Dynamic process and Uhlmann process: Incompatibility and dynamic phase of mixed quantum states

TL;DR

This paper investigates the accumulation of phases in mixed quantum states, revealing that the Uhlmann process generally conflicts with Hamiltonian evolution, leading to a dynamic phase that varies with temperature and spectral properties.

Contribution

It demonstrates the incompatibility of the Uhlmann process with Hamiltonian evolution and derives the dynamic phase expression for mixed states, highlighting differences from classical systems.

Findings

Uhlmann process often incompatible with density matrix evolution

Dynamic phase can take multiple discrete values at high temperature

Behavior varies with energy spectrum and presence of band gaps

Abstract

While a pure quantum state may accumulate both the Berry phase and dynamic phase as it undergoes a cyclic path in the parameter space, the situation is more complicated when mixed quantum states are considered. From the Ulhmann bundle, a mixed quantum state can accumulate the Ulhmann phase if the parallel-transport condition is satisfied. However, we show that the Ulhmann process is in general not compatible with the evolution equation of the density matrix governed by the Hamiltonian. Thus, a mixed quantum state usually accumulates a dynamic phase during its time evolution. We present the expression of the dynamic phase for mixed quantum states. In examples of quasi-static one-dimensional two-band models and simple harmonic oscillator, the dynamic phase can take multiple discrete values at infinitely high temperature due to the resonant points. However, the behavior differs if the energy spectrum is continuous without a band gap. Moreover, there is no natural analog of the dynamic phase in classical systems.