# Resonant diffusion of a gravitactic circle swimmer

Oleksandr Chepizhko  Thomas Franosch  Institut für Theoretische Physik,
Universität Innsbruck, Technikerstraße 21A, A-6020, Innsbruck, Austria

(August 27, 2024)

###### Abstract

We investigate the dynamics of a single chiral active particle subject to an
external torque due to the presence of a gravitational field. Our computer
simulations reveal an arbitrarily strong increase of the long-time diffusivity
of the gravitactic agent when the external torque approaches the intrinsic
angular drift. We provide analytic expressions for the mean-square
displacement in terms of eigenfunctions and eigenvalues of the noisy-driven-
pendulum problem. The pronounced maximum in the diffusivity is then
rationalized by the vanishing of the lowest eigenvalues of the Fokker-Planck
equation for the angular motion as the rotational diffusion decreases and the
underlying classical bifurcation is approached. A simple harmonic-oscillator
picture for the barrier-dominated motion provides a quantitative description
for the onset of the resonance while its range of validity is determined by
the crossover to a critical-fluctuation-dominated regime.

Active particles capable of self-propulsion by converting energy into directed
motion have come into research focus and are important from both a fundamental
and an applied point of view [1, 2, 3, 4, 5]. Examples for active agents
include various motile organisms, in particular bacteria [6, 7, 8] or algae
[9], as well as artificial realizations such as Janus rods [10], spheres [11],
or Quincke rollers [12]. Recently, significant advances in our understanding
of transport properties of active motion in homogeneous environments [13, 14,
15, 4, 16, 17, 18] and in media crowded with obstacles [19, 20, 21, 22, 23,
24, 25] have been achieved.

External fields, torques and gradients induce various forms of _taxis_ such as
chemotaxis [26, 27], magnetotaxis [28, 29], gravitaxis [30, 31], rheotaxis
[32], or viscotaxis [33] due to a combination of the persistence of motion and
different noise sources [1, 4]. Already the case of a homogeneous force field
such as gravity gives rise to counterintuitive dynamics by coupling to the
orientational motion. For systems of $L$-shaped chiral microswimmers [34] and
Janus rods [35], mass-anisotropic colloids [36, 37], and microorganisms [30,
*Roberts2010] gravitaxis has been demonstrated experimentally. In particular,
these experiments have discovered that chiral active particles subject to a
gravitational field can move upwards, which was also supported by simulation
studies in bottom-heavy microswimmers [38] and chiral swimmers [39].
Experimental studies of sedimenting active particles revealed an enhancement
of diffusivity by activity [40], while an analytical solution for the density
profile has been obtained only recently [41]. The sedimentation profile of
active particles was also studied in detail in computer simulations and
experiments with Janus particles [42] as well as for run-and-tumble particles
[43]. Yet, the temporal dynamics and transport properties such as the mean-
square displacement and the corresponding diffusivity have not been elucidated
in detail.

In this Letter, we demonstrate by computer simulation that gravitaxis of
circle swimmers displays resonant diffusivities for small orientational
diffusion as the graviational torque approaches the intrinsic angular drift
velocity. We then elaborate a complete analytical solution of gravitactic
motion for the model developed by ten Hagen _et al_ [34]. A formal expression
for the intermediate scattering function, encoding the spatio-temporal motion,
is derived. Using a time-dependent perturbative approach we extract the mean
drift and the mean-square displacement. Our analytic results reveal that the
resonance is encoded in the vanishing of the eigenvalues of the associated
Fokker-Planck operator. We rationalize the onset of the resonance within a
harmonic approximation and determine the growth of the maximum of the
resonance by crossover scaling.

#### Model.–

We rely on the model derived in Ref. [34] for an active chiral particle
subject to an external (gravitational) field. The particle moves at constant
speed $v$ along a direction
$\bm{\mathrm{u}}(t):=(\cos\vartheta(t),\sin\vartheta(t))^{T}$ parametrized by
a time-dependent angle $\vartheta(t)$ measured from the horizontal

$\dot{\mathbf{r}}(t)=v\bm{\mathrm{u}}(t)=v(\cos\vartheta(t),\sin\vartheta(t))\,.$
(1)

The evolution of $\vartheta(t)$ is governed by two contributions. The external
field results in an angle-dependent torque aligning the orientation in a
certain direction. The coating of the active particle is such that with this
orientation the self-propulsion is horizontal. Additionally, the anisotropy of
the particle induces an internal angular drift $\omega>0$ resulting in the
equation of motion

$\dot{\vartheta}(t)=\omega-\gamma\sin\vartheta(t)+\zeta(t)\,.$ (2)

Here, the torque $\gamma$ is proportional to the external force and $\zeta(t)$
is a centered Gaussian white noise
$\langle\zeta(t)\zeta(t^{\prime})\rangle=2D_{\text{rot}}\delta(t-t^{\prime})$,
where $D_{\text{rot}}$ is the bare orientational diffusion constant. The
equations of motion are rewritten with proper substitutions from Ref. [34],
where for simplicity we discard noise terms and additional drift terms due to
(anisotropic) translational diffusion in Eq. (1). The neglected terms can be
incorporated with little effort but do not change the overall picture of the
resonance phenomenon (see Supplemental Material [44]). Without the external
field the model reduces to the free circle swimmer [13, 45, 46, 17], while for
$\gamma>0$ the angular motion corresponds to Brownian motion in a tilted
washboard potential [47, *Reimann2002, 49]. Ignoring the noise in Eq. (2)
yields the classical dynamics of an overdamped driven pendulum displaying a
saddle-node bifurcation at the critical value $\gamma_{c}=\omega$ [50]. The
mapping of the gravitaxis problem to the noisy driven pendulum constitutes our
first result.

#### Simulation.–

The model is encoded in 4 parameters characterizing gravitaxis. We employ
$1/\omega$ as fundamental unit of time, while the radius of the circular
motion $v/\omega$ sets the unit of length. Then $\gamma/\omega$ is the
dimensionless torque and $D_{\text{rot}}/\omega$ quantifies the relative
importance of fluctuations. Stochastic simulations are performed and the
displacement $\Delta\bm{\mathrm{r}}(t):=\bm{\mathrm{r}}(t)-\bm{\mathrm{r}}(0)$
is monitored in the stationary state. In particular, we extract the mean
displacement $\langle\Delta\bm{\mathrm{r}}(t)\rangle$ and the variance
$\textsf{Var}[\Delta\bm{\mathrm{r}}(t)]:=\langle[\Delta\bm{\mathrm{r}}(t)-\langle\Delta\bm{\mathrm{r}}(t)\rangle]^{2}\rangle$.

In the stationary state the mean displacement grows linearly in time with the
average velocity
$v(\langle\cos\vartheta(t)\rangle,\langle\sin\vartheta(t)\rangle)$. Since the
stationary distribution of the angle $p^{\text{st}}(\vartheta)$ is elementary
[51] the mean drift can be readily obtained by quadrature. Here, we recall
that without noise, $D_{\text{rot}}=0$, the orientational angle is locked at
$0<\vartheta_{*}\leq\pi/2$ with $\sin\vartheta_{*}=\omega/\gamma$ provided the
torque fulfills $\gamma\geq\omega$. If the torque is weaker than the internal
drift, $\gamma<\omega$, the angular motion is periodic [50]. Directly at the
classical bifurcation, the particle moves upwards against the field [52]. Upon
reintroducing the noise the average horizontal motion is suppressed above the
bifurcation, while below the fluctuations enable a net drift (see also Fig.
S.3 in SM [44]). The drift against the field direction is always suppressed by
noise.

Fig. 1: Time-dependent variance $\textsf{Var}[\Delta\bm{\mathrm{r}}(t)]$ on
logarithmic scales for a rotational diffusivity of
$D_{\text{rot}}/\omega=0.005$222Unfortunately, the published version states a
wrong value. for different torques $\gamma$. Symbols correspond to simulation,
the solid lines represent the analytic solution. The dotted black lines are
evaluated within the harmonic oscillator approximation, Eq. (13b), for
$\gamma=1.1\omega$ and $\gamma=1.2\omega$. The thick black lines are power
laws serving as guide to the eye.

The variance $\textsf{Var}[\Delta\bm{\mathrm{r}}(t)]$ increases as $t^{2}$ for
small times, see Fig. 2, where the prefactor decreases drastically as the
torque is increased. Below the classical bifurcation characteristic
oscillations emerge, similar to the free circle swimmer [53, 17], while for
$\gamma>\omega$ the variance increases monotonically. The long-time behavior
is diffusive in the presence of noise, $D_{\text{rot}}>0$, and the diffusion
coefficient is enhanced close to the classical bifurcation. The extracted
long-time diffusion coefficients
$D:=\lim_{t\to\infty}(1/4)\mathrm{d}\textsf{Var}[\Delta\bm{\mathrm{r}}(t)]/\mathrm{d}t$
are displayed in terms of the known diffusivity without external torque [13,
17] $D_{0}=v_{0}^{2}D_{\text{rot}}/[2(D_{\text{rot}}^{2}+\omega^{2})]$ in Fig.
4. Close to the classical bifurcation a resonance emerges that becomes
narrower and more pronounced as the noise is decreased.

Fig. 2: Resonance of diffusivity. The diffusivity $D$ as a function of the
torque $\gamma$ for different rotational diffusivities $D_{\text{rot}}$. The
diffusivity is normalized to the diffusion coefficient of a free circle
swimmer $D_{0}=v^{2}D_{\text{rot}}/[2(D_{\text{rot}}^{2}+\omega^{2})]$.
Symbols correspond to simulation, the full lines are analytical results. The
black dotted line corresponds to the harmonic oscillator approximation, Eq.
(13b), for $D_{\text{rot}}=0.005\omega$. The right boundary of the shaded area
corresponds to the parametric curve
$\hat{D}_{\text{rot}}:=|\gamma/\omega-1|^{-3/2}D_{\text{rot}}/\omega=\text{const.}$
asymptotically intersecting the maxima of the diffusion coefficients. Inset:
Increase of the maximal diffusion coefficient $D_{\text{max}}$ for increasing
inverse noise $1/D_{\text{rot}}$ on logarithmic scales obtained from
theory.444Unfortunately, this inset contains wrong tic marks in the published
version.

