Giuseppe Fofﬁ, Laszl´ o Forr´ o, Sylvia Jeney´ Thomas ... · Thomas Franosch, Matthias Grimm,...

W W W. N A T U R E . C O M / N A T U R E | 1

SUPPLEMENTARY INFORMATIONdoi:10.1038/nature10498

Resonances arising from hydrodynamic memory in Brownianmotion

Thomas Franosch, Matthias Grimm, Maxim Belushkin, Flavio M. Mor,Giuseppe Foffi, Laszlo Forro, Sylvia Jeney∗

∗To whom correspondence should be addressed; E-mail: [email protected]

August 17, 2011

SUPPLEMENTARY INFORMATION

Section 1. Instrument set-up and characterization. Our Instrument consists of a custom-built, ultra-stable

inverted light microscope with a three-dimensional (3D) sample positioning stage, an infrared laser for optical

trapping, and a quadrant photodiode (QPD) for high-resolution and time-resolved position detection. The trapping

beam is produced by a diode-pumped, Gaussian beam, ultra-low noise Nd:YAG laser with a wavelength of λ =

1064nm (IRCL-500-1064-S, CrystaLaser, USA), and a maximal light power of 500mW in continuous wave mode.

Best trapping efficiency is achieved by expanding the effective laser beam diameter 10 times with a telecentric

lens system (Sill Optics, Germany). The laser is linearly polarized in the x direction, which results in Ky Kx

by maximally 15% as reported previously31. The position detector consists of an InGaAs quadrant photodiode30

(G6849, Hamamatsu Photonics, Japan) is 2.0 mm in diameter with a dead zone of 0.1 mm between the quadrants.

The photosensitivity is 0.67 A/W at 1064 nm. The QPD signals are fed into a custom-built preamplifier (Offner

MSR-Technik, Germany), which provides two differential signals between the segments and one signal that is

proportional to the total light intensity. This allows detecting the particle’s position in all three dimensions. Pre-

amplification of the QPD signals at 20 V/mA with 0.67 A/W photosensitivity leads to a voltage of 13.4 V/mW.

Subsequently, differential amplifiers (Offner MSR-Technik, Germany) adjust the preamplifier signals for optimal

digitalization by the data acquisition board with a dynamic range of 12 bits (NI-6110, National Instruments, USA).

Amplification of the QPD signal is chosen to span the maximal dynamic range of the acquisition card. Each of our

13 analogous amplifiers has a maximal gain of 500 and a built-in electronic low-pass filter with an adjustable cut-off

frequency, f3DB . It is a Butterworth-type filter of first order, which we have set at 1MHz. Data were acquired at

the sampling frequency of 1 MHz. Further details on the set-up are given in 19,32 and its online supplement.

In order to exclude effects from instrumental noise to be at the origin of the observed resonances, we determined

the noise floor of our set-up and its influence on the measurements. Fig. S1 displays the full, unbinned spectra

from 0.2 Hz to 500 kHz in a log-log representation and in all 3 dimensions of a bead with a diameter of 2.9 µm in a

strong optical trap (blue graphs). The directions lateral to the optical axis are denoted x and y, and z gives the axis

along the laser beam path.

First, no 1/f noise is observed below 100 Hz in any direction, pointing to a perfectly spherical particle (see also

Fig. S3a). This excludes particle asymmetry to be at the origin of the detected resonance in the PSD. However, we

observe at f ≈ 90 Hz a very sharp peak, which is unstable in intensity, can vary in frequency from approximately

80 to 100 Hz and was also observed by other authors33. As shown by the red graphs in Fig. S1a to c, the peak

disappears when switching off the laser, then only recording the noise originating from the QPD and the remaining

electronic circuits. Furthermore, we also detected this ≈ 90 Hz-peak in measurements with three other Nd:YAG

lasers each from different manufacturers. The origin of this disturbance resides, hence, most likely within the

laser circuit. As a consequence, under conditions where the resonant peak of Brownian motion is close to 100 Hz,

detection becomes seriously distorted by laser noise. At lower trap stiffness, when the plateau in the Brownian

fluctuation signal (blue graphs in Fig. S1a-c) is higher and the corner frequency much lower, the strength of the

laser peak has a proportionally less significant influence on the data.

