Phi-Ground-Any-4B: Four-Terminal Superconducting Device

• The paper introduces a four-terminal superconducting device on a 2D metal where flux-dependent quartet current inversion is the central focus.
• It distinguishes two competing channels: π-shifted three-terminal quartets and 0-shifted four-terminal split quartets, with their interplay controlled by perturbative orders.
• The study emphasizes the role of 2D quantum wake in graphene and shows how gate tuning impacts the relative contribution of higher-order quartet processes.

The designation Phi-Ground-Any-4B is not independently defined in the available source, but the surrounding context strongly suggests a flux-threaded, grounded, four-terminal superconducting structure of the type analyzed in "Inversion in a four-terminal superconducting device on the quartet line: I. Two-dimensional metal and the quartet beam splitter" (Mélin, 2020). The underlying system is a four-terminal superconducting device built on a 2D metal, modeled as graphene gated away from the Dirac point, and operated on the quartet line. Its defining theoretical problem is the flux dependence of the quartet critical current, especially the inversion between $\Phi/\Phi_0=0$ and $\Phi/\Phi_0=1/2$, which the paper interprets through interference between standard three-terminal quartets and nonstandard four-terminal split quartets.

1. Device geometry and quartet-line biasing

The device consists of four superconducting terminals, $S_a$, $S_b$, $S_{c_1}$, and $S_{c_2}$, attached to a 2D metal. The terminals $S_{c_1}$ and $S_{c_2}$ belong to the same grounded superconducting loop $S_c$, threaded by magnetic flux $\Phi$. The operating condition is the quartet line,

$\Phi/\Phi_0=1/2$0

so that $\Phi/\Phi_0=1/2$1 and $\Phi/\Phi_0=1/2$2 are biased at opposite voltages while the loop contacts are grounded. A convenient gauge choice is

$\Phi/\Phi_0=1/2$3

with $\Phi/\Phi_0=1/2$4 the common loop phase, $\Phi/\Phi_0=1/2$5 the flux quantum, and $\Phi/\Phi_0=1/2$6 the reduced flux. On the quartet line, the gauge-invariant static phase combination is

$\Phi/\Phi_0=1/2$7

(Mélin, 2020)

This geometry is specifically designed to isolate dc quartet transport under opposite bias. The loop flux shifts the phases of the two grounded contacts with opposite signs, so flux enters the quartet response as a controlled interferometric variable rather than as a passive perturbation. That phase structure is the basis for the inversion phenomenon emphasized in the paper.

2. Microscopic formulation and perturbative organization

The Hamiltonian contains BCS superconducting leads, a 2D metal Hamiltonian, and tunneling terms between the 2D metal and each superconducting lead. The tunnel Hamiltonian is written as

$\Phi/\Phi_0=1/2$8

with $\Phi/\Phi_0=1/2$9 and $S_a$0. The expansion parameter is $S_a$1, where $S_a$2 is the band scale and $S_a$3 is a representative tunneling amplitude. The current is organized through Dyson equations for Nambu Green's functions and extracted from Keldysh Green's functions via an expression of the form

$S_a$4

with the dc component taken at harmonic index $S_a$5. In the adiabatic limit $S_a$6, the perturbative hierarchy places three-terminal quartets at order $S_a$7 and four-terminal split quartets at order $S_a$8 (Mélin, 2020).

This ordering is central because it controls the relative visibility of competing coherent processes. The perturbative structure implies that the four-terminal contribution is parametrically higher order than the standard three-terminal channel, which in turn makes gate tuning of the effective coupling particularly consequential for the observed harmonic content.

3. Quartet channels and their sign structure

Two distinct quartet mechanisms contribute on the quartet line. The first is the standard three-terminal quartet process. In 3TQ$S_a$9, two Cooper pairs originating from $S_b$0 and $S_b$1 exchange partners and recombine into the same grounded terminal $S_b$2; in 3TQ$S_b$3, the same occurs into $S_b$4. Their current-phase relations are

$S_b$5

$S_b$6

with perturbative signs

$S_b$7

Both 3TQ channels are therefore $S_b$8-shifted. The second mechanism is the four-terminal split quartet (4TSQ), in which the two Cooper pairs are split so that one pair enters $S_b$9 and the other enters $S_{c_1}$0. Its leading current-phase relation is

$S_{c_1}$1

with dominant sign

$S_{c_1}$2

so the 4TSQ is $S_{c_1}$3-shifted (Mélin, 2020).

The physical distinction is not merely topological but algebraic. For 3TQ, the recombination of two Cooper pairs into a single grounded lead generates an overall minus sign, yielding the $S_{c_1}$4 shift. For 4TSQ, an additional exchange of two quasiparticles between $S_{c_1}$5 and $S_{c_1}$6 changes the sign structure. The abstract characterizes this mechanism as synchronization of two Josephson junctions by exchange of two quasiparticles "surfing" on the 2D quantum wake, and states that the mechanism is already operational at equilibrium. This sign opposition between $S_{c_1}$7-shifted 3TQ and $S_{c_1}$8-shifted 4TSQ is the decisive ingredient in the flux inversion argument.

