Abstract
Engineered spin-electric coupling enables spin qubits in semiconductor nanostructures to be manipulated efficiently and addressed individually. While synthetic spin-orbit coupling using a micromagnet is widely investigated for driving and entangling qubits based on single spins in silicon, the baseband control of encoded spin qubits with a micromagnet in isotopically purified silicon has been less well investigated. Here, we demonstrate fast singlet-triplet qubit oscillation (~100 MHz) in a gate-defined double quantum dot in 28Si/SiGe with an on-chip micromagnet with which we show the oscillation quality factor of an encoded spin qubit exceeding 580. The coherence time T2* is analyzed as a function of potential detuning and an external magnetic field. In weak magnetic fields, the coherence is limited by frequency-independent noise whose time scale is faster than the typical data acquisition time of ~100 ms, which limits the T2* below 1 μs in the ergodic limit. We present evidence of sizable and coherent coupling of the qubit with the spin states of a nearby quantum dot, demonstrating that appropriate spin-electric coupling may enable a charge-based two-qubit gate in a (1,1) charge configuration.
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Discover the latest articles, news and stories from top researchers in related subjects.Introduction
Balancing the manipulation speed and coherence time, which often play opposing roles, has been a major goal of semiconductor quantum dot-based quantum information processing platforms1,2,3 to maximize the qubit control fidelity. The electrical control of spin states is a representative example where, depending on the properties of the host material, either intrinsic4,5 or extrinsic6,7 spin-electric coupling methods have been explored. While strong spin-orbit coupling in compound semiconductors such as InAs and InSb enables fast Rabi oscillations4,5, the fluctuations of the nuclear bath or susceptibility to charge noise due to strong spin-orbit coupling limit the inhomogeneous coherence time T2* to the order of tens of nanoseconds. More recently, hole spins in group IV materials such as Ge (ref. 8) and Si (ref. 9) or electron spins in the Si-MOS structure10 have been attracting much attention due to a more favorable ratio between the spin-orbit-based control speed and coherence time.
The electrons in silicon, in particular in the Si/SiGe heterostructure, have small intrinsic spin-orbit coupling11; therefore, an extrinsic method such as a micromagnet is necessary to rapidly manipulate its spin states. For single-spin qubits, the placement of an on-chip micromagnet has proven effective for both natural7,12 and isotopically enriched silicon13 in Si/SiGe and Si-MOS structures, where the field gradient provides fast control while not severely compromising the spin coherence. In the case of the silicon-based two-electron singlet-triplet qubit, however, the efficiency of the technique involving a micromagnet has not been fully examined. Previous studies of singlet-triplet qubit operation either used a small field gradient14 or relied on the modulation of the exchange energy15 in natural silicon. Exploration of the micromagnet technique with a field gradient in the intermediate range in isotopically purified silicon would thus be important for optimizing spin-electric coupling. In addition, this approach would enable this route to be compared with other methods such as the recently demonstrated spin-valley-driven coherent singlet-triplet oscillation in silicon16,17.
Here, we demonstrate singlet-triplet qubit oscillation in a gate-defined double quantum dot in 28Si/SiGe. An on-chip micromagnet is used to generate a magnetic field gradient that is sufficient to allow fast manipulation (oscillation frequency fQ ~ 100 MHz), while benefiting from high spin coherence by isotopic enrichment. We measure the variation in the spin-electric coupling strength in the large valley-splitting regime (>175 μeV) in which an appropriate field gradient enables an encoded spin qubit to attain an oscillation quality factor over 580. We also present the analysis of the variation in T2* as a function of experimental parameters such as detuning ε, magnetic field Bz,ext, and gate tuning conditions, exploring the origin of the dominant noise source in the system. Moreover, we present evidence that the qubit engages in sizable and coherent coupling with the spin states of a nearby quantum dot, thereby demonstrating that the appropriate amount of spin-electric coupling may enable a different type of two-qubit gates of encoded spin qubits.
