Stochastic Syncing in Sinusoidally Driven Atomic Orbital Memory

Stochastically fluctuating multiwell systems are a promising route toward physical implementations of energy-based machine learning and neuromorphic hardware. One of the challenges is finding tunable material platforms that exhibit such multiwell behavior and understanding how complex dynamic input signals influence their stochastic response. One such platform is the recently discovered atomic Boltzmann machine, where each stochastic unit is represented by a binary orbital memory state of an individual atom. Here, we investigate the stochastic response of binary orbital memory states to sinusoidal input voltages. Using scanning tunneling microscopy, we investigated orbital memory derived from individual Fe and Co atoms on black phosphorus. We quantify the state residence times as a function of various input parameters such as frequency, amplitude, and offset voltage. The state residence times for both species, when driven by a sinusoidal signal, exhibit synchronization that can be quantitatively modeled by a Poisson process based on the switching rates in the absence of a sinusoidal signal. For individual Fe atoms, we also observe a frequency-dependent response of the state favorability, which can be tuned by the input parameters. In contrast to Fe, there is no significant frequency dependence in the state favorability for individual Co atoms. Based on the Poisson model, the difference in the response of the state favorability can be traced to the difference in the voltage-dependent switching rates of the two different species. This platform provides a tunable way to induce population changes in stochastic systems and provides a foundation toward understanding driven stochastic multiwell systems.


