Relativistic Quantum Mechanics

in

Trapped Ions

The two-dimensional Dirac Oscillator

Alejandro Bermudez Carballo

Director:Miguel Angel Martın-Delgado

Dpto. de Fısica Teorica IFacultad de Ciencias FısicasUniversidad Complutense

Madrid

Trabajo de Investigacion presentado para optar al tıtulo de Master.Programa de Master: Fısica Fundamental, Junio 2007.

i

Acknowledgements

Foremost, I would like to thank Miguel Angel Martın-Delgado, who gave methe opportunity to collaborate in a fascinating task: understanding “Quan-tumness”. I have learnt and enjoyed trough the many discussions during thelast two years, and I hope they last for ages. This work is unavoidable entan-gled with Enrique Solano, whose passion for physics is happily contagious. Imust also thank every teacher that has broadened my knowledge, and feedmy interests for physics.

On the non-scientific side, I thank all of those who showed me that lifeis mainly not quantum mechanical. My family, who think I have turned intoa kind of “domador de iones cautivos”, but still encourage me to keep on.Special thanks to my sister, who have revised my poor English and did notgive up in the first ~. I cannot forget about Valle, who makes my life worthliving, and forgives me for my continuous mental absences. To all of thosethat are happy to share a moment with me, thanks!.

“Logic will get you from A to B.Imagination will take you everywhere”

Albert Einstein

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Contents

1 Introduction 1

2 The Dirac Oscillator 42.1 Energy Spectrum and Eigenstates . . . . . . . . . . . . . . . . 52.2 Dynamical Zitterbewegung . . . . . . . . . . . . . . . . . . . . 72.3 Mapping onto a Jaynes-Cummings Model . . . . . . . . . . . . 9

3 Ion Trap Experimental Proposal 93.1 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . 103.2 Dirac Oscillator Hamiltonian . . . . . . . . . . . . . . . . . . . 113.3 Zitterbewegung Measurement . . . . . . . . . . . . . . . . . . . 123.4 Dirac Oscillator Chiral Partner . . . . . . . . . . . . . . . . . 13

4 Conclusions and Future Directions 13

5 Publications 15

iii

1 Introduction

The principle of relativity states that there does not exist any absolute ref-erence frame in nature, and therefore all the physical laws should be iden-tical in any inertial reference frame ( i.e. Lorentz covariant). Nevertheless,the fruitful theory of Quantum Mechanics is clearly not Lorentz covariant,which is the reason why a number of major physicists made a huge effortto merge Special Relativity and Quantum Mechanics. The first attempts bySchrodinger, Gordon and Klein lead to the Klein-Gordon equation [1, 2, 3],which correctly describes the relativistic dispersion relation for a free particle

E =√

p2c2 + m2c4, (1)

where p stands for the particle momentum, m for its mass and c for the speedof light. Nonetheless, this equation gave rise to various interpretational prob-lems, such as a negative probability distribution, and was early abandonedin favor of the Dirac equation [4]. Dirac sought a relativistic equation whichshould:

Describe the correct energy dispersion relation (1);

Lead to a probabilistic interpretation;

Show a manifest Lorentz Covariance.

Since the theory of Relativity treats Space and Time on an equal footing,they are inevitably merged in Minkowski Space-time [5]. This made Diracpropose a equation linear in both space and time coordinates

i~∂|Ψ〉∂t

= (cα · p + βmc2)|Ψ〉, (2)

where |Ψ〉 is the state vector describing the relativistic particle, ~ stands forthe Planck constant, and αi, β are certain parameters still to be determined.In order to recover the appropriate energy dispersion (1), these parametersmust satisfy the following relations

αjαk + αkαj = 2δjk,

αjβ + βαj = 0,(3)

1

known as a Clifford Algebra. Consequently, these parameters are compulsory4-dimensional matrices, which in the Standard Representation are

αi :=

[0 σi

σi 0

]β :=

[I 00 −I

], (4)

with σi as the usual Pauli matrices. The principal consequence of such con-straints (3) is that the state vector becomes a Dirac-spinor with four compo-nents |Ψ〉 := [|ψ1〉, |ψ2〉, |ψ3〉, |ψ4〉]t.

