communica ions physics A icle
A Na u e Po olio jou nal
h ps://doi.o g/10.1038/s42005-025-02106-0
Nonlinea s ochas ic and quan um mo ion
om Coulomb o ces
Check o upda es
Luca O nigo i ,Da enW.Moo e & Radim Filip
Con ollable nonlinea quan um in e ac ions a e a much sough a e a ge o mode n quan um
echnologies. They a e ypically di ficul and cos ly o enginee o bespoke pu poses. Howe e
con ollable nonlinea i ies may ha e always been in each ia he na u al and undamen al o ces
be ween quan um pa icles. The Coulomb in e ac ion be ween cha ged pa icles is he simples
example. We show ha a e elimina ing he ha monic pa o he Coulomb o ce by an auxilia y linea
o ce, he emaining ecip ocal nonlinea pa esul s in a di ec ly obse able non- ecip ocal nonlinea
e ec : inc ease o he signal- o-noise a io (SNR) o he cohe en displacemen o one pa icle, d i en
by he posi ion noise, o unce ain y in quan um egime, in ano he pa icle. This essen ial e idence o
nonlinea o ces is p esen ac oss la ge anges o ap equency and mass scales, as well as isible in
bo h s ochas ic and quan um egimes.
The con ol o quan um sys ems has p oceeded apace, and many expe i-
men al se ings possess p ecise con ol o e linea quan um sys ems1,2,asin
quan um op ics3,4and quan um op omechanics5,and hemos simple
nonlinea sys ems such as he Jaynes–Cummings model, exemplified in
ca i y-QED6o apped ion sys ems7. To cons uc a la ge-scale nonlinea
sys em om such simple nonlinea sys ems is a cu en challenge, and
he e o e, many expe imen s a e pushing in o exci ing nonlinea egimes
whe e s anda d modes o analysis ail, o en ca as ophically. Such nonlinea
egimes a e, as migh be expec ed, also he si e o ye unco e ed quan um
phenomena and absolu ely equi ed o he mos ad anced quan um
echnologies o come o ui ion8,9. Simila s a emen s can be made
ega ding ecen ly de eloped pla o ms ope a ing en i ely in he classical
egime and nea he classical-quan um bounda y, such as le i a ed nano-
objec s10–21. Such sys ems a e on he oad om being inhe en ly s ochas ic in
a high empe a u e en i onmen o g adually eaching semiclassical and
quan um domains. The esul ing classical- o-quan um ansi ion wi h
mac oscopic objec s will likely open access o unexplo ed physics once hey
en e nonlinea egimes, such as en anglemen -by-hea ing22, and wi h he
possibili y o applica ions in opics such as quan um sensing o pa ame e
es ima ion23,24 o o g a i y25, quan um he modynamics26,27 and e en
quan um compu ing28,29.
In ac , he mos p ospec i e sys ems wo k in a egime de i ed om he
na u al o ces be ween he pa icles. Pe haps he mos widelyexploi ed is he
di ec ecip ocal in e ac ion by he Coulomb o ce be ween a pai o cha ged
pa icles. Wi h s a e-o - he-a echnology, such in e ac ions can be con-
ollably ealised on la ge anges o mass and equency scales, om cha ged
le i a ed nano-pa icles30–32 all he way down o apped indi idual ions33
and e en ecen ly be ween pai s o elec ons34.Whenconfined by e ec i e
ha monic aps, he linea isa ion o he o ce esul s in coupled ha monic
oscilla ions, wi h each di ec ional mode decoupled om he o he s35.We
p omo e an app oach o nonlinea e ec s which show hemsel es in he
in e modal coupling achie able by going beyond he ha monic app ox-
ima ion. Such e ec s ha e al eady been obse ed in he o a ing wa e
app oxima ion o apped ion sys ems36–38. Howe e , he s eps beyond his
app oxima ion in mac oscopic mechanical sys ems ha e no been exploi ed
ye . On he o he hand, in he con ex o mic oscopic single-ion hea
engines, cubic in e ac ions be ween di e en adial and axial modes ha e
been enginee ed by ailo ing he ap geome y39,40.
As he fi s s ep, we in es iga e di ec ly obse able nonlinea e ec s
a ising om undamen al o ces beyond he ha monic app oxima ion and
ou side any o a ing wa e app oxima ions, in bo h classical s ochas ic and
quan um mechanical egimes. In essence, he nonlinea e ec s a e made
mani es non- ecip ocally h ough he noise o unce ain y s imula ed
p ope ies o he sys em, in ou case, a cohe en displacemen o one pa icle
d i en ia he noise o unce ain y in he o he . By defini ion, his can only
occu ia nonlinea in e ac ion o he pa icles. E en in an app oxima ion
whe e he in e modal e ec s along di e en di ec ions a e supp essed, his
e ec can be p edic ed by expanding he Coulomb in e ac ion only o hi d
o de , he fi s non i ial nonlinea e m, and ocusing on a single nonlinea
in e ac ion along he s aigh line be ween he pa icles. The in e ac ion is
compound, main aining he ecip oci y o he Coulomb o ce, bu does no
p e en he obse a ion o non- ecip ocal e ec s in classical and quan um
egimes.
