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Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

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Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion PHYSICAL REVIEW LETTERS week ending PRL 98, 253005 (2007) 22 JUNE 2007 Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion L. Lamata,1 J. Leo´n,1 T. Scha¨tz,2 and E. Solano3,4 1Instituto de Matema´ticas y Fı´sica Fundamental, CSIC, Serrano 113-bis, 28006 Madrid, Spain 2Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany 3Physics Department, ASC, and CeNS, Ludwig-Maximilians-Universita¨t, Theresienstrasse 37, 80333 Munich, Germany 4Seccio´nFı´sica, Departamento de Ciencias, Pontificia Universidad Cato´lica del Peru´, Apartado Postal 1761, Lima, Peru (Received 27 March 2007; published 22 June 2007) We present a method of simulating the Dirac equation in 3 1 dimensions for a free spin-1=2 particle in a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position and momentum of the Dirac particle are associated with the respective ionic variables. We show also how to simulate the simplified 1 1 case, requiring the manipulation of only two internal levels and one motional degree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung and Klein’s paradox, the transition from massless to massive fermions, and the relativistic and nonrelativistic limits, via the tuning of controllable experimental parameters. DOI: 10.1103/PhysRevLett.98.253005 PACS numbers: 31.30.Jv, 03.65.Pm, 32.80.Pj The search for a fully relativistic Schro¨dinger equation of freedom. We study some quantum-relativistic effects, gave rise to the Klein-Gordon and Dirac equations. P.A. M. like the Zitterbewegung and the Klein’s paradox, in terms Dirac looked for a Lorentz-covariant wave equation that is of measurable observables. Moreover, we discuss the tran- linear in spatial and time derivatives, expecting that the sition from massless to massive fermions, and from the interpretation of the square wave function as a probability relativistic to the nonrelativistic limit. Finally, we describe density holds. As a result, he obtained a fully covariant a possible experimental scenario. wave equation for a spin-1=2 massive particle, which in- We consider a single ion of mass M inside a Paul trap corporated ab initio the spin degree of freedom. It is known with frequencies x, y, and z, where four metastable [1] that the Dirac formalism describes accurately the spec- ionic internal states, jai, jbi, jci, and jdi, may be coupled trum of the hydrogen atom and that it plays a central role in pairwise to the center-of-mass (c.m.) motion in directions quantum field theory, where creation and annihilation of x, y, and z. We will make use of three standard interactions particles are allowed. However, the one-particle solutions in trapped-ion technology, allowing for the coherent con- of the Dirac equation in relativistic quantum mechanics trol of the vibronic dynamics [8]: first, a carrier interaction predict some astonishing effects, like the Zitterbewegung consisting of a coherent driving field acting resonantly on a and the Klein’s paradox. pair of internal levels, while leaving untouched the mo- In recent years, a growing interest has appeared regard- tional degrees of freedom. It can be described effectively ing simulations of relativistic effects in controllable physi- i ÿ ÿi by the Hamiltonian H @ e e , where cal systems. Some examples are the simulation of Unruh and ÿ are the raising and lowering ionic spin-1=2 effect in trapped ions [2], the Zitterbewegung for massive operators, respectively, and is the associated coupling fermions in solid state physics [3], and black-hole proper- strength. The phases and frequencies of the laser field ties in the realm of Bose-Einstein condensates [4]. could be adjusted so as to produce H @ xx, H Moreover, the low-energy excitations of a nonrelativistic x y @ , and H @ , where , , and are two-dimensional electron system in a single layer of graph- y y z z z x y z ite (graphene) are known to follow the Dirac-Weyl equa- atomic Pauli operators in the conventional directions x, y, tions for massless relativistic particles [5,6]. On the other and z. Second, we consider a Jaynes-Cummings (JC) in- hand, the fresh dialog between quantum information and teraction, usually called red-sideband excitation, consist- special relativity has raised important issues concerning ing of a laser field acting resonantly on two internal levels the quantum information content of Dirac bispinors under and one of the vibrational c.m. modes. Typically, a reso- Lorentz transformations [7]. nant JC