Hierarchical Schrödinger equations of motion for open quantum dynamics

We rigorously investigate the quantum non-Markovian dissipative dynamics of a system coupled to a harmonic-oscillator bath by deriving hierarchical Schrödinger equations of motion (HSEOM) and studying their dynamics. The HSEOM are the equations for wave functions derived on the basis of the Feynman-Vernon influence functional formalism for the density operator, $\langle q\left|\rho \left(t\right)\right|q^{\prime }\rangle$, where $\langle q|$ and $|q^{\prime }\rangle$ are the left- and right-hand elements. The time evolution of $\langle q|$ is computed from time 0 to $t$, and, subsequently, the time evolution of $|q^{\prime }\rangle$ is computed from time $t$ to 0 along a contour in the complex time plane. By appropriately choosing functions for the bath correlation function and the spectral density, we can take advantage of an HSEOM method to carry out simulations without the need for a great amount of computational memory. As a demonstration, quantum annealing simulation for a ferromagnetic $p$-spin model is studied.

Figure 1

The contour $C$ in the complex time plane.

Figure 2

The imaginary part of the Fourier transform of the first-order response function $R^{\left(1\right)}\left(\omega \right)$ for a spin-boson system with an Ohmic spectral density with a circular cutoff (red) and an exponential (exp.) cutoff (blue). Results for two cases, in which $\beta \hslash =3$ (solid curve) and $\beta \hslash \to \infty$ (zero-temperature case, dashed curve) are displayed. The unit of the frequency $\omega$ is set to the characteristic frequency of the system ${\omega }_0$ in Eq. (13).

Figure 3

The time evolution of the ground-state (red) and first-excited-state (blue) populations in the (i) weak $\hslash \zeta =0.01$ (solid line), (ii) intermediate $\hslash \zeta =0.1$ (dashed line), and (iii) strong $\hslash \zeta =0.5$ (dot-dashed line) coupling cases calculated from the quantum annealing simulation at the zero temperature. The unit of the time $t$ is set to $t_f$ in Eq. (25).