# Superconductor–semiconductor hybrid-circuit quantum electrodynamics

## Abstract

Light–matter interactions at the single-particle level have generally been explored in the context of atomic, molecular and optical physics. Recent advances motivated by quantum information science have made it possible to explore coherent interactions between photons trapped in superconducting cavities and superconducting qubits. In the context of quantum information, the study of coherent interactions between single charges and spins in semiconductors and photons trapped in superconducting cavities is very relevant, as the spin degree of freedom has a coherence time that can potentially exceed that of superconducting qubits, and cavity photons can serve to effectively overcome the limitation of short-range interaction inherent to spin qubits. Here, we review recent advances in hybrid ‘super–semi’ quantum systems, which coherently couple superconducting cavities to semiconductor quantum dots. We first present an overview of the physics governing the behaviour of superconducting cavities, semiconductor quantum dots and their modes of interaction. We then survey experimental progress in the field, focusing on recent demonstrations of cavity quantum electrodynamics in the strong-coupling regime with a single charge and a single spin. Finally, we broadly discuss promising avenues of future research, including the use of super–semi systems to investigate phenomena in condensed-matter physics.

## Key points

• Hybrid quantum systems integrate the most desirable properties of semiconductor spin qubits and superconducting quantum devices.

• Single electron charges can be coherently coupled to single microwave-frequency photons.

• Using spin–orbit interactions, a single electron spin can be coherently coupled to a single photon.

• Coherent charge–photon and spin–photon coupling may enable long-range qubit interactions that are mediated by microwave-frequency photons.

• Hybrid quantum devices are also finding utility as sensitive probes of Kondo and valley physics, and perhaps of Majorana fermions.

## Introduction

A remarkable experimental achievement of the 1980s and 1990s was the coherent coupling of a single atom existing in one of two states to a single photon trapped in a cavity1,2,3,4. These results showed that it is possible to create a quantum superposition of light and matter, and started the field of cavity quantum electrodynamics (QED)5. More complex systems such as superconductors and semiconductors can also be used as building blocks in hybrid systems on a larger scale. In the early 2000s, cavity QED experiments were realized in condensed-matter systems using self-assembled quantum dots (QDs) confined in photonic cavities6,7,8 or by placing a superconducting qubit inside a microwave cavity9,10. In these experiments the atom that is conventionally used in atomic cavity QED was replaced with a quantum device with discrete energy levels, whose energy separation can be matched to the energy of a cavity photon. Around the same time, it was conjectured that cavity QED could be performed using individual electrons trapped in gate-defined semiconductor QDs, using either charge or spin degrees of freedom to mimic the states of an atom11,12,13.

There are many reasons to study cavity QED in the context of condensed-matter systems, many of which are grounded in the rapidly growing field of quantum information science. On the heels of the discovery that a superconducting circuit can be coherently coupled to microwave photons10, two experiments showed that two spatially separated superconducting qubits can be coupled via a cavity14,15. The resulting subject of circuit QED has now expanded into a field of its own. Prominent advances include demonstrations of multi-qubit entanglement16,17,18,19,20, readout of quantum states21,22,23, generation of non-classical light24,25,26,27, development of error correction based on Schrödinger cat states28,29,30, quantum feedback31,32,33 and measurements of quantum trajectories34. Fundamentally, experiments involving superconducting quantum devices take advantage of a macroscopic superconducting condensate that is protected by an energy gap ΔBCS (for example, ΔBCS ~ 175 µeV in Al, roughly 20 times larger than the thermal energy of 100 mK ~ 8 µeV in typical experiments). However, it is also desirable to implement cavity QED with single charges and spins in semiconductor devices, where such protection is absent. The prospect of cavity QED studied with a single spin is especially intriguing35,36, as spin coherence times can exceed seconds in some solid-state systems37,38,39.

In this Review, we describe dramatic developments in the area of hybrid-circuit QED, in which gate-defined QDs are coupled to superconducting cavities in a ‘super–semi’ device architecture, including recent demonstrations of strong-coupling physics with single charges and spins confined in semiconductor QDs40,41,42,43,44. We begin by laying the theoretical groundwork underlying the experiments, with a description of the superconducting cavity, the ‘artificial atom’, which, in most experiments, consists of a semiconductor double quantum dot (DQD), their modes of interaction and the figures of merit that describe the quantum coherence of the system. We then survey experiments involving the charge degree of freedom, which interacts with the cavity electric field through the electric dipole interaction40,41. A combination of electric dipole coupling and spin–orbit coupling enables coherent spin–photon interactions, which we discuss next42,43,44. Lastly, we give examples of how semiconductor-circuit QED could impact fundamental science and engineering in areas that range from topological physics to surface microscopy and quantum technology.

