Deducing Phonon Scattering from Normal Mode Excitations.

While the quantum scattering theory has provided the theoretical underpinning for phonon interactions, the correspondence between the phonon modes and normal modes of vibrations has never been fully established; for example, the nature of energy exchange during elementary normal mode interactions remains largely unknown. In this work, by adopting a set of real asymmetric normal mode amplitudes, we first discriminate the normal and Umklapp processes directly from atomistic dynamics. We then demonstrate that the undulating harmonic and anharmonic potentials, which allow a number of interaction pathways, generate several total-energy-conserving forward and backward scattering events including those which are traditionally considered as quantum-forbidden. Although the normal mode energy is proportional to the square of the eigen-frequency, we deduce that the energy exchanged from one mode to another in each elementary interaction is proportional to the frequency - a quantum-like restriction. We anticipate that the current approach can be utilized profitably to discover unbiased scattering channels, many traditionally quantum forbidden, with complex anharmonicities. Our discovery will aid in the development of next-generation Peierls-Boltzmann transport simulations that access normal mode scattering pathways from finite temperature ab initio simulations.

Introduction

Normal modes are non-interacting, non-decaying collective excitations of atomic vibrations – these are the eigen-states of the interacting harmonic Hamiltonian. In a crystal lattice with translational symmetry, the normal modes have eigen-frequencies that correspond to well-defined wave-vectors. Analogous to photons that are quantum excitations of an electromagnetic field, phonons are the quantum excitations of the lattice waves in a crystal[1-3]. Unlike photons, which arise as a particle-wave representation of an exact harmonic gauge potential, phonons are conceptualized from materials-specific anharmonic potentials, which allows a large diversity in their interactions, including among themselves[4]. Unsurprisingly, phonon interactions form the bedrock of our understanding of crystal response to external disturbances, particularly in the area of thermal transport – recent advances showcase unprecedented strides in phononics[5-10], nanoscale and low-dimensional transport[11-20], materials discovery[21-23], and theoretical and simulation methods[24-34].

Phonon-phonon interactions are usually treated as scattering events within the time-dependent perturbation theory. A key assumption in deriving the transition rates is that the anharmonic Hamiltonians add only small perturbations, and higher order contributions get progressively weaker. This assumption has been examined recently from two angles. Historically, the three-phonon interactions have been assumed to dominate energy transport with little or no contributions from higher-order processes. Recent work through the iterative solution of Peierls-Boltzmann transport equation (PBTE), however, has upended this notion by demonstrating appreciable contributions from the four-phonon scattering processes in both bulk and 2D materials[32,35]. Although the probability for elementary four-phonon excitations is lower than that of three-phonons, the less-restrictive conservation rules can accord a larger phase space for four and higher order phonon scattering interactions. Perturbative expansion is also critiqued for its inability to account for strong anharmonic interactions that are comparable to the harmonic interactions[36-38]. Such situations typically arise at temperatures comparable to the Debye temperature (Θ), or near second order phase transitions[39] and dynamic/displacive instabilities. Large amplitude atomic/ionic oscillations[40] or off-center rattling[41,42] can also break the translational invariance and the assumption of a weak perturbation.

Many-body atomistic dynamics[43,44] with an arbitrarily accurate anharmonic Hamiltonian provides a direct approach for probing the interactions among the crystal normal modes. Loss of translation or lowering of symmetry associated with structural complexities can be accommodated, in principle, with atomistic dynamics to all orders of anharmonicity. Three serious limitations hamper the close correspondence between the phonons and normal modes. Phonons are bound by Bose-Einstein statistics[25,45] while atoms conform to the Boltzmann statistics. Semi-classical quantum baths[46,47] can potentially address this concern but their practicality is yet to be established fully[48]. Secondly, the traditional normal mode analysis (NMA) framework, which employs complex normal mode coordinates, does not distinguish the non-resistive normal (N) phonon interactions from the resistive Umklapp (U) phonon interactions[49] – a serious drawback in identifying quantum-like phonon interactions, given the overarching importance of the U processes in thermal energy transport[1]. The final limitation on the correspondence between the normal modes and phonon modes is right at the heart of quantum mechanics in that the energy transfer between quantum objects is always proportional to the corresponding frequencies, but the nature of energy exchange during elementary normal mode interactions remains largely unknown.