#### Theory.–

The goal of this part is to provide a theoretical explanation for the found
resonance and to derive scaling laws in the vicinity of the classical
bifurcation. The dynamical properties are encoded in the propagator
$\mathbb{P}(\bm{\mathrm{r}},\vartheta,t|\vartheta_{0})$, i.e. the conditional
probability distribution that the particle has displaced by $\bm{\mathrm{r}}$
and its instantaneous speed exhibits an orientation $\vartheta$ at lag time
$t$ given it started with orientation $\vartheta_{0}$. Angles are considered
to be $2\pi$-periodic. We focus on its spatial Fourier transform
$\tilde{\mathbb{P}}:=\tilde{\mathbb{P}}(\bm{\mathrm{k}},\vartheta,t|\vartheta_{0})=\int
d\mathbf{r}\exp(-\mathrm{i}\bm{\mathrm{k}}\cdot\bm{\mathrm{r}})\mathbb{P}(\bm{\mathrm{r}},\vartheta,t|\vartheta_{0})$
and derive by standard methods [51] the (Fourier transformed) Fokker-Planck
equation (see also SM [44])

$\displaystyle\partial_{t}\tilde{\mathbb{P}}$
$\displaystyle=-\partial_{\vartheta}[(\omega-\gamma\sin\vartheta)\tilde{\mathbb{P}}]+D_{\text{rot}}\partial_{\vartheta}^{2}\tilde{\mathbb{P}}-iv\bm{\mathrm{k}}\cdot\bm{\mathrm{u}}\tilde{\mathbb{P}}$
$\displaystyle=:(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})\tilde{\mathbb{P}}.$
(3)

Here the operator $\mathcal{L}$ encodes the motion of the angle, while
$\delta\mathcal{L}_{\bm{\mathrm{k}}}=-iv\bm{\mathrm{k}}\cdot\bm{\mathrm{u}}$
describes the coupling to the translational dynamics. The formal solution is
thus
$\tilde{\mathbb{P}}(\bm{\mathrm{k}},\vartheta,t)|\vartheta_{0})=\exp[(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t]\delta(\vartheta-\vartheta_{0})$.

From this quantity the intermediate scattering function (ISF)
$F(\mathbf{k},t)=\langle\exp(-\text{i}\bm{\mathrm{k}}\cdot\Delta\bm{\mathrm{r}}(t))\rangle$
is obtained in the stationary state by averaging over the initial angle and
integrating over the final one

$\displaystyle F(\bm{\mathrm{k}},t)$
$\displaystyle=\int_{0}^{2\pi}\\!\\!\mathrm{d}\vartheta\\!\\!\int_{0}^{2\pi}\\!\\!\mathrm{d}\vartheta_{0}\,\tilde{\mathbb{P}}(\bm{\mathrm{k}},\vartheta,t|\vartheta_{0})p^{\text{st}}(\vartheta_{0})$
$\displaystyle=\int_{0}^{2\pi}\\!\\!\mathrm{d}\vartheta\exp[(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t]p^{\text{st}}(\vartheta).$
(4)

Moments of the displacement can be extracted by a series expansion in the wave
vector $\bm{\mathrm{k}}$

$F(\bm{\mathrm{k}},t)=1-\mathrm{i}\bm{\mathrm{k}}\cdot\langle\Delta\bm{\mathrm{r}}(t)\rangle-\frac{1}{2}\langle[\bm{\mathrm{k}}\cdot\Delta\bm{\mathrm{r}}(t)]^{2}\rangle+\ldots\,.$
(5)

Here we follow the strategy of Ref. [17] (see also Ref. [54]), solve the
eigenvalue problem of the reference system $\mathcal{L}$, and apply time-
dependent perturbation theory for $\delta\mathcal{L}_{\bm{\mathrm{k}}}$. We
denote by $\\{|n\rangle:n\in\mathbb{Z}\\}$ the standard orthonormal basis in
the Hilbert space $L^{2}[0,2\pi]$ with real-space representation
$\langle\vartheta|n\rangle:=\exp(in\vartheta)/\sqrt{2\pi}$. In this basis
$\mathcal{L}$ is represented in terms of its matrix elements [51]

$\displaystyle\mathcal{L}_{mn}$ $\displaystyle=\langle
m|\mathcal{L}n\rangle:=\int_{0}^{2\pi}\frac{\mathrm{d}\vartheta}{2\pi}e^{-im\vartheta}\mathcal{L}e^{in\vartheta}$
$\displaystyle=(-D_{\text{rot}}m^{2}-im\omega)\delta_{mn}+\frac{\gamma}{2}m(\delta_{m,n+1}-\delta_{m,n-1}).$
(6)

In particular, the matrix representation is non-Hermitian and tridiagonal due
to the torque. Right and left eigenstates
$\mathcal{L}|r_{\lambda}\rangle=-\lambda|r_{\lambda}\rangle,\mathcal{L}^{\dagger}|l_{\lambda}\rangle=-\lambda^{*}|l_{\lambda}\rangle$
are readily obtained by numerically diagonalizing the matrix, Eq. (SM16),
yielding the expansion coefficients $\langle n|r_{\lambda}\rangle$, $\langle
l_{\lambda}|n\rangle$ as right and left eigenvector of the matrix
$\mathcal{L}_{mn}$. The corresponding real-space representation is obtained by
expansion
$r_{\lambda}(\vartheta)=\sum_{n\in\mathbb{Z}}\langle\vartheta|n\rangle\langle
n|r_{\lambda}\rangle$. By conservation of probability $\lambda=0$ is an
eigenvalue and the real-space representation of the associated eigenstates are
$r_{0}(\vartheta)=p^{\text{st}}(\vartheta)$ and $l_{0}(\vartheta)=1$.
Comparing with Eq. (SM11) we find the compact expression for the ISF

$\displaystyle F(\bm{\mathrm{k}},t)$ $\displaystyle=\langle
l_{0}|\exp[(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t]r_{0}\rangle.$
(7)

Since
$\delta\mathcal{L}_{\bm{\mathrm{k}}}=-iv\bm{\mathrm{k}}\cdot\bm{\mathrm{u}}$,
time-dependent perturbation theory (Born series) [55]

$\displaystyle e^{(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t}=$
$\displaystyle
e^{\mathcal{L}t}+\int_{0}^{t}\mathrm{d}s\,e^{\mathcal{L}(t-s)}\delta\mathcal{L}_{\bm{\mathrm{k}}}e^{\mathcal{L}s}$
$\displaystyle+\int_{0}^{t}\mathrm{d}s\int_{0}^{s}\mathrm{d}u\,e^{\mathcal{L}(t-s)}\delta\mathcal{L}_{\bm{\mathrm{k}}}e^{\mathcal{L}(s-u)}\delta\mathcal{L}_{\bm{\mathrm{k}}}e^{\mathcal{L}u}$
$\displaystyle+O(\delta\mathcal{L}_{\bm{\mathrm{k}}})^{3},$ (8)

directly yields the moments of the series expansion in $\bm{\mathrm{k}}$ as in
Eq. (5). In particular, expanding to second order in $\bm{\mathrm{k}}$ and
squeezing in the completeness relation
$\sum_{\lambda}|r_{\lambda}\rangle\langle l_{\lambda}|=\mathbbm{1}$, the time
integrals can be formally performed (see also SM [44]). Then we read off the
mean drift velocity and the variance along the direction
$\bm{\mathrm{n}}:=\bm{\mathrm{k}}/k$

$\displaystyle\bm{\mathrm{n}}\cdot\frac{\mathrm{d}}{\mathrm{d}t}\langle\Delta\bm{\mathrm{r}}(t)\rangle$
$\displaystyle=\frac{i}{k}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle,$ (9a)
$\displaystyle\textsf{Var}[\bm{\mathrm{n}}\cdot\Delta\bm{\mathrm{r}}(t)]$
$\displaystyle=\frac{2}{k^{2}}\sum_{\lambda\neq 0}\frac{1-\lambda
t-e^{-\lambda t}}{\lambda^{2}}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{\lambda}\rangle\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle,$ (9b)

and infer the associated long-time diffusion coefficient

$\displaystyle D_{\bm{\mathrm{n}}}$
$\displaystyle=\frac{-1}{k^{2}}\sum_{\lambda\neq 0}\frac{1}{\lambda}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{\lambda}\rangle\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle.$ (10)

The analytical expressions for the total variances as well as the
corresponding diffusion coefficient perfectly match the simulations, see Figs.
2 and 4. The exact expressions in terms of eigenfunctions and eigenvalues
suggest that at resonance the eigenvalues become smaller and smaller as the
rotational diffusion coefficient is decreased, which is confirmed by numerical
diagonalization (see SM [44]).

To gain further analytical insight, we evaluate Eqs. (9) and (10) within a
harmonic approximation for the motion close to the classical fixed point
$\vartheta_{*}$. For small $D_{\text{rot}}\ll\omega$ and well above the
bifurcation the fluctuations are anticipated to be small and the corresponding
linearized Langevin equation reads

$\displaystyle\dot{\vartheta}(t)=-\frac{1}{\tau}[\vartheta(t)-\vartheta_{*}]+\zeta(t),$
(11)

with vanishing relaxation rate $1/\tau=\sqrt{\gamma^{2}-\omega^{2}}\to 0$ as
$\gamma\downarrow\omega$. The associated eigenvalues of the overdamped
harmonic oscillator are then simply $\lambda_{n}=n/\tau,n\in\mathbb{N}_{0}$
[51], and indeed they approach zero for $\gamma\downarrow 0$ (see also SM
[44]).

