Abstract
Light–matter interactions at the singleparticle 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 shortrange 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 strongcoupling 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 condensedmatter 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 microwavefrequency 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 longrange qubit interactions that are mediated by microwavefrequency 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 cavity^{1,2,3,4}. 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 condensedmatter systems using selfassembled quantum dots (QDs) confined in photonic cavities^{6,7,8} or by placing a superconducting qubit inside a microwave cavity^{9,10}. 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 gatedefined semiconductor QDs, using either charge or spin degrees of freedom to mimic the states of an atom^{11,12,13}.
There are many reasons to study cavity QED in the context of condensedmatter 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 photons^{10}, two experiments showed that two spatially separated superconducting qubits can be coupled via a cavity^{14,15}. The resulting subject of circuit QED has now expanded into a field of its own. Prominent advances include demonstrations of multiqubit entanglement^{16,17,18,19,20}, readout of quantum states^{21,22,23}, generation of nonclassical light^{24,25,26,27}, development of error correction based on Schrödinger cat states^{28,29,30}, quantum feedback^{31,32,33} and measurements of quantum trajectories^{34}. Fundamentally, experiments involving superconducting quantum devices take advantage of a macroscopic superconducting condensate that is protected by an energy gap Δ_{BCS} (for example, Δ_{BCS} ~ 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 intriguing^{35,36}, as spin coherence times can exceed seconds in some solidstate systems^{37,38,39}.
In this Review, we describe dramatic developments in the area of hybridcircuit QED, in which gatedefined QDs are coupled to superconducting cavities in a ‘super–semi’ device architecture, including recent demonstrations of strongcoupling physics with single charges and spins confined in semiconductor QDs^{40,41,42,43,44}. 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 interaction^{40,41}. A combination of electric dipole coupling and spin–orbit coupling enables coherent spin–photon interactions, which we discuss next^{42,43,44}. Lastly, we give examples of how semiconductorcircuit 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 twolevel 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 atoms^{45}. These results were eventually extended to the visible spectral range^{46,47}.
Cavity QED can also be implemented using a wide variety of solidstate systems (Fig. 1b). Colour centres in diamond, such as nitrogen and silicon vacancy centres, have spinfull ground states and narrow, spinselective microwave and optical transitions, which allows the realization of cavity QED with integrated photonic structures^{48,49}. 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 fields^{50,51,52} and the spin degree of freedom with magnetic fields^{53} and the exchange interaction through changes to the confining potential^{54}. Superconducting circuits combine a capacitance C with a Josephson inductance L_{J} to create a quantum system with an anharmonic energylevel spectrum^{55,56,57,58,59,60}; cavity QED experiments involving superconducting quantum devices are reviewed in refs^{61,62}. In this Review, we focus on experiments involving semiconductor DQDs^{63,64}, whose electrical tunability opens the door to cavity QED using longlived spin states.
Charge–photon interaction
A single electron trapped in a DQD forms a fully tunable, twolevel 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 g_{c}. 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 E_{0} 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 photons^{47}. For the much larger quantumdot devices, typical energy scales are on the order of 20–40 µeV, so it becomes convenient to use superconducting resonators to trap microwavefrequency photons (1 GHz ~ 4.2 µeV). Cavities are never perfect and can have internal losses, described by a decay rate κ_{int}, and losses through the cavity ports, κ_{1} and κ_{2}. 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 a_{in} and the signal a_{out} exiting port 2 is measured.