Introduction
The development of neuromorphic hardware that is inherently adaptive and exhibits self-learning requires a deeper understanding of materials that show multi-well behavior (1).One route toward this end is to create materials systems with tunable multi-well energy landscapes, where local minima represent information (2).Such systems are mimicked by energy-based models in machine learning, for instance the Hopfield model or its stochastic analogue, the Boltzmann machine, where the energy landscape can be described by an Ising Hamiltonian (3,4).In these models, each spin represents an individual neuron and the coupling is represented by a synaptic, memory-bearing weight (5).To this end, there have been a number of tunable coupled spin arrays derived from atomic spins on surfaces, based on scanning tunneling microscopy (6)(7)(8)(9).While the coupling in such spin systems can be tuned, these platforms are often limited to double-well energy landscapes and thus to binary configurations (10,11).However for interesting neuromorphic computing, it was proposed that atomic spin ensembles require competing long range interactions (12), linked to the concepts of spin glasses (13)(14)(15)(16).Recently, it was shown that arrays of coupled binary orbital memory states emulate an atomicscale Boltzmann machine, by exploiting their dynamical properties (17,18).The multi-state dynamics emerging from the coupling between stochastic atoms directly maps onto the Hamiltonian of the Boltzmann machine, with tunable stochastic weights.A local gate voltage can be used to influence the system dynamics, and thus tune the stochastic weights.(17,19).
Fundamentally, it is still unclear how such stochastic multi-well systems respond to a dynamical drive signal.For an individual binary spin in the stochastic regime, the response to a periodic stimulus is often described in terms of phase-synchronization and stochastic resonance (SR) (20)(21)(22).Generally, SR manifests itself in nonlinear dynamical systems as an enhanced response to a weak input signal due to the presence of intermediate noise, compared to the response for low and high noise amplitudes.In systems with an activation threshold energy, SR arises when a weak, noisy signal periodically exceeds the threshold, thereby inducing dynamics.This is referred to as threshold SR (21).However, in Arrhenius-type double-well systems where the switching rates are always finite, similar phenomena can be observed due to a periodic modulation of the switching rates, known as dynamical SR (20,21).Recently, dynamical SR has also been reported in atomic-scale systems, where the dynamics of the system originates from quantum tunneling processes (23,24).It remains both theoretically and experimentally unclear how multi-state systems, where multiple Markov rates are present, respond to a periodic drive (25).Experimentally, coupled arrays of orbital memory present a promising route toward exploring stochastic response in the multi-well limit.However, since the dynamical complexity rapidly increases with the number of stochastic units (17), it is essential to first gain a fundamental understanding of how synchronization and frequency-induced population coding can be achieved with AC signals for an individual atom.
Here, we quantify the stochastic response of orbital memory derived from individual Fe and Co atoms on black phosphorus (BP) to sinusoidal input voltages.Using scanning tunneling microscopy (STM), we apply a chosen DC bias voltage and AC sinusoidal voltage and study the stochastic response of the orbital states of individual Fe and Co atoms as a function of frequency and amplitude.For individual Fe atoms, we observe a frequency-dependent change in the state-dependent residence shape of their charge density (Fig. 1A,B).We label each of these two distinguishable states as high/low for each atomic species.Below a threshold DC voltage, each orbital state remains indefinitely stable allowing for state identification.
Above an applied DC voltage (VDC), there is a finite probability of switching between the two orbital configurations for both individual Co and Fe atoms (18,27).The stochastic switching is recorded by measuring the tunneling current (It) as a function of time for a constant tip-sample separation and lateral tip position.Before each measurement, the tip is first stabilized on the BP substrate to a predefined tip-sample separation.Fig. 1C exemplifies the stochastic switching for an individual Fe atom (Felow/Fehigh).By recording a statistical number of stochastic switching events (> 1000 events), we extract the average residence time (̅ We selectively choose parameters to yield switching rates within the time resolution of the experiment (see Fig. S2) and avoid the strongly asymmetric regime for these experiments.
In addition to measuring the stochastic switching for a given  , we also measured the stochastic switching in response to an additional sinusoidal voltage  | |/2 sin(2π) added to  (Fig. S2).| | is subsequently defined as the peak-to-peak amplitude.The resultant values of It are modulated, as shown in Fig. 1D, for the case where | | = 40 mV and  = 2 Hz.The number of switching events is highest when | +  | is maximized, indicating that the switching probability of each state  is sensitive to the phase  of  .Moreover, the switching probability remains finite at any point of the waveform.This means that we consider synchronization of a thermodynamic double-well system to a sinusoidal input signal (20,21).
In Fig. 2A,B S9).Therefore, the change in   represents the time-averaged frequency response of the system.In contrast to the significant frequency response of both ̅ and  for an Fe atom, there is no significant frequency response in the various extracted variables for a Co atom, as plotted in Fig. 2E,F.We explain this difference below.
In order to understand the difference in the frequency response between Fe and Co atoms, we first quantified  / as a function of  without an additional  signal (Fig. 3A,B).The general dependent trend for both Co and Fe agrees with previous publications (18,27), where  / decreases for an increasing magnitude of | |.Phenomenologically, we can describe  low/high ( ) in the voltage range of interest using an exponential dependence, as shown with the fits in Fig. 3A,D.In addition, we also extracted the dependence of A on  and find that A( ) in the studied range strongly depends on the value of  for the case of Fe, but is nearly independent of  for Co.
Accordingly, we can relate the slope of A, which we define as    .We refer to  as the voltage susceptibility.From Fig. 3A,D, we observe that | Fe | ≫ | Co |, which, as we show below, is directly linked to the frequency response.Besides ̅ and A, the switching characteristics can be described in terms of the state occupations, defined as  low =  /   and  high =  /   .
We subsequently model the stochastic response of each atom type to the applied  .We assume that  in absence of  is described by a Poisson process with an exponential probability distribution.
For  0, the distribution of  can be described by average transition rates  low→high , which are extracted from Fig. 3A,D.Subsequently, we simulate the stochastic dynamics in response to a non-zero value of  , considering a Poisson point process with periodically changing , as introduced in ref. (28).Since A() is a time-averaged quantity, it can be simulated by solving the associated rate-equations (see Supplementary information).We find that the simulated A() (solid line) is in good agreement with the experimental data in Fig. 2D,F.To model the  histograms, stochastic data can be generated using the experimentally determined  low→high  and  high→low  with the aforementioned assumptions (see Materials and Methods).This approach allows us to model the  histograms (Fig. 2A,B), as well as  low and  high as a function of the modulation phase .
Based on this modeling, we considered the average  low ,  high ,  low→high , and  high→low within a period of the waveform.In Fig. 3B  Each of the cases shows a similarly sized change of A() with the previously observed sigmoidal shape.This is expected, given the nearly constant value of  in the voltage range of interest.
However, the sign change of A( ) in Fig. 4A introduces a vertical offset for the three cases.
Additionally, a shift of the A() curves in Fig. 4B along the frequency axis with varying  can be seen.This reflects the lower ̅ , of the system at higher  .
To further illustrate the tunability of the frequency response in the state occupation, we measured A()

