Filling an empty lattice by local injection of quantum particles

We study the quantum dynamics of filling an empty lattice of size $L$ by connecting it locally with an equilibrium thermal bath that injects noninteracting bosons or fermions. We adopt four different approaches, namely, (i) direct exact numerics, (ii) Redfield equation, (iii) Lindblad equation, and (iv) quantum Langevin equation, which are unique in their ways for solving the time dynamics and the steady state. In this simple setup we investigate open quantum dynamics and subsequent approach to thermalization. The quantities of interest that we consider are the spatial density profile and the total number of bosons and fermions in the lattice. The spatial spread is ballistic in nature and the local occupation eventually settles down owing to equilibration. The ballistic spread of local density admits a universal scaling form. We show that this universality is only seen when the condition of detailed balance is satisfied by the baths. The difference between bosons and fermions shows up in the early time growth rate and the saturation values of the profile. The techniques developed here are applicable to systems in arbitrary dimensions and for arbitrary geometries.

Figure 1

Schematic of our setup: an empty (blue) one-dimensional lattice of size $L$ is connected to a reservoir (red) at a particular site. The reservoir is coupled to the lattice at a particular site, represented by $m$. The intersite hopping parameter within the lattice (within the reservoir) is $g\left(t_B\right)$. The coupling between the lattice and the reservoir is denoted by $\gamma$. The reservoir is maintained in equilibrium at an inverse temperature $\beta$ and chemical potential $\mu$.

Figure 2

Behavior of total occupation $N\left(t\right)$ [Eq. (7)] versus $t$ for system size $L=40$ for the bosonic case, using direct exact numerics as described in Sec. 2a. The bath is connected at site number $m=21$. The linear early-time growth and exponential late-time saturation can be clearly seen. Red dashed line shows the steady-state value $N_{\text{SS}}^{\text{QLE}}$ obtained from exact quantum Langevin equation approach, described in Sec. 2d. The parameters are $g=0.5,\gamma =1,t_B=1,\beta =1$, and $\mu =-2.01$. Note that the fitting parameters for the early-time linear growth $\overline{a}t+\overline{b}$ are $\overline{a}=0.058,\overline{b}=1.082$. For the late-time behavior, the fitting parameters for the exponential relaxation $N_{\text{SS}}^{\text{QLE}}-\overline{c}{e}^{-dt}$ are $\overline{c}=1.359,\overline{d}=4.95\times {10}^{-4}$ with $N_{\text{SS}}^{\text{QLE}}$ chosen to be same as the steady-state value obtained from QLE $(N_{\text{SS}}^{\text{QLE}}=9.731)$. This implies that for $L=40$, the timescale to reach steady state is $t_{\text{SS}}=1/\overline{d}\sim 2000$. The inset shows the plot for $[ln(N_{\text{SS}}^{\text{QLE}}-N\left(t\right)\left)\right]$ vs $t$ (green dots) and it clearly demonstrates the long-time exponential relaxation in time towards the steady state. We choose $\gamma =1$ (nonperturbative regime in system-bath coupling) to ensure that the system relaxes towards steady state relatively fast. For this figure, the system size $L=40$ is chosen such that both the early-time linear behavior and long-time exponential behavior are clearly visible.

Figure 3

Total occupation $N\left(t\right)$ [Eq. (7)] versus $t$ for different system sizes using direct numerics as described in Sec. 2a. The parameters used are $g=0.5,t_B=1,\beta =1,\mu =-2.01$, and $\gamma =1$ are exactly the same as in Fig. 2 except the system size. It can be seen that the deviation from the linear growth starts at a timescale that scales with the system size $L$. The fitting parameters are $\overline{a}=0.058,\overline{b}=1.082$.

Figure 4

Local density profile $n_i\left(t\right)$ [Eq. (6)] for bosons with $L=20$ lattice sites with bath attached at a particular site $\left(m=11\right)$ for various time snapshots, using direct numerics (empty circles), as discussed in Sec. 2a. The long-time limit of this density profile agrees perfectly with that obtained from QLE (cross), as discussed in Sec. 2d. The parameters are $t_B=1,g=0.5,\beta =1,\mu =-2.01$, and $\gamma =1$. Note that apart from the system size the parameters chosen here are exactly the same as in Figs. 2 and 3. The system size $L=20$ is chosen keeping in mind computational feasibility and to ensure a relatively quick approach to steady state.

