Cavity QED with hybrid nanocircuits: from atomic-like physics to condensed matter phenomena

It is instructive to make a comparison of the physical ingredients involved in cavity, circuit and mesoscopic QED to identify the specificities of the light–nanocircuit interaction. Cavity QED focuses on the interactions between electrons in the atomic orbitals of a flying atom and the photons trapped inside a superconducting mirror cavity (see figure 3(a)). In these experiments, the effect of the cavity magnetic field on the atom can be disregarded for weak microwave amplitudes [75]. In most situations, one can consider that the cavity electric field is constant at the scale of the atom, because the atom is very small in comparison with the cavity. In this case, the light–matter interaction can be expressed quantum mechanically as $\hat{H} _{d}=\overrightarrow{\hat{E}}_{0}.\overrightarrow{\hat{d}}$, with $\overrightarrow{\hat{E}}_{0}$ being the quantized cavity electric field and $\overrightarrow{\hat{d}}= {\sum_{i}} q_{i}\overrightarrow{\hat{r}}_{i}$ the dipole associated to the atomic charges $q_{i}$ at position $\overrightarrow{\hat{r}}_{i}$. Note that this charge distribution includes not only electrons but also the ions of the atomic nucleus. However, the explicit description of these ions essentially grants the electroneutrality condition ${\sum_{i}} q_{i}=0$, which simplifies calculations. Cavity QED mainly focuses on electronic transitions between the atomic orbitals, induced by the cavity electric field.

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In circuit QED, the concept of orbital degree of freedom is no longer relevant, because only macroscopic collective degrees of freedom matter, due to the rigidity of the superconducting phase. For instance, in a Cooper pair box, which is a small superconducting island coupled to a superconducting reservoir through a Josephson junction, only the total excess number $\hat{n}$ of electrons on the island matters (see figure 3(b)). A second important difference from atomic cavity QED is that the cavity field cannot be considered as homogeneous on the scale of the superconducting quantum bit. Indeed, its spatial profile is strongly modified by the presence of the superconducting elements which tend to expel it. Figure 3(b) illustrates this situation in the case of a Cooper pair box embedded in a coplanar microwave cavity. The cavity electric field concentrates in capacitive areas between neighboring metallic elements, as represented by the darker pink areas. This capacitive coupling scheme is often described with a lumped element circuit model which discretizes the device into nodes with uniform photonic potential and superconducting phase, connected by capacitors, inductors or Josephson junctions. In the simplest picture, the cavity is modeled as a distributed (L,C) line, and the superconducting island in figure 3(b) corresponds to a single node contacted through capacitors and a Josephson junction to the rest of the circuit (see figure 2 of [76]). The cavity electric field shifts the island potential due to the presence of the capacitive coupling between the dot island and the cavity central conductor. Recently, more sophisticated lumped element circuit models have been introduced for a more realistic description of circuit QED devices [77, 78]. Note that Josephson circuits with superconducting loops may equally couple to the cavity magnetic field (not represented in figure 3(b)).

Mesoscopic QED represents an intermediate situation between cavity and circuit QED (see figure 3(c)). Indeed, due to the existence of small confined nanoconductor areas (like for instance quantum dots), there exist discrete electronic orbital levels which recall the atomic orbitals of cavity QED. However, the cavity field is strongly inhomogeneous on the scale of the nanocircuit, which rather recalls circuit QED. For instance, one can use ac gates often connected directly (see figure 3(c)) or sometimes capacitively (see figure 1) to the cavity central conductor to reinforce locally the coupling between the cavity electric field and the electrons in one small part of the nanocircuit. The area between the ac gate and the nanoconductor in figure 3(c) concentrates the electric field, as represented by the darker pink shade. This provides a capacitive coupling between the cavity central conductor and the nanoconductor. Field screening effects represent another source of field inhomogeneity. First, the cavity fields are confined between the superconducting cavity conductors, represented in blue in figure 3(c). This effect naturally goes together with a screening of the fields inside the cavity conductors. Second, the fermionic reservoirs in the nanocircuit can at least partially screen the cavity fields. These screening effects are due to electronic plasmonic modes, which are only implicitly taken into account in the usual descriptions of circuit QED, through current conservation. In mesoscopic QED, it is not a priori obvious to take into account plasmonic modes because one must take into account that fermionic reservoirs simultaneously host plasmonic modes and fermionic quasiparticle modes which cause quantum transport effects in the nanocircuit. These quasiparticle modes are coupled to the localized discrete electronic orbitals inside the nanoconductors through tunnel junctions. Tunneling is also at the heart of the Josephson coupling in superconducting circuits. However, tunneling from a normal metal reservoir involves the numerous quasiparticle modes in a reservoir on top of the nanoconductor levels. Therefore, the study of mesoscopic QED devices requires a description which combines physical ingredients from both cavity and circuit QED. In the following, we will follow the approach proposed by [80].

In order to take into account both quasiparticle tunneling and plasmonic screening in a minimal way, one can assume that the plasmonic screening charges on the cavity conductors or fermionic reservoirs have a frequency which is much higher than all the other relevant frequencies in the device. In particular, we assume that the plasmonic frequency is much higher than the tunnel rate between a reservoir and a nanoconductor, or than the tunnel hopping constant between two dots. Under this assumption, one can decompose the nanocircuit of figure 4(a) heuristically into two parts: an effective orbital nanocircuit represented in black in figures 4(b) and (d), in which tunneling physics prevails, and an effective plasmonic circuit made out of perfect conductors, represented in blue in figures 4(b) and (d). Below, we discuss this decomposition in more detail.

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In the black circuit of figure 4(d), the electronic orbital levels in the various circuit elements are connected through tunnel junctions (striped rectangles). In order to allow current conservation, the 'orbital' reservoirs have to be connected to the black voltage source and the ground, through wirings which necessarily host plasmonic modes. However, one can assume that these plasmonic modes do not have a significant influence on the value of the cavity field near the nanocircuit. The role of the black voltage source in figure 4(b) is essentially to ensure that the electronic levels in the nearby reservoir are filled up to the Fermi level plus a shift caused by the applied bias voltage.

The blue circuit of figure 4(c) is electrically disconnected from the black circuit. It represents the physical host of the screening charges which propagate together with the cavity photons. Some of these blue conductors directly correspond to the cavity central conductor and ground planes, or to the nanocircuit dc gates. The other conductors correspond to the nanocircuit reservoirs, and account at least qualitatively for the local screening of the cavity field in these reservoirs. This produces a renormalization of the cavity field, which can affect the coupling between the cavity photons and the quasiparticles in the black orbital circuit. Importantly, the blue conductors are connected to dc sources, drain, and gate voltage sources, similarly to the initial circuit of figure 4(a). This enables one to make a complete description of the cavity fields, including dc field contributions (this description will be implemented mathematically in section 3.1.3). Note that when a tunneling event occurs in the nanocircuit, displacement currents occur in order to effect the reorganization of the screening charges throughout the whole mesoscopic QED device. In the model of figure 4(b), these displacement currents are also carried by the blue plasmonic circuit. Importantly, the model of figure 4(b), which separates physically the plasmonic and fermionic modes of the nanocircuit, is only an effective model which we will use in next section to justify the form of the mesoscopic QED Hamiltonian. In practice, the plasmonic modes and tunneling quasiparticles are of course not spatially separated. To calculate in a realistic way how the spatial profile of the cavity field is renormalized by the screening charges of the nanocircuit, one should use microwave simulation software (which disregards tunneling physics). On the basis of the heuristic model discussed above, we will introduce in the next section a description of mesoscopic QED in which plasmons are not described explicitly. This approach is allowed by the large separation in the characteristic timescales associated with plasmons and tunneling.

In order to exploit the effective model of figure 4(b), we will first quantize the electromagnetic field outside the blue perfect conductors. Ultrafast plasmonic modes on these conductors will not be treated explicitly but included through boundary conditions on the blue conductors. These boundary conditions, which are disregarded in most descriptions of cavity QED, make the quantization of the electromagnetic field non-trivial. Some of the blue conductors are biased with a voltage $V_{i}$, such as for instance the electrostatic gates which are used to tune the positions of the energy levels in the nanocircuit. Some others are left floating with a constant charge $Q_{i}$, such as the central conductor of a coplanar waveguide cavity. To take this into account, we will decompose the total electric field $\vec {E}(\overrightarrow{r}, t)$ outside the blue conductors into a longitudinal component $\vec{E}_{\parallel}(\overrightarrow{r}, t)$, which has a finite gradient but no rotational, a transverse component $\vec{E}_{\perp }(\overrightarrow{r}, t)$, which has a finite rotational but no gradient, and a harmonic component $\vec{E}_{\rm harm}(\overrightarrow{r})$, which has none, such that:

$$\begin{align} \displaystyle \vec{E}(\overrightarrow{r},t)=\vec{E}_{\perp}(\overrightarrow{r},t)+\vec {E}_{\parallel}(\overrightarrow{r},t)+\vec{E}_{\rm harm}(\overrightarrow{r}). \label{y0} \nonumber \end{align} \tag{ 4 }$$

Then, it is very convenient to use the Coulomb gauge, defined by $\overrightarrow{\nabla}_{\overrightarrow{r}}.\vec{A}(\overrightarrow{r}, t)=0$. In this case, both the magnetic field in the system and the transverse electric field can be expressed in terms of the vector potential as

$$\begin{align} \displaystyle \overrightarrow{E}_{\perp}(\overrightarrow{r},t)=-\partial\vec{A} (\overrightarrow{r},t)/\partial t \label{y1} \nonumber \end{align} \tag{ 5 }$$

and

$$\begin{align} \displaystyle \vec{B}=\overrightarrow{\nabla}_{\overrightarrow{r}}\wedge\vec{A} (\overrightarrow{r},t). \label{y2} \nonumber \end{align} \tag{ 6 }$$

We now define scalar potentials $U_{\parallel}$ and $\Phi_{\rm harm}$ from the equations

$$\begin{align} \displaystyle \vec{E}_{\parallel}(\overrightarrow{r},t)=-\overrightarrow{\nabla }_{\overrightarrow{r}}U_{\parallel}(\overrightarrow{r},t) \label{y3} \nonumber \end{align} \tag{ 7 }$$

and

$$\begin{align} \displaystyle \vec{E}_{\rm harm}(\vec{r})=-\vec{\nabla}.\Phi_{\rm harm}(\vec{r}). \label{y4} \nonumber \end{align} \tag{ 8 }$$

