Nonlinear transport in an out-of-equilibrium single-site Bose-Hubbard model: Scaling, rectification, and time dynamics

Recent experiments in hybrid-quantum systems facilitate the potential realization of one of the most fundamental interacting Hamiltonian-reservoir systems, namely the single-site Bose-Hubbard model coupled to two reservoirs at different temperatures. Using Redfield equations in a Born-Markov approximation, we compute nonequilibrium average particle number, energy, and currents beyond linear response regime, both time dynamics and steady state, and investigate its dependence on various tunable parameters analytically. We find interesting scaling laws in high-temperature regimes that are independent of choice of bath spectral functions. We also demonstrate that the system shows very interesting particle and energy current rectification properties which can be controlled via the relative strength of interaction and temperatures, as well as via the degree of asymmetry in system-bath coupling. Specifically, we find inversion of direction of energy rectification as a function of the relative strength of the interaction strength and the temperatures. We also show that, in the limit of low-temperature and high interaction strength, our results are consistent with the nonequilibrium spin-Boson model. Our results are experimentally relevant not only to hybrid quantum systems but also in other areas such as molecular junctions.

Figure 1

The plot of population density (diagonal elements of density matrix in eigenbasis of system Hamiltonian) of SSBH model in equilibrium and under thermal bias with assymetric system-bath coupling $\left({\mathrm{\Gamma }}_1=0.4,{\mathrm{\Gamma }}_2=1.6\right)$ and ${\mathrm{\Omega }}_0=1$. The top and bottom panels are for interchanged hot and cold baths. The three dotted lines (in each plot) are the corresponding equilibrium distributions ($T_1=T_2=3.5$), i.e., Eq. (8) for the values of $\chi$ mentioned in the legend. The deviation from equilibrium is more prominent in the bottom panel. For large interaction strength $\left(\chi =4\right)$, only two levels have non-negligible probability, like a spin-boson model. All energy variables are measured in units of ${\mathrm{\Omega }}_0$.

Figure 2

The plot shows scaling behavior of average occupation and average energy of the single-site Bose-Hubbard model for fixed $r=T_2/T_1=1/3$. Here ${\omega }_0={\mathrm{\Omega }}_0+\chi$. The dotted plots correspond to the NESB model. For $\chi \gg {\mathrm{\Omega }}_0 \left({\mathrm{\Omega }}_0=1\right)$, there is data collapse over the entire range of temperatures. Even for small $\chi (=0.1)$, there is substantial deviation from linear $\left(\chi =0\right)$ behavior. For high temperatures, $\langle \widehat{N}\rangle$ scales as $\sqrt{T_1/{\omega }_0}$, while, $\langle {\widehat{H}}_S\rangle$ scales as $T_1$. Here the system-bath coupling is taken as symmetric: ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=1$. All energy variables are measured in units of ${\mathrm{\Omega }}_0$.

Figure 3

Plot of particle (top panel) and energy (bottom panel) currents as a function of interaction strength $\chi$ for transport under both forward and backward bias for Ohmic baths $[s=1$ in Eq. (5)] under asymmetric system bath coupling $\left({\mathrm{\Gamma }}_1=0.4,{\mathrm{\Gamma }}_2=1.6\right)$. Here the mean temperature $T_m=(T_1+T_2)/2=5$ for both plots. The two black dotted lines in each plot correspond to the NESB currents for $\mathrm{\Delta }T=\pm 5$. The currents from our model match the NESB currents for large $\chi$. Particle current decreases with $\chi$, while energy current behaves nonmonotonically with $\chi$. The forward and backward currents do not match, thereby showing rectification effect. The direction of rectification of energy current is reversed beyond a value of $\chi$. It follows that at this value of $\chi$, the energy current does not show rectification. The insets show the corresponding currents as a function of $\mathrm{\Delta }T=T_1-T_2$ for a chosen value of $\chi =3$ and $T_m=5$. It can be seen that $I$ and $J$ deviate from odd-function behavior, which is a signature of rectification. Other parameters are $\varepsilon =0.1,{\omega }_c=1000$. All energy variables are measured in units of ${\mathrm{\Omega }}_0$, and time is measured in units of ${\mathrm{\Omega }}_0^{-1}$.

Figure 4

Panel (a) and (b) show the scaling behavior of $I/\mathrm{\Delta }T$ and $J/\left(\mathrm{\Delta }T{\omega }_0\right)$ for Ohmic baths $[s=1$ in Eq. (5)] and for fixed $r=\frac{T_2}{T_1}=\frac{1}{3}$. The dotted lines show the corresponding NESB result. The horizontal dashed lines in panel (a) and panel (b) correspond to $I/\mathrm{\Delta }T=J/\left(\mathrm{\Delta }T{\omega }_0\right)=A={\varepsilon }^2{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2/({\mathrm{\Gamma }}_1+{\mathrm{\Gamma }}_2)$, which is the high-temperature result for the harmonic oscillator $\left(\chi =0\right)$. For $\chi \gg {\mathrm{\Omega }}_0 \left({\mathrm{\Omega }}_0=1\right)$, there is a data collapse for all temperatures. The NESB result matches with the nonequilibrium SSBH model for small temperatures. At higher temperatures, the NESB result shows nonmonotonicity, which is not seen in the nonequilibrium SSBH model which demonstrates a stark difference between the two models. In panel (a), $I/\mathrm{\Delta }T$ approaches a constant value at high temperatures irrespective of the strength of interaction strength. In panel (b), even for small interaction strength $\left(\chi =0.1\right)$, substantial deviation from linear $\left(\chi =0\right)$ behavior is noticeable. Panel (c) shows data collapse of $J/\left(I{\omega }_0\right)$ for all temperatures for $\chi \gg {\mathrm{\Omega }}_0$. Irrespective of choice of bath spectral function and for fixed $r(=\frac{1}{3}),J/\left(I{\omega }_0\right)$ shows a data collapse and goes as $\sim \sqrt{T_1/{\omega }_0}$ for $T_1\gg {\omega }_0$. All observations are also valid in the linear response regime (i.e., $r\approx 1$) and give the temperature scaling of conductance. Other parameters are as follows: $\varepsilon =0.1,{\omega }_c=1000,{\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=1$. All energy variables are measured in units of ${\mathrm{\Omega }}_0$, and time is measured in units of ${\mathrm{\Omega }}_0^{-1}$.

