# Coherent current correlations in a double-dot Cooper pair splitter

Bogdan R. Bułka  Institute of Molecular Physics, Polish Academy of Sciences,
ul. M. Smoluchowskiego 17, 60-179 Poznań, Poland

(Received )

###### Abstract

Exact analytical formulas are derived, by means of Keldysh Green functions,
for currents and current correlation functions in a Cooper pair splitter
modelled on a double quantum dot system coherently coupled to a superconductor
and two normal metallic electrodes. Confining to the subspace with the inter-
dot singlet we show perfect entanglement of split electrons in two separated
crossed Andreev reflection processes. The studies are focused on the noise
power spectrum in a whole bias voltage range. In particular, in the large
voltage limit shot noise dominates and its spectrum exhibits two extraordinary
side dips related to resonant inter-level current correlations caused by
coherent electron-hole recombination processes accompanied by emission and
absorption of photons. In the linear response limit we derived the frequency
dependent admittance which shows different interference patterns for the cross
and the auto current correlations.

##  I Introduction

Electronic transport through a nanoscopic system is stochastic in nature, and
therefore, in investigations one needs to measure and analyze the currents,
the second-order current correlation functions (noise power) and all other
high-order cumulants to get the full counting statistics (FCS) [1, 2, 3]. In
particular, the current-current correlations give insight into quantum noise
in nanoscopic electrical circuits [4], shot noise and dynamics of charge
transfers in the presence of interactions [5], entanglement of scattered
particles [6, 7, 8, 9, 10], their fermionic or bosonic nature [11] as well as
the effective charge in the fractional quantum Hall effect [12, 13, 14].

Ubbelhode et. al. [15] used a quantum point contact as a highly sensitive
counter of charge transport in a single-electron transistor. This setup
allowed them to determine directly the current statistics and timescales of
the current fluctuations (up to tens of $\mu$s) as well as the frequency-
dependent third order correlation function (the skewness). An active quantum
detector system was proposed [16, 17, 18] to measure quantum noise, and
observe zero-point fluctuations. Such a device was fabricated [19] on a double
quantum dot as a tunable two-level system and showed current fluctuations in
the capacitive coupled conductor over a very wide frequency range (up to
hundreds of GHz). An alternative quantum detector was based on a
superconductor-insulator-superconductor tunnel junction as an on-chip spectrum
analyzer [20], which allowed one to measure the nonsymmetrized current noise
arising from coherent charge oscillations in a superconducting charge qubit
[21, 22], or generated by the tunneling of quasiparticles across a Josephson
junction polarized in the vicinity of the superconducting gap [23, 24]. A
similar experimental setup was used to investigate the nonequilibrium dynamics
of many-body phenomena on the nanoscale, namely to measure the high frequency
current fluctuations of a carbon nanotube quantum dot in the Kondo regime
[25].

In electronic transport one can distinguish two regimes: sequential tunneling
and coherent (ballistic) transport. In the sequential regime following
tunneling events are independent and the transport is described within the
Markov approach, using the quantum master equation [26]. In particular, the
current correlation functions can be derived [27, 28, 29], one can get easily
FCS [30, 31, 32, 33] with all order cumulants for the zero-frequency case, as
well as, with some effort, frequency-dependent cumulants [34]. Using the
spectral decomposition of the noise power spectrum one can distinguish various
relaxation processes [27], related to local charge and spin fluctuations, as
well as see inter-channel current correlations, which can lead to the sub- or
super-Poissonian type of shot noise (showing anti-bunching or bunching of
transferred particles) [35, 36].

Noise power in the coherent regime was investigated by means of the scattering
approach by Büttiker and coworkers (in many seminal papers; see the review
[5]). The non-equilibrium Green function technique [37] is very efficient, it
can be used to determine zero-frequency cumulants within FCS [38, 39, 40]. The
spectral analysis of the current correlations plays a key role in electron
quantum optics, a new branch of nanoelectronics, where single electron packets
(the levitons) can propagate ballistically along the edge channels of the
quantum Hall effect and can be guided in optics like setup [41]. In
particular, the first and second-order correlation functions are used in
description of two-particle interferences in the Hanbury-Brown and Twiss (HBT)
geometry [42, 43] and in the electron analog the Hong, Ou and Mandel (HOM)
experiment [44] (see also [45]).

We are interested in quantum coherence processes in the noise power spectrum
and the Cooper pair splitter (CPS) seems appropriate for this issue. Such
system consists of three electrodes in a Y geometry, where the central
electrode is a superconductor and serves as a reservoir of Cooper pairs which
are injected into two normal metallic electrodes. [6, 7, 8, 46, 47] This is
the fermionic analog of two photon interference experiments: HBT and HOM,
where spin-entangled electrons, as Einstein-Podolsky-Rosen (EPR) pairs, are
spatially coherently separated into their entangled constituents. The device
allows one to test Bell inequalities in terms of current-current correlations
and to show their violation as evidence of entanglement of electrons [48, 49,
50]. Therefore, the CPS is considered as the solid state setup for quantum
communications [51].

The efficiency of CPS can increase when the superconductor is coupled by two
quantum dots (QD) with two normal metallic electrodes [8, 47]. If the intra-
dot charging energy is large, then two electrons of the Cooper pair are forced
to split into separate channels. Such devices with tunable QDs were fabricated
and shown to control the Cooper pair splitting with a very high efficiency
[52, 53, 54, 55, 56]; recently also with graphene QDs [57] and two
topologically non-trivial semiconducting nanowires [58].

Chevallier at al. [59] modeled the double-dot Cooper pair splitter (2QD-CPS)
and calculated the current as well as the current correlations by means of
non-equilibrium Green functions. Cross Andreev reflections (CAR) are relevant
for operation of CPS. In the CAR process an electron ejected from the normal
metallic electrode to the superconductor forms a Cooper pair and
simultaneously a hole of opposite spin is injected to the other metallic
electrode. Their excitation energies are less than the superconducting energy
gap. There are two other processes: direct Andreev reflection (DAR), when the
electron and the hole propagate to the same electrode, and electron
cotunneling (EC) related to electron transfers from one metallic electrode to
the other one through the superconductor. However, these processes spoil the
splitting efficiency. The anomalous electron-hole current correlations (for
CAR and DAR) are positive [60], which is related to bunching of tunneling
events [6]. This is in contrast to the EC processes which result in negative
cross correlations [60], as in the normal metallic Y splitter [5, 61]. The
interplay between these processes was demonstrated in the 2QD-CPS model with
two ferromagnetic electrodes [62].

The above mentioned studies focused solely on the zero-frequency coherent
current correlations. Our aim is to study the noise power spectrum in 2QD-CPS
to get insight into quantum coherence processes in transport, a relevant time
scale for Cooper pair splitting, local charge dynamics and relaxation
processes. Recently, Droste et al. [63] studied the frequency dependent noise
power in a two terminal hybrid system, composed of a single-level quantum dot
coherently coupled to a superconductor and a normal conducting electrode
(N-QD-S). They found that the noise power spectrum reflects the internal
spectrum of the proximized dot and shows extraordinary resonance dips at
frequencies corresponding to transitions between the Andreev bound state
(ABS). Our goal is to investigate a role of quantum interference in the
second-order current correlation functions, in particular, the origin of the
dips in the noise power spectrum. We predict that interference processes in
the noise power exhibit themselves in a different way than Young’s
interference patterns observed in the first-order correlations.

The paper is organized as follows. In Sec. II we present the model and the
Keldysh Green function method, following Chevallier at al. [59] with some
modifications which make the presentation more transparent. The considerations
are focused on the subgap energy region with the ABS reflections only. In Sec.
III we derive analytical formulas for the current and the frequency dependent
(cross and auto) current correlation functions. The analysis of the results is
presented in Sec. IV: first the zero-frequency case (Sec.IV.1) and next the
main results of the paper on the frequency dependence of the noise power
(Sec.IV.2). The results are presented for a whole range of the bias voltage:
the shot noise spectrum in the large voltage limit, down to the linear
response limit with the frequency dependent admittance. Supplemental materials
are placed in Appendixes A and B.

