Relativity on the Table-Top

Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

L. Lamata, J. Leon, T. Schatz, and E. Solano

PRL 98, 253005 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e253005

In this article, the authors demonstrate the experimental possibility of simulating a Dirac Hamiltonian in an atomic system. In short, they propose a table-top experiment that would probe the effects of a relativistic system.

The requirements are relatively modest. To simulate the (1+1)-D or (2+1)-D Dirac equation, one needs a two-level atomic system; for the (3+1)-D Dirac equation, one needs a 4-level system. Three types of couplings are also required:
• Carrier Interaction -- a resonant coupling between two of the internal atomic states (such as a laser tuned to the transition frequency).
• Jaynes-Cummings Interaction -- couples two internal states with a vibrational mode of the center of mass. An upward transition between internal states is accompanied by the destruction of a phonon.
• Anti-Jaynes-Cummings Interaction -- also couples two internal states with a vibrational mode, but the frequency is tuned so that an upward transition between is accompanied instead by the creation of a phonon.

When the phases of the laser fields used to generate the above interactions are appropriately tuned, the effective Hamiltonian for the two-level or four-level system is identical in form with the free Dirac equation. By varying the couplings, one can tune the effective particle mass and velocity of light.

The authors dedicate a lot of space to a discussion of Zitterbewegung, which is a rapid oscillatory motion of an electron about its mean position. It was predicted in 1930 by Schrodinger, but has not yet been observed experimentally. For electrons, the amplitude of the oscillations is on the order of 10^{-13} m, and the frequency is on the order of 10^{21} Hz. In addition, Zitterbewegung is an effect of the single-particle Dirac equation. There is some controversy over whether or not the effect persists in QED.

The authors point out that by controlling the effective electron mass and the effective speed of light, one should be able to bring the amplitude and frequency of the oscillations into an experimentally accessible range. Another point they don't mention: even if the effect does not occur for real electrons, the dynamics of their atomic system are (in theory) accurately described by a single-particle Dirac equation. Whether or not real electrons jitter around, the trapped ions on their desks should.

The authors mention a few other relativistic effects that might be simulated: by controlling the particle mass, one could observe something akin to the Higgs mechanism for mass generation; one could simulate the Klein paradox, in which particle-hole pairs are created in a strong potential; one could simulate a (1+1)-D axial anomaly, the lower-dimensional counterpart of the chiral anomaly in (3+1)-D.

The possibility of tuning the parameters of the Dirac equation is interesting. However, such simulations of relativistic systems is a bit perplexing. A similar situation exists in the theory of graphene and carbon nanotubes, where the low-energy excitations are described by a Dirac equation. On a very fundamental level, the simulation can't be right. So how far can one push the analogy?

How similar are a pseudospinor and a spinor? Electron spin has to do with angular momentum, but pseudospin has nothing to do with it. The simulated system can't have pseudospin-orbit coupling, but the Dirac equation can.

How effective is an effective speed of light? In the table-top simulation, nothing prohibits the electron or the center of mass of the ion system from exceeding the effective speed of light. What then? Would the authors claim they were simulating tachyons?

In the Klein paradox, where are the holes going to come from? No corresponding anti-cesium ion is going to materialize in the trap.

It would be quite interesting to probe these unphysical effects in a trap and see exactly how the effective Dirac physics breaks down.

A final note: This was a well-written paper, but I must criticize the authors for a particular choice of phrase. On page 2, they refer to the "notorious analogy" between the Dirac equation and the effective Hamiltonian of the trapped ion. Jesse James was notorious. The three-body problem is notoriously difficult. An analogy between Hamiltonians is not "famous or well known, typically for some bad quality or deed."