Input-output theory for spin-photon coupling in Si double quantum dots

The interaction of qubits via microwave frequency photons enables long-distance qubit-qubit coupling and facilitates the realization of a large-scale quantum processor. However, qubits based on electron spins in semiconductor quantum dots have proven challenging to couple to microwave photons. In this theoretical work we show that a sizable coupling for a single electron spin is possible via spin-charge hybridization using a magnetic field gradient in a silicon double quantum dot. Based on parameters already shown in recent experiments, we predict optimal working points to achieve a coherent spin-photon coupling, an essential ingredient for the generation of long-range entanglement. Furthermore, we employ input-output theory to identify observable signatures of spin-photon coupling in the cavity output field, which may provide guidance to the experimental search for strong coupling in such spin-photon systems and opens the way to cavity-based readout of the spin qubit.

Figure 1

Schematic illustration of the Si gate-defined DQD influenced by an homogeneous external magnetic field, $B_z$, and the inhomogeneous perpendicular magnetic field created by a micromagnet, with opposite direction at the positions of the two QDs, $\pm B_x$. The DQD is electric-dipole-coupled to the microwave cavity represented in blue. The cavity field is excited at the left and right ports via $a_{\mathrm{in},1}$ and $a_{\mathrm{in},2}$, and the output can be measured either at the left ($a_{\mathrm{out},1}$) or right port ($a_{\mathrm{out},2}$).

Figure 2

(a),(b) Energy levels $E_n(n=0,\cdots ,3)$ as a function of the DQD detuning parameter $\epsilon$. The dashed lines are the energy levels without a magnetic field gradient ($B_x=0$). They correspond to the bonding ($-$) and antibonding (+) orbitals with spin $\uparrow ,\downarrow$ in the $z$ direction, denoted by $\left|\pm ,\uparrow (\downarrow )\right.〉$. The arrow represents the transition driven by the probe field, at frequency ${\omega }_{\mathrm{R}}$. Here, we choose the parameters $B_z=24\mu \mathrm{eV}$ and $B_x=10\mu \mathrm{eV}$. For the tunnel coupling: (a) $t_c=15.4\mu \mathrm{eV}>B_z/2$ and (b) $t_c=10.2\mu \mathrm{eV}<B_z/2$. (c),(d) Schematic representation of the $\mathrm{\Lambda }$ system that captures the essential dynamics in (a) and (b), respectively (near $\epsilon =0$). If the orbital energy, $\mathrm{\Omega }=\sqrt{{\epsilon }^2+4t_c^2}$, is near $B_z$, the levels $\left|-,\uparrow \right.〉$ and $\left|+,\downarrow \right.〉$ hybridize into the states $\left|1\right.〉$ and $\left|2\right.〉$ due to the magnetic field gradient, while the ground state is approximately unperturbed $\left|0\right.〉\sim \left|-,\downarrow \right.〉$. The wavy lines represent charge decoherence with rate ${\gamma }_c$.

Figure 3

(a) Expected effective coupling $g_s/g_c=\left|d_{01\left(2\right)}\right|$, according to Eqs. (22) and (23) as a function of $t_c$ and $\epsilon$. The black dashed line corresponds to $\mathrm{\Omega }=B_z$ and separates the region where $g_s/g_c=\left|d_{0,1}\right|$ (above) from the region where $g_s/g_c=\left|d_{0,2}\right|$ (below). The most interesting region lies in between the two white dashed lines, where our approximations are accurate $[\sqrt{{(\mathrm{\Omega }-B_z)}^2+B_x^2}\ll (\mathrm{\Omega }+B_z)]$. We chose $B_x=1.62\mu \mathrm{eV}$ and $B_z=24\mu \mathrm{eV}$. (b) Spin-photon coupling strength $g_s/g_c$ and spin decoherence rate ${\gamma }_s/{\gamma }_c$ as a function of $t_c$ for $\epsilon =0$, $B_z=B_z^{\mathrm{res}}$, and $B_x=1.62\mu \mathrm{eV}$. Between the two blue vertical lines, the resonance cannot be achieved by tuning $B_z$. Inset: ratio $g_s/\sqrt{[{\gamma }_s^2+{(\kappa /2)}^2]/2}$ in the same range. The coupling is strong when this quantity is larger than one (dashed line). We have chosen ${\gamma }_c/2\pi =100\mathrm{MHz}$, $g_c/2\pi =40\mathrm{MHz}$, and $\kappa /2\pi =1.77\mathrm{MHz}$.

