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arxiv: 2606.22208 · v1 · pith:3KQDKF5Xnew · submitted 2026-06-20 · ❄️ cond-mat.mes-hall

Deterministic control of the probabilistic phase dynamics in injection-locked spin-torque nano-oscillators

Abderrazak Hakam , Chlo\'e Chopin , Nhat-Tan Phan , Nicolas Mollard , Franck Badets , Louis Hutin , Luana Benetti , Alex S. Jenkins
show 3 more authors
Ricardo Ferreira Ursula Ebels Philippe Talatchian
This is my paper

Pith reviewed 2026-06-26 11:27 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords spin-torque nano-oscillatorsinjection lockingphase dynamicsstochastic fluctuationsprobabilistic computingquasipotentialvortex oscillators
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The pith

A weak radio-frequency perturbation at the free-running frequency enables continuous deterministic control of phase occupation probabilities in injection-locked spin-torque nano-oscillators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Spin-torque nano-oscillators exhibit thermally driven phase fluctuations that make their dynamics stochastic. The paper demonstrates deterministic control of the probability of occupying each of two phase states separated by pi. This is achieved by adding a weak radio-frequency signal at the oscillator's free-running frequency, which tunes the rates of phase jumps between the states without eliminating the fluctuations. The phase of this bias signal selects which state is favored, while its amplitude controls the degree of imbalance, allowing probabilities to be set anywhere from 50 percent to nearly 0 or 1. A phase-reduced model based on an effective quasipotential explains these observations quantitatively.

Core claim

In a vortex-based STNO under second-harmonic injection-locking, the phase occupies two degenerate attractors separated by π with thermally activated jumps between them. Applying a weak RF perturbation at the free-running frequency tunes the jump rates between the attractors, achieving continuous probability control from the unbiased case to values approaching 0 or 1. The bias phase selects the favored attractor while the bias amplitude sets the imbalance strength.

What carries the argument

Effective quasipotential in the phase-reduced description that quantitatively accounts for the bias in phase-jump rates.

If this is right

  • STNOs function as programmable stochastic elements.
  • The bias phase and amplitude provide two complementary control knobs in a single device.
  • These devices serve as hardware primitives for probabilistic computing, Ising machines, and brain-inspired architectures.
  • The control preserves the intrinsic fluctuations while directing their statistics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Arrays of such controlled oscillators could form the basis for scalable probabilistic computing hardware.
  • Similar perturbation techniques might be applied to other bistable oscillator systems to achieve probability control.
  • Further work could test whether this control remains effective at higher temperatures or in coupled oscillator networks.

Load-bearing premise

The phase-reduced description based on an effective quasipotential quantitatively accounts for the observed changes in phase-jump rates.

What would settle it

If the measured occupation probabilities do not vary continuously with the amplitude of the perturbation or fail to match the predictions of the quasipotential model, the deterministic control claim would be falsified.

Figures

Figures reproduced from arXiv: 2606.22208 by Abderrazak Hakam, Alex S. Jenkins, Chlo\'e Chopin, Franck Badets, Louis Hutin, Luana Benetti, Nhat-Tan Phan, Nicolas Mollard, Philippe Talatchian, Ricardo Ferreira, Ursula Ebels.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic of the experimental implementation. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Experimental circuit schematic. A DC current [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Typical voltage-time traces and corresponding count [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Mean dwell times of the 0 and [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Occupation probabilities of the 0-phase state in the SHIL regime under RF bias excitation. (a) 0-phase state occupation [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Schematic quasipotential of an STNO in the SHIL [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Experimental power spectral density measured for [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Empirical cumulative distribution function (ECDF) [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Power spectral density corresponding to the mi [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
read the original abstract