4. Critical current, flux interference, and inversion

The quartet critical current on the quartet line is defined as the maximum, over the quartet phase, of the supercurrent on the quartet line. In the adiabatic regime, its flux dependence is governed by interference between the two 3TQ channels and the 4TSQ channel:

$S_{c_1}$9

At the two flux values emphasized in the paper, this yields

$S_{c_2}$0

$S_{c_2}$1

Accordingly,

$S_{c_2}$2

(Mélin, 2020)

This is the paper's central logical claim: because perturbation theory gives $S_{c_2}$3-shifted 3TQ and $S_{c_2}$4-shifted 4TSQ, the observation of unequal critical currents at $S_{c_2}$5 and $S_{c_2}$6 is interpreted as evidence for coexistence of those contributions, especially the 4TSQ channel. A common simplification would be to regard the inversion as a generic flux effect of any four-terminal interferometer. The paper argues against that reading by tying the inequality specifically to interference between even and odd partitions of Cooper pairs between the two grounded contacts, and by stating that "Observation of $S_{c_2}$7" implies "Evidence for the four-terminal 4TSQ" for finite bias voltage on the quartet line and arbitrary interface transparencies.

5. Conditions for four-terminal split quartets

The 4TSQ contribution is derived under several explicit conditions. The superconducting leads are assumed to be in the dirty limit,

$S_{c_2}$8

where $S_{c_2}$9 is the elastic mean free path and $S_{c_1}$0 is the zero-energy ballistic coherence length. In this regime, the relevant superconducting correlations are diffusion modes rather than independent single-particle propagators. The paper also distinguishes two contact-size regimes: if the contact radius satisfies $S_{c_1}$1, the effective scaling is proportional to $S_{c_1}$2, whereas if $S_{c_1}$3, the scaling is proportional to $S_{c_1}$4. The dirty-limit coherence length is given parametrically by

$S_{c_1}$5

(Mélin, 2020)

The dimensionality of the normal region is equally important. The paper states that 4TSQ is robust only in 2D, because the 2D Green's function has a long-range oscillatory tail,

$S_{c_1}$6

interpreted as a 2D quantum wake. In 1D or 3D, the corresponding averaged squared propagators vanish under multichannel averaging, suppressing the effect. Mesoscopic long-range coherence is also required: the effect survives when the separation between $S_{c_1}$7 and $S_{c_1}$8 is within the phase-coherent scale,

$S_{c_1}$9

with $S_{c_2}$0 the mesoscopic phase coherence length. This suggests that the grounded-loop geometry is not merely a circuit motif but a coherence-sensitive splitter whose operation depends on 2D propagation, diffusive superconducting correlations, and sufficiently extended phase coherence.

6. Gate dependence, generalized formulas, and experimental interpretation

The paper argues that gating the graphene sheet away from the Dirac point increases the density of states and effectively increases $S_{c_2}$1. Because 3TQ appears at order $S_{c_2}$2 while 4TSQ appears at order $S_{c_2}$3, gating can alter the relative weights of the harmonics. The phenomenological flux dependence is written as

$S_{c_2}$4

with $S_{c_2}$5 and

$S_{c_2}$6

The reported gate dependence favors larger $S_{c_2}$7 and smaller $S_{c_2}$8, hence more weight in the $S_{c_2}$9 harmonic relative to the $S_c$0 harmonic. Beyond lowest-order perturbation theory, the paper introduces an Ambegaokar-Baratoff-type structure. In the adiabatic case, the quartet current is expressed through real coefficients $S_c$1, with $S_c$2 counting Cooper pairs taken from $S_c$3 and $S_c$4, and $S_c$5 counting how many are transmitted into $S_c$6; this leads to an even/odd decomposition,

$S_c$7

$S_c$8

At finite bias, the same Ambegaokar-Baratoff-like structure is extended to coefficients $S_c$9, and the current is decomposed into a quasiequilibrium piece and a nonequilibrium piece using Keldysh Green's functions (Mélin, 2020).

These general formulas sharpen the interpretation of experiment. In the paper's reading of the recent Harvard graphene experiment on a four-terminal Josephson junction with a grounded loop, the salient observations are quartet anomalies on the quartet line $\Phi$0, oscillations with reduced flux $\Phi$1, an inversion in which the quartet signal is stronger at $\Phi$2 than at $\Phi$3, and gate dependence consistent with enhanced higher-order quartet processes. The authors conclude that these observations are naturally explained if the usual 3TQ channels are $\Phi$4-shifted and a nonstandard 4TSQ channel is present and $\Phi$5-shifted, enabled by the 2D quantum wake in graphene and the grounded-loop geometry. In that sense, the statement

$\Phi$6

is the paper's principal interpretive claim rather than a purely model-free identity.