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Eu . Phys. J. B manusc ip No.
(will be inse ed by he edi o )
Gene ic shape o mul ich oma ic esonance peaks
Ma ´ıa Lau a Oli e a-A encio1, Jes´us Casado-Pascual1, and Sigmund Kohle 2
1F´ısica Te´o ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, 41080 Se illa, Spain
2Ins i u o de Ciencia de Ma e iales de Mad id, CSIC, 28049 Mad id, Spain
Janua y 15, 2020
Abs ac . We in es iga e dissipa i e dynamical sys ems unde he in luence o an ex e nal d i ing wi h wo
o mo e equencies. Ou main quan i ies o in e es a e long- ime a e ages o expec a ion alues which
u n ou o exhibi uni e sal ea u es. In pa icula , esonance peaks in he icini y o commensu able
equencies possess a gene ic en eloping unc ion whose wid h is in e sely p opo ional o he a e aging
ime. While he uni e sal ea u es can be de i ed analy ically, he ansi ion om he speci ic sho - ime
beha io o he long- ime limi is illus a ed o he examples o a classical andom walk and a dissipa i e
wo-le el sys em bo h wi h biha monic d i ing. In hese models, he dependence o he ime-a e aged
esponse on he ela i e phase be ween he wo d i ing equencies changes wi h inc easing in eg a ion
ime. Fo sho imes, i exhibi s he 2πpe iodici y o he dynamic equa ions, while in he long- ime limi ,
he pe iod becomes a ac ion o his alue.
PACS.
1 In oduc ion
Dynamical sys ems d i en by ime-dependen o ces ep-
esen pa adigma ic models o non-equilib ium e ec s in
he ealm o bo h classical and quan um mechanics. The e
one may ind coun e -in ui i e phenomena such as s ochas-
ic esonance [1,2], synch oniza ion [3–7], and he a che
e ec by which di ec ed cu en s eme ge despi e he ab-
sence o any ne o ce [8–10]. The common physics be-
hind hese phenomena is an in e play o non-linea i ies
and non-equilib ium. D i en sys ems ha e been s udied
in he Hamil onian limi [11–13], as well as in he s eady
s a e o dissipa i e classical [8,9,14,15] and quan um me-
chanical models [16–18].
E en a om equilib ium, spa io- empo al symme-
ies may inhibi he eme gence o a dc esponse such as
a a che cu en . Many o hese symme ies a e based
on he ac ha a sinusoidal d i ing o ce changes i s
sign when ime is shi ed by hal a pe iod. Fo bich o-
ma ic o mul ich oma ic o ces, he pe iods o he a ious
o ces a e di e en and, hus, he symme y analysis has
o be e ised. Typically, one has o dis inguish wo cases,
namely hose o commensu able and incommensu able e-
quencies. In he o me case, he symme y depends on
he phase be ween he componen s o he d i ing, which
has been e i ied wi h anspo expe imen s in quan um
do s [19, 20]. In e e ence [19], i has also been demon-
s a ed heo e ically and expe imen ally ha o incom-
mensu able equencies, he symme y may be highe , de-
spi e ha he d i ing o ce is only quasi pe iodic and may
no possess any symme y o an i-symme y.
He e, we wo k ou gene ic p ope ies o dissipa i e dy-
namical sys ems unde mul i- equency d i ing, in pa ic-
ula he shape o he esul ing esonance peaks in he long-
ime limi . Mo eo e , we s udy how his limi eme ges.
The a icle is o ganized as ollows. In Sec ion 2, we o -
mula e he p oblem and in Sec ion 3 de i e how he inde-
pendence on he ini ial ime p o ides gene ic p ope ies.
In Sec ion 4, we conside ini e- ime e ec s o bich o-
ma ic d i ing and in Sec ions 5 and 6 show how he limi s
a e app oached o a classical andom walk model and o
a quan um mechanical wo-le el sys em, espec i ely. Fi-
nally, we summa ize and conclude in Sec ion 7.
2 Fo mula ion o he p oblem
Suppose ha he dynamical equa ions go e ning he ime
e olu ion o he sys em unde conside a ion depend on
ime h ough N ime-pe iodic unc ions o he o m
j( ) = Φj(Ωj +ϕj),(1)
wi h j= 1, . . . , N. In he abo e exp ession, Φja e 2π-
pe iodic unc ions [i.e., Φj(θ+ 2π) = Φj(θ)∀θ∈R] and
Ωjand ϕjdeno e he angula equency and he phase
o j( ), espec i ely. The p ecise na u e o he unc ions
j( ) is i ele an o ou pu poses, e.g., hey may ep-
esen ex e nal oscilla o y o ces, modula ing ampli udes,
e c.
Assume ha he sys em has been p epa ed in a s a e s0
a an ini ial ime 0. Depending on he case, s0may ep e-
sen he sys em’s densi y ope a o (in he case o quan um
2 Ma ´ıa Lau a Oli e a-A encio e al.: Gene ic shape o mul ich oma ic esonance peaks
sys ems), he one- ime p obabili y densi y (in he case o
classical s ochas ic sys ems), he alues o a ini e numbe
o s a e a iables o classical ields (in he case o classical
de e minis ic sys ems), e c. Fo ou pu pose, he only el-
e an ac is ha all he physical p ope ies o he sys em
a any subsequen ime ≥ 0a e uniquely de e mined
by his ini ial p epa a ion and he dynamical equa ions.
In pa icula , he (expec a ion) alue a ime ≥ 0o a
ce ain physical quan i y Qo his sys em, deno ed by Q ,
will depend on he speci ic alues aken by he pa ame-
e s appea ing in he unc ions j( ), as well as, on he
ini ial p epa a ion. When necessa y, his dependence will
be made explici by he no a ion Q (Ω,ϕ|s0, 0) wi h he
ec o no a ion Ω= (Ω1, . . . , ΩN) and ϕ= (ϕ1, . . . , ϕN).
