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Accumulation of Berry phase in the adiabatic evolution of an ion confined in an anisotropic Paul trap

M. Scala, Benedetto Militello, ANTONINO MESSINA

Published Sep 25, 2004• 0 citations
OpenAlex

Abstract

We show how the anisotropy in a bidimensional Paul trap is responsible for the impossibility of finding an exact resonance condition in the dynamics induced on an ion by a laser beam. The occurrence of a periodic hamiltonian model allows us to investigate the adiabatic evolution of the system. After calculating the geometric phase we elaborate a strategy to bring to light physical effects related to it and its connections to the presence of anisotropy in the trap.

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Physical Review A•Sep 24, 2008•8 citations
We show how the anisotropy in a bidimensional Paul trap is responsible for the impossibility of finding an exact resonance condition in the dynamics induced on an ion by a laser beam. The occurrence of a periodic hamiltonian model allows us to investigate the adiabatic evolution of the system. After calculating the geometric phase we elaborate a strategy to bring to light physical effects related to it and its connections to the presence of anisotropy in the trap.
The Berry phase acquired by an eigenstate that experienced a nonlinear adiabatic evolution is investigated thoroughly. The circuit integral of the Berry connection of the instantaneous eigenstate cannot account for the adiabatic geometric phase, while the Bogoliubov excitations around the eigenstates are found to be accumulated during the nonlinear adiabatic evolution and contribute a finite phase of geometric nature. A two-mode model is used to illustrate our theory. Our theory is applicable to Bose-Einstein condensate, nonlinear light propagation, and Ginzburg-Landau equations for complex order parameters in condensed-matter physics.
A classically chaotic system consisting of a Paul trapped ion and a sequences of standing laser pulses is treated quantum-mechanically. Under the circumstance of time-dependence, we derive the transition probability from the ion's motional state to ', and find, in the first-order approximation, the classically chaotic character disappears. Theoretical analysis and numerical calculations show that by regulating the phase parameter we can control the transition probability. When reaches some specific values, the transition from the state to ' is forbidden and, for some laser periods, resonance occurs, which leads to the corresponding transitions between different motional states. The time-evolution of an initial motional state just over one period is also studied in detail.
We investigate the quantum motion of two ions stored in a Paul trap and interacting with a time-periodic laser field. In the pseudopotential approximation and large detuning condition, we find that the relative motion is independent of the laser field, but the exact centre-of-mass motion is closely related to the laser field. By adjusting the laser intensity and frequency, we can well control the quantum motion of the centre-of-mass. We illustrate some physical properties described by the centre-of-mass states, such as the squeezed coherent property, the widths and heights of the wavepackets of probability density, the classical-quantum correspondence, the resonance ladders of expectation energy and the transition probabilities between time-dependent quantum levels.
We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.
We derive an effective Hamiltonian to account for the geometric phase that occurs in the adiabatic separation of center-of-mass and internal motion of a molecule moving in external fields. We then apply this Hamiltonian to the determination of the coherence time of molecules confined in a region of space by an anisotropic electric field. We show that the coherence time is expected to scale as the load temperature of the molecules squared and that milli-K temperatures are likely to be required for a trapped molecule experiment to compete with a more conventional Ramsey beam resonance experiment.