##  II Model description and Green function calculations

Figure 1: Schematic presentation of the Cooper pair splitter with two quantum
dots (1,2) coupled to the normal ($L,R$) electrodes and strongly coupled to
the superconductor ($S$) as a reservoir of Cooper pairs. Transport is due to
perfect cross Andreev reflections when an electron ($e$) is injected to the
normal electrode and a hole ($h$) with an opposite spin is simultaneously
ejected from the second metallic electrode.

We consider a Cooper pair splitter with two quantum dots (2QD), where each QD
is coupled to the normal $L$ or $R$ electrode and both are coupled the
superconducting BCS reservoir of Cooper pairs; see Fig.1. Moreover, we assume
the case of strong coupling to the $S$ electrode when due to proximity effect
the QDs behave like superconducting grains and entangled electrons can be
transported through the QDs. The model is described by the Hamiltonian

$H=\sum_{i=1,2;\sigma=\uparrow,\downarrow}\epsilon_{i}a^{{\dagger}}_{i\sigma}a_{i\sigma}+\sum_{\alpha=L,R,S}H_{\alpha}+H_{T}\;,$
(1)

where the first term corresponds to the 2QD system with a single level
$\epsilon_{i}$ available at the $i$-th QD. The electrode Hamiltonians

$\displaystyle H_{\alpha}=\sum_{k}\Psi^{{\dagger}}_{\alpha
k}(\varepsilon_{\alpha k}\sigma_{z}+\Delta_{\alpha}\sigma_{x})\Psi_{\alpha
k},$ (2)

are expressed in Nambu notation $\Psi^{{\dagger}}_{\alpha
k}=[c^{{\dagger}}_{\alpha k\uparrow},c_{\alpha\bar{k}\downarrow}]$,
$\bar{k}=-k$, $\sigma_{z}$, $\sigma_{x}$ are the Pauli matrices,
$\varepsilon_{\alpha k}$ and $\Delta_{\alpha}$ denote the electron energy and
the superconducting gap, which in the normal electrodes is taken as zero. The
transfer Hamiltonian is given by

$\displaystyle
H_{T}=\sum_{k}\big{(}\Psi^{{\dagger}}_{Lk}t_{L1}\sigma_{z}d_{1}+\Psi^{{\dagger}}_{Rk}t_{R2}\sigma_{z}d_{2}+$
$\displaystyle\Psi^{{\dagger}}_{Sk}t_{S1}\sigma_{z}d_{1}+\Psi^{{\dagger}}_{Sk}t_{S2}\sigma_{z}d_{2}+h.c.\big{)},$
(3)

where $d^{{\dagger}}_{i}=[a^{{\dagger}}_{i\uparrow},a_{i\downarrow}]$ and
$t_{\alpha i}$ describes electron hopping between the $\alpha$ electrode and
the $i$-th QD.

The charge current operator from the $\alpha$ electrode to the $i$-th dot is

$\displaystyle\hat{I}_{\alpha i}\equiv\left(\begin{array}[]{l}\hat{I}_{\alpha
ie}\\\ \hat{I}_{\alpha ih}\end{array}\right)=\imath
e\sum_{\alpha,k}(\Psi^{{\dagger}}_{\alpha k}t_{\alpha
i}\sigma_{z}d_{i}-d^{{\dagger}}_{i}t_{i\alpha}\sigma_{z}\Psi_{\alpha k}),$ (6)

where $\hat{I}_{\alpha ie}$ and $\hat{I}_{\alpha ih}$ denote contributions by
electrons and holes, respectively.

To calculate the average of the currents and their correlation functions we
use the Keldysh Green function method [59, 64, 65, 66]. Since the considered
model is for noninteracting particles one can easily write the equation of
motion for the Keldysh Green functions. In derivations the interdot singlet,
with $\langle
a^{{\dagger}}_{1\uparrow}a^{{\dagger}}_{2\downarrow}-a^{{\dagger}}_{1\downarrow}a^{{\dagger}}_{2\uparrow}\rangle\neq
0$, is only taken into account. We assume that intra-dot Coulomb interactions
are large and double occupancy of the dots is neglected, $\langle
a^{{\dagger}}_{i\uparrow}a^{{\dagger}}_{i\downarrow}\rangle=0$; thus, the
transfer of the Cooper pair through a single QD as a high energy process is
forbidden. In the first step the self-energy $\hat{\Sigma}_{S}$ (in the
Keldysh-Nambu space) describing the Cooper pair transfer from the S-electrode
is derived according to Eq.(26) in Ref.[59] – see Appendix A for more details.
Our derivations are confined to the subgap regime $|E|<\Delta$ and in the
limit $\Delta\rightarrow\infty$ in which the self-energy becomes

$\displaystyle\hat{\Sigma}_{S}=\frac{\gamma_{S}}{2}\left[\begin{array}[]{cccc}0&0&1&0\\\
0&0&0&-1\\\ 1&0&0&0\\\ 0&-1&0&0\end{array}\right],$ (11)

where $\gamma_{S}=\pi\rho_{S}t_{S1}t_{S2}$ and $\rho_{S}$ is the density of
states in the S-electrode in the normal state. Notice that $\gamma_{S}$
describes the exchange electron-hole coupling between the quantum dots. In
this way the S-electrode is integrated out and the Keldysh Green function
matrix, for the system with two normal electrodes and two proximized QDs, is
expressed as a product of two components

$\displaystyle\hat{G}_{L2QDR}=\hat{G}_{e\uparrow,h\downarrow}\otimes\hat{G}_{h\downarrow,e\uparrow},$
(12)

where

$\displaystyle\hat{G}_{e\uparrow,h\downarrow}\equiv\left[\begin{array}[]{cccc}\hat{G}_{Le\uparrow,Le\uparrow}&\hat{G}_{Le\uparrow,1e\uparrow}&\hat{G}_{Le\uparrow,2h\downarrow}&\hat{G}_{Le\uparrow,Rh\downarrow}\\\
\hat{G}_{1e\uparrow,Le\uparrow}&\hat{G}_{1e\uparrow,1e\uparrow}&\hat{G}_{1e\uparrow,2h\downarrow}&\hat{G}_{1e\uparrow,Rh\downarrow}\\\
\hat{G}_{2h\downarrow,Le\uparrow}&\hat{G}_{2h\downarrow,1e\uparrow}&\hat{G}_{2h\downarrow,2h\downarrow}&\hat{G}_{2h\downarrow,Rh\downarrow}\\\
\hat{G}_{Rh\downarrow,Le\uparrow}&\hat{G}_{Rh\downarrow,1e\uparrow}&\hat{G}_{Rh\downarrow,2h\downarrow}&\hat{G}_{Rh\downarrow,Rh\downarrow}\end{array}\right]=\left[\begin{array}[]{cccccccc}w_{L,11}^{--}&w_{L,11}^{-+}&t_{L1}&0&0&0&0&0\\\
w_{L,11}^{+-}&w_{L,11}^{++}&0&-t_{L1}&0&0&0&0\\\
t_{L1}&0&z_{1e}&0&\gamma_{S}/2&0&0&0\\\
0&-t_{L1}&0&-z_{1e}&0&-\gamma_{S}/2&0&0\\\
0&0&\gamma_{S}/2&0&z_{2h}&0&-t_{R2}&0\\\
0&0&0&-\gamma_{S}/2&0&-z_{2h}&0&t_{R1}\\\
0&0&0&0&-t_{R2}&0&w_{R,22}^{--}&w_{R,22}^{-+}\\\
0&0&0&0&0&t_{R2}&w_{R,22}^{+-}&w_{R,22}^{++}\end{array}\right]^{-1}.$ (25)

Here, we use the Keldysh-Nambu matrix notation. For example the Green function
for the $L$ electrode decoupled from the system is expressed as

$\displaystyle\hat{g}_{Le\uparrow,Le\uparrow}=\left[\begin{array}[]{cc}g_{Le\uparrow,Le\uparrow}^{--}&g_{Le\uparrow,Le\uparrow}^{-+}\\\
g_{Le\uparrow,Le\uparrow}^{+-}&g_{Le\uparrow,Le\uparrow}^{++}\end{array}\right],$
(28)

where its inverse elements derived in the wide-band approximation are:
$w_{L,11}^{--}=w_{L,11}^{++}=-2\imath\rho_{L}(f_{Le}-1/2)$,
$w_{L,11}^{-+}=2\imath\rho_{L}f_{Le}$,
$w_{L,11}^{+-}=-2\imath\rho_{L}(1-f_{Le})$. Similarly, the inverse elements of
$\hat{g}_{Rh\downarrow,Rh\downarrow}$ are:
$w_{R,22}^{--}=w_{R,22}^{++}=-2\imath\rho_{R}(f_{Rh}-1/2)$,
$w_{R,22}^{-+}=2\imath\rho_{R}f_{Rh}$,
$w_{R,22}^{+-}=-2\imath\rho_{R}(1-f_{Rh})$. $f_{\alpha
e}=\\{\exp[(E-\mu_{\alpha})/k_{B}T]+1\\}^{-1}$ and $f_{\alpha
h}=\\{\exp[(E+\mu_{\alpha})/k_{B}T]+1\\}^{-1}$ are the Fermi distribution
functions for electrons and holes in the $\alpha$ electrode with the chemical
potential $\mu_{\alpha}$. The chemical potential in the superconductor is
taken $\mu_{S}=0$. We also denoted $z_{1e}=E-\epsilon_{1}$ and
$z_{2h}=E+\epsilon_{2}$. Confining to the subgap regime one gets the inter-dot
coupling $\gamma_{S}=\pi\rho_{S}t_{S1}t_{S2}$, where $\rho_{S}$ is the density
of states in the S-electrode in the normal state.

