Stochastic dynamics of an electron in a Penning trap: phase flips correlated with amplitude collapses and revivals
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Stochastic dynamics of an electron in a Penningtrap: phase flips correlated with amplitudecollapses and revivals
S. Brouard and J. PlataJune 24, 2018
Departamento de Física Fundamental II, Universidad de La Laguna,La Laguna E38204, Tenerife, Spain.
Abstract
We study the effect of noise on the axial mode of an electron in a Pen-ning trap under parametric-resonance conditions. Our approach, basedon the application of averaging techniques to the description of the dy-namics, provides an understanding of the random phase flips detected inrecent experiments. The observed correlation between the phase jumpsand the amplitude collapses is explained. Moreover, we discuss the actualrelevance of noise color to the identified phase-switching mechanism. Ourapproach is then generalized to analyze the persistence of the stochasticphase flips in the dynamics of a cloud of N electrons. In particular, wecharacterize the detected scaling of the phase-jump rate with the numberof electrons. The research on electron traps has opened the way to significant advances infields ranging from Atomic Physics to Metrology [1]. For instance, the appli-cation of the trapping techniques has been crucial for achievements like thegeneration of antimatter atoms [2] or the realization of precision tests on funda-mental constants [3] Moreover, trapped electrons provide a controllable testingground for a variety of physical behaviors, predicted or experimentally identi-fied in other areas. Actually, the possibility of controlling the trapping setup,in particular, of varying its components, can allow the systematic characteriza-tion of different effects via their realization under well-defined conditions andin regimes unexplored in other contexts. In this way, problems like the emer-gence of nontrivial effects of noise [4], the preparation of Fock states [5], the1ppearance of squeezing in quantum-dissipation processes in nonlinear oscil-lators [6, 7, 8], or the implementation of proposals for quantum-informationalgorithms [9, 10, 11, 12] have been analyzed with different variations of thebasic trapping setup. Here, we focus on a novel effect detected in recent exper-iments on electrons in a Penning trap [4]. Namely, under parametric-resonanceconditions, the axial mode of a one-electron system was observed to present ran-dom amplitude collapses correlated with phase flips. That behavior was tracedto noise rooted in different elements of the practical arrangement. Indeed, byadding fluctuations in a controlled way, the dependence of the phase-jump rateon the noise strength was characterized. Remarkably, for increasing noise inten-sities, the correlation between the amplitude collapses and the phase jumps wasfound to disappear. The study of the persistence of those effects in the dynam-ics of a cloud of N electrons revealed a nontrivial behavior, in particular, theattenuation and eventual disappearance of the stochastic phase switching as thenumber of electrons was increased. Despite the advances in the characterizationof the observed dynamics [4, 13], a satisfactory explanation of the underlayingphysical mechanisms is still needed, as stressed in Ref. [4]. For example, theactual relevance of colored noise to the emergence of some of the detected effectsis an open question. Here, we present a description of the dynamics based onaveraging techniques applicable to stochastic systems. From our approach, theorigin of the experimental findings is uncovered and the elements of the systemwhich are essential for the appearance of the observed behavior are identified.Furthermore, our semi-analytical characterization of the dynamics provides uswith some clues to controlling the system response to the fluctuations. Thedirect implications of the study to the advances in the techniques of confine-ment and stabilization are evident. Additionally, because of the fundamentalcharacter of the physics involved, the analysis can be relevant to different con-texts, where conditions similar to those realized in the trapping scenario can beimplemented [14, 15].The outline of the paper is as follows. In Sec. II, we present our modelfor the stochastic dynamics of the axial mode of an electron in a Penning trap.Through the application of the averaging methods of Bogoliubov, Krylov, andStratonovich [16, 17], we derive an effective description in terms of a systemof stochastic differential equations for the amplitude and the phase. In Sec.III, the validity of our approach is confirmed through the simulation of themain experimental findings. Moreover, an analysis of the physical mechanismsresponsible for the observed behavior is presented. Sec. IV contains the studyof