## Cavity QED with DQDs

At a basic level, a typical cavity QED system (Fig. 1a) consists of just two components: a photonic cavity and a two-level quantum system with closely matched resonant frequencies. The first demonstrations of cavity QED in atomic physics used microwave transitions between Rydberg states of single caesium atoms45. These results were eventually extended to the visible spectral range46,47.

Cavity QED can also be implemented using a wide variety of solid-state systems (Fig. 1b). Colour centres in diamond, such as nitrogen and silicon vacancy centres, have spin-full ground states and narrow, spin-selective microwave and optical transitions, which allows the realization of cavity QED with integrated photonic structures48,49. Using nanofabrication techniques, it is possible to build mesoscopic semiconducting and superconducting devices that are quantum coherent. Semiconductor DQDs can be used to isolate single electrons; in these systems, the charge degree of freedom can be controlled with electric fields50,51,52 and the spin degree of freedom with magnetic fields53 and the exchange interaction through changes to the confining potential54. Superconducting circuits combine a capacitance C with a Josephson inductance LJ to create a quantum system with an anharmonic energy-level spectrum55,56,57,58,59,60; cavity QED experiments involving superconducting quantum devices are reviewed in refs61,62. In this Review, we focus on experiments involving semiconductor DQDs63,64, whose electrical tunability opens the door to cavity QED using long-lived spin states.

### Charge–photon interaction

A single electron trapped in a DQD forms a fully tunable, two-level system (Box 1). Electric dipole interactions couple the electron trapped in the DQD to the cavity photon with a strength described by the charge–photon coupling rate gc. The interdot spacing is typically on the order of d = 100 nm, which leads to a large electric dipole moment ed. The coupling rate is given by the product of this dipole moment with the root mean square of the vacuum electric field E0 of the cavity. Near zero detuning, the charge states are strongly hybridized, leading to the maximum in the charge–photon coupling rate. Away from zero detuning, the charge states are only weakly admixed.

In atomic physics, Fabry–Pérot cavities are typically employed to trap optical photons47. For the much larger quantum-dot devices, typical energy scales are on the order of 20–40 µeV, so it becomes convenient to use superconducting resonators to trap microwave-frequency photons (1 GHz ~ 4.2 µeV). Cavities are never perfect and can have internal losses, described by a decay rate κint, and losses through the cavity ports, κ1 and κ2. Cavity QED systems can be conveniently probed by measuring the transmission through (or reflection from) the cavity. For example, in Fig. 1a, port 1 is driven by a weak input field ain and the signal aout exiting port 2 is measured.

When a DQD is placed inside a superconducting microwave resonator, the electric field Eres inside the resonator tilts the energy landscape and the difference ε between the energy levels of the left and right dots becomes ε + eEresd. Because d is much smaller than the wavelength of the electromagnetic waves inside the resonator, we can apply the electric dipole approximation and consider Eres constant within the entire volume of the DQD. The quantized electric-field operator can be expressed in terms of creation and annihilation operators a and a of the electromagnetic field mode inside the resonator (these are equivalent to the ladder operators of the quantum harmonic oscillator), so that Eres = E0(a + a). The coupling of the single electron trapped in the DQD (which forms a charge qubit, Box 1) to the resonator mode is described by the Hamiltonian H = H0 + Hint, with Hint = gc(a + a)τz in units in which $$\hbar =1$$, with the charge-cavity coupling gc = eE0d and the quantum operator τz defined via $${\tau }_{z}| (1,0)\rangle =| (1,0)\rangle$$ and $${\tau }_{z}| (0,1)\rangle =-| (0,1)\rangle$$. The electric dipole of the DQD with one electron, edτz, can be probed via the microwave transmission through the cavity. Theoretically, this means that the DQD and cavity need to be treated as an open quantum system. The transmission can be efficiently calculated using input–output theory (Box 2).

Typical experiments operate in a regime in which the charge–cavity coupling gc is a weak perturbation compared with the bare energy scale of the qubit Ω. In this operating regime, it is convenient to first diagonalize H0 and then transform Hint into the eigenbasis of H0. Transforming into a frame rotating with the probe field frequency and neglecting fast oscillating terms within the rotating-wave approximation, one finds $$H=\frac{1}{2}\Omega {\tau }_{z}+{\widetilde{g}}_{{\rm{c}}}(a{\tau }_{+}+{a}^{\dagger }{\tau }_{-})+\varDelta {a}^{\dagger }a$$, where $${\widetilde{g}}_{{\rm{c}}}=\frac{{g}_{0}{t}_{{\rm{c}}}}{\Omega }$$, and where we have added the photon energy in the rotating frame $$\frac{\varDelta }{2\pi }={f}_{{\rm{c}}}-{f}_{{\rm{R}}}$$ (fR is the probe frequency; often fc = fR and, thus, Δ = 0). τ+ and τ are the raising and lowering operator for the charge qubit, respectively, and the τ operators are understood to act in the eigenbasis of H0. The Heisenberg–Langevin equations of motion for the photon operator and the electron coherence operator are then found to be (Box 2)

