Extracting dynamical maps of non-Markovian open quantum systems.

Area of Science:

• Quantum mechanics
• Condensed matter physics
• Quantum information theory

Background:

• Quantum evolution is described by dynamical maps (Λ̂(τ)), which are generally non-Markovian when systems couple to thermal baths.
• Assessing non-Markovian dynamics and extracting accurate dynamical maps is computationally challenging, especially without clear time-scale separation.
• The assumption of a unique steady state implies finite memory times in the baths.

Purpose of the Study:

• To develop a tensor network framework for directly and accurately extracting quantum dynamical maps (Λ̂(τ)) for systems coupled to thermal baths.
• To investigate the non-Markovian dynamics and establish memory time hierarchies for open quantum systems.
• To enable efficient computation of stationary states by identifying relevant timescales.

Main Methods:

• Utilized an orthogonal polynomial mapping and thermofield doubling to create a purified chain representation of infinite Fermi baths.
• Employed the Choi-Jamiolkowski isomorphism to reconstruct the dynamical map (Λ̂(τ)) from pure state calculations.
• Computed the time-local propagator (L̂(τ)) and analyzed fixed-point convergence to determine memory times (τmΛ, τmL).

Main Results:

• Successfully extracted the dynamical map (Λ̂(τ)) and time-local propagator (L̂(τ)) for interacting fermionic modes coupled to Fermi baths.
• Established a memory time hierarchy: τmL ≤ τmΛ ≤ τre, where τre is the characteristic relaxation time.
• Demonstrated significant speedups in determining stationary states for models like the Anderson impurity model when τre ≫ τm.

Conclusions:

• The developed tensor network approach accurately captures non-Markovian dynamics and provides insightful analyses of open quantum systems.
• The identified memory time hierarchy offers a pathway to efficiently compute long-time steady states, bypassing direct long-time simulations.
• This method is particularly advantageous in regimes where relaxation times significantly exceed bath memory times.