PHYSICAL REVIEW E 83, 046208 (2011)
Exci ed-s a e phase ansi ion and onse o chaos in quan um op ical models
P. P ´
e ez-Fe n´
andez,1A. Rela˜
no,2J. M. A ias,1P. Cejna ,3J. Dukelsky,4and J. E. Ga c´
ıa-Ramos5
1Depa amen o de F´
ısica A ´
omica, Molecula y Nuclea , Facul ad de F´
ısica, Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain
2G upo de F´
ısica Nuclea , Depa amen o de F´
ısica A ´
omica, Molecula y Nuclea , Uni e sidad Complu ense de Mad id, A enida
Complu ense s/n, E-28040 Mad id, Spain
3Ins i u e o Pa icle and Nuclea Physics, Facul y o Ma hema ics and Physics, Cha les Uni e si y, V Holeˇ
so iˇ
ck´
ach 2, P ague 18000,
Czech Republic
4Ins i u o de Es uc u a de la Ma e ia, CSIC, Se ano 123, E-28006 Mad id, Spain
5Depa amen o de F´
ısica Aplicada, Uni e sidad de Huel a, E-21071 Huel a, Spain
(Recei ed 14 Sep embe 2010; e ised manusc ip ecei ed 16 Decembe 2010; published 15 Ap il 2011)
We s udy he c i ical beha io o exci ed s a es and i s ela ion o o de and chaos in he Jaynes-Cummings
and Dicke models o quan um op ics. We show ha bo h models exhibi a chain o exci ed-s a e quan um phase
ansi ions dema ca ing he uppe edge o he supe adian phase. Fo he Dicke model, he signa u es o c i icali y
in exci ed s a es a e blu ed by he onse o quan um chaos. We show ha he eme gence o quan um chaos is
caused by he p ecu so s o he exci ed-s a e quan um phase ansi ion.
DOI: 10.1103/PhysRe E.83.046208 PACS numbe (s): 05.45.M , 05.30.R , 05.70.Fh, 42.50.Nn
I. INTRODUCTION
One o he goals o many-body physics is he unde s anding
o quan um c i ical phenomena [1]. A quan um phase ansi-
ion (QPT) appea s in sys ems wi h Hamil onian H(λ), which
show a sudden change o he g ound-s a e p ope ies when he
con ol pa ame e λ a ies ac oss a c i ical alue λc.In he
las ew yea s, he connec ion o a QPT wi h he en anglemen
[2–4] and wi h he eme gence o chao ic beha io [2,4,5] has
been in es iga ed, mo i a ed by he impo an ole hey could
play in he cu en esea ch ela ed o quan um in o ma ion
echnologies [6].
Ve y ecen ly, a quan um c i ical phenomenon o a new
ype— he one ela ed o exci ed s a es a he han o he g ound
s a e—has been discussed o se e al model sys ems [7–10].
An exci ed-s a e quan um phase ansi ion (ESQPT) ep esen s
a nonanaly ic e olu ion o indi idual exci ed s a es wi h he
con ol pa ame e . I can also be obse ed as a singula
a ia ion o he s a e densi y wi h ene gy and en ails d ama ic
dynamical consequences [11,12].
As ESQPTs ha e so a been iden i ied in simple, mos ly
in eg able sys ems, wo na u al ques ions a ise: Fi s , a e he
ESQPTs also gene ic in mo e complex, p e ailingly chao ic
sys ems? Second, how does he collapse o quan um ene gy
le els, a ypical ESQPT signa u e, a ec he le el epulsion,
which is inhe en in quan um chaos? The aim o his pape
is o add ess hese ques ions by analyzing a simpli ied model
o collec i e in e ac ions o ma e and ligh , known as he
Dicke model [13]. The Dicke model is a nonin eg able model,
howe e , in he o a ing wa e app oxima ion, i educes o he
in eg able Jaynes-Cummings model [14]. Bo h e sions ha e
ecen ly s i ed up g ea in e es , since he implemen a ion o a
unable ma e -ligh coupling ep esen s a ou e o s udy quan-
um c i ical e ec s [15–17]. The QPT o a supe adian phase
wi hin he Dicke model was s udied heo e ically [2,18–20],
and ecen ly ealized expe imen ally using a supe luid gas in
an op ical ca i y [17].
