Signatures of discrete breathers in coherentstate quantum dynamicsCite as: J. Chem. Phys. 138, 054104 (2013); https://doi.org/10.1063/1.4788618Submitted: 18 December 2012 . Accepted: 06 January 2013 . Published Online: 01 February 2013
Kirill Igumenshchev, Misha Ovchinnikov, Panagiotis Maniadis, and Oleg Prezhdo
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THE JOURNAL OF CHEMICAL PHYSICS 138, 054104 (2013)
Signatures of discrete breathers in coherent state quantum dynamicsKirill Igumenshchev,1,a) Misha Ovchinnikov,1,b) Panagiotis Maniadis,2,c)
and Oleg Prezhdo1,d)
1Department of Chemistry, University of Rochester, Rochester, New York 14627, USA2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Received 18 December 2012; accepted 6 January 2013; published online 1 February 2013)
Understanding the properties of quantum discretebreathers (QDBs) is important for applications in many areas,including chemistry,1–5 medicine,6–10 nano-materials,11–15
and quantum computers.16 From the classical mechanics per-spective, discrete breathers (DBs) are fairly well understoodphenomena. They occur as spatial energy localization in asystem for an infinite amount of time. The effect is due toan interplay of the nonlinearity of the site potentials and thecoupling between the sites. The DB phenomenon is differ-ent in quantum mechanics; the infinite-time localization isnot possible due to tunneling. Experimentally, one can ob-serve traces of QDBs in the form of anomalously long relax-ation times and structural stabilities.17 QDBs and their equiv-alents have been observed in proteins,7, 9, 10 π -conjugatedpolymers,18 crystals,14, 15, 17, 19 micro-mechanical systems,11
anti-ferromagnets,20 Bose-Einstein condensates,21 interact-ing Josephson junctions,16 and intramolecular vibrational en-ergy redistribution.3, 5 Advances in experimental techniquesand theoretical modeling promise occurrence of QDB in up-
coming fields. Theoretical descriptions of QDBs stem fromquantizing classical solutions for both integrable22 and non-integrable systems.23 Theoretically, DBs and their equivalentshave been studied in a number of systems of practical impor-tance, including biomolecules,6, 8 polymers12, 13 nanotubes,2
graphene,4 and water.1, 5 Higher order femtosecond spec-troscopy allows to observe dynamics of coherent states in an-harmonic potential,24–27 where QDB may occur when effectsof anharmonicity dominate normal-mode motion.
Investigating the quantum dynamics of a coherent statein the context of DBs targets two areas of research: bridg-ing classical and quantum descriptions of DBs and build-ing a foundation for numerical methods that use coher-ent states. A coherent state is a quantum equivalent ofa point in the classical phase space. Comparing the sig-natures of DBs in the dynamics of a classical point anda coherent state highlights the conceptual differences andsimilarities between the classical and quantum descriptionsof DBs. Numerically, large, experimentally interesting sys-tems are difficult to model using exact quantum dynam-ics methods. One needs approximations that represent thequantum effects and scale well with system size. Meth-ods that rely on time-evolution of coherent states are effec-tive for large systems. They include frozen,28 thawed,29 andthermal30, 31 Gaussians, the Herman-Kluk (HK) propagator32
054104-2 Igumenshchev et al. J. Chem. Phys. 138, 054104 (2013)
and its higher order extensions,33, 34 coupled coherentstates (CCS),35 matching-pursuit/split-operator Fourier trans-form (MP-SOFT),36, 37 coherent-state path-integrals (CSPI),38
multiple-spawning,39 multi-configuration Hartree,40, 41 quan-tized Hamilton dynamics (QHD),42 etc. Exact for harmonicsystems, these and related methods have been used success-fully for moderately nonlinear and anharmonic systems. Forinstance, the anharmonicity of the coupled quartic dimer stud-ied with the semi-classical methods43, 44 is too small to pro-duce DBs. Even if signatures of QDBs would appear in asemi-classical simulation, they might not be recognized assuch. The analysis presented here clearly identifies the QDBsignatures in the dynamics of a coherent state. Modelinghighly nonlinear phenomena, such as DBs, can be a chal-lenge. Few semi-classical techniques have been used for thispurpose. QHD45 and HK46 provide rare exceptions. Studyingthe quantum dynamics of a coherent state illustrates the ad-vantages and limitations of these techniques when applied toQDBs.
This study connects to the previous work in relatedfields in three distinct directions: modeling of DBs in sys-tems that may have non-negligible quantum effects, funda-mental theory on QDBs, and applicability of semi-classicalmethods to highly nonlinear systems. Due to the high com-putational cost of quantum simulation, in many nano-scaleand atomic-scale systems with DB solutions, quantum effectsare neglected.2, 4, 6, 8, 13 Contributing to the previous work inthis field,47–49 our work highlights the limitations of a classi-cal models of DBs by outlining the quantum effects seen inthe dynamics of a coherent state that provides the simplestextension of classical mechanics. Conventionally, QDBs arestudied by finding eigenstates of the system Hamiltonian.50
Previous theoretical work focused on the quantization ofintegrable22 and non-integrable23 dimers, the analysis of QDBmode splitting,51 and the existence of avoided crossings be-tween the energy levels.52 We extend the previous work byproviding an analysis of distribution of probability densitybetween sites within eigenstates and its targeted contribu-tion to coherent states. Previously, theory of quantum ergod-icity helped to understand the relation between delocalized(non-tunneling) and localized (tunneling) eigenstates in an-harmonic systems and their relation to the classical limit.47, 48
An influence of this phenomena on the dynamics of a coherentstate is useful for both, theory and experiment. Some dynam-ical properties of coherent state has been studied by meansof quantum phase space entropy.49 We offer an alternativeperspective by decomposing a coherent state into eigenstates.Spectra of this dynamics is useful for the comparison of quan-tum theory to experiment and more efficient semi-classicalmethods. Understanding the nature of the spectra lines is un-clear from previous work. To gain this understanding, weplot the distribution of the eigenstate wavefunction among thesites. Seeing this distribution allows us to have qualitative un-derstanding of the dynamics of a composition of eigenstatesthat overlap with coherent state. This procedure allows us tounderstand the tunneling versus non-tunneling nature of thelines in the spectrum and the full versus partial transfer of thecoherent state from one site to another. Since the energy ofthe eigenstates are not multiples of each other, the period of
the phase corresponding to each eigenstate will be also dif-ferent. The coherent state will evolve with time and the partthat overlaps with the tunneling modes will require up to aninfinite time to transfer from one site to another. By having aqualitative understanding of the anatomy of the dynamics ofthe coherent state, one can have an insight into the phenomenajust by analyzing a spectrum plot. Coherent state based path-integral semi-classical methods are more effective at generat-ing spectrum and can simulate systems with many-degrees offreedom and energies closer to the quasi-continuum. Our pre-vious studies showed that these semi-classical methods areeffective in representing a quantum effects in DB;45, 46 how-ever, a deeper analysis of the spectra was needed resulting inthis study.