Spin-torque nano-oscillators (STNOs) inherently exhibit thermally driven phase fluctuations that render their dynamics truly stochastic. Here, we demonstrate that, despite this intrinsic randomness, the probability of occupying each phase state can be deterministically and continuously programmed. We experimentally investigate a vortex-based STNO operating under second-harmonic injection-locking, where the oscillator phase settles into two degenerate attractors separated by $\pi$ and undergoes thermally activated phase jumps. By applying a weak radio-frequency perturbation at the free-running frequency, we tune the phase-jump rates between the two attractors without suppressing the fluctuations, achieving continuous probability control from the unbiased limit to values approaching 0 or 1. The bias phase selects which attractor is favored while the bias amplitude sets the strength of the imbalance, providing two complementary control knobs within a single nanoscale device. A phase-reduced description based on an effective quasipotential quantitatively accounts for the observations. These results establish injection-locked STNOs as programmable stochastic elements and provide a hardware primitive for probabilistic computing, Ising machines, and brain-inspired computing architectures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper experimentally demonstrates continuous, deterministic control of the stationary occupation probabilities of two degenerate phase attractors (separated by π) in a vortex-based spin-torque nano-oscillator under second-harmonic injection locking. A weak RF perturbation applied exactly at the free-running frequency is shown to tune the thermally activated phase-jump rates between the attractors without suppressing fluctuations; the bias phase selects the favored attractor and the bias amplitude sets the imbalance strength. A phase-reduced effective quasipotential model is reported to quantitatively account for the measured probabilities.

Significance. If the central experimental result and the quantitative model agreement hold, the work supplies a compact, electrically tunable stochastic element whose occupation probability can be programmed over the full [0,1] range. This directly addresses a hardware primitive needed for probabilistic computing, Ising machines, and neuromorphic architectures that exploit intrinsic noise rather than fighting it. The demonstration that a single nanoscale device can provide two orthogonal control knobs (phase and amplitude of the bias tone) while preserving fluctuation strength is a concrete advance over prior injection-locking studies.

major comments (1)
  1. [Abstract and modeling section] Abstract and modeling section: the central claim that the phase-reduced quasipotential 'quantitatively accounts for the observations' rests on the assumption that the free-running-frequency perturbation leaves amplitude fluctuations unchanged. In vortex STNOs the dynamics are governed by coupled amplitude-phase equations; a perturbation at the free-running frequency can in principle modulate amplitude via the nonlinear frequency shift. The manuscript must therefore include explicit data (or an analysis) demonstrating that the amplitude variance remains constant to within experimental precision when the bias RF is applied; otherwise the extracted escape rates and stationary probabilities cannot be attributed solely to the phase quasipotential.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting an important point regarding the validation of the phase-reduced model. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract and modeling section] Abstract and modeling section: the central claim that the phase-reduced quasipotential 'quantitatively accounts for the observations' rests on the assumption that the free-running-frequency perturbation leaves amplitude fluctuations unchanged. In vortex STNOs the dynamics are governed by coupled amplitude-phase equations; a perturbation at the free-running frequency can in principle modulate amplitude via the nonlinear frequency shift. The manuscript must therefore include explicit data (or an analysis) demonstrating that the amplitude variance remains constant to within experimental precision when the bias RF is applied; otherwise the extracted escape rates and stationary probabilities cannot be attributed solely to the phase quasipotential.

    Authors: We agree that an explicit check on amplitude variance is necessary to fully justify the phase-reduced quasipotential description. Although the perturbation is weak and applied under second-harmonic locking, the coupled amplitude-phase dynamics in vortex STNOs warrant direct verification. In the revised manuscript we will add experimental data (or a dedicated analysis section) comparing the amplitude variance with and without the bias RF tone, confirming that it remains constant within experimental precision across the explored bias amplitudes and phases. This addition will strengthen the attribution of the observed stationary probabilities to the phase quasipotential alone. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental demonstration with independent model support

full rationale

The paper reports experimental tuning of phase-jump rates and stationary probabilities via weak RF bias at the free-running frequency in vortex STNOs, with a phase-reduced quasipotential model stated to account for the data. No quoted equations or sections show probabilities or control parameters reducing by construction to fitted inputs, self-citations, or ansatzes imported from the same authors' prior work. The central claims rest on direct measurements and an effective model whose validity is presented as an external check rather than a definitional identity. This is the normal case of an experimental result with supporting theory that does not collapse into its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the quasipotential is presented as a modeling tool whose details are not supplied.