This pape is o ganized as ollows. In Sec. II we desc ibe he
models. In Sec. III we analyze he quan um phase ansi ions
in bo h models, pa icula y hose ela ed o exci ed s a es. In
Sec. IV we s udy he ela ionship be ween he exci ed-s a e
quan um phase ansi ions and he eme gence o chaos in
he Dicke model. Finally, in Sec. Vwe summa ize he main
conclusions.
II. THE MODELS
Bo h he Dicke and Jaynes-Cummings models assume a
se o wo-le el a oms in e ac ing by a dipole coupling o
s eng h λwi h a single-mode bosonic ield (ca i y pho ons).
The models a e exp essed ia he c ea ion and annihila ion
ope a o s b†and b, desc ibing a bosonic mode wi h equency
ω(wi h Nb=b†bbeing he numbe o pho ons), and he SU(2)
gene a o s {J±,Jz}desc ibing he ensemble o Na wo-le el
a oms wi h he le el spli ing ω0in e ms o a pseudospin o
leng h J=Na/2. A use ul ealiza ion o he SU(2) algeb a
can be buil h ough an a ay o spin-1
2pa icles loca ed on 2J
si es,
J+=
2J

i=1
a†
↑ia↓i=J†
−,J
z=1
2
2J

i=1
(a†
↑ia↑i−a†
↓ia↓i),(1)
whe e a†
↑io a↑i(a†
↓io a↓i) c ea e o annihila e spin-up (spin-
down) s a es o he e mion on si e i( he uppe and lowe
s a e o he i h a om), and he ladde ope a o s J±desc ibe
collec i e spin lips along he a ay. N↑≡Jz+Jcoun s he
numbe o a oms exci ed o he uppe le el. The sys em has
wo deg ees o eedom associa ed wi h, e.g., Nband Jz.
The Jaynes-Cummings Hamil onian [14]
H1(λ)=ω0Jz+ωb†b+λ
√4J[bJ++b†J−](2)
conse es he quan i y M/2=Nb+N↑, which oge he wi h
he Hamil onian a e he wo equi ed conse ed quan i ies
(in eg als o mo ion) o assu e he in eg abili y o he sys em.
We wo k wi h a ixed alue, M/2=Na, which implies a ini e
dimension o he Hilbe space, and se ω>ω
0.
046208-1
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P. P ´
EREZ-FERN ´
ANDEZ e al. PHYSICAL REVIEW E 83, 046208 (2011)
The Dicke Hamil onian [13]
H2(λ)=ω0Jz+ωb†b+λ
√4J[(b+b†)(J++J−)] (3)
conse es he pa i y =(−1)M/2, bu no he numbe M
i sel . The e o e, i is no in eg able. The Hilbe -space
dimension o any Nais in ini e, since s a es wi h unlimi ed
pho on numbe s Nba e coupled by he in e ac ion e m. F om
a p ac ical poin o iew, his means ha he pho on space
needs o be unca ed by a ce ain maximal alue Nmax
b.In he
nume ical calcula ions p esen ed below, we checked ha an
inc ease o Nmax
bdoes no cause no iceable changes in esul s.
In hese calcula ions, we se ω=ω0, which co esponds o he
esonance abso p ion and emission o pho ons by he a oms.
III. QUANTUM PHASE TRANSITIONS
These models se e as oy examples o he mase phase
ansi ion. Indeed, in he he modynamic limi , J→∞,
bo h Hamil onians yield a QPT o he second o de [2]in
which p ope ies a e ob ained om a ele an semiclassical
analysis [12]. The c i ical alues o he coupling s eng h a e
λc=(ω0−ω)2/2 o he Jaynes-Cummings model1and
λc=√ωω0/2 o he he Dicke model. Below he c i ical
poin , a no mal phase exis s, in which he g ound s a e is
simila o ha a λ=0, gi en by he pho on acuum (Nb=0)
combined wi h a maximally exci ed (Jz=+J) o a o ally
unexci ed (Jz=−J) s a e o he a om a ay. The i s case
is alid o he Jaynes-Cummings model wi h ω>ω
0and
M=4J, and he second one is alid o he Dicke model.
