Direct Measurement of Kramers Turnover with a Levitated Nanoparticle

TL;DR

This study provides the first direct experimental observation of the Kramers turnover phenomenon by measuring transition rates of a levitated nanoparticle in a bi-stable optical potential across different damping regimes.

Contribution

The paper presents the first direct, quantitative experimental verification of the Kramers turnover using a levitated nanoparticle in a bi-stable optical trap.

Findings

Measured transition rates across damping regimes

Confirmed the theoretical Kramers turnover behavior

Results agree with parameter-free analytical model

Abstract

Understanding the thermally activated escape from a metastable state is at the heart of important phenomena such as the folding dynamics of proteins, the kinetics of chemical reactions or the stability of mechanical systems. In 1940 Kramers calculated escape rates both in the high damping and the low damping regime and suggested that the rate must have a maximum for intermediate damping. This phenomenon, today known as the Kramers turnover, has triggered important theoretical and numerical studies. However, to date there is no direct and quantitative experimental verification of this turnover. Using a nanoparticle trapped in a bi-stable optical potential we experimentally measure the nanoparticle's transition rates for variable damping and directly resolve the Kramers turnover. Our measurements are in agreement with an analytical model that is free of adjustable parameters.

Equations

$$R_{AC}^{\mathrm{HD}}=\frac{1}{2\pi }\prod _{i\in \{x,y,z\}}\frac{{\omega }_i^A}{|{\omega }_i^B|}\left[\sqrt{|{\omega }_B^S{|}^2+\frac{{\Gamma }^2}{4}}-\frac{\Gamma }{2}\right]e^{-\frac{U_A}{k_{B}T}},$$

$$I(\delta )={\int }_{-\infty }^{\infty }\ln \left\{1-\exp \left[-\frac{\delta }{k_{B}T}(x^2+\frac{1}{4})\right]\right\}\frac{\mathrm{d}x}{x^2+\frac{1}{4}},$$

$$R(\Gamma )=\frac{{\rm Y}(\Gamma S_A){\rm Y}(\Gamma S_C)}{{\rm Y}(\Gamma S_A+\Gamma S_C)}\left[R_{AC}^{\mathrm{HD}}+R_{CA}^{\mathrm{HD}}\right],$$

$$U(\mathbf{r})=-k_{B}T\ln \left[\rho (\mathbf{r})\right],$$

$$U_{\mathrm{HO}}(\text{r})=U_{\alpha }+\frac{1}{2}\sum _{i,j}{\Lambda }_{i,j}^{\alpha }(r_i-r_i^{\alpha })(r_j-r_j^{\alpha }),$$

$${\Pi }_{\mathrm{fit}}(x,y,z)=\sum _{i,j,k}{\mu }_{i,j,k}{x}^i{y}^j{z}^k$$

$$S_i=4{\int }_{r_i}^{r_B}\sqrt{2m[U_B-U(r)]}\mathrm{d}r,$$

$$R^{\text{LD}}=\frac{S_A{S}_C}{S_A+S_C}\frac{\Gamma }{2\pi k_{B}T}\left({\omega }_{y^{\prime }}^A{e}^{-\frac{U_A}{k_{B}T}}+{\omega }_{y^{\prime }}^C{e}^{-\frac{U_C}{k_{B}T}}\right).$$

$$S_{qq}^{\mathrm{fit}}(\omega )=\sum _{i\in \{x,y,z\}}\frac{{\alpha }_{qi}}{({\omega }_i^2-{\omega }^2{)}^2+{\omega }^2{\Gamma }_i^2}+\beta ,$$

$${\mathcal{C}}_{\mathrm{calib}}^q(y)=\sqrt{\frac{{\alpha }^q(y)\pi m}{\Gamma k_{B}T}},$$

$$h_{A}(t)=\left\{\begin{array}[]{ll}1&\mbox{if the particle is measured in subspace $\mathcal{A}$ at time }t\\ 0&\mbox{if it is measured in $\mathcal{C}$ ,}\end{array}\right.$$

$$h_{A}(t)=\left\{\begin{array}[]{ll}1&\mbox{if \ }y(t)<0\\ 0&\mbox{else.}\end{array}\right.$$

