Heat rectification via a superconducting artificial atom
Jorden Senior, Azat Gubaydullin, Bayan Karimi, Joonas T. Peltonen, Joachim Ankerhold, Jukka P. Pekola
HHeat rectification via a superconducting artificial atom
Jorden Senior, Azat Gubaydullin, Bayan Karimi, JoonasT. Peltonen, Joachim Ankerhold, and Jukka P. Pekola QTF Centre of Excellence, Department of Applied Physics,Aalto University School of Science,P.O. Box 13500, 00076 Aalto, Finland Institute for Complex Quantum Systems and IQST,University of Ulm, 89069 Ulm, Germany a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug n miniaturising electrical devices down to nanoscales, heat transfer hasturned into a serious obstacle but also potential resource for future devel-opments, both for conventional and quantum computing architectures[1–3].Controlling heat transport in superconducting circuits has thus received in-creasing attention in engineering microwave environments for circuit quantumelectrodynamics (cQED)[4–6] and circuit quantum thermodynamics experiments(cQTD)[7, 8]. While theoretical proposals for cQTD devices are numerous[9–15],the experimental situation is much less advanced. There exist only relativelyfew experimental realisations[16–20], mostly due to the difficulties in developingthe hybrid devices and in interfacing these often technologically contrastingcomponents. Here we show a realisation of a quantum heat rectifier, a ther-mal equivalent to the electronic diode, utilising a superconducting transmonqubit coupled to two strongly unequal resonators terminated by mesoscopicheat baths. Our work is the experimental realisation of the spin-boson rectifierproposed by Segal and Nitzan [21]. A rectifier is a device in which the transport is directionally impeded, optimally to allowit only in one direction. In the charge regime, the ubiquitous rectifier, or diode, is oneof the most fundamental components in electronic circuits, and can be realised relativelysimplistically, for example as a device exploiting the depletion zone between electrons andholes in a semiconductor p-type/n-type junction.The implementation of devices for manipulating charge current is enabled by the discreteand polarised nature of the carrier, and their interaction with electromagnetic fields. Heatcurrents do not have this quality, and manipulating their flow usually relies on precise controlof the energy population distributions of the various materials involved in the device. In thesuperconducting regime, it is provided by the superconducting/semiconducting gap in onedirection, and/or asymmetric couplings of the heat baths to quasiparticle heat carriers, forexample by using metals with differing electron-phonon couplings.The two-level system of a transmon-type qubit coupled to two unequal resonators isa well-placed tool for studying asymmetric photonic transport, each element having engi-neered resonances and couplings to each other that can be designed for various modes ofoperation [23, 24]. In particular, it represents a minimal set-up to explore, under well-controlled conditions, the subtle phenomenon of heat rectification which requires both non-2inearities and symmetry breaking [21, 22]. By utilising a superconducting quantum in-terferometer (SQUID) as the non-linear element of the transmon, the Josephson energy E J (Φ) (cid:39) E J | cos( π Φ / Φ ) | can be tuned by an incident magnetic flux Φ. Here, Φ = h/ e isthe magnetic flux quantum. This in turn allows to control the excitation frequencies of thetransmon determined by E J (Φ) and the charging energy E C with the transition frequencies ω n,n +1 between levels n and n + 1 ( n = 0 , , , ... ) given by ω n,n +1 (Φ) = ω p (Φ) − ( n + 1) E C / (cid:126) , (1)where the plasma frequency ω p (Φ) = (cid:112) E C E J (Φ) / (cid:126) . The n -dependence in Eq. (1) produceshere the necessary non-linearity for rectification; this is the property that in fact makes thetransmon a qubit in cQED applications. On the other hand, the symmetry breaking is due tothe difference in effective qubit-bath couplings g and g , via the left and right resonators ofclearly different frequencies, ω < ω , respectively. This depends upon the spectral overlapbetween the corresponding resonator and the transmon and can be controlled by the appliedflux [25]. Assuming one of the temperatures is T = 1 / ( k B β ) > R of power P i to bath i in the forwards (+) andbackwards ( − ) direction [22] is obtained in the two level approximation for the transmon as R = | P + i || P − i | = g + g coth( β (cid:126) ω ) g coth( β (cid:126) ω ) + g . (2)Any value R (cid:54) = 1 corresponds to heat rectification while R = 1 describes completely sym-metric heat flux. By introducing the asymmetry in coupling factors δ = 1 − g /g , thisexpression can be simplified for | δ | (cid:28) R = 1 + e − β (cid:126) ω δ . (3)In contrast, if one considers a scheme in which the superconducting qubit is replaced bya harmonic oscillator or with a single-level quantum dot, the linearity or fermionic natureof such a device instead does not