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arXiv:2604.16647

Nonequilibrium Cooper quartet generation in superconducting devices

Luca Chirolli, Alessandro Braggio, Michele Governale

Published Apr 17, 2026• 0 citations
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Abstract

Cooper quartets are aggregates of four electrons that generalize the concept of Cooper pairs, and their study can unfold unexplored perspectives in correlated matter and many-body physics. We propose a method to isolate them in a double-quantum-dot system coupled to conventional superconducting and normal leads. By driving the system out of equilibrium, we show that a resonance between the vacuum $|0\rangle$ and the four-electron state $|4e\rangle$ emerges in the high bias voltage regime, which involves a two-Cooper pair exchange process and is characterized by finite quartet correlations. We study the transport properties of the system and show that a peak in the Andreev current at high bias voltage has a width that scales with the magnitude of the quartet coupling $Γ_{4e}$, which can be tuned by the phase of additional superconducting leads, yielding distinctive signatures. By further studying the current-current correlations and the Fano factor, we establish a regime characterized by equal auto- and cross-correlations, which we interpret as a definitive signature of fast coherent two-Cooper-pair oscillations between the dots and the superconducting leads. The proposed platform, experimentally accessible in a quantum solid-state laboratory, enables exploration of quartet correlations and multifermion-correlated states of matter.