The ans o ma ion p ope y o Q (Ω,ϕ|s0, 0) is de-
e mined by he j, as o any ime shi τ, he se o
pe iodic unc ions { j( )}j=1,...,N is in a ian unde he
ans o ma ion { , ϕ} 7→ { +τ, ϕ−Ωτ}. Consequen ly,
since he only explici ime-dependence o he dynamical
equa ions comes om he unc ions j( ), i eadily ollows
ha
Q (Ω,ϕ|s0, 0) = Q +τ(Ω,ϕ−Ωτ|s0, 0+τ).(2)
In pa icula , aking τ=− , one ob ains
Q (Ω,ϕ|s0, 0) = Q0(Ω,ϕ+Ω |s0, 0− ).(3)
Hence o h, we assume ha , a e a ce ain ansien
ime (o , mo e o mally, in he limi 0→ −∞), he ob-
se able unde in es iga ion eaches a s a iona y alue
Qs
(Ω,ϕ) = lim
0→−∞Q (Ω,ϕ|s0, 0).(4)
This assump ion has been ound in many dissipa i e sys-
ems. F om equa ion (3), i hen ollows ha
Qs
(Ω,ϕ) = Qs
0(Ω,ϕ+Ω ).(5)
The e o e, in he s a iona y egime, he ime e olu ion o
Qadmi s a desc ip ion in e ms o a ime-dependen phase
ec o o he o m ϕ+Ω .
He e, we a e in e es ed in he gene ic p ope ies o he
ime a e age
QT(Ω,ϕ) = 1
TZT
0
d Qs
(Ω,ϕ),(6)
and mo e speci ically on he in ini e- ime a e age
Q∞(Ω,ϕ) = lim
T→∞QT(Ω,ϕ).(7)
Since he unc ions Φj(Ωj +ϕj) a e 2π-pe iodic in he
phases ϕj, so will be Qs
(Ω,ϕ). The e o e, aking in o
accoun equa ion (5), i can be Fou ie expanded as
Qs
(Ω,ϕ) = X
k∈ZN
qk(Ω)eik·(ϕ+Ω ),(8)
whe e he cen e ed do deno es he usual scala p oduc
in RNand
qk(Ω) = Zπ
−π···Zπ
−π
e−ik·ϕ
(2π)NQs
0(Ω,ϕ)dNϕ.(9)
By inse ing equa ion (8) in o equa ion (6), we ob ain
QT(Ω,ϕ) = X
k∈ZN
qk(Ω)eik·(ϕ+ΩT/2)sinc k·ΩT
2,
(10)
whe e sinc(x) = sin(x)/x deno es he unno malized sinus
ca dinalis. Since limT→∞ sinc (αT/2) equals 1 i α= 0 and
0 o he wise, om equa ions (7) and (10) we ge
Q∞(Ω,ϕ) = X
k∈SΩ
qk(Ω)eik·ϕ,(11)
whe e SΩis he se o all o de ed N- uples o in ege s
o hogonal o Ω, i.e., SΩ=k∈ZN:k·Ω= 0.
I he Ncomponen s o he equency ec o Ωa e
incommensu able (i.e., i is no possible o exp ess one o
hem as a linea combina ion o he o he s wi h a ional
coe icien s), hen he se SΩ educes o he single elemen
k=0. In his case, om equa ion (11), i eadily ollows
ha
Q∞(Ω,ϕ) = q0(Ω) (12)
and, he e o e, he in ini e- ime a e age is independen o
ϕ. An expe imen al e i ica ion o his gene al s a emen
has been epo ed in e e ence [19].
In con as , i he Ncomponen s o Ωa e commensu-
able (i.e., i is possible o exp ess one o hem as a linea
combina ion o he o he s wi h a ional coe icien s), hen
he se SΩcon ains addi ional elemen s o he han k=0
and, acco ding o equa ion (11), Q∞(Ω,ϕ) depends on
he phases in ϕ. In pa icula , i he equencies a e pai -
wise commensu able, hen he e exis s a equency Ωsuch
ha Ω=Ωn, whe e n= (n1, . . . , nN), wi h njbeing
posi i e in ege s. Consequen ly, he condi ion k·Ω= 0
becomes equi alen o he Diophan ine equa ion k·n= 0.
The gene al solu ion o he la e equa ion can be ex-
p essed as an in ege linea combina ion o a se o N−1
gene a ing ec o s, g(1),...,g(N−1), each o which sa is-
ies he equa ion g(j)·n= 0 [15, 21, 22]. Thus, equa ion
(11) can be w i en as
Q∞(Ω,ϕ) = X
`∈ZN−1
qk(`)(Ω)eik(`)·ϕ,(13)
whe e k(`) = PN−1
j=1 `jg(j).
Le WCand WIbe he se s o all Ωwhose compo-
nen s a e commensu able and incommensu able, espec-
i ely. F om he abo e esul s, i can easily be shown ha
he in ini e- ime a e age Q∞(Ω,ϕ) is discon inuous on
he se WC. Indeed, since he se WIis dense in RN, o
any Ωc∈ WCone can ind a sequence {Ω(n)}n∈N⊂ WI
such ha limn→∞ Ω(n)=Ωc. I Q∞(Ω,ϕ) we e con inu-
ous a Ωc, hen
Q∞(Ωc,ϕ) = lim
n→∞Q∞(Ω(n),ϕ) = lim
n→∞q0(Ω(n)) (14)
and, as a esul , Q∞(Ωc,ϕ) would be independen o ϕ.