By combining the Maxwell equations with equations (4-8), one finds that $\Phi_{\rm harm}(\vec{r})$, $U_{\parallel}(\overrightarrow{r}, t)$ and $\vec{A}(\overrightarrow{r}, t)$ are set by separate equations. First, $\Phi_{\rm harm}$ is a static field set by the boundary conditions on the blue perfect conductors. More precisely, it fulfills the Poisson equation

$$\begin{align} \displaystyle \Delta_{\vec{r}}\Phi_{\rm harm}(\vec{r})=0 \nonumber \end{align} \tag{ 9 }$$

with boundary conditions corresponding to having charges $Q_{i}$ or voltages $V_{i}$ on the blue conductors (see [80] for details). Second, the potential $U_{\parallel}$ is instantaneously set by the charge distribution $\{e_{\alpha}, \overrightarrow{q}_{\alpha}\}$ in the black nanocircuit, i.e.

$$\begin{align} \displaystyle U_{\parallel}(\overrightarrow{r},t)= {\sum_{\alpha}} e_{\alpha}G(\vec{r},\overrightarrow{q}_{\alpha}). \label{pooisson} \nonumber \end{align} \tag{ 10 }$$

The function G above is the solution of the Poisson equation $\Delta _{\vec{r}}G(\vec{r}, \overrightarrow{r}^{\prime})=-\delta(\vec{r} -\overrightarrow{r}^{\prime})/\varepsilon_{0}$ with boundary conditions $Q_{i}=0$ and $V_{i}=0$ on the blue conductors. In other words, $G(\vec {r}, \overrightarrow{r}^{\prime})$ describes the electrostatic interaction between two charges at points $\vec{r}$ and $\overrightarrow{r}^{\prime}$, renormalized by the screening charges on the blue conductors. Finally, $\vec{A}$ follows the propagation equation

$$\begin{align} \displaystyle \left(\Delta_{\overrightarrow{r}}-\frac{1}{c^{2}}\frac{\partial^{2} }{\partial t^{2}}\right) \vec{A}(\overrightarrow{r},t)=-\mu_{0} \overrightarrow{j}_{\perp}(\overrightarrow{r},t) \label{field} \nonumber \end{align} \tag{ 11 }$$

with the same boundary conditions as equation (10). One can conclude that $\vec{A}$ corresponds to the photonic field which can be quantized. Interestingly, this picture is similar to the picture used for cavity QED, up to two differences. First, in cavity QED, the harmonic potential is generally omitted because the boundary conditions on the cavity conductors are not explicitly treated [79]. Second, in the simplest descriptions of cavity QED, the longitudinal potential $U_{\parallel}$ corresponds to the potential caused by the atom in vacuum [75], i.e. $U_{\parallel}(\overrightarrow{r}, t)= {\sum_{\alpha}} e_{\alpha}/4\pi\varepsilon_{0}\left\vert \vec{r}-\vec{q}_{\alpha}\right\vert$. In the case of mesoscopic QED, this potential is dressed by the screening charges on the blue conductors. This effect is taken into account by the G function in equation (10).

From the above section, the independent degrees of freedom in the mesoscopic QED device are the value of the cavity potential $\vec{A}$, and the position $\overrightarrow{q}_{\alpha}$ of the charges $e_{\alpha}$ in the black nanocircuit. In principle, the charge distribution $\{e_{\alpha}, \overrightarrow{q}_{\alpha}\}$ includes the crystalline background ${\bf C}$ and electrons from the valence bands ${\bf V}$ of the black nanocircuit. However, assuming that these charges are off resonant with the cavity, one can describe them with the mean field approximation. This requires taking into account that conduction electrons feel a confinement potential $V_{\rm conf}(\overrightarrow{r})=\left\langle {\sum_{\alpha\in{\bf C, V}}} e_{\alpha_{i}}G(\vec{r}_{\alpha_{i}}, \overrightarrow{q}_{\alpha_{i} })\right\rangle$, with $\left\langle {}\right\rangle$ a statistical average on the state of the charges of ${\bf C}$ and ${\bf V}$. This potential accounts for the transverse confinement of the conduction electrons inside the nanocircuit, and the tunnel barriers between the different nanocircuit elements. Then, by using a quantization procedure analogous to that used in cavity QED, one can express the mesoscopic QED Hamiltonian as (see [80] for details):

$$\begin{align} \displaystyle \hat{H}_{\rm tot} & =\int {\rm d}^{3}r\hat{\psi}^{\dagger}(\vec{r})\hat{h}_{\rho} (\vec{r})\hat{\psi}(\vec{r})+\hat{H}_{\rm Coul}+\omega_{0}\hat{a}^{\dagger}\hat {a}\label{1}\nonumber \\ & +\int {\rm d}^{3}r\left(\Delta(\vec{r}){\rm e}^{2\Phi(\vec{r})}\hat{\psi}_{\uparrow }^{\dagger}(\vec{r})\hat{\psi}_{\downarrow}^{\dagger}(\vec{r})+{\rm H.c.}\right) \nonumber \end{align} \tag{ 12 }$$

with

$$\begin{align} \displaystyle \hat{h}_{\rho}(\vec{r})=\frac{1}{2m}\left(\frac{\overrightarrow{\nabla }_{\vec{r}}}{i}+e\overrightarrow{\hat{A}}(\vec{r})\right) ^{2}-e\Phi _{\rm harm}(\overrightarrow{r})-eV_{\rm conf}(\overrightarrow{r})~{\rm{,}} \label{202} \nonumber \end{align} \tag{ 13 }$$

$$\begin{align} \displaystyle \hat{H}_{\rm Coul}=\frac{e^{2}}{2}\int {\rm d}^{3}r{\rm d}^{3}r^{\prime}\hat{\psi}^{\dagger }(\vec{r})\hat{\psi}^{\dagger}(\vec{r}^{\prime})G(\vec{r},\vec {r}^{\prime})\hat{\psi}(\vec{r}^{\prime})\hat{\psi}(\vec {r})~{\rm{,}} \label{coul} \nonumber \end{align} \tag{ 14 }$$

and

$$\begin{align} \displaystyle \overrightarrow{\hat{A}}(\vec{r})=\mathcal{\vec{A}}(\vec{r})i(\hat{a}-\hat {a}^{\dagger})~{\rm{.}} \label{6} \nonumber \end{align} \tag{ 15 }$$

We have introduced above the field operator $\hat{\psi}^{\dagger}(\vec {r})=(\hat{\psi}_{\uparrow}^{\dagger}(\vec{r}), \hat{\psi}_{\downarrow }^{\dagger}(\vec{r}))$ associated with the creation of conduction electrons in the black nanocircuit. The potential $V_{\rm conf}(\overrightarrow{r})$ can be treated on the same footing as the harmonic potential $\Phi_{\rm harm} (\overrightarrow{r})$. The term $\hat{H}_{\rm Coul}$ describes Coulomb interactions between electrons. This term depends on the function G because Coulomb interactions between the charges α of the black nanocircuit are renormalized by the screening charges on the blue conductors. In equation (13), the vector potential $\overrightarrow{\hat{A}}(\vec{r})$ ensures the gauge invariance of the single electron term. For simplicity, equation (15) expresses $\overrightarrow{\hat{A}}(\vec{r})$ by using only one cavity mode, corresponding to the creation operator $\hat{a}$, but equations (12)-(15) can be generalized straightforwardly to the multimode case, either to take into account several cavity modes or to describe the cavity bare linewidth with a bosonic bath. The second line of equation (12) is a pairing term which describes superconducting correlations in the nanocircuit. This term must include a phase factor $\Phi(\vec{r})$ which depends on the photonic operators, in order to ensure the gauge invariance of the Hamiltonian (see next section for details).

Different types of light–matter interaction appear in Hamiltonian (12). Indeed, equation (13) contains a linear term in $\overrightarrow{\nabla }_{\vec{r}}.\overrightarrow{\hat{A}}(\vec{r})$ and a non-linear term in $\hat{A}^{2}$. It also contains the exponential of the phase factor $\Phi (\vec{r})$ which is non-linear. The effect of the non-linear terms is not negligible, in principle (see appendix B of [80] for details). Therefore, in this section, we introduce a unitary transformation of the Hamiltonian $\hat{H}_{\rm tot}$ which simplifies the form of the light–matter interaction. For simplicity, we consider nanocircuits with standard dimensions and without loops, so that one can disregard magnetic effects induced by the photons. This means that one can use $\vec{\nabla}_{\vec{r}}\wedge \overrightarrow{\hat{A}}\simeq0$ on the scale of the whole nanocircuit. This assumption is valid for all the mesoscopic QED devices studied experimentally so far, except [35]. The more general case will be discussed elsewhere. When $\vec{\nabla}_{\vec{r}}\wedge\overrightarrow{\hat {A}}\simeq0$ it is possible to define a photonic pseudo-potential $V_{\perp }(\overrightarrow{r})$ such that

$$\begin{align} \displaystyle \vec{\nabla}_{\vec{r}} \cdot V_{\perp}(\overrightarrow{r})\simeq\omega _{0}\mathcal{\vec{A}}(\vec{r}) \nonumber \end{align} \tag{ 16 }$$

and

$$\begin{align} \displaystyle \hat{\Phi}(\vec{r})=e(\hat{a}-\hat{a}^{\dagger})V_{\perp}(\overrightarrow{r})/\omega_{0}. \nonumber \end{align} \tag{ 17 }$$

Then, one can apply to Hamiltonian (12) the unitary transformation $\tilde{H}_{\rm tot}=\mathcal{U}^{\dagger}H_{\rm tot}\mathcal{U}$ with

$$\begin{align} \displaystyle \mathcal{U}=\exp\left(\frac{(\hat{a}^{\dagger}-\hat{a})}{\omega_{0}} \hat{\mathcal{V}}\right) \label{UU} \nonumber \end{align} \tag{ 18 }$$

and

$$\begin{align} \displaystyle \hat{\mathcal{V}}=-e\int {\rm d}^{3}rV_{\perp}(\overrightarrow{r})\hat{\psi }^{\dagger}(\vec{r})\hat{\psi}(\vec{r}). \nonumber \end{align} \tag{ 19 }$$

This leads to the Hamiltonian

$$\begin{align} \displaystyle \tilde{H}_{\rm tot}=H_{0}+\hat{\mathcal{V}}(\hat{a}+\hat{a}^{\dagger})+\omega_{0} \hat{a}^{\dagger}\hat{a} \label{hi} \nonumber \end{align} \tag{ 20 }$$

with

$$\begin{align} \displaystyle H_{0} & =\int {\rm d}^{3}r\hat{\psi}^{\dagger}(r)\widetilde{h}_{\mathcal{\rho} }(\vec{r})\hat{\psi}(\vec{r})+\hat{H}_{\rm Coul}\nonumber \\ & +(\hat{\mathcal{V}}^{2}/\omega_{0})+\int {\rm d}^{3}r\left(\Delta(\vec{r})\hat{\psi}_{\uparrow}^{\dagger}(\vec{r})\hat{\psi}_{\downarrow}^{\dagger }(\vec{r})+{\rm H.c.}\right) \nonumber \end{align} \tag{ 21 }$$

and

$$\begin{align} \displaystyle \widetilde{h}_{\mathcal{\rho}}(\vec{r})=-\Delta_{\overrightarrow{r}} /2m-e\Phi_{\rm harm}(\overrightarrow{r})-eV_{\rm conf}(\overrightarrow{r}). \nonumber \end{align} \tag{ 22 }$$

In the Hamiltonian of equation (20), the light–matter interaction is greatly simplified since it involves a single linear term in $\hat{\mathcal{V}} (\hat{a}+\hat{a}^{\dagger})$. Interestingly, this Hamiltonian bridges between cavity QED and circuit QED. Indeed, the dipolar electric approximation of cavity QED corresponds to a photonic potential which evolves linearly in space i.e. $V_{\perp}(\overrightarrow{r})=\overrightarrow{\hat{E}}_{0} \cdot \overrightarrow{r}$, whereas circuit QED corresponds to a constant photonic potential inside each node of the circuit model.

