Unitary limit in crossed Andreev transport

One of the most promising approaches of generating spin- and energy-entangled electron pairs is splitting a Cooper pair into the metal through spatially separated terminals. Utilizing hybrid systems with the energy-dependent barriers at the superconductor-normal metal interfaces, one can achieve practically 100% efficiency outcome of entangled electrons. We investigate minimalistic one-dimensional model comprising a superconductor and two metallic leads and derive an expression for an electron-to-hole transmission probability as a measure of splitting efficiency. We find the conditions for achieving 100% efficiency and present analytical results for the differential conductance and differential noise.


INTRODUCTION
Quantum entanglement dating back to Einsten's and Schrödinger's seminal papers [1,2] has emerged as one of the most active research areas of contemporary condensed matter physics. Interest in this problem is motivated both, by the important promise of utilizing entanglement effects in communication and computation technologies [3][4][5], and by the intellectual appeal of dealing with the most fundamental issues of quantum mechanics [6]. One of the major experimental tasks to counter is the very creation of and subsequent manipulation with the entangled quantum states. A Cooper pair comprising two electrons endowed with a unique inseparable quantum state is a natural source of electrons with states inextricably linked to each other and remaining entangled with respect to spin and/or energy despite having become spatially separated [7,8]. Andreev reflection at the ideal NS interface. In one dimensional a case Cooper pair converts into two entangled electrons in normal metal wire with the unity probability. (b) Threedimensional crossed Andreev reflection (CAR) with two different exits to the normal metal. Cooper pair converts into two electrons in different normal wires with probability suppressed by the distance between normal contacts. (c) Onedimensional CAR to the opposite sides of NSN system is completely suppressed for the ideal NS boundaries. (d) Additional energy-dependent normal reflection at NS boundaries may increase the probability of the CAR up to unity.
The initial stage of splitting can occur via an Andreev reflection (AR) phenomenon [9], see Fig. 1(a), where the Cooper pair crosses the ideal normal metalsuperconductor (NS) interface and enters normal metal as two energy-and spin-entangled electrons (or, more precisely, electron-like quasiparticle enters and hole-like quasiparticle leaves normal metal) with the probability of unity. The further spatial separation of electrons, requires some more effort. To do so, one can apply the external magnetic field, which would take apart electrons and holes [10,11].
A different approach to splitting a Cooper pair and further handling the resulting separate electrons was proposed in [7], where the normal-metal fork with leads endowed with the different resonance energies would separate holes and electrons with the opposite excitation energies. Developing further this idea, one can utilize a crossed Andreev reflection (CAR) for Cooper pair splitting (CPS) using two-terminal configuration in which a Cooper pair generates electrons escaping through two separated normal terminals [12][13][14][15][16][17] as shown in Fig. 1(b). Efficiency of the two-terminal configuration for CPS can be improved by plugging a quantum dot into the each lead [8], utilizing Coulomb blockade for manipulating the electrons. However, the CAR-based splitters encounter an inherent problem following from the very nature of CAR. Since the initial separation of exiting electron is the size of the Cooper pair, the escape probability remains appreciable only for terminal separation not much exceed the coherence length, see Fig. 1(b). Moreover, this probability is exactly zero in a one-dimensional geometry with the ideal NS boundary, see Fig. 1(c). In higher dimensions, the amplitude of CAR decays exponentially with the distance L between the terminals. What more, it acquires a small prefactor ∝ 1/(k F L) D−1 for clean D-dimensional superconductor [8,14,18,19]. In three-dimensional disordered superconductors this factor is ∝ 1/k F √ lL [20]. Here k F is the Fermi wave vector and l is the mean-free path. Hence one expects low efficiency of the CAR-based CPS [21,22]. Remarkably, though, the experiments show somewhat better outcome [23].
In this article we solve the 'low-efficiency' problem for one-dimensional geometry with the initial zero CAR amplitude. We demonstrate that adding a double energydependent barrier to each terminal of scheme shown in Fig. 1(c) and properly tuning over the resonance levels in separately biased output terminals, we realize CPS with the hundred percent outcome.

PRELIMINARIES
We quantify the entangler efficiency by transmission probability i.e. by the probability for the incident electron to reach a superconductor through one terminal having created a Cooper pair in the superconductor and a hole that left the entangler through the other terminal. Our idea is to control the resonance structure of the Bogoliubov states by setting double-barrier potentials in such a way that for specific resonance energies in a hybrid structure [24,25] the electron-to-hole transmission probability became equal to unity, in full analogy with Fabry-Pérot interferometer. To gain a feeling how the perfect transmission can be reached, we consider a special resonance structure of the barriers associated with terminals. Let the left scatterer have a very narrow resonance E F + ε above Fermi energy E F , and the right one have the same resonance below Fermi energy at E F −ε, see Fig. 2. Suppose further that the transparency assumes the value of unity at these energies and is zero otherwise. The electron with the energy E F + ε incidents from the left. The electron-to-hole reflection (with amplituder eh ) and electron-to-electron transmission (t ee ) are blocked, so incident electron can ether reflect back as an electron (r ee ) or transmit as a hole (t eh ). The latter process occurs with the probability of unity for certain energy ε and the superconductor length L. The described configuration is, in fact, realization of Mach-Zehnder interferometer, involving just two trajectories.

