Longitudinal coupling between a Si/SiGe quantum dot and an off-chip TiN resonator

Quantum couplings between microwave resonators and quantum dot qubits are expected to form key components in scalable quantum computing architectures Majer et al. (2007); Nigg et al. (2017), with potential applications for both readout and couplings between distant qubits. Recent experiments have demonstrated spin-photon interactions Petersson et al. (2012); Mi et al. (2017); Landig et al. (2018); Samkharadze et al. (2018); Holman et al. (2021) and photon-mediated spin-spin interactions Borjans et al. (2020). Most of these have focused on “transverse” qubit-cavity couplings, which have some disadvantages, such as the required resonance between the qubit and cavity Burkard et al. (2020), which can be difficult to engineer and which causes rapid decoherence due to the Purcell effect Purcell (1946); Bienfait et al. (2016); Kjaergaard et al. (2020). Other types of coupling are therefore also of interest, including dispersive and dynamic longitudinal couplings. The latter are expected to be strong and tunable, enabling fast gates and quantum-nondemolition (QND) measurements Kerman (2013); Didier and Bourassa (2015); Beaudoin et al. (2016); Royer et al. (2017); Ruskov and Tahan (2019, 2021). Both static Eichler and Petta (2018); Dassonneville et al. (2020); Hazra et al. (2020); Toida et al. (2020) and dynamic longitudinal couplings Touzard et al. (2019); Ikonen et al. (2019) have been demonstrated experimentally in the superconducting qubit community. Longitudinal coupling schemes have also been proposed for quantum-dot qubits, and although initial experiments appear promising Bøttcher et al. (2022), their full potential is not yet known.

In this paper, we report significant enhancement of transmission through an off-chip microwave cavity, arising from the longitudinal coupling to an ac-driven double quantum dot. The device is tuned to a charge degeneracy point and driven at the cavity resonance frequency through non-standard cavity ports. The crucial signature of the longitudinal coupling mechanism is its change of sign in the resonator response, depending on which of the two dots is driven. The sign change also provides the first direct proof of longitudinal coupling. We further theoretically describe the longitudinal coupling and its effect on cavity transmission, for a range of detuning, tunnel coupling, and ac driving parameters, obtaining good agreement with the experimental results.

The sample used in this experiment is a 3D-integrated, high-impedance TiN coplanar waveguide resonator on a Si chip coupled to a Si/SiGe quantum-dot chip, as depicted schematically in Fig. 1, pictorially in Fig. 2, and explained in further detail in Ref. Holman et al. (2021). The 3D nature of the device relieves both wiring and fabrication constraints. Wiring is simplified by allowing more space on the qubit die for qubit gate lines and fanout; the low-impedance gates reduce cavity losses Holman et al. (2020). Fabrication is simplified because the qubit and cavity chips are no longer constrained to follow the same lithographic processes. Figure 2(d) shows a scanning electron micrograph of a device nominally identical to the one used in the experiment, in which the quantum dots are capacitively coupled to the resonator through a tap segment on the resonator chip, which is shunted by a large parallel plate capacitor on the quantum-dot chip. Galvanic contacts between the dies are achieved using underbump metal pads [Fig. 2(c)] formed at the voltage antinodes near the quantum dots, and at the end of the tap segment. SPICE simulations indicate that the fundamental resonant mode used in this work, with frequency $\omega_{r}/2\pi=1.304$ GHz, has weight at both these antinodes and in the tap segment.

Figures 3(a) and 3(b) show interdot charging transitions of a three-electron double dot, formed under plunger gates P2 and P3, tuned to its (2,1) or (1,2) charge occupations, as labelled. The data are obtained as a function of the dc voltages applied to gates P2 and P3, while simultaneously ac-modulating the voltages on gates S1 [Fig. 3(a)] or P3 [Fig. 3(b)] at the cavity resonant frequency. As explained in Appendix B, the recorded IQ signal is proportional to the microwave signal transmitted through the cavity, which is measured via a homodyne detection scheme. A dc offset in the homodyne IQ output, which was not calibrated away during experiments, has been subtracted from all data presented in this work.

