In situ Tuning of the Electric-Dipole Strength of a Double-Dot Charge Qubit: Charge-Noise Protection and Ultrastrong Coupling

Semiconductor quantum dots in which electrons or holes are isolated via electrostatic potentials generated by surface gates are promising building blocks for semiconductor-based quantum technology. Here, we investigate double-quantum-dot (DQD) charge qubits in GaAs capacitively coupled to high-impedance superconducting quantum interference device array and Josephson-junction array resonators. We tune the strength of the electric-dipole interaction between the qubit and the resonator in situ using surface gates. We characterize the qubit-resonator coupling strength, the qubit decoherence, and the detuning noise affecting the charge qubit for different electrostatic DQD configurations. We find all quantities to be systematically tunable over more than one order of magnitude, resulting in reproducible decoherence rates ${\mathrm{\Gamma }}_2/2\pi <5\mathrm{MHz}$ in the limit of high interdot capacitance. In the opposite limit, by reducing the interdot capacitance, we increase the DQD electric-dipole strength and, therefore, its coupling to the resonator. Employing a Josephson-junction array resonator with an impedance of approximately $4\mathrm{k}\mathrm{\Omega }$ and a resonance frequency of ${\omega }_r/2\pi \sim 5.6\mathrm{GHz}$, we observe a coupling strength of $g/2\pi \sim 630\mathrm{MHz}$, demonstrating the possibility to operate electrons hosted in a semiconductor DQD in the ultrastrong-coupling regime (USC). The presented results are essential for further increasing the coherence of quantum-dot-based qubits and investigating USC physics in semiconducting QDs.

Popular Summary

Electrons or holes isolated in semiconductor quantum dots by electrostatic potentials are promising building blocks for semiconductor-based quantum technology. Recently, quantum-dot-based qubits have been successfully coupled to a microwave resonator via their electric-dipolar interaction. While manipulating the qubit spin is of particular interest for quantum-information applications, charge noise in the host substrate remains a major limitation. Here, we develop a strategy to systematically modify in situ the magnitude of the electric dipole moment of a quantum-dot qubit, thereby controlling the qubit coherence and the coupling strength between the quantum dots and the superconducting resonator.

Our method relies on exploring different configurations of the double-dot confinement potential created by the metallic surface gates of the qubit. The approach is based on altering the magnitude of the interdot capacitance while maintaining the interdot tunneling rate close to the resonator frequency. Increasing the interdot capacitance lowers the electric-dipole strength of the double quantum dot and enhances its coherence. In the limit of a large electric-dipolar interaction, we move a step further by demonstrating the onset of the ultrastrong coupling regime in this hybrid platform for a double quantum dot coupled to a resonator.

These experiments break a frontier in the coupling strength of hybrid quantum-dot–superconducting-qubit devices. Also, our method for tuning the resonator-qubit coupling and qubit coherence is of particular interest for future spin-qubit applications.

Figure 1

Simplified circuit diagram and micrographs of the devices. (a) [(b)] False-colored SEM micrograph of a section of the SQUID [Josephson-junction] array resonator indicated by the light [dark] orange rectangle in (c) [(f)]. Josephson junctions in the array are highlighted in red. (c) False-colored optical micrograph of the measured device described in Sec. 2, with a SQUID array resonator (red), ground plane (light gray), fine (light gray), and coarse (gold) gates defining the DQD. (d) [(g)] Schematic of the device and control line indicating a simplified circuit diagram of the SQUID [Josephson-junction] array resonator (red), drive line (green), the DQD (cyan), and an external coil (black). $C_{\mathrm{RPG},2}$, $C_{\mathrm{RPG},1}$, $C_{\mathrm{\Sigma },2}$, $C_{\mathrm{\Sigma },1}$, and $C_m$ are the capacitance between the ${\mathrm{QD}}_2$ [${\mathrm{QD}}_1$] and the resonator, total capacitance of ${\mathrm{QD}}_2$ [${\mathrm{QD}}_1$], and interdot capacitance, respectively. (e) Scanning electron micrograph of the areas indicated by yellow rectangles in (c) and (f) showing the DQD fine gates (light gray) on the GaAs mesa (dark gray). The plunger gate galvanically connected to the resonator is highlighted in red. (f) False-colored optical micrograph of the measured device described in Sec. 3, showing the substrate (dark blue), the superconducting structures including the Al fine gate forming the DQD (light blue), the Josephson-junction array (red), and the microwave feedline (green).

