Quantum brachistochrone problem for spin-1 in a magnetic field

TL;DR

This paper investigates the quantum brachistochrone problem for a spin-1 system in a magnetic field, revealing that time-optimal evolution can involve state vectors leaving the subspace spanned by initial and final states, under constrained Hamiltonians.

Contribution

It demonstrates that, with constrained Hamiltonians, the minimal time evolution can occur outside the initial-final subspace, providing new insights into quantum speed limits and experimental implementations.

Findings

Existence of time-optimal evolution leaving the initial-final subspace.

Constrained Hamiltonians can still achieve minimal passage time.

Not all final states are accessible under constrained Hamiltonians.

Abstract

We study quantum brachistochrone problem for the spin-1 system in a magnetic field of a constant absolute value. Such system gives us a possibility to examine in detail the statement of papers [A. Carlini {\it et al.}, Phys. Rev. Lett. {\bf 96}, 060503 (2006), D. C. Brody, D. W. Hook, J. Phys. A {\bf 39}, L167, (2006)] that {\it the state vectors realizing the evolution with the minimal time of passage evolve along the subspace spanned by the initial and final state vectors.} Using explicit example we show the existence of quantum brachistochrone with minimal possible time, but the state vector of which, during the evolution {\em leaves} the subspace spanned by the initial and final state vectors. This is the result of the choice of more constrained Hamiltonian then assumed in the general quantum brachistochrone problem, but what is worth noting, despite that such evolution is more complicated it is still time optimal. This might be important for experiment, where general Hamiltonian with the all allowed parameters is difficult to implement, but constrained one depending on magnetic field can be realized. However for pre-constrained Hamiltonian not all final states are accessible. Present result does not contradict general statement of the quantum brachistochrone problem, but gives new insight how time optimal passage can be realized.