Ob iously, his con adic s he ac ha , since Ωc∈ WC,
Q∞(Ωc,ϕ) depends on ϕ. Hence, we can conclude ha
Ma ´ıa Lau a Oli e a-A encio e al.: Gene ic shape o mul ich oma ic esonance peaks 3
Q∞(Ω,ϕ) is discon inuous a Ωcand, hus, on he se
WC. Taking in o accoun ha WCis also dense in RN, his
las esul implies ha he in ini e- ime a e age Q∞(Ω,ϕ)
is a highly discon inuous unc ion o he equency ec o
Ω. In his con ex , he ollowing ques ions a ise: (i) Can
his discon inui y ac ually be obse ed? (ii) How does i
mani es i sel in p ac ice?
3 Long- ime asymp o ic beha io o QT
In eal si ua ions, he physical quan i y Qis known only in
a ini e ime-in e al. The e o e, he in ini e ime-a e age
Q∞can be calcula ed only app oxima ely by aking a su -
icien ly la ge alue o T. Suppose we a e in e es ed in an-
alyzing he dependence o QTon Ωnea a ixed equency
ec o Ω0. Mo e p ecisely, we ocus on alues o Ωsuch
ha |Ω−Ω0|is o he same o de o magni ude as T−1,
whe e |Ω−Ω0|= [PN
j=1(Ωj−Ω0,j)2]1/2. Then, as Tin-
c eases, he size o he egion o in e es becomes smalle
in in e se p opo ion o T.
By de ining he dimensionless equency ec o δ˜ω=
(Ω−Ω0)T, he ini e ime-a e age can be b ough o he
o m
QT(Ω,ϕ) = QT(Ω0+δ˜ω/T, ϕ)
=Qas(Ω0, δ˜ω,ϕ) + RT(Ω0, δ˜ω,ϕ),(15)
whe e
Qas(Ω0, δ˜ω,ϕ) = lim
T→∞QT(Ω0+δ˜ω/T, ϕ) (16)
is he leading-o de o he asymp o ic beha io o he unc-
ion QT(Ω0+δ˜ω/T, ϕ) o T→ ∞, while δ˜ωis held con-
s an . Fo he es RT(Ω0, δ˜ω,ϕ) = QT(Ω0+δ˜ω/T, ϕ)−
Qas(Ω0, δ˜ω,ϕ), i eadily ollows ha
lim
T→∞RT(Ω0, δ˜ω,ϕ)=0,(17)
and oge he wi h equa ion (7), ha
Qas(Ω0,0,ϕ) = Q∞(Ω0,ϕ).(18)
To ob ain an exp ession o Qas(Ω0, δ˜ω,ϕ), we se
Ω=Ω0+δ˜ω/T in equa ion (10), and inse he e-
sul ing exp ession in o equa ion (16). Then, using again
limT→∞ sinc (αT/2) = δα,0, one ob ains
Qas(Ω0, δ˜ω,ϕ) = X
k∈SΩ
qk(Ω0)eik·(ϕ+δ˜ω/2)sinc k·δ˜ω
2.
(19)
Using equa ion (11), one sees ha equa ion (19) can be
w i en in he mo e compac o m
Qas(Ω0, δ˜ω,ϕ) = Z1
0
dλ Q∞(Ω0,ϕ+λδ˜ω).(20)
The e o e, in he limi T→ ∞, he asymp o ic beha io
o QTin a neighbo hood o Ω0o size p opo ional o
T−1is comple ely de e mined by he in ini e- ime limi
Q∞a Ω0. No ice ha , acco ding o he abo e esul s,
no ma e how la ge T, i is always possible o ind a
neighbo hood o Ω0wi hin which QTis con inuous. Thus,
he discon inui y men ioned in he p e ious sec ion is a
ma hema ical idealiza ion ha , owing o he necessa ily
ini e measu emen ime, canno be obse ed in eali y.
F om equa ion (20) he e immedia ely ollows an in-
e es ing conclusion. I Ω0∈ WI(i.e., all Ncomponen s
a e incommensu able), hen Q∞(Ω0,ϕ) is independen o
ϕand Qas(Ω0, δ˜ω,ϕ) = Q∞(Ω0,ϕ) o all δ˜ω. Conse-
quen ly, om equa ions (7) and (16), one ob ains ha
lim
T→∞QT(Ω0+δ˜ω/T, ϕ)−QT(Ω0,ϕ)= 0 (21)
o all δ˜ω. In con as , i Ω0∈ WC(i.e., i s Ncomponen s
a e commensu able), hen Q∞(Ω0,ϕ) depends on ϕand,
in gene al,
lim
T→∞QT(Ω0+δ˜ω/T, ϕ)−QT(Ω0,ϕ)6= 0.(22)
This di e ence in beha io will be appa en in he nume -
ical calcula ions p esen ed below.
4 Bich oma ic d i ing
The analy ic conside a ions made so a p o ide he long-
ime limi o he non-linea esponse o a mul i- equency
o cing. In o de o in es iga e nume ically how his limi
is app oached, we ocus on bich oma ic d i ing, i.e., on dy-
namic equa ions which con ain e ms o he o m 1( ) =
A1cos(Ω1 +ϕ1) and 2( ) = A2cos(Ω2 +ϕ2), whe e ou
line o easoning s ill holds i he cosines a e eplaced by
any o he 2π-pe iodic unc ions.
4.1 Commensu able equencies and pe iodici y in ϕ
While he 2π-pe iodici y o { 1( ), 2( )}in ϕ1and ϕ2
is e iden , disc e e ime ansla ion symme y is p esen
only when Ω1and Ω2a e commensu able, i.e., o a io-
nal alues o Ω2/Ω1. Indeed, le us assume ha 1( ) and
2( ) ha e a common undamen al pe iod T. F om he pe-
iodici y o he cosine unc ion, i hen ollows ha he e
mus exis wo in ege s qand psuch ha Ω1T= 2πq and
Ω2T= 2πp. This implies ha Ω1=qΩ and Ω2=pΩ,
wi h Ω= 2π/T, and consequen ly, ha Ω2/Ω1=p/q. In
addi ion, he in ege s qand pmus be cop ime because
o he wise T/gcd(q, p)<T, would be he common unda-
men al pe iod o 1( ) and 2( ), wi h gcd(q, p) deno ing
he g ea es common di iso o qand p.