$$\dot{a}=-i\Delta a-\frac{\kappa }{2}a+\sum _{n}\sqrt{{\kappa }_{n}}{a}_{n,{\rm{in}}}(t)-i{\widetilde{g}}_{{\rm{c}}}{\tau }_{-},$$

and

$${\dot{\tau }}_{-}=-i\Omega {\tau }_{-}-\frac{\gamma }{2}{\tau }_{-}-i{\widetilde{g}}_{{\rm{c}}}a+{\tau }_{z}F(t),$$

where F(t) is a quantum noise operator with zero expectation value satisfying $$[F(t),{F}^{\dagger }(t{\prime} )]=\gamma \delta (t-t{\prime} )$$, a condition needed to satisfy fluctuation–dissipation relations65,66,67. Here, in addition to the coherent contributions from the quantum Heisenberg equations of motion, the incoherent terms take into account the cavity decay with rate $$\kappa ={\kappa }_{1}+{\kappa }_{2}+{\kappa }_{{\rm{int}}}$$ (photon loss at the two ports plus intrinsic losses), the charge qubit decay rate γ and the cavity input field an,in on port n. In the stationary limit $$\langle \dot{a}\rangle =\langle {\dot{\tau }}_{-}\rangle =0$$, the transmission coefficient through the cavity is

$$A=\frac{\langle {a}_{2,out}\rangle }{\langle {a}_{1,in}\rangle }=\frac{\sqrt{{\kappa }_{2}}\langle a\rangle }{\langle {a}_{1,in}\rangle }=\frac{-i\sqrt{{\kappa }_{1}{\kappa }_{2}}}{\varDelta -\frac{i\kappa }{2}+{\widetilde{g}}_{c}\chi },$$

with the single-electron DQD electric susceptibility

$$\chi =\frac{{\widetilde{g}}_{{\rm{c}}}}{-\Omega +i\gamma /2}.$$

For simplicity, we can consider a symmetric cavity without intrinsic losses (κint = 0), such that $${\kappa }_{1}={\kappa }_{2}=\kappa /2$$. The cavity is also often probed on resonance (Δ = 0). In the absence of a DQD, χ = 0 and microwaves are transmitted unhindered through the cavity (A = 1). Charge dynamics within the DQD results in an effective microwave admittance that loads the superconducting cavity, changing the cavity amplitude and phase response68,69,70,71,72. The electric susceptibility χ is greatest (and, thus, A smallest) for a symmetric DQD (ε = 0), because, in this configuration, the electron is most easily transferred between dots and the dipole moment is maximized.

### Quantum-coherent charge–photon coupling

The scale of the susceptibility, χgc, and transmission, $$A\propto 1/{g}_{{\rm{c}}}^{2}$$ (assuming κgc and ε = 0 in the case of a DQD), are both determined by the electric dipole and vacuum cavity electric field via the electron–dipole coupling strength (gc = eE0d). For a Rydberg atom in an optical cavity, E0 ≈ 1 mV m−1 and d ≈ 100 nm, leading to couplings gc roughly in the 10–100 kHz range73. This coupling can, in principle, be strengthened in two ways: by increasing the electric dipole through an increase in the DQD size or by increasing the vacuum electric field $${E}_{0}=\sqrt{h{f}_{c}/2\,{\varepsilon }_{0}V}$$, where V is the cavity mode volume. Although superconducting circuit microwave resonators typically have a slightly lower resonance frequency than 3D cavities for cavity QED based on Rydberg atoms, their mode volume can be thousands of times smaller than that of 3D cavities9,10. The vacuum electric field can, therefore, be several orders of magnitude stronger in superconducting resonators, which allows for qubit–resonator couplings of $$\frac{{g}_{{\rm{c}}}}{2\pi }\approx 1-10\,{\rm{MHz}}$$ for QDs (d ≈ 100 nm) and $$\frac{{g}_{{\rm{c}}}}{2\pi }\approx 10-100\,{\rm{MHz}}$$ for superconducting qubits (d ≈ 1 µm). Crucially, such large values of gc can easily exceed both the cavity linewidth κ and qubit decoherence rate γ. The limit in which gc (γ, κ) is called the strong-coupling regime of cavity QED4,74. Achieving strong coupling is significant because the qubit and photon degrees of freedom become directly entangled with each other4. In addition to being of fundamental interest, this entanglement can be exploited for applications in quantum information science75.

## Experimental demonstrations

Hybrid quantum devices comprising gate-defined QDs coupled via their electric dipole moment to microwave cavities have been successfully demonstrated using multiple material systems, including GaAs/AlGaAs heterostructures41,76,77,78,79, InAs nanowires67,80, graphene81,82, carbon nanotubes83,84 and Si/SiGe heterostructures40,42,43,85. The microwave cavity is often realized as a superconducting coplanar waveguide resonator9,10,14,24,86 (left panel of Fig. 2a). To maximize the quality factor of the cavity and the chance of reaching the strong-coupling regime, each gate line leading to the DQD is sometimes filtered by an on-chip, low-pass LC-filter to suppress photon leakage from the cavity85.

