Dual Shapiro Steps in Superconducting Circuits

• Dual Shapiro steps are quantized current plateaus occurring when Bloch oscillations in a charge-dominated superconducting device synchronize to an external drive, yielding I = 2ef plateaus.
• The phenomenon leverages charge–phase duality, where dual RSJ-type equations map quasicharge dynamics to phase dynamics, mirroring conventional Shapiro voltage steps.
• Experimental realizations in Josephson arrays and phase-slip junctions demonstrate precise current quantization, offering promising avenues for quantum current standards and metrological applications.

Dual Shapiro steps are quantized current plateaus that occur when Bloch oscillations in a charge-dominated superconducting element synchronize with a periodic drive. They are the charge–phase dual of conventional Shapiro steps: instead of voltage plateaus at $V_n = n\,hf/2e$ in a current-biased Josephson junction, the dual effect yields current plateaus at $I_n = n\,2ef$ in a phase-slip or Bloch regime device, with $f$ set by an external or internally generated oscillation (Arndt et al., 2018, Kaap et al., 2024). In modern treatments, the phenomenon is formulated both in dual RSJ-type equations for quasicharge and in full circuit-QED or kinetic descriptions that retain multi-band junction dynamics, finite-size electromagnetic environments, and fluctuation effects (Borletto et al., 2024, Resch et al., 6 Aug 2025).

1. Charge–phase duality and the definition of dual Shapiro steps

The conceptual basis of dual Shapiro steps is the canonical conjugacy of superconducting phase and charge. For a Josephson junction, the low-energy variables satisfy $[ \phi, Q ] = 2ie$, and the balance between their fluctuations is controlled by the ratio of Josephson and charging energies. In the usual phase-dominated regime $E_J/E_C \gg 1$, the phase is well defined, the AC Josephson relation gives $f_J = 2eV/h$, and microwave locking produces voltage steps. In the charge-dominated regime $E_C \gtrsim E_J$, quasicharge becomes the natural dynamical variable, the junction forms Bloch bands, and the dual synchronization condition gives current steps instead (Crescini et al., 2022, Kaap et al., 2024).

In the ideal dual experiment, Bloch oscillations occur in a small Josephson junction or phase-slip junction under effective current bias. The Bloch oscillation frequency is set by current, written either as $f_B = I/(2e)$ or equivalently $\omega_B = \pi I/e$. Phase-locking to a drive of frequency $\omega_0$ then yields

$I_n = n\,2ef$0

with $I_n = n\,2ef$1, defining the positions of dual Shapiro steps (Arndt et al., 2018, Kaap et al., 2024).

This duality is often summarized operationally. Conventional Shapiro steps are constant-voltage plateaus in a current-biased Josephson junction under microwave irradiation. Dual Shapiro steps are constant-current plateaus in a voltage-biased or nearly current-biased phase-slip/Bloch device under periodic drive, with transport quantized by the tunneling of a Cooper pair per drive period on the first step (Crescini et al., 2022).

2. Minimal models and synchronization mechanisms

A standard starting point is the Hamiltonian

$I_n = n\,2ef$2

which supports either phase dynamics or Bloch-band dynamics depending on $I_n = n\,2ef$3. In the phase-slip description, the relevant nonlinear constitutive relation is the voltage–charge relation

$I_n = n\,2ef$4

with critical voltage $I_n = n\,2ef$5. In the lowest-band approximation, the quasicharge $I_n = n\,2ef$6 evolves in a periodic potential, directly dual to the washboard potential for phase in the standard RSJ model (Arndt et al., 2018).

This leads to a dual RSJ equation. In one common form,

$I_n = n\,2ef$7

where $I_n = n\,2ef$8 is the DC voltage, $I_n = n\,2ef$9 is the AC drive, and $f$0 is the charge through the junction. This equation is the exact dual of the classical equation of motion for phase in the RCSJ model, under the mapping $f$1, $f$2, $f$3, and $f$4 (Crescini et al., 2022). In the noise-free case with sinusoidal drive, synchronized solutions give the quantized current plateaus $f$5 (Kaap et al., 2024).

The same synchronization logic extends beyond sinusoidal drive. In a pulsed regime that is explicitly described as dual to the single flux quantum mode of Josephson oscillations, short asymmetric pulses generate asymmetric dual-step patterns: positive sawtooth pulses enhance the positive current step and suppress the negative step, while negative sawtooth pulses do the opposite (Kaap et al., 2024). This establishes that the dual phenomenon is not limited to monochromatic locking but belongs to the broader family of nonlinear synchronization effects in charge space.

