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Symmetries and Supersymmetries in Trapped Ion Hamiltonian Models

ANTONINO MESSINA, Benedetto Militello, A. G. Nikitin

Published Jan 1, 2004• 0 citations

Abstract

The implementation of a vibronic coupling in the context of trapped ions, and its diagona- lization are reported. The specific spin-boson-like Hamiltonian model is studied exploiting its symmetries and supersymmetries. Applications are discussed. The study of symmetries plays a central role in the analysis of trapped ion dynamical prob- lems. Indeed, in most of the cases, symmetries provide an elegant and effective way for finding Hamiltonian eigensolutions. In this paper we describe a trapped ion physical situation described by a Hamiltonian model possessing interesting and useful symmetries. In detail, conservation of total excitation num- ber and rotational invariance are found. Moreover a Second Order Hidden Supersymmetry is discovered. Symmetry based diagonalization provides the possibility of writing down the time evolution concerning simple initial conditions. Interesting nonclassical behaviors like the gene- ration of GHZ (7) states are predicted.

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The implementation of a vibronic coupling in the context of trapped ions, and its diagona- lization are reported. The specific spin-boson-like Hamiltonian model is studied exploiting its symmetries and supersymmetries. Applications are discussed. The study of symmetries plays a central role in the analysis of trapped ion dynamical prob- lems. Indeed, in most of the cases, symmetries provide an elegant and effective way for finding Hamiltonian eigensolutions. In this paper we describe a trapped ion physical situation described by a Hamiltonian model possessing interesting and useful symmetries. In detail, conservation of total excitation num- ber and rotational invariance are found. Moreover a Second Order Hidden Supersymmetry is discovered. Symmetry based diagonalization provides the possibility of writing down the time evolution concerning simple initial conditions. Interesting nonclassical behaviors like the gene- ration of GHZ (7) states are predicted.

Abstract A novel approach is proposed to analyze a rather vast counter-rotating Hamiltonian interaction in the context of quantum optics.The method relies upon the supersymmetric mapping of the corresponding rotating interaction and allows the analysis of thedynamics in the counter-rotating system in a fully general and exact analytical manner by diagonalizing the correspondingHamiltonian. Intriguing features of the counter-rotating system are revealed through the simple supersymmetric transformation.In turn, such interesting attributes have an important range of potential technological applications. In this way, supersymmetryemerges as a useful tool to both connect and construct exactly solvable photonic systems in quantum optics, as well as toanalyze the corresponding physical consequences and their possible technology implementations.

Trapped ions arranged in Coulomb crystals provide us with the elements to study the physics of a single spin coupled to a boson bath. In this work, we show that optical forces allow us to realize a variety of spin-boson models, depending on the crystal geometry and the laser configuration. We study in detail the ohmic case, which can be implemented by illuminating a single ion with a traveling wave. The mesoscopic character of the phonon bath in trapped ions induces effects such as the appearance of quantum revivals in the spin evolution.

Trapped-ion arrays offer interesting possibilities for quantum simulation. We show that a proper arrangement of elliptical micro-traps combined with the external driving of the micro-trap frequencies allows, without the need of any precise fine-tuning, for the robust realization of non-Abelian vibron dynamics. We show that this non-Abelian nature may be readily probed experimentally in a simple plaquette arrangement. This study opens interesting perspectives for the study of non-Abelian spin-orbit coupling with motional excitations in two- and three-dimensional ion arrays.

We investigate how symmetries and conserved quantities relate to the occurrence of the boundary time crystal (BTC) phase in a generalized spin model with Lindblad dissipation. BTCs are a nonequilibrium phase of matter in which the system, coupled to an external environment, breaks the continuous time translational invariance. We perform a detailed mean-field study aided by a finite-size analysis of the quantum model of a

p,q

-spin-interaction system, a generalized

p

-spin-interaction system, which can be implemented in fully connected spin-1/2 ensembles. We find the following conditions for the observation of the BTC phase. First, the BTC appears when the discrete symmetry held by the Hamiltonian,

{\mathbb{Z}}_{2}

in the considered models, is explicitly broken by the Lindblad jump operators. Second, the system must be coupled uniformly to the same bath in order to preserve the total angular momentum during the time evolution. If these conditions are not satisfied, any oscillatory behavior appears only as a transient in the dynamics and a time-independent stationary state is eventually reached. Our results suggest that these two elements may be general requirements for the observation of a stable BTC phase relating symmetries and conserved quantities in arbitrary spin models.

An algebraic view of supersymmetric quantum mechanics is taken, emphasising the couplings between bosonic and fermionic modes in the supercharges. A class of model Hamiltonians is introduced wherein the fermionic (bosonic) operators are canonical and the bosonic (fermionic) ones satisfy a Lie algebra (superalgebra) whose representation theory permits the complete solution of the model in principle. The kinematical symmetry of such models is also described. The examples of one and two bosonic models, with 5U(2) and 5U(3) dynamical algebras respectively, are analysed in detail.