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arxiv: 2305.19260 · v2 · submitted 2023-05-30 · ⚛️ physics.app-ph

Simulations and theory of power spectral density functions for time dependent and anharmonic Langevin oscillators

AbdAlGhaffar K. Amer , F. Robicheaux This is my paper

Pith reviewed 2026-05-24 08:24 UTC · model grok-4.3

classification ⚛️ physics.app-ph
keywords power spectral densityLangevin oscillatorPaul trapMathieu equationanharmonic potentialtime-dependent frequencynanoparticlethermal noise
0
0 comments X

The pith

Power spectral densities for Paul-trap nanoparticles deviate from harmonic Langevin predictions by more than a factor of two once the Mathieu parameter q grows.

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

The paper develops both analytic expressions and numerical simulations for the power spectral density functions of particles whose potentials include either time-dependent frequency shifts or a quartic anharmonic term, all under standard thermal damping and fluctuating forces. It focuses on the two PSDs commonly extracted in trapped-nano-oscillator experiments and extends the treatment to a spherical nanoparticle inside a Paul trap. For the Paul-trap case the authors show that the usual simple-harmonic Langevin formulas remain accurate only at small values of the Mathieu parameter q; the discrepancy exceeds a factor of two at larger q. Numerical tests further demonstrate that a permanent electric dipole on the nanosphere produces substantial changes to the x- and y-coordinate spectra.

Core claim

The authors derive and numerically test the PSDs that result from two time-dependent frequency perturbations and from a time-independent quartic extension of the harmonic potential. For a nanoparticle in a Paul trap they find that the simple-harmonic Langevin PSDs approximate the x- and y-motion spectra well only when the Mathieu parameter q is small, while the difference grows beyond a factor of two as q increases; they also show numerically that a permanent electric dipole on the particle significantly modifies those same PSDs.

What carries the argument

The Mathieu equation governing Paul-trap motion together with the Langevin equation augmented by explicit time-dependent frequency terms or a quartic anharmonic term.

If this is right

  • Experimental extraction of trap parameters or temperature from PSDs must incorporate q-dependent corrections once q is no longer small.
  • Presence of a permanent dipole must be included in any model of the x- and y-coordinate spectra.
  • Time-dependent frequency perturbations require their own PSD formulas rather than the unperturbed harmonic expressions.
  • Simulations confirm that the derived analytic PSDs accurately reproduce the motion statistics for the three perturbations examined.

Where Pith is reading between the lines

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

  • The same q-dependent deviations are likely to appear in any periodically driven trap whose stability diagram contains a Mathieu parameter of comparable size.
  • Precision force or torque measurements that rely on calibrated PSDs will inherit systematic errors if the dipole contribution is neglected.
  • Extensions to other noise spectra or to two-dimensional coupling could be tested by repeating the same numerical protocol with modified forcing terms.

Load-bearing premise

The derivations and numerical tests assume the specific functional forms chosen for the time-dependent frequency changes and the quartic term, plus the standard white-noise thermal model.

What would settle it

Measure the x- and y-PSDs of a levitated nanosphere in a Paul trap at a known large value of q and check whether the observed spectra differ from the simple-harmonic prediction by more than a factor of two.

Figures

Figures reproduced from arXiv: 2305.19260 by AbdAlGhaffar K. Amer, F. Robicheaux.