To leading order we replace the perturbing operator
$\delta\mathcal{L}_{\bm{\mathrm{k}}}$ by the complex number
$iv\bm{\mathrm{k}}\cdot\bm{\mathrm{u}}_{*}$ with the fixed orientation
$\bm{\mathrm{u}}_{*}=(\cos\vartheta_{*},\sin\vartheta_{*})$. In this
approximation, the drift velocity, is non-fluctuating and assumes its
classical value. In the same approximation the variance and diffusion
coefficient vanish by orthogonality of the eigenstates, consistent with a
purely deterministic and non-chaotic motion. To leading non-trivial order we
replace

$\displaystyle\delta\mathcal{L}_{\bm{\mathrm{k}}}\simeq-
iv\bm{\mathrm{k}}\cdot\bm{\mathrm{u}}_{*}+(\vartheta-\vartheta_{*})\left.\frac{\partial}{\partial\vartheta}\delta\mathcal{L}_{\bm{\mathrm{k}}}\right|_{\vartheta=\vartheta_{*}}.$
(12)

Then within the harmonic oscillator approximation, also the off-diagonal
matrix elements can be evaluated analytically, in particular, their magnitude
is proportional to the angular oscillator width
$\langle(\vartheta-\vartheta_{*})^{2}\rangle=\sqrt{D_{\text{rot}}\tau}$ (see
SM [44] for details). Furthermore, the angular position operator
$\vartheta-\vartheta_{*}$ induces only non-vanishing transition matrix element
in Eqs. (9b),(10) coupling the ground state to the first excited state.
Therefore the sums reduce to a single term and we find the compact expressions

$\displaystyle\textsf{Var}[\bm{\mathrm{n}}\cdot\Delta\bm{\mathrm{r}}(t)]\simeq$
$\displaystyle\,2D_{\bm{\mathrm{n}}}\tau\left(\frac{t}{\tau}-1+e^{-t/\tau}\right),$
(13a) $\displaystyle D_{\bm{\mathrm{n}}}\simeq$
$\displaystyle\,(v\tau)^{2}D_{\text{rot}}(n_{x}\sin\vartheta_{*}-n_{y}\cos\vartheta_{*})^{2}.$
(13b)

Within the harmonic oscillator approximation the variance is strictly
proportional to $D_{\text{rot}}$ suggesting that for sufficiently small
orientational fluctuations the curves in Figs. 2 and 4 should approach a
master curve. The corresponding curves are included in Figs. 2 and 4 as black
dotted lines and are in quantitative agreement for small
$D_{\text{rot}}\ll\omega$ and $\gamma>\omega$ not too close to the
bifurcation. The emergence of the resonance is thus rationalized in terms of
the softening of the harmonic relaxation rate $1/\tau\to 0$ as the bifurcation
is approached.

The harmonic picture suggests that the diffusion coefficient becomes infinite
directly at the bifurcation while the simulation and the full analytic
expression predict a rounding with a maximal diffusivity. The picture of the
harmonic oscillator should hold provided the barrier is sufficiently high such
that Kramers’ escape rate [51] is much smaller than the harmonic relaxation
rate. In terms of the effective potential
$D_{\text{rot}}U(\vartheta)/k_{B}T=-\omega\vartheta-\gamma\cos\vartheta$ the
barrier height $\Delta U:=U(\pi-\vartheta_{*})-U(\vartheta_{*})$ reduces to

$\displaystyle\frac{\Delta
U}{k_{B}T}=\frac{4\sqrt{2}}{3}\frac{\omega}{D_{\text{rot}}}\epsilon^{3/2}[1+O(\epsilon)],$
(14)

where we introduced the separation parameter
$\epsilon:=(\gamma-\omega)/\omega$ for the distance to the bifurcation (see SM
[44] for details). This observation suggests introducing the reduced
rotational diffusion coefficient
$\hat{D}_{\text{rot}}:=|\epsilon|^{-3/2}D_{\text{rot}}/\omega$ such that for
$\hat{D}_{\text{rot}}\ll 1,\epsilon>0$ the harmonic approximation holds, while
for $\hat{D}_{\text{rot}}\gg 1$ the barrier can be crossed readily by
fluctuations and the precise height of the barrier should be irrelevant. By
matching the critical fluctuation to the harmonic oscillator we predict that
the maximal diffusivity should occur at $\hat{D}_{\text{rot}}=O(1)$ or
$D_{\text{rot}}\propto|\epsilon|^{3/2}$. For small $\epsilon>$, the relaxation
time diverges as $\tau\propto|\epsilon|^{-1/2}$ and from Eq. (13b) we infer
the scaling law for the maximal diffusivity

$\displaystyle D_{\text{max}}/D_{0}\propto|\epsilon|^{-1}\propto
D_{\text{rot}}^{-2/3}\qquad\text{for }D_{\text{rot}}\to 0,$ (15)

where the we used that $D_{0}\propto D_{\text{rot}}$ as $D_{\text{rot}}\to 0$.
The numerical values nicely follow the prediction asymptotically as shown in
Fig. 4.

#### Summary and Conclusion.–

We have demonstrated that the long-time translational diffusivity of a chiral
active Brownian particle displays a resonance for the external torque
approaching the intrinsic angular drift. The resonance originates from an
underlying bifurcation of the classical driven pendulum. There are certain
similarities with the giant diffusion in tilted washboard potentials [47,
*Reimann2002], yet our approach of decomposition into eigenfunctions is rather
complementary and allows calculating the entire time-dependence of low-order
moments and is in spirit closer to Ref. [49]. The connection of the resonance
to the vanishing low-lying eigenvalues is uncovered and an intuitive picture
in terms of a competition between barrier-dominated and critical-fluctuation-
dominated regime is developed.

The harmonic approximation is surprisingly accurate for the variance as well
as for the diffusivity, despite ignoring the rare activation processes over
the barrier. We conclude that far above the bifurcation, the orientation
performs only small fluctuations most of the time close to the minimum of the
effective potential. Then the diffusion coefficient for the translational
diffusion remains also small. Yet, approaching the bifurcation these
fluctuations become more significant as the confining angular potentials
becomes softer yielding a significant enhancement of the diffusivity. For too
large angular fluctuations the harmonic approximation breaks down as barrier-
crossing events become important. After such barrier crossings the orientation
quickly completes a full turn until getting stuck again. From the accurateness
of the harmonic oscillator description we conclude that these fast events do
not significantly contribute to the diffusivity. The essence of the resonance
is thus due to the enhancement of small fluctuations as provided by the
orientational diffusion coefficient exploring a softening potential as the
bifurcation is approached.

Our theoretical work makes detailed predictions for active motion of chiral
agents in external fields that can be tested in experiments, both for
artificial and biological asymmetric microswimmers. The method is readily
extended to calculate higher moments such as the skewness, the non-Gaussian
parameter, or the complete intermediate scattering function. While we
explicitly considered gravitaxis, the underlying equations are rather generic
and the analysis and methods presented should transfer with suitable
adjustments to other forms of _taxis_ at the microscale such as durotaxis
[56], chemotaxis [27], thermotaxis [57], or topotaxis [58]. Also, the results
could be valuable for the transport rigid polymer solutions in a flow near a
wall [59].

The resonance is of interest not only for single-particle transport in
external fields, but also has implications for the collective motion of chiral
active particles with alignment interactions [60, 61, 62, 63] provided
effective mean-field equations can be derived [64, 65, 66, 67] which would be
similar in structure to the equations of motion studied here.

Last our work has theoretical ramifications on the interplay of the critical
slowing down of classical transport close to bifurcations and their smearing
by random fluctuations. The evolution of the eigenspectrum also for other
bifurcations, including the pitchfork or transcritical bifurcations, should be
relevant for various branches of science, such as the physics of the Josephson
junction [50] or collective (Kuramoto) synchronization [68].

###### Acknowledgements.

#### Acknowledgments.–

We thank Christina Kurzthaler for constructive criticism on the manuscript. OC
is supported by the Austrian Science Fund (FWF): M 2450-NBL. TF acknowledges
funding by FWF: P 35580-N. The computational results presented have been
achieved in part using the HPC infrastructure LEO of the University of
Innsbruck.

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  * Note [1] Since $\mathcal{L}$ is non-Hermitian one has to carefully indicate whether operators act to the right or the left.

SUPPLEMENTAL MATERIAL

##  I Simplifying the equations of motion and mapping to the noisy driven
pendulum

The stochastic equations of motion for an asymmetric circle swimmer in the
presence of an external (gravitational) field were proposed by ten Hagen _et
al._ in Ref. [1]. The goal of this section is to show that the model greatly
simplifies upon a simple rotation to the principal axis of the system.
Furthermore the essence of gravitaxis can be discussed by discarding
additional terms connected to translational diffusion and the associated
noise. In this approximation the model reduces to the paradigmatic noise
driven pendulum. Explicitly the equations of motion of Ref. [1] read

Fig. SM1: a) Illustration of the coordinate system used. The orientation of
the principal axis of the particle is along
$\bm{\mathrm{u}}^{\parallel}=(\cos\varphi,\sin\varphi)$, the drift velocity
$\bm{\mathrm{u}}=(\cos\vartheta,\sin\vartheta)$ is tilted by the Hall angle
$\varphi_{H}$ to the particle’s principal axes. The torque $\varphi_{H}$ tends
to align the particle’s direction with the Hall direction while the intrinsic
angular drift rotates the direction counter-clockwise.

