Cooper Quartets: Charge‑4e Correlations

• Cooper quartets are four-fermion objects (formed by two Cooper pairs) that enable charge‑4e transport in multiterminal superconducting systems.
• They are characterized by connected four-point correlators and phase-sensitive Josephson harmonics, revealing non-classical coherent interactions.
• Experimental platforms such as nanowires, graphene junctions, and quantum dots demonstrate quartet signatures, advancing many-body superconductivity research.

Cooper quartets are correlated four-fermion objects built from two Cooper pairs. In multiterminal Josephson systems, the term denotes a phase-coherent four-electron transport process involving three superconducting terminals, in which two Cooper pairs are exchanged collectively and the relevant phase variable is a combination of several superconducting phases rather than a single phase difference. In many-body settings, the same term denotes a genuine four-body correlated state or condensate, often discussed as the charge-$4e$ analogue of ordinary $2e$ Cooper pairing. Across these settings, the defining feature is that the coherent object is not reducible to independent two-electron pair amplitudes (Gupta et al., 2023, Chirolli et al., 2024, Guo et al., 20 Jan 2026).

1. Definition and conceptual scope

In the multiterminal Josephson literature, a Cooper quartet is a phase-coherent four-electron transport process involving three superconducting electrodes. Two Cooper pairs are coherently correlated among all three terminals, so the coupling is intrinsically non-classical: it cannot be represented as a simple network of independent two-terminal Josephson junctions. In this usage, the quartet is a transport process and a phase-sensitive Josephson harmonic at charge $4e$, rather than a bound state in isolation (Gupta et al., 2023, Mélin et al., 2014).

In interacting hybrid systems and quartet-superfluid theories, the same expression refers to a genuine four-body correlated state. A standard diagnostic is the connected four-point correlator

$$Q=\langle d_{1\downarrow}d_{1\uparrow}d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{1\uparrow}\rangle\langle d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{2\uparrow}\rangle\langle d_{1\uparrow}d_{2\downarrow}\rangle,$$

constructed to vanish for Wick-factorizable states. A nonzero $Q$ therefore isolates quartet correlations beyond ordinary pair physics (Chirolli et al., 2024, Chirolli et al., 17 Apr 2026).

A recurrent quartet state in double-dot and effective many-body formulations is the coherent superposition

$$|\phi^\pm_{\rm Q}\rangle=\frac{1}{\sqrt 2}(|0\rangle \pm |4e\rangle),$$

where $|0\rangle$ is the vacuum and $|4e\rangle$ is a fully occupied four-electron configuration. In that form, total charge fluctuates in units of $4e$, and the quartet sector is characterized by maximal four-body correlations with negligible or vanishing pair correlators under the conditions stated in the relevant models (Chirolli et al., 2024, Chirolli et al., 17 Apr 2026).

2. Multiterminal Josephson mechanism and phase structure

The canonical transport realization of a Cooper quartet is a three-terminal Josephson junction biased so that the phase combination associated with a two-Cooper-pair process becomes stationary. For voltages $(0,V,-V)$, or equivalently $2e$0 with the middle terminal used as the phase reference, the quartet phase

$2e$1

is time independent even though the individual superconducting phases evolve. This permits a d.c. Josephson-like current at finite bias, carried by a three-terminal Andreev bound state or an effective quartet Andreev bound state after averaging over the fast phase variable (Cohen et al., 2016, Mélin et al., 2014).

The quartet current-phase relation is predicted to be generically $2e$2-shifted relative to ordinary Josephson transport. In a four-terminal circuit-theory treatment, the quartet contribution is written as

$2e$3

so the quartet term appears as a sign-reversed multipair Josephson harmonic. In the PbTe three-terminal junction model, the same physics is represented by a negative three-terminal coupling $2e$4, which yields a $2e$5-shifted harmonic in the Josephson energy and a field dependence

$2e$6

The additional $2e$7 term is the characteristic non-classical harmonic used to account for field-enhanced critical current and non-convex diffraction patterns (Mélin et al., 2023, Gupta et al., 2023).

This phase structure is central to the broader multiterminal program. An $2e$8-terminal junction has $2e$9 independent phase differences, and the corresponding Andreev spectrum can host Weyl crossings and higher-order Chern numbers. The quartet term is important in that context because it demonstrates a genuinely multiterminal Josephson interaction through a single coherent scattering region, especially in the few-mode regime required by topological proposals (Gupta et al., 2023).

