Josephson current and thermal noise in a junction between two topological superconductors

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First proposals of Cooper pair splitters

The process of Andreev Reflection (AR) has already been introduced in Section 1.2 of Chapter 1. When a superconductor is connected to a normal metal, the constituent electrons of a CP can be transmitted through AR: a hole from the normal metal incident on the superconductor is reflected as an electron in this same metal. If now two metallic contacts are connected to a superconductor, the reflection can occur accross the two contacts if they are separated by a distance smaller than the superconducting coherence length (Fig. 2.2a). This process called Crossed AR (CAR) is responsible for CP splitting: the pair exits the superconductor and its two constituent electrons propagate in two different metallic leads forming a nonlocal state with spin and orbital degrees of freedom which are entangled. Evidently the local AR refered as Direct AR (DAR) can still occur inside each metallic lead (Fig. 2.2b) and spoils the efficiency of the device operating as a CP splitter. Another competing nonlocal process will occur in such a three-terminal device: Electron Cotunneling (EC) [62] refers to the transmission of an electron from one metallic lead to the other through the superconductor (Fig. 2.2c). EC is not based on AR and therefore no real CP production/destruction is involved in this tunnel effect but it requires a virtual excited quasiparticle state in the superconductor. EC is reminiscent of the normal behavior of the injecting material (when superconductivity is quenched by temperature or magnetic field) and spoils the efficiency of the device operating as an electron entangler. Let us emphasize that the transmission of the electrons originated from CPs inside the superconductor to the normal material, through AR processes, is driven here by voltage biases (non-equilibrium setup) and CAR signatures are expected in nonlocal/cross-junction (linear or differential) conductance measurements as well as in the cross correlations of the currents flowing in the two outgoing leads. An early theoretical proposal [63] for the manifestation of CAR relies on the measurement of nonlocal current in a nanoscale two-contact tunneling device (Fig. 2.3a). Ferromagnets with opposite polarizations (spin filtering for CAR enhancement to the detriment of EC) as contacts were proposed [64] and a first experimental CP splitting signature was obtained in an aluminium sample [65] (Fig. 2.3b). Nonlocal linear conductance Gcross is different between parallel ( = +) and antiparallel ( = 􀀀) alignement configurations of the spin valve. Below the critical.

Cooper pair splitting in an equilibrium setup

All the experimental works highlighted in Section 2.1 rely on non-equilibrium measurements. Noise cross correlation measurements achieved in [90] represent a considerable ordeal due to the poor signal to noise ratio, and no attempt has been made so far to reproduce them. In order to circumvent these difficulties a Josephson (equilibrium) geometry was proposed [18]: two QDs are placed between two superconductors and materialize two spatially separated conduction channels for the constituent electrons of a CP which can tunnel from one superconductor to the other driven by an applied phase difference. Fig. 2.7 illustrates the different possible processes: the two electrons can either pass both through a given QD or they can transit through different QDs and realize CP splitting through a CAR process at the boundary of the superconducting source (the recombination on the superconducting drain involves a second CAR process). This double Josephson junction with embedded QDs realizes an AB interferometer and we will refer to this setup as the nanoSQUID CP splitter [19]. In the presence of a magnetic flux = 20 (0 = h/e is the flux quantum), electrons experience a path dependent phase shift. When a CP tunnels as a whole through a single QD, it accumulates a phase shift ( depending on which QD) in addition to the phase difference ϕ between the two superconductors. However, when the Cooper pair is delocalized on the two QDs, one electron gets a phase shift + 2 while the other one gets a phase shift 􀀀 2 , the pair accumulating as a result no additional phase shift. The Josephson current at the lowest order in the transmission probabilities (where only the three processes illustrated in Fig. 2.7 are relevant) can be written as J(ϕ; ) = I1 sin (ϕ + ) + I2 sin (ϕ 􀀀 ) + ICAR sin ϕ .

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Path integral formulation of the partition function

In the following ℏ = kB = e = 1. The device is illustrated in Fig. 2.1. For simplicity, the two leads (labeled j = L;R) consist of the same superconducting material with chemical potential and gap energy Δ. ^ y jk denotes the creation operator for an electron with momentum k and spin = »; # in the superconductor j. We introduce the Nambu spinors and the Pauli matrices i (i = x; y; z) that act in Nambu space, useful to write the BCS Hamiltonian
of the lead j as The two nanotube/nanowire QDs (labeled a = U;D) are placed in the nanogap between the two electrodes. The energies « a of the QDs can be monitored via gate voltages. Non zero charging energies of these QDs originate in Coulomb on-site repulsions Ua. ^ dy a denotes the creation operator for an electron with spin = »; # on the QD a. It is convenient to conserve the Nambu structure and we introduce then.

Table of contents :

Introduction
1 Conventional superconductors 
1.1 Microscopic description of s-wave superconductivity
1.1.1 BCS variational method
1.1.2 Bogoliubov mean-field method
1.2 A new conduction channel: Andreev Reflection
1.3 Andreev bound states and Josephson current
1.4 junction
1.5 MAR processes
2 Cooper pair splitting in a Josephson junction geometry 
2.1 First proposals of Cooper pair splitters
2.2 Cooper pair splitting in an equilibrium setup
2.3 Path integral formulation of the partition function
2.4 Free energy and Josephson current
2.5 Numerical results
2.6 Conclusion and perspectives
3 Current and noise characteristics of multiple Cooper pair resonances 
3.1 Multipair production in superconducting bijunctions
3.2 Model
3.2.1 Hamiltonian formulation
3.2.2 Green’s functions in the Keldysh formalism
3.2.3 Self energy of the quantum dots
3.2.4 Double Fourier representation and Dyson equation
3.3 Current correlations
3.3.1 Current operator and its average value
3.3.2 Current correlations
3.3.3 Josephson current and noise at a MCPR
3.4 Numerical results
3.4.1 Resonant dots regime
3.4.2 Metallic junction regime
3.5 Conclusion and perspectives
4 Topological superconductors 
4.1 From particle physics to condensed matter physics
4.2 Kitaev modelization of p-wave superconductivity
4.2.1 Bulk properties of the Kitaev model
4.2.2 Topological phase transition
4.2.3 Majorana mode expansion in the Kitaev model
4.2.4 Topological protection of Majorana end states
4.3 Practical realization of p-wave superconductivity
4.4 Non-Abelian statistics
5 Josephson current and thermal noise in a junction between two topological superconductors 
5.1 Unified scattering approach to quantum transport in S-S and TS-TS junctions .
5.1.1 Unified Hamiltonian description of S-S and TS-TS junctions
5.1.2 Hamiltonian diagonalization
5.1.3 Current operator and statistics
5.2 Scattering states
5.2.1 First consequences of the matching condition
5.2.2 Continuum wavefunctions
5.2.3 Andreev bound states
5.2.4 Majorana states in a TS-TS junction
5.3 Andreev sector
5.4 Non-resonant frequency noise
5.4.1 Continuum-continuum transitions
5.4.2 Andreev-continuum transitions
5.4.3 Total non-resonant noise
5.5 Conclusion and perspectives
Conclusion
Bibliography

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