The paper is organized as follows. The system Hamilto-nian, theoretical methods and numerical procedures are de-scribed in Sec. II. Section III starts with an analysis of thespectrum of the coherent state and relates the energy eigen-states to the localized and delocalized dynamics. Section III Bfocuses in detail on the mixed region, in which the dynamicsswitches from the delocalized to the localized regime, with in-creasing energy. Section III C discusses additional effects, in-cluding avoided crossings between tunneling (localized) andnon-tunneling (delocalized) modes, the existence and proper-ties of higher order tunneling modes (HOTM), control overHOTM by choice of initial conditions, and behavior of theauto-correlation function (ACF) corresponding to the energyspectrum. The paper concludes with a summary of the keyresults.
II. SYSTEM AND METHODS
Breathers appear in a wide range of systems.1–5 In orderto focus on the fundamental concepts of QDBs and to guaran-tee an accurate quantum mechanical description, we considera simple model that is computationally accessible and con-ceptually simple. The model consists of two anharmonic sites,represented by quartic oscillators, and bi-linear coupling. It iscongruent with realistic systems. Interactions between atomsor molecules are governed by anharmonic potential, for ex-ample, Lennard-Jones potential. Generalizing to other an-harmonic potentials (ion traps, mechanical cantilever arrays,and others), a system with strong anharmonicities and lossof resonance between the sites would have conceptually thesame dynamics as our model. The Hamiltonian of our modelsystem is
H = ch
(P 2
1 + P 22 + X2
1 + X22
) + ca
(X4
1 + X42
) + ccX1X2,
(1)
where P1, P2, X1, and X2 are either classical coordinates orquantum mechanical operators, ch is the harmonic constantequal to 1/2, ca is the anharmonicity parameter, and cc is thelinear coupling coefficient.
The anharmonicity and coupling terms of the Hamil-tonian govern the DB phenomenon, since energy localiza-tion arises due to anharmonicity, while energy exchange anddelocalization occur due to coupling. For fixed values ofthe ca and cc constants, the ratio of the anharmonic and
054104-3 Igumenshchev et al. J. Chem. Phys. 138, 054104 (2013)
coupling terms is determined by the X1 and X2 coordinate val-ues, which, in turn, depend on the energy. When the energy islow, the quadratic term dominates over the quartic term, themotion is harmonic, the energy is exchanged between the os-cillators, and the eigenstates are delocalized between the sites.When the energy is high, the quartic term dominates over thequadratic term, the eigenstates become localized, and the en-ergy is not exchanged. If one of the oscillators is excited, itnever shares the excitation energy with the second oscillatorin classical mechanics, while in quantum mechanics the en-ergy transfer is very slow and occurs by tunneling.
For our purposes, it is convenient to represent the Hamil-tonian in the occupation number basis set of harmonic oscil-lators, also known as Fock states. The occupation basis set ofthe system is a direct product of the occupation states of alloscillators: |N1N2 . . . Ni〉, where Ni is the occupational state onthe ith oscillator. For our calculations, i = 2. The occupationnumber operator N is closely related to the creation a† andannihilation a operators, N = a†a. Using the creation and an-nihilation operators, the system Hamiltonian is expressed as
H = chHh + caHa + ccHc, (2a)
Hh =∑
i
a†i ai + aia
†i , (2b)
Ha =∑
i
aiaiaiai + aiaiaia†i + aiaia
†i ai + · · · + a
†i a
†i a
†i a
†i ,
(2c)
Hc =∑
i
(aiai+1+aia†i+1+a
†i ai+1+a
†i a
†i+1 + · · · +a
†i+1a
†i ).
(2d)
The number of levels in the basis set and the size of thematrix required for the calculation depend on the initial con-ditions, such as the displacement of the coherent state fromthe ground state, and anharmonicity of the system. The basisset can be truncated at high energies, where the overlap be-tween the initial state and the energy eigenstates is small. Forexample, if ca ≤ 0.05 and a coherent state is initially displacedby 1.0 a.u., the basis set can be truncated after as few as 10occupation states. The size of the total basis set is given bythe number of states for a single oscillator to the power of thenumber of the oscillators.
A coherent state is a state of the minimum uncertainty.53
A coherent state of a single site is expressed in the occupationnumber basis |i〉 as
|α(x0)〉 =∑
i
ci(x0)|i〉, (3a)
ci(x0) = e− |x0 |22
|x0|i√i!
, (3b)
where ci(x0) represents the contribution from the ith energylevel and x0 is the displacement of the coherent state from theground energy eigenstate. The basis set for the current modelis a direct product of the occupation states of two harmonic
oscillators
|ψ(x0)〉 =∑
i,j
ci(x0i)cj (x0j )|ij 〉, (4a)
ci(x0i) = e− |x0i |22
|x0i |i√i!