3. Experimental and interferometric signatures

Several platforms use distinct observables to separate quartet transport from ordinary pair supercurrent, higher harmonics, or dissipative multiple Andreev reflection.

Platform or diagnostic	Reported or predicted quartet signature	Source
Three-terminal Al–InAs nanowire junction	Sharp conductance ridge along $4e$0, positive zero-frequency current cross-correlation, suppression when one arm is pinched off, absence when crossed Andreev reflection is suppressed, estimated $4e$1	(Cohen et al., 2016)
Four-terminal graphene Josephson junction with flux loop	Magneto-oscillation of the quartet critical supercurrent with $4e$2 periodicity, plus non-monotonic bias dependence attributed to Landau-Zener transitions between Floquet bands	(Huang et al., 2020)
Three-terminal SAG PbTe nanowire junction	Finite differential resistance near zero bias at $4e$3, zero-resistance regions at finite $4e$4, non-convex magnetic diffraction pattern, fitted negative $4e$5, and few-mode conductance features including $4e$6, $4e$7, $4e$8, and $4e$9 at $$Q=\langle d_{1\downarrow}d_{1\uparrow}d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{1\uparrow}\rangle\langle d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{2\uparrow}\rangle\langle d_{1\uparrow}d_{2\downarrow}\rangle,$$0 T	(Gupta et al., 2023)
Four-terminal critical current contour proposal	Zero-field nonconvex critical current contours and reentrant pockets when $$Q=\langle d_{1\downarrow}d_{1\uparrow}d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{1\uparrow}\rangle\langle d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{2\uparrow}\rangle\langle d_{1\uparrow}d_{2\downarrow}\rangle,$$1	(Mélin et al., 2023)
Quartet-SQUID interferometer	Flux periodicity $$Q=\langle d_{1\downarrow}d_{1\uparrow}d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{1\uparrow}\rangle\langle d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{2\uparrow}\rangle\langle d_{1\uparrow}d_{2\downarrow}\rangle,$$2 and a phase lapse caused by superposition of an odd quartet supercurrent and an even phase-MAR current, making the TTJ a generic $$Q=\langle d_{1\downarrow}d_{1\uparrow}d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{1\uparrow}\rangle\langle d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{2\uparrow}\rangle\langle d_{1\uparrow}d_{2\downarrow}\rangle,$$3-junction	(Mélin et al., 2023)

These signatures are not equivalent. In the biased three-terminal nanowire geometry, positive current cross-correlations were used to distinguish quartet supercurrent from nonlocal MAR, which is expected to yield negative correlations. In the interferometric proposals and graphene experiment, the central observable is the effective $$Q=\langle d_{1\downarrow}d_{1\uparrow}d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{1\uparrow}\rangle\langle d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{2\uparrow}\rangle\langle d_{1\uparrow}d_{2\downarrow}\rangle,$$4 flux periodicity. In the PbTe device and in the critical-current-contour proposal, the relevant hallmark is the $$Q=\langle d_{1\downarrow}d_{1\uparrow}d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{1\uparrow}\rangle\langle d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{2\uparrow}\rangle\langle d_{1\uparrow}d_{2\downarrow}\rangle,$$5-shifted harmonic, which produces non-convex field or current-space responses instead of a conventional convex Josephson boundary (Cohen et al., 2016, Huang et al., 2020, Gupta et al., 2023, Mélin et al., 2023, Mélin et al., 2023).

A further recurring requirement is few-mode transport. In the PbTe selective-area-grown geometry, supercurrent coexists with plateau-like gate-dependent conductance and non-monotonic conductance evolution, supporting the few-mode regime required for non-classical multiterminal couplings and for effective Hamiltonians with accessible Andreev-band topology (Gupta et al., 2023).

