on dynamical tunneling and classical resonanceshome.iitk.ac.in/~srihari/skpaps/ks_16.pdf · cal...

On dynamical tunneling and classical resonancesSrihari Keshavamurthya!

Max-Planck-Instiut für Physik komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany

sReceived 4 October 2004; accepted 3 February 2005; published online 22 March 2005d

This work establishes a firm relationship between classical nonlinear resonances and thephenomenon of dynamical tunneling. It is shown that the classical phase space with its hierarchy ofresonance islands completely characterizes dynamical tunneling and explicit forms of the dynamicalbarriers can be obtained only by identifying the key resonances. Relationship between the phasespace viewpoint and the quantum mechanical superexchange approach is discussed innear-integrable and mixed regular-chaotic situations. For near-integrable systems with sufficientanharmonicity the effect of multiple resonances, i.e., resonance-assisted tunneling, can beincorporated approximately. It is also argued that the presumed relation of avoided crossings tononlinear resonances does not have to be invoked in order to understand dynamical tunneling. Formolecules with low density of states the resonance-assisted mechanism is expected to bedominant. ©2005 American Institute of Physics. fDOI: 10.1063/1.1881152g

I. INTRODUCTION

Intramolecular vibrational energy redistributionsIVRd ina molecule, from an initially prepared nonstationary state, isat the heart of chemical reaction dynamics. The timescalesand mechanisms involved in this process of energy flow havebeen investigated in great detail by experiments and theory.1

Although considerable progress has been made over theyears, there are many aspects of this phenomenon, the under-standing of which still eludes us. Nevertheless, it is expectedthat there is a part of the IVR mechanism which can beusefully understood based on classical dynamics alone andthere is a part of IVR which is intrinsically quantum me-chanical. Both classical and quantum mechanisms coexist ina given molecule and give rise to complicated spectral pat-terns and splittings.

It is now well established that a specific overtone exci-tation in a molecule will undergo IVR if the initial nonsta-tionary state, zeroth-order bright statesZOBSd, is coupled toother “dark” zeroth-order states via strong anharmonic reso-nances. In such cases classical dynamics, at various levels ofdetail and sophistication, can provide useful insights into thetimescales and mechanisms of IVR.2 On the other hand, ifthe ZOBS is not close to any resonance then classically oneexpects no IVR and hence no fractionation of spectral lines.However, it is still possible for IVR to occur via quantumroutes. In other words, the initial state will mix with otherstates giving rise to spectral splittings and, possibly, compli-cated eigenstates. Since the mixing is classically “forbidden”it would be appropriate to associate some kind of tunnelingwith such quantum routes to energy flow. Indeed such a sug-gestion was made over two decades ago3 and the term “dy-namical tunneling” was coined. The notion of tunneling ismeaningless without one form of a barrier or other and theterm “dynamical” is prefixed to distinguish from the usual

coordinate space tunneling through static potential barriers.The barriers in dynamical tunneling, in general, are moresubtle to identify and they exist in the phase space.

Dynamical tunneling can have important consequencesfor the interpretation of molecular spectra since the finger-prints of IVR are spectrally encoded in the form of intensi-ties and splittings. Traditionally, dynamical tunneling in themolecular context has been associated with the occurence oflocal mode4 stretches in symmetric molecules. However, it isimportant to emphasize that the concept of dynamical tun-neling is more general—any flow of quantum probabilitybetween regions which are classically disconnected is dy-namical tunneling. The identification of classically discon-nected regions requires the knowledge of the phase spacetopology and hence it is not very surprising that dynamicaltunneling is inevitably linked to the underlying classical dy-namics. In the context of local mode doublets in symmetricsystems, important work by Jaffé and Brumer5 and Kellman6

provided classical phase space perspectives on the normal tolocal transition. One of the reasons for the interest in thenear-degenerate local doublets has to do with their long life-times. Thus provided one can experimentally prepare a highovertone, of say the OH-stretch in H2O, then the extremelysmall splitting provides a large window of time to performmode selective chemistry. The main problem with such anidea is that at such high vibrational excitations other Fermiresonances and rovibrational interactions could destroy thedegeneracies.7 We hope to shed some light on these issues inthis work wherein the system of interest does exhibit closedegeneracies despite multiple resonances and a classicalphase space which is mixed regular-chaotic. Interestingly,similar issues have been addressed in the context of the ex-istence of discrete breathers in a network of nonlinearoscillators.8

One of the earliest examples in symmetric systems arethe local mode doublets observed in the water moleculewhich were explained by Lawton and Child as due to dy-

adPermanent address: Department of Chemistry, Indian Institute of Technol-ogy, Kanpur, U.P. 208016, India.

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http://dx.doi.org/10.1063/1.1881152

namical tunneling.9 The experiments10 of Kerstel et al. onunusually slowshundreds of picosecondsd intramolecular vi-brational relaxation of CH-stretch excitation insCH3d3CCCHmolecule has also been ascribed to dynamical tunneling byStuchebrukhov and Marcus.11 In this case, the extremelyslow IVR out of the CH-stretch is an instance of dynamicaltunneling in nonsymmetric systems. In a later experimentMcIlroy and Nesbitt noted multiquantum state mixing due tovery small couplingss0.1 cm−1 or lessd in the CH-stretchexcitation in propyne.12 Recent experiments have shownsimilar effects in the CH-stretch excitations of triazine13 and1-butyne14 and there are reasons to expect dynamical tunnel-ing to manifest in general rovibrational spectra.15–19

Much of the theoretical understanding of dynamical tun-neling has come about from the analysis of bent triatomicABA molecules. The focus primarily has been on vibrationsand this work is no exception in this regard. However, wenote the important work by Lehmann18 wherein the non-trivial effects due to coupling between local modes and ro-tations are studied in detail. Davis and Heller3 emphasized aphase space picture and implicated classical resonances inthe phase space as the agents of dynamical tunneling. In thisapproach classical trajectories trapped in one region of thephase space were imagined to be separated by dynamicalbarriers, due to the resonances, from symmetrically equiva-lent regions of the phase space. The degeneracy is then bro-ken by dynamical tunneling. A clear demonstration of therole of isolated resonances was provided by Ozorio deAlmeida as well.20 A slightly different picture was providedby Sibert, Reinhardt, and Hynes21 in their work on energyflow and local mode splittings in the water molecule. Thesetting was again in terms of classical nonlinear resonancesbut the dynamical barrier was identified in angle space in-stead of in action space. Later Hutchinson, Sibert andHynes22 provided an explanation for the quantum energyflow in terms of high-order perturbation theory.23 Stuche-brukhov and Marcus24 reanalyzed the ABA system in termsof chains of off-resonance virtual statess“vibrational super-exchange”d connecting any two degenerate local modestates. An important result was the equivalence between thevibrational superexchange approach and the usual WKB ex-pression for splittings in a doublewell system. This equiva-lence between perturbative and the WKB approaches wasshown for a classically integrable case. Demonstration ofsuch an equivalence in more general situations is an openproblem. Despite seemingly different approaches a crucialingredient to any description of dynamical tunneling in-volves the various anharmonic resonances. Depending on thelevel of excitation, the underlying phase space can exhibitseveral sneard equivalent regions separated by nonlinearresonancessnear-integrable phase spaced or chaossmixedphase spaced. For large molecules with sufficient density ofstates, in the near-integrable regime, Heller has conjectured25

that a nominal 10−1–10−2 cm−1 broadening of spectroscopi-cally prepared zeroth-order states is due to dynamical tunnel-ing between remote regions of phase space facilitated bydistant resonances.