Figure 1
Figure 1. Figure 1: The PSD for a linear ω(t) Eq. (17) uniformly spreads over the frequencies from 99 to 101 Hz for Γ = 1.0 s −1 , shown here in the dashed orange curve. A plot of the PSD from Eq. (15) (solid green curve) as well as a plot using the analytic expression in Eq. (B14) obtained in App.B 1 (dotted purple curve) are shown for comparison 0.0 0.5 1.0 1.5 −0.50 −0.25 0.00 0.25 0.50 QPSD (10−19 m 4 sec) f (Hz) Γ = 0.2 … view at source ↗
Figure 3
Figure 3. Figure 3: The numerical simulation’s PSD for the case of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Same as Fig. 3 but with [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Same as Fig. 3 but for QPSD. The numerical QPSD [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 11
Figure 11. Figure 11: Same as Fig. (9) except with q∼ 0.6. 0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 0.2 0.4 0.6 0.8 QPSD percentage difference q [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The QPSD percentage difference between the for [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗
Figure 10
Figure 10. Figure 10: Same as Fig. (8), except with q∼ 0.6. 0.0 1.0 2.0 3.0 4.0 −0.50 −0.25 0.00 0.25 0.50 QPSD (10−24 m 4 sec) f (Hz) Numerical QPSD Mathieu QPSD SHLO QPSD [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 13
Figure 13. Figure 13: The PSD for the z-component for an electric dipole of 16eR, where e is the elementary charge and R is the radius of the nanosphere. The numerical PSD shown by the dotted black curves doesn’t deviate from SHO PSD Eq. (15) for the two value of Γ = 0.5 and 1.0 s −1 represented by the pink and the orange solid curves respectively. the numerical answer for both the QPSD and the PSD still match the analytic for… view at source ↗
Figure 14
Figure 14. Figure 14: The PSD for the x-component for the different values of the electric dipole p =8eR, 12eR and 16eR in figures 14(a),14(b) and 14(c) respectively, where e is the elementary charge and R is the radius of the nanosphere. The numerical PSD represented by the dashed pink and orange curves ,for the two value of Γ = 0.5 and 1.0 s −1 respectively, deviates greatly as the electric dipole p increases from the Paul t… view at source ↗
Figure 15
Figure 15. Figure 15: Same as Fig. 14, but for the QPSD and only for [PITH_FULL_IMAGE:figures/full_fig_p010_15.png] view at source ↗
read the original abstract

Simulations and theory are presented for the power spectral density functions (PSDs) of particles in time dependent and anharmonic potentials including the effects of a thermal environment leading to damping and fluctuating forces. We investigate three one dimensional perturbations to the harmonic oscillator of which two are time dependent changes in the natural frequency of the oscillator, while the other is a time independent extension of the quadratic potential to include a quartic term. We investigate the effect of these perturbations on two PSDs of the motion that are used in experiments on trapped nano-oscillators. We also derive and numerically test the PSDs for the motion of a spherical nanoparticle in a Paul trap. We found that the simple harmonic Langevin oscillator's PSDs are good approximations for the $x$-and $y$-coordinates' PSDs for small values of the parameter $q$ of the Mathieu equation, but the difference can be more than a factor of two as '$q$' increases. We also numerically showed that the presence of a permanent electric dipole on the nanosphere can significantly affect the PSDs in the $x$-and $y$-coordinates.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript develops analytic expressions and numerical simulations for the power spectral density (PSD) functions of one-dimensional Langevin oscillators subject to three classes of perturbation: two forms of time-dependent frequency modulation (leading to Mathieu equations) and a time-independent quartic anharmonicity. It extends the analysis to the x- and y-motion of a spherical nanoparticle in a Paul trap, including the additional effect of a permanent electric dipole moment. The central quantitative claim is that the PSDs of the unperturbed harmonic Langevin oscillator remain good approximations to the x/y PSDs only for small values of the Mathieu parameter q, with deviations exceeding a factor of two at larger q; the dipole is shown numerically to produce significant changes in the same PSDs.

Significance. If the numerical results hold under the stated conditions, the work supplies concrete, experimentally relevant bounds on the validity of the harmonic Langevin PSD model for levitated or trapped nano-oscillators. The explicit demonstration that a permanent dipole can alter the PSDs by a large factor is a useful cautionary result for precision measurements. The combination of derivation from the Langevin equation plus independent numerical checks is a positive feature, though the lack of reported parameter ranges and error analysis limits immediate quantitative use.

major comments (2)
  1. [Abstract / numerical tests] Abstract and numerical-results section: the claim that deviations 'can be more than a factor of two as q increases' is the central quantitative result, yet no specific range of q, number of trajectories, integration time, or statistical uncertainty is stated; without these the factor-of-two statement cannot be verified or reproduced from the given information.
  2. [Theory section] Derivations of the PSDs for the Mathieu-driven case: the manuscript states that the derivations start from the Langevin equation and Mathieu equation, but the text does not supply the intermediate steps or the explicit functional form assumed for the time-dependent frequency; this omission makes it impossible to confirm that the reported PSD expressions are free of hidden fitting parameters.
minor comments (2)
  1. [Introduction] The three perturbation forms are introduced without a clear statement of why these particular functional forms were chosen over other common time dependences (e.g., sinusoidal or stochastic).
  2. [Figures] Figure captions and axis labels should explicitly indicate the value of q (or the corresponding perturbation amplitude) used in each panel so that the factor-of-two claim can be read directly from the plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Abstract / numerical tests] Abstract and numerical-results section: the claim that deviations 'can be more than a factor of two as q increases' is the central quantitative result, yet no specific range of q, number of trajectories, integration time, or statistical uncertainty is stated; without these the factor-of-two statement cannot be verified or reproduced from the given information.