4. Interaction-driven quartets in hybrid quantum-dot systems

A distinct route to Cooper quartets does not begin from multiterminal phase bias, but from interactions in hybrid mesoscopic systems. In a double quantum dot coupled to an ordinary superconducting lead, an attractive interdot interaction $$Q=\langle d_{1\downarrow}d_{1\uparrow}d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{1\uparrow}\rangle\langle d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{2\uparrow}\rangle\langle d_{1\uparrow}d_{2\downarrow}\rangle,$$6 can stabilize a quartet ground state when $$Q=\langle d_{1\downarrow}d_{1\uparrow}d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{1\uparrow}\rangle\langle d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{2\uparrow}\rangle\langle d_{1\uparrow}d_{2\downarrow}\rangle,$$7 and the resonance condition

$$Q=\langle d_{1\downarrow}d_{1\uparrow}d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{1\uparrow}\rangle\langle d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{2\uparrow}\rangle\langle d_{1\uparrow}d_{2\downarrow}\rangle,$$8

brings the vacuum and four-electron configurations close to degeneracy. Projecting onto that low-energy subspace yields an effective two-level Hamiltonian with coupling

$$Q=\langle d_{1\downarrow}d_{1\uparrow}d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{1\uparrow}\rangle\langle d_{2\downarrow}d_{2\uparrow}\rangle -\langle d_{1\downarrow}d_{2\uparrow}\rangle\langle d_{1\uparrow}d_{2\downarrow}\rangle,$$9

and the ground state becomes the quartet superposition $Q$0. At resonance, the quartet correlator reaches

$Q$1

while the pair correlator is zero on the quartet states. In the same framework, a three-terminal extension exhibits a $Q$2-periodic current under suitable conditions, a nonlocal phase response, and a quartet Andreev qubit interpretation for the two low-lying hybridized states (Chirolli et al., 2024).

A nonequilibrium variant uses a double quantum dot coupled to a common superconducting lead, two normal leads, and optionally extra superconducting terminals. There the quartet sector is not the equilibrium ground state; instead, a high-bias resonance is generated between $Q$3 and $Q$4. The effective coupling is

$Q$5

and additional superconducting phases make $Q$6 phase tunable. The paper identifies several transport diagnostics: a high-bias Andreev-current peak whose linewidth scales with $Q$7, a finite quartet correlator with a double-peak structure around resonance, and a noise regime in which auto- and cross-correlations become equal, interpreted as fast coherent two-Cooper-pair oscillations between the dots and the superconducting leads (Chirolli et al., 17 Apr 2026).

Taken together, these double-dot results shift the meaning of “Cooper quartet” from a phase-biased Josephson transport mode to an interaction-generated many-body state. The common element is still charge-$Q$8 coherence, but the operative control parameters are now the interdot interaction, dot detuning, and induced local and crossed Andreev amplitudes rather than only superconducting phase bias (Chirolli et al., 2024, Chirolli et al., 17 Apr 2026).

5. Quartet condensation and charge-$Q$9 order in extended matter

Beyond mesoscopic devices, quartet physics has been formulated as a genuine many-body condensation problem. In an electron-hole liquid, a quartet-BCS variational treatment accommodates exciton-like Cooper pairs and biexciton-like Cooper quartets on equal footing. The quartet amplitude enters the density and the equation of state directly, the low-density limit satisfies

$$|\phi^\pm_{\rm Q}\rangle=\frac{1}{\sqrt 2}(|0\rangle \pm |4e\rangle),$$0

and the main conclusion is that biexciton-like quartets survive at high density as a BCS-like quartet condensate that lowers the ground-state energy and enlarges the excitation gap (Guo et al., 2022).

For a one-dimensional four-component Fermi gas, quartet superfluidity has been analyzed through both a quartet BCS variational ansatz and a generalized Nambu-Gor’kov formalism. The decisive methodological result is that the multiple-infinite-product ansatz reproduces the exact four-body dilute-limit result, whereas the single-infinite-product ansatz does not. The same work shows that the generalized Nambu-Gor’kov gap equation is consistent with the variational quartet equation, and that the single-particle spectral function evolves from a coherent BCS-like branch in weak coupling to a strongly damped, continuum-dominated spectrum in strong coupling while the quartet order parameter remains nonzero. A complementary effective theory of dilute spin-$$|\phi^\pm_{\rm Q}\rangle=\frac{1}{\sqrt 2}(|0\rangle \pm |4e\rangle),$$1 quartet condensation derives a quartic mean-field Hamiltonian, a quartet gap equation, a sixteen-dimensional occupation-space picture for excitations, and a superfluid fraction $$|\phi^\pm_{\rm Q}\rangle=\frac{1}{\sqrt 2}(|0\rangle \pm |4e\rangle),$$2 at $$|\phi^\pm_{\rm Q}\rangle=\frac{1}{\sqrt 2}(|0\rangle \pm |4e\rangle),$$3 within the dilute quartet approximation (Guo et al., 20 Jan 2026, Xu et al., 28 May 2026).

