Coupling a Germanium Hut Wire Hole Quantum Dot to a

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Coupling a Germanium Hut Wire Hole Quantum Dot to a Superconducting Microwave Resonator Yan Li, Shu-Xiao Li, Fei Gao, Hai-Ou Li, Gang Xu, Ke Wang, Di Liu, Gang Cao, Ming Xiao, Ting Wang, Jianjun Zhang, Guang-Can Guo, and Guo-Ping Guo Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b00272 • Publication Date (Web): 22 Feb 2018 Downloaded from http://pubs.acs.org on February 25, 2018

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Coupling a Germanium Hut Wire Hole Quantum Dot to a Superconducting Microwave Resonator Yan Li,1,2,# Shu-Xiao Li,1,2,# Fei Gao,3 Hai-Ou Li,1,2,* Gang Xu,1,2 Ke Wang,1,2 Di Liu,1,2 Gang Cao,1,2 Ming Xiao,1,2 Ting Wang,3 Jian-Jun Zhang,3 Guang-Can Guo,1,2 and Guo-Ping Guo1,2,* 1

Key Laboratory of Quantum Information, CAS, University of Science and Technology of China, Hefei, Anhui 230026, China

2

Synergetic Innovation Center of Quantum Information & Quantum Physics,

University of Science and Technology of China, Hefei, Anhui 230026, China 3

National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China #

these authors contributed equally to this work

*

Emails: [email protected]; [email protected] Fax: 86-0551-63606043

ABSTRACT: Realizing a strong coupling between spin and resonator is an important issue for scalable quantum computation in semiconductor systems. Benefiting from the advantages of strong spin–orbit coupling strength and long coherence time, the Ge hut wire, which is proposed to be site–controlled grown for scalability, is considered as a promising candidate to achieve this goal. Here we present a hybrid architecture in which an on-chip superconducting microwave resonator is coupled to the holes in a Ge quantum dot. The charge stability diagram can be obtained from the amplitude and phase response of the resonator independently from the DC transport measurement. Furthermore, we estimate the hole-resonator coupling rate of gc ⁄2π=148 MHz in the

single quantum dot-resonator system, and estimate the spin–resonator coupling rate

gs ⁄2π to be in the range 2–4 MHz. We anticipate that strong coupling between hole spins and microwave photons in a Ge hut wire is feasible with optimized schemes in the future. KEYWORDS: Ge hut wire, hole quantum dot, resonator, microwave 1 ACS Paragon Plus Environment

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On-chip microwave resonators have been playing an important role, in recent years, for sensitive metrology and coupling distant qubits in quantum electrodynamic circuit architecture.1 To date, hybrid architectures, in which quantum dots (QDs) coupled to microwave resonators, have been extensively investigated on a variety of materials such as GaAs,2-5 silicon,6-8 carbon nanotube,9-12 graphene,13-15 InAs nanowire,16, 17 and InSb nanowire.18 In these studies, realizing the strong coupling region between spin and resonator is a major focus for the applications in coupling and entanglement of distant spin qubits. Very recently, strong coupling of charge states in QDs and microwave resonators has been realized.3, 6, 12 Previous work has also realized readout of the spin state in InAs nanowire QD using spin–orbit coupling.16 However, strong coupling of spin states and resonators have been theoretically proposed19, 20 but as yet has not been achieved until now. Moreover, the spin–resonator coupling rate obtained by directly coupling a single spin magnetic dipole to the magnetic field of a resonator in the circuit quantum electrodynamic (cQED) circuit architecture is gM /2π ≈ 10 Hz, which is too weak for applications in quantum information processing.21 To realize strong coupling of a single spin to a microwave resonator and spin qubits with tunable long-range coupling, one needs a material with longer coherence time or larger spin–orbit coupling strength16,

22-25

, which has been theoretically

predicted and experimentally demonstrated in holes in Ge/Si core/shell nanowire.26, 27 In 2012, a novel type of Ge nanowire with non-cylindrical symmetry and large light– heavy hole splitting28-31, the Ge hut wire, was grown using the Stranski–Krastanow (SK) growth mode.32 Compared with core/shell nanowire, the Ge hut wire has a similar spin-orbit coupling strength and an even longer coherence time,28, 29 and is considered to be a promising candidate for achieving all-electrically controlled spin qubit33 and strong spin-resonator coupling.16 Notably, the positions of Ge hut wires are proposed to be well controlled by means of site-controlled technique,34, 35 which would avoid the additional step of precise transfer or chemical vapor deposition

