\section{Introduction}
Nuclear spins are promising candidates for quantum information applications due to their long coherence times \cite{muhonen2014,pla2013} that can even be observed up to room temperature \cite{saeedi2013}.
However, the small gyromagnetic ratio of nuclear spins that renders them well isolated from the environment and thus underlies their robust coherence also leads to long gate operation times compared to those reported for electron spins. Therefore, nuclear spins will most likely find their application as quantum memories \cite{maurer2012,steger2012}, either for pure storage or as buffers for quantum computation purposes. In many cases, it is beneficial to perform readout directly on the nuclear spin instead of coherently transferring the information to another system, e.g. an electron spin \cite{freer2017}, before this system is read out. Particularly with regard to scalable quantum computing devices, readout relying on electrical means is favourable over methods depending on ac-magnetic fields.  

To a large extent driven by the microelectronics industry, the manufacturing of nanoscale semiconductor devices has matured during the last decades, and silicon-based devices have a particularly high potential for scaling. Moreover, isotopically purified $^{28}\mathrm{Si}$ material containing predominantly nuclear spin $0$ atoms can be produced and thus provides an excellent host material for spin qubits based on the electron spin or single impurity nuclear spins \cite{loss1998,kane1998,tyryshkin2012,zwanenburg2013,awschalom2013,veldhorst2014,morton2008,freer2017}.

Cavity quantum electrodynamics (cQED) has been successfully used for charge-photon  \cite{mi2017,stockklauser2017,bruhat2018} and spin-photon coupling \cite{kubo2010,laflamme2012,tosi2014,mi2018,benito2017,landig2018,samkharadze2018,cubaynes2019}, as well as the detection of photons \cite{schuster2007}. 
Moreover, cQED and gate reflectometry lend themselves for qubit readout \cite{blais2004,didier2015,didier2015a,govia2017,mi2017,danjou2019,zheng2019,west2019,crippa2019}. Nevertheless, it is an open question whether or not single nuclear spins could be detected via a cavity or coupled to cavity photons.
Among the before mentioned achievements in cQED it is particularly noteworthy that the strong coupling regime is accessible for the spin of a single electron in a $\mathrm{Si}$ double quantum dot (DQD) subject to a magnetic field gradient where spin photon coupling emerges due to
\begin{figure}[H]
	\includegraphics[width=\columnwidth]{Figure1}
	\caption{ 
	QD-donor system coupled to a single-mode microwave cavity.  
	The cavity transmission $A_c=\langle a_{\mathrm{out}}\rangle/\langle a_{\mathrm{in}}\rangle$ reveals the nuclear spin state. The QD-donor energy levels are detuned by an amount $\epsilon$ and hybridized by tunnel coupling $t_c$. The spin of the confined electron is subject to a homogeneous magnetic field $B_z$ and a gradient field $b_x$ perpendicular to $B_z$, while the electron charge is coupled to a single mode ($\omega_c$) of a microwave cavity with electric-dipole coupling strength $g_c$. The electron interacts with the nuclear spin of the implanted donor via the hyperfine interaction $A$.}
	\label{fig:model}
\end{figure}
\noindent an effective spin orbit coupling caused by the the combination of the electric dipole interaction and spin-charge hybridization \cite{mi2018,benito2017}. The same mechanism can also be used to realize a flopping-mode spin qubit with full electrical spin control via electric dipole spin resonance  (EDSR) \cite{benito2019,croot2020}. In this system, a longitudinal magnetic field gradient leads to a shift of the phase and amplitude response of the cavity transmission depending on the strength of the field gradient \cite{croot2020}. 
Motivated by this observation we consider a lateral architecture consisting of a quantum dot (QD) in a planar Si/SiGe structure and a single $^{31}\mathrm{P}$ donor implanted in the Si host material. While this system has been successfully operated in the multi electron regime \cite{harvey-collard2017}, we consider the single electron regime to form a flopping-mode electron spin qubit (Fig.~\ref{fig:model}).



As a consequence, if the electron is confined to the donor, it
couples to the donor nuclear spin via the hyperfine interaction. In this configuration, we expect the donor to generate a nuclear spin state dependent Overhauser field. This field constitutes a longitudinal magnetic field gradient that leads to a nuclear spin state and detuning dependent shift of the electron spin transition frequency. Therefore, the cavity response essentially probing the EDSR frequency is expected to shift accordingly.
Our detailed discussion of the expected characteristics in the cavity transmission indicates that the observable signature of the strong electron spin photon coupling \cite{benito2017,mi2018} is indeed significantly altered by the state of the nuclear spin and could therefore be used for nuclear spin state readout. This prediction is verified by calculating the cavity transmission using input-output theory.

Moreover, we investigate the effective excitation conserving nuclear spin photon coupling and find that our suggested method for nuclear spin readout does not require strong nuclear spin photon coupling. 

This article is organized as follows:  The following Sec.~\ref{sec:model} contains a discussion of the model of the QD-donor system coupled to a cavity mode. In Sec.~\ref{sec:nuclearspinreadout} we predict the impact of the donor nuclear spin state on the cavity transmission, and verify our expectation by calculating the cavity transmission using input-output theory. Section~\ref{sec:nuclear_spin-photon_coupling} contains the derivation of an effective Hamiltonian describing the nuclear spin dynamics followed by a discussion of the emerging effective nuclear spin photon coupling. Finally, we summarize our results in Sec.~\ref{sec:conclusion}.

