Dipole coupling of a tunable hole double quantum dot in germanium hut wire to a microwave resonator

Germanium (Ge) hole quantum dot system is a promising candidate for the realization of strong coupling of spin to superconducting resonators and scalable for multiple qubits coupling through resonators, benefiting from its strong and tunable spin orbit coupling, long coherence time due to the special structure of Ge hut wire, and site controlled large scale hut wire positioning. We have coupled a reflection linear resonator to the hole double quantum dot (DQD) formed in a Ge hut wire. The amplitude and phase responses from the microwave resonator revealed the charge stability diagrams of the DQD which in good agreement with that obtained from transport measurements. The DQD interdot tunneling rate is determined ranging from 6.2 GHz to 8.5 GHz, which shows the tunability of hole resonator interaction. Furthermore, we achieved a hole-resonator coupling rate up to 15 MHz, with the charge qubit decay rate {\gamma} of approximately 0.28 GHz. Meanwhile the hole spin resonator coupling rate was estimated to be 0.5 MHz. These results suggest that holes of DQD in a Ge hut wire are dipole coupled to microwave photons, potentially enabling tunable coherent hole spin-photon interactions in Ge with an inherent spin-orbit coupling.


I. INTRODUCTION
Spin-based qubits formed by electrons trapped in silicon quantum dots have been recognized as a highly promising system for quantum computing because of its long coherent time and its compatibility with mature semiconductor technology [1,2]. Recent advances include fault-tolerant control fidelities for single-qubit gates and high-fidelity for two-qubit gate [3][4][5][6] and strong spinphoton coupling [7][8][9]. These results are believed to enable the construction of a quantum computer based on spin qubits with photonic interconnects for long range coupling and hybrid integration to other quantum systems [10,11]. However, since the spin-orbit coupling in silicon is small, an integrated component like a micro-magnet [4,5] or a strip line [3,6] has to be incorporated, which will complicate the whole fabrication process in such devices for spin control and spin-charge hybridization.
Hole quantum dots in Ge are regarded as an alternative option, due to the strong spin-orbit coupling in Ge [12][13][14], which could acquire spin-charge hybridization, thus spin-qubit manipulation and spin-photon coupling may be implemented directly without any additional components [15][16][17].
As a group-IV element, Ge also has the capability to obtain nuclear spin-0 isotopes like silicon and thus a weak hyperfine interaction. Therefore, the hole spin qubit in Ge is expected to possess a long coherence time [18]. Ge/Si core/shell nanowire, as a hole material grown by chemical vapor deposition, has been investigated for years [19][20][21]. However, recently a new type of hole material named Ge hut wires grown by molecular beam epitaxy [22] attracts research attention [22][23][24][25][26][27]. Compared to Ge/Si core/shell nanowire, Ge hut wire is an entire different hole system rely on its distinctive growth mechanism. Firstly, the cylindrical geometry of the Ge/Si core/shell nanowire leads to a mixture of heavy holes (HH) and light holes (LH). As a consequence, the hyperfine interaction is not of Ising type, leading to the reduction of spin coherence time [18]. In comparison, Ge hut wire exhibits a triangular cross-section with a height of about 2 nm above the wetting layer and is fully strained, which sharply reduce the HH and LH mixing. Therefore, it reduces the non-Ising type coupling to nuclear spins that negatively affected the coherence time [22,23]. Latest work demonstrated that the hole spin-orbit qubit in Ge hut wire with dephasing time * of 130 ns [27], which is comparable to the dephasing time estimated in Ge/Si core/shell nanowire, about 180ns [20], and twice the dephasing time reported for holes in Si MOSFET, about 60ns [28]. Furthermore, due to special growth mechanism, which is called Stranski−Krastanow (SK) growth mode, Ge hut wire system allows site-controlled large-scale hut wire positioning on patterned substrate via planar epitaxial growth, presenting the important advantage of scalability for quantum computation. Previous advances in Ge hut wires have shown transport measurements [23,24] single-shot reflectometry of hole spin states [25] and hole spin-orbit qubit [27].
Moreover, we have recently demonstrated the coupling of a Ge-hut-wire single quantum dot (SQD) to a microwave resonator [26]. Compared with the SQD system, the energy-level splitting in the double quantum dot (DQD) is able to be tuned by the gate-induced detuning [29], and could be tuned to an energy scale close to that of a photon inside a microwave resonator. Therefore, to realize a tunable coherent spin-photon interaction, the architecture of a DQD dipole coupled to a resonator is preferable.
This coupling scheme has a strong microwave dipole interaction of the resonator with two charge states in which a hole is on either the left or right quantum dot. Such hybrid devices have been achieved with a variety of materials, such as Si [7,8], GaAs [9,30], carbon nanotube [31,32], graphene [33,34], InAs nanowire [35,36], InSb nanowire [37], and very recently Ge/Si core/shell nanowire [38].
Here, we report an on-demand demonstration of a Ge-hut-wire hole DQD dipole coupled to a microwave resonator. The charge stability diagrams of DQD were determined from the amplitude and phase of a microwave tone reflected from the resonator. The characteristic parameters of the hybrid device can be extracted from the interdot charge transfer line, including the hole-resonator coupling rate, charge decay rate, and interdot tunneling rate. In addition, the middle gate dependent tunneling rate was obtained, which shows the tunability of the DQD interdot coupling, and the spin-resonator coupling rate was estimated.

