title: Non-Markovian Stochastic Resonance of Light in a Microcavity
authors: K. J. H. Peters Z. Geng K. Malmir J. M. Smith S. R. K. Rodriguez
text: 
  
  
  
  Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands
  
  
  Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands
  
  
  Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK
  
  
  Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK
  
    s.rodriguez@amolf.nl
  Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands
  
  
  We report the first observation of non-Markovian stochastic resonance, i.e., noise-assisted amplification of a periodic signal in a system with memory. Our system is an oil-filled optical microcavity which, driven by a continuous wave laser, has memory in its nonlinear optical response.  By modulating the cavity length while adding noise to the driving laser, we observe a peak in the transmitted signal-to-noise ratio as a function of the noise variance. Our experimental observations are reproduced by numerical simulations, which further reveal that the stochastic resonance bandwidth is enlarged by the memory time of the nonlinearity. This frequency range available for noise-assisted amplification is 10^8 times greater in our oil-filled cavity than in a Kerr nonlinear cavity. Our results pave the way for exploring the interplay of nonlinearity, noise, and memory, in oil-filled cavity arrays, where non-Markovian dynamics could enhance noise-assisted transport and synchronization effects.
  
  
  Non-Markovian Stochastic Resonance of Light in a Microcavity
      S. R. K. Rodriguez
      December 30, 2023
  ============================================================
  
  
  
  
  Stochastic resonance (SR) is a phenomenon wherein an optimum amount of noise amplifies the response of a nonlinear system to a periodic signal <cit.>. The essence of SR can be recognized in the behavior of a Brownian particle in a double well potential (DWP). As explained by Kramers <cit.>, a Brownian particle escapes from a potential well in a characteristic time τ_esc∝exp(Δ V / k_BT),  with Δ V the energy barrier  and  k_BT the energy of thermal fluctuations. If the potential is modulated with period T_mod=2τ_esc, the particle's Brownian motion can synchronize with the modulation. This results in two barrier crossings per period — stochastic resonance.
  
  SR was first proposed as an explanation for recurrent ice ages <cit.>. Since then, SR has been extensively observed in physics <cit.>,  chemistry <cit.>, biology <cit.>, ecology <cit.>, psychophysics <cit.>, climate science <cit.>, finance <cit.> and social science <cit.>. Applications of SR to imaging <cit.> and mehanical engineering <cit.> have also emerged. To date, all observations of SR have been described within the Markov approximation neglecting memory effects. While non-Markovian dynamics have been experimentally observed in various systems <cit.> and are theoretically expected to modify SR <cit.>, non-Markovian SR has not been experimentally reported.
  
  
  In this Letter, we report the first observation of non-Markovian stochastic resonance.  We investigate an oil-filled optical microcavity with memory in its nonlinear optical response. In contrast to the widely studied class of non-Markovian systems introduced by Mori <cit.> and described by generalized Langevin equations <cit.>, the memory of our cavity is unrelated to its dissipation. Moreover, non-Markovian dynamics emerge even for Gaussian white noise. Here, we  evidence SR  in this non-Markovian regime by driving our oil-filled cavity with a continuous wave laser to which we add a controlled amount of Gaussian white noise.  By periodically modulating the cavity length we imprint a signal on the laser, and we measure the transmitted signal-to-noise ratio (SNR) as a function of the added noise variance. SR manifests as a peak in the SNR at a certain noise variance. Our experimental observations are reproduced by numerical simulations based on a recently developed model accounting for the memory time of the nonlinearity <cit.>. Simulations show how the memory time of the nonlinearity enlarges the SR bandwidth, making it 8 orders of magnitude greater for our oil-filled cavity than for conventional Kerr nonlinear cavities. This memory-enhanced robustness of SR against changes in signal frequency opens new perspectives for harvesting energy from fluctuations across an unprecedentedly large bandwidth.
  
  
  Figure <ref>(a) illustrates our experimental setup, comprising a tunable microcavity filled with macadamia oil. The cavity is made by a planar and a concave mirror. The planar mirror is a 60 nm thick silver layer on a glass substrate. The concave mirror (7 μm diameter, 12 μm radius of curvature) is  made by milling a glass substrate with a focused ion beam <cit.>, and subsequently coating it with a distributed Bragg reflector (DBR). The DBR has a peak reflectance of 99.9% at 530 nm. We use a six degree-of-freedom piezoelectric actuator to align and position the concave mirror, and a single high-frequency piezoelectric actuator to periodically displace the planar mirror. We can probe a single optical mode when displacing the planar mirror across several nanometers because the micron-scale dimensions of the concave mirror strongly confine the optical modes. The cavity is driven by a 532 nm single-mode continuous wave laser, which heats the oil and causes its refractive index to decrease. A similar  intensity-dependent refractive index underlies the observation of SR in Kerr nonlinear cavities at cryogenic temperatures <cit.>, but with two important differences.  Our oil-filled cavity has a strong nonlinearity at room temperature, and non-instantaneous thermal relaxation endows our system with memory, thereby enabling us to explore non-Markovian regimes. Our excitation and collection objectives both have 10× magnification and a numerical aperture of 0.25. In all experiments here presented, we drive the 15^th longitudinal cavity mode with a laser power of 7.8 mW at the excitation objective. The transmitted light is measured with a photodetector and an oscilloscope.
  
  
  In a frame rotating at the laser frequency ω, the light field α in our single-mode oil-filled cavity satisfies the following stochastic integro-differential equation:
  
      
      i α̇(t) = [-Δ- iΓ/2 + U ∫_0^t ds K(t-s)(|α(s)|^2-1)]α(t) + i √(κ_L)F + D/√(2)[ξ_1(t) + i ξ_2(t) ].
  
  
  
   
  
  Δ=ω-ω_0 is the laser-cavity detuning, with ω_0 the resonance frequency. Γ=γ+κ_L+κ_R is the total loss rate, with γ the absorption rate and  κ_L,R the input-output rates through the left and right mirror.  U is the strength of the cubic nonlinearity, with non-instantaneous character captured by the  time-integral in Eq. <ref>. The integral contains the  memory kernel K(t)=exp(-t/τ)/τ with  τ the thermal relaxation time. This form of the kernel ensures that steady-states (α̇=0 in Eq. <ref>)   are the same as for an instantaneous nonlinearity <cit.>. F is the laser amplitude.  ξ_1(t) and ξ_2(t) provide Gaussian white noise in the laser amplitude and phase; the combined variance of these stochastic processes is D^2. Note that for τ→ 0 and D=√(Γ/2), Eq. <ref>  describes a Kerr nonlinear cavity influenced by quantum fluctuations within the so-called truncated Wigner approximation <cit.>. In Supplemental Material we provide further details about our calculations based on the xSPDE toolbox <cit.> for Matlab, and the values of the model parameters we used <cit.>.
  
