Josephson Harmonics: Non-Sinusoidal Phase Dynamics

• Josephson harmonics are defined as the higher-order Fourier components in the current–phase relation that capture non-sinusoidal superconducting transport.
• They originate from microscopic transport phenomena, interface symmetry breaking, and circuit-level effects such as series inductance, leading to observable features like fractional Shapiro steps.
• Applications include tuning qubit spectra, designing superconducting diodes, and engineering parametric amplifiers by exploiting modified phase dynamics.

Josephson harmonics are the higher-order Fourier components of the current–phase relation (CPR) and, equivalently, of the energy–phase relation of a Josephson element. In the ideal tunnel-junction limit, the CPR is sinusoidal, $I(\varphi)=I_c\sin\varphi$, and the Josephson potential is $-E_J\cos\varphi$. More generally, one writes

$$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi+\chi_n),\qquad U(\varphi)=-\sum_{n\ge 1} E_n \cos(n\varphi),$$

with $I_n=(2e/\hbar)\,nE_n$. Higher harmonics arise from microscopic transport beyond the extreme tunnel limit, from symmetry breaking and spin-active interfaces, from mesoscopic fluctuations, and from circuit dressing such as series inductance or nonlinear elimination of internal phases in composite elements. Their observable consequences include modified interference patterns, fractional Shapiro steps, altered qubit spectra, nonreciprocal critical currents, and tunable $\pi$-periodic or charge-$4e$ regimes (Tsarev et al., 26 May 2025, Willsch et al., 2023, Kim et al., 10 Jul 2025).

1. Harmonic expansion and Josephson energetics

The most general description used across the literature is a Fourier expansion of the supercurrent in the gauge-invariant phase difference. In one common notation,

$I(\varphi)=\sum_{n\ge1} I_n\sin(n\varphi+\chi_n),$

where $I_n$ are harmonic amplitudes and $\chi_n$ are symmetry-breaking phase shifts. In interferometric settings, the same structure can be written as

$$I(\varphi)=\sum_{n\ge1} I_n\sin[n(\varphi-\phi_0)] =\sum_{n\ge1}(A_n\sin n\varphi+B_n\cos n\varphi),$$

so that phase offsets and cosine components become explicit. The energy relation is fixed by

$-E_J\cos\varphi$0

which leads to $-E_J\cos\varphi$1 and $-E_J\cos\varphi$2 (Leblanc et al., 2023, Kim et al., 10 Jul 2025).

A two-harmonic truncation is especially prominent,

$-E_J\cos\varphi$3

or, in dimensionless notation,

$-E_J\cos\varphi$4

with $-E_J\cos\varphi$5. This form is sufficient to generate the full hierarchy of fractional Shapiro steps in the overdamped RSJ model and to produce polarity asymmetry when a relative phase shift is present (Tsarev et al., 26 May 2025).

The harmonic viewpoint is not restricted to deterministic CPRs. In quasi-one-dimensional diffusive magnetic Josephson junctions with transparent contacts, the fluctuational regime yields

$-E_J\cos\varphi$6

with sample-dependent amplitudes and independent phase shifts. In that setting, all harmonics are present, each harmonic has its own sample-dependent amplitude and phase shift with no correlation between different harmonics, and the junction can realize either a $-E_J\cos\varphi$7 or a $-E_J\cos\varphi$8 junction depending on the magnetic weak link (Ioselevich et al., 2018).

These expansions make clear that “Josephson harmonics” are not merely a correction to a sinusoidal law. They are the natural language for describing non-sinusoidal superconducting transport, whether the non-sinusoidality originates in microscopic Andreev physics, interfacial symmetry breaking, mesoscopic randomness, or effective circuit reduction.

2. Microscopic and circuit origins

Several distinct mechanisms generate higher harmonics in the CPR. In overdamped Josephson junction transport, higher harmonics appear when transport channels are highly transparent, when the Andreev bound-state spectrum departs from a pure sine, or when inversion symmetry is broken so that odd and even harmonics mix (Tsarev et al., 26 May 2025). In hybrid superconductor–semiconductor weak links, high transparency produces a nonsinusoidal CPR through Andreev bound states, and coherent transfer of $-E_J\cos\varphi$9 Cooper pairs at a time generates $$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi+\chi_n),\qquad U(\varphi)=-\sum_{n\ge 1} E_n \cos(n\varphi),$$0 terms (Leblanc et al., 2023, Banszerus et al., 2024).