A simplified schematic of the hybrid device is depicted in the middle panel of Fig. 2a. At the fundamental resonance frequency fc, the vacuum fluctuation of the cavity generates a half-wavelength λ/2 electromagnetic standing wave10,87. At a voltage antinode of the standing wave, a delocalized electron occupying the molecular bonding and antibonding states of the DQD50,51,63 couples to the electric field of the cavity via the electric dipole interaction67,76. A circuit representation of the device is shown in the right panel of Fig. 2a. Here, the cavity is modelled as a parallel LC-oscillator having an effective inductance Lc, effective capacitance Cc and resonance frequency $${f}_{{\rm{c}}}=\frac{\sqrt{\frac{1}{{L}_{{\rm{c}}}{C}_{{\rm{c}}}}}}{2\pi }$$ (refs88,89). The DQD is mutually coupled via a capacitance Cm and dot 2 is capacitively coupled to the cavity via Cg. The system is connected to an input port via capacitor C1 and to an output port via capacitor C2, allowing for measurements of the cavity transmission amplitude A = |a2,out/a1,in| and phase φ = −arg(a2,out/a1,in) using homodyne or heterodyne detection techniques10,21,90.

Because the charge–photon coupling rate gc scales linearly with $$\sqrt{{Z}_{{\rm{c}}}}$$, where $${Z}_{{\rm{c}}}=\sqrt{{L}_{{\rm{c}}}/{C}_{{\rm{c}}}}$$ is the characteristic impedance of the cavity9,58, it is desirable to increase Zc beyond the range 20–200 Ω, which is the typical limit of coplanar waveguide cavities89. One way of increasing the impedance is to define the microwave cavity using a linear array of superconducting quantum interference devices (SQUIDs) made from Al Josephson junctions, leading to Lc ≈ 1.5 kΩ by virtue of the large Josephson inductance of each SQUID41. Another approach, discussed in the next section, utilizes the large kinetic inductance of a nanowire made from NbTiN43,91.

Detailed scanning electron micrographs of cavity-coupled DQDs made with different host materials are shown in Fig. 2b. In the case of GaAs or Si, one or three layers of surface gate electrodes are directly patterned on top of the buried quantum well of a GaAs/AlGaAs or Si/SiGe heterostructure76,85. For InAs nanowires, graphene and carbon nanotubes, the host material is transferred to a Si substrate before the patterning of gate electrodes67,81,92. To maximize gc, an electrode is often galvanically connected to the centre pin of the superconducting cavity40,67,76,77,81.

### Strong charge–photon coupling

Achieving the strong-coupling regime for DQD charge qubits is generally challenging owing to their large decoherence rates γc, which commonly fall between a few hundred MHz and several GHz50,52,67,76,77,78,81,84,93,94,95. These values often exceed the coherent charge–photon coupling rate gc by one or more orders of magnitude67,76,77,78,79,81,84,94,95. Therefore, a significant reduction in γc or a significant increase in gc is needed to access the strong-coupling regime gc > (γc, κ). Both approaches have recently been successful in two experiments that we discuss here40,41.

A first step towards the determination of the charge–photon coupling rate is the detection of the charge states within the DQD. This is traditionally accomplished by measuring the conductance of a proximal quantum point contact that is sensitive to the charge distribution within the QDs96,97,98. Charge-state detection may also be performed by measuring the transmission properties of the cavity, which are sensitive to the tunnel-rate-dependent complex admittance of the QDs68,69,72. An example is shown in Fig. 3a: here, the cavity transmission amplitude A/A0 (A0 is a normalization constant) at a fixed drive frequency f = fc is measured as a function of gate voltages VP1 and VP2 (see Si/SiGe in Fig. 2b), which control the chemical potentials of dot 1 and dot 2, respectively. The charge-stability diagram characteristic of a few-electron DQD is clearly visible63. (N1,N2) denotes a charge state with N1 electrons in dot 1 and N2 electrons in dot 2. To determine gc, A/A0 is measured around an interdot charge transition (N1 + 1,N2) − (N1,N2 + 1)63 (bottom panel of Fig. 3a). Here, a pair of minima are observed along the DQD detuning axis ε at locations where Ω/h = fc. At these detunings, the charge qubit is strongly hybridized with the cavity photons40,67,76. Detailed fitting of the response A(ε)/A0 to input–output theory allows the charge–photon coupling rate gc to be extracted from this type of measurement40,67,76.