the persistence of the stochastic phase switching in the dynamics of a cloudof N electrons. Finally, some general conclusions are summarized in Sec. V. An electron in a Penning trap is usually described in terms of three coupledmodes (magnetron, axial, and cyclotron) with widely different time scales [1].Here, we concentrate on the axial coordinate z . In general, because of the inter-2ode coupling, the axial dynamics can be quite complex: a variety of behaviorscan emerge depending on the considered regime of experimental parameters.However, under standard conditions, different approximations can be made andthe description simplifies considerably. Namely, the (slow) magnetron motioncan be adiabatically treated and its influence on the dynamics can be reducedby sideband cooling [4]. Additionally, by damping the (fast) cyclotron modeto its fundamental state, its effect on the axial coordinate is minimized [4]. Intypical realizations, z is coupled to a measuring external circuit, which intro-duces resistive damping and noise in the mode; moreover, the trap potentialthat is usually applied is approximately harmonic. Hence, in the basic scheme, z corresponds to a dissipative harmonic oscillator which can be described clas-sically. Axial outputs qualitatively different from that basic realization can begenerated by introducing different driving fields and controlled fluctuations inthe practical setup [1, 18, 19, 20]. Apart from directly affecting the dynam-ics, those external elements can indirectly enhance the effect of the nonlinearcorrections to the trap potential. Here, we aim at explaining the experimentsof Ref. [4]. In them, a nontrivial axial dynamics was uncovered in a variationof the basic setup which incorporated a driving field at parametric resonanceand noise of controllable intensity. Those experimental conditions are simulatedin our approach. Specifically, we consider that z is parametrically driven at afrequency ω d which is nearly twice the characteristic resonant frequency ω z , i.e., ω d = 2( ω z + ǫ ) with the restriction ǫ ≪ ω z for the detuning ǫ . Moreover, ourmodel incorporates a stochastic force η ( t ) with the characteristics of the noisepresent in the practical setup. Residual nonlinear terms of the confining poten-tial, which are known to account for the stabilization of the noiseless versionof the system in the parametric-resonance regime [18, 19], are also included inthe model. Accordingly, we consider that the axial coordinate, (normalized to atypical trap length, and, therefore, dimensionless), is described by the equation ¨ z + γ z ˙ z + ω z [1 + h cos ω d t ] z + λ ω z z + λ ω z z = η ( t ) , (1)where γ z is the friction coefficient, h characterizes the amplitude of the drivingforce, and, λ and λ are coefficients which determine the magnitude of thenonlinear terms of the confining potential. By now, we deal with a one-electronsystem; later on, we will tackle the dynamics of a cloud of N electrons.Crucial to the applicability of our model to the considered experiments is theappropriate modeling of the stochastic force η ( t ) . Two types of fluctuations arerelevant to the experimental scheme. First, the system presents “internal” noiserooted in different elements of the practical setup. It has been argued that thosefluctuations are well modeled by broadband noise and centered narrow-bandfluctuations. Second, white noise, more intense than the internal fluctuations,was injected in the experimental realization of Ref. [4]. Indeed, it was theaddition of this “external” noise that allowed studying the dependence of theswitching mechanism on the noise strength. Here, in order to account for both,the broadband internal fluctuations and the added white noise, we considerthat η ( t ) has general Gaussian wideband characteristics [21]. Specifically, we3ssume that the correlation function k η ( t ′ − t ) ≡ h η ( t ) η ( t ′ ) i − h η ( t ) i has ageneric functional form and that the correlation time is much shorter than anyother relevant time scale in the system evolution. The intensity coefficient D = R ∞−∞ k η ( τ ) dτ will be used to characterize the noise strength [17]. (The white-noise limit, defined by k η ( t ′ − t ) = 2 Dδ ( t − t ′ ) , is included in our analysis.)Additionally, a zero mean value, h η ( t ) i = 0 , is assumed. (Notice that a nonzero h η ( t ) i can be simply incorporated into the model as an effective deterministiccontribution.) A more elaborate noisy input should be added to tackle the effectof residual colored noise. However, that generalization of our approach is notnecessary for the objectives of the present paper: it will be shown that thedetected behavior can be simply traced to broadband-noise characteristics.Our approach to deal with Eq. (1) is based on the averaging methods de-veloped by Krylov and Bogoliubov for