A particularly explicit microscopic route is barrier inhomogeneity in nominal tunnel junctions. In Al/AlO$$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi+\chi_n),\qquad U(\varphi)=-\sum_{n\ge 1} E_n \cos(n\varphi),$$1/Al transmons, the tunnel barrier is modeled as many channels with transparencies $$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi+\chi_n),\qquad U(\varphi)=-\sum_{n\ge 1} E_n \cos(n\varphi),$$2. The single-channel current

$$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi+\chi_n),\qquad U(\varphi)=-\sum_{n\ge 1} E_n \cos(n\varphi),$$3

has a Fourier expansion whose coefficients decay slowly as $$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi+\chi_n),\qquad U(\varphi)=-\sum_{n\ge 1} E_n \cos(n\varphi),$$4 grows. Realistic transparency distributions then generate percent-level higher harmonics in standard AlO$$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi+\chi_n),\qquad U(\varphi)=-\sum_{n\ge 1} E_n \cos(n\varphi),$$5 junctions, sufficient to alter transmon spectroscopy and charge dispersion (Willsch et al., 2023).

Spin-active and magnetic interfaces provide another route. In NbN/GdN/NbN spin-filter junctions, the CPR can become purely second harmonic,

$$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi+\chi_n),\qquad U(\varphi)=-\sum_{n\ge 1} E_n \cos(n\varphi),$$6

once the GdN barrier is sufficiently magnetic. The observed halved $$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi+\chi_n),\qquad U(\varphi)=-\sum_{n\ge 1} E_n \cos(n\varphi),$$7 periodicity becomes independent of barrier thickness after high spin polarization is established, and the data were interpreted as a robust second harmonic associated with spin filtering, spin mixing, and coherent transport of two triplet pairs through asymmetric spin-active interfaces (Pal et al., 2014).

Higher harmonics can also be induced at the circuit level even when each individual junction is intrinsically sinusoidal. One robust example is series inductance. If the total phase across a branch is

$$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi+\chi_n),\qquad U(\varphi)=-\sum_{n\ge 1} E_n \cos(n\varphi),$$8

then, for small

$$I(\varphi)=\sum_{n\ge 1} I_n \sin(n\varphi+\chi_n),\qquad U(\varphi)=-\sum_{n\ge 1} E_n \cos(n\varphi),$$9

an ideal junction develops an effective second harmonic,

$I_n=(2e/\hbar)\,nE_n$0

In nearly symmetric SQUIDs, spectroscopy showed that the observed second harmonic scaled with junction size in a way consistent with this inductive dressing, with intrinsic Andreev harmonics below $I_n=(2e/\hbar)\,nE_n$1 of the fundamental and a fitted series inductance $I_n=(2e/\hbar)\,nE_n$2 pH (Kim et al., 10 Jul 2025).

A related circuit-synthesis mechanism occurs when two conventional junctions are placed in series. For Josephson energies $I_n=(2e/\hbar)\,nE_n$3 and $I_n=(2e/\hbar)\,nE_n$4, the effective CPR becomes

$I_n=(2e/\hbar)\,nE_n$5

with $I_n=(2e/\hbar)\,nE_n$6, $I_n=(2e/\hbar)\,nE_n$7, and effective transparency

$I_n=(2e/\hbar)\,nE_n$8

Symmetric tuning $I_n=(2e/\hbar)\,nE_n$9 maximizes $\pi$0 and produces strong higher harmonics, while asymmetric tuning suppresses them (Banszerus et al., 2024).

Non-centrosymmetric multi-orbital superconductors provide yet another origin. There, inversion breaking and spin–orbit coupling produce a momentum-space texture of the spin-triplet $\pi$1-vector, and the resulting junction CPR can display dominant high harmonics and a $\pi$2-junction ground state without time-reversal-symmetry breaking (Fukaya et al., 2022).