  
  For vanishing noise (D=0) and a sufficiently large F making the mean-field interaction energy larger than the losses (i.e., U |α|^2 ≳Γ), Eq. <ref> predicts optical bistability: two steady-states with different intracavity intensity |α|^2 at a single driving condition. To observe bistability, we measure the transmitted laser power at constant F while opening and closing the cavity. The cavity length maps to Δ/Γ, with Γ the linewidth at low power. For example,  Fig. <ref>(b) shows transmission measurements where bistability occurs for 2 ≲Δ/Γ≲ 5.5. The measurements also display an overshoot at Δ/Γ∼ 1.5, due to the oil's non-instantaneous thermal relaxation <cit.>. Steady-state calculations assuming instantaneous nonlinearity [gray curves in Fig. <ref>(b)]  do not reproduce this overshoot. In contrast, dynamic simulations [purple curves in Fig. <ref>(b)] based on Eq. <ref> including memory effects reproduce all our observations, including the overshoot. The width of the overshoot is determined by T_mod/τ. In  Supplemental Material we show that  τ=10 μs in our oil-filled cavity <cit.>.
  
  
  
  For D ≠ 0,  light in a bistable optical cavity behaves like a Brownian particle in a DWP. For constant F and Δ/Γ yielding bistability, α randomly switches between states at the Kramers rate τ_esc^-1∝exp(D). This switching behavior can be observed in  stochastic trajectories of |α|^2 shown in Supplemental Material <cit.>. Based on multiple trajectories α(t), we calculate a probability density function (PDF)  in the complex α-plane. Figures <ref>(c,d) show this PDF for two values of Δ/Γ. Notice the bimodal distributions reminiscent of a Brownian particle in a DWP.
  The matching symbols in Fig. <ref>(b) and Fig. <ref>(c) indicate the correspondence between  high & low intensity states of the bistability and  the peaks in the PDF. It follows that by modulating Δ/Γ and for the right D, SR can emerge in the transmission of light through our microcavity.
  
  
  To demonstrate  SR, we measure the transmitted laser signal while modulating the cavity length.  The cavity acts as a transducer, converting a mirror displacement to an optical signal.  We set a subthreshold modulation amplitude of 12 nm, which is insufficient for deterministic switching between states given our laser power. Using the setup  illustrated in Fig. <ref>(a), we add uncorrelated amplitude and phase noise by passing the laser through electro-optic modulators connected to different waveform generators. In Supplemental Material we show that the noise power is approximately constant up to 200 kHz <cit.>, well-above our cavity modulation frequency f_mod=42.5Hz and above the thermal relaxation rate τ^-1=100 kHz. Effectively, this corresponds to white noise. We also show that the standard deviation of the noise D is proportional to the peak-to-peak voltage V_pp in the waveform generators <cit.>.
  
  
  Figures <ref>(a)-(d) and  <ref>(e)-(h) show measurements of the transmitted signal and the corresponding power spectral density (PSD), respectively, for different V_pp.  For the smallest V_pp,  the  transmitted power is modulated by the changing cavity length while noise plays a secondary role [Fig. <ref>(a)]. The corresponding PSD [Fig. <ref>(e)] shows a  peak at f_mod. For increased V_pp, noise makes the transmitted intensity switch between high and low values sometimes. These switches occur close to the maximum modulation amplitude [Fig. <ref>(b)]. In the corresponding PSD [Fig. <ref>(f)], this manifests as a greater noise floor and a reduced peak at f_mod.  For greater V_pp, the switching synchronizes with the modulation [Fig. <ref>(c)] and the peak in the PSD grows [Fig. <ref>(g)]. Further increasing V_pp makes the transmitted signal  more noisy  [Fig. <ref>(d)]. The noise floor and the peak at f_mod barely change [Fig. <ref>(h)], but peaks at higher harmonics substantially decrease (see Supplemental Material <cit.>).   Overall, these measurements demonstrate the signature of  SR: the signal power is a non-monotonic function of the noise strength.
  
  
  Next, we analyze the measured transmitted signal-to-noise ratio. We define the SNR as the power in the first six harmonics including the fundamental f_mod, relative to the noise floor. Figure <ref>(a) shows the SNR versus V_pp over an extended range. We can distinguish three regimes. For small V_pp, the SNR decreases with increasing V_pp. This is due to an increasing number of random switching events and a growing  noise floor. For intermediate V_pp, the SNR increases with V_pp. Here switching events increasingly synchronize with the modulation.  Finally, for large V_pp the SNR decreases with V_pp. This effect can be observed in the reduced amplitude of the signal harmonics as shown in Supplemental Material <cit.>. In combination, these three regimes result in a SR peak in the SNR around V_pp=70 V.
  
  
  
  We further validate our model by reproducing our SNR measurements. This will enable us to confidently simulate the effect of the memory time τ on SR  over an extended range,  which is not easily done experimentally. Figure <ref>(b) shows results of simulations corresponding to the measurements in Fig. <ref>(a). We consider three values of τ,  keeping the ratio  T_mod/τ constant since it determines the detuning range for which bistability occurs <cit.>. For  τ>Γ^-1, we observe a SNR peak growing with τ and remaining at approximately constant D/Γ. This behavior is due to two competing effects. On one hand, increasing T_mod makes the SNR peak grow and shift to smaller D; the same is true in standard Markovian SR. On the other hand, increasing τ slightly reduces the SNR peak and shifts it to larger D; this effect is unique to our non-Markovian system. We attribute the peak SNR reduction to the overshoot following each switching [see Fig. <ref>(b)], which makes the signal further deviate from a pure sinusoidal. The shift  to larger D  is associated with non-exponential distributions of residence times in the metastable states of the cavity <cit.>, which is the hallmark of non-Markovian dynamics. Essentially, thermal relaxation imposes a high-frequency cut-off for switching events at τ^-1. This results in SR at larger input noise variance than in the Markovian case.
  