Similarly one can express $\hat{G}_{h\downarrow,e\uparrow}^{-1}$ replacing the
indexes $\\{L,1,\uparrow\\}\leftrightarrow\\{R,2,\downarrow\\}$ in Eq.(25).

##  III Currents and their correlation functions

Using the Keldysh Green function (25) one can calculate the currents as

$\displaystyle
I_{Le\uparrow}=I_{Rh\downarrow}=-\frac{e}{\hbar}\int\frac{dE}{2\pi}(f_{Le}-f_{Rh})\mathcal{T}_{Aeh}(E),$
(29) $\displaystyle
I_{Lh\downarrow}=I_{Re\uparrow}=-\frac{e}{\hbar}\int\frac{dE}{2\pi}(f_{Re}-f_{Lh})\mathcal{T}_{Ahe}(E),$
(30)

where the transmission probability is expressed as

$\displaystyle\mathcal{T}_{Aeh}(E)=\frac{16\gamma_{L}\gamma_{R}\gamma_{S}^{2}}{|4(z_{1e}-\imath\gamma_{L})(z_{2h}-\imath\gamma_{R})-\gamma_{S}^{2}|^{2}},$
(31)
$\displaystyle\mathcal{T}_{Ahe}(E)=\frac{16\gamma_{L}\gamma_{R}\gamma_{S}^{2}}{|4(z_{1h}-\imath\gamma_{L})(z_{2e}-\imath\gamma_{R})-\gamma_{S}^{2}|^{2}}$
(32)

and the couplings are $\gamma_{L}=\pi\rho_{L}|t_{L1}|^{2}$,
$\gamma_{R}=\pi\rho_{R}|t_{R2}|^{2}$. It is seen that $\mathcal{T}_{Aeh}(E)$
and $\mathcal{T}_{Ahe}(E)$ have a resonant shape with two peaks, associated
with the Andreev bound states (ABS), at the poles
$E_{\pm}=[\epsilon_{1}-\epsilon_{2}\pm\imath(\gamma_{L}+\gamma_{R})\pm\sqrt{(\epsilon_{1}+\epsilon_{2}\pm\imath(\gamma_{L}-\gamma_{R}))^{2}+\gamma_{S}^{2}}]/2$
and
$E_{\pm}=[-\epsilon_{1}+\epsilon_{2}\pm\imath(\gamma_{L}+\gamma_{R})\pm\sqrt{(\epsilon_{1}+\epsilon_{2}\mp\imath(\gamma_{L}-\gamma_{R}))^{2}+\gamma_{S}^{2}}]/2$,
respectively. Although $\mathcal{T}_{Aeh}(E)$ and $\mathcal{T}_{Ahe}(E)$ can
be different, they have the relative electron-hole symmetry, and therefore the
currents $I_{Le\uparrow}$ and $I_{Lh\downarrow}$ are equal.

To study dynamical correlations of the currents we introduce the operator
$\Delta\hat{I}_{\alpha i}(t)\equiv\hat{I}_{\alpha i}(t)-\langle
I_{\alpha}\rangle$ describing current fluctuation from its average value, and
next define the non-symmetrized correlation function (following [32])

$\displaystyle S_{\alpha
i,\alpha^{\prime}i^{\prime}}(\omega)=2\int_{-\infty}^{\infty}dt\;e^{-i\omega
t}\langle\Delta\hat{I}_{\alpha
i}(0)\Delta\hat{I}_{\alpha^{\prime}i^{\prime}}(t)\rangle$ (33)

for the current in the junction $\alpha i$ and the current in the junction
$\alpha^{\prime}i^{\prime}$. To measure quantum noise, one needs a quantum
detector [16, 19, 4], which can absorb or emit energy from noise, for positive
and negative frequencies, respectively.

First we derive the cross current correlation function
$S_{Le\uparrow,Rh\downarrow}$. Using the current operators, Eq.(6), one gets
the function as a sum of two particle averages, which are decoupled by means
of Wick’s theorem to products of single particle averages, and next we express
them by the Green functions. The result is

$\displaystyle
S_{Le\uparrow,Rh\downarrow}(\omega)=\frac{2e^{2}}{\hbar}t_{L1}t_{R2}\int_{-\infty}^{\infty}\frac{dE}{2\pi}\big{[}\;G^{+-}_{Rh\downarrow,Le\uparrow}(E)G^{-+}_{1e\uparrow,2h\downarrow}(E+\hbar\omega)-G^{+-}_{2h\downarrow,Le\uparrow}(E)G^{-+}_{1e\uparrow,Rh\downarrow}(E+\hbar\omega)$
$\displaystyle-G^{+-}_{Rh\downarrow,1e\uparrow}(E)G^{-+}_{Le\uparrow,2h\downarrow}(E+\hbar\omega)+G^{+-}_{2h\downarrow,1e\uparrow}(E)G^{-+}_{Le\uparrow,Rh\downarrow}(E+\hbar\omega)\big{]}.$
(34)

Similarly, the auto-correlation function is expressed as

$\displaystyle
S_{Le\uparrow,Le\uparrow}(\omega)=\frac{2e^{2}}{\hbar}t_{L1}^{2}\int_{-\infty}^{\infty}\frac{dE}{2\pi}\big{[}\;G^{+-}_{Le\uparrow,Le\uparrow}(E)G^{-+}_{1e\uparrow,1e\uparrow}(E+\hbar\omega)-G^{+-}_{1e\uparrow,Le\uparrow}(E)G^{-+}_{1e\uparrow,Le\uparrow}(E+\hbar\omega)$
$\displaystyle-G^{+-}_{Le\uparrow,1e\uparrow}(E)G^{-+}_{Le\uparrow,1e\uparrow}(E+\hbar\omega)+G^{+-}_{1e\uparrow,1e\uparrow}(E)G^{-+}_{Le\uparrow,Le\uparrow}(E+\hbar\omega)\big{]}.$
(35)

This approach is equivalent to the one presented by Chevallier et al. [59],
but it is more transparent; one needs to determine only the Keldysh Green
function with the electrodes, as that one (25).

Notice that in Eq.(12) the Keldysh Green function matrix is decoupled, which
means that there are two uncorrelated Andreev scattering channels:
(e$\uparrow$,h$\downarrow$) and (h$\downarrow$,e$\uparrow$); and the
corresponding currents are uncorrelated as well - similarly to the case in the
paramagnetic system where the currents for electrons with the spin $\uparrow$
and $\downarrow$ are uncorrelated for a noninteracting case [35]. Thus, the
cross correlations from two different subspaces are: $S_{\alpha
e\uparrow,\alpha e\downarrow}=S_{\alpha h\uparrow,\alpha h\downarrow}=0$, and
$S_{Le\sigma,Re\sigma}=S_{Lh\sigma,Rh\sigma}=0$. This means that the direct
Andreev reflections (DARs) as well as the single particle transfers (EC) are
absent, and transport is only due to perfect cross Andreev reflections (CARs).
The situation would be different for a large transport window, when high
energy processes (with Cooper pair transfers through a single QD) should be
taken into account. Further, we will consider the (e$\uparrow$,h$\downarrow$)
channel only, and to simplify the notation we will omit the spin indices
$\sigma=\uparrow,\downarrow$, which are related to the electron ($e$) and the
hole ($h$), respectively.