SCATTERING MATRIX APPROACH
Let us consider an electron-like quasiparticle with the energy E F + ε incident at the one-dimensional X-S-X structure shown in Fig. 2. Both X-parts stand for the energy-dependent barriers realized via the insulatornormal metal-insulator (INI) scatterers, which exhibit sharp resonances with the unity transparency for identical δ-function-like insulators. To find the transmission probabilityT eh = |t eh | 2 of such an X-S-X system, we solve Bogoliubov-de Gennes (BdG) [26][27][28] equations with energy ε below the superconducting gap ∆, 0 ε ∆. We take the piecewise potential in BdG equations so that ∆(x) = ∆ in the superconductor and∆(x) = 0 outside. The electron-(u) and hole-like (v) components of the wave function of the superconductor are then where dots stand for some constants. The normal state solution of BdG between superconductor and left (L) [right (R)] scatterer is given by the linear combination of plane waves: The central superconducting part with the ideal NS boundaries couples the incident and reflected states, with nonzero amplitudes . (2) Hereafter we will be using subscript e(h) to denote electron (hole) component of the wave function (e.g. t eh is the electron-to-hole transmission amplitude). The transmission amplitudes t ee(hh) and corresponding transparencies T ee(hh) = sin 2 α sin 2 α + sinh 2 (qL) describe co-tunnetilng [29,30] in an ideal N-S-N contact. The inverse coherence length q and the wave vector p naturally appear from the solution of BdG equation with the fixed modulus of the superconducting gap ∆ and are defined as p 2 − q 2 = k 2 F and 2pq = (2m/ 2 )∆ sin α, where α = arccos(ε/∆) is the auxiliary phase α ∈ [0 . . . π/2], k F = √ 2mE F / is the Fermi wave vector, and m is the mass of the electron. For ∆ E F one finds q ≈ (k F ∆/2E F ) sin α and p ≈ k F . We count the energy ε from the Fermi energy E F .
The left and right hand side energy-dependent X barriers form energy dependent barriers with scattering matrices S L(R) = diag{S L(R) e , S L(R) h }. The electron and hole subparts are given by where t e(h) = t(±ε) and r e(h) = r(±ε). Experimentally, one can control the position of the resonances with respect to the Fermi energy by the external gate voltage. Note, that the symmetry of the BdG equations (ε, u, v) → (−ε, −v * , u * ) leads to the following relations among the transmission coefficients,t hh =t * ee , t he = −t * eh ,r he = −r * eh , andr hh =r * ee , where ' ' stands for (ε, ε L , ε R ) → −(ε, ε L , ε R ). Hereafter we concentrate on the electron-to-hole transmission amplitudet eh and transparencyT eh as a measure of the entangler efficiency.

IDEAL TRANSPARENCY CASE
Now we are equipped for finding components of the scattering matrix of the X-S-X system. We consider a narrow resonance with the width much smaller then ε and spacings between resonances being much lager then ∆. If t e L = 1, r h L = e iϕ h L , r e R = e iϕ e R , and t h R = 1, the X-S-X transparency becomes ideal,T eh = 1. In this case, both, X-S-X transmission and reflection amplitudes, are defined by two pairs of paths shown in Fig. 2, and the corresponding expression fort eh assumes a simple form Combining this relation with Eq. (2) one finds the X-S-X transparency as where θ = pL + (ϕ e R − ϕ h L )/2. We see that the transparency becomes ideal,T eh = 1, provided θ = πn and sinh(qL) = sin α. Since both qL and α are energy dependent, the latter equality implicitly defines the energy at which the transparency becomes unity. The maximal transparencyT ε eh = max ε {T eh } as a function of the dimensionless parameter L/ξ 0 , where ξ 0 = v F /∆ and v F = k F /m, is shown in Fig. 3. Note thatT eh is small in both limits of (i) a short superconductor, L ξ 0 , where electron-to-hole reflection amplitudes are small r eh(he) ∝ L/ξ 0 and electron passes the superconductor freely and of (ii) a long superconductor, L ξ 0 , because the transmission amplitude of the NSN part decays exponentially, t ee(hh) ∝ e −L/ξ0 . The unity valueT ε eh = 1 is achieved in the interval L/ξ 0 ∈ [arcsinh 1 . . . 1]. According to Eq. (2), each point at the flat top corresponds to the different energy ε and is a result of the competition between q(ε) and α(ε) dependencies.