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Works this paper leans on

77 extracted references · 9 canonical work pages

  1. [1]

    S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em- ley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Microwave oscillations of a nanomagnet driven by a spin- polarized current, Nature425, 380 (2003)

    2003

  2. [2]

    W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Direct-current induced dynamics in CoFe/NiFe point contacts, Phys. Rev. Lett.92, 027201 (2004)

    2004

  3. [3]

    H. S. Choi, S. Y. Kang, S. J. Cho, I.-Y. Oh, M. Shin, H. Park, C. Jang, B.-C. Min, S.-I. Kim, S.-Y. Park,et al., Spin nano–oscillator–based wireless communication, Sci- entific reports4, 5486 (2014)

    2014

  4. [4]

    Litvinenko, A

    A. Litvinenko, A. Sidi El Valli, V. Iurchuk, S. Louis, V. Tyberkevych, B. Dieny, A. N. Slavin, and U. Ebels, Ultrafast ghz-range swept-tuned spectrum analyzer with 20 ns temporal resolution based on a spin-torque nano- 18 Toy model parameters used to fit experiments using Eq.(4) Constant parameters n= 2 δ= -3.02 MHz Ω = 21.86 MHz T= 300 K Fitting paramete...

    2022

  5. [5]

    K. Zhu, M. Carpentieri, L. Zhang, B. Fang, J. Cai, R. Verba, A. Giordano, V. Puliafito, B. Zhang, G. Finoc- chio,et al., Nonlinear amplification of microwave signals in spin-torque oscillators, Nature Communications14, 2183 (2023)

    2023

  6. [6]

    Grollier, D

    J. Grollier, D. Querlioz, K. Camsari, K. Everschor-Sitte, S. Fukami, and M. Stiles, Neuromorphic spintronics, Nat. Electron.3, 360 (2020)

    2020

  7. [7]

    Finocchio, M

    G. Finocchio, M. Di Ventra, K. Y. Camsari, K. Everschor-Sitte, P. K. Amiri, and Z. Zeng, The promise of spintronics for unconventional computing, Journal of Magnetism and Magnetic Materials521, 167506 (2021)

    2021

  8. [8]

    R. Ma, A. Sidi El Valli, M. Kreißig, G. Di Pendina, F. Protze, U. Ebels, G. Prenat, A. Chavent, V. Iurchuk, R. Sousa,et al., Microwave functionality of spintronic devices implemented in a hybrid complementary metal oxide semiconductor and magnetic tunnel junction tech- nology, Electronics Letters57, 264 (2021)

    2021

  9. [9]

    Berger, Emission of spin waves by a magnetic mul- tilayer traversed by a current, Phys

    L. Berger, Emission of spin waves by a magnetic mul- tilayer traversed by a current, Phys. Rev. B54, 9353 (1996)

    1996

  10. [10]

    J. C. Slonczewski, Current-driven excitation of magnetic multilayers, J. Magn. Magn. Mater.159, L1 (1996)

    1996

  11. [11]

    D. C. Ralph and M. D. Stiles, Spin transfer torques, J. Magn. Magn. Mater.320, 1190 (2008)

    2008

  12. [12]

    Slavin and V

    A. Slavin and V. Tiberkevich, Nonlinear auto-oscillator theory of microwave generation by spin-polarized current, IEEE Trans. Magn.45, 1875 (2009)

    2009

  13. [13]

    Rippard, M

    W. Rippard, M. Pufall, S. Kaka, T. J. Silva, S. E. Russek, and J. A. Katine, Injection locking and phase control of spin transfer nano-oscillators, Physical review letters95, 067203 (2005)

    2005

  14. [14]