Fo a coupling s eng h below λcin he he modynamic limi ,
he λ=0 g ound-s a e o m is p ese ed wi h expec a ion
alues Nb=0, Jz=±J, and g ound-s a e ene gy Eno m
0=
±Jω
0. When c ossing he c i ical alue λc, he g ound s a e
e en ually lips o a o m wi h expec a ion alues Nb>0
and −J<Jz<+J, and dec easing ene gy E0(λ)<E
no m
0,
in which bo h he pho on ield and he a omic a ay acqui e pa -
ial, mac oscopic exci a ions. This egime can be in e p e ed
asasupe adian phase [2,18]. The quan i ies Jz(o N↑)
and Nb ep esen sui able o de pa ame e s o cha ac e ize
he supe adian phase ansi ion.
The g ound-s a e QPT in bo h models is ollowed o λ>
λcby a chain o exci ed-s a e phase ansi ions loca ed a he
c i ical ene gy, coinciding app oxima ely wi h he g ound-s a e
ene gy o he no mal phase, Ec=Eno m
0. A de ailed discussion
o he ESQPT e ec s and hei semiclassical oo s is gi en
elsewhe e [12]. The beha io o he o de pa ame e Jzclose
o he c i ical ene gy Ecis gi en by
Jz=Jzc+A|E−Ec|α,(4)
which is cha ac e ized by a c i ical exponen α. Resul s o he
Jaynes-Cummings model wi h 1000 a oms a e plo ed in Fig. 1
o λ=1.5. The ed poin s co espond o he expec a ion alue
o Jzin each eigens a e wi h eigen alue E. The con inuous
1No e ha we a e desc ibing a ansi ion wi h a ixed numbe o
pa icles, he e o e his c i ical alue di e s om ha o Re . [2]. Fo
mo e de ails, see [12].
0.5 11.5
E/J
0.2
0.4
0.6
<J
z
>/J
Na=1000; λ=1.5
y= A |x-xc| α + C
α = 0.33
FIG. 1. (Colo online) Scaled a omic in e sion as a unc ion o
ene gy o he Jaynes-Cummings model, wi h calcula ions o ω0=1,
ω=2, λ=1.5, and J=500.
black cu e shows a i using Eq. (4) wi h he c i ical exponen
α=0.33. The cusp singula i y a Ec/J =1 is cha ac e ized
by he alue Jzc=J, hence N↑=Na(no seen in he
igu e). This can be desc ibed as he λ<λ
cg ound-s a e
s uc u e p opaga ing h ough he spec um along he line
E=Ec, whe e we indeed obse e mul iple a oided c ossings
o indi idual le els [12].
Simila esul s o he Dicke model wi h 60 a oms and
λ=1.5 a e plo ed in Fig. 2. The hin ed oscilla ing line
ep esen s nume ical da a o Jzob ained as an a e age o e
20 eigens a es a ound he eigen alue E; he poin s ha e been
joined by a line o easie isualiza ion. The hick black cu e
co esponds o a i by Eq. (4). The numbe o a oms is much
smalle han ha used in he p e ious calcula ion because o
a a he la ge alue o Nmax
bneeded o ge con e gence o
he le els abo e he c i ical ene gy. In pa icula , esul s o
E/J > 1 ( he la pa o he nume ical dependence in Fig. 2)
a e no con e ged.
A compa ison wi h Fig. 1shows ha he esul s o he
Dicke model a e uzzie han hose o he Jaynes-Cummings
model. They exhibi sizable luc ua ions a ound a smoo h
-0.25
-0.2
-0.15
-0.1
-0.05
0
-4 -3 -2 -1 0 1 2
<Jz>/J
E/J
N=60; λ=1.5
α1=0.25; α2=1.5
FIG. 2. (Colo online) Scaled a omic in e sion as a unc ion o
ene gy o he Dicke model, wi h calcula ions o ω0=ω=1, λ=
1.5, J=30, and esul s smoo hed o e 20 poin s.
046208-2
EXCITED-STATE PHASE TRANSITION AND ONSET OF ... PHYSICAL REVIEW E 83, 046208 (2011)
dependence. Ano he peculia i y o he Dicke model is ha he
o de pa ame e is bimodal wi h wo c i ical exponen s: α1=
0.25 o E<E
c, and α2=1.5 o E>E
c. Despi e hese
di e ences, Fig. 2shows a quali a i ely simila dependence
as Fig. 1, demons a ing a ela ed ype o singula i y in
he Dicke model a he c i ical ene gy Ec/J =−1. We
he e o e conclude ha o λ>λ
c, bo h models exhibi an
ESQPT a Ec=Eno m
0. This c i ical ene gy e mina es he
domain o he supe adian phase p esen a low empe a u es
[18].