lead to rectification (see supplementary material).Figure 1 presents a diagram of the realised device, consisting of a transmon qubit coupledto two resonators, designed at 2.8 GHz and 6.7 GHz frequencies, respectively. Each resonatoris shunted at the current maximum with a copper thin-film resistor, shown in the left inset asa colourised scanning electron micrograph. This element provides the thermal bath, which isheated and measured by superconducting aluminium probes, grown using an electron beam3vaporator with the Dolan bridge technique. The thin insulator is formed by the nativeoxide of aluminium, grown by an in-situ oxidation prior to the deposition of the copper.The SQUID of the transmon qubit is shown on the right inset, utilising a similar fabricationprocedure. A simple diagrammatic model for understanding the system is also presented,with the role of the resonator-qubit-resonator structure represented by a diode.Due to the definition of resistance being dissipative, normal metal baths used in this wayas terminations to superconducting resonators lead to a substantial decrease in the qualityfactor of the resonator, from an intrinsic value in the range 10 , to a value of order 10,as verified in the measurement in Ref. [26]. Lower resistance baths would result in higherquality resonators with larger asymmetry in the qubit-resonator coupling, and thus strongerrectification, but at the expense of lower power transfer.Temperature differences between the normal metal baths can be controllably achievedby local Joule heating of each bath independently, with the tendency to thermally populatetheir corresponding resonator. We define the “forward” to be the direction from the lowfrequency (2.8 GHz) to high frequency (6.7 GHz) resonator, and declare the heated bath tobe the source with the heat flowing to the target bath. Identical but opposite temperaturebias can be applied in the “reverse” direction. The measured power on the target bathis shown in Figure 2a for various source temperatures between 380 mK and 420 mK. Thehorizontal axis depicts the static magnetic flux Φ on the transmon qubit, normalised withrespect to the flux quantum Φ , applied to tune its transition frequency.We observe qualitatively different magnetic flux-dependences of the heat transport basedupon the directionality. Domains of strong differences between forward and backward heattransfer with pronounced sub-structures alternate with those of weak discrepancies andrelatively smooth traces. To better understand this directionality, one can extract the recti-fication by extracting the ratio of transmitted power in the forward direction to the reversedirection, see Figure 2b. The flux-tunable rectification is isolated by a subtraction of thenon-tunable contribution R min . The origin for non-zero R min is likely to originate from theuncertainty of the temperature bias, which we estimate to be ± T and T + ∆ T with ∆ TT (cid:28) R ( T ) / R ( T + ∆ T ) ≈ − ∆ TT δβ (cid:126) ω e − β (cid:126) ω . This argument also applies to more complex structures. The flux de-pendence is directly related to the energy levels of the resonator-qubit-resonator compounddue to the prevailing resonator-qubit coupling compared to the resonator-bath interaction.Figure 3 displays these levels of the measured structure using realistic parameters fromindependent measurements, plotted against Φ / Φ . We use the Hamiltonian and the ba-sis states given in the supplementary material. Domains of more regular behavior in thespectrum alternate with those of high densities of avoided crossings when the flux is tuned.The latter appear at half-integer values of Φ / Φ . Consequently, we expect to observe aweak flux dependence of rectification away from these domains. The strong flux dependencenear these degeneracies can be directly attributed to the fast varying transition spectrumof the transmon. Physically, heat transfer occurs via photon transfer between hot and coldreservoirs with the resonator-qubit-resonator structure acting as nonlinear medium. Morespecifically, since the resonators’ frequencies are strongly de-tuned far beyond their respec-tive linewidths, finite transmission is favoured if ω is tuned inbetween ω and ω . In thissituation, heat rectification is extremely sensitive to the transmon’s level structure. Whilethe first transition frequency Eq. (1) changes relatively smoothly with applied flux (see Fig-ure 3), higher lying levels experience avoided crossings in narrow ranges of flux. In theseregimes the spectral overlap with only one of the resonators changes abruptly, thus givingrise to pronounced peaks in the rectification. In this sense, the variations in the rectificationcoefficient can also be understood as the opening and closing of photonic transfer channelsof a nonlinear medium between hot and cold reservoirs by varying the magnetic flux. Exper-imental spectra of the resonator-qubit-resonator structure by a two-tone measurement areshown in this figure as well. They demonstrate the higher frequency resonator at ∼ ∼ . ∼ .