DOI
10.48550/arxiv.2604.16647

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Cooper quartets are aggregates of four electrons that generalize the concept of Cooper pairs, and their study can unfold unexplored perspectives in correlated matter and many-body physics. We propose a method to isolate them in a double-quantum-dot system coupled to conventional superconducting and normal leads. By driving the system out of equilibrium, we show that a resonance between the vacuum ∣0⟩|0\rangle∣0⟩ and the four-electron state ∣4e⟩|4e\rangle∣4e⟩ emerges in the high bias voltage regime, which involves a two-Cooper pair exchange process and is characterized by finite quartet correlations. We study the transport properties of the system and show that a peak in the Andreev current at high bias voltage has a width that scales with the magnitude of the quartet coupling Γ4eΓ_{4e}Γ4e​, which can be tuned by the phase of additional superconducting leads, yielding distinctive signatures. By further studying the current-current correlations and the Fano factor, we establish a regime characterized by equal auto- and cross-correlations, which we interpret as a definitive signature of fast coherent two-Cooper-pair oscillations between the dots and the superconducting leads. The proposed platform, experimentally accessible in a quantum solid-state laboratory, enables exploration of quartet correlations and multifermion-correlated states of matter.
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We propose and analyze a spin-entangler for electrons based on an s-wave superconductor coupled to two quantum dots, each of which is coupled to normal Fermi leads. We show that in the presence of a voltage bias and in the Coulomb blockade regime two correlated electrons provided by the Andreev process can coherently tunnel from the superconductor via different dots into different leads. The spin singlet coming from the Cooper pair remains preserved in this process, and the setup provides a source of mobile and nonlocal spin-entangled electrons. The transport current is calculated and shown to be dominated by a two-particle Breit-Wigner resonance that allows the injection of two spin-entangled electrons into different leads at exactly the same orbital energy, which is a crucial requirement for the detection of spin entanglement via noise measurements. The coherent tunneling of both electrons into the same lead is suppressed by the on-site Coulomb repulsion and/or the superconducting gap, while the tunneling into different leads is suppressed through the initial separation of the tunneling electrons. In the regime of interest the particle-hole excitations of the leads are shown to be negligible. The Aharonov-Bohm oscillations in the current are shown to contain single- and two-electron periods with amplitudes that both vanish with increasing Coulomb repulsion albeit differently fast.
Recent experiments on Cooper pair splitters using superconductor-quantum dot hybrids have embarked on creating entanglement in the solid-state, by engineering the sub-gap processes in the superconducting region. Using the thermoelectric Cooper pair splitter setup [Nat. Comm., 12, 21, (2021)] as a prototype, we present a comprehensive analysis of the fundamental components of the observed transport signal, aiming to critically clarify the operating regimes and confirm the nonlocal and nonclassical nature of correlations arising from crossed Andreev processes. By making a nexus with quantum discord, we identify operating points of nonlocal quantum correlations in the CPS device—information that cannot be extracted from the transport signal alone. A notable consequence of our analysis is the finding that contact-induced level broadening of the quantum dot’s discrete energy spectrum, along with its hybridization with the superconducting segment, can lead to shifted resonances in the crossed Andreev process as well as a parity reversal in the thermoelectric current. Our work thereby provides detailed insights into the gate voltage control of the quantum correlations in superconducting-hybrid Cooper pair splitters, revealing new avenues for harnessing quantum correlations in solid-state systems.
A three-terminal Josephson junction biased at opposite voltages can sustain a phase-sensitive dc current carrying three-body static phase coherence, known as the ``quartet current.'' We calculate the zero-frequency current noise cross correlations and answer the question of whether this current is noisy (like a normal current in response to a voltage drop) or noiseless (like an equilibrium supercurrent in response to a phase drop). A quantum dot with a level at energy ${\ensuremath{\epsilon}}_{0}$ is connected to three superconductors Sa,Sb{S}_{a},\phantom{\rule{0.16em}{0ex}}{S}_{b}Sa​,Sb​, and Sc{S}_{c}Sc​ with gap $\mathrm{\ensuremath{\Delta}}$, biased at ${V}_{a}=V,\phantom{\rule{0.16em}{0ex}}{V}_{b}=\ensuremath{-}V$, and Vc=0{V}_{c}=0Vc​=0, and with intermediate contact transparencies. At zero temperature, nonlocal quartets (in the sense of four-fermion correlations) are noiseless at subgap voltage in the nonresonant dot regime ${\ensuremath{\epsilon}}_{0}/\mathrm{\ensuremath{\Delta}}\ensuremath{\gg}1$, which is demonstrated with a semianalytical perturbative expansion of the cross correlations. Noise reveals the absence of granularity of the superflow splitting from Sc{S}_{c}Sc​ towards (Sa,Sb)({S}_{a},{S}_{b})(Sa​,Sb​) in the nonresonant dot regime, in spite of finite voltage. In the resonant dot regime ${\ensuremath{\epsilon}}_{0}/\mathrm{\ensuremath{\Delta}}\ensuremath{\lesssim}1$, cross correlations measured in the (Va,Vb)({V}_{a},{V}_{b})(Va​,Vb​) plane should reveal an ``anomaly'' in the vicinity of the quartet line Va+Vb=0{V}_{a}+{V}_{b}=0Va​+Vb​=0, related to an additional contribution to the noise, manifesting the phase sensitivity of cross correlations under the appearance of a three-body phase variable. Phase-dependent effective Fano factors ${F}_{\ensuremath{\varphi}}$ are introduced, defined as the ratio between the amplitudes of phase modulations of the noise and the currents. At low bias, the Fano factors ${F}_{\ensuremath{\varphi}}$ are of order unity in the resonant dot regime ${\ensuremath{\epsilon}}_{0}/\mathrm{\ensuremath{\Delta}}\ensuremath{\lesssim}1$, and they are vanishingly small in the nonresonant dot regime ${\ensuremath{\epsilon}}_{0}/\mathrm{\ensuremath{\Delta}}\ensuremath{\gg}1$.
We study the critical Josephson current flowing through a double quantum dot weakly coupled to two superconducting leads. We use analytical as well as numerical methods to investigate this setup in the limit of small and large bandwidth leads in all possible charging states, where we account for on-site interactions exactly. Our results provide clear signatures of nonlocal spin-entangled pairs, which support interpretations of recent experiments [R. S. Deacon, A. Oiwa, J. Sailer, S. Baba, Y. Kanai, K. Shibata, K. Hirakawa, and S. Tarucha, Nat. Commun. 6, 7446 (2015)]. In addition, we find that the ground state with one electron on each quantum dot can undergo a tunable singlet-triplet phase transition in the regime where the superconducting gap in the leads is not too large, which gives rise to an additional new signature of nonlocal Cooper-pair transport.
Abstract Entanglement of electrons is studied by means of current–current correlations in two Cooper pair splitter devices: with one and two proximized quantum dots (1QD and 2QD), in presence of intra- and inter-dot Coulomb interactions, and weakly coupled with metallic electrodes. The 1QD system, where Cooper pairs can be transmitted to the same or split to different normal electrodes, is contrasted with the 2QD device, where double occupancy of a single quantum dot is forbidden and transport is only through an inter-dot singlet due to non-local crossed Andreev reflection processes deep in the superconducting energy gap. Separating the current correlation function into components for partial currents of electrons and holes through various Andreev bound states, one can see bunching and antibunching of split particles: inter-level components between electron and hole currents flowing to different electrodes are positive, while intra-level electron–electron or hole–hole components are negative, respectively. Spectral decomposition of the frequency-dependent current cross-correlation is performed to get better insight into mechanisms of entanglement and dynamics of split Cooper pairs, and to extract various charge fluctuation processes with different relaxation times, related to electron and hole currents flowing through the Andreev bound states. Only low frequency polarization fluctuations are seen in the current cross-correlations, while various negative and positive high frequency (charge fluctuations) components compensate each other in the symmetric system.