    Quinsat, J

    M. Quinsat, J. Sierra, I. Firastrau, V. Tiberkevich, A. Slavin, D. Gusakova, L. Buda-Prejbeanu, M. Zarud- niev, J.-P. Michel, U. Ebels,et al., Injection locking of tunnel junction oscillators to a microwave current, Ap- plied Physics Letters98, 182503 (2011)

    2011

  15. [15]

    Dussaux, A

    A. Dussaux, A. Khvalkovskiy, J. Grollier, V. Cros, A. Fukushima, M. Konoto, H. Kubota, K. Yakushiji, S. Yuasa, K. Ando,et al., Phase locking of vortex based spin transfer oscillators to a microwave current, Applied Physics Letters98, 132506 (2011)

    2011

  16. [16]

    Hamadeh, N

    A. Hamadeh, N. Locatelli, V. Naletov, R. Lebrun, G. De Loubens, J. Grollier, O. Klein, and V. Cros, Per- fect and robust phase-locking of a spin transfer vortex nano-oscillator to an external microwave source, Applied Physics Letters104, 022408 (2014). 19

    2014

  17. [17]

    Pikovsky, M

    A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza- tion: a universal concept in nonlinear science (2002)

    2002

  18. [18]

    Tamaru, H

    S. Tamaru, H. Kubota, K. Yakushiji, S. Yuasa, and A. Fukushima, Extremely coherent microwave emission from spin torque oscillator stabilized by phase locked loop, Scientific reports5, 1 (2015)

    2015

  19. [19]

    Kreissig, R

    M. Kreissig, R. Lebrun, F. Protze, K. Merazzo, J. Hem, L. Vila, R. Ferreira, M. Cyrille, F. Ellinger, V. Cros, et al., Vortex spin-torque oscillator stabilized by phase locked loop using integrated circuits, AIP advances7, 056653 (2017)

    2017

  20. [20]

    Wittrock, M

    S. Wittrock, M. Kreißig, B. Lacoste, A. Litvinenko, P. Talatchian, F. Protze, F. Ellinger, R. Ferreira, R. Le- brun, P. Bortolotti,et al., Stabilization of phase noise in spin torque nano oscillators by a phase locked loop, arXiv preprint arXiv:2110.13073 (2021)

    arXiv 2021

  21. [21]

    Martins, A

    L. Martins, A. S. Jenkins, J. Borme, J. Ventura, P. P. Fre- itas, and R. Ferreira, Second harmonic injection locking of coupled spin torque vortex oscillators with an individ- ual phase access, Communications Physics6, 72 (2023)

    2023

  22. [22]

    N.-T. Phan, N. Prasad, A. Hakam, A. S. E. Valli, L. Anghel, L. Benetti, A. Madhavan, A. S. Jenkins, R. Ferreira, M. D. Stiles, U. Ebels, and P. Talatchian, Unbiased random bitstream generation using injection- locked spin-torque nano-oscillators, Physical Review Ap- plied21, 034063 (2024)

    2024

  23. [23]

    Mohseni, P

    N. Mohseni, P. L. McMahon, and T. Byrnes, Ising ma- chines as hardware solvers of combinatorial optimization problems, Nature Reviews Physics4, 363 (2022)

    2022

  24. [24]

    P. L. McMahon, A. Marandi, Y. Haribara, R. Hamerly, C. Langrock, S. Tamate, T. Inagaki, H. Takesue, S. Ut- sunomiya, K. Aihara,et al., A fully programmable 100- spin coherent ising machine with all-to-all connections, Science354, 614 (2016)

    2016

  25. [25]

    Inagaki, Y

    T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Ta- mate, T. Honjo, A. Marandi, P. L. McMahon, T. Umeki, K. Enbutsu,et al., A coherent ising machine for 2000- node optimization problems, Science354, 603 (2016)

    2000

  26. [26]

    Dutta, A

    S. Dutta, A. Khanna, A. S. Assoa, H. Paik, D. G. Schlom, Z. Toroczkai, A. Raychowdhury, and S. Datta, An ising hamiltonian solver based on coupled stochastic phase-transition nano-oscillators, Nature Electronics4, 502 (2021)