Le us now ocus on a equency ec o Ω0≡(q, p)Ω,
wi h some cop ime in ege s pand q, hence o h e e ed
o as (q, p)- esonance. Using equa ion (8), i can easily be
seen ha he s a iona y alue Qs
(Ω0,ϕ) is a T-pe iodic
unc ion o ime. A u he in e es ing ac is ha he
in ini e- ime a e age Q∞(Ω0,ϕ) displays a highe sym-
me y in ϕ han he ob ious ϕ1→ϕ1+ 2πand ϕ2→
4 Ma ´ıa Lau a Oli e a-A encio e al.: Gene ic shape o mul ich oma ic esonance peaks
ϕ2+ 2π[14]. To see ha his is so, no e ha in he
p esen case he condi ion k·Ω0= 0 becomes equi alen
o k1q+k2p= 0. The gene al solu ion o his Diophan ine
equa ion is `g, wi h he gene a ing ec o g≡(−p, q) and
any in ege `. Thus, in his case, equa ion (13) becomes
Q∞(Ω0,ϕ) = X
`∈Z
q`g(Ω0)ei`g·ϕ.(23)
Since g·ϕ=−pϕ1+qϕ2, om equa ion (23) i eadily
ollows ha Q∞(Ω0,ϕ) is 2π/p-pe iodic in ϕ1and 2π/q-
pe iodic in ϕ2.
4.2 The neighbo hood o commensu able equencies
Le us now u n ou a en ion o he icini y o a (q, p)-
esonance. Mo e speci ically, and ollowing he gene al ap-
p oach ou lined in Sec ion 3, we ocus on alues o Ω
such ha |Ω−Ω0|= [(Ω1−qΩ)2+ (Ω2−pΩ)2]1/2is
o he same o de as T−1. Then, in oducing he no a ion
δω=δ˜ω/T =Ω−Ω0, om equa ion (15), (17), and
(19), i ollows ha
QT(Ω0+δω,ϕ)∼X
`∈Z
q`g(Ω0)ei`g·(ϕ+T δω/2)
×sinc `Tg·δω
2,(24)
p o ided ha Tis la ge enough so ha he e m RTcan
be neglec ed.
Two conclusions can be d awn om equa ion (24).
Fi s , he ime a e age QT(Ω0+δω,ϕ) is 2π/p-pe iodic in
ϕ1and 2π/q-pe iodic in ϕ2. The e is, howe e , an impo -
an di e ence o he exac pe iodici y o he in ini e- ime
a e age Q∞(Ω0,ϕ) p esen ed in Sec ion 4.1. He e he pe-
iodici y holds only o su icien ly la ge alues o Tand
o Ωsu icien ly close o Ω0— o be p ecise, o |δω|=
O(T−1). Second, unde hese same es ic ions, he ela-
i e heigh ∆QT(Ω0+δω,ϕ)≡QT(Ω0+δω,ϕ)−q0(Ω0)
anishes when all sinc unc ions in equa ion (24) a e ze o
o all `6= 0, i.e., when g·δωis a nonze o in ege mul iple
o 2π/T. In pa icula , i we se δω1= 0 and a y δω2,
∆QT(Ω0+δω,ϕ) anishes when δω2is a nonze o in e-
ge mul iple o 2π/(qT). Analogously, i we se δω2= 0
and a y δω1,∆QT(Ω0+δω,ϕ) anishes when δω1is a
nonze o in ege mul iple o 2π/(pT). Hence, in he icin-
i y o he (q, p)- esonance, 2π/(qT) and 2π/(pT) ep esen
he equency scales on which ∆QT(Ω0+δω,ϕ) a ies.
The main aim o he nex wo sec ions is o p o ide
quan i a i e in o ma ion on how hese asymp o ic p op-
e ies can ac ually be obse ed in p ac ice. No e, o ex-
ample, ha a u he limi a ion may s em om he ac
ha he se o commensu able equencies ec o s is dense
in R2and, consequen ly, any (q, p)- esonance may be dis-
u bed by ano he esonance ha lies a bi a ily close o
i . I will u n ou , howe e , o all cases in es iga ed, p ac-
ically only esonances wi h a he small qand pma e .
Mo eo e , o su icien ly la ge alues o T, he shape o
he esonance peak is mainly go e ned by he e m wi h
`= 1 in equa ion (24) and, he e o e, hei en eloping
unc ion appea s as a a he clean sinc unc ion.
5 Classical andom walk model
In he model p esen ed in his sec ion, he mo ion o a
pa icle in a pe iodic subs a e is gi en by a andom walk
on a one-dimensional la ice. The la ice si es a e loca ed
a xn=na, whe e nis any in ege and a he dis ance
be ween wo neighbo ing si es. The e olu ion o he p ob-
abili ies pn( ) ha he pa icle is a si e nis go e ned by
he mas e equa ion [23]
˙pn( ) = −[ +( ) + −( )] pn( )
+ +( )pn−1( ) + −( )pn+1( ),(25)
whe e +( ) and −( ) a e he ansi ion a es om si e n
o si e n+1 and n−1, espec i ely. They a e assumed o be
independen o nand o ollow he Van’ Ho -A henius
law [24]
±( ) = 0e−β[E0±∆E( )],(26)
whe e 0is a p e ac o wi h he dimension o an in e se
ime, β= (kBΘ)−1is he in e se empe a u e, and E0+
∆E( ) and E0−∆E( ) a e, espec i ely, he ac i a ion
ene gies o he o wa d and backwa d s eps. These ac i-
a ion ene gies oscilla e a ound a cons an alue E0wi h
ime-dependen ampli udes ∆E( ) and −∆E( ), espec-
i ely.