\section{Theoretical Model \label{sec:model}}
We consider a lateral QD-donor system fabricated in isotopically enriched $^{28}$Si. QD and donor are aligned along the $z$-axis, such that a single electron can either be localized in the QD on the left or the  donor on the right by adjusting the  QD-donor energy level detuning $\epsilon$. QD-donor tunnel coupling $t_c$ results in charge hybridization near $\epsilon = 0$. The proposed experimental setup, including the various interactions, is sketched in Fig.~\ref{fig:model}. 
The detuning $\epsilon$ determined by the energy difference between QD and donor can be controlled by applying an electric field in $z$-direction and by tuning the gates defining the QD confinement potential.
In the presence of a homogeneous magnetic field $B_z$ and a magnetic field gradient $b_x$ perpendicular to $B_z$, the
QD-donor system in the single electron configuration can be modelled by the Hamiltonian
\begin{eqnarray}
\widetilde{H}_0=\tfrac{1}{2}\left(\epsilon \widetilde{\tau}_z +2 t_c \widetilde{\tau}_x+B_z \sigma_z + b_x \sigma_x \widetilde{\tau}_z\right),
\label{eq:H0tilde}
\end{eqnarray}
with $\widetilde{\tau}_i$ and $\sigma_i$ the Pauli operators in position and electron spin space, respectively. The interaction between the nuclear spin and the magnetic field is neglected because it is roughly three orders of magnitude smaller than all other relevant energy scales \cite{tosi2017}. The magnetic fields $B_z$ and $b_x$ are given in units of energy and energy units are chosen such that $\hbar=1$. The donor ground and first excited state are energetically separated by $\gtrsim 2.5\,\mathrm{meV}$ taking into account strain effects due to the Si/SiGe interface \cite{usman2015}. On the other hand, a low-lying excited state, the excited valley state, is present in Si QDs. However, valley splittings $\gtrsim50\,\upmu\mathrm{eV}$ observed in recent devices \cite{mi2017b,borjans2021} together with the possibility to operate the QD-donor system at temperatures $\lesssim 50\,\mathrm{mK}$ allow for a negligible population of the excited valley state. Hence, the valley degree of freedom can be neglected in our model.

Electric dipole interactions allow to couple the electron in the
QD-donor system to microwave resonator photons, described by the coupling Hamiltonian
\begin{eqnarray}
\widetilde{H}_I=g_c\widetilde{\tau}_z(a+a^{\dagger}),
\end{eqnarray}
where $a$ and $a^{\dagger}$ are the bosonic cavity photon annihilation and creation operators of the relevant cavity mode, respectively. 
The charge-photon coupling strength for a DQD has been found to be on the order of $g_c/2\pi \approx 30$ to $40\,\mathrm{MHz}$ \cite{mi2018,petersson2012a}. In the QD-donor scenario we expect it to be $\approx 1/3$ of the DQD value, as discussed in Appendix~\ref{appendix:QD-donor system}.
The Hamiltonian for the cavity mode with frequency $\omega_c$ is given by $\widetilde{H}_{\mathrm{cav}}=\omega_c a^{\dagger}a$.
If the electron is confined to the donor, electron spin and $^{31}\mathrm{P}$ donor nuclear spin couple via the hyperfine interaction. The hyperfine interaction strength $A=117 \,\mathrm{MHz}$ \cite{steger2011,feher1959} present in bulk Si is significantly reduced to $A\approx 25\,\mathrm{MHz}$ in the Si quantum well of a Si/Si$_{0.7}$Ge$_{0.3}$ heterostructure due to strain effects caused by the Si- and SiGe-lattice mismatch \cite{huebl2006,usman2015}. On the other hand, the donor is ionized and the electron does not interact with the donor nuclear spin if it occupies the left QD. Therefore, we can represent the electron spin nuclear spin interaction as
\begin{eqnarray}
\widetilde{H}_{e-n}=\frac{A}{8}\vec{\sigma}\cdot \vec{\nu}\cdot (1-\widetilde{\tau}_z),
\label{eq:He-n}
\end{eqnarray}
with $\vec{\nu}=(\nu_x,\nu_y,\nu_z)^T$ and $\nu_i$ the nuclear spin Pauli operators. The factor $(1-\widetilde{\tau}_z)/2$ is a projection on the subspace with the electron bound to the donor. 

 


\begin{figure}[ht]
	\centering
	\includegraphics[width=\columnwidth]{energyspectrum_system}
	\caption{Energy spectrum of the QD-donor Hamiltonian $\widetilde{H}_{0}$ as a function of the
	QD-donor detuning parameter $\epsilon$ for $B_z=14.5\,\upmu\mathrm{eV}$. The dashed blue lines show the energy levels for $t_c=b_x=0$ with the corresponding eigenstates denoted by the electronic position in the
	QD-donor system, $L$ and $R$, and its spin projection $(\downarrow/\uparrow)$ along the $z-$axis. Switching on the tunnel coupling $t_c$ (here $t_c=9\,\upmu\mathrm{eV}$) mixes the $L$ and $R$ states with the same spin resulting in the bonding $(-)$ and antibonding $(+)$ molecular orbital states, whose energies are given by the dashed black lines. Finally, the dotted red energy levels include the hyperfine coupling to the nuclear spin (eigenenergies $E_0,\ldots,E_7$ of $\widetilde{H}_{\rm sys}$). For this plot, we have used exaggerated values $b_x=5\,\upmu\mathrm{eV}$ and $A=1\,\mathrm{GHz}$ to emphasize the effects of the magnetic field gradient and the hyperfine interaction.}
	\label{fig:energylevels}
\end{figure}




Signatures of the electron spin-photon coupling can be observed in the cavity transmission \cite{mi2018,benito2017}. We now investigate whether these signatures will be altered in the presence of a nuclear spin interacting with the electron spin via the hyperfine interaction and whether the combined spin-photon and hyperfine interactions have a potential application for nuclear spin readout. 