Figure 5

Local density profile $n_i\left(t\right)$ [Eq. (6)] for bosons with $L=100$ sites with bath attached at $m=51$ from direct exact numerics, described in Sec. 2a. We choose relatively short times to clearly demonstrate the ballistic growth of the density profile. We truncate the $y$ axis to highlight the propagation of the density front. Note that this front of the density profiles spreads with velocity $c=2g$, where $g$ is the intersite hopping within the lattice. This figure clearly indicates the presence of scaling which is demonstrated in Fig. 6. The parameters are $t_B=1,g=0.25,\beta =1,\mu =-2.01$, and $\gamma =1$. Note that the parameters chosen here are exactly the same as in Figs. 2, 3, 4 except the value of $g$. $g=0.25$ is chosen here to in order to illustrate ballistic spreading over a computationally feasible system size $L$.

Figure 6

A scaled plot of $ln\left(n_i\left(t\right)\right)$ [Eq. (6)] for $L=100$ bosonic sites with bath attached at $m=51$ from direct exact numerics to demonstrate the ballistic spread of spatial density profile $n_i\left(t\right)$ [Eq. (6)]. The parameters are $t_B=1,g=0.25,\beta =1,\mu =-2.01$, and $\gamma =1$. The logarithm mentioned in this plot is with base 10.

Figure 7

Local density profile $n_i\left(t\right)$ [Eq. (6)] for bosons with $L=21$ lattice sites with bath attached at a particular site $\left(m=11\right)$ for various time snapshots $\left(t=400,t=4000\right)$ using direct exact numerics (cross), as discussed in Sec. 2a. We show good agreement of these results with that obtained using the Redfield equation approach (squares), discussed in Sec. 2b. The inset shows total occupation $N\left(t\right)$ [Eq. (7)] for $t=1000,2000,3000,4000$ with both direct exact numerics (Sec. 2a) and Redfield (Sec. 2b) approaches. The slope obtained from this plot perfectly matches with the slope extracted following Eq. (47). The parameters are $t_B=1,g=0.5,\beta =1,\mu =-2.01$, and $\gamma =0.01$. Note that as $\gamma =0.01$ and $g=0.5$, the Redfield approach is well suited for comparison with the direct exact numerics. Also, as a consequence of low value of $\gamma$, the time to reach steady state is very long and much higher than the times presented in this figure. This further implies that the steady value of local density is far from the values presented here.

Figure 8

Behavior of total occupation $N\left(t\right)$ [Eq. (7)] versus $t$ for system size $L=21$ for the bosonic case, comparing direct exact numerics, Redfield, and Lindblad approaches. The bath is connected at site number $m=11$. The parameters are $g=0.5,\gamma =0.01,t_B=1,\beta =1$, and $\mu =-2.01$. This plot shows that the Redfield approach agrees with exact numerics whereas the local Lindblad deviates. This is due to relatively large intersite hopping $g$.

Figure 9

Behavior of total occupation $N\left(t\right)$ [Eq. (7)] versus $t$ for system size $L=40$ for the bosonic case, comparing exact numerics, Redfield, and Lindblad approaches. This figure is analogous to Fig. 2 in the main text. The bath is connected at site number $m=21$. The red dashed line shows the steady-state value $N_{\text{SS}}^{\text{QLE}}$ obtained from the quantum Langevin equation approach. The parameters are $g=0.5,\gamma =1,t_B=1,\beta =1$, and $\mu =-2.01$. The inset shows the plot for total occupation $N\left(t\right)$ versus $t$ for the Redfield approach demonstrating exponential growth of total occupation number. This plot shows that neither of the perturbative approaches agrees with exact numerics, implying the invalidity of these approaches in this parameter regime.

Figure 10

Behavior of total occupation $N\left(t\right)$ [Eq. (7)] versus $t$ for system size $L=41$ for the bosonic case, comparing exact numerics, Redfield, and Lindblad approaches. The bath is connected at site number $m=21$. The parameters are $g=0.05,\gamma =0.01,t_B=1,\beta =1$, and $\mu =-2.01$. This plot shows that both of the perturbative approaches agree well with exact numerics, implying the validity of these approaches in this parameter regime.

Figure 11

Initial rate of growth of total occupation as a function of $g$, keeping $\gamma$ fixed for system size $L=5$ with bosons, comparing exact numerics, Redfield, and Lindblad approaches. The bath is connected at site number $m=3$. The parameters are $\gamma =0.05,t_B=1,\beta =1$, and $\mu =-2.01$.

Figure 12

Initial rate of growth of total occupation as a function of $\gamma$, keeping $g$ fixed for system size $L=5$ with bosons, comparing exact numerics, Redfield, and Lindblad approaches. The bath is connected at site number $m=3$. The parameters are $g=0.5,t_B=1,\beta =1$, and $\mu =-2.01$.