Additionally to further noisy peaks, the green graphs in Fig. S1a to c also display a 1/f2 decrease between 0.2 Hz

and 2 kHz, which can be attributed to the diffusive motion of the laser spot on the QPD and gives an indication

of its pointing stability. At frequencies above 2 kHz the noise attains a plateau. The green graph represents our

noise floor. The signal-to-noise ratio of our set-up (S/N) can hence be defined as the difference between the signal

arising from the position fluctuations of the Brownian sphere (blue) and the laser noise (green). Low frequency

noise below 5 Hz arises from mechanical drift and the limited measurement time leading to low statistics.

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3 analogous amplifiers has a maximal gain of 500 and a built-in electronic low-pass filter with an adjustable cut-off

frequency, f3DB . It is a Butterworth-type filter of first order, which we have set at 1MHz. Data were acquired at

the sampling frequency of 1 MHz. Further details on the set-up are given in 19,32 and its online supplement.

In order to exclude effects from instrumental noise to be at the origin of the observed resonances, we determined

the noise floor of our set-up and its influence on the measurements. Fig. S1 displays the full, unbinned spectra

from 0.2 Hz to 500 kHz in a log-log representation and in all 3 dimensions of a bead with a diameter of 2.9 µm in a

strong optical trap (blue graphs). The directions lateral to the optical axis are denoted x and y, and z gives the axis

along the laser beam path.

First, no 1/f noise is observed below 100 Hz in any direction, pointing to a perfectly spherical particle (see also

Fig. S3a). This excludes particle asymmetry to be at the origin of the detected resonance in the PSD. However, we

observe at f ≈ 90 Hz a very sharp peak, which is unstable in intensity, can vary in frequency from approximately

80 to 100 Hz and was also observed by other authors33. As shown by the red graphs in Fig. S1a to c, the peak

disappears when switching off the laser, then only recording the noise originating from the QPD and the remaining

electronic circuits. Furthermore, we also detected this ≈ 90 Hz-peak in measurements with three other Nd:YAG

lasers each from different manufacturers. The origin of this disturbance resides, hence, most likely within the

laser circuit. As a consequence, under conditions where the resonant peak of Brownian motion is close to 100 Hz,

detection becomes seriously distorted by laser noise. At lower trap stiffness, when the plateau in the Brownian

fluctuation signal (blue graphs in Fig. S1a-c) is higher and the corner frequency much lower, the strength of the

laser peak has a proportionally less significant influence on the data.

Additionally to further noisy peaks, the green graphs in Fig. S1a to c also display a 1/f2 decrease between 0.2 Hz

and 2 kHz, which can be attributed to the diffusive motion of the laser spot on the QPD and gives an indication

of its pointing stability. At frequencies above 2 kHz the noise attains a plateau. The green graph represents our

noise floor. The signal-to-noise ratio of our set-up (S/N) can hence be defined as the difference between the signal

arising from the position fluctuations of the Brownian sphere (blue) and the laser noise (green). Low frequency

noise below 5 Hz arises from mechanical drift and the limited measurement time leading to low statistics.

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SUPPLEMENTARY INFORMATION RESEARCH

Fig. S1: a,b,c, Comparison of the measured PSD in a log-log plot (computed from overlapping windows of 222

points) in all 3 dimensions x, y respectively z for trapped a bead immersed in acetone (R = 1.45 µm). The elec-

tronic noise comprising the noise from the QPD, the pre-amplifier circuit, the amplifier and the data acquisition

card is shown in red. The green graphs represent the PSD in each dimension, when no bead is trapped and the laser

beam is directly impinging on the QPD, determinung our noise floor. The blue graphs show the contribution of the

Brownian particle’s fluctuations to the PSD. The corner frequencies give trapping stiffnesses of Kx = 226µN/m,

Ky = 212µN/m, Kz = 40 µN/m. d, Direct comparison of the noise floor and signal-to-noise ratio in each dimen-

sion.

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A direct comparison of the S/N between each dimension is given in Fig. S1d. In the frequency range be-

tween 0.1 and 500 kHz, relevant to our experiments, the S/N is more than 2 orders of magnitude better in x and

y compared to z, which has also the weakest trap stiffness. Therefore, subtleties in Brownian motion are clearly

detectable in x and y, but remain hidden by noise in z. As a rule of thumb, the detection threshold for inertial

effects requires a S/N spanning over more than 6 decades in signal.

