We study scarring phenomena in open quantum systems. We show numerical
evidence that individual resonance eigenstates of an open quantum system
present localization around unstable short periodic orbits in a similar way as
their closed counterparts. The structure of eigenfunctions around these
classical objects is not destroyed by the opening. This is exposed in a
paradigmatic system of quantum chaos, the cat map.Comment: 4 pages, 4 figure

Diego Wisniacki

Gabriel G. Carlo

P. Gaspard

English

arXiv

   We study scarring phenomena in open quantum systems. We show numerical evidence that individual resonance eigenstates of an open quantum system present localization around unstable short periodic orbits in a similar way as their closed counterparts. The structure of eigenfunctions around these classical objects is not destroyed by the opening. This is exposed in a paradigmatic system of quantum chaos, the cat map.Fil: Wisniacki, Diego Ariel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Carlo, Gabriel Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentin

Wisniacki, Diego Ariel

Carlo, Gabriel Gustavo

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

Scarring in open quantum systemsDiego Wisniacki1 and Gabriel G. Carlo21Departamento de Física, FCEyN, UBA, Pabellón 1 Ciudad Universitaria, C1428EGA Buenos Aires, Argentina2Departamento de Física, Comisión Nacional de Energía Atómica, Avenida del Libertador 8250, C1429BNP, Buenos Aires, ArgentinaReceived 30 November 2007; published 15 April 2008We study scarring phenomena in open quantum systems. We show numerical evidence that individualresonance eigenstates of an open quantum system present localization around unstable short periodic orbits ina similar way as their closed counterparts. The structure of eigenfunctions around these classical objects is notdestroyed by the opening. This is exposed in a paradigmatic system of quantum chaos, the cat map.DOI: 10.1103/PhysRevE.77.045201 PACS numbers: 05.45.Mt, 03.65.SqOpen quantum systems are very important in many areasof physics. For example, they play a central role in the studyof quantum to classical correspondence 1, microlasers2,3, quantum dots 4, chaotic scattering 5, and more.However, there are several properties of these systems thatare less known if compared to those of closed ones.Quantum evolution in open systems is given by nonuni-tary matrices, whose eigenstates resonances are nonor-thogonal and the eigenvalues are complex with modulus lessthan or equal to one. One of the main conjectures about theproperties of the spectrum is that the mean density of reso-nances follows the fractal Weyl law 6. This law predictsthat the number of eigenstates that have a finite decay rategoes as N−d−1, where d is a fractal dimension of theclassical strange repeller. This result has been tested in somesystems 7,8. As a consequence, the majority of the eigen-functions become degenerate with their eigenvalue modulustending to zero as the size of the opening the number ofdecay channels relative to  increases. These are the short-lived eigenstates, which cannot be associated to any classicaltrapped set instead, they can be related to the trajectoriesthat escape from the system before the Ehrenfest time. Onthe other hand, the number of remaining eigenstates thelong-lived ones tends to zero. However, they contain themost relevant classical information, resembling the classicalrepeller. This was noticed in 9, where they were coinedquantum fractal eigenstates. Moreover, this investigation wasrecently extended 10,11 by looking at the right and leftresonances of the open baker’s map. It was found that theprobability density averaged for several right eigenstates issupported by the classical Cantor set the repeller, showingself-similarity both in the q and p representation. Finally, inthe more specific context of optical microcavities, the forma-tion of long-lived scarred modes has been observed 3. Thisbehavior has been associated to avoided resonance crossings.Nevertheless, almost nothing else is known about the mor-phology of individual resonances.We are interested in the study of quantum systems whichare classically chaotic. In closed quantum chaotic systems,the morphology of the eigenfunctions has been extensivelystudied. One of the most important and striking properties isscarring 12. This consists of the localization, i.e., the prob-ability enhancement of given individual eigenfunctions alongshort unstable periodic orbits POs. This effect has beendiscovered in the Bunimovich stadium billiard 13 and agreat amount of work has been done since then 14, givingrise to what is known as “scar theory.”In this Rapid Communication we explore quantitativelythe localization properties of resonances. We have studiedthe overlaps of wave functions highly localized on