To calculate the cavity response, we first transform the total Hamiltonian to the eigenbasis $|\pm\rangle$ 
of $\epsilon \widetilde{\tau}_z/2 +t_c \widetilde{\tau}_x$,
with the electron position expressed in terms of antibonding ($+$) and bonding ($-$) molecular orbital states because these basis states are a good approximation for the eigenstates $|n\rangle$ with corresponding energies $E_n$ of $\widetilde{H}_{\rm sys}=\widetilde{H}_0+\widetilde{H}_{\mathrm{e-n}}$ as illustrated in Fig.~\ref{fig:energylevels}. Then, the Hamiltonian $\widetilde{H}=\widetilde{H}_{\rm sys}+\widetilde{H}_{I}+\widetilde{H}_{\mathrm{cav}}$ can be written as the sum of a diagonal part $H_0$ and an off-diagonal perturbation $V$ as $H=H_0+V$, with 
\begin{align}
H_0=&\tfrac{1}{2}\left(\Omega \tau_z + B_z \sigma_z  \right)+ 
\tfrac{1}{8} A \sigma_z \nu_z \left(1-\sin\theta\, \tau_z\right) + \omega_c a^{\dagger} a,
\label{eq:H0}\\
V=&\frac{b_x}{2}\left(\sin\theta\,\tau_z -\cos\theta\, \tau_x \right)\sigma_x + \frac{A}{8}\cos\theta(\sigma_z \nu_z) \,\tau_x \nonumber\\
&+\frac{A}{8}(\sigma_x \nu_x + \sigma_y \nu_y) \left(1- 
\sin\theta\,\tau_z + \cos\theta\,\tau_x\right)\nonumber\\
&+ g_c (\sin\theta\,\tau_z-\cos\theta\,\tau_x)(a + a^{\dagger}),
\label{eq:V}
\end{align}
where $\tau_i$ are Pauli operators acting on the space of bonding ($+$) and antibonding ($-$) orbitals, i.e. $\tau_z |\pm\rangle=\pm |\pm\rangle$.
Moreover, we introduce the orbital energy $\Omega=\sqrt{\epsilon^2+
	4t_c^2}$ and the orbital mixing angle $\theta=\arctan(\epsilon /2 t_c)$. 
$H_0$ is diagonal with respect to the basis $\{|\pm,\downarrow (\uparrow), \Downarrow (\Uparrow),n\rangle\}$ indicating the orbital state of the electron $(\pm)$, the electron spin state ($\downarrow,\uparrow$), the nuclear spin state ($\Downarrow,\Uparrow$) and the number of photons in the cavity mode $(n)$, respectively, while $V$ is purely off-diagonal in this basis. 
In order to predict the impact of the nuclear spin on the cavity transmission, we derive an effective Hamiltonian for the lower orbital subspace defined by the projection operator 
$P_0=(1-\tau_z)/2$,
that projects on the subspace spanned by the states 
${|-,\downarrow,\Uparrow,n\rangle},{|-,\downarrow,\Downarrow,n\rangle},{|-,\uparrow,\Uparrow,n\rangle},{|-,\uparrow,\Downarrow,n\rangle}$ with $n=0,1,2,\ldots $.
 