$$\Gamma=\left\{\begin{array}[]{ll}P_{\text{gas}}^{\text{gauge}}/51\leavevmode\nobreak\ \text{Hz/Pa}&\mbox{for \ }P_{\text{gas}}<5\times 10^{3}\leavevmode\nobreak\ \text{Pa}\\ \Gamma_{\text{PSD}}&\mbox{for \ }P_{\text{gas}}\geq 5\times 10^{3}\leavevmode\nobreak\ \text{Pa}\\ \end{array}\right.$$

$$P_{\text{gas}}=51\text{leavevmode}\text{Pa/Hz}\text{leavevmode}\Gamma .$$

Full text

Methods

Experimental setup

The optical double well trap is generated using two cross-polarized, and frequency shifted (40 MHz) beams from a continuous-wave infrared laser ($\lambda=1064$ nm, Coherent Mephisto). Before entering the chamber, the two orthogonal polarizations are recombined using a polarizing beam splitter (PBS), and focused by a N.A.=0.8 objective (Nikon). The relative position of the two traps can be changed using a steering mirror (SM, Newport Fast Steering Mirror).

The particle position is recorded using an additional, weakly focused, green laser ($\lambda=532$ nm, Coherent) and three balanced detectors (Newport, 1817-FC). The signal is filtered (low pass filter at 1 MHz) before recording on a fast data acquisition card (Gage, Oscar CS4342). The 3D particle trajectory is reconstructed from the measured signal using a calibration function, determined independently. Additional details on the experimental setup and the measurement calibration can be found in the Supplementary Sections A and B.

Determination of physical parameters

1. Positions of the saddle and stable points : The optical potential is derived from the particle distribution function $\rho(\mathbf{r})$ obtained by histogramming the particle position time trace. Thus, the optical potential $U$ at each point $\mathbf{r}$ follows from

[TABLE]

where $k_{B}$ is the Boltzmann constant and $T=300$ K is the temperature of the gas. By differentiation we obtain the effective force field $\mathbf{F}(\mathbf{r})=-\mathrm{\bf grad}\ U(\mathbf{r})$. The force vanishes at the two stable points $A$ and $C$, and at the saddle point $B$.

2. Energy barriers and potential curvature : Near the saddle point and the stable points, the potential energy can be approximated by a harmonic function,

[TABLE]

where $\alpha=A,B,$ or $C$, $\textbf{r}^{\alpha}$ is the spatial position of point $\alpha$, $U_{\alpha}$ is the potential energy at point $\alpha$, and $i,j\in\{x,y,z\}$.

A local fit of the measured potential $U_{\mathrm{HO}}$ yields the matrix $(\Lambda^{\alpha}_{i,j})$ for each of the three spatial points. The three eigenvalues $\lambda^{\alpha}_{k}$ of each matrix correspond to the curvatures along the three local normal axes $k\in\{x^{\prime},y^{\prime},z^{\prime}\}$ of the potential, $\omega_{k}^{\alpha}=\sqrt{{\lambda^{\alpha}_{k}}/{m}}$, where $m$ is the mass of the particle, and the primed coordinates denote the local coordinate system. Alternatively, the full potential can be fitted by

[TABLE]

to derive the curvature in any point of the potential. Good convergence is typically obtained for a polynomial at order $\sim 10$. The results of the two different fitting procedures agree within a few percent.

3. Particle action : The particle action over one oscillation is approximated by the action over the minimum energy path, given by

[TABLE]

where $i=A,C$.

The integration is carried out along the minimum energy path, which is the path between the two stables points $A$ and $C$ that minimizes the energy. We determine the minimum energy path using a steepest gradient algorithm. This path is shown in Fig. 1c. As a guide for the eye, the path in this figure has been continued beyond $A$ and $C$, although there is no minimum energy path outside of the $A$ to $C$ range. Finally, note that the factor $4$ in Eq. (7) originates from measuring only one fourth of the full oscillation period.

The evolution of recorded parameters as a function of pressure is presented in the Supplementary Information section (Fig S7).

Low and high damping limiting cases.

Figure 3 presents the theoretical limiting cases, corresponding to the low and the high damping regimes (blue dash-dotted lines). The high damping rates have been computed using eq (1), while the low damping rates correspond to the equation melnikov91 :

[TABLE]

Note that this equation is an approximation for the low damping regime in one dimension that should be a good approximation for our three-dimensions case since the three normal modes are only weakly coupled hershkovitz97 .