In this work, we show  that the theoretical limitation of the traditional NMA analyses can be removed by adopting a set of real asymmetric normal mode amplitudes, which can distinguish lattice waves moving in opposite wave vector (i.e. in +q and −q) directions – a virtue that immediately endows the ability to discriminate N and U processes and the phase space associated with their interactions (see Methods section). Although the normal modes exchange energy continuously, they are localized in the reciprocal space and engage in discrete interactions. Crucially, we observe that the harmonic and anharmonic energies do not remain constant in time. The oscillating energies, which appear to be generic to all anharmonic systems, allow interaction pathways that have not been identified before including those which are traditionally considered as quantum-forbidden. Finally, using controlled cubic and quartic anharmonicity, we infer that the energy transferred from one normal mode to another in an elementary interaction is proportional to the corresponding frequencies – a surprising similarity to energy interchanges at the quantum level.

Results

Atomistic systems

We select linear Fermi-Pasta-Ulam (FPU)[50-52] lattices with third and fourth order anharmonicities for illustrating the emergence of N and U modes. With the vibrations constrained to a single dimension, the lattices with cubic (FPU-α) and quartic (FPU-β) anharmonicities are specifically intended to probe the third and fourth order scattering processes, respectively. Historically, FPU lattices have been analyzed in the context of quasi-periodic or recurrence phenomena but in this work we have selected the parameters such that the lattices are ergodic and fully amenable to thermalization (see Methods section). The FPU models thus serve as surrogates of more sophisticated anharmonic systems. To illustrate the broader capability of the current approach, we also examine the modal interactions in graphene, which is modeled through an optimized Tersoff potential[53] that includes anharmonicities of all orders. While the phonon dispersion is reasonably captured by the optimized Tersoff potential, the higher order anharmonicities of this potential may not be accurate. The choice of the models reflects our central purpose of demonstrating the scattering of normal modes and evaluating the extent to which it concurs with or diverges from the established quantum phonon scattering theory.

Conservation of crystal momentum – N and U modes

In Fig. 1 we delineate the excited normal modes (x-axis) following perturbation of a single mode (y-axis) in a FPU-α lattice (top) and a FPU-β lattice (bottom) – the details are given in the Methods section and in Supplemental Information A. With all the atoms initially at their equilibrium positions without any kinetic energy, a single mode is perturbed first, and the resultant excited normal modes, which are measured by the magnitude of the modal energies, are tracked during a short observation time window of 0.4 units. As expected, the most prominent excited mode is N0, which corresponds to the perturbing wavevector q. The next prominent excitations are the overtones of the primary modes: q + q → 2q; q + 2q → 3q for the FPU-α lattice (with only cubic anharmonicity) and q + q + q → 3q for the FPU-β lattice (with only quartic anharmonicity); these excitations are analogous to the three and four phonon merging events, respectively. More strikingly, the corresponding U-type processes (2q + g and 3q + g), also emerge. Thus by faithfully capturing the expected N and U processes, which is made possible by the asymmetric normal mode coordinates, we provide the first key correspondence between the normal mode interactions and the phonon modes. Since only a single mode is perturbed initially, merging events are dominant in the short time of observation (0.4 units). At longer times, or with more complex perturbations involving multiple modes, both Class I merging and Class II splitting events[4] can be identified (results not shown). A curiously peculiar set of excitations involves the negative wavevectors, which does not fall in the aforesaid classification. The −2q mode for the FPU-α lattice, and the −q and −3q modes for the FPU-β lattice conserve crystal momentum and they are similar to the overtone modes previously described, however, with a negative sign. Similar overtones and Umklapp reflection at the zone boundary are observed in graphene, which is governed by a fully anharmonic interatomic potential (Supplemental Information B). Interestingly, negative modes appear prominently in graphene too as shown in Supplemental Information C and D.

