# Mimicking quantum self-propulsion creates a generic route to quantum active
matter

Yuanjian Zheng  Institut für Theoretische Physik II: Weiche Materie, Heinrich-
Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany
Benno Liebchen  Technische Universität Darmstadt, Hochschulstraße 8, D-64289
Darmstadt, Germany  Hartmut Löwen  Institut für Theoretische Physik II: Weiche
Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225
Düsseldorf, Germany

###### Abstract

We introduce a generic framework for describing quantum active matter that
involves mimicking the role of self-propulsion through an external trapping
potential that is moving along imposed trajectories traced by classical active
dynamics. This approach in the presence of dissipation, not only recovers
essential dynamical behavior of classical activity, including the ballistic to
diffusive cross-over of its mean-square displacement, but also reveals
additional features of activity that is of pure quantum origin. These quantum-
active features are most clearly revealed in non-dissipative systems, and
manifest as novel exponents of the mean-square displacement at short time
scales, that are intricately dependent on the geometry of the trapping
potential.

Over the past two decades, non-equilibrium physics has been profoundly shaped
by two remarkable developments: the rise of active matter, which has unveiled
a plethora of out of equilibrium self-organization phenomena, and the
advancement of methods enabling the precise control and manipulation of
individual inherently quantum objects.

The first development – active matter, concerns the description of agents that
are capable of using energy from their environment to propel themselves
MarchettiSimha2013 ; Ramaswamy2010 ; Gompper2020  – either autonomously
paxton2004catalytic ; howse2007self ; BechingerVolpe2016  – or through the
actuation with time-dependent external fields DreyfusBibette2005 ;
TiernoSagues2008 ; DeseigneChate2010 ; LimJaeger2019 . Such agents, called
active particles were first realized at the micron-scale about two decades ago
paxton2004catalytic , and induce a spectacular plethora of non-equilibrium
phenomena. For a single particle, this includes the emergence of a mean-square
displacement (MSD) beyond the generic Stokes-Einstein relation and
characteristic ballistic motion that enables the usage of active particles as
efficient micro-motors for targeted drug delivery. At the many-body level, the
local driving of active systems further opens up a world of spectacular
phenomena. This includes the spontaneous emergence of a center of mass motion
induced by non-reciprocal interactions of isotropic particles soto2014self ;
ivlev2015statistical ; niu2018dynamics ; schmidt2019light ; saha2020scalar ;
grauer2021active ; SurowkaBanerjee2023 , and the emergence of novel types of
phase transitions that lead to so-called flocks where particles align over
large distances and move collectively in a spontaneously chosen direction.
These flocks exploit non-equilibrium fluctuations to bypass the Mermin-Wagner
theorem and exhibit true long-range order in two-dimensions toner1995long ;
toner1998flocks , showing giant number fluctuations that violate the central
limit theorem, making it impossible to measure (predict) the density in a
finite domain narayan2007long . Other remarkable phenomena include motility-
induced phase separation where an ensemble of purely repulsively interacting
particles spontaneously phase separate into a co-existing gas-like and a
liquid-like phase cates2015motility , as well as living clusters that
dynamically form, break-up and reform recursively theurkauff2012dynamic ;
palacci2013living ; buttinoni2013dynamical ; ginot2018aggregation . Many of
these phenomena have been show in the past decade to occur at a broad range of
scales, ranging from macroscopic robots, birds and drones Whitesides2018 ;
LeymanVolpe2018 ; VasarhelyiVicsek2018 ; AraujoVolpe2023 , down to the micro-
and larger nanoscale where bacteria, molecular motors and synthetic colloidal
microswimmers thrive Ebbens2016 ; ShaebaniRieger2020 ; MognettiFrenkel2013 ;
Klapp2016 ; LeunissenBlaaderen2005 ; ZiepkeFrey2022 .

