Non-Markovian dynamics of a qubit coupled to an Ising spin bath

We study the analytically solvable Ising model of a single qubit system coupled to a spin bath. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the system qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs particularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed post-Markovian master equation performs comparably to the time-convolutionless master equation for a properly chosen memory kernel, and outperforms all the approximation methods considered here at long times. Our findings shed light on the applicability of master equations to the description of reduced system dynamics in the presence of spin baths.

Figure 1

(Color online) Comparison of the exact solution at $\beta =1$ and $\beta =10$ for $N=100$.

Figure 2

(Color online) Comparison of the exact solution at $\beta =1$ and $\beta =10$ for $N=4$.

Figure 3

(Color online) Comparison of the exact solution at $\beta =1$ and $\beta =10$ for $N=100$ for randomly generated $g_n$ and ${\Omega }_n$.

Figure 4

(Color online) Comparison of the exact solution at $\beta =1$ and $\beta =10$ for $N=4$ for randomly generated $g_n$ and ${\Omega }_n$.

Figure 5

(Color online) Comparison of the exact solution, NZ2, NZ3, and NZ4 at $\beta =1$ and $\beta =10$ for $N=100$. The exact solution is the solid (blue) line, NZ2 is the dashed (green) line, NZ3 is the dot-dashed (red) line, and NZ4 is the dotted (cyan) line.

Figure 6

(Color online) Comparison of the exact solution, NZ2, NZ3, and NZ4 at $\beta =1$ and $\beta =10$ for $N=4$. The exact solution is the solid (blue) line, NZ2 is the dashed (green) line, NZ3 is the dot-dashed (red) line, and NZ4 is the dotted (cyan) line.

Figure 7

(Color online) Comparison of the exact solution, TCL2, TCL3, and TCL4 at $\beta =1$ and $\beta =10$ for $N=100$. The exact solution is the solid (blue) line, TCL2 is the dashed (green) line, TCL3 is the dot-dashed (red) line, and TCL4 is the dotted (cyan) line. Note that for $\beta =1$, the curves nearly coincide.

Figure 8

(Color online) Comparison of the exact solution, TCL2, TCL3, and TCL4 at $\beta =1$ and $\beta =10$ for $N=4$. The exact solution is the solid (blue) line, TCL2 is the dashed (green) line, TCL3 is the dot-dashed (red) line, and TCL4 is the dotted (cyan) line. Note that for $\beta =1$, TCL3, TCL4, and the exact solution nearly coincide.

Figure 9

(Color online) Comparison of TCL2 and the exact solution to demonstrate the validity of the TCL approximation for $N=4$ and $\beta =1$. The solid (blue) line denotes the exact solution and the dashed (green) line is TCL2. Note that the time axis here is on a linear scale. TCL2 breaks down at $\alpha t\approx 0.9$, where it remains flat, while the exact solution has a recurrence.

Figure 10

(Color online) Comparison of the exact solution, NZ4, TCL4, and PM at $\beta =1$ and $\beta =10$ for $N=100$. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line, and TCL4 is the dotted (cyan) line. Note that for $\beta =1$, TCL4, PM, and the exact solution nearly coincide for short and medium times. Only PM captures the recurrences of the exact solution at long times.

Figure 11

(Color online) Comparison of the exact solution, NZ4, TCL4, and PM at $\beta =1$ and $\beta =10$ for $N=4$. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line, and TCL4 is the dotted (cyan) line. Note that for $\beta =1$, TCL4 and the exact solution nearly coincide for short and medium times.

Figure 12

(Color online) Comparison of the exact solution, NZ4, TCL4, and PM at $\alpha t=0.1$ for $N=100$ for different $\beta ∊[0.01,10]$. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line, and TCL4 is the dotted (cyan) line.

Figure 13

(Color online) Comparison of the exact solution, NZ4, TCL4, and PM at $\alpha t=0.5$ for $N=4$ for different $\beta ∊[0.01,10]$. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line, and TCL4 is the dotted (cyan) line.

Figure 14

(Color online) Comparison of the exact solution, NZ4, TCL4, and PM at $\beta =1$ and $\beta =10$ for $N=100$ for random values of $g_n$ and ${\Omega }_n$. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line, and TCL4 is the dotted (cyan) line. Note that for $\beta =1$ and $\beta =10$, TCL4, PM, and the exact solution nearly coincide.

Figure 15

(Color online) Comparison of the exact solution, NZ4, TCL4, and PM at $\beta =1$ and $\beta =10$ for $N=4$ for random values of $g_n$ and ${\Omega }_n$. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line, and TCL4 is the dotted (cyan) line. Note that for $\beta =1$, TCL4, PM, and the exact solution nearly coincide for short and medium times.

Figure 16

(Color online) Comparison of the exact solution and the optimal coarse-graining approximation for $N=50$ and $\beta =1$. The exact solution is the solid (blue) line and the coarse-graining approximation is the dashed (green) line. Note the linear scale time axis.