IV. CHAOS AND EXCITED-STATE QUANTUM PHASE
TRANSITIONS
We know ha he Dicke model is nonin eg able and pa ly
chao ic. The supe adian ansi ion a ze o empe a u e was
shown [2] o be co ela ed wi h a c osso e om o de ed
o chao ic beha io . Hence one may ask whe he quan um
chaos is also somehow ela ed o he c i ical beha io o
exci ed s a es. I can be an icipa ed ha chao ic p ope ies o
he spec um in oduce la ge luc ua ions ha pa ly hide he
singula dependence a he c i ical poin in Fig. 2. Howe e ,
chaos and ESQPTs ha e some p ope ies which a e di icul
o concilia e. On one hand, he mos signi ican ea u e o
quan um chaos is he le el epulsion, which en ails a null
p obabili y o inding wo le els a he same ene gy [21]. On he
o he hand, an ESQPT as a apid es uc u ing o exci ed s a es
is ypically connec ed wi h a a he close app oach o le els
(nume ous sha p a oided c ossings), o en wi h a singula
accumula ion o le els a E=Ec[7–9,11]. The e o e, he
ela ion be ween he le el epulsion and spec al signa u es o
an ESQPT cons i u es an in e es ing heo e ical challenge.
Le us conside he Dicke Hamil onian wi h a alue o λ
abo e λc, whe e he sys em is pa ly chao ic [2]. To analyze
how chaos and he ESQPT can dwell oge he , we calcula e
he spacing dis ibu ion P(s), whe e sis a no malized (s=1)
dis ance be ween wo neighbo ing le els, on bo h sides o he
c i ical ene gy Ec. This dis ibu ion is known o in e pola e
be ween he Poissonian and Wigne o ms, PP=e−sand
PW=π
2se−πs2/4, espec i ely, as he sys em ans o ms om
a egula o a chao ic egime [21]. In o de o ha e a la ge
numbe o le els below he c i ical ene gy, we p esen esul s
o λ=3; di e en choices lead o simila pic u es.
In Fig. 3we show he P(s) dis ibu ions calcula ed o
bo h subc i ical (E<E
c) and supe c i ical (E>E
c) pa s o
he spec um, and o a ious a om numbe s (see cap ion o
de ails). In all cases, we used he same numbe o le els o
building he his og ams on bo h sides o he c i ical ene gy.
Fo E>E
c, he spec al s a is ics closely ollow he Wigne
su mise. In con as , o E<E
c, he shape o he his og ams
is no clea : while o he Na=46 case, i is a he close o
he Poissonian dis ibu ion, o lowe a om numbe s, i yields
nei he he Wigne no he Poissonian o m. The mos ele an
ac is ha below Ecwe obse e P(s=0) >0, while abo e
Ecwe ob ain P(s=0) ≈0. In o he wo ds, he le els s a o
epel each o he when c ossing he ESQPT c i ical ene gy.
To ob ain a mo e quan i a i e desc ip ion, we analyze how
he spec al s a is ics change as one mo es in he spec um.
To inc ease he accu acy o he p ocedu e, we ely on he
accumula ed spacing dis ibu ion F(s)=s
0dxP(x). Gi en a
0
0.5
0 2
s
0
0.5
P(s)
0
0.5
0
0.5
0 2
FIG. 3. Nea es -neighbo spacing dis ibu ion P(s) o heDicke
model wi h λ=3. His og ams a e shown o E<E
c(le column)
and E>E
c( igh column). The ows om bo om o op co espond
o Na=22, 30, 38, and 46. The sho dashed lines co espond o
he Poisson dis ibu ion, and he long dashed lines co espond o he
Wigne -Dyson dis ibu ion.
nume ical sequence o spacings {si}, we measu e i s “dis ance”
om he Wigne su mise by means o he ollowing quan i y:
=i[FW(si)−F(si)]2
i[FW(si)−FP(si)]2,(5)
whe e FWand FPa e accumula ed dis ibu ions de i ed om
PWand PP, espec i ely. The e o e, =0 i he nume ical
sequence ollows he Wigne dis ibu ion, while =1 o a
Poissonian sequence. We ha e cons uc ed sequences o 200
consecu i e spacings and, using Eq. (5), we ha e calcula ed
he dis ance o each sequence om he Wigne s a is ics.