When a DQD is placed inside a superconducting microwave resonator, the electric field E_{res} inside the resonator tilts the energy landscape and the difference ε between the energy levels of the left and right dots becomes ε + eE_{res}d. Because d is much smaller than the wavelength of the electromagnetic waves inside the resonator, we can apply the electric dipole approximation and consider E_{res} constant within the entire volume of the DQD. The quantized electricfield 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 E_{res} = E_{0}(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 = H_{0} + H_{int}, with H_{int} = g_{c}(a + a^{†})τ_{z} in units in which \(\hbar =1\), with the chargecavity coupling g_{c} = eE_{0}d and the quantum operator τ_{z} 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, edτ_{z}, 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 g_{c} is a weak perturbation compared with the bare energy scale of the qubit Ω. In this operating regime, it is convenient to first diagonalize H_{0} and then transform H_{int} into the eigenbasis of H_{0}. Transforming into a frame rotating with the probe field frequency and neglecting fast oscillating terms within the rotatingwave 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 \(\frac{\varDelta }{2\pi }={f}_{{\rm{c}}}{f}_{{\rm{R}}}\) (f_{R} is the probe frequency; often f_{c} = f_{R} 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 H_{0}. The Heisenberg–Langevin equations of motion for the photon operator and the electron coherence operator are then found to be (Box 2)
and
where F(t) is a quantum noise operator with zero expectation value satisfying \([F(t),{F}^{\dagger }(t{\prime} )]=\gamma \delta (tt{\prime} )\), a condition needed to satisfy fluctuation–dissipation relations^{65,66,67}. 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 a_{n,in} on port n. In the stationary limit \(\langle \dot{a}\rangle =\langle {\dot{\tau }}_{}\rangle =0\), the transmission coefficient through the cavity is
with the singleelectron DQD electric susceptibility
For simplicity, we can consider a symmetric cavity without intrinsic losses (κ_{int} = 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 response^{68,69,70,71,72}. 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.
Quantumcoherent charge–photon coupling
The scale of the susceptibility, χ ∝ g_{c}, and transmission, \(A\propto 1/{g}_{{\rm{c}}}^{2}\) (assuming κ ≪ g_{c} 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 (g_{c} = eE_{0}d). For a Rydberg atom in an optical cavity, E_{0} ≈ 1 mV m^{−1} and d ≈ 100 nm, leading to couplings g_{c} roughly in the 10–100 kHz range^{73}. 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 cavities^{9,10}. The vacuum electric field can, therefore, be several orders of magnitude stronger in superconducting resonators, which allows for qubit–resonator couplings of \(\frac{{g}_{{\rm{c}}}}{2\pi }\approx 110\,{\rm{MHz}}\) for QDs (d ≈ 100 nm) and \(\frac{{g}_{{\rm{c}}}}{2\pi }\approx 10100\,{\rm{MHz}}\) for superconducting qubits (d ≈ 1 µm). Crucially, such large values of g_{c} can easily exceed both the cavity linewidth κ and qubit decoherence rate γ. The limit in which g_{c} ≫ (γ, κ) is called the strongcoupling regime of cavity QED^{4,74}. Achieving strong coupling is significant because the qubit and photon degrees of freedom become directly entangled with each other^{4}. In addition to being of fundamental interest, this entanglement can be exploited for applications in quantum information science^{75}.
Experimental demonstrations
Hybrid quantum devices comprising gatedefined QDs coupled via their electric dipole moment to microwave cavities have been successfully demonstrated using multiple material systems, including GaAs/AlGaAs heterostructures^{41,76,77,78,79}, InAs nanowires^{67,80}, graphene^{81,82}, carbon nanotubes^{83,84} and Si/SiGe heterostructures^{40,42,43,85}. The microwave cavity is often realized as a superconducting coplanar waveguide resonator^{9,10,14,24,86} (left panel of Fig. 2a). To maximize the quality factor of the cavity and the chance of reaching the strongcoupling regime, each gate line leading to the DQD is sometimes filtered by an onchip, lowpass LCfilter to suppress photon leakage from the cavity^{85}.
A simplified schematic of the hybrid device is depicted in the middle panel of Fig. 2a. At the fundamental resonance frequency f_{c}, the vacuum fluctuation of the cavity generates a halfwavelength λ/2 electromagnetic standing wave^{10,87}. At a voltage antinode of the standing wave, a delocalized electron occupying the molecular bonding and antibonding states of the DQD^{50,51,63} couples to the electric field of the cavity via the electric dipole interaction^{67,76}. A circuit representation of the device is shown in the right panel of Fig. 2a. Here, the cavity is modelled as a parallel LCoscillator having an effective inductance L_{c}, effective capacitance C_{c} and resonance frequency \({f}_{{\rm{c}}}=\frac{\sqrt{\frac{1}{{L}_{{\rm{c}}}{C}_{{\rm{c}}}}}}{2\pi }\) (refs^{88,89}). The DQD is mutually coupled via a capacitance C_{m} and dot 2 is capacitively coupled to the cavity via C_{g}. The system is connected to an input port via capacitor C_{1} and to an output port via capacitor C_{2}, allowing for measurements of the cavity transmission amplitude A = a_{2,out}/a_{1,in} and phase φ = −arg(a_{2,out}/a_{1,in}) using homodyne or heterodyne detection techniques^{10,21,90}.
