Jou nal o Ma hema ical Chemis y (2023) 61:288–295
h ps://doi.o g/10.1007/s10910-022-01343-w
ORIGINAL PAPER
Sudden exci a ions o ha monic no mal modes
I. Aldazabal1,2 ·I. Nagy2,3
Recei ed: 25 Janua y 2022 / Accep ed: 2 Ma ch 2022 / Published online: 4 Ap il 2022
© The Au ho (s) 2022
Abs ac
The N-ha monium boson sys em, i.e., a comple ely in eg able model o Npa icles
whe e bo h he ex e nal con inemen and he wo-pa icle in e ac ion a e ha monic, is
in es iga ed unde he ac ion o sudden ime-dependen pe u ba ion. This quench-
like ex e nal pe u ba ion o con inemen has a quad upola space-cha ac e . The
ime-independen ansi ion p obabili ies, which cha ac e ize he impac o quench
as a e age occupa ion numbe s, o m a comple e dis ibu ion in he sense o p ob-
abili y heo y. The quench-gene a ed ene gy shi Ein he co ela ed many-body
sys em, and a pu i y- ype Rényi en opy Sα=2a e calcula ed. Challenging ein e p e-
a ions o such an ene gy change in e ms o a iables o a classical he modynamical
sys em o N(N−1)/2 pai s a e gi en as well. As in he case o he g ound-s a e co -
ela ed sys em, an en opy could cha ac e ize a global link o ene ge ically op imized
independen -pa icle models.
Keywo ds Co ela ion ·En opy ·Exci a ions
1 Su ey o he unpe u bed model sys em
Ad ances in op ical apping o cold a oms ha e allowed o an unp eceden ed manip-
ula ion o e he size o hese quan um sys ems such ha he numbe No a oms being
apped can be [1] p ecisely speci ied. In gene al, ope a ing on quan um many-body
sys ems p o ides a way o unde s anding [2–5]. In pa icula , o p ecisely speci ied
in e ac ing quan um sys ems, ime-dependen uning (quench) o ex e nal ha monic
This s udy on analogies be ween many-body sys ems is dedica ed o he memo y o János Pipek.
BI. Aldazabal
[email p o ec ed]
1Cen o de Física de Ma e iales (CSIC-UPV/EHU)-MPC, P. Manuel de La dizabal 5, 20018 San
Sebas ián, Spain
2Donos ia In e na ional Physics Cen e , P. Manuel de La dizabal 4, 20018 San Sebas ián, Spain
3Depa men o Theo e ical Physics, Ins i u e o Physics, Budapes Uni e si y o Technology and
Economics, 1521 Budapes , Hunga y
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Jou nal o Ma hema ical Chemis y (2023) 61:288–295 289
con inemen seems o be an easily ealizable expe imen al ool o gene a e dynam-
ics. Rema kably, his ex e nal con ol can be a con inuous one, in con as o he
disc e e manipula ion o many-body ( a ge ) sys ems wi h ex e nal p ojec ile-impac .
Mo i a ed by he easibili y o expe imen al access o a e ages o d i en many-body
sys ems, he e we apply a well-analyzed model sys em o an ene ge ic s udy. Such a
s udy on a model sys em can be conside ed as a clinical a emp o impo an ene ge ic
and s a is ical de ails, which migh gene a e u he e o s.
Following ea lie wo ks [6,7], he e we ake a p o o ype one-dimensional sys em
o Niden ical pa icles, bosons, wi h mass mand scala coo dina es xi, whe e i=
1,2, .....N. The Hamil onian, in oduced by Heisenbe g as he simples many-body
o m o his s udies
HN=
N
i=1p2
i
2m+1
2mω2
0x2
i−1
2mω2
0
1≤i≤j≤N
(xi−xj)2,(1)
is sepa able (a = 0) and his ac esul s in independen no mal modes. Thus
he basic expec a ion alue in quan um mechanics, he g ound-s a e ene gy, becomes
addi i e
E=1
2ω1+(N−1)
2ω2(2)
whe e, wi hou loss o gene ali y, we ake uni s de ined by m=1 and =1.
The equencies o ha monic no mal modes a e [6,7] gi en by ω1=ω0and
ω2=ω0√1−N. No ice ha he s abili y, o epulsi e in e pa icle in e ac ion
( > 0),isma kedby heN<1 condi ion. The e is no such cons ain o he
a ac i e, like in nuclea physics, case.