In his sec ion, we will conside ha he ole o Q is
played by he mean pa icle eloci y V , which is de ined
as he ime de i a i e o he mean pa icle posi ion X =
aPn∈Znpn( ). Using equa ions (25) and (26), i can be
seen easily ha
V =a[ +( )− −( )] = sinh [ ( )] ,(27)
whe e = 2a 0e−βE0and ( ) = −β∆E( ). Hence o h,
we will assume ha
( ) = A1cos(Ω1 +ϕ1) + A2cos(Ω2 +ϕ2),(28)
wi h A1and A2being wo dimensionless cons an s, which
can be aken as posi i e by sui able choice o he phases ϕ1
and ϕ2. No e ha , in he p esen model, he e is no di e -
ence be ween he s a iona y Vs
and V because he mean
pa icle eloci y is independen o he ini ial p epa a ion.
In addi ion, om equa ions (27) and (28), i immedia ely
ollows ha V sa is ies he symme y p ope ies
V− (Ω,−ϕ) = V (Ω,ϕ),(29)
V (Ω,ϕ+π) = −V (Ω,ϕ),(30)
whe e π≡(π, π).
Now he Fou ie expansion in equa ion (8) akes he
o m
V (Ω,ϕ) = X
k∈Z2
keik·(ϕ+Ω ),(31)

Ma ´ıa Lau a Oli e a-A encio e al.: Gene ic shape o mul ich oma ic esonance peaks 5
wi h
k= Zπ
−πZπ
−π
e−ik·ϕ
(2π)2sinh (A1cos ϕ1+A2cos ϕ2)d2ϕ.
(32)
Since he Fou ie expansion is unique, using equa ions (29)
and (31), i is easy o see ha −k= k. Fo he same ea-
son, om equa ions (30) and (31), i eadily ollows ha
k=−ei(k1+k2)π k. In addi ion, om equa ion (32), i is
clea ha ∗
k= −k, whe e he as e isk deno es complex
conjuga ion. In conclusion, all he coe icien s ka e eal
and anish when k1+k2is an e en in ege .
Wi h he abo e esul s, le us now examine he de-
pendence o he in ini e- ime a e age eloci y on Ω. I Ω1
and Ω2a e incommensu able, om equa ion (12) one con-
cludes ha V∞(Ω,ϕ) = 0= 0. I , by con as , Ω1and
Ω2a e commensu able, aking in o accoun ha k= −k
and ha 2k= 0, i is easy o see om equa ion (23) ha
V∞(Ω0,ϕ)=2 ∞
X
`=0
(2`+1)gcos [(2`+ 1)g·ϕ].(33)
Using he de ini ion o gand he ac ha k anishes
when k1+k2is e en, i can be e i ied wi h equa ion (33)
ha V∞(Ω0,ϕ) = 0 when q−pis e en. The e o e, as
a consequence o he symme y p ope y (30), only he
(q, p)- esonances wi h q−podd a e p esen in his model.
The in eg al in equa ion (32) can be e alua ed explic-
i ly by expanding he hype bolic sine in o a powe se ies.
Then, a e some calcula ions, we ob ain
k= ∞
X
j=0
2j+1
X
`=0
`
X
m=0
2j+1−`
X
n=0
δk1,2m−`δk2,2n+`−2j−1
×A`
1A2j+1−`
2
22j+1m!n!(`−m)!(2j+ 1 −`−n)!.(34)
I can be e i ied ha he coe icien s kgi en by equa-
ion (34), as could no be o he wise, sa is y he condi ions
discussed abo e. In addi ion, since A1and A2a e posi-
i e, all he coe icien s ka e clea ly non-nega i e. Thus,
om equa ion (33), i ollows ha he maximum alue o
V∞(Ω0,ϕ) is
V∞,M(Ω0)=2 ∞
X
`=0
(2`+1)g,(35)
and i occu s when g·ϕis an in ege mul iple o 2π, i.e.,
when
qϕ2−pϕ1= 2πn (36)
wi h nbeing any in ege . Assuming ha ϕ1and ϕ2sa is y
his condi ion o maximum esonance, om equa ion (24)
oge he wi h −k= k, i ollows ha
VT(Ω0+δω,ϕ)∼2∞
X
`=0
(2`+1)gsinc [(2`+ 1)Tg·δω],
(37)
Fig. 1. Dependence o he dimensionless a e age eloci y
VT/ on Ω2/Ω1 o A1=A2= 1, ϕ1=ϕ2= 0, and T=
104/Ω1. The esonance peaks co esponding o he ac ions
0 (only pa ially shown o be e isibili y o he emaining
peaks) and 1/2 a e clea ly isible. The ones co esponding o
1/4 and 2/3 a e also isible bu conside ably smalle . O he es-
onances, such as he ones co esponding o 2/5 and 6/7 shown
in Figu e 3 canno be app ecia ed on his scale. The inse s
show a zoomed-in iew a ound Ω2/Ω1= 1/4 (le inse ) and
Ω2/Ω1= 2/3 ( igh inse ) in e ms o δω/Ω1=Ω2/Ω1−p/q.
p o ided ha |δω|=O(T−1) and ha Tis la ge enough
such ha he e m RTcan be neglec ed. No e ha , in
his case, VT(Ω0+δω,ϕ) anishes whene e g·δωis a
nonze o in ege mul iple o π/T. Thus, i we se δω1=
0 and a y δω2,VT(Ω0+δω,ϕ) anishes when δω2is
a nonze o in ege mul iple o π/(qT), whe eas i we se
δω2= 0 and a y δω1,VT(Ω0+δω,ϕ) anishes i δω1is
a nonze o in ege mul iple o π/(pT).
