparticle transistor as a sensitive microwave detector

We report on the on-chip detection of microwaves in the frequency range around 100GHz. For the purpose of detection, we employ a discrete transport channel triggered in a superconducting single-electron transistor by photon-assisted tunneling of quasiparticles. The technique is successfully applied to observe the spectrum of the dressed states of a model cQED system consisting of a superconducting coplanar resonator coupled to a quantum Josephson oscillator. The dressed states appear as typical resonance anticrossing exhibiting, in our case, an expectedly wide frequency splitting corresponding to the Jaynes-Cummings coupling strength, g/pi~10GHz. Due to the high decay rate, gamma~20-40GHz, in the very transparent Josephson junctions used, the strong coupling limit, g>>gamma, which is required for qubit operation, is not achieved, and the photon population in the resonator is low,~1. Remarkably, the continuous readout of the low population states demonstrates the high microwave sensitivity of the detector.

Strong coupling between a microwave resonator and a quantum oscillator is an important prerequisite for clear observation of circuit/cavity quantum electrodynamics (cQED/CQED) effects [1,2]. One of the significant advantages of superconducting circuit QED systems over cavity QED with real atoms is a much stronger coupling between the components. For example, for a Cooper pair box coupled to a coplanar waveguide (CPW) resonator with a typical length l of 10 mm to 20 mm and mode frequency f r of 5 GHz to 10 GHz, the coupling strength g, which appears in the interaction term of the Jaynes-Cummings Hamiltonian [3], routinely achieves the level g ∼ 2π×100 MHz (see, e.g., Refs. [1,4,5]).
Even stronger coupling should be expected in shorter resonators, l ∼ 1 mm, with the coupling strength, g ∝ e ω r /2C ∝ l −1 [4,6], increasing linearly with the frequency, ω r = 2πf r = 1/ √ LC ∝ l −1 due to the decrease in the effective inductance/capacitance L, C ∝ l. This could justify an interest in the compact quantum systems operating in an upper microwave frequency range up to 100 GHz. On the other hand, a substantial increase in the loss rate can critically impact the quantum coherence as the result of a stronger capacitive coupling to the electromagnetic environment at elevated frequencies and more intensive generation of non-equilibrium quasiparticles (QP) (cf., e.g., Refs. [7,8]). For example, in a recent experiment [9], a non-coherent model approach was used successfully to describe an Aluminum SQUID array up to the microwave frequencies f from 50 GHz to 210 GHz, exceeding the Cooper pair breaking energy, 2∆ ≈ 400 µeV ≈ h×100 GHz. An important basic question is therefore whether cQED behavior can be observed at high microwave frequencies. However, the required measurement is technically challenging, and dedicated microwave detection techniques [9][10][11][12][13] are necessary for a mK-setup of the dilution fridge at frequencies beyond the standard scale, f r ∼ 10 GHz.
In this Letter, we report our observations concerning the dressed states of a CPW resonator directly coupled to a Josephson oscillator using the frequency range f ∼ 100 GHz. In order to access this range, we implement an on-chip test bench, shown in Fig. 1(a), based on Al/AlO x /Al-junctions and fully controlled via dc signals from room temperature electronics. As a microwave source, we use an overdamped Josephson junction (JJ source), and for the photon detection, we utilize a quasiparticle sensing regime based on a photon-assisted tunneling (PAT) effect in a superconducting single-electron transistor (SSET), shown in Fig. 1(b) (cf. Refs. [12,13]). A significant gain in microwave sensitivity appears in our detection circuit owing to an intrinsic photon-electron multiplying mechanism described below.
The experimental circuit was fabricated using the shadow evaporation technique [14] and studied in a shielded DC setup at T = 15 mK with an integration constant, τ int ∼ 0.5 s, that was sufficiently long for measuring the sub-pA currents. All superconducting components were integrated into a single evaporation mask and included three successive layers of aluminum. Mild (10 min at oxygen pressure 1 Pa) and heavy (15 min, 25 Pa) oxidation steps were performed for the first and the second Al layers, respectively, to form the tunnel junctions for the JJ source/Josephson oscillator and for the highohmic SSET device. As a resistive shunt attached to the JJ source (see the right hand part of Fig. 1(a)), we used a finite-loss AuPd coplanar transmission line (TL) with a specific high-frequency impedance of Z s ≈ 50 Ω. The line was 2.3 mm long and it was terminated at the opposite end by means of a 6 Ω section of the AuPd film used as a cold part of the DC biasing circuitry. The full DC load resistance seen by the Josephson junction was R L ≈ 25 Ω, which was low enough for stabilization of the Josephson voltage, V J = hf J /2e, and the base frequency f J of the Josephson generation (cf. Ref. [15]).