Results
The triple quantum dot system
Figure 1a shows a multiple quantum dot device fabricated on top of a 28Si/SiGe heterostructure (see Methods for details of the material structure and device fabrication). We focus on a two-electron singlet-triplet (ST0) qubit formed by the gate electrodes near the left Ohmic contact and a global top gate (not shown) while the regions beneath the other electrodes are fully accumulated. The general Hamiltonian H of the ST0 qubit can be expressed as H = J(ε)σz + ΔBzσx, where J(ε) is a ε-dependent exchange interaction with the Pauli matrix σi=x,y,z. ΔBz is the magnetic field difference between the quantum dots constituting the qubit and is denoted in the frequency unit Hz using TgμΒ/h, where g, μB, and h are the Lande g-factor of the electrons in silicon, the Bohr magneton, and Planck’s constant, respectively.
Additionally, we formed a third, many-electron quantum dot next to the ST0 qubit to study the capacitive interaction between them. The design of the micromagnet on top of the device is similar to the ones used previously18. High frequency and synchronous voltage pulses, combined with the DC voltage through bias tees, were input to gates V1, V2, and VT. Fast RF reflectometry19,20 was performed by injecting a carrier signal with a frequency of approximately 125 MHz and power of –100 dBm at the Ohmic contact of the RF single-electron transistors on the left (see Fig. 1a). The reflected power was monitored through a chain of cryogenic and room temperature amplification and subsequent homodyne detection. The device was operated in a dilution refrigerator with a base temperature of approximately\(\approx \,7mK\), with Bz,ext ranging from –400 mT to 400 mT applied in the direction shown in Fig. 1a.
Figure 1b shows the charge stability diagram of the ST0 qubit coupled with a many-electron quantum dot. The full specification of the number of electrons in the left, middle, and right quantum dots (QDL, QDM, QDR, see green circles in Fig. 1a. For the position estimation sequence, see Supplementary Note 1) are denoted as (n,m,l), whereas the (n,m) notation is used whenever we focus on the ST0 qubit only. A voltage pulse with a width of approximately 10 ns and rise time of 0.5 ns is input to V1 and V2 in the directions indicated by the red arrows in Fig. 1b. Near the charge transition from (0,2,N) to (1,1,N), the pulse abruptly changes the Hamiltonian to the form H = ΔBzσx, where the spin state initialized to the singlet rotates around the x-axis on the Bloch sphere (Larmor oscillations11), thereby resulting in a non-zero triplet state probability PT. The discrimination of the resultant excited state population is conventionally performed by Pauli spin-blockade (PSB)-based spin-to-charge conversion2,11,21 where the singlet and triplet spin states are mapped to the (0,2) and (1,1) charge configurations, respectively. However, ΔBz produced by the micromagnet facilitates relaxation of the transient triplet (1,1) to the singlet (0,2) by mixing with the singlet (1,1) state, which makes a high-fidelity single-shot readout problematic22.
To circumvent the problem, we adopted one of the latched-PSB techniques that maps the triplet state to a long-lived metastable charge configuration23,24,25. Pioneered in a similar experiment performed in a GaAs triplet quantum dot system24, the version we used converts the triplet state (1,1) to the (1,2) state by rapidly loading an electron from the reservoir, which is expected to be located between QDM, QDR and the gate displayed as the horizontal gray line in Fig. 1a, at a tunneling rate greater than the sensor bandwidth of 10 MHz (Fig. 1c, middle panel). On the other hand, tunneling to the reservoir on the left is tuned to be of the order of 10 Hz. At this rate, the metastable (1,2) state can relax to a singlet (0,2) state only by indirect and slow tunneling of an electron from QDL to the reservoir on the right (Fig. 1c, rightmost panel). Along with the higher signal contrast of a charge of one electron compared with conventional PSB, the prolonged relaxation time of the triplet states enables fast and high-fidelity single-shot measurement. The fast measurement capability is also important to examine the extent to which the variation in the qubit coherence time depends on the total data acquisition time to determine the effect of slow charge noise26,27,28, as discussed in detail below.
Figure 1d shows the magnetospectroscopy measurements of the valley splitting29 for QDL and QDM. By observing the crossover of the ground state from the singlet to the triplet by measuring the dependence of the energy required to add the second electron to each dot on Bz,ext, we obtain the valley splitting ~175 μeV (257 μeV) in QDL (QDM). The result confirms that the valley splitting in our device is at the largest energy scale of at least twice that of the Zeeman splitting at the maximum Bz,ext applied in this study. Thus, we ignore the valley degree of freedom in this work and focus only on the ΔBz-driven ST0 qubit dynamics.