 As a next step, we apply a Schrieffer-Wolff transformation to decouple the subspaces defined by the projection operators $P_0$ and $Q_0=1-P_0$ \cite{bravyi2011}, 
to find the effective Hamiltonian $H_{\mathrm{eff}}=e^SHe^{-S}$,
and follow the perturbative method presented in \cite{bravyi2011} to determine the block off-diagonal and antihermitian generator $S$ defining the unitary transformation $e^S$.
If one chooses the ansatz $S=\sum_{n=1}^{\infty}S_n$ with $S_n\sim V^n$, the first contribution $(S_1)$ must obey the relation \cite{bravyi2011}
$[H_0,S_1]= P_0VQ_0+Q_0VP_0$.
This relation together with the commutation relations of the Pauli operators and the bosonic photon operators allows us to determine $S_1$.
The knowledge of $S_1$ is in turn sufficient to compute the effective Hamiltonian for the subspace defined by $P_0$ up to second order in the perturbation $V$ \cite{bravyi2011},
\begin{align}
H_{e}=&P_0H_0P_0+P_0VP_0\nonumber\\
&+\tfrac{1}{2}P_0[S_1,P_0VQ_0+Q_0VP_0]P_0.
\label{eq:Heffeformula}
\end{align}
The explicit form of $H_{e}$ is presented in Appendix \ref{appendix:effective_Hamiltonian}. However, for the following discussion it is essential to determine transition frequencies as precisely as possible. To this end, we transform $H_{e}$ to a basis accounting for the electron spin mixing due to the magnetic field gradient with the basis states
\begin{align}
|\tilde{\downarrow},\Uparrow(\Downarrow)\rangle=&\cos \frac{\phi}{2}|-,\downarrow,\Uparrow(\Downarrow)\rangle
-\sin\frac{\phi}{2}|-,\uparrow,\Uparrow(\Downarrow)\rangle,\\
|\tilde{\uparrow},\Uparrow(\Downarrow)\rangle=&\sin\frac{\phi}{2}|-,\downarrow,\Uparrow(\Downarrow)\rangle
+\cos\frac{\phi}{2}|-,\uparrow,\Uparrow(\Downarrow)\rangle,
\end{align}
defined by the electron spin mixing angle $\phi$ via
\begin{align}
\tan\phi=\frac{-b_x \sin \theta}{B_z\left( 1-\cos^2\theta\left(\tfrac{A^2}{8}+b_x^2\right)\tfrac{1}{\Omega^{2}-B_z^{2}}\right)}.
\end{align}
Since, here, $b_x\ll B_z$ the electron spin mixing angle is small and therefore the states $|\tilde{\downarrow}(\tilde{\uparrow})\rangle$ are predominantly the electron spin states $|\downarrow(\uparrow)\rangle$ up to small contributions of the opposite electron spin state. Hence, in the following we refer to $|\tilde{\downarrow}(\tilde{\uparrow})\rangle$ as the electron spin states. 
The diagonal part of the transformed Hamiltonian reads
\begin{align}
H_{e,0}=& \tfrac{1}{2}\left(E_{\tilde{\sigma}}+\delta E_{\tilde{\sigma}}\nu_z\right)\tilde{\sigma}_z+ \tfrac{1}{2} E_{\nu}\nu_z+\widetilde{\omega}_c a^\dagger a,
\label{eq:Heffdiagonalmain}
\end{align}
with the Pauli operators $\widetilde{\sigma}_i$ operating on the $|\tilde{\downarrow}(\tilde{\uparrow})\rangle$ states and 
\begin{align}
    &E_{\tilde{\sigma}}=\sqrt{{B_z^2}\left(1-\frac{\left(A^2+8b_x^2\right)\cos^2\theta}{8(\Omega^{2}-B_z^{2})}\right)^2+\left(b_x\sin\theta\right)^2},\\
    %
    &\delta E_{\tilde{\sigma}}=\left(\frac{A}{4}(1\!+\!\sin\theta)
    +\frac{\Omega}{B_z}E_\nu\right)\cos\phi
    \nonumber\\
    &\quad\quad\quad -\frac{Ab_x\cos\theta B_z}{8(\Omega^{2}-B_z^{2})}\sin\phi,\\
    %
    &E_{\nu}=\frac{A^2\cos^2\theta\,{B_z}}{{16(\Omega^{2}}-B_z^{2})},\\
    %
    &\widetilde{\omega}_c=\omega_c-2\frac{g_c^2\cos^2\theta\,B_z}{\Omega^{2}-\omega_c^{2}},
\end{align}
as derived in Appendix \ref{appendix:effective_Hamiltonian}. 
Since the signatures of the electron spin-photon coupling that we expect to change  due to the nuclear spin are observed close to resonance between the electron spin transition and the resonator \cite{benito2017,mi2018}, it is justified to assume $E_{\tilde{\sigma}}\approx\widetilde{\omega}_c$. Under this assumption we can apply the rotating wave approximation (RWA) retaining terms rotating with frequencies $\ll E_{\tilde{\sigma}}\approx\widetilde{\omega}_c$ and find that the nondiagonal part of the transformed Hamiltonian comprises interactions between the electron spin and the nuclear spin of the  QD-donor system and the cavity mode
\begin{align}
H_{e,\mathrm{cav-int}}=
&- g_{\tilde{\sigma}\nu}\sin^2\frac{\phi}{2}\left(\tilde{\sigma}^+ \nu^+ a +\tilde{\sigma}^- \nu^- a^{\dagger}\right)\nonumber\\
&+g_{\tilde{\sigma}\nu}\cos^2\frac{\phi}{2}\left(\tilde{\sigma}^+ \nu^- a +\tilde{\sigma}^- \nu^+ a^{\dagger}\right)\label{eq:HeffeintRWA}\\
&+\left(g_{\tilde{\sigma}}\cos\phi+\delta g_{\tilde{\sigma}}\sin\phi\,\nu_z\right)\left(\tilde{\sigma}^+a+\tilde{\sigma}^- a^{\dagger}\right),\nonumber
\end{align}
with the explicit forms of the spin-photon couplings $g_{\tilde{\sigma}\nu}, g_{\tilde{\sigma}}, \delta g_{\tilde{\sigma}}$
given in Appendix \ref{appendix:effective_Hamiltonian}.
The interaction terms in \eqref{eq:HeffeintRWA} are of particular interest since one can expect to see signatures of these interactions in the transmission. However, the terms in the first line are negligible for $\phi\ll 1$. 

The terms in the second line of \eqref{eq:HeffeintRWA} incorporate a flip of both the nuclear spin and the electron spin if the two are antialigned  with the concomitant creation or annihilation of a cavity photon. This coupling emerges due to the combined effect of the dipole operator coupling the states $|-,\uparrow,\Downarrow(\Uparrow)\rangle$ and $|+,\uparrow,\Downarrow(\Uparrow)\rangle$, and the hyperfine interaction between the states $|+,\uparrow,\Downarrow\rangle$ and $|-,\downarrow,\Uparrow\rangle$. Thus, the interaction persists in the absence of the magnetic field gradient and has already been observed and analyzed in setups without such a gradient. The interaction can be used to control the flip-flop qubit and to construct gates between two such qubits \cite{tosi2017}, while the combination with an oscillating magnetic field allows for controlling the nuclear spin qubit and implementing a nuclear spin two-qubit gate \cite{tosi2018}. 

\begin{figure}[ht]
	\centering
	\includegraphics[width=\columnwidth]{energyspectrum_Heffdiag}
	\caption{Hyperfine-split electronic energy levels (\ref{eq:energyexpectationvaluesdown}) and (\ref{eq:energyexpectationvaluesup}) corresponding to the four states $|\tilde{\downarrow},\Uparrow\rangle$, $|\tilde{\downarrow},\Downarrow\rangle$, $|\tilde{\uparrow},\Uparrow\rangle$ and $|\tilde{\uparrow},\Downarrow\rangle$ as a function of the QD-donor detuning parameter $\epsilon$. The highlighted transition frequencies determine the location of the observable signatures in Fig.~\ref{fig:cavityTransmission}. The  parameters used for this plot are $B_z=\omega_c/2\pi=3.5\,\mathrm{GHz}$, $g_c/2\pi=13\,\mathrm{MHz}$, $t_c=9\,\upmu\mathrm{eV}$, $b_x=1.62\,\upmu\mathrm{eV}$, and $A=25\,\mathrm{MHz}$.}
	\label{fig:energyspectrumheffdiag}
\end{figure}