The hallmark of the strong-coupling regime is the vacuum Rabi splitting, which is the emergence of a pair of distinct resonance peaks in the cavity transmission spectrum for a fixed detuning, at which the qubit and a cavity photon become equal in frequency6,7,10,47. The data must be carefully acquired with a weak drive tone, such that the intracavity photon number nc is much smaller than 1 (ref.10). The measurement of A/A0 as a function of f and ε, taken with a Si-based DQD tuned to 2tc/h = fc, is shown in the top-left panel of Fig. 3b (ref.40). With ε = 6 µeV, the cavity transmission spectrum exhibits a single peak (bottom-left panel of Fig. 3b). Here, the qubit–photon frequency detuning is large (white dashed lines) and the full width at half maximum of A2/A02 gives a bare cavity loss rate κ/2π = 1.0 MHz. At ε = 0 µeV, the DQD splitting is Ω = 2tc = hfc and the charge qubit becomes entangled with a single photon, leading to an avoided crossing in the eigenenergies of the charge–photon system (white solid lines). Correspondingly, vacuum Rabi splitting is observed in the cavity transmission spectrum (bottom-left panel of Fig. 3b), where the two resonance modes are separated by a vacuum Rabi frequency 2gc/2π = 13.4 MHz. The top-right panel of Fig. 3b shows A/A0 as a function of f and ε measured with 2tc/h < fc: a pair of avoided crossings are observed when Ω/h = fc at finite ε. The clear resolution of the vacuum Rabi splitting suggests that the regime of strong charge–photon coupling has been achieved for this Si-based device, a conclusion further supported by a charge decoherence rate γc/2π = 2.6 MHz that was independently determined using microwave spectroscopy in the dispersive regime40,99. The charge decoherence rate of this device is about two to three orders of magnitude lower than that of typical DQD charge qubits50,52,67,76,77,78,79,81,84,93. Environmental factors leading to charge noise are the subject of ongoing investigation. It is believed that charge noise originates from fluctuating two-level systems in the device (in oxides and at interfaces). Recent work suggests that reducing the gate oxide thickness may result in a reduction of charge noise100. Finally, the sensitivity to charge noise might be reduced by engineering optimal gate lever arms101 or designing qubits with ‘sweet spots’ that are first-order-insensitive to charge noise56,102,103,104,105,106.

Strong charge–photon coupling has also been achieved in GaAs DQD devices41. These devices implement a novel frequency-tunable SQUID array cavity design. The SQUID array yields a large inductance L ~ 50 nH, leading to a cavity impedance of 1.8 kΩ, which is substantially larger than the conventional 50 Ω of coplanar waveguide resonators10. Owing to the higher impedance of the microwave cavity, a large charge–photon coupling rate gc/2π = 119 MHz is obtained (bottom-right panel of Fig. 3b), which allows the strong-coupling regime to be accessed despite comparatively large values of γc/2π = 40 MHz and κ/2π = 12 MHz. These experiments have been extended to include coherent charge-state readout and control107, as well as cavity-mediated interactions between two GaAs charge qubits108. Vacuum Rabi splitting has also been observed in carbon nanotubes coupled to a cavity mode, in which a two-transition scheme involving the K/K′ valleys was invoked to fit the data101.

## Spin–photon coupling

The quantum coherence of the spin-½ of individual electrons in QDs or defects in silicon typically lasts between tens of microseconds and several milliseconds109,110,111,112 and can, in some cases, even approach a second37, whereas nuclear spin coherence times approach 1 minute113. Spin is, therefore, the primary choice as a qubit for quantum information processing in semiconductors35,36. Because the exchange interaction is short ranged54, this naturally leads to the question of how to couple two electron spins that are separated by a large distance using spin–electric coupling to a common cavity mode. This might seem extremely challenging because the spin of an electron does not directly couple to the electric field of the cavity. However, there are several techniques to hybridize the spin and charge degrees of freedom (qubits) of an electron. All these methods endow the spin with an effective electric dipole that enables its interaction with the electric field of the cavity. For multi-electron spin qubits, the Pauli exclusion principle provides a way to couple orbital and spin degrees of freedom12,114,115 (a recent experiment using this method has attained strong spin-qubit–photon coupling44). One mechanism that works for single electron spins is the natural built-in spin–orbit coupling due to relativistic effects, which is sizeable in a number of semiconductor materials13,116,117. The intrinsic spin–orbit coupling may work particularly well for holes in the valence band of some semiconductors118. Without relying on such intrinsic effects, one can engineer a spin–electric interaction using controlled magnetic fields, either time-dependent fields that induce electron spin resonance11,119,120,121 or static but spatially varying fields produced by an on-chip microscale ferromagnet42,43,111,122,123,124,125,126,127.