Figure 1

Normal mode excitations (q′, x-axis) from externally perturbed modes (q, y-axis) on a FPU lattice with only cubic anharmonicity (FPU-α, top) and only quartic anharmonicity (FPU-β, bottom); the scale on the right denotes the energy of the corresponding modes. With all the atoms initially at their equilibrium positions and without any initial kinetic energy, a single mode q is first perturbed with an excess energy E. This external perturbation engenders additional interactions or excitations. The perturbation protocol is then repeated sequentually for several values of wavevectors (see Methods section). The excitations comprise of both N (yellow) and U (green) processes associated with the elementary merging overtones: q + q → 2q + g; q + 2q → 3q + g for the FPU-α lattice with only cubic (III) anharmonicity, and q + q + q → 3q + g for the FPU-β lattice with only quartic (IV) anharmonicity. The ability to capture the N and U modes separately arises through the use of real asymmetric normal mode amplitudes. Although not shown, splitting modes emerge at later times. Modes with negative wavevectors that do not fall in the merging or splitting categories also appear (−q, −2q, −3q). The numerical protocol for generating the excited modes is detailed in Supplemental Information A.

Time-varying harmonic and anharmonic energies

To understand the origin of the excited negative modes, we first examine the temporal behavior of the harmonic energies in Fig. 2 for FPU lattices. We simulate these systems with small kinetic energies such that they are nearly harmonic. Without using an external thermal bath or rescaling, both FPU systems are allowed to attain a steady temperature T of 0.01, which is sufficiently close to 0 K and almost two orders smaller than ΘD, the Debye temperature. The left panel in Fig. 2 depicts the temporal behavior of the instantaneous harmonic and total energy from a single run as well as the harmonic energy averaged over time and different initial phases. The right panel shows the instantaneous and average behavior of the anharmonic energy (H3/H4) for the FPU-α/β systems.

Figure 2

(Left) Variation of harmonic energy (H2) and total energy (H) in time for the FPU-α/β systems at a temperature of 0.01. Although the total energy remains a constant in time, the cubic (α) and quartic (β) harmonic Hamiltonians show significant fluctuations. Only with increasing number of time or phase-space averages (the number in the parenthesis depicts the number of averages from different initial conditions), the harmonic energy approaches a constant value. The right panel shows the temporal variation of the instantaneous anharmonic energy αH3(t) = αH(t) − αH2(t) from a single phase space trajectory along with the (running) time-averaged αH3 for the FPU-α lattice. The corresponding anharmonic energy βH4(t) = βH(t) − βH2(t) and the (running) time average for the FPU-β lattice are shown in the inset. Note that αH4(t) and βH3(t) are both zero, by construct.

Most notably, the harmonic energy (H2) is not constant in time even at low temperatures. Although H2 approaches a steady value when averaged over time or a large number of initial phases, the fluctuations are conspicuously large for a single run as shown in Fig. 2 (left panel). Since the normal modes interact in any given phase trajectory, the oscillating harmonic energy deserves more attention because the theory of quantum phonon interactions, which is largely formulated through the time-dependent perturbation theory or through renormalization[36], assumes a constant harmonic energy for all singular events such as three-phonon or four-phonon interactions. Further, the standard quantum phonon theory prescribes constant anharmonic energies as well without allowing any exchange between the harmonic and anharmonic Hamiltonians. Normal mode energies in atomistic systems, contrarily, portray a time varying behavior that is distinctly different. For any given initial phase, H2 serves as an energy bank from which the anharmonic Hamiltonians can borrow or return small amounts of energy. The instantaneous fluctuations of the anharmonic energies H3 and H4 (right panel of Fig. 2) reflect this energy transfer between H2 and H3, and H2 and H4, respectively. Expectedly, with time or phase space averaging, the energy fluctuations become small. Since βH4 is always positive, βH2 is lower than the total energy for FPU-β system. Although αH3 can take positive or negative values, αH3 for the FPU-α system that is investigated here is negative. We will return to the limiting constancy of harmonic and anharmonic energies on phase-space/time averaging in the context of energy conservation in phonon-phonon interactions within the quantum framework.

The oscillatory nature of energies is not unique to the particular systems that are investigated here; rather it appears to be universal to all vibratory anharmonic atomistic systems as there are no inherent physical mechanisms to maintain harmonic and anharmonic energies that are invariant in time. The undulatory nature of the Hamiltonians, however, confers additional freedom for the normal modes to interact in ways not possible with constant Hamiltonians. Since the total energy is constant, we infer that portions of harmonic energy are continuously interchanged with the anharmonic components.

