Adiabatic Cooper pair splitter

Recent experiments have observed Cooper pair splitting in quantum dots coupled to superconductors, and efficient schemes for controlling and timing the splitting process are now called for. Here, we propose and analyze an adiabatic Cooper pair splitter that can produce a regular flow of spin-entangled electrons in response to a time-dependent and periodic gate voltage. The splitting process is controlled by moving adiabatically back and forth along an avoided crossing between the empty state and the singlet state of two quantum dots that are coupled to a superconductor, followed by the emission of the split Cooper pairs into two normal-state drains. The scheme does not rely on fine-tuned resonance conditions and is therefore robust against experimental imperfections in the driving signal. We identify a range of driving frequencies, where the output currents are quantized and proportional to the driving frequency combined with suppressed low-frequency noise. We also discuss the main sources of cycle-missing events and evaluate the statistics of electrons emitted within a period of the drive as well as the distribution of waiting times between them. Realistic parameter estimates indicate that the Cooper pair splitter can be operated in the gigahertz regime.

Figure 1

Adiabatic Cooper pair splitter. (a) The device consists of a nanowire (green) with gate-defined quantum dots coupled to a superconductor ($S$, blue). The amplitude for Cooper pair splitting is denoted by $\gamma$, while $\mathrm{\Gamma }$ is the rate at which electrons are emitted into the normal-state electrodes ($N$, red). A time-dependent gate voltage, $V_g\left(t\right)$, is used to control the left quantum dot level. (b) The superconducting gap is denoted by $\mathrm{\Delta }$, and ${\varepsilon }_L$ and ${\varepsilon }_R$ are the tunable level positions. (c) Current as a function of the level positions for a static device with $\hslash \mathrm{\Gamma }/\gamma =0.01$. In our scheme, we move the left level back and forth across the peak in the current. (d) The amplitude for Cooper pair splitting leads to an avoided crossing between the singlet state $|S\rangle$ and the empty state $|0\rangle$. We move back and forth along the avoided crossing, and a split Cooper pair is emitted into the drains after each crossing.

Figure 2

Adiabatic driving scheme, average current, and low-frequency noise. (a) The period of the drive is divided into two splitting phases and two emission phases, each of duration ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$, respectively, such that $\mathcal{T}=2({\mathcal{T}}_1+{\mathcal{T}}_2)$. During the splitting phases, the left level is moved across the resonance, ${\varepsilon }_L=-{\varepsilon }_R$, and a Cooper pair is split. During the emission phases, the levels are kept far off resonance, and the split Cooper pair tunnels into the drains. (b) Average current as a function of the driving frequency. The parameters are $\kappa =\gamma$; ${\varepsilon }_1=50\gamma$; ${\varepsilon }_2=100\gamma$; ${\varepsilon }_R=-100\gamma$; and $\hslash \mathrm{\Gamma }=0.001\gamma$ (red), $0.002\gamma$ (green), and $0.003\gamma$ (blue), and we have defined $f_0=\alpha \gamma /2\pi \hslash$ with $\alpha ={\mathcal{T}}_1/\mathcal{T}=0.01$. The adiabatic regime, where the current should take on the value $I_{\ell }=2f$, is indicated for the red curve by the shaded area according to Eq. (3). (c) The Fano factor, $F_{\ell }=S_{\ell }/I_{\ell }$, as a function of the driving frequency. The three circles in panel (b) indicate the frequencies used in Figs. 3 and 4.

Figure 3

Time-dependent currents. We show the time-dependent currents corresponding to the three points marked with circles in Fig. 2. (a) At low frequencies, more than one electron is emitted per half-period, and emissions occur already in the splitting phase (see inset). (b) In the adiabatic regime, one electron is emitted at every half-period, and the leakage current in the splitting phase is suppressed. (c) At high frequencies, the regularity is gradually lost, and there is always a finite current running.

Figure 4

Distribution of waiting times. We show distributions corresponding to the three points marked with circles in Fig. 2. (a) At low frequencies, a peak develops at short waiting times. (b) In the adiabatic regime, a single peak at half the period shows that Cooper pairs are being split periodically. (c) At high frequencies, cycle-missing events give rise to several peaks.