In wha ollows, we will make hese p oposi ions mo e p ecise by
in oducing he simples model, p esen ing es able esul s, and hen
demons a ing how such e ec s can be unde s ood in e ms o a sui able
app oxima ion. The de ec ion o nonlinea mo ion, quan ified by a noise/
unce ain y-d i en inc ease in he signal- o-noise a io (SNR) o he
Depa men o Op ics, Palacký Uni e si y Olomouc, 17. lis opadu 1192/12, Olomouc, 77900, Czech Republic. e-mail: [email p o ec ed]
Communica ions Physics | (2025) 8:195 1
1234567890():,;
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momen um displacemen , demons a es he ac i i y o he nonlinea pa o
he Coulomb in e ac ion. These e ec s a e shown o be obse able ac oss a
ange o expe imen ally ele an pa ame e s. This o ms a p oo -o -
p inciple s ep owa ds expe imen al e ifica ion in he s ochas ic and, la e ,
he quan um egime. We expec ha he obse a ion o such e ec s will
open u he di ec ions o u he in es iga ions beyond Gaussian
en anglemen 30,31.
Resul s
Nonlinea mo ion om he Coulomb o ce
A pai o equally cha ged pa icles o masses m
i
,i∈{1, 2}, confined o a
h ee-dimensional ha monic ap and in e ac ing ia he Coulomb po en-
ial, can be a anged along he z-axis by ha ing a much g ea e ap e-
quency in ha di ec ion15,20,21. The axes can hen be chosen so ha he
equilib ium poin s a e dðiÞ¼00zi;0
. E en beyond he ha monic
app oxima ion, whe e he mo ional axes a e no decoupled, he in e ac ions
depend upon he dis ances be ween he wo pa icles along he espec i e
axes. I mo ion along he xand yaxes is su ficien ly cooled so ha fluc-
ua ions along hese axes a e small, hen he in e ac ions be ween xo y,and
za e supp essed. The Hamil onian o he sys em can hen be w i en as
H¼1
2X
i
p2
i
miþmiω2
iðzizi;0Þ2
þκ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðz1z2dÞ2
q;ð1Þ
whe e ω
i
a e he ap equencies, κ=q
1
q
2
/4πϵ
0
is he coupling s eng h o
he ecip ocal in e ac ion calcula ed ia he cha ge q
i
and he elec ic
pe mi i i y in acuum ϵ
0
,andd=z
1,0
−z
2,0
is he ini ial dis ance be ween
he pa icles’equilib ium posi ions. The se up is illus a ed in Fig. 1a. In he
ha monic app oxima ion, expanding a ound he dis ance be ween he
pa icles in each di ec ion o second o de , esul s in an ex a displacemen
e m in he zdi ec ion o bo h pa icles. The mo ion o he cha gedpa icles
is hen desc ibed by ha monically coupled oscilla o s wi h modified e-
quencies. The coupling induced be ween he pa icles is in amodal, such
ha he modes along di e en axes do no alk o each o he . The hi d-o de
e m in oduces nonlinea and in e modal in e ac ions. To isola e hem, a
compensa ing o ce elimina ing he lowe -o de con ibu ions is necessa y.
The cons an o ce can be compensa ed ia linea il h ough an
elec os a ic o ce41, whe eas he linea o ce can be compensa ed by
pa ame ic con ol, such as eedback42–44. In he egime o op imal
compensa ion, assumed h oughou he manusc ip , he s ill ecip ocal
in e ac ion Hamil onian can be app oxima ed as
H3κ
d4ðz1z2Þ3¼κ
d4z3
1z3
2þ3ðz1z2
2z2
1z2Þ
:ð2Þ
A non-op imal compensa ion s ill esul s in he non- ecip ocal nonlinea
e ec , albei educed in isibili y, see Supplemen a y Ma e ial (SM) No e 3.