Supplementary Information

.1 Experimental Setup

An optical double well trap is generated using two cross-polarized beams from a continuous-wave infrared laser ($\lambda=1064$ nm), split by a polarizing beam splitter (PBS) (see Fig. S1). One polarization is frequency shifted by 40 MHz with an acousto-optic modulator (AOM) to suppress effects related to interferences of the two trapping beams. Before entering the chamber, the two orthogonal polarizations are recombined using a second PBS, and focused by a N.A.=0.8 objective. The transverse position of the focus depends on the beam angle at the objective back-focal-plane. The relative angle of the two trapping beams can be easily changed using a telecentric $4f$ setup combined with a steering mirror (SM, Newport Fast Steering Mirror). In addition, the optical power of each trapping beam can be adjusted using half-wave plates (HWP). Therefore, by changing the angle and adjusting the power we can tune the symmetry and the depth of the optical double-well potential.

An additional green laser ($\lambda=532$ nm) is used for measurement. It is combined with the trapping beams using a dichroic mirror (DM). The beam size of the measurement laser at the objective back-aperture is chosen such that the effective NA for this beam is roughly $0.2$. This helps to reduce the nonlinearities in the position measurement over the whole optical double trap. The measurement beam is then recollimated with an aspheric lens (AsL), and isolated from the trapping beam using a 532 nm band pass filter (BPF).

The detection scheme is based on a common path interferometer and is described in Ref. gieseler12SI . In short, the motion of the particle is imprinted in the phase of the re-scattered light, which interferes with the non-scattered transmitted light beam. The phase modulation is thus converted into an amplitude modulation that is directly recorded using fiber coupled 80 MHz balanced detectors (1807-FC, Newport). For a particle close to the focus of the measurement beam, recording the raw signal provides a signal proportional to the particle’s $z$-motion. Similarly, by splitting the beam vertically [or horizontally] with a D-shape mirror (DsM), a signal proportional to the $x$ [or $y$] motion is obtained. As discussed in the next section, for large displacement of the particle from the optical axis, non linearities have to be taken into account in the measurement scheme.

.2 Detection scheme calibration

A voltage is recorded on the three detectors. To reconstruct the dynamics of the particle the recorded voltage has to be expressed as a distance in nanometers through a calibration factor $\mathcal{C}_{\mathrm{calib}}$. Recent experiments involving levitated particles in vacuum work in a regime where the recorded signal is simply linear with the particle position and, as a consequence, the calibration factor $\mathcal{C}_{\mathrm{calib}}$ is constant. In the present work, the particle displacement is large and the detection scheme is no longer linear. To account for these nonlinearities we introduce the calibration functions $\mathcal{C}_{\mathrm{calib}}^{q}(\vec{r})$, where $q\in\{x,y,z\}$ is the considered axis, and $\vec{r}$ the position of the particle.

To experimentally determine the $\mathcal{C}_{\mathrm{calib}}^{q}$ functions, a particle is trapped using a single trapping beam. The particle is then scanned through the focal plane of the measurement beam by displacing the trap with a fast steering mirror (Fig. S2-(a)). At each point in the focal plane the particle dynamics is recorded for 2 s using a fixed measurement beam. Our telecentric measurement scheme allow us to keep the axes of the particle’s trapping potential aligned with the axes of the measurement beam. The power spectral density (PSD) is computed from the recorded time traces for the 3 measurement axes $x$, $y$ and $z$. Each of the PSDs can features 3 peaks at the three eigenfrequencies $\omega_{x}$, $\omega_{y}$, and $\omega_{z}$, which correspond to the three trap axes. The presence of three frequencies is due to cross coupling between the different measurement axes. Indeed, under certain circumstances the $x$-detector may, for example, detect a signal coming from the motion along to $y$ and/or $z$ axis (Fig. S2-(b)). To completely characterise the detection nonlinearities and associated axes cross-coupling, each of the three peaks is fitted by the function

[TABLE]

where $\alpha_{qi},\beta,~\Gamma_{i}$ and $\omega_{i}$ are adjustable parameters, and $q$ is the measurement axis. As an example, a PSD $S_{xx}$ recorded for the particle away from the measurement beam’s centre is shown in Fig. S2. The measurement is overlaid with analytical fitting functions according Eq. (9).