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(CVD) growth applied in host nanowire materials. It would present the important advantage of scalability for quantum information processing. Here, we fabricated a hybrid device coupling Ge single quantum dot (SQD) to a microwave resonator. Quantum transport properties of the holes in the Ge QD were investigated by amplitude and phase of the resonator. Measuring mechanism was theoretically explained and confirmed by experimental observation that both frequency shift and frequency width broadening of resonator were modulated by QD. We experimentally studied the hole-resonator coupling rate of Ge SQD based on Coulomb blockade phenomenon,47-49 and further estimated the spin-resonator coupling rate.16, 23 In the sample structure (Figure 1a), the patterning of the electrodes was performed by electron beam lithography. Source and drain electrodes were metallized with a 30-nm palladium layer following a buffered hydrofluoric acid (HF) etch of the silicon oxide layer. After a 25-nm alumina dielectric layer was grown by atomic layer deposition, a gate electrode was subsequently fabricated with 3-nm titanium and 25-nm palladium layers. Then, alumina in the proximity of source electrodes was etched to allow a connection between the electrodes and the resonator. Finally, the superconducting resonator was patterned by optical lithography and deposited with 200-nm aluminum. We employed a half-wavelength reflection line resonator, which was designed without the ground plane to avoid extra energy leakage.36 The two superconducting lines of the resonator were separately connected to the source electrodes of four QDs, but only one was considered in this experiment (Figure 1b). The QD was placed at the position of antinode of the voltage standing wave, maximizing the coupling rate between the QD and resonator. The typical QD structure is shown in Figure 1c. And the circuit schematic for the hybrid device is shown in Figure 1d, a continuous microwave signal is split into two components of opposite phases by a 180° hybrid and then applied into two striplines, and the reflected signal is detected by a network analyzer. The sample was mounted in a dilution refrigerator with base temperature of about 26 mK. The reflection spectrum 3 ACS Paragon Plus Environment

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was measured by a network analyzer with a cryogenic HEMT and room-temperature amplifiers. The terminal power applied to the resonator was below −110 dBm, to avoid thermal or nonlinear effects. (More details about the sample structure and measurement setup are given in Supporting Information.) Our scheme could be extended for double quantum dots, single-atom laser, and other multifarious circuits. 37-40

Figure 1. (a) Optical micrograph of a half-wavelength reflection line resonator integrated with four QDs. The resonator consists of two superconducting striplines, which are connected to AC pads by finger capacitance, and two AC pads are used for the input and output of the microwave. One of the QDs is located in the red square region. (b) False-color SEM image of a QD coupled to the striplines of the resonator through the source electrode. (c) Zoom in SEM image of a typical three-terminal structure of the QD. Source and drain electrodes are used to drive the DC current; the top gate serves to tune the potential of the QD. (d) Circuit schematic of the hybrid device. A microwave signal is split into two components of opposite phases by a 180-degree hybrid device, and then two opposite phases are applied in the two striplines of the resonator. The reflected signal is detected using a network analyzer with a primary cryogenic amplifier and a secondary room-temperature amplifier. 4 ACS Paragon Plus Environment

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The amplitude and phase signals from the resonator as functions of frequency were extracted using network analyzer, with the QD tuned in Coulomb valley region (Figures 2a and b, black lines). An amplitude decrease and a phase reversal were observed near the resonance frequency. A λ/2 open-circuit microstrip resonator model41 was applied to describe the hybrid system of Ge QDs and resonator. The reflection coefficient can be expressed as

j(ω0 - ω)  2(κi - κe ) 1

S11 (ω)=−

j(ω0 -ω)  2(κi +κe ) 1

,

(1)

which determine the amplitude and phase, A=∣S11∣ and ϕ=arctan(S11). Equipped

with this model the resonance frequency is ω0 ⁄2π= 5.972 GHz, the internal loss

κi /2π= 5.18 MHz, the external loss κe /2π= 2.19 MHz, and the total photon loss rate

κ⁄2π=(κi +κe )/2π=7.37 MHz, corresponding to average life time (1/κ) of photons

trapped in the resonator of more than 130 ns. In addition, the quality factor (Q) equals 810; a higher quality factor can be obtained if fewer QDs are coupled to the resonator. In contrast, with the QD was tuned at Coulomb resonance region, obvious variations in both amplitude and phase signals were plotted (Figure 2a,b; red lines). Both δ and δA were obtained resulting from dispersive and dissipative coupling,13, 14 which indicating that the microwave resonator and the QD were coupled and the resonator was able to detect the changes in the hole states. Having demonstrated good QD–resonator coupling, we then characterized the QD and measured its properties by simultaneously recording the reflected signal of the resonator and the DC transport signal through the source/drain leads. With the probe frequency fixed at the resonance frequency, these measurements were in response to changes of gate voltage (VG) and source/drain bias voltage (VSD). Similar Coulomb diamond diagrams were obtained via DC and amplitude signals (Figure 2c,d). From these characteristic diagrams, the charging energy (EC) of the single quantum dot (SQD) is found to be approximately 4–6 meV, and the lever arm α of the gate was extracted to be approximately 0.20–0.25 eV/V. To investigate the corresponding relationship between DC and resonator response 5 ACS Paragon Plus Environment