In he ene ge ically-op imal (e), independen -pa icle modeling one ge s o he
ene gy
Ee=N
2ωe=N
2ω01−(N−1), (3)
which is based on he addi i e s uc u e o Eq. (1) wi hou xixjp oduc - e ms. Fo
>0 he equency-o de ing becomes ω2<ω
e<ω
0, and o <0 he o de ing is
ω2>ω
e>ω
0, The di e ence Ec=(E−Ee)is, acco ding o Wigne [8] pionee ing
de ini ion, he co ela ion ene gy. I is ins uc i e (c. ., nex pa ag aph) o in es iga e he
small-coupling (→0) limi o he co ela ion ene gy. By s aigh o wa d expansion
one a i es a , in ou uni s
Ec( << 1)=−
N(N−1)
2
2
8ω0.(4)
The second de i a i e, in coupling, o he di e ence o wo a ia ional quan i y is
nega i e. This obse a ion on sign is in acco d wi h gene al s a emen s in quan um
chemis y [9].
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290 Jou nal o Ma hema ical Chemis y (2023) 61:288–295
In he second pa o his su ey, we de i e a challenging co espondence o
Ec( << 1)by using p ecise esul on he one-pa icle educed densi y ma ix
[6,7] o he in e ac ing boson sys em, and i s well-known [10,11] o mal equi alence
wi h he s a is ical densi y ma ix o an ideal sys em o Noscilla o s wi h equency
¯ω(ω0,,N)in he mal equilib ium a empe a u e T(ω0,,N). These oscilla o s do
no in e ac wi h each o he , bu only wi h he hea ba h. We no e ha in ield- heo e ic
a emp s o black-hole physics [10,11], he usual associa ion is based on pho ons.
The e, he acing ou o high-ene gy deg ees o eedom yields a low-ene gy e ec-
i e ield heo y wi h an accompanying s a is ical measu e o black-body-like en opy
which may be conside ed as an in o ma ion loss.
Fo he equi alen he modynamical sys em one has E=F+TS
N, whe e F
is Helmhol z’s ee ene gy, and he e is a hea -like p oduc o he empe a u e T
and he on Neumann en opy SN. These a e he a iables in he pa h based on an
ideal canonical-ensemble o he modynamics [12]. Employing p ecise [6] mappings
be ween he o mally equi alen one-ma ices, we de i e o a di ec compa ison a
weak coupling
TS
N≡T(ω0,,N)SN(ω0,,N)=ω0N(N−1)
2
2
8,(5)
whe e common loga i hmic ac o s cancel ou in he lhs p oduc . Thus, based on o mal
equi alence o wo densi y ma ices, we ge as pa ial co espondence TS
N=−Ec.
No ice ha a N||<< 1 in he quan um-mechanical case, i.e., a (T/ω0)<<1in
he he mal case, he he mal pa o he Helmhol z ee ene gy, i.e., he pa beyond i s
ze o-poin ene gy (1/2)N¯ω, is exponen ially small. To ou bes knowledge, he abo e-
de i ed o mal co espondence is no el on a p o o ype many-body sys em. Besides,
he quad a ic-in-and he linea -in-numbe -o -pai cha ac e s o he en opy-based
pa o he o de ed-p oduc in Eq. (5) sugges s ha he co espondence ound migh
hold independen ly o he s a is ics in weakly co ela ed sys ems. We s ess ha a
p opo ionali y be ween −Ecand on Neumann en opy SNis known in quan um
chemis y as conjec u e [13].
2 Time-dependen pe u ba ion o he model sys em
The selec ed de ails o he p e ious Sec ion signal ha he comple e o hono mal se s
o independen no mal modes a e oscilla o wa e unc ions
φn(ω, u)=ω
π1/41
√2nn!e−1
2ωu2Hn(√ωu). (6)
As Eq. (2) shows, he e is one mode wi h ω1=ω0, and (N−1)mode wi h ω2.In
he ideal, i.e., nonin e ac ing (=0) case all (N)modes a e equi alen (ω1=ω2=
ω0), and ha sys em has ze o in o ma ion- heo e ic en opy since i s one-ma ix is
idempo en . No ice he e, ha below we will use, o simpli y ma hema ics, a common
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Jou nal o Ma hema ical Chemis y (2023) 61:288–295 291
no a ion ω o hese equencies whe e his is possible and e u n o ω1and ω2channel-
no a ion whe e i is needed o physics.