  
  
  
  In Fig. <ref> we calculate the average number of switches per cycle n_avg as a function of T_mod, for different values of τ. We consider a fixed noisy environment (constant D≠ 0), and calculate n_avg as explained in Supplemental Material <cit.>. Recall that SR corresponds to n_avg≈2. For an instantaneous nonlinearity (τ→ 0), the black circles in Fig. <ref> show how n_avg simply increases linearly with T_mod. In our system, this Markovian limit is reached  for τ≪Γ^-1. More interesting behavior arises in the non-Markovian regime where τ > Γ^-1. Taking  Γτ = 4 and Γτ = 10 as examples, Fig. <ref> shows the emergence of a `plateau' at the SR condition n_avg≈2. This plateau represents an enlargement of the signal frequency range in which SR can be achieved, i.e., the SR bandwidth. Thus, thanks to the slow nonlinearity, SR becomes robust to changes in signal frequency. In Supplemental Material we show that the SR bandwidth increases linearly with τ <cit.>.   By extrapolating the SR bandwidth calculated as a function of τ up to our experimental value τ = 10  μs, and considering  Γ^-1 =4 ps in our experiments, we deduce that the SR bandwidth is enlarged by a factor of 10^8 in our oil-filled cavity relative to a Kerr nonlinear cavity.
  
  The enlarged SR bandwidth in our non-Markovian system is associated with the suppression of fast switching events by the non-instantaneous nonlinearity. Indeed, for fast modulations satisfying Γ T_mod≲ 100, bistability disappears because the nonlinearity does not have time to build up <cit.>. This high-frequency regime is characterized by n_avg<2, as shown in the inset of Fig. <ref>. In the opposite regime of slow modulations,  n_avg increases linearly with T_mod as in the Markovian case, albeit with a smaller slope. The smaller slope further enlarges the SR bandwidth.
  
  In summary, we demonstrated noise-assisted transmission of light through an optical microcavity with non-instantaneous nonlinearity — non-Markovian SR. Through numerical simulations we found that the frequency range available for SR grows with the memory time of the nonlinearity. Extensions of our work could investigate the effects of colored or non-Gaussian noise, which were previously shown to enhance Markovian SR in certain regimes <cit.>. Our findings can have important applications to nonlinear energy harvesting <cit.>. Typically, energy harvesting is limited to a narrow frequency bandwidth by the (instantaneous) response of the device. In contrast, a device with   non-instantaneous  nonlinearity could be used to extract energy from environmental fluctuations across an unprecedentedly large bandwidth.  Our results also provide a plausible explanation for the occurrence of various noise-assisted processes at different frequencies within a single noisy environment, as observed in Nature.  Beyond single-resonator physics,  our  work paves the way  to study various noise-assisted processes (e.g. noise-assisted transport <cit.>) in nonlinear and non-Markovian cavity arrays. Such arrays could also be used to explore how complex networks can be made resilient against breakdown of synchronization thanks to non-Markovian effects <cit.>.  
  
  
  
  
  
  § ACKNOWLEDGMENTS
  
  This work is part of the research programme of the Netherlands Organisation for Scientific Research (NWO). We thank Ewold Verhagen for critical comments on our manuscript, Allard Mosk, Femius Koenderink, and Sanli Faez for stimulating discussions, and Ricardo Struik and Niels Commandeur for technical support. S.R.K.R. acknowledges a NWO Veni grant with file number 016.Veni.189.039.
  
  
  
  
  
  
  
  
  
  
  