Due o he nonlinea i y o he Coulomb o ce, a ecip ocal cubic in e ac ion
eme ges along he z-axis. This is a minimal model o he nonlinea Coulomb
in e ac ion be ween wo pa icles. While simila nonlinea in e ac ions can
be cons uc ed o ions di ec ly ia he ap geome y o hyb id adial and
axial modes39,40, he e he nonlinea i y eme ges di ec ly om he Coulombic
in e ac ion be ween he pa icles. Bo h he ap and he compensa ion
Fig. 1 | Noise and unce ain y-induced momen-
um displacemen ia a nonlinea Coulomb
in e ac ion. a(i), (ii) Illus a ion o he p inciple in
classical and quan um egimes. Two ha monically
confined pa icles (blue, yellow) expe ience a non-
linea Coulomb in e ac ion ( ed sp ing) along a
single axis, oge he wi h a compensa ing o ce
(g een). In he classical egime, he s ochas ic pa -
icle 1 is p epa ed in oscilla o equilib ium s a es ia
dissipa ion o he he mal en i onmen a em-
pe a u e T
1
, while pa icle 2 is ini ially cooled o
T0¼10 mK in a oom empe a u e en i onmen
T
1
=T
2
= 300 K. In he quan um egime, he pa i-
cles a e p epa ed in g ound s a es, and he only
fluc ua ions a ise om he quan um unce ain ies. A
weak linea damping wi h a a e Γ, ac ing only on
pa icle 2, is p esen in o de o ensu e he s abili y o
he e ec . The ap equency modula ion, and mass
disp opo ion o pa icle 1, as shown in he able, a e
used o gene a e unidi ec ional flow o fluc ua ions
o pa icle 2, hus a oiding backac ions. b(i), (ii)
Time e olu ion o mean momen um hpz2i, no -
malised o he ini ial s anda d de ia ion ( op) and
signal- o-noise a io SNRpz2(bo om). Fo la ge
noise ( ull ci cles), he SNR ¼1=ffiffiffi
2
pis quickly
eached by all egimes, bu uning equency (blue)
allows o be e noise con ol. A lowe noise
(emp y ci cles), he pa ame ic symme y is he only
egime eaching he SNR bound (g ey). In he
quan um egime (b(ii)), he g ound s a e fluc ua-
ions (emp y ci cles) a e equally ha nessed by all
egimes, whe eas an ini ial unce ain y amplifica-
ion, by ee- all ( ull ci cle), allows he SNR bound
o be eached by all egimes. Symme ic (g ey) and
uning equency (blue) u he expe ience he as e
unce ain y g ow h (SNR d op), no isible o
uning mass (o ange), which is also he bes
egime he e.
h ps://doi.o g/10.1038/s42005-025-02106-0 A icle
Communica ions Physics | (2025) 8:195 2
ope a e in he ha monic app oxima ion, and he e o e canno con ibu e o
he nonlinea e ec s ou lined below. Di e en ly han o op ical cubic
po en ials45, he in e ac ion in Eq. (2) combines compe ing cubic
nonlinea i ies. Tha is, he cubic single pa icle po en ial z3
1;z3
2,and he
cubic in e pa icle po en ial z1z2
2;z2
1z2. Thei ying na u e may limi he
di ec obse a ion o he in e pa icle nonlinea noise o unce ain y-
induced phenomena. Quan isa ion can be accomplished by p omo ing he
canonical a iables o ope a o s sa is ying he commu a ion ela-
ions [z
i
,p
i
]=i.
Classical noise-induced momen um. Nonlinea in e ac ions such as
z2
1z2allow o he possibili y o cohe en ly displace he momen um o one
pa icle ia inc eases in he ini ial posi ion noise o he o he . Heu -
is ically, he comple e ecip ocal in e ac ion e m o Eq. (2) indica es ha
o ini ially unco ela ed s a es he mean momen um displacemen in
one mode is apidly d i en by he noise in he o he i.e.
hpz2ihpz2;0i3κðhz2
1;0i2hz2;0ihz1;0iþhz2
2;0iÞ =d4. The ecip ocal
na u e o he in e ac ion Hamil onian o Eq. (2) sugges s ha a sepa a e
asymme ical manipula ion pe o med solely on one o he wo pa icles
can enhance he in e pa icle non- ecip ocal noise-induced nonlinea
e ec obse ed on he o he . Fo pa ame ically symme ic in e ac ions,
m
1
=m
2
and ω
1
=ω
2
, he minimal asymme ical manipula ion is
accomplished by unbalancing he ini ial dis ibu ion o he mal noise.
Tha is, p epa e he ini ial he mal s a e a empe a u e T= 300 K o
pa icle 2 a an e ec i e empe a u e o T0¼10 mK by cooling, while
pa icle 1 is p epa ed in an oscilla o he mal-equilib ium s a e a oom
empe a u e T
1
= 300 K. These a e ze o-mean Gaussian s a es wi h a -
iances in posi ion σ2
z¼kBT=mωand momen um σ2
p¼mkBT, whe e Tis
he e ec i e empe a u e o mode zand k
B
is he Bol zmann cons an .
Du ing he dynamics, he modes a e imme sed in he mal en i onmen s
wi h T
1
,T
2
= 300 K. This ini ial he mal dis ibu ion imbalance be ween
T
1
and T0minimises he he mal fluc ua ions hp2
z2;0i, which o he wise
obscu e he noise-induced e ec , while simul aneously minimising any
unwan ed back-ac ion e ec s on he pa icle whose noise d i es he
noise-induced e ec . While he echnical de ails o his p epa a ion
depend s ongly on he chosen pla o m, we sugges a p oo -o -p inciple
s a e p epa a ion scheme in SM No e 4.