, (4b)
cj (x0j ) = e− |x0j |22
|x0j |j√j !
. (4c)
The dynamics of classical and quantum DBs can be com-pared by displacing one of the two oscillators away from thelowest energy state.
The energy spectrum represents the dynamics of the sys-tem in the frequency domain. The spectrum of a bound quan-tum system consists of discrete energy values. The energyeigenstates and eigenvalues are found by diagonalization ofthe Hamiltonian expressed in the occupation number basis,Eq. (2). The diagonalization is performed using the symmet-ric QR routine from GMM++ library providing an interfaceto LAPACK. The projection of the initial state onto the energyeigenstate
ck = 〈ψinitial |ek〉〈ψinitial |ψinitial〉 (5)
determines the intensity of the corresponding spectral line.Here, |ek〉 is the kth energy eigenstate.
The ACF represents the system dynamics in the time-domain. It is defined as the projection of the initial state ontothe state at time t > 0. By expressing the initial state and thetime-propagation operator e−iH t in the energy eigenstate ba-sis, one obtains the following expression for the ACF:
〈ψt |ψ0〉 =∑
k
|ck|2e−iEkt . (6)
Fourier transformation of the ACF provides an alternativeroute for computing the spectral line intensities, Eq. (5).
III. RESULTS AND DISCUSSION
A. Tunneling and non-tunneling modes
Signatures of breathers are observed in the system spec-trum, where one can identify two types of eigenstates, cor-responding to non-tunneling and tunneling modes. Tunnel-ing modes arise when the coupling between the oscillatorsis not sufficiently strong to create eigenstates that are sharedbetween the sites. The particle has to tunnel from one site tothe other. In the classical limit, the tunneling time approachesinfinity, and the tunneling modes become DB modes. Eigen-states shared between the sites correspond to an alternativetype of motion – non-tunneling modes. These modes occur inthe low energy region, where coupling dominates. Besides thelimiting cases, where there is either a small or a large amountof density shared between the sites, it is sometimes hard todistinguish between tunneling and non-tunneling modes. InSecs. III B and III C, we show that the transition between
054104-4 Igumenshchev et al. J. Chem. Phys. 138, 054104 (2013)
00.020.040.060.080.1
0.120.140.160.18
0 5 10 15 20 25 30
Energy (a.u.)
cc = 0.2, ca = 0.02
00.050.1
0.150.2
0.250.3
0.35
Spec
tral
Inte
nsity
cc = 0.0, ca = 0.02
0
0.05
0.1
0.15
0.2
0.25
cc = 0.2, ca = 0.00
FIG. 1. The spectrum of a coherent state (x0i = 3.5 and x0j = 0.0 in Eq. (4)) ina system with the Hamiltonian in Eq. (2). The top panel is the harmonic limit(ca = 0). All of the spectral lines are non-tunneling modes. The spectral linesin the middle panel are tunneling modes – the anharmonic limit (cc = 0). Thebottom panel shows modes in a coupled anharmonic potential. The spectrum(bottom) has modes that resemble both limiting cases (top and middle). Themodes that resemble the delocalized modes are non-tunneling modes and themodes that resemble the localized modes are tunneling modes.
these two types of modes is smooth. To have a better under-standing of how the two coexist in the same energy region, letus first focus on the limiting cases: coupled harmonic oscilla-tors with no anharmonicity (Figure 1 (top)) and anharmonicoscillators with no coupling (Figure 1 (middle)). We then ana-lyze the case where the potential combines both anharmonic-ity and coupling (Figure 1 (bottom)). The system is gov-erned by Hamiltonian in Eq. (2) with initial state described byEq. (4), where x0i = 3.5 and x0j = 0.0.
In the coupled case (Figure 1 (top)), we see a splittingof the energy levels. All of the eigenstates in this case arenon-tunneling. For example, at 2 a.u., a system of uncoupledharmonic oscillators would have the second lowest eigenstate|1〉. Since there is a coupling between the oscillators, these |1〉states of each oscillator combine to form a linear combinationof symmetric and anti-symmetric states: |10〉 + |01〉 and |10〉− |01〉. For the third lowest states (|02〉, 2 × |11〉, and |02〉),we also see splitting around 3 a.u., corresponding to the en-ergy of |2〉 of a system of uncoupled harmonic oscillators. Foreach eigenvalue of a 1D oscillator, in the case of coupled os-cillators, we get a group of states. Each state in that groupis a linear combination of states that have the same energy;for example, |05〉, |14〉, |23〉, |32〉, |41〉, and |50〉. This is anexample of what we call a same-quanta group – a group ofstates that have the same energy. We extend this definition tononlinear cases by defining a same-quanta group as a groupof states that become degenerate when nonlinearities of thepotential are decreased to zero.
The other limiting case is when there is anharmonicityand no coupling (Figure 1 (middle)). The spectrum is simi-lar to that of a 1D oscillator. There is no splitting between
the states of the different sites. The ground state energy is at1 a.u.; this is because the energy of each of the two oscillatorsis ≈0.5 a.u. Since there is no coupling, all of the modes inFigure 1 (middle) are tunneling modes in the limit of infinitetunneling time.
In a coupled harmonic case, the levels split due to cou-pling. A similar situation occurs in the case of a coupled an-harmonic potential. Once there is a coupling between tunnel-ing modes, they split. The difference is that in the anharmoniccase only two states can be involved in splitting; for example,|90〉 and |09〉 are involved and |81〉, |72〉, |63〉, . . . , |18〉 arenot involved in the same splitting. In the harmonic coupledcase, all of the levels with the same number of quanta splittogether.
