Ultralong-distance quantum correlations in three-terminal Josephson junctions

In the paper, we address nonlocality and quantum correlations in three-terminal Josephson junctions, where the BCS superconductors $S_a$, $S_b$ and $S_c$ are voltage-biased at $(V_a,V_b,V_c)=(V,-V,0)$ and $V$ is a significant fraction of the gap. The constituting two-terminal $S_a$-dot-$S_c$ and $S_c$-dot-$S_b$ are connected at arbitrary distance $R_0$ on the grounded $S_c$. The proposed interpretation of the numerical experiments relies on the interplay between the time-periodic Floquet-Josephson dynamics, Cooper pair splitting and the long-range Tomasch effect. We find cross-over between the "Floquet-Andreev quartets" (if $R_0<\xi_0$ is smaller than the superconducting coherence length), and the "ultralong-distance Floquet-Tomasch clusters of Cooper pairs" if $R_0 < l_\varphi$, where $l_\varphi\gg\xi_0$ is the mesoscopic coherence length of the BCS quasiparticles. Analytical theory is presented for the simplest cluster at voltage $eV>\Delta/2$, i.e. the ultralong-distance Floquet-Tomasch octets, where $\Delta$ is the superconducting gap. The range of the effect is conjectured to be the same as in the Tomasch experiment, i.e. the junctions can be remotely separated by the mesoscopic $R_0\approx 30\,\mu$m which is orders of magnitude larger than the zero-energy BCS coherence length $\xi_0$. Our results go beyond the paradigm of classical synchronization in the macroscopic Josephson circuits. The effect can be detected with dc-transport and zero-frequency quantum current-noise cross-correlation experiments, and it can be used for fundamental studies of superconducting quasiparticle quantum coherence in the circuits of quantum engineering.Comment: 12 pages, 9 figures, a reference was adde

Enhancement of Cooper pair splitting by multiple scattering

In three-terminal NSN hybrid structures the influence of additional barriers on the nonlocal conductance and on current cross-correlations is studied within a scattering theory. In metallic systems with additional barriers and phase averaging, which simulate disordered regions, local processes can be enhanced by reflectionless tunneling but this mechanism has little influence on nonlocal processes and on current cross-correlations. Therefore Cooper pair splitting cannot be enhanced by reflectionless tunneling. On the contrary, in ballistic systems, additional barriers lead to Fabry-Perot resonances and allow to separate the different contributions to the conductance and to the current cross-correlations. In particular, crossed Andreev processes can be selectively enhanced by tuning the length or the chemical potential of the interbarrier region.Comment: 18 pages, 18 figures, 1 tabl

Absence of split pairs in the cross-correlations of a highly transparent normal metal-superconductor-normal metal electron beam splitter

The nonlocal conductance and the current cross-correlations are investigated within scattering theory for three-terminal normal metal-superconductor-normal metal (NSN) hybrid structures. The positive cross-correlations at high transparency found by M\'elin, Benjamin and Martin [Phys. Rev. B 77, 094512 (2008)] are not due to crossed Andreev reflection. On the other hand, local processes can be enhanced by reflectionless tunneling but this mechanism has little influence on nonlocal processes and on current cross-correlations. Therefore Cooper pair splitting cannot be enhanced by reflectionless tunneling. Overall, this shows that NSN structures with highly transparent or effectively highly transparent interfaces are not suited to experimentally producing entangled split pairs of electrons.Comment: 11 pages, 6 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:1211.534

Partially resummed perturbation theory for multiple Andreev reflections in a short three-terminal Josephson junction

In a transparent three-terminal Josephson junction, modeling nonequilibrium transport is numerically challenging, owing to the interplay between multiple Andreev reflection (MAR) thresholds and multipair resonances in the pair current. An approximate method, coined as "partially resummed perturbation theory in the number of nonlocal Green's functions", is presented that can be operational on a standard computer and demonstrates compatibility with results existing in the literature. In a linear structure made of two neighboring interfaces (with intermediate transparency) connected by a central superconductor, tunneling through each of the interfaces separately is taken into account to all orders. On the contrary, nonlocal processes connecting the two interfaces are accounted for at the lowest relevant order. This yields logarithmically divergent contributions at the gap edges, which are sufficient as a semi-quantitative description. The method is able to describe the current in the full two-dimensional voltage range, including commensurate as well as incommensurate values. The results found for the multipair (for instance quartet) current-phase characteristics as well as the MAR thresholds are compatible with previous results. At intermediate transparency, the multipair critical current is much larger than the background MAR current, which supports an experimental observation of the quartet and multipair resonances. The paper provides a proof of principle for addressing in the future the interplay between quasiparticles and multipairs in four-terminal structures.Comment: 18 pages, 10 figures, improvements in the presentation, Eur. Phys. J. B in pres

Positive current cross-correlations in a highly transparent normal-superconducting beam splitter due to synchronized Andreev and inverse Andreev reflections

Predictions are established for linear differential current-current cross-correlations dSab/dV in a symmetrically biased three-terminal normal metal-superconductor-normal metal (NSN) device. Highly transparent contacts turn out to be especially interesting because they feature positive dSab/dV. At high transparency, processes based on Crossed Andreev Reflection (CAR) contribute only negligibly to the current and to dSab/dV. Under these circumstances, current-current cross-correlations can be plausibly interpreted as a coherent coupling between the two NS interfaces in the form of synchronized Andreev and inverse Andreev reflections, corresponding to the process where a pair of electron-like quasi-particles and a pair of hole-like quasi-particles arrive from the normal electrodes and annihilate in the superconductor. Hence, positive dSab/dV does not automatically imply CAR. For tunnel contacts, dSab/dV is positive because of CAR. In between these two extremities, at intermediate transparencies, dSab/dV is negative because both processes which cause positive correlations, occur only with small amplitude. We use scattering theory to obtain analytic expressions for current and noise, and microscopic calculation using a tight binding model in order to obtain a clear interpretation of the physical processes.Comment: 15 pages, 11 figures; Revised manuscript, analytical BTK-calculation, results not change