In the case of a static magnetic field gradient $${\nabla }_{x}B$$ produced by a micromagnet, an applied electric field Eac shifts the electron position in a single QD by

$${x}_{{\rm{E}}}=\frac{e{E}_{{\rm{ac}}}{a}_{0}^{2}}{{E}_{{\rm{orb}}}},$$

where Eorb and a0 denote the energy-level spacing and size of the QD, respectively. For an oscillatory electric field, this means that the magnetic field seen by the electron also becomes oscillatory, $$B({x}_{0}+{x}_{{\rm{E}}}{\rm{\sin }}\,\omega t) \sim B({x}_{0})+{\nabla }_{x}B{x}_{{\rm{E}}}{\rm{\sin }}\,\omega t$$, allowing for electric dipole spin resonance. For the quantized cavityfield, $${E}_{{\rm{ac}}}{\rm{\sin }}\,\omega t$$ is replaced by Eres = E0(a + a), resulting in a spin–photon coupling $${g}_{{\rm{s}}} \sim \frac{e{E}_{0}{\nabla }_{x}B\,{a}_{0}^{2}}{{E}_{{\rm{orb}}}}$$. The spin–phonon coupling gs in a DQD can be much larger and more controllable than in a single QD128. For a symmetric DQD at ε = 0, one finds $${g}_{{\rm{s}}} \sim \frac{e{E}_{0}{\nabla }_{x}B{d}^{2}}{\Omega } \sim \frac{{g}_{{\rm{c}}}\varDelta {B}_{x}}{\Omega }$$, where d is the distance between the two dots, $$\varDelta {B}_{x}={\nabla }_{x}Bd$$ the change in magnetic field (measured in energy units) from one dot to the other and the DQD energy splitting Ω can be tuned by the interdot tunnel coupling and the external magnetic field122. Because d > a0 and Ω Eorb, gs is much larger in a DQD than in a single QD128,129.

To study the combined charge and spin dynamics of a single electron in a DQD, one can employ the 4 × 4 Hamiltonian in the basis $$| (\uparrow ,0)\rangle ,| (\downarrow ,0)\rangle ,| (0,\uparrow )\rangle ,| (0,\downarrow )\rangle$$

$${H}_{0}=\frac{1}{2}\left(\begin{array}{cccc}\varepsilon +{B}_{z} & \Delta {B}_{x} & 2{t}_{{\rm{c}}} & 0\\ \Delta {B}_{x} & \varepsilon -{B}_{z} & 0 & 2{t}_{{\rm{c}}}\\ 2{t}_{{\rm{c}}} & 0 & -\varepsilon +{B}_{z} & -\Delta {B}_{x}\\ 0 & 2{t}_{{\rm{c}}} & -\Delta {B}_{x} & -\varepsilon -{B}_{z}\end{array}\right)$$

which includes the Zeeman coupling $${H}_{Z}={\bf{S}}\cdot {\bf{B}}({\bf{r}})$$ of the spin S to an external magnetic field B(r) (in energy units)122. A magnetic field Bz pointing in the z-direction leads to an energy splitting between the spin-up and spin-down states. As long as the field has the same strength in both dots, that is, B does not depend on the position r, the spin and charge qubits are completely separate. In this case, only the charge qubit interacts with the electric field of the cavity, whereas the spin qubit is decoupled from it. However, as soon as a magnetic field difference ΔBx perpendicular to the homogeneous field component is applied, the charge and spin qubits are hybridized, allowing for a coupling of the spin qubit to the cavity electric field. The coupling to the cavity is again obtained by replacing ε with ε + eEresd. In this way, we obtain the Hamiltonian H = H0 + Hint, as discussed above. The four relevant energy levels $$| n\rangle$$ of the DQD are found by diagonalizing the matrix H0, whereas the electric dipole transition matrix elements dnm can be determined by transforming Hint into the eigenbasis of H0,

$${H}_{{\rm{int}}}={g}_{{\rm{c}}}(a+{a}^{\dagger }){\sum }_{n,m=0}^{3}{d}_{nm}| n\rangle \langle m| .$$

For the understanding of the most important mechanisms for the spin–photon interaction, it is sufficient to consider an effective two-level model. Introducing the rotating-wave approximation, one arrives at the Jaynes–Cummings model $$H=\frac{1}{2}{\Delta }_{{\rm{s}}}{\sigma }_{z}+{g}_{{\rm{s}}}(a{\sigma }_{+}+{a}^{\dagger }{\sigma }_{-})$$, where the Pauli operators σz act on the low-energy hybridized spin states, and $${\Delta }_{{\rm{s}}}={B}_{z}-h{f}_{{\rm{c}}}$$ is the detuning of the spin Zeeman splitting Bz from the photon energy hfc. For a symmetric DQD with ε = 0, one finds a spin–photon coupling rate $${g}_{{\rm{s}}}\cong \frac{{g}_{{\rm{c}}}\Delta {B}_{x}}{2\delta }$$, where δ = 2tc − hfc can be controlled by adjusting the DQD tunnel coupling tc. Although this two-level model explains the vacuum Rabi splitting that has been observed experimentally, there are more subtle effects such as the asymmetry of the Rabi peak heights that require a three-level model to be explained122.

