Abstract
We design a quantum simulator for the Majorana equation, a non-Hamiltonian relativistic wave equation that might describe neutrinos and other exotic particles beyond the standard model. Driven by the need of the simulation, we devise a general method for implementing a number of mathematical operations that are unphysical, including charge conjugation, complex conjugation, and time reversal. Furthermore, we describe how to realize the general method in a system of trapped ions. The work opens a new front in quantum simulations.
- Received 22 April 2011
DOI:https://doi.org/10.1103/PhysRevX.1.021018
This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Popular Summary
The theoretical language of quantum mechanics contains concepts like charge conjugation, complex conjugation, and time reversal—commonly used mathematical operations on the wave functions or fields describing quantum systems. They are essential to fundamental classifications of quantum systems, but they are unphysical: For example, time cannot be reversed in the real world. Simulating these operations in a real physical setting then seems out of the question. In this paper, we show, however, that the seemingly impossible is in fact possible: by creating a set of conceptual designs for implementing these unphysical operations, and by presenting a concrete proposal for simulating the Majorana equation—one of the seminal relativistic quantum-mechanical equations—using real tools fashioned out of the conceptual designs.
The Majorana equation embodies charge conjugation, which in turn involves complex conjugation. As a first step toward simulating the equation, we develop a general concept for how to implement the unphysical complex conjugation physically. The trick is to map the complex wave function of the equation to a pair of real wave functions. In a surprisingly simple way, the mapping then turns the complex conjugation on the former into a physical operation on the latter, which can be implemented in an experimental setting. A reverse mapping from the pair of changed wave functions to their corresponding complex counterpart finally completes the implementation of the complex conjugation. In the same spirit, physical implementations of charge conjugation and time reversal can also be accomplished, adding more pieces to the conceptual toolbox.
Simulating the actions of the unphysical operations needs a transformation of our conceptual designs into real physical implementations within the platform of a physical system. We go, therefore, a step further. We choose a system of two trapped ions as our platform and present a concrete proposal for simulating the Majorana equation. Here, the Majorana wave function is mapped onto both the internal and the motional quantum states of the two trapped ions and the unphysical operations are turned into manipulations of the interactions between the ions and applied laser pulses.
Looking ahead, we believe that this work also opens the door to the physical implementations of other related fundamental symmetry-based operations in quantum mechanics and adds a novel toolbox to the recently emerged interdisciplinary field of quantum simulations.