To illus a e ou heo e ical esul s, we ha e calcu-
la ed VT(Ω,ϕ) using equa ion (10) wi h QT(Ω,ϕ) =
VT(Ω,ϕ), and qk(Ω) = k. The coe icien s kappea -
ing in ha equa ion ha e been e alua ed using equa ion
(34). To ensu e ha bo h d i ings ha e oughly he same
in luence, we ha e ocused on he mos symme ic case o
equal d i ing ampli udes. Speci ically, in all he igu es o
his sec ion we ha e aken A1=A2= 1. In addi ion, o
maximize he heigh o he esonance peaks, we ha e e-
s ic ed ou analysis o alues o ϕ1and ϕ2 ha sa is y
he condi ion o maximum esonance in equa ion (36).
In Figu e 1, we plo he dimensionless ime-a e age e-
loci y VT/ as a unc ion o Ω2/Ω1 o ϕ1=ϕ2= 0 and
T= 104/Ω1. Acco ding o ou heo e ical esul s, he e
should eme ge esonance peaks when Ω2/Ω1is equal o a
a ional numbe . Fu hemo e, only he esonances co e-
sponding o he ac ions 0, 1/4, 1/2, and 2/3 a e isible.
O he esonances, such as he ones co esponding o 2/5
and 6/7 s udied below, canno be app ecia ed in he ig-
u e. This absence o esonances is no due o he use o
a ini e a e aging ime. In ac , hese peaks would be im-
pe cep ible e en in he limi T→ ∞, because hey a e
e y small compa ed o he smalles peak isible in Fig-
6 Ma ´ıa Lau a Oli e a-A encio e al.: Gene ic shape o mul ich oma ic esonance peaks
u e 1. Indeed, using equa ion (35), i is easy o e i y ha
a Ω2/Ω1= 1/4 ( he smalles peak isible in Figu e 1),
V∞,Mis app oxima ely equal o 3 ×10−3 , whe eas a
Ω2/Ω1= 2/5 and Ω2/Ω1= 6/7, V∞,Mis app oxima ely
equal o 7 ×10−5 and 7 ×10−11 , espec i ely.
To analyze in mo e de ail he beha io o VTin he
icini y o a (q, p)- esonance, we now conside ha one
equency, say Ω1, is kep ixed, while he o he equency,
Ω2, is a ied a ound he alue pΩ1/q. In e ms o ou
p e ious no a ion, his co esponds o se ing δω1= 0 and
δω2=Ω2−pΩ1/q. Hence o h, o no a ional simplici y,
we will w i e δω ins ead o δω2. In addi ion, o acili a e
compa ison wi h he asymp o ic exp ession (37), we will
use he dimensionless a iable qδωT, which he e is no hing
bu Tg·δω.
In Figu es 2 and 3, we depic he dependence o VT/
on qδωT o he esonances (q, p) = (4,1) (le column
in Figu e 2), (q, p) = (3,2) ( igh column in Figu e 2),
(q, p) = (5,2) (le column in Figu e 3), and (q, p) = (7,6)
( igh column in Figu e 3). To analyze how hese peaks
eme ge as he a e aging ime inc eases, di e en alues
o Tha e been conside ed, which a e indica ed in he
panels. In addi ion, in each panel, qcu es ha e been
plo ed, co esponding o he alues ϕ2= 2πn/q, wi h
n= 0, . . . , q −1. Since ϕ1= 0, all hese alues sa is y
he condi ion o maximum esonance in equa ion (36).
The esul s in Figu es 2 and 3 co obo a e he heo e i-
cal p edic ion ha , o su icien ly la ge alues o T, he q
cu es con e ge o he asymp o ic esul in equa ion (37).
Su p isingly, he a e aging imes necessa y o each he
asymp o ic egime a e huge in compa ison o he pe iod
o he d i ing T= 2πq/Ω1.
6 Quan um mechanical wo-le el sys em
Le us now conside a dissipa i e quan um mechanical
model wi h he Hamil onian H( ) = H0+HD( ), whe e
H0=
2(σxcos θ+σzsin θ) (38)
and he bich oma ic d i ing
HD( ) = A1σxcos(Ω1 +ϕ1) + A2σzcos(Ω2 +ϕ2).
(39)
To ensu e ha bo h d i ings ha e oughly he same im-
pac , we ocus on he mos symme ic case θ=π/4 and
equal d i ing ampli udes, A1=A2≡A.
Fo he conside a ion o dissipa ion, one may s a
om a sys em-ba h model o ob ain an equa ion o mo ion
o he educed densi y ope a o o he dissipa i e sys em.
Then one can show ha gene ally dissipa ion is quan i a-
i ely a ec ed by he d i ing [25, 26]. He e howe e , we
a e in e es ed in he gene ic esponse o bich oma ic d i -
ings and, hus, we ollow a less in ol ed pa h which allows
an e icien nume ical solu ion o a he long p opaga ion
imes. In doing so, we employ a Lindblad mas e equa-
ion o he densi y ope a o [27], ˙ρ=−i[H0+HD( ), ρ]+
Fig. 2. Dependence o he dimensionless a e age eloci y
VT/ on he dimensionless a iable qδωT o he esonances
(q, p) = (4,1) (le column) and (q, p) = (3,2) ( igh column),
and he alues o he a e aging imes displayed in he panels.
Fo all he cu es ϕ1= 0 and A1=A2= 1. In each panel,
he e a e qcu es co esponding o he alues ϕ2= 2πn/q, o
n= 0,...,q−1 ( om blue o ed), which a e ob ained om
he condi ion o maximum esonance in equa ion (36). As ex-
pec ed om he heo e ical analysis, wi h inc easing he alue
o T, he qcu es con e ge o he asymp o ic cu e gi en by
equa ion (37).
Fig. 3. The same as in Figu e 2 bu o he esonances (q, p) =
(5,2) (le column) and (q, p) = (7,6) ( igh column).
γD(ρ) (in uni s wi h ~= 1) wi h a dissipa o [27]
Dρ= ˜σ−ρ˜σ+−1
2˜σ+˜σ−ρ−1
2ρ˜σ+˜σ−,(40)
whe e ˜σ−=|ϕ0ihϕ1|=σ†
+induces dissipa i e decay om
he exci ed s a e |ϕ1io he und i en Hamil onian H0 o
he co esponding g ound s a e |ϕ0i.
