Simulating the Physics of a Free Dirac Particle

Christian Roos

[This is an invited article based on a recently published work by the author and his collaborators from Austria and Spain -- 2Physics.com]

Author: Christian Roos

Affiliation: Institut für Experimentalphysik, Universität Innsbruck, Austria
and
Institute for Quantum Optics and Quantum Information
Austrian Academy of Sciences

By the mid 1920s physicists had established the dynamics of quantum particles in the non-relativistic limit. The celebrated Schrödinger equation established a framework that allowed tackling a vast range of problems in atomic, molecular and solid state physics. However, the equation is limited to the regime of particles with velocities that are small compared to the speed of light. In 1928, Dirac put forward an equation to describe electrons in a way that successfully reconciles quantum physics with special theory of relativity. The Dirac equation provides a natural explanation of spin as an intrinsic property of the electron. It has not only positive energy solutions but also solutions with negative energies which led to the prediction of anti-matter.

In 1930, at a time when the interpretation of solutions to Dirac equation was still debated, Schrödinger noticed another peculiar feature: the equation admits solutions where the centre-of-mass of a quantum particle exhibits a trembling motion, called Zitterbewegung, in the absence of external forces [1]. This effect is surprising because according to Newton’s first law, a particle that experiences no forces should move in a straight line. In real quantum particles, such as electrons, this trembling motion would have a very small amplitude (10^-13m) and an extremely high frequency (10²¹ Hz). Moreover, it arises only as an interference effect in solutions comprised of positive and negative energy components. Such solutions, which might seem irrelevant, arise, however, in the presence of external fields. For free electrons, this phenomenon does to seem to be experimentally accessible.

It is, however, possible to engineer other quantum systems such that they mimic the physics of the Dirac equation. One such system is an ion held in an ion trap and cooled and manipulated by laser light [2]. How can such a trapped non-relativistic quantum particle simulate the physics of a free Dirac particle? To answer this question, it is helpful to look first at the case of a classical particle held in a harmonic potential. The motion of this particle is described by a circle in phase space. For a particle that is resonantly excited by an external driving force, its phase space trajectory will turn into a helix. In a frame where the phase space coordinates rotate at the resonance frequency of the particle, the helix turns into a straight line which the particle follows with constant velocity, i.e. the particle looks like a free particle in the absence of forces.

The same approach can be followed in the case of a relativistic quantum particle. Using a trapped ion, internal energy levels of the ion can be used for encoding the four spinor components representing the particle’s wave function. The term that couples the particle’s momentum operator and the spinor components in the Dirac equation can be simulated in the trapped-ion case by laser beams coupling the ion’s internal states with its motion. The term representing the ion’s rest energy is simulated by another laser-ion interaction that modifies the internal-state energies. In this way, a perfect match is achieved between the form of the Dirac equation and the Schrödinger equation describing the quantum physics of the trapped ion.

In an experiment reported in the Nature issue of the 7th January [3], this proposal is realized using a single trapped ⁴⁰Ca⁺ held in a linear ion trap (see Fig.1).

Fig.1: Experimental setup. An ion trap set up in a ultra-high vacuum system is used to store a ⁴⁰Ca⁺ ion. The ion is illuminated by laser light that serves to laser-cool, manipulate and detect the particle. (Image Credit: C. Lackner, IQOQI)

The goal of the experiment consists in observing the trembling motion predicted by Schrödinger. For this, the ion’s motion is first laser-cooled to the lowest energy state in which the ion is localized to a space of about 10 nm, the uncertainty in the position being due to the Heisenberg uncertainty relation. Then, for a certain amount of time, a suitable combination of laser beams is switched on to simulate the physics of the Dirac equation. The final step consists in a measurement that detects the change in the ion’s position. These three basic steps take no longer than 20 ms to carry out. They are repeated over and over again in order to measure the ion motion as a function of time. In perfect agreement with Schrödinger’s prediction, we indeed observe a trembling motion which is shown in Fig. 2.

Fig.2: Measured ‘Zitterbewegung’. (a) Average position of the ion as a function of time. The ion motion is composed of a uniform motion on top of which the trembling motion appears. (b) Time evolution of the ion’s wave function. Its two spinor components are shown in red and blue. The trembling motion disappears as soon as the two spinor components are no longer spatially overlapped.

Why can this experiment be called a quantum simulation? In the 1980's Richard Feynman and others proposed a new method for approaching quantum mechanical problems that are too hard to solve on ordinary computers. Their idea was to use a more accessible quantum system to simulate quantum effects of interest. To date, only a few quantum systems can be controlled well enough to act as a quantum simulator. In our experiment, we have performed a quantum simulation of a free Dirac particle using a single trapped ion manipulated with laser light. In this case, the quantum-mechanical state space has no more than 100 dimensions, a size that can be handled perfectly well by any current desktop computer. So the experiment is far from outperforming computers. But the small size of the quantum system is also an advantage because it allows us to compare experiment and theoretical prediction and in this way test the concept of a quantum simulator. The hope is that in the future systems of trapped ions or neutral atoms held in optical lattices might be used to simulate and study quantum phenomena that can no longer be analyzed by computer simulations.

References
[1] “Über die kräftefreie Bewegung in der relativistischen Quantenmechanik”, Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. 24, 418–428 (1930).
[2] “Robust Dirac equation and quantum relativistic effects in a single trapped ion”, L. Lamata, J. León, T. Schätz, E. Solano. Phys. Rev. Lett. 98, 253005 (2007). Abstract.
[3] “Quantum simulation of the Dirac equation”, R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, C. F. Roos, Nature 463, 68 (2010). Abstract.