Discussion
In conclusion, we studied the stochastic response of the orbital memory states derived from individual Fe and Co atoms to a sinusoidal drive.For individual Fe and Co atoms, we observed phasesynchronization for residence times that coincide with the inverse of the applied drive frequencies.
This is manifested by a deviation from a Poisson distribution, compared to the DC limit.Notably, this effect can be observed for frequencies much higher than the inverse average residence time.Based on stochastic modeling, we can relate the amplitude of the synchronization effect to the evolution of the switching rates as a function of bias voltage.As a second effect, we observed a change in the state favorability, namely the asymmetry, as a function of frequency.In contrast to the response of individual Fe atoms, this effect is absent for individual Co atoms.Based on modeling and measuring the state favorability and lifetimes as a function of applied DC voltage, in the absence of a sinusoidal drive, we found that both the synchronization effect and the frequency-dependent asymmetry are derived from the underlying voltage-dependent transition rates.This is an important consideration, when using a sinusoidal drive and the resultant change in asymmetry, as a measure of the mean residence time of a two-state system, when the telegraph processes cannot be resolved.In other words, a time-averaged frequency response can only be observed if the modulated switching rates exhibit a significant asymmetry change in the modulation range of interest.If no change in asymmetry with frequency is observed, the system may still show dynamics and synchronization in that frequency range.We showed that the asymmetry and residence times of single-atom orbital memory can be controlled by tuning the tip position, VDC and atom type.This in turn enables tuning the frequency response of orbital memory to an AC stimulus.The characteristic frequency is dictated by the transition rates and can be tuned by the injection current It (27).The magnitude of the frequency response depends on  in the modulation range and can be tuned by choice of the atom type, tip position, VDC, and VAC.This is advantageous when probing multi-well behavior, as for example seen in the atomic Boltzmann machine.As coupled orbital memory exhibit multi-well behavior (17), these studies provide a starting basis for future studies where the stochastic response of multi-state systems to complex waveforms can be interrogated.We expect coupling atoms that switch stochastically at different time scales to respond in multiple frequency regimes.

Simulating the state occupations for AC voltage input
The transition rates in the orbital memory are strongly bias dependent (18,27).Accordingly, VAC modulates the transition rates in the bistable orbital memory periodically.In order to simulate the state occupation of the orbital memory in the AC regime, we developed a model of a Poisson point process with transition rates changing periodically in time.The stepwise approach of simulating the system dynamics and time-averaged state occupations from this stochastic model will be described in the following.
First, we derive the voltage dependence of the transition rates  → between the binary orbital memory states.Fig. 4A exemplarily shows the mean state lifetimes of a single Fe atom for varying DC bias VDC.The transition rates are derived from the mean value of the measured state lifetimes:  → , ( = high/low).We perform a weighed exponential fit in order to derive a functional dependence of the transition rates, on the applied bias voltage (see Figure S4A): In the fitting procedure, the values of  → for each bias value are weighed with the inverse of their squared standard deviation .

Simulations of stochastic real-time data for AC bias input
The DC residence time statistics are accurately described by exponential distributions, corresponding to homogeneous Poisson point processes.The switching dynamics with fixed rates (constant voltage) can therefore be simulated as a repetition of two consecutive random steps, where exponentially distributed waiting times are generated with the state-dependent rates.
The non-homogeneous case, corresponding to a finite VAC, is in principle not much more complex.A Poisson process with a time-dependent rate can be simulated by noticing that the integrated rate      is a random variable that is exponentially distributed according to   exp  .
The idea is then to generate a random value for  and invert   to get a random sample of the residence time.As explained in the previous section, the switching rates   are fitted to an exponential of the voltage signal.For the AC case, the integrated rate function is inverted numerically via a lookup table.The fact that the rate is a periodic function of time is exploited for efficiency.
The synthetic stochastic data reproduces the experimental data for both Fe and Co atoms, as detailed in Figs.S5 and S6.

S1. Simulations of state occupation for AC bias input
Fig. S4B shows the result of a simulation for an Fe atom on BP under experimental conditions as in Fig. 2A of the main paper with an applied AC frequency of f = 10 Hz.It can be seen that the expectation value of  quickly converges to a steady state where it oscillates in time with frequency f.
In order to deduce the time averaged state asymmetry    shown in Fig. 2D, we calculate the average state occupations (broken lines) within two full periods of the AC signal in the equilibrium region (shaded area).

S2. Simulations of stochastic real-time data for AC bias input
Fig. S5 and S6 show that the synthetic stochastic data (black solid lines) reproduces the synchronization observed for both, the Fe and the Co atom (Fig. S5A, S6A), as well as the different frequency response in the phase-resolved state occupations (Fig. S5B, S6B).Furthermore, it can be seen that the simulation, which is solely based on DC switching rates, reproduces the experimentally measured phase-resolved switching rates very well, underlining the feasibility of the description as a Poisson process with varying transition rates.