    Authors: We agree that the numerical parameters underlying the factor-of-two claim must be stated explicitly. In the revised manuscript we will add, in both the abstract and the numerical-results section, the precise range of q values examined, the number of independent trajectories, the integration time per trajectory, and the statistical uncertainties on the computed PSDs. This will allow direct verification of the reported deviations. revision: yes

  2. Referee: [Theory section] Derivations of the PSDs for the Mathieu-driven case: the manuscript states that the derivations start from the Langevin equation and Mathieu equation, but the text does not supply the intermediate steps or the explicit functional form assumed for the time-dependent frequency; this omission makes it impossible to confirm that the reported PSD expressions are free of hidden fitting parameters.

    Authors: The referee is correct that the intermediate algebraic steps and the explicit form of the time-dependent frequency are not shown. We will expand the theory section to include the full derivation, beginning from the Langevin equation with the Mathieu-type modulation and arriving at the reported PSD expressions, thereby confirming the absence of fitting parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations start from Langevin/Matthieu equations with independent numerical checks

full rationale

The paper presents derivations of PSDs directly from the Langevin equation with time-dependent or anharmonic perturbations (including the Mathieu equation for Paul traps) and reports numerical simulations as independent tests of those derivations. No load-bearing steps reduce to fitted parameters renamed as predictions, self-citations that justify uniqueness, or ansatzes smuggled via prior work. The central claims (approximation quality for small q, factor-of-two deviations at larger q, and dipole effects) are framed as empirical/numerical outcomes for the chosen models rather than tautological outputs. This matches the default case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. Full text would be required to populate the ledger.

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Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages

  1. [1]

    The effects of higher order nonlinear instabilities [20] aren’t included in the following results because the trap potential Eq

    Paul trap without dipole The potential inside the trap V (x,y,z ) is given by [14],[19] V (x,y,z ) = kVend z2 − x2+y2 2 z2 0 − VRF x2 − y2 2r2 0 cos(ΩRFt) (2) whereVend is the potential on the end caps in the z-axis, k is a dimensionless constant, z0 and r0 are constants of length dimension, and VRF is the potential on the rods oscillating with frequency ...

    work page

  2. [2]

    is quadratic. The above potential produces forces on the three coordinates {x,y ,z} of the trapped particle of the form[14]: Fi/m = − Ω2 RF 4 (ai − 2qi cos(ΩRFt)) xi(t) (3) where the index i runs over the three components {x, y,z }, with −az/ 2 = ax = ay = −k 4QVend mΩ2 RF z2 0 , qx = −qy = 2QVRF mΩ2 RF r2 0 andqz = 0. The notation for {ax,a y,a z} is the...

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  3. [3]

    This will introduce an additional force on the nano-particle which has the form − ⃗∇ (−⃗ p·⃗E) with ⃗E(x,y,z ) = − ⃗∇ V (x,y,z ) the electric field at the nanoparticle

    Paul trap with dipole In this section, we give the equations of motion with a nonzero permanent electric dipole ⃗ p= {px,p y,p z} on the nanoparticle. This will introduce an additional force on the nano-particle which has the form − ⃗∇ (−⃗ p·⃗E) with ⃗E(x,y,z ) = − ⃗∇ V (x,y,z ) the electric field at the nanoparticle. This changes Eq. (

    work page

  4. [4]

    by replacing xi(t) on the right hand side with xi(t) +pi(t)/Q for the three components {x, y,z }. Since the nanoparticle is spinning, the direction of the electric dipole moment vector changes over time following the differential equation [22] ˙⃗ p=⃗ η× ⃗ p (7) with⃗ η= {ηx,η y,η z} whereηx,η y andηz are the angular frequencies of the spherical nanoparticl...

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  5. [5]

    So, in order to simulate the effect of the Fth term, we adjust the ve- locity v after each time step dt as follows

    Both the first and the second term are in the acceleration calculator of the RKQS integrator, how- ever the thermalization term Fth is a stochastic term and isn’t directly expressible as an acceleration. So, in order to simulate the effect of the Fth term, we adjust the ve- locity v after each time step dt as follows. v → v +δv (9) where the second term δv ...

