Tuning energy transport using interacting vibrational modes

We study energy transport in a chain of quantum harmonic and anharmonic oscillators where the anharmonicity is induced by interaction between local vibrational states of the chain. Using adiabatic elimination and numerical simulations with matrix product states, we show how strong interactions significantly slow down the relaxation dynamics (with the emergence of a different time scale) and can alter the properties of the steady state. We also show that steady-state properties are completely different depending on the order in which the limits of infinite time and infinite interaction are taken.

Figure 1

Depiction of the system under study: A chain of oscillators in which the ones at the extremities are local harmonic oscillators while those in the middle are anharmonic. The local oscillators are described by bosonic modes with a local potential ${\mu }_l$ and interaction strength $U_l$. Vibrations in each oscillator can tunnel to neighboring sites via the tunneling $J_l$. The harmonic oscillators are in contact with an environment through couplings ${\mathrm{\Lambda }}_l^{\sigma }$.

Figure 2

(a) Diagonal elements of the density matrix of the second site at steady state ${\rho }_{n,2}^{ss}$. The continuous black line represents the theoretical prediction; numerical results are represented by the symbols: black circle $\circ$ for $U=0$, red (gray) $\times$ for $U=3J$, and green (gray) $+$ for $U=6J$. The dissipation on the left sites are $\hslash {\mathrm{\Lambda }}_1^{+}=3J,\hslash {\mathrm{\Lambda }}_1^{-}=6J$. The dashed lines show the distribution for a local thermal distribution for $U=3J$ (green [gray] circles) and $U=6J$ (blue [gray] squares). (b) Probability of occupying level 2 of the oscillator on site 2, $\langle P_{2,2}\rangle$, vs time. The parameters used are $U=30J,\hslash {\mathrm{\Lambda }}_1^{+}=3J,\hslash {\mathrm{\Lambda }}_1^{-}=9J$ for the pink (light gray) stars; $U=40J,\hslash {\mathrm{\Lambda }}_1^{+}=3J,\hslash {\mathrm{\Lambda }}_1^{-}=9J$ for the red (gray) solid squares; $U=50J,\hslash {\mathrm{\Lambda }}_1^{+}=3J,\hslash {\mathrm{\Lambda }}_1^{-}=9J$ for the purple (gray) circles; $U=50J,\hslash {\mathrm{\Lambda }}_1^{+}=4J,\hslash {\mathrm{\Lambda }}_1^{-}=12J$ for the green (gray) $\times$; and $U=50J,\hslash {\mathrm{\Lambda }}_1^{+}=5J,\hslash {\mathrm{\Lambda }}_1^{-}=15J$ for the light blue (light gray) $+$.

Figure 3

(a) ${\overline{n}}_2$ and (b) local fluctuations ${\overline{\kappa }}_2$ as a function of $U_2$: The blue (gray) triangles are results from the numerical t-MPS method while the red (gray) circles are analytic predictions by adiabatic elimination. The dissipation coefficients used on the first and third sites are $\hslash {\mathrm{\Lambda }}_1^{+}=5J,\hslash {\mathrm{\Lambda }}_1^{-}=21.66J$ and $\hslash {\mathrm{\Lambda }}_3^{+}=5J$ and $\hslash {\mathrm{\Lambda }}_3^{-}=55J$. The insets represent the values of the density or fluctuations minus their asymptotic values for $U\to \infty ,{\overline{n}}_2^{\infty }$, and ${\overline{\kappa }}_2^{\infty }$. The blue (gray) dashed lines represent the numerical results while the red (gray) continuous lines are power-law fit with the exponent $-2$.

Figure 4

Local density ${\langle {\widehat{n}}_l\rangle }_{ss}$ (a) and local energy ${\langle {\widehat{h}}_l\rangle }_{ss}$ (b) vs the site number for a chain of 4 sites. The values of the interaction $U_2=U_3$ are $U_2=0$ red (gray) stars, $U_2=5J$ black circles, $U_2=10J$ blue (gray) diamonds, $U_2=25J$ purple (gray) squares, and $U_2=35J$ brown (dark gray) triangles. The dashed line represent the results from an hardcore bosons study. (c) Particle current $j^n$ (blue [gray] circles) and energy current $j^h$ (red [gray] diamonds) vs the site number for a chain of 4 sites. The red (gray) dotted and the blue (gray) continuous lines represent the hardcore bosons limit respectively for the energy and the particle currents. Other parameters are $\hslash {\mathrm{\Lambda }}_1^{+}=J,\hslash {\mathrm{\Lambda }}_1^{-}=4.33J,\hslash {\mathrm{\Lambda }}_3^{+}=J,\hslash {\mathrm{\Lambda }}_3^{-}=11J$ and number of sites $L=4$.