The system that includes both coupling and anharmonic-ity (Figure 1 (bottom)) resembles both limiting cases: ca
= 0 (Figure 1 (top)) and cc = 0 (Figure 1 (middle)). Thenon-tunneling modes dominate at lower energy, and tunnel-ing modes dominate at higher energy. A classical mechani-cal description of DBs separates the localized and delocalizedmodes by a separatrix in the energy domain. Non-tunnelingmodes are the quantum equivalent of delocalized modes. Theycan be found higher in the energy spectrum compared to thedelocalized modes of classical mechanics. This is becausethere exists no clear separation between the regions with non-tunneling and tunneling modes in quantum mechanics. As en-ergy increases, tunneling modes slowly emerge out of non-tunneling modes. We call the region in energy, where bothtunneling and non-tunneling modes co-exist, a mixed region(Figure 1 (bottom), a region of 10–15 a.u.). The tunnelingmodes appear as isolated peaks that resemble the modes in anuncoupled quartic potential. The tunneling modes can be seenclearly at energies higher than the mixed region. The analysispresented below shows that the wave function of the tunnel-ing modes is localized on the sites. These modes are equiv-alent to localized modes in a classical mechanical descrip-tion of DBs. The non-tunneling modes resemble the modes ina coupled harmonic potential. They have a significant wave-function density between the sites and appear lower in energywhere nonlinear contributions are not significant. The non-tunneling modes are an equivalent of delocalized modes inthe classical mechanics description of DBs.
We choose the values for the ca and cc coefficients, andthe position of the coherent state on site 1 (q1) with the in-tention to show clearly the mixed region and the interplay be-tween the non-tunneling and tunneling modes. In some cases,DBs are seen with low energy eigenstates. For example, theanharmonicity dominates the coupling already in the secondexcited state of the C–H stretch.54 Our choice of the systemparameters and initial conditions gives a general understand-ing of the dynamics. Similar to our model, the displaced statespans a large number of eigenstates in heavy oscillators, suchas nano-mechanical cantilever arrays or large molecules withhydrogen bonding.
The spectrum shows the overlap of the initial state withthe eigenstates; the intensity of each line is determined bya projection of an initial state on the corresponding eigen-state. A displaced initial state shows little intensity in thelow and high energy regions of the spectrum. The majority
054104-5 Igumenshchev et al. J. Chem. Phys. 138, 054104 (2013)
FIG. 2. The schematic shows how to interpret a contour plot of an eigenstateof a Hamiltonian in Eq. (2); specifically, the eigenstate 237 of Eq. (2). Eachaxis represents Fock states for the two harmonic oscillators (ccx
2i , i = 1, 2).
The value on the axis is the eigenvalue of the harmonic oscillator Fock state.The schematics show that to portray an eigenstate, we need to change thebasis set from the anharmonic (ccx
2i + cax
4i ) to the harmonic (ccx
2i ). Using
this representation, we can distinguish tunneling eigenstates by identifyingthat the localization of the wave-function (red) is towards the axis.
of the intense states are at the energy H(q1 = 3.5, q2 = 0.0)(Figure 1 (bottom)). At lower energy, non-tunneling eigen-states have a considerable amount of overlap with the coher-ent state. Anharmonicity prevails at higher energies, and theinitial state overlaps strongly with tunneling eigenstates. Thewave-functions of tunneling eigenstates localize on individ-ual oscillators and is not shared between them. These tunnel-ing eigenstates correspond to the highest energy states in thegroup arising from a single, degenerate level of uncoupledoscillators.
Contour plots illustrate the energy distribution within aneigenstate among the oscillators (Figure 2). Quantum dynam-ics of the coherent state result from the projection of the ini-tial state onto the eigenstates. By visualizing and comparingthe density distribution of the initial wave function and eigen-states, one can understand how the spectrum forms. For ex-ample, the wave-function of tunneling eigenstates has higherdensity concentrating on the sites; it can be seen as higheroverlap with the |i, j〉 states, where |i − j| ≈ i + j. Therefore,the initial state, which is localized on the sites rather than de-localized between the sites, will have a higher intensity.
The contour plots are obtained using the following pro-cedure. The Hamiltonian matrix is written in the Fock basis,which is a direct product of the occupation states of each os-cillator. The matrix is diagonalized. The eigenstates expressedin the Fock basis are represented as a 2D matrix. The columnsof the matrix are enumerated according to the Fock states ofthe first oscillators: |0, i〉,|1, i〉, . . . , |n, i〉; where i = 0 . . . n.The rows are enumerated according to the Fock states of thesecond oscillator: |i, 0〉,|i, 1〉, . . . , |i, n〉. The matrix elementsrepresent the projections of the eigenstate onto the Fock statebasis. The contour plot value at a coordinate (x, y) corre-sponds to the component of the |x, y〉 basis vector. Figure2 gives an example of such a contour plot. In the plot, onecan see 237th eigenstate of the Hamiltonian in Eq. (2) in theearlier mentioned Fock basis set of two harmonic oscillators(Hih = a
†i ai + aia
†i ). The figure also explains how to inter-
pret the plot in terms of the occupation levels of the harmonicoscillators.
The contour plots (Figure 2) help to visualize and distin-guish localized states from delocalized states. In addition, thecontour plots of the eigenstates are well suited to illustrate thetransition between localized and delocalized modes, and toidentify modes that have signatures of both regimes. Finally,the contour plots help to describe the origin of the higher or-der breathers, the small satellite peaks that are slightly lowerin energy than the localized modes in Figure 1 (bottom).
Figure 3 provides examples of tunneling ((d) and (e)) andnon-tunneling modes ((a), (b), and (c). Tunneling mode con-tour plots show a non-zero density next to the axis, where thevalue on one axis is high and on the other axis is low. Thisdensity distribution indicates that one of the oscillators is inlower energy states, and the other oscillator is in higher energystates. Another way to identify the tunneling modes is to ob-serve that the value of the plotted eigenstate along the increas-ing, y = x diagonal is close to zero. Non-tunneling modes,on the other hand, have higher values of the wave-functionon the increasing diagonal, meaning that the eigenstate wave-function is shared between both oscillators.