ASYMMETRIC INI PARTS AND ARBITRARY RESONANCE POSITIONS
To understand how robust the unitary limit of CPS is and derive to which degree one can deviate from the ideal resonances condition still maintaining the nearly unitary limit, let us model the energy-dependent X parts as non-ideal dots (asymmetric INI double barriers) with arbitrary resonance positions. We first choose INI double barriers such that both of them had identical pairs of inner and outer point scatterers as shown in Fig. 2. The inner point scatterers are described by transmission t i and reflection r i amplitudes, and outer point scatterers have t o and r o correspondingly. Then the transmission and reflection coefficients for each INI part assume the form , where k e(h) = 2m(E F ± ε)/ and d L The transmission amplitude of the hybrid INI-S-INI system is given bỹ For ideal conditions of two isolated resonances, as in The analysis of the transparencyT eh as a function of ε L and ε R for different energies of an incident electron ε is presented in a form of the color plots in Fig. 4. We choose resonance half-widths Γ L = Γ R = 0.1∆ in Fig. 4(a) and Γ L = Γ R = 0.045∆ in Fig. 4(b) to be smaller than the superconducting gap. Typically, the transparency for each energy ε has a pronounced peak at (ε L , ε R ) ∼ (ε, −ε). The peaks at energies ε/∆ = 0.01, 0.2, 0.4, 0.6, 0.8, and 0.99 are well separated, so one can absorb all the dependencies ofT eh of ε L and ε R in the same figure. These peaks are marked with white circles. For energies near the Fermi level ε ∼ 0 the resonance is expected to be at (ε L , ε R ) ∼ (0, 0), but, as follows from Eq. (6), at this point ε L = ε R = ε = 0 and as for symmetric barriers r i = r o , the transparency is suppressed,T eh = 0. For larger energies 0 ε ∆ the locus of resonances is about the 'diagonal' (ε L , ε R ) ∼ (ε, −ε). At (ε L , ε R ) = (ε, −ε) and sym-  parencyT ε eh = max ε {T eh } behavior. Figure 5(a) demonstrates color plot ofT ε eh as function of ε L and ε R for symmetric INI barriers with Z i(o) = 15 (Γ L(R) = 0.045∆). One sees the permanent resonance along the tilted white line corresponding to the ideal case defined by Eq. (5). In Fig. 5(c) the asymmetric case is shown for the inner barrier strength Z i = 5 smaller then outer barrier strength Z o = 15 (Γ L(R) = 0.22∆): the maximum is determined not by the internal INI resonance, but by the resonance between the outer walls. The reversed situation, Z i = 15 and Z o = 5 is presented in Fig. 5(d). One observes additional resonances, e.g, loci ofT ε eh form a cross-like configuration comprising the diagonal and the segment of the line ε L − ε R = 2∆ in Fig. 5(d). While at the 'diagonal' the resonances (along line ε L = −ε R ) originate from the ideal case (4), the additional 'crossbar' loci of maximal transparencies stem from the 'off-diagonal' resonances in Fig. 4. The regimes with strong and nearly symmetric resonances are stable against added randomness so thatT eh as a function of ε, ε L , and ε R has a maximum value of unity. Finally, we analyze the experimentally measurable quantity, differential conductivity dI/dV = hg/e 2 with g =T eh −T ee . The color plot Fig. 5(b) of the maximal conductivity for the exemplary case Z i(o) = 15 repeats features ofT eh [ Fig. 5(a)]. Differential conductivity has slightly sharper peaks and is almost suppressed at ε R > 0. Appendix A presents more color plots ofT ε eh and g ε for variety of parameters.

CONCLUSION
We demonstrated that the outcome of Cooper pair splitting via the crossed Andreev transport in a onedimensional hybrid system, comprising a superconductor sandwiched between the two normal metal terminals endowed with the double point scatterers, can achieve a robust unitary limit stable against asymmetry of the scatterers. We found that the electron-to-hole transmission probability (per a conducting channel) across such a system achieves its maximum if the width of the superconductor is of the order of the coherence length. Our study opens a route to reliable high-outcome procedure for creating entangled electrons.
We thank P. Hakonen and D. S. Golubev for illuminating discussions which to large extent motivated this research. The work was supported by the U.S. Department of Energy, Office of Science, Materials Sciences and Engineering Division (I.S., V.V., and, partly, G.L. via Materials Theory Institute) and by the RFBR Grant No. 14-02-01287 (G.L.).