In Fig. 2a we show he noise-induced mo ion o his pa ame ically
symme ic case (g ey). The classical simula ions a e ca ied ou wi h he
pa ame e s ω
i
=50kHz,m
1
=m
2
=8×10
−17 kg, κ=2.3×10
−24 Nm
2,and
d=3μm, inspi ed by op ical le i a ion15,20,21,32 and en iched wi h magne ic
le i a ion16,17 pla o ms in mind as hey ope a e wi h a wide apping e-
quency ange. No displacemen occu s in z
1
, howe e , he noise g ows
apidly. The mean momen um o he second pa icle, howe e , expe iences
a posi i e sha p inc ease away om ze o as well as an inc ease in noise. This
displacemen is no c i ically ou pe o med by noise, as seen in he SNR in
Fig. 1b(i) (emp y ci cles). This is a sign o a di ec noise-induced cohe en
e ec . Impo an ly, he SNR g ows wi h inc easing σz1;0( ull ci cles) and can
sa u a e he maximum o 1=ffiffiffi
2
pa he cos o la ge noise in he ini ial
posi ion o z
1
.
This e ec can be explained by examining he Lange in equa ions o
mo ion o he hi d-o de Coulomb e m. These a e gi en by
m2
€
z2ð Þþm2Γ_
z2ð Þ3κ
d4z2
1ð Þþz2
2ð Þ
m2ω2
2þ6κ
d4z1ð Þ
z2ð Þ
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ΓkBT2
pξ2ð Þ;
m1
€
z1ð Þþm1Γ_
z1ð Þ3κ
d4z2ð Þ2þz2
1ð Þ
m1ω2
16κ
d4z2ð Þ
z1ð Þ
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ΓkBT1
pξ1ð Þ;
ð3Þ
whe e Γis he d ag coe ficien whose alue and phenomenological o igin
depends upon implemen a ion, and ξ
1
,ξ
2
a e independen ze o-mean
Gaussian whi e noises wi h hξið 0Þξið 00Þi¼ δð 0 00Þ. In wha ollows, we
ocus on he unde damped egime, wi h Γ=10
−4Hz,asin heo e damped
egime, he nonlinea e ec in momen um pz2is no isible (see SM No e 1).
Fo pa ame ic symme y, i is use ul o discuss he dynamics using he mean
alue app oxima ion, by educing he wo-body in e ac ion in o a one-body
p oblem by i ue o he e ec i e po en ials
~
Vðz2Þ¼3κhz1i2z2=d4þ
~
ω2z2
2κz3
2=d4and
~
Vðz1Þ¼3κhz2i2z1=d4þ~
ω1z2
1þκz3
1=d4.In his
amewo k, he unwan ed back-ac ion o z
2
on z
1
is unde s ood as he mean
displacemen 〈z
2
〉which (i) gene a es a d i in z
1
,as isible om hefi s
e m in
~
Vðz1Þ, and (ii) modifies he equencyo heha monicconfinemen
o z
1
ia ~
ω1¼m1ω2
1=23κhz2i=d4. The imbalance in he ini ial noise
p ope ies minimises bo h back-ac ion con ibu ions a sho ansien s
<20μs. Tha is, low empe a u e T0makes he noise-induced shi
gene a ed by he cubic po en ial in
~
Vðz2Þnegligible, keeping he posi ion
below he c i ical alue o 〈z
2
〉≈1μm (calcula ed o he pa ame e s used in
nume ical simula ion) a e which he e ec i e equency ~
ω1becomes
nega i e and he ha monic confinemen becomesanin e edquad a ic
po en ial, leading o uns able di e ging ajec o ies. Howe e , he noise in z
1
is
s ill subs an ially inc easing in ime, and can, in gene al, complica e bo h
p edic ions and applica ions o nonlinea i ies. Fo la ge ini ial noise, as isible
in Fig. 2a (g ey), he highe -o de nonlinea e ms o he Coulomb in e ac ion
make o a posi i e back-ac ion z
2
on z
1
. I esul s in a dec ease o he noise o
z
1
, e en below ha o i s ini ial he mal s a e.