Qubit dynamics driven by the field gradient
With the calibrated π/2 pulse obtained from the Larmor oscillation measurement at the pulse amplitude |ΔV1 | = |ΔV2 | = 270 mV, we construct a three-step pulse sequence for Ramsey interferometry (Fig. 2a). During the second step, at the pulse amplitude of free evolution Vevol = |ΔV1 | = |ΔV2 | , the qubit evolves around the axis of the Bloch sphere determined by the ratio of J(Vevol) and ΔBz. Figure 2b shows a representative quantum oscillation at Vevol = 770 mV under the representative tuning conditions. This demonstrates a record-high11,14,15 oscillation quality factor Q* = fQ × T2* = 116.25 MHz × 4.8 μs = 558 of a ΔBz-driven ST0 qubit rotation in the deep (1,1) charge configuration. Although the Q* tends to decrease as J(Vevol) increases, the Q* remains above 100 for J(Vevol) < 20 MHz, the regime where we expect that the high-fidelity two-axis control of ST0 qubit can be implemented using AC driving15. In addition, high-resolution measurement (10,000 shots per data point with a single-shot readout time of 20 μs) of the first few oscillations (Fig. 2b, left panel) shows a readout visibility of ~85% (see Supplementary Note 2 for details on the signal-to-noise ratio).
To more fully understand spin-electric coupling and its effect on the coherence time, we mapped the dependence on the free evolution time tevol and Vevol of the Ramsey interference at Bz,ext = 300 mT, as shown in Fig. 2c. The oscillation observed for Vevol < 0.1 V shows the fast but short-lived oscillation driven by J (marked as ● in Fig. 2d and the corresponding schematic diagram in Fig. 2e), whereas the oscillations driven by ΔBz exhibit prolonged T2* (white dashed contour in Fig. 2c) for Vevol > 0.1 V due to the lower charge noise susceptibility dfQ/dVevol. In this regime, the data in the time and frequency domains exhibit the following main features. First, fQ is generally linearly dependent on Vevol, which arises from the presence of the micromagnet (see ■ and ★ in Fig. 2d), and this is consistent with the previously observed linear shift of the single-spin resonance frequency in silicon in the presence of the synthetic field gradient13. Second, T2* depends non-monotonically on Vevol. In particular, a significant decrease in T2* is observed in the vicinity of Vevol = 0.35 V (near ⬢ in Fig. 2d). Third, fQ undergoes an abrupt frequency shift of about ΔfQ ~ 1.7 MHz at approximately Vevol = 0.45 V (▲ in Fig. 2d). Estimated from the calibrated lever arm of 0.023, the cross-talk effect of Vevol = 0.45 V on QDR shifts the chemical potential of QDR to the Fermi-level of the right contact EF where the ground state charge transition occurs. Therefore, the observed ΔfQ ~ 1.7 MHz per one electron change is the measurement of the capacitive coupling between the ST0 qubit and QDR. We additionally verified this interpretation by adjusting the DC tuning of the plunger gate of QDR and by observing the systematic shifts of the point ▲ (see Supplementary Note 3).
In general, the charge fluctuation in QDR adversely affects the coherence of the capacitively coupled qubit. However, we note that Vevol = 0.35 V (near ⬢ in Fig. 2d) showing the lowest T2* occurs below Vevol = 0.45 V (▲ in Fig. 2d) where QDR experiences the maximum charge fluctuation. Assuming that non-negligible spin-dependent coupling occurs between the ST0 qubit and the Zeeman-split ground and excited spin states in QDR occupied by N electrons (Eg and Ee, respectively), the qualitative interpretation of this phenomenon is as follows (See Supplementary Note 4 for data supporting spin-dependent coupling). For detuning near ◼, the spin state in QDR remains in the ground state with high fidelity as both Eg and Ee are well below EF, and the spin-dependent charge fluctuation is low. From point ◼ to ▲, as Ee approaches and passes EF, fast tunneling between QDR and the reservoir through Ee leads to enhanced spin-dependent charge number fluctuation. This fluctuation reduces T2* and produces a small frequency shift of ΔfQ ~ 0.4 MHz (red arrows in Fig. 2d). In this regime, the spin state in QDR is expected to be a mixed state because of the non-negligible average occupation in Ee. The maximum charge fluctuation, hence the minimum T2*, is expected to occur when Ee approximates EF, which corresponds to the point ⬢. Finally, at point ▲, Eg aligns with EF, resulting in the change in the full electron number in QDR and the appearance of the kink in time-averaged fQ measurement.