Negative normal modes

The origin of the negative modes is rooted in the time variation of the harmonic/anharmonic energies, and is best illustrated by examining the theoretical structure of the Hamiltonians with the real asymmetric normal mode amplitudes in three dimensions. For simplicity, we focus only on the cubic term (the full derivation is given in Supplemental Information E), which is given by:

The above time-dependent anharmonic Hamiltonian H3(t) is summed over all polarizations p (not shown explicitly above) and wavevectors that satisfy q1 + q2 + q3 = g (for the FPU lattices, the above expression reduces to a single dimension). Equation (1) is completely analogous to the anharmonic expression involving the creation and annihilation operators[3,4]. The first term allows simultaneous creation and annihilation of three normal modes while the remaining three cover all possible combinations for the merging and splitting processes. Interestingly, the two spontaneous processes, which correspond to the first term, are deemed impossible quantum mechanically as they violate the H3 energy constancy (for specific three-phonon interactions) but this restriction is true only if H3 is taken as a constant. For anharmonic atomistic systems, the Hamiltonians H2 and H3 are time varying as discussed previously, and thus both simultaneous creation and destruction events can occur without violating total energy (H) conservation. For example, if we consider the −2q mode in the FPU-α system (see Fig. 1) – this mode does not correspond to either a merging or a splitting mode, but it can arise through a simultaneous creation N process: 0 → q + q + (−2q). In the traditional viewpoint, each of the partitions of the total Hamiltonian (H2, H3, H4 …) is assumed to be constant; with such a restriction, energy is conserved for each partition of the Hamiltonian. The simulations on FPU chains (and graphene), however, show that the partitions of total Hamiltonian are not constant, but they fluctuate with time. Thus the anharmonic Hamiltonians (H3, H4 …) can borrow or return small amounts of energy to harmonic energy (H2) bank as mentioned previously. This energy exchange allows the process: 0 → q + q + (−2q), which is traditionally considered as a simultaneous creation process. We emphasize here that simultaneous creation or annihilation processes do not violate the energy conservation law since at all times the total Hamiltonian (H) remains constant. The corresponding U process: 0 → q + q + (−2q) + g is also discernible and its magnitude is comparable to the N process. Very feeble −q modes, which do not appear on the chosen scale in Fig. 1, also emerge in the FPU-α lattice. They can either be formed through a splitting process q → 2q + (−q) or through a decay process −2q → (−q) + (−q). Thus the number of possible normal mode interactions is greatly enhanced by the time-varying Hamiltonians.

For the FPU-β lattice, the −q mode, which is quite notable occurs through the decay process q → q + q + (−q). Thus the backward scattering normal mode −q appears prominent largely from the simplicity of the combination rule for a quartic Hamiltonian. The less conspicuous −3q mode, similar to the −2q mode in the FPU-α lattice, can arise from a spontaneous creation event: 0 → q + q + q + (−3q). The magnitude of the corresponding U processes for the negative modes with the FPU-β chain is insignificant for the short window of observation. With due passage of time, secondary events emerge leading to a plethora of normal mode interactions. Interestingly, negative −q (longitudinal acoustic) and −q (longitudinal optic) modes emerge in graphene following an excitation of a longitudinal LA mode (see Supplemental Information C and D). These modes ensue from four-phonon creation events: and , which are remarkably identical to those observed with the FPU-β lattice that interacts only through a quartic potential. Such simultaneous creation (and annihilation) events are facilitated only through the time varying harmonic and anharmonic Hamiltonians, and appear to be generic and not specific to a particular kind of interatomic potential or dimensionality.

Frequency of the excited energy oscillations

The temporal behavior of the anharmonic Hamiltonian is regarded to have only a weak dependence on the rate of change of the amplitude A(q, t). Inspection of Eq. (1) reveals that the rate of energy that is exchanged in a normal mode interaction will then be proportional to the corresponding cosine term. For example, the energy corresponding to the most prominent 2q mode in the FPU-α lattice that is formed by the merging event: q + q → 2q + g, is expected to oscillate with a distinctive frequency: 2w(q) − w(2q). Similarly, the expected frequency of the energy oscillations for the −2q mode will be: 2w(q) + w(2q). It is possible, therefore, to estimate the anticipated frequencies of all the modes that can be rationalized from crystal momentum conservation. Numerically, the excited frequencies of the energy exchange rate can be computed by evaluating the normalized Fourier transform of the deviation of the energy associated with each mode involved in any given interaction. In Fig. 3, we compare the numerical and the theoretical values of energy oscillations for the 2q mode (left) and the −2q mode (right). Impressive agreement is observed between the theoretical predictions and the results from the simulations for the particular frequency set: 2w(q) ± w(q), which corresponds to ±2q modes. Although not shown, we have verified that the conformity is excellent for the other pathways in the FPU systems and graphene. Thus we can identify the frequencies of the energy oscillations of the normal modes that participate in different anharmonic interactions using simulations, as well as predict them with reasonable fidelity from the structure of the anharmonic Hamiltonian that is cast using the real asymmetric normal mode amplitudes. While the conservation of crystal momentum can allow several possibilities, the frequency analysis of the anharmonic energy oscillations can assist in isolating the pertinent normal mode interactions.

