Coherent Spin-Spin Coupling Mediated by Virtual Microwave Photons

We report the coherent coupling of two electron spins at a distance via virtual microwave photons. Each spin is trapped in a silicon double quantum dot at either end of a superconducting resonator, achieving spin-photon couplings up to around $g_s/2\pi =40\mathrm{MHz}$. As the two spins are brought into resonance with each other, but detuned from the photons, an avoided crossing larger than the spin linewidths is observed with an exchange splitting around $2J/2\pi =20\mathrm{MHz}$. In addition, photon-number states are resolved from the shift $2{\chi }_s/2\pi =-13\mathrm{MHz}$ that they induce on the spin frequency. These observations demonstrate that we reach the strong dispersive regime of circuit quantum electrodynamics with spins. Achieving spin-spin coupling without real photons is essential to long-range two-qubit gates between spin qubits and scalable networks of spin qubits on a chip.

Popular Summary

Semiconductor quantum-dot-based spin qubits are rapidly developing as a promising technology for quantum computing. Their interaction range is limited to nearest neighbors roughly a few hundreds of nanometers apart. Therefore, researchers are investigating ways to increase the qubit connectivity or to bridge modules of finite size together through circuit quantum electrodynamics (QED). Here, we overcome previous challenges to this goal using a high-impedance resonator to increase the spin-photon interaction strength.

Previous experiments have demonstrated the resonant interaction of two separated electron spins connected through a microwave photon in a superconducting resonator. This is analogous to the way superconducting qubits can be coupled through a resonator “quantum bus.” However, the interaction-to-decoherence ratio was not strong enough to observe spin-spin interaction using only detuned virtual photons, an essential feature of modern circuit QED. Indeed, the resonant regime does not allow for a straightforward implementation of a two-qubit entangling gate. In this new demonstration, we couple the spins coherently without populating the resonator with real photons, a key step toward two-qubit gates. Furthermore, we observe photon number splitting, a feature of the regime of strong dispersive coupling of circuit QED.

These experiments break a new frontier in the coherence of hybrid spin-superconducting qubit devices. The new coupling regime achieved can enable long-range two-qubit gates between distant spin qubits or improve qubit readout.

Figure 1

Spin-spin coupling device. (a) Optical image of the resonator and QD areas with colored overlays. The ground plane, resonator, microwave (MW) in and out ports, and dc tap are thin superconducting NbTiN. The substantial kinetic inductance and resonator impedance result in a 6.916-GHz resonator that is only $250\mu \mathrm{m}$ long. (b) Colorized angled scanning electron microscope image of a (nominally identical) DQD gate structure and micromagnets. The NbTiN resonator end is contacted with an Al plunger electrode of the DQD, which is biased through the dc tap. (Image rotated 90°.) (c) Schematic of the DQD heterostructure and gate stack using the same color scheme as in panel (b). A photon couples to an electron spin through spin-orbit (micromagnet) and orbital-photon (voltage zero-point fluctuations $V_{\mathrm{zpf}}$) interactions. (d) Micromagnets are tilted $\pm 15°$, allowing one to fine-tune each spin’s Zeeman energy through the external magnetic field angle $\varphi$. Here, ${\mathit{B}}_{\mathrm{ext}}=(B_r,\theta =-90°,\varphi )$.

Figure 2

Resonant spin-photon-spin coupling. (a,b) Spin-photon vacuum Rabi splitting of DQD1 (a) and DQD2 (b) for tunnel splittings $(2t_c^1,2t_c^2)/h=(13.2,13.7)\mathrm{GHz}$ and a field angle of $\varphi =10.5°$. When the gate voltage is set on the zero-charge-detuning point, which corresponds to the middle of the (0,1)-(1,0) interdot charge transition, the charge-photon interaction $g_c/2\pi =192\mathrm{MHz}$ between the DQD charge and the resonator dispersively shifts the resonator frequency (insets) and enables the spin-photon interaction through the artificial spin-orbit interaction. The dashed lines are the Hamiltonian model transitions. The probe power is $-117.5\mathrm{dBm}$. (c) When both dots interact simultaneously with the resonator, the dispersive shifts are additive, and the vacuum Rabi splitting is enhanced by a factor of approximately $\sqrt{2}$, an effect of the coherent collective spin-photon interaction. Model transitions are adjusted to the individual interaction data and used to predict the simultaneous interaction data.

Figure 3

Spin-spin coupling via virtual photons. (a,b) Dependence of the independently interacting spin transition frequencies as a function of the external magnetic field angle $\varphi$ at $B_r=52\mathrm{mT}$. See Appendix pp3 for the reconstruction procedure responsible for the vertical stripes. (c) When both spins interact simultaneously with the resonator, an avoided crossing is observed. The spin states hybridize with a splitting $2J/2\pi =19.0\mathrm{MHz}$, larger than any of the transition widths $2{\mathrm{\Gamma }}_s/2\pi \le (13.2\pm 4.2)\mathrm{MHz}$. A dark state is observed along the upper branch, an effect caused by the symmetry of the coherently hybridized spin states. (d) Plot showing how the exchange interaction is reduced when the spin-photon detuning is increased.

Figure 4

Photon-number-resolved spin dispersive shift. (a) Spin transition frequency of DQD1 showing an extra dip below the main transition for larger probe power. The dip matches well the prediction from the Hamiltonian model (dashed lines) for $|\downarrow ,n⟩\leftrightarrow |\uparrow ,n⟩$ transitions, with the first two transitions visible in this plot. (b) Line cut of the data in panel (a) for $\varphi =10.55°$, and fit to four Lorentzian dips (first three shown with dashed lines). A dispersive shift $2{\chi }_s/2\pi =-13.1\pm 2.2\mathrm{MHz}$ is extracted. (c) Relative area of the dips for $\varphi =10.55°$, and comparison with thermal [$P_{\text{thermal}}(n)={\overline{n}}_r^n/({\overline{n}}_r+1{)}^{n+1}$] and coherent [$P_{\text{coherent}}(n)=e^{-{\overline{n}}_r}{\overline{n}}_r^n/n!$] photon-number distributions. The error bar represents the 95% confidence interval. (d) Relative area for $\varphi =11.6°$. At this angle, the resonator has a smaller ${\kappa }_r^{\prime }$, resulting in a larger steady-state photon number. The coherent-state distribution has better agreement.