Since tunneling physics is at the heart of quantum transport, it is useful to re-express Hamiltonian (20) to describe tunneling explicitly. For this purpose, one needs to decompose the field operator $\hat{\psi}^{\dagger} (\vec{r})$ associated with quasiparticle modes of the black circuit on the ensemble of the creation operators $\hat{c}_{n}^{\dagger}$ for electrons in an orbital n with energy $\varepsilon_{n}$ of a given circuit element (reservoir, dot,...). At lowest order in tunneling, one can use $\hat{\psi }^{\dagger}(\vec{r})=\varphi_{n}^{\ast}\hat{c}_{n}^{\dagger}$ [81]. Then, Hamiltonian (20) directly gives

$$\begin{align} \displaystyle \hat{H}_{\rm tot}^{\rm tun}=\hat{H}_{0}^{t}+\hat{h}_{\rm int}(\hat{a}+\hat{a}^{\dagger })+\omega_{0}\hat{a}^{\dagger}\hat{a} \label{Ht} \nonumber \end{align} \tag{ 23 }$$

with

$$\begin{align} \displaystyle \hat{H}_{0}^{t}=\sum_{n}\varepsilon_{n}\hat{c}_{n}^{\dagger}\hat{c}_{n} +\sum_{n\neq n^{\prime}}(t_{n,n^{\prime}}\hat{c}_{n^{\prime}}^{\dagger}\hat {c}_{n}+{\rm H.c.}) \nonumber \end{align} \tag{ 24 }$$

$$\begin{align} \displaystyle \hat{h}_{\rm int}=\sum_{n}g_{n}\hat{c}_{n}^{\dagger}\hat{c}_{n}+\sum_{n\neq n^{\prime}}(\gamma_{n,n^{\prime}}\hat{c}_{n^{\prime}}^{\dagger}\hat{c} _{n}+{\rm H.c.})~{\rm{.}} \label{hint2} \nonumber \end{align} \tag{ 25 }$$

$$\begin{align} \displaystyle g_{n}=-e {\int} {\rm d}r^{3}\left\vert \varphi_{n}(\vec{r})\right\vert ^{2}V_{\perp}(\vec {r})~{\rm{.}} \label{jj1} \nonumber \end{align} \tag{ 26 }$$

and

$$\begin{align} \displaystyle \gamma_{n^{\prime},n}=-e {\int} {\rm d}r^{3}\varphi_{n}^{\ast}(\vec{r})\varphi_{n^{\prime}}(\vec{r})V_{\perp} (\vec{r}). \label{jj2} \nonumber \end{align} \tag{ 27 }$$

Above, $\hat{H}_{0}^{t}$ is the Anderson-like Hamiltonian of the nanocircuit, with $t_{n, n^{\prime}}$ the tunnel coupling between orbitals n and $n^{\prime}$, which is finite only if n and $n^{\prime}$ correspond to two orbitals in two different circuit elements coupled through a tunnel junction. The term $\hat{h}_{\rm int}(\hat{a}+\hat{a}^{\dagger})$ describes the interaction of the nanocircuit with the cavity. Cavity photons can have different effects. First, they can shift the energy of orbital n due to the term in $g_{n}$. In the limit where $V_{\perp}(\vec{r})$ can be considered as constant inside a given circuit element, $g_{n}$ can be considered to be the same for all the orbitals of this element (at zeroth order in tunneling). In this limit, the coupling of the cavity to the element can be seen as a capacitive coupling due to the finite capacitance between this element and the cavity central conductor. This recalls the case of a superconducting charge quantum bit in circuit QED, where the qubit island potential is modulated due to the capacitive coupling between the island and the cavity. In principle, cavity photons can also produce a direct coupling between two different orbitals of the nanocircuit, due to the term in $\gamma_{n^{\prime}, n}$. This corresponds to two physically different situations. First, there can be photo-induced transition terms between two different orbitals of the same circuit element, which recalls the orbital transitions inside an atom, which are used in cavity QED. Second, there can also be a photo-induced tunneling term between two different circuit elements. Nevertheless, the terms $\gamma_{n^{\prime}, n}$ are expected to be weak because $\varphi_{n}(\vec{r})$ and $\varphi _{n^{\prime}}(\vec{r})$ have a small matrix element in the tunnel theory.

In this review, we will mainly discuss the effects of the $g_{n}$ elements which are expected to be dominant in most mesoscopic QED devices, and have been sufficient so far to interpret experiments. For standard coplanar cavities similar to that of [26], the pseudo photonic potential $V_{\perp}(\vec{r})$ typically varies by $V_{\rm rms}\simeq1~{\rm \mu eV}=240~{\rm MHz}$ from the cavity central conductor to the ground plane, across a spatial gap of $5~{\rm \mu m}$. From one nanocircuit site to another, $V_{\perp}(\vec{r})$ can vary by a significant fraction of V_rms, especially if ac gates are fabricated to concentrate the cavity voltage drop between these sites. Consequently, the value of $g_{n}$ can strongly depend on the orbital n considered. Even for a given quantum dot, the value of $g_{n}$ can vary strongly from one orbital to the other due to variations in the spatial profile of $\varphi_{n}(\vec{r})$. For instance, in a single quantum dot with multiple gates made in a carbon nanotube, $g_{n}/2\pi$ was found to vary from $55~{\rm MHz}$ to $120~{\rm MHz}$ [37]. This could mean that $V_{\perp}(\vec{r})$ cannot be considered as homogeneous on the scale of this dot.

So far, in mesoscopic QED, most experiments have focused on the modification of the cavity microwave transmission or reflection due to the presence of the nanocircuit. To describe such a measurement, one can use the general Hamiltonian

$$\begin{align} \displaystyle \hat{H} & =\hat{H}_{0}^{t}+\hat{h}_{\rm int}(\hat{a}+\hat{a}^{\dagger})+\omega _{0}\hat{a}^{\dagger}\hat{a}+H_{\rm diss}\label{HH}\nonumber \\ & +{\rm i}\hat{a}^{\dagger}\varepsilon_{\rm in}{\rm e}^{-{\rm i}\omega_{\rm RF}t}-{\rm i}\hat{a}\varepsilon _{\rm in}^{\ast}{\rm e}^{{\rm i}\omega_{\rm RF}t}\,,\nonumber \end{align} \tag{ 28 }$$

which is a generalization of equation (23). Above, we describe cavity dissipation with a bosonic bath

$$\begin{align} \displaystyle H_{\rm diss}=\int {\rm d}\epsilon\eta\left(\omega_{\epsilon}\hat{b}_{\epsilon}^{\dagger }\hat{b}_{\epsilon}+\beta\hat{b}_{\epsilon}^{\dagger}\hat{a}+\beta^{\ast}\hat {a}^{\dagger}\hat{b}_{\epsilon}\right) \label{hdiss} \nonumber \end{align} \tag{ 29 }$$

with $\hat{b}_{\epsilon}^{\dagger}$ the creation operator for a bosonic mode at energy , and η the density of modes. For simplicity, we assume that the coupling constant β between the cavity and the bath is energy-independent. This term was not included in equation (23), but it can be added by generalizing equations (15) and (18)^3,4. We also use a drive term with amplitude $\varepsilon_{\rm in}$ which describes the effect of the continuous microwave tone which is injected at the input of the cavity. Following the discussion in section 3.1.6, we assume that the light–matter interaction is well approximated by

$$\begin{align} \displaystyle \hat{h}_{\rm int}= {\sum_{n}} g_{n}\hat{c}_{n}^{\dagger}\hat{c}_{n}. \label{hint} \nonumber \end{align} \tag{ 30 }$$

From equations (28) and (29), one has

$$\begin{align} \displaystyle \frac{\rm d}{{\rm d}t}\hat{a}={\rm i}[\hat{H},\hat{a}]=-{\rm i}\omega_{0}\hat{a}-{\rm i}\hat{h} _{\rm int}-\varepsilon_{\rm in}{\rm e}^{-{\rm i}\omega_{\rm RF}t}-\Lambda_{0}\hat{a} \label{eom} \nonumber \end{align} \tag{ 31 }$$

with $\Lambda_{0}=\pi\eta\left\vert \beta\right\vert ^{2}$ the cavity mode decay rate. Note that the full width at half maximum (FWHM) of the bare cavity transmission corresponds to the parameter $\varkappa=2\Lambda_{0}$ used in many studies. In most experiments performed so far, a large number of cavity photons has been used, i.e. $\left\langle \hat{a}^{\dagger}\hat{a}\right\rangle >10$. In this case, it is sufficient to treat $\hat{a}$ as a classical quantity i.e. $\hat{a}\simeq\left\langle \hat{a}\right\rangle =a$. In the case $\omega_{\rm RF}\simeq\omega_{0}$, one can furthermore use the resonant approximation:

$$\begin{align} \displaystyle \left\langle \hat{a}\right\rangle \simeq\bar{a}{\rm e}^{-{\rm i}\omega_{\rm RF}t}. \label{aaa} \nonumber \end{align} \tag{ 32 }$$

In the above, $\left\vert \bar{a}\right\vert ^{2}$ corresponds to the average cavity photon number $\left\langle \hat{a}^{\dagger}\hat{a}\right\rangle$. From the linear response theory one obtains

$$\begin{align} \displaystyle \left\langle \hat{c}_{n}^{\dagger}\hat{c}_{n}\right\rangle (t) & =p_{n}+ {\sum_{n^{\prime}}} g_{n^{\prime}}\chi_{n,n^{\prime}}(\omega_{\rm RF})\bar{a}{\rm e}^{-{\rm i}\omega_{\rm RF} t}\label{nn}\nonumber \\ & + {\sum_{n^{\prime}}} g_{n^{\prime}}\chi_{n,n^{\prime}}(-\omega_{\rm RF})\bar{a}^{\ast}{\rm e}^{{\rm i}\omega_{\rm RF} t}.\nonumber \end{align} \tag{ 33 }$$