The second development revolves around the evolution of cold atom physics,
that was invigorated by the first observation of Bose-Einstein condensation in
dilute atomic gases BEC1 ; BEC2  in 1995, the very same year as the inception
of active matter research Vicsek1995 ; toner1995long . Ever since, remarkable
progress has been achieved regarding the developments of techniques for
cooling, trapping and the controlled manipulation of quantum objects both at
the single hu1994observation ; leibfried2003quantum  and the many-particle
level lewenstein2007ultracold ; bloch2008many , even in non-equilibrium
situations polkovnikov2011colloquium ; langen2015ultracold ;
schneider2012fermionic ; heyl2018dynamical .

This co-discovery of fundamentally new phenomena in active matter and novel
methods for controlling atoms in the deep quantum regime, begs the question of
whether novel active non-equilibrium phase transitions and pattern formation
could also be realized at a scale where quantum effects are non-negligible. In
particular, one may wonder if there is a quantum-analog to active phenomena
like motility-induced phase separation, dynamic clustering or giant density
fluctuations. However, progress in understanding this interplay has been
limited thus far, since it is yet unknown how a self-propelled particle could
be realized in the deep quantum regime. Accordingly, existing attempts to
unify quantum physics and active matter are limited to a relatively small
number of pioneering works either in identifying and exploring original formal
analogies between descriptions of quantum systems and classical active
behavior LopezClenebtt2015 ; GuoCheng2018 ; teVrugtWittkowski2023 ;
SoneAshida2019 ; LoeweGoldbart2018 , or examining innovative models involving
classical swimmers embedded in quantum media such as fermionic liquids
AvronOaknin2006 , Bose-Einstein condensates Saito2015  and super-fluid He4
ShuklaPandit2016 ; GiuriatoKrstulovic2019 ; KolmakovAranson2021 . In all these
semi-classical approaches, the driven agent itself is not quantum mechanical
in nature but remains strictly a classical object. Very recently, the role of
activity AdachiKawaguchi2022 ; YamagishiObuse2023  has also been considered in
quantum many-body systems with non-hermitian Hamiltonians
MatsoukasDelcampo2023 ; quantumflocks . These highly original approaches allow
for the first time, exploration of activated dynamics on a lattice, which does
however restrict them from exploring generic active matter phenomena in
continuum space.

In this work, we present a generic framework that is orthogonal to these
existing approaches. This allows us to _bypass the challenge of actualizing
self-propelled quantum particles when realizing active phenomena in the
quantum regime_. The central idea is to _mimic self-propulsion_ by guiding
particles that are deep in the quantum regime with a moving external trapping
potential along the path of an active particle (Fig.1). We show that this
driven quantum particle, despite not self-propelling autonomously, recovers
all desirable properties of a quantum active particle (Fig.1). That is, our
approach leads, for the first time, to a fully quantized Hamiltonian that
describes a quantum particle that

  1. i.

breaks local detailed balance

  2. ii.

moves in continuum space

  3. iii.

leads to a MSD that is quadratic at intermediate timescales in the presence of
dissipation BechingerVolpe2016 .

  4. iv.

leads to a MSD that is linear at long timescales (active diffusion
BechingerVolpe2016 ).

Figure 1: [Schematics and dynamics of dissipative quantum and classical active
systems]. Mimicking self-propulsion in the presence of dissipation produces
ballistic and diffusive regimes in the MSD at various time scales that are
characteristic of classical active particles. Signature of quantum-activity
beyond classical dynamics is captured by an anomalous $t^{6}$ scaling of the
MSD at short time scales

In particular, this approach is beyond semi-classical and lattice-based
realizations of active matter and directly leads to new quantum active effects
even at the single particle level that are accentuated in the absence of
dissipation.