  
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  @bib@innerbibempty
  
  [Gammaitoni et al.(1998)Gammaitoni, Hänggi, Jung, and  Marchesoni]Gammaitoni_RMP1998
    author author L. Gammaitoni, author P. Hänggi, author P. Jung,  and author F. Marchesoni,  title title Stochastic resonance, https://doi.org/10.1103/RevModPhys.70.223 journal journal Rev. Mod. Phys. volume 70, pages 223 (year 1998)NoStop
  [Kramers(1940)]Kramers
    author author H. Kramers, title title Brownian motion in a
    field of force and the diffusion model of chemical reactions, https://doi.org/https://doi.org/10.1016/S0031-8914(40)90098-2 journal journal Physica volume 7,  pages 284  (year 1940)NoStop
  [Benzi et al.(1981)Benzi,
    Sutera, and Vulpiani]Benzi_JPhysA1981
    author author R. Benzi, author A. Sutera, and  author A. Vulpiani, title title The mechanism of stochastic resonance, https://doi.org/10.1088/0305-4470/14/11/006 journal journal J. Phys. A volume 14, pages
    L453 (year 1981)NoStop
  [Nicolis(1993)]nicolis1993
    author author C. Nicolis, title title Long-term climatic
    transitions and stochastic resonance, https://link.springer.com/article/10.1007/BF01053950 journal
    journal J. Stat. Phys. volume 70,  pages 3 (year 1993)NoStop
  [Fauve and Heslot(1983)]Fauve83
    author author S. Fauve and author F. Heslot,  title title Stochastic resonance in a bistable
    system, https://doi.org/https://doi.org/10.1016/0375-9601(83)90086-5 journal journal Phys. Lett. A volume
    97, pages 5  (year 1983)NoStop
  [McNamara et al.(1988)McNamara, Wiesenfeld, and Roy]McNamara_PRL1988
    author author B. McNamara, author K. Wiesenfeld, and author R. Roy, title title Observation of stochastic
    resonance in a ring laser, https://doi.org/10.1103/PhysRevLett.60.2626 journal journal Phys. Rev. Lett. volume 60, pages 2626 (year 1988)NoStop
  [Hibbs et al.(1995)Hibbs,
    Singsaas, Jacobs, Bulsara,
    Bekkedahl, and Moss]hibbs1995
    author author A. Hibbs, author A. Singsaas,
    author E. Jacobs, author A. Bulsara, author
    J. Bekkedahl, and author
    F. Moss, title title Stochastic resonance in a superconducting loop with a Josephson
    junction, https://aip.scitation.org/doi/abs/10.1063/1.358720
    journal journal J. Appl. Phys. volume 77, pages 2582 (year
    1995)NoStop
  [Lee et al.(2003)Lee,
    Lia, Kosko, and Zhou]Lee03
    author author I. Y. Lee, author X. Lia, author B. Kosko, and author
    C. Zhou, title title Nanosignal processing: Stochastic resonance in carbon nanotubes that
    detect subthreshold signals, https://doi.org/10.1021/nl0348239
    journal journal Nano Lett. volume 3, pages 1683 (year 2003)NoStop
  [Abbaspour et al.(2014)Abbaspour, Trebaol, Morier-Genoud,
    Portella-Oberli, and Deveaud]Abbaspour_PRL2014
    author author H. Abbaspour, author S. Trebaol,
    author F. Morier-Genoud, author M. T. Portella-Oberli, and  author B. Deveaud, title title Stochastic resonance in collective
    exciton-polariton excitations inside a GaAs microcavity, https://doi.org/10.1103/PhysRevLett.113.057401 journal
    journal Phys. Rev. Lett. volume 113,  pages 057401 (year 2014)NoStop
  [Abbaspour et al.(2015)Abbaspour, Trebaol, Morier-Genoud,
    Portella-Oberli, and Deveaud]Abbaspour_PRB2015
    author author H. Abbaspour, author S. Trebaol,
    author F. Morier-Genoud, author M. T. Portella-Oberli, and  author B. Deveaud, title title Spinor stochastic resonance, https://doi.org/10.1103/PhysRevB.91.155307 journal journal Phys. Rev. B volume 91, pages 155307 (year 2015)NoStop
  [Venstra et al.(2013)Venstra, Westra, and Van
    Der Zant]Venstra13
    author author W. J. Venstra, author H. J. Westra, and author H. S. Van
    Der Zant, title title Stochastic switching of
    cantilever motion, https://www.nature.com/articles/ncomms3624
    journal journal Nat. Commun. volume 4, pages 1 (year 2013)NoStop
  [Monifi et al.(2016)Monifi,
    Zhang, Özdemir, Peng,
    Liu, Bo, Nori, and  Yang]Monifi_NatPhot2016
    author author F. Monifi, author J. Zhang,
    author S. Özdemir, author B. Peng, author
    Y. Liu, author F. Bo, author F. Nori, and author L. Yang,  title title Optomechanically induced stochastic
    resonance and chaos transfer between optical fields, https://doi.org/10.1038/nphoton.2016.73 journal journal Nat. Photon. volume 10, pages 399 (year 2016)NoStop
  [Ricci et al.(2017)Ricci,
    Rica, Spasenović, Gieseler, Rondin, Novotny, and  Quidant]Ricci17
    author author F. Ricci, author R. A. Rica,
    author M. Spasenović,
    author J. Gieseler, author L. Rondin, author
    L. Novotny, and author
    R. Quidant, title title Optically levitated nanoparticle as a model system for stochastic
    bistable dynamics, https://www.nature.com/articles/ncomms15141
    journal journal Nat. Commun. volume 8, pages 1 (year 2017)NoStop
  [Chowdhury et al.(2017)Chowdhury, Barbay, Clerc, Robert-Philip, and Braive]Chowdhury17
    author author A. Chowdhury, author S. Barbay,
    author M. G. Clerc, author I. Robert-Philip, and author R. Braive, title
    title Phase stochastic resonance in a forced
    nanoelectromechanical membrane, https://doi.org/10.1103/PhysRevLett.119.234101 journal
    journal Phys. Rev. Lett. volume 119,  pages 234101 (year 2017)NoStop
  [Wagner et al.(2019)Wagner,
    Talkner, Bayer, Rugeramigabo,
    Hänggi, and Haug]Wagner19
    author author T. Wagner, author P. Talkner,
    author J. C. Bayer, author E. P. Rugeramigabo, author P. Hänggi, and author R. J. Haug, title
    title Quantum stochastic resonance in an ac-driven
    single-electron quantum dot, https://www.nature.com/articles/s41567-018-0412-5 journal
    journal Nat. Phys. volume 15,  pages 330 (year 2019)NoStop
  [Leonard and Reichl(1994)]leonard1994
    author author D. S. Leonard and author L. Reichl, title title Stochastic resonance in a
    chemical reaction, https://journals.aps.org/pre/abstract/10.1103/PhysRevE.49.1734 journal journal Phys. Rev. E volume
    49, pages 1734 (year 1994)NoStop
  [Guderian et al.(1996)Guderian, Dechert, Zeyer, and  Schneider]guderian1996
    author author A. Guderian, author G. Dechert,
    author K.-P. Zeyer, and  author F. Schneider, title title Stochastic resonance in chemistry. 1. the
    Belousov-Zhabotinsky reaction, https://pubs.acs.org/doi/abs/10.1021/jp952243x journal
    journal J. Phys. Chem. volume 100,  pages 4437 (year 1996)NoStop
  [Förster et al.(1996)Förster, Merget, and Schneider]forster1996
    author author A. Förster, author M. Merget, and author F. Schneider, title title Stochastic resonance in
    chemistry. 2. the peroxidase-oxidase reaction, https://pubs.acs.org/doi/abs/10.1021/jp952244p journal
    journal J. Phys. Chem. volume 100,  pages 4442 (year 1996)NoStop
  [Hohmann et al.(1996)Hohmann, Müller, and Schneider]hohmann1996
    author author W. Hohmann, author J. Müller, and author F. Schneider, title title Stochastic resonance in
    chemistry. 3. the minimal-bromate reaction, https://pubs.acs.org/doi/abs/10.1021/jp953269r journal
    journal J. Phys. Chem. volume 100,  pages 5388 (year 1996)NoStop
  [Douglass et al.(1993)Douglass, Wilkens, Pantazelou, and  Moss]Douglass_Nat1993
    author author J. K. Douglass, author L. A. Wilkens, author E. Pantazelou, and author F. Moss, title title Noise enhancement of
    information transfer in crayfish mechanoreceptors by stochastic resonance,  https://www.nature.com/articles/365337a0 journal
    journal Nature volume 365, pages 337 (year 1993)NoStop
  [Bezrukov and Vodyanoy(1995)]bezrukov1995
    author author S. M. Bezrukov and author I. Vodyanoy, title title Noise-induced
    enhancement of signal transduction across voltage-dependent ion channels,  https://www.nature.com/articles/378362a0 journal
    journal Nature volume 378, pages 362 (year 1995)NoStop
  [Jaramillo and Wiesenfeld(1998)]jaramillo1998
    author author F. Jaramillo and author K. Wiesenfeld, title title Mechanoelectrical