Because the charge–photon coupling rate g_{c} scales linearly with \(\sqrt{{Z}_{{\rm{c}}}}\), where \({Z}_{{\rm{c}}}=\sqrt{{L}_{{\rm{c}}}/{C}_{{\rm{c}}}}\) is the characteristic impedance of the cavity^{9,58}, it is desirable to increase Z_{c} beyond the range 20–200 Ω, which is the typical limit of coplanar waveguide cavities^{89}. 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 L_{c} ≈ 1.5 kΩ by virtue of the large Josephson inductance of each SQUID^{41}. Another approach, discussed in the next section, utilizes the large kinetic inductance of a nanowire made from NbTiN^{43,91}.
Detailed scanning electron micrographs of cavitycoupled 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 heterostructure^{76,85}. For InAs nanowires, graphene and carbon nanotubes, the host material is transferred to a Si substrate before the patterning of gate electrodes^{67,81,92}. To maximize g_{c}, an electrode is often galvanically connected to the centre pin of the superconducting cavity^{40,67,76,77,81}.
Strong charge–photon coupling
Achieving the strongcoupling regime for DQD charge qubits is generally challenging owing to their large decoherence rates γ_{c}, which commonly fall between a few hundred MHz and several GHz^{50,52,67,76,77,78,81,84,93,94,95}. These values often exceed the coherent charge–photon coupling rate g_{c} by one or more orders of magnitude^{67,76,77,78,79,81,84,94,95}. Therefore, a significant reduction in γ_{c} or a significant increase in g_{c} is needed to access the strongcoupling regime g_{c} > (γ_{c}, κ). Both approaches have recently been successful in two experiments that we discuss here^{40,41}.
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 QDs^{96,97,98}. Chargestate detection may also be performed by measuring the transmission properties of the cavity, which are sensitive to the tunnelratedependent complex admittance of the QDs^{68,69,72}. An example is shown in Fig. 3a: here, the cavity transmission amplitude A/A_{0} (A_{0} is a normalization constant) at a fixed drive frequency f = f_{c} is measured as a function of gate voltages V_{P1} and V_{P2} (see Si/SiGe in Fig. 2b), which control the chemical potentials of dot 1 and dot 2, respectively. The chargestability diagram characteristic of a fewelectron DQD is clearly visible^{63}. (N_{1},N_{2}) denotes a charge state with N_{1} electrons in dot 1 and N_{2} electrons in dot 2. To determine g_{c}, A/A_{0} is measured around an interdot charge transition (N_{1} + 1,N_{2}) − (N_{1},N_{2} + 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 = f_{c}. At these detunings, the charge qubit is strongly hybridized with the cavity photons^{40,67,76}. Detailed fitting of the response A(ε)/A_{0} to input–output theory allows the charge–photon coupling rate g_{c} to be extracted from this type of measurement^{40,67,76}.