On the other hand, the combined effect of the magnetic field gradient, giving rise to the coupling between the states $|+,\uparrow,\Downarrow(\Uparrow)\rangle$ and $|-,\downarrow,\Downarrow(\Uparrow)\rangle$, and the dipole operator leads to the terms in the third line that describe a flip of the electron spin  accompanied by the annihilation or creation of a cavity photon, while the state of the nuclear spin remains unchanged. These two different types of interaction cause a hybridization of the QD-donor system and the cavity mode when the transition in the QD-donor system is close to resonance with the cavity mode. Since the resulting hybrid states have a significant impact on the cavity transmission we inspect the energy expectation values of the QD-donor system states involved in the respective transitions. 
The energy expectation values of the four basis states defining the lower orbital subspace can be easily read off from \eqref{eq:Heffdiagonalmain}:
\begin{align}
&E_{|\tilde{\downarrow},\Uparrow(\Downarrow)\rangle}=-\frac{E_{\tilde{\sigma}}}{2}\varmp\frac{\delta E_{\tilde{\sigma}}}{2}\varpm\frac{E_{\nu}}{2},\label{eq:energyexpectationvaluesdown}\\
&E_{|\tilde{\uparrow},\Uparrow(\Downarrow)\rangle}= \frac{E_{\tilde{\sigma}}}{2}\varpm\frac{\delta E_{\tilde{\sigma}}}{2}\varpm\frac{E_{\nu}}{2}, 
\label{eq:energyexpectationvaluesup}
\end{align}
and we immediately find the transition frequencies for electron spin flips with a fixed nuclear spin state
\begin{align}
E_{|\tilde{\uparrow},\Uparrow(\Downarrow)\rangle}-E_{|\tilde{\downarrow},\Uparrow(\Downarrow)\rangle}=E_{\tilde{\sigma}}\varpm\delta E_{\tilde{\sigma}},
\label{eq:electrontransitionfrequency}
\end{align}
as well as the transition frequency for the electron spin-nuclear spin flip-flop
\begin{align}
E_{|\tilde{\uparrow},\Downarrow\rangle}-E_{|\tilde{\downarrow},\Uparrow\rangle}=E_{\tilde{\sigma}}-E_{\nu}.
\label{eq:ekectronnuclearflipflop}
\end{align}
\begin{figure*}[ht]
	\includegraphics[width=\textwidth]{Transmission}
	\caption{Cavity transmission $|A_c|$ on resonance, i.e., for $\omega=\omega_c$ with $\omega_c/2\pi=3.5\,\mathrm{GHz}= 14.5\,\upmu\mathrm{eV}$, as a function of the magnetic field $B_z$ and the QD-donor detuning $\epsilon$ for different populations of the energy levels with (a) both nuclear spin states  equally populated, (b)  the nuclear spin up state  populated, (c)  the nuclear spin down state  populated. The  dashed lines indicate the points in the $(\epsilon,B_z)$ parameter plane at which the transitions indicated in Fig.~\ref{fig:energyspectrumheffdiag} fulfill one of the resonance conditions \eqref{eq:resonance_spin_flip} or \eqref{eq:resonance_flipflop}. The remaining system parameters are	$t_c=9\,\upmu\mathrm{eV}$, $b_x=1.62\,\upmu\mathrm{eV}$, $g_c/2\pi=13\,\mathrm{MHz}$, ${2\kappa_1/2\pi=2\kappa_2/2\pi=\kappa/2\pi=1.3\,\mathrm{MHz}}$, and $A=25\,\mathrm{MHz}\approx 0.1\,\upmu\mathrm{eV}$. For the phonon induced decoherence entering the decoherence superoperator $\gamma$ we assume the parameters $d=37\,\mathrm{nm}$, $\omega_0/2\pi=30\,\mathrm{GHz}$, $c_b=4000\,\mathrm{m/s}$ and $J_0$ determined by $J(2t_c=5.4\,\mathrm{GHz},d=120\,\mathrm{nm})=35\,\mathrm{MHz}$. The strength of the quasistatic charge noise affecting the detuning paramter $\epsilon$ is chosen to be $\sigma_{\epsilon}=1\,\upmu\mathrm{eV}$.}
	\label{fig:cavityTransmission}
\end{figure*}
\begin{figure}[ht]
\includegraphics[width=\columnwidth]{goodreadoutpoints}
    \caption{Difference of the cavity transmission $|A_c|$ between the nuclear spin down case  $|A_c|_{\Downarrow}$, and  the nuclear spin up case $|A_c|_{\Uparrow}$. The dot-dashed lines in (a) indicate the positions of the linecuts shown in (b). Parameters and dashed lines are as in Fig.~\ref{fig:cavityTransmission}. }
    \label{fig:readoutdifference}
\end{figure}
\begin{figure}[ht]
    \centering
    \includegraphics[width=\columnwidth]{deltaomegaEpsilonFive}
    \caption{Cavity transmission $|A_c|_{\Uparrow}$, $|A_c|_{\Downarrow}$ and readout contrast, ${|A_c|_{\Downarrow}-|A_c|_{\Uparrow}}$, for $\epsilon=10\,\upmu\mathrm{eV}$ and $B_z=14.57\,\upmu\mathrm{eV}$, i.e. the point where the red and the second dashed orange line from the left in Fig.~\ref{fig:readoutdifference} intersect, as a function of the probe pulse detuning $\delta\omega$. The remaining parameters are as in Fig.~\ref{fig:cavityTransmission}.}
    \label{fig:deltaomega}
\end{figure}
The energy expectation values (\ref{eq:electrontransitionfrequency}) as a function of the QD-donor detuning $\epsilon$ and the various transition frequencies are presented in Fig.~\ref{fig:energyspectrumheffdiag}. 
Both Figure~\ref{fig:energyspectrumheffdiag} and Eq.~\eqref{eq:electrontransitionfrequency} show that the electron spin flip transition frequency depends on the state of the nuclear spin. More precisely, for a small electron spin mixing angle $\phi \ll 1$ the transition frequency with the nuclear spin in the states $\Uparrow$ and $\Downarrow$ differs by 
\begin{align}
\Delta=2\delta E_{\tilde{\sigma}}
\approx\frac{A}{2}\left(1+\sin\theta\right)+\frac{A^2\cos^2\theta\,\Omega}{8(\Omega^{2}-B_z^{2})}.
\label{eq:transitionfrequencydifference}
\end{align}
Hence, in the limits of large positive, zero, and large negative DQD detuning $\epsilon$, the shift in the resonance frequency $\Delta$ takes the values (note that $t_c>0$),
\begin{align}
\epsilon\gg 2t_c: & & \lim_{\theta\rightarrow \pi/2} \Delta\approx A, \nonumber\\
\epsilon=0:& & \Delta\approx \frac{A}{2}+\frac{A^2\Omega}{8(\Omega^{2}-B_z^{2})}, \\
\epsilon\ll - 2t_c: & & \lim_{\theta\rightarrow -\pi/2} \Delta\approx 0. \nonumber
\end{align}
The increasing impact of the nuclear spin on $\Delta$ with increasing QD-donor detuning is intuitively easy to understand: For $\epsilon\ll -2 t_c$ the electron is  localized in the left QD and therefore decoupled from the nuclear spin, at $\epsilon=0$ it is completely delocalized between the left QD and the donor, while it is trapped in the donor with a high probability for $\epsilon\gg 2 t_c$ such that the coupling to the nuclear spin is maximized. 

