Tunable Josephson Harmonics

• Tunable Josephson harmonics are engineered multi-harmonic current–phase relations achieved via gate voltage, magnetic flux, or geometry adjustments, altering periodicity and symmetry.
• They underpin innovative superconducting circuit functionalities such as parity-protected qubits, superconducting diodes, and parametric devices by precisely controlling Fourier components.
• Experimental methods like phase-bias interferometry and Shapiro step analysis, combined with advanced theoretical models, facilitate scalable, reconfigurable designs in quantum technology.

Tunable Josephson harmonics refer to the engineered multi-harmonic content of the current–phase relation (CPR) in Josephson junctions, where the amplitudes and relative phases of the Fourier components can be dynamically controlled via local gates, magnetic flux, or device geometry. These tunable harmonics underlie diverse circuit phenotypes, including parity-protected qubits, Josephson diodes, flux phase batteries, and parametric nonlinearities, and have enabled superconducting elements with programmable transport characteristics well beyond the archetypal $\sin\varphi$ dependence. In advanced superconducting circuits, precise control of these harmonics determines both the periodicity and symmetry of the Josephson potential landscape, providing new tools for quantum coherence protection, microwave engineering, and device non-reciprocity.

1. Multi-Harmonic Current–Phase Relations: Formalism and Phenomenology

The general Josephson CPR in a weak link is expressed as a Fourier series:

$I(\varphi) = \sum_{n=1}^{N} I_n \sin(n\varphi)$

where $I_n$ is the amplitude of the $n$-th harmonic, corresponding to coherent transfer of $2ne$ charge. Tunneling-dominated SIS junctions are well described by $I_1$ only, yielding a $2\pi$-periodic, sinusoidal CPR. For highly transparent junctions, such as SNS or hybrid superconductor–semiconductor junctions, higher harmonics ($n>1$) become significant due to enhanced Andreev processes and coherent multi-pair tunneling, enabling nonsinusoidal and, under suitable conditions, even $\pi$-periodic CPRs. The presence and relative amplitude of harmonics dictate the junction’s periodicity (e.g., $2\pi$, $\pi$, $4\pi$) and the symmetry of the Josephson potential.

2. Device Architectures for Tunable Harmonics

Tunable Josephson harmonics are implemented by combining circuit elements whose harmonics can be independently tuned and by engineering interference effects:

• Parallel and Series Junctions: In double-SQUIDs or series-configured hybrid junctions, gate voltages and flux control the transparency and phase offset of each component, enabling programmable superpositions of multiple harmonics (Leblanc et al., 23 May 2024, Banszerus et al., 18 Feb 2024).
• Multi-terminal Networks: Three- and four-terminal junctions with high-transmission arms (as in InAs/Al or Ge/SiGe structures) permit not only higher-harmonic generation but also nontrivial phase-dependence of each harmonic through multidimensional flux control. Each branch’s amplitude and phase offset is flux-tunable, resulting in a CPR of the form $$I(\varphi)=\sum_n I_n(\{\Phi_i, V_j\})\sin[n\varphi+\delta_n(\{\Phi_i,V_j\})]$$ (Gupta et al., 2022, Coraiola et al., 2023, Chandrasekhar, 10 Jun 2025).
• Hybrid Superconductor–Semiconductor Junctions (JoFETs): Gate-tunable junctions based on high-mobility quantum well heterostructures, such as SiGe/Ge/SiGe or InAs 2DEGs, support strong higher harmonics. JoFETs in SQUID and double-SQUID geometries enable in situ control of I$_2$/I$_1$ and suppression or enhancement of odd/even harmonics (Leblanc et al., 23 May 2024, Leblanc et al., 2023).
• Flux-Tunable Circuits: Josephson elements with internal SQUID loops or full-shell nanowires display harmonics that can be modulated by external flux through interference of Andreev bound state trajectories and Little–Parks oscillations (Giavaras et al., 7 Mar 2025, Shagalov et al., 9 Dec 2025).