Here, $\chi_{n, n^{\prime}}(\omega_{\rm RF})$ is by definition the charge susceptibility expressing how the occupation of level n responds at first order to a classical modulation of the energy of level $n^{\prime}$, in stationary conditions. We denote by $p_{n}$ the average occupation of state n for $\hat{h}_{\rm int}=0$. One has, in the framework of the linear response theory,

$$\begin{align} \displaystyle \chi_{n,n^{\prime}}(t-t^{\prime})=-{\rm i}\theta(t)\left\langle [\hat{c}_{n}^{\dagger }(t)\hat{c}_{n}(t),\hat{c}_{n^{\prime}}^{\dagger}(t^{\prime})\hat{c}_{n^{\prime} }(t^{\prime})]\right\rangle _{\hat{h}_{\rm int}=0}, \label{Chi} \nonumber \end{align} \tag{ 34 }$$

where $\left\langle {}\right\rangle _{\hat{h}_{\rm int}=0}$ denotes the statistical averaging for $g_{n}=0$ for any n. Inserting equations (32) and (33) into equation (31), and keeping only resonant terms, one gets

$$\begin{align} \displaystyle \bar{a}=\frac{\varepsilon_{\rm in}}{\omega_{\rm RF}-\omega_{0}+{\rm i}\Lambda_{0}-\Xi (\omega_{\rm RF})} \label{aa} \nonumber \end{align} \tag{ 35 }$$

with

$$\begin{align} \displaystyle \Xi(\omega_{\rm RF})= {\sum_{n,n^{\prime}}} g_{n}g_{n^{\prime}}\chi_{n,n^{\prime}}(\omega_{\rm RF}) \label{iiii} \nonumber \end{align} \tag{ 36 }$$

the global charge susceptibility of the nanocircuit. Note that in the general case the indices $n, n^{\prime}$ in equation (35) can belong to the nanoconductors as well as the fermionic reservoirs. From equation (35), the presence of the nanocircuit modifies the apparent frequency and linewidth of the cavity. For most of the experiments reported in the review, the cavity response is measured at $\omega_{\rm RF}=\omega_{0}$. In the rest of this section, we will also assume that Ξ(ω₀) can be considered as constant for ω₀ – Λ₀ ≲ ω_RF ≲ ω₀ + Λ₀, which occurs for instance if the electronic relaxation rates in the nanocircuit are much larger than the photon relaxation rate Λ₀. In this case, the cavity frequency and linewidth shifts caused by the presence of the nanocircuit are, from equation (35),

$$\begin{align} \displaystyle \Delta\omega_{0}={\rm Re}[\Xi(\omega_{0})] \label{j1} \nonumber \end{align} \tag{ 37 }$$

and

$$\begin{align} \displaystyle \Delta\Lambda_{0}=-{\rm Im}[\Xi(\omega_{0})]. \label{j2} \nonumber \end{align} \tag{ 38 }$$

Note that this type of measurement has been pioneered by works in which a coplanar waveguide resonator [82] or a lumped element resonator [83] was coupled to an array of $10^{5}$ isolated mesoscopic rings. The resonator frequency and linewidth shifts revealed the global electric and magnetic response of the rings to the resonator field, at frequency $\omega_{0}\sim350~$ MHz. More recently, the admittance of a single double quantum dot was measured at frequencies $\omega_{0} <400~$ MHz by using a lumped element (L,C) resonator [20, 21]. One important advantage of the circuit QED architecture is the higher frequency of the cavity. One has typically $\omega_{0}\sim5~$ GHz, which is higher than the cryogenic temperatures obtained with a dilution fridge $k_{\rm b}T\simeq25~$ mK ≃$0.5~$ GHz. Therefore, the quantum regime with a low number of cavity photons is accessible. In this regime, the semiclassical approximation used in this section is no longer valid. However, understanding the semiclassical regime of mesoscopic QED is an important prerequisite to the realization of quantum experiments. Most experiments realized so far correspond to this regime and can be well understood with the semiclassical picture.

Above, we have discussed how a nanocircuit induces modifications $\Delta \omega_{0}$ and $\Delta\Lambda_{0}$ of the cavity apparent frequency and linewidth. For a full description of a mesoscopic QED experiment, it is necessary to describe how one can measure these quantities. For that purpose, one has to take into account the existence of the input and output ports of the cavity, through which the cavity is excited and measured. These ports correspond to the pieces of waveguide connected capacitively to the cavity, on both sides of the cavity central conductor (see figure 1(a)). In the semiclassical limit, the incident, transmitted and reflected photon fluxes in these ports can be characterized by complex amplitudes b_in, $b_{t}$ and $b_{r}$. We assume that the cavity is excited through its input port, which can be, for instance, the left port in figure 1. The transmission $b_{t}/b_{\rm in}$ of the cavity can be obtained experimentally by measuring the microwave amplitude $b_{t}$ going out through the output port, which can be the right port in figure 1. In the semiclassical limit, the correspondence between the approach of section 3.2.1 and the input–output formalism of circuit QED [84] gives the relations of figure 5 between b_in, $b_{t(r)}$, $\varepsilon_{\rm in}$ and $\bar{a}$ [37]. There, $\Lambda_{L(R)}$ corresponds to the contribution of the left (right) port to the cavity linewidth, which implies $\Lambda_{L} +\Lambda_{R}\leqslant\Lambda_{0}$. From equation (35) and figure 5, the cavity microwave transmission can be expressed as

$$\begin{align} \displaystyle \frac{b_{t}}{b_{\rm in}}=\frac{2\sqrt{\Lambda_{L}\Lambda_{R}}}{\omega_{\rm RF} -\omega_{0}+{\rm i}\Lambda_{0}-\Xi(\omega_{\rm RF})}. \label{ii} \nonumber \end{align} \tag{ 39 }$$

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Hence, in the semiclassical linear limit, the cavity transmission amplitude is set by the charge susceptibility of the nanocircuit [36, 37, 48, 85, 133]. A similar result can be obtained for the cavity reflection amplitude. Note that above, $b_{t}/b_{\rm in}$ is expressed with the usual quantum mechanics convention for the definition of Fourier transforms, i.e. $g(\omega)= {\int_{-\infty}^{+\infty}} {\rm d}t~g(t){\rm e}^{{\rm i}\omega t}$, which is used throughout this review. In order to interpret experiments, one has to take into account that microwave equipment uses the electrical engineering Fourier transform convention, which is complex conjugated to the former, so that $(b_{t}/b_{\rm in}){\hspace{0pt}}^{\ast}$ is obtained experimentally.

In practice, the experimental signals which are directly measured are the transmission phase shift $\Delta\varphi$ and amplitude shift $\Delta A$ defined by

$$\begin{align} \displaystyle \frac{b_{t}}{b_{\rm in}}=(A_{0}+\Delta A){\rm e}^{{\rm i}(\varphi_{0}+\Delta\varphi)} \label{iii} \nonumber \end{align} \tag{ 40 }$$

Using (37) and (38) valid for a poorly coherent nanocircuit, and assuming that $\omega_{\rm RF}=\omega_{0}$ and $\left\vert \Delta\omega _{0}\right\vert, \left\vert \Delta\Lambda_{0}\right\vert \ll\left\vert \omega_{0}\right\vert, \left\vert \Lambda_{0}\right\vert$, one finds that the cavity signals are directly related to the cavity parameters' shifts, i.e.

$$\begin{align} \displaystyle \Delta\varphi=\frac{\Delta\omega_{0}}{\Lambda_{0}}={\rm Re} [\Xi(\omega_{0})]\frac{1}{\Lambda_{0}} \nonumber \end{align} \tag{ 41 }$$

and

$$\begin{align} \displaystyle \Delta A=-\frac{\Delta\Lambda_{0}A_{0}}{\Lambda_{0}}={\rm Im} [\Xi(\omega_{0})]\frac{A_{0}}{\Lambda_{0}}, \label{deltaA} \nonumber \end{align} \tag{ 42 }$$

so that $\Delta\varphi$ and $\Delta A$ correspond to the dispersive and dissipative parts of the signal. Beyond this limit, the experimental data can be understood by combining equations (39) and (40).

Depending on the regime of parameters fulfilled by the nanocircuit, and in particular the order of magnitude of the tunnel rates between the dots and reservoirs, different calculation techniques can be used to calculate $\Xi(\omega_{0})$. We will discuss several possibilities in the coming sections. In this review, we will only consider cases where the summation on indices n and $n^{\prime}$ in equation (36) can be restricted to internal sites of the nanoconductor. This requires that the coupling between the cavity and the nanoconductor sites is much larger than the coupling between the cavity and the reservoirs. This is not obvious a priori, since the nanoconductor is much smaller than the reservoirs and thus tends to have a smaller capacitance towards the cavity resonator. However, this feature can be compensated by using for instance ac top gates which reinforce the coupling between the nanoconductor and the cavity. In this limit, $\Xi(\omega_{0})$ corresponds to the charge susceptibility of the nanoconductor at frequency $\omega_{0}$. In the opposite limit where the coupling between the cavity and the source/drain of the nanocircuit is dominant, it has been observed experimentally that the cavity signals essentially show replicas of the conductance signal [54].

The case of a double quantum dot embedded in a microwave cavity has received a lot of experimental attention [36, 51, 52, 56, 59, 60, 90–98]. The intrinsic level separation $\Delta_{o}$ between the orbitals of one dot (see equation (1)) is usually very large in comparison with the other energy scales of the device. Therefore, it is sufficient to consider a single orbital with energy $\varepsilon_{L(R)}$ in dot $L(R)$. These two orbitals are coupled with a hopping constant t. In practice, each dot is also contacted to a normal metal reservoir, which enables one to control and measure the double dot charge. One can use the energy diagram of figure 7, where the Fermi levels in the reservoirs are filled up to the Fermi energy E_F and the orbital levels of the two dots have an energy separation $\varepsilon=\varepsilon_{L}-\varepsilon_{R}$. This last parameter can be controlled with the gate voltages $V_{\rm g}^{L}$ and $V_{\rm g}^{R}$ shown in figure 8.

