Phonon bottleneck in the low-excitation limit

The phonon-bottleneck problem in the relaxation of two-level systems (spins) via direct phonon processes is considered numerically in the weak-excitation limit where the Schrödinger equation for the spin-phonon system simplifies. The solution for the relaxing spin excitation $p\left(t\right)$, emitted phonons $n_{\mathbf{k}}\left(t\right)$, etc., is obtained in terms of the exact many-body eigenstates. In the absence of phonon damping ${\Gamma }_{\mathrm{ph}}$ and inhomogeneous broadening, $p\left(t\right)$ approaches the bottleneck plateau $p_{\infty }>0$ with strongly damped oscillations, the frequency being related to the spin-phonon splitting $\Delta$ at the avoided crossing. For any ${\Gamma }_{\mathrm{ph}}>0$, one has $p\left(t\right)\to 0$, but in the case of strong bottleneck, the spin relaxation rate is much smaller than ${\Gamma }_{\mathrm{ph}}$ and $p\left(t\right)$ is nonexponential. Inhomogeneous broadening exceeding $\Delta$ partially alleviates the bottleneck and removes oscillations of $p\left(t\right)$. The linewidth of emitted phonons as well as $\Delta$ increase with the strength of the bottleneck, i.e., with the concentration of spins.

Figure 1

Frequency eigenvalues ${\Omega }_{\mu }$ plotted vs their appropriately shifted number for different values of the bottleneck parameter $B$.

Figure 2

(Color online) Spinness of the spin-phonon eigenstates vs ${\Omega }_{\mu }$ for different values of $B$.

Figure 3

(Color online) Spin-phonon eigenstates ${\Omega }_{\mu }$ vs the average phonon detuning ${⟨{\omega }_{\mathbf{k}}-{\omega }_0⟩}_{\mu }$ that shows splitting of the spin-phonon eigenstates. The solid red line is the result of Eq. (30).

Figure 4

Spinness vs the average phonon detuning ${⟨{\omega }_{\mathbf{k}}-{\omega }_0⟩}_{\mu }$ that shows how both spin-phonon branches gradually change from pure phonon modes to pure spin modes.

Figure 5

Phonon dispersion of Eq. (62) showing the resonance phonon scattering on spins.

Figure 6

Bottlenecked relaxation of the spin excitation for different values of the bottleneck parameter $B$. The number of spins used in simulations are indicated, and the number of phonon modes were chosen appropriately. The dashed horizontal line is the large-time asymptote for $B=0.1$.

Figure 7

Bottleneck plateau $p_{\infty }$ vs $N_S∕(N_S+N_{\Gamma })=B∕(1+B)$. The asymptote $p_{\infty }=1-1∕\sqrt{2B}$ for $B⪢1$ is shown by the dashed line.

Figure 8

Bottlenecked relaxation of the spin excitation for $B=10$ and different values of the ad hoc phonon damping ${\Gamma }_{\mathrm{ph}}$.

Figure 9

Nonexponential relaxation of the spin excitation for $B=300$ and ${\Gamma }_{\mathrm{ph}}=10\Gamma$ in logarthmic scale.

Figure 10

Distribution of emitted phonons in the final state for different values of $B$: (a) in natural units; (b) normalized.

Figure 11

The bottleneck plateau in the case of inhomogeneous broadening with the Gaussian line shape vs $\delta {\omega }_0$.

Figure 12

The bottleneck plateau in the case of strong $(\delta {\omega }_0⪢\Gamma ,\Delta )$ inhomogeneous broadening with the Gaussian line shape vs the bottleneck parameter $B_{{\omega }_0}$.

Figure 13

Time dependence of the spin excitation $p\left(t\right)$ for the spins with frequencies around ${\omega }_0$ in the case of strong inhomogeneous broadening.

Figure 14

The same in the case of damped phonons. Again, for $B⪢1$, the effective spin relaxation rate is much smaller than the phonon relaxation rate ${\Gamma }_{\mathrm{ph}}$.