3. Methods for Extraction and Control of Harmonics

Harmonic Extraction

A variety of experimental protocols exist for measuring and quantifying Josephson harmonics:

• Phase-bias Interferometry: Devices are embedded in a SQUID or circuit geometry that allows the phase across the junction of interest to be swept via external flux. The critical current oscillations $I_C(\Phi)$ are Fourier transformed to extract the amplitudes $A_n$ of distinct harmonics (Leblanc et al., 23 May 2024, Banszerus et al., 18 Feb 2024).
• Shapiro Step Analysis: Under microwave irradiation, the CPR’s harmonic content directly dictates the quantization of step voltages. The presence of significant second harmonic ($I_2$) is revealed by half-integer Shapiro steps, whose relative weight tracks $I_2/I_1$ (Leblanc et al., 2023).
• Nonreciprocal Critical Current Characterization: Multi-terminal geometries feature asymmetric $I_C^+$ and $I_C^-$ due to higher harmonics with nonzero phase offsets. The diode efficiency parameter, $\eta = \frac{I_C^+ - I_C^-}{I_C^+ + I_C^-}$, quantifies the net influence of the harmonics (Gupta et al., 2022, Coraiola et al., 2023).

Active Tuning

• Gate Voltage Control: In semiconducting hybrid junctions, individual gate voltages set the local transparency and thus the Josephson coupling energy of each microscopic channel, enabling fine adjustment of the amplitude ratios $\rho = E_{J,1}/E_{J,2}$ and hence of the effective transparency:

$\tau_\text{eff} = \frac{4\rho}{(1+\rho)^2}$

This parameter directly determines the multi-harmonic CPR shape (Banszerus et al., 18 Feb 2024, Amani et al., 22 Nov 2024).

• Magnetic Flux: Threading flux through SQUID or multi-loop networks allows destructive or constructive interference between harmonics. For symmetric junctions at half-flux quantum ($\Phi=\Phi_0/2$), the first harmonic can be suppressed, isolating the $\sin 2\varphi$ term, and achieving a nearly ideal $\pi$-periodic CPR (Leblanc et al., 23 May 2024).
• Circuit Geometry: Asymmetry in 0–$\pi$ segment lengths or critical current densities in spatially heterogeneous junctions generates an intrinsic, tunable second harmonic, analyzable via a spatially averaged sine-Gordon model (Goldobin et al., 2011, Lipman et al., 2012).
• Multi-terminal Phase Control: In four-terminal diffusive SNS junctions, relative harmonics of $2\pi$ and $4\pi$ periodicity are continuously tunable by adjusting a control phase difference $\psi$ between electrodes:

$$I_s(\phi,\psi)\approx I_{4\pi}(\psi)\sin\left(\frac{\phi}{2}\right) + I_{2\pi}(\psi)\sin\phi$$

(Chandrasekhar, 10 Jun 2025).

4. Experimental Demonstrations and Figures of Merit

Experimental work has realized tunable Josephson harmonics in multiple platforms:

• High-Purity $\pi$-Periodic Elements: Gate-tunable Ge-based JoFETs in double-SQUID architectures achieve second-harmonic purity $P_2 = A_2/\sum_{n=1}^4 A_n$ exceeding 95%, with prospects for $>99\%$ purity via slight gate or inductance optimization. Suppression of $I_1$ by more than two orders of magnitude at $\Phi_2=\Phi_0/2$ directly confirms the realization of nearly pure $\sin(2\varphi)$ transport (Leblanc et al., 23 May 2024).
• Control Over Single-to-Multipair Supercurrent: Full-shell InAs/Al nanowire junctions exhibit a crossover from a conventional single well (cos ϕ) Josephson potential to a double-well (cos 2ϕ) regime via axial flux. The first harmonic $E_J^{(1)}(\Phi)$ vanishes and $E_J^{(2)}(\Phi)$ peaks away from integer flux quantum, confirming active harmonic control (Giavaras et al., 7 Mar 2025).
• Superconducting Diode Effect: Multi-terminal devices show field- and gate-tuned sign reversals and efficiency maxima in nonreciprocal $I_C$ measurements, a direct manifestation of controlled higher harmonics and their relative phase. In some devices, the diode efficiency $\eta$ can be tuned up to 34% (Coraiola et al., 2023).
• Tunable Harmonics in Double-Junction Circuits: A transmon geometry comprising a large tunnel junction in series with a symmetric SQUID achieves a second harmonic contribution up to 10% of the first harmonic; this harmonic content is adjustable via flux (Shagalov et al., 9 Dec 2025).