the vicin-ity of POs scar functions 15–17 with the eigenfunctionsof an open quantum system in order to unveil the quantummechanical manifestation of short POs. These values becomehigher than when the overlap is calculated with the eigen-functions of the closed system. The  smoothed fractal na-ture does not destroy structures of this kind. This effect iseven greater when the area of the opening grows, thus itcannot be ascribed to a perturbative origin. We provide onewith an interpretation of these results.One of the most studied open systems corresponds to two-dimensional torus maps, where a band along the q or p di-rection is cut by means of a projection. The correspondingquantum dynamics is given by a nonunitary matrix M̄ = PMor equivalently, M̄=MP, which is related to M̄ by a timereversal operation, where M is the closed map and P is theprojector on the complement of the opening. This quantumevolution is characterized by decaying eigenstates i, whosecorresponding eigenvalues zi have complex energies. It isusual to define zi2=exp−i, where i0 is called the de-cay rate. We have studied a paradigmatic model of quantumchaos, the cat map, which is a linear automorphism on thetwo-torus generated by the 22 symplectic matrix M,modulus 1. We have usedM = 2 31 2 . 1When quantizing the torus we have a finite Hilbert space ofdimension N=1 /2 and a discrete N lattice of position andmomenta in the unit interval. The quantum cat map in theposition representation is given by the matrix M whose ele-ments are 18Mkj =  in1/2exp2ink2 − jk + j2	 . 2Finally, we choose to apply the projector P after M to obtainthe nonunitary matrix M̄ that gives the evolution of the opencat map.The main resource that we use to investigate localizationis the scar function, which not only applies to maps but alsoto general flows. These functions have been deeply studied inPHYSICAL REVIEW E 77, 045201R 2008RAPID COMMUNICATIONS1539-3755/2008/774/0452014 ©2008 The American Physical Society045201-1the literature 15–17. They are wave functions highly local-ized on the stable and unstable manifolds of POs, and on theenergy given by a Bohr-Sommerfeld quantization conditionon the trajectory. We are going to use a formulation suitablefor a Poincaré surface of section, or more directly for mapsof the two-torus examples of this can be found in 16. Wedefine the “tube functions” for maps, tubemaps, as a sum ofcoherent states centered at the fixed points of a given PO 	,each one having a phase 17. Then, a dynamical average isperformed, and we have the following expression for the scarfunction:scarmaps = l=−TTeiS	l/ cosl2TMltubemaps , 3where T stands for the number of iterations of the map up tothe Ehrenfest time TE=ln  /  is the Lyapunov exponent,and S	 is the classical action of 	. We have used Eq. 3 toconstruct functions highly localized on the vicinity of theperiodic points of the closed cat map. In Fig. 1a we can seethe structure of the scar function corresponding to the POgiven by q , p= 0.5,0.5, one of the shortest of this map,for N=225. The maximum probabilities correspond to thedarkest regions. Panel 1b of the same figure displays thisfunction in a logarithmic scale of gray, showing the way itextends along the stable and unstable manifolds of the cor-responding orbit.In the following we are going to describe the behavior oflocalization in the open system by means of the maximumoverlaps of the scar function with its resonances. We exploredifferent values of N and two different shapes of the projec-tor P. For simplicity we define a map M̄a= PaM, where Pacorresponds to the projection on the complement of a bandparallel to the p direction of width q and centered at q=q0. We also define the map M̄b= PbM, where we now con-sider two symmetric bands each one having a q /2 width,and centered at q=q0 and q=1−q0. This is shown in the leftand right insets of Fig. 3. In Figs. 1c and 1d we show theright and left eigenstates of the M̄a map that have maximumoverlap with the scar function displayed in Fig. 1a hereq0=0.225 and q=0.25. The left resonance localizes on theunstable manifold, while the right one does it on the stablemanifold. The same can be found in panel 1e but for the M̄bmap q0=0.1625. We can see that the symmetric cut local-izes the resonance on the stable and unstable manifolds ofthe trajectory. Finally, in panel 1f the eigenfunction of theclosed cat map with maximum overlap with the scar functionof panel 1a is shown.First, we systematically analyze the behavior of localiza-tion as a function of . For that purpose, in Fig. 2 we showthe maximum overlaps of the scar function with the right andleft eigenstates of the open cat map, as a function of N forclarity of the exposition we show the running average ofthese values in a window of size N=10. It is evident thatthese values are greater for the open