Simulating open quantum dynamics with time-dependent variational matrix product states: Towards microscopic correlation of environment dynamics and reduced system evolution

We report the development of an efficient many-body algorithm for simulating open quantum system dynamics that utilizes a time-dependent variational principle for matrix product states to evolve large system-environment states. Capturing all system-environment correlations, we reproduce the nonperturbative, quantum-critical dynamics of the zero-temperature spin-boson model, and then exploit the many-body information to visualize the complete time-frequency spectrum of the environmental excitations. Our “environmental spectra” reveal correlated vibrational motion in polaronic modes which preserve their vibrational coherence during incoherent spin relaxation, demonstrating how environment information could yield valuable insights into complex quantum dissipative processes.

Figure 1

(a)–(c) All possible normalizations of $A$. The straight line represents the identity matrix. (d)–(g) Center matrices (orange) and corresponding network normalization for one site.

Figure 2

All effective local Hamiltonians needed for the TDVP with OBB.

Figure 3

Comparison of CPU time per sweep for TDVP without OBB and $d_k=200,d_k=500$ (lines) and for TDVP with OBB and $d_k=500$ (crosses).

Figure 4

(a) Weak ($\alpha <{\alpha }_c$) and (b) strong coupling for Ohmic ($s=1, {\alpha }_c=1$) spectral density at different $\alpha$. Panel (b) was obtained at $\mathrm{\Delta }t=0.01$.

Figure 5

(a), (b) Weak and (c), (d) strong coupling (below and above ${\alpha }_c$) for sub-Ohmic spectral density ($s=0.5, {\alpha }_c=0.107$) at different couplings $\alpha$. Initial state is (a), (c) the product state and (b), (d) the coupled state.

Figure 6

Weak and strong coupling (below and above ${\alpha }_c$) for sub-Ohmic spectral densities [(a), (b): $s=0.25, {\alpha }_c=0.022$; (c), (d): $s=0.75, {\alpha }_c\approx 0.3, {\alpha }_{CI}\approx 0.22$] at different couplings $\alpha$. Initial state is the product [(a), (c)] or the coupled state [(b), (d)].

Figure 7

(a) Phonon modes ${\omega }_k$ occupation for an Ohmic environment with $\alpha =0.2$ exhibits a strong resonance at the renormalized tunneling amplitude ${\mathrm{\Delta }}_r\approx 0.057$ and a damping of oscillations. (b) Simulated renormalized tunneling ${\mathrm{\Delta }}_r$ versus analytic expression (lines) shows good agreement. Polaron theory predicts ${\mathrm{\Delta }}_r=\mathrm{\Delta }{e}^{-\frac{\alpha }{s-1}}\text{for}s=3$. (c) The projected phonon displacement $f_k^{\uparrow }$ for $s=1,\alpha =0.4$ oscillates even after spin relaxation at $t\approx 350$.

Figure 8

The occupation of the chain for $s=1,\alpha =0.2$ shows a clear constant offset on the first few sites indicating the dressing and renormalization of the spin. Additionally the chain exhibits reflections from the end at ${\omega }_{c}t\approx 380$ leading to artifacts in the inverse chain mapping. The open system dynamics are correct until the arrival of the reflected waves at ${\omega }_{c}t\approx 760$.

Figure 9

(a) The occupation $\langle n_k\rangle$ and (b) the spin-projected displacements $f_k^{\uparrow /\downarrow }$ of the mode ${\omega }_k=0.325$ with initial product state.

Figure 10

The phonon occupation for $s=0.75,\alpha =0.4$ with initial product state oscillates even after the relaxation of the spin oscillations at ${\omega }_{c}t\approx 150$ due to the finite $\langle {\sigma }_z\rangle \ne 0$.