The aforementioned conjecture is based on the notionthat phase space is the correct setting for an understanding of

dynamical tunneling. It is thus natural to expect the splittingsto be sensitive to the structure of the phase space. It alsofollows that given the phase space structure one ought to beable to compute the associated splittings. General forms forsplittings cannot be written down easily since explicit formsof the dynamical barriers in the phase space, separatingquantum states localized in distant regions of the phasespace, can only be provided upon identification of the keynonlinear resonances. This is also tantamount to identifyingthe mechanism of dynamical tunneling and hence IVR. Twofactors make this a difficult problem. First, our understand-ing of global structure of multidimensional classical phasespace is still in its infancy. Thus without a global view of thephase space at energies corresponding to the doublets it isdifficult to identify the main nonlinear resonances and per-haps the presence of appreciable stochastic layers. Second,and related to the first factor, important work over the lastdecade has established that dynamical tunneling is sensitivenot only to the nonlinear resonances26,27 but also to the clas-sical stochasticity.28 In the near-integrable regimes dynami-cal tunneling can be enhanced by many orders of magnitudedue to the various nonlinear resonances. This is calledresonance-assisted tunneling and it has been suggested29 thatthe nonlinear resonances play an important role in mixedphase space scenario as well. In cases where an appreciablechaotic region separates the two symmetry-related regularzones one has the phenomenon of chaos-assisted tunneling.The hallmark of chaos-assisted tunneling is the erratic fluc-tuations of the splittings28 due to “chaotic” states havingnonzero overlaps with the regular doublets. In particular, thesplittings show algebraic dependence on" as opposed to theintegrable exps−1/"d scaling. Experiments30 on a wide vari-ety of systems have highlighted the role of the phase spacestructures in dynamical tunneling.

Is the evidence for the sensitivity of dynamical tunnelingto the phase space structures already present in the molecularspectra? The answer to this question, apart from a fundamen-tal viewpoint, is also potentially relevant to mode-specificchemistry and control. In this context, it is significant to notethat most of the earlier works in the molecular context havefocused on dynamical tunneling where only one resonancewas involved with no chaos. A notable exception is the workof Davis and Heller3 wherein a hint to the role played byclassical chaos was provided. The sensitivity of dynamicaltunneling to the underlying phase space structure can haveimportant ramifications in the molecular context. Phasespace of molecular systems at higher energies, correspondingto high overtone transitions, are typically a mixture of regu-lar and chaotic regions. The presence of stochasticity canlead to the dynamical tunneling being enhancedsor sup-pressedd by several orders of magnitude and hence highlymixed states and complicated spectral patterns. Even in thenear-integrable regimes tiny induced resonances can arise asa result of the primary resonances which can enhance orsuppress dynamical tunneling between symmetry-relatedmodes. Alternatively, molecule-field interactions can31 leadto the creationsdestructiond of nonlinear resonances whichcan lead to significant enhancement or suppression of energyflow.

114109-2 Srihari Keshavamurthy J. Chem. Phys. 122, 114109 ~2005!


The purpose of this work is to establish the role of clas-sical resonances in the near-integrable phase space for dy-namical tunneling and the nontrivial effect of higher-order-induced resonances and perhaps chaos. We focus on thesymmetric case in this work but the role played by the non-linear resonances is expected to hold in the nonsymmetricsituations as well. In the near-integrable regime it is possiblefor more than one nonlinear resonance zone to manifest inthe phase space. In such instances, the single rotorpicture20,21 is not sufficient to account for dynamical tunnel-ing. Yet it is shown that hopping across the various reso-nance islands is an approximate picture that yields qualita-tive sand perhaps quantitatived insights into the splittingpatterns. We start by motivating the model Hamiltonian inSec. II. In Sec. III the concept of dynamical tunneling andquantum viewpoints are illustrated. The phase space view-point and relations to the quantum viewpoint are discussed inSec. IV by focusing on specific cases. In Sec. IV B two is-sues regarding avoided crossings and influence of very smallinduced resonances are addressed briefly. Complications dueto multiple resonances in the near-integrable limit and theresulting resonance-assisted tunneling are illustrated in Sec.IV C. The possibility of chaos-assisted tunneling occuring inthe system is explored in Sec. IV D. A brief summary isprovided in Sec. V.

II. MODEL HAMILTONIAN

As discussed in the introduction attempts to understanddynamical tunneling, from a classical phase space viewpoint,in systems with three or more degrees of freedom is ambi-tious without studying the detailed correspondence in lowerdimensions with multiple resonances. In this regard the ef-fective spectroscopic Hamiltonian for water due to Baggott32

provides a good model system. The Hamiltonian is givenby33

HC = H0 + g8V1:1s12d + gV2:2

s12d +b

2Î2sV2:1

s1bd + V2:1s2bdd s1d

with g8=g+l8sn1+n2+1d+l9nb and

H0 = vssn1 + n2d + vbnb + xssn12 + n2

2d

+ xbnb2 + xsbnbsn1 + n2d + xssn1n2 s2d

describing the anharmonic local stretchess1, 2d and bendsbd.The various parameter values arevs=3885.57,vb=1651.72,xs=−81.99,xb=−18.91,xsb=−19.12, andxss=−12.17 cm−1.The j th mode occupancy isnj =aj

†aj with saj†,ajd being the

harmonic oscillator creation and destruction operators. Theperturbations, anharmonic resonances,

Vp:qsi j d = sai

†dqsajdp + saj†dpsaidq s3d

connect zeroth-order statesunl, un8l with uni8−niu=q andunj8−nju=p. The strengths of the resonances areg=−56.48,l8=3.02,l9=−0.18,g=−0.91, andb=26.57 cm−1. In orderto perform detailed studies on the above Hamiltonian in thiswork we also analyze subsystems obtained by retaining spe-cific resonances. These subsystems, denoted byA andB, aredescribed by the HamiltoniansHA=H0+bsV2:1

s1bd+V2:1s2bdd and

HB=HA+gV2:2s12d, respectively. The full system with all the

resonances will be denoted byC. The reason for this specificchoice of subsystems has to do with the fact that classicallyHA andHB exhibit near-integrable dynamics, whereasHC has

mixed regular-chaotic dynamics.H is effectively two dimen-sional due to the existence of the conserved polyadP=sn1

+n2d+nb/2, i.e., fH , Pg=0. Spectroscopic Hamiltonians canbe obtained by a fit to the high resolution spectra or from aperturbative analysis of an existing high qualityab initiopotential energy surface.34 In any case, such Hamiltonians,and variants thereof, have proved quite useful in studying thehighly excited vibrations of many small molecules.35 Cou-plings to rotations and large amplitude modes are certainlyimportant18 but a detailed phase space analysis of such gen-eral rovibrational Hamiltonians is beyond the scope of thiswork.