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in detail, we measured the amplitude and phase as well as DC signal simultaneously, as functions of gate voltage with zero bias voltage (VSD =0 mV). Figure 2e shows the Coulomb oscillations diagram with fluctuating heights of the DC signal. The peaks of the DC curve have corresponding features in the amplitude curve as well as phase curve (Figure 2f-g). Nevertheless, several peaks in the phase curve are missing possibly because the signal-to-noise ratio (SNR) is low. The peak heights in the amplitude and phase curves are not in general proportional to those of the DC curve. While signals in DC curve are not apparent, indicated as the two encircled regions and yellow region, peaks were still obvious in the amplitude and phase curves (Figure 2e-g).

Figure 2. (a-b) Measured amplitude (a) and phase (b) response of the resonator while the QD is tuned in the Coulomb valley region (black line) and the Coulomb resonance region (red line), corresponding to positions B and A, respectively, marked in (c). (c-d) Color scale plots of the DC transport (c) and amplitude (d) signals versus VG and VSD measured simultaneously while the probe frequency was fixed at 5.972 GHz. The corresponding Coulomb diamond patterns are typical for quantum transport of the QD. (e-g) DC transport, amplitude, and phase signal versus VG measured at the same time with the probe frequency fixed at 5.972 GHz and VSD = 0 mV. Corresponding Coulomb oscillations diagrams are shown in (e-g). Inset: Zoom in 6 ACS Paragon Plus Environment

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views where the signal is difficultly macroscopic in the DC curve. The coincidence of the DC and reflected resonator signals demonstrates that the microwave resonator indeed functions as a charge detector. The microwave readout appears to be more sensitive compared with the traditional DC transport measurement, while the DC transport signals become too small to be measured. It also enables the measurement of devices with weak couplings to leads that avoid the decoherence caused by coupling with fermionic reservoirs.42 The sensitive microwave resonator would be applied to detect the rich physics into few-hole/electron region, such as measuring the valley splitting and charge dephasing rates in the single-electron region.5, 8 Furthermore, we employed the admittance model2, 43, 44 and a quantum model of the hybrid system13 to analyze quantitatively the hybrid system according to the scattering theory45 and input-output theory33, respectively. Inspired by a previous studies of the phase response performed on carbon nanotubes,9, 10, 42 we used the amplitude and phase responses to analyze the change in hole states of the Ge QD in this work. The resonator could be equivalent to a RLC circuit,41 and QD is equivalent

to a RC circuit with complex admittance Ydot (ω) ≈ 1/Rdot + jωCdot (Figure 3a). With the coupling between the resonator and QD taken into account, the effective impedance of the resonator can be written as

where f0 + j

1

= f0 + j

2RresCres

1

2RresCres

−

Cdot f + 2Cres 0

j



2CresRdot

−

f0 Cdot 2Q Cres

,

(2)

is related to the bare resonator, and the rest terms are contributed

by the QD. Cdot leads to dispersive coupling between the QD and the resonator and

results in a frequency shift δfR . While Rdot leads to a dissipative coupling and

results in a frequency broadening δfD reflected in both amplitude and phase signals,5, 9, 42, 46

expressed as δfR ≈−Cdot f0 /(2Cres ) and δfD ≈1/(2Cres Rdot ) with 1/Rdot =dI/dV.

(More details about frequency shift and frequency broadening is given in Supporting Information.)

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Figure 3. (a) Equivalent electronic circuit of a QD coupled to a resonator. (b, c) Simulated influence of frequency shift and broadening on phase (b) and amplitude (c) response. Reference resonance (without δfR and δfD , black line), shifted resonance (blue line) with a finite δfR , and broadened resonance (red line) with a finite δfD ; (d-e) Simulated influence of δfR (green line) and δfD (orange line) on phase (d) and amplitude (e) variations. Experimental variations contributed by δfR (blue dots) and δfD (red dots), measured at VG=−0.638V, correspond to the simulated results. (f-g) Extracted values δfR (blue dots) and δfD (red triangles) as functions of gate voltage. Both δfR and δfD modulate similar to the Coulomb blockade peaks. From reflection coefficient of equation (1), the phase ϕ and amplitude A are determined as ϕ=arctan

κe (ω0 -ω)

ω0 -ω + 0.25 κi 2 -κe 2  2

,

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(3)

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 ω0 -ω2 +0.25 κi 2 -κe 2  +κe 2 ω0 -ω2 2

A=

 ω0 -ω +0.25 κi 2 -κe 2  2

2

.

