Thermal transport in out-of-equilibrium quantum harmonic chains

We address the problem of heat transport in a chain of coupled quantum harmonic oscillators, exposed to the influences of local environments of various nature, stressing the effects that the specific nature of the environment has on the phenomenology of the transport process. We study in detail the behavior of thermodynamically relevant quantities such as heat currents and mean energies of the oscillators, establishing rigorous analytical conditions for the existence of a steady state, whose features we analyze carefully. In particular, we assess the conditions that should be faced to recover trends reminiscent of the classical Fourier law of heat conduction and highlight how such a possibility depends on the environment linked to our system.

Figure 1

Schematic representation of the system. A chain of coupled harmonic oscillators interacting according to an RWA model.

Figure 2

Distribution of mean occupation numbers for the elements of a chain of two lengths. The (green) diamond-shaped points correspond to a chain of $n=25$ with ${\overline{N}}_A=2{\overline{N}}_B=10{\overline{N}}_k=100$. The (blue) square points are for $n=50$ with ${\overline{N}}_A=2{\overline{N}}_B=10{\overline{N}}_k=100$. Finally, the (violet) dots are for $n=50$ with ${\overline{N}}_A={\overline{N}}_B=10{\overline{N}}_k=100$. The remaining parameters are $\Omega /\omega =1/2, \zeta /\omega ={\zeta }_A/\omega ={\zeta }_B/\omega =1/10$, and $\hslash =1$. For all the three plots $k=1,..,n$.

Figure 3

Currents across the chain. The (green) diamond-shaped points correspond to a chain of $n=25$ with ${\overline{N}}_A=2{\overline{N}}_B=10{\overline{N}}_k=100$. The (blue) square points are for $n=50$ with ${\overline{N}}_A=2{\overline{N}}_B=10{\overline{N}}_k=100$. Finally, the (violet) dots are for $n=50$ with ${\overline{N}}_A={\overline{N}}_B=10{\overline{N}}_k=100$. The remaining parameters are as in Fig. 2.

Figure 4

Mean value of energy (left) and currents (right) for fixed-position oscillators [the first $(k=1)$ and the ${10}^{\text{th}}(k=10)]$, the midpoint (bulk) $k=n/2$, and the last oscillator $k=n$. We consider ${\overline{N}}_A=2{\overline{N}}_B=10{\overline{N}}_k=100$; the remaining parameters are the same as in Fig. 2.

Figure 5

Distribution of mean occupation numbers for the elements of a chain without the diffusive reservoirs. Diamonds (green): chain with $n=25$ oscillators and temperatures of the baths given in terms of ${\overline{N}}_1=2{\overline{N}}_n=100$ and ${\overline{N}}_k=30$. Squares (blue): chain with $n=50$ oscillators and ${\overline{N}}_1=2{\overline{N}}_n=100$ and ${\overline{N}}_k=30$. Circles (violet): $n=50$ and ${\overline{N}}_1={\overline{N}}_n=100$ and ${\overline{N}}_k=30$. The remaining parameters are the same as in Fig. 2 except for ${\zeta }_A$ and ${\zeta }_B$, which are taken to be zero and for all three plots $k=2,..,n-1$.

Figure 6

Currents across the chain without the diffusive reservoirs. The color-and-symbol code is the same as in Fig. 5, although we have taken $k=2,\cdots ,n-1$, i.e., we have excluded the positive currents from the first and last reservoir.

Figure 7

Currents across a chain of $n=25$ oscillators with ${\overline{N}}_8=30$ and ${\overline{N}}_k=10,\forall k\ne 8$. The remaining parameters are the same as in Fig. 2. Inset: Mean occupation number of the oscillators in the chain.

Figure 8

Distribution of mean occupation numbers for the elements of a chain of $n=25$ oscillators attached only to two end-chain diffusive reservoirs. We sample the dynamics at the instants of time $t=1,10,20,30$. The chain is initially prepared in its vacuum state. We have taken ${\zeta }_{A,B}=1/10$ and ${\overline{N}}_A=2{\overline{N}}_B=100$. As for the other parameters of the system, we have $\Omega /\omega =1/2$ and $\hslash =1$.

Figure 9

Distribution of mean occupation numbers for the elements of a chain connected to two ordinary end-chain reservoirs and $n$ dephasing reservoirs. Diamonds (green): chain with $n=25$ oscillators and bath mean occupation numbers ${\overline{N}}_1=2{\overline{N}}_n=100$. Squares (blue): chain with $n=50$ oscillators and ${\overline{N}}_1=2{\overline{N}}_n=100$. Circles (violet): $n=50$ and ${\overline{N}}_1=4{\overline{N}}_n/3=100$. All other parameters are as in Fig. 2.

Figure 10

Current across the chain with two ordinary end-chain reservoirs, plotted as a function of the number of oscillators $n$ and the number of dephasing reservoirs $k$. We highlight the curves describing the extreme cases $k=0$ and $k=n$, which are the functions reported, respectively, in Eq. (43) with ${\tilde{N}}_1=2{\tilde{N}}_2=100$ and in Eq. (46) with ${\overline{N}}_1=2{\overline{N}}_2=100$. The dephasing coupling is $\gamma /\omega =0.5$ while the remaining parameters are the same as in Fig. 2.

Figure 11

Distribution of mean occupation numbers for the elements of a chain with distinct coupling constants. Inset: Set of coupling constants used in the simulation. The remaining parameters are as Fig. 2.

Figure 12

Currents across the chain for the same situation depicted in Fig. 11.

Figure 13

Mean value of energy (left) and currents (right) for fixed position oscillators [the first $(k=1)$ and the ${10}^{\text{th}}(k=10)]$, the midpoint (bulk) $k=n/2$, and the last oscillator $k=n$ for a chain with distinct constant couplings. Bottom panel: Set of coupling constants used in the simulation. We consider ${\overline{N}}_A=2{\overline{N}}_B=10{\overline{N}}_k=100$. The remaining parameters are the same as in Fig. 2.

Figure 14

Distribution of mean occupation numbers for the elements of a chain of two lengths for the SMC Hamiltonian. Diamonds (green): chain of $n=25$ with ${\overline{N}}_A=2{\overline{N}}_B=10{\overline{N}}_k=100$. Squares (blue): chain of $n=50$ with ${\overline{N}}_A=2{\overline{N}}_B=10{\overline{N}}_k=100$. Circles (violet): $n=50$ with ${\overline{N}}_A={\overline{N}}_B=10{\overline{N}}_k=100$. The remaining parameters are $\kappa /\omega =1/2, \zeta /\omega ={\zeta }_A/\omega ={\zeta }_B/\omega =1/10$, and $\hslash =1$.

Figure 15

Currents across the chain for the SMC Hamiltonian. We used the same parameters as in Fig. 14.