Relaxation Dynamics of the Holstein Polaron

Keeping the full quantum nature of the problem, we compute the relaxation time of the Holstein polaron in one dimension after it was driven far from the equilibrium by a strong oscillatory pulse. Just after the pulse, the polaron’s kinetic energy increases and subsequently exhibits a relaxation-type decrease with simultaneous emission of phonons. In the weak coupling regime, partial tunneling of the electron from the polaron self-potential is observed. The inverse relaxation time is for small values of electron-phonon coupling $\lambda$ linear with $\lambda$, while it deviates downwards from the linear regime at $\lambda \gtrsim 0.1/{\omega }_0$. The imaginary part of the equilibrium self-energy shows good agreement with the inverse relaxation time obtained from nonequilibrium simulations.

Figure 1

(a) Expectation values of different parts of the Hamiltonian vs time at $\lambda =0.1$ and ${\omega }_0=1.0$ after the action of $\varphi (t)={\Phi }_0\theta (t)$ for ${\Phi }_0=\pi$. (b) $\Delta E_{\mathrm{kin}}(t)$ vs $t$ for different ${\Phi }_0=\pi$, $3\pi /4$, $\pi /2$, $\pi /4$, while other parameters are the same as in (a). (c) Expectation values of different parts of the Hamiltonian vs time at $\lambda =0.1$ and ${\omega }_0=1$ after the action of the pulse, as given in Eq. (1), with $A=3$, where ${\omega }_p=1.5$ is set close to the maximum of the optical conductivity, while parameters $t_c=5$ and $t_p=2$ remain unchanged throughout this Letter. (d) $\Delta E_{\mathrm{kin}}(t)$ vs $t$ for different values of $A=1$, 3, 5; the rest is the same as in (c). The inset represents the difference between kinetic energy and $E_{\mathrm{kin}}(t\to \infty )$ for the same pulse amplitudes as the main figure. (e) Energy of the different parts of the Hamiltonian vs $t$ for $\lambda =1.0$ and ${\omega }_p=2.5$. (f) $\Delta E_{\mathrm{kin}}(t)$ vs $t$ for different values of $A=1$, 3, 5, while the rest is the same as in (e).

Figure 2

Electron-phonon correlation function $\gamma (x)$ for pulse amplitude $A=3$: (a) $\lambda =0.1$ and the pulse frequency ${\omega }_p=1.5$ and (b) $\lambda =1.0$ and ${\omega }_p=2.5$, at different times. Note that there are different scales on both figures.

Figure 3

Inverse relaxation time $1/\tau$ (squares) vs $\lambda$ for ${\omega }_0=1.0$ using a Gaussian pulse, Eq. (1). The dashed line represents $1/\tau =2{\omega }_0\lambda$; see the discussion in the text. The diamonds represent the imaginary part of the averaged equilibrium self-energy $\mathrm{Im}[\tilde{\Sigma }]$. The circles represent $1/\tau$ after the instantaneous pulse $\varphi (t)={\Phi }_0\theta (t)$, where pulse strength ${\Phi }_0=\pi$ was used. The horizontal dotted line indicates the inverse pulse width. The inset presents $1/\tau$ vs $\lambda {\omega }_0$, using the Gaussian pulse, Eq. (1), for ${\omega }_0=1$ (solid squares) and ${\omega }_0=0.7$ (open squares).