The hallmark of the strongcoupling 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 frequency^{6,7,10,47}. The data must be carefully acquired with a weak drive tone, such that the intracavity photon number n_{c} is much smaller than 1 (ref.^{10}). The measurement of A/A_{0} as a function of f and ε, taken with a Sibased DQD tuned to 2t_{c}/h = f_{c}, is shown in the topleft panel of Fig. 3b (ref.^{40}). With ε = 6 µeV, the cavity transmission spectrum exhibits a single peak (bottomleft panel of Fig. 3b). Here, the qubit–photon frequency detuning is large (white dashed lines) and the full width at half maximum of A^{2}/A_{0}^{2} gives a bare cavity loss rate κ/2π = 1.0 MHz. At ε = 0 µeV, the DQD splitting is Ω = 2t_{c} = hf_{c} 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 (bottomleft panel of Fig. 3b), where the two resonance modes are separated by a vacuum Rabi frequency 2g_{c}/2π = 13.4 MHz. The topright panel of Fig. 3b shows A/A_{0} as a function of f and ε measured with 2t_{c}/h < f_{c}: a pair of avoided crossings are observed when Ω/h = f_{c} 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 Sibased device, a conclusion further supported by a charge decoherence rate γ_{c}/2π = 2.6 MHz that was independently determined using microwave spectroscopy in the dispersive regime^{40,99}. The charge decoherence rate of this device is about two to three orders of magnitude lower than that of typical DQD charge qubits^{50,52,67,76,77,78,79,81,84,93}. Environmental factors leading to charge noise are the subject of ongoing investigation. It is believed that charge noise originates from fluctuating twolevel 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 noise^{100}. Finally, the sensitivity to charge noise might be reduced by engineering optimal gate lever arms^{101} or designing qubits with ‘sweet spots’ that are firstorderinsensitive to charge noise^{56,102,103,104,105,106}.
Strong charge–photon coupling has also been achieved in GaAs DQD devices^{41}. These devices implement a novel frequencytunable 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 resonators^{10}. Owing to the higher impedance of the microwave cavity, a large charge–photon coupling rate g_{c}/2π = 119 MHz is obtained (bottomright panel of Fig. 3b), which allows the strongcoupling regime to be accessed despite comparatively large values of γ_{c}/2π = 40 MHz and κ/2π = 12 MHz. These experiments have been extended to include coherent chargestate readout and control^{107}, as well as cavitymediated interactions between two GaAs charge qubits^{108}. Vacuum Rabi splitting has also been observed in carbon nanotubes coupled to a cavity mode, in which a twotransition scheme involving the K/K′ valleys was invoked to fit the data^{101}.
Spin–photon coupling
The quantum coherence of the spin½ of individual electrons in QDs or defects in silicon typically lasts between tens of microseconds and several milliseconds^{109,110,111,112} and can, in some cases, even approach a second^{37}, whereas nuclear spin coherence times approach 1 minute^{113}. Spin is, therefore, the primary choice as a qubit for quantum information processing in semiconductors^{35,36}. Because the exchange interaction is short ranged^{54}, 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 multielectron spin qubits, the Pauli exclusion principle provides a way to couple orbital and spin degrees of freedom^{12,114,115} (a recent experiment using this method has attained strong spinqubit–photon coupling^{44}). One mechanism that works for single electron spins is the natural builtin spin–orbit coupling due to relativistic effects, which is sizeable in a number of semiconductor materials^{13,116,117}. The intrinsic spin–orbit coupling may work particularly well for holes in the valence band of some semiconductors^{118}. Without relying on such intrinsic effects, one can engineer a spin–electric interaction using controlled magnetic fields, either timedependent fields that induce electron spin resonance^{11,119,120,121} or static but spatially varying fields produced by an onchip microscale ferromagnet^{42,43,111,122,123,124,125,126,127}.
In the case of a static magnetic field gradient \({\nabla }_{x}B\) produced by a micromagnet, an applied electric field E_{ac} shifts the electron position in a single QD by
where E_{orb} and a_{0} denote the energylevel 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 E_{res} = E_{0}(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 g_{s} in a DQD can be much larger and more controllable than in a single QD^{128}. 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 field^{122}. Because d > a_{0} and Ω ≪ E_{orb}, g_{s} is much larger in a DQD than in a single QD^{128,129}.
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 \( (\uparrow ,0)\rangle , (\downarrow ,0)\rangle , (0,\uparrow )\rangle , (0,\downarrow )\rangle \)
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 B_{z} pointing in the zdirection leads to an energy splitting between the spinup and spindown 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 ΔB_{x} 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 + eE_{res}d. In this way, we obtain the Hamiltonian H = H_{0} + H_{int}, as discussed above. The four relevant energy levels \( n\rangle \) of the DQD are found by diagonalizing the matrix H_{0}, whereas the electric dipole transition matrix elements d_{nm} can be determined by transforming H_{int} into the eigenbasis of H_{0},
For the understanding of the most important mechanisms for the spin–photon interaction, it is sufficient to consider an effective twolevel model. Introducing the rotatingwave 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 σ_{z} act on the lowenergy hybridized spin states, and \({\Delta }_{{\rm{s}}}={B}_{z}h{f}_{{\rm{c}}}\) is the detuning of the spin Zeeman splitting B_{z} from the photon energy hf_{c}. 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 = 2t_{c} − hf_{c} can be controlled by adjusting the DQD tunnel coupling t_{c}. Although this twolevel 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 threelevel model to be explained^{122}.