Figure 2

DQD charge stability diagrams. (a) A schematic of a DQD charge stability diagram for a configuration with a large mutual capacitance $C_m$, resulting in $\eta \sim 0.10$. The black areas (lines) represent interdot (${\mathrm{QD}}_i\text{−}{\mathrm{lead}}_i$) charge degeneracy regions. The dipole strength $\eta$ is determined directly from the charge stability diagrams. $\mathrm{\Delta }{V}_m$ and $\mathrm{\Delta }{V}_g$ are the voltage distance between the two triple points and QD-lead energy degeneracies, respectively. (b) The same as (a) but for smaller $C_m$, resulting in $\eta \sim 0.70$. (c)–(f) Measured DQD charge stability diagrams obtained for four different DQD configurations in correspondence of four distinct values of $C_m$ [decreasing from (c) to (f)]. Each charge stability diagram is measured by monitoring the change in the phase $\mathrm{\Delta }\varphi$ of the resonator reflectance in response to the DQD gate voltages. The axes scales of the LSG and RSG gate voltages are kept the same in the four panels for ease of comparison.

Figure 3

The dependence of the coupling strength $g$ and DQD coherence rates ${\mathrm{\Gamma }}_R$ and ${\mathrm{\Gamma }}_2$ for DQD configurations with dipole strengths $\eta =0.42$, 0.17, 0.10. (a)–(c) Resonator reflectance amplitude $|S_{11}|$ versus DQD detuning $\epsilon$ for three representative values of the dipole strength $\eta \sim 0.42\pm 0.08$ (blue curve), $\eta \sim 0.17\pm 0.08$ (green curve), and $\eta \sim 0.10\pm 0.07$ (red curve) [corresponding to the DQD charge stability diagrams in Figs. 2, 2, and 2, respectively]. (d) Resonator amplitude response $|S_{11}|$ (dots) versus probe frequency ${\omega }_p/2\pi$ at $\epsilon =0$ [see the black arrow in (a)–(c)], displaying well-resolved vacuum Rabi mode splittings. The solid line is a fit to the sum of two Lorentzian lines. The quoted ${\mathrm{\Gamma }}_R$ is computed as an average of the two linewidths. (e) Squared qubit linewidth $\delta {\nu }_q^2$ (dots) versus spectroscopy drive power $P_s$, measured via two-tone spectroscopy [10]. The dashed lines are linear fits. The zero-power linewidths ${\mathrm{\Gamma }}_2$ are given. (f) Qubit linewidth $\delta {\nu }_q$ (dots) versus $d(\hslash \omega )/d\epsilon$ extracted from two-tone spectroscopy [10]. The dashed lines are linear fits. Their slopes define ${\sigma }_{\epsilon }$ according to Eq. (11).