3. Bias-driven dynamics, synchronization, and harmonic generation

The dynamical consequences of Josephson harmonics are most transparent under dc and ac bias. In the overdamped RSJ model with monochromatic drive,

$\pi$3

or, in normalized variables,

$\pi$4

The time-averaged voltage obeys the Josephson relation $\pi$5. Shapiro steps occur at

$\pi$6

equivalently $\pi$7 with coprime integers $\pi$8 (Tsarev et al., 26 May 2025).

For a single harmonic CPR, $\pi$9, the overdamped RSJ model produces only integer Shapiro steps. No fractional steps occur in that case. With two harmonics, however, the interplay of $4e$0, $4e$1, and the ac drive generates the full set of resonances $4e$2, $4e$3, in successively higher orders of perturbation theory. In the weak-ac, small-$4e$4 regime, the main series obeys

$4e$5

while in the large-dc regime the denominator $4e$6 first appears at a perturbative order determined by exponents $4e$7 satisfying

$4e$8

This establishes that two harmonics suffice to produce all fractional Shapiro steps within the overdamped, current-driven RSJ framework (Tsarev et al., 26 May 2025).

Higher harmonics also structure synchronization in networks. In one-dimensional series arrays with unequal resistances, charge conservation enforces collective oscillation at the network frequency $4e$9, where $I(\varphi)=\sum_{n\ge1} I_n\sin(n\varphi+\chi_n),$0 is the total dc voltage. Because the current and voltage are periodic but not purely sinusoidal, the synchronized state contains harmonics at $I(\varphi)=\sum_{n\ge1} I_n\sin(n\varphi+\chi_n),$1 and subharmonics under external drive, yielding Shapiro structures at rational ratios $I(\varphi)=\sum_{n\ge1} I_n\sin(n\varphi+\chi_n),$2 (Ovchinnikov et al., 2013).

Under dc voltage bias, harmonic generation can couple to collective superconducting modes. In transparent single-band $I(\varphi)=\sum_{n\ge1} I_n\sin(n\varphi+\chi_n),$3-wave junctions, the Higgs or amplitude mode is resonantly driven when

$I(\varphi)=\sum_{n\ge1} I_n\sin(n\varphi+\chi_n),$4

and the resulting order-parameter modulation mixes with the phase dynamics. The paper on AC Josephson Higgs signatures found that this mechanism strongly enhances the second harmonic at $I(\varphi)=\sum_{n\ge1} I_n\sin(n\varphi+\chi_n),$5, and in asymmetric-gap junctions the second harmonic may eclipse the first (Lahiri et al., 2024).

Josephson harmonics can also be generated as radiation harmonics of a driven extended junction. In the Josephson radiation comb generator, periodic crossings of critical nodes in $I(\varphi)=\sum_{n\ge1} I_n\sin(n\varphi+\chi_n),$6 force $I(\varphi)=\sum_{n\ge1} I_n\sin(n\varphi+\chi_n),$7-jumps of the superconducting phase. The voltage–phase relation then converts each jump into a pulse of quantized area,

$I(\varphi)=\sum_{n\ge1} I_n\sin(n\varphi+\chi_n),$8

and periodic driving produces a comb-like sequence with harmonics at integer multiples of the low-frequency magnetic drive. Simulations predicted up to hundreds of harmonics and, for $I(\varphi)=\sum_{n\ge1} I_n\sin(n\varphi+\chi_n),$9 Nb/AlO$I_n$0/Nb junctions, power up to $I_n$1 pW at $I_n$2 GHz for a $I_n$3 MHz drive in the rectangular geometry (Solinas et al., 2015).

4. Experimental realizations and diagnostic signatures

Experimental evidence for Josephson harmonics spans tunnel junctions, hybrid semiconductor weak links, ferromagnetic barriers, and composite superconducting circuits. The signatures differ by platform, but several recurring diagnostics appear: half-periodic SQUID modulation, halved Fraunhofer periods, kinks near half flux quantum in diffraction patterns, half-integer Shapiro steps, spectroscopy requiring non-cosine potentials, and flux-dependent nonreciprocity.

