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arXiv:1510.07854 · Physical Review A

Adiabatic excitation of a confined particle in one dimension with a variable infinitely sharp wall

Sho Kasumie, Manabu Miyamoto, Atushi Tanaka

Published Apr 5, 2016• 7 citations
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Abstract

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.

DOI
10.1103/physreva.93.042105

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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.
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.
We show that an adiabatic cycle excites Bose particles confined in a one-dimensional box. During the adiabatic cycle, a wall described by a δ -shaped potential is applied and its strength and position are slowly varied. When the system is initially prepared in the ground state, namely, in the zero-temperature equilibrium state, the adiabatic cycle brings all Bosons into the first excited one-particle state, leaving the system in a nonequilibrium state. The absorbed energy during the cycle is proportional to the number of Bosons.
An adiabatic cycle involving the δ-function potential is shown to excite Bose particles confined in a one-dimensional box.We consider a cycle, during which a wall described by a δ potential is applied and its strength and position are slowly varied.When the system is initially prepared in the equilibrium ground state, the adiabatic cycle brings all bosons into the first excited one-particle state, leaving the system in a nonequilibrium state.The absorbed energy during the cycle is proportional to the number of bosons.
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.
The evolution of a driven quantum system is said to be adiabatic whenever the state of the system stays close to an instantaneous eigenstate of its time-dependent Hamiltonian. The celebrated quantum adiabatic theorem ensures that such pure state adiabaticity can be maintained with arbitrary accuracy, provided one chooses a small enough driving rate. Here, we extend the notion of quantum adiabaticity to closed quantum systems initially prepared at finite temperature. In this case adiabaticity implies that the (mixed) state of the system stays close to a quasi-Gibbs state diagonal in the basis of the instantaneous eigenstates of the Hamiltonian. We prove a sufficient condition for the finite temperature adiabaticity. Remarkably, it turns out that the finite temperature adiabaticity can be more robust than the pure state adiabaticity with respect to increasing the system size. This can be the case for one-body systems with large Hilbert spaces, such as a particle in a large box, as well as for certain many-body systems. In particular, we present an example of a driven many-body system where, in the thermodynamic limit, the finite temperature adiabaticity is maintained, while the pure state adiabaticity breaks down. On the other hand, for generic many-body systems the scaling of the finite temperature adiabatic condition with the system size is exponential, analogously to pure state adiabatic conditions.
An adiabatic time evolution of a closed quantum system connects eigenspaces of initial and final Hermitian Hamiltonians for slowly driven systems, or, unitary Floquet operators for slowly modulated driven systems. We show that the connection of eigenspaces depends on a topological property of the adiabatic paths for given initial and final points. An example in slowly modulated periodically driven systems is shown. These analysis are based on the topological analysis of the exotic quantum holonomy in adiabatic closed paths.