3. Circuit engineering and environmental constraints

Observation of dual Shapiro steps is conditioned by the electromagnetic environment. A central difficulty is that parasitic capacitance $f$6 shunting the junction suppresses the amplitude of quasicharge oscillations and thereby endangers Bloch oscillations. In a realistic circuit with a large off-chip resistor $f$7, parasitic capacitance $f$8, and a superinductance $f$9, the relevant scale is the characteristic impedance

$[ \phi, Q ] = 2ie$0

In the finite-$[ \phi, Q ] = 2ie$1 regime, the effective critical voltage is renormalized to

$[ \phi, Q ] = 2ie$2

so dual Shapiro steps are exponentially suppressed when $[ \phi, Q ] = 2ie$3 is too small. A practical criterion identified for observability is

$[ \phi, Q ] = 2ie$4

together with overdamped dynamics and sufficiently low effective temperature (Arndt et al., 2018).

The role of the superinductance is twofold. It screens the junction from the parasitic shunt and allows the large resistor to be placed off-chip, reducing local heating. Analytical results were derived both in the high-impedance regime and in a ground-state approximation, yielding explicit step-height formulas and showing that realistic parasitic capacitances do not preclude observing dual Shapiro steps if $[ \phi, Q ] = 2ie$5 is sufficiently large (Arndt et al., 2018).

A closely related development replaces external microwave injection by an on-chip AC source. In this scheme, a second Josephson junction, DC biased so that it generates AC Josephson oscillations, is coupled to the Bloch junction through an LC transconductance. The effective charge equation then contains a drive at the LC resonance frequency, while a back-action term captures how Bloch oscillations perturb the source junction. A weak-back-action regime is identified by $[ \phi, Q ] = 2ie$6 and $[ \phi, Q ] = 2ie$7, where the source acts as a nearly rigid on-chip microwave drive; stronger back action can even enhance step widths, although that regime is experimentally difficult because it tends to require $[ \phi, Q ] = 2ie$8 while the quasiclassical Bloch regime needs $[ \phi, Q ] = 2ie$9 (Scheer et al., 2023).

4. Experimental realizations in Josephson arrays and small junctions

The first clear contemporary evidence in a Josephson-junction-array platform embedded an ultrasmall Josephson junction in a high-impedance array of larger junctions. In that Bloch-array architecture, the central SQUID operated in the charge-dominated regime while the surrounding arrays formed superinductances with $E_J/E_C \gg 1$0 and total inductance $E_J/E_C \gg 1$1. Under microwave drive, the experiment detected a narrow “Bloch mode” in transmission exactly at the drive frequency and observed flat current plateaus just before each $E_J/E_C \gg 1$2 current peak, with plateau current equal to $E_J/E_C \gg 1$3 to within about $E_J/E_C \gg 1$4 at $E_J/E_C \gg 1$5. For $E_J/E_C \gg 1$6, the lowest-voltage plateau followed $E_J/E_C \gg 1$7, and fitting the measured plateau current versus frequency gave $E_J/E_C \gg 1$8, consistent with $E_J/E_C \gg 1$9 (Crescini et al., 2022).

A subsequent demonstration in small Al/AlO$f_J = 2eV/h$0/Al Josephson junctions showed dual Shapiro steps over a broad frequency range without relying on a few discrete array resonances. The devices used standard shadow-evaporated junctions in a high-impedance environment built from granular Al superinductors and TiO$f_J = 2eV/h$1 resistors. Synchronizing Bloch oscillations to sinusoidal drives with frequencies from $f_J = 2eV/h$2 to $f_J = 2eV/h$3, the experiment observed dual Shapiro steps up to $f_J = 2eV/h$4. In the pulsed-drive regime, a similar asymmetric pattern was observed, dual to single-flux-quantum operation in voltage standards. Linear fits of step position versus frequency yielded $f_J = 2eV/h$5 and $f_J = 2eV/h$6, both in excellent agreement with the elementary charge (Kaap et al., 2024).

These experiments established two points of lasting importance. First, the quantization law $f_J = 2eV/h$7 can be realized in integrated Josephson circuits using currently accessible fabrication methods. Second, the plateau morphology depends strongly on the microwave waveform and on the detailed environment, so step visibility is inseparable from circuit engineering (Crescini et al., 2022, Kaap et al., 2024).

5. Beyond ideal duality: multi-band dynamics, fluctuations, and quantum–classical crossover

Ideal charge–phase duality treats the nonlinear element as a phase-slip junction with a sinusoidal voltage–charge relation. A more complete treatment keeps the actual Josephson Hamiltonian, its full Bloch-band structure in quasicharge, and an explicit multimode electromagnetic environment. In a circuit-QED formulation with a finite-size high-impedance transmission line resonator, the small Josephson junction is modeled by the full multi-band Hamiltonian and coupled to a finite set of resonator modes. Mean-field equations reproduce the dissipative classical dual-RCSJ result when the number of transmission-line modes is large enough, but the full calculation shows that plateau widths are modified by multi-band effects even when step positions remain near $f_J = 2eV/h$8 (Borletto et al., 2024).

