Phase-dependent noise correlations in normal-superconducting structures

We study nonequilibrium noise correlations in diffusive normal-superconducting structures in the presence of a supercurrent. We present a parametrization for the quasiclassical Green’s function in the first order of the counting field $\chi$. This we employ to obtain the voltage and phase dependence of cross and autocorrelations and to describe the role played by the setup geometry. We find that the low-voltage behavior of the effective charge $q_{\mathrm{eff}}$ describing shot noise is a result of a competition between the anticorrelation of Andreev pairs due to proximity effect and the depression of the local density of states. Furthermore, we show that the noise correlations are independent of the sign of the supercurrent.

Figure 1

Setup schematic studied in this work.

Figure 2

(Color online) Effective charge $q_{\mathrm{eff}}$ vs voltage and $\Delta \varphi$. The voltage at which the minimum $q_{\mathrm{eff}}$ is obtained follows the phase-dependent minigap in the SNS system (bold dashed black curve). Note that the junction undergoes a $\pi$ transition, in which the supercurrent changes its sign, at $eV\approx 2E_T$ (Refs. 25, 26).

Figure 3

(Color online) Effective charge $q_{\mathrm{eff}}$ vs $eV$ for $\Delta \varphi =0$ (main figure) and $\Delta \varphi =2\pi ∕3$ (inset). The cross section $A_c$ is close to 0 (blue dotted), $A∕4$ (blue solid), $A$ (black bold 엯), and $4A$ (red dashed) and for these curves $L_{1,2,3}=L$. The values for $L_3$ are $1∕8$ (green dash-dotted ◻), $1∕2$ (green solid ∗), 2 (magenta solid ▿), and 4 (magenta dashed +) times $L_{1,2}=L$ with $A_c=A$; $E_T=\hslash D∕L^2$. For $\Delta \varphi =2\pi ∕3$, the minima of $q_{\mathrm{eff}}$ occur at lower $V$ and except for the curves for $A_c=0$ and $L_3=L∕8$, the dips are deeper than for $\Delta \varphi =0$. Note the different energy scales on the $x$ axes in the inset and in the main figure.

Figure 4

(Color online) Effective charge $q_{\mathrm{eff}}$ for different values of phase difference in up-down symmetric $(s=1)$ and asymmetric $(s=1∕4)$ structures.

Figure 5

(Color online) Normalized differential cross correlations $(1∕2e)(d{S}_{23}∕d{\overline{I}}_3)$ vs $eV$, for $\Delta \varphi =0$ (black solid), $\Delta \varphi =\pi ∕2$ (blue dotted), $\Delta \varphi =2\pi ∕3$ (red dashed), and $\Delta \varphi =8\pi ∕9$ (green dash dotted) and $s=1∕4,1∕2,3∕4,1$ for the curves from top to bottom. The cross correlations related to the opposite wire are directly obtained from $S_{13}=-S_{23}-S_{33}$. The results for $S_{23}$ and the symmetry parameter $s$ are identical for those for $S_{13}$ and $2-s$ by the symmetry of $S_{ij}$ under the reversal of the phase gradient.