The classical limit33 of the above Hamiltonian can beobtained by using the Heisenberg correspondence,aj ↔ÎI j expsiu jd, and results in a nonlinear multiresonantHamiltonian

HsI ,ud = H0sI d + gc8ÎI1I2cossu1 − u2d + gcI1I2 coss2u1 − 2u2d

+ bcIbfÎI1cossu1 − 2ubd + ÎI2cossu2 − 2ubdg s4d

with sI j ,u jd corresponding to the action-angle variables asso-ciated with the modej . Note that the actionsI and thezeroth-order quantum numbersn are related by the corre-spondenceI → sn+1/2d" for simple vibrations. The variousresonant couplings are related to the quantum couplingstrengths bygc8=2g8, gc=2g, andbc=2b. We note that an-other reason for the choice of this system has to do with thefact that fairly detailed classical-quantum correspondencestudies of the highly excited eigenstates have already beenperformed.33,36

In this study the focus is on a specific set of molecularparameterssprovided aboved which are representative of theH2O molecule.32,33 However, at the outset, we emphasizethat this choice is by no means special for the analysis andconclusions of this work. The goal of this work is not toprovide accurate quantitative estimates for the splittings—something that can be trivially calculated quantum mechani-cally for the model Hamiltonian. The emphasis here is onestablishing and understanding, qualitatively, the importantrole of nonlinear resonances in dynamical tunneling and forthis purpose the above model Hamiltonian provides a goodparadigm. In particular we choose the states corresponding tothe polyad P=6 which lie in an energy range off16600,21300g cm−1 above the ground state. There are twomain reasons for such a choice. First, the motivation is tounderstand aspects of dynamical tunneling for excited vibra-tional states in the presence of both nonlinear resonances andchaos.33 Second, the exhaustive compilation37 of the experi-mental energy levels of H2O by Tennysonet al. does showthree doublets in this energy region. In local mode notationthese doublets ares60d±0, s50d±2, ands51d±0. A few otherlocal modes are also reported with only one of the paritystates such ass40d−4. Although the effective Hamiltonianused here is perhaps not the best it still provides some cor-respondence to the patterns observed in the experimental re-

114109-3 On dynamical tunneling and classical resonances J. Chem. Phys. 122, 114109 ~2005!


sults. Moreover at such high levels of excitation, certainlyfor the bending mode, there can be significant interpolyadmixings38 which require analyzing a system with three de-grees of freedom. Thus we use the effective Hamiltonian as amodel which would guide further, more rigorous, studies infull dimensionality. To give an idea about the validity of theeffective Hamiltonian in the chosen energy range, in Fig. 1we show the energy levels as compared to the experimentallevels and the agreement is satisfactory.

III. DYNAMICAL TUNNELING: ILLUSTRATIONAND QUANTUM VIEWPOINT

In the rest of the paper, the zeroth-order states will bedenoted byun1n2l since the bend quantum numbernb is fixedby the polyadP. In order to illustrate the concept of dynami-cal tunneling, we consider zeroth-order degenerate statesu20land u02l with only the 2:1 resonances present, i.e.,g8=g

=0 in Eq. s1d. As k20uV2:1s1bdu02l=0=k20uV2:1

s2bdu02l, the stateu20l has no direct coupling to the symmetric counterpartu02l.Nonetheless there is an indirect coupling via the 2:1 reso-nances. However, as shown in Fig. 2sad, the zeroth-orderstate is quite far from the region of state space which canpotentially come under the influence of the 2:1 anharmonicresonances. A classical trajectory initiated with initial condi-tions corresponding to the stateu20l will continue to stay inthat region of the phase space forever without reaching thesymmetric region corresponding to the stateu02l. One suchclassical trajectory, for about 0.25 ns, in phase space isshown in Fig. 2sbd sinsetd. This classical trapping occurs de-spite no apparent energetic barriers since in Fig. 2sad weshow the classical actions satisfyingH0<E20

0 which connectsthe two states. However, contrary to the classical observa-tion, in Fig. 2sbd the quantum survival probability of the stateu20l shows transfer of population tou02l with a period ofabout 0.20 ns. In a two-state approximation this correspondsto a splitting of the degenerate levelsD2<0.17 cm−1. Thisis the phenomenon of dynamical tunneling. Previousdiscussions21,22,24 of dynamical tunneling have used the ef-fective Hamiltonian with only the primary 1:1 present. Here

we delibrately show this effect with the 2:1s only to empha-size the role of induced resonances.

A possible explanation of the tunneling arises from thevibrational superexchange22,24 perspective wherein thezeroth-order states coupled locally by the 2:1 perturbationsto u20l and u02l are considered. One then constructs all pos-sible perturbative chains connecting the statesu20l and u02l.An example of such a chain, shown in Fig. 2sad, is u20l→ u10l→ u11l→ u01l→ u02l. The contribution to the splittingis

b4k20uVs1bdu10lk10uVs2bdu11lk11uVs1bdu01lk01uVs2bdu02lsDE10

0 d2sDE110 d

s5d

with DEn1n2

0 ;E200 −En1n2

0 . In principle, there are an infinitenumber of chains that connect the two degenerate states. Inpractice, due to the energy denominators it is sufficient toconsider the minimal length chains.39 By minimal lengthchains one means inclusion of all possible perturbative termsat the lowest order of perturbation necessary. For instance, inthe example above, it is necessary to go to at least the fourthorder in perturbation theory, thus scaling likeb4, to get afinite value for the splitting. In our case, there are six mini-mal chains and summing the contributions, one obtains asplitting of about 0.18 cm−1 which compares well with theexact splitting.

More generally, splitting for any set of degenerate statesur0l and u0rl at minimal order can be calculated as39

Dr = oabc

bagbgcom

DrsGabcsmd d s6d

with m being an index of all possible chainsGabc for a par-ticular choice ofa, b, c satisfying the constrainta+2b+4c=2r. Indeed, such a calculation can be done on our modelsystem and the resulting perturbative estimates are comparedto the exact splittings in Fig. 3. For further discussions, thesuperexchange calculations have been compared to the exactresults for casesA, B, and C. At the outset note that theperturbative calculations essentially reproduce, to quantita-

FIG. 1. A portion of the energy levels of the model effective Hamiltoniancorresponding to polyadP=6. The full system is denoted by C and some ofthe experimental energy levels are shown as open circles. B and A show theenergy levels for the subsystems with specific resonances removed. Some ofthe closely spaced doublets are indicated by *. In C the box around the topthree levels indicates levels with 4, 5, and 6 quanta of excitation in thestretch mode.