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It is possible to tune $V_{\rm g}^{L}$ and $V_{\rm g}^{R}$ such that there is a single electron in the double dot due to Coulomb blockade. In the spin-degenerate case, the spin degree of freedom can be disregarded to describe this situation since the two spin species play the same role and are not present simultaneously in the double dot. Therefore, the only internal degree of freedom relevant to describe the internal dynamics of the double dot in this limit is the left/right charge degree of freedom. In this framework, the double dot Hamiltonian writes

$$\begin{align} \displaystyle \hat{H}_{0}^{t}=\frac{\varepsilon}{2}(\hat{c}_{L}^{\dagger}\hat{c}_{L}-\hat {c}_{R}^{\dagger}\hat{c}_{R})+t\hat{c}_{L}^{\dagger}\hat{c}_{R}+t^{\ast}\hat{c} _{R}^{\dagger}\hat{c}_{L}+\hat{H}_{\rm Coul} \label{HH1} \nonumber \end{align} \tag{ 47 }$$

where $\hat{H}_{\rm Coul}$ forbids the double occupation of the double dot. Using the basis of bonding and antibonding states of the double dot, one gets

$$\begin{align} \displaystyle \hat{H}_{0}^{t}\simeq\frac{\omega_{\rm DQD}}{2}(\hat{c}_{-}^{\dagger}\hat{c}_{-} -\hat{c}_{+}^{\dagger}\hat{c}_{+})+\hat{H}_{\rm Coul} \nonumber \end{align} \tag{ 48 }$$

with $\omega_{\rm DQD}=\sqrt{\varepsilon^{2}+4t^{2}}$ the double dot transition frequency. Above, we have used the creation operators

$$\begin{align} \displaystyle \hat{c}_{+}^{\dagger}=\cos[\theta/2]\hat{c}_{L}^{\dagger}+\sin[\theta/2]\hat{c} _{R}^{\dagger} \nonumber \end{align} \tag{ 49 }$$

and

$$\begin{align} \displaystyle \hat{c}_{-}^{\dagger}=-\sin[\theta/2]\hat{c}_{L}^{\dagger}+\cos[\theta/2]\hat{c} _{R}^{\dagger} \nonumber \end{align} \tag{ 50 }$$

for bonding and antibonding states in the dot, and the parameter $\theta=\arctan[2t/\varepsilon]$.

In most experiments designed so far, the samples have been designed with an asymmetric coupling of the two dots to the cavity, in order to modulate the parameter ε with the cavity electric field (see for instance figure 8(a)). In this case, following section 3.1.6, the interaction term between the double dot and the cavity can be expressed as

$$\begin{align} \displaystyle \hat{h}_{\rm int}(\hat{a}+\hat{a}^{\dagger})=\left(g_{L}\hat{c}_{L}^{\dagger}\hat {c}_{L}+g_{R}\hat{c}_{R}^{\dagger}\hat{c}_{R}\right) (\hat{a}+\hat{a}^{\dagger}) \nonumber \end{align} \tag{ 51 }$$

with $g_{L}\neq g_{R}$. Using a rotating wave approximation, this term can be expressed as

$$\begin{align} \displaystyle \hat{h}_{\rm int}(\hat{a}+\hat{a}^{\dagger})\simeq g_{t}(\hat{a}^{\dagger}\hat{c} _{-}^{\dagger}\hat{c}_{+}+\hat{a}\hat{c}_{+}^{\dagger}\hat{c}_{-}) \label{HH2} \nonumber \end{align} \tag{ 52 }$$

with

$$\begin{align} \displaystyle g_{t}=\frac{g_{R}-g_{L}}{2}\sin[\theta]. \label{gt} \nonumber \end{align} \tag{ 53 }$$

This coefficient depends on the differential coupling $g_{R}-g_{L}$ because the coupling to the cavity occurs through the modulation of the dot orbital detuning ε.

As we have seen in the previous section, in order to obtain the strong coupling regime, one needs to have a small enough $\Gamma_{2}^{\ast}$. The coherence of a charge double dot is mainly limited by charge noise due to charge fluctuators which move in the vicinity of the double dot. This induces fluctuations of the parameters which are electrically controlled, i.e. ε in the present case. This effect should be minimal at the charge noise sweet spot $\varepsilon=0$, where $\partial\omega_{\rm DQD}/\partial \varepsilon=0$, similarly to what has been done for early days charge superconducting quantum bits—which are also affected by this problem [99]. Hence, it would be interesting to perform a systematic study of the figures of merit of the cavity/double dot resonance when the double dot parameters, and in particular ε, are varied.

In principle, the coupling of a double-dot dot to a microwave cavity can be mediated by other variables than ε, depending on the sample design. The first manipulations of the quantum state of a double dot were performed by modulating the parameter ε with a strong classical drive [22]. Recently, similar experiments were performed by modulating the interdot tunnel parameter t [71–73], following the theory proposal of [74]. One could push this idea further by building ac gates connecting the double dot barrier to the cavity central conductor, in order to modulate the interdot tunnel parameter t with the cavity electric field. In principle, this could enable one to obtain double dots with a better coherence, since the electric (and charge noise) control of the variable ε can be shunted, in this case. In section 4.4, we will discuss an alternative strategy to control electrically t, which involves using a superconducting contact.

This section shows how to calculate the cavity charge susceptibility $\Xi(\omega_{\rm RF})$ of the double dot when a microwave cavity is coupled to a single transition $i\leftrightarrow j$ corresponding to the left/right charge degree of freedom of a double quantum dot. The dynamics of this mesoscopic QED device can be described by using a master equation approach (or Lindbladt formalism) already widely used for cavity or circuit QED [75]. From equations (28), (47) and (52), one gets

$$\begin{align} \displaystyle \frac{\rm d}{{\rm d}t}\hat{a}=-{\rm i}\omega_{r}\hat{a}-{\rm i}g_{t}\hat{c}_{-}^{\dagger}\hat{c} _{+}-\Lambda_{0}\hat{a}+\varepsilon_{\rm in}{\rm e}^{-{\rm i}\omega_{\rm RF}t} \label{v1} \nonumber \end{align} \tag{ 54 }$$

$$\begin{align} \displaystyle \frac{\rm d}{{\rm d}t}\hat{c}_{-}^{\dagger}\hat{c}_{+}=-{\rm i}\omega_{\rm DQD}\hat{c}_{-}^{\dagger} \hat{c}_{+}+ig_{t}(\hat{n}_{+}-\hat{n}_{-})\hat{a}-\Gamma_{2}^{\ast}\hat {c}_{-}^{\dagger}\hat{c}_{+} \label{v2} \nonumber \end{align} \tag{ 55 }$$

with $\Gamma_{2}^{\ast}$ the total decoherence rate of the double dot transition, which includes relaxation and dephasing effects. Importantly, the above equations are valid for $\Gamma_{2}^{\ast}\ll k_{\rm B}T$. In the semiclassical limit with $\omega_{\rm RF}\simeq\omega_{0}$, one can use equation (32) and the resonant expression

$$\begin{align} \displaystyle \hat{c}_{-}^{\dagger}\hat{c}_{+}\simeq\left\langle \hat{c}_{-}^{\dagger}\hat{c} _{+}\right\rangle _{0}+\overline{\hat{c}_{-}^{\dagger}\hat{c}_{+}}{\rm e}^{-{\rm i}\omega _{\rm RF}t} \nonumber \end{align} \tag{ 56 }$$

Hence, equations (54) and (55) give

$$\begin{align} \displaystyle \bar{a}=\frac{\varepsilon_{\rm in}}{\left(\omega_{\rm RF}-\omega_{0}+{\rm i}\Lambda _{0}-\frac{g_{t}^{2}(n_{-}-n_{+})}{\omega_{\rm RF}-\omega_{\rm DQD}+{\rm i}\Gamma_{2}^{\ast }}\right) } \nonumber \end{align} \tag{ 57 }$$

with $n_{-}$ and $n_{+}$ the average occupation numbers of the bonding and antibonding states. A comparison with equation (35) gives

$$\begin{align} \displaystyle \Xi(\omega_{\rm RF})=\frac{g_{t}^{2}(n_{-}-n_{+})}{\omega_{\rm RF}-\omega _{\rm DQD}+{\rm i}\Gamma_{2}^{\ast}}. \label{XiDQD} \nonumber \end{align} \tag{ 58 }$$

The above equation corresponds to a well known result in the dispersive regime $\omega_{\rm RF}-\omega_{\rm DQD}\gg\Gamma_{2}^{\ast}(V_{\rm g}^{L}, V_{\rm g}^{R})$ where $\Gamma_{2}^{\ast}$ can be disregarded (see for instance [76]). In this limit, depending on whether the nanocircuit is in the state $+$ or −, the cavity shows a frequency pull $\Delta\omega_{0}=\pm g_{t}^{2} /(\omega_{\rm RF}-\omega_{\rm DQD})$. This can be used to read out the state of the nanocircuit in a nondestructive way, since in this limit, $\chi(\omega_{\rm RF})$ accounts for second order processes which do not change the state of the nanocircuit. This method is widely used to read out the state of superconducting quantum bits [26]. In section 4.2, we consider double dots with no voltage bias, and we also assume that the dot levels are not resonant with the normal metal reservoirs, so that the electron which is trapped in the double dot cannot escape. We also assume that the power of the microwave tone applied to the cavity is too low to excite the transition between the bonding and antibonding states. In this case, at equilibrium, one has $n_{-}=1$ and $n_{+}=0$, which leads to equation (44).

Weak coupling limit. 

Strong coupling limit.

Very recently, the strong coupling regime was reached simultaneously in three experiments based on different types of charge double quantum dots [30–32]. In this regime, for a low number of photons $\left\langle \tilde{n}\right\rangle \rightarrow0$, the cavity transmission (or reflection) amplitude versus the frequency excitation $\omega_{\rm RF}$ shows a double peak, due to the strong hybridization between the cavity and the L/R charge degree of freedom of the double dot (see figure 9(d)). To reach this regime, the differential light–matter coupling $g_{L}-g_{R}$ must be sufficiently large in comparison with the decoherence rate of the L/R degree of freedom, which is typically dominated by dephasing due to charge noise. In this case, the dephasing rate $\Gamma _{\varphi}$ of the charge double dot takes the form [34]

$$\begin{align} \displaystyle \Gamma_{\varphi}\simeq\frac{\partial\omega_{\rm DQD}}{\partial\varepsilon} E_{c}A+\frac{1}{2}\frac{\partial^{2}\omega_{\rm DQD}}{\partial\varepsilon^{2} }E_{c}^{2}A^{2} \label{noise} \nonumber \end{align} \tag{ 59 }$$

where $E_{c}$ is the local charging energy of one dot (we disregard the mutual charging energy between the dots). Above, A is the dimensionless prefactor in the noise spectrum $A^{2}/f$ which adds up to the reduced gate charge $C_{g}^{L(R)}V_{\rm g}^{L(R)}/e$ with $C_{g}^{L(R)}$ the capacitance between dot $L(R)$ and its gate voltage source with voltage $V_{\rm g}^{L(R)}$. From equations (46), (53) and (59), in order to have $Q_{\rm e{-}ph}>1$, three different technical strategies are possible: either use a nanoconductor technology with an intrinsically lower charge noise (i.e. decrease A) [30], or use a double dot with larger capacitances (i.e. decrease $E_{c}$) in order to shunt the effect of charge noise [31], or change the cavity technology in order to increase the cavity electric field and thus the $g_{L}-g_{R}$ factor [32]. These three strategies have already been implemented experimentally, as discussed in the paragraphs below. In all cases, using a device with $\omega_{0}=2t$ should be advantageous, so that $\Gamma_{\varphi}$ is only a second order effect in A when the device is tuned at the anticrossing between the cavity and the double dot (one has $\varepsilon=0$ and thus $\partial\omega_{\rm DQD}/\partial\varepsilon =0$).