    2021

  27. [27]

    Honjo, T

    T. Honjo, T. Sonobe, K. Inaba, T. Inagaki, T. Ikuta, Y. Yamada, T. Kazama, K. Enbutsu, T. Umeki, R. Kasa- hara,et al., 100,000-spin coherent ising machine, Science advances7, eabh0952 (2021)

    2021

  28. [28]

    M. A. Iftakher, H. Levices, K.-E. Harabi, A. Re- naudineau, M.-C. Faye, C. Bouchard, F. Disdier, B. Viala, E. Vianello, P. Talatchian,et al., Intrinsic an- nealing in a hybrid memristor-magnetic tunnel junction ising machine, Nature Communications (2026)

    2026

  29. [29]

    N. A. Aadit, A. Grimaldi, M. Carpentieri, L. Theogara- jan, J. M. Martinis, G. Finocchio, and K. Y. Camsari, Massively parallel probabilistic computing with sparse ising machines, Nature Electronics5, 460 (2022)

    2022

  30. [30]

    Mizrahi, T

    A. Mizrahi, T. Hirtzlin, A. Fukushima, H. Kubota, S. Yuasa, J. Grollier, and D. Querlioz, Neural-like com- puting with populations of superparamagnetic basis func- tions, Nat. Comm.9, 1533 (2018)

    2018

  31. [31]

    M. W. Daniels, A. Madhavan, P. Talatchian, A. Mizrahi, and M. D. Stiles, Energy-efficient stochastic computing with superparamagnetic tunnel junctions, Physical re- view applied13 3(2019)

    2019

  32. [32]

    R. Zand, K. Y. Camsari, S. D. Pyle, I. Ahmed, C. H. Kim, and R. F. DeMara, Low-energy deep belief networks us- ing intrinsic sigmoidal spintronic-based probabilistic neu- rons, inProceedings of the 2018 Great Lakes Symposium on VLSI(2018) pp. 15–20

    2018

  33. [33]

    Kaiser, W

    J. Kaiser, W. A. Borders, K. Y. Camsari, S. Fukami, H. Ohno, and S. Datta, Hardware-aware in situ learning based on stochastic magnetic tunnel junctions, Physical Review Applied17, 014016 (2022)

    2022

  34. [34]

    Faria, K

    R. Faria, K. Y. Camsari, and S. Datta, Low-barrier nano- magnets asp-bits for spin logic, IEEE Magnetics Letters 8, 1 (2017)

    2017

  35. [35]

    Sutton, K

    B. Sutton, K. Y. Camsari, B. Behin-Aein, and S. Datta, Intrinsic optimization using stochastic nanomagnets, Sci- entific reports7, 44370 (2017)

    2017

  36. [36]

    K. Y. Camsari, R. Faria, B. M. Sutton, and S. Datta, Stochastic p-bits for invertible logic, Phys. Rev. X7, 031014 (2017)

    2017

  37. [37]

    Chowdhury, A

    S. Chowdhury, A. Grimaldi, N. A. Aadit, S. Niazi, M. Mohseni, S. Kanai, H. Ohno, S. Fukami, L. Theogara- jan, G. Finocchio,et al., A full-stack view of probabilistic computing with p-bits: Devices, architectures, and algo- rithms, IEEE Journal on Exploratory Solid-State Com- putational Devices and Circuits9, 1 (2023)

    2023

  38. [38]

    A. Z. Pervaiz, L. A. Ghantasala, K. Y. Camsari, and S. Datta, Hardware emulation of stochastic p-bits for in- vertible logic, Scientific reports7, 10994 (2017)

    2017

  39. [39]

    W. A. Borders, A. Z. Pervaiz, S. Fukami, K. Y. Cam- sari, H. Ohno, and S. Datta, Integer factorization using stochastic magnetic tunnel junctions, Nature573, 390 (2019)

    2019

  40. [40]