We note that in a DQD architecture with the second QD overlapping with an isoelectric $^{29}$Si nuclear spin, readout of the nuclear spin state has been realized for the maximal coupling scenario  $\epsilon\gg 2 t_c$ by probing the electron spin resonance frequency with frequency-selective ac magnetic field pulses \cite{hensen2020}. Even though the hyperfine interaction is as low as a few hundred kHz in such a device, we expect that alternatively our suggested readout method can be used as discussed in detail in Appendix \ref{appendix:hyperfineresolution}.


\section{Nuclear spin readout via the electron spin \label{sec:nuclearspinreadout}}

We now describe how the nuclear-spin dependent shift $\Delta$ of the electron-spin resonance frequency Eq.~\eqref{eq:transitionfrequencydifference} allows for a read-out of the nuclear spin.
The last term in \eqref{eq:Heffdiagonalmain} identifies the cavity resonance frequency including shifts of the empty cavity frequency $\omega_c$ due to the interaction with the QD-donor system. Thus, the cavity mode is resonant with the electron spin flip transition for a fixed nuclear spin state if
\begin{align}
E_{|\tilde{\uparrow},\Uparrow(\Downarrow)\rangle}-E_{|\tilde{\downarrow},\Uparrow(\Downarrow)\rangle}=\widetilde{\omega}_c,
\label{eq:resonance_spin_flip}
\end{align} and resonant with the electron spin-nuclear spin flip-flop transition if 
\begin{align}
E_{|\tilde{\uparrow},\Downarrow\rangle}-E_{|\tilde{\downarrow},\Uparrow\rangle}=\widetilde{\omega}_c.
\label{eq:resonance_flipflop}
\end{align}
 We expect a signature of the respective coupling in the cavity transmission in the vicinity of system parameters $\epsilon$, $t_c$, $B_z$, $b_x$, and $\omega_c$ for which one of these relations is fulfilled.
 
In order to verify our prediction we calculate the cavity transmission $A_c$ using input-output theory (Appendix~\ref{sec:InputOutput}) and compare the system parameters for which characteristic features emerge with those satisfying the resonance conditions derived above. The calculation of $A_c$ takes charge relaxation processes due to the phonon environment and quasi-static charge noise affecting the detuning parameter $\epsilon$ into account (see Appendix~\ref{appendix:DecoherenceModel} for details).
Figure~\ref{fig:cavityTransmission} shows the absolute value of the cavity transmission $|A_c|$ for three different populations of the hyperfine levels where (a) the two lowest energy levels are equally populated approximating the thermal equilibrium state for $T\gtrsim 30\,\mathrm{mK}$, i.e., the
QD-donor system is with equal probability in the states $|0\rangle$ and $|1\rangle$ which, up to small corrections, correspond to the nuclear spin up and down states 
$|\tilde{\downarrow},\Uparrow\rangle$ and $|\tilde{\downarrow},\Downarrow\rangle$, respectively; (b) only the ground state $\approx |\tilde{\downarrow},\Uparrow\rangle$ is populated; (c) only the excited state $\approx |\tilde{\downarrow},\Downarrow\rangle$ is populated. We point out that a single measurement will always be represented by the Figs.~\ref{fig:cavityTransmission}$(b)$ or $(c)$, while Fig.~\ref{fig:cavityTransmission}$(a)$ corresponds to the average over many measurements if the system is initialized with equal probability in the states $|0\rangle\approx |\tilde{\downarrow},\Uparrow\rangle$ and $|1\rangle \approx |\tilde{\downarrow},\Downarrow\rangle$ before the measurement. We find that the emerging characteristic features, given by a significantly reduced transmission due to the interaction of the cavity mode with the QD-donor system appear in the immediate vicinity of the parameters fulfilling the resonance conditions Eqns. \eqref{eq:resonance_spin_flip} and \eqref{eq:resonance_flipflop}, as indicated by the dashed lines in Fig.~\ref{fig:cavityTransmission}. 
One also observes that the signatures are less pronounced for $|\epsilon| \gg 2 t_c$. The last line of Eq.~\eqref{eq:V} shows that the electric dipole moment of the $|+\rangle\leftrightarrow|-\rangle$ transition is proportional to $\cos\theta$ and therefore decreases with increasing $|\epsilon/2t_c|$, which, in turn leads to the weakening of the effective couplings responsible for the observed signatures.