As an obse able Q, we may choose any combina ion o
Pauli ma ices. Gene ic ea u es, howe e , will no depend
on he pa icula choice such ha wi hou loss o gene -
ali y, we conside Qs
≡ hσzi . To ensu e independence
Ma ´ıa Lau a Oli e a-A encio e al.: Gene ic shape o mul ich oma ic esonance peaks 7
0.5 0.6 0.7 0.8 0.911.1
−0.5
−0.25
0
0.25
Ω2/Ω1
hσziT
T= 500/Ω1
T= 1000/Ω1
1
1
1
2
4
73
5
2
3
5
7
3
4
5
6
Fig. 4. Time-a e aged esponse o he bich oma ically d i en
wo-le el sys em a e a ansien s age a e aged o e a ious
imes T. When Ω2/Ω1is close o a a ional p/q, peaks wi h
gene ic shape eme ge (labeled by he co esponding ac ion).
Pa ame e alues a e: =√2Ω1,θ=π/4, A1=A2=Ω1, and
γ= 0.2Ω1. Fo g aphical easons, he cu e o T= 500/Ω1is
e ically shi ed.
o de ails, we ha e e i ied all nume ical esul s by using
also sligh ly di e en se ups and obse ables. The calcula-
ions a e pe o med by nume ically in eg a ing he Lind-
blad mas e equa ion s a ing a ime = 0 in he g ound
s a e o H0 o ob ain he densi y ope a o ρ and, hus,
Q ≡ (ρ σz). Since we a e in e es ed in s a iona y expec-
a ion alues, we exclude he ansien s age and compu e
he ime a e ages in an in e al o du a ion Ts a ing a
ime 10/γ.
We s a by ske ching he global pic u e o he esponse
(Figu e 4) which shows hσziT o wo di e en a e aging
imes Tas a unc ion o Ω2and o a ious phases ϕ2
(in his sec ion, we always ake ϕ1= 0). As expec ed,
hσziTexhibi s esonance peaks a simple a ional alues
o Ω2/Ω1which sha pen wi h inc easing T. The size o he
peaks as well as he shape o he backg ound depend on de-
ails. Gene ally i is such ha he (1,1) esonance is a he
p ominen , which can be unde s ood by equency mix-
ing: Fo wo equal d i ing equencies, a ze o- equency
esponse can be ob ained al eady in second-o de pe u -
ba ion heo y. Fo all o he alues o (q, p), one has o
go o highe o de . Hence o h we ocus on he uni e sal
ea u es in na ow egions a ound p onounced peaks.
As smalle peaks end o be less comp omised by com-
ponen s wi h `6= 1, we s udy he eme gence o a peak
wi h inc easing p opaga ion ime T o (q, p) = (5,3).
Figu e 5 shows hσziT o he 20 equally spaced phases
ϕ2= 2πn/20 wi h n= 0,1,...,19. Fo he ela i ely sho
p opaga ion ime T= 150/Ω1, all cu es a e signi ican ly
di e en om each o he , while a clea peak s uc u e is
s ill missing. Wi h inc easing Tand s a ing a he cen e
δω = 0, cu es o phases ha di e by 2π/q coincide,
such ha e en ually 20/q = 4 g oups o cu es eme ge.
This e lec s he 2π/q pe iodici y in ϕ2de i ed abo e o
δω = 0. Mo eo e , i con i ms he gene aliza ion conjec-
u ed om equa ion (24), namely ha he pe iodici y o
−4π−2π0 2π4π
−0.5
−0.475
q δω T
hσziT
T= 5000/Ω1
−0.5
−0.475
hσziT
T= 1000/Ω1
−0.5
−0.475
hσziT
T= 750/Ω1
−0.5
−0.475
hσziT
T= 250/Ω1
−0.5
−0.475
hσziT
T= 150/Ω1
Fig. 5. Resonance peak (q, p) = (5,3) o he phases ϕ1= 0
and ϕ2= 2πn/(4q), wi h n= 0,1, .., 4q−1 ( om blue o ed),
showing he ansi ion om 2π-pe iodici y o 2π/q-pe iodici y.
No ice ha he abscissa is scaled wi h he a e aging ime T.
All o he pa ame e s a e as in Figu e 4.
a good app oxima ion holds in a whole neighbo hood o
he (q, p)-peak.
To unde line his esul , we also plo he cu es wi hin
one 2π/q pe iod and hose o ϕ2equal o mul iples o
2π/q sepa a ely, bu now o he (q, p) = (7,5) esonance,
see Figu e 6. The le column con ains he esul s o al-
ues o ϕ2in he ange [0,2π/q]. They show ha only o a
su icien ly la ge T, he en eloping unc ion is domina ed
by he `= 1 componen and esembles he sinc conjec-
u ed in equa ion (24). Mo eo e , he i s and he las
cu e smoo hly connec o each o he , which depic s how
he 2π/q-pe iodici y eme ges. Acco dingly, he cu es o
ϕ2a mul iples o 2π/q e en ually coincide, as can be ap-
p ecia ed in he igh column o Fig. 6
As a emnan o ini e p opaga ion ime T, we wi ness
in all panels o Figu e 6 an inclina ion o he esonance
peak, which o a smalle ex en is no iceable also in Fig-
u e 5. I s ems om he global backg ound o hσziT isible
in Figu e 4. Owing o he scaling o he abscissa, i dimin-
ishes wi h inc easing Tand, in acco dance wi h equa ion
(24), i e en ually anishes.