Figure 13

Behavior of total occupation at steady state $N_{\text{SS}}$ as a function of $g$, keeping $\gamma$ fixed for system size $L=5$ with bosons, comparing exact numerics, QLE, Redfield, and Lindblad approaches. The bath is connected at site number $m=3$. The parameters are $\gamma =0.05,t_B=1,\beta =1$, and $\mu =-2.01$.

Figure 14

Behavior of total occupation at steady state $N_{\text{SS}}$ as a function of $\gamma$, keeping $g$ fixed for system size $L=5$ with bosons, comparing exact numerics, QLE, Redfield, and Lindblad approaches. The bath is connected at site number $m=3$. The parameters are $g=0.5,t_B=1,\beta =1$, and $\mu =-2.01$.

Figure 15

Spatial density profile $n_i\left(t\right)$ [Eq. (6)] for bosons for $L=41$ sites with the bath attached at $m=21$ for two different time snapshots following direct exact numerics (Sec. 2a), and analytical result based on Lindblad approach [Eqs. (57) and (58)] discussed in Sec. 2c. We notice excellent agreement between the two approaches. The parameters chosen are $t_B=1,g=0.05,\beta =1,\mu =-2.01$, and $\gamma =0.01$. The plots are shown for time snapshot $t=90$ and 180. Note that for the analytical case, no data are presented at $m=21$ as the analytical expression in Eq. (58) is expected to only hold away from the lattice site where the bath is attached. Recall that we have chosen weak intersite hopping $g$ to ensure the validity of local Lindblad approach. The inset shows the plot for total occupation $N\left(t\right)$ vs $t$ for the same parameter values. The slope in the inset preciously turns out to be $2{\mathrm{\Gamma }}_G$ and therefore matches with the prediction in Eq. (70) (Sec. 2c).

Figure 16

Spatial scaled density profile $n_i\left(t\right)$ [Eq. (6)] for bosons for $L=41$ sites with the bath attached at $m=21$ for two different time snapshots following analytical result based on Lindblad approach (57) and (58) and the scaling form (61) discussed in Sec. 2c. We notice excellent agreement between the two results. The parameters chosen are exactly the same as those of Fig. 15, i.e., $t_B=1,g=0.05,\beta =1,\mu =-2.01$, and $\gamma =0.01$. The plots are shown for time snapshot $t=90$ and 180. Recall that we have chosen weak intersite hopping $g$ to ensure the validity of the Lindblad approach.

Figure 17

Comparison between fermions and bosons for total occupation $N\left(t\right)$ for $L=40$ sites with bath attached at $m=21$, following direct exact numerics discussed in Sec. 2a. It is clear from the plot that the eventual saturation value for fermions is smaller than that of bosons. The inset denotes the zoomed version of the dynamics at relatively short times and clearly shows that the rate of growth for the fermionic case is smaller than that of the bosonic case. The parameters chosen are $t_B=1,g=0.5,\beta =1,\mu =-2.01$, and $\gamma =1$. Note that although we have presented the results only from direct exact numerics Sec. 2a, similar trends can be obtained following other methods.

Figure 18

Local density profile $n_i\left(t\right)$ [Eq. (6)] for fermions for $L=20$ lattice sites with bath attached at a particular site $\left(m=11\right)$ for various time snapshots, using direct numerics (empty circles), as discussed in Sec. 2a. The long-time limit of this density profile agrees perfectly with that obtained from QLE (cross), as discussed in Sec. 2d. The parameters are $t_B=1,g=0.5,\beta =1,\mu =-2.01$, and $\gamma =1$. Note that the parameters chosen here are exactly the same as in Fig. 4.

Figure 19

Local density profile $n_i\left(t\right)$ [Eq. (6)] for fermions for $L=100$ sites with bath attached at $m=51$ from direct exact numerics, described in Sec. 2a. We choose relatively short times to clearly demonstrate the ballistic growth of the density profile. We truncate the $y$ axis to highlight the propagation of the density front. Note that this front of the density profiles spreads with velocity $c=2g$, where $g$ is the intersite hopping within the lattice. This figure clearly indicates the presence of scaling which is demonstrated in Fig. 6. The parameters are $t_B=1,g=0.25,\beta =1,\mu =-2.01$, and $\gamma =1$.

Figure 20

A scaled plot of $ln\left(n_i\left(t\right)\right)$ [Eq. (6)] for fermions with $L=100$ sites with bath attached at $m=51$ to demonstrate the ballistic spread of spatial density profile $n_i\left(t\right)$ [Eq. (6)]. The parameters are $t_B=1,g=0.25,\beta =1,\mu =-2.01$, and $\gamma =1$. The logarithm mentioned in this plot is with base 10.