Using the Green function, Eq.(25), one can derive the current correlation
function

$\displaystyle
S_{\alpha,\alpha^{\prime}}(\omega)=\frac{2e^{2}}{\hbar}\sum_{\nu,\nu^{\prime}=Le,Rh}\int_{-\infty}^{\infty}\frac{dE}{2\pi}\;\mathscr{S}_{\alpha,\alpha^{\prime}}^{\nu,\nu^{\prime}}(E,E+\hbar\omega)\times$
$\displaystyle[1-f_{\nu}(E)]f_{\nu^{\prime}}(E+\hbar\omega)\;,$ (36)

where its the density elements are

$\displaystyle\mathscr{S}_{Le,Le}^{LeLe}(E,E$
$\displaystyle+\hbar\omega)=1-r(E)r^{*}(E+\hbar\omega)-$ $\displaystyle
r^{*}(E)r(E+\hbar\omega)+\mathcal{R}(E)\mathcal{R}(E+\hbar\omega),$ (37)
$\displaystyle\mathscr{S}_{Le,Le}^{RhRh}(E,E$
$\displaystyle+\hbar\omega)=\mathcal{T}_{Aeh}(E)\mathcal{T}_{Aeh}(E+\hbar\omega),$
(38) $\displaystyle\mathscr{S}_{Le,Le}^{RhLe}(E,E$
$\displaystyle+\hbar\omega)=\mathcal{T}_{Aeh}(E)\mathcal{R}_{Aeh}(E+\hbar\omega),$
(39) $\displaystyle\mathscr{S}_{Le,Le}^{LeRh}(E,E$
$\displaystyle+\hbar\omega)=\mathcal{R}_{Aeh}(E)\mathcal{T}_{Aeh}(E+\hbar\omega)$
(40)

for the auto-correlation function and

$\displaystyle\mathscr{S}_{Le,Rh}^{LeLe}(E,E+\hbar\omega)=$ $\displaystyle
t^{*}(E)t(E+\hbar\omega)$
$\displaystyle[1-r_{LL}^{*}(E+\hbar\omega)r_{LL}(E)],$ (41)
$\displaystyle\mathscr{S}_{Le,Rh}^{RhRh}(E,E+\hbar\omega)=$ $\displaystyle
t(E)t^{*}(E+\hbar\omega)$
$\displaystyle[1-r_{RR}^{*}(E)r_{RR}(E+\hbar\omega)],$ (42)
$\displaystyle\mathscr{S}_{Le,Rh}^{LeRh}(E,E+\hbar\omega)=$
$\displaystyle\mathscr{S}_{Le,Rh}^{RLeh}(E,E+\hbar\omega)=$ $\displaystyle
t^{*}(E)r_{LL}(E)t(E+\hbar\omega)r^{*}_{LL}(E+\hbar\omega)$ (43)

for the cross-correlation function, respectively. One can check that the
formulas for $S_{\alpha,\alpha^{\prime}}(\omega)$ have the same structure as
those ones derived by means of the scattering matrix approach for the single
quantum dot (1QD) attached to two normal electrodes [67, 68, 69]. However, in
our case the scattering matrix is different and its elements are

$\displaystyle
t(E)=\frac{4\imath\sqrt{\gamma_{L}\gamma_{R}\gamma_{S}^{2}}}{4(z_{1e}-\imath\gamma_{L})(z_{2h}-\imath\gamma_{R})-\gamma_{S}^{2}},$
(44) $\displaystyle
r_{LL}(E)=\frac{4(z_{1e}+\imath\gamma_{L})(z_{2h}-\imath\gamma_{R})-\gamma_{S}^{2}}{4(z_{1e}-\imath\gamma_{L})(z_{2h}-\imath\gamma_{R})-\gamma_{S}^{2}},$
(45) $\displaystyle
r_{RR}(E)=\frac{4(z_{1e}+\imath\gamma_{L})(z_{2h}-\imath\gamma_{R})-\gamma_{S}^{2}}{4(z_{1e}+\imath\gamma_{L})(z_{2h}+\imath\gamma_{R})-\gamma_{S}^{2}}.$
(46)

and $\mathcal{T}_{Aeh}(E)=t(E)t^{*}(E)$,
$\mathcal{R}_{Aeh}(E)=r_{LL}(E)r_{LL}^{*}(E)=r_{RR}(E)r_{RR}^{*}(E)=1-\mathcal{T}_{Aeh}(E)$.

##  IV Results

The preceding section has given the general analytical formulas for the
current and their correlation. Let us now consider the results for special
cases in detail.

###  IV.1 Zero frequency noise

First we present the zero-frequency current correlation function

$\displaystyle S_{Le,Le}(0)=S_{Le,Rh}(0)=$
$\displaystyle\frac{2e^{2}}{\hbar}\int_{-\infty}^{\infty}\frac{dE}{2\pi}\;\big{\\{}\mathcal{T}_{Aeh}^{2}(E)[f_{Le}(1-f_{Le})+f_{Rh}(1-f_{Rh})]$
$\displaystyle+\mathcal{T}_{Aeh}(E)[1-\mathcal{T}_{Aeh}(E)][f_{Le}(1-f_{Rh})+f_{Rh}(1-f_{Le})]\big{\\}}.$
(47)

The auto and cross correlations are equal due to charge current conservation.
Moreover, this means a lack of direct Andreev reflections (DAR) and coherent
cross Andreev reflections (CAR) are only responsible for transport through the
system – there is a perfect entanglement of Cooper pairs. The formula (IV.1)
is similar to the noise power in a two-terminal conductor [70, 71, 67, 68, 5],
as well as in a hybrid structure in the presence of the electron-hole Andreev
scattering [60, 72] (see also the review [73, 5]). Note that the sign of the
cross correlation function is positive because we consider the correlations
between the electron and hole currents [60, 72]. This in contrast to the case
in the normal metallic Y splitter, where the cross correlation function
corresponds to the electron-electron current correlations and its sign is
negative [5, 61]. Our calculations give an exact analytical expression
(31)-(32) for the transmission probabilities $\mathcal{T}_{Aeh}$ and
$\mathcal{T}_{Ahe}$.

In the linear response limit ($eV\rightarrow 0$) one can easily find the
conductance and the spectral functions for the current correlations [74, 75].
Let us consider the large bias voltage regime for the splitter bias
configuration (with the chemical potentials $\mu_{L}=\mu_{R}=eV$ in both
normal electrodes and $\mu_{S}=0$ in the superconductor), i.e. for
$f_{Le}=f_{Re}=0$ and $f_{Lh}=f_{Rh}=1$. In this case transport is
unidirectional through both Andreev states, electrons are transmitted to the
normal electrodes and holes in the opposite direction. There is no
backscattering. The current can be determined as

$\displaystyle
I_{Le}=\frac{e}{\hbar}\int_{-\infty}^{\infty}\frac{dE}{2\pi}\mathcal{T}_{Aeh}(E)=\frac{e}{\hbar}\frac{\gamma\gamma_{S}^{2}}{(\epsilon_{1}+\epsilon_{2})^{2}+4\gamma^{2}+\gamma_{S}^{2}}.$
(48)

We consider only the single channel transport and therefore, the coefficient
$e/\hbar$ appears. The zero frequency cross correlation function is expressed
as

$\displaystyle
S_{Le,Rh}(0)=\frac{2e^{2}}{\hbar}\int_{-\infty}^{\infty}\frac{dE}{2\pi}\mathcal{T}_{Aeh}(E)[1-\mathcal{T}_{Aeh}(E)]$
$\displaystyle=2eI_{Le}-\frac{2e^{2}}{\hbar}\frac{\gamma\gamma_{S}^{4}[(\epsilon_{1}+\epsilon_{2})^{2}+20\gamma^{2}+\gamma_{S}^{2}]}{2[(\epsilon_{1}+\epsilon_{2})^{2}+4\gamma^{2}+\gamma_{S}^{2}]^{3}}\;.$
(49)

Here, to simplify the presentation we have assumed the symmetric coupling
$\gamma_{L}=\gamma_{R}=\gamma$. The integration was performed using the
residue theorem with the poles in the upper half plane:
$E_{\pm}=\epsilon_{\pm}+\imath\gamma$, where
$\epsilon_{\pm}=\\{(\epsilon_{1}-\epsilon_{2})\pm\delta\\}/2$ denotes the
position of ABS;
$\delta=\epsilon_{+}-\epsilon_{-}=\sqrt{(\epsilon_{1}+\epsilon_{2})^{2}+\gamma_{S}^{2}}$
is the separation between them. One can also write the Fano factor

$\displaystyle
F_{Le,Rh}\equiv\frac{S_{Le,Rh}(0)}{2|eI_{Le}|}=1-\frac{\gamma_{S}^{2}[(\epsilon_{1}+\epsilon_{2})^{2}+20\gamma+\gamma_{S}^{2}]}{2[(\epsilon_{1}+\epsilon_{2})^{2}+4\gamma^{2}+\gamma_{S}^{2}]^{2}}\;.$

It is seen that its minimal value $\min[F_{Le,Rh}]=7/32$ at
$\gamma=\sqrt{3/20}\gamma_{S}$ and $\epsilon_{1}=-\epsilon_{2}$.

