The quantum motion of an ion in a quadrupole radio-frequency trap is explicitly solved in terms of coherent states for the dynamical symplectic group in connection with the classical motion. The coherent states are built over extremal and nonextremal weight states. The geometrical Berry phases and the quasienergy spectrum are also obtained.
The quantum motion of an ion in a quadrupole radio-frequency trap is explicitly solved in terms of coherent states for the dynamical symplectic group in connection with the classical motion. The coherent states are built over extremal and nonextremal weight states. The geometrical Berry phases and the quasienergy spectrum are also obtained.
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 firstly discuss classical stability for a dynamical system of two ions levitated in a 3D Radio-Frequency (RF) trap, assimilated with two coupled oscillators. We obtain the solutions of the coupled system of equations that characterizes the associated dynamics. In addition, we supply the modes of oscillation and demonstrate the weak coupling condition is inappropriate in practice, while for collective modes of motion (and strong coupling) only a peak of the mass can be detected. Phase portraits and power spectra are employed to illustrate how the trajectory executes quasiperiodic motion on the surface of torus, namely a Kolmogorov-Arnold-Moser (KAM) torus. In an attempt to better describe dynamical stability of the system, we introduce a model that characterizes dynamical stability and the critical points based on the Hessian matrix approach. The model is then applied to investigate quantum dynamics for many-body systems consisting of identical ions, levitated in 2D and 3D ion traps. Finally, the same model is applied to the case of a combined 3D Quadrupole Ion Trap (QIT) with axial symmetry, for which we obtain the associated Hamilton function. The ion distribution can be described by means of numerical modeling, based on the Hamilton function we assign to the system. The approach we introduce is effective to infer the parameters of distinct types of traps by applying a unitary and coherent method, and especially for identifying equilibrium configurations, of large interest for ion crystals or 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 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."
Quantum-mechanical analysis of ion motion in a rotating-radio-frequency (rrf) trap or in a Penning trap with a quadrupole rotating field is carried out. Rrf traps were introduced by Hasegawa and Bollinger [Phys. Rev. A 72, 043404 (2005)]. The classical motion of a single ion in this trap is described by only trigonometric functions, whereas in the conventional linear radio-frequency (rf) traps it is by the Mathieu functions. Because of the simple classical motion in the rrf trap, it is expected that the quantum-mechanical analysis of the rrf traps is also simple compared to that of the linear rf traps. The analysis of Penning traps with a quadrupole rotating field is also possible in a way similar to the rrf traps. As a result, the Hamiltonian in these traps is the same as the two-dimensional harmonic oscillator, and energy levels and wave functions are derived as exact results. In these traps, it is found that one of the vibrational modes in the rotating frame can have negative energy levels, which means that the zero-quantum-number state (``ground'' state) is the highest energy state.
Dynamical decoupling techniques are a versatile tool for engineering quantum states with tailored properties. In trapped ions, nested layers of continuous dynamical decoupling by means of radio-frequency field dressing can cancel dominant magnetic and electric shifts and therefore provide highly prolonged coherence times of electronic states. Exploiting this enhancement for frequency metrology, quantum simulation or quantum computation, poses the challenge to combine the decoupling with laser-ion interactions for the quantum control of electronic and motional states of trapped ions. Ultimately, this will require running quantum gates on qubits from dressed decoupled states. We provide here a compact representation of nested continuous dynamical decoupling in trapped ions, and apply it to electronic S and D states and optical quadrupole transitions. Our treatment provides all effective transition frequencies and Rabi rates, as well as the effective selection rules of these transitions. On this basis, we discuss the possibility of combining continuous dynamical decoupling and Mølmer-Sørensen gates.