The value of the wave-function for non-tunneling andtunneling eigenstates oscillates differently along the decreas-ing diagonal – the line connecting |0, n〉 and |n, 0〉. The wave-function of tunneling modes varies slightly. It either does notchange sign at all along the decreasing diagonal for the sym-metric modes, or changes the sign only once for the anti-symmetric modes. For the non-tunneling modes on the other
0 5 10 15 200
5
10
15
20
H2
(a.u
.) (a)
0 5 10 15 20
(b)
0 5 10 15 20H1 (a.u.)
(c)
0 5 10 15 20
(d)
0 5 10 15 20-0.2-0.15-0.1-0.0500.050.10.150.2
(e)
FIG. 3. Plots (a), (b), and (c) are examples of non-tunneling modes. They correspond to the 21st, 50th, and 95th energy levels, with the energies of 7.1494,11.6861, and 16.3442 a.u., respectively. Plots (d) and (e) are examples of tunneling modes. They are the quantum equivalent of classical breathers. The twoshown plots are anti-symmetric and symmetric states corresponding to the 93rd (16.3329 a.u.) and 94th (16.3332 a.u.) energy levels, respectively.
054104-6 Igumenshchev et al. J. Chem. Phys. 138, 054104 (2013)
0 5 10 15 200
5
10
15
20
H2
(a.u
.) (a)
0 5 10 15 20
(b)
0 5 10 15 20H1 (a.u.)
(c)
0 5 10 15 20
(d)
0 5 10 15 20-0.2-0.15-0.1-0.0500.050.10.150.2
(e)
FIG. 4. Contour plots (see Figure 2) of delocalized ((b) and (d)) and tunneling ((a), (c), and (e)) modes within the mixed region (12–23 a.u.). The plots are inthe order of increasing energy: plot (a) is the 55th state with the energy 12.1386 a.u., (b) – 75th – 14.454 a.u., (c) – 110th –17.7796 a.u., (d) –145th – 20.82 a.u.,and (e) – 163rd – 22.2542 a.u. The figure shows that in the mixed region, the tunneling and the non-tunneling modes alternate.
hand, the wave function varies quickly. This variation is seenas blue and red peaks alternating along the diagonal.
The contour plot of the tunneling modes (Figures 3(d)and 3(e)) correspond to the pair of peaks at 16.33 a.u. found inthe spectrum (Figure 1 (bottom)). The pair appears as a singlepeak; the two states are only 0.003 a.u. apart. The eigenstatethat is higher in energy is symmetric, and the lower energyeigenstate is anti-symmetric. The small splitting between thesymmetric and anti-symmetric states of the tunneling modespair is consistent with the classical formulation of DBs. Ifboth symmetric and anti-symmetric states are occupied, thenthe density on one of the oscillators cancels out, and the en-ergy is localized to the other oscillator. A superposition ofsymmetric and anti-symmetric states will evolve in time, andat some point the energy will transfer between the oscillators.Since the splitting is very small, the energy transfer time isvery long.
Investigating the intensity and distribution of energy be-tween the sites allows us to study in detail the mixed region,the evolution of non-tunneling modes into tunneling modes,the influence of the variation of coupling and initial condi-tions on the spectrum, and the properties of HOTM. Contourplots provide an important tool to distinguish between local-ized and delocalized modes. They are particularly importantfor cases when the density of states increases and the spec-trum becomes complex; for example, in soft potentials or forhigh energy initial conditions.
B. The mixed region
Most of the systems in nature, such as molecules, crys-tals, and nano-materials, consist of a network of coupled sites.At higher energies, the coupling between sites of the systembecomes negligible when compared to the nonlinearity of on-site potential. At lower energies, the on-site potential can bemodeled with a linear approximation. The approximation issimilar to the sine approximation for the motion of a pendu-lum and may be familiar to the readers from their introduc-tory physics class. With an increase in energy, the nonlinear-ity becomes more important and the system dynamics changefrom delocalized modes to localized modes. This transitionis important because the behavior of the system changes sig-nificantly. In the delocalized regime, the system transfers theenergy from one site to the next site. In the localized regime,quantum, and classical mechanics differ in how they modelthe distribution of energy between the sites. In classical me-
chanics, the energy does not transfer between the sites. Quan-tum mechanics allows the energy to transfer from one site toanother through tunneling. Without the loss of generality, wemodel this transition in the simplest potential – two coupledquartic oscillators.
In addition to the difference in describing localizedmodes, the transition from the non-tunneling region to the tun-neling region also differs between classical and quantum me-chanics. In classical mechanics, the two regions are dividedby a point called the separatrix. Such a sharp division doesnot exist in quantum mechanics. As mentioned earlier, the en-ergy does not localize but takes an almost infinite amount oftime to transfer by tunneling. Therefore, in contrast to the lo-calization seen with classical mechanics, quantum tunnelingmodes slowly evolve from the non-tunneling modes as energyincreases. Understanding the quantum equivalent of the tran-sition between delocalized and localized modes will help inunderstanding the limitations of a classical model, while pro-viding insight into semi-classical dynamics and complement-ing experiments on QDBs.
Depending on the system’s Hamiltonian and the initialconditions, the mixed region can contain part or all of the co-herent state. In the mixed region, non-tunneling and tunnelingmodes co-exist. Figure 1 (bottom) shows that as the energyincreases, these modes alternate. Non-tunneling modes canhave higher energy than tunneling modes and vice versa. Tun-neling modes are the highest energy modes found in same-quanta groups. The mixed region may contain a number ofsame-quanta groups. It can also contain non-tunneling statesfollowed by the tunneling states of the same-quanta group.They are then followed by non-tunneling states and tunnel-ing states of the next same-quanta group. Figure 4 gives anexample of alternating tunneling and non-tunneling modes.
