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Quasienergy operators and general squeezed states for systems of trapped ions

Bogdan M. Mihalcea

Published Aug 26, 2021• 0 citations
OpenAlex

Abstract

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 ${\cal {SL}}(2, \mathbb R)$, 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.

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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.
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.
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.
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.
"We investigate quantum dynamics for an ion confined within an oscillating quadrupole field, starting from two well known and elegant approaches. It is established that the Hamilton equations of motion, in both Schrodinger and Heisenberg representations, are equivalent to the Hill equation. One searches for a linear independent solution associated to a harmonic oscillator (HO). An adiabatic invariant, which is also a constant of motion, is introduced based on the Heisenberg representation. Thus, the state of the non-autonomous system can be determined at any subsequent moment of time. The quantum states for trapped ions are demonstrated to be Fock (number) states, while the exact solutions of the Schrodinger equation for a trapped ion are exactly the quasienergy states. Semiclassical dynamics is also investigated for many-body systems of trapped ions, where the wavefunction associated to the Schrodinger equation is prepared as a Gauss package multiplied by a Hermite polynomial. We also discuss time evolution for the system under investigation and supply the propagator."
Using Lie algebra methods, we find a new class of Bogoliubov transformations which generalize the notion of squeezed states. The Hamiltonians for the simple harmonic and anharmonic oscillators, turn out to be the generators of a Lie group, whose other generators may be found exactly, or up to any desired order of the perturbation parameter. An element of this Lie group, which is realized as the multi-photon operator, transforms the anharmonic Hamiltonian to the harmonic one. The transformation of the ordinary annihilation and creation operators under this unitary transformation leads to the introduction of multi-photon coherent states. We specifically consider four-photon coherent states in detail and study the time dependent position and momentum uncertainties in these states.
An ultracold two-level ion experiencing the standing wave of a resonant laser in a Paul trap is investigated in the Lamb-Dicke limit and weak excitation regime, with full consideration of the time-dependence of the trapping potential. The analytical forms of the wave functions of the system can be described with our approach, and the time evolution of the pseudo-energy of the system as well as the squeezing property of the quadrature components is studied in comparison with the treatment of harmonic oscillator model.