Theory of Adiabatic Evolution

Jie Liu, Sheng-Chang Li, Libin Fu +1 more

Da-Jian Zhang, P. Z. Zhao, Guofu Xu

We unveil that even an infinitesimal deviation of the initial state from the eigenstates of the initial Hamiltonian can yield a significant change in the geometric phase accompanying adiabatic evolution without degeneracies of the energy levels. This leads to the counterintuitive result that the values of two geometric phases computed from the general definition of the Berry phase [Samuel and Bhandari, Phys. Rev. Lett. 60, 2339 (1988)] can dramatically differ from each other in the adiabatic limit even if their evolution paths are sufficiently close in a pointwise manner. The result of this work indicates that the widely accepted theory of geometric phases is not very consistent in the context of adiabatic evolution involving superposed eigenstates.

Alain Joye, Charles‐Edouard Pfister

Qi Zhang, Jiangbin Gong, Biao Wu

Quantum adiabatic evolution is a dynamical evolution of a quantum system under slow external driving. According to the quantum adiabatic theorem, no transitions occur between nondegenerate instantaneous energy eigenstates in such a dynamical evolution. However, this is true only when the driving rate is infinitesimally small. For a small nonzero driving rate, there are generally small transition probabilities between the energy eigenstates. We develop a classical mechanics framework to address the small deviations from the quantum adiabatic theorem order by order. A hierarchy of Hamiltonians is constructed iteratively with the zeroth-order Hamiltonian being determined by the original system Hamiltonian. The k th-order deviations are governed by a k th-order Hamiltonian, which depends on the time derivatives of the adiabatic parameters up to the k th-order. Two simple examples, the Landau–Zener model and a spin-1/2 particle in a rotating magnetic field, are used to illustrate our hierarchical theory. Our analysis also exposes a deep, previously unknown connection between classical adiabatic theory and quantum adiabatic theory.

Jie Liu, Sheng-Chang Li, Libin Fu +1 more

Yi Zuo, Qinghong Yang, Bang‐Gui Liu +1 more

Keldysh field theory, based on adiabatic assumptions, serves as a widely used framework for addressing nonequilibrium many-body systems. Nonetheless, the validity of such adiabatic assumption when addressing interacting Gibbs states remains a topic of contention. Interestingly, the knowledge of work statistics developed in nonequilibrium thermodynamics helps us to quantitatively explore this problem. Consequently, we deduce a universal theorem delineating the characteristics of evolutions that transition an initial Gibbs state to another. Based on this theorem, we analytically ascertain that adiabatic evolutions fail to transition a noninteracting Gibbs state to its interacting counterpart. However, the adiabatic evolution remains a superior approximation relative to its nonadiabatic counterparts. Numerics verifying our theory and predictions are also provided. Furthermore, our findings render insights into the Gibbs state preparation within the domain of quantum computation.