For the experimental realization of nuclear spin state readout it is essential to obtain a strong contrast between the signal for nuclear spin $\Uparrow$ and $\Downarrow$. In order to identify suitable readout points, we calculate the difference of the cavity transmission $|A_c|$ obtained for the excited state populated, $|A_c|_{\Downarrow}$, and the one with only the ground state populated, $|A_c|_{\Uparrow}$, i.e., Fig.~\ref{fig:cavityTransmission}$(b)$ is subtracted from Fig.~\ref{fig:cavityTransmission}$(c)$. The result presented in Fig.~\ref{fig:readoutdifference}$(a)$ unveils extended regions providing a high signal contrast for nuclear spin readout in the vicinity of the three resonance conditions, the two resonances \eqref{eq:resonance_spin_flip} and the resonance \eqref{eq:resonance_flipflop}, and weak QD-donor detuning in the range between $\epsilon=-10\,\upmu\mathrm{eV}$ and $\epsilon=15\,\upmu\mathrm{eV}$. The linecuts in Fig.~\ref{fig:readoutdifference}$(b)$ show that, within this range of QD-donor detuning, maximal contrast is achieved for points in the immediate vicinity of the resonance for nuclear spin $\Uparrow$. The amplitude difference of the readout contrast between the resonances for $\Uparrow$ and $\Downarrow$ can be attributed to a shift of the cavity resonance frequency caused by the interaction with the QD-donor system (see Appendix~\ref{appendix:hyperfineresolution} for more details).

Moreover, we can check the sensitivity of the readout contrast with respect to the cavity detuning from the probe field
\begin{align}
    \delta\omega=\omega-\omega_c,
\end{align}
for good readout points. To do so, we calculate 
$|A_c|_{\Uparrow}$, $|A_c|_{\Downarrow}$ and the readout contrast, $|A_c|_{\Downarrow}-|A_c|_{\Uparrow}$, for the point in Fig.~\ref{fig:readoutdifference} where the red and the second dashed orange line from the left intersect, as a function of the detuning $\delta\omega$. The result is presented in Fig.~\ref{fig:deltaomega} and shows a readout contrast larger than $0.2$ for $|\delta\omega|<1.5\,\mathrm{MHz}$. In addition, the figure allows one to identify the origin of reduced transmission in Fig.~\ref{fig:cavityTransmission}$(b)$ and the  resulting good readout contrast:
At the chosen readout point, the electron spin flip transition for nuclear spin $\Uparrow$ is close to resonance with the cavity, while the electron spin flip transition for nuclear spin  $\Downarrow$ is off-resonant. Due to the strong electron spin photon-coupling one observes Rabi splitting for nuclear spin $\Uparrow$ ($|A_c|_{\Uparrow}$ in Fig.~\ref{fig:deltaomega}), whereas $|A_c|_{\Downarrow}$ shows a single resonance located between the Rabi split modes of $|A_c|_{\Uparrow}$.  

To further characterize the nuclear spin measurement, we go beyond the input-output theory and  inspect Eqs.~\eqref{eq:Heffdiagonalmain} and \eqref{eq:Heintfull} describing the effective electronic Hamiltonian $H_{e,0}+H_{e,\mathrm{int}}$ in order to assess the expected measurement back-action. We note that since $[H_{e,0},\nu_z]=0$, the main part
of the hyperfine coupling leads to a nuclear spin readout in the form of a quantum non-demolition (QND) measurement \cite{braginsky1992}.  In general, $[H_{e,\mathrm{int}},\nu_z]\neq 0$ leading to small corrections to the QND behavior. However, for an adiabatic transfer of the electron from the left QD to the delocalized configuration between the QD and the donor and back under continuous transmission of a microwave field at constant frequency, we expect a recovery of the QND readout because the pure nuclear spin states are adiabatically transferred to eigenstates of $H_{e,0}+H_{e,\mathrm{int}}$.
Away from the resonance, the analogous argument holds with the non-resonant Hamiltonian \eqref{eq:effectiveHamiltonianTransmission}.

For the experimental verification of the suggested method for nuclear spin readout, we envision the following protocol: The cavity transmission is measured at one of the suitable readout points. Then, a nuclear spin resonance $\pi$-pulse is performed before the cavity transmission is probed again. Following the above discussion, successful nuclear spin readout is achieved if there is a significant difference in the absolute value of the transmission, and, depending on this value for the respective measurement, the state of the nuclear spin at the time of each measurement can be assigned.