Fig. S2: a, Log-log plot of the measured (coloured circles) and fitted (black line) normalized PSD multiplied by

the gain GB(f) of the respective low-pass filter for the x direction. b, Lin-log representations of the same datasets

to highlight the resonant peak. Insets, Corresponding data in the y direction. All data were computed from over-

lapping windows of 222 points and blocked in 20 bins per decade. Experimental conditions were the same as in

Fig. S1.

We have also studied the effect of our electronic low-pass filter, when changing from f3DB = 1MHz down

to 50 or 20 kHz as shown in Fig. S2 for x and y. Data were fitted accordingly taking the gain, |GB(f)|2 =

1/[1 + (f/f3dB)]2, of the first order Butterworth-type filter into account21. Fig. S2 clearly highlights that the

4



low-pass filter has no effect on the resonant peak. As a consequence, strong filtering and signal preprocessing is

unnecessary to distinguish colour when instrumental noise is as low as in the present set-up.

All in all, the graphs in Fig. S1 indicate that the trapping laser is the strongest noise source and point to the im-

pressive S/N ratio in x and y of our detection system even at maximal trap stiffness, when the corner frequency is

high and the plateau in the Brownian fluctuation signal very low and closer to the noise floor.

Section 2. Materials. Melamine resin microspheres (ρp = 1510 kg/m3, R = 1.5, 1.45, 1.35 or 1 µm,

microparticles GmbH, Germany and Sigma-Aldrich, Analytix 5, 2001) were suspended in high purity acetone

(η = 0.32cP, ρf = 790 kg/m3) or water (η = 0.95cP, ρf = 1000 kg/m3) at minimal concentrations to allow trap-

ping and observation of exclusively one particle. The sample chambers were made of a microscope slide having

a ground-in polished cavity in its center (18mm in diameter and 0.8mm in depth, Marienfeld GmbH, Germany).

The cavity was filled with the bead-acetone or bead-water mixture and sealed with a glass coverslip (thickness

∼ 130µm) stuck by high viscosity vacuum grease to prevent evaporation. Special care was taken not to introduce

bubbles into the sample chamber. We verified the sphericity of the used beads by high resolution electron mi-

croscopy as shown in Fig. S3a.

Fig. S3: a, High resolution electron micrograph of a 2.9 µm sphere. b, Absorption spectrum of water (blue line)

and acetone (red line).

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In order to exclude heating effects of acetone through laser absorption, we measured the absorbance of acetone

at λ = 1064 nm. Fig. S3b highlights that the absorbance of acetone is more than twice smaller than the one of

water. The maximal laser power we employed in our experiments was measured to be 200 mW at the focus. This

corresponds to an increase in temperature of less than 2C in water according to 34 and consequently much less

than 1C in acetone. Therefore, we assumed a temperature of ∼ 21C, our laboratory temperature, for experi-

ments in acetone and ∼ 22C for experiments in water. The corresponding solvent viscosities were taken from

www.ddbst.com en online Online Calc visc Form.php.

Material-related effects of the trapped particles on the resonance were excluded by performing the same exper-

iments with a different bead material. Fig. S4 shows a comparison between silica (ρp = 2000 kg/m3, Bangs

Laboratories, Inc.) and resin in acetone. As expected, both datasets (coloured circles) fit the theory (black lines)

with very high accuracy and display significant deviations from the Lorentzian spectra (coloured lines).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

102

103

104

Silica

Resin

Acetone

PS

D(f

)/P

SD(0

)

Frequency f [Hz]

Fig. S4: Semilogarithmic plot of the PSD normalized to its zero-frequency value for optically trapped micropar-

ticles of different material but same size (R 1.45 µm) and at the same incoming laser power of ∼ 200 mW

at the focus: melamine resin (blue circles, K = 260 µN/m, τf = 5.2µs, τK = 33.5µs), silica (green circles:

K = 85 µN/m, τf = 4.8µs, τK = 98.6µs). The black lines in each plot correspond to the full hydrodynamic theory

including inertial effects and the coloured lines are Lorentzian spectra for reference.