A. Experimental setup
The hybrid architecture consists of Ge-hut-wire DQD and a half-wavelength refection line resonator (Figure 1a), the resonator was fabricated with 200 nm thick aluminum. Compared to transmission line resonator, there is no ground plane. The two ends of our resonator are both free to couple qubits via a symmetric, differential excitation, which can potentially have a larger coupling strength and immunity to common-mode noise [39]. The source and drain contacts (Figures 1b, c) are used to drive current through the DQD, and a series of gate electrodes are tuned to create potentials confining the DQD. A dielectric layer of alumina separates the gates and contacts. In the measurement setup and equivalent circuit of the DQD (Figure 1d), a continuous microwave signal is split into two differential components with opposite phases and then applied in the strip lines to establish an electromagnetic field which oscillates with voltage res (t). Microwave reflectometry is performed by a network analyzer together with a primary cryogenic amplifier and a secondary room-temperature amplifier, which probes the variation of the DQD states. Only the DQD2 is considered in this study, all the gates of the DQD1 on the opposite strip line are grounded. The sample is anchored in the mixing chamber of a dilution refrigerator with a base temperature of 18 mK.

B. Resonator response and charge stability diagrams of the DQD
We first characterized the microwave resonator by analyzing the amplitude and phase of the reflection frequency spectrum of the bare resonator. During the measurements, all gates of the DQD1 were grounded, and the microwave power applied to the input port was below −110 dBm. The measured reflected microwave signals as a function of driving frequency was plotted in (Figures 2a,   b; red lines), from which the resonance frequency was found to be 6.038 GHz. On the basis of the λ/2 open-circuit micro-strip resonator model [40,41], we determined the internal loss rate, external loss rate, and total loss rate, (κ i , κ e , κ)~(2.6, 4.0, 6.6) MHz, with a quality factor Q of 820.
Next, we employed the microwave resonator to probe the properties of the Ge-hut-wire hole DQD.
The probe frequency was fixed at the resonance frequency of the resonator. When the charge states of the DQD changed, the reflected amplitude and phase of the microwave signal changed correspondingly on account of dispersive coupling between the DQD and resonator. When the DQD enters the Coulomb blockade (CB) regime, the reflected microwave signal behaves as a bare resonator. When the DQD is During the measurements, the DC transport signal through the source and drain contacts was recorded with a multimeter after a low-noise pre-amplifier at a source-drain bias voltage of 0.1mV.
Figures 2d and e display the charge stability diagrams measured using the amplitude and phase signals independently from the DC transport measurement (Figure 2c). The three diagrams show similar honeycomb structures, indicating that a microwave signal can be used to detect the charge states of the Ge-hut-wire DQD. However, the amplitude and phase responses were found to behave differently from the transport measurements at the same boundaries of the honeycomb structures. From DC transport measurement (Figure 2c), the resonance to the right lead is obvious while to the left lead is poor visibility. However, in the corresponding microwave measurement, the resonance to the left lead is more pronounced (Figure 2d, e).This indicates that the physical origin of the signal is different for the two measurement techniques. For the transport measurement, the tunnel rate to the left lead is smaller than that to the right lead. Therefore, the current can only be observed along two of the boundaries of the hexagon. For the microwave measurement, the opposite phenomenon is likely explained by that the DQD is coupled to the microwave resonator through left lead, which indicates that the capacitively coupling strength of the resonator to the left dot is stronger than that of the resonator to the right dot or lead [42].