More quantitatively, we compared the experimental results with those of the numerical simulation30 using the following phenomenological Hamiltonian and Lindblad operators, which were built based on the systematic analysis of Ramsey interferometry of the ST0 qubit discussed in the next section.
Here, the Hamiltonian describes the two interacting qubits, ST0 qubit, and the two spin states of the nearby N-electron quantum dot, QDR. More specifically, the Hamiltonian of the ST0 qubit is constructed as J(Vevol)σz + ΔBz(Vevol)σx, where the background ΔBz(Vevol) = 47.7 Vevol + 61.3 (0.25 V < Vevol < 0.7 V) is estimated from Fig. 2d. We assumed that the Hamiltonian of QDR is analogous to that of the ST0 qubit, such that the diagonal term β(Vevol) is an exponential function of Vevol, and the off-diagonal term γ(Bz,ext) is a linear function of Bz,ext. Furthermore, the coupling we consider is the spin-electric coupling induced from the spatial distribution of the orbital wavefunction of QDR depending on its spin states. In view thereof, we chose the spin-electric coupled eigenstate of QDR as the σz basis and introduced the interaction between the ST0 qubit and QDR, which was assumed to be in the form of \({\sigma }_{x}\otimes {{\sigma }}_{z}\). This coupling term is multiplied by a phenomenological Fermi−Dirac distribution with proper constant c, η and coupling strength Jint(Vevol) and to incorporate the change in the charge state of the nearby N-electron quantum dot (see Supplementary Note 5 for details of the simulation). Additionally, the phenomenological Lindblad operators for the ST0 qubit (L1) and two-level system in QDR (L2) are introduced with the proportionality constant τ1 (τ2) to reflect the experimentally observed decoherence.
The inset in Fig. 2c shows the simulation result which consistently reproduces the sudden kink in the frequency near Vevol = 0.45 V, the significant decrease in T2* near Vevol = 0.35 V, and the subsequent recovery of T2* near the kink. Overall, by comparing the result of the simulation with that of the experiment, we concluded that the kink in the frequency and the drop in T2* indicate capacitive coupling between the ST0 qubit and QDR during the charge transition of QDR.
Coherent coupling between ST0 qubit and many-electron spin states
We further substantiated the validity of the above analysis by showing that the experimental and simulation results were consistently comparable under different quantum dot tuning conditions. Specifically, we fine-tuned the gate voltage levels to induce significant deviations in the overall coupling strength and the decoherence rates compared with the previous tuning condition. To establish the desired tuning condition, we primarily adjusted the gate VR of Fig. 1a. In detail, while maintaining the optimal experimental parameters for qubit readout, we applied more negative voltage to the gate VR, which is expected to increase Jint by decreasing the distance between QDM and QDR. In this new tuning, we observed the characteristic beating of the quantum oscillation below Vevol < 0.42 V, as shown in Fig. 3a. Notably, the coherence of the oscillation markedly diminished when the coupling between the two qubits became appreciable. Figure 3b enables a more detailed examination of these results and provides the line cuts that offer a clearer comparison between the oscillation traces in the uncoupled (Vevol = 0.6 V, top trace in Fig. 3b) and coupled (Vevol = 0.2 V, bottom trace in Fig. 3b) regimes.