system, both for theright dotted and left dashed resonances. On the otherhand, we can see the insets, where the order number max ofthe eigenstate with maximum overlap with the scar functionis plotted vs N they were ordered in ascending eigenvaluemodulus. It is clear that the maximum overlap correspondsto resonances with the smaller decay rates larger eigenvaluemoduli. This guarantees that we are looking at wave func-tions which have a support on the classical repeller and donot belong to the null subspace. In Fig. 2a we can see theresults for the M̄a map, while in 2b they correspond to M̄b,where q0 and q values are taken the same as in the particu-lar case of Fig. 1. The difference between the two maps turnsout to be very important. In fact, the greater overlaps wereobtained when opening in a symmetric fashion rather thanwith a single strip. Finally, we mention that the overlaps ofthe scar function with the left or the right resonances of theopen map differ. These eigenstates are supported by differenttrapped classical sets, so in principle, there is no reason forthem to coincide. Anyway, we think that the detailed expla-0110ppq 10q 10(b)(a)(c) (d)(f)(e)FIG. 1. a Scar function of the PO given by q , p= 0.5,0.5for N=225. The horizontal axis corresponds to the position q 0,1 and the vertical axis to the momentum p 0,1 coordi-nate. b Logarithmic version of a. Black lines correspond to thestable and unstable manifolds of the orbit. c Right eigenvector ofthe M̄a map, which has the maximum overlap with the scar functionshown in a. d Same as previous panel but for the left eigenstate.e Same as c in the M̄b map case. f Eigenfunction of the closedsystem having maximum overlap with the scar function of panela. See the text for more details.DIEGO WISNIACKI AND GABRIEL G. CARLO PHYSICAL REVIEW E 77, 045201R 2008RAPID COMMUNICATIONS045201-2nation of this difference is an interesting open problem.But then a natural question arises: how does the relation-ship between the shape and the size of the projection influ-ence the intensity of scarring? For instance, this is relevant ifwe want to obtain highly localized resonances with the mini-mum amount of losses. This happens in many applications,the cases of two-dimensional billiards that can be used asoptical microcavities for lasers or that can be attached toperfect leads, being some examples. To answer this we havefurther investigated the behavior of localization by fixing the value, and studying how the width of the opening influ-ences it for both Pa and Pb operators. The results are shownin Fig. 3, where we display the average of xmax taken fromN=350 to N=360, as a function of the width of the openingq. In all cases we take q0=0.225 for M̄a and q0=0.1625 forM̄b. The overlaps were calculated with the right dotted andleft dashed eigenstates. The lower curves correspond to M̄a,while the upper ones correspond to M̄b. We have found thatnot only the overlap in general increases with the size of theopening, but also that this effect is greater due to the sym-metrization.But this seemingly greater scarring effect in open systemsshould be interpreted in the proper context. In order to dothis we will analyze the weight that these long-lived reso-nances have in the whole spectrum, and relate it with typicaltime scales of the system. This is given by a connectionbetween the fractal Weyl law and the Ehrenfest time T0=lnO / with O the number of open channels, and =1.31 in our case, first obtained in 7. There, it was foundFIG. 4. Color online Fraction of eigenstates N /N whose de-cay rate  is smaller than  f =0.71, as a function of N. The linescorresponds to the theoretical expression O−1/Td see the text fordetails. The upper curve corresponds to M̄a with an opening de-fined by q0=0.125 and q=0.05, the middle one by q0=0.225 andq=0.25, and finally, the lower one corresponds to M̄b with q0=0.1625 and q=0.25.(b)(a)maxNνmaxNνFIG. 2. Color online Maximum overlap xmax of the scar func-tion with the eigenstates of the open cat map as a function of Nrunning average in a window N=10. In the insets we show theorder number max in ascending eigenvalue modulus of the rightresonance with maximum overlap, as a function of N for the leftresonance, results are similar. Panel a corresponds to the M̄amap, while the M̄b map results are shown in b. In all panels, thesolid black lines correspond to the maximum overlap for the closedmap, and the dotted and dashed lines correspond to the maximumoverlaps with the right and left resonances of the open maps,respectively.qpFIG. 