Impulsive energy decay at a displaced lattice atom: robustness of Schwinger's 1990 phonon-dynamics predictions beyond the separable, narrow-band model

When a single atom of a crystal in its phonon vacuum is impulsively displaced and released, how does the deposited energy leave the displaced site? In a 1990 note, Julian Schwinger answered this question exactly within a highly idealized model: a separable phonon dispersion linear in the wave-vector components, a thermodynamic limit, and a narrow frequency spectrum. The predictions are striking - a long-time envelope with $[\sin(\pi u t/2a)]^6$ structure whose overall scale decays as an inverse sixth power of time, complete return of the site energy to zero at integer multiples of the transit time $2a/u$, and a decomposition of the deposited energy into eight pulses, one per octant, each carrying one eighth of the energy. The physically important regime in which the displacement lies outside the small-amplitude range, where lattice anharmonicity enters, was explicitly deferred. No later work has addressed the deferred anharmonic regime or the robustness of the model's predictions under realistic non-separable dispersions, multiple phonon branches, finite crystal size, or wide spectra. The problem remains open.

Background

Phonons - the normal-mode quanta of the harmonic vibration of a crystal - admit two equivalent descriptions: delocalized modes labeled by a wave vector $\mathbf{k}$ and a branch index, or localized constituents in which each atom carries oscillator variables. In a pair of notes, Schwinger developed the localized description and used it to analyze a sharply posed dynamical question. An atomic lattice, initially in its phonon vacuum (the ground state of the harmonic crystal), is excited by rapidly displacing one atom through a vector distance $D$ and releasing it; the energy $E$ thereby deposited spreads through the lattice, and the quantity of interest is $E_a(t)$, the energy remaining at the displaced site after time $t$.

Within the harmonic theory the answer is exact and reduces to spectral averages over the phonon dispersion relation $\omega(\mathbf{k})$. Concretely, $E_a(t)$ is the harmonic energy assigned to the displaced constituent (mass $M$) in the localized description, evaluated on the exact solutions $(r - r_0)(t)_a = D\,\langle\cos\omega t\rangle$ and $p(t)_a = M D\,\frac{d}{dt}\langle\cos\omega t\rangle$ of the lattice equations of motion (Schwinger 1990, Eqs. 7-8), so that $E_a(0) = E = \frac{1}{2} M D^2 \langle\omega^2\rangle$. To obtain closed forms, Schwinger restricted the model in three ways. First, the displacement $D$ is taken large on the scale of vacuum fluctuations, and the regime in which $D$ lies outside the small-amplitude range - where the harmonic approximation to the interatomic forces fails - was expressly "deferred for later study". Second, the dispersion is taken separable and linear in the wave-vector components, $\omega(\mathbf{k}) = \bar\omega + u(|k_x| + |k_y| + |k_z|)$, with the allowed wave vectors forming a grid of $\nu^3 = N$ points that is then taken to the large-$\nu$ (thermodynamic) limit. Third, the spectrum is narrow, $\langle\omega\rangle \gg u/a$, where $a$ is the lattice constant and angle brackets denote averages over the allowed wave vectors. Under these restrictions the long-time behavior of $E_a(t)/E$ is governed by an envelope that combines a periodic factor $[\sin(\pi u t/2a)]^6$, vanishing at integer multiples of the transit time $2a/u$, with an overall inverse-sixth-power decay in $t$; $E_a(t)$ itself returns to zero at integer multiples of $2a/u$. The deposited energy propagates as $2^3 = 8$ phonon pulses, one per octant of the body diagonals, each carrying energy $E/8$ at speed $\sqrt{3}\,u$.

The underlying physical question - how quickly, and in what form, energy deposited at a single lattice site leaves that site - is broader than this model, and a substantial neighboring literature studies energy spreading from localized initial excitations in harmonic, disordered, and anharmonic chains, as well as the decay of quantum emitters coupled to structured baths. Schwinger's treatment is distinguished by being fully quantum-mechanical, exact within the linear theory, and explicit in the time domain at the injecting site itself, rather than in the spatial energy profile. His subsequent note of 1991 recast the energy-input relation of the 1990 analysis as an exact vacuum expectation value of the source-theory Green's function, independent of the large-$D$ assumption, and connected it with the recoilless (Mössbauer) emission of a nucleus in a lattice, but did not revisit the time-dependent decay. The three notes were written during the episode in which Schwinger was pursuing a phonon-based treatment of the reported cold-fusion claims; further papers in the series were rejected and never published, and the program ended with his death in 1994. What remains is the question that the 1990 note left precisely posed but answered only under its idealizations.

Problem Statement

Determine the time dependence of the energy $E_a(t)$ remaining at the site of an atom that has been impulsively displaced by a vector $D$ and released in a crystal initially in its phonon vacuum (the ground state of the harmonic crystal), beyond the idealizations of Schwinger's 1990 analysis. Throughout, $E = \frac{1}{2} M D^2 \langle\omega^2\rangle$ is the total energy deposited by the displacement ($M$ is the atomic mass), $a$ is the lattice constant, $u$ is the velocity parameter of the idealized separable dispersion $\omega(\mathbf{k}) = \bar\omega + u(|k_x| + |k_y| + |k_z|)$, and $\langle\omega\rangle$ is its band-averaged frequency. Concretely: (1) Robustness within the harmonic theory. For realistic, non-separable phonon dispersion relations with multiple branches, and for finite crystals of $N$ atoms, establish which of the model's predictions - the long-time envelope with $[\sin(\pi u t/2a)]^6$ structure and inverse-sixth-power decay in $t$, the return of $E_a(t)$ to zero at integer multiples of the transit time $2a/u$, the decomposition into eight pulses of energy $E/8$ along the body diagonals, and the numerical prefactors - survive, in what modified form, and which are artifacts of the separable, narrow-band model; extend the analysis outside the narrow-spectrum regime $\langle\omega\rangle \gg u/a$. (2) The deferred anharmonic regime. Analyze $E_a(t)$ when $D$ lies outside the small-amplitude range, where lattice anharmonicity invalidates the linear theory, as Schwinger explicitly deferred. Determine quantitative laws or tight asymptotic characterizations of $E_a(t)$ in these regimes - non-separable and multi-branch harmonic lattices, finite crystals of $N$ atoms, the wide-spectrum regime outside $\langle\omega\rangle \gg u/a$, and the anharmonic regime - and identify which qualitative features of the idealized model are universal consequences of harmonic lattice dynamics and which are not.

References

1. Julian Schwinger, "Phonon representations.", Proceedings of the National Academy of Sciences 87(18):6983-6984, 1990. DOI: 10.1073/pnas.87.18.6983, https://doi.org/10.1073/pnas.87.18.6983
2. Julian Schwinger, "Phonon dynamics.", Proceedings of the National Academy of Sciences 87(21):8370-8372, 1990. DOI: 10.1073/pnas.87.21.8370, https://doi.org/10.1073/pnas.87.21.8370
3. Julian Schwinger, "Phonon Green's function.", Proceedings of the National Academy of Sciences 88(15):6537-6539, 1991. DOI: 10.1073/pnas.88.15.6537, https://doi.org/10.1073/pnas.88.15.6537
4. Kimball A. Milton, "My years with Julian Schwinger: From source theory through sonoluminescence", in Julian Schwinger Centennial Conference (World Scientific, 2019), pp. 221-235; preprint 2018. arXiv:1808.01981, DOI: 10.1142/9789811213144_0015, https://arxiv.org/abs/1808.01981