Towards a microscopic understanding of the phonon bottleneck

The problem of the phonon bottleneck in the relaxation of two-level systems (spins) to a narrow group of resonant phonons via emission-absorption processes is investigated from first principles. It is shown that the kinetic approach based on the Pauli master equation is invalid because of the narrow distribution of the phonons exchanging their energy with the spins. This results in a long-memory effect that can be best taken into account by introducing an additional dynamical variable corresponding to the nondiagonal matrix elements responsible for spin-phonon correlation. The resulting system of dynamical equations describes the phonon-bottleneck plateau in the spin excitation, as well as a gap in the spin-phonon spectrum, for any finite concentration of spins. On the other hand, it does not accurately render the line shape of emitted phonons and still needs improving.

Figure 1

(Color online) The phonon bottleneck happens for $B\gtrsim 1$, and it results in the reabsorption of the initially emitted phonons by spins. Fast phonon relaxation supresses the bottleneck.

Figure 2

Numerical solution of the dynamical equations for the phonon bottleneck with nonrelaxing phonons at $T=0$ for different values of the bottleneck parameter $B$. (a) Low initial excitation of spins, $p_0=0.1$. Asymptotes $p\left(\infty \right)$ given by Eq. (74) with $n_0=0$ are shown by dashed horizontal lines on the right. (b) High spin excitation, $p_0=1$. In the case (b) the solution diverges for $B\gtrsim 100$. The bottleneck plateau $p\left(\infty \right)$ grows with $B$ but it should not exceed $1∕2$.

Figure 3

Numerical solution for the spin excitation $p\left(t\right)$ of the dynamical equations with relaxing phonons, Eqs. (60), at $T=0$ for $B=10$ and different values of the phonon relaxation rate ${\Gamma }_{\mathrm{ph}}$. Note that ${\Gamma }_{\mathrm{ph}}⪢\Gamma$ is needed to suppress the bottleneck if $B⪢1$.

Figure 4

Numerical solution for $p\left(t\right)$ with relaxing phonons and $k_{B}T∕\left(\hslash {\omega }_0\right)=2$. Here the spins are initially in the ground state, $p_0=0$, and they absorb the energy from phonons. For ${\Gamma }_{\mathrm{ph}}=0$, depletion of the phonons prevents reaching the thermal equilibrium value of $p$ (the dotted horizontal line).

Figure 5

(Color online) Emitted phonons in the absence of the phonon relaxation at $T=0$ for a moderate bottleneck, $B=0.1$. (a) Time dependence $n_{\mathbf{k}}\left(t\right)$ at resonance remains oscillating long after the spin excitation $p\left(t\right)$ reached the bottleneck plateau. (b) Line shape of the emitted phonons (at the maximal time) is narrow but not everywhere positive, which indicates that Eqs. (57) needs improving.