Figure 3

Comparison between the theoretical and computed frequencies of the modal energy oscillations in a FPU-α lattice. The left panel corresponds to the energy of mode 2q due to the merging interaction: q + q → 2q + g and the right panel corresponds to the energy of mode −2q from the simultaneous creation of three normal modes: 0 → q + q + (−2q) + g. The expected theoretical frequencies are depicted by the open circles while the computed frequencies are delineated by the short horizontal line segments, which are mostly coincident with the open circles. The color scale corresponds to the magnitude of the normalized Fourier transform of the deviation of the energy associated with the 2q mode (left) and the −2q mode (right).

Quantum-like energy exchange among the normal mode interactions

The structure of the anharmonic Hamiltonians also embodies information on energy exchange between different modes. With Eq. (1) as reference, it is evident that for H3(t) to remain finite on an average, the arguments of the cosine functions should tend to become zero. The simulations demonstrate that the mean value of H3(t) is finite and nearly a constant when averaged over a certain time period or different initial phases (see Fig. 2); in this regard, the simulated FPU systems are ergodic with the chosen parameters across the period of observation. With vanishing cosine arguments we observe the emergence of quantum-like frequency conditions: w1 ± w2 ± w3 = 0. Interestingly, this so-called resonant condition cannot be satisfied for finite wave-vectors for a FPU-α lattice along a particular phase space trajectory. However, the normal mode interactions, when averaged over time (or phase space), can mimic the distinctive energy conservation rule for quantum phonon interactions.

One final question that remains is on the energy interchange between the discrete normal mode interactions in any given phase space trajectory. Taking advantage of the symmetry but not the equivalence between ±nq modes, we now proceed to make a quantitative comparison of the ratio of energies exchanged in specific sets of normal mode interactions. We now go back to the protocol that has generated the modes shown in Fig. 1 where each mode q is independently perturbed by excess energy E. The externally perturbed mode q can hold on to this energy E only momentarily as plane wave modes cannot be sustained indefinitely in an anharmonic lattice. When the crystal momentum conditions are satisfied, the modes interact through the time-dependent anharmonic Hamiltonian that takes finite values. Since the normal modes exchange energy continuously, only a global energy balance can be performed; we therefore choose the most dominant modes following the excitation. The energy transfer for the dominant ±2q modes in the FPU-α lattice, and ±q and ±3q modes in the FPU-β lattice is depicted pictorially in Fig. 4. For the FPU-α lattice, the energy conservation is stated as: E = E + ΔE(q) − ΔE(q), where ΔE(q) and ΔE(q) are the energy gained by mode q in creating the −2q mode, and the energy lost by the mode q in establishing a 2q mode, respectively. The energies of ±nq modes are not the same with the real asymmetric normal mode amplitudes but the symmetry of the dispersion relationship grants the condition: λ = ΔE(2q)/ΔE(q) = ΔE(−2q)/ΔE(q). Note that we have only assumed that the energy exchanged by a particular normal mode is a unique function of its frequency. We thus can express λ, which is the ratio of energy exchanged during the modal interactions as: λ−1 = (E − E)/(ΔE(−2q) − ΔE(2q)). The question that we are seeking now is whether this ratio is proportional to the ratio of the corresponding frequencies as prescribed by the quantum theory.

Figure 4

Pictorial representation of the energy transfer for the dominant ±2q modes in the FPU-α lattice (left), and ±q and ±3q modes in the FPU-β lattice (right), immediately following a single mode (q) perturbation.