\section{Nuclear spin photon coupling \label{sec:nuclear_spin-photon_coupling}}

 It has been shown that the nuclear spin of a QD-donor system can be controlled with a classical electric field \cite{boross2018}. However, this does not allow coherent information transfer between the nuclear spin and photons.
In order to assess the potential of the system for coherent coupling of the nuclear spin to cavity photons, we derive a Hamiltonian describing the effective dynamics of the nuclear spin interacting with the resonator mode while the remaining parts of the system are near the ground state.
More precisely, we investigate the dynamics of the subspace determined by the projection operator
\begin{align}
P_0=\frac{1-\tau_z}{2}\,\frac{1-\sigma_z}{2},
\end{align}
that defines the subspace spanned by the states ${|-,\downarrow,\Downarrow,n\rangle}$, 
${|-,\downarrow,\Uparrow,n\rangle}$, with $n=0,1,2,...$ .
 To do this, we apply a Schrieffer-Wolff transformation to decouple the subspaces defined by the projection operators $P_0$ and $Q_0=1-P_0$ \cite{bravyi2011}. Following the  procedure sketched in Sec. \ref{sec:nuclearspinreadout}, we determine $S_1$ and $S_2$, where $S_2$ is defined by 
$[H_0,S_2]=  -[P_0VP_0+Q_0VQ_0,S_1]$,
to obtain the effective Hamiltonian for the subspace defined by $P_0$ up to third order in the perturbation $V$ \cite{bravyi2011},
\begin{align}
H_{n}=&P_0H_0P_0+P_0VP_0+\tfrac{1}{2}P_0[S_1,P_0VQ_0+Q_0VP_0]P_0\nonumber\\
&+\tfrac{1}{2}P_0[S_2,P_0VQ_0+Q_0VP_0]P_0.
\label{eq:Heffformulanu}
\end{align}
In particular, the diagonal part of the effective Hamiltonian reads
\begin{align}
    H_{n,0}=&\frac{E_{\Bar{\nu}}}{2}\nu_z+\Bar{\omega}_ca^\dagger a+\frac{\delta E_{\Bar{\nu}}}{2}\nu_za^\dagger a,
\end{align}
with the expressions for ${E_{\Bar{\nu}},\,\Bar{\omega}_c}$ and $\delta E_{\Bar{\nu}}$ presented in Appendix \ref{appendix:EffectiveHnuclearspin}. We find that ${|\delta E_{\Bar{\nu}}|\ll |E_{\Bar{\nu}}|\ll \Bar{\omega}_c}$ if the electron is not entirely confined to the left QD. Thus, the microwave resonator and the donor nuclear spin flip transition cannot be tuned to resonance.
The coherent excitation exchange between these two subsystems is described by the term, 
\begin{align}
   
    g_{\nu}\left(\nu^{+}a^{\dagger}+\nu^{-}a\right),
    \label{eq:nuplusadager}
\end{align} 
within $H_n$ given in \eqref{eq:Heff}. We note that ${E_{|\Downarrow,n\rangle}>E_{|\Uparrow,n\rangle}}$ because ${E_{\Bar{\nu}}\approx-\tfrac{A}{4}(1+\sin(\theta))+\mathcal{O}(V^2)}$ such that Eq.~\eqref{eq:nuplusadager} is an excitation conserving interaction term.
The explicit form of the coupling constant $g_{\nu}$ in terms of the system parameters is given in Appendix~\ref{appendix:EffectiveHnuclearspin} and we find that nuclear spin to photon coupling strengths of  ${g_\nu\approx 0.5}$ MHz can be achieved. 
Given realistic values for the nuclear-spin 
and cavity loss rates, $\gamma\ll \kappa \approx 1$ MHz,
we note that the strong coupling regime for nuclear spin cavity QED ($g_\nu\gg \kappa, \gamma$) should be within reach. However, the coherent excitation exchange between these two subsystems is suppressed by the large detuning from resonance.

The raising or lowering of the nuclear spin state along with the creation or annihilation of a cavity photon results from the combined effect of the hyperfine interaction, the magnetic field gradient and the electric dipole interaction. 
The fundamental problem preventing resonant coupling is that, in principle, the energy splitting between the orbital states ($+$ and $-$) and the energy splitting between the electron spin states ($\uparrow$ and $\downarrow$) can simultaneously be tuned close to resonance with the microwave resonator, while, at the same time, the energy splitting between the nuclear spin states ($\Uparrow$ and $\Downarrow$) is far off-resonant because the nuclear gyromagnetic ratio is $\approx 1000$ times smaller than the one of the electron spin.


\section{Conclusion \label{sec:conclusion}}
In conclusion, we have investigated a system composed of a donor nuclear spin coupled to the spin of a single electron in a QD-donor architecture via the hyperfine interaction. The electron is subject to a homogeneous magnetic field and a magnetic field gradient perpendicular to the homogeneous component, while it is also dipole coupled to a microwave resonator.
We demonstrate that the effective excitation-conserving nuclear spin-photon interaction resulting from the combined effect of the hyperfine interaction, the electric dipole interaction, and the magnetic field gradient cannot directly be tuned to resonance. 

Nevertheless, we show that the signature of the strong electron spin-photon coupling \cite{benito2017} in the cavity transmission is altered due to the hyperfine interaction. We find well separated signatures for the electron spin-photon coupling with the nuclear spin in the states $\Uparrow$ and $\Downarrow$, whereby the splitting of the two signatures is determined by the hyperfine interaction strength $A$. For a $^{31}\mathrm{P}$ donor in the strained Si quantum well with $A\approx 25\,\mathrm{MHz}$ we expect that recent experimental setups are able to resolve the split signatures individually. Moreover, we identified good readout points at which one finds a high contrast between the measurement signal for the two opposing nuclear spin polarizations. Therefore, the cavity transmission allows for a readout of the nuclear spin state and for the measurement of the hyperfine interaction strength. 


\section*{Acknowledgments}
We thank M\'{o}nica Benito and N. Tobias Jacobson for helpful discussions. This work has been supported by ARO grant number W911NF-15-1-0149.


