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Physical Review

Phonon-Phonon Interactions in Bose Systems

Mark Nelkin

Published Aug 15, 1962• 7 citations
Publisher

Abstract

The stability of phonons in Bose systems is considered. It is noted that the phonons in helium II are stable against decay into two or more phonons, in contrast to the low-density Bose gas where the lifetime for decay of one phonon into two is finite and has been calculated by several authors. The consequences of phonon stability for allowable mechanisms of phonon-phonon interaction are considered. It is concluded that virtual three-phonon processes may still occur although energy-conserving three-phonon processes cannot take place. The inclusion of such virtual processes makes it plausible that the interactions between excitations can be weak, even though the linewidths measured in neutron scattering are larger than the line shifts. The quantum hydrodynamics of Landau and Khalatnikov including nonlinear terms is considered as a calculational model. In this theory the stability of the phonons is indeterminate and must be legislated. If the phonons are assumed unstable against decay, the phonon lifetime in the low-density and long-wavelength limit can be calculated, and is in agreement with the result for the atomic theory of the low-density Bose gas. This lends considerable plausibility to the quantitative validity of the Landau-Khalatnikov calculation of phonon-phonon scattering in helium II via an almost energy-conserving intermediate one-phonon state. The paper concludes by comparing the results of an idealized neutron scattering experiment as calculated from the Feynman-Cohen point of view, and from the point of view of the theory of the low-density Bose gas.