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Section 3. Data analysis. Since the location of the trap center is not known with high precision, the position

autocorrelation function, PAF(t) = 〈x(t)x(0)〉 cannot be computed directly. Rather, the PAF is derived from the

mean-square displacement, MSD(t), via the relation PAF(t) = kBT/K−MSD(t)/2. The MSD is computed from

the position time trace x(t) as

MSD(t) = 〈∆x2(t)〉 =1N

∑j

[x(t + tj) − x(tj)]2, (1)

with j = 0, ..., N , tj = j∆t, and fS = 1/∆t = 1MHz our sampling frequency. Using the long-time limit of the

MSD determined by the confinement of the trap, MSD(t → ∞) = 2kBT/K, the normalized PAF reads

PAF(t)/PAF(0) = 1 − MSD(t)/MSD(∞). (2)

We decomposed x(t) into Fourier modes yielding the Fourier transform

xT (f) =∫ T /2

−T /2

x(t) exp(2πift)dt, (3)

where the frequencies f are integer multiples of 1/T . The power spectral density is then obtained for long obser-

vation times as

PSD(f) =⟨|xT (f)|2

⟩/T . (4)

When plotting the PAF and the PSD on a logarithmic time, respectively, frequency scale, further data processing

can be performed to extract smooth functions from noisy data35. The data are collected in equidistant time steps

and since both PSD(f) and PAF(t) vary essentially only on a logarithmic horizontal axis, the number of points

in a log-plot will increase for long times or low frequency. Therefore, data are commonly averaged by ’blocking’

consecutive data points, resulting in equidistantly distributed points on the logarithmic scale. The data scattering

around their average value gives the standard error on the mean.

Section 4. Theory. The Langevin equation for a particle subject to Brownian motion in a harmonic potential22

reads

mpx(t) = Ffr(t) − Kx(t) + Fth(t), (5)

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where x(t) is the displacement of the bead relative to the trap center. The mass of the particle is denoted by mp,

Ffr(t) the deterministic friction forces, K represents the stiffness of the trap, and Fth(t) is the thermal noise. Since

the hydrodynamic flow pattern has to build up first, the friction force is retarded Ffr(t) = −∫

γ(t − t′)x(t′)dt′,

where the kernel γ(t) vanishes for t < 0 by causality. By the well-known fluctuation dissipation theorem36, the

time correlation function of the random forces is determined by the same friction kernel

〈Fth(t)Fth(t′)〉 = kBTγ(|t − t′|), (6)

and kBT is the thermal energy.

It is convenient to perform a temporal Fourier transform, x(ω) =∫

eiωtx(t)dt, where ω = 2πf is the angular

frequency. The Langevin equation is then equivalent to G(ω)−1x(ω) = Fth(ω), where

G(ω) = [−mpω2 − iωγ(ω) + K]−1, (7)

denotes the Green function. For the correlation function of the random forces in the frequency domain, one obtains

by Fourier transform

〈Fth(ω)Fth(ω′)∗〉 = 4πkBTRe[γ(ω)]δ(ω − ω′). (8)

For finite observation times T , one replaces 2πδ(ω − ω′) → T , and infers the power spectral density of the forces

as PSDth(ω) = 2kBTRe[γ(ω)]. The colour of Brownian motion becomes manifest in the frequency dependence

of the friction kernel. The power spectral density (PSD) corresponding to the positional fluctuations (PAF) of the

bead is then calculated to

PSD(ω) = PSDth(ω)|G(ω)|2 = 2kBTIm[G(ω)]

ω. (9)

By the Wiener-Khinchin theorem the PAF(t) = 〈x(t)x(0)〉 is then obtained from the corresponding PSD by an

inverse Fourier transform

PAF(t) =∫

dω

2πe−iωt PSD(ω). (10)

Einstein’s theory is recovered by assuming a Gaussian white noise behavior at the frequencies of interest for

the power spectral density of the forces PSDth(ω) ≈ 2kBTγ, with γ = γ(ω = 0). This corresponds to a time

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correlation function that is short-lived 〈Ffr(t)Ffr(t′)〉 = 2kBTγδ(t− t′) as is used in many introductory textbooks.

For a spherical particle and no-slip boundary conditions the Stokes friction coefficient equals γ = 6πηR, where

R is the radius of the bead. The assumption that the friction force is given by Stokes’ law is incorrect, since the

long-range flow profile has to build up first.