7 Conclusions
We ha e s udied he asymp o ic limi o mul ich oma i-
cally d i en, dissipa i e dynamical sys ems. I u ned ou
8 Ma ´ıa Lau a Oli e a-A encio e al.: Gene ic shape o mul ich oma ic esonance peaks
−5π0 5π
−0.43
−0.42
q δω T
hσziT
T= 2000/Ω1
−5π0 5π
q δω T
T= 2000/Ω1
−0.43
−0.42
hσziT
T= 500/Ω1T= 500/Ω1
−0.43
−0.42
hσziT
ϕ2= 0
ϕ2= 2π9
10q
T= 250/Ω1
ϕ2= 0
ϕ2= 2πq−1
q
T= 250/Ω1
Fig. 6. Resonance peak (q, p) = (7,5) o he a e aging imes
displayed in he g aphics, while all o he pa ame e s a e as
in Figu e 4. Le column: Resul o 10 equally spaced phases
ϕ2= 2πn/(10q) o n= 0,1,...,9 showing ha o su icien ly
la ge T, he cu e o ϕ2= 0 (n= 0, blue) smoo hly connec s
o he one o ϕ= 2π/q. Righ column: The same bu o
ϕ= 2πn/q,n= 0,1,...,q−1. Wi h inc easing T, he cu es
s a o coincide.
ha a s ic dis inc ion be ween commensu able and in-
commensu able equencies equi es an in ini e p opaga-
ion ime. Ne e heless, he e exi s a no iceable di e ence
be ween he wo cases, namely ha only o commensu-
able equencies he phases o he d i ing ields may ma -
e . Mo eo e , he phase dependence gene ally has a lowe
pe iodici y han he nai ely expec ed 2π. While his im-
plies non-gene ic ea u es o he esonances, he esul ing
peaks upon a ia ion o he phase exhibi a gene ic o m
gi en by sinc unc ions.
While he limi ing beha io can be de i ed analy i-
cally, we ha e pe o med nume ical s udies o see how
he limi s a e app oached. Fo a classical andom walk
on a la ice wi h bich oma ically ime-dependen ansi-
ion a es, he eloci y has been ob ained analy ically up
o he nume ical compu a ion o a sum. The esponse as
a unc ion o he wo d i ing equencies shows how eso-
nances eme ge a ound a ional alues o Ω2/Ω1.
The case o a dissipa i e wo-le el sys em has been
ea ed ully nume ically. I e ealed how wi h inc easing
p opaga ion ime, he gene ic ea u es o esonance peaks
eme ge, namely he sinc shape and he sub 2πpe iodici y
in he phase shi . The wid h o he esonance peaks a
a ional equency quo ien s sh inks wi h inc easing a e -
aging ime, such ha he backg ound e en ually appea s
la and he peaks become p onounced. The alue o he
esponse depends on he ela i e phase o he wo d i ings,
while in i s icini y, he esponse becomes phase indepen-
den .
An impo an poin in p ac ical calcula ions is ha
commensu able equency a ios wi h a he la ge nume -
a o o denomina o imply la ge pe iods. Then owing o
he necessa ily ini e p opaga ion ime, his pe iodici y
may s ill no be mani es in he esul . In o he wo ds, up
o such ini e ime, he sys em beha es as i i we e quasi-
pe iodic, i.e., as i he equencies we e incommensu able.
Howe e , in pa icula o he andom-walk model, i may
ake e en conside ably longe un il he peaks assume hei
gene ic shape. Quan i a i e s a emen s abou his issue
s ill ep esen a challenge o u u e in es iga ions.
This wo k was suppo ed by he Spanish Minis y o Science,
Inno a ion, and Uni e si ies h ough he CSIC Resea ch Pla -
o m on Quan um Technologies PTI-001 and ia g an s No.
MAT2017-86717-P and FIS2017-86478-P.
Au ho con ibu ion s a emen
JCP has de i ed he analy ical esul s, MLOA and SK
ha e pe o med he nume ical calcula ions o he an-
dom walk model and he wo-le el sys em, espec i ely.
All au ho s ha e con ibu ed o w i ing he manusc ip .
Re e ences
1. L. Gammai oni, P. H¨anggi, P. Jung, F. Ma chesoni, Re .
Mod. Phys. 70, 223 (1998)
2. J. Casado-Pascual, J. G´omez-O d´o˜nez, M. Mo illo,
P. H¨anggi, Phys. Re . Le . 91, 210601 (2003)
3. V. Anishchenko, A. Neiman, A. As akho , T. Vadia aso a,
L. Schimansky-Geie , Chao ic and S ochas ic P ocesses in
Dynamic Sys ems (Sp inge , Be lin, 2002)
4. J.A. F eund, L. Schimansky-Geie , P. H¨anggi, Chaos 13,
225 (2003)
5. B. Lindne , J. Ga cia-Ojal o, A. Neiman, L. Schimansky-
Geie , Phys. Rep. 392, 321 (2004)
6. J. Casado-Pascual, J. G´omez-O d´o˜nez, M. Mo illo,
J. Lehmann, I. Goychuk, P. H¨anggi, Phys. Re . E 71,
011101 (2005)
7. I. Goychuk, J. Casado-Pascual, M. Mo illo, J. Lehmann,
P. H¨anggi, Phys. Re . Le . 97, 210601 (2006)
8. P. Reimann, Phys. Rep. 361, 57 (2002)
9. P. H¨anggi, F. Ma chesoni, Re . Mod. Phys. 81, 387 (2009)
10. D. Cube o, F. Renzoni, B ownian Ra che s: F om S a is i-
cal Physics o Bio and Nano-mo o s (Camb idge Uni e -
si y P ess, Camb idge, 2016)
11. S. Flach, O. Ye ushenko, Y. Zolo a yuk, Phys. Re . Le .
84, 2358 (2000)
12. D. Cube o, F. Renzoni, Phys. Re . E 97, 062139 (2018)
13. D. Cube o, G.R. Robb, F. Renzoni, Phys. Re . Le . 121,
213904 (2018)
14. J. Casado-Pascual, D. Cube o, F. Renzoni, Phys. Re . E
88, 062919 (2013)