All analytical results were verified for $\omega=0$ by the full counting
statistics (FCS) procedure [64, 40], when counting fields $\bm{\lambda}$ were
introduced to the transfer Hamiltonian (II), to the Keldysh Green function
(25) and a proper adiabatic potential $\mathcal{U}(\bm{\lambda})$ was derived
and used to calculate currents and their variances.

###  IV.2 Frequency dependence of current correlations

####  IV.2.1 Large bias

Figure 2: Frequency dependence of the Fano factor: $F_{Le,Rh}$ (blue) and
$F_{Le,Le}$ (red) for the cross- and auto-current correlation function,
calculated for the large voltage limit. The solid curves represent the weak
coupling $\gamma=0.05$, while the dashed curves are for the strong coupling
$\gamma=0.2$. The other parameters are: $\gamma_{S}=1$,
$\epsilon_{1}=\epsilon_{2}=0$, for which the separation between ABS is
$\delta=1$ and it is taken as the unit for the energy scale.

In the large bias voltage regime both ABS are in the voltage window and
transport is unidirectional, backscattering processes are absent. For the
splitter voltage configuration the Fermi distribution functions
$f_{Le}(E)=f_{Re}(E)=0$ and $f_{Lh}(E)=f_{Rh}(E)=1$. Moreover we assume
moderate frequencies, $\hbar|\omega|\ll e|V|$, for which
$f_{Le}(E+\hbar\omega)=f_{Re}(E+\hbar\omega)=0$ and
$f_{Lh}(E+\hbar\omega)=f_{Rh}(E+\hbar\omega)=1$. In this case the current
correlation function is expressed only by the shot noise term

$\displaystyle
S_{Le,Rh}(\omega)=\frac{2e^{2}}{\hbar}\int_{-\infty}^{\infty}\frac{dE}{2\pi}\;\mathscr{S}_{Le,Rh}^{LeRh}(E,E+\hbar\omega)$
(51)

with the density function $\mathscr{S}_{Le,Rh}^{LeRh}$, given by Eq.(III). For
the symmetric coupling one can derive (using the residuum theorem)

$\displaystyle
S_{Le,Rh}(\omega)=\frac{2e^{2}}{\hbar}\frac{\gamma\gamma_{S}^{2}}{\delta^{2}+4\gamma^{2}}\bigg{[}\frac{2\gamma^{2}(2\delta^{2}-\gamma_{S}^{2})}{\delta^{2}(4\gamma^{2}+\hbar^{2}\omega^{2})}$
$\displaystyle-\frac{\gamma^{2}\gamma_{S}^{2}}{\delta^{2}(\delta^{2}+4\gamma^{2})}\frac{\delta^{2}-4\gamma^{2}+2\delta(\delta-\hbar\omega)}{(\delta-\hbar\omega)^{2}+4\gamma^{2}}$
$\displaystyle-\frac{\gamma^{2}\gamma_{S}^{2}}{\delta^{2}(\delta^{2}+4\gamma^{2})}\frac{\delta^{2}-4\gamma^{2}+2\delta(\delta+\hbar\omega)}{(\delta+\hbar\omega)^{2}+4\gamma^{2}}\bigg{]}.$
(52)

We have performed the spectral decomposition, which allows insight into local
charge fluctuations and find their characteristic frequencies related to
relaxation processes [36, 35]. Here, the method shows the interplay of the
currents flowing through the higher (+) and lower (-) ABS and their
contribution to the correlation function (see Appendix B for details). The
first term is positive and represents the correlation function,
$S_{Le,Rh}^{++}$ and $S_{Le,Rh}^{--}$, for the currents flowing through the
same ABS. The other two terms correspond to inter-level current correlations,
$S_{Le,Rh}^{-+}$ and $S_{Le,Rh}^{+-}$, respectively. These functions are
negative and are related to emission and absorption of energy $\hbar\omega$ by
the system. For $\omega=0$ the result obviously coincides with Eq.(49). Notice
that $S_{Le,Rh}(\omega)=S_{Lh,Re}(\omega)$ and they depend only on
$|\epsilon_{1}+\epsilon_{2}|$, which means that fluctuations on both Andreev
states are equivalent.

In a similar way we derive shot noise in the current auto-correlation
function:

$\displaystyle
S_{Le,Le}(\omega)=\frac{2e^{2}}{\hbar}\frac{\gamma\gamma_{S}^{2}}{\delta^{2}+4\gamma^{2}}\bigg{[}1-\frac{2\gamma^{2}\gamma_{S}^{2}}{\delta^{2}(4\gamma^{2}+\hbar^{2}\omega^{2})}$
$\displaystyle-\frac{\gamma^{2}\gamma_{S}^{2}}{\delta^{2}(\delta^{2}+4\gamma^{2})}\frac{\delta^{2}-4\gamma^{2}+2\delta(\delta-\hbar\omega)}{(\delta-\hbar\omega)^{2}+4\gamma^{2}}$
$\displaystyle-\frac{\gamma^{2}\gamma_{S}^{2}}{\delta^{2}(\delta^{2}+4\gamma^{2})}\frac{\delta^{2}-4\gamma^{2}+2\delta(\delta+\hbar\omega)}{(\delta+\hbar\omega)^{2}+4\gamma^{2}}\bigg{]}.$
(53)

The first term of $S_{Le,Le}(\omega)$ corresponds to the Schottky noise, which
is frequency independent and describes uncorrelated transfers of particles
with a Poissonian distribution function of time intervals between transfer
events [5]. The Pauli principle and the symmetry of electron wave functions
result in the negative current correlations, which is also a signature of
anti-bunching of the transfer events, as occurs in our case for the second
term in Eq.(IV.2.1).

In contrast the first term in $S_{Le,Rh}(\omega)$, Eq.(IV.2.1), is positive,
which means that the electron and hole transfers, in the Andreev scattering,
are bunched. The two last terms are the same in $S_{Le,Rh}(\omega)$ and
$S_{Le,Le}(\omega)$. They are related to correlation of the currents flowing
through the lower and the upper ABSs, which is maximal at
$\hbar\omega=\pm\delta$ when the resonant peaks of $\mathcal{T}_{Aeh}(E)$ and
$\mathcal{T}_{Aeh}(E\pm\delta)$ overlap with each other. The sign of these
correlations is negative which means that inter-level transfers are anti-
bunched. The similar situation can occur in the multilevel quantum dot system,
when a transmission coefficient has many resonant peaks, which can lead to
negative and sometimes positive current correlations.

Figure 3: Map of the density function:
$\mathcal{S}^{LeRh}_{Le,Rh}(E,E+\hbar\omega)$ – (a) and
$\mathcal{S}^{LeRh}_{Le,Le}(E,E+\hbar\omega)$ – (c), plotted vs the energy $E$
and the frequency $\hbar\omega$. Panels (b) and (d), present the plots of the
density functions vs $E$ at $\hbar\omega=0$ and $\hbar\omega=1$, the red and
the blue curves, respectively. The plots are performed for the weak coupling
$\gamma_{L}=\gamma_{R}=0.05$ and the other parameter are the same as in Fig.2.