The frequency-dependent friction coefficient γ(ω) corresponding to a sphere performing small-amplitude os-

cillations in an incompressible fluid with no-slip boundary conditions was already calculated by Stokes γ(ω) =

6πηR(1 + α + α2/9). Here α =√−iωτf and the square roots are to be taken with positive real part, and

τf = R2ρf/η denotes the vortex diffusion time. Since 6πηRτf/9 = mf/2, the last term can be interpreted as

effectively increasing the mass of the particle by half the mass of the displaced fluid,

γ(ω) = 6πηa[1 +

√−iωτf

]− iωmf/2. (11)

The square root describes the evolution of the back flow by vortex diffusion and is non-analytic in frequency. The

corresponding force is known as the Basset force and reads 37

Ffr(t) = −6πηRx(t) − mf

2x(t) − 6πηR

√τf

π

∫ t

−∞

x(t′)dt′√t − t′

. (12)

Whereas the first two terms are local in time, the last contribution is retarded and persists for all times. The force

can be written as −∫ t

−∞ γ(t − t′)x(t′)dt′ by a suitable kernel γ(t). A heuristic derivation for γ(t) for positive

times t > 0 goes as follows: Assume first a velocity x(t) = u0ϑ(t), i.e. switching on a uniform motion at time

t = 0. Then, for t > 0, Ffr(t) = −6πηRu0−6πηRu0

√τf/πt = −u0

∫ t

0γ(t′)dt′. Differentiating yields the tail in

the retarded friction γ(t) = −6πηR√

τf/4πt−3/2 , for t > 0. In particular, by the fluctuation-dissipation theorem,

Eq. (9) the force autocorrelation displays a tail

〈Fth(t)Fth(t′)〉 = −6πηRkBT

√τf

4π|t − t′|−3/2, |t − t′| > 0, (13)

The last relation holds within the hydrodynamic model of an incompressible fluid and deviations are expected only

for times where sound waves become important.

In multi-particle collision dynamics, the large compressibility of the simulated fluid gives rise to an additional

time scale τs = R/c0, with c0 the isothermal sound velocity, which comes close to the vortex diffusion time

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τf. Furthermore, the simulations employ effectively full-slip boundary conditions, since no angular momentum is

transferred to the bead during the collisions with the solvent particles. The corresponding frequency-dependent

friction coefficient was calculated in38-40

γ(ω) =4πηR

3(1 + λ)(18 + 18α + 3α2 + α3) + 4(1 + α)λ2

(2 + 2λ + λ2)(3 + α) + 2(1 + α)λ2/α2. (14)

Here λ = Rω/√

−c20 − iω(ζ + 4η/3)/ρf accounts for the sound propagation, with ζ the bulk viscosity and ρf the

mass density of the fluid. One readily checks that γ(ω = 0) = 4πηR assumes the correct value for the stationary

friction of a sphere for full slip.

Section 5. Influence of particle inertia on the hydrodynamic peak. By Eq. (12), the PSD of the trapped

bead, PSD(ω) = 2kBTRe[γ(ω)]|G(ω)|2 involves mass terms, which are relevant only in the inertial regime. In

the hydrodynamic regime τK τf, τp the particle’s mass and the added mass of the fluid are negligible.

|PA

F(t

)/PA

F(0

)|

Time t [s]

PAF(t)PAF(0)

τK

Anticorrelations

t−3/2

fK fK

Colorednoise

PS

D(f

)/P

SD(0

)

Frequency f [Hz]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

10-2

10-1

100

103 10410-510-4

10-3

10-4 10-3

10-4 10-310-5

0.0

0.2

0.4

0.6

0.8

a b

Fig. S5: a, Normalized PAF and b, PSD of the data presented in Fig. 2. Coloured circles: data; black line: full

hydrodynamic theory (Eq. 12); red line: full hydrodynamic theory with mp = 0, mf = 0 ; green and blue lines:

overdamped harmonic oscillators, with mp = 0, for comparison.

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To emphasize the separation between the inertial and hydrodynamic regimes, a comparison of the data with theory

is shown in Fig. S5, when neglecting the particle’s mass mp in the propagator G(ω) = [−mpω2 − iωγ(ω)+K]−1

and the added fluid mass mf/2 in the dynamic friction γ(ω) = 6πηR[1 +√−iωτf]− iωmf/2, but still accounting

for the vortex diffusion τf = 0 (red lines). Deviations are strongest at short times and decrease with τf/tauK.

