Quantum brachistochrone problem for a spin-1 system in a magnetic field

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 et al., Phys. Rev. Lett. 96, 060503 (2006), D. C. Brody, D. W. Hook, J. Phys. A 39, L167, (2006)] that 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 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.Recently Carlini et al.[1] generalized the classical brachistochrone problem to the quantum case. The quantum brachistochrone problem can be formulated in the following way:

The quantum brachistochrone problem for an arbitrary spin in a magnetic field

“…Let us take the initial states as the eigenstates of S z |1 , |0 and | − 1 [7]. Then using (22), we finally find…”

The quantum brachistochrone problem for two spins-$\frac{1}{2}$ with anisotropic Heisenberg interaction

“…When φ = which is allowed by Hamiltonian (1) and finite energy condition (9). Initial state | ↑↑ can reach ortogonal final states | ↓↓ during the time t = π 2ω…”

“…For a more detailed discussion on this subject see [4,5]. The quantum brachistochrone problem for a spin-1 system in the magnetic field was solved in [9].…”

Preparation of an arbitrary two-qubit quantum gate on two spins with an anisotropic Heisenberg interaction

“…For instance, the conditions for time-optimal evolution of a spin-1 2 in the magnetic field were obtained in [27,28,29,30]. Similar problems for the spin-1 and arbitrary spin were solved in [31] and [32], respectively. Also it was widely studied the implementation of quantum gate on two spins with different types of interaction [33,34,35,36,37,38,39].…”