75 GHz, all interacting with the qubit. The couplingenergy of the resonators and the qubit is of order 50 MHz given by the seperation at theanticrossing of the modes. The quality factor of the resonator, limited by the bath-phononcoupling, is estimated to be of order 10, based on measurements performed in [26].In conclusion, we present wireless flux-tunable thermal rectification up to 10% via thephoton channel in a superconducting artificial atom. This phenomenon arises due to the5nharmonicity of the system and its asymmetric coupling to two microwave resonators,thermally populated by mesoscopic resistive baths. The device can be integrated withexisting superconducting circuit architectures, in particular with superconducting qubitsand Josephson logic, with potential applications in directionally manipulating heat flow insuperconducting devices, for example in the fast initialisation of a qubit. Additionally, thisdevice may be used as a platform for exploring coherent caloritronics, and the frontiers ofquantum thermodynamics.
A. Methods
Fabrication protocols
These devices are fabricated on a high resistivity silicon wafer, upon which a 20 nm-thickalumimina film has been grown by atomic layer deposition, followed by a susbsequentlysputtered 200 nm-thick niobium film. Larger features, namely the microwave resonators(measured spectroscopically at 2 . . µ m-wide centrelinespaced by 10 µ m with respect to the ground plane. The tunnel junction elements in the su-perconducting probes and interferometer are also patterned by electron beam lithography,then transferred onto the wafer by physical vapour deposition of aluminium in an electronbeam evaporator with an in situ oxidation step, resulting in a per junction resistance of22 ± ± in situ Ar ion plasma millingis performed before the aluminium deposition process step. After liftoff in acetone and clean-ing in isopropyl alcohol, the substrates are diced to 7 × Measurements
The measurements are performed in a custom-made plastic dilution refrigerator with thecooling stage set at 150 mK for these measurements. The bonded chip is enclosed by twobrass Faraday shields, and is readout at room-temperature via 1 m of Thermocoax filtered6ryogenic lines, resulting in an effective signal bandwidth of 0 −
10 kHz, for low-impedanceloads. The magnetic-flux-tuning of the energy level spacing of the transmon is achievedby a superconducting solenoid encompassing the entire sample stage assembly, inside ofa high-permeability magnetic shield. It is mounted inside the refrigerator at a tempera-ture of 4 K. Electronic characterisation is applied by programmable function generators atroom temperature and measured by a low noise amplification chain (room-temperature low-noise amplifiers FEMTO Messtechnik GmbH DLPVA-100) into a DC multimeter (for DCmeasurements), and additionally into a lock-in amplifier synchronised to the square-wavemodulation (22 Hz) of the voltage bias of the heated thermal reservoir, in the configurationshown in Figure 4. By applying a current bias between the superconducting probes close tothe superconducting energy gap, hot electrons from the normal metal can tunnel into thesuperconducting probes, and a voltage signal can be measured, with a well characteriseddependence on the temperature of the electrons in the metal in the range 100 mK - 500mK, of interest to this experiment. By assuming that the heat is well localised due to thenear-perfect Andreev mirrors at the superconductor-normal metal interface, and that themain relaxation channel for the heat is via the electron-phonon coupling, the power can beestimated by P el − ph = Σ V ( T − T ), where Σ is the material dependent electron-phononcoupling constant, V is volume of the normal metal bath, and T el and T ph correspond to theelectron temperature and phonon temperature respectively. As the experiment is performedunder steady-state conditions, we can assume that the phonon temperature is in equilibriumwith the cryostat base temperature, measured by a ruthenium oxide thermometer that hasbeen calibrated against a Coulomb blockade thermometer. Additionally, by voltage bias-ing the SINIS structure sufficiently above the superconducting energy gap, Joule heatingof the baths can be applied (for small voltage biasing below the gap, evaporative coolingcan also occur). Hence, by utilising four superconducting probes, we can both engineer thetemperature of the baths, and measure this induced temperature in parallel [27]. I. ACKNOWLEDGEMENTS
This work was funded through Academy of Finland grant 312057 and from the EuropeanUnion’s Horizon 2020 research and innovation programme under the European ResearchCouncil (ERC) programme and Marie Sklodowska-Curie actions (grant agreements 7425597nd 766025). J.A. acknowledges financial support from the IQST and the German ScienceFoundation (DFG) under AN336/12-1 (For2724). This work was supported by QuantumTechnology Finland (QTF) at Aalto University. We acknowledge the facilities and technicalsupport of Otaniemi research infrastructure for Micro and Nanotechnologies (OtaNano), andVTT technical research centre for sputtered Nb films.We acknowledge L.B. Wang for technical help and thank Y.-C. Chang, A. Ronzani, D.Golubev, G. Thomas, and R. Fazio for useful discussions.