S3. Data statistics
The tunability of the switching rates of the orbital memory states can provide a lot of variability to the frequency response of the system, as exemplified in Fig. 4. In Fig. S7, we present additional data sets on different Fe atoms, taken with identical measurement conditions as Fig. 2 and 3. We note that these measurements were not taken with an identical tip apex.In each case, small variations in the DC switching rates (A-C) impact the frequency dependent residence times (D-F), but the overall trend is consistent between all data sets.
Fig. 3D.As a result, the frequency-induced change of A() (Fig. 2E,F) is zero for Co atoms.Based on the comparison between the dynamical properties of Fe and Co atoms and supported by stochastic modeling, we can identify the requirements to observe a time-averaged frequency response and link it to the measured  low/high ( ).Namely, the frequency-induced changes of A() can only be observed if  low and  high are modulated for  ≪  * .This can only be obtained when  low→high  and  high→low  evolve such that A() exhibits significant changes in the voltage range of  | |/2 (i.e. must be finite).Furthermore, we can expect the change in A() to occur for  * 1/̅ av .To generalize the relation between  low/high ( | 0) and the resulting frequency response, we studied the stochastic response of individual Fe atoms for different input signals.Firstly, we apply an input voltage with a fixed | | = 40 mV around three different values of  , as illustrated in Fig. 4A.
variable | |.We probed the frequency response for | | = 20, 30, 40 and 50 mV, while oscillating around the same bias point of  = -510 mV, as indicated in Fig.4C.We find that the total change of A() keeps increasing as | | increases (Fig.4D).Interestingly, A( ≪  * ) saturates around 0 for each | |.These observations suggest that the system adopts an average A with an effective bias   .The saturated A( ≫  * indicates that the application of  mimics the application of a  , where  shifts towards shorter  low/high with increasing .The relation between the  low/high ( ) and the time-averaged frequency response was statistically confirmed by measuring the change of A() for multiple Fe atoms using similar conditions (see Fig.S7).Additionally, we could obtain varying trends of A() by modifying the rate evolution with the tip position,  and the atom type (see Fig.S8).Each of these data sets could be reproduced by simulating A() based on a rate-equation model, as exemplified in Fig.4B,D.
distribution is described by an exponential function with decay constant ⁄ , where  ⁄ is the average residence time ⁄ ∑  , ⁄ .This exponential distribution is the characteristic of a Poisson point process.To quantify uncertainties in the measurement of , we define a standard deviation  of  by averaging random sets of 500  , ⁄ acquired under equivalent experimental conditions and calculating the standard deviation of these averaged values.Error bars throughout the paper represent 1.

Fig. S4B shows
Fig.S4Bshows the time evolution of the state occupations obtained with this model.

Fig. 1 .
Fig. 1.Stochastically switching orbital memory states of Fe and Co atoms on black phosphorus.(A,B) Constant-current STM images of (A) an individual Fe atom and (B) an individual Co atom adsorbed in the hollow site of BP, imaged for both orbital memory states (low/high) (VDC = -400 mV, (A) It = 10 pA and (B) 50 pA, scale bar = 1 nm).(C,D) Telegraph noise recorded at the center of an Fe atom in constant height for: (C) VAC = 0 mV, VDC = -500 mV and (D) VAC = 40 mV, VDC = -500 mV,  = 2 Hz.The tip was stabilized on the substrate at VDC = -400 mV and It = 100 pA, before the feedback loop was opened.

Fig. 2 .
Fig. 2. Frequency dependence of mean residence times.(A,B) Histogram plots of the detected  for (A) Felow (red) and Fehigh (blue), and (B) Colow (red) and Cohigh (blue) for two frequencies, measured with (A) VDC = -500 mV and (B) VDC = 480 mV with an amplitude of |VAC| = 40 mV (peak-to-peak).The histograms present ≥ 28 extracted residence times per bin, i.e. per data point (plotted with an artificial vertical offset for each state and frequency).Histograms of synthesized stochastic data based on a Poisson point process are shown in black.(C,D) Time-averaged residence time  low/high and deduced state asymmetry  of Felow (red) and Fehigh (blue) as a function of the frequency of VAC.(E,F) Timeaveraged residence time  low/high and deduced state asymmetry  of Colow (red) and Cohigh (blue) as a function of the frequency of VAC.Solid lines in (D,F) represent the simulated frequency response, based on rate-equations (see Materials and Methods).The tip was stabilized on the substrate at VDC = -400 mV and (A,C,D) It = 100 pA or (B,E,F) It = 30 pA.