C. Characteristic parameters extracted from the interdot charge transfer line
We have studied in more details about the phase of the reflected microwaves in one particular interdot charge transfer line. To account for photon exchange between the microwave field and the DQD as well as to investigate the interdot tunneling via the phase response, we employed the Jaynes-Cummings model [43] in which the resonance frequency ω 0 , probe frequency ω, internal and external resonator dissipation rates κ and κ e , DQD interdot tunneling rate 2 , the total decay rate of charge qubit γ= dot. The DQD is coupled to the resonator with coupling strength and the detuning dependent dipole moment of the DQD has an admittance that loads the cavity. In Figure 3c, the DQD forms a two-level system with an energy splitting of Ω= 2 , where is the detuning. Interdot tunnel coupling hybridizes the charge states around ~0, resulting in a tunnel splitting of 2 [35]. The coefficient of reflection is expressible as [33,34]: The phase and amplitude are extracted from ϕ =arg(S11) and A=|S11|. In Eq. (1), χ eff j ω characterizes the DQD susceptibility to the microwave field. Here, Ω= 2 , eff is the effective coupling strength between the DQD and the resonator. The measured phase shifts along the detuning line are plotted in Figure 3d, along with the best-fit line obtained using Eq. (1). Therefore, we are able to determine that the interdot tunneling rate 2 is around 6.20 GHz. The hole-resonator coupling strength is ~15 MHz and charge qubit decay rate γ ~0.28 GHz, it is an order of magnitude lower than that in Ge/Si core/shell nanowire system (γ~4-6 GHz) [38], and comparable to previous reports in GaAs [44] carbon nanotube [31] and graphene DQD systems [34].

A. Estimation of the spin-photon coupling strength
Since the electron spin states cannot be directly coupled to an electric field, and spin-orbit interaction enables electrical control by acting on the orbital component of the electron wave function.
Spin-orbit interaction mixes spin and orbital degrees of freedom, resulting in spin states that have some orbital characters [45,46]. The spin-orbit interaction would be a feasible way to achieve spinphoton coupling [15,47]. Coherent and fast electrical control of spin states in quantum dots has been demonstrated in Ge-hut-wires with strong spin-orbit coupling [27]. For a spin in a quantum dot, the spin-resonator coupling rate is expressed as [35,47]: Where is the hole-resonator coupling rate, is the Zeeman splitting of the spin states, is the orbital level spacing, is the quantum dot size and is the spin-orbit length which characterizes the strength of the spin-orbit interaction. On the basis of our results, we can estimate the effective spin-resonator coupling rate. The hole-resonator coupling rate was demonstrated as ~15 MHz. Considering the spin-orbit interaction induced spin photon coupling, the photon energy should be close to Zeeman splitting, which is assumed to be of the same order of magnitude as the energy of the lowest cavity mode [47], ω 0~2 5 GHz, corresponding with B~0.5 T and g factor~4 [24]. The orbital level spacing Δ is about 1meV extracted from previous work [24] and the size of Ge-hut-wire QD is calculated to be =ћ m * Δ ⁄~50 nm, with the heavy-hole effective mass m * =0.28m e [48]. The spin-orbit length is determined by ћ 2m * Δ ⁄ , the spin-orbit coupling strength has been demonstrated in Ge nanocrystal quantum dot as ~40 μeV [49], which has same characteristic with Ge hut wire, and the value is similar to that obtained in our unpublished work of the spin blockade in the double quantum dots, thus is calculated to be ~40 nm. Therefore, the spin-resonator coupling rate is estimated to be ~0.5 MHz, which is of the same level as the spinresonator coupling rate ~0.2 MHz obtained in InAs nanowire [15,33]. Moreover, the latest work demonstrated that the hole spin-orbit qubit in Ge hut wire with dephasing times of 130 ns [27], which is larger than that of electron or hole qubit in the InAs nanowire [50] and natural Si [28]. coupling rate is anticipated to be enhanced several times or one order of magnitude. And in combination with long dephasing time, the strong spin-resonator coupling and long-distance coupling of spin qubits via a resonator [53] could be achieved in the future work.