The Ramsey interferometry of the ST0 qubit reveals the structure of the multielectron state in QDR. The observed beating oscillation of the ST0 qubit and the split of the corresponding FFT peak into two suggest that the multielectron state of QDR is either a superposition or a mixed state of two different eigenstates. The interferometry of the ST0 qubit under different Bz,ext (see Supplementary Note 4 for the low Bz,ext result) shows that the degree of beating decreases and eventually disappears as Bz,ext decreases, revealing that the degree of mixing of the two QDR states is inversely proportional to Bz,ext. Furthermore, the dependence of the beating of the oscillation on Vevol suggests that the degree of mixing of the eigenstates of QDR, whose orbital wavefunction directly couples with fQ through spin-electric coupling, is a function of Vevol. The significant drop in T2* in the regime of sizeable coupling is again likely a consequence of the interplay between the interqubit coupling and the dephasing effect discussed in the previous section. Nonetheless, the interqubit coupling rate is faster than the decoherence rate, suggesting the possibility of entangling an ST0 qubit with the multielectron level in QDR. Additionally, the coupling strength with the maximum value of approximately 10 MHz also exhibits a dependence on Vevol, highlighting the electrical tunability of the interqubit coupling strength. The phenomenological Hamiltonian we introduced in the previous section summarizes these findings.
This experimental result was compared with the numerical simulation, which employed the identical Hamiltonian and Lindblad operators introduced in the previous section, whose parameters were appropriately adjusted to reflect the different tuning conditions. One of the key adjustments involves the parameters of β(Vevol), which effectively transform the eigenstate of QDR into the superposition of the σz-eigenstates for Vevol < 0.4 V, which gives rise to the observed beating of the oscillation through the spin-electric coupling \({\sigma }_{x}\otimes {\sigma }_{z}\) term. The numerical calculation consistently reproduces the experimental results, including the characteristic kink near Vevol = 0.48 V, the emergence of the beating, and the significant reduction in T2* throughout the coupling regime. These observations can be attributed to the mixed eigenstate of QDR and the Lindblad operators, which effectively mimic the aforementioned dephasing effect. Overall, our spin-electric coupling scenario convincingly reproduces the experimental results for various coupling parameters.
External field dependence of ST0 qubit coherence
We turn to discuss the dominant noise source limiting the coherence of ST0 qubit by investigating the variation in fQ and dfQ/dVevol as a function of Bz,ext. The magnitude of the field gradient |ΔBz| is determined by measuring fQ of the ΔBz-dominated Ramsey oscillations at Vevol = 800 mV for various values of Bz,ext from 400 mT to –400 mT (Fig. 4a). Generally, |ΔBz| was positively correlated with Bz,ext, which likely originated from the formation of multiple domains due to the demagnetization of the Co micromagnet at low Bz,ext. The calculated value expected for |ΔBz| by simulation of the magnetic field using the Object Oriented Micromagnetic Framework (OOMMF)31,32 was in qualitative agreement with the experimental observation. (see Supplementary Note 6 for details of the micromagnetic simulation).
The controllability of |ΔBz| via Bz,ext paved the way to test whether a decrease in |ΔBz| could lead to a smaller dfQ/dVevol and, consequently, an improved T2* at low Bz,ext. Figure 4b shows the dependence of dfQ/dVevol on fQ, extracted at various levels of Bz,ext and Vevol. Unexpectedly, a strong correlation did not exist between fQ and dfQ/dVevol. Depending on the experimental iteration, dfQ/dVevol was widely dispersed even at similar fQ controlled by Vevol. We again attribute this to the nanoscale formation of multiple domains in the micromagnet, which generates a locally inhomogeneous field distribution.
Figure 4c shows T2* and Q* as functions of Bz,ext at several Vevol. Generally, the decreasing Q* is predominantly the result of the rapid decrease in fQ as the applied magnetic field Bz,ext weakens, whereas T2* varies at most by a factor of two as a function of Bz,ext. The latter finding is also consistent with the observation that fQ and dfQ/dVevol are not strongly correlated. Although T2* tends to increase in the presence of strong Bz,ext, we argue that this is because of the interplay between the experimental data acquisition time and dominant noise band, which shifts to the low frequency at stronger Bz,ext. We confirm that extending the total data acquisition time significantly affects T2* at Bz,ext = 400 mT, indicating that slow charge noise compared to a given measurement time plays an important role (see Supplementary Note 7). Moreover, near Vevol = 350 mV, where T2* is limited by strong coupling with the spin states in QDR and fast charge noise is therefore presumed to dominate the noise spectrum, T2* is nearly constant as a function of Bz,ext. In this respect, we assume that T2* at low Bz,ext, which approximates 1 μs regardless of the value of Vevol, is entirely dominated by the noise spectrum, which is faster than the measurement time. This indicates that the coherences of the ST0 qubit are closer to the ergodic limit.