Section 6. Modulation of the trap strength. The Langevin equation that governs the displacements x(t) of

the bead from the center of a harmonic trap that is periodically modulated as K(t) = K[1 + |g| cos(Ωt + δ)] with

small relative amplitude |g|, an angular frequency Ω = 2πfexc and a phase δ, is given by

mpx(t) +∫ t

−∞γ(t − t′)x(t′)dt′ + K(t)x(t) = Fth(t). (15)

In the Fourier domain the equation of motion reads

G(ω)−1x(ω) = Fth(ω) − (K/2)[gx(ω + Ω) + g∗x(ω − Ω)], (16)

where g = |g| exp(iδ). We iterate this equation to second order in g, and since the Fourier transform of the noise

correlator is diagonal with respect to the frequencies, Eq. (11), the power spectral density is calculated to

PSD(ω) =2kBTRe[γ(ω)]|G(ω)|2∣∣∣∣1 +

|g|2K2

4

[G(ω + Ω) + G(ω − Ω)

]G(ω)

∣∣∣∣2

+ 2kBTRe[γ(ω + Ω)]|g|2K2

4|G(ω + Ω)|2|G(ω)|2

+ 2kBTRe[γ(ω − Ω)]|g|2K2

4|G(ω − Ω)|2|G(ω)|2 + O(g3). (17)

Interestingly, the PSD(ω) is independent of δ, unlike the positional autocorrelation 〈x(t)x(t′)〉, which is expected

to depend on the value of the phase24,27,41,42.

Section 7. Simulations. The simulations are based on the method of multi-particle collision dynamics (MPC)26,43.

This mesoscale simulation technique models the solvent as a set of point particles of mass mf and density ρf with

continuous coordinates r and velocities v. At every simulation step the solvent particles undergo propagation and

collision. In the propagation step, the coordinates of each particle are updated according to r(t+δt) = r(t)+v(t)δt.

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The particles are then sorted into collision cells, and their velocities are rotated with respect to the center-of-mass

velocity of the respective collision cell by a fixed angle ϑ around a random axis. The direction of the rotation

axis is the same within each collision cell, but is statistically independent between different cells and for any cell

at different times. The standard grid-shifting procedure to guarantee Galilean invariance is employed. Mass is

measured in units of the solvent particle mass mf, length in units of the collision cell size b, energy in units of

the thermal energy kBT and time in units of t0 =√

mfb2/kBT . Our model consists of a single colloidal particle

of mass m = 335mf embedded in a sea of solvent particles with density ρf = 10mf/b3 within a simulation box

with a lateral size L = 32b with periodic boundary conditions. The solute-solvent particle coupling is given by

a truncated Lennard-Jones potential44 with the interaction radius σ = 2b. Since no angular momentum is trans-

ferred to the particle by the collisions, we simulate effectively full-slip boundary conditions. In order to reproduce

fluid-like behavior, we use the collision time step 0.05t0 and collision angle45 ϑ = 130. The kinematic viscosity

is then calculated26 to be ν = η/ρf = 1.67b2/t0, and the Schmidt number to Sc = 65. The diffusion coefficient

D of the colloidal particle measured simultaneously through its mean-square displacement and the integral of the

velocity auto-correlation function is D = 0.003(2)b2/t0. Then, the effective hydrodynamic radius was determined

to R = 1.8b by the Stokes-Einstein relation taking into account appropriate corrections, which arise due to the

finite size of the simulation box as described in detail in 43. Accordingly, the vortex diffusion time scale is cal-

culated to τf = 1.94t0, whereas the momentum relaxation time scale τp = m/γ evaluates to τp = 1.0t0. The

colloidal diffusion time τD = a2/D = 841t0 is much larger in agreement with the experimental conditions. Due

to the high compressibility of the solvent, the sonic time τs = R/c0 = 1.59t0, with the isothermal sound velocity

c0 =√

kBT/mf, becomes comparable to vortex diffusion. Ignoring the sonic scale, the resulting order of time

scales follows the right physical hierarchy τp < τf τD present in the experimental system, as prescribed in 43.

The force exerted on the colloidal particle by the optical trap is taken to be of a harmonic form, Ftrap(t) = −Kx(t),

where K is the spring constant of the trap and x(t) is the displacement of the particle relative to the center of the

trap. The trapping potential gives rise to an additional characteristic time scale τK = γ/K. In the simulations

for parametric excitation, the spring constant of the trap is modulated as K(t) = K[1 + g cos(2πfexct)], with a

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relative modulation amplitude g = 0.5. To prevent heating, the system is thermalized with a time step ttherm = t0

by rescaling the velocities of all particles such that the temperature of the system corresponds to the thermal energy

kBT .