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  6. [6]

    The Langevin oscillators have been studied in the literature

    Simple Harmonic Langevin Oscillator For a harmonic Langevin oscillator, Fx/m = −w2 0x withw0 being the natural frequency of the oscillator. The Langevin oscillators have been studied in the literature

    work page

  7. [7]

    In some applications it is convenient to compute the PSD relative to its average value at w = w0 [6] which we term in this paper as the Quadrature Power Spec- tral Density (QPSD)

    and its power spectral density (PSD) as defined in Appendix A 1 is known to take the analytic form: Sx(w) = 2ΓkbT/πm (w2 − w2 0)2 + Γ2w2 (15) This PSD has a maximum value of (2 kbT )/ ( πm Γw2 0 ) at w = ±w0. In some applications it is convenient to compute the PSD relative to its average value at w = w0 [6] which we term in this paper as the Quadrature Po...

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  8. [8]

    Oscillator with a linear ω(t) This is the case for which the force Fx is given by Fx/m = −ω 2(t)x where ω (t) is given by ω (t) = w0 ( 1 − δ + 2δ τ t ) (17) 4 In this system, the frequency of the oscillator drifts from ω 0(1 − δ) to w0(1 +δ) over the period from t = 0 to t =τ and δ ≪ 1. We derived the PSD for this case in Appendix B 1 and it approximately...

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  9. [9]

    as will be later dis- cussed in more detail in Section III A 1

    work page

  10. [10]

    For this case the particle develops side bands and os- cillates at multiple frequencies ω n

    Oscillator with an oscillating ω(t) This is the case for which the force Fx is given by Fx/m = −ω 2(t)x where ω (t) is given by ω (t) = ω 0 + ∆ω cos(Ωt) (19) The parameters ∆ω and Ω control the amplitude and the frequency of the oscillating part in ω (t) respectively. For this case the particle develops side bands and os- cillates at multiple frequencies ...

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  11. [11]

    ( 3), where the constants {a,q } are functions of the trap parameters and deter- mine the stability of the solutions

    Paul trap This is the case in which the oscillator’s force leads to the Mathieu equation, Eq. ( 3), where the constants {a,q } are functions of the trap parameters and deter- mine the stability of the solutions. We do not know of a derivation of the PSD and QPSD for this case; they are derived in Appendix C and the PSD is Sx(ω ) = 4 (⟨ ¯c2⟩ ( Γ2 + 4ω 2) +...

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  12. [12]

    Oscillator with a linear ω(t) In this section, we investigate the case when the oscil- lation frequency drifts linearly in time, Eq. ( 17). For the simulations presented here, Figs. 1 and 2, ω 0 = 100 × 2π rad/s,δ = 0. 01, and τ = 100/ Γ with Γ = 1 s− 1. This represents an oscillator’s frequency linearly drifting by 2% over a time interval of 100 / Γ arou...

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  13. [13]

    Oscillator with oscillating ω(t) In this section we consider the situation where the fre- quency oscillates in time, Eq. ( 19). For our simulations ω 0 is taken to be 100 × 2π rad/sec and ∆ ω is taken to be ξω 0 whereξ is the fractional change in the frequency amplitude ω (t) = 100 × 2π (1 +ξ cos(Ωt +φ)) (23) where φ is a phase factor that is chosen from ...

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  14. [14]

    This would result if the harmonic potential was only the first order expansion of an even potential near its minimum [26]

    Non-linear oscillator In this section, the case of having an extra non- harmonic term in the potential is considered. This would result if the harmonic potential was only the first order expansion of an even potential near its minimum [26]. The potential is taken to be V (x) = 1 2mw2 0x2 ( 1 +α mw2 0 kT x2 ) (24) with α being a dimensionless parameter givi...

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  15. [15]

    To first order in the pertur- bation parameter α , the oscillation frequency blue shifts from ω 0 to ω 0(1 +α 3 2 ) [27]

    This change to the PSD and QPSD arises from the effect of the quartic perturbation on the oscillation frequency. To first order in the pertur- bation parameter α , the oscillation frequency blue shifts from ω 0 to ω 0(1 +α 3 2 ) [27]. This explains the blue shift in the PSD as well as the rough size of the shift, 1.5 Hz. To explain the behaviour in the QPSD...

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  16. [16]

    Since the quartic term blue shifts the frequency, the peak height of the QPSD will decrease

    for the QPSD. Since the quartic term blue shifts the frequency, the peak height of the QPSD will decrease. In Fig. 7 the perturbation parameter α is equal to 0.01 leading to approximately a 15% decrease in the peak value of the QPSD from that of the SHO. Our simulations showed that as α increases, the peak value in 7 the QPSD decreases. However, the perce...