(4)

When finite δfR and δfD are taken into consideration,   0=ω0+δωR , ̃ i=κi+2δωD , with δωR =2πδfR , δωD =2πδfD , S11 becomes

11(ω) = −

j[(ω0 +δωR )-ω]+2[(κi +2δωD )-κe ] 1

j[(ω0 +δωR)-ω]+ [(κi +2δωD )+κe ] 1 2

.

(5)

Therefore phase and amplitude variation could be acquired δϕ = arctan(  11) − arctan(S11), and δA=│  11│−│S11│. Figure 3b (c) shows the calculated phase

(amplitude) influenced by a finite δfR and δfD . The black curve serves as a standard reference without δfR and δfD , a finite pure δfR shifts resonance frequency and a

finite pure δfD broadens resonance frequency width. Figure 3d (e) shows the simulated phase (amplitude) variation influenced by δfR and δfD , with VG=−0.638V. δfR influences the even resonant part of phase variation (Figure 3d, green line), and also effects the odd part of amplitude variation (Figure 3e, green line). δfD influences the odd resonant part of phase variation (Figure 3d, orange line), and also effects the even part of amplitude variation (Figure 3e, orange line). The corresponding experimental variations effected by δfR (blue dots) and δfD (red dots) indicate quantitative agreements with the simulated results. We extract the value of δfR from integrating the even (odd) resonant part of phase (amplitude) variation curve, and extract δfD by integrating the odd (even) resonant part of phase (amplitude) variation curve with VG=−0.638V. (See the Supporting Information for details.) Figure 3 (f) and (g) respectively show the δfR and δfD extracted with different values of gate voltages, and VG ranges from −0.65 V to −0.59 V in steps of 1 mV. Both δfR and δfD exhibit similar oscillations with Coulomb blockade peaks, 9 ACS Paragon Plus Environment

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indicating that both frequency shift and frequency broadening are modulated by QD. These experimental results verify theoretical interpretation of the probing mechanism above. So far, the dipole charging coupling of DQD and resonator has attracted considerable attention for years, meanwhile few studies have been focused upon SQD-resonator architectures. Very recently, several studies have revealed the differences between SQD- and DQD-resonator systems, and developed two approaches and theories to investigate the coupling rate between the resonator and carbon nanotube electron SQD.

47-49

In this letter, we applied these approaches and

theories in analyzing the hole-resonator coupling rate based on Ge hut wire SQD. In DQD-resonator system, the coupling rate is studied based on the dipole between left and right dots. However, the dipole in SQD-resonator system corresponds to the tunneling between the dot and leads. The dot is more or empty and the reservoir has a conversely more or less excess of hole, thus creating the dipole. To date, using this kind of dipole to directly study charge-resonator coupling in SQD-resonator system have not been reported in the theoretical and experimental works. The coupling is studied by means of the photonic potential but not the dipole. The coupling rate is calibrated based on the relation between frequency shift and compressibility by means of Coulomb blockade phenomenon (Figure 4a).47-49 In a Coulomb valley (Figure 4b), the hole tunneling is frozen, leading to the absence of compressibility variation, phase and amplitude responses as the signals of bare resonator (black lines in Figure 4d). In

a Coulomb resonance (Figure 4c), with the finite values of chemical potential  , hole tunnels between the QD and leads, therefore altering the number of holes N and  

compressibility of the hole gas as χ= . Different compressibility corresponds to different resonance frequency of the resonator, and leads to a frequency shift (blue lines in Figure 4d). A finite compressibility of the hole gas shifts the resonance frequency and lead to a frequency shift δfR = 

simulated with χ=



ћg ! " # 47

%$. The frequency shift δfR =

ћg ! " #

%$is

using equation of motion (EOM) theory (soild lines in Figure 4f 10 ACS Paragon Plus Environment

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and 4g),9, 10, 46 which is based on the experimental data of tunneling rate, charging energy, and chemical potential. By fitting the simulated δfR to experimental δfR extracted from phase (blue dots in Figure 4f) and amplitude (red triangles in Figure 4g) signals, the hole-resonator coupling rate is gc ⁄2π=148 MHz, which is similar to coupling rates obtained in Carbon nanotube electron SQD.9, 10, 42, 49

 

Figure 4. (a) The compressibility of hole gas in QD is associated with χ= , with N

the number of holes and  the chemical potential, is simply the density of states at

the Fermi energy. 47(b) In a Coulomb valley, the hole cannot tunnel between dot and leads, which corresponds to the absence of compressibility (the ‘off’ state). (c) In a 11 ACS Paragon Plus Environment