When he ecip oci y in he nonlinea e ec is u he b oken by ei he
uning he mass (a fixed equency) o equency (a fixed mass) o pa icle
1, he fluc ua ions in z
1
a e modified and he p ope ies o he cohe en
mo ion a e al e ed. The ini ial he mal s a e o z
1
is de e mined by i s
dynamical a iables m
1
,ω
1
, and local en i onmen al empe a u e T
1
ia he
Fig. 2 | Analysis o he ime e olu ion o a iables
unde going nonlinea mo ion. Analysis o he ime
e olu ion o posi ion z
1
and momen um pz2a di -
e en ini ial fluc ua ions o z
1
. The shaded a ea
ep esen s he s anda d de ia ion a ound he mean
e olu ion (solid). All quan i ies a e no malised o
he s anda d de ia ion o hei espec i e ini ial
s a es. Symme y b eaking by mass uning (o ange)
and equency uning (blue) allows o con ol
di e gence in pz2in bo h mean and s anda d
de ia ion. In he classical egime (a), he mass uning
isibly pe o ms be e han he o he s a egies as i
p oduces la ge momen um d i hpz2i. Fo he
quan um egime (b), he symme ic (g ey) and e-
quency uned (blue) ou pe o m he mass uned
wi h he same me ic. I confi ms he esul p e-
sen ed in Fig. 1b(i), (ii).
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Communica ions Physics | (2025) 8:195 3
s anda d de ia ion σz1;0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kBT=m1ω1
p. Changing he mass o equency
o pu sue he symme y-b eaking echniques esul s in a modifica ion o he
ini ial fluc ua ions. We he e o e fix he noise p ope ies o σz1;0¼30 nm o
obse e he e ec a ising om he nonlinea in e ac ion wi h cons an ini ial
noise ac oss di e en pa ame e egimes. Tha is, z
1
is no p epa ed in an
equilib ium s a e o i s local oscilla o bu a he in an ou -o -equilib ium
he mally squeezed s a e.
T apping a massi e pa icle m
1
≫m
2
minimises i s kine ic e m
p2
z1=2m1, and as a esul , he posi ion does no mo e away om i s ini ial
mean condi ion z1
z1;0
. Tha is, he back-ac ion is negligible o
sho ansien s, as is isible. Di e en ly han pa ame ic symme y, he
addi ional imbalance o he mass eshapes he noise e olu ion o z
1
, keeping
i close o i s ini ial dis ibu ion σz1;0 o low ini ial fluc ua ions. Howe e , as
isible in Fig. 2a (o ange), la ge ini ial noise p omo es a posi i e back-ac ion
loop o e ime, dec easing he fluc ua ions o z
1
below he ini ial he mal
s a e. I is simila o he pa ame ic symme y (g ey) back-ac ion e ec ,
howe e , i s e ec on pa icle 2 showcases a la ge displacemen , al hough
wi h la ge fluc ua ions (o ange shaded a ea). The esul ing SNR in
Fig. 1b(i), while inc easing e en a low fluc ua ions (emp y ci cles), only
sa u a es he bound o la ge ini ial noise σz1;0( ull ci cles). The sho
ansien o p
2
e ol es as pz2ð 0Þpz2;0þ3κR 0
0ds0z2
1ðs0Þ=d4m2ω2
2z2;0 0
in he limi o ze o damping Γ= 0. Fo mass uning, i leads o he ollowing
momen s
hp2ð Þi 3κ σ2
z1;0
d4;SNR ¼hp2i
σp21
ffiffiffi
2
p:ð4Þ
Al e na i ely, apping pa icle 1 in a s i e ha monic po en ial
ω
1
≫ω
2
confines i s noise dynamics o ha o a ha monic oscilla o o
sho ansien s. I s e olu ion is desc ibed by cohe en oscilla ions,
app oxima ely desc ibed by z1z1;0cosðω1 Þunde he assump ion o
anishing ini ial eloci y _
z1;0¼0. As isible in Fig. 2a (blue), his oscilla o y
e olu ion domina es o e he back-ac ion o imes o a ew en hs o
mic oseconds. This added imbalance o equency nega i ely impac s he
dynamics, gene a ing a lowe momen um d i hpz2i,andalowe SNR
ou pu Fig. 1b(i) (blue) a small ini ial noise (emp y ci cles), bu i oo
sa u a es he 1=ffiffiffi
2
pbound a la ge ini ial noise σz1;0( ull ci cles). Fo uning
equencies, he momen s o momen um p
2
app oach
hpz2ð Þi 3κ
4d4ω1
2θþsinð2θÞ
½
σ2
z1;0;SNR 1
ffiffiffi
2
p;ð5Þ
whe e θ=ω
1
.