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  17. [17]

    For the case shown in Fig

    but with different values of the oscillation frequency w0 and the friction coefficient Γ. For the case shown in Fig. 7, a fit of the numerically obtained QPSD gave w0 approximately 1.044 times the actual value of w0 and a friction coefficient, Γ, approxi- mately 1.03 instead of 1. For many cases, the QPSD can approximate the physical parameters to an accuracy o...

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  18. [18]

    Without an electric dipole In this section, the motion of a particle in a Paul trap is simulated using the equation of motion Eq. ( 3). Since the motion of the z-component follows a SHLO equation of motion, the numerical results for the PSDs matched perfectly the analytic expressions in Eqs. (

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  19. [19]

    We only show the PSD and QPSD for the x-component because the x-and y-components follow the same equa- tion of motion with a phase difference

    and ( 16). We only show the PSD and QPSD for the x-component because the x-and y-components follow the same equa- tion of motion with a phase difference. The PSD and the QPSD for the x-motion are given in Eqs. ( C16) and ( C21). Here we compare those ex- pressions and the ones for the simple harmonic Langevin oscillator(SHLO) in Eqs. (

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  20. [20]

    and ( 16) to the results we obtained from the numerical simulations. We found that for some trap parameters the SHLO PSDs are a good approximation for the Paul trap PSDs, however the dif- ferences can be much more than a factor of two for a wide range of parameters. For the trap parameters above, the ax ∼ 0. 0014 and qx ∼ 0. 14. These values lead to an x-...

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  21. [21]

    The Mathieu equation PSD Eq

    go exactly like 1/ Γ. The Mathieu equation PSD Eq. ( 21) only goes approximately like 1/ Γ for Γ ≪ ω 0 which is the case for

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  22. [22]

    1< Γ< 1. For the temperature dependence, because both the SHLO and the Mathieu equation’s PSD depend linearly on the temperature at their peak values, the temperature dependence cancels out in the relative difference between Eq. (

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  23. [23]

    and Eq. ( 21). Similarly both the SHLO and the Mathieu equation’s QPSD depend quadratically on the temperature, so the relative difference will not depend on the temperature. While the Paul trap parameters for Figs. (

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  24. [24]

    and ( 9) gave small difference from the results of a simple har- monic oscillator, the difference can be much larger for different parameters. In Figs. 10 and 11, VRF is in- creased from 200 V to approximately 870 V leading to an increase in value of the parameter q of the Mathieu equation from q ∼ 0. 14 to q ∼ 0. 6. This value for q is still within the stab...

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  25. [25]

    With an electric dipole In this section, the effect of a permanent electric dipole on the sphere is included in the equations of motion. If the Coulomb interaction is neglected, the N charges are distributed randomly on the surface of the sphere giv- ing a dipole roughly √ N × eR, where R is the radius of 8 0.0 0.5 1.0 1.5 224.6 224.7 224.8 224.9 PSD (10−1...

    work page 2000

  26. [26]

    Even with the electric dipole increased to 16 eR, 9 0.00 0.25 0.50 0.75 1.00 133.1 133.2 133.3 133.4 133.5 133.6 PSD (10−10 m2 sec) f (Hz) Γ = 0.5 Γ = 1.0 Figure 13

    and (16). Even with the electric dipole increased to 16 eR, 9 0.00 0.25 0.50 0.75 1.00 133.1 133.2 133.3 133.4 133.5 133.6 PSD (10−10 m2 sec) f (Hz) Γ = 0.5 Γ = 1.0 Figure 13. The PSD for the z-component for an electric dipole of 16eR, where e is the elementary charge and R is the radius of the nanosphere. The numerical PSD shown by the dotted black curve...

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  27. [27]

    by 23% for an electric dipole of 8eR with Γ = 0 . 5 s − 1. Also, as the electric dipole in- creases, the peak value of the numerical simulation’s PSD decreases and the peak shifts to smaller frequencies, Fig

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  28. [28]

    ( 21) is bigger the smaller the friction coefficient

    We observed that the percentage decrease in the peak value from the Paul trap’s PSD Eq. ( 21) is bigger the smaller the friction coefficient. From our simulations, for an electric dipole of 16 eR, when Γ = 0 . 5 s− 1 the de- crease was about 60% while for Γ = 1 . 0s− 1 the decrease was only about 40% as shown in Fig. 14(c). However, the QPSD only slightly d...