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Coulomb resonance, the hole tunnels between dot and leads, which corresponds to a finite compressibility (the ‘on’ state). (d) Microwave resonator has different phase and amplitude responses with different compressibility of the QD, and corresponds to the variation of the resonance frequency. A finite compressibility of the hole gas shifts the resonance frequency and lead to a frequency shift δfR =

ћg ! " #

%$. (e) Capacitance model

of the hybrid SQD-resonator device, CAC is the coupling capacitive between the QD and resonator. (f-g) The blue dots represent the δfR extracted from phase variation, and the red triangles represent the δfR extracted from amplitude variation. Solid lines ћg ! "#

is the simulated δfR =



%$, with χ=

 

obtained using EOM theory.

There is an alternative approach to evaluate the hole-resonator coupling rate on the basis of the variation of coupling capacitance between the QD and resonator, CAC

(Figure 4e). The variation of compressibility in the QD alters the coupling capacitance by(()* +)$

 

. CAC influences the resonance frequency of the resonator in the

electronic model, and results in a frequency shift, as δfR =(()* +)$



 0

f /(2Cres ), with

(AC =CAC ⁄C∑ is capacitive lever arm between CAC and the QD capacitance C∑ . 9, 10, 42, 46

According to previous studies performed on SQD-resonator system,

9, 42, 47, 49

the

hole-resonator coupling rate can be studied by gc ⁄2π = e(AC Vrms , where e is the elementary charge, Vrms =-

ћω0 2Cres

corresponds to the root-mean-square voltage of a

single photon in the resonator model. It is worthwhile to note that this expression of δfR=(()* +)$

 

f0 /(2Cres ) can be equivalently transferred to δfR =

ћg! " #

%$. (AC is 0.32

by fitting simulated δfR to the experimental results. Since Cres =0.53 pF and ω0 /2π=

5.972 GHz, gc is evaluated to be the same value as the first approach. (More details about hole-resonator coupling strength and EOM theory are given in Supporting Information.) For a spin in a single quantum dot, previously theoretical work by M. Trif et al 12 ACS Paragon Plus Environment

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gives an approximate prediction of effective spin-resonator coupling rate as gs = gc (Ez /∆E0 )(l//so ).16, 23 Dot size l is calculated to be 70 nm in this device. In our previous work, orbital level spacing ∆E0 is approximately 1 meV and g-factor is around 4.3.29 Assuming perpendicular magnetic field of 50 mT applied, the Zeeman splitting is calculated as Ez =gµB B=13.1 µeV. The spin-orbit length /so is determined

by λso=ћ⁄0(2m∗ ∆so) with the bulk heavy-hole effective mass m*=0.28me .16, 22, 23, 27 Since the spin-orbit energy ∆so was measured in the range of 35-50µeV in our

unpublished work of spin blockade in double quantum dot, /so is obtained to be in

the range of 28-57 nm. Consequently, the spin–resonator coupling rate gs ⁄2π is estimated to be 2–4 MHz, which is five orders of magnitude larger than the coupling rate obtained by directly coupling a single spin magnetic dipole and magnetic field of the resonator. 21 The distinguishing feature of strong coupling is that the coherent coupling rate

gs ⁄2π exceeds the photon loss rate κ⁄2π and decoherence rate γ⁄2π. The estimated

gs ⁄2π is near but smaller compared with κ⁄2π and γ⁄2π , therefore the strong coupling limit could not be reached in our actual device. The photon loss rate is measured to be 7.4 MHz in this work. Since the larger light–heavy hole splitting in the Ge hut wire would reduce the non-Ising type coupling to nuclear spins and result in a longer decoherence time compared to Ge/Si core/shell nanowire,

29, 30

the

decoherence rate in the latter material is determined to be 6 MHz, so Ge hut wire is expected to be smaller than 6 MHz.50,

51

There are optimized schemes in the

prospective investigation. The photon loss rate κ⁄2π could be decreased by optimizing the substrate and resonator. As group IV material, isotropic purified

germanium52, 53might have even smaller decoherence rate γ⁄2π, and this technique has been performed on silicon.54, 55 With these optimized schemes in the future work,

we anticipate that the strong spin-resonator coupling is viable in hole QD based on Ge hut wire system. 13 ACS Paragon Plus Environment