$\displaystyle\dot{\bm{\mathrm{r}}}(t)$
$\displaystyle=\frac{P^{*}}{b}[\textsf{D}_{T}(t)\cdot\bm{\mathrm{u}}^{\perp}(t)+\ell\bm{\mathrm{D}}_{C}(t)]+\beta\textsf{D}_{T}(t)\cdot\bm{\mathrm{F}}+\bm{\zeta}_{\bm{\mathrm{r}}}(t),$
(SM1a) $\displaystyle\dot{\varphi}(t)$ $\displaystyle=\frac{P^{*}}{b}[\ell
D_{\text{rot}}+\bm{\mathrm{D}}_{C}(t)\cdot\bm{\mathrm{u}}^{\perp}(t)]+\beta\bm{\mathrm{D}}_{C}(t)\cdot\bm{\mathrm{F}}+\zeta(t),$
(SM1b)

for the time-dependent position $\bm{\mathrm{r}}(t)$ and the orientational
angle $\varphi(t)$ of the particle. The two unit vectors
$\bm{\mathrm{u}}^{\parallel}(t)=(\cos\varphi(t),\sin\varphi(t))$ and
$\bm{\mathrm{u}}^{\perp}(t)=(-\sin\varphi(t),\cos\varphi(t))$ are associated
with the orientational angle $\varphi(t)$, see Fig. SM1 for illustration.
Furthermore, $\bm{\mathrm{F}}=(0,-mg\sin\alpha)=(0,-F)$ denotes the
gravitational force,
$\textsf{D}_{T}(t)=D_{\parallel}\bm{\mathrm{u}}^{\parallel}(t)\otimes\bm{\mathrm{u}}^{\parallel}(t)+D^{\perp}_{\parallel}[\bm{\mathrm{u}}^{\parallel}\otimes\bm{\mathrm{u}}^{\perp}(t)+\bm{\mathrm{u}}^{\perp}(t)\otimes\bm{\mathrm{u}}^{\parallel}(t)]+D_{\perp}\bm{\mathrm{u}}_{\perp}(t)\otimes\bm{\mathrm{u}}_{\perp}(t)$,
$\bm{\mathrm{D}}_{C}(t)=D_{C}^{\parallel}\bm{\mathrm{u}}^{\parallel}(t)+D_{C}^{\perp}\bm{\mathrm{u}}^{\perp}(t)$
encodes the translational diffusion and the translation-rotation coupling[2],
$\beta=1/k_{B}T$ refers to the inverse (effective) temperature, $\ell$ and $b$
are characteristic lengths, and $P^{*}$ is the strength of the self
propulsion. Last, $\bm{\zeta}_{\bm{\mathrm{r}}},\zeta_{\varphi}$ are
independent centered Gaussian white noises of variance
$\langle\bm{\zeta}_{\bm{\mathrm{r}}}(t)\otimes\bm{\zeta}_{\bm{\mathrm{r}}}(t^{\prime})\rangle=2\textsf{D}_{T}(t)\delta(t-t^{\prime}),\langle\zeta(t)\zeta(t^{\prime})\rangle=2D_{\text{rot}}\delta(t-t^{\prime})$
respectively.

Without restriction we assume the unit vectors
$\bm{\mathrm{u}}^{\parallel}(t),\bm{\mathrm{u}}^{\perp}(t)$ to be chosen as
the principal axes of the the tensor $\underline{\textbf{D}}_{T}$. In this
frame the tensor becomes diagonal and we henceforth ignore the couplings
$D^{\perp}_{\parallel}$ and refer to the orientation of the particle as the
angle of the principal axis of the larger eigenvalue $D^{\parallel}$ with
respect to some space-fixed reference axis.

Evaluating the scalar products in the angular equation of motion yields

$\displaystyle\dot{\varphi}(t)$ $\displaystyle=\frac{P^{*}}{b}(\ell
D_{\text{rot}}+D_{C}^{\perp})-\beta
F(D^{\parallel}_{C}\cos\varphi(t)+D^{\perp}_{C}\sin\varphi(t))+\zeta(t).$
(SM2)

Using trigonometric addition formulas this simplifies to

$\displaystyle\dot{\varphi}(t)$
$\displaystyle=\omega-\gamma\sin(\varphi(t)-\varphi_{H})+\zeta(t),$ (SM3)

with the angular drift velocity $\omega$, angular driving $\gamma$, and the
_Hall angle_ $\varphi_{H}$ given by

$\displaystyle\omega$ $\displaystyle=\frac{P^{*}}{b}(\ell
D_{\text{rot}}+D_{C}^{\perp}),\qquad\gamma=\beta
F\sqrt{(D_{C}^{\parallel})^{2}+(D_{C}^{\perp})^{2}},$ (SM4a)
$\displaystyle\sin\varphi_{H}$ $\displaystyle=-\beta
FD_{C}^{\perp}/\gamma,\qquad\cos\varphi_{H}=\beta FD_{C}^{\parallel}/\gamma.$
(SM4b)

Similarly the translational equation of motion simplifies to

$\displaystyle\dot{\bm{\mathrm{r}}}(t)$
$\displaystyle=\frac{P^{*}}{b}[D_{\perp}\bm{\mathrm{u}}^{\perp}(t)+\ell
D_{C}^{\parallel}\bm{\mathrm{u}}^{\parallel}(t)+\ell
D_{C}^{\perp}\bm{\mathrm{u}}^{\perp}(t)]+\beta
D_{\parallel}[\bm{\mathrm{F}}\cdot\bm{\mathrm{u}}^{\parallel}(t)]\bm{\mathrm{u}}^{\parallel}(t)\cos\varphi(t)+\beta
D_{\perp}[\bm{\mathrm{F}}\cdot\bm{\mathrm{u}}^{\parallel}(t)]\bm{\mathrm{u}}^{\perp}(t)\cos\varphi(t)+\bm{\zeta}_{\bm{\mathrm{r}}}$
$\displaystyle=:v\begin{pmatrix}\cos(\varphi(t)-\varphi_{H})\\\
\sin(\varphi(t)-\varphi_{H})\end{pmatrix}+v_{1}\begin{pmatrix}-\sin\varphi(t)\\\
\cos\varphi(t)\end{pmatrix}-v_{2}\begin{pmatrix}\sin(2\varphi(t))\\\
\cos(2\varphi(t))\end{pmatrix}-v_{3}\begin{pmatrix}0\\\
1\end{pmatrix}+\bm{\zeta}_{\bm{\mathrm{r}}}$
$\displaystyle=:\bm{\mathrm{V}}(t)+\bm{\zeta}_{\bm{\mathrm{r}}}(t),$ (SM5)

with $v=P^{*}\ell\gamma/b\beta F$, $v_{1}=P^{*}D_{\perp}/b$, $v_{2}=\beta
F(D_{\parallel}-D_{\perp})/2$, $v_{3}=\beta F(D_{\parallel}+D_{\perp})/2$.
Without circular motion and ignoring both rotational as well as translational
thermal fluctuations, a stationary solution with $\varphi(t)=\varphi_{H}$
exists, the particle moves with constant velocity perpendicularly to the
external field, see Fig. SM1. We abbreviate
$\vartheta(t)=\varphi(t)-\varphi_{H}$ and find the e.o.m.

$\displaystyle\dot{\vartheta}(t)$
$\displaystyle=\omega-\gamma\sin\vartheta(t)+\zeta(t),\qquad\dot{\bm{\mathrm{r}}}(t)=\bm{\mathrm{V}}(t)+\bm{\zeta}_{\bm{\mathrm{r}}}(t),$
(SM6)

with independent centered Gaussian white noise characterized by
$\langle\zeta(t)\zeta(t^{\prime})\rangle=2D_{\text{rot}}\delta(t-t^{\prime})$,
$\langle\bm{\zeta}_{\bm{\mathrm{r}}}(t)\otimes\bm{\zeta}_{\bm{\mathrm{r}}}(t^{\prime})=2\textsf{D}_{T}(t)\delta(t-t^{\prime})$.
Ignoring the terms induced by translational diffusion, the e.o.m. for the
translational motion simplifies to the one discussed in the main text

$\displaystyle\dot{\vartheta}(t)$
$\displaystyle=\omega-\gamma\sin\vartheta(t)+\zeta(t),\qquad\dot{\bm{\mathrm{r}}}(t)=v\bm{\mathrm{u}}(t):=v\begin{pmatrix}\cos\vartheta(t)\\\
\sin\vartheta(t)\end{pmatrix}.$ (SM7)

##  II Fokker-Planck equation, intermediate scattering function and
perturbation theory

###  II.1 Fokker-Planck equation

By standard techniques [3] one derives the Fokker-Planck equation for the
conditional probability density
$\mathbb{P}=\mathbb{P}(\bm{\mathrm{r}},\vartheta,t|\vartheta_{0})$ to find the
particle at speed orientation $\vartheta$ and displaced by $\bm{\mathrm{r}}$
at time $t$ given the orientation at time $t=0$ is prescribed to
$\vartheta_{0}$:

$\displaystyle\partial_{t}\mathbb{P}=-\partial_{\vartheta}[(\omega-\gamma\sin\vartheta)\mathbb{P}]+D_{\text{rot}}\partial_{\vartheta}^{2}\mathbb{P}-\frac{\partial}{\partial\bm{\mathrm{r}}}\cdot(\bm{\mathrm{V}}\mathbb{P})+\frac{\partial}{\partial\bm{\mathrm{r}}}\cdot\left(\textsf{D}_{T}\cdot\frac{\partial}{\partial\bm{\mathrm{r}}}\mathbb{P}\right).$
(SM8)

The initial condition is provided by
$\mathbb{P}(\bm{\mathrm{r}},\vartheta,t=0|\vartheta_{0})=\delta(\bm{\mathrm{r}})\delta(\vartheta-\vartheta_{0})$.