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In [30], a double quantum dot in an undoped Si/SiGe heterostructure was used (see figure 9(b)). When the double dot is off resonant with the cavity, a single resonance is visible along the $\omega_{\rm RF}$ axis, which corresponds to the bare cavity resonance (see blue line in figure 9(c)). When $\omega_{0}=2t$ is used, and when the double dot is tuned near its sweet spot ($\varepsilon\sim0$), the vacuum Rabi splitting is observed, i.e. a double cavity resonance is visible along the $\omega_{\rm RF}$ axis (see red line in figure 9(c)). In this device, the charge–photon coupling $(g_{L}-g_{R})/2\pi=13.4~{\rm MHz}$ is comparable to what has been obtained with other charge double dots (see table 1). The microwave cavity is also similar to the ones used in previous experiments, with $\Lambda_{0}=1.0~{\rm MHz}$. The vacuum Rabi splitting is achieved thanks to an unusually small decoherence rate $\Gamma_{2}^{\ast} =2.6~{\rm MHz}$ of the left/right charge degree of freedom in the double dot. This gives light–matter coupling ratios $Q_{\rm e{-}ph}=4$ and $C_{\rm e{-}ph}=34$. In the Si/SiGe two-dimensional structure used for this experiment, the dot charging energies are typically of the order of $E_{c}\simeq7~{\rm meV}$ [33]. In comparison, the GaAs/AlGaAs structure of [38] has smaller charging energies $E_{c} \sim1~{\rm meV}$, but it is far from the strong coupling regime. This suggests that the low $\Gamma_{2}^{\ast}$ value in [30] might be due to a much lower intrinsic charge noise in Si/SiGe devices. In agreement with this, in GaAs/AlGaAs devices, one of the smallest reported values of charge noise is [101] is $A=2.10^{-4}$, whereas the values $2.3~10^{-6}<A<1~10^{-5}$ have been reported [102] for doped Si/SiGe heterostructures. Undoped Si/SiGe heterostructures might have an even lower charge noise [103]. The value of charge noise in Si/SiGe deserves a thorough investigation in order to confirm this picture.

Stockklauser et al [32] have used a GaAs/AlGaAs heterostructure similar to that in [38]. However, the coplanar waveguide architecture has been modified by replacing the central resonator of the cavity by an array of 32 superconducting quantum interference devices (SQUIDs). This increases by a factor  ∼10 the couplings $g_{L(R)}$. One has $(g_{L}-g_{R})/2\pi=238~{\rm MHz}$, and $\Gamma_{2}^{\ast }=93~{\rm MHz}$ which gives $Q_{\rm e{-}ph}=2$ and $C_{\rm e{-}ph}=25$. These figures of merit are very close to those of [30]. However, they have not been obtained at the double dot sweet spot, since $\omega_{0}=1.22\ast2t$ was used. This could suggest that the above figures of merit are not the optimal ones for this setup. Interestingly, with the SQUID array architecture, the cavity frequency can be tuned by using an external magnetic field. However, the cavity decoherence rate is stronger with this architecture ($\Lambda _{0}=6.2~{\rm MHz}$).

Alternatively, [31] has reached the strong coupling regime to the left/right charge degree of freedom of a double dot by using a device with small charging energies. However, the double dot has a fundamentally different architecture in this experiment, since it comprises a superconducting contact, and since the coupling to the cavity photons seems to occur through the variable $\varepsilon_{L}+\varepsilon_{R}$ instead of $\varepsilon_{L}-\varepsilon_{R}$. Therefore, we will discuss this experiment in section 4.5.

The electronic spin degree of freedom draws a lot of interest in nanoconductors because it could be a good means to encode quantum information. Indeed, spins are expected to have a long coherence time in nanoconductors because they are more weakly coupled to their environment than charges. The counterpart of this immunity is that the natural magnetic coupling $g_{m}$ between a spin and a standard coplanar microwave cavity is only a few 10Hz, which is not sufficient for manipulation and readout operations. It is possible to circumvent this difficulty by using a large number of spins, as demonstrated recently with several types of crystals coupled to coplanar microwave cavities [104–106]. However, in this case, the anharmonicity which is inherent in a two level system is lost, so that the spin ensemble can only be used as a quantum memory. To remain at the single spin level, it has been suggested to include in the microwave cavity a nanometric constriction to concentrate the cavity field, which would yield $g_{m}\sim10~$ kHz [107, 108]. Alternatively, various theories have suggested to use a weak hybridization between the spin and charge degrees of freedom of a quantum dot circuit [9, 109–112], provided by a real or artificial spin–orbit coupling. To understand this effect, let us assume that the state of the dot circuit can be decomposed on a basis of pure spin eigenstates $\left\vert \varphi_{n\uparrow }\right\rangle$ and $\left\vert \varphi_{n\downarrow}\right\rangle$ with $n\in\mathbb{N}$ an orbital index. In the presence of a spin–orbit coupling term the Hamiltonian of the dot circuit will be written:

$$\begin{align} \displaystyle \hat{H}_{0}^{t} =\,\,& {\sum_{n,\sigma}} \left(E_{n}+\frac{E_{z}\sigma}{2}\right) \left\vert \varphi_{n\sigma }\right\rangle \left\langle \varphi_{n\sigma}\right\vert \nonumber \\ & + {\sum_{n,n^{\prime}}} \left(h_{nn^{\prime}}\left\vert \varphi_{n^{\prime}\downarrow}\right\rangle \left\langle \varphi_{n\uparrow}\right\vert +{\rm H.c.}\right). \nonumber \end{align} \tag{ 60 }$$

In the above, $E_{n}$ is the orbital energy of state n, $E_{z}$ is the external Zeeman field applied to the circuit, and $h_{nn^{\prime}}$ corresponds to the matrix elements of the spin–orbit interaction on the basis of states $\left\vert \varphi_{n\sigma}\right\rangle =c_{n\sigma}^{\dagger}\left\vert \varnothing\right\rangle$. The photonic pseudo potential $V_{\bot}$ is spin-conserving, so that the light–matter interaction given by equation (20) is

$$\begin{align} \displaystyle \hat{h}_{\rm int}= {\sum_{n,\sigma}} V_{\bot}^{n,n^{\prime}}\left\vert \varphi_{n^{\prime}\sigma}\right\rangle \left\langle \varphi_{n\sigma}\right\vert. \nonumber \end{align} \tag{ 61 }$$

Hence, at first order in spin–orbit coupling, the eigenstates

$$\begin{align} \displaystyle \left\vert \widetilde{\varphi_{n\uparrow(\downarrow)}}\right\rangle =\left\vert \varphi_{n\uparrow(\downarrow)}\right\rangle + {\sum_{n^{\prime}}} \frac{h_{nn^{\prime}}^{(\ast)}}{E_{n\uparrow(\downarrow)}-E_{n\downarrow ({\hspace{0pt}}^{\prime}\uparrow)}}\left\vert \varphi_{n^{\prime}\downarrow({\hspace{0pt}}^{\prime }\uparrow)}\right\rangle \nonumber \end{align} \tag{ 62 }$$

of $H_{0}$ are coupled to the cavity with the matrix element

$$\begin{align} \displaystyle & \left\langle \widetilde{\varphi_{n\downarrow}}\right\vert \hat{h} _{\rm int}\left\vert \widetilde{\varphi_{n\uparrow}}\right\rangle \label{couplSO} \nonumber \\ & = {\sum_{n^{\prime}}} \left(\frac{h_{nn^{\prime}}}{E_{n}-E_{n^{\prime}}+E_{z}}-\frac {h_{nn^{\prime}}^{\ast}}{E_{n}-E_{n^{\prime}}-E_{z}}\right) V_{\bot }^{n,n^{\prime}}.\nonumber \end{align} \tag{ 63 }$$

One can imagine to build a qubit by using two states $\left\vert \widetilde{\varphi_{n\uparrow}}\right\rangle$ and $\left\vert \widetilde{\varphi_{n\downarrow}}\right\rangle$ which can be considered as quasi-spin states if the spin–charge hybridization is small. Due to this hybridization, this qubit will be sensitive to charge noise. Therefore, one has to choose a design which establishes a good compromise between having a small enough decoherence and a strong enough light–matter-coupling.

In principle, from equation (63), a single quantum dot with a natural Rashba or Dresselhaus spin–orbit coupling could already offer a spin–photon interaction. Indeed, the spin of a quantum dot was manipulated by using a large ac drive applied directly on the dot gate and coupled to the spin through the spin–orbit coupling [24, 39]. One can imagine replacing the direct ac drive by the cavity field. Then, the indices n, $n^{\prime}$ in the above equations can correspond to the natural sub-bands in the dot spectrum. However, for most quantum dots, the spin–orbit interaction is too weak to enable the strong coupling regime with a single quantum dot circuit. For instance, for GaAs quantum dots, it has been shown theoretically that the effect of spin–orbit coupling is limited by the small spatial extension of the quantum dot [113–115]. As we will see in section 4.3.3, an alternative approach is to engineer extrinsically an artificial spin–orbit coupling by using a quantum dot circuit with ferromagnetic contacts, which induce local effective Zeeman fields such as those of equation (2). Then, it is not necessary to invoke the existence of several levels in each dot. For instance, in the case of a double quantum dot, the indices n, $n^{\prime}$ can be restricted to a pair of bonding and antibonding states, formed by the coherent coupling of left and right orbitals of the double dot. In principle, this should enable one to tune the value of the spin–orbit interaction, thanks to the electric control of the orbital energy detuning ε. Another interesting possibility could be to use designs which exploit stray fields from micromagnets [116, 117].