II. AUTHOR CONTRIBUTIONS
J.S. and A.G. designed, fabricated, and measured the samples. Modelling of the workwhich is detailed in the supplementary material was done by B.K. and J.P.P. Technical sup-port in fabrication, low-temperature setups, and measurements were provided by J.T.P. Allauthors have been involved in the analysis, and discussion of scientific results and implica-tions of this work. The manuscript was written by J.S., B.K., J.P.P. and J.A.
III. COMPETING FINANCIAL INTERESTS
The authors declare no competing financial interests.
IV. DATA AVAILABILITY
The data that support the plots within this article are available from the correspondingauthor upon reasonable request. [1] Erdman, P.A., et al.
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Quantum-circuit refrigerator. Nat. Comm. , 15189 (2017).[7] Pekola, J. P. Towards Quantum Thermodynamics in electronic circuits. Nat. Phys. , 118(2015).[8] Vinjanampathy, S. & Anders, J. Quantum thermodynamics. Contemp. Phys. , 545 (2016).[9] Mart´ınez-P´erez, M. J. & Giazotto, F. The Josephson heat interferometer, Nature , 401(2012).[10] Hwang, S.Y., Giazotto, F., & Sothmann, B. Phase-coherent heat circulator based on multi-terminal Josephson junctions. Physical Review Applied, , 044062. (2018)[11] Ordonez-Miranda, J. et al. Photonic thermal diode based on superconductors, J. Appl. Phs. , 093105 (2017).[12] Pereira, E. Thermal rectification in classical and quantum systems: Searching for efficientthermal diodes. EPS , 1 (2019)[13] Barzanjeh, S., Aquilina, M. & Xuereb, A. Manipulating the Flow of Thermal Noise in Quan-tum Devices, Phys. Rev. Lett. , 060601 (2018)[14] Li, B. Negative differential thermal resistance and thermal transistor, Appl. Phys. Lett. ,143501 (2006)[15] Kosloff, R. & Levy, A. Quantum heat engines and refrigerators: continuous devices. AnnualRev. Phys. Chem. , 365 (2014).[16] Chang, C.W. et al. Solid-state Thermal Rectifier, Science , 1121 (2006).[17] Mart´ınez-P´erez, M.J., Fornieri, A., & Giazotto, F. Rectification of electronic heat current bya hybrid thermal diode, Nature Nanotechnology et al. Tunable photonic heat transport in a quantum heat valve, Nature Physics , 991-995 (2018).[19] Roßnagel, J. et al. single atom heat engine. Science , 325 (2016).[20] Scheibner, R. et al Quantum dot as thermal rectifier, New J. Phys. , 034301 (2005)
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Rectification of heat currents across nonlinear quantum chains: a versatileapproach beyond weak thermal contact, New Journal of Physics , (2018).[23] Schmidt, D. R., Schoelkopf, R. J. & Cleland, A. N. Photon-mediated thermal relaxation ofelectrons in nanostructures. Phys. Rev. Lett. , 045901 (2004).[24] Meschke, M. & Guichard, W. & Pekola, J. P. Single-mode heat conduction by photons. Nature , 187 (2006).[25] Koch, J. et al. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev.A , 042319 (2007).[26] Chang, Y.-C. et al. Utilization of the Superconducting Transition for Characterizing Low-Quality-Factor Superconducting Resonators, Appl. Phys. Lett. , 022601 (2019).[27] Giazotto, F. et al.
