Microwave spectroscopy of a Cooper pair beam splitter

This article discusses how to demonstrate the entanglement of the split Cooper pairs produced in a double-quantum-dot based Cooper pair beam splitter (CPS), by performing the microwave spectroscopy of the CPS. More precisely, one can study the dc current response of such a CPS to two on-phase microwave gate irradiations applied to the two CPS dots. Some of the current peaks caused by the microwaves show a strongly nonmonotonic variation with the amplitude of the irradiation applied individually to one dot. This effect is directly due to a subradiance property caused by the coherence of the split pairs. Using realistic parameters, one finds that this effect has a measurable amplitude.

Figure 1

Scheme of a Cooper pair splitter made out of a carbon nanotube. The two quantum dots $L$ and $R$ are defined by the normal metal contacts (in green) and the superconducting contact (in blue) deposited on top of the carbon nanotube (in light blue). The dot $L\left(R\right)$ is capacitively coupled to a dc gate voltage $V_g^{L\left(R\right)}$ and microwave gate voltage $V_{\mathrm{ac}}^{L\left(R\right)}$. The superconducting contact is connected to ground and the normal metal contacts are biased with a voltage $V_b$.

Figure 2

(a) Energies $E_1$, $E_2$, and $E_{-}$ of the states $|V_1\rangle$, $|V_2\rangle$, and $|T_{-}\rangle$ as a function of $\delta$. (b) Transition frequencies ${\omega }_{V_1{T}_{-}}$, ${\omega }_{T_{-}{V}_2}$ and ${\omega }_{V_1{V}_2}$ of the CPS as a function of $\delta$. (c) Dynamics of the CPS near the working point $\delta =2{\Delta }_r$. We consider a bias voltage regime such that the tunnel transitions between the different CPS states (blue arrows) occur together with the transfer of one electron towards the normal contacts. A microwave irradiation can induce transitions between the states $|V_1\rangle$ and $|V_2\rangle$, $|V_1\rangle$ and $|T_{-}\rangle$, or $|T_{-}\rangle$ and $|V_2\rangle$ without any transfer of electrons between the CPS and the leads (red wavy arrows). We have used $t_{\text{eh}}/{\Delta }_{\mathrm{so}}=1/3$ and ${\Delta }_{K/K^{\prime }}/{\Delta }_{\mathrm{so}}=6$ in (a) and (b).

Figure 3

Coefficients $v_{1\left(2\right)}^2$ and probabilities $P_{V_1}$, $P_{V_2}$, and $P_{T_{-}}$ of the CPS states $|V_1\rangle$, $|V_2\rangle ,$ and $|T_{-}\rangle$ as a function of $\delta$. We have used $t_{\text{eh}}/{\Delta }_{\mathrm{so}}=1/3$, ${\Delta }_{K/K^{\prime }}/{\Delta }_{\mathrm{so}}=3$, $2\pi \hslash {\Gamma }_N/{\Delta }_{\mathrm{so}}=1.37{10}^{-3}$, $e{v}_{\mathrm{ac}}^L/{\Delta }_{\mathrm{so}}=1/15$, $e{v}_{\mathrm{ac}}^R/{\Delta }_{\mathrm{so}}=8/15$, ${\alpha }_L={\alpha }_R=3.{10}^{-4}$, ${\kappa }_L={\kappa }_R={10}^{-2}$, and ${\omega }_{\mathrm{RF}}=3t_{\text{eh}}$.

Figure 4

(a) Current $I_0$ in the absence of any microwave irradiation as a function of $\delta$. (b) Difference between the current $I$ for a finite microwave irradiation and the current $I_0$, as a function of ${\omega }_{\mathrm{RF}}$ and $\delta$. (c) and (d) Current difference $I-I_0$ as a function of $\delta$ for $\hslash {\omega }_{\mathrm{RF}}=1.6t_{\mathrm{eh}}$ and $\hslash {\omega }_{\mathrm{RF}}=3t_{\mathrm{eh}}$. The other parameters used are the same as in Fig. 2.

Figure 5

Amplitude of the current peaks $\Delta I_{V_1\leftrightarrow T_{-}}$ (top panel) and $\Delta I_{V_1\leftrightarrow V_2}$ (bottom panel) as a function of $v_{\mathrm{ac}}^R$ for a constant value of $v_{\mathrm{ac}}^L$, i.e., $e{v}_{\mathrm{ac}}^L/{\Delta }_{\mathrm{so}}=2/3$. The other parameters used are the same as in Fig. 2.