Thermofield-based chain-mapping approach for open quantum systems

We consider a thermofield approach to analyze the evolution of an open quantum system coupled to an environment at finite temperature. In this approach, the finite-temperature environment is exactly mapped onto two virtual environments at zero temperature. These two environments are then unitarily transformed into two different chains of oscillators, leading to a one-dimensional structure that can be numerically studied using tensor network techniques. Compared to previous approaches using a single chain mapping, our strategy offers the advantage of an exact description of the initial state at arbitrary temperatures, which results in a gain in computational efficiency and a reduced truncation error.

Figure 1

(a) The initial problem described with (1) of an OQS coupled to a harmonic oscillator reservoir at finite temperature. (b) The thermofield-transformed problem (2), in which the finite temperature of the reservoir is encoded in two different reservoirs at zero temperature. (c) The chain representation of the latter.

Figure 2

Evolution of the mean value of $\langle {\sigma }_x\left(t\right)\rangle$ for $a=b=1/\sqrt{2}$ for $\eta =0.1$ and ${\omega }_S=0$. The top panels correspond to the Ohmic model $\left(s=1\right)$ for $\beta =5$ (left) and $\beta =1$ (right). The bottom panels correspond to a sub-Ohmic model with $s=1/2$ (left) and a super-Ohmic model with $s=3/2$ (right), both for $\beta =1$. The exact solution is given by the solid black curves. For the MPS, we consider a varying $M$: purple, green, and blue curves with purple squares, green triangles, and blue circles correspond to $M=2,10,40$ respectively, in all plots. The last plot includes a curve with orange diamonds with $M=120$. Bond dimension is $D=20$, and maximum occupation numbers in the harmonic oscillator basis is $n=3$.

Figure 3

Comparison of ME (solid curves) with MPS results for $\langle {\sigma }_x\left(t\right)\rangle$ considering $\eta =0.01,{\omega }_S=0.1$, and $M=100$ for both chains. We consider $\beta =10$ (blue squares), $\beta =50$ (orange triangles) and $\beta =1000$ (red circles). We observe that the MPS results are converged with maximum population per oscillator $n=4$.

Figure 4

Evolution of $\langle {\sigma }_x\left(t\right)\rangle$ for (top) $\beta =10$ (with maximum bond dimension $D=40$) and (bottom) $\beta =50$ (with $D=20$) for $\eta =0.1,{\omega }_S=0.1$, and $M=100$ for both chains. The solid black curve corresponds to the solution for the ME. Considering $n_1$ the dimension of the first two oscillators in the chain and $n_2$ the dimension of the following ones, the curves with green squares correspond to $(n_1=5,n_2=4)$, and the ones with blue diamonds correspond to $(n_1=6,n_2=5)$ (bottom panel) and $(n_1=7,n_2=6)$ (top panel). The curve with orange triangles in the top panel corresponds to $(n_1=8,n_2=7)$.

Figure 5

Evolution of $\langle n_{\uparrow }\rangle$ (green triangles) and $\langle n_{\downarrow }\rangle$ (purple squares) for the ME (solid lines) and the t-DMRG (symbols). We have considered (top) $\beta =1$ and (bottom) $\beta =10$ with $U=0.2,V=-U/2,{\omega }_c=15,t=0.01$, and $M=100$ oscillators in the chain.