Dirac theory also meets the requirement of Lorentz covariance, intro-ducing a particular transformation on the spinor degrees of freedom thatcorresponds to the usual Lorentz transformations in Space-time [5]. Further-more, it yields the appropriate non-relativistic limit predicting the electronspin operator S := ~σ/2, and also accounting for the correct gyromagneticratio gs = 2, which has been accurately demonstrated in QED experimentsgQED ≈ 2.0011614 . Another renown triumph of this relativistic quantum the-ory is the exact prediction of the Hydrogen atom energy spectrum, where theinvolved relativistic corrections have been experimentally tested with greataccuracy. Actually, even the loopholes of the theory, have turned into suc-cessful predictions. Such is the case of negative energy states of free fermionsE = −

√p2c2 + m2c4, which become a prediction of a fundamental symme-

try in nature: an antiparticle is associated to every existing particle [7]. Thismotivated the discovery of the positron e+.

As any other physical theory, Relativistic Quantum Mechanics cannotprecisely describe phenomena at every scale. Since it is mainly a one-bodytheory, it shall not account for processes where the particle number is notconserved, such as e− − e+ creation or annihilation. Therefore, we mustdistinguish between regimes where the one-body theory predictions are reli-able, and situations where we must move on to a Relativistic Quantum FieldTheory. This cutoff is known as the Compton wavelength

λc =2π~mc

, (5)

and ensures that as long as the relativistic particle position is not localizedbelow such wavelength ∆x > λc, no pair production occurs, and RelativisticQuantum Mechanics describes adequately the underlying physical phenom-ena.

2

Therefore, one should be careful and always check that the predictionsof the theory do not go beyond its range of validity. Such is the case ofthe Zitterbewegung effect in the free Dirac equation (2), first discussed bySchrodinger [6], where a free relativistic electron experiences a tremblingmotion around the usual free-particle trajectory as shown in Figure 1.

Figure 1: Zitterbewegung of a free fermion extracted from [8]

These wiggling oscillations are associated to the interference betweenpositive- and negative-energy solutions of the free Dirac equation (2), havingthus a clear relativistic origin. Nonetheless, their amplitude lies below theCompton wavelength and we must not rely on this relativistic prediction.Actually, the Zitterbewegung has never been experimentally observed, and isconsequently believed to be a paradox of the Dirac equation.

In this project we have tried to solve this paradox, studying the Zitter-bewegung of a relativistic system where no fermion production occurs, theso-called Dirac Oscillator [9, 10] in two dimensions. Interestingly enough,the Zitterbewegung can be identified with some oscillations in the spin degreeof freedom, arising due to the interference of positive- and negative-energysolutions. In addition, we derive an exact mapping of the 2+1 Dirac oscilla-tor onto the Jaynes-Cummings model [11], an archetypical quantum opticalsystem. This opens up a fresh dialog between these two seemingly unrelatedfields, Quantum Optics and Relativistic Quantum Mechanics.

3

Recently, there has been a growing interest in simulating quantum rela-tivistic effects in other physical systems, such as black hole evaporation inBose-Einstein condensates [12] and the Unruh effect in an ion chain [13]. TheZitterbewegung has been discussed in the context of condensed matter sys-tems [14] and the free-particle Dirac equation [15]. In this work, we proposethe simulation of this relativistic 2+1 Dirac Oscillator in a single trappedion, a physical setup possessing outstanding coherence properties [16]. Wepropose how the Zitterbewegung spin oscillations can be reproduced and ac-curately measured in an ion trap table-top experiment. The experimentaltrapped ion set-up can be considered as an instance of a Relativistic Quan-tum Simulator, capable of mimicking well-known relativistic quantum effectswith an outstanding degree of control, and predict some others which havenot been realized yet.

And I’m not happy with all the analyses that go with just theclassical theory, because nature isn’t classical, dammit, and if

you want to make a simulation of nature, you’d better make itquantum mechanical, and by golly it’s a wonderful problem,

because it doesn’t look so easy.Richard Feynman

2 The Dirac Oscillator

The Dirac oscillator was presented as a relativistic version of the usual quan-tum harmonic oscillator, with the following coupling being introduced in theDirac equation (2)

i~∂|Ψ〉∂t

=

[3∑

j=1

cαj

(pj − imβωrj

)+ βmc2

]|Ψ〉, (6)

where ω is the oscillator frequency. The Dirac oscillator looks like a non-minimal coupling p → p− e

cA for a vector potential A linear in space coor-

dinates, but the presence of the i and the β matrix makes a crucial difference.Apart from its exact solvability, this system presents several interesting prop-erties from a physical and mathematical point of view.