In classical mechanics, the transition from localized todelocalized modes happens instantly. Quantum mechanicsshows this transition to happen smoothly. Figure 5 showsthe highest energy eigenstates from the same-quanta groupswithin the mixed region. In this figure, one can observean iterative transformation from non-tunneling to tunnelingmodes. This phenomenon is quantum in nature and stronglycontrasts with the step-like transition from delocalized to lo-calized modes observed in classical mechanics.
Unlike in a classical mechanical perspective, the transi-tion from tunneling modes to non-tunneling modes is smooth.Figure 5 shows this transition. The last two plots on the right(d− and d+) are tunneling modes; there is no shared density
054104-7 Igumenshchev et al. J. Chem. Phys. 138, 054104 (2013)
0
4
8
12
-0.2-0.15-0.1-0.0500.050.10.150.2
(a)
(a)
+ (b) + (c) + (d) +
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0 4 8 120
4
8
12H2
(a.u
.)
-0.2-0.15-0.1-0.0500.050.10.150.2
− (b)− (c)− (d)−
0 4 8 12H1 (a.u.)
-0.2-0.15-0.1-0.0500.050.10.150.2
0 4 8 12
-0.2-0.15-0.1-0.0500.050.10.150.2
0 4 8 12
-0.2-0.15-0.1-0.0500.050.10.150.2
FIG. 5. The smooth transition from non-tunneling modes (a) into tunneling modes (d). Vertical pairs of plots show symmetric (+) and anti-symmetric (−)eigenstates. From the same-quanta group, these states have the highest localization. As the energy of the states increases (a → d), the wave-function movestowards the axis. The presence of the wave-function between the axes, which corresponds to the density between the sites, gradually decreases; (d) has negligibleamount of density wave-function between the sites – the wave-function has to tunnel from one site to the other. The corresponding values of energy (E) andintensity (I) for the plots are: (a+) E = 9.46539 a.u. I = 0.175297, (a−) E = 9.50399 a.u. I = 0.135935, (b+) E = 12.1386 a.u. I = 0.161693, (b−) E = 12.1478a.u. I = 0.151122, (c+) E = 14.9101 a.u. I = 0.139205, (c−) E = 14.9113 a.u. I = 0.138124, (d+) E = 17.7795 a.u. I = 0.11148, and (d−) E = 17.7796 a.u. I= 0.111422.
between the sites and the wave-function has to tunnel through.The major contribution to the tunneling eigenstates comesfrom basis states that have only one of the sites excited; forexample, |14, 0〉 and |0, 14〉. The two plots on the left of Fig-ure 5 (a− and a+) are a pair of eigenstates that resemble thetunneling mode. The pair of states b− and b+ (Figure 5) havefeatures similar to the tunneling modes, but they are more de-localized than the tunneling mode. There is significant densityon the increasing diagonal, meaning that there is an overlapwith basis set states that have the density delocalized betweenthe sites. The pair of states in the middle is between the twocases.
C. Additional observations
1. Effects of coupling variation on discretebreather signatures
Slight changes in the setup of an experiment can varythe shape of the potential. For example, one can change partsof the molecules to be heavier or lighter, or use a differentsolvent. These perturbations can control the interplay betweenthe nonlinearity and coupling in the potential.
Figure 1 illustrates the signatures of breathers in a spec-trum. We extend those results by investigating how these sig-natures depend on the parameters of the potential. First, weanalyze the evolution of the eigenvalues with respect to thecoupling coefficient in the Hamiltonian (Eq. (2)). Startingfrom zero coupling (cc = 0), we increase it to the point, wherethe coupling is much larger than the anharmonicity, so thatthe system approaches the limit of coupled harmonic oscil-lators. We focus on eigenvalues of the same-quanta group ofstates between 9.1 and 11.4. The coefficient of anharmonicityis constant: ca = 0.02.
Figure 6 shows the dependence of anharmonic oscilla-tor (Eq. (2)) eigenvalues on coupling (cc). The plot showsthe variation of eigenvalues from one extreme of two uncou-
pled anharmonic oscillators (cc = 0.0) to another extremethat has coupling (cc = 0.3) much larger than anharmonic-ity (ca = 0.02). With this plot, we would like to focus on agroup of same-quanta states. At cc = 0, the eigenstates ofthat group have energy in the interval ≈(10, 10.6) a.u. Thefigure clearly shows how an increase in coupling causes de-generate states to split. The tunneling states, which are thetop states of the same-quanta group, are the last ones to split.Keeping in mind that the size of splitting is inversely propor-tional to the transfer time between the sites and noting thatthe size of the splitting between tunneling states approachesthe size of the splitting between lower lying non-tunnelingmodes, one can conclude that the transition from tunnelingto non-tunneling states is smooth from a quantum mechani-cal perspective. This observation complements our result inSec. III B on the smooth transition between tunneling andnon-tunneling modes.
We observe avoided crossings for states from the same-quanta group. At cc ≈ 0.8, the top (symmetric) state that startsat ≈10.34 avoids crossing with the low state that starts at≈10.14. Another avoided crossing happens at cc ≈ 0.04 forthe next lowest pair of states. In the harmonic case, there is nocrossing since the degenerate states split linearly and do notcross. Previously, avoided crossing was observed in an anhar-monic trimer.52 The avoided crossing is expected since theeigenstates belong to the same symmetry group of irreduciblerepresentations. The practical importance of this result showsup in systems with many degrees of freedom. In multidimen-sional systems, the avoided crossings become conical inter-sections that allow for radiationless transitions between theenergy surfaces.55
2. Higher order tunneling modes
In the spectrum plots (Figure 1), one can notice thatbeside tunneling modes, there are eigenstates of smaller
054104-8 Igumenshchev et al. J. Chem. Phys. 138, 054104 (2013)
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
10.8
11
11.2
11.4
0 0.05 0.1 0.15 0.2 0.25 0.3
Ene
rgy
(a.u
.)