As shown in section 4.2, internal tunnel hopping of charges inside a double quantum dot modifies the cavity signals. This property can be used to detect with a dc current measurement the spin state of a pair of electrons trapped in a double quantum dot, thanks to a spin-rectification effect induced by Pauli spin blockade, which has been widely studied through current measurements [118]. This effect was recently exploited in a mesoscopic QED device, based on a singlet–triplet qubit in an InAs double quantum dot, with one electron in each dot [56]. The readout of this qubit requires discriminating the two states $\left\vert S_{1, 1}\right\rangle =(\left\vert L\uparrow, R\downarrow\right\rangle -\left\vert L\uparrow, R\downarrow\right\rangle)/\sqrt{2}$ and $\left\vert T_{0}\right\rangle =(\left\vert L\uparrow, R\downarrow\right\rangle +\left\vert R\uparrow, L\downarrow\right\rangle)/\sqrt{2}$. When the orbital detuning ε between the left and right dot is modulated by the cavity electric field, transitions to the state $\left\vert L\uparrow, L\downarrow\right\rangle$ are possible only if the double dot initially occupies the state $\left\vert S_{1, 1}\right\rangle$, due to the spin-conserving character of interdot tunneling. This leads to $\Xi(\omega_{\rm RF})\neq0$. In contrast, if the double dot is in the state $\left\vert T_{0}\right\rangle$, one has $\Xi(\omega_{\rm RF})=0$ due to the Pauli exclusion principle. Therefore, the spin state of the double dot can be detected through the cavity signals. It is nevertheless important to point out that a direct spin–photon coupling was not implemented in the experiment of [56]. The state of the double dot was manipulated by applying a strong microwave drive directly to the dot gates, to rotate the spins thanks to spin–orbit coupling. The cavity was used only to perform the charge readout of the spin qubit. To date, no experiment has been able to detect a spin–cavity coupling caused by intrinsic spin–orbit coupling in a nanocircuit.

It was recently suggested that the coupling of equation (63) could be realized by using a double quantum dot with two ferromagnetic contacts magnetized in non-collinear directions [9], represented in figure 10(a). These contacts cause a spin-mixing of the double dot eigenstates, which can be viewed as an artificial spin–orbit coupling. This effect occurs due to intradot effective Zeeman fields similar to those of equation (2). By tuning the orbital detuning ε, one can in principle control the degree of delocalization of the electron between the two dots, in order to tune the magnitude of the artificial spin–orbit interaction.

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A first version of this device has been realized recently, by using a double quantum dot made in a single wall carbon nanotube on top of which two ferromagnetic PdNi contacts are deposited [34]. When the microwave transmission amplitude of the cavity is measured versus ε and the external magnetic field B applied to the double dot, three resonant lines appear (see figure 10(b)). Various features suggest that the spin degree of freedom is an important ingredient in this pattern. First, the resonances split and strongly move with the external magnetic field B, with a maximum of contrast/coherence for a finite value of B. Second, the black point of figure 10(b) corresponds to a coupling $\ g_{s} =1.3~{\rm MHz}$ and a double dot decoherence rate $\Gamma_{2}^{\ast} /2\pi=2.5~{\rm MHz}$. This last number is about 1/200 of the charge decoherence rate determined for a similar carbon nanotube device (see section 4.2.3). One has $Q_{\rm e{-}ph}=0.81$ which means that this device is almost in the strong coupling regime.

To understand the contribution of the spin degree of freedom to the cavity signals better, one can use equation (43) which is a generalization of equation (58), valid if the different transition frequencies $\omega_{ij}$ of the nanocircuit are well separated. To calculate $\omega_{ij}$ and the couplings $g_{ij}$, one has to use a double dot Hamiltonian which takes into account not only the existence of the left/right and spin degrees of freedom of the double dot, but also the K/K' local orbital degree in each dot (or valley degree of freedom), which is due to the fact that electrons can rotate clockwise or anticlockwise around the carbon nanotube. The linewidth of the resonances can be modeled by taking into account the effect or charge noise. This gives figure 10(c), which reproduces well the behavior of figure 10(b). The two strongest resonances mainly correspond to spin transitions with a conserved K/K' index. These two resonances are slightly split, due to a small lifting of the K/K' degeneracy. The third weaker resonance mainly corresponds to a transition where both the spin and the K/K' index are reversed. In figure 10(c), this transition is less visible than the two others because the K/K' degree of freedom is only weakly coupled to cavity photons, probably due to weak microscopic disorder in the carbon nanotube structure. However, this resonance is very interesting in the light of recent works which investigate the coupling between the valley degree of freedom of a silicon dot and a microwave cavity [128, 129].

Remarkably, the coherence (or, visually, the contrast) of the three transitions is maximal along the green dashed line in figure 10(c). This is because the derivative of the transition frequencies $\omega_{ij}$ with respect to ε vanishes along this line, which is a charge noise sweet line. This behavior also occurs in the data, which confirms that charge noise is an important source of decoherence in this device. It may be possible to enhance these performances by reducing the spin–charge hybridization to decrease decoherence due to charge noise. It is expected that $\Gamma _{2}^{\ast}$ will decrease more quickly than $g_{s}$ with ε, so that the strong coupling regime is accessible with this geometry, in principle [9].

Various theories have suggested coupling the spin degree of freedom to the cavity electric field by using two or three electron states in a quantum dot circuit. For that purpose, one can use a multi-quantum dot circuit with proper spin-symmetry breaking ingredients, in order to transduce the charge–photon into a spin–photon coupling. For instance, in a double dot with finite interdot hopping, the transition between the singlet and triplet spin states $\left\vert S_{1, 1}\right\rangle$ and $\left\vert T_{0} \right\rangle$ is coupled to cavity photons due to the presence of a Zeeman field with constant direction but a different amplitude in the two dots [29, 119–121]. This field can correspond to an Overhauser field due to nuclear spins in a two-dimensional electron gas, or to stray fields from a ferromagnet. In the case of a triple quantum dot, it is possible to use three electron states from the $(S=1, S_{z}=1/2)$ subspace, with S the total spin of the dots, to define the resonant exchange qubit [123–125]. In this case, the spin–photon coupling can be obtained with a homogeneous Zeeman field if the spatial symmetry of the triple dot is adequately broken. Note that the above setups do not involve any real or effective spin–orbit interaction. On the contrary, they consider devices where the individual spin of electrons would be conserved in the single electron regime. At present, the multiparticle spin–photon coupling of [29, 119–125] is awaiting an experimental realization.

From equation (58), current transport in the double dot can modify the cavity signals by modifying the populations $n_{-}$ and $n_{+}$ of the bonding and antibonding states. The transport configuration can be tuned electrically through the double dot gate voltages $V_{\rm g}^{L}$ and $V_{\rm g}^{R}$ and source-drain bias voltage V_b. Figure 15 presents results obtained with a carbon nanotube double quantum dot with a finite Coulomb interaction [36]. As observed usually, the current I through the double dot is finite only inside some triangles in the $V_{\rm g}^{L} -V_{\rm g}^{R}$ plane where the bonding or antibonding states are inside the transport window opened by the source-drain voltage V_b (figure 15(a)). The cavity signal $\Delta\varphi$ is maximum along the line $\varepsilon=0$ where $\omega_{\rm DQD}-\omega_{0}$ is minimum (figures 15(b)), and it takes a different value along the transport triangles, because the populations of $n_{-}$ or $\ n_{+}$ are modified by transport. In figure 15(c), this behavior is well reproduced by using equation (58), with $n_{-}$ and $n_{+}$ calculated with a master equation approach at lowest order in the light–matter coupling ($g_{t}=0$) (see details in [36]). In the regime $\Gamma_{L(R)}\ll k_{\rm b}T$, this approach is sufficient to take into account Coulomb blockade, which essentially affects the structure of the nanocircuit state space and tunnel rates. The inclusion of Coulomb interactions in the Keldysh formalism is a more complex task [148]. Note that however, in the non-interacting case, the Keldysh approach also reproduces well the fact that $\Delta\varphi$ is affected by current transport through a modification of $n_{-}-n_{+}$ (see appendix B). Remarkably, the dc current through the double dot and the cavity signals are qualitatively different, since the cavity signals directly probe $n_{-}-n_{+}$ whereas the current I is a more complex combination of $n_{-}$, $n_{+}$ and the double dot parameters. Therefore, the simultaneous study of the two signals can again enable a more accurate characterization of the double dot parameters.

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The splitting between the bonding and antibonding states of the double dot can become resonant with the cavity if ε is tuned properly. In this regime, it is possible to obtain photon emission due to the dc voltage bias. This phenomenon has been investigated in two dimensional electron gas structures and semiconducting nanowires [38, 39]. In [38], the number P of photons emitted by the cavity per unit time was measured as a function of the double dot detuning ε for a constant interdot detuning t (see figure 16(a)). For $2t>\omega_{0}$, two resonances appear in P, for orbital detunings $\varepsilon=\pm\varepsilon_{0}$ with $\varepsilon_{0}=\sqrt{\omega_{0} ^{2}-4t^{2}}$. The resonance for $\varepsilon<0$ is less pronounced than the resonance for $\varepsilon>0$ because it requires that the electrons tunnel from the left reservoir to an antibonding state which has little extension on the left dot (see figure 16(1)). Figure 16(b) shows the measured value of $\varepsilon_{0}$ versus t, which can be tuned via gate voltages in two dimensional electron gas structures. As expected, $\varepsilon_{0}$ vanishes for $2t>\omega_{0}$, because it is not possible to satisfy the resonant condition $\omega _{\rm DQD}=\omega_{0}$ in this case. In this limit, the cavity shows a single resonance centered on $\varepsilon=0$ (not shown). The light–matter coupling in this experiment can be estimated from the parameters $g/2\pi =11~{\rm MHz}$, $\Gamma_{2}^{\ast}/2\pi=250~{\rm MHz}$ and $\Lambda _{0}=1.7~{\rm MHz}$, which corresponds to $Q_{\rm e{-}ph}=0.068$ and $C_{\rm e{-}ph}=0.29$. The data were interpreted with a master equation approach similar to that of [36] (see section 5.4.1). In the out-of-equilibrium regime, the rate of photon emission in the cavity by the double dot is $\Gamma_{e}\sim2\pi\times0.3~{\rm MHz}$. This number is similar to the performances obtained with the normal metal/dot/superconductor bijunction of section 5.3.