The main goal below is o conside he ene ge ic-impac o ime-dependen , passing,
pe u ba ions wi h quad a ic space-cha ac e in no mal coo dina es. In ac , he N=2
case, wi h such a pe u ba ion o di e en sign, was al eady in es iga ed ecen ly in
o de o shed ligh on he sign-dependence o ene gy shi , a well-known p oblem in
swi p o on and an ip o on close-impac on ine helium a om [14]. Besides, ha s udy
add essed ew ala ming p oblems inhe en in he ime-dependen densi y- unc ional
me hod [15] whe e one wo ks wi h auxilia y (densi y-op imal) o bi als ins ead o
p ecise independen modes.
In sho , wi h pe u ba ions o ini e du a ion one can calcula e he ene gy shi by
using Di ac’s a ia ion o cons an me hod ins ead o ollowing in ime he e ol ing
wa e unc ions, since we know he Hamil onian a he beginning and a he end o
a passing pe u ba ion. The e o e, he ene gy shi can be calcula ed in his case by
conside ing he exci a ion p obabili ies as occupa ion numbe s which cha ac e ize he
ansi ions om a gi en (in ou case: g ound) s a e o o he elemen s o he o hono -
mal comple e se s. The sum o hese p obabilis ic occupa ion numbe s sa is y he
no maliza ion condi ion, as i should be. They weigh he mode-ene gies in summa-
ion o e quan um numbe n o ge he o al ene gy change. Fo a passing ( anishing a
→±∞) pe u ba ion he expec a ion alue o he Hamil onian wi h e ol ing s a es
esul s in he same [14] ime-independen ene gy change.
Which s ill emains o ou enume a ion o ools, is he conc e iza ion o he abo e-
ou lined occupa ion numbe s. Bu an insigh ul me hod o ha conc e iza ion is,
o una ely, also well-documen ed due o es ablished wo ks [16,17]. In ac , a com-
ple e Chap e [18] w i en by expe s is de o ed o simila p oblems. B ie ly, ha
insigh ul me hod es s on an asymp o ic analysis ia a cle e a iable-change o map
ime-dependence in o a s a iona y sca e ing p oblem. In ou compa a i e s udy we
employ he exp essions, deduced o a single oscilla o [16–18], o ou case wi h inde-
penden modes. We s ess, howe e , ha we conside N-mode sys ems wi h p ecise
and ene ge ically-op imized modes. In o he wo ds, we in es iga e he in e play o
inhe en co ela ion and ex e nal pe u ba ion o quench-cha ac e . We belie e ha
he such-ob ained esul s could con ibu e o unde s anding.
The equi ed s a is ical weigh s o ene gy a e aging, i.e., he occupa ion ( ansi ion)
p obabili ies W2n,0a e gi en by [16–18] he ollowing exp ession
W2n,0(R)=(2n)!
22n(n!)2√1−R(R)n=1
√π
(n+1/2)
(n+1)√1−R(R)n,(7)
which e lec s he selec ion ule o allowed (upwa d) ansi ion wi h a quad a ic
pe u ba ion. This comple e dis ibu ion unc ion is no malized since in gene al
1
(1−x)η=∞
n=0
(n+η)
n!(η) xn.
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292 Jou nal o Ma hema ical Chemis y (2023) 61:288–295
The e lec ion coe icien Ro he men ioned (auxilia y) sca e ing p oblem (see,
abo e) can be calcula ed in he knowledge o ime-de ails on a quad upola pe u ba ion
[14,16].
In ou wo k on a con ined boson sys em we es ic ou sel es o he quench-like
si ua ion. In his ab up case a =0, whe e ω2⇒ω2
=(ω2+λω2
0), one ge s [17]
o he e lec ion
R(ω, λ) =ω−ω
ω+ω 2
in e ms o ini ial (ω) and inal (ω ) equencies which cha ac e ize he no mal modes
be o e and a e he quench, espec i ely. I should be no ed ha he occupa ion num-
be s in Eq. (7) a e now simply he squa es o expansion coe icien s ob ained by
expanding a gi en s a iona y g ound-s a e φ0(ω, u)in e ms o a comple e se o s a-
iona y o hono mal φn(ω ,u) unc ions. The selec ion ule men ioned is based on
pa i y-conside a ion in expansion.
