Quantum simulation of the Dirac equation

Abstract

The Dirac equation, proposed by Paul Dirac in 1928 to describe the behaviour of relativistic quantum particles, merges quantum mechanics with special relativity. A number of peculiar effects emerge from the equation, including a rapid quivering motion or 'Zitterbewegung', well established in theory but difficult to observe in real particles. Christian Roos and colleagues have developed a proof-of-principle quantum simulation of the Dirac equation using a single trapped ion set to behave as a free relativistic quantum particle. The high level of control of trapped-ion experimental parameters in this system makes it possible to simulate and study Zitterbewegung and other textbook examples of relativistic quantum physics. The Dirac equation successfully merges quantum mechanics with special relativity. It predicts some peculiar effects such as 'Zitterbewegung', an unexpected quivering motion of a free relativistic quantum particle. This and other predicted phenomena are key fundamental examples for understanding relativistic quantum effects, but are difficult to observe in real particles. Here, using a single trapped ion set to behave as a free relativistic quantum particle, a quantum simulation of the one-dimensional Dirac equation is demonstrated. The Dirac equation1 successfully merges quantum mechanics with special relativity. It provides a natural description of the electron spin, predicts the existence of antimatter2 and is able to reproduce accurately the spectrum of the hydrogen atom. The realm of the Dirac equation—relativistic quantum mechanics—is considered to be the natural transition to quantum field theory. However, the Dirac equation also predicts some peculiar effects, such as Klein’s paradox3 and ‘Zitterbewegung’, an unexpected quivering motion of a free relativistic quantum particle4. These and other predicted phenomena are key fundamental examples for understanding relativistic quantum effects, but are difficult to observe in real particles. In recent years, there has been increased interest in simulations of relativistic quantum effects using different physical set-ups5,6,7,8,9,10,11, in which parameter tunability allows access to different physical regimes. Here we perform a proof-of-principle quantum simulation of the one-dimensional Dirac equation using a single trapped ion7 set to behave as a free relativistic quantum particle. We measure the particle position as a function of time and study Zitterbewegung for different initial superpositions of positive- and negative-energy spinor states, as well as the crossover from relativistic to non-relativistic dynamics. The high level of control of trapped-ion experimental parameters makes it possible to simulate textbook examples of relativistic quantum physics.