The dominance of high-frequency noise in our system is also supported by measurement of the spin-echo time Techo and echoed quality factor Qecho at various Bz,ext and Vevol (Fig. 4d). Notably, the spin-echo enables only a minor improvement in the coherence time by a factor of at most two compared with T2* at low Bz,ext of ~100 mT, thereby indicating that the major source of noise in this regime is in the high-frequency band. Similar ineffectiveness of the spin-echo was observed in a NatSi/SiGe-based singlet-triplet qubit for non-negligible J (ref. 33). Although a previous study34 pointed out that the increased flip-flop motion of residual 29Si nuclear spins at low Bz,ext leads to the reduction of T2*, we rule out this possibility since we expect that the nuclear spin flip-flop rate is suppressed below 10 Hz by the presence of on-chip micromagnet35,36. Moreover, this type of noise is more likely to occur under RF excitations needed for single-spin qubit manipulation. The absence of such control in this experiment also indicates that the mechanism of the dominant noise source at low Bz,ext in our experiment differs from that in the previous study. The spin-echo more effectively enhances the coherence time at high Bz,ext > 300 mT, which is consistent with our scenario that, in this regime, the dominant noise primarily stems from the low-frequency band. Similar to the behavior of T2*, the spin-echo is not effective when the ST0 qubit is strongly coupled with QDR (see Fig. 4d, third panel).
We additionally compared the power spectral density (PSD) of noise for strong and weak magnetic fields, Bz,ext (400 and 50 mT, respectively) obtained by the single-shot measurement-based rapid Bayesian estimation method26,37,38 (see also Supplementary Note 8) as shown in Fig. 5a. Although both of these spectra exhibit a larger white noise component compared to previous studies28,33, the PSD of the spectrum at Bz,ext = 50 mT is about two orders of magnitude larger across the entire range of frequencies with different exponent α of the 1/fα-like power spectrum compared to the PSD at Bz,ext = 400 mT. Assuming that the frequency-independent noise extends to frequencies beyond the experimentally measured limit of ~40 Hz, the result explains the overall ineffective noise refocusing via the spin-echo technique in our system, in particular at low Bz,ext. In addition, the tendency of the white noise floor to increase with decreasing Bz,ext along with the change in α (Fig. 5b), which is generally indicative of a relative increase in the portion of fast charge noise, is consistent with the variation in T2* and Techo, as presented in Fig. 4.
Discussion
The origin of the rather high white noise floor in our system, which further increases at low Bz,ext, remains an open question. Although a more comprehensive understanding of the dominant noise source would require further experiments, the ineffective coherence recovery using the spin-echo technique due to relatively fast noise indicates that the noise does not predominantly originate from the increased flip-flop rate of the residual 29Si nuclear spins. Based on our investigation of the signature of the nanoscale multi-domain structure as the micromagnet demagnetizes at Bz,ext < 200 mT, we speculate that the fast noise could have stemmed from the interplay between the field inhomogeneity induced by the magnetic domain structure and charge noise. This could be clarified by studying multiple devices containing micromagnets with various magnetic properties. A potential approach could involve the use of different techniques for micromagnet fabrication; for example, the deposition of magnetic material in the presence of an applied magnetic field, which is known to induce a preferential magnetization axis and hence a significantly modified hysteresis loop39. This technique may enable the magnetic structure to be more stably controlled in a weak magnetic field, which would allow an investigation of the transduced noise with varying magnetic properties. Moreover, the technique could also be useful for other applications such as semiconductor-superconductor hybrid circuits40,41 for long-range coupling where operation in a weak magnetic field is beneficial.