    transduction assisted by Brownian motion: a role for noise in the auditory
    system, https://www.nature.com/articles/nn0998_384 journal journal Nat. Neurosci. volume
    1, pages 384 (year 1998)NoStop
  [Russell et al.(1999)Russell, Wilkens, and Moss]Russell_Nat1999
    author author D. F. Russell, author L. A. Wilkens, and author F. Moss, title title Use of behavioural
    stochastic resonance by paddle fish for feeding, https://www.nature.com/articles/46279 journal journal Nature volume 402, pages
    291 (year 1999)NoStop
  [Mori and Kai(2002)]Mori02
    author author T. Mori and author S. Kai,  title title Noise-induced entrainment and
    stochastic resonance in human brain waves, https://doi.org/10.1103/PhysRevLett.88.218101 journal
    journal Phys. Rev. Lett. volume 88,  pages 218101 (year 2002)NoStop
  [Blarer and Doebeli(1999)]Blarer1999
    author author Blarer and author Doebeli, title title
    Resonance effects and outbreaks in ecological time series, https://doi.org/10.1046/j.1461-0248.1999.00067.x journal
    journal Ecol. Lett. volume 2,  pages 167 (year 1999)NoStop
  [Collins et al.(1996)Collins, Imhoff, and Grigg]collins1996
    author author J. J. Collins, author T. T. Imhoff, and author P. Grigg, title title Noise-enhanced tactile
    sensation, https://www.nature.com/articles/383770a0.pdf?origin=ppub journal journal Nature  (year
    1996)NoStop
  [Simonotto et al.(1997)Simonotto, Riani, Seife, Roberts, Twitty, and Moss]simonotto1997
    author author E. Simonotto, author M. Riani,
    author C. Seife, author M. Roberts, author
    J. Twitty, and author
    F. Moss, title title Visual perception of stochastic resonance, https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.78.1186
    journal journal Phys. Rev. Lett. volume 78, pages 1186 (year
    1997)NoStop
  [Ward et al.(2002)Ward,
    Neiman, and Moss]ward2002
    author author L. M. Ward, author A. Neiman, and  author F. Moss, title title Stochastic resonance in psychophysics and in
    animal behavior, https://link.springer.com/article/10.1007/s00422-002-0328-z journal journal Biol. Cybern. volume
    87, pages 91 (year 2002)NoStop
  [Ganopolski and Rahmstorf(2002)]Ganopolski_PRL2002
    author author A. Ganopolski and author S. Rahmstorf, title title Abrupt glacial climate
    changes due to stochastic resonance, https://doi.org/10.1103/PhysRevLett.88.038501 journal
    journal Phys. Rev. Lett. volume 88,  pages 038501 (year 2002)NoStop
  [Xiao-Ming et al.(2002)Xiao-Ming, Kai, and Qi]xiao2002
    author author M. Xiao-Ming, author S. Kai,  and author O. Qi, title title Stochastic resonance in a financial model,  https://iopscience.iop.org/article/10.1088/1009-1963/11/11/303/meta
    journal journal Chinese Phys. volume 11, pages 1106 (year
    2002)NoStop
  [Krawiecki and Hołyst(2003)]krawiecki2003
    author author A. Krawiecki and author J. Hołyst, title title Stochastic resonance as
    a model for financial market crashes and bubbles, https://www.sciencedirect.com/science/article/pii/S0378437102013754
    journal journal Physica A volume 317, pages 597 (year
    2003)NoStop
  [Wallace et al.(1997)Wallace, Wallace, and Andrews]wallace1997
    author author R. Wallace, author D. Wallace,  and author H. Andrews,  title title AIDS, tuberculosis, violent crime,
    and low birthweight in eight US metropolitan areas: public policy,
    stochastic resonance, and the regional diffusion of inner-city markers,  https://journals.sagepub.com/doi/abs/10.1068/a290525 journal journal Environ. Plan A volume
    29, pages 525 (year 1997)NoStop
  [Marks et al.(2002)Marks, Thompson, El-Sharkawi,
    Fox, and Miyamoto]Marks2002
    author author R. J. Marks, author B. Thompson,
    author M. A. El-Sharkawi,
    author W. L. J. Fox, and  author R. T. Miyamoto,  title title Stochastic resonance of a threshold
    detector: image visualization and explanation, in https://ieeexplore.ieee.org/abstract/document/1010507 booktitle IEEE Symp. Circ. Syst., Vol. volume 4  (year 2002) pp. pages IV–IVNoStop
  [Rallabandi(2008)]rallabandi2008
    author author V. S. Rallabandi, title title Enhancement of
    ultrasound images using stochastic resonance-based wavelet transform,  https://www.sciencedirect.com/science/article/pii/S0895611108000190
    journal journal Comput. Med. Imag. Graph.  volume 32, pages 316 (year
    2008)NoStop
  [Dylov and Fleischer(2010)]dylov2010
    author author D. V. Dylov and author J. W. Fleischer, title title Nonlinear
    self-filtering of noisy images via dynamical stochastic resonance, https://www.nature.com/articles/nphoton.2010.31 journal
    journal Nat. Photon. volume 4,  pages 323 (year 2010)NoStop
  [Niaoqing et al.(2003)Niaoqing, Min, and Xisen]Niaoqing
    author author H. Niaoqing, author C. Min, and  author W. Xisen, title title The application of stochastic resonance theory for
    early detecting rub-impact fault of rotor system, https://doi.org/https://doi.org/10.1006/mssp.2002.1470 journal journal Mech. Syst. Signal Pr. volume 17, pages 883  (year
    2003)NoStop
  [Li et al.(2013)Li,
    Chen, and He]li2013
    author author J. Li, author X. Chen, and  author Z. He, title title Multi-stable stochastic resonance and its
    application research on mechanical fault diagnosis, https://www.sciencedirect.com/science/article/pii/S0022460X13005324
    journal journal J. Sound. Vib. volume 332, pages 5999 (year
    2013)NoStop
  [Hanggi et al.(1985)Hanggi,
    Mroczkowski, Moss, and McClintock]Hanggi85
    author author P. Hanggi, author T. J. Mroczkowski, author F. Moss,  and author P. V. E. McClintock, title title Bistability driven by
    colored noise: Theory and experiment, https://doi.org/10.1103/PhysRevA.32.695 journal journal Phys. Rev. A volume 32, pages 695 (year 1985)NoStop
  [Liebovitch and Sullivan(1987)]Liebovitch87
    author author L. Liebovitch and author J. Sullivan, title title Fractal analysis of a
    voltage-dependent potassium channel from cultured mouse hippocampal
    neurons, https://doi.org/https://doi.org/10.1016/S0006-3495(87)83290-3 journal journal Biophys. J. volume
    52, pages 979  (year 1987)NoStop
  [Mercik and Weron(2001)]Mercik01
    author author S. Mercik and author K. Weron,  title title Stochastic origins of the long-range
    correlations of ionic current fluctuations in membrane channels, https://doi.org/10.1103/PhysRevE.63.051910 journal journal Phys. Rev. E volume 63, pages 051910 (year 2001)NoStop
  [Houlihan et al.(2004)Houlihan, Goulding, Busch, Masoller, and Huyet]Houlihan04
    author author J. Houlihan, author D. Goulding,
    author T. Busch, author C. Masoller, and author G. Huyet, title
    title Experimental investigation of a bistable system in the
    presence of noise and delay, https://doi.org/10.1103/PhysRevLett.92.050601 journal
    journal Phys. Rev. Lett. volume 92,  pages 050601 (year 2004)NoStop
  [Liu et al.(2011)Liu,
    Li, Huang, Li, Guo, Laine, Breuer, and Piilo]Piilo11
    author author B.-H. Liu, author L. Li, author Y.-F. Huang, author
    C.-F. Li, author G.-C. Guo, author E.-M.  Laine, author H.-P. Breuer, and author J. Piilo, title title Experimental control of the
    transition from Markovian to non-Markovian dynamics of open quantum
    systems, https://www.nature.com/articles/nphys2085 journal journal Nat. Phys. volume
    7, pages 931 (year 2011)NoStop
  [Madsen et al.(2011)Madsen,
    Ates, Lund-Hansen, Löffler, Reitzenstein, Forchel, and  Lodahl]madsen2011
    author author K. H. Madsen, author S. Ates,
    author T. Lund-Hansen, author A. Löffler, author
    S. Reitzenstein, author
    A. Forchel, and author
    P. Lodahl, title title Observation of non-Markovian dynamics of a single quantum dot in a
    micropillar cavity, https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.106.233601