Figure 4

Coupling strength and decoherence parameters extracted for 11 DQD gate bias voltage configurations. (a) Normalized coupling ${\overline{g}}_{\perp }$ [see Eq. (12)] of the DQD charge qubit to the resonator versus the dipole strength $\eta$. (b) Qubit linewidth ${\mathrm{\Gamma }}_2$ versus $\eta$. The linewidth is extracted as in Fig. 3. (c) Effective detuning noise of the DQD charge qubit ${\sigma }_{\epsilon }$ versus $\eta$, obtained as in Fig. 3. For two configurations in correspondence with $\eta =0.123$ and 0.709, we cannot extract ${\sigma }_{\epsilon }$ due to either spurious resonances or enhanced sensitivity to detuning noise, respectively. (d) DQD linewidth ${\mathrm{\Gamma }}_2$ versus the normalized coupling. The data in (a)–(d) are fitted to a linear model plotted as dashed lines, and the fit parameters are stated in the panels. The dark [light] blue area represents the one- [two-]sigma confidence interval. (e) The quality factor ${\overline{g}}_{\perp }/{\mathrm{\Gamma }}_2$ versus $\eta$. (f) Visibility of the vacuum Rabi modes (at resonance) $(1-|S_{11}|)=2{\kappa }_{\mathrm{ext}}/({\kappa }_{\mathrm{ext}}+{\kappa }_{\mathrm{int}}+2{\mathrm{\Gamma }}_2)$ versus $\eta$. The inset shows an example of a vacuum Rabi mode splitting, with the black arrow indicating the visibility of a Rabi mode at the resonance.

Figure 5

Investigation of a bias configuration approaching the ultrastrong-coupling regime for a DQD coupled to a JJ array resonator. (a) Charge stability diagram of the DQD measured by monitoring the change in resonator reflectance amplitude $|S_{11}|$ for the extracted dipole strength $\eta =0.72\pm 0.08$. (b) Resonator amplitude response $|S_{11}|$ taken by varying the DQD detuning $\epsilon$ along the gray line indicated in (a) by applying appropriately chosen voltages to the two side gates. The red (blue) line represents a fit to the Rabi (JC) model (see Appendix pp6). (c) Measured resonator reflectance $|S_{11}|$ (dots) versus probe frequency ${\omega }_p$ extracted at resonance for $\epsilon /h=0.15\mathrm{GHz}$ [black arrows in (b)], displaying a vacuum Rabi mode splitting. The orange line represents a fit to a Rabi master equation model. The JJ array resonator losses are ${\kappa }_{\mathrm{int}}/2\pi =19.5\pm 0.1\mathrm{MHz}$ and ${\kappa }_{\mathrm{ext}}/2\pi =5.7\pm 0.1\mathrm{MHz}$.

Figure 6

Parameter comparison between the 11 configurations analyzed in the main text. (a) DQD capacitances $C_{\mathrm{\Sigma },1}$, $C_{\mathrm{\Sigma },2}$, and $C_m$. (b) Dipole strength $\eta$. (c) ${\kappa }_{\mathrm{ext}}$ and ${\kappa }_{\mathrm{int}}$, extracted by fitting the reflectance of the bare SQUID array to a Lorentzian with the DQD deep in Coulomb blockade. (d) Interdot tunneling rates $\mathrm{\Delta }/h$ obtained from the JC model [see dashed lines in Figs. 3]. In (c), the data are ordered according to the resonator frequency. In the remaining panels, the $x$ axis is the configuration index.

Figure 7

An example of a DQD charge stability diagram. It shows the phase response of the resonator reflectance while changing the voltage of gates RSG and LSG [see Fig. 1]. The six voltage differences indicated allow one to extract the QDs capacitances and the dipole strength $\eta$.

Figure 8

Comparison between a SQUID and a junction array resonator. (a) False-colored SEM micrograph of a section of a SQUID array. The dashed red line encloses the unit cell of the SQUID array. (b) False-colored SEM micrograph of a section of a Josephson-junction array. The dashed green line encloses the unit cell of the array, with a single $0.5\times 0.9\mu {\mathrm{m}}^2$ Josephson junction. (c) Schematic circuit for a $\lambda /4$ SQUID array resonator. $C_D=C_c+C_{\mathrm{RPG}}+C_g$ represents the capacitive coupling between the resonator array and the microwave feedline, the DQD device, and the rest of the DQD gates. The other end of the array is grounded. (d) Circuit schematic of the unit cell of the SQUID array. $L_S$ and $L_J^{\star }$ represent the inductance of each SQUID junction (red) and of the extra Josephson junction (blue) connected in series, while $C_S$ (red) and $C_J^{\star }$ (blue) represent their junction capacitance. $C_0$ and $C_0^{\star }$ are their respective capacitances to the ground. (e) Schematic circuit for a $\lambda /4$ JJ array. (f) Circuit schematic of the JJ array’s unit cell. $L_J$ represents the Josephson inductance, while $C_J$ and $C_0$ are the junction capacitance and the stray capacitance to ground, respectively.