Section 8. Force harmonicity in a strong optical trap. The bead is exposed to an optical field resulting in a

time-averaged force on the particle that consists of a conservative gradient force and a scattering contribution46.

For applications the hookean spring model of the trap gives an accurate description of the interaction of the bead

with the light. Only recently the deviations of the forces from this model have been scrutinized47 and arrive at the

conclusion that the corrections are tiny. In principle, all fields and stresses can be calculated within the Mie theory

and toolboxes are available to determine the forces for arbitrarily shaped beam profiles48.

For a Gaussian beam trap, forces display significant nonlinearities on the scale of the laser wavelength, which

were observed by Jahnel et al.49. They compared their experimental results with numerical calculations obtained

by using the Matlab routines presented by Nieminen48 and found remarkable agreement. The nonlinear effects

become prominent on a scale larger than hundreds of nanometers. For the strong trap strengths, K > 100 µN/m,

employed in the present experiments, typical excursions (standard deviation) of the particle are of the order of

√kBT/K 10nm, much smaller than the scale of trapping force non-linearities.

Another phenomenon arising from a non-conservative component of the optical force may influence the dynamics

of the trapped particle, yet only for motion parallel to the optical axis, as was recently described by Roichman et

al.50. According to their Eqs. (8) and (9), the dimensionless mean circulation rate Ω0τK serves as a quantitative

indication of the non-conservative effects, and decays as 1/√

K with increasing trap stiffness. As in the present

work, experiments necessarily employ very strong optical traps, the contributions from non-conservative forces are

negligible. Also Pesce et al.51 confirm that these effects are only relevant for weak traps. To test the relevance of

non-conservative forces in our approach, we implemented the force field proposed by 50 in computer simulations:

Fx(r) = −Kxx, Fy(r) = −Kyy, Fz(r) = −Kzz + f1 exp(−r2/2σ2) (18)

The parameters describing the non-conservative part of the force are chosen to be in line with 50 as σ = 3 > R,

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RESEARCH

ε = f1/Kzσ = 0.1. Thus, the contribution of the non-conservative component of the force is 10% relative to the

conservative component. We also introduce an asymmetry of the optical trap with Ky = 0.9Kx and Kz = 0.15Kx.

The resulting power spectral density of the particle in the lateral dimensions, x or y (Fig. S6, symbols) completely

agrees with the power spectral density measured in simulations as presented in the manuscript (lines). Therefore,

non-conservative forces are shown to have no consequences for the present measurements, and have no effect on

the hydrodynamic peak.

0.00

0.02

0.04

0.06

0.08

0.10

10-3

10-2

10-1

100

PS

D[

]

Frequency f [√

kBT/mb2]

Fig. S6: Computer simulation results for the PSD corresponding to the position fluctuations of a Brownian particle

in the lateral plane confined by an asymmetric optical trap for a weaker (green) and a stronger (blue) trap. The case

when nonconservative forces are taken into account (solid symbols) is indistinguishable from simulations with a

symmetric trap of a corresponding stiffness with purely conservative forces (solid lines).

Section 9. Influence of a solid boundary on the hydrodynamic peak. In all our measurements the micro-

sphere was brought by the 3D piezo stage at least 40 µm away from the top and bottom glass surface of our more

than 100 µm thick sample chambers. A very systematic study on the influence of a solid boundary on hydrody-

namic backflow is published in 6,52. Accordingly, for the present experimental configuration, Fig. S7 shows the

14



measured and theoretical PSD of a 2.9 µm sized sphere trapped at different distances from the surface. It clearly

shows that the presence of a wall drastically damps the propagation of the hydrodynamic backflow, and compro-

mises resonance.

0.2

0.4

0.6

0.8

1.0

1.2

102

103

104

PS

D(f

)/P

SD(0

)

40µ20µ10µ5.0µ2.5µ

Frequency [ ]

Fig. S7: Log-linear representation of the normalized PSD of a resin sphere (R = 1.45 µm) in acetone either at

40 µm (blue) or 2.5 µm (red) from the surface of the coverslip. The continuous lines correspond to the theoretical

fits to the data (circles). Theoretical graphs according to 52 are also plotted for intermediate distances to highlight

the gradual disappearance of the resonance upon approaching the boundary.

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