Phase-sensitive transport at a normal metal-superconductor interface close to a Josephson junction

Phase- and voltage bias-sensitive quasiparticle transport at a double $NIS_1IS_2$ interface is considered. The barriers $I$ range from tunnel to transparent, and the intermediate region $S_1$ has a width comparable to the superconducting coherence length. A phase difference $\varphi$ is applied to the Josephson junction $S_1IS_2$. The normal and Andreev reflections at the $NIS_1$ interface become $\varphi$-sensitive, and transport is governed by interferences within the narrow $S_1$ region, both in the normal and anomalous channels. The subgap conductance is separately (energy $E$)- and (phase $\varphi$)- symmetric. Above the superconducting gap, the conductance is in general not symmetric even if $(E,\varphi)$ is changed in $(-E,-\varphi)$, but the symmetry is restored by averaging Fermi oscillations. The Tomasch oscillations are amplified by the phase difference. The subgap conductance exhibits a resonant structure at the energy of the Andreev bound states (ABS) of the $S_1IS_2$ junction, providing a side-spectroscopy of such states. Depending on the relative transparencies of the junctions, the resonance can increase or reduce the conductance, and it can even vanish for $\varphi=\pi$, featuring total reflection of quasiparticles at $NS_1$ by the ABS at $S_1S_2$.Comment: 8 pages, 10 figures, 1 tabl

A quantum interferometer for quartets in superconducting three-terminal Josephson junctions

An interferometric device is proposed in order to analyze the quartet mode in biased three-terminal Josephson junctions (TTJs), and to provide experimental evidence for emergence of a single stationary phase, the so-called quartet phase. In such a quartet-Superconducting Quantum Interference Device (quartet-SQUID), the flux sensitivity exhibits period ${hc}/{4e}$, which is the fingerprint of a transient intermediate state involving two entangled Cooper pairs. The quartet-SQUID provides two informations: an amplitude that measures a total ``quartet critical current'', and a phase lapse coming from the superposition of the following two current components: the quartet supercurrent that is odd in the quartet phase, and the phase-sensitive multiple Andreev reflection (phase-MAR) quasiparticle current, that is even in the quartet phase. This makes a TTJ a generically "$\theta$-junction". Evidence for phase-MARs plays against conservative scenarii involving synchronization of AC Josephson currents, based on ``adiabatic'' phase dynamics and RSJ-like models.Comment: 6 pages, 2 figures, revised manuscript (minor modifications

Designability, thermodynamic stability, and dynamics in protein folding: a lattice model study

In the framework of a lattice-model study of protein folding, we investigate the interplay between designability, thermodynamic stability, and kinetics. To be ``protein-like'', heteropolymers must be thermodynamically stable, stable against mutating the amino-acid sequence, and must be fast folders. We find two criteria which, together, guarantee that a sequence will be ``protein like'': i) the ground state is a highly designable stucture, i. e. the native structure is the ground state of a large number of sequences, and ii) the sequence has a large $\Delta/\Gamma$ ratio, $\Delta$ being the average energy separation between the ground state and the excited compact conformations, and $\Gamma$ the dispersion in energy of excited compact conformations. These two criteria are not incompatible since, on average, sequences whose ground states are highly designable structures have large $\Delta/\Gamma$ values. These two criteria require knowledge only of the compact-state spectrum. These claims are substantiated by the study of 45 sequences, with various values of $\Delta/\Gamma$ and various degrees of designability, by means of a Borst-Kalos-Lebowitz algorithm, and the Ferrenberg-Swendsen histogram optimization method. Finally, we report on the reasons for slow folding. A comparison between a very slow folding sequence, an average folding one and a fast folding one suggests that slow folding originates from a proliferation of nearly compact low-energy conformations, not present for fast folders.Comment: 24 pages, 10 figures, 2 table

Berry phase in superconducting multiterminal quantum dots

We report on the study of the non-trivial Berry phase in superconducting multiterminal quantum dots biased at commensurate voltages. Starting with the time-periodic Bogoliubov-de Gennes equations, we obtain a tight binding model in the Floquet space, and we solve these equations in the semiclassical limit. We observe that the parameter space defined by the contact transparencies and quartet phase splits into two components with a non-trivial Berry phase. We use the Bohr-Sommerfeld quantization to calculate the Berry phase. We find that if the quantum dot level sits at zero energy, then the Berry phase takes the values $\varphi_B=0$ or $\varphi_B=\pi$. We demonstrate that this non-trivial Berry phase can be observed by tunneling spectroscopy in the Floquet spectra. Consequently, the Floquet-Wannier-Stark ladder spectra of superconducting multiterminal quantum dots are shifted by half-a-period if $\varphi_B=\pi$. Our numerical calculations based on Keldysh Green's functions show that this Berry phase spectral shift can be observed from the quantum dot tunneling density of states.Comment: 15 pages, 7 figures. Supplemental Material as ancillary file (3 pages, 5 figures), manuscript in final for