These dis ances a e plo ed in Fig. 4as a unc ion o he
mean ene gy o he espec i e sequence. We can see ha a
qui e ab up ansi ion om ini e alues o  o ∼0 akes
place jus below he c i ical ene gy Ec. Al hough he ange o
he co e ed alues o Nais no la ge enough o app oach he
he modynamic limi , i su ices o con i m ha he c i ical
beha io in he o de pa ame e is accompanied by a change in
he spec al s a is ics. Fu he mo e, as his change is so ab up
in ini e size sys ems, we a e led o a ibu e i o he p ecu so s
o he ESQPT.
We a e now able o go beyond he hypo hesis o Re . [2],
s a ing ha he spec um o he Dicke model wi h λ>λ
c
is egula a low ene gies. F om ou nume ical esul s, we
conclude ha a ansi ion o chaos akes place a ound he
c i ical ene gy Ec. A compac pic u e o a ious dynamical
egimes implici in he Dicke model can he e o e be s a ed as
ollows: We s a a λ=0, wi h he g ound s a e ha ing Nb=
N↑=0. Inc easing λlea es he g ound s a e unpe u bed un il
we c oss he c i ical poin λc o he supe adian ansi ion,
whe e he expec a ion alues Nband N↑s a o inc ease.
Fo any alue o λ>λ
c, he e exis s a egion abo e he g ound
s a e in which we ind no le el epulsion, P(s=0) >0. This
seems o be a common ea u e o almos he en i e supe adian
domain in he λ×Ephase diag am. I he ene gy is inc eased
046208-3
P. P ´
EREZ-FERN ´
ANDEZ e al. PHYSICAL REVIEW E 83, 046208 (2011)
0
0.1
0.2
0.3
0.4
0.5
-10 -5 0 5 10
Δ
E/J
FIG. 4. (Colo online) Onse o chaos in he Dicke model, wi h
λ=3 measu ed by he dis ance (5) o ac ual le el s a is ics om
he Wigne dis ibu ion. The dis ance () is plo ed as a unc ion
o he mean scaled ene gy o he ele an ac ion o he spec um,
wi h he e ical line indica ing he c i ical ene gy. Calcula ions a e
done o a ious a om numbe s: Na=22 ( e y ligh yellow do ed
line), 30 (ligh g een dashed-do ed line), 38 (da k ed dashed line),
and 46 ( e y da k blue solid line).
closely below he c i ical alue Ec=−Jω
0, whe e N↑d ops
sha ply, he le el epulsion se s in, leading he spec um o he
Wigne ype o s a is ics. Abo e Ec, he sys em becomes ully
chao ic.
V. CONCLUSIONS
In conclusion, we ha e demons a ed he exis ence o
an exci ed-s a e quan um phase ansi ion in wo models
desc ibing he collec i e ma e -ligh in e ac ion. In he in-
eg able Jaynes-Cummings model, he ESQPT leads o a nea
nonanaly ici y o he o de pa ame e Jza he c i ical ene gy
Ec. The nonin eg able Dicke model exhibi s a simila ype o
ESQPT, bu he signa u es a e blu ed by he onse o chao ic
beha io in he spec um. Le el epulsion (a undamen al
ea u e o quan um chao ic sys ems) and a cumula ion o
sha p a oided c ossings (a ypical signa u e o an ESQPT)
a e di icul o concilia e in gene al. Howe e , ou nume ical
calcula ions show ha a c osso e , om he egime wi h no
le el epulsion o he one wi h he Wigne le el s a is ics,
akes place p ecisely a ound he c i ical ene gy. Mo eo e ,
ou esul s a e compa ible wi h he hypo hesis ha his ab up
eme gence o le el epulsion is caused by he p ecu so s o
he ESQPT, in a simila way as discussed in Re . [2] o
he g ound s a e. We an icipa e he exis ence o a simila
quali a i e beha io in o he nonin eg able sys ems wi h
ESQPTs. Mo eo e , o he dynamical e ec s o ESQPTs, such
as, o ins ance, anomalous decohe ence ac o s p e iously
ob ained in he Lipkin model [11], a e expec ed o ake place
in a uzzie manne in quan um chao ic sys ems.
