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Journal of Shaanxi Normal University

Existence and structures of a class of quantum states satisfying adiabatic approximation

YU Baomi

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

To discuss the sufficient condition that the state of a quantum system can be approximated by an eigenstate of the Hamiltonian,the functions f(t)satisfying the condition that〈f(t)|f′(t)〉=0(t∈[0,∞))and∫∞0‖|f′(t)〉‖dteare characterized.At first the general form of a qubit is given,then the exact structures and a necessary and sufficient condition of the components a qubit which satisfying the above adiabatic approximation conditions are proved.Finally some necessary conditions and sufficient conditions of the components of n-dimensional quantum state satisfying the adiabatic approximation are obtained.

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To discuss the sufficient condition that the state of a quantum system can be approximated by an eigenstate of the Hamiltonian,the functions f(t)satisfying the condition that〈f(t)|f′(t)〉=0(t∈[0,∞))and∫∞0‖|f′(t)〉‖dteare characterized.At first the general form of a qubit is given,then the exact structures and a necessary and sufficient condition of the components a qubit which satisfying the above adiabatic approximation conditions are proved.Finally some necessary conditions and sufficient conditions of the components of n-dimensional quantum state satisfying the adiabatic approximation are obtained.
Abstract The adiabatic theorem was proposed about 90 years ago and has played an important role in quantum physics. The quantitative adiabatic condition constructed from eigenstates and eigenvalues of a Hamiltonian is a traditional tool to estimate adiabaticity and has proven to be the necessary and sufficient condition for adiabaticity. However, recently the condition has become a controversial subject. In this paper, we list some expressions to estimate the validity of the adiabatic approximation. We show that the quantitative adiabatic condition is invalid for the adiabatic approximation via the Euclidean distance between the adiabatic state and the evolution state. Furthermore, we deduce general necessary and sufficient conditions for the validity of the adiabatic approximation by different definitions.
We examine the quantitative condition which has been widely used as a criterion for the adiabatic approximation but was recently found insufficient. Our results indicate that the usual quantitative condition is sufficient for a special class of quantum mechanical systems. For general systems, it may not be sufficient, but it, along with additional conditions, is sufficient. The usual quantitative condition and the additional conditions constitute a general criterion for the validity of the adiabatic approximation, which is applicable to all N-dimensional quantum systems. Moreover, we illustrate the use of the general quantitative criterion in some physical models.
Two linear In this letter, we prove the following conclusions by introducing a function Fn(t): (1) If a quantum system S with a time-dependent non-degenerate Hamiltonian H(t) is initially in the n-th eigenstate of H(0), then the state of the system at time t will remain in the n-th eigenstate of H(t) up to a multiplicative phase factor if and only if the values Fn(t) for all t are always on the circle centered at 1 with radius 1; (2) If a quantum system S with a time-dependent Hamiltonian H(t) is initially in the n-th eigenstate of H(0), then the state of the system at time t will remain-uniformly approximately in the n-th eigenstate of H(t) up to a multiplicative phase factor if and only if the values Fn(t) for all t are always outside of the circle centered at 1 with radius 1 . Moreover, some quantitative sufficient conditions for the state of the system at time t to remain-uniformly approximately in the n-th eigenstate of H(t) up to a multiplicative phase factor are established. Lastly, our results are illustrated by a spin-half particle in a rotating magnetic field.
A gapped quantum system that is adiabatically perturbed remains approximately in its eigenstate after the evolution. We prove that, for constant gap, general quantum processes that approximately prepare the final eigenstate require a minimum time proportional to the ratio of the length of the eigenstate path to the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We also give a necessary condition for the adiabatic approximation that depends on local properties of the path, which is appropriate when the gap varies.
Adiabaticity occurs when, during its evolution, a physical system remains in the instantaneous eigenstate of the Hamiltonian. Unfortunately, existing results, such as the quantum adiabatic theorem based on a slow down evolution $[H(ϵt),ϵ\ensuremath{\rightarrow}0]$, are insufficient to describe an evolution driven by the Hamiltonian H(t)H(t)H(t) itself. Here we derive general criteria and exact bounds, for the state and its phase, ensuring an adiabatic evolution for any Hamiltonian H(t)H(t)H(t). As a corollary, we demonstrate that the commonly used condition of a slow Hamiltonian variation rate, compared to the spectral gap, is indeed sufficient to ensure adiabaticity but only when the Hamiltonian is real and nonoscillating (for instance, containing exponential or polynomial but no sinusoidal functions).