In nuclear matter, quartet-BCS theory has been applied to infinite symmetric matter and to finite $$|\phi^\pm_{\rm Q}\rangle=\frac{1}{\sqrt 2}(|0\rangle \pm |4e\rangle),$$4 nuclei. In infinite matter, the framework describes coexistence of pair and quartet correlations and a hierarchical structure of in-medium cluster formation in which low-momentum nucleons favor $$|\phi^\pm_{\rm Q}\rangle=\frac{1}{\sqrt 2}(|0\rangle \pm |4e\rangle),$$5-like quartet correlations. In finite nuclei, a local-density analysis combined with Skyrme Hartree-Fock densities for $$|\phi^\pm_{\rm Q}\rangle=\frac{1}{\sqrt 2}(|0\rangle \pm |4e\rangle),$$6, $$|\phi^\pm_{\rm Q}\rangle=\frac{1}{\sqrt 2}(|0\rangle \pm |4e\rangle),$$7, and $$|\phi^\pm_{\rm Q}\rangle=\frac{1}{\sqrt 2}(|0\rangle \pm |4e\rangle),$$8 finds large quartet condensate fractions at the nuclear surface and estimates a nucleon-quartet contribution to the Wigner term that is about one order of magnitude smaller than the empirical strength. A separate Ginzburg-Landau proposal models isospin symmetry breaking of nuclear Cooper quartets by coupling a quartet field to a pion-meson field and $$|\phi^\pm_{\rm Q}\rangle=\frac{1}{\sqrt 2}(|0\rangle \pm |4e\rangle),$$9 gluons (Guo et al., 2021, Guo et al., 10 Mar 2025, Miron, 2021).

A structurally different realization of charge-$|0\rangle$0 order appears in frustrated Josephson junction dice arrays. At frustration $|0\rangle$1, the second harmonic $|0\rangle$2 of the bulk current-phase relation becomes dominant and is about five times larger than $|0\rangle$3 on the sixfold-coordinated sublattice. Monte Carlo scaling gives

$|0\rangle$4

consistent with a half-vortex BKT transition, while tensor-network data show exponentially decaying $|0\rangle$5 correlations and algebraically decaying $|0\rangle$6 correlations below the transition. In that system, Cooper quartet superconductivity is organized by zero-energy domain walls and half-vortex deconfinement rather than by a mesoscopic Andreev mechanism (Weerda et al., 3 Jun 2026).

6. Distinctions, related usages, and common misconceptions

Not every use of the word quartet in superconductivity or many-body theory refers to a Cooper quartet in the charge-$|0\rangle$7 sense. In a mean-field theory of coupled condensates, a quartet can mean a set of four order parameters whose cyclic matrix product closes to $|0\rangle$8, so that any three induce the fourth as a hidden order parameter. That notion concerns algebraic constraints among condensates and may include kinetic terms, particle-hole orders, and superconducting orders; it is explicitly not a four-fermion bound state or a quartet supercurrent (Varelogiannis, 2013).

A second source of confusion is the terminology of orbital quintet pairing. “High orbital-moment Cooper pairs by crystalline symmetry breaking” studies ordinary two-electron Cooper pairs whose orbital sector has quintet angular momentum structure and momentum-space $|0\rangle$9-shifted domains. The paper states explicitly that it is not about Cooper quartets in the usual sense of a four-fermion bound state or four-electron correlated condensate (Mercaldo et al., 2023).

The literature therefore contains at least three non-equivalent objects: a multiterminal quartet supercurrent, an interaction-driven four-body state with nonzero connected correlator $|4e\rangle$0, and unrelated “quartets” of order parameters. This suggests that the most precise usage of Cooper quartet is reserved for settings where the operative coherent object carries charge $|4e\rangle$1 and is diagnosed either by a genuine four-point correlator, by a stationary multiterminal phase combination such as $|4e\rangle$2, or by long-range coherence of $|4e\rangle$3 in the absence of ordinary $|4e\rangle$4 order (Chirolli et al., 2024, Cohen et al., 2016, Weerda et al., 3 Jun 2026).