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In conclusion, we coupled an on-chip resonator to holes in the Ge hut wire QD. The microwave resonator can be utilized as a useful tool for charge state readout of the QD. The measuring mechanism was theoretically analyzed based on the admittance model and a quantum model of the hybrid system. We applied two approaches and theories in the analysis of SQD-resonator system, and estimated the

hole-resonator coupling rate gc ⁄2π=148 MHz. Furthermore, the spin–resonator coupling rate is estimated to be in the range 2–4 MHz, which is near but smaller than the photon loss rate and decoherence rate, suggesting that strong coupling limit could not be reached in our actual device. Our work launches the first step and indicates the feasibility to achieve strong spin-resonator coupling in Ge hut wire hole QD with the optimized schemes in the future work. Note added: After submission of our Letter, we became aware of three related works in which the strong coupling between spin and resonator has been demonstrated based on silicon and GaAs quantum dot.56-58

Acknowledgements: We thank M. R. Delbecq for fruitful discussions of charge-resonator coupling strength in SQD-resonator system. And we thank T. Mircea, K. D. Petersson and C. Kloeffel for fruitful discussions of estimating spin-resonator coupling strength. This work was supported by the National Key Research and Development Program of China (Grant No.2016YFA0301700), the National Natural Science Foundation of China (Grants No. 61674132, 11674300, 11575172, 11574356 and 11625419), the Anhui initiative in Quantum information Technologies (Grants No. AHY080000) and this work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.

Supporting Information: Further information on sample structure and measurement setup; frequency shift and frequency broadening; equation of motion theory (EOM); hole-resonator coupling strength. 14 ACS Paragon Plus Environment

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References: (1) Xiang, Z.-L.; Ashhab, S.; You, J. Q.; Nori, F. Rev. Mod. Phys. 2013, 85, 623-653. (2) Frey, T.; Leek, P. J.; Beck, M.; Faist, J.; Wallraff, A.; Ensslin, K.; Ihn, T.; Büttiker, M. Phys. Rev. B 2012, 86, 115303. (3) Stockklauser, A.; Scarlino, P.; Koski, J. V.; Gasparinetti, S.; Andersen, C. K.; Reichl, C.; Wegscheider, W.; Ihn, T.; Ensslin, K.; Wallraff, A. Phys. Rev. X 2017, 7, 011030. (4) Frey, T.; Leek, P. J.; Beck, M.; Blais, A.; Ihn, T.; Ensslin, K.; Wallraff, A. Phys. Rev. Lett. 2012, 108, 046807. (5) Basset, J.; Jarausch, D. D.; Stockklauser, A.; Frey, T.; Reichl, C.; Wegscheider, W.; Ihn, T. M.; Ensslin, K.; Wallraff, A. Phys. Rev. B 2013, 88, 125312. (6) Mi, X.; Cady, J. V.; Zajac, D. M.; Deelman, P. W.; Petta, J. R. Science 2017, 355,156-158 . (7) Mi, X.; Cady, J. V.; Zajac, D. M.; Stehlik, J.; Edge, L. F.; Petta, J. R. Appl. Phys. Lett. 2017, 110, 043502. (8) Mi, X.; Péterfalvi, C. G.; Burkard, G.; Petta, J. R. Phys. Rev. Lett. 2017, 119, 176803. (9) Delbecq, M. R.; Schmitt, V.; Parmentier, F. D.; Roch, N.; Viennot, J. J.; Fève, G.; Huard, B.; Mora, C.; Cottet, A.; Kontos, T. Phys. Rev. Lett. 2011, 107, 256804. (10) Delbecq, M. R.; Bruhat, L. E.; Viennot, J. J.; Datta, S.; Cottet, A.; Kontos, T. Nat. commun. 2013, 4, 1400. (11) Viennot, J. J.; Dartiailh, M. C.; Cottet, A.; Kontos, T. Science 2015, 349, 408-411. (12) Bruhat, L. E.; Cubaynes, T.; Viennot, J. J.; Dartiailh, M. C.; Desjardins, M. M.; Cottet, A.; Kontos, T. arXiv:1612.05214 2016. (13) Deng, G.-W.; Wei, D.; Li, S.-X.; Johansson, J. R.; Kong, W.-C.; Li, H.-O.; Cao, G.; Xiao, M.; Guo, G.-C.; Nori, F.; Jiang, H.-W.; Guo, G.-P. Nano Lett. 2015, 15, 6620-6625. (14) Deng, G.-W.; Wei, D.; Johansson, J. R.; Zhang, M.-L.; Li, S.-X.; Li, H.-O.; Cao, G.; Xiao, M.; Tu, T.; Guo, G.-C.; Jiang, H.-W.; Nori, F.; Guo, G.-P. Phys. Rev. Lett. 2015, 115, 126804. (15) Zhang, M.-L.; Wei, D.; Deng, G.-W.; Li, S.-X.; Li, H.-O.; Cao, G.; Tu, T.; Xiao, M.; Guo, G.-C.; Jiang, H.-W.; Guo, G.-P. Appl. Phys. Lett. 2014, 105, 073510. (16) Petersson, K. D.; McFaul, L. W.; Schroer, M. D.; Jung, M.; Taylor, J. M.; Houck, A. A.; Petta, J. R. Nature 2012, 490, 380-383. (17) Liu, Y. Y.; Petersson, K. D.; Stehlik, J.; Taylor, J. M.; Petta, J. R. Phys. Rev. Lett. 15 ACS Paragon Plus Environment