The frequency dependence of the Fano factor $F_{Le,Rh}(\omega)\equiv
S_{Le,Rh}(\omega)/2eI_{Le}$ and $F_{Le,Le}(\omega)\equiv
S_{Le,Rh}(\omega)/2eI_{Le}$ for the cross and auto-correlations are shown in
Fig.2. In this case normalization is given by
$2eI_{Le}=(2e^{2}/\hbar)\gamma\gamma_{S}^{2}/(\delta^{2}+4\gamma^{2})$.

A similar frequency dependence was found by Droste et al. [63] for the
correlation function in the two terminal hybrid system, with one quantum dot
(N-QD-S), which was derived by means of a real-time diagrammatic perturbation
expansion in the tunnel-coupling to the normal electrode. The authors argue
[63], that the dips arise from a coherent destructive interference between
ABS. However, our studies show that the origin of the dips is different. The
transport in both the models is coherent, but there is no evidence for
destructive interference in the transmission coefficient
$\mathcal{T}_{Aeh}(E)$ and in the differential conductance. The spectral
decomposition analysis shows that the side dips are due to the negative
current correlations related with photon assisted electron transfers through
the different ABS. Such structure of the noise power spectrum can occur also
for coherent transport through multilevel quantum dots [76, 77, 78]. In
particular, Ref. [79] showed that electron-hole recombination processes can
result in resonant inter-level current correlations with emission/absorption
of energy in the system of two quantum dots in a T-geometry coupled to the
normal metallic electrodes.

Our results are in contrast to those in Ref.[80] where the noise power
spectrum in 2QD-CPS was determined in the sequential tunneling regime based on
a diagonalized master equation (DME). The method gives reliably the voltage
dependence of the current, which reflects the internal spectrum of the
proximized 2QDs with ABSs. The spectral decomposition analysis showed
contributions of current correlations through various ABSs and the interplay
between different local charge relaxation processes. However, the DME method
neglects coherent electron-hole recombination processes, and therefore, any
extraordinary side dips have not been found in the noise power spectrum.

To have a better insight into the role of quantum interference we analyze the
density functions $\mathscr{S}_{Le,Rh}^{\nu\nu^{\prime}}$ and
$\mathscr{S}_{Le,Le}^{\nu\nu^{\prime}}$ for the cross- and the auto-current
correlation, respectively. In the shot noise regime the relevant components
are:

$\displaystyle\mathscr{S}^{LeRh}_{Le,Rh}=t^{*}(E)r_{LL}(E)t(E+\hbar\omega)r^{*}_{LL}(E+\hbar\omega)$
(54)

and

$\displaystyle\mathscr{S}^{LeRh}_{Le,Le}=\mathcal{T}_{Aeh}(E+\hbar\omega)[1-\mathcal{T}_{Aeh}(E)].$
(55)

Figure 3 presents their plots as maps in the energy and frequency space,
$(E,\hbar\omega)$. The function $\mathscr{S}^{LeRh}_{Le,Le}$ has two peaks
shifted in energy by $\delta$, which is related to the spectrum of ABS seen in
$\mathcal{T}_{Aeh}(E+\hbar\omega)$ and $[1-\mathcal{T}_{Aeh}(E)]$. The density
function $\mathscr{S}^{LeRh}_{Le,Rh}$ is more interesting, which can be
positive or negative. For example, it is positive for $\hbar\omega=0$ and
equal to $\mathcal{T}_{Aeh}(1-\mathcal{T}_{Aeh})$, while it is negative for
whole energy range at $\hbar\omega=\pm 1$ \- see Fig.3b.

####  IV.2.2 Moderate bias voltage

For $V\rightarrow\infty$ we have been able to derived the analytical formulas,
but for a finite voltage one should use the general formula for the
correlation functions, Eq.(III), taking into account all components of the
density functions, Eqs.(III)-(III). The integrals have been calculated
numerically for the temperature $T=0$ and the results for the frequency
dependent current correlation functions are presented in Fig.4. Comparing with
Fig.2 (for $V\rightarrow\infty$) one sees that the plots vanish for the
positive $\hbar\omega$ when the voltage $eV$ becomes smaller. As expected with
decreasing voltage less noise power is emitted by the system. On the other
hand the other components, those related to thermal fluctuations on the same
junctions, become more relevant, and they result in an increase of the noise
power spectrum, in particular for $\hbar\omega<0$. The plots show the
pronounced dips at $\hbar\omega=-1$ (in particular that for $S_{Le,Rh}$),
which are related to negative inter-level current correlations with energy
absorption.

Figure 4: Frequency dependence of the real part of the cross- and the auto-
correlation function (black thick curves) with their components:
$S_{Le,Rh}^{LeRh}$ and $S_{Le,Le}^{LeRh}$ (red), $S_{Le,Rh}^{RhLe}$ and
$S_{Le,Le}^{RhLe}$ (blue), $S_{Le,Rh}^{LeLe}=S_{Le,Rh}^{RhRh}$ and
$S_{Le,Le}^{LeLe}$ (green), $S_{Le,Le}^{RhRh}$ (orange), for $eV=0.7$ [(a) and
(b)] and $eV=0.1$ [(c) and (d)]. The plots are calculated at temperature $T=0$
and normalized to
$2e\max[I_{Le}]=(2e^{2}/\hbar)\gamma\gamma_{S}^{2}/(\delta^{2}+4\gamma^{2})$.
The other parameter are the same as in Fig.3.

####  IV.2.3 Linear response limit

For a small bias voltage one can use the Kubo theory of linear response and
establish the relation

$\displaystyle
S_{\alpha\alpha^{\prime}}(\omega)-S_{\alpha^{\prime}\alpha}(-\omega)=-4\hbar\omega\mathcal{G}^{\prime}_{\alpha\alpha^{\prime}}(\omega)=$
$\displaystyle\frac{4e^{2}}{h}\int
dE\;\mathscr{S}_{\alpha\alpha^{\prime}}^{\text{eq}}(E,E+\hbar\omega)[f(E)-f(E+\hbar\omega)]\;,$
(56)

between the noise power spectrum and the real part of the admittance
$\mathcal{G}_{\alpha\alpha^{\prime}}(\omega)$ (responsible for dissipation)
[32]. Here, we denoted
$\mathscr{S}_{\alpha\alpha^{\prime}}^{\text{eq}}\equiv\sum_{\eta,\eta^{\prime}=L,R}\mathscr{S}_{\alpha\alpha^{\prime}}^{\eta\eta^{\prime}}$,
which can be explicitly expressed as

$\displaystyle\mathscr{S}_{LL}^{\text{eq}}=2-r_{LL}^{*}(E)r_{LL}(E+\hbar\omega)-r_{LL}(E)r_{LL}^{*}(E+\hbar\omega),$
(57)
$\displaystyle\mathscr{S}_{LR}^{\text{eq}}=t(E)t^{*}(E+\hbar\omega)+t^{*}(E)t(E+\hbar\omega)$
(58)

for the auto-and cross-correlations, respectively. [The coefficients $t(E)$
and $r_{LL}(E)$ are given by Eqs.(44)-(45).] For $T=0$ one gets

$\displaystyle
4\hbar|\omega|\mathcal{G}^{\prime}_{\alpha\alpha^{\prime}}(\omega)=\frac{4e^{2}}{h}\int_{E_{F}}^{E_{F}+\hbar|\omega|}dE\;\mathscr{S}_{\alpha\alpha^{\prime}}^{\text{eq}}(E,E+\hbar\omega)\;,$
(59)

where the Fermi energy $E_{F}=0$. Notice that in this limit the noise is
purely quantum:
$S_{\alpha\alpha^{\prime}}(\omega)=-4\hbar\omega\mathcal{G}^{\prime}_{\alpha\alpha^{\prime}}(\omega)\theta(-\omega)$,
where $\theta$ denotes the Heaviside step function. Figure 5 presents the
admittance, $\mathcal{G}^{\prime}_{Le,Rh}$ (a) and
$\mathcal{G}^{\prime}_{Le,Le}$ (b), plotted in the space $(\eta,\hbar\omega)$,
where $\eta\equiv(\epsilon_{1}-\epsilon_{2})/2$ and
$\epsilon\equiv(\epsilon_{1}+\epsilon_{2})/2=0$. Although for $\hbar\omega=0$
one gets
$\mathcal{G}^{\prime}_{Le,Rh}(0)=\mathcal{G}^{\prime}_{Le,Le}(0)=(2e^{2}/h)\mathcal{T}_{Aeh}(E_{F})$,
but for finite frequencies these admittances exhibit different interference
patterns. It is seen that the cross admittance,
$\mathcal{G}^{\prime}_{Le,Rh}$, decreases with $\hbar|\omega|$ and becomes
negative, while the auto admittance, $\mathcal{G}^{\prime}_{Le,Le}$, is always
positive and can be larger than $2e^{2}/h$ due to current fluctuations on the
same tunnel junction. These results show that displacement currents play a
different role in the cross and the auto correlations.

