Quantum and tunneling capacitance in charge and spin qubits

We present a theoretical analysis of the capacitance of a double quantum dot in the charge and spin qubit configurations probed at high frequencies. We find that, in general, the total capacitance of the system consists of two state-dependent terms: the quantum capacitance arising from adiabatic charge motion and the tunneling capacitance that appears when repopulation occurs at a rate comparable or faster than the probing frequency. The analysis of the capacitance lineshape as a function of externally controllable variables offers a way to characterize the qubits' charge and spin state as well as relevant system parameters such as charge and spin relaxation rates, tunnel coupling, electron temperature, and electron $g$ factor. Overall, our analysis provides a formalism to understand dispersive qubit-resonator interactions which can be applied to high-sensitivity and noninvasive quantum-state readout.

Figure 1

Charge and spin qubits in double quantum dots. (a) A schematic of a DQD system composed of QDs 1 and 2 coupled to a gate electrode G by capacitances $C_{\text{G1}}$ and $C_{\text{G2}}$, respectively. (b) Equivalent circuit of (a) including the geometrical and parametric capacitance in parallel. The resonator, with characteristic inductance $L_{\text{r}}$ and capacitance to ground $C_{\text{r}}$ creates an oscillatory voltage on the gate of the DQD. (c) Energy level spectrum of a DQD as a function of detuning in the single electron regime. Mixing of the states forms a finite gap of magnitude ${\mathrm{\Delta }}_{\text{c}}$ at $\varepsilon =0$. (d) The energy level spectrum of a DQD in the two-electron regime. Split-off triplet states, ${\mathrm{T}}^0, {\mathrm{T}}^{-}$, and ${\mathrm{T}}^{+}$ must be considered in addition to the mixed singlet states.

Figure 2

Charge qubit. (a) Schematic energy level diagram of the DQD indicating the electron dynamics in the slow (${\mathrm{\Gamma }}_1^{\text{c}}\ll {\omega }_{\text{r}}$) and fast (${\mathrm{\Gamma }}_1^{\text{c}}\gg {\omega }_{\text{r}}$) relaxation regime. Only in the fast relaxation regime the system can track the instantaneous thermal population of the system exemplified by the vertical black arrows. (b) FWHM of the parametric capacitance lineshape in the slow (black solid line) and fast relaxation regime (red solid line). The slow regime tends asymptotically to $\mathrm{FWHM}=2{\mathrm{\Delta }}_{\text{c}}$ (black dashed line) and the fast regime to $\mathrm{FWHM}=3.53k_{\text{B}}T$ (red dashed line). Parametric capacitance lineshape as a function of detuning for $k_{\text{B}}T/{\mathrm{\Delta }}_{\text{c}}=0.1,0.5,1,2$ (red, black, dark blue, light blue solid lines) in the slow (c) and fast (d) relaxation regimes. Insets indicate the maximum parametric capacitance as a function of increasing temperature for both regimes.

Figure 3

Charge qubit relaxation. (a) Parametric capacitance lineshape as a function of detuning for ${\mathrm{\Gamma }}_1^{\text{c}}/{\omega }_{\text{r}}=0,1$, and 10 (red, black, blue lines).(b) FHWM of the parametric capacitance peak as a function of reduced charge relaxation rate ${\mathrm{\Gamma }}_1^{\text{c}}/{\omega }_{\text{r}}$ for $k_{\text{B}}T/{\mathrm{\Delta }}_{\text{c}}=0.1,0.5,1$ and 2 (black, red, green and blue lines, respectively). (c) Change in FWHM with respect to temperature as a function of reduced charge relaxation rate ${\mathrm{\Gamma }}_1^{\text{c}}/{\omega }_{\text{r}}$. The dashed horizontal lines at 0 and 3.53 indicate the rate of change in the slow and fast relaxation limits, respectively.

Figure 4

Spin qubit in the slow relaxation regime. (a) Schematic of the electron dynamics at $B=0$ (top panel) and $B=2{\mathrm{\Delta }}_{\text{c}}/g{\mu }_B$ (bottom panel). At zero field, the electron tracks the curvature of the singlet ground state. At finite fields the electron performs diabatic transitions at the singlet-triplet crossing and no relaxation occurs in the timescale of the resonator. (b) Parametric capacitance lineshape as a function of detuning for $k_{\text{B}}T/{\mathrm{\Delta }}_{\text{c}}=0.1,0.25,0.5,1.0,5$ (black, red, green, dark blue, light blue solid lines). The insets indicate the maximum parametric capacitance as a function of increasing temperature. Parametric capacitance as a function of detuning and magnetic field at $k_{\text{B}}T/{\mathrm{\Delta }}_{\text{c}}=0.1$ (c) and $k_{\text{B}}T/{\mathrm{\Delta }}_{\text{c}}=1$ (d). The dot-dashed line in (c) indicates the position in the $\varepsilon -B$ plane of the singlet-triplet crossing point.

Figure 5

Spin qubit in the fast relaxation regime. (a) Schematic of the electron dynamics at $B=2{\mathrm{\Delta }}_{\text{c}}/g{\mu }_B$. The electron performs adiabatic transitions at the singlet-triplet crossing since relaxation occurs much faster than the timescale of the resonator drive. (b) Parametric capacitance lineshape as a function of detuning for $k_{\text{B}}T/{\mathrm{\Delta }}_{\text{c}}=0.1,0.3,0.5,1.0,1.65$ (black, red, green, dark blue, light blue solid lines). The insets indicate the maximum parametric capacitance as a function of increasing temperature. Parametric capacitance as a function of detuning and magnetic field at $k_{\text{B}}T/{\mathrm{\Delta }}_{\text{c}}=0.1$ (c) and $k_{\text{B}}T/{\mathrm{\Delta }}_{\text{c}}=1$ (d). The white dot-dashed line in (c) indicates the singlet-triplet crossing point.

Figure 6

Spin qubit in the intermediate regime. (a) Schematic of the relaxation rates. Relaxation between states with same spin occurs within the timescale of the resonator whereas between states with different spin it does not. (b) Parametric capacitance lineshape as a function of detuning for $k_{\text{c}}T/{\mathrm{\Delta }}_{\text{c}}=0.1,0.2,0.5,1,2$ (black, red, green, dark blue, and light blue lines) and ${\mathrm{\Gamma }}_1^{\text{c}}=10$. (c) Parametric capacitance as a function of detuning and magnetic field at $k_{\text{c}}T/{\mathrm{\Delta }}_{\text{c}}=1$ and ${\mathrm{\Gamma }}_1^{\text{c}}/{\omega }_{\text{r}}=10$. (d) Parametric capacitance lineshape as a function of detuning and reduced relaxation rate for $k_{\text{B}}T/{\mathrm{\Delta }}_{\text{c}}=1$ and $B=0$.