The amplitudes $\alpha_{qi}$ of the three peaks present in the PSD provide information about a detector’s sensitivity to the particle motion, both along the axis of interest and along the two orthogonal axes (cross-coupling). These amplitudes, normalized by the overall maximum amplitude $\alpha_{0}$, for the three trap frequencies and for each detector are shown in Fig. S3. As expected, we observe that the detection sensitivity is constant only near the centre of the measurement beam. For larger displacements the sensitivity drops and cross-couplings come into play. Such a behaviour is theoretically expected and has already previously been used to extend the spatial range of particle dynamics studies perrone08SI .

In our experiment, the two potential wells are aligned along the $y$-axis. Consequently, large displacements are recorded along the $y$-axis, whereas displacements along the $x$ and $z$-axes remain small (blue dotted rectangular area in figure S3) we assume that cross-couplings can be neglected and that the detection sensitivity depends only on $y$-axis. By virtue of the equipartition theorem, the integrated area under the power spectral density is proportional to the bath temperature $k_{B}T$ gieseler12SI . Therefore, from the fit of the PSD we extract the calibration function for each axis,

[TABLE]

which converts the measured voltage on the detector to particle displacement in meters. To reduce noise, this calibration function is integrated along the $x$-axis in the blue dotted rectangular area of figure S3. The obtained calibration functions for the three different axes are shown in figure S4. These functions are fitted by a 10th order polynomial (red solid line) that is used to calibrate the particle position from the recorded time traces. From these calibration functions we estimate the sensitivity of our measurement. Typical noise on the detectors is on the order of 300 $\mu$V. Given that the conversion factor for $x$-and $y$-axes is better than 50 nm/mV, we deduce that the position is known with a precision better than 15 nm. For the $z$-axis, the sensitivity is almost ten times worse due to the use of a weakly focused measurement beam. However, errors in the $z$-axis position only weakly affect the theoretical rate estimate, since the $z$-axis is orthogonal to the trapping axis.

Measuring the PSD of the particle dynamics allows us to also retrieve information on the gas damping $\Gamma$. The particle, while trapped in the centre of the measurement beam, is recorded at different gas pressures and the PSD is computed (Fig. S5-(a)).

From the fits we retrieve the value of the gas damping $\Gamma$ and verify its linear relationship with the pressure $P_{\mathrm{gas}}$ (Fig. S5-(b)). As expected from kinetic theory for a 68 nm radius particle gieseler12SI ; beresnev90SI , we experimentally measure $\Gamma/P_{\mathrm{gas}}\approx 51$ Hz/Pa.

.3 Rate Measurements

To determine the particle jumping rate we split the space into two sub-spaces $\mathcal{A}$ and $\mathcal{C}$. These two sub-spaces are separated by a dividing surface, which sets the boundary between the two traps $A$ and $C$. Transition rates are extracted from the time autocorrelation function $\langle\delta h_{A}(0)\delta h_{A}(t)\rangle$, where $h_{A}$ is the binary population function defined as

[TABLE]

and $\delta h_{A}=h_{A}-\langle h_{A}\rangle$ is the fluctuation of $h_{A}$.

With this approach, the resulting rate constant is not affected by the exact position of the dividing surface. Therefore, for simplicity, we choose the $x\!-\!z$ plane as a separating surface. Thus, the population binary function is obtained by assigning to each measurement point $\{x(t),y(t),z(t)\}$ of the particle motion, the value

[TABLE]

An example of the time evolution of the population function $h_{A}$ is shown in figure S6.

The time autocorrelation $\langle\delta h_{A}(0)\delta h_{A}(t)\rangle$ of a particle jumping between two traps is characterized, in general, by a transient for short times (due to correlated recrossings) followed by an exponential decay dellago09SI ; chandler78SI . An exponential fit $e^{-Rt}$ of the long-time behaviour yields the jumping rate $R=R_{\mathrm{AC}}+R_{\mathrm{CA}}$.

.4 Evolution of Physical Parameters

As mentioned in the main text, the optical potential is affected by slow drifts. Therefore, the parameters of the optical potential have to be re-evaluated for each setting of the vacuum pressure. Figure S7 shows the variations of normal mode frequencies and energy barriers as a function of gas pressure.

.5 Estimation of the damping rate

The damping rate is central to calculating the transition rates. For a levitated particle the damping rate is proportional to the gas pressure inside the experimental chamber gieseler12SI ; beresnev90SI . Therefore, in principle, we can directly measure the damping rate using our pressure gauge (Pirani gauge, Instrutech), once the proportionality factor between pressure and damping factor is know. As shown in the calibration subsection (Figure S5), a fit to the PSD of the particle dynamics provides an estimate of the damping term, and is used to determine the relation between damping and pressure at pressure.