FIG. 2. Classical and quantum viewpoints on dynamical tunneling for sub-systemA. The states areu02l andu20l. sad The state space for polyadP=6 isshown with the regions strongly influenced by the 2:1 anharmonic reso-nances. An example of a minimalssolidd superexchange chain connectingthe states is shown. The energy contour satisfyingH0<E20

0 is shown con-necting the two states.sbd Quantum survival probabilties foru20l and u02lssquaresd are shown for 0.25 ns. Note the almost coherent transfer of popu-lation with a period of 0.20 ns. The inset shows a classical trajectory initi-ated nearu20l for a time period of about 0.25 ns. The time variations of theactionssI1−1/2d and sI2−1/2d are shown as filled and unfilled circles, re-spectively. The actions stay trapped for the entire period.



tive accuracy, almost all of the splittings. This degree ofaccuracy, given high orders of perturbation and number ofchains, is pleasantly surprising. For instance the inset to Fig.3 shows that 130 contributions of varying signs, some aslarge as 10−4, conspire to yield the correct splitting of about10−7 for D6. Two important observations can be made at thisjuncture. First, the significant role of the tiny 2:2 resonanceis clear in that addition of this 2:2 to the 2:1s enhances thesplittings by many orders of magnitude—something which isnot readily apparent from Fig. 1. For even values ofr, espe-cially r =2, this is expected since the 2:2 provides a dominantcoupling. In fact for r =2 one expectsD2<4g=3.6 cm−1

which compares very well with the exact value of 3.8 cm−1.However, it is significant to note that such a calculation forr =6 is too small by a factor of five and that the oddr stateswould not split at all in the presence of a 2:2 coupling alone.Second, in all three casesA, B, andC the splittings more orless monotonically decrease with increasing stretch excita-tions. The one exception is in the full systemscaseCd for thestate withr =4. Later it is shown that this is an instance ofsuppression of tunneling due to an avoided crossing betweenthe doublet and a third “intruder” state. Incidentally, in theexperimental data37 pertaining tor =4 among the doublets,denoted in the local mode basis ass40d±4, only the states40d−4 is reported.40 This, given our Hamiltonian model,could be a pure accident!

It is important to note that the superexchange approachinvokes the 2:1 resonant terms without any reference to theclassical phase space. This surprisingly good accuracy seemsto hold whenever the phase space is integrable or near-integrable and is perhaps related to the equivalence of thesuperexchange to the WKB approach.24 If this observation istrue in multidimensional systems, then a phase space per-spective on dynamical tunneling should be able to providesome insights. Apart from providing a phase space analog tothe superexchange approach, it would also be possible to

study the extent of sensitivity of dynamical tunneling to thevarious classical structures. The rest of the paper is dedicatedto uncovering precisely such a phase space picture.

IV. DYNAMICAL TUNNELING: PHASE SPACEVIEWPOINT

A. Induced resonances and dynamical barriers

In what follows, the classical Poincaré surface of sec-tions are plotted infsI1− I2d /2 ,su1−u2dg;sK1/2 ,2f1d atconstant values of the sectioning anglef=u1+u2−4ub withf.0. In this representation the normal-mode resonant re-gions appear aroundI1, I2. Local-mode regions correspondto above and below the normal-mode regions and large 2:1bend/local-stretch resonant islands appear in the local-moderegions.

We begin by asking the question as to what possiblestructure in the classical phase space is mediating the dy-namical tunneling in Fig. 2sbd. The answer, based onearlier3,21–24 works, should involve a nonlinear resonancezone in the phase space. Indeed, the Poincaré surface of sec-tion atE=E20

0 <18641 cm−1 shown in Fig. 4 indicates a reso-nance zone juxtaposed between the zeroth-order states asseen in the phase space. The surface of section is consistentwith the state space view in Fig. 2sad in that the primary 2:1resonances does not appear in the phase space. The 2:1 reso-nance zones, if present, would show up in Fig. 4 aroundK1/2< ±2.6. Hence, a direct involvement of the 2:1 reso-nances is ruled out but surely the existence of the 2:1s mustbe giving rise to a dynamical barrier, in the form of theobserved resonance zone, mediating the tunneling. In orderto confirm that this nonlinear resonance is mediating the dy-namical tunneling, it is necessary to extract an explicit ex-pression for the dynamical barrier and obtain the splitting.For this purpose we apply methods of nonlineardynamics34,41 on the classical HamiltonianfEq. s4dg with g=0=g andbÞ0. The analysis below is not new and can befound in many of the earlier works including Ozorio deAlmeida20 and Sibert.23 For the sake of completeness onlyessential results are explicitly given and technical aspects ofclassical perturbation theory will not be discussed.

FIG. 3. Comparison of the exact quantumssymbolsd dynamical tunnelingsplittings D in cm−1 to the superexchange resultsslinesd. Shown are thesplitting between the statesun1=r ,n2=0l and u0rl. The r =1, 2 cases are notshown for the full system since on inclusion of the large 1:1 they are notlocal modes but resonant normal modes with large splittings. The experi-mental values for the splitting forr =5, 6 are shown as filled circles forcomparison. The arrow indicates the substantial deviation of the superex-change result from the exact value forr =4. The inset shows the contributionfrom 130 terms for the superexchange calculation ofD6 in case A.

FIG. 4. Classical Poincaré surface of section forHA at E<E200 showing the

induced 1:1 resonance island due to the two stretch-bend 2:1 resonances.



To start with a canonical transformation is performedsI ,ud→ sJ ,cd using the generating function

F = su1 − 2ubdJ1 + su2 − 2ubdJ2 + ubN. s7d

The choice of the generating function impliesck=uk−2ub,Jk= Ik for k=1, 2 andc3=ub, N=2sI1+ I2d+ Ib. The resultingHamiltonian

HsJ,c;Nd = H0sJ;Nd + ebcsN − 2J1 − 2J2d

3sÎJ1cosc1 + ÎJ2cosc2d s8d

is ignorable in the anglec3 implying the conserved quantityN=P+5/2 which is the classical analog of the polyad num-ber P. The above two-dimensional Hamiltonian contains the2:1 resonances and hence is nonintegrable. However, thephase space from Fig. 4 suggests that atE<E2,0,12

0 the 2:1sdo not have a direct role to play. Another way of stating thisis that the nonlinear frequenciesVk;]H0sI d /]Ik are faraway from the conditionV1,2−2Vb=0 for the states in con-sideration. For the system parameters in this study theu2,0,8l state is detuned by almost 800 cm−1. Consequently, aformal parametere has been introduced with the aim of per-turbatively removing the 2:1 resonances, characterized byc1,2 in Eq. s8d, to Osed. This can be done by invoking thegenerating function

G = J1c1 + J2c2 + esg1 sinc1 + g2 sinc2d s9d

where the functionsg1,2=g1,2sJ1, J2d are determined by thecondition of the removal of the primary 2:1s toOsed. The

angles conjugate toJ are denoted byc. The new variables

sJ ,cd are related to the variablessJ ,cd through the relations

J1,2=]G/]c1,2 and c1,2=]G/]J1,2. Using the identities

cossezsinswdd = J0sezd + 2olù1

J2lsezdcoss2lwd, s10d

sinsezsinswdd = 2olù0

J2l+1sezdsinfs2l + 1dwg s11d

with Jk being the Bessel functions, it can be shown that

HsJ,c;Nd = H0sJ;Nd + ok

ekHksJ,c;Nd s12d

where

H0 = VssJ1 + J2d + asssJ12 + J2

2d + a12J1J2 s13d

with Vs=vs−2vb+sxsb−4xbdN, ass=xs+4xb−2xsb and a12

=8xb−4xsb+xss. At Osed,

H1sJ,c;Nd = ok=1,2

fbcJbÎJk + gksJdVkgcosck s14d

where we have denotedJb;N−2sJ1+ J2d andVk;]H0/]Jk.