the analysis of deterministic nonlinearoscillations as they were generalized by Stratonovich to the study of stochasticprocesses [16, 17]. Those averaging techniques can be applied to generic wide-band fluctuations with sufficiently short correlation time. In this approach, theamplitude A and the phase Ψ of the oscillations are defined through the equa-tions z = A cos [( ω z + ǫ ) t + Ψ] and ˙ z = − ( ω z + ǫ ) A sin [( ω z + ǫ ) t + Ψ] . Withthese changes, Eq. (1) is reduced to a system of two first-order equations in standard form [17], i.e., with the structure of a harmonic oscillator perturbedby deterministic and stochastic terms. For ω z ≫ γ z , ǫ , the average of the de-terministic perturbative elements over the period τ ef = 2 π/ ( ω z + ǫ ) = 4 π/ω d isreadily carried out. Moreover, for a noise correlation time much smaller thanthe relaxation times of the amplitude and the phase, the coarse graining of thestochastic terms over τ ef can be applied following the procedure presented inRef. [17]. Accordingly, we obtain that, to first-order, the averaged equationsare [17, 20, 22] ˙ A = − γ z − hh T sin 2Ψ] A + D eff A + ξ ( t ) , (2) ˙Ψ = − ǫ + 38 λ ω z A + 516 λ ω z A + 14 ω z h cos 2Ψ + ξ ( t ) A , (3)where we have introduced h T ≡ γ z /ω z . (The meaning of h T as a thresh-old amplitude of the driving field will be evident shortly.) Additionally, ξ ( t ) and ξ ( t ) are effective Gaussian white-noise terms defined by h ξ i ( t ) i = 0 , and h ξ i ( t ) ξ j ( t ′ ) i = 2 D eff δ i,j δ ( t − t ′ ) , i, j = 1 , , with D eff = κ η ( ω z + ǫ ) / [4( ω z + ǫ ) ] . (Note that D eff , which determines the strength of the (uncorrelated)effective noise terms, is obtained from the power spectral density κ η ( ω ) ≡ R ∞−∞ e iωτ k η ( τ ) dτ of the original noise η ( t ) at the frequency ω z + ǫ . Here, we mustremark that, from the broadband characteristics assumed for η ( t ) , a smoothform of κ η ( ω ) can be inferred. Indeed, a completely flat spectrum occurs in thewhite-noise limit.) Whereas the noise term in Eq. (2), ξ ( t ) , is additive, thefluctuations enter Eq. (3) through the term ξ ( t ) /A , and, therefore, have multi-plicative character. Moreover, it is important to take into account the presenceof the noise-induced “deterministic” term D eff /A in Eq. (2): its appearance4ill be shown to account for the partial character of the amplitude collapsesdetected in the experiments. It is worth emphasizing that our use of averagedequations is specially appropriate for the considered experimental setup, where,because of the specific characteristics of the detection scheme, the registereddata do actually correspond to averaged magnitudes.In order to trace the response of the system to noise, we must clearly definethe deterministic scenario into which the fluctuations enter. The noiseless dy-namics of the system, described by Eq. (1) without the random term η ( t ) , and,consequently, by Eqs. (2) and (3) with D eff = 0 , has been intensively studied[18, 19]. From the averaged equations, it is straightforwardly shown that para-metric amplification, i.e., exponential growth of the amplitude, takes place fora driving amplitude h larger than the threshold value h T and for a detuning ǫ within the excitation range, namely, for ǫ − < ǫ < ǫ + ( ǫ ± = ± ω z p h − h T .)The experimental conditions on which we focus correspond to this parametric-amplification regime. In the absence of nonlinear terms in the trap potential, theamplitude would grow monotonously. However, the nonlinear corrections, char-acterized by the coefficients λ and λ , which must be included in the descriptionto simulate the actual potential applied in the practical setup, do arrest the am-plitude growth, allowing the stabilization of the motion. The system presentstwo stationary states with the same amplitude and with phase values differingin π radians. Specifically, the stationary amplitude A SS is obtained from theequation ǫ + − ǫ + λ ω z A SS + λ ω z A SS = 0 , and the two π -differing valuesof the equilibrium phase Ψ SS are given by Ψ SS = arcsin( h T h ) . [Important forthe discussion of some of the noisy features is to notice that, for the values ofthe nonlinear parameters applied in the experiments, ( λ = 0 and λ < ), A SS increases with ǫ + − ǫ .] The stationary states correspond to attractors in thephase space. Depending on the initial conditions, the system eventually reachesone or other attractor. The objective of the next section is the explanation ofthe effects of noise on this deterministic scenario. In the analysis of the noisy dynamics, we proceed by showing first that Eqs.(2) and (3) provide a satisfactory description of the behavior observed in theexperiments. Then, once its validity has been confirmed, our approach will beapplied to uncover the physics underlying the detected features.