    Soumah, L

    L. Soumah, L. Desplat, N.-T. Phan, A. S. E. Valli, A. Madhavan, F. Disdier, S. Auffret, R. C. Sousa, U. Ebels, M. D. Stiles,et al., Entropy-assisted nanosec- ond stochastic operation in perpendicular superparam- agnetic tunnel junctions, Physical Review Applied24, L011002 (2025)

    2025

  41. [41]

    Hutin, P

    L. Hutin, P. Talatchian, P. I. Vaz, A. S. El Valli, A. Bazzi, N.-T. Phan, and F. Andrieu, Importance of stochastic components for ising hamiltonian-based invertible logic, in2022 IEEE Silicon Nanoelectronics Workshop (SNW) (IEEE, 2022) pp. 1–2

    2022

  42. [42]

    Danouchi, L

    K. Danouchi, L. Soumah, C. Bouchard, F. Disdier, A. Fassatoui, N.-T. Phan, M. Ezzadeen, B. Delaet, B. Viala, G. Prenat,et al., Designing networks of resistively-coupled stochastic magnetic tunnel junctions for energy-based optimum search, in2023 International Electron Devices Meeting (IEDM)(IEEE, 2023) pp. 1–4

    2023

  43. [43]

    N. S. Singh, K. Kobayashi, Q. Cao, K. Selcuk, T. Hu, S. Niazi, N. A. Aadit, S. Kanai, H. Ohno, S. Fukami, et al., Cmos plus stochastic nanomagnets enabling het- erogeneous computers for probabilistic inference and learning, Nature Communications15, 2685 (2024)

    2024

  44. [44]

    Wang and J

    T. Wang and J. Roychowdhury, Oim: Oscillator-based ising machines for solving combinatorial optimisation problems, inInternational Conference on Unconventional Computation and Natural Computation(Springer, 2019) pp. 232–256

    2019

  45. [45]

    Raimondo, E

    E. Raimondo, E. Garz´ on, Y. Shao, A. Grimaldi, S. Chi- appini, R. Tomasello, N. Davila-Melendez, J. A. Katine, M. Carpentieri, M. Chiappini,et al., High-performance and reliable probabilistic ising machine based on simu- lated quantum annealing, Physical Review X15, 041001 (2025). 20

    2025

  46. [46]

    Raimondo, A

    E. Raimondo, A. Grimaldi, V. Tyberkevych, R. Tomasello, A. Giordano, M. Carpentieri, A. Slavin, M. Chiappini, and G. Finocchio, Adaptive ising ma- chine based on phase locking of an auto-oscillator to a biharmonic external driving with noise, Physical Review Letters136, 10.1103/jv9m-4mny (2026)

    work page doi:10.1103/jv9m-4mny 2026

  47. [47]

    E. M. H. Ekanayake, N. Khan, and N. Shukla, Configur- ing oscillator ising machines as p-bit engines, Communi- cations Physics9, 10.1038/s42005-026-02492-z (2026)

    work page doi:10.1038/s42005-026-02492-z 2026

  48. [48]

    Litvinenko, P

    A. Litvinenko, P. Sethi, C. Murapaka, A. Jenkins, V. Cros, P. Bortolotti, R. Ferreira, B. Dieny, and U. Ebels, Analog and digital phase modulation and signal transmission with spin-torque nano-oscillators, Physical Review Applied16, 024048 (2021)

    2021

  49. [49]

    J. Hem, L. D. Buda-Prejbeanu, and U. Ebels, Power and phase dynamics of injection-locked spin torque nano- oscillators under conservative and dissipative driving sig- nals, Physical Review B100, 054414 (2019)

    2019

  50. [50]

    N. T. Phan,Leveraging stochastic properties of spintronic nanodevices for unconventional computing, Ph.D. thesis, Universit´ e Grenoble Alpes [2020-....] (2025)

    2020

  51. [51]

    Adler, A study of locking phenomena in oscillators, Proceedings of the IRE34, 351–357 (1946)

    R. Adler, A study of locking phenomena in oscillators, Proceedings of the IRE34, 351–357 (1946)

    1946

  52. [52]