Figure 5:  Map of the real part of the admittance (a)
$\mathcal{G}^{\prime}_{Le,Rh}$ and (b) $\mathcal{G}^{\prime}_{Le,Le}$ (in
units $2e^{2}/h$) plotted in the space $(\eta,\hbar\omega)$, where
$\eta=(\epsilon_{1}-\epsilon_{2})/2$. We take
$\epsilon\equiv(\epsilon_{1}+\epsilon_{2})/2=0$, $E_{F}=0$, temperature $T=0$,
and the other parameters are the same as in Fig.3.

##  V Summary

Let us summarize the main results. We analyzed a Cooper pair splitter model,
where two entangled electrons were transferred through the inter-dot singlet
state on two proximized QDs into two normal electrodes. Since the derivations
were confined to the subspace with the inter-dot singlet, where high energy
processes were forbidden, we were able to show the separation of the cross
Andreev reflections: for an electron-hole (e$\uparrow$,h$\downarrow$) and a
hole-electron (h$\downarrow$,e$\uparrow$) scattering [see Eq.(12)]. This means
that the transfer of an electron with the spin $\uparrow$ from the first QD to
the left electrode is coherently coupled (fully entangled) with the transfer
of a hole with the spin $\downarrow$ from the right electrode to the second
QD, and this process is independent from the (h$\downarrow$,e$\uparrow$)
transfer. Consequently, the zero frequency cross and auto correlation
functions are equal and positive, $S_{Le,Rh}(0)=S_{Le,Le}(0)>0$.

The other important result is the derivation of the analytical formulas for
the frequency dependent cross and auto-correlation functions,
$S_{Le,Rh}(\omega)$ and $S_{Le,Le}(\omega)$ in the large bias limit,
Eq.(IV.2.1) and (IV.2.1). Here, quantum coherence manifests itself in the
negative inter-level current correlations as the dips at
$\hbar\omega=\pm\delta$, when the system absorbs or emits the energy. The
intra-level current correlations are seen at the low frequency limit and they
are responsible for the central dip in $S_{Le,Le}(\omega)$ as well as the
central peak in $S_{Le,Rh}(\omega)$. These low-frequency features are well
known in sequential and coherent transport through quantum dots [67, 68, 5,
28, 35].

We also considered the admittance (derived by means of the linear response
theory) for the cross- and the auto-correlation case, $\mathcal{G}_{Le,Le}$
and $\mathcal{G}_{Le,Rh}$, respectively. Although at $\omega=0$ they are equal
and coincide with the conductance, for the finite $\omega$ they have different
interference patterns. Our results show that quantum interference manifests
itself in the second-order current correlation functions in a different way
than Young’s interference patterns in the first-order correlations.

We studied the noise power spectrum in the non-symmetrized form to get a
direct relation with a quantum noise measurement. To verify our predictions
one can use the experimental setup as in Ref. [19, 20, 21, 22, 23, 24, 25],
with a quantum detector sensitive in a large frequency range to detect inter-
level transitions between ABSs. This seems to be an experimental challenge.

###### Acknowledgements.

The research was financed by National Science Centre, Poland, Project No.
2016/21/B/ST3/02160.

##  Appendix A  Self-energies for N and S-electrode

We use the equation of motion to derive the Green functions in the Keldysh and
Nambu spaces. In this method important quantities are the self-energies which
describe coupling of the system with the normal and superconducting
electrodes. For the normal electrode coupling one simply gets the self-energy

$\displaystyle\hat{\Sigma}_{\alpha i}=\frac{\imath\gamma_{\alpha
i}}{2}\left[\begin{array}[]{cccc}2f_{\alpha e}-1&2f_{\alpha e}&0&0\\\
2(f_{\alpha e}-1)&2f_{\alpha e}-1&0&0\\\ 0&0&2f_{\alpha h}-1&2f_{\alpha h}\\\
0&0&2(f_{\alpha h}-1)&2f_{\alpha h}-1\end{array}\right],$ (64)

where the wide flat-band approximation is used, and $\gamma_{\alpha
i}=\pi\rho_{\alpha}|t_{\alpha i}|^{2}$, $\rho_{\alpha}$ denotes the density of
states in the electrode, $f_{\alpha
e}=\\{\exp[(E-\mu_{\alpha})/k_{B}T]+1\\}^{-1}$ and $f_{\alpha
h}=\\{\exp[(E+\mu_{\alpha})/k_{B}T]+1\\}^{-1}$ are the distribution functions
for electrons and holes, respectively. For the coupling of the S-electrode
with the 1 and 2 QD the self-energy is expressed as (following [59, 64, 65,
66]):

$\displaystyle\hat{\Sigma}_{S}=\frac{\imath\gamma_{S}}{2}\times$
$\displaystyle\left[\begin{array}[]{cccc}(2f_{Se}-1)\beta^{{}^{\prime}}_{S}+\imath\beta^{"}_{S}&2f_{Se}\beta^{{}^{\prime}}_{S}&-[(2f_{Se}-1)\beta^{{}^{\prime}}_{S}+\imath\beta^{"}_{S}]\Delta/E&-2f_{Se}\beta^{{}^{\prime}}_{S}\Delta/E\\\
2(f_{Se}-1)\beta^{{}^{\prime}}_{S}&(2f_{Se}-1)\beta^{{}^{\prime}}_{S}+\imath\beta^{"}_{S}&2f_{Se}\beta^{{}^{\prime}}_{S}\Delta/E&[(2f_{Se}-1)\beta^{{}^{\prime}}_{S}+\imath\beta^{"}_{S}]\Delta/E\\\
((2f_{Sh}-1)\beta^{{}^{\prime}}_{S}-\imath\beta^{"}_{S})\Delta/E&2(f_{Sh}-1)\beta^{{}^{\prime}}_{S}\Delta/E&(2f_{Sh}-1)\beta^{{}^{\prime}}_{S}-\imath\beta^{"}_{S}&2f_{Sh}\beta^{{}^{\prime}}_{S}\\\
-2(f_{Sh}-1)\beta^{{}^{\prime}}_{S}\Delta/E&-[(2f_{Sh}-1)\beta^{{}^{\prime}}_{S}-\imath\beta^{"}_{S}]\Delta/E&2(f_{Sh}-1)\beta^{{}^{\prime}}_{S}&(2f_{Sh}-1)\beta^{{}^{\prime}}_{S}-\imath\beta^{"}_{S}\end{array}\right],$
(69)

where $\gamma_{S}=\pi\rho_{S}t_{L1}t_{2R}$, $\beta^{{}^{\prime}}_{S}$ and
$\beta^{"}_{S}$ are the real and the imaginary part of the function

$\displaystyle\beta_{S}=\frac{|E|\theta(|E|-\Delta)}{\sqrt{E^{2}-\Delta^{2}}}-\imath\frac{E\theta(\Delta-|E|)}{\sqrt{\Delta^{2}-E^{2}}}\;.$
(71)

Notice that in the limit $\Delta\rightarrow 0$ one gets $\hat{\Sigma}_{S}$ as
for the normal electrode, Eq.(64). For the sub-gap regime $|E|<\Delta$ the
function $\beta_{S}=-\imath E/\sqrt{\Delta^{2}-E^{2}}$ and in the limit
$\Delta\rightarrow\infty$ Eq.(A) reduces itself to Eq.(11) in Section II.

Let us stress that in a similar way one can derive the self-energy
$\hat{\Sigma}_{S1}$ and $\hat{\Sigma}_{S2}$ related to the intra-dot pairing.
However, in our considerations the intra-dot Coulomb interactions are assumed
to be large and these terms are neglected as they correspond to high energy
tunneling processes.