In he low noise limi , he added symme y b eaking lowe s he SNR
ela i e o he symme ic case (see Fig. 1b(i), emp y ci cle). To each i , an
ex a cos o inc easing he ini ial noise σz1;0is equi ed. Specifically, o
uning s i ness, he same SNR is eached a a smalle displacemen (see
Fig. 2a, blue), while o uning mass, he SNR sa u a ion is ob ained wi h a
much la ge displacemen (o ange), making i a a ou able s a egy. This is
u he highligh ed in Fig. 3a, whe e a a ge SNR ¼1=ffiffiffi
2
pis fixed, and he
ou pu momen um displacemen hpz2i( op) and s anda d de ia ion σpz2
(bo om) a e plo ed agains inpu noise cos σz1;0. I shows ha o ini ial
noise below σz1;0≲100 nm, he pa ame ic symme y (g ey) ha nesses he
noise o z
1
h ough he Coulomb in e ac ion mo e e ficien ly. Howe e , o
ini ial noise inpu beyond σz1;0≳100 nm, uning equency (blue) gene a es
he same SNR wi h smalle ou pu noise and momen um d i , while
uning mass (o ange) eaches i wi h la ge momen um d i . Fig. 3shows
ha he bes s a egy, o la ge ini ial noise, is o b eak he symme y by
uning he mass o each he a ge signal- o-noise wi h he la ges
momen um d i .
Quan um unce ain y-induced momen um. As we dec ease he ini ial
noise o he g ound s a e ex ension, ha is σz1;0¼0:01 nm, he nonlinea
e ec in he s ochas ic amewo k desc ibed by Eq. (3) anishes (see SM
No e 2). Ope a ing in he quan um egime o Eq. (3), using pu e ini ial
s a es, an analogue o he p e ious noise-induced phenomena comes
di ec ly om he quan um fluc ua ions in z
1
. As expec ed, i is su ficien
o p oduce momen um displacemen on z
2
, as isible in Fig. 2b.
In his sec ion, he nume ical simula ions a e pe o med di ec ly on
Eq. (3), see SM No e 2, using he same pa ame e s ou lined in he p e ious
Fig. 3 | Noise/Unce ain y-induced momen um
unde noise confinemen . a The ou pu displace-
men hpz2i(i), and s anda d de ia ion σpz2(ii) a he
a ge signal- o-noise a io SNRpz2¼1=ffiffiffi
2
pa e
plo ed o he s ochas ic classical dynamics agains
he inpu noise σz1;0. A low ini ial inpu noise, he
pa ame ic symme y (g ey) always eaches he
a ge . A la ge inpu noise, all egimes each he
a ge , bu b eaking he symme y ia uning e-
quency (blue) p o ides he leas noise ou pu , and
hus e en smalle momen um displacemen .
B eaking symme y by uning mass (o ange) is
use ul only be ween noise inpu 60 ≲σz1;0≲100 nm.
bThe ou pu displacemen hpz2i(i), and s anda d
de ia ion σpz2(ii) a he a ge SNR o di e en
ini ial unce ain y σz1;0. All egimes each he a ge ,
bu a di e en imes. Fo pa ame ic symme y
(g ey) and uning equency (blue), he a ge is
eached a ≈1μs, while o uning mass (o ange),
he a ge is eached a la ge imes ≈2μs.
Rega dless o he equi ed ime, when he a ge is
eached, all egimes p oduce he same displacemen
and s anda d de ia ion ou pu , making he pa a-
me ic symme y (g ey) he p e e ed s a egy o
each he a ge wi h he minimum noise cos . No e,
b eaking symme y equi es ex a squeezing o
each he same ini ial noise inpu . The dashed filled
ci cles eco d he alue o momen um displacemen
and s anda d de ia ion when he a ge SNR is no
eached.
h ps://doi.o g/10.1038/s42005-025-02106-0 A icle
Communica ions Physics | (2025) 8:195 4
sec ion. All quan i ies a e escaled by he s anda d de ia ion o he ini ial
s a e. Fo pa ame ic symme y, he quan um fluc ua ions o z
1
induce a
small d i in he momen um, u he enhanced by he quan um fluc ua ion
o z
2
h ough he cubic nonlinea i y, eaching hpz2i3κðσ2
z1;0þσ2
z2;0Þ =d4.
The unwan ed back-ac ion o ce is oo small o induce ins abili y, he e o e,
he noise o z
1
does no inc ease la ge han i s e e ence s a e. I d i es he
momen um hpz2iwi h an inc easing SNR (Fig. 3panel b(i), emp y ci cles)
ha does no each he maximum o 1=ffiffiffi
2
p,as o imes >2μs, he cubic
nonlinea i y domina es bo h dynamics and he back-ac ion s ongly d i es
z
1
o ins abili y. I enhances he noise o p
2
beyond he ini ial g ound s a e,
hus esul ing in a d op o he SNR. The conse a i e symme y-b eaking
s a egies, in oduced o he s ochas ic dynamics, esul in a quali a i ely
simila ime e olu ion (see Fig. 2b, blue and o ange).
The ini ial s anda d de ia ion σz1;0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
_=ð2m1ω1Þ
pis calcula ed using
he same pa ame e s o he s ochas ic sys em. Fo pa ame ic symme y, i.e.
m
1
=8×10
−17 kg and ω
1
= 50 kHz, he ini ial s anda d de ia ion esul s in
σz1;0¼0:01 nm. Fo uning equency, i.e. ω
1
= 2500 kHz, and uning mass
m
1
=8×10
−16 kg, he ini ial s anda d de ia ions assume di e en alues.