coupling induces an excitation in the internal levels In this Letter, we propose the simulation of the Dirac while producing a deexcitation of the motional harmonic equation for a free spin-1=2 particle in a single trapped ion. oscillator, and vice versa. The resonant JC Hamiltonian can @ ~ ir ÿ y ÿir We show how to implement realistic interactions on four be written as Hr ae a e , where a y ionic internal levels, coupled to the motional degrees of and a are the annihilation and creation operatorsp associ- freedom, so as to reproduce this fundamental quantum- ated with a motional degree of freedom. k @=2M is relativistic wave equation. We propose also the simulation the Lamb-Dicke parameter [8], where k is the wave num- of the Dirac equation in 1 1 dimensions, requiring only ber of the driving field. Third, we consider an anti-JC the control of two internal levels and one motional degree (AJC) interaction, consisting of a JC-like coupling tuned 0031-9007=07=98(25)=253005(4) 253005-1 © 2007 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 98, 253005 (2007) 22 JUNE 2007 2 to the blue motional sideband with Hamiltonian Hb 0 c ~ p~ÿimc H D 2 : (5) @ ~ ayeib ÿaeÿib . In this case, an internal c ~ p~imc 0 level excitation accompanies an excitation in the consid- Here, the 44 matrix ~ : ; ; off-diag ;~ ~ ered motional degree of freedom, and vice versa. x y z : All of these interactions could be applied simultaneously is the velocity operator, off-diag ÿi12;i12, and and addressed to different pairs of internal levels coupled ~ 2 @ c : 2 ;mc: ; (6) to different c.m. modes. For example, it is possible to ad- just field phases to implement a simultaneous blue- and are the speed of light and the electron rest energy, respec- red-sideband excitation scheme to form the Hamilton- tively. The notorious analogy between Eqs. (3) and (5) px @ ~ y ÿ ~ shows that the quantum-relativistic evolution of a ian Hx i x xx ax ax 2xpx xxpx, with y spin-1=2 particle can be fully reproduced in a tabletop i ax ÿ ax=2 xpx=@. Here, x : @=2Mx is the spread in position along the x axis of the zero-point wave ion-trap experiment, allowing the study of otherwise inac- cessible physical regimes and effects, as shown below. function and px the corresponding dimensioned momen- p In the Dirac formalism, the spin-1=2 degree of freedom tum operator. The physics of H x cannot be described x ab initio anymore by Rabi oscillations. In turn, it yields a condi- is incorporated . Moreover, the Dirac bispinor in tional displacement in the motion depending on the inter- Eq. (1) is built by componentsp associated with positive and 2 2 2 4 nal state, producing the so-called Schro¨dinger cat states negative energies, ED p c m c . This descrip- [9,10]. By further manipulation of laser field directions and tion is the source of diverse controversial predictions, as py ~ the Zitterbewegung and the Klein’s paradox. phases, we can also implement Hy 2yy yypy and pz ~ The Zitterbewegung is a known quantum-relativistic Hx 2zz zxpz. This kind of interaction has already been produced in the lab, under resonant [11] and disper- effect consisting of a helicoidal motion of a free Dirac sive conditions [12]. particle, a natural consequence of the noncommutativity of c i x y z We define the wave vector associated with the four ionic its velocity operator components, i, with , , .It internal levels as can be proved straightforwardly [1] that the time evolution 0 1 of the position operator r~ x; y; z in the Heisenberg @ a picture, following dr=dt~ r;~ HD=i , reads B C B b C 2 2 ~ 2 ~ ~ @ j i : ajai bjbi cjci djdi@ A: 4 p~ 2 p~ i c r~ tr~ 0 t ~ ÿ HD HD HD d @ (1) e2iHDt= ÿ 1: (7) We may apply simultaneously different laser pulses, with proper directions and phases, x y z, Here, the first two terms on the right-hand side account for ~ ~ ~ ~ x y z, x y z, x y the classical kinematics of a free particle, while the last oscillating term is responsible for a transversal ‘‘quiver- z, to compose the following Hamiltonian acting on j i, ing’’ motion. If we consider a bispinor state with a peaked ~ ad bc ~ ad bc HD 2 x x px 2 y ÿ y py momentum around p0, j 0ijaiexpÿ p ÿ p 2=22 , the Zitterbewegung frequency associated with 2 ~ ac ÿ bdp @ ac bd: (2) 0 p x x z y y the measurable quantity hr~ ti can be estimated as We rewrite Eq. (2) in the suitable matrix form q @ 2 2 ~ 2 2 @2 2 !!ZB 2jEDj= 2 4 p0= ; (8) 02 ~ ~ p~ÿi@ HD ; where E hH i is the average energy. Similarly, we can 2 ~ ~ p~i@0 D D estimate from Eq. (7) the Zitterbewegung amplitude asso- (3) ciated with hr~ ti as where each entry represents a 2 2 matrix.
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