We also introduce the spatial Fourier transform

$\displaystyle\tilde{\mathbb{P}}(\bm{\mathrm{k}},\vartheta,t|\vartheta_{0})=\int\mathrm{d}^{2}r\exp(-i\bm{\mathrm{k}}\cdot\bm{\mathrm{r}})\mathbb{P}(\bm{\mathrm{r}},\vartheta,t|\vartheta_{0}),$
(SM9)

with corresponding e.o.m.

$\displaystyle\partial_{t}\tilde{\mathbb{P}}$
$\displaystyle=-\partial_{\vartheta}[(\omega-\gamma\sin\vartheta)\tilde{\mathbb{P}}]+D_{\text{rot}}\partial_{\vartheta}^{2}\tilde{\mathbb{P}}-i\bm{\mathrm{k}}\cdot\bm{\mathrm{V}}\tilde{\mathbb{P}}-D_{\perp}k^{2}\tilde{\mathbb{P}}-(D_{\parallel}-D_{\perp})(\bm{\mathrm{u}}^{\parallel}\cdot\bm{\mathrm{k}})^{2}\tilde{\mathbb{P}}$
$\displaystyle=:(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})\tilde{\mathbb{P}}.$
(SM10)

Here the operator $\mathcal{L}$ encodes the motion of the angle, while
$\delta\mathcal{L}_{\bm{\mathrm{k}}}=-i\bm{\mathrm{k}}\cdot\bm{\mathrm{V}}-D_{\perp}k^{2}-(D_{\parallel}-D_{\perp})(\bm{\mathrm{u}}^{\parallel}\cdot\bm{\mathrm{k}})^{2}$
describes the coupling to the translational dynamics. The formal solution is
thus
$\tilde{\mathbb{P}}(\bm{\mathrm{k}},\vartheta,t)|\vartheta_{0})=\exp[(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t]\delta(\vartheta-\vartheta_{0})$.
From this quantity the intermediate scattering function (ISF)
$F(\mathbf{k},t)=\langle\exp(-i\bm{\mathrm{k}}\cdot\Delta\bm{\mathrm{r}}(t))\rangle$
is obtained in the stationary state by averaging over the initial angle and
summing over the final one

$\displaystyle F(\bm{\mathrm{k}},t)$
$\displaystyle=\int_{0}^{2\pi}\\!\\!\mathrm{d}\vartheta\\!\\!\int_{0}^{2\pi}\\!\\!\mathrm{d}\vartheta_{0}\,\tilde{\mathbb{P}}(\bm{\mathrm{k}},\vartheta,t|\vartheta_{0})p^{\text{st}}(\vartheta_{0})=\int_{0}^{2\pi}\\!\\!\mathrm{d}\vartheta\exp[(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t]p^{\text{st}}(\vartheta),$
(SM11)

where $p^{\text{st}}(.)$ is the non-trivial distribution of the orientation in
the steady state. We will derive low-order spatial moments by series expansion
in powers of the wavevector $\bm{\mathrm{k}}$.

It is favorable to rely on the isomorphism of periodic square-integrable
functions $f(\vartheta)\in L^{2}[0,2\pi]$ and abstract states
$|f\rangle\in\mathcal{H}$ [4]. The relation is made explicit by introducing
generalized eigenstates $|\vartheta\rangle$ such that
$f(\vartheta)=\langle\vartheta|f\rangle$. For this to hold we require

$\displaystyle\langle
f|g\rangle=\int_{0}^{2\pi}\mathrm{d}\vartheta\,f(\vartheta)^{*}g(\vartheta)=\int_{0}^{2\pi}\mathrm{d}\vartheta\,\langle
f|\vartheta\rangle\langle\vartheta|g\rangle,$ (SM12)

for all $f,g\in L^{2}[0,2\pi]$. We therefore infer the completeness relation

$\displaystyle\int_{0}^{2\pi}\mathrm{d}\vartheta|\vartheta\rangle\langle\vartheta|=\mathbbm{1}.$
(SM13)

Furthermore from

$\displaystyle\mathbbm{1}^{2}=\int_{0}^{2\pi}\mathrm{d}\vartheta\int_{0}^{2\pi}\mathrm{d}\vartheta_{0}\,|\vartheta\rangle\langle\vartheta|\vartheta_{0}\rangle\langle\vartheta_{0}|\stackrel{{\scriptstyle!}}{{=}}\mathbbm{1},$
(SM14)

we conclude the generalized orthogonality

$\displaystyle\langle\vartheta|\vartheta_{0}\rangle=\delta(\vartheta-\vartheta_{0}).$
(SM15)

We write $\\{|n\rangle:n\in\mathbb{Z}\\}$ for the standard orthonormal basis
(ONB) in $\mathcal{H}$ with real-space representation
$\langle\vartheta|n\rangle=\exp(in\vartheta)/\sqrt{2\pi}$. We define the (non-
Hermitian) operator $\mathcal{L}$ in $\mathcal{H}$ via its matrix elements
$\langle m|\mathcal{L}n\rangle$ 555Since $\mathcal{L}$ is non-Hermitian one
has to carefully indicate whether operators act to the right or the left.  of
the Fokker-Planck operator [3]

$\displaystyle\mathcal{L}_{mn}=\langle
m|\mathcal{L}n\rangle=\int_{0}^{2\pi}\frac{\mathrm{d}\vartheta}{2\pi}e^{-im\vartheta}\mathcal{L}e^{in\vartheta}=(-D_{\text{rot}}m^{2}-im\omega)\delta_{mn}+\frac{\gamma}{2}m(\delta_{m,n+1}-\delta_{m,n-1}).$
(SM16)

In particular the matrix representation becomes tridiagonal. We solve for
right and left eigenstates of the Fokker-Planck operator

$\displaystyle\mathcal{L}|r_{\lambda}\rangle$
$\displaystyle=-\lambda|r_{\lambda}\rangle,\qquad\mathcal{L}^{\dagger}|l_{\lambda}\rangle=-\lambda^{*}|l_{\lambda}\rangle,$
(SM17)

by expansion in the ONB

$\displaystyle|r_{\lambda}\rangle=\sum_{n}|n\rangle\langle
n|r_{\lambda}\rangle,\qquad|l_{\lambda}\rangle=\sum_{n}|n\rangle\langle
n|l_{\lambda}\rangle.$ (SM18)

Then the expansion coefficients $\langle n|r_{\lambda}\rangle$ and $\langle
n|l_{\lambda}\rangle$ diagonalize the matrix $\langle m|\mathcal{L}n\rangle$

$\displaystyle\sum_{n}\langle m|\mathcal{L}n\rangle\langle
n|r_{\lambda}\rangle=-\lambda\langle
m|r_{\lambda}\rangle,\qquad\sum_{n}\langle
m|\mathcal{L}^{\dagger}n\rangle\langle n|l_{\lambda}\rangle=-\lambda\langle
m|l_{\lambda}\rangle.$ (SM19)

In particular, the left and right eigenstates can be chosen to be orthonormal
and we assume that they span the entire Hilbert space

$\displaystyle\langle l_{\mu}|r_{\lambda}\rangle$
$\displaystyle=\delta_{\mu\lambda},\qquad\sum_{\lambda}|r_{\lambda}\rangle\langle
l_{\lambda}|=\mathbbm{1},$ (SM20)

where $\sum_{\lambda}$ means that we are summing over all eigenspaces. By
conservation of probability the left eigenstate to eigenvalue $0$ is trivial:
$\langle l_{0}|=\langle 0|$, see Eq. (SM16). The corresponding right
eigenstate $|r_{0}\rangle$ is the stationary state with real space
representation $\langle\vartheta|r_{0}\rangle=p^{\text{st}}(\vartheta)$, where
the normalizing factor is chosen such that $\langle l_{0}|r_{0}\rangle=1$.

The angular propagator $P(\vartheta,t|\vartheta_{0})$ encoding the probability
to find an angle $\vartheta$ at lag time $t$ given the initial orientation at
time $t=0$ was $\vartheta_{0}$ can then be written conveniently in abstract
form as

$\displaystyle P(\vartheta,t|\vartheta_{0})$
$\displaystyle=e^{\mathcal{L}t}\delta(\vartheta-\vartheta_{0})=\sum_{\lambda}e^{-\lambda
t}\langle\vartheta|r_{\lambda}\rangle\langle
l_{\lambda}|\vartheta_{0}\rangle=\langle\vartheta|\sum_{\lambda}e^{\mathcal{L}t}r_{\lambda}\rangle\langle
l_{\lambda}|\vartheta_{0}\rangle=\langle\vartheta|e^{\mathcal{L}t}\vartheta_{0}\rangle.$
(SM21)

###  II.2 Perturbation theory

The full propagator $\tilde{\mathbb{P}}(\bm{\mathrm{k}}\vartheta
t|\vartheta_{0})$ solves the Fokker-Planck equation

$\displaystyle\partial_{t}\tilde{\mathbb{P}}=(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})\tilde{\mathbb{P}},$
(SM22)

with the perturbation

$\displaystyle\delta\mathcal{L}_{\bm{\mathrm{k}}}=$
$\displaystyle-i\bm{\mathrm{k}}\cdot\bm{\mathrm{V}}-D_{\perp}k^{2}-(D_{\parallel}-D_{\perp})(\bm{\mathrm{u}}^{\parallel}\cdot\bm{\mathrm{k}})^{2}.$
$\displaystyle=$ $\displaystyle-ik_{x}v\cos\vartheta-
ik_{y}v\sin\vartheta+ik_{x}v_{1}\sin(\varphi_{H}+\vartheta)-ik_{y}v_{1}\cos(\vartheta_{H}+\vartheta)+ik_{x}v_{2}\sin[2(\varphi_{H}+\vartheta)]+ik_{y}v_{2}\cos[2(\varphi_{H}+\vartheta)]$
$\displaystyle+iv_{3}k_{y}-\frac{D_{\parallel}-D_{\perp}}{2}[(k_{x}^{2}-k_{y}^{2})\cos(2(\varphi_{H}+\vartheta))+2k_{x}k_{y}\sin(2(\varphi_{H}+\vartheta))]-\frac{D_{\parallel}+D_{\perp}}{2}k^{2}.$
(SM23)