Figure 4

Cavity transmission spectrum, $\left|A\right|$, as a function of $B_z$ and $\epsilon$ at zero driving frequency detuning ${\mathrm{\Delta }}_0=0$. The other parameters are $t_c=15.4\mu \mathrm{eV}$, $B_x=1.62\mu \mathrm{eV}$, ${\gamma }_c/2\pi =(100+150|sin\theta \left|\right)\mathrm{MHz}$, $g_c/2\pi =40\mathrm{MHz}$, $\kappa /2\pi =1.77\mathrm{MHz}$, and ${\omega }_{\mathrm{c}}/2\pi =5.85\mathrm{GHz}\approx 24\mu \mathrm{eV}$.

Figure 5

(a) Cavity transmission spectrum $\left|A\right|$ as a function of $B_z$ and ${\mathrm{\Delta }}_0$. (b) [(c)] shows $\left|A\right|$ as a function of $B_z$ (${\mathrm{\Delta }}_0$) for the value of ${\mathrm{\Delta }}_0$ ($B_z$) indicated by the black (blue) dashed line. In (a) $t_c=15.4\mu \mathrm{eV}$, while in (b) and (c) we show the result for this value (solid line) and for $t_c=13.9\mu \mathrm{eV}$ (dashed line) and $t_c=19.7\mu \mathrm{eV}$ (dotted line). The other parameters are $\epsilon =0$, $B_x=1.62\mu \mathrm{eV}$, ${\gamma }_c/2\pi =100\mathrm{MHz}$, $g_c/2\pi =40\mathrm{MHz}$, $\kappa /2\pi =1.77\mathrm{MHz}$, and ${\omega }_{\mathrm{c}}/2\pi =5.85\mathrm{GHz}\approx 24\mu \mathrm{eV}$.

Figure 6

Cavity transmission $\left|A\right|$ as a function of ${\mathrm{\Delta }}_0$ close to the resonant field for different values of the charge-cavity coupling $g_c/2\pi =\{40,80,160\}\mathrm{MHz}$. The magnetic field has been slightly detuned from the resonance condition to make the relative heights of the vacuum Rabi split modes the same. The rest of the parameters are $t_c=15.4\mu \mathrm{eV}$, $\epsilon =0$, $B_x=1.62\mu \mathrm{eV}$, ${\gamma }_c/2\pi =100\mathrm{MHz}$, $g_c/2\pi =40\mathrm{MHz}$, $\kappa /2\pi =1.77\mathrm{MHz}$, and ${\omega }_{\mathrm{c}}/2\pi =5.85\mathrm{GHz}\approx 24\mu \mathrm{eV}$.

Figure 7

Strength of the spin-cavity coupling $g_s/\mathrm{\Gamma }$ according to Eq. (45) as a function of $t_c$ and $\epsilon$. $B_z$ has been adjusted to the resonance condition Eq. (44). Note that the strong-coupling regime ($g_s/\mathrm{\Gamma }>1$) is achieved away from the black dashed line, $\mathrm{\Omega }={\omega }_{\mathrm{c}}=24\mu \mathrm{eV}$, in agreement with Fig. 3. The other parameters are $B_x=1.62\mu \mathrm{eV}$, ${\gamma }_c/2\pi =(100+150|sin\theta \left|\right)\mathrm{MHz}$, $g_c/2\pi =40\mathrm{MHz}$, and $\kappa /2\pi =1.77\mathrm{MHz}$.

Figure 8

(a) Expected effective coupling $g_s/g_c=\left|d_{01\left(2\right)}\right|$, according to Eqs. (C9) and (C10) as a function of $t_c$ and $\epsilon$. The black dashed line corresponds to ${\mathrm{\Omega }}_{+}+{\mathrm{\Omega }}_{-}=2B_z$ and separates the region where $g_s/g_c=\left|d_{0,1}\right|$ (above) from the region where $g_s/g_c=\left|d_{0,2}\right|$ (below). (b) Vertical cuts for $\epsilon =0$ (black) and $\epsilon =20\mu \mathrm{eV}$ (blue) for different values of ${\mathrm{\Delta }}_z$. We chose $B_x=1.62\mu \mathrm{eV}$, $B_z=24\mu \mathrm{eV}$, and the values of ${\mathrm{\Delta }}_z$ indicated on the figures.