The resonator response observed in Fig. 3(a) is primarily achieved through a longitudinal curvature coupling ($g_{\|}^{\text{dy}}>0$) induced by driving the qubit at the cavity resonant frequency. A line cut through the charging transition, shown in Fig. 3(c), provides a clearer view of the suppressed transmission amplitude as a function of the double-dot detuning $\varepsilon_{0}$, as computed using the measured lever arm $\alpha_{\mathrm{P2,\varepsilon}}=0.11$ eV/V (see Appendix D1). A dispersive shift ($\delta\omega$) of the cavity resonance frequency, arising from the energy curvature of the double-dot energy levels, further contributes to suppression of the signal. However, through the analysis below, we find its effect to be much weaker than that of the longitudinal coupling. Note here that by adiabatically sweeping the gate voltages in this stability diagram, we ensure that the qubit remains in its ground state at all times. Thus here, and throughout this work, it is the magnitude of qubit-cavity coupling, not the qubit state itself, that changes near the charging transition.

The behavior observed in Fig. 3(b), and its corresponding line cut in Fig. 3(d) is extraordinary for being opposite of the conventional behavior, and clearly inconsistent with a dispersive frequency shift. Instead, it indicates an enhancement of the stationary field in the resonator. Remarkably, the observed qubit-cavity coupling ($g_{\|}^{\text{dy}}<0$) not only has the opposite sign as the response in Fig. 3(a), it is also much larger in magnitude and provides a much cleaner signal. This is because P3’s detuning lever arm $\alpha_{\mathrm{P3,\varepsilon}}=0.09$ eV/V is more than twice as large as S1’s, $\alpha_{\mathrm{S1,\varepsilon}}=0.04$ eV/V. Therefore, modulating the voltage on P3 applies a stronger drive to the qubit, which increases the magnitude of the coupling and the measured resonator response. Aside from our choice of driving gate, there are no other electrostatic changes in the double-dot detuning or tunnel coupling between Figs. 3(a) and 3(b). Hence, the heightened response in Fig. 3(b) ¹¹1The different background noise levels observed in the experimental configurations of Figs. 3(a) and 3(b) can be partially explained by the different numbers of microwave photons in the resonator (see Appendix E2), since the noise level scales as $1/\sqrt{\langle n\rangle}$ MacDonaldBook, which we show below to be proportional to the amplitude of the qubit drive, has immediate and important consequences for qubit readout.

To understand these different effects, we construct a Hamiltonian for our qubit system. As argued in Appendix A, since we work in an operating regime very near the charging transition, $\varepsilon_{0}=0$, and since the data in Figs. 3(c) and 3(d) appear symmetric about $\varepsilon_{0}=0$, it should be a good approximation to ignore other nearby level crossings. We therefore treat the system as a simple charge qubit, in which an excess electron moves between two sides of a double dot, atop a fixed (1,1) charge configuration. The resulting qubit Hamiltonian is given by $H_{q}=(\varepsilon/2)\sigma_{z}+t_{c}\sigma_{x}$, where the double-dot detuning parameter $\varepsilon=\varepsilon_{0}+\varepsilon_{q}\text{cos}(\omega t)$ includes both static and driving terms, $t_{c}$ is the tunnel coupling between the two dots, and $\sigma_{x}$ and $\sigma_{z}$ are Pauli operators acting on the charge basis of left or right-localized states. Since $\sigma_{z}$ is proportional to the charge-dipole operator of the double dot and $(a+a^{\dagger})$ corresponds to the quantum field of the resonator ($a^{\dagger}$ and $a$ are photon creation and annihilation operators), the full Hamiltonian of the coupled qubit-cavity system is given by $H=H_{q}+\hbar\omega_{r}a^{\dagger}a+2\varepsilon_{r}\cos(\omega_{r}t)(a+a^{\dagger})+\hbar g_{0}\sigma_{z}(a+a^{\dagger})$, where $g_{0}$ is the bare coupling between the resonator and double dot, and we have included a resonant cavity-driving term, with amplitude $2\varepsilon_{r}$. In the absence of coupling ($g_{0}=0$) or qubit driving ($\varepsilon_{q}=0$), the bare qubit energy is given by $E_{q0}=\hbar\omega_{q0}=\sqrt{\varepsilon_{0}^{2}+4t_{c}^{2}}$.