    journal journal Phys. Rev. Lett. volume 106, pages 233601 (year
    2011)NoStop
  [Hoeppe et al.(2012)Hoeppe,
    Wolff, Küchenmeister, Niegemann, Drescher, Benner, and  Busch]hoeppe2012
    author author U. Hoeppe, author C. Wolff,
    author J. Küchenmeister,
    author J. Niegemann, author M. Drescher, author
    H. Benner, and author
    K. Busch, title title Direct observation of non-Markovian radiation dynamics in 3D
    bulk photonic crystals, https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.043603
    journal journal Phys. Rev. Lett. volume 108, pages 043603 (year
    2012)NoStop
  [Lu et al.(2020)Lu,
    Zhang, Liu, Nori,
    Fan, and Pan]lu2020
    author author Y.-N. Lu, author Y.-R. Zhang,
    author G.-Q. Liu, author F. Nori, author
    H. Fan, and author
    X.-Y. Pan, title title Observing information backflow from controllable non-Markovian
    multichannels in diamond, https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.124.210502
    journal journal Phys. Rev. Lett. volume 124, pages 210502 (year
    2020)NoStop
  [Neiman and Sung(1996)]Neiman96
    author author A. Neiman and author W. Sung,  title title Memory effects on stochastic
    resonance, https://doi.org/https://doi.org/10.1016/S0375-9601(96)00742-6 journal journal Phys. Lett. A volume
    223, pages 341  (year 1996)NoStop
  [Goychuk and Hänggi(2003)]Goychuk03
    author author I. Goychuk and author P. Hänggi, title title Non-Markovian
    stochastic resonance, https://doi.org/10.1103/PhysRevLett.91.070601 journal
    journal Phys. Rev. Lett. volume 91,  pages 070601 (year 2003)NoStop
  [Prager and Schimansky-Geier(2003)]Prager03
    author author T. Prager and author L. Schimansky-Geier, title title Stochastic
    resonance in a non-Markovian discrete state model for excitable systems,  https://doi.org/10.1103/PhysRevLett.91.230601 journal
    journal Phys. Rev. Lett. volume 91,  pages 230601 (year 2003)NoStop
  [Mori(1965)]Mori
    author author H. Mori, title title Transport, collective
    motion, and Brownian motion, https://doi.org/10.1143/PTP.33.423
    journal journal Prog. Theor. Phys. volume 33, pages 423 (year
    1965)NoStop
  [Hänggi(1978)]Hanggi78
    author author P. Hänggi, title title Correlation functions
    and masterequations of generalized (non-Markovian) Langevin equations,  https://doi.org/10.1007/BF01351552 journal journal Z. Phys. volume 31, pages
    407 (year 1978)NoStop
  [Geng et al.(2020)Geng,
    Peters, Trichet, Malmir,
    Kolkowski, Smith, and Rodriguez]Geng_PRL2020
    author author Z. Geng, author K. J. H. Peters,
    author A. A. P. Trichet,
    author K. Malmir, author R. Kolkowski, author
    J. M. Smith, and author
    S. R. K. Rodriguez, title
    title Universal scaling in the dynamic hysteresis, and
    non-Markovian dynamics, of a tunable optical cavity, https://doi.org/10.1103/PhysRevLett.124.153603 journal
    journal Phys. Rev. Lett. volume 124,  pages 153603 (year 2020)NoStop
  [Trichet et al.(2015)Trichet, Dolan, Coles, Hughes, and Smith]Trichet15
    author author A. A. P.  Trichet, author P. R.  Dolan, author D. M. Coles, author G. M. Hughes,  and author J. M. Smith,  title title Topographic control of open-access
    microcavities at the nanometer scale, https://doi.org/10.1364/OE.23.017205 journal journal Opt. Express volume 23, pages 17205 (year 2015)NoStop
  [Carusotto and Ciuti(2013)]CarusottoRMP
    author author I. Carusotto and author C. Ciuti, title title Quantum fluids of light,  https://doi.org/10.1103/RevModPhys.85.299 journal
    journal Rev. Mod. Phys. volume 85,  pages 299 (year 2013)NoStop
  [Kiesewetter et al.(2016)Kiesewetter, Polkinghorne, Opanchuk, and  Drummond]xSPDE
    author author S. Kiesewetter, author R. Polkinghorne, author B. Opanchuk, and author P. D. Drummond, title title xSPDE: Extensible
    software for stochastic equations, http://www.sciencedirect.com/science/article/pii/S2352711016000030
    journal journal SoftwareX volume 5, pages 12  (year 2016)NoStop
  [sup()]supp
    @noop  note See Supplemental Material for details about
    the calculations, the thermal relaxation time, the switching behavior in
    simulated trajectories, the input noise properties, the measured power
    spectral density for different noise variances, the method used for detecting
    switching events, and the scaling of the stochastic resonance bandwidth with
    the memory time of the nonlinearity.Stop
  [Nozaki et al.(1999)Nozaki,
    Mar, Grigg, and Collins]Nozaki99
    author author D. Nozaki, author D. J. Mar,
    author P. Grigg, and author J. J. Collins, title title Effects of colored noise on stochastic resonance
    in sensory neurons, https://doi.org/10.1103/PhysRevLett.82.2402
    journal journal Phys. Rev. Lett. volume 82, pages 2402 (year
    1999)NoStop
  [Fuentes et al.(2001)Fuentes, Toral, and Wio]Fuentes01
    author author M. Fuentes, author R. Toral,  and author H. S. Wio,  title title Enhancement of stochastic resonance:
    the role of non Gaussian noises, https://doi.org/https://doi.org/10.1016/S0378-4371(01)00062-0 journal journal Physica A volume
    295, pages 114  (year 2001)NoStop
  [Castro et al.(2001)Castro,
    Kuperman, Fuentes, and Wio]Castro01
    author author F. J. Castro, author M. N. Kuperman, author M. Fuentes,  and author H. S. Wio,  title title Experimental evidence of stochastic
    resonance without tuning due to non-Gaussian noises, https://doi.org/10.1103/PhysRevE.64.051105 journal journal Phys. Rev. E volume 64, pages 051105 (year 2001)NoStop
  [Cottone et al.(2009)Cottone, Vocca, and Gammaitoni]Cottone09
    author author F. Cottone, author H. Vocca,  and author L. Gammaitoni,  title title Nonlinear energy harvesting, https://doi.org/10.1103/PhysRevLett.102.080601 journal
    journal Phys. Rev. Lett. volume 102,  pages 080601 (year 2009)NoStop
  [Plenio and Huelga(2008)]Plenio08
    author author M. B. Plenio and author S. F. Huelga, title title Dephasing-assisted
    transport: quantum networks and biomolecules, https://doi.org/10.1088/1367-2630/10/11/113019 journal
    journal New J. Phys. volume 10,  pages 113019 (year 2008)NoStop
  [Caruso et al.(2009)Caruso,
    Chin, Datta, Huelga, and  Plenio]Caruso_JChemPhys_2009
    author author F. Caruso, author A. W. Chin,
    author A. Datta, author S. F. Huelga, and author M. B. Plenio, title
    title Highly efficient energy excitation transfer in
    light-harvesting complexes: The fundamental role of noise-assisted
    transport, https://doi.org/10.1063/1.3223548 journal
    journal J. Chem. Phys. volume 131,  pages 105106 (year 2009)NoStop
  [Viciani et al.(2015)Viciani, Lima, Bellini, and Caruso]Caruso15
    author author S. Viciani, author M. Lima,
    author M. Bellini, and author F. Caruso, title
    title Observation of noise-assisted transport in an all-optical
    cavity-based network, https://doi.org/10.1103/PhysRevLett.115.083601 journal
    journal Phys. Rev. Lett. volume 115,  pages 083601 (year 2015)NoStop
  [Ramirez-San Juan et al.(2020)Ramirez-San Juan, Mathijssen, He,
    Jan, Marshall, and Prakash]Ramirez20
    author author G. R. Ramirez-San Juan, author A. J. Mathijssen, author M. He,
    author L. Jan, author
    W. Marshall, and author
    M. Prakash, title title Multi-scale spatial heterogeneity enhances particle clearance in
    airway ciliary arrays, https://www.nature.com/articles/s41567-020-0923-8 journal
    journal Nat. Phys. , pages 1 (year
    2020)NoStop
  [Lin et al.(2020)Lin,
    Feng, Tang, Liu,
    Xu, Hui, and Lai]Lin20
    author author Z.-H. Lin, author M. Feng, author M. Tang, author
    Z. Liu, author C. Xu, author P. M.  Hui, and author Y.-C.  Lai, title title Non-Markovian
    recovery makes complex networks more resilient against large-scale
    failures, https://www.nature.com/articles/s41467-020-15860-2
    journal journal Nat. Commun. volume 11, pages 1 (year 2020)NoStop
  