The actionJb is essentially the number of bend quanta in thesystem. The choice of the functions

gksJd = − bcJb

ÎJk

Vk

s15d

eliminates the 2:1s toOsed and hence theOse2d term in thetransformed Hamiltonian Eq.s12d is determined to be

H2sJ,c;Nd = ok=1,2

lks1 + cos 2ckd + l±cossc1 ± c2d.

s16d

The term cossc1−c2d is easily identified with a 1:1 reso-nance between the modes 1 and 2 and the strength of thisinduced 1:1 resonance is obtained as

l−sJ;Nd =1

2bc

2F2sV1 + V2d + a12Jb

V1V2

GJbÎJ1J2. s17d

Expressions for the strengthsl1,2 andl+ are not needed forour analysis hereafter and hence are not explicitly given inthis work.

At this stage the transformed Hamiltonian toOse2d in

Eq. s16d still depends on both the anglesc1 andc2 and henceis nonintegrable. In order to isolate the induced 1:1 reso-nance we perform a canonical transformation to the variables

sK ,fd using the generating functionG=sc1−c2dK1/2

+sc1+c2dK2/2 and average the resulting Hamiltonian overthe fast anglef2. The resonance centerK1

r =0 approximationis invoked resulting in a pendulum Hamiltonian describingthe induced 1:1 resonance island structure seen in the surfaceof section shown in Fig. 4. Within the averaged approxima-

tion the actionK2= J1+ J2 is a constant of the motion and canbe identified as the 1:1 polyad associated with the secondaryresonance. The resulting integrable Hamiltonian is given by

HsK1,f1;K2,Nd =1

2M11K1

2 + 2gindsK2,Ndcos 2f1 s18d

where

M11 = 2sa12 − 2assd−1, s19ad

gindsK2,Nd =bc

2

2fsK2,NdsN − 2K2dK2, s19bd

with

fsK2,Nd =4sVs + assK2d + a12N

f2sVs + assK2d + a12K2g2 . s19cd

Note, and this is important for the discussion in Sec. IV B,the averaging proceduredoes notremove the higher harmon-ics of the induced 1:1. Thus by focusing on the induced 1:1alone in Eq.s18d we have neglected all induced resonancesof the formq:q. In terms of the zeroth-order quantum num-bers K1=n1−n2 and K2=n1+n2+1;m+1. Note that Eq.s19bd, more generally Eq.s17d, is an analytic expression forthe dynamical barrier separating any two localized states interms of the zeroth-order actionssI1,I2,Ibd.

One can now use Eq.s18d to calculate the dynamicaltunneling splitting of the degenerate modesun1=r ,n2=0,nb

=2sP−rdl and un1=0,n2=r ,nb=2sP−rdl via24



Drsc

2= gind p

k=−sr−2d

sr−2dgind

ER0srd − ER

0skd= gind

sM11ginddr−1

2sr−1dfsr − 1d!g2

s20d

where ER0skd=k2/2M11 is the zeroth-order energy. For our

example withr =2, m=2 using the parameters of the Hamil-tonian we find M11<1.32310−2 and gind<3.93 cm−1.Within the pendulum approximation the half-width of theresonance zone can be estimated as 2Î2M11gind<0.64. Thisestimate, realizing thatK1<sI1− I2d, is in good agreementwith the surface of section in Fig. 4. The resulting splittingD2

sc<0.20 cm−1 agrees well with the exact splitting. Thisproves that the induced 1:1 resonance arising from the inter-action of the two primary 2:1 resonances is mediating thedynamical tunneling between the degenerate states. More-over, from a superexchange perspective it is illuminating tonote that the splitting can be calculated trivially by recogniz-ing the secondary phase space structuresa viewpoint empha-sized in Ref. 27 as welld. This observation emphasizes thesuperior nature of a phase space viewpoint on dynamicaltunneling. The analysis above also sheds light on the relationbetween the quantum superexchange viewpoint and WKBapproximation. Note thatgind,b2 and thus the phase spaceviewpoint, withD2,gind

2 is equivalent to the superexchangeapproach withD2,b4. In this near-integrable situation thiscorrespondence generalizes the observation by Stuche-brukhov and Marcus about the equivalence of superexchangeand WKB methods for integrable systems.24

B. Higher harmonics and level crossings

In this section we address two issues relevant in the con-text of dynamical tunneling. The first issue has to do with therole and relation of eigenvalue avoided crossings25,42 to non-linear resonances and hence the phenomenon of dynamicaltunneling. Since we have just argued for the involvement ofthe 2:1 resonances in mediating the dynamical tunneling be-tween degenerate states, it is interesting to ask if the statesare also involved in an avoided crossing as the 2:1 strengthbis varied. The answer to this question is negative and Fig. 5provides evidence that the statesu20l± do not undergo anyidentifiable avoided or exact crossings with varyingb. Infact the splittingssFig. 5 insetd predicted from Eq.s20d agreefairly well over the entire range ofb. It is also worth notingthat for the systemHA the splittingsD1,2,3 are described quitewell by a single induced 1:1 resonance. Therefore, in gen-eral, it is not necessary that dynamical tunneling betweentwo zeroth-order statesun1n2l and un1+p,n2−ql be mediatedby a p:q nonlinear resonance alone.

A second issue has to do with the role of higher harmon-ics or Fourier components in dynamical tunneling.25 Thedoublet splittingD2 for HA was seen to arise from an induced1:1 resonance. The perturbative analysis, on the other hand,even in the averaged approximation contains all the higherharmonics of the induced 1:1. These Fourier components ap-pear at higher orders ine and have very small, but finite,contribution toD2. If for some reason the leading order in-duced 1:1 were to vanish, then the higher Fourier compo-

nents become important. For the present example the 2:2component would be especially significant. In order to dem-onstrate this effect consider the Hamiltonian

H = H0 − uguV1:1s12d + bsV2:1

s1bd + V2:1s2bdd. s21d

A 1:1 resonance has been added toHA with a strength oppo-site in sign to the induced 1:1 resonance. The strengthg isvaried for fixedb and there is a specific value ofg=g0 forwhich the primary and the induced 1:1 resonance cancel eachother out. The value ofg0 can be estimated by recognizingthe integrable pendulum approximation to the above Hamil-tonian as

H <1

2M11K1

2 + s− uguK2 + 2ginddcos 2f1. s22d

Thus aroundugu0<2gind/K2 the above leading order approxi-mation would predict a shutdown of dynamical tunneling.From the estimates provided earlier it is easy to obtain thevalue ugu0<2.62 cm−1 for D2.