Linear coupling coefficient (cc)
tunneling modes
FIG. 6. Eigenenergy of a quartic dimer (Eq. (2)) with ch = 0.5 and ca = 0.02. Dashed crossing lines indicate examples of the avoided crossings.
intensity in the anharmonic regime, where there are not anynon-tunneling modes – the energy region beyond 16 a.u. (Fig-ure 1 (bottom)). These eigenstates do not classify as non-tunneling modes. In this higher energy regime the influence ofcoupling is small, and contrary to the case with non-tunnelingmodes, they increase in intensity as they increase in energy.The eigenstates also do not fit our previous definition of tun-neling modes. They do not have the highest intensity in theirsame-quanta groups, and they are not a linear combination ofmodes that have one of the sites in the ground state and theother site occupying a high state.
The anomaly becomes clear when we analyze the con-tour plots of these eigenstates (Figure 7). In these eigenstates,the energy has to tunnel from one site to the other in a similarmanner to the tunneling eigenstates. The contour plots evi-dence a distinction of these eigenstates from tunneling modes– the majority of the wave function density is not on the axis.These two observations motivate us to define them as HOTM.To be more precise, we define HOTM as a group of eigen-states that satisfy the following conditions: (1) the energyfrom the displaced oscillator can transfer to the other oscil-lator(s) only through tunneling, and (2) the undisplaced os-
0 8 16 24H1 (a.u.)
0
8
16
24
H2
(a.u
.)
(a)
0 8 16 240
8
16
24
-0.2-0.15-0.1-0.0500.050.10.150.2
(b)
FIG. 7. An example of higher order tunneling modes, specifically the secondorder tunneling modes. (a) and (b) Contour plots of the 90th (E = 16.01) and91th (E = 16.05) energy levels.
cillator has the highest wave-function density on non-groundstate(s). Figure 7, specifically, is an example of second ordertunneling modes (SOTM). In the basis set of a direct productof anharmonic oscillators, they are a linear combination of|n − 1, 1〉 and |1, n − 1〉 states. Figure 1 shows other HOTMthat are even smaller in intensity than SOTM and manifest atan even higher energy limit. Next, we compare HOTM andfirst order tunneling modes and outline a couple of other in-teresting points.
To identify the relation of HOTM to tunneling modes,we would like to point out a few similarities. As seen fromFigure 7, HOTM are a pair of symmetric and anti-symmetricstates with a higher probability of being on the oscillatorsthan states shared between them. In the figures, one can seethe lack of density on the increasing diagonal; the densitycloser to the axis. As seen in Figure 6 at the energy 10.33 a.u.,HOTM start as degenerate states but further along, they splitinto a pair, similar to the first order tunneling states. Since weare interested at a lower value of coupling, we can see HOTMat 10.33, rather than at 16 a.u.
To emphasize that HOTM should be treated differentlyfrom the first order tunneling modes, we would like to pointout some key differences. The probability density in HOTMshifts towards the other oscillator by one occupational level.Figure 7 shows this shift in density for the second order tun-neling modes; they correspond to |n − 1, 1〉 and |1, n − 1〉occupational states. The maxima of the wave-functions arenot on the axes anymore but are closer to the increasing di-agonal. Compared to the energy region 10.3–10.6 a.u. of thetunneling modes in Figure 6, the SOTM have larger splittingbetween symmetric and anti-symmetric modes than the firstorder modes. The distance between the top left and bottomright density concentrations in the contour plot is related tothe tunneling probability; therefore, it is rational to assumethat with a decrease in this distance the splitting and the tun-neling probability increases.
054104-9 Igumenshchev et al. J. Chem. Phys. 138, 054104 (2013)
0.01
0.1
0.01
0.1
0 5 10 15 20 25
Log
arithm
ofSp
ectr
alIn
tens
ity
(arb
.u.
)
Energy (a.u.)
not symmetric: q1 = 2.5√2, q2 = 0
symmetric: q1 = 1.5√2, q2 = 1.5√
2
FIG. 8. The top plot is a spectrum from Figure 1 with a log scale. The bottom plot is the same as the top plot but for different initial conditions – both of thesites have a symmetrically displaced coherent state.
3. Influence of the initial conditions
The initial state determines the intensities in the spec-trum. We show that when both of the sites are displaced ini-tially, HOTM have higher intensity than tunneling or non-tunneling modes. The impact of this observation is twofold:we gain insight into experiments with displaced coherentstates on all sites and we understand the influence of HOTMon the dynamics. Depending on the experimental systemsetup, the initial conditions can be different. When an exper-imental setup cannot target a specific site, multiple sites willbe displaced. An example of this would be exciting a numberof sites in a system from a ground state to an excited state,where the excited state is displaced. One application may bean understanding of DNA breaking under THz radiation.56
Emphasizing the presence of the HOTM in the dynamics, onthe other hand, results in a new type of dynamics – the energyis neither localized on sites nor distributed between the sites.
Figure 8 illustrates the effects of displacing the coherentstates symmetrically on both sites (the bottom panel), in con-trast to displacing just one of the sites (the top panel). Thefigure shows the spectrum of a coherent state (x0i = q1 andx0j = q2 in Eq. (4)) in Eq. (2). In the lower energy region(<6 a.u.), the effect of displacing both sites is clear. At 2 a.u.,for example, the case with both sites displaced (BSD) has onlyone line corresponding to a symmetric eigenstate; the casewith one site displaced (OSD) has both symmetric and anti-symmetric modes. In the case of BSD, the pattern continuesat higher energies – only the symmetric eigenstates have in-tensity. Furthermore, in the same-quanta group the highest en-ergy eigenstate has the highest intensity. In the mixed region(10–15 a.u.), SOTM start to have the highest intensity. TheOSD case shows the opposite trend. At lower energies, thehighest intensity is in the middle of the same-quanta group.In the mixed region, tunneling modes start to dominate. How-ever, the most interesting contrast is in the anharmonic region,
where in the OSD case, tunneling modes dominate. In BSD,SOTM, and other HOTM dominate.