  
  
  
  
  
  
  
  
  
  § SUPPLEMENTARY INFORMATION
  
  
  
  
  A. Calculation details 
  
  In this section we provide details about the calculations discussed in the main text, and explain how parameter values were determined. First, we consider the steady-state calculations presented in Fig. 1(b) of the main text. We express Eq. 1 of the main text as two coupled ordinary differential equations by defining w=U ∫_0^t ds K(t-s)(|α(s)|^2-1). Then
  
      iα̇(t)    = (-Δ-iΓ/2+w(t) )α(t)+i√(κ_1)F
      ẇ(t)    = { U[|α(t)|^2-1]-w(t)}/τ
  
  The steady-state solutions are then obtained by requiring α̇=ẇ=0. The stability of the steady states is assessed by analyzing the spectrum of small fluctuations around the steady-state solution.
  
  For the dynamic simulations based on xSPDE, we use a fourth-order Runge-Kutta algorithm and set a time increment Δ t=Γ^-1/30. The stochastic terms ξ_1(t) and ξ_2(t)  each have zero mean [i.e., ⟨ξ_1 ⟩ = ⟨ξ_2 ⟩ = 0], and they are delta-correlated with unit variance [i.e., ⟨ξ_1(t) ξ_1(t+t') ⟩=  ⟨ξ_2(t) ξ_2(t+t') ⟩ =  δ(t')]. Moreover,  ξ_1(t) and ξ_2(t)  are mutually uncorrelated.
  
  The parameters in Eq. <ref> were set to model our experiments as follows. First note that Δ and D are control parameters, fixed in experiments by the cavity length and the peak-to-peak voltage in the waveform generators driving the modulators. Γ is fixed  via measurements of the resonance linewidth at low power. The value of F only matters in relation to the critical amplitude for bistability F_c.  As explained in Ref. Geng_PRL2020, by  performing measurements at several powers we identify (F/F_c)^2 = 1 as the minimum power for which bistability can be observed. The thermal relaxation time τ, retrieved experimentally as shown in  Supplementary Section B, is far too long to be implemented in our simulations given the memory limitations of our computer. However, this is not a problem because all dynamical effects scale with the ratio T_mod/τ <cit.>. Therefore, by fixing this ratio to T_mod/τ=10^3, as in our experiments, our numerical simulations reproduce our experiments. Finally, the only `free' parameter is U. The determination of this parameter is not trivial, since it requires precise knowledge of the intracavity photon number. However, in the absence of quantum effects (as in our experiments), the physics is entirely determined by the ratio  U|α|^2/Γ. Changing the value of U simply changes the intracavity intensity and the value of F for which bistability is observed, but the spectral lineshape and the PDF remain unchanged provided that F/F_c and Δ/Γ remain unchanged.
  
