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arXiv:0709.0931 · Physical Review A

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

Andrzej Frydryszak, V. M. Tkachuk

Published Jan 15, 2008• 31 citations

Abstract

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.

DOI: 10.1103/physreva.77.014103

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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.

Recently, Bender et al. have considered the quantum brachistochrone problem for the non-Hermitian PT -symmetric quantum system and have shown that the optimal time evolution required to transform a given initial state |ψ i into a specific final state |ψ f can be made arbitrarily small.Additionally, it has been shown that finding the shortest possible time requires only the solution of the two-dimensional problem for the quantum system governed by the effective Hamiltonian acting in the subspace spanned by |ψ i and |ψ f .In this paper, we study a similar problem for the generic non-Hermitian Hamiltonian, focusing our attention on the geometric aspects of the problem.

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.

The quantum brachistochrone problem -- to find the time-optimal transition between given initial and final quantum states -- is investigated in this bachelor's thesis. First the quantum equivalent of distance (the Fubini-Study metric) is formulated, using geometry of spheres. Together with constraints, it is used to create the quantum action. Functional derivatives are then taken of said action to find equations of motion (eoms). For the unconstrained case (excluding finite energy), these eoms are solved in closed form, whilst general to be solved formulae are obtained for the constrained case. The latter is used to explicitly solve an example quantum brachistochrone problem: a spin-1/2 particle in a controllable magnetic field constrained to an x-y plane. Finally, the link with quantum computing is illustrated through the time-optimization of unitary transformations that the quantum brachistochrone yields us.

We present a general framework for finding the time-optimal evolution and the optimal Hamiltonian for a quantum system with a given set of initial and final states. Our formulation is based on the variational principle and is analogous to that for the brachistochrone in classical mechanics. We reduce the problem to a formal equation for the Hamiltonian which depends on certain constraint functions specifying the range of available Hamiltonians. For some simple examples of the constraints, we explicitly find the optimal solutions.

Minimum-time quantum control protocols can be obtained from the quantum brachistochrone formalism [Carlini, Hosoya, Koike, and Okudaira, Phys. Rev. Lett. 96, 06053, (2006)]. We point out that the original treatment implicitly applied the variational calculus with fixed boundary conditions. We argue that the genuine quantum brachistochrone problem involves a variational problem with a movable endpoint, contrary to the classical brachistochrone problem. This formulation not only simplifies the derivation of the quantum brachistochrone equation but introduces an additional constraint at the endpoint due to the boundary effect. We present the general solution to the full quantum brachistochrone equation and discuss its main features. Using it, we prove that the speed of evolution under constraints is reduced with respect to the unrestricted case. In addition, we find that solving the quantum brachistochrone equation is closely connected to solving the dynamics of the Lagrange multipliers, which is in general governed by nonlinear differential equations. Their numerical integration allows generating time-extremal trajectories. Furthermore, when the restricted operators form a closed subalgebra, the Lagrange multipliers become constant and the optimal Hamiltonian takes a concise form. The new class of analytically solvable models for the quantum brachistochrone problem opens up the possibility of applying it to many-body quantum systems, exploring notions related to geometry such as quantum speed limits, and advancing significantly the quantum state and gate preparation for quantum information processing.

Recently Bender, Brody, Jones and Meister found that in the quantum brachistochrone problem the passage time needed for the evolution of certain initial states into specified final states can be made arbitrarily small, when the time-evolution operator is taken to be non-Hermitian but PT-symmetric. Here we demonstrate that such phenomena can also be obtained for dissipative systems. We conclude that the effect of a tunable passage time can be attributed to the non-Hermitian nature of the time-evolution operator rather than to its PT-symmetry.