    J.-V. Kim, V. Tiberkevich, and A. N. Slavin, Generation linewidth of an auto-oscillator with a nonlinear frequency shift: Spin-torque nano-oscillator, Phys. Rev. Lett.100, 017207 (2008)

    2008

  53. [53]

    A. H. Nayfeh,Introduction to Perturbation Techniques (John Wiley & Sons, Ltd, 2011)

    2011

  54. [54]

    S. H. Strogatz,Nonlinear Dynamics and Chaos(CRC Press, 2018)

    2018

  55. [55]

    Grimaldi, A

    E. Grimaldi, A. Dussaux, P. Bortolotti, J. Grollier, G. Pillet, A. Fukushima, H. Kubota, K. Yakushiji, S. Yuasa, and V. Cros, Response to noise of a vortex based spin transfer nano-oscillator, Phys. Rev. B89, 104404 (2014)

    2014

  56. [56]

    D. I. Albertsson, M. Zahedinejad, A. Houshang, R. Khymyn, J. ˚Akerman, and A. Rusu, Ultrafast ising machines using spin torque nano-oscillators, Applied Physics Letters118, 10.1063/5.0041575 (2021)

    work page doi:10.1063/5.0041575 2021

  57. [57]

    Hakam, C

    A. Hakam, C. Chopin, L. Martins, E. Aubouin, S. de Wergifosse, C. de Barbarin, N. Mollard, E. Yildiz, L. Hutin, F. Badets, L. Benetti, A. Jenkins, R. Fer- reira, L. D. Buda-Prejbeanu, F. A. Araujo, P. Talatchian, and U. Ebels, A programmable Ising machine using vor- tex spin-torque nano-oscillator, inSpintronics and Nano- magnetism, Vol. 14081, edited by ...

    2026

  58. [58]

    N. G. Berloff, M. Silva, K. Kalinin, A. Askitopoulos, J. D. T¨ opfer, P. Cilibrizzi, W. Langbein, and P. G. Lagoudakis, Realizing the classical XY Hamiltonian in polariton sim- ulators, Nature Materials16, 1120 (2017)

    2017

  59. [59]

    Honari-Latifpour and M.-A

    M. Honari-Latifpour and M.-A. Miri, Optical Potts ma- chine through networks of three-photon down-conversion oscillators, Physical Review Research2, 043335 (2020)

    2020

  60. [60]

    Chalupnik, A

    M. Chalupnik, A. Singh, J. Leatham, M. Lonˇ car, and M. Soltani, Nanophotonic phased array XY Hamiltonian solver, APL Photonics9, 031306 (2024)

    2024

  61. [61]

    Prasad, P

    N. Prasad, P. Mukim, A. Madhavan, and M. D. Stiles, Associative memories using complex-valued Hopfield net- works based on spin-torque oscillator arrays, Neuromor- phic Computing and Engineering2, 034003 (2022)

    2022

  62. [62]

    Lebrun, A

    R. Lebrun, A. Jenkins, A. Dussaux, N. Locatelli, S. Tsunegi, E. Grimaldi, H. Kubota, P. Bortolotti, K. Yakushiji, J. Grollier, A. Fukushima, S. Yuasa, and V. Cros, Understanding of phase noise squeezing under fractional synchronization of a nonlinear spin transfer vortex oscillator, Phys. Rev. Lett.115(2015)

    2015

  63. [63]

    Urazhdin, P

    S. Urazhdin, P. Tabor, V. Tiberkevich, and A. Slavin, Fractional synchronization of spin-torque nano- oscillators, Phys. Rev. Lett.105, 104101 (2010)

    2010

  64. [64]

    Mallick, M

    A. Mallick, M. K. Bashar, Z. Lin, and N. Shukla, Com- putational models based on synchronized oscillators for solving combinatorial optimization problems, Physical Review Applied17, 10.1103/physrevapplied.17.064064 (2022)

    work page doi:10.1103/physrevapplied.17.064064 2022

  65. [65]