##  Appendix B  Spectral decomposition

Let us present the spectral decomposition approach which allows to separate
various contributions of currents flowing through ABSs and their components in
noise power. Our derivations are performed in the large voltage regime, in
which the current

$\displaystyle
I_{Le}=-\frac{e}{\hbar}\int_{-\infty}^{\infty}\frac{dE}{2\pi}\mathcal{T}_{Aeh}(E)$
(72)

and we want to express it as $I_{Le}=I_{Le}^{+}+I_{Le}^{-}$, separating the
current flowing through the upper ($+$) and the lower ($-$) ABSs. To this end
the transmission coefficient $\mathcal{T}_{Aeh}$, Eq.(31), is decomposed into
partial fractions and the terms are grouped into those corresponding to
transmission through the upper and the lower state:

$\displaystyle\mathcal{T}^{\pm}_{Aeh}(E)=$
$\displaystyle\mp\frac{\imath\gamma\gamma_{S}^{2}}{2\delta(\delta^{2}+4\gamma^{2})}\Big{[}\frac{-2\imath\gamma\pm\delta}{E-\epsilon_{\pm}-\imath\gamma}-\frac{2\imath\gamma\pm\delta}{E-\epsilon_{\pm}+\imath\gamma}\Big{]}$
$\displaystyle=$
$\displaystyle\mp\frac{2\gamma^{2}\gamma_{S}^{2}(E-\epsilon_{\pm}\mp\delta/2)}{\delta(4\gamma^{2}+\delta^{2})[(E-\epsilon_{\pm})^{2}+\gamma^{2}]},$
(73)

where $\epsilon_{\pm}$ denotes the position of the ABS. Thus, the current is
$I_{Le}^{+}=I_{Le}^{-}=(e/\hbar)\gamma\gamma_{S}^{2}/[2(4\gamma^{2}+\delta)]$.
Here, we assume the symmetric coupling, $\gamma_{L}=\gamma_{R}=\gamma$.

Similarly, we perform spectral decomposition of the cross correlation function
$S_{Le,Rh}^{LeRh}(\omega)$, where the corresponding density
$\mathscr{S}_{Le,Rh}^{LeRh}(E,\omega)=t^{*}(E)r_{LL}(E)t(E+\hbar\omega)r^{*}_{LL}(E+\hbar\omega)=g(E)g^{*}(E+\hbar\omega)$,
Eq.(III). To this end the function $g(E)\equiv\imath t^{*}(E)r_{LL}(E)$ is
decomposed into partial fractions and next, the terms are grouped for those
for transfers through $\epsilon_{+}$ and $\epsilon_{-}$. The result is

$\displaystyle
g_{\pm}(E)=\pm\frac{\gamma\gamma_{S}}{2\delta(\delta^{2}+4\gamma^{2})}\Big{[}\frac{(\epsilon_{1}+\epsilon_{2}\pm\delta)(-2\imath\gamma\pm\delta)}{E-\epsilon_{\pm}-\imath\gamma}$
$\displaystyle\>\;\;-\frac{(\epsilon_{1}+\epsilon_{2}\mp\delta)(2\imath\gamma\pm\delta)}{E-\epsilon_{\pm}+\imath\gamma}\Big{]}=\mp\frac{\gamma\gamma_{S}}{\delta(\delta^{2}+4\gamma^{2})}\times$
$\displaystyle\frac{(E-\epsilon_{\pm})\delta^{2}\pm
2\delta\gamma^{2}-2\imath(\epsilon_{1}+\epsilon_{2})(E-\epsilon_{\pm}\mp\delta/2)}{(E-\epsilon_{\pm})^{2}+\gamma^{2}}.$
(74)

Thus, the function

$\displaystyle\mathscr{S}_{Le,Rh}^{LeRh}(E,\omega)=\sum_{\nu,\nu^{\prime}=\pm}\mathscr{S}_{Le,Rh}^{LeRh,\nu\nu^{\prime}}(E,\omega),$
(75)

where its density
$\mathscr{S}_{Le,Rh}^{LeRh,\nu\nu^{\prime}}(E,\omega)=g_{\nu}(E)g_{\nu^{\prime}}^{*}(E+\hbar\omega)$.

After integration one gets the spectral components of
$S_{Le,Rh}^{LeRh}(\omega)$:

$\displaystyle
S_{Le,Rh}^{LeRh,++}(\omega)=\frac{2e^{2}}{\hbar}\frac{\gamma^{2}\gamma_{S}^{2}[(2\delta^{2}-\gamma_{S}^{2})\delta+\imath(\epsilon_{1}+\epsilon_{2})\delta]}{\delta^{2}(4\gamma^{2}+\delta^{2})(4\gamma^{2}+\hbar^{2}\omega^{2})},$
(76) $\displaystyle
S_{Le,Rh}^{LeRh,+-}(\omega)=-\frac{2e^{2}}{\hbar}\frac{\gamma^{3}\gamma_{S}^{4}[\delta^{2}-4\gamma^{2}+2\delta(\delta+\hbar\omega)]}{\delta^{2}(4\gamma^{2}+\delta^{2})^{2}[4\gamma^{2}+(\delta+\hbar\omega)^{2}]},$
(77) $\displaystyle
S_{Le,Rh}^{LeRh,-+}(\omega)=-\frac{2e^{2}}{\hbar}\frac{\gamma^{3}\gamma_{S}^{4}[\delta^{2}-4\gamma^{2}+2\delta(\delta-\hbar\omega)]}{\delta^{2}(4\gamma^{2}+\delta^{2})^{2}[4\gamma^{2}+(\delta-\hbar\omega)^{2}]},$
(78) $\displaystyle
S_{Le,Rh}^{LeRh,--}(\omega)=\frac{2e^{2}}{\hbar}\frac{\gamma^{2}\gamma_{S}^{2}[(2\delta^{2}-\gamma_{S}^{2})\delta-\imath(\epsilon_{1}+\epsilon_{2})\delta]}{\delta^{2}(4\gamma^{2}+\delta^{2})(4\gamma^{2}+\hbar^{2}\omega^{2})}.$
(79)

Let us perform spectral decomposition of $S_{Le,Le}^{LeRh}(\omega)$, for which
the density is expressed as
$\mathscr{S}_{Le,Le}^{LeRh}(E,\omega)=\mathcal{T}_{Aeh}(E+\hbar\omega)-\mathcal{T}_{Aeh}(E+\hbar\omega)\mathcal{T}_{Aeh}(E)$.
Since $\mathcal{T}_{Aeh}(E+\hbar\omega)$ gives the frequency independent
Schottky term in the noise power, we focus on the second term which can be
written as
$\mathcal{T}_{Aeh}(E+\hbar\omega)\mathcal{T}_{Aeh}(E)=\sum_{\nu,\nu^{\prime}=\pm}\mathcal{T}^{\nu}_{Aeh}(E+\hbar\omega)\mathcal{T}^{\nu^{\prime}}_{Aeh}(E)$.
Using Eq.(B) and integrating one can find the spectral components of the
frequency dependent part of $S_{Le,Le}^{LeRh}(\omega)$:

$\displaystyle\frac{2e^{2}}{\hbar}\bigg{\\{}$
$\displaystyle-\frac{2\gamma^{3}\gamma_{S}^{4}}{\delta^{2}(4\gamma^{2}+\hbar^{2}\omega^{2})}$
$\displaystyle-\frac{\gamma^{3}\gamma_{S}^{4}[\delta^{2}-4\gamma^{2}+2\delta(\delta+\hbar\omega)]}{\delta^{2}(\delta^{2}+4\gamma^{2})^{2}[4\gamma^{2}+(\delta+\hbar\omega)^{2}]}$
$\displaystyle-\frac{\gamma^{3}\gamma_{S}^{4}[\delta^{2}-4\gamma^{2}+2\delta(\delta-\hbar\omega)]}{\delta^{2}(\delta^{2}+4\gamma^{2})^{2}[4\gamma^{2}+(\delta-\hbar\omega)^{2}]}\bigg{\\}}.$
(80)

Here, the first term comes from the $(++)$ and $(--)$ component, while the
second and third term comes from the inter-level $(+-)$ and $(-+)$
correlation, and they are equal to $S_{Le,Rh}^{LeRh,+-}(\omega)$ and
$S_{Le,Rh}^{LeRh,-+}(\omega)$, Eqs.(77) and (78), respectively.

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