While the focus of this work is on introducing a minimal, generic framework of
quantum active particles, and accordingly, exploration of collective phenomena
is beyond its scope, we emphasize that this approach allows for an almost
direct realization of single and collective active matter phenomena with
quantum particles. Our approach can be realized using ultra-cold atoms in
time-dependent optical traps BlochNascimbene2012 ; Kaufman2021 ; Lahaye2020 ;
SchaferTakahashi2020 ; MorigiWineland2015 , in which the positions of the
potential minimum can be externally programmed almost at will and at very high
speeds PesceVolpe2020 ; Ashkin1970 ; ChenMuga2010 ; AbahLutz2012 ;
RossnagelLutz2014 . Our work thus opens a door to a new class of materials,
namely that of quantum active matter.

## Mimicking quantum self-propulsion –

We now consider a time-dependent (harmonic) trapping potential of a quantum
particle that is uniquely parameterized by its minimum position $x_{c}(t)$.
This position is externally programmed to follow random paths drawn from an
ensemble of trajectories generated by classical active dynamics
SabhapanditMajumdar2024 ; UhlenbeckOrnstein1930 ; BechingerVolpe2016 ;
RomanczukGeier2012 , such that the instantaneous time-dependent Hamiltonian is
written as

$\hat{H}(t)=\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\left(\hat{x}-x_{c}(t)\right)^{2}$
(1)

where $\hat{p}=-i\hbar\partial_{x}$. We further assume that this particle is
weakly coupled to its environment such that dynamics of its density matrix
$\hat{\rho}(t)$ is governed by the Lindblad equation RivasHuelga2011 :

$\dot{\hat{\rho}}=-\frac{i}{\hbar}\left[\hat{H}(t),\hat{\rho}\right]+\hat{\mathcal{D}}(\hat{\rho},t)$
(2)

The (Markovian) dissipator is given by

$\hat{\mathcal{D}}(\hat{\rho},t)=\nu_{+}\left(\hat{a}^{\dagger}\hat{\rho}\hat{a}-\frac{1}{2}\left\\{\hat{a}\hat{a}^{\dagger},\hat{\rho}\right\\}\right)+\nu_{-}\left(\hat{a}\hat{\rho}\hat{a}^{\dagger}-\frac{1}{2}\left\\{\hat{a}^{\dagger}\hat{a},\hat{\rho}\right\\}\right)$
(3)

where $\hat{a}$ ($\hat{a}^{\dagger}$) is the creation (annihilation) operator,
while $\nu^{+}$ and $\nu^{-}$ are the gain and loss rates for the excitations
of the quantum particle in the trap respectively, and correspond to an
equilibrium temperature ($T$) of the thermal bath through
$\nu_{+}/\nu_{-}=e^{-\hbar\omega}/k_{b}T$ where $\omega$ is the frequency of
the harmonic trapping potential, $k_{B}$ is the Boltzmann constant and $\hbar$
is the reduced Planck constant.

Figure 2: [MSD of a dissipative quantum-active particle]. (a) MSD at various
time scales for $D=0.01$, $\tau=10$ and $\nu_{-}=10^{-2}$. (b) Best fits of
the MSD (blue solid lines) to MSD for $\nu_{+}=10^{-8}$ indicating diffusive-
ballistic-diffusive behavior reminiscent of classical active matter is
recovered by the quantum-active framework

Now, in this work, we write the quantum mean-square displacement (MSD) or the
instantaneous spread of the wavefunction $\psi_{t}$ is written as the
expectation value:
$\langle\psi_{t}|\hat{x}^{2}|\psi_{t}\rangle-\langle\psi_{t=0}|\hat{x}^{2}|\psi_{t=0}\rangle$.
Since Eq. (3) is linear in $\hat{\rho}$, contributions to the MSD from terms
in $\hat{\mathcal{D}}(\hat{\rho})$ are additive and thus can be considered
independently from the Hamiltonian time evolution in (2). In particular, at
short time scales, we show in the SI that the MSD is determined by the gain
and loss term and is linear in $t$. We also show in the SI that this result
holds more generically for arbitrary trapping potentials, which we discuss
subsequently in more details for non-dissipative scenarios. Hence, unless the
system is fully isolated from its environment or at $T=0$, the short-time MSD
scales diffusively, suggesting that this present framework successfully
captures short-time behavior of classical activity BechingerVolpe2016 .