The detection principle is based on the remarkably high quasiparticle sensitivity of the tunnel current I SSET at low bias voltages, V b < 4∆/e, where single-electron tunneling is suppressed. As discussed in Refs. [16,17] quasiparticle tunneling (SQPT) and Cooper pair -Electron (CPE) cotunneling transfers (see the cycle depicted in Fig. 1(c)). The SQPT/CPE sequence persists provided there is at least one unpaired QP hosted by the central island. It starts with a low-rate SQPT process (see the estimate below) transferring an electron into the island and pairing it with the odd QP, followed by CPE cotunneling, a much faster step discharging the island, but leaving behind another unpaired QP that will be incorporated into the next cycle. The density of QPs (and thus the rate of SQPT) depends crucially both on the parity effect (see, e.g., Ref. [18]) and on the intensity of the incident or ambient microwaves (see, e.g., Refs. [7,[19][20][21]). The microwave photon energy E ph must exceed the activation threshold for photon-assisted single-electron tunneling, Here, n g ≡ C g V g /e is the gate charge, C g (V g ) denotes the gate capacitance (voltage), E C ≡ e 2 /2C Σ is the charging energy, C Σ is the total capacitance of the central island, and m is an even integer number.
Further details of the detector operation can be illustrated by using the data in Fig. 2, which was obtained for a test sample without a quantum oscillator. Without irradiation, see Fig. 2(a), a clear sub-pA current pattern appears in the odd gate domains, m + 1/2 ≤ n g ≤ m+3/2, with one unpaired QP residing in the island most of the time. By contrast, the current landscape is almost suppressed in the even domains, m−1/2 ≤ n g ≤ m+1/2, with all QPs being paired. The current appears at the voltage thresholds V b 2E C (±2n g − 1 + 2m)/e, for SQPT, and V b [4∆ + 2E C (1 ∓ 2n g + 2m)] /3e, for the CPE cotunneling. The expectation time τ CPE of the CPE cotunneling greatly exceeds the time τ SQPT for SQPT. Indeed, the CPE/CPE cotunneling current, which is onset in the triangles (color online: red) at the top of the diagram in Fig. 2(a) (see also Ref. [17]) and reaches the values I SSET ∼ 10 pA beyond the diagram scope, provides an estimate τ CPE ∼ 25 ns. On the other hand, a much lower CPE/SQPT current measured, I SSET ≈ 0.2-0.5 pA, corresponds to the cycle period on the scale of a microsecond. This period is obviously limited by the rate of SQPT, Γ −1 SQPT = τ SQPT ≈ 0.7-1.6 µs.
The high τ SQPT /τ CPE ratio provides the conditions for an intrinsic mechanism of current triggering due to photon absorption. In charge dynamics, the PAT process shown in Fig. 1(b) is followed directly by the charge relaxation act via CPE cotunneling, shown in Fig. 1(c). The resulting state with two unpaired QPs in the island invokes the cyclic SQPT/CPE transport similar to that in the autonomous mode. Accordingly, the tunnel current should consist of the trains of CPE/SQPT pulses, persisting until the unpaired QP escapes from the island in advance of the CPE event. The most probable escape  Fig. 1(b). Fig. 1(c), has the rate close to Γ SQPT , competing with the rate of CPE cotunneling, τ −1 CPE . The number of cycles in the train N p is thus defined by the times ratio, N p ∼ τ SQPT /τ CPE ∼ 50, and the train duration is an N p -multiple of the cycle time, τ QP ∼ τ SQPT × τ SQPT /τ CPE ∼ 60 µs. A more accurate description of QP dynamics (which is beyond the scope of this Letter) should include a QP recombination rate in the island and requires the master equation approach [22].

process, shown by the dashed arrow in
As expected, the detector signal is found to increase significantly (see Fig. 2(b)), due to microwave irradia-tion arriving at the detector via the CPW resonator.
The current diagram appears to be almost 1e-periodic, by exhibiting similar currents in the odd and even domains, which indicates a QP population about unity in the island. The signal dependence on the Josephson frequency (i.e., microwave photon energy), which is shown in Fig. 2(c), confirms a clear resonant peak structure at f J ≈ 108 GHz, but also reveals an inferior pattern of stray box resonances and higher Josephson harmonics. Similar to Refs. [12,13], we roughly estimated the rate of photon-assisted tunneling in SSET irradiated under the conditions of the resonant peak and the gate tuned to the sensitive point, n g ≈ −0.25. The obtained value, Γ ph ∼ 10 4 s −1 , corresponds to an extremely low level of energy dissipation from the CPW into the detector, W D = Γ ph × hf J ∼ 0.7 aW ≈ -151 dBm. Furthermore, relating Γ ph to the signal peak value, I SSET ≈ 0.6 pA ∼ 4 × 10 6 electrons per second, it was possible to obtain a reasonable estimate, N p ∼ 200 cycles per photon.