### Strong spin–photon coupling

Compared with charge–photon coupling, reaching the strong-coupling regime of spin–photon interaction faces a distinct challenge: the direct magnetic-dipole coupling rate gs between a single electron spin and a single photon is mostly limited to 10–500 Hz, which is too slow to overcome single-spin dephasing rates or cavity-loss rates130,131,132,133,134,135. As such, a robust scheme for spin–charge hybridization is necessary to increase gs to the MHz range, where strong coupling becomes feasible11,12,13,114,115,122,127,128,129,136,137. At the same time, a low level of charge noise is required of the device because spin–charge hybridization subjects the electron spin to charge-noise-induced spin dephasing. A cavity-coupled carbon nanotube DQD has been used to hybridize spin and charge via ferromagnetic leads to achieve gs/2π = 1.3 MHz but remained in the weak-coupling regime, owing to a relatively large spin decoherence rate γs/2π = 2.5 MHz138. More recently, two experiments using Si-based DQDs have successfully attained the strong-coupling regime between a single spin and a single photon42,43. In this section, we review these experiments, as well as a separate experiment that coupled a three-electron spin state in a GaAs triple QD to a single photon44.

It is remarkable that strong spin–photon coupling via spin–charge hybridization does not require strong charge–photon coupling. This follows from the different scaling of the spin–photon coupling gs and induced spin decoherence rate γs on the degree of spin–charge hybridization controlled by the ratio of magnetic field gradient ΔBx and the controllable energy detuning δ = 2tc − hfc: whereas $$\frac{{g}_{{\rm{s}}}}{{g}_{{\rm{c}}}}\propto \frac{\Delta {B}_{x}}{\delta }$$ is linear in the spin–charge hybridization, $$\frac{{\gamma }_{{\rm{s}}}}{{\gamma }_{{\rm{c}}}}\propto (\frac{\Delta {B}_{x}}{\delta }{)}^{2}$$, thus allowing for $$\frac{{g}_{{\rm{s}}}}{{\gamma }_{{\rm{s}}}} > \frac{{g}_{{\rm{c}}}}{{\gamma }_{{\rm{c}}}}$$ at sufficiently small $$\frac{\Delta {B}_{x}}{\delta }$$.

The setup for the first experiment we discuss42 is illustrated in Fig. 4a. The device is a gate-defined DQD on top of a Si/SiGe heterostructure coupled to a coplanar waveguide cavity, similar to devices used in previous work on strong charge–photon coupling40 but including a micron-sized Co magnet on top of the DQD gate electrodes109,125,139. The micromagnet creates a large magnetic field gradient, such that the quantization axis of the electron spin is dependent on its location, hybridizing the spin and charge degrees of freedom13,122,127,128,129. A plot of the DQD energy levels including the spin degree of freedom is provided in Fig. 4b.

To search for spin–photon coupling, the frequency of the single-spin qubit EZ/h = BBtot/h is tuned into resonance with the cavity by changing the external magnetic field Bzext. Here, EZ is the Zeeman energy, g the g-factor of the electron and µB the Bohr magneton. Btot is the total magnetic field spatially averaged over the electron’s wave function, having contributions from both the externally applied field and the intrinsic field of the micromagnet. The cavity transmission amplitude as a function of f and Bzext, taken at ε = 0, is shown in Fig.4c. A clear avoided crossing is observed around Bzext = 92.2 mT, where the resonance condition EZ/h = fc is met. A vacuum Rabi splitting 2gs/2π = 11.0 MHz is obtained, indicating strong coupling between the single electron spin and a cavity photon. The spin–photon coupling rate gs/2π = 5.5 MHz exceeds direct magnetic-dipole coupling rates by four to five orders of magnitude130,131,132,133,134,135. A second experiment, also involving a DQD defined on a Si/SiGe heterostructure, used a high-impedance cavity design composed of a thin NbTiN nanowire with a large kinetic inductance43. Strong spin–photon coupling was also achieved in this device, as shown by the avoided crossing in Fig. 4d.