To verify this predicted short time diffusive behavior, we numerically solve
Eq. (2) and calculate the ensemble averaged MSD (averaged over the underlying
random path $x_{c}(t)$) in Fig.2, using a finite-difference method with
discrete time step of $dt=10^{-2}$, where $\sqrt{\hbar/m\omega}$ and
$\omega^{-1}$ are units of length and time respectively. A total of $5\times
10^{2}$ stochastic trajectories $x_{c}(t)$ were used for each $\nu_{+}$
considered (see legend) where $D=1$, $\tau=10$ and $\nu_{-}=10^{-2}$. Indeed,
in Fig.2(a) we observe the universal presence of a short-ranged diffusive (MSD
$\sim t$) regime for any non-zero $\nu_{+}$ (or equivalently $T\neq 0$),
indicating that any dissipation necessarily results in short time diffusive
behavior. At intermediate time scales ($t\sim\tau$), the quantum-active
dynamics also exhibits a distinct ballistic regime (MSD $\sim t^{2}$), as in
the classical case BechingerVolpe2016 , before eventually entering a diffusive
regime (MSD $\sim t$) at much longer times ($t\gg\tau$) as in the classical
case BechingerVolpe2016 . We note that while the cross-over time scales is
highly dependent on strength of the dissipation (i.e on $\nu_{-}$) such that
the ballistic regime is not clearly visible at too high temperatures, power-
law fits of the MSD in various dynamical regimes for $\nu_{+}=10^{-8}$ in
Fig.2(b) shows a clear $t^{2}$ regime. As such, our quantum-active framework
reproduces all defining dynamical features of classical activity.

Moreover, besides these active dynamical features with well established
classical analogs, we observe in Fig.2(a), the emergence of an additional
quantum-active regime at short time scales where the MSD increases $\sim
t^{6}$ or thereabout (Fig.2(a)). This regime has no classical equivalent. The
temporal extension of this anomalous dynamical regime (marked as cross-over in
Fig.2(b)) is observed to be increasing with decreasing gain rates $\nu_{+}$
(i.e lower $T$), by a progressively shortening of the initial diffusive
regime. This suggests that the anomalous regime has a quantum origin, as pure
quantum effects originating from the Hamiltonian dynamics are enhanced with
lower dissipation rates. In fact, the initial diffusive regime of the MSD
completely vanishes (black dashed line in Fig.2(a)) in the absence of
dissipation (i.e $\nu_{+}=0$ ) or at zero temperature, producing instead, a
short time power-law dependence MSD $\sim t^{6}$. Hence, to elucidate purely
quantum features beyond classical active matter, we now focus on discussing
the dynamical contribution from the Hamiltonian time-evolution term of (2) to
the MSD, which in itself can also be regarded as a non-dissipative quantum-
active particle.

To this end, we first treat the short-time behavior of the MSD using time-
dependent perturbation theory. In such an approach, the MSD to first order,
can be written as a stochastic average (denoted by an over-bar) over an
ensemble of active trajectories $\Gamma:\\{x_{c}(t)\\}$ (see SI for
derivation)

$\overline{\langle\hat{x}^{2}(t)\rangle}-\langle\hat{x}^{2}_{t=0}\rangle\sim\overline{\left[\int^{t}_{0}dt^{\prime}x_{c}(t)\right]^{2}}=\int_{0}^{t}\int_{0}^{t}dt^{\prime}dt^{\prime\prime}\overline{x_{c}(t^{\prime})x_{c}(t^{\prime\prime})}$
(4)

which reduces to a double time-integral of the position auto-correlation
function of the active ensemble. We stress the significance of this result
that essentially expresses a quantum mechanical expectation value as a
classical position auto-correlation function of the underlying imposed
dynamics, which fundamentally reveals the presence of a subtle quantum-active
coupling in the current framework.