In Ref. [4], the presence of noise was shown to significantly alter the deter-ministic picture. The system was not longer stabilized in one of the attractors.Instead, it was observed to display amplitude collapses and revivals correlatedwith abrupt changes in the phase. Those experimental findings are reproducedby our approach. Figs. 1a and 1b respectively depict results for A and Ψ asobtained from Eqs. (2) and (3). There, the correlation between the phase flips5nd the collapses and revivals of the amplitude is evident. Actually, in agree-ment with the experimental results, it is found that the amplitude collapsesare almost always followed by phase flips. In other words, the system rarelystays in the same basin of attraction once a collapse in A has occurred. Oneshould notice that, in the inter-jump intervals, Ψ is strongly localized aroundits equilibrium values; in contrast, a significant dispersion in A is observed.Our study reproduces the detected partial character of the collapses. Alreadynoticeable in Fig. 1a, this feature is particularly evident in Fig. 2, where wedepict a typical noisy trajectory, which includes different flips between the twobasins. There, it is apparent that the unstable point defined by A = 0 is neverreached. As stressed in Ref. [4], these features cannot be understood with asimple activation-process model. ph a s e time (s) a m p lit ud e (a)(b) Figure 1: Time evolution of the phase (a) and of the amplitude (b) as obtainedfrom Eqs. (2) and (3). [ ω z / π = 61 . M Hz , λ = 0 , λ = − . , γ z =(10 ms ) − , ǫ + / π = 100 Hz , ǫ/ π = 50 Hz , D = 10 − (arbitrary units).] (Thesame set of parameters is used throughout the paper.)6igure 2: Phase space diagram for a particular noisy trajectory.In the analysis of the experimental results, a histogram for the residencetime, (i.e., the time interval between phase flips), served to obtain the averagejump rate Γ . Actually, Γ was found to be well approximated by an exponen-tial function of the noise strength, namely, Γ ∼ exp( − E/D ) , where E denotesthe effective activation energy. That characterization is reproduced by our ap-proach. In Fig. 3, we represent a histogram for the residence time as obtainedfrom Eqs. (2) and (3). Moreover, in Fig. 4, we plot the jump rate as a func-tion of the noise strength. There, the validity of the exponential fit is patent.(In our calculations, we have directly worked with D eff as noise strength: theused arbitrary units include the ratio between the actual strength of the orig-inal noise D and D eff .) Apart from the dependence on the fluctuations, thejump rate incorporates, through the effective activation energy, the influenceon the process of elements like the frequency and strength of the driving field,the characteristics of the trapping potential, or the damping coefficient. Thedescription of the role played by those deterministic components of the systemin the noisy dynamics is crucial for understanding the process of phase switch-ing. In the experiments, the dependence of E on those system parameters wastraced via the systematic variation of the practical conditions. In particular,an approximately exponential dependence of the jump rate Γ on the detuning,expressed as ǫ + − ǫ , was reported. That behavior is also simulated with ourapproach, as shown in Fig. 5. 7 nu m b e r o f f li p s Figure 3: Histogram for the time interval between phase flips.