Opportunities for mesoscopics in thermometry and refrigeration: Physicsand applications. Rev. Mod. Phys (1) 217 (2006). upplementary material (Dated: August 15, 2019) I. RECTIFICATION IN A TWO-LEVEL SYSTEMA. Qubit
Consider a qubit coupled to two baths as shown in Fig. 1a. The transition rates are given by
FIG. 1: (a) A two-level system coupled to two baths, together with the associated transition rates. (b) As (a) but qubit replacedby a harmonic oscillator. Γ (1)in = g ω q e β ~ ω q − , Γ (2)in = g ω q e β ~ ω q − (1)out = g ω q − e − β ~ ω q , Γ (2)out = g ω q − e − β ~ ω q , (1)where g i is the coupling to bath i = 1 , ~ ω q denotes the energy level separation of the qubit, and β i = 1 /k B T i is theinverse temperature of each bath. In steady state the population of the excited state, ρ e = 1 − ρ g reads ρ e = Γ in Γ in + Γ out , (2)where Γ in , out = Γ (1)in , out + Γ (2)in , out and ρ g is the population of the ground state of the qubit. The expression for powerto bath i is then P i = ~ ω q ( ρ e Γ ( i )out − ρ g Γ ( i )in ) . (3)The thermal rectification by definition is given by R = | P + i P − i | , (4)where ± refers to the sign of the temperature bias. Consider case when one of the bath temperatures is much smallerthan the other with k B T = 1 /β the higher temperature. In this case the rectification ratio is given by R = g + g coth( β ~ ω q ) g coth( β ~ ω q ) + g . (5) | P i + / P i - | - n | P i + / P i - | - g / g g / g ) (a) (b) FIG. 2: Rectification ratio. (a)
R − n . The parameters are: g /g = 1 /
3, black circles β ~ ω q = 0 . β ~ ω q = 4 .
8, blue circles β ~ ω q = 0 . β ~ ω q = 4 .
8, and green circles β ~ ω q = 1 . β ~ ω q = 4 .
8. Stars at n = 2are from Eq. (4). (b) R − n = 2. Blue circles are from the numerics and red ones from Eq. (4). Theparameters are: β ~ ω q = 0 . β ~ ω q = 4 . For small asymmetry δ = 1 − g /g , | δ | (cid:28)
1, one can expand the rectification ratio into
R − e − β ~ ω q δ. (6)The inset of Fig. 2b shows this result by solid line for the corresponding temperature. II. MULTILEVEL SYSTEM
Consider n -level system with constant energy spacing ~ ω q . In the limit of n → ∞ it represents a linear harmonicoscillator as shown in Fig. 1b. The transition rates between levels k and k ± ( i ) k → k ± = 1 ~ |h k | ˆ Q | k ± i| S i ( ∓ ω q ) . (7)Here i = 1 , Q = i q ~ Z (ˆ a † − ˆ a ) and S i ( ω ) = 2 R i ~ ω − e − βi ~ ω are charge operator and voltage noisewhen applied to a circuit, respectively. Here, Z is a characteristic impedance of the system, ˆ a (ˆ a † ) is annihilation(creation) operator of the ladder of the system, and R i is the resistance of the bath. In this case only the transitionsbetween the nearest levels are allowed according to Γ ( i ) k → k − = k Γ ( i )out Γ ( i ) k − → k = k Γ ( i )in . (8)The steady state population of each level reads ρ i = S i /S , where S = n Y k =1 Γ k → k − , S j = j Y k =1 Γ k − → k n Y k = j +1 Γ k → k − , S n = n Y k =1 Γ k − → k , S = n X i =0 S i , (9)where 1 ≤ j ≤ n − k → k ± = Γ (1) k → k ± + Γ (2) k → k ± . The expression of power to bath i is then given by P i = ~ ω q n X k =1 (Γ ( i ) k → k − ρ k − Γ ( i ) k − → k ρ k − ) . (10)As seen in Fig. 2a, the rectification vanishes exponentially when the number of equidistant levels increases. III. ENERGY LEVELS OF THE EXPERIMENTAL DEVICE
The Hamiltonian ˆ H is given byˆ H = ~ ω L ˆ a † L ˆ a L + ~ ω q ˆ a † q ˆ a q + ~ ω R ˆ a † R ˆ a R + g (ˆ a q ˆ a † L + ˆ a † q ˆ a L + ˆ a q ˆ a † R + ˆ a † q ˆ a R ) + ˜ g (ˆ a L ˆ a † R + ˆ a † L ˆ a R ) . (11)Here, ~ ω L , ~ ω q , and ~ ω R are the energies of the left resonator, qubit and the right resonator, respectively, g is thecommon coupling constant of the qubit to the two resonators, and ˜ g is the cross-coupling between the resonators.In the eleven-level basis of | i , | i , | i , | i , | i , | i , | i , | i , | i , | i , | i , | i , where theentries in each state refer to the left resonator, the qubit, and the right resonator, respectively, the matrix form of theHamiltonian can be written H = ~ ω − a/ γ ˜ γ γ r γ γ γ a/ − a/ r γ ˜ γ √ γ γ γ √ γ γ γ r + 1 + a/ r √ γ