3. Color online Maximum overlap xmax average fromN=350 to N=360 of the scar function with the right dotted andleft dashed eigenstates of the open cat map as a function of thesize of the opening q. The lower curves correspond to M̄a, whilethe upper ones correspond to M̄b. The solid horizontal line standsfor the value corresponding to the closed cat map. Left and rightinsets illustrate the projectors Pa and Pb.SCARRING IN OPEN QUANTUM SYSTEMS PHYSICAL REVIEW E 77, 045201R 2008RAPID COMMUNICATIONS045201-3that the fraction of resonances with decay rate  smaller thana fixed value  f 1 /T0 behaves like N /N=exp−T0 /Td=O−1/Td, where Td=N /O is the so-called “dwell time” Tdlarge. We have numerically confirmed the validity of thisprediction in our system by fitting the data with the expres-sion N /N=aN−b. We show three cases in Fig. 4, where wehave taken  f =0.71 in all of them. The upper curve corre-sponds to M̄a with an opening defined by q0=0.125 andq=0.05, showing a fitted bf =0.032 that agrees with thetheoretical bth=0.038. The middle curve corresponds to q0=0.225 and q=0.25 with bf =0.181, and the lower one cor-responds to M̄b with q0=0.1625 and the same q with bf=0.2, being bth=0.191 for both. In all cases this fraction goesto zero, leaving a small amount of classically meaningfuleigenstates. This directly implies a persistent localization ef-fect on the few remaining “fractal eigenfunctions.” However,these greater overlaps could correspond to a normalizationdifference with respect to the closed system. In fact, the ef-fective size of the Hilbert space of the open system issmaller. Then, if we can go further and claim that these re-sults mean a true enhancement of scarring, remains to bedetermined 19.In summary, we have found that there is a greater overlapof the scar functions with the resonances of an open systemcompared to the closed one. The fractal structure of theeigenstates has been widely studied, motivating their de-nomination as “quantum fractal eigenstates.” However, thissignificant alteration of the morphology of the eigenfunctionswith respect to the analog closed system does not destroy thelocalization around POs. We think that this is due to the factthat the pruning of orbits that escape through the openingsbefore the Ehrenfest time leaves parts of the stable and un-stable manifolds. These remaining parts are enough to sup-port the quantum structure associated to the scar function.However, the way they interfere in order to construct thesame object as the smooth manifolds of the closed systemremains unknown. Also, the scarring grows with the size ofthe opening ruling out any perturbative explanation for this.Moreover, this phenomenon is persistent, in the sense that itsurvives in the vanishing fraction of long-lived resonances asN grows. In future studies 19 we will focus on these openquestions.The authors gratefully acknowledge support fromCONICET PIP-6137, UBACyT X248, and ANPCyT. Wehave benefited from fruitful discussions with M. Saracenoand E. Vergini.1 W. H. Zurek and J. P. Paz, Phys. Rev. Lett. 72, 2508 1994.2 W. Fang, A. Yamilov, and H. Cao, Phys. Rev. A 72, 0238152005; J. U. Nöckel and D. A. Stone, Nature London 385,45 1997; T. Harayama, P. Davis, and K. S. Ikeda, Phys. Rev.Lett. 90, 063901 2003; J. Wiersig and M. Hentschel, Phys.Rev. A 73, 031802R 2006; Phys. Rev. Lett. 100, 0339012008.3 J. Wiersig, Phys. Rev. Lett. 97, 253901 2006.4 R. Akis, D. K. Ferry, and J. P. Bird, Phys. Rev. Lett. 79, 1231997.5 P. Gaspard, Chaos, Scattering and Statistical Mechanics Cam-bridge University Press, Cambridge, 1998; C. Jung and T. H.Seligman, Phys. Rep. 285, 77 1997.6 W. T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91,154101 2003.7 H. Schomerus and J. Tworzydlo, Phys. Rev. Lett. 93, 1541022004; S. Nonnenmacher et al., J. Phys. A 38, 10683 2005.8 D. L. Shepelyansky, Phys. Rev. E 77, 015202R 2008.9 G. Casati et al., Physica D 131, 311 1999.10 J. P. Keating, M. Novaes, S. D. Prado, and M. Sieber, Phys.Rev. Lett. 97, 150406 2006.11 S. Nonnenmacher and M. Rubin, Nonlinearity 20, 13872007.12 E. J. Heller, Phys. Rev. Lett. 53, 1515 1984.13 S. W. McDonald, LBL Report No. 14837, 1983 unpublished.14 L. Kaplan and E. J. Heller, Ann. Phys. N.Y. 264, 171 1998;E. B. Bogomolny, Physica D 31, 169 1988; M. V. Berry,Proc. R. Soc. London, Ser. A 423, 219 1989; J. P. Keatingand S. D. Prado, ibid. 457, 1855 2001; D. A. Wisniacki, E.Vergini, R. M. Benito, and F. Borondo, Phys. Rev. Lett. 94,054101 2005. D. A. Wisniacki, E. Vergini, R. M. Benito, andF. Borondo, ibid. 97, 094101 2006.15 G. G. de Polavieja, F. Borondo, and R. M. Benito, Phys. Rev.Lett. 73, 1613 1994.16 E. G. Vergini and G. G. Carlo, J. Phys. A 34, 4525 2001; E.G. Vergini and D. Schneider, ibid. 38, 587 2005; A. M. F.Rivas, ibid. 40, 11057 2007.17 F. Faure, S. Nonnenmacher, and S. De Bievre, Commun. Math.Phys. 239, 449 2003.18 J. H. Hannay and M. V. Berry, Physica D 1, 267 1980.19 D. A. Wisniacki and G. G. Carlo unpublished.DIEGO WISNIACKI AND GABRIEL G. CARLO PHYSICAL REVIEW E 77, 045201R 2008RAPID COMMUNICATIONS045201-4

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