Some of its exceptional physical properties are listed bellow. Its non-relativistic limit yields a quantum harmonic oscillator with an strikingly

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strong spin-orbit coupling . The energy spectrum leads to bound states withE ≶ 0, which can be associated to particles and antiparticles and makes theDirac oscillator an appealing phenomenological potential for quark confine-ment in mesons and baryons( see [19] and references therein). It also showsan intriguing Zitterbewegung in the 3+1 and 1+1 cases [17, 18]. The Diracoscillator also presents some unusual mathematical properties, such as itsinfinite accidental degeneracy associated to a certain Lie symmetry [20], theexistence of a hidden supersymmetry [21], or its exact solvability through aFoldy-Wouthuysen transformation.

In this paper, we shall restrict our attention to the Dirac oscillator in2+1 dimensions, since it is in this setting where we can establish a preciseequivalence with the Jaynes-Cummings (JC) model [11]. In two spatial di-mensions, the solution to the Clifford algebra (3) is given by the 2× 2 Paulimatrices: αx = σx, αy = σy, β = σz. In this case, |Ψ〉 can be described by a2-component spinor which mixes spin up and down components with positiveand negative energies. In particular, the Dirac oscillator model now takesthe form

i~∂|Ψ〉∂t

=

[2∑

j=1

cσj

(pj − imσzωrj

)+ σzmc2

]|Ψ〉, (7)

which will be studied in detail along the following sections. In Section 2.1 weshall derive the energy spectrum and the associated eigenstates. A thoroughdescription of the dynamical evolution, and the appearance of the Zitterbewe-gung in the spin degree of freedom will be dealt with in Section 2.2. Finally,the mapping of the relativistic Hamiltonian onto a Jaynes-Cummings modelwill be discussed in Section 2.3.

2.1 Energy Spectrum and Eigenstates

Considering the spinor |Ψ〉 := [|ψ1〉, |ψ2〉]t, equation (7) becomes a set ofcoupled equations

(E −mc2)|ψ1〉 = c [(px + imωx)− i(py + imωy)] |ψ2〉,(E + mc2)|ψ2〉 = c [(px − imωx) + i(py − imωy)] |ψ1〉. (8)

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In order to find the solutions, it is convenient to introduce the following chiralcreation and annihilation operators

ar := 1√2(ax − iay), a†r := 1√

2(a†x + ia†y),

al := 1√2(ax + iay), a†l := 1√

2(a†x − ia†y),

(9)

where ax, a†x, ay, a

†y are the usual annihilation and creation operators of the

harmonic oscillator a†i = 1√2

(1∆

ri − i∆~ pi), and ∆ =

√~/mω represents the

oscillator’s ground state width. The orbital angular momentum may also beexpressed as

Lz = ~(a†rar − a†l al), (10)

which leads to a physical interpretation of a†r and a†l . These operators createa right or left quantum of angular momentum respectively, and are henceknown as circular creation-annihilation operators. Equations (8) can berewritten in the language of these circular operators

|ψ1〉 = i2mc2√

ξE−mc2

a†l |ψ2〉,|ψ2〉 = −i2mc2

√ξ

E+mc2al |ψ1〉,

(11)

where ξ := ~ω/mc2 controls the non-relativistic limit. In order to find theenergy spectrum we shall solve the associated Klein-Gordon equation, whichcan be derived from Eqs. (11) as follows

(E2 −m2c4)|ψ1〉 = 4m2c4ξ a†l al |ψ1〉,(E2 −m2c4)|ψ2〉 = 4m2c4ξ (1 + a†l al)|ψ2〉.

(12)

These equations can be simultaneously diagonalized by writing the spinorcomponents in terms of the left chiral quanta basis

|nl〉 =1√nl!