In all of the experiments reported here, the device is operated in the adiabatic coupling regime $\omega_{r}\ll\omega_{q0}$, in which the resonator is unable to excite the qubit. We further consider the weak-driving, dispersive limit, $\varepsilon_{q},\hbar g_{0}\ll t_{c}$. As explained in Appendix A, an effective Hamiltonian can be derived under these assumptions. Moving to a frame rotating at the cavity frequency, performing a rotating-wave approximation, and retaining coupling terms up to order ${\cal O}[\varepsilon_{q}g_{0},g_{0}^{2}]$, the system Hamiltonian reduces to

where the tildes on the operators $\tilde{\sigma}_{z}$, $\tilde{a}$, and $\tilde{a}^{\dagger}$ indicate that they are defined in the rotating frame of the qubit energy-basis states (see Appendix A). The effective coupling terms are given by

Such couplings can be observed as effective shifts in the cavity field or cavity energy that depend on the state of the qubit. They are known as energy-curvature or quantum-capacitance couplings Ruskov and Tahan (2019, 2021), and we refer to $g_{\|}^{\text{dy}}$ and $\delta\omega$ here as the dynamic longitudinal and dispersive couplings, respectively. It can be seen that $\delta\omega$ differs from the conventional definition Blais et al. (2021) of the dispersive coupling ($\chi=g_{\perp}^{2}/\omega_{q}=\delta\omega/2$) by a factor of 2, due to our experiment being performed in the ultra-dispersive or “adiabatic” regime, $\omega_{r}\ll\omega_{q}$ (see Appendix A).

Although the microwave resonator cannot excite the qubit in the present scheme, the qubit has a measurable effect on the resonator through $\delta\omega$ and $g_{\|}^{\text{dy}}$, with the latter being proportional to the ac drive applied to the qubit detuning parameter, $\varepsilon_{q}$. It is crucial to note that an ac drive applied through gate S1 in Fig. 2(d) primarily affects the left dot, while an ac drive on gate P3 primarily affects the right dot. The two drives therefore act on $\varepsilon_{q}$ with opposite sign. On the other hand, the two driving channels act on the cavity (and therefore $\varepsilon_{r}$) with the same sign. Here, S1 drives the cavity directly, while P3 drives the cavity indirectly through crosstalk between the two gates. As explained in Appendix A, this difference in signs is crucial for observing opposing behaviors in Figs. 3(a) and 3(b). Indeed, for future qubit readout experiments, it could be preferable to drive the qubit on the same side as the resonator coupling (i.e., through P2 rather than P3), so that $\delta\omega$ and $g_{\|}^{\text{dy}}$ affect the resonator in tandem, rather than opposition, to achieve a stronger total effect.

Finally, we provide an expression for the output of our homodyne detector (see Appendix B), which converts the signal transmitted through the microwave cavity into a dc voltage:

Here, the minus sign in the subscript indicates that the qubit is always in its ground state, $c$ is a proportionality constant, relating the signal transmitted through the microwave cavity to the output voltage from the IQ mixer, and $\kappa/2\pi$ is the photon loss rate from the cavity.

Equations (1)-(3) are used to quantitatively fit the data in Figs. 3(c) and 3(d), to determine the different coupling parameters. The fitting procedure is explained in detail in Appendix C1, and the results are shown here as yellow curves. After fitting, we subtract the asymptotic background signal in IQ_-, which varies around 10% between the two drive configurations. Note that the proportionality constant $c$ in Eq. (3) is related to the measurement setup rather than the qubit-cavity coupling, and cannot be measured directly. Since $\varepsilon_{r}$ and $g_{\|}^{\text{dy}}$ are both multiplied by $c$, their values cannot be determined independently. However, the tunnel coupling $t_{c}$ can be obtained without ambiguity from the fitting procedure, and the bare qubit-cavity coupling $g_{0}/2\pi=5.5$ MHz can be determined from measured device parameters Appendix D2. Combining these results with knowledge of the applied drive power allows us to estimate $\varepsilon_{r}$ and $g_{\|}^{\text{dy}}$ in the current experiments, as described below.

We now further explore the physics of $g_{\|}^{\text{dy}}$ by varying different device parameters that affect the coupling strength. (For all remaining measurements reported here, the ac drive is applied to gate P3.) Figure 4(a) shows $\varepsilon_{0}$ line cuts labeled (i)-(iii), for three different ac driving amplitudes $\varepsilon_{q}$. The IQ units for this plot are normalized to the height of the tallest peak [(iii)], and the plots are shifted vertically for clarity. The reported $\varepsilon_{q}$ units are also relative, since we cannot determine this parameter precisely. We see that the transmission peaks increase significantly in height with $\varepsilon_{q}$. For comparison, we note that these data were acquired at a similar device tuning as Fig. 3, where the relative $\varepsilon_{q}$ value was 3.5.

