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All fractional Shapiro steps in the RSJ model with two Josephson harmonics
by Pavel N. Tsarev, Yakov V. Fominov
Submission summary
| Authors (as registered SciPost users): | Yakov Fominov · Pavel Tsarev |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2505.20502v2 (pdf) |
| Date accepted: | Nov. 17, 2025 |
| Date submitted: | Nov. 6, 2025, 10:13 a.m. |
| Submitted by: | Pavel Tsarev |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Synchronization between the internal dynamics of the superconducting phase in a Josephson junction (JJ) and an external ac signal is a fundamental physical phenomenon, manifesting as constant-voltage Shapiro steps in the current-voltage characteristic. Mathematically, this phase-locking effect is captured by the Resistively Shunted Junction (RSJ) model, an important example of a nonlinear dynamical system. The standard RSJ model considers an overdamped JJ with a sinusoidal (single-harmonic) current-phase relation (CPR) in the current-driven regime with a monochromatic ac component. While this model predicts only integer Shapiro steps, the inclusion of higher Josephson harmonics is known to generate fractional Shapiro steps. In this paper, we show that only two Josephson harmonics in the CPR are sufficient to produce all possible fractional Shapiro steps within the RSJ framework. Using perturbative methods, we analyze the amplitudes of these fractional steps. Furthermore, by introducing a phase shift between the two Josephson harmonics, we reveal an asymmetry between positive and negative fractional steps - a signature of the Josephson diode effect.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Dear Editor,
We are resubmitting our manuscript. We have taken into account all the Referee reports, and revised our manuscript accordingly (as detailed in our replies to Referees and in the List of changes).
Additionally, we have received a number of constructive comments to our preprint from our other colleagues on how to better illustrate our results. Inspired by those comments, we have made several additions to our manuscript: 1) Added Fig. 2 in order to illustrate the result of Eq. (42). 2) Added new Section 3.5 (including Fig. 3) in order to illustrate possible relation between the amplitudes of the first nontrivial fractional Shapiro steps, $\Delta j_{1/3}$ and $\Delta j_{2/3}$. 3) Added Fig. 4 in order to illustrate the Josephson diode effect in fractional Shapiro steps. 4) Lines 380-391: Commented upon difference between our work and Ref. [18]. Added discussion of implications of our results for the actively debated field of Majorana bound states and their predicted $4\pi$-periodic contribution to the Josephson current [35-38].
Although the latter changes were not requested by Referees, we feel that our manuscript definitely benefits from them, becoming more accessible to readers. It is important to underline that those changes do not alter any of our results and conclusions, and only serve the purpose of illustration and better presentation. We hope that you will find such modifications permissible.
Best regards, Pavel Tsarev and Yakov Fominov.
List of changes
Changes in response to the report by Referee 1: - Lines 23-24: Corrected description of $R$. - Equation (5): Corrected $I_2$ in the second term. - Lines 111-122: Added explanation of the perturbation theory with feedback.
Changes in response to the report by Referee 2: - Lines 77-78, 150-152, 198-200: Clarified the meaning and role of the phase shifts $\beta$ and $\Delta\varphi$. - Line 141 and footnote 2: Provided definition of phase-locking (synchronization). - Lines 360-361: Clarified that the phase shift between the two Josephson harmonics requires breaking of the time-reversal symmetry.
Changes in response to the report by Referee 3: - Added new section 3.4 describing the stability analysis. - Corrected articles throughout the manuscript.
Additionally, in response to comments from other colleagues, in order to better illustrate our results, we have made the following changes: - Added Fig. 2 in order to illustrate the result of Eq. (42). - Added new Section 3.5 (including Fig. 3) in order to illustrate possible relation between the amplitudes of the first nontrivial fractional Shapiro steps, $\Delta j_{1/3}$ and $\Delta j_{2/3}$. - Added Fig. 4 in order to illustrate the Josephson diode effect in fractional Shapiro steps. - Lines 380-382: Commented upon difference between our work and Ref. [18]. - Lines 383-391: Added discussion of implications of our results for the actively debated field of Majorana bound states and their predicted $4\pi$-periodic contribution to the Josephson current. - Added relevant references [35-38,40]. - Corrected a misprint in the last term in Eq. (44) (this was just a misprint in the text, no mistakes in the actual calculations).
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)