Thus, o one mode we ha e ω=ω1and o he o he (N−1)modes we ha e
ω=ω2. The pa ame e λmeasu es he s eng h o a sudden-change in ex e nal
con inemen . I can ha e bo h sign, wi hin he s abili y ange [ω2>−|λ|ω2
0]o
he sys em. Fu he mo e, in he s abili y ange, he e is a duali y in R(λ) unde he
ma hema ical cons ain o ω1ω2=ω2
. Unde such a special cons ain , he magni ude
o e lec ion Rcan no dis inguish be ween physical cases wi h co esponding λ>0
o λ<0, i.e., up- o down- uning.
This duali y clea ly signals, simila ly o ea lie obse a ions [19,20] wi h eigen-
alues o one-ma ices o he unpe u bed sys em, ha simple p obabilis ic measu es
alone can no cha ac e ize comple ely he physics. We add he e based on Eq. (7)
(mode i, wi h Ri, whe e i=1,2,e) he so-called pu i y (R), a equen ly [19]
applied in o ma ion measu e
(R)=∞
n=0[W2n,0(R)]2=(1−R)2
πK(R2)≤1,(8)
whe e K(x)is he comple e ellip ic in eg al o he i s kind. An o he measu e, he so-
called Rényi’s min-en opy [21], is gi en by Sα=∞(R)=ln[1/(1−R)].HisSα=2(R)
is ela ed o a pu i y ia =exp(−S2)in mode i. Such a connec ion was conside ed
ea lie [22] as a p omising pa h o S2 ia an expe imen al es ima ion o . In Rényi’s
classi ica ion Sαis a measu e o o de αo he amoun o in o ma ion. We conside
[23] such ma hema ical measu es as po en ially use ul ones e en o no -scale-less
p oblems, and e u n o physics.
Despi e he abo e-men ioned duali y in occupa ion numbe s (s a is ical weigh s),
he o al inal ene gy (E ) o he sys em a e quench, and hus he E=(E −E)
o al ene gy shi , e lec he in o ma ions (ene gy scales) encoded in he Hamil onian.
Keeping in mind he ema k a Eq. (6) on simpli ica ion in no a ions, we con inue
wi h he de e mina ion o channel-con ibu ions, deno ed by Eiwhe e i=1,2. To
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Jou nal o Ma hema ical Chemis y (2023) 61:288–295 293
E=(E1+E2)we ob ain
E1(ω1,R1)=ω1
∞
n=0
(2n)W2n,0(R1)=1
2ω1
2R1
1−Ri
,(9)
E2(ω2,R2)=(N−1)ω
2
∞
n=0
(2n)W2n,0(R2)=(N−1)
2ω2
2R2
1−R2
.(10)
In he knowledge o his p ecise esul , we add he one, deno ed by Ee, which is
based on an ene ge ically (e)p e-op imized independen -pa icle modeling ou lined
in Sec ion I, unde he impac o he same change (∼λω2
0)in ex e nal con inemen .
We a i e a
Ee(ωe,Re)=Nωe
∞
n=0
(2n)W2n,0(Re)=N
2ωe
2Re
1−Re
.(11)
Mo i a ed by Wigne ’s de ini ion o co ela ion ene gy Ec() =[E() −Ee()]
in he g ound-s a e si ua ion, we a e emp ed o in oduce a quench- ela ed e m de ined
as
Ec(, λ) =[E(, λ) −Ee(, λ)]
which also e lec s he di e ence be ween exac and ene ge ically p e-op imized
independen -pa icle desc ip ions. Now we ake he pe u ba i e limi whe e λ→0
a ixed N,ω0and (small) , hus all Ri<< 1. To a use ul compa ison wi h Eq. (4)
we ob ain
Ec(, λ) =+N(N−1)
2
2λ2
64 ω0(12)
The posi i i y is expec ed on physical g ounds since he quench ac s as an ex e nal
agen which ies o diminish igid indi idual beha io s (i.e., di e ence in modes)
e lec ed in Eq. (4) in o he di ec ion o a common beha io (i.e., simila , ene ge ically
op imized modes). P ecisely, i is his obse a ion which sugges s us o make inally
a somewha ca alie conjec u e ia Ec(, λ) =[TS
N]. By such a conjec u e,
which is mo i a ed by he i s law o mac oscopic he modynamics as well, we a e
emp ed o iew Ecas he esul o ce ain hea - ans e (Q) o a classical sys em
o pai s.