To include quantum and thermal fluctuations, that work goes beyond mean field using a truncated Wigner approach. The resulting picture is markedly asymmetric between direct and dual steps: direct Shapiro steps remain comparatively robust, whereas dual steps are very sensitive to fluctuations. In the dual case, the calculations identify very high impedance as essential; zero-point fluctuations already destroy the dual plateaus for $f_J = 2eV/h$9, while $E_C \gtrsim E_J$0 can preserve a reduced first step. The same study indicates that useful dual steps require $E_C \gtrsim E_J$1, $E_C \gtrsim E_J$2, moderate $E_C \gtrsim E_J$3, and sufficiently large inter-band gaps relative to the inductive energy scale (Borletto et al., 2024).

A complementary kinetic theory unifies dual and classical Shapiro steps within a single framework for a small Josephson junction in an $E_C \gtrsim E_J$4–$E_C \gtrsim E_J$5 environment. The crossover is governed by a single effective relaxation time $E_C \gtrsim E_J$6. In the adiabatic low-frequency, low-power regime,

$E_C \gtrsim E_J$7

the rate equation reduces to a Bloch-oscillation picture and yields dual Shapiro steps. In the opposite limit,

$E_C \gtrsim E_J$8

the same kinetic equation reduces to a Tien–Gordon-type description and produces classical Shapiro steps. The model predicts that both types of steps can occur in the same sample, with the large inductor in the bias circuit increasing $E_C \gtrsim E_J$9 and filtering high-frequency noise so as to protect the dual regime (Resch et al., 6 Aug 2025).

6. Related phenomena, distinctions, and metrological significance

The term “dual Shapiro steps” should be distinguished from several superficially similar phenomena. In a dynamic axion insulator Josephson junction, the reported effect is “doubled Shapiro steps,” not charge–phase-dual Shapiro steps. There the intrinsic current–phase relation remains $f_B = I/(2e)$0, but antiferromagnetic resonance generates a dynamic axion field with $f_B = I/(2e)$1, which produces an axion-induced current $f_B = I/(2e)$2. The resulting phase modulation occurs at $f_B = I/(2e)$3, and the junction shows only even Shapiro steps, with step positions $f_B = I/(2e)$4 and complete suppression of odd steps in the usual $f_B = I/(2e)$5 normalization (Li et al., 2024). This is a frequency-doubling effect in a voltage-step experiment, not the charge–phase-dual current quantization of Bloch devices.

Other internally driven step structures are likewise distinct. In a tunnel Josephson junction containing a molecular nanomagnet, spin nutation modulates the Josephson coupling and produces Shapiro-like voltage steps at $f_B = I/(2e)$6, with widths set by the nutation frequency and heights controlled by the nutation amplitude (Abdollahipour et al., 2015). Such steps broaden the general notion of synchronization-induced staircase structure, but they do not implement the dual current standard relation $f_B = I/(2e)$7.

A further interpretive caution concerns missing or fractional steps. In highly transparent but topologically trivial SNS junctions, strong higher harmonics in the CPR and multiphoton processes can generate “fractional” Shapiro steps and even apparently missing fundamental steps without any intrinsic $f_B = I/(2e)$8-periodic Josephson effect. By explicit duality, the same caution applies to dual-step experiments: fractional-looking or missing current plateaus are not, by themselves, proof of altered fundamental periodicity in charge space (Galaktionov et al., 2021).

Metrologically, dual Shapiro steps are pursued because they provide a direct current–frequency relation complementary to the Josephson voltage standard. Both the array experiment and the small-junction demonstration frame the effect as a route toward closing the quantum metrology triangle (Crescini et al., 2022, Kaap et al., 2024). A recent extension goes further by proposing a dc-driven Bloch transistor in which Bloch oscillations and Josephson oscillations mutually phase lock without external microwaves. In that device, current steps appear at $f_B = I/(2e)$9, while the transconductance takes the fundamental value

$\omega_B = \pi I/e$0

suggesting an alternative superconducting quantum standard of resistance operating without a strong magnetic field (Zorin, 13 May 2026).

Dual Shapiro steps therefore occupy a specific place within superconducting nonlinear dynamics. They are not merely any current staircase under periodic forcing, nor a synonym for doubled or missing voltage steps. They are the experimentally and theoretically sharpened signature of synchronized Bloch oscillations in a high-impedance, charge-dominated superconducting circuit, and they remain central to ongoing efforts to realize a frequency-based quantum current standard (Arndt et al., 2018, Kaap et al., 2024).