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arXiv:1107.0856

Time Dependent Variational Principle and Coherent State Orbits for a Trapped Ion

Bogdan M. Mihalcea

Published Jul 5, 2011• 1 citation
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Abstract

Spectral properties of the Hamiltonian function which characterizes a trapped ion are investigated. In order to study semiclassical dynamics of trapped ions, coherent state orbits are introduced as sub-manifolds of the quantum state space, with the Kähler structure induced by the transition probability. The time dependent variational principle (TDVP) is applied on coherent states' orbits. The Hamilton equations of motion on Kähler manifolds of the type of classical phase spaces naturally arise. The associated classical Hamiltonian is obtained from the expected values on symplectic coherent states of the quantum Hamiltonian. Spectral information is thus coded within the phase portrait. We deal with the bosonic realization of the Lie algebra of the SU(1,1) group, which we particularize for the case of an ion confined in a combined, Paul and Penning trap. This formalism can be applied to Hamiltonians which are nonlinear in the infinitesimal generators of a dynamical symmetry group, such as the case of ions confined in electrodynamic traps. Discrete quasienergy spectra are obtained and the corresponding quasienergy states are explicitly realized as coherent states parameterized by the stable solutions of the corresponding classical equations of motion. A correspondence between quantum and classical stability domains is thus established, using the Husimi representation.

DOI
10.48550/arxiv.1107.0856

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Spectral properties of the Hamiltonian function which characterizes a trapped ion are investigated. In order to study semiclassical dynamics of trapped ions, coherent state orbits are introduced as sub-manifolds of the quantum state space, with the Kähler structure induced by the transition probability. The time dependent variational principle (TDVP) is applied on coherent states' orbits. The Hamilton equations of motion on Kähler manifolds of the type of classical phase spaces naturally arise. The associated classical Hamiltonian is obtained from the expected values on symplectic coherent states of the quantum Hamiltonian. Spectral information is thus coded within the phase portrait. We deal with the bosonic realization of the Lie algebra of the SU(1,1) group, which we particularize for the case of an ion confined in a combined, Paul and Penning trap. This formalism can be applied to Hamiltonians which are nonlinear in the infinitesimal generators of a dynamical symmetry group, such as the case of ions confined in electrodynamic traps. Discrete quasienergy spectra are obtained and the corresponding quasienergy states are explicitly realized as coherent states parameterized by the stable solutions of the corresponding classical equations of motion. A correspondence between quantum and classical stability domains is thus established, using the Husimi representation.
Quasiclassical dynamics of trapped ions is described by applying the time dependent variational principle (TDVP) on coherent state orbits, introduced as sub-manifolds of the quantum state space. The classical Hamilton functions associated to the system are obtained as the expected values of the quantum Hamiltonian on symplectic coherent states. Such formalism can be applied to Hamilton functions which are nonlinear in the infinitesimal generators of a dynamical symmetry group. We give the explicit expression of the quantum Hamilton function in case of nonlinear electromagnetic traps. Discrete quasienergy spectra result, and the corresponding quasienergy states are explicitly realized as coherent states parameterized by the stable solutions of the corresponding classical equations of motion. A dequantization algorithm is also suggested. The method is then applied to combined (Paul and Penning) and ideal Paul traps. We obtain the explicit expressions of the classical Hamilton functions for such particular traps, as well as the classical Hamilton equations of motion in complex coordinates for an octupole trap. In the pseudopotential approximation, the points of minimum of the Hamiton function define equilibrium configurations for trapped ions, of interest for quantum logic.
We investigate the spectral properties of the Hamiltonian function that describes an ion confined within a nonlinear trap. The Hamiltonian is then particularized for the case of dynamic traps and we introduce algebraic models with the aim to characterize the associated dynamics in such traps. The coherent states formalism for dynamic groups and the time-dependent variational principle are applied in order to study the semiclassical behaviour of the confined ion. We deal with the bosonic realization of the Lie algebra of the SU(1,1) group, which we particularize for the case of an ion confined in a combined (Paul and Penning) trap. We infer the equations of motion for the ion in a semiclassical approach. We suggest an algorithm by means of which we can associate a classical Hamiltonian with the quantum Hamiltonian that describes the ion. The classical Hamiltonian implicitly contains spectral information on the quantum system, which means we can dequantify the system.
The evolution of squeezed coherent states (CSs) of motion for trapped ions is investigated by applying the time dependent variational principle (TDVP) for the Schrödinger equation. The method is applied in case of Paul and combined traps, for which the classical Hamiltonian and equations of motion are derived. Hence, CS provide a natural framework to: (a) engineer quantum correlated states for trapped ions intended for ultraprecise measurements, (b) explore the mechanisms responsible for decoherence, and (c) investigate the quantum-classical transition.
Collective many-body dynamics for time-dependent quantum Hamiltonian functions is investigated, for a dynamical system that exhibits multiple degrees of liberty, that is a combined (Paul and Penning) trap. We introduce the generators of the Lie algebra of the symplectic group SL(2,R){\cal {SL}}(2, \mathbb R)SL(2,R), which we use to build the coherent states (CS) associated to the system under investigation. The trapped ion is treated as a harmonic oscillator (HO), to which we associate the quantum Hamilton function. We obtain the kinetic and potential energy operators as functions of the Lie algebra generators, and supply the expressions for the classical coordinate, momentum, kinetic and potential energy, as well as the total energy. We also infer the dispersions for the coordinate and momentum, the asymmetry and the flatness parametre for the distribution. The system interaction with the laser radiation is also examined for a system of identical two-level atoms. The Hamilton function for the Dicke model is obtained. The optical system is modeled as a HO (trapped ion) that undergoes interaction with an external laser field. Such an approach enables one to build CS in a compact and smart manner by use of the group theory.
The time-dependent variational principle associates to a Hamiltonian quantum system a set of trajectories running on a classical phase space with canonical equations of motion. When the quantum observables generate a Lie group G and the states are taken as functions on an appropriate homogeneous quotient space space := G/G 0 under under G , they can be equipped with a classical Poisson bracket which reproduces the commutators of the Lie group. The quantum system maps into a classical system whose equations of motion are governed by the expectation value of the quantum Hamiltonian H. We consider examples of this construction and show that the analysis of generalized classical systems provides insight into quantum many-body dynamics like chemical reactions.