Platform	Harmonic content	Principal signature
NbN/GdN/NbN spin-filter junctions	Purely second harmonic	Halved $I_n$4 period
Planar Al/InAs and Sn/InAs junctions	Large $I_n$5 term	Half-periodic SQUID oscillations, kinks, half-integer Shapiro steps
Ge-based JoFET SQUIDs	Up to three harmonics	FFT of $I_n$6, flux-tunable half-integer steps
Al/AlO$I_n$7/Al transmons	Percent-level higher harmonics	Multi-transition spectroscopy beyond cosine model
Nearly symmetric Al/AlO$I_n$8 SQUIDs	Second harmonic dominated by series inductance	Double-well spectroscopy near half flux

In spin-filter NbN/GdN/NbN junctions, the central phase-sensitive evidence came from the magnetic interference pattern $I_n$9. For sufficiently thick magnetic GdN barriers, both the first lobe width $\chi_n$0 and the least-distorted second lobe width $\chi_n$1 converged to $\chi_n$2, rather than the conventional $\chi_n$3, while the lobe-amplitude ratio recovered to the Fraunhofer value $\chi_n$4. The second harmonic emerged once the barrier thickness exceeded approximately $\chi_n$5 nm and the spin polarization exceeded $\chi_n$6, reaching a representative $\chi_n$7 at $\chi_n$8 K (Pal et al., 2014).

Planar Al/InAs and Sn/InAs superconductor–semiconductor junctions showed an unusually large second harmonic. In a representative dc-SQUID, a two-component CPR

$\chi_n$9

with $$I(\varphi)=\sum_{n\ge1} I_n\sin[n(\varphi-\phi_0)] =\sum_{n\ge1}(A_n\sin n\varphi+B_n\cos n\varphi),$$0 reproduced lifted nodes, kinks near minima, half-periodic features, and gate-controlled asymmetry. Single-junction diffraction patterns displayed kinks near $$I(\varphi)=\sum_{n\ge1} I_n\sin[n(\varphi-\phi_0)] =\sum_{n\ge1}(A_n\sin n\varphi+B_n\cos n\varphi),$$1, and Shapiro maps at $$I(\varphi)=\sum_{n\ge1} I_n\sin[n(\varphi-\phi_0)] =\sum_{n\ge1}(A_n\sin n\varphi+B_n\cos n\varphi),$$2 GHz showed half-integer steps at $$I(\varphi)=\sum_{n\ge1} I_n\sin[n(\varphi-\phi_0)] =\sum_{n\ge1}(A_n\sin n\varphi+B_n\cos n\varphi),$$3 while $$I(\varphi)=\sum_{n\ge1} I_n\sin[n(\varphi-\phi_0)] =\sum_{n\ge1}(A_n\sin n\varphi+B_n\cos n\varphi),$$4 and $$I(\varphi)=\sum_{n\ge1} I_n\sin[n(\varphi-\phi_0)] =\sum_{n\ge1}(A_n\sin n\varphi+B_n\cos n\varphi),$$5 steps were absent (Zhang et al., 2022).

Ge-based JoFET SQUIDs made the harmonic content directly tunable. In an asymmetric SQUID used for CPR extraction, FFT analysis of $$I(\varphi)=\sum_{n\ge1} I_n\sin[n(\varphi-\phi_0)] =\sum_{n\ge1}(A_n\sin n\varphi+B_n\cos n\varphi),$$6 showed clear first and second harmonic peaks and a smaller third harmonic. In the highly accumulated regime the reported ratios reached $$I(\varphi)=\sum_{n\ge1} I_n\sin[n(\varphi-\phi_0)] =\sum_{n\ge1}(A_n\sin n\varphi+B_n\cos n\varphi),$$7 and $$I(\varphi)=\sum_{n\ge1} I_n\sin[n(\varphi-\phi_0)] =\sum_{n\ge1}(A_n\sin n\varphi+B_n\cos n\varphi),$$8, while depletion drove the CPR toward a sinusoid. Under microwave irradiation, half-integer Shapiro steps tracked the relative weight of the first two harmonics, and in a balanced symmetric SQUID the odd harmonics were suppressed at $$I(\varphi)=\sum_{n\ge1} I_n\sin[n(\varphi-\phi_0)] =\sum_{n\ge1}(A_n\sin n\varphi+B_n\cos n\varphi),$$9, yielding a $-E_J\cos\varphi$00-periodic, charge-$-E_J\cos\varphi$01 regime (Leblanc et al., 2023).