In Fig. 6sad the variation of the exact splittingD2 isshown as a function ofg. The results clearly show the de-struction of tunneling nearg0 but a more significant obser-vation is that atg0, labeled by regionM in Fig. 6sad, theactual value of the splitting is far from zero. This is in con-trast to the prediction of shutdown of dynamical tunnelingbased on the integrable pendulum approximation above. Thecorresponding phase space in regionM in Fig. 6sad indeedshows that the 1:1 resonance zone has almost vanished. Sowhat is mediating the dynamical tunneling in regionM? Ar-guments based on some residual 1:1 coupling are quicklydispelled by noting that the width of the resonance zone inregionM in Fig. 6sad is nowhere close to the required value.Chances of a strong avoided crossing leading to the en-hanced splitting are ruled out by inspecting Fig. 6sbd. Per-haps there is a broad avoided crossing, as suggested by theexact crossing of the doubletsu20l± in Fig. 6sbd sinsetd, lead-ing to the two dips inD2 fcf. Fig. 6sadg, with the upper statewhich happens to be a normal mode state. It is relevant to

FIG. 5. Variation of theu20l± doublet energies with the 2:1 resonancestrengthb. Results correspond toHA and no exact or avoided crossings areobserved. The inset compares the splittingsD2 calculated semiclassicallyfopen circles, Eq.s20dg to the exact quantumssolid lined over the same rangeof b variation. Note thatD2 increases by a factor of 108 which is capturedwell by the induced 1:1 resonance. All quantities are in cm−1.



note that the energy separation between the doublet and nor-mal mode state is almost equal to the mean level spacing.Moreover a superexchange calculation ofD2 at M includingthe required minimal chainsfcf. Eq. s6dg G400, G020, andG210,yields a value 1.70310−3 cm−1 which is in excellent agree-ment with the exact value of 1.65310−3 cm−1. All this sug-gests that in the regionM a tiny, but leading order, 2:2 withstrengthg=D2/4<4.1310−4 cm−1 is mediating the dynami-cal tunneling. A few important observations provide furthersupport and we briefly outline them. First, the splitting inregionM are very sensitive to adding a perturbation ±gV2:2

s12d

to the Hamiltonian in Eq.s21d whereas the regionsL andRare far less sensitive, as shown in Fig. 6sad. Second, theclassical perturbation analysis carried out to higher ordersreveals the emergence of an induced 2:2, scaling asb2g atOse3d, with strength comparable tog. Indeed, the exactD2 inregion M for different values ofb scales linearly withb2g.The superexchange calculation also shows that the contribu-tion from the minimal familyG210, i.e., b2g nearly, but notexactly, balances the contribution fromb4, andg2 families.The analysis here is an example of what Pearman andGruebele43 call “phase cancellation” effect in perturbativechains. Evidently, the analogy of this effect in phase spacehas to do with the destruction of nonlinear resonances. Al-though only ther =2 case has been shown here, similar ef-fects arise for the higher excitations and can be more com-plicated to interpret.

C. Multiple resonances: Resonance-assistedtunneling

One of the key lessons learnt from the preceeding dis-cussion is that nonlinear resonances have to be identified inorder to obtain a clear phase space picture of dynamical tun-neling. Given the near-integrable phase space ofHA it istempting to apply Eq.s20d to calculate the splitting of theother doublets corresponding to increasing stretch excita-tions. In particular, the monotonic decrease with increasingstretch excitations, seen in Fig. 3, seems to fit well with thenotion of a single resonancesthe induced 1:1 hered mediatingthe tunneling. The naivety of such a viewpoint is illustratedin Fig. 7sad. Clearly for r =1, 2, 3 the expectation holds butdeviations arise already atr =4 and the splittings are manyorders of magnitude smaller than the quantum results forr=5, 6. The reason for this large error is immediately clearfrom the surface of sections shown in Figs. 7sbd–7sdd. Theprimary 2:1 resonance has appeared in the phase space andcoexists with the induced 1:1. The states corresponding tor =4, 5 are “resonant” local modes and a Husimi representa-tion of the eigenstate in the classical phase spacefcf. Fig.10sadg confirms their nature. Ther =5 doublets are in factlocalized in the large 2:1 resonance islands. This can also beanticipated from the state space location of the resonancezones shown in Fig. 2sad. Thus a single rotor integrable ap-proximation is insufficient to describe the dynamical tunnel-ing for such cases. Notice the rich structure of the phasespaces in Figs. 7sbd–7sdd resulting from the presence of two“distant” and “nonoverlapping”sin the Chirikov44 sensedresonances—a clear manifestation of the nontrivial nature ofthe near-integrable systems.

Is it possible to use the phase space structure, for ex-ample in Fig. 7sdd, and calculate the splittingD6? The phasespace shows the existence of the primary 2:1s, the induced1:1, and a thin multimode resonance. For now we will ignorethe multimode resonance and consider the primary 2:1s andthe induced 1:1 to be independent. This is a strange approxi-

FIG. 6. sad Variation of the exactD2 with the primary 1:1 strengthg. Notethe two dips and the relatively flat region denoted byM. The arrow indicatesthe strengthg0. The effect of adding a small 2:2 with strengthg< ±4.1310−4 cm−1 at specific places are shown as triangles. Inset shows the linearscaling of exactD2 spointsd with b2ugu0. In sbd the variation of the relevantenergy levels are shown. Inset shows the details for the doubletu20l± whichundergo two exact crossings. The three panels denoted byL, M, andR showthe surface of sections corresponding to the regions labeled insad. Thesections are computed using the full classical limit Hamiltonian correspond-ing to HA− uguV1:1

s12d. Note the re-organization of the phase space in goingfrom L to R.

FIG. 7. sad Comparison of the exactsopen trianglesd, superexchangessolidlined, and the phase-space resonance-basedsfilled trianglesd splittings forHA

versus excitation quanta in the OH-stretch. Multiple resonance correctionsto r =5, 6 are shown by shaded circles. See text for details. Insbd, scd, andsdd the surface of sections at energies corresponding tor =4, 5, and 6, re-spectively, are shown. The axes ranges forsbd andsdd are identical to thoseshown inscd. In sdd one possible dynamical tunneling sequence is shown byarrows.



mation given that both the induced 1:1 and the multimodearise due to the 2:1s. However, since they are “isolated” fromeach other and lacking any other simple approach we takethis viewpoint and compute the splitting. This approach hasbeen advocated earlier by Brodier, Schlagheck, and Ullmo intheir analysis27 of resonance-assisted tunneling in kickedmaps. In the present case the effective" is too large to be inthe semiclassical limit and hence quantitative accuracies arenot expected. At the same time this is a blessing in disguisesince we do not have to deal with the the myriad of reso-nance zones in the phase space! Schematically we imaginethe following fcf. Fig. 7sddg:

u60l→beff

u40l→gind

u04l→beff

u06l s23d

wherebeff is the effective coupling across the 2:1 islands andgind is the effective 1:1 coupling derived in the previous sec-tion. It is important to note that thegind value appropriate tor =6 needs to be used, i.e., the resonance zone width as seenin the surface of section in Fig. 7sdd. The effective 2:1 cou-plings can be extracted by approximating the 2:1 resonances,for instance, the top island in Fig. 7sdd, by a pendulumHamiltonian

