Spatiotemporal evolution of polaronic states in finite quantum systems

We study the quantum dynamics of small polaron formation and polaron transport through finite quantum structures in the framework of the one-dimensional Holstein model with site-dependent potentials and interactions. Combining Lanczos diagonalization with Chebyshev moment expansion of the time evolution operator, we determine how different initial states, representing stationary ground states or injected wave packets, after an electron-phonon interaction quench, develop in real space and time. Thereby, the full quantum nature and dynamics of electrons and phonons is preserved. We find that the decay out of the initial state sensitively depends on the energy and momentum of the incoming particle, the electron-phonon coupling strength, and the phonon frequency, whereupon bound polaron-phonon excited states may emerge in the strong-coupling regime. The tunneling of a Holstein polaron through a quantum wall or dot is generally accompanied by strong phonon number fluctuations due to phonon emission and reabsorption processes.

Figure 1

Time dependence of the (top) particle density and (bottom) phonon number at the deformable “molecular crystal” site 9, starting out from the free-electron ground state of a 17-site chain with OBCs. The phonon frequency ${\omega }_0=0.3$. The EP coupling ${\varepsilon }_{p,9}$ is switched on at $t=0$, and different curves belong to ${\varepsilon }_{p,9}$ values increased by 0.1. In the numerical calculations we take into account up to $M=100$ phonons and $N_{\mathrm{C}}=10$ Chebyshev moments, and we use a time step $\Delta t/{\tau }_e=0.01$.

Figure 2

Time dependence of the (top) particle density and (bottom) phonon number at molecular crystal site 9 in the intermediate EP coupling adiabatic regime. The initial state is the same as in Fig. 1; ${\varepsilon }_{p,9}$ now is increased in steps of 0.04.

Figure 3

Time dependence of the (top) particle density and (bottom) phonon number at site 9 in the strong EP coupling adiabatic regime. Initial state is as in Fig. 1; ${\varepsilon }_{p,9}$ is increased in steps of 0.32.

Figure 4

Contribution of the $m$-phonon state to $|\psi (t)\rangle$ at different times $t$ (given in units of ${\tau }_e$). Results for various ${\varepsilon }_{p,9}$ were compared to the phonon distribution function of the system’s ground state (${\varepsilon }_{p,9}>0\forall t$, no interaction quench). The phonon frequency is ${\omega }_0=0.3$.

Figure 5

Electron densities on the one-dimensional chain at different times $t/{\tau }_e$. Initial state is as before; ${\omega }_0=0.3$.

Figure 6

Time dependence of the (top) particle density and (bottom) phonon number at molecular crystal site 9 in the weak EP coupling antiadiabatic regime. The phonon frequency ${\omega }_0=8$. The EP coupling is switched on at $t=0$; different curves belong to ${\varepsilon }_{p,9}$ values increased by 0.1.

Figure 7

Time dependence of the (top) particle density and (bottom) phonon number at molecular crystal site 9 in the strong EP coupling antiadiabatic regime.

Figure 8

Time dependence of the (top) particle density and (bottom) phonon number at molecular crystal site 9 in the extremely strong EP coupling antiadiabatic regime. The EP interaction is increased in steps of ${\omega }_0$. For $N=21$ the particle and phonon numbers are displayed at site 11.

Figure 9

Contribution of $m$-phonon states to $|\psi (t)\rangle$ at different EP couplings and points of time. Results were compared to the stationary phonon distribution function of the system’s true ground state. The phonon frequency is ${\omega }_0=8$.

Figure 10

Spatiotemporal evolution of a free electron Gaussian wave packet injected with wave vector $K=\pi /2$ at $t=0$ into an 18-site molecular crystal chain with PBCs. Model parameters are ${\varepsilon }_p=1$ and ${\omega }_0=0.5$. Displayed is the time evolution (in units of ${\tau }_e$) of the local particle densities $\langle n_i\rangle$ (black solid lines, open circles), phonon numbers $\langle b_i^{†}{b}_i\rangle$ (red dot-dashed lines, stars), and local EP correlations ${\chi }_{i,i}$ (blue dashed lines).

Figure 11

Spatiotemporal evolution of a free electron Gaussian wave packet injected with (top) wave vector $K=\pi /9$ and (bottom) $K=\pi /2$ at $t=0$. Model parameters are ${\varepsilon }_p=3$ and ${\omega }_0=2$. Notation is as in Fig. 10.

Figure 12

(top) Time dependence of the total number of phonons excited in the molecular crystal chain as the free electron wave packet evolves into a polaron. (bottom) Time evolution of the kinetic energy part $E_{\mathrm{kin}}$.

Figure 13

Quantum dynamics of polaron formation and polaron tunneling through a potential barrier ${\Delta }_{12}={\Delta }_w=2$. Model parameters are ${\varepsilon }_p=1$, ${\omega }_0=2$. The barrier or quantum dot is located at site 12 and has a total EP coupling $({\varepsilon }_p+{\varepsilon }_{p,w})$. OBCs were used at sites 1 and 18. Notation is the same as in Fig. 11. For further explanation, see text.

Figure 14

Time dependence of the (top) particle density and (bottom) phonon number at the site of the wall.

Figure 15

Cumulated particle density to the right of the potential wall at (top) different wave vectors $K$ and (bottom) various EP couplings ${\varepsilon }_p$ for the same setup as in Fig. 13.