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  29. [29]

    From our simula- tions results with the same parameters as before and for a dipole of 8 eR, Γ = 1

    and ( 22) at smaller dipole moments more than for small q cases. From our simula- tions results with the same parameters as before and for a dipole of 8 eR, Γ = 1 . 0, and q ∼ 0. 6, the simulation’s QPSD is lower by approximately 8% from Eq. (

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  30. [30]

    Also, the numerical PSD has a peak value that is approximately 40% lower than the Paul trap PSD Eq

    near the peak. Also, the numerical PSD has a peak value that is approximately 40% lower than the Paul trap PSD Eq. ( 21). IV. CONCLUSION The numerical as well as the analytic expression for the PSD and QPSD of oscillators in several time depen- 0.0 1.0 2.0 3.0 224.4 224.6 224.8 225.0 225.2 (a) PSD (10−11 m2 sec) f (Hz) Γ = 0.5 Γ = 1.0 0.0 1.0 2.0 3.0 224....

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  31. [31]

    We obtained ana- lytic expressions that matched the numerical results

    For a slowly oscillating frequency the PSD was altered significantly from that of the pure harmonic os- cillator, however the only change in the QPSD was the appearance of two minor side peaks. We obtained ana- lytic expressions that matched the numerical results. The time independent case we considered was in the form of adding a quartic term to the poten...

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  32. [32]

    PSD We present here the convention we use for the PSDs of a position signal x(t) over a finite time interval τ. We chose Sx(ω ) = lim τ →∞ 2 τ |xτ (ω )|2 (A1) 11 where xτ (ω ) is the finite Fourier transform of x(t) over the time interval 0 <t<τ and is given by xτ (ω ) = 1√ 2π ∫ τ 0 x(t)eiωtdt (A2)

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  33. [33]

    This involves doing the calculation in a frame rotating with the frequency w0 [12],[6]

    QPSD The QPSD is the PSD relative to its average value at w = w0. This involves doing the calculation in a frame rotating with the frequency w0 [12],[6] . To this end the motionx(t) is decomposed into two parts xc(t) and xs(t) with: xc(t) +ixs(t) = 2x(t)ei¯ωt (A3) where ¯ω is the peak oscillation frequency of the oscillator, ω 0, or close to it. Next, filt...

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  34. [34]

    The exponentially decaying term in Eq

    PSD for a linearly drifting ω(t) This is the case for which ω (t) is taken to be ω (t) = w0 ( 1 − δ + 2δ τ t ) (B10) where the frequency drifts from ω 0(1 − δ) to w0(1 + δ) over the period from t = 0 to t =τ and δ <<1. The exponentially decaying term in Eq. ( B9) guar- anties that for each term in the sum, ( t − tl) is of order 12 1 Γ ≪ τ. Thus, the ω (t)...

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  35. [35]

    ( B9) φ(t)−φ(tl) = ω 0(t−tl)+ ∆ω Ω sin(Ω(t−tl)+ξl)− ∆ω Ω sin(ξl) (B16) Simplifying sin [φ(t) − φ (tl)] in Eq

    PSD for a slowly oscillating ω(t) For this case we have ω (t) = ω 0 + ∆ω cos(Ω(t − tl) +ξl) (B15) where ξl = Ωtl and was intruduced to simplify the ex- pression for φ(t) − φ(tl) in Eq. ( B9) φ(t)−φ(tl) = ω 0(t−tl)+ ∆ω Ω sin(Ω(t−tl)+ξl)− ∆ω Ω sin(ξl) (B16) Simplifying sin [φ(t) − φ (tl)] in Eq. ( B9) using sin(a − b) = sin(a) cos(b) − cos(a) sin(b), the te...

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  36. [36]

    PSD It is straightforward to get the PSD for Eq. ( C14). We just substitute with its Fourier transform in Eq. ( A1). However, since the interest is in the peaks near ω 0, we only include the terms oscillating at such frequency when taking the Fourier transform. Sx(ω ) ≃ lim τ →∞ 2 τ ⏐ ⏐ ⏐ ⏐ ⏐ ∑ l 1 √ 2π ∫ τ 0 eiωte− Γ 2 t ( sl sin(ω 0t) +cl cos(ω 0t) ) dt...

    work page

  37. [37]

    QPSD For the QPSD we will follow the same steps discussed in Sec. II C 1. First, xc(t) and xs(t) are calculated from the term in x(t) that is oscillating at a frequency w0. This gives: xc(t) = ∑ l ( sl sin(ψ l(t)) +cl cos(ψ l(t)) ) e− Γ(t− tl)/2Θ (t − tl) xs(t) = ∑ l ( sl cos(ψ l(t)) − cl sin(ψ l(t)) ) e− Γ(t− tl)/2Θ (t − tl) (C18) where ψ l(t) = ( ω 0 − ...