3 Summa y, ema ks and ou look
In his wo k he N-ha monium boson sys em, i.e., a comple ely in eg able model
o Npa icles whe e bo h he ex e nal con inemen and he wo-pa icle in e ac ion
a e ha monic, is in es iga ed unde he ac ion o a quench-like-in- ime pe u ba ion.
This ex e nal pe u ba ion o con inemen has a quad upola space-cha ac e . The
123
294 Jou nal o Ma hema ical Chemis y (2023) 61:288–295
ime-independen ansi ion p obabili ies, which cha ac e ize he impac o quench as
a e age occupa ion numbe s, a e used o calcula e analy ically he ene gy shi E
in he many-body sys em, and he pu i y- ela ed Rényi’s en opy Sα=2. Challenging
ein e p e a ions, Eqs. (4) and (12), o cha ac e is ic ene gy di e ences, in e ms o
a iables o a classical he modynamical sys em o N(N−1)/2 pai s, a e gi en as
well o bo h he g ound- and exi ed-s a e si ua ions.
We s ess, as i s ema k, ha ou con ollable quench is in he ex e nal [14] con ine-
men and no in he pa icle-pa icle in e ac ion. Thei coupling (∼) is no changed.
Howe e , con ollable quench in ha coupling could also be in e es ing o a gene al,
de ailed unde s anding. Indeed, such a change is in he ocus o e o s in [2–5]a
ixed con inemen . Ou p e ious expe ience [23] wi h such a quench in he simple
wo-pa icle (N=2) ha monic model sugges s ha a simila connec ion as he one in
Eq. (12) o he impo an di e ence o s a iona y ene gies, i.e., quan um mechanical
expec a ion alues, can be ound as well. De ails due o di e en quenches, and hei
possible in e play, equi e a dedica ed s udy.
We add o comple eness, ha he ime-dependence (a e sudden quenches a
=0) o he e ol ing wa e unc ions and associa ed ime-dependen one-ma ices
[23] con ain, ia hei ime-dependen eigen alues, use ul p obabilis ic in o ma ion on
inhe en dynamics in isola ed in e ac ing sys ems. Compa ison o he such-ob ained
ime-dependen en opic measu es, say a ime-dependen sys em pu i y [23], wi h s a-
iona y quan i ies cha ac e ized in his s udy based on independen modes, could allow
impo an [14] conclusions on obse able quan i ies. As a inal ema k, we no e ha i
would be in e es ing o ex end he p esen app oach wi h ab up con inemen - uning,
o cases wi h o he pa icle-pa icle in e ac ion. Say, o he con ac in e ac ion which
seems o be ealis ic and ex e nally unable ( ia Feshbach esonances) in Bose sys ems
o ha monically con ined a oms [24,25].
As an ou look we u n o a eally heo e ical challenge. Acco ding o ea lie insigh s
on he black-hole aspec o ma e [26], he e one also has a la ge numbe o unob-
se able in e nal con igu a ions which may e lec , ia an en opy, he end o ce ain
p ocesses. Ou quench-media ed changes, gene a ed in a gi en closed en angled sys-
em, in i s ene gy and an associa ed en opy, al eady sugges ha one may iden i y
ealis ic p ocesses, and hei modula ing ole, in ha ascina ing ield as well. Fo
ins ance, a p ocess in which he e is a suddenly cap u ed cloud o ma e which changes
he in e nal ene gy o a black-hole. In ou modeling his would co espond o si ua ion
wi h o al ( )ene gy E( )(ω0,,N1+N2)≡[E1(ω0,,N1)+E2(ω0,,N2)]≡
F( )+T( )S( )
N. A u u e analysis o his si ua ion, along he second law o he mo-
dynamics, is desi able. Indeed, he " he maliza ion p ocess" om subsys em’s T1and
T2 o a common Tis qui e challenging.
Acknowledgemen s One o us (I.N.) is indeb ed o he deceased P o esso János Pipek o his ea lie
ad ices in many de ails o unde lying ma hema ics behind his s udy. This wo k was suppo ed in pa by
G an PID2019-105488GB-I00 unded by MCIN/AEI/10.13039/501100011033.
Funding Open Access unding p o ided hanks o he CRUE-CSIC ag eemen wi h Sp inge Na u e.
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Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps
and ins i u ional a ilia ions.
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