In the following we discard all terms due to translational motion and keep
only

$\displaystyle\delta\mathcal{L}_{\bm{\mathrm{k}}}=-iv\bm{\mathrm{k}}\cdot\bm{\mathrm{u}}=-ik_{x}v\cos\vartheta-
ik_{y}v\sin\vartheta,$ (SM24)

with corresponding matrix elements

$\displaystyle(\delta\mathcal{L}_{\bm{\mathrm{k}}})_{mn}=\langle
m|\delta\mathcal{L}_{\bm{\mathrm{k}}}n\rangle=-\frac{ik_{x}v}{2}(\delta_{m,n+1}+\delta_{m,n-1})-\frac{k_{y}v}{2}(\delta_{m,n+1}-\delta_{m,n-1}).$
(SM25)

The formal solution is then given by

$\displaystyle\tilde{\mathbb{P}}(\bm{\mathrm{k}}\vartheta
t|\vartheta_{0})=e^{(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t}\delta(\vartheta-\vartheta_{0})=\langle\vartheta|e^{(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t}\vartheta_{0}\rangle.$
(SM26)

Then the intermediate scattering function in the stationary state is obtained
as

$\displaystyle F(\bm{\mathrm{k}},t)$
$\displaystyle=\int_{0}^{2\pi}\mathrm{d}\vartheta\int_{0}^{2\pi}\mathrm{d}\vartheta_{0}\langle
l_{0}|\vartheta\rangle\langle\vartheta|e^{(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t}\vartheta_{0}\rangle\langle\vartheta_{0}|r_{0}\rangle=\langle
l_{0}|e^{(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t}r_{0}\rangle.$
(SM27)

We use the Dyson representation [4] for the time evolution operator

$\displaystyle
e^{(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t}=e^{\mathcal{L}t}+\int_{0}^{t}\mathrm{d}s\,e^{\mathcal{L}(t-s)}\delta\mathcal{L}_{\bm{\mathrm{k}}}e^{(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})s},$
(SM28)

and expand to second order in the perturbation

$\displaystyle
e^{(\mathcal{L}+\delta\mathcal{L}_{\bm{\mathrm{k}}})t}=e^{\mathcal{L}t}+\int_{0}^{t}\mathrm{d}s\,e^{\mathcal{L}(t-s)}\delta\mathcal{L}_{\bm{\mathrm{k}}}e^{\mathcal{L}s}+\int_{0}^{t}\mathrm{d}s\int_{0}^{s}\mathrm{d}u\,e^{\mathcal{L}(t-s)}\delta\mathcal{L}_{\bm{\mathrm{k}}}e^{\mathcal{L}(s-u)}\delta\mathcal{L}_{\bm{\mathrm{k}}}e^{\mathcal{L}u}+O(\delta\mathcal{L}_{\bm{\mathrm{k}}})^{3},$
(SM29)

and sandwich it between the states $\langle l_{0}|$ and $|r_{0}\rangle$. Note
further that $\langle l_{0}|e^{\mathcal{L}t}=\langle
e^{\mathcal{L}^{\dagger}t}l_{0}|=\langle l_{0}|$ by conservation of
probability and $e^{\mathcal{L}t}|r_{0}\rangle=|r_{0}\rangle$ by the
definition of the stationary state. This yields

$\displaystyle F(\bm{\mathrm{k}},t)$
$\displaystyle=1+\int_{0}^{t}\\!\mathrm{d}s\,\langle
l_{0}|e^{\mathcal{L}(t-s)}\delta\mathcal{L}_{\bm{\mathrm{k}}}e^{\mathcal{L}s}r_{0}\rangle+\int_{0}^{t}\mathrm{d}s\int_{0}^{s}\\!\mathrm{d}u\langle
l_{0}|e^{\mathcal{L}(t-s)}\delta\mathcal{L}_{\bm{\mathrm{k}}}e^{\mathcal{L}_{\bm{\mathrm{k}}}(s-u)}\delta\mathcal{L}_{\bm{\mathrm{k}}}e^{\mathcal{L}u}r_{0}\rangle+O(|\bm{\mathrm{k}}|^{3})$
$\displaystyle=1+t\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+\sum_{\lambda}\int_{0}^{t}\\!\mathrm{d}s\int_{0}^{s}\\!\mathrm{d}u\,\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}e^{\mathcal{L}(s-u)}r_{\lambda}\rangle\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+O(|\bm{\mathrm{k}}|^{3})$
$\displaystyle=1+t\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+\sum_{\lambda}\int_{0}^{t}\\!\mathrm{d}s\int_{0}^{s}\\!\mathrm{d}u\,e^{-\lambda(s-u)}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{\lambda}\rangle\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+O(|\bm{\mathrm{k}}|^{3})$
$\displaystyle=1+t\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+\sum_{\lambda}\frac{e^{-\lambda
t}+\lambda t-1}{\lambda^{2}}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{\lambda}\rangle\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+O(|\bm{\mathrm{k}}|^{3}).$
(SM30)

For the cumulant-generating function this implies

$\displaystyle\ln F(\bm{\mathrm{k}},t)$ $\displaystyle=t\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+\sum_{\lambda}\frac{e^{-\lambda
t}+\lambda t-1}{\lambda^{2}}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{\lambda}\rangle\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle-\frac{t^{2}}{2}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+O(|\bm{\mathrm{k}}|^{3})$
$\displaystyle=t\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+\sum_{\lambda\neq
0}\frac{e^{-\lambda t}+\lambda t-1}{\lambda^{2}}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{\lambda}\rangle\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+O(|\bm{\mathrm{k}}|^{3}).$
(SM31)

On the other hand the cumulant expansion yields

$\displaystyle\ln F(\bm{\mathrm{k}},t)$
$\displaystyle=\ln\langle\exp(-i\bm{\mathrm{k}}\cdot\Delta\bm{\mathrm{r}}(t))\rangle=\ln\left\langle
1-i\bm{\mathrm{k}}\cdot\Delta\bm{\mathrm{r}}(t)-\frac{1}{2}[\bm{\mathrm{k}}\cdot\Delta\bm{\mathrm{r}}(t)]^{2}+O(|\bm{\mathrm{k}}|^{3})\right\rangle$
$\displaystyle=-i\bm{\mathrm{k}}\cdot\langle\Delta\bm{\mathrm{r}}(t)\rangle-\frac{1}{2}\left\\{\langle[\bm{\mathrm{k}}\cdot\Delta\bm{\mathrm{r}}(t)]^{2}\rangle-[\langle\bm{\mathrm{k}}\cdot\Delta\bm{\mathrm{r}}(t)\rangle]^{2}\right\\}.$
(SM32)

In particular, we read off the mean drift velocity along the direction
$\bm{\mathrm{n}}:=\bm{\mathrm{k}}/k$

$\displaystyle\bm{\mathrm{n}}\cdot\frac{\mathrm{d}}{\mathrm{d}t}\langle\Delta\bm{\mathrm{r}}(t)\rangle=\frac{i}{k}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle,$ (SM33)

as well as the variance

$\displaystyle\textsf{Var}[\bm{\mathrm{n}}\cdot\Delta\bm{\mathrm{r}}(t)]$
$\displaystyle=\langle[\bm{\mathrm{n}}\cdot\Delta\bm{\mathrm{r}}(t)]^{2}\rangle-[\langle\bm{\mathrm{n}}\cdot\Delta\bm{\mathrm{r}}(t)\rangle]^{2}$
$\displaystyle=\frac{2}{k^{2}}\sum_{\lambda\neq 0}\frac{1-\lambda
t-e^{-\lambda t}}{\lambda^{2}}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{\lambda}\rangle\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle.$ (SM34)

In particular for the corresponding diffusion coefficient

$\displaystyle
D_{\bm{\mathrm{n}}}:=\lim_{t\to\infty}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\textsf{Var}[\bm{\mathrm{n}}\cdot\Delta\bm{\mathrm{r}}(t)]=\frac{-1}{k^{2}}\sum_{\lambda\neq
0}\frac{1}{\lambda}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{\lambda}\rangle\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle.$ (SM35)

For short times the variance grows as $t^{2}$ since

$\displaystyle\textsf{Var}[\bm{\mathrm{n}}\cdot\Delta\bm{\mathrm{r}}(t)]$
$\displaystyle=\frac{1}{k^{2}}t^{2}\sum_{\lambda\neq 0}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{\lambda}\rangle\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+O(t^{3})$
$\displaystyle=\frac{1}{k^{2}}t^{2}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle-\frac{1}{k^{2}}t^{2}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle\langle
l_{0}\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle+O(t^{3}).$ (SM36)

###  II.3 Numerical results for the eigenspectrum

Fig. SM2: Absolute value of the first non-vanishing eigenvalue,
$|\lambda_{1}|$, vs. diffusivity $D_{\text{rot}}$ for decreasing separation
parameter $\epsilon=(\gamma-\omega)/\omega=\pm 10^{-n/3},n=2,\ldots,8$. The
red lines are for $\epsilon>0$, while blue ones correspond to $\epsilon<0$.
The color becomes darker as the absolute value of the separation parameter
approaches zero. Inset: Rescaled eigenvalues
$|\lambda_{1}|/\omega\sqrt{2|\epsilon|}$ vs. reduced rotational diffusion
coefficient $\hat{D}_{\text{rot}}:=|\epsilon|^{-3/2}D_{\text{rot}}/\omega$.
The straight solid line indicates a power law
$\propto\hat{D}_{\text{rot}}^{1/3}$ and serves as guide to the eye.