Respec i ely σz1;0¼0:001 nm, and σz1;0¼0:003 nm. To obse e only he
e ec s o he nonlinea in e ac ion gi en by he dynamics, unde he same
ini ial noise condi ions, he ini ial s a e o z
1
is squeezed by a ac o ξ¼
logðσ g ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m1ω1=_
pÞ=2 o each he unified a ge s anda d de ia ion o
σ
g
= 0.01 nm. Tha is, he posi ion a iance is amplified σz1;0¼
ξffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
_=ð2m1ω1Þ
pby ξ, while he momen um a iance is a enua ed by he
in e se amoun σpz1;0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
_m1ω1=2
p=ξ.
When he ini ial g ound s a e o z
1
is u he squeezed in momen um, i
ealises a la ge d i eaching a SNR ¼1=ffiffiffi
2
pa sho ansien s (Fig. 1,
panel b(ii)). No ice ha pa ame ic symme y (g ey) and uning equency
(blue) a e subjec ed o ins abili y o imes la ge han ≈1μs, esul ing in a
d op o he SNR, while uning mass (o ange) is no ye a ec ed by i .
Fo uning mass m
1
≫m
2
, he sho ansien o he momen s o p
2
app oach
hpz2i3κ σ2
z1;0
d4;
SNRpz21
ffiffiffi
2
p1þm2
2ω4
2d8σ2
z2;0
18κ2σ4
z1;0þ
d8σ2
pz2;0
18κ2σ4
z1;0 2
"#
1
2
:ð6Þ
Fo sho ansien s, he g ound s a e momen um noise σ2
pz2;0p e en s
he SNR om eaching he 1=ffiffiffi
2
pbound. A la ge imes i becomes neg-
ligible, lea ing he posi ion noise σ2
z2;0as he dominan limi ing e m o he
e olu ion. Fo fixed m
2
,ω
2
,anamplifica ion o posi ion noise σ2
z1;0¼ξσ2
z1;0
by ξallows o each a SNR ¼1ffiffiffi
2
pas isible in Fig. 1b(ii), ull o ange ci cle.
Tha is, he ini ial s a e is u he squeezed in momen um. Squeezing he
posi ion noise σ2
z2;0can, in p inciple, imp o e upon he SNR o Eq. (6).
Howe e , he complemen a y amplifica ion o momen um noise σ2
pz2;0
inc eases he back-ac ion o pa icle 1 a la ge imes, hus leading o he
di e gence quicke .
Fo uning equency ω
1
≫ω
2
, henoiseo z
1
is confined in a s i e
ha monic bound, and i s dynamics is desc ibed as z1z1;0cosðθÞ,wi h
θ=ω
1
.In his egime, hedynamicse ol essimila ly o ha o hepa a-
me ic symme y as isible in Fig. 1b(ii), blue and g ey ci cles. I s momen a
e ol e in sho ansien s acco ding o
hpz2i3κσ2
z1;0
4d4ω1
2θþsinð2θÞ
½
;
SNRpz21
ffiffiffi
2
p1þ
8ω2
2d8σ2
pz2;0þ8ω2
1d8 2σ2
z2;0
9κ2σ4
z1;02θþsinð2θÞ
½
2
2
43
5
1
2
:
ð7Þ
Fo an ini ial g ound s a e σ2
z1;0¼_=ð2m1ω1Þ, he momen um and
posi ion noise o he ini ial s a e o pa icle 2, i.e. σ2
z2;0;σ2
pz2;0hinde s he
unce ain y-induced e ec om eaching he SNR ¼1=ffiffiffi
2
pbound a sho
ansien s. Fo a longe ime, he ins abili y om he cubic po en ial, no
accoun ed o in Eq. (7), domina es he dynamics, esul ing in a d op o he
SNR. Simila o uning mass, o each he signal- o-noise bound o sho
ansien s, he amplifica ion o posi ion noise σ2
z1;0is equi ed, as isible in
Fig. 1b(ii), ull blue ci cles. Mo eo e , squeezing he posi ion noise σz2;0
esul s in as e di e gence, simila o he case o he uning mass.
The hidden cos o pa ame ic symme y b eaking lies in he p e-
pa a ion o he ini ial s a e, which equi es squeezing o he g ound s a e.
Tha is, o ha e compa able noise in posi ion o σz1;0¼0:01 nm a e uning
mass and equency, he g ound s a e mus be squeezed, by ξ=1.15and
ξ= 1.96, espec i ely. Fo he la ge noise case, a u he amplifica ion by ee-
all is hen used o each he posi ion noise in all egimes o σz1;0¼0:08 nm.