In Fig. 5 (left), we plot λ for different excited modes along with the expected frequency ratio ζ = w(2q)/2w(q) = w(−2q)/2w(q) that is accessible from the phonon dispersion relationship. We can observe an impressive agreement between the ratios λ and ζ. We further test the proportionality of energy interchange in a FPU-β lattice, for which ±3q and ±q modes are most prominent (see Figs 1 and 4). The energy balance now has to account for both the interactions that have similar energies and can be written as: E = E + ΔE1(q) + ΔE2(q) − ΔE(q), where ΔE1(q) and ΔE2(q) are the energies gained by mode q in forging the −3q mode and −q mode, respectively, and ΔE(q) is the energy lost by mode q in establishing a 3q mode. As before, by letting λ = ΔE(3q)/ΔE(q) = ΔE(−3q)/ΔE1(q), and ΔE2(q)/ΔE(−q) = 1, we arrive at: λ−1 = (E − E − ΔE(−q))/(ΔE(−3q)  − ΔE(3q)). In Fig. 5 (right panel), we compare the value of λ with ζ = w(3q)/3w(q) = w(−3q)/3w(q); Both ratios are accessible, λ from the simulations and ζ from the phonon dispersion relationship. The ratios again show a near-perfect agreement, which highlights the quantum-like proportionality of energy exchange between the normal modes for the FPU-β lattice. If the observation window is extended, more interactions would take place and a similar analysis would need a careful consideration of all the dominant processes. The remarkable agreement between the two ratios λ and ζ for all excited modes (q) reveals a rather curious nature of normal mode interactions. Although the energy of the normal modes is proportional to the square of the eigen frequencies, the energy exchanged during discrete interaction events is proportional to the corresponding frequencies. Thus the ratio of energy interchange among the participating normal modes appears in the same proportion as that stipulated by quantum rules.

Figure 5

(Left) Ratio (λ) of the energy gained by mode 2q in the Class I merging process: q + q → 2q + g to that lost by mode q, or of the energy gained by mode −2q in the simultaneous creation event: 0 → q + q + (−2q) + g to that gained by mode q, in a FPU-α lattice. The quantum-like expected ratio (ζ) is depicted by the open squares. (Right) Corresponding ratios for the FPU-β lattice with the dominant processes q + q + q → 3q + g, 0 → q + q + q + (−3q) and q → q + q + (−q). The agreement between the energy ratio and the expected frequency ratio ζ shows a surprising quantum-like energy exchange between the normal modes.

Discussion

While phonons are doubtless quantum objects, their roots are embedded in the vibrational normal modes that exchange energy continuously. A fundamental question naturally arises: under what conditions can the normal modes be treated as perfect quantum entities? In the quantum scattering theory, energy is not continuously exchanged and the discrete events that materialize from interactions are not necessarily well-defined. Several approximations are made in the scattering theory – the main one being the existence of well-defined quantum states before and after scattering. Therefore, the scattering theory presumes a well-defined set of plane waves changing sharply to another following an idealized scattering event, conserving both energy and momentum[3]. The theory, however, does not stipulate the nature of the event or the time duration for such an event to take place; thus multiple virtual states that can violate energy conservation are naturally accommodated within a composite scattering event that includes several multi-phonon processes. Although the elementary phonon interactions are usually limited to first order in the perturbing Hamiltonian, higher order perturbation expansions can lead to multi-phonon processes. For example, an overall or composite four-phonon interaction can take place by combining two three-phonon interactions via an intermediate state[3]. While energy is conserved for the composite event, it is not, in general, for the virtual or intermediate states.

As highlighted in this work, the Hamiltonians (H2, H3, H4, etc.) of an anharmonic vibratory system, even at low thermal energies, are time-dependent by nature, and can generate a large number of pathways for normal modes to interact. A straightforward incorporation of the time-dependent Hamiltonians, say, from high-fidelity ab initio atomistic dynamics simulations at finite temperatures, into the quantum phonon scattering theory can provide a direct correspondence to the normal mode interactions. However, by taking advantage of the near-constancy of the Hamiltonians (H2, H3, etc.) over time or with different phase-space averaging – both are equivalent for ergodic systems – an alternate approach can be construed that is identical to the current method of using time-independent Hamiltonians. The interpretation of a phonon scattering event then becomes different. Instead of an elementary phonon event that is characterized by a first order process, say, a three phonon merging/decaying process that can be described with an exact time-independent Hamiltonian H3, the composite phonon interaction with an ensemble-averaged constant Hamiltonian