1

2M21sJ1 − J1

r d2 + 2bp cosc1, s24d

with M21=s2assd−1 and bp=bcsN−2J2−2J1r dÎJ1

r /2. Theresonance centerJ1

r =−M21s2Vs+a12d /2<5.7. The effectivecoupling is thus estimated as

beff <bp

2

E60 − E5

0 , s25d

with En0=f"sn1+1/2d−J1

r g2/ s2M21d. The splitting is then cal-culated via the expression

D6 < S beff

E60 − E4

0D 3 D4sc3 S beff

E60 − E4

0D , s26d

where D4sc is the splitting forr =4 calculated assuming the

induced 1:1 resonance alone. Using the numerical values forthe various parameters, we findD6<4.8310−8 cm−1 ascompared to the exact value of 1.4310−7 cm−1. This simpleestimate is quite encouraging if one also considers the factthat the induced 1:1 alone would give,10−13 cm−1 and thesuperexchange value is 6.5310−7 cm−1. Similar calculationfor r =5 doublet also improves the result, as shown in Fig.7sad. This simple-minded scheme has been tested for otherparameters and the results are comparable to the presentcase. Further support for such an approach comes from con-sidering ther =6 doublet splitting with an additonal 2:2 reso-nance present, i.e., described by the HamiltonianHB. FromFig. 3 it is clear that the addition of the 2:2 results in aboutfour orders of magnitude increase inD6. The phase space isstill near-integrable and a calculation based on the 2:2 aloneunderestimates the exact result of 3.7310−3 cm−1 by nearlya factor of four. Once again employing the approach of hop-ping across the 2:1 usingbeff, followed by the 2:2 connectingu40l to u04l, and hopping across the symmetric 2:1 islandimproves the result yielding a splitting of about 2.8

310−3 cm−1. Interestingly, the superexchange calculationhas a dominant contribution of similar magnitude from thefamily G402 which is readily identified with theb4g2 “path”in the phase space outlined above. The analysis here indi-cates that it is conceivable that nonlinear resonances can leadto dynamical tunneling between energetically similar regionsof phase space even in nonsymmetric situations.

D. Mixed phase space: Chaos-assisted tunneling?

Up until this point dynamical tunneling was investigatedwith the underlying phase space being integrable or near-integrable. We now look at the full systemHC which exhibitsmixed chaotic-regular phase space. The phenomenon ofchaos-assistedssuppressedd tunnelingsCATd has to do withthe coupling of quantum states, localized on two symmetry-related regular regions of phase space, with one or more“irregular” states delocalized over the chaotic sea.28 Clearly,such processes cannot be understood within a two-state pic-ture and the sensitivity of the nature of the chaotic states toparametric variations is manifested in the splitting fluctua-tions of the tunneling doublets. It is also reasonable to expectthat the effective" in the system needs to be sufficientlysmall in order for CAT to manifest itself. Perhaps it is usefulto note a recent analysis45 by Mouchet and Delande whereinthey argue that even in cold atom tunneling experiments theeffective" is not small enough to observe CAT. The systeminvestigated herein is nowhere close to such limits and, at theoutset, we do not expect to see CAT. In other words, for ourfull system described byHC only resonance-assisted tunnel-ing sRATd mechanisms should be sufficient to explain therelevant splittings.45

We focus on the doubletsr =4, 5, 6 since the phase spaceshows substantial chaos at the corresponding energies. InFig. 3 it is seen that ther =4 case showed marked deviationsfrom monotonicity. Moreover the superexchange calculationpredicted a much higherD4. In Fig. 8sad we show the varia-tion of energy levels with the 1:1 strengthg and a clearavoided crossing between one of the tunnel doubletsswith 2parityd and a normal mode state is observed. The inset in Fig.8sad implicates the avoided crossing in the observed exactcrossing between the tunnel doublets themselves leading tothe suppression of tunneling. In Fig. 9 the phase spaceHusimi46 representationssrefer to Ref. 33 for details regard-ing the computation of the Husimi distributionsd of the three

FIG. 8. Variation of eigenenergies with the 1:1 strength parameterg. sad r=4 doublet involved in an avoided crossing with a2 parity normal modestate. Inset shows the behavior ofD4 with varying g. sbd Same as insad forthe r =5 doublets. All parameters are in cm−1. Note that the inset x axisrange forg is from 0 to −60 cm−1. Chaos is visible in the phase space forugu.30 cm−1.



states are shown with the corresponding phase space. Noticethat the normal modes“intruder”d state has a much moreregular Husimi in the phase space as compared to the doubletHusimis themselves. In particular the2 doublet is exten-sively delocalized over the phase space. Given the value ofthe effective", it is not possible to declare the2 paritydoublet to be a chaotic state. Note, however, that the doubletscorresponding tor =4 are essentially living in the chaoticregions. For the near-integrable subsystemsHA and HB ther =4 doublets live on the separatrix associated with the 2:1resonance. So this is an instance where a regular intruderstate is affecting the tunneling between irregular doublets.No attempt is made here to calculate the tunneling splittingsemiclassically. However we note that the exact valueD4

<0.1 cm−1 as compared to the superexchange value of2.3 cm−1. Interestingly with further increase ofg, resulting inmore chaos, the superexchange calculation is only a factor oftwo higher than the exact value.

On the other hand, in Fig. 8sbd the variation of ther=5 doublet energies are shown as a function of the 1:1 pa-rameterg. This case does not show any avoided crossing.The Husimis for this doublet corresponding to the systemsHA, HB snear-integrabled, andHC smixedd are shown in Figs.10sad–10scd, respectively. It is notable that the splitting, ingoing fromHA to HB increases by more than three orders ofmagnitudescf. Fig. 3d. The Husimis and surface of sectionsin Figs. 10sad and 10sbd are quite similar except for the 2:2resonance zone clearly visible in Fig. 10sbd. However, the2:2 alone cannot mediate dynamical tunneling between thelocalized states in the 2:1 zone. As in the previous section,

excitation out to the separatrix stateu40l, followed by tun-neling across the 2:2 zone tou04l, and finally coupling to thesymmetricu05l state captures quantitatively the tunneling en-hancement. This is a clear manifestation of RAT in the sys-tem. Adding the primary 1:1 resonance, resulting inHC, thedoublet splitting is further enhanced by two orders of mag-nitude scf. Fig. 3d. The phase space as shown in Fig. 10 isclearly of a mixed nature. The large 1:1 resonance zones arevisible and, tentatively, one assigns the enhancement in split-ting to the 1:1 coupling. The 1:1 alone can account for theone-order-of-magnitude enhancement. However of all thepossible resonant subsystems considered, quantum mechani-cally or semiclassically, the best estimate is a factor of twotoo small as compared to theD5 for the full system. In thiscontext an inspection of Fig. 10 Husimi for the full casereveals amplitude spreading through the classically chaoticregion. This is in stark contrast to the well localized Husimisin the near-integrable cases. Again the effective" is too largeto implicate the classical stochasticity for part of the en-hancement inD5. Although the case forr =6 is not shownhere,D6 is enhancedscf. Fig. 3d by four orders of magnitudein going from HA to HB. This enhancement could again becaptured quantitatively by the semiclassical analysis. How-ever the best integrable estimate based on the 1:1 and the 2:2resonances is almost an order of magnitude too small for thefull systemHC.