400 500 600(noise strength) -1 ph a s e -f li p r a t e ( s - ) Figure 4: Phase-jump rate Γ ( s − ) versus the inverse noise-strength D − (arbi-trary units). 8 ε + - ε )/2 π (Hz)0.11 ph a s e -f li p r a t e ( s - ) Figure 5: Phase-jump rate Γ ( s − ) as a function of the detuning expressed as ( ǫ + − ǫ ) / π ( Hz ). ( ǫ + / π = 100 Hz ).In Ref. [4], no results are presented for the dependence of the activation en-ergy on the friction coefficient. However, as this aspect of the system behaviorwill be an important element of our discussion of the phase switching mecha-nism, it is pertinent to describe it here using our approach. Indeed, since earlytheoretical studies were set up from a Hamiltonian approximation to the dy-namics, it is worthwhile to inquire into the actual significance of the dissipativecharacter of the system. Let us first recall some aspects of the purely deter-ministic (dissipative) dynamics which are relevant to the resulting stochasticscenario. Namely, from the found threshold amplitude, given by h T = 2 γ z /ω z ,it is evident that the generation of oscillations is inhibited as the friction coef-ficient γ z increases. Additionally, we must take into account that, as found inRef. [11], the relaxation of the system from any initial conditions to the equilib-rium states becomes faster for larger γ z . Therefore, we can conjecture that, as γ z grows, simply because of the deterministic inhibition of the oscillations andof the enhanced stability of the system, the role of noise in activating the phasejumps must be hindered. This conjecture is confirmed by our results for thedependence of E on γ z . We have found that there is an approximately expo-nential decrease of the jump rate Γ with γ z , as shown in Fig. 6. Consequently,from the expression Γ ∼ exp( − E/D ) , a nearly linear increase of E with γ z isderived. Specifically, we can write E ≈ C + C γ z , where the coefficients C and C incorporate the dependence of E on other parameters of the system. Someimplications of these results will be considered in the forthcoming discussion.By now, we anticipate that the analysis of the persistence in the N -electronsystem of the found dependence of E on γ z will be central to our understandingof the detected scaling of the phase jump rate with N .9 γ z (s -1 )0.010.11 ph a s e -f li p r a t e ( s - ) Figure 6: Phase-jump rate Γ ( s − ) versus the friction coefficient γ z ( s − ). In our discussion of the physics that underlies the observed features, we proceedgradually: we start with a simplified picture of the dynamics, which will beimproved by successively incorporating the different elements of the completesystem. Our scheme is summarized in the following steps.(i) A zero-order approximation to the random response detected in the ex-periments is provided by the artificial decoupling of Eqs. (2) and (3). Indeed,some clues to the origin of prominent features of the complete system are givenby the analysis of the “independent” behaviors of A and Ψ that respectivelyfollow from fixing the phase in Eq. (2) and the amplitude in Eq. (3).For a constant value of Ψ , Eq. (2) describes fluctuations of A around itsequilibrium position. Interestingly, because of the term D eff /A , the equilibriumamplitude is larger than its counterpart in the absence of noise. Noise-inducedexcursions to the region of small A can be predicted. Due to the effective“deterministic” term D eff /A , the value A = 0 is not reached, i.e., a completecollapse never takes place. As in the description of the deterministic systemat parametric resonance, to account in our approach for the limited growth of A observed in practice, the coupling to the phase equation, which contains thenonlinear terms of the confining potential, is necessary.Conversely, for a fixed A , Eq. (3) describes a process of phase diffusion ina tilted periodic potential [17]. The bias, given by − ǫ + λ ω z A + λ ω z A ,is determined by the detuning and by the artificially fixed amplitude. The po-tential presents two minima separated by π radians.When the bias is smallerthan the height of the periodic potential, Ψ evolves only because of the fluctu-ations. Actually, in the regime considered in the experiments, it is noise that10eads to phase jumps between the minima. Since the magnitude of the (mul-tiplicative) random term ξ ( t ) /A increases as A diminishes, the jumps becomemore frequent for smaller A .