Current correlations of Cooper-pair tunneling into a quantum Hall system

We study Cooper-pair transport through a quantum point contact between a superconductor and a quantum Hall edge state at integer and fractional filling factors. We calculate the tunneling current and its finite-frequency noise to the leading order in the tunneling amplitude for dc and ac bias voltage in the limit of low temperatures. At zero temperature and in the case of tunneling into a single edge channel both the conductance and differential shot noise vanish as a result of the Pauli exclusion principle. In contrast, in the presence of two edge channels, this Pauli blockade is softened and a nonzero conductance and shot noise are revealed.

Figure 1

Schematic representation of the system: a QPC with tunneling amplitude $\tau$ connects a superconductor (SC) to the chiral edge states of an integer quantum Hall (QH) phase at filling factor $\nu$. At $\nu =1$ both electrons of the Cooper pair would have to occupy the same state, leading to a Pauli blockade, while at $\nu =2$ the electrons can enter different states. The bias is applied between the chiral edge channel and the superconductor.

Figure 2

The normalized tunneling current $I_{dc}\left(V\right)/I_0$ for filling factor $\nu =1$ oscillates with the dimensionless bias $2V\xi /v_F$ around an Ohmic behavior, with nonzero temperature damping the oscillation [see Eqs. (10) and (13)].

Figure 3

At filling factor $\nu =2$, the normalized tunneling current oscillates both with the dimensionless bias $2V\xi /v_F$ and the interference factor argument $k\xi$. These oscillations are shown at finite temperature, $2\xi T/v_F=0.5$, and tunneling probability $p=0.5$ [see Eq. (14)]. Black lines indicate integer values of the current.

Figure 4

When the electrons tunnel into two interacting QH edge states, we see a decrease in the magnitude of the normalized tunneling current, as well as longer oscillation periods with applied bias, for stronger interactions. The figure shows the case for finite temperature $2\xi T/v_F=0.5$ with an interaction parameter $v_2/v_1$ which has $v_2=v_F$ [see Eq. (21)], and where $v_2/v_1=1.0$ is the noninteracting case.

Figure 5

The normalized finite-frequency nonsymmetrized noise of the dc tunneling current $S_{dc}(\omega ,V)/I_0$ at filling factor $\nu =2$ as a function of the dimensionless frequency $\omega \xi /v_F$ at applied voltages $2V\xi /v_F=4$ (blue lines) and $2V\xi /v_F=6$ (red lines). Here, the solid lines correspond to finite temperature $2T\xi /v_F=0.5$ and the dashed lines correspond to zero temperature. Notably, we find no resonances (singularities). We have $p=0.5$ and $k\xi =2\pi /3$. See Eqs. (30) and (31).

Figure 6

The normalized finite-frequency nonsymmetrized noise of the tunneling current in the presence of periodic time-dependent (ac) bias, $\tilde{V}\left(t\right)=V+V_{1}cos\left(\mathrm{\Omega }t\right)$ at zero temperature for different $2V\xi /v_F$ [see Eq. (42)]. Again we find no resonances. In this figure we show the case of filling factor $\nu =1$ and choose $S_{dc}(\omega ,V)$ from Eq. (31). We set $p=0$ or 1, $2V_1\xi /v_F=2$, and $\mathrm{\Omega }\xi /v_F=1$.