The upshot of the preceding analysis is that no integrablesubsystem can account for the splittings ofHC and one atleast needs some chaos to be present. Even subsystems withchaos, for instance, 2:1s+1:1, on exact diagonalizationyield splittings which are about a factor of two in error. This

FIG. 9. Husimi distributions of ther =4 doublets and a close-lying normalmode state for the full systemHC. The axes ranges are identical to thoseshown in Fig. 7sbd. The dark regions correspond to large values while thelight regions signify smaller values. The top row shows the Husimis for the2 and1 parity states while the bottom left is for the2 parity normal modestate. The bottom right panel shows the phase space with clearly identified1:1 and 2:1 resonance islands. Note the delocalized Husimis of the doubletsin contrast to that of the normal mode state.

FIG. 10. The top panels show the Husimis superimposed on phase space forthe1 symmetric partner of ther =5 doublet in the near integrable cases. Theaxes ranges are identical to those shown in Fig. 7scd. Top left is forHA s2:1onlyd and top right is forHB=HA+2:2. The Husimis in both instances arelocalized in the 2:1 islands. The bottom left panel shows the Husimi for thefull caseHC and the corresponding phase space is shown in the bottom right.Note the small isthumus across the chaotic sea connecting the two islands.



points to a fairly subtle involvement by all the resonances,irrespective of their strengths, and is not understood well atthis point in time. Reducing the effective" in the system is atheoretical tool to clarify the picture. For our system, a non-scaling one, it is not very easy to perform"-scaling compu-tations. Nevertheless, some preliminary calculations indicatethat the near-integrable cases show exponential falloff with"−1, whereas the mixed cases exhibit fluctuations. In particu-lar the r =4 doublet splitting exhibits an algebraic depen-dence"−a rather than the usual exps−1/"d dependence.

V. CONCLUSIONS

In this work an intimate connection between dynamicaltunneling and the resonance structure of the classical phasespace is established. The order and widths of the resonancesdetermine the explicit dynamical barriers in the zeroth-orderaction squantum numberd space. The current study indicatesthat it is safer to “blame” dynamical tunneling on the non-linear resonances without invoking further connections be-tween avoided crossings and the resonances. If accidentalavoided crossings occur, as they do in near-integrable,mixed, and chaotic systems, then dynamical tunneling can beenhanced or suppressed further. This study also lends supportto earlier suggestions25 that even very small resonant cou-plings can be the cause of the experimentally observed nar-row spectral clusters. The resonances responsible for suchspectral clusters can be primary or induced resonances and,with small but significant amount of stochasticity in thephase space, the consequences for energy flow can be non-trivial.

In near-integrable situations, generic to lower energy re-gimes in large molecules, a combination of more than oneresonance zone can control dynamical tunneling. Not only isit important to identify the relevant resonance zones but it isalso necessary to prescribe a route to calculate the splittings.It is significant to note that neither of these problems are easyto solve. Again, and not surprisingly, the lack of understand-ing of the structure of multidimensional phase space is themain bottleneck. In this regard the superexchange approachworks well but such an accuracy for high dimensional sys-tems, with mixed regular-chaotic phase spaces, is not guar-anteed. As shown in the present work and in a previouswork,39 the superexchange approach is prone to errors, atleast by an order of magnitude, whenever states are involvedin avoided crossings. It remains to be seen if going beyondthe minimal path prescription, with some clever “resumma-tion” tricks, would lead to quantitative improvements. Wehave shown that treating the resonance zones as pendulumsand coupling across them to connect the two nearly degen-erate states provide an approximate route to calculate thetunnel splittings. This chain of resonance zones, in somesense, provides a semiclassical basis for the success of thevibrational superexchange approach. The current work hasshown the close correspondence between various terms con-tributing in the superexchange approach and the underlyingresonances in the phase space. The superexchange mecha-nism itself has its roots in the field of electron transfer inmolecular systems.47 In this context it is interesting to note

the analogy between the role played by the “bridge” statesconnecting the donor and acceptor sites for electrontransfer47 and the resonance zones connecting twosymmetry-related distant states localized in phase space.Some similarities, perhaps formal, to this work, showing thepromotion of dynamical tunneling by multiple resonances,and a recent work,48 demonstrating promotion of deep tun-neling through molecular barriers by electron-nuclear cou-pling, further exemplify the close parallels between the phe-nomena of off-resonant electron transfer and dynamicaltunneling. Regarding the various mechanisms of dynamicaltunneling in molecules, this work suggests that for smallmolecules with low density of statesslarge effective"d thedominant mechanism would be resonance-assisted. For mol-ecules with sufficiently high density of statesssufficientlysmall effective"d, both chaos and resonances can mediatedynamical tunneling, and the dominant mechanism is de-cided by the energy of the doublets and the location of thedoublets in the corresponding phase space.

Although the analysis was on coupled three-mode sys-tems, the conclusion remains valid in general, for systemswhich exhibit local mode behavior. This is based on ouranalysis, along similar lines, for many other systems whichare described more naturally by localsD2O,H2Sd or normalmodesSO2d limits. For systems with small or vanishing an-harmonicities, where the issue of the existence of localizedmodes itself is moot, the various resonances are still impor-tant, but any calculation based on a pendulum approximationis clearly invalid. Thus spectra of molecules involving lightatom stretch-bend modes are good candidates to observe thefingerprints of dynamical tunneling. In particular we suggestthe high overtone spectral regions of the water molecule asone possibility. This suggestion is tentative since a modelHamiltonian has been utilized for the analysis and it is prob-able that the polyad picture will breakdown at such highenergies. One possibility is to do a detailed analysis on thebest potential-energy surface available and/or explicitlybreak the polyad by further adding weak resonances. Bothapproaches result in genuine phase spaces with three degreesof freedom, and thus a straightforward extension is not easy.Apart from the limitation in visualizing the global phasespace structures, this has to do with the fact that there areclassical phenomena41 in three or more degrees of freedomthat could modify certain aspects of the lower dimensionalanalysis. For instance, in systems with three or higher de-grees of freedom two long-time phenomena are controlled bynonlinear resonances—Arnol’d diffusion41,49 and dynamicaltunneling. Precious little is known about either of these phe-nomena and their possible spectral manifestations in suchsituations, and either needs further study.50 As a final note,we comment on the possibility of observing long-lived localexcitations in systems with multiple resonances. This workshows that additional resonances can modify the near-degeneracies via CAT and RAT. But this modification, by thesame additional resonances and resulting chaos, can go eitherway due to multistate interactions! Moreover, as shown inthis work, additional resonances and, presumably, couplingto rotations18,19can also conspire to increase the degeneracy.Coherent manipulation of tunneling using external fields has



been demonstrated before31,51 and there are reasons to thinkthat such ideas can be utilized in the present context as well.

ACKNOWLEDGMENTS

It is a pleasure to thank P. Schlagheck for critical andilluminating discussions. I am grateful to Professor KlausRichter for the hospitality and support at the Universität Re-gensburg, where part of this work was done.

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