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arXiv:1106.6222 · New Journal of Physics

Relativistic quantum mechanics with trapped ions

Lucas Lamata, Jorge Casanova, R. Gerritsma, Christian F. Roos, Juan José García‐Ripoll, E. Solano

Published Sep 6, 2011• 79 citations
arXivHTMLPDF

Abstract

We consider the quantum simulation of relativistic quantum mechanics, as described by the Dirac equation and classical potentials, in trapped-ion systems. We concentrate on three problems of growing complexity. Firstly, we study the bidimensional relativistic scattering of single Dirac particles by a linear potential. Secondly, we explore the case of a Dirac particle in a magnetic field and its topological properties. Finally, we analyze the problem of two Dirac particles that are coupled by a controllable and confining potential. The latter interaction may be useful to study important phenomena such as the confinement and asymptotic freedom of quarks.

DOI
10.1088/1367-2630/13/9/095003

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We consider the quantum simulation of relativistic quantum mechanics, as described by the Dirac equation and classical potentials, in trapped-ion systems. We concentrate on three problems of growing complexity. Firstly, we study the bidimensional relativistic scattering of single Dirac particles by a linear potential. Secondly, we explore the case of a Dirac particle in a magnetic field and its topological properties. Finally, we analyze the problem of two Dirac particles that are coupled by a controllable and confining potential. The latter interaction may be useful to study important phenomena such as the confinement and asymptotic freedom of quarks.
We propose the quantum simulation of the Dirac equation with potentials, allowing the study of relativistic scattering and Klein tunneling. This quantum relativistic effect permits a positive-energy Dirac particle to propagate through a repulsive potential via the population transfer to negative-energy components. We show how to engineer scalar, pseudoscalar, and other potentials in the 1+11+11+1 Dirac equation by manipulating two trapped ions. The Dirac spinor is represented by the internal states of one ion, while its position and momentum are described by those of a collective motional mode. The second ion is used to build the desired potentials with high spatial resolution.
We report on quantum simulations of relativistic scattering dynamics using trapped ions. The simulated state of a scattering particle is encoded in both the electronic and vibrational state of an ion, representing the discrete and continuous components of relativistic wave functions. Multiple laser fields and an auxiliary ion simulate the dynamics generated by the Dirac equation in the presence of a scattering potential. Measurement and reconstruction of the particle wave packet enables a frame-by-frame visualization of the scattering processes. By precisely engineering a range of external potentials we are able to simulate text book relativistic scattering experiments and study Klein tunneling in an analogue quantum simulator. We describe extensions to solve problems that are beyond current classical computing capabilities.
We present a method of simulating the Dirac equation in 3+1 dimensions for a free spin-1/2 particle in a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position and momentum of the Dirac particle are associated with the respective ionic variables. We show also how to simulate the simplified 1+1 case, requiring the manipulation of only two internal levels and one motional degree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung and Klein's paradox, the transition from massless to massive fermions, and the relativistic and nonrelativistic limits, via the tuning of controllable experimental parameters.
Quantum transition probabilities and quantum entanglement for two-qubit states of a four-level trapped ion quantum system are computed for time-evolving ionic states driven by Jaynes-Cummings Hamiltonians with interactions mapped onto a $\text{SU}(2)\ensuremath{\bigotimes}\text{SU}(2)$ group structure. Using the correspondence of the method of simulating a 3+13+13+1 dimensional Dirac-like Hamiltonian for bispinor particles into a single trapped ion, one preliminarily obtains the analytical tools for describing ionic state transition probabilities as a typical quantum oscillation feature. For Dirac-like structures driven by generalized Poincar\'e classes of coupling potentials, one also identifies the $\text{SU}(2)\ensuremath{\bigotimes}\text{SU}(2)$ internal degrees of freedom corresponding to intrinsic parity and spin polarization as an adaptive platform for computing the quantum entanglement between the internal quantum subsystems which define two-qubit ionic states. The obtained quantum correlational content is then translated into the quantum entanglement of two-qubit ionic states with quantum numbers related to the total angular momentum and to its projection onto the direction of the trapping magnetic field. Experimentally, the controllable parameters simulated by ion traps can be mapped into a Dirac-like system in the presence of an electrostatic field which, in this case, is associated to ionic carrier interactions. Besides exhibiting a complete analytical profile for ionic quantum transitions and quantum entanglement, our results indicate that carrier interactions actively drive an overall suppression of the quantum entanglement.
Abstract The two-dimensional Dirac oscillator is one of the most transparent exactly solvable models in relativistic quantum mechanics. Its Hamiltonian is linear in both momentum and posi- tion and therefore has no direct elementary-particle realization in ordinary harmonic traps. This extended proposal develops two complementary laboratory routes. The first is an optical trapped-ion implementation in which the electronic two-level system encodes the Dirac bispinor and two transverse motional modes encode the two spatial degrees of freedom. Bichromatic red- and blue-sideband laser fields synthesize the Jaynes-Cummings and anti-Jaynes-Cummings cou- plings required by the oscillator. The second route is a microwave-based proposal. Two meanings of “microwave implementation” are distinguished: a microwave-driven trapped-ion implementa- tion using magnetic-gradient-induced spin-motion coupling, and a microwave-resonator/tight- binding analogue based on coupled dielectric resonators. The former remains a genuine single- ion quantum simulator and can, in principle, implement the same effective Hamiltonian with improved phase stability and reduced optical spontaneous-emission errors. The latter is experi- mentally simpler for spectral measurements and can serve as a second proposal, but it is a wave analogue rather than a single-particle trapped-ion quantum simulation. We do not claim the first mapping of the 2 + 1 Dirac oscillator onto trapped-ion dynamics, which is already established in the literature; rather, the contribution is a consolidated, experimentally oriented roadmap that compares the optical trapped-ion, microwave-driven trapped-ion, and microwave-resonator routes, with explicit feasibility conditions, an observable strategy, a corrected graphical map- ping, an explicit separation of the ideal Hamiltonian from its leading correction terms, a worked order-of-magnitude estimate for a 171 Yb+ ion, and an explicit quantitative cross-walk to the trapped-ion and microwave experiments on which the scheme builds. We identify the advan- tages and limitations of each route and propose a staged experimental program for measuring Zitterbewegung, the relativistic square-root spectrum, and chirality-dependent dynamics.
The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.