  The PDFs in Fig. 1(c) and (d) of the main text were constructed as follows. For the selected F and Δ/Γ, we first calculated steady-state solutions and assessed their stability. Next we performed 10^4 stochastic simulations with the value of α(t=0) Gaussian-distributed around the unstable state. The standard deviation of this distribution was 10% of the distance in phase space between the stable states. Finally, the PDFs were constructed on basis of all 10^4 trajectories.
  
  
  
  B. Thermal relaxation time 
  
  
  Here we present a measurement of the thermal relaxation time in our oil-filled microcavity. Our measurement scheme consists of swiftly driving the cavity into the upper branch of the bistability, and subsequently measuring the relaxation time to a steady state. To this end, we measure the cavity transmission while modulating the laser power in a step-like fashion using a chopper; the cavity length is fixed during this measurement. Figure <ref>(a) shows the chopped input power as a red curve, and the  corresponding cavity transmission as a black curve. When the chopper blocks the laser light, the cavity transmission is negligible. Once the laser passes through an opening in the chopper, the transmission first rises to a transient low-intensity state. Next, there is a build-up phase during which the temperature of the oil, and hence the nonlinearity strength, increases. Once the nonlinearity has built up, the transmission suddenly increases. This leads to a sharp overshoot peak, which is followed by thermal relaxation to a high-intensity steady-state.
  
  
  
  In Fig. <ref>(b) we reproduce the experimentally observed behavior through  numerical simulations based on Eq. 1 in the main manuscript. In the simulations,  we modulate the driving amplitude F in a step-like fashion as done by the chopper in experiments. We performed similar simulations for many values of the model parameters, all in the nonlinear regime. We found that the full width at half maximum (FWHM) of the overshoot is always ∼τ/2 with τ the memory time of the nonlinearity. Therefore, by analyzing the FWHM of the overshoot in experiments we estimate a memory (thermal relaxation) time τ=10 μs for our optical microcavity filled with macadamia oil. This value of τ is close to the one deduced from dynamic hysteresis measurements in a microcavity filled with olive oil, which was found to be τ=16 μs in Ref. Geng_PRL2020.  
  
  
  
  
  
  
  
  C. Switching behavior 
  
  
  Figure <ref> shows simulated trajectories of |α|^2 for constant F and three different Δ/Γ, all in the bistability. In all three cases, we observe noise-induced switching between high and low intensity metastable states at random times. The simulations also show how the mean residence times in the metastable states depend on  Δ/Γ. The cavity is increasingly biased towards the low intensity state as Δ/Γ increases. This, in turn, changes the shape of the PDFs as shown in Figs. <ref>(c) and (d) of the main text. 
  
  
  
  
  
  
  
  
  
  D. Noise properties 
  
  
  Here we present a characterization of the noise imprinted on the driving laser by the electro-optic modulators (EOMs), to which we supply a peak-to-peak voltage V_pp. To this end, we measured the laser power directly after the second EOM while adding noise of variable strength. Figure <ref>(a) shows the noise spectrum for two different V_pp. The measurement time was 50 seconds, and the sampling rate was 1 MS/s. We observe an approximately flat spectrum corresponding to white noise, and a greater noise floor for greater V_pp. Figure <ref>(b) shows the  standard deviation of the noise as a function of the V_pp provided to the EOMs by the waveform generators plus amplifiers. The measurement time was 1 second, and the sampling rate was 10 MS/s. We observe a linear increase of the standard deviation of the noise with V_pp.  
  
  
  
  
  
  
  
  E. Decrease in output power 
  
  Here we explain why the SNR corresponding to the trajectory in Fig. 2(d) of the main text is lower than the SNR corresponding to the trajectory in Fig. 2(c). In Fig. <ref> we plot the power spectral density PSD corresponding to the  measurements in Figs. 2(c) and (d). The PSDs show peaks corresponding to the first six harmonics. The height of these peaks is lower for the transmitted signal at V_pp=200 V, most notably for the third and sixth harmonics. This implies a lower total signal power at V_pp=200 V. Hence,  although Figs. 2(c) and (d), show  a similar noise floor for V_pp=70 V and V_pp=200 V, the SNR is lower for V_PP=200 V.
  
  
  
  
  
  
  
  
  F. Detection of switching events 
  
  To compute the average number of switching events per modulation cycle n_avg, we used the standard Matlab function `findchangepts'. Given an N-point time series { x_1,...,x_N }, the function determines the K change points { x_k} minimizing the cost function
  
      J(K)=∑_r=0^K-1∑_i=k_r^k_r+1-1δ[x_i;χ({ x_k_r,...,x_k_r+1-1})]+β K.
  
  Here, k_r is the index of the r^th change point. δ gives the difference between a point x_i∈{ x_k_r,...,x_k_r+1-1} and the mean χ of the subtrajectory { x_k_r,...,x_k_r+1-1}. β is a fixed penalty which is added for each change point.
  
  
  
  
  The number of detected change points generally depends on the value of β. If β is too small (large), the algorithm returns too many (few) change points. Therefore, the right value of β needs to be determined for each data set. To avoid false or missing change points related to the value of β, we implemented the following algorithm. First we set a relatively small value of β and get all true change points and extra ones. Next, we compute the mean and standard deviation of each subtrajectory. If the difference in the mean of two consecutive subtrajectories is less than the sum of their standard deviations, we remove the change point that separates the two subtrajectories. This process is repeated until no change points are removed anymore.
  
  
  Figure <ref> illustrates the analysis of three representative trajectories using our algorithm. The mean and standard deviation of each subtrajectory is indicated by a red line and shaded box, respectively. Figures <ref>(a),  <ref>(b), and  <ref>(c) show trajectories for which n_avg<2, n_avg∼ 2 and  n_avg>2, respectively. In Fig. <ref>(c) we show fewer modulation periods to make the multiple switching events per cycle visible in the plot. Overall, the results in Fig. <ref> demonstrate that our algorithm successfully detects all true change points and does not give any false change points. 
  
  
  
  
  
  
  G. Memory-enhanced stochastic resonance bandwidth 
  
  Figure <ref> in the main text shows the emergence of a `plateau' at n_avg≈2 as τ increases. That `plateau'  represents an enlargement of the frequency range in which stochastic resonance (SR) can be achieved, i.e., the SR bandwidth. Here we show that the SR bandwidth  increases linearly with τ. To this end, in  Fig. <ref> we plot the range of modulation periods δ T_mod for which n_avg∈[2-ϵ,  2+ϵ]. Results of numerical simulations (circles) are fitted by a line, demonstrating that δ T_mod increases linearly with the memory time τ. We confirm that the relation between δ T_mod and τ remains linear for any ϵ≪ 2.
  
  
  
  
  
  
  