Excited-state phase transition and onset of chaos in quantum optical models

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 1539-3755/2011/83(4)/046208(4) ©2011 Ame ican Physical Socie y 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. 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