Figure 5

Composite optical image of the device chip without wire bonds.

Figure 6

Schematic of the measurement setup.

Figure 7

Fitting of zero charge detuning. (a) Example of a single frame of data used in a vacuum Rabi splitting reconstruction. (b) Cut along the black line from panel (a) corresponding to the bare resonator frequency to extract the zero-detuning gate voltage $x_0$. An empiric fitting function is used to find the center of the signal dip due to the dispersive shifts. (c) Cut along the fitted zero-detuning value $x_0$ (white line) and fit to a single Fano resonance.

Figure 8

Dispersive regime transitions, and spin-spin interaction versus DQD detuning. (a) Schematic representation of the transition energy shifts in the dispersive regime. See Appendix pp4 for details. (b,c) Measurement of spin transitions for DQD1 (b) and DQD2 (c) as a function of the charge detuning $\epsilon$ with dispersive readout, with $2t_c/h\approx 8.7\mathrm{GHz}$ and $B_r=52\mathrm{mT}$. A pump tone is sent down the gate lines to excite the spins, resulting in a signal dip when the pump and spin frequencies match. Here, $(f_{\text{probe}}^1,f_{\text{probe}}^2,f_{\text{probe}}^{12})\approx (6.9038,6.9061,6.8981)\mathrm{GHz}$ for $\varphi =10.8°$. The spin transition frequency at $\epsilon =0\mu \mathrm{eV}$ is marked with a white dashed line for comparison between plots. (d) When the spins are simultaneously interacting with the resonator, the spin states hybridize depending on their energy difference. In the nonresonant case, $\varphi =13.0°$, their energies are minimally perturbed, as can be seen by the model lines and the white guides. In the resonant case, $\varphi =10.8°$, the two spins hybridize due to the exchange interaction $2J/2\pi =20.2\mathrm{MHz}$ mediated by virtual resonator photons. The upper state is hardly visible because its symmetry makes it dark to the resonator probe, which is expected for a coherent spin-spin interaction.

Figure 9

Example of data used to extract model parameters. For this set, $2t_c/h=13.5\mathrm{GHz}$, $B_r=52\mathrm{mT}$, and $\varphi =10.5°$. The first column shows vacuum Rabi splitting reconstructions for individually and simultaneously interacting spins. The second column shows dispersive spin sensing (the data near spin-photon resonance are scrambled because of the close proximity of the levels). The third column shows the angular dependence of the spin levels with dispersive spin sensing. The two spinlike and the photonlike transitions are shown together. These data, and more, are used to calibrate the micromagnet parameters for use in the Hamiltonian model Eq. (d2). For these tunnel splittings and spin-photon detunings, the model predicts $g_s/2\pi =11.4\mathrm{MHz}$ and $2J/2\pi =3.3\mathrm{MHz}$. Here, $2J/2\pi$ is too small to be resolved. Note that transition linewidths are power broadened.

Figure 10

Extended spin-spin hybridization data versus spin-photon detuning. (a)–(e) Dispersive spin-spin hybridization measured for different spin-photon detuning values. The probe power is larger for these data than for the data in Fig. 3; the shadow visible below the main transition is caused by the photon-induced dispersive shift of the spin. The apparent jitter of the transition energy as a function of $\varphi$ is caused by jitter of the DQD2 tunnel coupling over the time it takes to reconstruct all angles. The $2t_c/h\approx 8.7\mathrm{GHz}$ is retuned between plots and yields $g_s/2\pi \approx 34\mathrm{MHz}$ for both spins. The model fully captures the observed transition frequencies within plots and between the plots, and the only adjustment is the experimentally measured small variations in $t_c$. (f) Splittings $2J/2\pi$ extracted from the model as a function of spin-photon detuning (through $B_r$). Comparison with the linewidth is impaired by the photon-number broadening of the transitions. As expected, $2J/2\pi$ decreases as $|{\mathrm{\Delta }}_{2s}/2\pi |$ increases.

Figure 11

Extended spin-spin hybridization data versus spin-photon coupling strength. (a)–(d) Dispersive spin-spin hybridization measured for decreasing spin-photon coupling values. The $g_s$ is adjusted through its dependence on $t_c$ and is approximately the same for both spins, while $B_r=52\mathrm{mT}$ is fixed. The ${\mathrm{\Delta }}_{2s}/2\pi \approx 78\mathrm{MHz}$ is approximately unchanged between plots because changes in ${\chi }_{\mathrm{SO}}$ are approximately compensated by changes in ${\chi }_c$. The model fully captures the observed transition frequencies within plots and between the plots, and the only adjustment is the experimentally measured $t_c$. As expected, $2J/2\pi$ decreases as $g_s/2\pi$ decreases.

Figure 12

Extended photon-number-dependent spin dispersive shift data. (a) Photon-number-dependent spin dispersive shift of DQD1 (same as in the main text). (b) Line cut of the data in panel (a) for a different angle $\varphi =11.6°$, and fit to four Lorentzian dips (first three shown with dashed lines). A dispersive shift $2{\chi }_s/2\pi =-12.5\mathrm{MHz}$ is extracted. The coherent-state distribution has better agreement than the thermal distribution; see the main text for number distribution results.