We can extract the quantum couplings $\delta\omega$ and $g_{\|}^{\text{dy}}$ at different driving strengths $\varepsilon_{q}$ by fitting the transmission peaks to Eq. (3), following the procedure described in Appendix C. Three such fits are shown in yellow in Fig. 4(a). Figure 4(b) shows the full set of fitting results for $\delta\omega$, and the product parameter $cg_{\|}^{\text{dy}}$ (see Appendices B and C), evaluated at the $\varepsilon_{0}=0$ transmission peak, as a function of $\varepsilon_{q}$. The points corresponding to the three data sets in Fig. 4(a) are indicated by arrows. Because we are unable to estimate parameters $c$ and $g_{\|}^{\text{dy}}$ independently in this fitting procedure, we scale $\delta\omega$ and $cg_{\|}^{\text{dy}}$ by their values at $\varepsilon_{q}=1$ in the top panel. (In the bottom panel, $\delta\omega$ is also reported in dimensionful units.) We note that the $cg_{\|}^{\text{dy}}$ results in Fig. 4(b) follow a parabolic scaling law with respect to $\varepsilon_{q}$ (dashed gray curve). Since $c\propto\varepsilon_{q}$ for our homodyne detection scheme, as explained in Appendix B, we infer that $g_{\|}^{\text{dy}}\propto\varepsilon_{q}$, as consistent with Eq. (2). This confirms the parametric nature of $g_{\|}^{\text{dy}}$, and has important implications for qubit tunability, as we discuss below.

The dispersive-like coupling term $\delta\omega$ is not expected to exhibit $\varepsilon_{q}$ dependence. Unfortunately, the peak-fitting procedure described above does not enable a direct measurement of $\delta\omega$ because of its small size compared to $g_{\|}^{\text{dy}}$; however it does allow us to determine $t_{c}$. In Fig. 4(b), we therefore plot $\delta\omega$ computed from Eq. (2). Alternatively, we can also obtain direct measurements of $\delta\omega$ in a slightly different tuning regime, as described in Appendix G, obtaining results very similar to Fig. 4(b). These combined results indicate that $\delta\omega$ does not depend on the driving strength $\varepsilon_{q}$, as consistent with our theoretical understanding, and they confirm that the dispersive-like coupling is very small in the ultra-dispersive regime, with $\delta\omega/2\pi\approx 2.6$-6.2 kHz.

We can also explore the variation of $g_{\|}^{\text{dy}}$ with tunnel coupling. In this case, Fig. 4(c) indicates that the transmission peak amplitude decreases with $t_{c}$. (Here, we only vary $V_{\text{B3}}$, keeping all other device parameters fixed.) Note that this experiment was performed on a different date, resulting in different device tunings than Figs. 3, 4(a), and 4(b), and a different value of $g_{0}/2\pi=4.1$ MHz (see Appendix D2), due to electrostatic drift.

To analyze the data, we first extract the peak heights. This is done using a Lorentzian fitting form, to avoid assumptions about the physical model; however results are essentially identical to fits using Eq. (3). The Lorentzian peak heights are plotted in Fig. 4(c) after normalizing the data to the tallest peak, labeled (iv). We also fit the IQ peaks to Eq. (3). The detailed fitting procedure is described in Appendix C2. Three example fits are shown in Fig. 4(d), corresponding to the three labeled points (iv)-(vi) in Fig. 4(c). The peak heights are replotted in Fig. 4(c) as a function $t_{c}/h$, as determined through this procedure. The behavior is consistent with our expectations from Eqs. (2) and (3), that $\text{peak amplitude}\sim g_{\|}^{\text{dy}}\propto 1/t_{c}$.

The global parameter $c$ in Eq. (3) precludes being able to specifically determine $\varepsilon_{r}$ or $\varepsilon_{q}$ in our fitting procedure. However, by taking into account the full microwave power budget, including input power and all significant attenuation sources, we estimate P3 driving amplitudes as large as $\varepsilon_{q}/h\approx 152$ MHz in these experiments (see Appendix C). Indeed, we expect P3 to provide the largest longitudinal interaction, due to its strong capacitive coupling to the dot. Using this estimate and fitting results for $t_{c}$, we use Eq. (2) to estimate $|g_{\|}^{\text{dy}}/2\pi|\approx 12$-68 kHz in the current experiments, including Figs. 3 and 4. Since $g_{\|}^{\text{dy}}$ is proportional to $\varepsilon_{q}$, we emphasize that it should be possible to further enhance the coupling through stronger driving, while still remaining in the weak-driving regime. Although we did not explore this possibility here, it remains an important direction for future work. To estimate coupling values that could potentially be achieved in the present device, we take $\varepsilon_{q}/t_{c}=0.1$ as an upper bound on weak driving, yielding a maximum value of $|g_{\|}^{\text{dy}}/2\pi|=0.28$ MHz, which is much larger than our estimates for $\delta\omega$, described above.

