We study the quantum brachistochrone problem for a spin-1 system in a magnetic field of constant absolute value. Such a system gives us the possibility to examine in detail the statement that the state vectors realizing evolution with the minimal time of passage evolve along the subspace spanned by the initial and final state vectors [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006); Brody and Hook, J. Phys. A 39, L167 (2006)]. Considering an explicit example, we show the existence of a quantum brachistochrone with the minimal possible time; however, the state vector we study leaves the subspace spanned by the initial and final state vectors during evolution. This is the result of our choice of a more constrained Hamiltonian than the one assumed in the general quantum brachistochrone problem. It is worth noting that such an evolution, being more complicated, is time optimal but with larger time than in the general case. This might be important for experiments, where a general Hamiltonian with all the allowed parameters is difficult to implement, but a constrained one, depending on the magnetic field, can be realized. However, for the preconstrained Hamiltonian not all final states are accessible. The present result does not contradict the general statement of the quantum brachistochrone problem, but gives additional insight into possible realization of the time-optimal passage.We study the quantum brachistochrone problem for a spin-1 system in a magnetic field of constant absolute value. Such a system gives us the possibility to examine in detail the statement that the state vectors realizing evolution with the minimal time of passage evolve along the subspace spanned by the initial and final state vectors [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006); Brody and Hook, J. Phys. A 39, L167 (2006)]. Considering an explicit example, we show the existence of a quantum brachistochrone with the minimal possible time; however, the state vector we study leaves the subspace spanned by the initial and final state vectors during evolution. This is the result of our choice of a more constrained Hamiltonian than the one assumed in the general quantum brachistochrone problem. It is worth noting that such an evolution, being more complicated, is time optimal but with larger time than in the general case. This might be important for experiments, where a general Hamiltonian with all the allowed parameters is difficult to implement, but a constrained one, depending on the magnetic field, can be realized. However, for the preconstrained Hamiltonian not all final states are accessible. The present result does not contradict the general statement of the quantum brachistochrone problem, but gives additional insight into possible realization of the time-optimal passage.

arXiv:0709.0931Physical Review AJan 15, 200831 citations

The Problem of the Electron Spin within a Magnetic Field

IMPERIAL COLLEGE PRESS eBooksJul 7, 20150 citations

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

Andrij Romanovych Kuzmak, V M Tkachuk

We study the quantum brachistochrone evolution for a system of two spins- $\frac{1}{2}$ described by an anisotropic Heisenberg Hamiltonian without $zx$ , $zy$ interacting couplings in magnetic field directed along the z-axis. This Hamiltonian realizes quantum evolution in two subspaces spanned by $\|\uparrow\uparrow\rangle$ , $\|\downarrow\downarrow\rangle$ and $\|\uparrow\downarrow\rangle$ , $\|\downarrow\uparrow\rangle$ separately and allows to consider the brachistochrone problem on each subspace separately. Using the evolution operator for this Hamiltonian we generate quantum gates, namely an entangler gate, SWAP gate, iSWAP gate et al.We study the quantum brachistochrone evolution for a system of two spins-$\frac{1}{2}$ described by an anisotropic Heisenberg Hamiltonian without $zx$, $zy$ interacting couplings in magnetic field directed along the z-axis. This Hamiltonian realizes quantum evolution in two subspaces spanned by $\|\uparrow\uparrow\rangle$, $\|\downarrow\downarrow\rangle$ and $\|\uparrow\downarrow\rangle$, $\|\downarrow\uparrow\rangle$ separately and allows to consider the brachistochrone problem on each subspace separately. Using the evolution operator for this Hamiltonian we generate quantum gates, namely an entangler gate, SWAP gate, iSWAP gate et al.

arXiv:1211.2549Journal of Physics A Mathematical and TheoreticalApr 3, 201317 citations

Classical spin in a potential field

Garnet N. Ord, A. S. Deakin

International Journal of Theoretical PhysicsSep 1, 19970 citations

Geometrical and dynamical description of two interacting spins under the XXZ-type Heisenberg model

Jamal Elfakir, Brahim Amghar, Mohammed Daoud A

We consider a physical system consisting of two interacting spins governed by the [Formula: see text]-type Heisenberg Hamiltonian in an external magnetic field. We investigate the quantum evolution and the Riemannian geometry of the two-spin state space by means of the relevant Fubini–Study metric. The geometrical phase accumulated by the two-spin state is also examined under arbitrary and cyclic evolutions. By computing the evolution speed and the corresponding geodesic distance, we solve the quantum brachistochrone problem. The entanglement between the two spins is also studied via two approaches: the first one deals with the entanglement effect on the Fubini–Study metric and the geometrical phase, while the second one treats the entanglement effect on the evolution speed and the corresponding geodesic distance. Finally, we solve the quantum brachistochrone problem using the entanglement degree.We consider a physical system consisting of two interacting spins governed by the [Formula: see text]-type Heisenberg Hamiltonian in an external magnetic field. We investigate the quantum evolution and the Riemannian geometry of the two-spin state space by means of the relevant Fubini–Study metric. The geometrical phase accumulated by the two-spin state is also examined under arbitrary and cyclic evolutions. By computing the evolution speed and the corresponding geodesic distance, we solve the quantum brachistochrone problem. The entanglement between the two spins is also studied via two approaches: the first one deals with the entanglement effect on the Fubini–Study metric and the geometrical phase, while the second one treats the entanglement effect on the evolution speed and the corresponding geodesic distance. Finally, we solve the quantum brachistochrone problem using the entanglement degree.

International Journal of Geometric Methods in Modern PhysicsAug 26, 20224 citations

Spinning Quantum System in an External Magnetic Field

Arno Böhm, Ali Mostafazadeh, Hiroyasu Koizumi +2 more

Springer eBooksJan 1, 20030 citations

The quantum brachistochrone problem for two spins-

\frac{1}{2}

with anisotropic Heisenberg interaction

We study the quantum brachistochrone evolution for a system of two spins-

\frac{1}{2}

described by an anisotropic Heisenberg Hamiltonian without

zx

zy

interacting couplings in magnetic field directed along the z-axis. This Hamiltonian realizes quantum evolution in two subspaces spanned by

\|\uparrow\uparrow\rangle

\|\downarrow\downarrow\rangle

and

\|\uparrow\downarrow\rangle

\|\downarrow\uparrow\rangle

separately and allows to consider the brachistochrone problem on each subspace separately. Using the evolution operator for this Hamiltonian we generate quantum gates, namely an entangler gate, SWAP gate, iSWAP gate et al.