    D. E. Nikonov, A. Kurtoglu, T. A. Nirschl, J. Skinner, M. M. A. Faruque, and I. A. Young, Polychronous oscil- latory cellular neural networks for solving graph coloring problems, IEEE Transactions on Circuits and Systems I: Regular Papers (2023)

    2023

  66. [66]

    Cheng and Z

    Y. Cheng and Z. Lin, From ising to potts: Physics- inspired potts machines of coupled oscillators for low- energy sampling and combinatorial optimization, arXiv 10.48550/arXiv.2507.18379 (2025)

    work page doi:10.48550/arxiv.2507.18379 2025

  67. [67]

    Faria, K

    R. Faria, K. Y. Camsari, and S. Datta, Implementing bayesian networks with embedded stochastic mram, AIP Adv.8, 045101 (2018)

    2018

  68. [68]

    M. W. Daniels, A. Madhavan, P. Talatchian, A. Mizrahi, and M. D. Stiles, Energy-efficient stochastic comput- ing with superparamagnetic tunnel junctions, Phys. Rev. Appl.13, 034016 (2020)

    2020

  69. [69]

    Csaba and W

    G. Csaba and W. Porod, Coupled oscillators for comput- ing: A review and perspective, Applied physics reviews 7(2020)

    2020

  70. [70]

    Todri-Sanial, C

    A. Todri-Sanial, C. Delacour, M. Abernot, and F. Sabo, Computing with oscillators from theoretical underpin- nings to applications and demonstrators, Npj unconven- tional computing1, 14 (2024)

    2024

  71. [71]

    Wittrock, P

    S. Wittrock, P. Talatchian, M. Romera, S. Menshawy, M. Jotta Garcia, M.-C. Cyrille, R. Ferreira, R. Lebrun, P. Bortolotti, U. Ebels, J. Grollier, and V. Cros, Be- yond the gyrotropic motion: Dynamic c-state in vor- tex spin torque oscillators, Applied Physics Letters118, 10.1063/5.0029083 (2021)

    work page doi:10.1063/5.0029083 2021

  72. [72]

    Naaman and J

    O. Naaman and J. Aumentado, Poisson transition rates from time-domain measurements with a finite band- width, Phys. Rev. Lett.96, 100201 (2006)

    2006

  73. [73]

    A. A. Thiele, Steady-state motion of magnetic domains, Phys. Rev. Lett.30, 230 (1973)

    1973

  74. [74]

    Khalsa, M

    G. Khalsa, M. D. Stiles, and J. Grollier, Critical current and linewidth reduction in spin-torque nano-oscillators by delayed self-injection, Applied Physics Letters106, 242402 (2015)

    2015

  75. [75]

    Abreu Araujo, C

    F. Abreu Araujo, C. Chopin, and S. de Wergifosse, Ampere–oersted field splitting of the nonlinear spin- torque vortex oscillator dynamics, Scientific Reports12, 10.1038/s41598-022-14574-3 (2022)

    work page doi:10.1038/s41598-022-14574-3 2022

  76. [77]

    Barra and P

    A. Dussaux, A. V. Khvalkovskiy, P. Bortolotti, J. Grol- lier, V. Cros, and A. Fert, Field dependence of 21 spin-transfer-induced vortex dynamics in the nonlin- ear regime, Physical Review B86, 10.1103/phys- revb.86.014402 (2012)

    work page doi:10.1103/phys- 2012

  77. [78]

    Quinsat, D

    M. Quinsat, D. Gusakova, J. F. Sierra, J. P. Michel, D. Houssameddine, B. Delaet, M.-C. Cyrille, U. Ebels, B. Dieny, L. D. Buda-Prejbeanu, J. A. Katine, D. Mauri, A. Zeltser, M. Prigent, J.-C. Nallatamby, and R. Som- met, Amplitude and phase noise of magnetic tun- nel junction oscillators, Applied Physics Letters97, 10.1063/1.3506901 (2010)

    work page doi:10.1063/1.3506901 2010