It is crucial to note that the current experiments are performed in the far-detuned regime, $\omega_{q0}/\omega_{r}\gtrsim 7$, which we refer to as “adiabatic” here, because the resonator cannot excite the qubit. To our knowledge, previous longitudinal coupling experiments have never accessed this extreme-detuning regime, either in superconducting Touzard et al. (2019); Ikonen et al. (2019) or quantum dot qubit implementations Bøttcher et al. (2022). The adiabatic regime has several appealing features: (1) Purcell decoherence is strongly suppressed, since we are far from resonance; (2) there are no engineering constraints related to satisfying a resonance condition; (3) the coupling can be turned on and off via the driving term $\varepsilon_{q}$. Focusing on this final point, we note that when $\varepsilon_{q}=0$, the total coupling is reduced to the small, residual dispersive coupling $\delta\omega$. When larger couplings are desired, they can be achieved by increasing $\varepsilon_{q}$ up to the upper bound on weak driving. A useful figure of merit is therefore given by the tunability ratio, $|g_{\|}^{\text{dy}}/\delta\omega|=\varepsilon_{q}/2\hbar g_{0}$, which can be large when the driving amplitude $\varepsilon_{q}$ is much larger than the zero-point fluctuations of the cavity, as felt by the dot. Estimates for the current device indicate a tunability factor of 18.4. However, we expect that tunability factors as large as 79 could be achieved for the tunnel couplings measured here, if the drive was increased to the weak-driving limit. These estimates differ considerably from recent experiments in GaAs Bøttcher et al. (2022), where much larger values of the dispersive coupling $\delta\omega/2\pi\sim 0.1$-0.25 MHz were obtained ²²2Here, we compare to the dispersive coupling definition $\chi_{B}$ used in Bøttcher et al. (2022), which is unconventional Blais et al. (2021), and differs from our definition of $\delta\omega$ by a factor of 2., due to the large bare resonator coupling $g_{0}$. A larger longitudinal coupling $g_{\|}^{\text{dy}}/2\pi\sim 1$ MHz was also observed in Bøttcher et al. (2022), yielding figures of merit in the range of 1-5, much smaller than those reported here. These differences emphasize the inherent competition between the strength ($\propto$$g_{0}$) and the tunability ($\propto$$g_{0}^{-1}$) of the longitudinal coupling, where $g_{0}$ is an engineered variable.

In conclusion, we have explored the energy curvature couplings $\delta\omega$ and $g_{\|}^{\text{dy}}$ between a Si/SiGe double-dot charge qubit and an off-chip, high-impedance microwave resonator. By varying the tunnel coupling and applying the drive through different channels and at different strengths, we are able to prove the existence of the dynamic longitudinal coupling. We estimate its potential strength here as $|g_{\|}^{\text{dy}}/2\pi|\approx 68$ kHz, which can be increased to 0.28 MHz upon stronger driving, even in the far-detuned regime. We compare $g_{\|}^{\text{dy}}$ to the dispersive coupling $\delta\omega$, finding $|g_{\|}^{\text{dy}}/\delta\omega|>18$ here, and potentially up to 79 under optimal conditions. Moreover such strong coupling ($|g_{\|}^{\text{dy}}/\delta\omega|\gg 1$) is achieved here in the ultra-dispersive (“adiabatic”) regime, which has not been previously explored. Crucially, such strong, dispersive couplings reduce engineering constraints and suppress Purcell decoherence. We conclude that such longitudinal driving forms an important tool for future quantum-dot qubit experiments.

The authors thank HRL for support and L. F. Edge for providing the Si/SiGe heterostructure used in this work. Research was sponsored in part by the Army Research Office (ARO) under Grant Number W911NF-17-1-0274 and by the Department of Defense. JC acknowledges support from the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1747503 and the Graduate School and the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation. We acknowledge the use of facilities supported by NSF through the UW-Madison MRSEC (DMR-1720415) and the MRI program (DMR–1625348). Work done at MIT Lincoln Laboratory was funded in part by the Assistant Secretary of Defense for Research & Engineering under Air Force Contract No. FA8721-05-C-0002. The views, conclusions, and recommendations contained in this document are those of the authors and are not necessarily endorsed nor should they be interpreted as representing the official policies, either expressed or implied, of the Army Research Office (ARO) or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.