Nevertheless, we successfully demonstrated coherent ST0 oscillations with outstanding Q*. This was enabled by using an on-chip micromagnet technique in an isotopically purified 28Si/SiGe heterostructure where fQ is tunable in the (1,1) charge configuration due to the dependence of the magnetization on Bz,ext. Our findings reveal that capacitive coupling can facilitate coherent interactions between two quantum systems: the two-electron ST0 qubit and the many-electron quantum dot. Moreover, by formulating Hamiltonians for these quantum systems and their interactions, we effectively reproduced the coherent ST0 qubit oscillation observed in our experiments through numerical simulation. Our work also suggests areas for improvement. Even though our device was designed to allow us to coarsely tune the chemical potential of QDR, independent control of the quantum states of QDR was challenging because of the limited number of control lines in the current single-gate layer structure. Enhanced control over individual quantum dots and precise coupling strength modulation could be attained by adopting an overlapped gate structure42, which may enable different two-qubit gate schemes for encoded spin qubits in silicon.
Methods
Material structure and device fabrication
The 28Si/SiGe heterostructure wafer was grown by a molecular beam epitaxy growth method. An isotopically purified silicon source (with a residual 29Si concentration of approximately 800 ppm) was used for the strained quantum well with a thickness of 12 nm. The design of the surface gate electrode resembles that of GaAs spin qubit devices where both quantum dot confinement and barrier gates reside in the same layer and a global accumulation gate is used for electrostatic doping. The dimensions of the accumulation gate were maintained below 2 × 2 μm2 to minimize the parasitic capacitance20, enabling proper impedance matching conditions for radio frequency (RF) reflectometry. A Co micromagnet was deposited above the accumulation gate using an e-beam evaporator, with a Au cap for the antioxidation layer.
Measurement setup
The sample was cooled to the base temperature, ~7 mK, with a cryogen-free dilution refrigerator (Oxford Instruments Triton-500). A sensing dot based on an RF single-electron transistor was used to detect the change in the charge state of QDL, QDM, and QDR in our system. An onboard inductor of 1500 nH and a parasitic capacitance on the order of 1 pF formed an LC-tank circuit with a resonance frequency at ~125 MHz, which was used for RF reflectometry. Two arbitrary waveform generators (HDAWG and Operator-X+ by Zurich Instruments and Quantum Machines, respectively) were used to synchronize the multi-channel voltage pulses and timing marker generation. A high-frequency lock-in amplifier (Zurich Instruments, UHFLI) was used as a carrier generator and demodulator for homodyne detection. At room temperature, a carrier power of –40 dBm was generated which was further attenuated by –50 dB by the cryogenic attenuators and the directional coupler. The reflected signal is initially amplified by 50 dB with the cryogenic amplifier (Caltech Microwave Research Group, CITLF2 x2 in series), and then additionally amplified by 20 dB at room temperature using a custom-built RF amplifier. We used the QUA (Quantum Machines) language framework for scripting experimental sequences, performing single-shot readouts, and signal conditioning.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIT) (No. 2019M3E4A1080144, No. 2019M3E4A1080145, No. 2019R1A5A1027055, RS-2023-00283291, SRC Center for Quantum Coherence in Condensed Matter RS-2023-00207732, No. 2023R1A2C2005809, and RS-2024-00413957) and the core center program grant funded by the Ministry of Education (No. 2021R1A6C101B418). The work on the 28Si/SiGe growth was supported by JST Moonshot R&D grant No. JPMJMS226B and JSPS Grant-in-Aid for Scientific Research (KAKENHI) grant No. JP21H01808. The authors thank Susan Coppersmith for fruitful discussions. Correspondence and requests for materials should be addressed to D.K. (dohunkim@snu.ac.kr).
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D.K. conceived and supervised the project. Y.S. and J.K. fabricated the device. Y.S., J.Y., and H.J. performed the measurements and analyzed the data with W.J. J.P., M.C., and H.S. built the experimental setup and configured the measurement software. S.M., N.U., and K.M.I. synthesized and provided the 28Si/SiGe heterostructure. All the authors contributed to the preparation of the manuscript.
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Supplementary information for Coherence of a field-gradient-driven singlet-triplet qubit coupled to many-electron spin states in 28Si/SiGe
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Song, Y., Yun, J., Kim, J. et al. Coherence of a field gradient driven singlet-triplet qubit coupled to multielectron spin states in 28Si/SiGe. npj Quantum Inf 10, 77 (2024). https://doi.org/10.1038/s41534-024-00869-y
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DOI: https://doi.org/10.1038/s41534-024-00869-y
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