(a†l

)nl |vac〉, (13)

where nl = 0, 1, ... The energies can be expressed as

(E2nl−m2c4)|nl〉 = 4m2c4ξnl |nl〉,

(E2n′l−m2c4)|n′l〉 = 4m2c4ξ (1 + n′l)|n′l〉.

(14)

Since both components |ψ1〉 and |ψ2〉 belong to the same solution, the energiesmust be the same En′l = Enl

. This physical requirement creates a constraint

6

on the quantum numbers nl =: n′l + 1. Note that, following (10), the state|nl〉 corresponds to a negative angular momentum. The energy spectrum canbe described as

E = ±Enl= ±mc2

√1 + 4ξnl. (15)

In order to find the corresponding eigenstates, we must go back to Eq. (11).After normalization, we obtain the positive and negative energy eigenstates

| ± Enl〉 =

√Enl

±mc2

2Enl|nl〉

∓i√

Enl∓mc2

2Enl|nl − 1〉

, (16)

where the quantum number is now restricted to nl = 1, 2, ... We have thussolved the two-dimensional Dirac oscillator by describing the energy spec-trum and the eigenstates in terms of circular quanta. The distinction be-tween Dirac and Klein-Gordon eigenstates is an important point in order tounderstand the dynamics of the 2+1 Dirac oscillator and its realization inan ion trap.

The eigenstates of the 2D Dirac oscillator can be expressed more trans-parently in terms of 2-component Pauli spinors |χ↑〉 and |χ↓〉

|+ Enl〉 = αnl

|nl〉|χ↑〉 − iβnl|nl − 1〉|χ↓〉,

| − Enl〉 = βnl

|nl〉|χ↑〉+ iαnl|nl − 1〉|χ↓〉, (17)

where αnl:=

√Enl

+mc2

2Enland βnl

:=√

Enl−mc2

2Enlare real. From this expression

we may conclude that the energy eigenstates present entanglement betweenthe orbital and spin degrees of freedom.

2.2 Dynamical Zitterbewegung

The presence of entanglement in the oscillator eigenstates (17) is a crucialproperty since the initial state

|Ψ(0)〉 := |nl − 1〉|χ↓〉 = iβnl|+ Enl

〉 − iαnl| − Enl

〉 (18)

superposes states with positive and negative energies, and this is the funda-mental ingredient that leads to Zitterbewegung in relativistic quantum dy-namics.

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Now, the evolution of this initial state can be expressed in the energy-eigenstate basis as

|Ψ(t)〉 = iβnl|+ Enl

〉e−iωnlt − iαnl

| − Enl〉eiωnl

t, (19)

where

ωnl:=

Enl

~=

mc2

~√

1 + 4ξnl (20)

describes the frequency of oscillations. Expressing this evolved state in thelanguage of Pauli spinors,

|Ψ(t)〉 =

(cos ωnl

t +i√

1 + 4ξnl

sin ωnlt

)|nl − 1〉|χ↓〉+

+

(√4ξnl

1 + 4ξnl

sin ωnlt

)|nl〉|χ↑〉,

(21)

we observe oscillatory dynamics between |nl〉|χ↑〉 and |nl − 1〉|χ↓〉. The ini-tial state, |nl − 1〉|χ↓〉, which has spin-down and nl − 1 quanta of left orbitalangular momentum, evolves by exchanging a quantum of angular momen-tum from the spin to the orbital motion. The dynamics described in (21)bears great resemblance to atomic Rabi oscillations in the Jaynes-Cummingsmodel, though due to a completely different reason. Whereas the Rabi os-cillations are caused by the interaction of a quantized electromagnetic fieldwith a two-level atom, the relativistic oscillations are caused by the interfer-ence of positive and negative energy states and therefore constitute a clearsignature of Zitterbewegung.

To further clarify this issue, we proceed to calculate the time evolutionof the following physical observables, which capture the pure essence of thesystem dynamics,

〈Lz〉t = −(nl − 1)~− 4ξnl

1+4ξnl~ sin2 ωnl

t,

〈Sz〉t = −~2

+ 4ξnl

1+4ξnl~ sin2 ωnl

t,

〈Jz〉t = ~(12− nl),

(22)

where Jz = Lz + Sz stands for the z-component of the total angular mo-mentum. The above relations describe an oscillation in the spin and orbitalangular momentum, while the total angular momentum is conserved due tothe existing invariance under rotations around the z-axis. It must be carefully

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stressed that this relativistic interference effect is not associated to tremblingoscillations below the Compton wavelength ∆x < λc, and therefore no in-consistency of the one-body theory is apparent.

