Multistage dynamics of the spin-lattice polaron formation

We follow the formation of a spin-lattice polaron after a quantum quench that simulates absorption of the pump pulse in the time-resolved experiments. We discover a two-stage relaxation where spin and lattice degrees of freedom represent an integral part of the relaxation mechanism. In the first stage, the kinetic energy of the spin-lattice polaron relaxes towards its ground-state value. In the second, typically much longer stage, a subsequent energy transfer between lattice and spin degrees of freedom via the charge carrier emerges. The multistage relaxation is generic and occurs even when carriers couple to different subsystems with comparable strengths.

Figure 1

The evolution after the $t_0$ quench: time dependence of the change of the kinetic energy of the hole $\Delta E_{\mathrm{kin}}=\langle \psi \left(t\right)|H_{\mathrm{kin}}|\psi \left(t\right)\rangle -\langle \psi \left(0\right)|H_{\mathrm{kin}}|\psi \left(0\right)\rangle$ (note that we plot $-\Delta E_{\mathrm{kin}}$), total spin energy $\Delta E_{\mathrm{spin}}=\langle \psi \left(t\right)|H_J+H_h|\psi \left(t\right)\rangle -\langle \psi \left(0\right)|H_J+H_h|\psi \left(0\right)\rangle$, phonon energy $\Delta E_{\mathrm{phon}}=\langle \psi \left(t\right)|H_{\mathrm{ph}}+H_{\mathrm{EP}}|\psi \left(t\right)\rangle -\langle \psi \left(0\right)|H_{\mathrm{ph}}+H_{\mathrm{EP}}|\psi \left(0\right)\rangle$ for different values of $\lambda =g^2/2{\omega }_0{t}_0=0.0,0.025,0.1,0.2,0.3,$ and 0.4, $J=0.3$, $h=0.7J$, and ${\omega }_0=1.0$. Inset in (b) represents $\Delta {\tilde{E}}_{\mathrm{kin}}=\langle \psi \left(t\right)|H_{\mathrm{kin}}|\psi \left(t\right)\rangle -\langle {\psi }_{\mathrm{G}}|H_{\mathrm{kin}}|{\psi }_{\mathrm{G}}\rangle$, where $|{\psi }_{\mathrm{G}}\rangle$ is the respective ground-state wave function of the quenched Hamiltonians (with $t_0=1$). Vertical arrows in (a) indicate ${\tau }_{\mathrm{form}}$. Open slanted arrows indicate ordering of data according to increasing values of $\lambda$. In this and all subsequent figures, we measure time in units of $1/t_0=1$.

Figure 2

Hole-spin correlation function $C_s(t,j)$ for $J=0.3$, $h=0.7J$ and different values of $\lambda$. Full line indicates maximal free electron velocity $v_{\mathrm{free}}=2$ and the dashed line represents the magnon velocity $v_{\mathrm{mag}}$.

Figure 3

Density plots of the hole-phonon number correlation function $C_{\mathrm{ph}}(t,j)$ for different values of $\lambda$. The rest of parameters are the same as in Fig. 2. Dashed and dotted lines connect points of selected constant values of $C_{\mathrm{ph}}(t,j)$. Full line indicates maximal free electron velocity $v_{\mathrm{free}}=2$ and open circles present $j_{\mathrm{ave}}\left(t\right)$ as explained in the text.

Figure 4

$-\Delta E_{\mathrm{kin}}$ as defined in the main paper for $J=0.3$ and $\lambda =0.4$. The insert represents the deviation of $-\Delta E_{\mathrm{kin}}$ from results obtained using the largest Hilbert space with $N_h=19$, $N_{\mathrm{phmax}}=6$, and roughly $1.2\times {10}^7$ states. Note also the division of the time scale.