Numerical results for the magnitude of the first nontrivial eigenvalue
$\lambda_{1}$ for decreasing noise are shown in Fig. SM2 on double-logarithmic
scales. For $D_{\text{rot}}\to 0$ they approach finite values which become
smaller and smaller as the separation parameter
$\epsilon=(\gamma-\omega)/\omega$ decreases. Empirically we find that
$\lambda_{n}\propto\sqrt{\epsilon}$ in the limit of vanishing noise. The
shapes of the curves above the bifurcation become identical as the bifurcation
is approached, $\epsilon\to 0$, and similarly below the bifurcation. This
observation holds generally for the low-lying eigenvalues
$\lambda_{n},n=1,2,\ldots$. The curves can be superimposed upon shifting
vertically by their low-noise values and proper rescaling of the rotational
diffusion constant. We achieve data collapse using the reduced rotational
diffusion coefficient
$\hat{D}_{\text{rot}}\propto|\epsilon|^{-3/2}D_{\text{rot}}$ suggesting the
scaling law

$\displaystyle\lambda_{n}/\omega=\sqrt{2|\epsilon|}\Lambda_{n,\pm}(\hat{D}_{\text{rot}}),$
(SM37)

with dimensionless scaling functions $\Lambda_{n,\pm}(.)$ for
$\epsilon\gtrless 0$ (it turns out that the factor $2$ in the square root
renders $\Lambda_{n,+}(\hat{D}_{\text{rot}}\to 0)=n$).

The inset of Fig. SM2 exhibits the scaling behavior in the vicinity of the
bifurcation for small noise. By construction both scaling functions saturate
for small reduced rotational diffusion coefficient $\hat{D}_{\text{rot}}\ll
1$, while for large $\hat{D}_{\text{rot}}\gg 1$ they behave as a power law
with an exponent of $1/3$:

$\displaystyle\Lambda_{n,\pm}(\hat{D}_{\text{rot}})\sim
C_{n}\hat{D}_{\text{rot}}^{1/3}\qquad\text{for
}\hat{D}_{\text{rot}}\to\infty,$ (SM38)

with constants $C_{n}$ independent of the sign of $\epsilon$. In the next
subsection the scaling behavior of the low-noise limit as well as the proper
choice of $\hat{D}_{\text{rot}}$ is rationalized in terms of a harmonic-
oscillator picture.

###  II.4 Harmonic approximation

a)  b)

Fig. SM3: a) Titled washboard potential $U(\vartheta)$. b) Zoom close to the
classical bifurcation.

We rewrite the stochastic e.o.m. in terms of an effective potential
$U(\vartheta)$

$\displaystyle\dot{\vartheta}=-\frac{D_{\text{rot}}}{k_{B}T}\partial_{\vartheta}U+\zeta(t).$
(SM39)

Here $D_{\text{rot}}/k_{B}T$ is then the mobility for the overdamped motion of
the angle and the potential reads

$\displaystyle\frac{D_{\text{rot}}}{k_{B}T}U(\vartheta)=-\omega\vartheta-\gamma\cos\vartheta.$
(SM40)

For $\gamma\geq\omega$ the potential displays a minimum at
$\vartheta_{*}\in[0,\pi/2]$ with $\sin\gamma_{*}=\omega/\gamma$ while a
maximum is displayed at $\pi-\vartheta_{*}\in[\pi/2,\pi]$, see Fig. SM3 for
illustration. Let’s make a harmonic approximation for the potential close to
the minimum,

$\displaystyle\frac{D_{\text{rot}}}{k_{B}T}U(\vartheta)$
$\displaystyle=\frac{D_{\text{rot}}}{k_{B}T}U(\vartheta_{*})+\frac{(\vartheta-\vartheta_{*})^{2}}{2}\gamma\cos\vartheta_{*}+O(\vartheta-\vartheta_{*})^{3}.$
(SM41)

For the stochastic e.o.m. this results in a linear Langevin equation

$\displaystyle\dot{\vartheta}$
$\displaystyle=-\frac{1}{\tau}(\vartheta-\vartheta_{*})+\zeta(t),$ (SM42)

with relaxation rate

$\displaystyle\frac{1}{\tau}=\gamma\cos\vartheta_{*}=\sqrt{\gamma^{2}-\omega^{2}}.$
(SM43)

The eigenvalues of the overdamped harmonic oscillator are simply

$\displaystyle\lambda_{n}=\frac{1}{\tau}n.$ (SM44)

The picture of the harmonic oscillator is expected to be valid for the low
eigenvalues if the escape rate for diffusing over the barrier is small
compared to the relaxation rate. For large barriers $\Delta
U:=U(\pi-\vartheta_{*})-U(\vartheta_{*})$ the escape is exponentially
suppressed. $\propto\exp(-\Delta U/k_{B}T)$, by Kramers’ law, correspondingly
we require that the barrier is high with respect to the thermal energy

$\displaystyle 1\ll\frac{\Delta U}{k_{B}T}$
$\displaystyle=\frac{1}{k_{B}T}[U(\pi-\vartheta_{*})-U(\vartheta_{*})]$
$\displaystyle=\frac{1}{D_{\text{rot}}}[2\omega\arcsin(\omega/\gamma)-\pi\omega+2\sqrt{\gamma^{2}-\omega^{2}}].$
(SM45)

The condition is certainly fulfilled for $D_{\text{rot}}\to 0$ for fixed
$\gamma>\omega$. Here we want to investigate the scaling limit of
$D_{\text{rot}}/\omega\to 0$ and $\epsilon:=(\gamma-\omega)/\omega\to 0$. Then
upon series expansion the condition of high barrier translates to

$\displaystyle 1\ll\frac{\Delta
U}{k_{B}T}=\frac{4\sqrt{2}}{3}\frac{\omega}{D_{\text{rot}}}\epsilon^{3/2}+O(\ldots).$
(SM46)

This is the condition, Eq. (14), of the main text.

### Matrix elements of the harmonic oscillator

For small $D_{\text{rot}}$ and $\gamma>\omega$ we perform the harmonic
oscillator approximation around the minimum $\vartheta_{*}$,
$x:=\vartheta-\vartheta_{*}$

$\displaystyle\mathcal{L}$
$\displaystyle=D_{\text{rot}}\frac{\partial^{2}}{\partial\vartheta^{2}}-\frac{\partial}{\partial\vartheta}(\omega-\gamma\sin\vartheta)=D_{\text{rot}}\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial}{\partial x}[\omega-\gamma\sin(\vartheta_{*}+x)]$
$\displaystyle\approx D_{\text{rot}}\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial}{\partial
x}(\omega-\gamma\sin\vartheta_{*}-x\gamma\cos\vartheta_{*}).$ (SM47)

Since $\gamma\sin\vartheta_{*}=\omega$ and
$\gamma\cos\vartheta_{*}=\sqrt{\gamma^{2}-\omega^{2}}=:1/\tau$ this simplifies
to

$\displaystyle\mathcal{L}=D_{\text{rot}}\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial}{\partial x}\left(\frac{x}{\tau}\right).$ (SM48)

This Fokker-Planck operator corresponds to the effective potential

$\displaystyle\frac{D_{\text{rot}}U(x)}{k_{B}T}=\frac{x^{2}}{2\tau}.$ (SM49)

The angular oscillator width
$\langle(\vartheta-\vartheta_{*})^{2}\rangle^{1/2}=\sqrt{\langle
x^{2}\rangle}=\sqrt{D_{\text{rot}}\tau}$ quantifies the mean-square
fluctuations. For $D_{\text{rot}}\to 0$ the eigenfunctions become more and
more localized.

For the perturbation we expand also to linear order

$\displaystyle\delta\mathcal{L}_{\bm{\mathrm{k}}}$
$\displaystyle=-ik_{x}v\cos\vartheta-
ik_{y}\sin\vartheta=-ik_{x}v\cos(\vartheta_{*}+x)-ik_{y}v\sin(\vartheta_{*}+x)$
$\displaystyle=-ik_{x}v\cos\vartheta_{*}-ik_{y}v\sin\vartheta_{*}+(ik_{x}v\sin\vartheta_{*}-ik_{y}v\cos\vartheta_{*})x+O(x^{2}).$
(SM50)

Thus to leading order only the diagonal components are non-vanishing

$\displaystyle\langle
l_{\mu}|\delta\mathcal{L}_{\bm{\mathrm{k}}}|r_{\lambda}\rangle=(-ik_{x}v\cos\vartheta_{*}-ik_{y}v\sin\vartheta_{*})\delta_{\mu\lambda}+O(.).$
(SM51)

This yields for the mean drift

$\displaystyle\bm{\mathrm{n}}\cdot\frac{\mathrm{d}}{\mathrm{d}t}\langle\Delta\bm{\mathrm{r}}(t)\rangle=\frac{i}{k}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}|r_{0}\rangle\to(n_{x}v\cos\vartheta_{*}+n_{y}v\sin\vartheta_{*}),$
(SM52)

with $\bm{\mathrm{n}}=\bm{\mathrm{k}}/k$. This reproduces the classical
result.

