Steady-state quantum transport through an anharmonic oscillator strongly coupled to two heat reservoirs

We investigate the transport properties of an anharmonic oscillator, modeled by a single-site Bose-Hubbard model, coupled to two different thermal baths using the numerically exact thermofield based chain-mapping matrix product states (TCMPS) approach. We compare the effectiveness of TCMPS to probe the nonequilibrium dynamics of strongly interacting system irrespective of the system-bath coupling against the global master equation approach in Gorini-Kossakowski-Sudarshan-Lindblad form. We discuss the effect of on-site interactions, temperature bias as well as the system-bath couplings on the steady-state transport properties. Last, we also show evidence of non-Markovian dynamics by studying the nonmonotonicity of the time evolution of the trace distance between two different initial states.

Figure 1

(a) A single site Bose Hubbard model is coupled to two baths at different temperatures: left bath temperature is $T_L$ and the right bath temperature is $T_R$; (b) schematics of thermofield-based transformation: The bath on each side is discretized and is mapped to two baths at zero temperature labeled as solid and hollow circles. The system is coupled to all bath modes on each side. (c) Star-to-chain mapping: The total system is mapped to one single chain on each side with next-nearest neighbor tunnelings.

Figure 2

Time evolution of system's occupation $\langle n_S\rangle$ for (a) different ratios of temperature biases $T_L/T_R=2,2.5$, and 4.0, where $T_R=0.125$. Lighter to darker colors represent the ratios $T_L/T_R$ from low to high. (b) Time evolution for low, intermediate, and high temperature regimes with the same temperature bias $T_L/T_R=2\text{:}{T}_L=0.125,T_R=0.0625,T_L=0.25,T_R=0.125$, and $T_L=0.5,T_R=0.25$. Lighter to darker colors represent the temperatures of both baths from low to high. Other parameters for (a) and (b) are $U=1.5$ and $\gamma =0.1225$. (c) Time evolution for low to high system interactions: $U=0,U=0.25$, and $U=2.25$ with the coupling $\gamma =0.105625,T_L=0.5$, and $T_R=0.25$. Lighter to darker colors represent $U$ from low to high. (d) Time evolution for different system-bath coupling constants $\gamma =0.09,0.1225,0.16$ for an interaction $U=1.5,T_L=0.5$, and $T_R=0.25$. Lighter to darker colors represent the system-bath coupling constants from low to high. Dotted lines correspond to results obtained using the global master equation, see Appendix pp2.

Figure 3

Time evolution of energy current (a) and (b) and particle current (c) and (d). Left panels (a) and (c) are for low and high temperature regimes with the same temperature bias $T_L/T_R=2$. Darker-colored curves are for $T_L=0.5$ and $T_R=0.25$, and lighter-colored curves are for $T_L=0.125$, and $T_R=0.0625$. The system-bath coupling strength is $\gamma =0.1225$. Right panels (b) and (d) are for different system-bath coupling strengths $\gamma$. Darker-colored curves are for $\gamma =0.16$, and lighter-colored curves are for $\gamma =0.09$. The bath temperature is as follows: $T_L=0.5$, and $T_R=0.25$. In all panels, solid lines are for currents from the left bath, and dot-dashed lines are for currents from the right bath. Other parameters used are as follows: $U=1.5$, and ${\omega }_c=2.5$.

Figure 4

Effect of interaction $U$ on (a) and (b) system occupation, (c) and (d) particle current, and (e) and (f) energy current for different magnitudes of the system-bath coupling constant $\gamma$. Left panels (a), (c), and (e) are for low temperatures $T_L=0.125,T_R=0.0625$ and right panels (b), (d), and (f) for high temperatures $T_L=0.5,T_R=0.25$. Curves from lighter to darker color in all panels correspond to smaller to larger system-bath coupling strengths with values of $\gamma =0.04,0.09$, and 0.16, respectively. Other parameters used are $\mu =1,{\omega }_c=2.5$.

Figure 5

Steady-state energy current as a function of system-bath coupling strength $\gamma$ for (a) low temperatures $(T_L=0.125,T_R=0.0625)$, (b) intermediate temperatures $(T_L=0.25,T_R=0.125)$, and (c) high temperatures $(T_L=0.5,T_R=0.25)$. The ratio of bath temperature bias is the same $(T_L/T_R=2)$ for all panels. In each panel, solid curves from lighter to darker green color indicate smaller to larger strength of interaction: $U=0,0.75$ and 2.25. The red dot-dashed lines indicate a linear fit for $U=0$ and weak system-bath coupling $\gamma <0.04$. In all panels, we used ${\omega }_c=2.5$.

Figure 6

Time evolution of trace distance of systems density matrix. (a) Role of system-bath coupling strength: The red dashed line represents smaller ($\gamma =0.04$), and the green solid line represents larger ($\gamma =0.18$) system-bath coupling strength. Here, $U=1.125$. (b) The role of interaction $U$: The red dashed line is for smaller interaction $(U=0.5)$, and the green solid line is for larger interaction $(U=1.4)$. Here, $\gamma =0.16$. In panels (a) and (b), $T_L=0.5,T_R=0.25$, and ${\omega }_c=1.5$. (c) The role of bath frequency cutoff ${\omega }_c$: The red dashed line is for lower cutoff $({\omega }_c=1.5)$, and the green solid line is for higher cutoff $({\omega }_c=1.75)$. Here, $T_L=0.125,T_R=0.0625,U=1.25$, and $\gamma =0.16$.

Figure 7

Time evolution of the system occupation $\langle n_S\rangle$ for different initial conditions: The green solid line is for one boson in the system $\left(\right|1\rangle )$ whereas the red dashed line is for an empty system at initial time $\left(\right|0\rangle )$. Other parameters used are as follows: $T_L=0.3125,T_R=0.125,\gamma =0.1225,U=1.5$, and ${\omega }_c=2.5$.