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In principle, when a double dot with $\Gamma_{\varphi}^{\ast}\gg\Gamma_{1}$ is resonant with a cavity ($\omega_{ij}=\omega_{0}$), it is possible to obtain a lasing effect, which corresponds to an emission of a coherent microwave radiation by the double dot, if $C_{\rm e{-}ph}\gtrsim1/2$ [149–156]. When $C_{\rm e{-}ph}<1/2$, it is possible to reach the lasing regime by coupling several double quantum dots to the cavity. This was recently realized with two double dots made in InAs nanowires [40, 41]. Figure 17 shows the in-phase and quadrature phase components I and Q of the output field of the cavity, measured when only one of the dots fulfills $\omega_{\rm DQD}=\omega_{0}$ (panel (b)) or when the two dots satisfy $\omega_{\rm DQD}=\omega_{0}$ (panel (c)). In the first case, the cavity photons show a thermal distribution, because the device is below the lasing threshold. In the second case, the 'ring' in the tomography reveals a coherent photonic emission, because the device is above the lasing threshold. An average photon number in the cavity $\left\langle \hat{a}^{\dagger}\hat{a}\right\rangle \simeq8000$ is estimated in this last case. The rate of photon emission by the double dot can be estimated as $\Gamma_{e}\simeq\left\langle \hat {n}\right\rangle \omega_{0}/2Q_{0}=10~{\rm GHz}$, which is significantly larger than in [37, 38]. Charge noise limits the maser linewidth. However, a linewidth narrowing by more than a factor of ten can be obtained by using a microwave input tone that stabilizes the frequency of laser emission by triggering stimulated emission [41]. This technique is known as injection locking in the field of conventional lasers [157]. Note that in the lasing regime, the linear theory of section 3.2 fails. We refer interested readers to [159] for a simple theoretical description of this limit. Interestingly, there is a close analogy between the lasing effect produced in a microwave cavity by an out-of-equilibrium quantum dot, and that obtained with a Josephson superconducting transistor with a finite voltage bias [160]. In this second case, the lasing transition corresponds to a change in the number of Cooper pairs inside a superconducting island. In [160], a photon emission rate inside the cavity $\Gamma_{e}\simeq2~{\rm GHz}$ has been obtained, which is comparable to the performance of [41].

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The Kondo effect has been observed experimentally since the 30s (see for instance [161]). Although the resistivity of bulk metals is expected to decrease with temperature, for some metals containing a small amount of magnetic impurities an increase was observed below the Kondo temperature T_K. Thirty years later, Kondo suggested that this behavior is due to spin-flip scattering processes between the itinerant electrons of the metal and the magnetic impurities [162]. These processes give birth to a Kondo cloud, which screens the spins of the impurities and reduces the conductivity of the metal.

More recently, it was shown that quantum dots can also be used to study the Kondo effect if they are subject to a strong Coulomb interaction [163–165]. In this case, the spin-flip scattering processes give rise to an increase of the dot zero-bias conductance. This effect depends on the value of the charging energy U to add a second electron in a dot level. If U is large, and the dot orbital energy $\varepsilon_{d}$ fulfills $-U<\varepsilon_{d}<0$, there can only be a single electron in the dot orbital, whose electronic spin simulates a local magnetic impurity. Then, if the dot is coupled to normal metal contacts with a high enough tunnel rate $\Gamma_{N}$, the local spin in the dot can fluctuate due to even order tunnel processes, which change the dot spin but not its charge. These processes involve intermediate virtual dot states with a different charge (dot orbital empty or doubly occupied), which are energetically forbidden but quantum mechanically allowed for a very short amount of time  ∼$\hbar/U$. Therefore, the Kondo effect should remain transparent to a microwave cavity which is only sensitive to charge fluctuations at frequencies $\omega\lesssim \omega_{0}$. Importantly, this test requires that the cavity is mainly coupled to the dot level, so that the cavity signals are set by the dot charge susceptibility. Conversely, if the cavity modulates the potentials of the source and drain reservoirs asymmetrically, the cavity signals can reveal resonances similar to the Kondo conductance peak, as found experimentally in [54, 166].

The charge susceptibility of a Kondo dot has been studied recently in a carbon nanotube quantum dot coupled to the cavity through the dot energy level alone [167]. Both the Kondo regime and the Coulomb blockade regime have been studied with the same sample. In the Coulomb blockade regime, the conductance peaks through the dot are also visible in the cavity signals because they correspond to real charge fluctuations which are visible by cavity photons. However, in the Kondo regime, the low energy Kondo peaks are visible in the dot conductance but not in the cavity signals. Hence, the Kondo effect corresponds to conduction through essentially frozen charges in the dot (for a detector with a frequency cutoff $\omega_{0}\ll T_{\rm K}$). This illustrates the decoupling of the spin and charge dynamics in the Kondo effect.

The spatial separation of spin-entangled electrons from a Cooper pair is an interesting goal in the context of quantum computation and communication. In principle, a Cooper pair beam splitter (CPS) connected to a central superconducting contact and two outer normal metal (N) contacts could facilitate this process [10]. The spatial splitting of Cooper pairs has been demonstrated experimentally from an analysis of the CPS average currents, current noise, and current cross-correlations, in devices made out of a carbon nanotube [168–173] or a semiconducting InAs quantum wire [174]. However, new tools appear to be necessary to investigate further the Cooper pair splitting dynamics, and in particular its coherence, which has not been demonstrated experimentally so far in the N/dot/S/dot/N geometry. This coherence has two intimately related aspects: the coherence of Cooper pair injection and the conservation of spin entanglement. The first aspect is due to the fact that Cooper pair injection into the CPS is a coherent crossed Andreev process, which produces a coherent coupling between the initial and final states of the Cooper pair in the superconducting contact and the double dot. In this context, coupling the CPS to a microwave cavity would be very interesting because it would enable one to perform the spectroscopy of the CPS and identify anticrossings in the CPS spectrum, which are due to the coherence of the injection process [43]. Detecting the conservation of spin entanglement represents an even greater challenge. In principle, microwaves couple to transitions between the states of the CPS with matrix elements which keep signatures of the coherent superposition of spin states displayed by a singlet state. Therefore, a microwave cavity could help to characterize split singlet states [42]. Interestingly, a supercurrent was recently observed in Josephson junction made out of two self-assembled quantum dots coupled in parallel to two superconducting contacts [175]. The observation of a supercurrent necessarily implies a non-dissipative and thus coherent pair injection process. However, even in this case, the use of a microwave cavity would be very interesting to characterize the device further, i.e. perform its spectroscopy and check whether the coherent Cooper pair injection really goes together with a spin-entangled double dot state.

Majorana quasiparticles are among the most intriguing excitations predicted in condensed matter physics [176]. By definition, the creation operator $m^{\dagger}$ of a Majorana quasiparticle is self adjoint, i.e. $m^{\dagger}=m$. This property offers possibilities of non-Abelian statistics [177] and topologically protected quantum computation [178] in condensed matter systems. It has been found that different types of hybrid electronic circuits could enclose Majorana quasiparticles. In particular, hybrid structures combining a semiconducting nanowire in contact with a superconductor raise a lot of attention [179–185]. It has been predicted that in some situations a single pair of overlapping Majorana bound states $(\hat{m}_{L}, \hat{m}_{R})$ could appear inside a semiconducting nanowire, with an overlap which can be switched off with an external magnetic field or a gate voltage, in order to obtain two isolated Majorana bound states [186, 187]. Recently, pairs of conductance peaks with a splitting oscillating and decaying with the magnetic field were observed, in striking agreement with these predictions [183, 184]. However, so far, mainly dc conductance measurements have been used, which reveal essentially the DOS of the nanowire. This gives only a very indirect access to the property $m^{\dagger}=m$. A microwave cavity could represent an interesting tool to test this property more directly, since the self-adjoint property affects the structure of the light–matter coupling. Here, we will mainly focus on the proposal of [49], which considers a nanowire contacted to a superconducting contact and two normal metal tunnel probes (figure 18(a)). In practice, it is possible to measure $\Delta\Lambda_{0}$ from the cavity response and the DOS of the nanowire simultaneously by using the tunnel probes. The Keldysh theory was used to calculate these two quantities, by using a coarse grained description of the nanowire (see equation (67) and figures 18(b) and (c)). To understand how a Majorana pair affects physical signals, it is convenient to reexpress the degree of freedom associated with the Majorana pair by defining an ordinary fermion operator $\hat{\gamma}_{1}^{\dagger}=(\hat{m}_{L}-{\rm i}\hat{m}_{R})/\sqrt{2}$ which fulfills $\{\hat{\gamma}_{1}^{\dagger}, \hat{\gamma}_{1}^{\dagger}\}=1$. At low energies, in the subspace spanned by $\hat{\gamma}_{1}^{\dagger}$, one gets the nanocircuit Hamiltonian (23) with $\hat{H}_{0}=\varepsilon\hat{\gamma }_{1}^{\dagger}\hat{\gamma}_{1}$ and $\hat{h}_{\rm int}=\beta\hat{\gamma}_{1}^{\dagger }\hat{\gamma}_{1}$ (with the conventions of equations (64) and (65), one has $\varepsilon=E_{1}$ and $\beta=M_{11}$). This means that the Majorana pair corresponds to a fermionic state which can be split into two fully independent parts for $\varepsilon=0$. In the simplest situation, when the nanowire is driven to its topological phase, ε tends to zero, by showing or not an oscillatory behavior, depending on the length of the nanowire. The electron–hole conjugated pair $(\hat{\gamma} _{1}^{\dagger}, \hat{\gamma}_{1})$ appears in the DOS of the nanowire as a pair of resonances at ε and $-\varepsilon$ (see figure 18(b)). However, no transition should be visible in the cavity signals at $\omega _{0}=2\varepsilon$ because from equation (66), the cavity photons cannot induce transitions between a pair of electron–hole conjugated states (see figure 18(c)). Nevertheless, the light–Majorana coupling has physical consequences, since it can induce a step at $\omega_{0}=\varepsilon$ in the $\Delta\Lambda_{0}$ signal. This feature is due to photo-assisted tunneling between the Majorana pair and the residual zero-energy DOS in the imperfect superconducting reservoir. It can be used to check that the low-energy doublet is well coupled to cavity photons, so that the absence of a cavity resonance at $\omega_{0}=2\varepsilon$ is not due to $\beta=0$. Then, the simultaneous presence[absence] of the step at $\omega_{0}=\varepsilon \lbrack2\varepsilon]$ would represent a good indication that the low energy doublet in the nanowire results from the combination of a non-degenerate electron–hole pair, which is a natural precursor for a Majorana pair. Importantly, this information cannot be directly obtained from the DC current. Importantly, this protocol requires measuring the cavity response and the dc current through the tunnel probes simultaneously. Although the spectroscopic measurement described above is probably the most straightforward measurement to perform with a cavity, many other effects are expected by combining Majorana fermions and cavities. Hence, the direct electric coupling between Majorana bound states and cavity photons has raised a lot of attention recently [44–48]. The indirect coupling of Majorana fermions to cavities through a superconducting quantum bit also raises interest for quantum computation purposes [188–196].

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