In standard Al/AlO$-E_J\cos\varphi$02/Al transmons, spectroscopy of the first several transitions showed that the cosine-only Hamiltonian systematically failed, with deviations in $-E_J\cos\varphi$03 often exceeding $-E_J\cos\varphi$04 MHz, whereas harmonic-inclusive fits reduced residuals to the $-E_J\cos\varphi$05 MHz measurement accuracy. Across KIT, ENS, and Köln devices, typical fitted values were $-E_J\cos\varphi$06 to $-E_J\cos\varphi$07, with $-E_J\cos\varphi$08 to $-E_J\cos\varphi$09, while several IBM devices exhibited much larger odd harmonics, including $-E_J\cos\varphi$10 and $-E_J\cos\varphi$11 in one example (Willsch et al., 2023).

Circuit spectroscopy can also separate intrinsic from emergent harmonics. In capacitively shunted, nearly symmetric Al/AlO$-E_J\cos\varphi$12 SQUIDs, the measured ratio $-E_J\cos\varphi$13 scaled linearly with $-E_J\cos\varphi$14 as

$-E_J\cos\varphi$15

with intercept $-E_J\cos\varphi$16 and slope corresponding to $-E_J\cos\varphi$17 pH. This matched 3D inductance extraction and identified series inductance, rather than intrinsic Andreev physics, as the dominant source of the observed second harmonic (Kim et al., 10 Jul 2025).

A further example of engineered harmonic control is the tunable double-junction transmon, where a tunnel junction in series with a SQUID yielded a flux-tunable harmonic expansion. Spectroscopy of the first four transitions required inclusion of the internal mode and revealed a second harmonic up to $-E_J\cos\varphi$18 of the fundamental harmonic (Shagalov et al., 9 Dec 2025).

5. Nonreciprocity, protected regimes, and superconducting-circuit applications

Josephson harmonics are central to nonreciprocal transport. In a two-harmonic CPR with a phase shift,

$-E_J\cos\varphi$19

the amplitudes of positive and negative fractional Shapiro steps become unequal. In the weak-ac, small-$-E_J\cos\varphi$20 regime, the asymmetry of the main $-E_J\cos\varphi$21 series is

$-E_J\cos\varphi$22

and in the large-dc limit analogous asymmetries remain proportional to $-E_J\cos\varphi$23. This provides a direct microwave signature of the Josephson diode effect (Tsarev et al., 26 May 2025).

Flux and gate tuning can turn this principle into a device functionality. In Ge-based symmetric SQUIDs, the diode efficiency

$-E_J\cos\varphi$24

reached up to $-E_J\cos\varphi$25. The same platform could be tuned to a balanced point at half flux, where odd harmonics destructively interfered and the effective element became $-E_J\cos\varphi$26-periodic with strong half-integer Shapiro steps (Leblanc et al., 2023).

Multi-terminal devices generate even harmonics and phase shifts synthetically. In the three-terminal InAs device, the network reduction led to an effective CPR containing both $-E_J\cos\varphi$27 and $-E_J\cos\varphi$28 terms, enabling a superconducting diode effect without invoking material-specific exotic pairing. The reported diode efficiencies were $-E_J\cos\varphi$29 in one device and $-E_J\cos\varphi$30 in another, together with nonlinear dc intermodulation and simultaneous opposite-polarity rectification (Gupta et al., 2022).

Capacitance and noise modify higher-harmonic diode phenomenology in asymmetric SQUIDs. In the minimal model with one sinusoidal arm and one arm containing a second harmonic, capacitance generally weakens the diode effect in the resistive branch, but it also produces asymmetric retrapping currents and can open a regime of single-sided hysteresis. Thermal fluctuations then generate exponentially asymmetric subcritical voltages because the forward and backward barrier heights differ, and ac irradiation can induce asymmetric switching currents through both thermal activation and Josephson plasma resonances (Seleznev et al., 2024).