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2014, 113, 036801. (18) Wang, R.; Deacon, R. S.; Car, D.; Bakkers, E. P. A. M.; Ishibashi, K. Appl. Phys. Lett. 2016, 108, 203502. (19) Hu, X.; Liu, Y.-x.; Nori, F. Phys. Rev. B 2012, 86, 035314. (20) Jin, P.-Q.; Marthaler, M.; Shnirman, A.; Schön, G. Phys. Rev. Lett. 2012, 108, 190506. (21) Imamoğlu, A. Phys. Rev. Lett. 2009, 102, 083602. (22) Kloeffel, C.; Trif, M.; Loss, D. Phys. Rev. B 2011, 84, 195314. (23) Trif, M.; Golovach, V. N.; Loss, D. Phys. Rev. B 2008, 77, 045434. (24) Joe, S.; Mengyang, T.; Sven, R.; Dimitrie, C. Nanotechnology 2016, 27, 244001. (25) Hung, J.-T.; Marcellina, E.; Wang, B.; Hamilton, A. R.; Culcer, D. Phys. Rev. B 2017, 95, 195316. (26) Hao, X.-J.; Tu, T.; Cao, G.; Zhou, C.; Li, H.-O.; Guo, G.-C.; Fung, W. Y.; Ji, Z.; Guo, G.-P.; Lu, W. Nano Lett. 2010, 10, 2956-2960. (27) Higginbotham, A. P.; Kuemmeth, F.; Larsen, T. W.; Fitzpatrick, M.; Yao, J.; Yan, H.; Lieber, C. M.; Marcus, C. M. Phys. Rev. Lett. 2014, 112, 216806. (28) Watzinger, H.; Kloeffel, C.; Vukušić, L.; Rossell, M. D.; Sessi, V.; Kukučka, J.; Kirchschlager, R.; Lausecker, E.; Truhlar, A.; Glaser, M.; Rastelli, A.; Fuhrer, A.; Loss, D.; Katsaros, G. Nano Lett. 2016, 16, 6879-6885. (29) Li, S.-X.; Li, Y.; Gao, F.; Xu, G.; Li, H.-O.; Cao, G.; Xiao, M.; Wang, T.; Zhang, J.-J.; Guo, G.-P. Appl. Phys. Lett. 2017, 110, 133105. (30) Wang, D. Q.; Klochan, O.; Hung, J.-T.; Culcer, D.; Farrer, I.; Ritchie, D. A.; Hamilton, A. R. Nano Lett. 2016, 16, 7685-7689. (31) Marcellina, E.; Hamilton, A. R.; Winkler, R.; Culcer, D. Phys. Rev. B 2017, 95, 075305. (32) Zhang, J. J.; Katsaros, G.; Montalenti, F.; Scopece, D.; Rezaev, R. O.; Mickel, C.; Rellinghaus, B.; Miglio, L.; De Franceschi, S.; Rastelli, A.; Schmidt, O. G. Phys. Rev. Lett. 2012, 109, 085502. (33) Li, R.; You, J. Q.; Sun, C. P.; Nori, F. Phys. Rev. Lett. 2013, 111, 086805. (34) Katsaros, G.; Tersoff, J.; Stoffel, M.; Rastelli, A.; Acosta-Diaz, P.; Kar, G. S.; Costantini, G.; Schmidt, O. G.; Kern, K. Phys. Rev. Lett. 2008, 101, 096103. (35) Zhang, J. J.; Stoffel, M.; Rastelli, A.; Schmidt, O. G.; Jovanović, V.; Nanver, L. K.; Bauer, G. Appl. Phys. Lett.2007, 91, 173115. (36) Zhang, M.-L.; Deng, G.-W.; Li, S.-X.; Li, H.-O.; Cao, G.; Tu, T.; Xiao, M.; Guo, G.-C.; Jiang, H.-W.; Siddiqi, I.; Guo, G.-P. Appl. Phys. Lett. 2014, 104, 083511. 16 ACS Paragon Plus Environment