Fig. 3 .
Fig. 3. Voltage dependence of state asymmetry.(A) Average residence times  low/high for Felow (red) and Fehigh (blue) and extracted asymmetry  (lower panel) as a function of bias voltage  DC , measured with  AC = 0 mV on top of an individual Fe atom (see inset).(B) Phase-resolved state occupation  low (simulation: solid black line) and  high (simulation: broken black line) and (C) conditional switching rates  low→high and  high→low for different frequencies (artificially offset), averaged over many periods.| AC | is indicated by the gray shaded area and the phase φ is defined at the top of (A).Tip height was stabilized on the substrate at VDC = -400 mV and It = 100 pA.(D) Average residence times  low/high for Colow (red) and Cohigh (blue) and extracted asymmetry  (lower panel) as a function of bias voltage  DC , measured with  AC = 0 mV next to an individual Co atom (see inset).(E) Phase-resolved state occupation  low (simulation: solid black line) and  high (simulation: solid black line) and (F) conditional switching rates  low→high and  high→low for different frequencies (artificially offset), averaged over many periods.| AC | is indicated by the gray shaded area and the phase φ is defined at the top of (D).Tip height was stabilized on the substrate at VDC = -400 mV and It = 30 pA.

Fig. 4 .
Fig. 4. Frequency response in dependence of DC offset and AC amplitude.(A) Average residence times  low/high for Felow (red) and Fehigh (blue), and extracted asymmetry (lower panel) as a function of bias voltage VDC, with |VAC| = 0 mV.(B) Average state asymmetry  as a function of frequency with a constant |VAC| = 40 mV and three different VDC values, as indicated in (A) with the yellow lines.(C) Average residence times  low/high for Felow (red) and Fehigh (blue), and extracted asymmetry (lower panel) as a function of bias voltage VDC, measured with  AC = 0 mV.(D) Average state asymmetry  as a function of frequency, measured with a constant |VDC| = -510 mV and four different |VAC| values, as indicated in (C) with the gray shaded areas.Solid lines in (B) and (D) represent the simulated frequency response based on the rate-equation model and the exponential fits (solid lines in (A) and (C)).Tip height was stabilized on the substrate at VDC = -400 mV and It = 100 pA.

Further variation was found
by using different values of VDC and It, as shown in the additional data sets in Fig. S8.Moreover, Fig. S8F demonstrates that an individual Co atom can exhibit a (small) frequency response when  is finite (compared to Fig. 2 and 3 in the manuscript).

Figure S1 :
Figure S1: Constant-current STM image of the surface of black phosphorus with Co and Fe atoms adsorbed on the top binding site after deposition.To obtain well-isolated Fe and Co atoms, atoms can be laterally manipulated along the [010] direction and contaminated species (18) as well as BP vacancies (29) are avoided (VDC = -400 mV, It = 10 pA, scale bar = 5 nm).

Figure S2 :
Figure S2: (A) Sketch of the experimental setup to measure with and without VAC.(B) Example of a high-resolution time trace of VDC+AC on Channel 1 with a frequency f = 5000 Hz and It on Channel 2, recorded with a sampling rate of 100000 samples/s for each channel.Note that the AC modulation has the correct amplitude (|VAC| = 40 mV) and that no significant phase delay is detected between the two channels at f = 5000 Hz.

Figure S3 :
Figure S3: (A) Current trace of stochastic switching of a single Fe atom on BP measured with VDC = -500 mV (VAC = 0 mV).Single residence times  , ⁄ for the low and high state are exemplarily indicated.The tip height was stabilized on the substrate at VDC = -400 mV and It = 100 pA.(B) Residence time histogram for Felow (red) and Fehigh (blue).The plots present a total of 28589 residence times, separated into 25 bins per state.Solid lines represent exponential functions with ̅ , where amplitude parameter A is fitted.

Figure S4 :
Figure S4: (A) DC transition rates  → with exponential fits and (B) simulated state occupation as a function of time for an Fe atom on BP, according to the experimental conditions in Fig. 2C: VDC = -500 mV, |VAC| = 40 mV (peak-to-peak), tip stabilized on the substrate at VDC = -400 mV and It = 100 pA.Here, the state occupation in the case of f = 10 Hz is exemplarily shown.The simulation was initialized at t = 0 in   .The time-averaged occupations (broken lines) in equilibrium conditions are averaged over exactly two periods of the AC signal (shaded area).