In quantum circuits, Josephson harmonics alter qubit Hamiltonians and can be used constructively. In Al/AlO$-E_J\cos\varphi$31 transmons, engineered harmonics were shown to reduce charge dispersion by about an order of magnitude while preserving anharmonicity; one explicit engineered set kept $-E_J\cos\varphi$32 and $-E_J\cos\varphi$33 fixed while enhancing anharmonicity using $-E_J\cos\varphi$34, $-E_J\cos\varphi$35, and $-E_J\cos\varphi$36 (Willsch et al., 2023). In the tunable double-junction transmon, flux control of the harmonic content also enabled a sweet spot where the dispersive shift vanished through cancellation between the qubit and internal-mode dispersive couplings (Shagalov et al., 9 Dec 2025).

Parametric amplifiers provide another application domain. A plasma-resonance-engineered JTWPA used Josephson plasma oscillations to phase-match three-wave mixing and suppress pump conversion to higher harmonics, achieving gain of $-E_J\cos\varphi$37 dB with a $-E_J\cos\varphi$38 GHz bandwidth in simulation (Rizvanov et al., 2024). A separate numerical study showed that explicitly adding a second-harmonic CPR contribution to a JTWPA changed the gain profile, phase-space structure, Poincaré sections, and Fourier spectra, with an optimal harmonic weighting yielding gains up to $-E_J\cos\varphi$39 dB in a device without dispersion engineering (Guarcello et al., 2 Feb 2025).

6. Ambiguities, limitations, and open questions

The interpretation of Josephson harmonics is often non-unique, and several recurring caveats appear in the literature. A central conceptual distinction is between intrinsic junction harmonics and effective circuit harmonics. The nearly symmetric SQUID spectroscopy study demonstrated that a second harmonic can arise almost entirely from trace inductance, so half-flux double-well behavior or an extracted $-E_J\cos\varphi$40 term does not by itself establish intrinsic Andreev physics (Kim et al., 10 Jul 2025).

Likewise, half-integer Shapiro steps are an important but not exclusive signature of a second harmonic. In planar Al/InAs junctions they were consistent with a large $-E_J\cos\varphi$41 term, but the same study explicitly noted alternative explanations for individual observations, including nonuniform current density, finite-voltage artifacts, phase locking of Josephson vortices, and quasiparticles. The authors concluded that no single alternative accounted for the full body of data, yet the microscopic origin of the unusually large second harmonic, with suppressed higher terms, remained unresolved (Zhang et al., 2022).

Model dependence is another limitation. In the overdamped RSJ model, a single-harmonic CPR yields only integer steps, whereas in the capacitive RCSJ model even a single harmonic can generate subharmonics and a devil’s staircase because of inertial dynamics. Consequently, observing fractional locking does not by itself distinguish harmonic mixing from capacitance-driven nonlinear dynamics (Tsarev et al., 26 May 2025). In asymmetric higher-harmonic SQUIDs, capacitance also suppresses resistive-state nonreciprocity while creating new hysteretic asymmetries, so the operative diode metric depends strongly on whether one probes the stationary, resistive, or thermally activated regime (Seleznev et al., 2024).

Several experimental systems still lack a complete microscopic description. In spin-filter junctions, the primary phase-sensitive evidence for a pure second harmonic came from $-E_J\cos\varphi$42, while direct Shapiro-step confirmation was not obtained because sufficient microwave power could not be coupled into the devices (Pal et al., 2014). In Al/AlO$-E_J\cos\varphi$43 transmons, the mesoscopic barrier model explained much of the spectroscopy but underestated the slow decay of higher harmonics in some IBM devices, and more sophisticated transparency distributions were identified as a needed next step (Willsch et al., 2023). In planar semiconductor junctions, the second harmonic could be fitted phenomenologically, but the reason it appears so prominently without comparably strong third and higher terms remains open (Zhang et al., 2022).

These limitations do not diminish the central role of Josephson harmonics. Rather, they show that harmonic content is jointly controlled by microscopic transport, symmetry, damping, fluctuation physics, and circuit embedding. The literature therefore treats Josephson harmonics not as a single mechanism, but as a unifying description of non-sinusoidal superconducting phase dynamics across junction materials, interferometer geometries, and quantum-circuit architectures.