The quantum oscillator coupled to the CPW resonator was designed as a Josephson DC SQUID with a loop of area A ≈ 40 µm 2 and two highly transparent, 0.25 µm 2 Josephson tunnel junctions with a total critical current I C (B = 0) = I C1 + I C2 ≈ 8 µA and a sum tunnel capacitance C J ≈ 22 fF (a fitted value, see below). As shown in Fig. 3(a), the magnetic field B applied to the SQUID loop periodically modulates the microwave signal transmitted to the detector. The measured period, ∆B ≈ 14 µT < Φ 0 /A ≈ 50 µT , is smaller than that expected for the loop area presumably due to a flux concentration effect in the slots of the CPW line. On the other hand, the actual flux and field values, Φ =B × A, can be calibrated directly using the plot periodicity, ∆Φ = Φ 0 . The plasma frequency of the SQUID, ω p (Φ) = 2eI C (Φ)/ C J , is estimated to vary in a wide range up to ω p (0) ∼ 2π×170 GHz and the Josephson-to-charging energy ratio approaches the values, E J /E CJ ∼ 4000, well within the limit of the transmon-type qubits [23].
Due to the high detector sensitivity, we succeeded in observing the dressed states [4,6] in the superconducting resonator system as a well-pronounced anticrossing, see Fig. 3(b), between the microwave and qubit resonances at low detuning, δ = |ω p − ω r | g. The uncoupled resonant frequencies ω r and ω p vary along the dashed lines and, on the state diagram in Fig. 3(c), correspond to the resonant transitions from the ground state |g, 0 to the single-photon state |g, 1 or to the lowest excited state of the qubit |e, 0 , respectively. The symmetric and antisymmetric dressed states appear for the coupled system as the coherent superpositions, |± ≈ 1/ √ 2(|g, 1 ±|e, 0 ) (1/ √ 2 is an exact prefactor at δ = 0) and give rise to resonances at [6]: shown by the solid lines in Fig. 3(b). The lines are fitted to the plot by adjusting the values of C J and the SQUID asymmetry factor, d = |I C1 − I C2 |/(I C1 + I C2 ) ≈ 0.16. The frequency splitting interval, equal to g/π = e ω r /hC ≈ 9.2 GHz, was calculated directly, based on the resonator geometry [24]. The signal peaks, observed at Φ ≈ 0.42 Φ 0 and ≈ 0.58 Φ 0 , appear in the vicinity of the zero detuning points, presumably due to the resonant increase in the qubit impedance, causing more efficient matching to the microwave source. Finally, we note that the anticrossing pattern was studied for two different CPW resonators with the microwave parameters summarized in Table I. As compared to the unloaded test samples, with the detected linewidth of the CPW resonances, ∆f ≈ 1.7 GHz, and the Josephson generation linewidth of the source, ∆f J ≈ 0.4 GHz [25], a significant broadening of the res-onance is observed for the samples including the SQUID junctions. We explain this, on a qualitative level, as being the result of a strong quasiparticle subgap leakage due to the high junction transparency (see, e.g., Refs. [26,27]) on the one hand, and the PAT effect [28] on the other hand. The higher-frequency peak exhibits a stronger broadening, thus emphasizing the significance for the leakage current of the photon energy approaching the Cooper pair breaking threshold, 2∆/h ≈ 100 GHz.
Under continuous irradiation, the total loss rate in the coupled CPW-plus-oscillator system, γ = 2π(∆f − ∆f J ), is counteracted by the power input from the JJ source, P in ∼ 0.1 pW. Due to the intensive losses, the average number n of photons in our CPW resonator, n = P in /(γ ω r ), is estimated to be lower than unity (see Table I). On the one hand, this may be a detector's figure of merit, for its clear signal indicates the proper sensitivity of the resonator readout. On the other hand, since the decay rate γ exceeds Rabi frequency g/2π, as both are in the GHz-range, no Rabi oscillation can exist under the present conditions. Much lower dissipation (and much more pronounced anharmonicity) can be expected for a SQUID oscillator with sub-100 nm junctions of the same high barrier transparency (and thus of the frequency ω r > 100 GHz). Furthermore, a more detailed study is envisaged of the loss rate in a wide range of resonator frequencies.
To conclude, on the basis of a superconducting singleelectron transistor, we have developed a highly sensitive on-chip detection technique for microwave frequencies on the scale of 100 GHz. An important element of this technique is a mechanism of photon-activated current triggering that facilitates a batch electron transfer for each absorbed photon. The detection technique was used to observe the two lowest dressed states in a quantum system with a CPW resonator coupled to a Josephson junction oscillator. This observation demonstrated a detector sensitivity down to very low photon populations in the resonator, n < 1. Further applications of the detector for on-chip studies of mesoscopic devices are in progress.
We would like to acknowledge the technical support of V. Rogalya and T. Weimann. This work was funded in part by the PARAWAVE Joint Research Project. This project has received funding from the EMPIR program co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation program. The measurement data for this paper is available at: https://doi.org/10.7795/720.20190617 (Lotkhov et al., 2019).