ACKNOWLEDGMENTS
This wo k was suppo ed by he Czech Science Founda ion
(G an No. 202/09/0084), Czech Minis y o Educa ion (G an
No. MSM 0021620859), Spanish Minis e io de Ciencia e
Inno aci´
on and Eu opean egional de elopmen und (G an s
No. FIS2009-07277, No. FIS2008-04189, No. FIS2009-
11621-C02-01, and No. FPA2007-63074), CPAN-Ingenio
(G an No. CSPD-2007-00042-Ingenio 2010), and Jun a de
Andaluc´
ıa (G an s No. FQM160, No. FQM318, No. P05-
FQM437, and No. P07-FQM-02962). The wo k o P. P.-F. is
unded by a FPU g an o he Spanish Minis e io de Educaci´
on,
and he wo k o A. R. is unded by he p og am CPAN
Consolide -Ingenio 2010.
[1] S. Sachde , Quan um Phase T ansi ions (Camb idge Uni e si y
P ess, Camb idge, 1999).
[2] C. Ema y and T. B andes, Phys. Re . Le . 90, 044101 (2003);
Phys. Re . E 67, 066203 (2003); N. Lambe , C. Ema y, and
T. B andes, Phys.Re .Le .92, 073602 (2004).
[3] L. Amico, R. Fazio, A. Os e loh, and V. Ved al, Re . Mod. Phys.
80, 517 (2008).
[4] E. A. Chagas and K. Fu uya, Phys. Le . A 372, 5564 (2008).
[5] T. Deguchi and P. K. Ghosh, Phys. Re . E 80, 021107
(2009); T. Deguchi, P. K. Ghosh, and K. Kudo, Phys. Re . E
80, 026213 (2009).
[6] Quan um Compu e s, Algo i hms and Chaos, edi ed by
G. Casa i, D. L. Shepelyansky, P. Zolle , and G. Benen i,
In e na ional School o Physics “En ico Fe mi,” Vol. 162 (IOS,
Ams e dam, 2006).
[7] P. Cejna , M. Macek, S. Heinze, J. Jolie, and J. Dobeˇ
s, J. Phys.
A39, L515 (2006).
[8] M. A. Cap io, P. Cejna , and F. Iachello, Ann. Phys. (NY) 323,
1106 (2008).
[9] P. Cejna and P. S ´
ansk´
y, Phys.Re .E78, 031130 (2008).
[10] P. Ribei o, J. Vidal, and R. Mosse i, Phys. Re . Le . 99, 050402
(2007); Phys. Re . E 78, 021106 (2008).
[11] A. Rela˜
no, J. M. A ias, J. Dukelsky, J. E. Ga c´
ıa-Ramos, and
P. P ´
e ez-Fe n´
andez, Phys.Re .A78, 060102 (2008); P. P ´
e ez-
Fe n´
andez, A. Rela˜
no, J. M. A ias, J. Dukelsky, and J. E. Ga c´
ıa-
Ramos, Phys. Re . A 80, 032111 (2009).
[12] P. P´
e ez-Fe n´
andez, P. Cejna , J. M. A ias, J. Dukelsky, J. E.
Ga c´
ıa-Ramos, and A. Rela˜
no, Phys. Re . A 83, 033802 (2011).
[13] R. H. Dicke, Phys. Re . 93, 99 (1954).
[14] E. T. Jaynes and F. W. Cummings, P oc. IEEE 51, 89 (1963);
M. Ta is and F. W. Cummings, Phys. Re . 170, 379 (1968).
[15] F. Illumina i, Na u e Phys. 2, 803 (2006).
[16] A. D. G een ee, C. Tahan, J. H. Cole, and L. C. L. Hollenbe g,
Na u e Phys. 2, 856 (2006).
[17] K. Baumann, C. Gue lin, F. B ennecke, and T. Esslinge , Na u e
(London) 464, 1301 (2010).
[18] K. Hepp and E. H. Lieb, Ann. Phys. (NY) 76, 360
(1973).
[19] D. Nagy, G. K´
onya, G. Szi mai, and P. Domokos, Phys. Re .
Le . 104, 130401 (2010).
[20] B. M. Rod ´
ıguez-La a and Ray-Kuang Lee, J. Op . Soc. Am. B
27, 2443 (2010).
[21] H.-J. S ¨
ockmann, Quan um Chaos: An In oduc ion (Camb idge
Uni e si y P ess, Camb idge, 1999).
046208-4