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arXiv:0704.2117 · Physical Review Letters

Quasienergy Anholonomy and its Application to Adiabatic Quantum State Manipulation

Atushi Tanaka, Manabu Miyamoto

Published Apr 19, 2007• 28 citations
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Abstract

The parametric dependence of a quantum map under the influence of a rank-1 perturbation is investigated. While the Floquet operator of the map and its spectrum have a common period with respect to the perturbation strength lambda, we show an example in which none of the quasienergies nor the eigenvectors obey the same period: After a periodic increment of lambda, the quasienergy arrives at the nearest higher one, instead of the initial one, exhibiting an anholonomy, which governs another anholonomy of the eigenvectors. An application to quantum state manipulations is outlined.

DOI
10.1103/physrevlett.98.160407

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In their Letter [A. Tanaka and M. Miyamoto, Phys. Rev. Lett. 98, 160407 (2007)], Tanaka and Miyamoto introduce a kicked spin model, for which they point out the generic existence of exotic eigenvalue anholonomy. They proceed to show the potential utility of the model for quantum state manipulation and quantum information processing. While we believe their model to be vary useful and their argument to be revealing, In our view, there are three missing elements in the article, without which their discussion seems to be incomplete and the prospect of the application limited. They are on dynamical phase, gauge potential, and on adiabaticity.

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The parametric dependence of a quantum map under the influence of a rank-1 perturbation is investigated. While the Floquet operator of the map and its spectrum have a common period with respect to the perturbation strength lambda, we show an example in which none of the quasienergies nor the eigenvectors obey the same period: After a periodic increment of lambda, the quasienergy arrives at the nearest higher one, instead of the initial one, exhibiting an anholonomy, which governs another anholonomy of the eigenvectors. An application to quantum state manipulations is outlined.
Anholonomies in the parametric dependences of the eigenvalues and the eigenvectors of Floquet operators that describe unit time evolutions of periodically driven systems, e.g., kicked rotors, are studied. First, an example of the anholonomies induced by a periodically pulsed rank-1 perturbation is given. As a function of the strength of the perturbation, the perturbed Floquet operator of the quantum map and its spectrum are shown to have a period. However, we show examples where each eigenvalue does not obey the periodicity of the perturbed Floquet operator and exhibits an anholonomy. Furthermore, this induces another anholonomy in the eigenspaces, i.e., the directions of the eigenvectors, of the Floquet operator. These two anholonomies are previously observed in a family of Hamiltonians [T. Cheon, Phys. Lett. A 248, 285 (1998)] and are different from the phase anholonomy known as geometric phases. Second, the stability of Cheon's anholonomies in periodically driven systems is established by a geometrical analysis of the family of Floquet operators. Accordingly, Cheon's anholonomies are expected to be abundant in systems whose time evolutions are described by Floquet operators. As an application, a design principle for quantum state manipulations along adiabatic passages is explained.
In their Letter [A. Tanaka and M. Miyamoto, Phys. Rev. Lett. 98, 160407 (2007)], Tanaka and Miyamoto introduce a kicked spin model, for which they point out the generic existence of exotic eigenvalue anholonomy. They proceed to show the potential utility of the model for quantum state manipulation and quantum information processing. While we believe their model to be vary useful and their argument to be revealing, In our view, there are three missing elements in the article, without which their discussion seems to be incomplete and the prospect of the application limited. They are on dynamical phase, gauge potential, and on adiabaticity.
Quantum adiabaticity is the evolution of a quantum system that remains close to an instantaneous eigenstate of a time-dependent Hamiltonian. Using Floquet formalism, we derive a rigorous sufficient condition for adiabaticity in closed, finite-dimensional periodically driven systems that is valid for arbitrarily many driving periods. The condition is stroboscopic and geometric, depending only on single-cycle information: the Fubini--Study length of the instantaneous eigenray and a quasienergy-separation measure extracted from the Floquet operator. We also formulate a state-targeted refinement that reduces conservativeness when only one adiabatic branch is relevant. Rather than synthesizing control pulses, the result provides a certification criterion for a given periodic protocol. We illustrate the criterion and contrast it with conventional instantaneous-gap conditions in three representative examples.
A quantum system driven by a high-frequency periodic perturbation is studied. By using the asymptotic expansions in terms of inverse powers of the driving frequency, a class of unitary time-dependent canonical transformations is defined which renders the transformed Hamiltonians time independent. One representative of those Hamiltonians is the quasienergy operator, which is explicitly derived up to the fourth order. The classical limit of the theory and the possibility of separating the mean-motion Hamiltonian is discussed. It is shown that this separation can consistently be carried out to higher orders only in the case of a uniform external force. The application of the method to unperturbed systems with regular confining and singular potentials is discussed by considering the examples of harmonic oscillator, particle in the double-well potential, and particle in the Coulomb potential.