5. Theoretical Models and Physical Mechanisms

Microscopic and circuit-level approaches clarify the emergence and tunability of harmonics:

• SNS and S-Sm-S Models: The Beenakker formula for single-mode SNS junctions

$$I(\varphi) = \frac{e\Delta}{2\hbar} \frac{\tau\sin\varphi}{\sqrt{1-\tau\sin^2(\varphi/2)}}$$

naturally yields higher harmonics at finite $\tau$; series and parallel network constructions combine channel transparencies to shape the CPR (Banszerus et al., 18 Feb 2024, Leblanc et al., 2023).

• Inhomogeneous 0–$\pi$ Models: Spatial averaging across 0 and $\pi$ segments with distinct $j_c$ produces negative second harmonics and $H \cos\psi$ terms, with analytical control via segment lengths and critical currents (Goldobin et al., 2011, Lipman et al., 2012).
• Andreev Bound State Interference: In multiterminal and full-shell nanowire junctions, interference effects and trajectory-dependent phase accumulation lead to crossover between $2\pi$ and $4\pi$ periodic harmonics, with full tunability by phase bias or flux (Chandrasekhar, 10 Jun 2025, Giavaras et al., 7 Mar 2025).
• Peak Harmonic Tunability: In symmetric two-junction series circuits, the maximal $I_2/I_1$ ratio approaches 0.4 (ideal), set by matching Josephson energies; practical gate tunability in measured devices spans $A_{4e}/A_{2e}$ from ≲0.01 up to ≈0.3 (Banszerus et al., 18 Feb 2024).

6. Applications and Prospects in Superconducting Circuits

Programmable Josephson harmonics are foundational for advanced circuit functionalities:

• Parity-Protected and 0–$\pi$ Qubits: Dominant $\sin(2\varphi)$ or $\cos(2\varphi)$ potentials localize qubit states in distinct charge and phase-parity sectors, exponentially suppressing single-Cooper-pair transitions and leading-order decoherence pathways (Leblanc et al., 23 May 2024, Giavaras et al., 7 Mar 2025).
• Nonreciprocal Devices: Tunable harmonics and nontrivial phase offsets are a generic mechanism for superconducting diode behavior, with swift electronic switching reducible to nanoscale control operations (Gupta et al., 2022, Coraiola et al., 2023).
• Parametric and Frequency-Converted Devices: Josephson circuits with engineered three- and four-wave mixing terms controlled by external flux or gates allow nondegenerate parametric amplification and tunable nonlinearity, supporting bandwidths and gains optimized for quantum-limited signal processing (Roch et al., 2012).
• Reconfigurable Potential Engineering: Multi-terminal, multitunable harmonics provide synthetic access to diverse Josephson energy landscapes for rapid single-flux-quantum logic, protected memory elements, and couplers with on-the-fly adjustable Hamiltonians (Chandrasekhar, 10 Jun 2025).

7. Limitations, Optimizations, and Outlook

Key technical parameters and challenges include:

• Purity and Robustness: Achievable second-harmonic purities are set by device symmetry, flux noise, kinetic inductance, and the residual imbalance in junction parameters. Ideal designs minimize self-inductance and variance in channel transparency for maximum harmonic isolation (Leblanc et al., 23 May 2024, Shagalov et al., 9 Dec 2025).
• Scalability and Control Speed: Gate and flux tuning in JoFETs and multiterminal geometries can be performed on nanosecond timescales, supporting dynamic reconfiguration in situ for logic and qubit operations (Leblanc et al., 23 May 2024).
• Flux and Charge Noise: Double-well (cos 2ϕ) qubit designs are limited by higher-order flux noise, while the suppression of single-Cooper-pair transport reduces charge dispersion and poisoning (Giavaras et al., 7 Mar 2025).
• Integration: CMOS compatibility and wafer-scale production is realized in Ge-based platforms, promising scale-up for quantum devices and hybrid systems (Leblanc et al., 2023).
• Tunable Hamiltonian Realization: The continuous tunability of periodicity between $2\pi$ and $4\pi$ in diffusive SNS junctions by control phase $\psi$ enables the realization of circuit elements with adjustable energy wells and tailored anharmonicities, establishing new directions for Hamiltonian engineering in superconducting quantum circuits (Chandrasekhar, 10 Jun 2025).

A broad consensus in recent research establishes tunable Josephson harmonics as a central enabling concept for next-generation superconducting quantum hardware, nonreciprocal microwave circuitry, and Hamiltonian programmability (Leblanc et al., 23 May 2024, Shagalov et al., 9 Dec 2025, Coraiola et al., 2023, Giavaras et al., 7 Mar 2025).