To arrive at the explicit time-dependence for the active-quantum MSD, we
further assume that the stochastic trajectories are drawn from an ensemble of
particles that obey active Ornstein-Uhlenbeck dynamics, since it is the
simplest non-trivial model of a classical active particle
UhlenbeckOrnstein1930 ; Szamel2014 ; MartinvanWijland2021 ;
CapriniVulpiani2019 . More precisely, we assume that the velocity auto-
correlation function of $x_{c}(t)$ is given by Bonilla2019 :

$\overline{\dot{x}_{c}(t^{\prime})\dot{x}_{c}(t^{\prime\prime})}=\frac{D}{\tau}e^{-|t^{\prime}-t^{\prime\prime}|/\tau}$
(5)

where $D$ and $\tau$ are the diffusion coefficient and persistence time
respectively, and $\dot{x}_{c}$ represent time derivatives of $x_{c}(t)$. It
then follows (see SI for derivation) that the lowest possible leading order
contribution to the ensemble-averaged quantum MSD in the perturbative regime
is $t^{4}$, regardless of the functional form of the trapping potential
$V(x)$.

Hence, the leading order in $t$ contribution to the MSD originating from the
Hamiltonian is always strictly of higher order than the linear contribution
from Markovian dissipation, such that the initial diffusive regime observed in
Fig.2 is always recovered in the presence of weak coupling to the environment.
However, note that while the leading order dependence is generically $t^{4}$,
Fig.2(a) for $\nu_{+}=0$ (black-dashed line) suggests that the actively driven
harmonic potential produces instead a $t^{6}$ quantum-active scaling. We show
in the SI that this can be understood as the consequence of a non-trivial
contribution from 2nd-order perturbation theory specific to the harmonic
potential, that exactly negates the 1st-order $t^{4}$ contribution to the MSD,
such that the leading behavior is $t^{6}$.

To further examine this quantum-active regime, we generalize the harmonic
driving potential to a family of power-law trapping potentials –
$\hat{V}(\hat{x})=\frac{1}{2}m\omega^{q}|\hat{x}-x_{c}(t)|^{q}$. The
Hamiltonians of these anharmonic potentials produce eigenenergy spectra that
are qualitatively different from the harmonic oscillator, and are thus
instructive in revealing the subtle interplay between quantized energy levels
and activity. In particular, unlike the uniformly distributed eigenenergies of
the harmonic oscillator ($q=2$), energy level spacings of these Hamiltonians
are distributed either increasing ($q>2$) or decreasing ($q<2$) monotonically.

Figure 3: [MSD of a non-dissipative quantum-active particle]. (a,c) MSD at
short times for $D=1$, $\tau=10$, $dt=10^{-2}$ reveals quantum-active scaling
with no classical counterparts. (b) Log-log plot of the absolute values of MSD
indicate $\sim t^{4}$ scaling for $q\neq 2$ and $\sim t^{6}$ for $q=2$. (d,e)
Diffusive regime at long times for $D=1$, $\tau=1$, $dt=10^{-1}$. (c,e)
Comparison of quantum to classical active MSD.

In Fig.3, we explore how such non-uniform spectra influence the short time MSD
for $q=1.5$, $q=2$, and $q=4$ of the non-dissipative, actively driven quantum
particle. The numerics are performed in real space using finite difference
discrete time steps of $dt=10^{-1}-10^{-2}$ and spatial resolution of at least
$dx=0.2$, while the MSD is averaged over $5\times 10^{2}$ stochastic
trajectories for each value of $q$ considered. In Fig.3(a,b), we see that
unlike $q=2$, the short time quantum-active scaling of the MSD is indeed $\sim
t^{4}$. Surprisingly, however, we find that the coefficient of the MSD for a
quartic trap ($q=4$) is negative at short time scales, indicating that the
quantum-active particle is transiently more localized than its ground state.
This cannot occur for a classical stochastic particle with well defined
position as we shall see explicitly further below. Conversely, the MSD for
$q=1.5$ scales positively as $\sim t^{4}$, growing more rapidly than $\sim
t^{6}$, i.e. faster than for the harmonic oscillator, due to monotonically
decreasing energy level structure which increasingly eases transitions to
higher excited states. At longer time scales, as expected, the MSD is purely
determined by the trap motion and the non-dissipative quantum particles enters
the diffusive regime regardless of $q$ (see Fig.3(c)).