Strong spin–photon coupling
Compared with charge–photon coupling, reaching the strongcoupling regime of spin–photon interaction faces a distinct challenge: the direct magneticdipole coupling rate g_{s} between a single electron spin and a single photon is mostly limited to 10–500 Hz, which is too slow to overcome singlespin dephasing rates or cavityloss rates^{130,131,132,133,134,135}. As such, a robust scheme for spin–charge hybridization is necessary to increase g_{s} to the MHz range, where strong coupling becomes feasible^{11,12,13,114,115,122,127,128,129,136,137}. At the same time, a low level of charge noise is required of the device because spin–charge hybridization subjects the electron spin to chargenoiseinduced spin dephasing. A cavitycoupled carbon nanotube DQD has been used to hybridize spin and charge via ferromagnetic leads to achieve g_{s}/2π = 1.3 MHz but remained in the weakcoupling regime, owing to a relatively large spin decoherence rate γ_{s}/2π = 2.5 MHz^{138}. More recently, two experiments using Sibased DQDs have successfully attained the strongcoupling regime between a single spin and a single photon^{42,43}. In this section, we review these experiments, as well as a separate experiment that coupled a threeelectron spin state in a GaAs triple QD to a single photon^{44}.
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 g_{s} and induced spin decoherence rate γ_{s} on the degree of spin–charge hybridization controlled by the ratio of magnetic field gradient ΔB_{x} and the controllable energy detuning δ = 2t_{c} − hf_{c}: 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 discuss^{42} is illustrated in Fig. 4a. The device is a gatedefined 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 coupling^{40} but including a micronsized Co magnet on top of the DQD gate electrodes^{109,125,139}. 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 freedom^{13,122,127,128,129}. 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 singlespin qubit E_{Z}/h = gµ_{B}B_{tot}/h is tuned into resonance with the cavity by changing the external magnetic field B_{z}^{ext}. Here, E_{Z} is the Zeeman energy, g the gfactor of the electron and µ_{B} the Bohr magneton. B_{tot} 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 B_{z}^{ext}, taken at ε = 0, is shown in Fig.4c. A clear avoided crossing is observed around B_{z}^{ext} = 92.2 mT, where the resonance condition E_{Z}/h = f_{c} is met. A vacuum Rabi splitting 2g_{s}/2π = 11.0 MHz is obtained, indicating strong coupling between the single electron spin and a cavity photon. The spin–photon coupling rate g_{s}/2π = 5.5 MHz exceeds direct magneticdipole coupling rates by four to five orders of magnitude^{130,131,132,133,134,135}. A second experiment, also involving a DQD defined on a Si/SiGe heterostructure, used a highimpedance cavity design composed of a thin NbTiN nanowire with a large kinetic inductance^{43}. 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 onchip micromagnet include the use of materials with strong spin–orbit coupling, such as holes in germanium^{118,140}, or the use of the Pauli exclusion principle in multispin qubits, such as the resonantexchange qubit^{115}. The resonantexchange qubit, like other exchangeonly qubits, uses a degenerate spin subspace (decoherencefree subspace) of three or more electrons in which allelectrical control is possible^{114,141,142,143,144}. The idea is based on the availability of two qubit states with identical spin quantum numbers (for example, S = 1/2 and S_{z} = +1/2), between which spinconserving, electrically controlled exchange interactions can operate^{145}. An experiment with a resonantexchange qubit formed in a triple QD in GaAs filled with three electrons and embedded into a NbTiN highimpedance cavity demonstrated a coupling between the resonantexchange qubit and cavity photons of g_{s}/2π = 31 MHz^{44}. With a qubit decoherence rate of γ_{s}/2π = 20 MHz and a cavity decay rate of κ/2π = 47 MHz, this experiment achieved the strongcoupling regime, resolving the two vacuum Rabi peaks, because 2g_{s} > κ/2 + γ_{s}. The resonantexchange qubit offers an extended tunability of g_{s} and γ_{s} via two electrostatic control parameters that control the dipolar (linear tilt) and quadrupolar (centre dot energy) component of the tripledot potential. By tuning these two parameters, the resonantexchange qubit can be operated at a sweet spot where the qubit energy splitting becomes insensitive to small electrostatic fluctuations due to external charge noise^{145}.