In Fig. 3b, he a ge SNR ¼1=ffiffiffi
2
pis eached equally by all egimes o
ini ial inpu noise σz1;0≳0:06 nm, bu a di e en imes. Ul ima ely, he
pa ame ic symme y eme ges as he bes egime, as i does no equi e any
ex a cos s, i.e. he ini ial squeezing o σz1;0¼0:01nm,asis hecaseo he
mass and equency uning egimes.
Conclusion
Expe imen s in ol ing he in e ac ion o apped cha ged pa icles
ypically ope a e in he ha monic app oxima ion30,31,35, whe e he mo ion
decouples in o oscilla ions along each coo dina e axis. This can only
esul in Gaussian e ec s: single pa icle and mul ipa icle squeezing, and
po en ially, en anglemen in he quan um egime. He e we ha e shown a
p oo -o -p inciple me hod o go beyond his app oxima ion, showing
ha he inhe en ly nonlinea e ec o noise-induced cohe en mo ion
can be seen and explained using he fi s non i ial nonlinea e m in an
expansion o he ecip ocal Coulomb in e ac ion be ween wo cha ged
pa icles. This is obse ed in Fig. 1b(i) by he non- ecip ocal e ec :
imp o emen in SNR o one pa icle due o inc eased posi ion he mal
noise in ano he nonlinea ly coupled pa icle. This e ec pe sis s in o he
quan um egime, whe e now he quan um unce ain y a he han
classical noise d i es he dynamics o he es pa icle o he same SNR
bound o classical dynamics, a smalle momen um displacemen . I is
isible in Fig. 1b(ii).
This me hod p o ides he fi s s ep o analysing one o he mos basic
wo-pa icle nonlinea e ec s, as isible in Fig. 3. Noise con ol is an
impo an basic ool o con ol o echnologies ha a e inhe en ly s o-
chas ic. Ad an ageously, his can be done au onomously h ough na u ally
occu ing o ces, lowe ing he enginee ing equi emen s o make such
e ec s isible ac oss a b oad ange o pa ame e s and pla o ms. I should be
no ed ha nano-pa icles possess a na u al ad an age compa ed o ions as
hei mass-cha ge a io a ou s obse a ion o he sho - ime nonlinea
e ec s. Tha is, hei high cha ge and high mass ange capabili ies allow o a
s onge in e ac ion unabili y and slowe di e gence, espec i ely.
To da e, he mos in-dep h s udies o mo ional nonlinea sys ems a e o
single-mode nonlinea i ies, possibly linked o o he linea sys ems.
Inc easingly de ailed s udies o expe imen ally accessible wo-pa icle
nonlinea in e ac ions will likely un eil many unexpec ed e ec s. B eaching
u he in o genuinely mul ipa i e sys ems combines he complexi y o he
many possible configu a ions (chains, iangles, la ices, o clus e s)46–48 wi h
ha o a nonlinea in e ac ion dis ibu ed ac oss mul iple pa icles, likely
hiding exci ing nonlinea e ec s. This hen opens he possibili y o u he
exploi such nonlinea i y in na u ally occu ing in e ac ions.
Code a ailabili y
Code a ailable upon easonable eques .
Recei ed: 20 Sep embe 2024; Accep ed: 22 Ap il 2025;
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Acknowledgemen s
We acknowledge he p ojec GA23-06224S o he Czech Science Founda ion,
EU and MEYS Czech Republic No. CZ.02.01.01/00/22_008/0004649
(QUEENTEC). R.F. also acknowledges unding om he MEYS o he Czech
Republic (G an Ag eemen 8C22001). P ojec SPARQL has ecei ed unding
om he Eu opean Union’s Ho izon 2020 Resea ch and Inno a ion P o-
g amme unde G an Ag eemen no. 731473 and 101017733 (Quan ERA).
h ps://doi.o g/10.1038/s42005-025-02106-0 A icle
Communica ions Physics | (2025) 8:195 6
Au ho con ibu ions
L.O. pe o med he nume ical simula ions and analy ical solu ions o he
classical dynamics, and D.M. pe o med he quan um mechanical
calcula ions. Bo h ecei ed heo y inpu s om R.F. R.F. concei ed and
supe ised he p ojec . All au ho s con ibu ed o he analysis o he esul s
and composi ion o he a icle.
Compe ing in e es s
The au ho s decla e no compe ing in e es s.
Addi ional in o ma ion
Supplemen a y in o ma ion The online e sion con ains
supplemen a y ma e ial a ailable a
h ps://doi.o g/10.1038/s42005-025-02106-0.
Co espondence and eques s o ma e ials should be add essed o
Da en W. Moo e.
Pee e iew in o ma ion Communica ions Physics hanks An on
Zaseda ele and he o he , anonymous, e iewe (s) o hei con ibu ion o
he pee e iew o his wo k.
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