DOI
10.1103/physrev.127.979

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The stability of phonons in Bose systems is considered. It is noted that the phonons in helium II are stable against decay into two or more phonons, in contrast to the low-density Bose gas where the lifetime for decay of one phonon into two is finite and has been calculated by several authors. The consequences of phonon stability for allowable mechanisms of phonon-phonon interaction are considered. It is concluded that virtual three-phonon processes may still occur although energy-conserving three-phonon processes cannot take place. The inclusion of such virtual processes makes it plausible that the interactions between excitations can be weak, even though the linewidths measured in neutron scattering are larger than the line shifts. The quantum hydrodynamics of Landau and Khalatnikov including nonlinear terms is considered as a calculational model. In this theory the stability of the phonons is indeterminate and must be legislated. If the phonons are assumed unstable against decay, the phonon lifetime in the low-density and long-wavelength limit can be calculated, and is in agreement with the result for the atomic theory of the low-density Bose gas. This lends considerable plausibility to the quantitative validity of the Landau-Khalatnikov calculation of phonon-phonon scattering in helium II via an almost energy-conserving intermediate one-phonon state. The paper concludes by comparing the results of an idealized neutron scattering experiment as calculated from the Feynman-Cohen point of view, and from the point of view of the theory of the low-density Bose gas.
This work investigates the theoretical and experimental decoherence of phonons in weakly and strongly-interacting Bose-Einstein Condensates (BECs). The theoretical analysis treats phonons as open quantum systems where the environment comprises all other quasi-particle modes of the BEC. The phonons are assumed to be Gaussian states and the time in which they decohere is estimated from the evolution of their purity and nonclassical depth in the dissipative channel. The calculations are performed for various BEC systems and it is found that the excited phonon states will always decohere much more rapidly than the rate at which they relax back to equilibrium with the environment. Part II of this work considers how the decoherence of the phonons can be measured experimentally. The experiment that is currently being investigated uses a strongly-interacting [superscript]6Li BEC where the scattering length can be varied with an external magnetic field. In a strongly-interacting Bose gas the mutual interaction between the condensed and noncondensed components plays a greater role than in weakly-interacting gases and results in distinct absorption images. Understanding the effects from this mutual interaction is vital to model the in-situ absorption images of a strongly-interacting BEC in order to extract accurate information such as that which will facilitate the experimental measurement of decoherence. Three theoretical models that could be used to fit in-situ absorption images of a strongly-interacting gas are analysed. These are the bi-modal, semi-ideal and Hartree-Fock models, which will be fit against the absorption images of the [superscript]6Li BEC for various scattering lengths. The validity of these models is also investigated to determine when beyond mean-field effects may be observed in the Bose gas. Controlling decoherence is essential to the operation and physical realisation of many quantum information tasks and quantum technologies. Recently, new technologies have emerged from relativistic quantum information science that, in principle, are more precise than their non-relativistic counterparts. The practical setup of these devices utilizes phonons of BECs but the decoherence of the phonons has not been considered. In this work, one of the BEC systems used to estimate the decoherence time of the phonons has been chosen to be based on these devices. This calculation is expected to inform the practical realization of these devices and inspire future related studies. Analogue gravity investigations based on BECs also utilize phonons, for example, in the analogue of Hawking radiation. The quantum properties of these states is of particular interest in these studies and understanding how they decohere, and at what rate, could potentially inform the theory of black hole physics as well as dictate what is possible to measure experimentally.
In a one-dimensional Bose gas, there is no nontrivial scattering channel involving three Bogoliubov quasiparticles that conserves both energy and momentum. Nevertheless, we show that such three-wave mixing processes (Beliaev and Landau damping) account for their decay via interactions with thermal fluctuations. Within an appropriate time window where the Fermi golden rule is expected to apply, the occupation number of the initially occupied mode decays exponentially and the rate takes a simple analytic form. The result is shown to compare favorably with simulations based on the truncated Wigner approximation. It is also shown that the same processes slow down the exponential growth of phonons induced by a parametric oscillation.
At any finite temperature, the correlation function $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(\mathcal{r})\ensuremath{\psi}(0)〉$ of a two-dimensional Bose system is known to approach zero as the separation r\mathcal{r}r approaches infinity. This paper is devoted to supporting the point of view that superfluid phenomena may still persist in such a system. To describe the low-temperature behavior of a condensed two-dimensional Bose system, we use a coherent-state representation of the low-lying phonon excitations. The expectation of the particle field operator $〈\ensuremath{\psi}(\mathcal{r})〉$ in a coherent phonon state of the system is shown to be a well-defined function. This function has only small fluctuations in its magnitude and, therefore, it is possible to make the usual arguments for superfluid behavior based on the existence of the condensate wave function. The large phase fluctuations of the function, when the coherentphonon states are collected into the thermal ensemble, give rise to the spatial decay of the correlation function or reduced density matrix $〈{\ensuremath{\psi}}^{+}(\mathcal{r})\ensuremath{\psi}(0)〉$. It is concluded that two-dimensional Bose systems are superfluid at sufficiently low temperatures, even in the inifinite-volume limit. An analogous argument would indicate the possibility of superconductivity in a two-dimensional Fermi system.
Abstract We study the interactions among phonons and the phonon lifetime in a pair‐condensed Fermi gas in the BEC‐BCS crossover in the collisionless regime. To compute the phonon‐phonon coupling amplitudes we use a microscopic model based on a generalized BCS Ansatz including moving pairs, which allows for a systematic expansion around the mean field BCS approximation of the ground state. We show that the quantum hydrodynamic expression of the amplitudes obtained by Landau and Khalatnikov apply only on the energy shell, that is for resonant processes that conserve energy. The microscopic model yields the same excitation spectrum as the Random Phase Approximation, with a linear (phononic) start and a concavity at low wave number that changes from upwards to downwards in the BEC‐BCS crossover. When the concavity of the dispersion relation is upwards at low wave number, the leading damping mechanism at low temperature is the Beliaev‐Landau process 2 phonons ↔ 1 phonon while, when the concavity is downwards, it is the Landau‐Khalatnikov process 2 phonons ↔ 2 phonons. In both cases, by rescaling the wave vectors to absorb the dependence on the interaction strength, we obtain a universal formula for the damping rate. This universal formula corrects and extends the original analytic results of Landau and Khalatnikov [ZhETF 19, 637 (1949)] for the 2↔2 processes in the downward concavity case. In the upward concavity case, for the Beliaev 1↔ 2 process for the unitary gas at zero temperature, we calculate the damping rate of an excitation with wave number q including the first correction proportional to q7 to the q5 hydrodynamic prediction, which was never done before in a systematic way.