The variance $\textsf{Var}_{\bm{\mathrm{n}}}[\Delta\bm{\mathrm{r}}(t)$
vanishes as $D_{\text{rot}}\to 0$. To obtain the leading order we need to
evaluate the matrix elements $\langle l_{\mu}|x|r_{\lambda}\rangle$ for
$\mu\neq\lambda$. The fastest way is to use the gauge transformation to map
the Fokker-Planck operator to a Schrödinger Hamiltonian [3] Define the
Hamiltonian by the gauge transform

$\displaystyle\mathcal{H}$
$\displaystyle=-e^{U(x)/2k_{B}T}\mathcal{L}e^{-U(x)/2k_{B}T}=-e^{x^{2}/4D_{\text{rot}}\tau}\mathcal{L}e^{-x^{2}/4D_{\text{rot}}\tau}$
$\displaystyle=-D_{\text{rot}}\frac{\partial^{2}}{\partial
x^{2}}+\frac{x^{2}}{4D_{\text{rot}}\tau^{2}}-\frac{1}{2\tau},$ (SM53)

which is Hermitian in $L^{2}(\mathbb{R})$. Adjoining yields

$\displaystyle\mathcal{H}$
$\displaystyle=-e^{-U(x)/2k_{B}T}\mathcal{L}^{\dagger}e^{U(x)/2k_{B}T}.$
(SM54)

We substitute
$D_{\text{rot}}\mapsto\hbar^{2}/2m,1/2D_{\text{rot}}\tau^{2}\mapsto
m\omega^{2}$ and find the shifted harmonic oscillator

$\displaystyle\mathcal{H}=\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}+\frac{m\omega^{2}}{2}x^{2}-\frac{\hbar\omega}{2},$ (SM55)

with eigenvalues $\lambda_{n}=n\hbar\omega=n/\tau$. In particular the quantum
oscillator length $\sqrt{\hbar/2m\omega}$ maps to the angular oscillator width
$\sqrt{D_{\text{rot}}\tau}$. The matrix elements of $x$ are thus expected to
be of order $\sqrt{D_{\text{rot}}\tau}$

The corresponding mapping for the eigenfunctions is then

$\displaystyle r_{n}(x)$ $\displaystyle=e^{-U(x)/2k_{B}T}\psi_{n}(x),$ (SM56)
$\displaystyle l_{n}(x)$ $\displaystyle=e^{U(x)/2k_{B}T}\psi_{n}(x),$ (SM57)

with $\psi_{n}(x)$ the eigenfunctions of $\mathcal{H}$. The matrix elements of
an operator $\mathcal{V}$ are obtained as

$\displaystyle\langle
l_{m}|\mathcal{V}r_{n}\rangle=\int\mathrm{d}x\,l_{m}(x)^{*}\,\mathcal{V}\,r_{n}(x)=\int\mathrm{d}x\,\psi_{m}(x)^{*}[e^{U(x)/2k_{B}T}\mathcal{V}e^{-U(x)/2k_{B}T}]\psi_{n}(x).$
(SM58)

For the case of $\mathcal{V}=x$ the exponentials cancel, and we read off (with
harmonic oscillator creation and annihilation operators $a,a^{\dagger}$)

$\displaystyle\langle l_{m}|x|r_{n}\rangle$
$\displaystyle=\langle\psi_{m}|x|\psi_{n}\rangle=\langle\psi_{m}|\sqrt{\frac{\hbar}{2m\omega}}\left(a^{\dagger}+a\right)|\psi_{n}\rangle$
$\displaystyle=\sqrt{\frac{\hbar}{2m\omega}}\left(\delta_{m,n+1}\sqrt{n+1}+\delta_{m,n-1}\sqrt{n}\right)\mapsto\sqrt{D_{\text{rot}}\tau}\left(\delta_{m,n+1}\sqrt{n+1}+\delta_{m,n-1}\sqrt{n}\right).$
(SM59)

We thus read off the relevant matrix elements for the variance/diffusion
coefficient

$\displaystyle\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{\lambda}\rangle$
$\displaystyle=\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle=iv\sqrt{D_{\text{rot}}\tau}(k_{x}\sin\vartheta_{*}-k_{y}\cos\vartheta_{*})\delta_{\lambda,1}.$
(SM60)

The picture of the harmonic oscillator is valid only for
$\hat{D}_{\text{rot}}\ll 1$. In this regime the formulas for the variance and
diffusion simplify since only a single term contributes

$\displaystyle\textsf{Var}[\bm{\mathrm{n}}\cdot\Delta\bm{\mathrm{r}}(t)]$
$\displaystyle=\frac{2}{k^{2}}\sum_{\lambda\neq 0}\frac{1-\lambda
t-e^{-\lambda t}}{\lambda^{2}}\langle
l_{0}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{\lambda}\rangle\langle
l_{\lambda}|\delta\mathcal{L}_{\bm{\mathrm{k}}}r_{0}\rangle$
$\displaystyle=2\left(\frac{t}{\tau}-1+e^{-t/\tau}\right)(v\tau)^{2}D_{\text{rot}}\tau(n_{x}\sin\vartheta_{*}-n_{y}\cos\vartheta_{*})^{2},$
(SM61)

and infer the associated long-time diffusion coefficient

$\displaystyle D_{\bm{\mathrm{n}}}$
$\displaystyle=(v\tau)^{2}D_{\text{rot}}(n_{x}\sin\vartheta_{*}-n_{y}\cos\vartheta_{*})^{2}.$
(SM62)

In particular, for small $D_{\text{rot}}\ll\omega$, $D_{0}\approx
v^{2}D_{\text{rot}}/2\omega^{2}$ in the harmonic oscillator approximation

$\displaystyle
D_{\bm{\mathrm{n}}}/D_{0}=2(\omega\tau)^{2}(n_{x}\sin\vartheta_{*}-n_{y}\cos\vartheta_{*})^{2}\approx\frac{1}{\epsilon}(n_{x}\sin\vartheta_{*}-n_{y}\cos\vartheta_{*})^{2}.$

This result yields prefactors to the scaling law derived in the main text.

##  III Supplementary figure on mean drift velocity

The mean drift velocity is evaluated directly from the stationary distribution

$\displaystyle v\begin{pmatrix}\langle\cos\vartheta\rangle\\\
\langle\sin\vartheta\rangle\end{pmatrix}=\int_{0}^{2\pi}v\begin{pmatrix}\cos\vartheta\\\
\sin\vartheta\end{pmatrix}p^{\text{st}}(\vartheta)\mathrm{d}\vartheta.$ (SM63)

Fig. SM4: Mean drift velocity in the stationary state along the horizontal
(left) and vertically against the field (right) as a function of the torque
$\gamma$ for various noise strength $D_{\text{rot}}$. Symbols correspond to
simulation, full lines to numerical results.

The stationary distribution is known explicitly [3]

$\displaystyle p^{\text{st}}(\vartheta)=$
$\displaystyle\mathcal{N}e^{-U(\vartheta)/k_{B}T}\left[\int_{\vartheta}^{2\pi}e^{U(\bar{\vartheta})/k_{B}T}\mathrm{d}\bar{\vartheta}+e^{[U(0)-U(2\pi)]/k_{B}T}\int_{0}^{\vartheta}e^{U(\bar{\vartheta})/k_{B}T}\mathrm{d}\bar{\vartheta}\right],$
(SM64)

with the potential

$\displaystyle\frac{D_{\text{rot}}U(\vartheta)}{k_{B}T}=-\omega\vartheta-\gamma\cos\vartheta.$
(SM65)

The normalization factor $\mathcal{N}$ is fixed by imposing
$\int_{0}^{2\pi}p^{\text{st}}(\vartheta)\mathrm{d}\vartheta=1$.

Numerical results for the average sine and cosine are displayed in Fig. SM4
but similar figures can also be found in Ref. [3].

##  IV Supplementary figure on influence of translational diffusion on the
resonance

The techniques elaborated in the main text can be readily extended to include
the additional drift terms and the translational diffusion in Eq. (I). The
resonance for the full model emerges in the simultaneous limit of vanishing
separation parameter and all noise terms approaching zero $D_{\text{rot}}\to
0,D_{\perp}\to 0,D_{\parallel}\to 0$ such that the ratios
$D_{\perp}/D_{\text{rot}},D_{\parallel}/D_{\text{rot}}$ remain fixed.
Simulation as well as numerical results are displayed in Fig. SM5. Although
diffusion broadens the resonance, the general picture remains valid.

Fig. SM5: Diffusion coefficient $D$ as a function of the torque $\gamma$ for
decreasing orientational diffusion coefficients $D_{\text{rot}}$ and
translational diffusion coefficients $D_{\perp}$ ($D_{\parallel}=2D_{\perp}$).
Symbols correspond to simulation, full lines to numerical results. The values
of the perpendicular component of the translational diffusivity $D_{\perp}$
and rotational diffusivity $D_{\mathrm{rot}}$ are specified in the legend. The
parallel component always is $D_{\parallel}=2D_{\perp}$, for simplicity
$\varphi_{H}=0$, and we choose parameters such that $v_{1}=v_{2}=v_{3}/3=v$.

## Bibliography

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B. ten Hagen, F. Kümmel, R. Wittkowski, D. Takagi, H. Löwen, and C. Bechinger,
Gravitaxis of asymmetric self-propelled colloidal particles, Nature
Communications 5, 4829 (2014).

  2. [2]

The use of the hydrodynamic expressions for active propulsion is
controversial, see Ref. [6].

  3. [3]

H. Risken, The Fokker-Planck Equation (Springer Berlin Heidelberg, 1989).

  4. [4]

J. Sakurai and J. Napolitano, Modern quantum mechanics. San Fransico (CA:
Addison-Wesley, 2011).

  5. [5]

Since $\mathcal{L}$ is non-Hermitian one has to carefully indicate whether
operators act to the right or the left.

  6. [6]

B. U. Felderhof, Comment on “Circular motion of asymmetric self-propelling
particles”, Physical Review Letters 113 (2014); F. Kümmel, B. ten Hagen, R.
Wittkowski, D. Takagi, I. Buttinoni, R. Eichhorn, G. Volpe, H. Löwen, and C.
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