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(37)Basset, J.; Stockklauser, A.; Jarausch, D. D.; Frey, T.; Reichl, C.; Wegscheider, W.; Wallraff, A.; Ensslin, K.; Ihn, T. Appl. Phys. Lett. 2014, 105, 063105. (38) Lambert, N.; Nori, F.; Flindt, C. Phys. Rev. Lett. 2015, 115, 216803. (39) Neill, L.; Christian, F.; Franco, N. EPL (Europhys. Lett.) 2013, 103, 17005. (40) Ashhab, S.; Johansson, J. R.; Zagoskin, A. M.; Franco, N. New J. Phys. 2009, 11, 023030. (41) Pozar, D. M. Microwave Engineering. (2004,Wiley) (42) Viennot, J. J.; Delbecq, M. R.; Bruhat, L. E.; Dartiailh, M. C.; Desjardins, M. M.; Baillergeau, M.; Cottet, A.; Kontos, T. C. R. Phys. 2016, 17, 705-717. (43) Chorley, S. J.; Wabnig, J.; Penfold-Fitch, Z. V.; Petersson, K. D.; Frake, J.; Smith, C. G.; Buitelaar, M. R. Phys. Rev. Lett. 2012, 108, 036802. (44) Cottet, A.; Mora, C.; Kontos, T. Phys. Rev. B 2011, 83, 121311. (45) Prêtre, A.; Thomas, H.; Büttiker, M. Phys. Rev. B 1996, 54, 8130-8143. (46) Delbecq, M. R. Coupling quantum dot circuits to microwave cavities. 2012. (47)Desjardins, M. M.; Viennot, J. J.; Dartiailh, M. C.; Bruhat, L. E.; Delbecq, M. R.; Lee, M.; Choi, M. S.; Cottet, A.; Kontos, T. Nature 2017, 545, 71. (48)Audrey, C.; Matthieu, C. D.; Matthieu, M. D.; Tino, C.; Lauriane, C. C.; Matthieu, D.; Jérémie, J. V.; Laure, E. B.; Benoit, D.; Takis, K. J. Phys.: Condens. Matter 2017, 29, 433002. (49)Bruhat, L. E.; Viennot, J. J.; Dartiailh, M. C.; Desjardins, M. M.; Kontos, T.; Cottet, A. Phys. Rev. X 2016, 6, (2), 021014. (50) Hu, Y.; Kuemmeth, F.; Lieber, C. M.; Marcus, C. M. Nat. Nanotechnol. 2012, 7, 47-50. (51) Higginbotham, A. P.; Larsen, T. W.; Yao, J.; Yan, H.; Lieber, C. M.; Marcus, C. M.; Kuemmeth, F. Nano Lett. 2014, 14, 3582-3586. (52)Churbanov, M. F.; Gavva, V. A.; Bulanov, A. D.; Abrosimov, N. V.; Kozyrev, E. A.; Andryushchenko, I. A.; Lipskii, V. A.; Adamchik, S. A.; Troshin, O. Y.; Lashkov, A. Y.; Gusev, A. V. Cryst. Res. Technol. 2017, 52, 1700026. (53)Adamchik, S. A.; Bulanov, A. D.; Sennikov, P. G.; Churbanov, M. F.; Sozin, A. Y.; Chernova, O. Y.; Andryushchenko, I. A.; Troshin, O. Y. Inorg. Mater. 2010 47(7):694-696. (54) Yoneda, J.; Takeda, K.; Otsuka, T.; Nakajima, T.; Delbecq, M. R.; Allison, G.; Honda, T.; Kodera, T.; Oda, S.; Hoshi,Y. ; Usami, N.; Itoh,K. M.; Tarucha, S. In arXiv:1708.01454, 2017. (55) Takeda, K.; Kamioka, J.; Otsuka, T.; Yoneda, J.; Nakajima, T.; Delbecq, M. R.; 17 ACS Paragon Plus Environment

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Amaha, S.; Allison, G.; Kodera, T.; Oda, S.; Tarucha, S. Sci. Adv. 2016, 2, e1600694 . (56) Mi, X.; Benito, M.; Putz, S.; Zajac, D. M.; Taylor, J. M.; Burkard, G.; Petta, J. R. arXiv:1710.03265 2017. (57) Landig, A. J.; Koski, J. V.; Scarlino, P.; Mendes, U. C.; Blais, A.; Reichl, C.; Wegscheider, W.; Wallraff, A.; Ensslin, K.; Ihn, T. arXiv:1711.01932 2017. (58) Samkharadze, N.; Zheng, G.; Kalhor, N.; Brousse, D.; Sammak, A.; Mendes, U. C.; Blais, A.; Scappucci, G.; Vandersypen, L. M. K. arXiv:1711.02040 2017.

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