Figure 9

Extracted figures of merit of light-matter hybridization. (a) System cooperativity $C={\overline{g}}_{\perp }^2/[{\mathrm{\Gamma }}_2({\kappa }_{\mathrm{ext}}+{\kappa }_{\mathrm{int}})]$. (b) Visibility of the vacuum Rabi modes at resonance $(1-|S_{11}|)=2{\kappa }_{\mathrm{ext}}/({\kappa }_{\mathrm{ext}}+{\kappa }_{\mathrm{int}}+2{\mathrm{\Gamma }}_2)$ versus the DQD-SQUID array coupling strength ${\overline{g}}_{\perp }$.

Figure 10

(a) (Left axis) $g/2\pi$ extracted by measuring a Rabi mode splitting for the DQD qubit in resonance at $\epsilon =0$ with the SQUID array fundamental mode, for different resonator frequency. (Right axis) system cooperativity $g^2/(\kappa {\mathrm{\Gamma }}_2)$ at different resonator frequency. During these measurements, the DQD system is kept at the sweet spot $\epsilon =0$. (b) Comparison of the extracted coupling strengths corrected only for the mixing angle $g_0=g\mathrm{\Delta }/{\omega }_r$ with normalized ${\overline{g}}_{\perp }\propto g_0[5\mathrm{GHz}/({\omega }_r/2\pi )]$ and ${\overline{g}}_{\perp }^{\prime }\propto g_0\sqrt{[5\mathrm{GHz}/({\omega }_r/2\pi )]}$.

Figure 11

Investigation of a configuration approaching the ultrastrong-coupling regime for a DQD with $\eta \sim 0.50\pm 0.14$ coupled to a JJ array. (a) Charge stability diagram of the DQD measured by monitoring the change in resonator reflectance amplitude $|S_{11}|$ for the extracted dipole strength $\eta \sim 0.50\pm 0.14$. (b) Resonator amplitude response $|S_{11}|$ taken by varying the DQD detuning $\epsilon$ along the gray line indicated in (a) by applying properly chosen voltages to the two side gates. The red (blue) lines are independent fits to the Rabi (JC) model (see Appendix pp6). (c) Line cut representing $|S_{11}|({\omega }_p/2\pi )$ taken along the black arrows in (b). The orange line represents a fit to a JC master equation model. The resonator losses are ${\kappa }_{\mathrm{int}}/2\pi =19.5\mathrm{MHz}$ and ${\kappa }_{\mathrm{ext}}/2\pi =4.3\pm 0.1\mathrm{MHz}$.

Figure 12

Response of the SQUID array resonator reflectance amplitude $|S_{11}|$ versus DQD detuning $\epsilon$ in correspondence of three distinct dipole strengths (a) $\eta \sim 0.42\pm 0.08$ (blue), (b) $\eta \sim 0.17\pm 0.08$ (green), and (c) $\eta \sim 0.10\pm 0.07$ (red) [the corresponding DQD charge stability diagrams are reported in Figs. 2, 2, and 2(c), respectively]. The three resonant spectrums are obtained by tuning the SQUID array in resonance with the DQD charge excitation frequency for $\epsilon =0$. Data are already reported in Figs. 3.

Figure 13

Response of the Josephson-junction array resonator reflectance amplitude $|S_{11}|$ versus DQD detuning $\epsilon$ in correspondence of a DQD configuration characterized by parameter (a) $\eta \sim 0.50\pm 0.14$ and (b) $\eta \sim 0.72\pm 0.08$. Data are already reported in Figs. 5 and 11.