00 0 0 0 √ γ √ γ − a √ γ − a + r √ γ √ γ − a/ , (12)where ω = ω R + ω L , a = ω R − ω L ω , γ = g ~ ω , ˜ γ = ˜ g ~ ω , and r = ~ ω q ~ ω . Here ω L = 2 π × . ω R = 2 π × . ~ ω q = p E J E C | cos( π Φ / Φ ) | − E C like in the anharmonic Josephson potential. IV. STATISTICS AND RECTIFICATION
Rectification (in a two-level system) depends on the statistics of transition rates. In particular let us considerfermions vs bosons. The latter case was discussed earlier in this Supplementary Material.We may write the transition rate into the system via contact i for fermions (+) and bosons (-) asΓ ( i )in = g i e β i E ± . (13)For fermions, we may take a single level quantum dot, where g i is determined by the barrier, and E is the energylevel position of the dot with respect to the fermi level, controlled by gate as shown in Fig. 3a. For bosons, we maytake as above a qubit coupled to a dissipative environment: g i = ω /Q i , where E = ~ ω q is the level splitting of thequbit, and Q i the quality factor of the environment i (Fig. 3b). For both cases, the detailed balance condition holdsΓ ( i )out = e β i E Γ ( i )in (14)for the rate out from the system.Steady-state occupation on the excited state of the system is given by ρ e = Γ (1)in + Γ (2)in Γ (1)in + Γ (2)in + Γ (1)out + Γ (2)out . (15)Let us assume for simplicity that bath 1 has inverse temperature β and bath 2 is at zero temperature. Then we findthe power to bath 2 as P = ρ e E Γ (2)out . (16)For fermions we find P = g g g + g Ef ( E ) , (17)where f ( E ) = 1 / ( e βE + 1). Due to the symmetry with respect to indices 1 and 2, this system does not rectify heat.On the contrary, as we saw already earlier, for bosons the situation is different, and we obtain P = g g g coth( βE/
2) + g En ( E ) , (18)where n ( E ) = 1 / ( e βE − g = g . Γ in(1) Γ out (1) TT T =0 T =0 Γ out (2) | ۧ𝑔 | ۧ𝑒 Γ in (1) Γ out(2) (a) (b) Γ out(1) EE FIG. 3: Single-level quantum dot in (a) and a qubit in (b). igure 1: Diagram of the sample, consisting of a centrally located Xmon typesuperconducting qubit cross-coupled to two superconducting co-planar waveguideresonators at 2.8 GHz and 6.7 GHz, respectively. Each resonator is terminated with athin-film copper microstrip resistor, acting as a mesoscopic thermal bath, one of whichshown in the left inset by a colourised scanning electron micrograph of the coppermicrostrip resistor (orange), with 4 superconducting aluminium probes (green) (separatedfrom the copper by an insulator, not visible) for temperature control and readout, and twosuperconducting aluminium contacts to the co-planar waveguide resonator (blue). Thesuperconducting quantum interferometer is shown similarly on the right inset. Thetopmost diagram represents a simple model of the system, with theresonator-qubit-resonator structure represented as a diode, and the forward and reversedirections drawn in purple and green respectively. A third electrode can be seen on theXmon island, which in non-dissipative variants of the device used for spectroscopy,connects the Xmon to a readout resonator15igure 2: (a)Power transmitted between the two baths at three voltage (heating) biaspoints, with the subplots corresponding to 420 mK (1000 fW), 400 mK (750 fW), and 380mK (600 fW) source temperatures (powers), in descending order. The bath temperature iskept fixed at 150 mK. Purple is the forward direction, and green the reverse, as shown inFigure 1. (b) Rectification ratio of traces from (a), with the non-tunable contributionremoved.16igure 3: Two-tone spectroscopic readout of resonator-qubit-resonator structure performedusing a tertiary readout resonator coupled to a third electrode of the qubit. Here, thecopper terminations are not present. We observe qubit-resonator couplings at 2.75 GHz,corresponding to the low frequency resonator, 5.5 GHz, corresponding to the second modeof this resonator, and at 7.05 GHz, corresponding to the high frequency resonator. Wenote, however, that there is broadening of these lines when the copper termination ispresent, as reported in [26]. The parameters used in the calculated energy spectra, shownin the upper right figure, are: E J /h = 45 GHz and E C /h = 0 .
15 GHz, which give ω (Φ = 0) / π = 7 ..