Optimal time evolution for pseudo-Hermitian Hamiltonians
- PDF / 474,015 Bytes
- 13 Pages / 612 x 792 pts (letter) Page_size
- 32 Downloads / 190 Views
OPTIMAL TIME EVOLUTION FOR PSEUDO-HERMITIAN HAMILTONIANS W. H. Wang,∗ Z. L. Chen,† Y. Song,‡ and Y. J. Fan§
If an initial state |ψI and a final state |ψF are given, then there exist many Hamiltonians under whose action |ψI evolves into |ψF . In this case, the problem of the transition of |ψI to |ψF in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if |ψI and |ψF are orthogonal. But for a P T -symmetric Hamiltonian, this time can be arbitrarily small, which seems amazing. We discuss the optimum time evolution for pseudo-Hermitian Hamiltonians and obtain a lower bound for the evolution time under the condition that the Hamiltonian is bounded. The optimum evolution time can be attained in the case where two quantum states are orthogonal with respect to some inner product. The results in the Hermitian and pseudo-Hermitian cases coincide if the evolution is unitary with some well-defined inner product. We also analyze two previously studied examples and find that they are consistent with our theory. In addition, we give some explanations of our results with two examples.
Keywords: optimum time, Hermitian Hamiltonian, pseudo-Hermitian Hamiltonian, inner product, unitary evolution DOI: 10.1134/S0040577920080048
1. Introduction The past few years have witnessed a growing interest in optimal time evolution (i.e., the quantum brachistochrone) problem [1], which is stated as follows. Given two different fixed quantum states, an initial state |ψI and a final state |ψF , in a Hilbert space, find the least possible time for which |ψI transforms into |ψF under the action of some Hamiltonian in accordance with the dynamics of states in the time evolution process. As is known, the time evolution in conventional quantum mechanics is unitary, and the problem is equivalent to finding a family of unitary operators {U (t)} depending on the time t such that |ψF = U (t)|ψI ∗
School of Ethnic Nationalities Education, Shaanxi Normal University, Xi’an, China, e-mail: [email protected]. †
School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, China.
‡
School of Computer Science, Shaanxi Normal University, Xi’an, China, e-mail: [email protected] (corresponding author). §
School of Mathematics and Information Science, North Minzu University, Yinchuan, China. This research is supported by the National Natural Science Foundation of China (Grant Nos. 11871318, 11771009, 11701011, 11601300, 11571213, and 61602291), the FRF for the Central Universities (Grant No. GK202003093), and the State Scholarship Fund of China Scholarship Council. Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 2, pp. 211–225, August, 2020. Received December 14, 2019. Revised February 20, 2020. Accepted March 30, 2020. 1020
c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2042-1020
in the least possible time t = Tmin . Moreover, this kind of unit
Data Loading...
