Long-range Cooper pair splitter with high entanglement production rate

Cooper pairs in the superconductor are a natural source of spin entanglement. The existing proposals of the Cooper pair splitter can only realize a low efficiency of entanglement production, and its size is constrained by the superconducting coherence length. Here we show that a long-range Cooper pair splitter can be implemented in a normal metal-superconductor-normal metal (NSN) junction by driving a supercurrent in the S. The supercurrent results in a band gap modification of the S, which significantly enhances the crossed Andreev reflection (CAR) of the NSN junction and simultaneously quenches its elastic cotunneling. Therefore, a high entanglement production rate close to its saturation value can be achieved by the inverse CAR. Interestingly, in addition to the conventional entangled electron states between opposite energy levels, novel entangled states with equal energy can also be induced in our proposal.

where the H(x) is defined by Eq. (1) in the main text. The Nambu representation is Φ = (ψ ↑ ,ψ ↓ ,ψ † ↑ ,ψ † ↓ ) T with "T" representing a matrix transpose. The four-component operator Φ satisfies (Φ † ) T = τ x Φ with τ x the Pauli matrix operating on the particle-hole components. By applying this restriction on Eq. (S.1), one obtains the particle-hole transformation invariance of the Hamiltonian as where Ξ = τ x K is the particle-hole transformation operator and K represents the complex conjugate. As a result, if ψ is an eigenstate of H with energy E as then we obtain by using Eq. (S.2). This means Ξψ(E) = ψ(−E) is an eigenstate of H as well with energy −E.
Next we prove that the particle-hole symmetry of the Hamiltonian H results in the particle-hole symmetry of the scattering matrix. For the translation invariant system, the eigenstate is a plane wave as where η, σ are the particle-hole and spin indexes, respectively. Performing the particle-hole transformation on the wave function leads to (S.6) which means the particle-hole transformation results in a flip of both particle-hole and spin components. Importantly, the moving direction of the quasiparticle is invariant after the particle-hole transformation, for the momentum and energy change their signs simultaneously. Therefore, an incident (outgoing) wave remains an incident (outgoing) wave after the particle-hole transformation.
In general, the incident and outgoing waves can be expressed as whereã αησ (E),b α η σ (E) are the wave amplitudes of the incident and outgoing waves. We have omitted the momentum index, for its sign corresponds to the incident or outgoing waves. The amplitudes of the incident and outgoing waves are related by the scattering matrix asb which can be obtained by solving the Bogoliubov-de Gennes equation. S α α η η,σ σ (E) is the scattering amplitude for an incident wave in the α lead with a particle-hole component η and spin σ being scattered into an outgoing wave in the α lead with a particle-hole component η and spin σ .
Performing the particle-hole transformation on the wave functions (S.7) results in (S.9) Since the above wave functions still satisfy the same Bogoliubov-de Gennes equation, the amplitudes of the incident and outgoing waves are also related by the same scattering matrix as Note that the sign of the energy in the matrix elements is reversed, for the wave functions after transformation possess an energy of −E. Comparing Eq. (S.8) and Eq. (S.10), we arrive at the particle-hole symmetry of the matrix as which means that the amplitude for a spin-up (spin-down) incident electron (hole) with an energy E being scattered into a spinup (spin-down) outgoing electron (hole) equals the amplitude for a spin-down (spin-up) incident hole (electron) with an energy −E being scattered into a spin-down (spin-up) outgoing hole (electron). Specifically, by adopting the notation of the scattering amplitudes in the main text, which correspond to the matrix el- (S.12)

DERIVATION OF THE ENTANGLED STATES
In order to obtain the entangled states generated via the inverse CAR, we start with the many-body incident state of |Ψ in = 0<E<|eV | γ i † L↑,E γ i † L↓,E |0 , where the incident holes occupy the energy window from the Fermi level to |eV | in the left N region. The many-body outgoing wave can be obtained by expressing the operators of the incident wave by that of the outgoing wave, i.e., Eq. (4) in the main text, which leads to Then we introduce a new vacuum state |0 , which is related to the original one through |0 = eV <E<0 c o † L↑,E c o † L↓,E |0 . Inserting this equality into the above equation results in (S.14)