    work page

  38. [38]

    Search for millicharged particles using optically levitat ed microspheres

    David C Moore, Alexander D Rider, and Giorgio Gratta. Search for millicharged particles using optically levitat ed microspheres. Physical review letters , 113(25):251801, 2014

    work page 2014

  39. [39]

    Electromagnetic traps for charged and neutral particles

    Wolfgang Paul. Electromagnetic traps for charged and neutral particles. Reviews of modern physics , 62(3):531, 1990

    work page 1990

  40. [40]

    Force and acceleration sensing with optically levitated nanogram masses at microkelvin temperatures

    Fernando Monteiro, Wenqiang Li, Gadi Afek, Chang-ling Li, Michael Mossman, and David C Moore. Force and acceleration sensing with optically levitated nanogram masses at microkelvin temperatures. Physical Review A , 101(5):053835, 2020

    work page 2020

  41. [41]

    Short-range force detection using optically cooled levitated microspheres

    Andrew A Geraci, Scott B Papp, and John Kitch- ing. Short-range force detection using optically cooled levitated microspheres. Physical review letters , 105(10):101101, 2010

    work page 2010

  42. [42]

    Testing wave-function-collapse models using parametric heating of a trapped nanosphere

    Daniel Goldwater, Mauro Paternostro, and PF Barker. Testing wave-function-collapse models using parametric heating of a trapped nanosphere. Physical Review A , 94(1):010104, 2016

    work page 2016

  43. [43]

    Ultranarrow-linewidth levitated nano-oscillator for tes t- ing dissipative wave-function collapse

    A Pontin, NP Bullier, M Toroˇ s, and PF Barker. Ultranarrow-linewidth levitated nano-oscillator for tes t- ing dissipative wave-function collapse. Physical Review Research, 2(2):023349, 2020

    work page 2020

  44. [44]

    Searching for new physics using optically levitated sensors

    David C Moore and Andrew A Geraci. Searching for new physics using optically levitated sensors. Quantum Science and Technology, 6(1):014008, 2021

    work page 2021

  45. [45]

    Searching for new physics with a levitated-sensor-based gravitational-wave detector

    Nancy Aggarwal, George P Winstone, Mae Teo, Masha Baryakhtar, Shane L Larson, Vicky Kalogera, and An- drew A Geraci. Searching for new physics with a levitated-sensor-based gravitational-wave detector. Phys- ical review letters , 128(11):111101, 2022

    work page 2022

  46. [46]

    Detecting high-frequency gravitational waves with optically levi- tated sensors

    Asimina Arvanitaki and Andrew A Geraci. Detecting high-frequency gravitational waves with optically levi- tated sensors. Physical review letters , 110(7):071105, 2013

    work page 2013

  47. [47]

    Introduction to non-linear mechanics

    Nikolai Mitrofanovich Krylov and Nikolai Nikolaevich Bogoliubov. Introduction to non-linear mechanics . Num- ber 11. Princeton university press, 1950

    work page 1950

  48. [48]

    Asymptotic methods in the theory of non-linear oscilla- tions, volume 10

    Yu A Mitropolsky and IUri ˘ ı Alekseevich Mitropolski ˘ ı. Asymptotic methods in the theory of non-linear oscilla- tions, volume 10. CRC Press, 1961

    work page 1961

  49. [49]

    Energy spectrum of a langevin oscillator

    Y Mishin and J Hickman. Energy spectrum of a langevin oscillator. Physical Review E , 94(6):062151, 2016

    work page 2016

  50. [50]

    Thermal nonlinearities in a nanomechanical oscillator

    Jan Gieseler, Lukas Novotny, and Romain Quidant. Thermal nonlinearities in a nanomechanical oscillator. Nature physics , 9(12):806–810, 2013

    work page 2013

  51. [51]

    Trapped Charged Particles-A Graduate Textbook with Problems and Solutions

    Martina Knoop, Niels Madsen, and Richard C Thomp- son. Trapped Charged Particles-A Graduate Textbook with Problems and Solutions . World Scientific, 2016

    work page 2016

  52. [52]

    Characterisation of a charged particle levitated nano-oscillator

    NP Bullier, A Pontin, and PF Barker. Characterisation of a charged particle levitated nano-oscillator. Journal of Physics D: Applied Physics , 53(17):175302, 2020

    work page 2020

  53. [53]