Figure 5

The plot shows particle rectification $R_I$ (top) and energy rectification $R_J$ (bottom) as a function of asymmetry parameter $\gamma$ for various values of interaction strength $\chi$ and Ohmic baths $[s=1$ in Eq. (5)]. $R_I$ and $R_J$ are as defined in Eq. (38). The black dotted line corresponds to NESB. The rectfications become maximum when $\gamma \approx 0.6$. Also, rectification of both particle and energy currents show a nonmonotonic change with $\chi$. Rectification for large $\chi$ matches with the NESB result. Energy rectification changes direction with increase in $\chi$. Other parameters are as follows: $\varepsilon =0.1,{\omega }_c=1000$. All energy variables are measured in units of ${\mathrm{\Omega }}_0$, and time is measured in units of ${\mathrm{\Omega }}_0^{-1}$.

Figure 6

The plot shows rectification of particle current (top panel) and of energy current (bottom panel) of SSBH model out-of-equilibrium for fixed $r=\frac{T_2}{T_1}=\frac{1}{3}$, for Ohmic baths $[s=1$ in Eq. (5)]. ${\omega }_0={\mathrm{\Omega }}_0+\chi$. $R_I$ and $R_J$ are as defined in Eq. (38). The vertical dashed line indicates the positions of $T_1/{\omega }_0=1$. The dotted plots correspond to the NESB model. For $\chi \gg {\mathrm{\Omega }}_0,({\mathrm{\Omega }}_0=1$), there is data collapse. Both $R_I$ and $R_J$ has a maximum for ${\omega }_0<T_1$. $R_J=0$ for ${\omega }_0\approx T_1$. $R_J$ shows reversal in direction of rectification beyond this point. Other parameters are as follows: $\varepsilon =0.1,{\omega }_c=1000$. All energy variables are measured in units of ${\mathrm{\Omega }}_0$ and time is measured in units of ${\mathrm{\Omega }}_0^{-1}$.

Figure 7

The figure shows time dynamics of (a) average occupation, $\langle N\left(t\right)\rangle$; (b) average energy, $\langle H_s\left(t\right)\rangle$; (c) particle current from left bath, $I_1\left(t\right)$; and (d) energy current from left bath, $J\left(t\right)$, for various values of interaction strength and for a temperature bias $T_1=4$ and $T_2=2$ for Ohmic baths $[s=1$ in Eq. (5)]. All physical quantities demonstrate a nonunitary evolution towards a steady state. The approach to steady state is faster for higher interactions. Since physical quantities plotted here are diagonal in the eigenbasis of the system Hamiltonian, none of them show oscillations with time. Other parameters are ${\mathrm{\Omega }}_0=1,\varepsilon =0.1,{\mathrm{\Gamma }}_1=0.4,{\mathrm{\Gamma }}_2=1.6,{\omega }_c=1000$. All energy variables are measured in units of ${\mathrm{\Omega }}_0$, and time is measured in units of ${\mathrm{\Omega }}_0^{-1}$.

Figure 8

The figure shows a log-log plot of time to reach NESS, $t_{\mathrm{ss}}$ [as defined in Eq. (41)] as a function of interaction strength $\chi$ for Ohmic baths $[s=1$ in Eq. (5)]. The horizontal dash-dotted line corresponds to ${\left[{\varepsilon }^2{\sum }_{\ell =1}^2{\mathcal{J}}_{\ell }\left({\mathrm{\Omega }}_0\right)\right]}^{-1}$, which is the $t_{\mathrm{ss}}$ for the linear $\left(\chi =0\right)$ system. For large $\chi ,t_{\mathrm{ss}}\approx {\left[{\varepsilon }^2\chi \left({\mathrm{\Gamma }}_1+{\mathrm{\Gamma }}_2\right)\right]}^{-1}$, and this is indicated by the dashed line. For intermediate $\chi ,t_{\mathrm{ss}}$ depends on the temperatures of the hot and cold baths. Other parameters are ${\mathrm{\Omega }}_0=1,\varepsilon =0.1,{\mathrm{\Gamma }}_1=0.4,{\mathrm{\Gamma }}_2=1.6,{\omega }_c=1000$. All energy variables are measured in units of ${\mathrm{\Omega }}_0$, and time is measured in units of ${\mathrm{\Omega }}_0^{-1}$.