(ii) A comparative analysis of the structure of the two averaged equationsuncovers the qualitatively different effects of noise on the two variables. Sincethe amplitude has no strong confining potential, it is continuously forced outof equilibrium by the stochastic force. Indeed, a significant dispersion in A isapparent in Fig. 1a. In contrast, as shown in Fig. 1b, the periodic potentialleads to a remarkable concentration of Ψ around its equilibrium values. Thisphase locking is interrupted by noise-induced flips. Attention must also be paidto some characteristics of the coupling between Eqs. (2) and (3). As previouslydiscussed, the presence of A in Eq. (3) is crucial for the evolution of Ψ . Inparticular, the magnitude of A determines the frequency of the phase flips viathe random term ξ ( t ) /A . In contrast, a much weaker effect of the phase flipson the evolution of the amplitude is apparent: given that the phase enters Eq.(2) through sin 2Ψ , the π jumps in the phase hardly alter the dynamics of theamplitude.(iii) By combining the ideas contained in the above points, the observedcorrelation between amplitude collapses and phase jumps can be explained.Noise can induce an appreciable reduction in the amplitude, which leads to asignificant increase of the random term ξ ( t ) /A in the equation for the phaseevolution, and, in turn, to a stochastic phase jump. Because of the “determin-istic” term D eff /A , the unstable point with A = 0 and undefined Ψ is avoidedin the switching. In fact, as A never reaches a zero value, the phase is alwayswell-defined. Additionally, the fast regrowth of the amplitude after each (par-tial) collapse is rooted in the term D eff /A and in the friction-induced stabilityof the underlying deterministic attractors. In the experiments, the correlationbetween the amplitude collapses and the phase jumps was observed to decay forincreasing noise intensities. This feature can be understood taking into accountthat, as the noise strength increases, the random term ξ ( t ) /A can be strongenough to lead to phase flips even without an appreciable reduction in A . Fur-thermore, the inhibition of the jumps for increasing γ z is linked to the higherstability of the (deterministic) stationary amplitude A SS and to the consequentless probable exploration by the system of the small-amplitude region. Then,we can understand that, as shown in Fig. 6, the phase jumps dwindle as γ z isenhanced, and, correspondingly, that the effective activation energy increaseswith γ z . A similar argument qualitatively explains the dependence of the fliprate on the detuning, reflected in Fig. 5. As previously pointed out, A SS growswith ǫ + − ǫ ; consequently, for increasing ǫ + − ǫ , the collapse region is less easilyreached, and, in turn, the phase jumps become less probable.From the above discussion, the essential components of the mechanism re-sponsible for the stochastic phase-switching can be identified. The periodicpotential, rooted in the driving field at parametric resonance, allows the stronglocalization of Ψ , and, therefore, the well-defined character of the jumps inphase. Additionally, the random term ξ ( t ) /A accounts for the correlation be-tween amplitude collapses and phase flips. This stochastic link can be traced11o two fundamental characteristics of the system. First, it is rooted in the ad-ditive character of the input noise η ( t ) : with the change of representation, from z to A and Ψ , the fluctuations become multiplicative and the random connec-tion does appear. Second, its specific compact form ξ ( t ) /A , uncovered by theapplication of the averaging methods, is a consequence of the broadband-noisecharacteristics, which guarantee the applicability of that methodology. Giventhe generality of its origin, this random link can be expected to be relevant toquite generic stochastic oscillators. In fact, it has been previously characterizedin pioneering work on nonlinear self-excited oscillations in electronic devices [17].Its intense differential effect on the current scenario results from its combinationwith the driving field: as the phase is strongly confined, its noise-induced evo-lution, (enhanced for small values of the amplitude), occurs basically throughnoticeable jumps between approximate equilibrium values. A comment on therole played by the nonlinearity of the system is also pertinent. It is importantto emphasize that the nonlinear terms of the trap potential, which are necessaryfor the stabilization of the amplitude in the parametric-resonance regime, arenot essential components of the phase-switching mechanism. It is the nonlin-earity induced by the parametric driving field, i.e., the phase bi-stability, thatreally counts in the process. Also, it is interesting to examine what can beextracted from our approach about the actual relevance of colored noise. Werecall