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arXiv:1106.4067 · Europhysics Letters (EPL)

Eigenvalue and eigenspace anholonomies in hierarchical systems

Atushi Tanaka, Sang Wook Kim, Taksu Cheon

Published Sep 20, 2011• 3 citations
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Abstract

An adiabatic cycle of parameters in a quantum system can yield the quantum anholonomies, non-trivial evolution not just in the phase of states, but also in eigenvalues and eigenstates. Such exotic anholonomies imply that an adiabatic cycle rearranges eigenstates even without spectral degeneracy. We show that an arbitrarily large quantum circuit generated by recursive extension can also exhibit the eigenvalue and eigenspace anholonomies.

DOI
10.1209/0295-5075/96/10005

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An adiabatic cycle of parameters in a quantum system can yield the quantum anholonomies, non-trivial evolution not just in the phase of states, but also in eigenvalues and eigenstates. Such exotic anholonomies imply that an adiabatic cycle rearranges eigenstates even without spectral degeneracy. We show that an arbitrarily large quantum circuit generated by recursive extension can also exhibit the eigenvalue and eigenspace anholonomies.
An adiabatic change of parameters along a closed path may interchange the (quasi-)eigenenergies and eigenspaces of a closed quantum system.Such discrepancies, induced by adiabatic cycles are referred to as the exotic quantum holonomy, which is an extension of the geometric phase."Small" adiabatic cycles induce no change on eigenspaces, whereas some "large" adiabatic cycles interchange eigenspaces.We explain the topological formulation for the eigenspace anholonomy, where the homotopy equivalence precisely distinguishes the larger cycles from smaller ones.An application to two level systems is explained.We also examine the cycles that involve the adiabatic evolution across an exact crossing, and the diabatic evolution across an avoided crossing.The latter is a nonadiabatic example of the exotic quantum holonomy.
An interplay of an exotic quantum holonomy and exceptional points is examined in one-dimensional Bose systems. The eigenenergy anholonomy, in which Hermitian adiabatic cycle induces nontrivial change in eigenenergies, can be interpreted as a manifestation of eigenenergy's Riemann surface structure, where the branch points are identified as the exceptional points which are degeneracy points in the complexified parameter space. It is also shown that the exceptional points are the divergent points of the non-Abelian gauge connection for the gauge theoretical formulation of the eigenspace anholonomy. This helps us to evaluate anti-path-ordered exponentials of the gauge connection to obtain gauge covariant quantities.
It is shown that adiabatic cycles excite a quantum particle, which is confined in a one-dimensional region and is initially in an eigenstate. During the cycle, an infinitely sharp wall is applied and its strength and position are varied. After the completion of the cycle, the state of the particle arrives at another eigenstate. It is also shown that we may vary the final adiabatic state by choosing the parameters of the cycle. With a combination of these adiabatic cycles, we can connect an arbitrary pair of eigenstates. Hence, these cycles may be regarded as the basis of the adiabatic excitations. A detailed argument is provided for the case that the particle is confined by an infinite square well. This is an example of exotic quantum holonomy in Hamiltonian systems.
The correspondence between exotic quantum holonomy, which occurs in families of Hermitian cycles, and exceptional points (EPs) for non-Hermitian quantum theory is examined in quantum kicked tops. Under a suitable condition, an explicit expression of the adiabatic parameter dependencies of quasienergies and stationary states, which exhibit anholonomies, is obtained. It is also shown that the quantum kicked tops with the complexified adiabatic parameter have a higher-order EP, which is broken into lower-order EPs with the application of small perturbations. The stability of exotic holonomy against such bifurcation is demonstrated.
The parametric deformations of quasienergies and eigenvectors of unitary operators are applied to the design of quantum adiabatic algorithms. The conventional, standard adiabatic quantum computation proceeds along eigenenergies of parameter-dependent Hamiltonians. By contrast, discrete adiabatic computation utilizes adiabatic passage along the quasienergies of parameter-dependent unitary operators. For example, such computation can be realized by a concatenation of parameterized quantum circuits, with an adiabatic though inevitably discrete change of the parameter. A design principle of adiabatic passage along quasienergy was recently proposed: Cheon's quasienergy and eigenspace anholonomies on unitary operators is available to realize anholonomic adiabatic algorithms [A. Tanaka and M. Miyamoto, Phys. Rev. Lett. 98, 160407 (2007)], which compose a nontrivial family of discrete adiabatic algorithms. It is straightforward to port a standard adiabatic algorithm to an anholonomic adiabatic one, except an introduction of a parameter $|v\ensuremath{\rangle}$, which is available to adjust the gaps of the quasienergies to control the running time steps. In Grover's database search problem, the costs to prepare $|v\ensuremath{\rangle}$ for the qualitatively different (i.e., power or exponential) running time steps are shown to be qualitatively different.
Adiabatic excitation of a confined one-dimensional particle with a variable δ\deltaδ-wall
It is shown that adiabatic cycles excite a quantum particle, which is confined in a one-dimensional region and is initially in an eigenstate. During the cycle, a δ\deltaδ-wall is applied and varied its strength and position. After the completion of the cycle, the state of the particle arrives another eigenstate. It is also shown that an arbitrary pair of eigenstates can be connected through a combination of adiabatic cycles. A detailed argument is provided for the case that the particle is confined by an infinite square well. This is an example of exotic quantum holonomy in Hamiltonian systems.