To apply spin–photon cavity QED devices to quantuminformationprocessing tasks, such as the implementation of longrange twoqubit gates, it is necessary to rapidly switch on and off the spin–photon coupling rate g_{s}. This allows the spin qubit to be manipulated in an isolated state (g_{s} ≈ 0), in which it is protected from cavityinduced Purcell decay^{133,146,147,148}, and read out via the cavity when the coupling is back on. One way to tune g_{s} is by tilting the DQD potential (Fig. 4c). As ε is increased from zero, a strong decrease of spin–photon coupling from g_{s}/2π = 5.5 MHz (ε = 0) to g_{s}/2π ≪ 1 MHz (ε = 40 µeV) is observed. This change is due to the fact that, with ε ≫ t_{c}, 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 pm^{42}. 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 longdistance spin–spin coupling mediated by microwave photons^{149,150}. Resonant interactions between two spins separated by 4 mm have recently been achieved using circuit QED^{151}. With device improvements, this coupling could be employed to perform twoqubit gates between spins separated by several millimetres. One should keep in mind that 1 mm is a very long distance compared with the 80nm separation of nearestneighbour spin qubits in Si^{152}. Their small footprint on a semiconductor chip is one characteristic feature of semiconductor spin qubits, which makes them strong contenders for a scalable quantuminformationprocessing platform. In addition to enabling the entanglement of distant spin qubits, nonlocal twoqubit gates may facilitate quantum error correction in the framework of a faulttolerant quantumcomputing architecture. The possibility of creating a network of spin qubits with engineered coupling may be very useful for realizing a surface code^{153}. Moreover, the possibility of creating a network of spin qubits with coupling geometries ranging from local to ‘alltoall’ opens interesting perspectives for the quantum simulation of interacting quantum manybody systems^{154,155}.
Spin–photon coupling has important implications beyond the generation of longrange 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 qubit^{42,67} and lays the foundation for the development of quantum nondemolition^{156,157} and singleshot readout methods^{158}. Because the spin–photon coupling rate g_{s} depends strongly on the detuning ε, electrically switching on the cavity coupling of each spin qubit^{40} may allow for selective readout in large arrays of spin qubits (Fig. 5b). Moreover, the superconducting qubit community has adopted the use of frequency multiplexing^{23,159} for quantumstate readout of multiqubit devices. A similar approach could be adopted for spins^{160}.
Looking beyond spin qubits, the nascent field of hybridcircuit QED could have a major impact on condensedmatter physics as a whole. Cavity measurements have been proposed to investigate Majorana modes^{161} and provide an alternative to the somewhat ambiguous measurements of zerobias conductance peaks^{162,163,164,165}. It has even been suggested that microwave cavities could be used to detect the braiding of Majorana fermions^{166} (Fig. 5c). Scanningprobe versions of superconducting cavities (Fig. 5d) could be used to probe valley physics in silicon^{167,168,169,170} and perhaps, more broadly, to investigate 2D quantum materials^{171}. Lastly, superconducting cavities have been shown to provide an alternative means to investigate Kondo physics^{172,173,174} (Fig. 5e) and electron–phonon coupling^{175,176} (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 liquids^{177,178,179} and as THz probes of topological phases of matter^{180}.
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Acknowledgements
Supported by Army Research Office grant W911NF1510149, DARPA grant no. D18AC0025 and the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4535.
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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.
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Burkard, G., Gullans, M.J., Mi, X. et al. Superconductor–semiconductor hybridcircuit quantum electrodynamics. Nat Rev Phys 2, 129–140 (2020). https://doi.org/10.1038/s4225401901352
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