Possibilities to circumvent the requirement for a strong magnetic field gradient generated by an on-chip micromagnet include the use of materials with strong spin–orbit coupling, such as holes in germanium118,140, or the use of the Pauli exclusion principle in multi-spin qubits, such as the resonant-exchange qubit115. The resonant-exchange qubit, like other exchange-only qubits, uses a degenerate spin subspace (decoherence-free subspace) of three or more electrons in which all-electrical control is possible114,141,142,143,144. The idea is based on the availability of two qubit states with identical spin quantum numbers (for example, S = 1/2 and Sz = +1/2), between which spin-conserving, electrically controlled exchange interactions can operate145. An experiment with a resonant-exchange qubit formed in a triple QD in GaAs filled with three electrons and embedded into a NbTiN high-impedance cavity demonstrated a coupling between the resonant-exchange qubit and cavity photons of gs/2π = 31 MHz44. With a qubit decoherence rate of γs/2π = 20 MHz and a cavity decay rate of κ/2π = 47 MHz, this experiment achieved the strong-coupling regime, resolving the two vacuum Rabi peaks, because 2gs > κ/2 + γs. The resonant-exchange qubit offers an extended tunability of gs and γs via two electrostatic control parameters that control the dipolar (linear tilt) and quadrupolar (centre dot energy) component of the triple-dot potential. By tuning these two parameters, the resonant-exchange qubit can be operated at a sweet spot where the qubit energy splitting becomes insensitive to small electrostatic fluctuations due to external charge noise145.

To apply spin–photon cavity QED devices to quantum-information-processing tasks, such as the implementation of long-range two-qubit gates, it is necessary to rapidly switch on and off the spin–photon coupling rate gs. This allows the spin qubit to be manipulated in an isolated state (gs ≈ 0), in which it is protected from cavity-induced Purcell decay133,146,147,148, and read out via the cavity when the coupling is back on. One way to tune gs is by tilting the DQD potential (Fig. 4c). As ε is increased from zero, a strong decrease of spin–photon coupling from gs/2π = 5.5 MHz (ε = 0) to gs/2π 1 MHz (ε = 40 µeV) is observed. This change is due to the fact that, with |ε| tc, the electron wave function becomes strongly localized within one dot (Fig. 4b), and interdot tunnelling is largely suppressed. Here, the displacement of the electron wave function by the cavity photon is limited to about 3 pm42. The effective magnetic field generated by a cavity photon is, therefore, very small, effectively turning off the spin–photon coupling.

## Conclusions

The strong and controllable coupling between individual spin qubits embedded in a superconducting microwave resonator allows for long-distance spin–spin coupling mediated by microwave photons149,150. Resonant interactions between two spins separated by 4 mm have recently been achieved using circuit QED151. With device improvements, this coupling could be employed to perform two-qubit gates between spins separated by several millimetres. One should keep in mind that 1 mm is a very long distance compared with the 80-nm separation of nearest-neighbour spin qubits in Si152. Their small footprint on a semiconductor chip is one characteristic feature of semiconductor spin qubits, which makes them strong contenders for a scalable quantum-information-processing platform. In addition to enabling the entanglement of distant spin qubits, non-local two-qubit gates may facilitate quantum error correction in the framework of a fault-tolerant quantum-computing architecture. The possibility of creating a network of spin qubits with engineered coupling may be very useful for realizing a surface code153. Moreover, the possibility of creating a network of spin qubits with coupling geometries ranging from local to ‘all-to-all’ opens interesting perspectives for the quantum simulation of interacting quantum many-body systems154,155.

Spin–photon coupling has important implications beyond the generation of long-range quantum entanglement (Fig. 5a). The coupling of the electron spin to an electromagnetic cavity also allows for the dispersive readout of the quantum state of a spin qubit42,67 and lays the foundation for the development of quantum non-demolition156,157 and single-shot readout methods158. Because the spin–photon coupling rate gs depends strongly on the detuning ε, electrically switching on the cavity coupling of each spin qubit40 may allow for selective readout in large arrays of spin qubits (Fig. 5b). Moreover, the superconducting qubit community has adopted the use of frequency multiplexing23,159 for quantum-state readout of multi-qubit devices. A similar approach could be adopted for spins160.

Looking beyond spin qubits, the nascent field of hybrid-circuit QED could have a major impact on condensed-matter physics as a whole. Cavity measurements have been proposed to investigate Majorana modes161 and provide an alternative to the somewhat ambiguous measurements of zero-bias conductance peaks162,163,164,165. It has even been suggested that microwave cavities could be used to detect the braiding of Majorana fermions166 (Fig. 5c). Scanning-probe versions of superconducting cavities (Fig. 5d) could be used to probe valley physics in silicon167,168,169,170 and perhaps, more broadly, to investigate 2D quantum materials171. Lastly, superconducting cavities have been shown to provide an alternative means to investigate Kondo physics172,173,174 (Fig. 5e) and electron–phonon coupling175,176 (Fig. 5f). These possible applications are just the beginning, and there are many unopened areas of investigation, including, for example, the use of hybrid circuit QED for the detection of spin–charge separation in Luttinger liquids177,178,179 and as THz probes of topological phases of matter180.

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## Acknowledgements

Supported by Army Research Office grant W911NF-15-1-0149, DARPA grant no. D18AC0025 and the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4535.

## Author information

G.B, X.M. and J.R.P. researched data for the article and discussed the content. All authors contributed to the writing and editing of the manuscript.

Correspondence to Jason R. Petta.

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