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arXiv:0811.2116 · Physical Review Letters

Confining Stationary Light: Dirac Dynamics and Klein Tunneling

Johannes Otterbach, Razmik G. Unanyan, Michael Fleischhauer

Published Feb 9, 2009• 50 citations
arXivHTMLPDF

Abstract

We discuss the properties of 1D stationary pulses of light in an atomic ensemble with electromagnetically induced transparency in the limit of tight spatial confinement. When the size of the wave packet becomes comparable or smaller than the absorption length of the medium, it must be described by a two-component vector which obeys the one-dimensional two-component Dirac equation with an effective mass m;{*} and effective speed of light c;{*}. Then a fundamental lower limit to the spatial width in an external potential arises from Klein tunneling and is given by the effective Compton length lambda_{C}=variant Planck's over 2pi/(m;{*}c;{*}). Since c;{*} and m;{*} can be externally controlled and can be made small, it is possible to observe effects of the relativistic dispersion for rather low energies or correspondingly on macroscopic length scales.

DOI
10.1103/physrevlett.102.063602

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We discuss the properties of 1D stationary pulses of light in an atomic ensemble with electromagnetically induced transparency in the limit of tight spatial confinement. When the size of the wave packet becomes comparable or smaller than the absorption length of the medium, it must be described by a two-component vector which obeys the one-dimensional two-component Dirac equation with an effective mass m;{} and effective speed of light c;{}. Then a fundamental lower limit to the spatial width in an external potential arises from Klein tunneling and is given by the effective Compton length lambda_{C}=variant Planck's over 2pi/(m;{}c;{}). Since c;{} and m;{} can be externally controlled and can be made small, it is possible to observe effects of the relativistic dispersion for rather low energies or correspondingly on macroscopic length scales.
Dirac's equation with static external fields is considered as an equation of relativistic quantum theory, which describes the quantum dynamics of an electron in four-dimensional space-time. It is shown that the Dirac electron has only one intrinsic degree of freedom -- spin. The quantum ensemble of an electron with a given energy consists of two inseparable from each other subensembles of heavy and light spin-1/2 quasiparticles to have opposite (relative) intrinsic parities. In the case of electrical scalar potential nonuniform in the direction of a particle motion, these subensembles behave as analogs of the electrical and magnetic components of the plane monochromatic electromagnetic wave propagating in a dispersive medium, nonuniform in the direction of a wave motion. The role of states with negative energies and the properties of the velocity operator are explained.
Dirac's equation with static external fields is considered as an equation of relativistic quantum theory, which describes the quantum dynamics of an electron in four-dimensional space-time. It is shown that the Dirac electron has only one intrinsic degree of freedom -- spin. The quantum ensemble of an electron with a given energy consists of two inseparable from each other subensembles of heavy and light spin-1/2 quasiparticles to have opposite (relative) intrinsic parities. In the case of electrical scalar potential nonuniform in the direction of a particle motion, these subensembles behave as analogs of the electrical and magnetic components of the plane monochromatic electromagnetic wave propagating in a dispersive medium, nonuniform in the direction of a wave motion. The role of states with negative energies and the properties of the velocity operator are explained.
Electromagnetically-induced transparency (EIT) with Rydberg atoms enables strong interactions between atoms to be controlled and measured by pulses of light. Consequently, Rydberg dark polaritons are a promising platform for quantum simulations. Of particular interest are simulations of one-dimensional systems, known to exhibit unique properties compared to their higher-dimensional counterparts. One limitation of standard EIT is that reducing the polariton group velocity to bring the dark polaritons to rest also implies activating strong Rydberg interactions. To achieve independent control of polariton speed and interaction strength, we propose a stationary light-scheme to bring Rydberg polaritons to a stand-still. This allows them to remain primarily photonic and, therefore, non-interacting, before Rydberg interactions are turned on. Once activated, in the dilute regime and in the presence of strong transverse confinement, the strong polarizabilities of Rydberg atoms give rise to an effective contact interaction between polaritons. By tuning the various parameters of the stationary-light scheme, the effective one-dimensional scattering length may be adjusted to yield various kinds of physics, including the regime of a Tonks-Girardeau gas, when polaritons exhibit fermionic behavior. We outline a protocol to observe the resulting spatial correlations between neighboring polaritons.
We study the dynamics of ultracold atoms in tailored bichromatic optical lattices. By tuning the lattice parameters, one can readily engineer the band structure and realize a Dirac point, i.e., a true crossing of two Bloch bands. The dynamics in the vicinity of such a crossing is described by the one-dimensional Dirac equation, which is rigorously shown beyond the tight-binding approximation. Within this framework we analyze the effects of an external potential and demonstrate numerically that it is possible to demonstrate Klein tunneling with current experimental setups.
We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like Ψ(r,t)=ψ(r)e−iEt/ℏΨ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}Ψ(r,t)=ψ(r)e−iEt/ℏ, and define m=E/c2m=E/c^2m=E/c2, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.
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.