In Fig.3(d,e), we show the behavior of the MSD for corresponding classical
particles trapped and driven actively in power-law potentials that are
initially at rest ($\dot{x}=0$) and are centered at $x=0$ for the various $q$
considered. The presence of quantum-active effects can also be inferred
through the departure in the MSD from that of the classical counterpart at
both shorter and longer length scales. Specifically, we note that the
agreement in the $t^{6}$ dependence of the MSD between classical and quantum
setups at short times is only valid for the particular case of harmonic traps
($q=2$). More generally, the $t$-dependence of the MSD between classical and
quantum active particles are different for $q\neq 2$. We summarize our
findings of the scaling behavior for various dynamical regimes and setups for
the non-dissipative particle in Fig.4.

Figure 4: [Schematics and dynamics of non-dissipative quantum and classical
active systems]. Trap geometry dependent quantum active effect on the MSD with
no classical equivalent.

In this work, we introduced a generic approach for exploring quantum-active
matter in continuum space. This approach involves mimicking the role of self-
propulsion in active particles through imposing a trapping potential that is
moving along imposed trajectories governed by classical active stochastic
dynamics. We show that this framework for a single quantum degree of freedom,
captures essential dynamical features of classical self-propelled active
particles in the presence of any dissipation. These features include, a mean-
square displacement (MSD) that scales diffusively ($\sim t$) at short times,
ballistically ($\sim t^{2}$) at intermediate times, and again diffusively
($\sim t$) at times much longer than the persistence time $\tau$ of the active
dynamics. Notably, this approach also reveals quantum-active dynamical
features with no classical equivalent and that are most pronounced in the
absence of dissipation. These features manifest as anomalous time-dependence
of the MSD at the shortest time scales. Most remarkably, the MSD can even
transiently acquire negative values for certain trap geometries, representing
an increased localization of the wavefunction with time transiently. This
phenomenon cannot emerge classically, and is highly counter-intuitive given
that active systems are typically super-diffusive at intermediate time scales.

We stress that this work focuses on establishing a conceptually new and
generic approach to study quantum-active matter that is meant to inspire
future simulations and experiments. This approach is highly generalizable, and
can be combined with or extended to interacting forms of classical active
matter. This may include alternate forms of microscopic activity including
run-and-tumble or active Brownian dynamics SolonCates2015 ; CatesTailleur2013
; FreyKroy2005  or extending the framework to interacting many-body systems
where collective effects such as motility induced phase separation may emerge.
On the other hand, the effect of inherently quantum phenomena such as
exchange-symmetry of indistinguishable quantum particles (e.g. fermions or
bosons) HyrkasManninen2013 ; AbrahamBonitz2012 ; ZhengPoletti2015 ;
MyersDeffner2020 , or collective behavior of exotic quantum many-body ground-
states (e.g. Bose-Einstein condensates) GreinerBloch2002 , on the dynamics of
an externally activated process could also be interesting avenues to explore
within the current context presented. Importantly, our theoretical predictions
can be tested using state of the art experimental techniques in ultra-cold
atoms confined to optical traps where the trap minimum can be programmed on
prescribed stochastic trajectories. As such, our framework of quantum active
matter could provide a paradigm for uncovering phenomenology involving the
non-trivial coupling of quantum fluctuations with non-equilibrium features of
activity.

###### Acknowledgements.

We thank I. Bloch, A. Hemmerich, P. Schmelcher, A. Widera, and R. Wittmann for
helpful discussions. YZ acknowledges support from the Alexander-von-Humboldt
Stiftung.

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