Quantum Trajectories III: More Examples

Walter T. Strunz, Lajos Diósi, Nicolas Gisin

We present a theory of non-Markovian quantum trajectories that describe the dynamics of a quantum system coupled to an environment of harmonic oscillators. When averaged over the noise, they reproduce the standard reduced density matrix without any approximation, in particular without Markov approximation. The time evolution of the environment can also be determined from these non-Markovian quantum trajectories. We present a simple example to demonstrate the power of our approach. As a remarkable feature of non-Markovian quantum trajectories, we show that for strong coupling to a dissipative environment with memory, the asymptotic state can be reached in a finite time. Further examples are described in the text.

Chia‐Chun Chou, Róbert E. Wyatt

One-dimensional time-independent scattering problems are investigated in the framework of the quantum Hamilton-Jacobi formalism. The equation for the local approximate quantum trajectories near the stagnation point of the quantum momentum function is derived, and the first derivative of the quantum momentum function is related to the local structure of quantum trajectories. Exact complex quantum trajectories are determined for two examples by numerically integrating the equations of motion. For the soft potential step, some particles penetrate into the nonclassical region, and then turn back to the reflection region. For the barrier scattering problem, quantum trajectories may spiral into the attractors or from the repellers in the barrier region. Although the classical potentials extended to complex space show different pole structures for each problem, the quantum potentials present the same second-order pole structure in the reflection region. This paper not only analyzes complex quantum trajectories and the total potentials for these examples but also demonstrates general properties and similar structures of the complex quantum trajectories and the quantum potentials for one-dimensional time-independent scattering problems.

Miguel Orszag