In classical mechanics if both sites have the same amountof the initial energy, there is no transfer of energy. In quantummechanics, a coherent state is a superposition of eigenstatesso the dynamics are more complex. To get a brief understand-ing, let us analyze the dynamics around 20 a.u. In the case ofOSD, the tunneling mode pair has the most intensity. As wementioned earlier, the energy is transferred from one site tothe other. In the case of BSD, the intensity is distributed onlyamong the symmetric modes – there is no transfer of energybetween the sites. However, the energy is distributed almostevenly among HOTM. HOTM have some density shared be-tween the sites. With time, the density oscillates from beinglocalized on the sites to being partially shared between thesites.
4. Auto-correlation of the displaced coherent state
A number of time-domain methods are based on prop-agation of coherent states. Examples include HK,32 QHD,42
CSPI,38 CCS,35 and MP-SOFT.36, 37 Earlier we calculatedeigenstates to analyze dynamic properties of the system.Eigenstates represent stationary properties of dynamics. Tocompare our results to time-domain methods, we calculate theACF. Besides showing evolution of the initial state in time,the ACF is used to calculate spectroscopic properties, diffu-sion rate, and reaction rate constants. Figure 9 shows the ACF(Eq. (6)) for different values of anharmonicity in the potentialEq. (2). The system is the same as in Figure 1 with cc = 0.2.
The top panel shows the dynamics of coupled harmonicoscillators. The wave-function returns to the previous statewith a period of 65 a.u. Once there is anharmonicity (cc
= 0.01, middle panel), the eigenstates have frequencies thatare not multiples of each other anymore. The condition for
054104-10 Igumenshchev et al. J. Chem. Phys. 138, 054104 (2013)
00.20.40.60.8
1
0 10 20 30 40 50 60 70 80 90 100
time (a.u.)
ca = 0.05
00.20.40.60.8
1C
orre
lati
on ca = 0.01
00.20.40.60.8
1ca = 0.00
FIG. 9. Auto-correlation functions 〈φ(t)|φ(0)〉 for different values of the anharmonicity coefficient ca: ca = 0.00 (top), ca = 0.01 (center), and ca = 0.05(bottom). The effects of increased the anharmonicity and loss of resonance can be seen as ca increases. In all cases, the wave packet on the first oscillator isdisplaced to x0 = 1.767.
resonance is lost, and the tails of the coherent state move atdifferent frequencies. The coherent state spreads with time.The wave function does not come back to the same state.The case with ca = 0.05 (bottom panel) is an even more ex-treme example. The coherent state spreads even faster. Cal-culating rapidly spreading coherent states is difficult. This re-sult shows the challenge faced by coherent state based path-integral methods – they have to track quickly spreading tailsof the coherent state. Nevertheless, the time-window that weused proves to be sufficiently large to get satisfying results.In addition, a quick decrease of the ACF in time justifies us-ing shorter propagation times. Time-domain methods that arebased on propagation of a coherent state will prove to be aneffective and accurate tool to model signatures of QDBs inmultidimensional systems.
IV. CONCLUSION
Understanding signatures of QDBs in the quantum dy-namics of coherent states is a step towards elucidating thelocalization phenomenon in large scale systems and bridg-ing the gap between the concepts of quantum and classicalDBs. We have analyzed tunneling modes, a quantum coun-terpart of classical DBs, in the spectrum of a quartic dimer,and characterized in detail the properties of the eigenstateswithin the mixed region where the dynamics switch from thedelocalized to localized regime. Additional important obser-vations include the following: tunneling modes avoid crossingwith non-tunneling modes arising from the same states of theuncoupled oscillators; new types of modes, HOTM appear athigher energies; controlling initial conditions allows us to en-hance the contribution of HOTM to the spectrum and fasterACF decay with increasing nonlinearity shortens the time re-quired for modeling DBs.
Tying our results to the classical mechanical interpreta-tion of DBs, we show that tunneling and non-tunneling modesare the quantum mechanical counterpart of the localized and
delocalized modes that appear in classical mechanics. Sincecoherent states span a large number of eigenstates, they simul-taneously contain both tunneling and non-tunneling modes. Inthe classical mechanical analogue, the initial energy localizedon one of the sites cannot completely transfer to the othersite. In quantum mechanics, the energy tunnels to the othersite; however, as the size of the system approaches a classicallimit, the tunneling time approaches infinity.
From the analysis of the mixed region, we concluded thattunneling and non-tunneling modes alternate with increasingenergy, and that the transition from non-tunneling to tunnel-ing modes is gradual. Both of these conclusions are in con-trast with the classical mechanical model of breathers, wherelocalized and non-localized modes are separated sharply at acertain energy. The convergence of the mixed region in quan-tum mechanics to the separatrix point in classical mechanicscan be reached by scaling the quantum result to the classicallimit.
Many semi-classical methods that are capable of describ-ing large systems rely on the propagation of coherent states.We show that both tunneling and non-tunneling modes arepresent in the coherent state. Therefore, a coherent state rep-resentation is capable of reflecting both types of dynamicssimultaneously: part of the coherent state would stay local-ized and part would be delocalized. It is likely that coherentstate based methods would have difficulty tracing the widthor other metrics of the coherent state, especially in the mixedregion, where both types of states have a nearly equal contri-bution. Nevertheless, coherent state based methods should beable to identify the location of the mixed region, and representnon-tunneling, tunneling, and HOTM.
ACKNOWLEDGMENTS
The research was funded by the National Science Foun-dation (NSF) Grant No. CHE-1050405 and NSF CAREERAward No. 0645340.
054104-11 Igumenshchev et al. J. Chem. Phys. 138, 054104 (2013)
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