that the possibility of tracing some of the observed features to noise-colorcharacteristics was pointed out in early discussions of the experiments. In thissense, our study conclusively shows that the experimental features reported inRef. [4] can be induced purely by broadband noise. Even more, we have foundthat simple white noise can account for those effects. Finally, we remark that,as stressed in Ref. [4], a simple activation-process model [21] fails to provide anappropriate picture of the observed dynamics. Although the introduction of aneffective activation energy is useful in the characterization of the phase-jumprate, a simple activation-process description misses the sequence of combinedeffects in the evolution of the amplitude and phase that leads to the distinctivecharacteristics of the phase switching. In the above, we have considered a mono-electronic system. Now, we turn toanalyze the dynamics of a cloud of N electrons. We aim at explaining thenontrivial features of the evolution of the center-of-mass coordinate Z ( Z = N P Ni =1 z i ) uncovered by the experiments of Ref. [4], specifically, the observedslow-down and eventual disappearance of the random phase jumps for increasingnumber of electrons.Some preliminary general considerations on the dynamics of the electroniccloud are in order. First, we recall that, in previous work on damping of apolyelectronic system in a Penning trap, the friction coefficient of Z , γ ( N ) z ,was shown to be well approximated as γ ( N ) z = N γ z [18, 19]. Second, wemust take into account that the random force on the center-of-mass coordi-12ate is given by η ( N ) ( t ) = N P Ni =1 η i ( t ) , where η i ( t ) denotes the noise on eachindividual electron. The statistical characterization of η ( N ) ( t ) is straightfor-ward. As given by a linear superposition of Gaussian fluctuations, η ( N ) ( t ) hasalso Gaussian-noise characteristics. From the zero-mean values of the mono-electronic stochastic forces, one trivially obtains (cid:10) η ( N ) ( t ) (cid:11) = 0 . Additionally,assuming that the individual random forces are completely uncorrelated, i.e., h η i ( t ) η j ( t ′ ) i = 2 Dδ ij δ ( t − t ′ ) , i, j = 1 . . . N , we obtain (cid:10) η ( N ) ( t ) η ( N ) ( t ′ ) (cid:11) =2 DN δ ( t − t ′ ) ≡ D ( N ) δ ( t − t ′ ) . Hence, the individual fluctuations average toa weaker noise in the collective coordinate. (In order to present our argumentsin simple terms, we have considered here the white-noise limit. The generaliza-tion to generic broadband noise is direct.) It follows that the main novelties inthe description of Z with respect to the previously described mono-electronicscenario are the presence of a larger damping coefficient and of a reduced noisestrength. Our analysis will focus on the implications of those differential char-acteristics for the scaling of the phase-flip rate with N . Since, as shown by thestudy of the one-electron oscillator, the mechanism of stochastic phase switchingdoes not essentially depend on the anharmonicity of the potential, we can ne-glect the nonlinear terms in the analysis. Accordingly, we consider the evolutionof Z as approximated by ¨ Z + γ ( N ) z ˙ Z + ω z [1 + h cos ω d t ] Z = η ( N ) ( t ) . (4)Given that Eq. (4) has the same structure as Eq. (1), the methodologypreviously presented for the study of the mono-electronic system can also beused here. Indeed, this parallelism allows us to extrapolate some of the previousresults. In particular, the flip rate, which, for a one-electron system was found toscale as exp( − E/D ) , can be expected now to have the form exp( − E ( N ) /D ( N ) ) ,where D ( N ) is the (effective) strength of η ( N ) ( t ) and E ( N ) is the activationenergy for Z . Crucial to the analysis is to take into account that, both, E ( N ) and D ( N ) , depend on N . Actually, as pointed out in the above applicationof our approach, the activation energy increases with the damping constant.Therefore, as the friction coefficient is γ ( N ) z = N γ z , we conclude that E ( N ) increases with N , and, consequently, that the phase flips are hindered as N grows. An additional contribution to the inhibition of the phase switching isrooted in the curbed fluctuations in the center-of-mass coordinate: the decreaseof the effective noise strength D ( N ) = D/N with N contributes also to thehindrance of the phase jumps for growing electron number. We can go further inthe analytical characterization of the dependence of Γ ( N ) on N : by combiningthe equations Γ ( N ) ∼ exp( − E ( N ) /D ( N ) ) , E ( N ) = C + C N γ z , and D ( N ) = D/N , we obtain Γ ( N ) ∼ exp( − N ( C + C