Figure S5 :
Figure S5: (A) Histograms of the residence times of Felow (red) and Fehigh (blue) for different frequencies (artificially offset).(B) Phase-resolved state occupation and (C) switching rate for different frequencies (artificially offset), averaged over many periods.The inset in (B) shows one period of the oscillatory bias voltage and defines the phase φ on the x-axis.Time-averaged properties of this data set are presented in Fig. 2D.The tip height was stabilized on the substrate at VDC = -400 mV and It = 100 pA.Solid lines represent the results of stochastic simulations based on the DC switching rates  →  .

Figure S6 :
Figure S6: (A) Histograms of the residence times of Colow (red) and Cohigh (blue) for different frequencies (artificially offset).(B) Phase-resolved state occupation and (C) switching rate for different frequencies (artificially offset), averaged over many periods.The inset in (B) shows one period of the oscillatory bias voltage and defines the phase φ on the x-axis.Time-averaged properties of this data set are presented in Fig. 2D.The tip height was stabilized on the substrate at VDC = -400 mV and It = 30 pA.Solid lines represent the results of stochastic simulations based on the DC switching rates  →  .

Figure S7 :
Figure S7: (A-C) Average residence times and extracted asymmetry, measured with VAC = 0 mV for different Fe atoms and tips, and (D-F) the corresponding frequency response of the average residence times.The data sets are taken with measurement conditions comparable to the data set presented in Fig. 2B; each data set was measured with |VAC| = 40 mV and VDC = -500 mV, and the tip height was stabilized on the substrate at VDC = -400 mV and It = 100 pA.

Figure S8 :
Figure S8: (A-C) Average residence times and extracted asymmetry (VAC = 0 mV) and (D-F) the corresponding frequency response of the average residence times.These data sets were measured (A,D) on top of an individual Fe atom using VDC = -470 mV, (B,E) on top of an individual Fe atom using VDC = -485 mV, and (C,F) next to an individual Co atom using VDC = 580 mV.Each data set was measured with |VAC| = 40 mV and with a different tip and atom.The tip height was stabilized on the substrate at VDC = -400 mV and (A,D) It = 130 pA, (B,E) It = 100 pA (C,F) It = 70 pA.

Figure S9 :
Figure S9: Comparison between the frequency-induced change in (A) the state occupation and (B)the time-averaged current.Data set corresponds to the data presented in Fig.2(A,C,D) and Fig.3(A- , we illustrate histograms of the state-resolved  for a set of frequencies, for both atom types (| | = 40 mV).Each data point represents the number of detected  's within a given time interval.The histograms contain the exponential decay behavior, expected for a Poisson process.In (28) plot the phase-resolved  low  (red) and  high  (blue), for various frequencies.Each data point is extracted by subdividing one period of the waveform into 100 phase segments, calculating  low ,  high ,  low→high , and  high→low for each segment and averaging over many periods.For  ≪  * ,  low and  high are clearly modulated, directly following the changes of A  in  low and  high becomes less pronounced and the maximum of  high shifts to larger , indicating that  low and  high exhibit a delayed response in time to  .For  ≫  * ,  low and  high show a flat response as a function of , converging to a time-averaged A. We find that the evolution of  low and  high as a function of  is reproduced.The phase-resolved conditional transition rates,  low→high  and  high→low  , can be determined from the switching rates within each phase segment, divided by the occupation of the corresponding state in that phase segment(28).As shown in Fig.3C, most of the switches occur around  /2, as expected from  / in the voltage range of  | |/2, and  low→high  and  high→low  are approximately constant as a function of f.The phase-resolved analysis for individual Co atoms reveals that  low→high  and  high→low  (Fig.3F) are clearly modulated, while  low and  high (Fig.3E) shows negligible changes within one period of the waveform.
Fig. 3A.For instance, at  3/2  and  = 1 Hz, the Fehigh state is more favorable, partially compensating for the larger  low in the first half of the waveform.With increasing , the modulation of This can be understood as a direct consequence of the parallel evolution of  and  , namely  0 for the two Co states in The choice of an exponential function is purely phenomenological.We constrain the fit on the bias interval [VDC -VAC/2, VDC + VAC/2] that is relevant for simulating the AC response.Following the exponential bias dependence, we model the time dependent transition rates →  .