B. The V -dependent tunneling rate
Fine control of the DQD tunnel coupling is also critical for achieving strong spin-photon coupling.
We further studied about the phase of the reflected microwave signals near one particular interdot charge transfer line for different interdot tunnel coupling energy, which was tuned by the middle gate voltage V . The phase responses in the same interdot charge transfer line were plotted for several values of V (Figures 4a-d). From the reflection spectra, we observed clear phase shift of the interdot charge transition lines, the line width also broadened when V was increased. By fitting the phase as a function of detuning (inset of Figure 4e), we obtained the middle gate dependent tunneling rate [34,43], which ranges from 6.2 GHz to 8.5 GHz. This implies the interdot coupling strength in our device is tunable and could be tuned close to the resonance frequency for further studies of the spinphoton coupling [54]. The hole temperature of about 125 mK is estimated from linewidth of the Coulomb peak in our system. For the phenomenon of the saturation in 2 (~6GHz) at V 1.4V ( Fig. 4e), the hole temperature is not the main factor [55]. A reasonable explanation is our DQD is imperfect, there are some traps and disorder nearby the working area, which could suppress the ability to tune 2 .

IV. CONCLUSIONS
In conclusion, we have demonstrated the dipole coupling of a Ge hut wire hole DQD to a refection line microwave resonator. The charge stability diagrams of DQD were determined from the amplitude and phase of a microwave tone reflected from the resonator and we obtained a tunable interdot tunneling rate ranging from 6.2 GHz to 8.5 GHz. Moreover, the interdot transfer line was analyzed by using the Jaynes-Cummings model. A hole-resonator coupling strength up to 15 MHz is achieved in the experiment with extracted charge decoherence rate γ of approximately 0.28 GHz, and the spinresonator coupling rate is estimated to be 0.5 MHz. Looking ahead, the DQD has a spin-statedependent dipole moment arising from the spin blockade allowing read-out of spin states via a superconducting resonator [56]. When an external magnetic field is applied, the measurements of the resonator response can be implemented for spin qubit read-out with the internal strong spin-orbit interaction and long dephasing time [27]. The above results indicate that our architecture based on Ge hut wire offers a novel way to probe hole spin system in the microwave regime and potentially enable the strong hole spin-photon coupling in future work.

ACKNOWLEDGEMENTS:
This Note added in proof: During submission of our manuscript, we became aware of a related work based on Ge/Si core/shell nanowire [38]. Figure 5 (a) and (b) shows the Ge hut wire [22] we study here. A 100 nm thick Si buffer layer was firstly grown on the intrinsic Si substrate, following by 6.5 Å thick Ge at the deposition temperature of 540 °C, to form Ge hut cluster structures on the surface. After an annealing process of 8 hours at 530 °C, these clusters transform into hut wires with lengths ranging from several hundred nanometers to approximately 1 µm. In the last step of the growth process, a 3.5 nm thick Si capping layer is deposited at 330 °C to protect the Ge wire from oxidation. Figure 5 (c) shows a simplified threedimensional schematic of a Ge hut wire double quantum dot with five gates. The DC transport signal through source/drain leads is measured by a multimeter after passing through a pre-amplifier. Five top gates voltage can be changed to tune the double quantum dot.

APPENDIX B: GE DOUBLE QUANTUM DOT
We show a stability diagram of a double quantum dot coupled to the resonator by source lead in (1) which determine the amplitude and phase, A=|S11|and ϕ=arg(S11). Equipped with this model, we can obtain resonance frequency is = 6.038 GH, ω is the probe frequency, the internal loss κ i = 2.6 MHz, the external loss κ e = 4.0 MHz, and the total photon loss rate =κ i +κ e = 6.6 MHz, and the quality factor Q is 820.
Consider the DQD, our system can be seen as a quantum two-level dipole coupled to a resonator, and it can be interpreted by master equation based on the Jaynes-Cummings model [43]. First, we write down the total Hamiltonian of the system which reads: ( Here the is the resonance frequency of the resonator, and ω is the probe frequency, eff is the effective coupling strength between the DQD and the resonator. Ω= 2 , 2 is the inter-dot tunneling rate of the DQD, ∆ ω, ∆ Ω . For the whole system, the dissipation of energy mainly consists of internal and external resonator dissipation rates κ and κ e , and the charge decay rate . We use a Markovian master equation approach to describe the dynamics of the system: with which is named in the literature as the Lindblad operator, κ κ e . We obtain: in our measurement, we assume that the quantum dot stays near its lower energy state with high probability, therefore → 1. Making a rotating wave approximation for ω , we find: According to the in-put out-put theory [35] constraint condition: κ e , we obtain: Here χ j ω ，characterizes the DQD susceptibility to the microwave wave field, and eff . After obtaining the parameters κ i , and κ e , we can extract the remaining parameters , 2 , and by further fitting ∆ϕ as a function of detuning ε. This fitting method has been used in previous experimental studies of graphene systems [33,34], which shows the accuracy.