Tunable hole spin-photon interaction based on $\mathtt{g}$-matrix modulation

We consider a spin circuit-QED device where a superconducting microwave resonator is capacitively coupled to a single hole confined in a semiconductor quantum dot. Thanks to the strong spin-orbit coupling intrinsic to valence-band states, the gyromagnetic $\mathtt{g}$ matrix of the hole can be modulated electrically. This modulation couples the photons in the resonator to the hole spin. We show that the applied gate voltages and the magnetic-field orientation enable a versatile control of the spin-photon interaction, whose character can be switched from fully transverse to fully longitudinal. The longitudinal coupling is actually maximal when the transverse one vanishes and vice versa. This “reciprocal sweetness” results from geometrical properties of the $\mathtt{g}$ matrix and protects the spin against dephasing or relaxation. We estimate coupling rates reaching $\sim 10\mathrm{MHz}$ in realistic settings and discuss potential circuit-QED applications harnessing either the transverse or the longitudinal spin-photon interaction. Furthermore, we demonstrate that the $\mathtt{g}$-matrix curvature can be used to achieve parametric longitudinal coupling with enhanced coherence.

Figure 1

(a) Sketch of the setup considered here. A single hole quantum dot (red shape) is confined in a rectangular nanowire with sides $L_y>L_z$ by a partly overlapping gate (light blue) with length $L_x$ and DC bias $V=V_G^0$. The hole is coupled to photons in a microwave coplanar resonator connected to the gate. In this configuration, the shape of the quantum dot (dashed lines) is primarily controlled by the in-plane component ${\mathcal{E}}_y$ of the electric field of the gate [45, 46, 47], which modulates the Larmor vector ${\mathit{\omega }}_L$ of the hole spin owing to spin-orbit coupling. (b) As a consequence, quantized voltage fluctuations $\delta V$ originating from the resonator photon field can change the norm of ${\mathit{\omega }}_L$ (longitudinal coupling $g_{\parallel }$) and/or its orientation (spin-flip transverse coupling $g_{\perp }$).

Figure 2

Maps of (a) the effective $\mathtt{g}$ factor, (b) the perpendicular ${\beta }_{\perp }$ and (c) the parallel $|{\beta }_{\parallel }|$ susceptibilities as a function of the magnetic field orientation $\mathbf{b}=\mathbf{B}/\left|\mathbf{B}\right|$. We consider an idealized silicon device with hard-wall confinement along $y$ and $z$ ($L_z=15\mathrm{nm}$, $L_y=40\mathrm{nm}$), harmonic confinement along $x$ (harmonic length ${\ell }_x=11.5\mathrm{nm}$, as defined in Ref. [47]), and a homogeneous static electric field ${\mathcal{E}}_y=0.12\mathrm{mV}/\mathrm{nm}$ (we assume $\partial {\mathcal{E}}_y/\partial V=-1/L_y$ [62]). The $\mathtt{g}$ matrices are numerically computed from the solutions of the four bands Luttinger-Kohn Hamiltonian [46, 47, 62]. The longitudinal susceptibility $|{\beta }_{\parallel }|$ is maximum for specific orientations (dephasing hot spots) where the transverse susceptibility ${\beta }_{\perp }$ is zero and vanishes along dephasing sweet lines. The transverse susceptibility ${\beta }_{\perp }$ is maximum (fastest Rabi oscillations) very close to such a sweet line.

Figure 3

(a) Maximum parallel and perpendicular susceptibilities ${\beta }_{\parallel }^{max}$ (red) and ${\beta }_{\perp }^{max}$ (blue) as a function of the static electric field ${\mathcal{E}}_y$ for the same parameters as in Fig. 2. They are computed at the magnetic field orientations that maximize, respectively, ${\beta }_{\parallel }$ and ${\beta }_{\perp }$ for each ${\mathcal{E}}_y$. The dashed line pinpoints the electric field ${\mathcal{E}}_y^{*}$ that optimizes ${\beta }_{\parallel }^{max}$ and ${\beta }_{\perp }^{max}$. (b) Second-order longitudinal susceptibility ${\gamma }_{\parallel }$ at the optimal ${\mathcal{E}}_y=0$ and $\mathbf{B}\parallel \mathbf{y}$ as a function of $L_y$. The dots are the numerical calculations with the parameters of Fig. 2, and the line is an analytical fit ${\gamma }_{\parallel }^{max}=0.14m_{\parallel }^h{L}_y^2{e}^2/\left({\pi }^3{\hslash }^2\mathrm{\Delta }\right)$.