However, it is important to note that this approach is only valid when the non-linearities of the potential are weak compared to the damping, i.e for pressures above $\approx 500$ Pa for a single beam trap gieseler13SI .

In the following we estimate the damping constant from the particle motion when the particle is trapped in a double well potential. A double trap setup is intrinsically non-linear. To overcome this limitation we isolate the dynamics of the particle when localised in one of the traps. Using a threshold algorithm, we segment the timetrace of the particle motion along the $x$-axis (orthogonal to the double trap) and only keep the parts when the particle is exclusively inside trap $A$ (resp. trap $C$). The PSD calculated from this sub time traces are fitted using equation (9) to determine the linewidth $\Gamma^{\text{PSD}}$ (See also section S2 where we did this for a particle trapped in a single well potential). The obtained linewidths, for the particle localised in both trap $A$ and trap $C$ are presented in Figure S8. We distinguish three interesting regimes.

• •

For pressures below $10^{3}$ Pa, the intrinsic non-linearities of the trapping potential become dominant, and the PSD linewidth is not an appropriate measure of the damping $\Gamma$. Since we know that there is a linear relationship between the pressure and the damping $\Gamma$ and the pressure gauge is well calibrated in this regime, we can calculate the damping as $\Gamma=P^{\text{gauge}}/51$ Hz/Pa.

• •

For pressures between $10^{3}$ and $5\times 10^{3}$ Pa,the pressure gauge is still well calibrated and nonlinearities in the optical potential do not broaden the PSD. As a consequence, we use this regime to calibrate the proportionality factor between damping and pressure.

• •

Above $5\times 10^{3}$ Pa, nonlinearities in the pressure gauge overestimate the actual pressure in the vacuum chamber. This leads to a deviation from the linear relationship. In contrast to the low pressure regime nonlinearities in the optical potential do not broaden the PSD and the fit to the PSDs provide a good estimate for the damping $\Gamma$.

Note that, since we are working with a double trap the non-linearities are higher than in the case of the single trap (see Figure S5). Consequently, figure S5 and figure S8 only agrees above 1000 Pa.

To summarize, we determine the experimental damping as

[TABLE]

Conversely, the real pressure is given by

[TABLE]

.6 Validity of the model

Our quantitative analysis is limited by experimental drifts. Drifts impact both, the shape of the potential and its measurement.

In the present case, we experimentally measure independently for each pressure the potential parameters, as previously discussed (see Methods and Figure S7). The estimation errors depends on:

• •

the errors arising from fitting the potential. This error is estimated to be below 5 % for the characteristic frequencies and a few percent for the energy barriers.

• •

the inaccuracy of our non-linear measurement, which is much harder to quantify.

To determine the impact of these two effects, we propose to compare, in Figure S9-(a) (blue curve), the error between the measured rate $R$ and the theoretically expected rate using the measured parameters at each pressure (Figure S7).

At low pressure, in a regime where drifts are negligible, we observe an almost constant error of less than $\approx 20$ %. This error can be completely understood in terms of errors on potential parameters. In addition, the fact that this error is almost constant, is a strong indication for systematic errors in the estimation of potential parameters.

Above a few thousands Pascals, we observe that drifts start to play an increasing role. This is illustrated by the evolution of the trap and the saddle point positions, shown in Figure S9-(b). These drifts, not only change the potential, but they also impact the measurement. For example, the regions of the potential that are further away from the center yields stronger non-linearities of the calibration functions (c.f. Figure S4). These leads to an error in the estimation of the potential energy barriers. We believe that this effect is responsible for the fast change in the measured energy barriers at the highest pressures (c.f. Figure S7-(b)). To test this hypothesis, we compute the theoretical rate from the average energy barriers over the first ten pressures (up to 1050 Pa). This method reduces the difference between the theoretical and the experimental rate (Figure S9-(b), black). We therefore anticipate that improving the stability of the experiment should allow to measure the transition rates to within a few percent.

Finally, note that the error in the measurement also affects the characteristic frequencies. However, this effect is weaker. Firstly because their measurement is local, in an area well calibrated, and secondly because errors in the energy barriers propagate more strongly because the rate depends exponentially on the energy barriers (see Eq. (1) of the main text).