N γ z ) /D ) . (5)This expression is the key element in our explanation of the observed scalingof the jump rate. The experimental procedure reported in Ref. [4] includeddifferent variations of the system parameters. Specially revealing of the mech-anism responsible for the jump inhibition is the analysis of the experimental13un corresponding to a simultaneous variation of N and γ z with constant N γ z .Indeed, the observed decrease of the jump rate with N points to a mechanismnot linked to the friction term γ ( N ) z , which, in fact, is kept constant in this run.From our approach, we can conjecture that, in this case, it is the decrease ofthe effective noise strength that leads to the detected slow-down of the phaseflips. More specifically, the measured linear dependence of the exponent of Γ ( N ) on N can be traced to the term N C /D in our analytical characterization of Γ ( N ) given by Eq. (5). Additional insight is provided by the experimental re-sults corresponding to a mere variation of N , with constant γ z . In this case,both, a reduction in D ( N ) and an increase of γ ( N ) z take place. Since, again, anapproximately linear dependence of the exponent of Γ ( N ) on N was found, wecan conjecture that, in the regime studied in the experiments, the reduced noisestrength is the dominant element in the mechanism responsible for the inhibitionof the phase switching. Following the report of the experimental findings, ourdiscussion in this section has focused on the persistence of the stochastic flipsin the polyelectronic system. For a more complete description of the dynamicsof the electronic cloud, the access to additional experimental data is necessary. Our description of the stochastic dynamics of the one-electron Penning-trap os-cillator explains the experimental findings of Ref. [4]. The physical mechanismresponsible for the observed random phase switching has been traced to thecombination of a driving field at parametric resonance and broadband noise en-tering additively the axial-mode equation. The driving field allows the stronglocalization of the phase around two equilibrium positions, and, therefore, theabrupt character of the changes in phase. Additionally, the fluctuations estab-lish a link between the amplitude and the phase which results in a significantenhancement of the effects of noise on the phase for small values of the ampli-tude. Our analysis uncovers the generality of this mechanism, and, therefore,its relevance to different contexts. Indeed, we have reported its previous char-acterization in studies on the appearance of selfexcited nonlinear oscillations inelectronic devices [17]. Our work proves that the detected characteristics of thesystem response do not specifically depend on the presence of residual colorednoise in the practical setup. In fact, it has been shown that the emergence of theobserved features can be simply traced to broadband fluctuations. Finally, froma generalization of our approach, we have shown that the observed attenuationof the phase flips in the dynamics of a cloud of N electrons can be explainedas rooted in the effective reduction of the noise strength in the center-of-masscoordinate. 14 eferences [1] For a review, see L. S. Brown and G. Gabrielse, Rev. Mod. Phys. , 233(1986), and references therein.[2] G. Gabrielse et al. , Phys. Rev. Lett. , 113001 (2008).[3] D. Hanneke et al. , Phys. Rev. Lett. , 120801 (2008).[4] L. J. Lapidus et al. , Phys. Rev. Lett. , 899 (1999).[5] S. Peil and G. Gabrielse, Phys. Rev. Lett. , 1287 (1999).[6] D. Enzer and G. Gabrielse, Phys. Rev. Lett. , 1211 (1997).[7] S. Brouard and J. Plata, Phys. Rev. A , 063405 (2001).[8] B. Vestergaard and J. Javanainen, Phys. Rev. A , 1537 (1998).[9] J. Goldman, and G. Gabrielse, Phys. Rev. A , 052335 (2010).[10] J. L. Lamata et al. , Phys. Rev. A , 022301 (2010).[11] C. H. Tseng et al. , Phys. Rev. A , 2094 (1999).[12] G. Ciaramicoli et al. , Phys. Rev. A , 032301 (2004).[13] M. I. Dykman et al. , Phys. Rev. E , 5202 (1998).[14] C. Stambaugh and H. B. Chan, Phys. Rev. B , 172302 (2006).[15] H. B. Chan and C. Stambaugh, Phys. Rev. Lett. , 060601 (2007).[16] N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theoryof Non-Linear Oscillations (Gordon and Breach, New York, 1961).[17] R. L. Stratonovich,
Topics in the Theory of Random Noise (Gordon andBreach, New York, 1963).[18] J. Tan and G. Gabrielse, Phys. Rev. Lett. , 3090 (1991).[19] J. Tan and G. Gabrielse, Phys. Rev. A , 3105 (1993).[20] S. Brouard and J. Plata, Phys. Rev. A , 053412 (2002).[21] H. Risken, The Fokker-Planck Equation (Springer-Verlag, New York, 1989)[22] J. Plata, Phys. Rev. E59