    The fluctuation-dissipation theorem

    Rep Kubo. The fluctuation-dissipation theorem. Reports on progress in physics , 29(1):255, 1966

    work page 1966

  54. [54]

    Statistical physics II: nonequilibrium statistical mecha n- ics, volume 31

    Ryogo Kubo, Morikazu Toda, and Natsuki Hashitsume. Statistical physics II: nonequilibrium statistical mecha n- ics, volume 31. Springer Science & Business Media, 2012

    work page 2012

  55. [55]

    Statistical Physics: Volume 5 , volume 5

    Lev Davidovich Landau and Evgenii Mikhailovich Lif- shitz. Statistical Physics: Volume 5 , volume 5. Elsevier, 2013

    work page 2013

  56. [56]

    Atomic physics , volume 7

    Christopher J Foot. Atomic physics , volume 7. OUP Oxford, 2004

    work page 2004

  57. [57]

    The non- linear resonance ion trap

    Yang Wang, J Franzen, and KP Wanczek. The non- linear resonance ion trap. part 2. a general theoretical analysis. International journal of mass spectrometry and ion processes, 124(2):125–144, 1993

    work page 1993

  58. [58]

    Quadrupole ion trap mass spectrometry

    Raymond E March and John F Todd. Quadrupole ion trap mass spectrometry. John Wiley & Sons, 2005

    work page 2005

  59. [59]

    Clas- sical mechanics, 2002

    Herbert Goldstein, Charles Poole, and John Safko. Clas- sical mechanics, 2002

    work page 2002

  60. [60]

    Introduction to electrodynamics, 2005

    David J Griffiths. Introduction to electrodynamics, 2005

    work page 2005

  61. [61]

    Numerical recipes 3rd edi- tion: The art of scientific computing

    William H Press, Saul A Teukolsky, William T Vetter- ling, and Brian P Flannery. Numerical recipes 3rd edi- tion: The art of scientific computing . Cambridge univer- sity press, 2007

    work page 2007

  62. [62]

    Robust and efficient configurational molecular sampling via langevin dynamics

    Benedict Leimkuhler and Charles Matthews. Robust and efficient configurational molecular sampling via langevin dynamics. The Journal of chemical physics , 138(17):05B601 1, 2013

    work page 2013

  63. [63]

    Loading a linear paul trap to satura- tion from a magneto-optical trap

    JE Wells, Reinhold Bl¨ umel, JM Kwolek, DS Goodman, and WW Smith. Loading a linear paul trap to satura- tion from a magneto-optical trap. Physical Review A , 95(5):053416, 2017

    work page 2017

  64. [64]

    Meso- scopic physics of nanomechanical systems

    Adrian Bachtold, Joel Moser, and MI Dykman. Meso- scopic physics of nanomechanical systems. arXiv preprint arXiv:2202.01819, 2022

    work page arXiv 2022

  65. [65]

    Higher order non-linear resonances in a paul trap

    R Alheit, S Kleineidam, F Vedel, M Vedel, and G Werth. Higher order non-linear resonances in a paul trap. In- ternational Journal of Mass Spectrometry and Ion Pro- cesses, 154(3):155–169, 1996

    work page 1996

  66. [66]

    Compu- tation of mathieu stability plot for an arbitrary toroidal ion trap mass analyser

    Appala Naidu Kotana and Atanu K Mohanty. Compu- tation of mathieu stability plot for an arbitrary toroidal ion trap mass analyser. International Journal of Mass Spectrometry, 414:13–22, 2017

    work page 2017

  67. [67]

    Parametric ex- citations of trapped ions in a linear rf ion trap

    X Zhao, VL Ryjkov, and HA Schuessler. Parametric ex- citations of trapped ions in a linear rf ion trap. Physical Review A , 66(6):063414, 2002

    work page 2002

  68. [68]

    Sur les ´ equations diff´ erentielles lin´ eaires ` a coefficients p´ eriodiques

    Gaston Floquet. Sur les ´ equations diff´ erentielles lin´ eaires ` a coefficients p´ eriodiques. In Annales scientifiques de l’ ´Ecole normale sup´ erieure, volume 12, pages 47–88, 1883

    work page

  69. [69]

    Floquet-based analysis of general responses of the mathieu equation

    Gizem Acar and Brian F Feeny. Floquet-based analysis of general responses of the mathieu equation. Journal of Vibration and Acoustics , 138(4), 2016

    work page 2016