It is also important to highlight that these oscillations are of a purelyrelativistic nature. In the non-relativistic limit ξ ¿ 1, these oscillationsbecome vanishingly small

〈Lz〉t = −(nl − 1)~− 4ξnl~ sin2 Ωnlt +O(ξ2),

〈Sz〉t = −~2

+ 4ξnl~ sin2 Ωnlt +O(ξ2),

(23)

where Ωnl:= mc2(1+2ξnl)/~ represents the oscillation frequency in the non-

relativistic limit. In this limit, the negative energy components are negligibleand therefore the Zitterbewegung disappears.

2.3 Mapping onto a Jaynes-Cummings Model

The results discussed so far allow us to precisely map two unrelated models:the Jaynes-Cummings model of Quantum Optics and the 2D Dirac oscillator.Starting from Eq. (11), we may write the Dirac oscillator Hamiltonian as

H = 2imc2√

ξ(a†l |ψ2〉〈ψ1| − al|ψ1〉〈ψ2|

)+ mc2σz =

= ~(gσ−a†l + g∗σ+al) + mc2σz,(24)

where σ+, σ− are the spin raising and lowering operators, and g := 2imc2√

ξ/~is the coupling strength between orbital and spin degrees of freedom. InQuantum Optics, this Hamiltonian describes a detuned Jaynes-Cummingsinteraction which has been studied in trapped ions and cavity QED [16, 23],among others. From this novel perspective, the electron spin can be asso-ciated with a two-level atom, and the orbital circular quanta with the ionquanta of vibration, i.e., phonons. As we will see below, the central resultof Eq. (24) allows both physical systems, the JC model and the 2D Diracoscillator, to exchange a wide range of important applications.

3 Ion Trap Experimental Proposal

Current technology has allowed the implementation of the paradigmatic non-relativistic quantum harmonic oscillator in a single trapped ion [16], one

9

of the most fundamental toy models in any quantum mechanical textbook.However, its relativistic version, the so-called Dirac oscillator , remains stillfar from any possible experimental consideration for technical reasons. Belowwe describe how available experimental tools may allow the implementationof the relativistic Dirac oscillator in a single non-relativistic trapped ion.In Section 3.1 we describe the ion trap experimental set-up which allowsan efficient simulation of the Dirac Oscillator Hamiltonian, as immediatelydiscussed in Section 3.2. In Section 3.3 we discuss the details of the Zit-terbewegung experimental realization, giving full details of the measurementtechnique. Finally, in Section 3.4 we point out the properties of a right-handed chiral partner of the Dirac oscillator.

3.1 Experimental Set-up

We will now show how to implement the dynamics of Eq. (7) in a single ioninside a Paul trap, which was shown to follow the dynamics of Eq. (24). TheDirac spinor is described by means of two metastable internal states, |g〉 and|e〉, as follows

|Ψ〉 := |ψ1〉|e〉+ |ψ2〉|g〉 (25)

while the circular angular momentum modes will be represented by two ionicvibrational modes, ax and ay. Current technology allows an overwhelminglycoherent control of ionic internal and external degrees of freedom [16]. Thereare three paradigmatic interactions: the carrier, red- and blue-sideband ex-citations, which can be implemented independently or simultaneously [24].With appropriately tuned lasers it is possible to produce the following inter-actions

HJCi = ~ηiΩi

[σ+aie

iφ + σ−a†ie−iφ

]+ ~δiσz,

HAJCi = ~ηiΩi

[σ+a†ie

iϕ + σ−aie−iϕ

],

(26)

where ai, a†i, with i = x, y, are the phonon annihilation and creation op-

erators in directions x and y, νi are the natural trap frequencies, ηi :=ki

√~/2Mνi are the associated Lamb-Dicke parameters depending on the

ion mass M and the wave vector k, δi and Ωi are the excitation couplingstrengths and φ, ϕ, the red- and blue-sideband phases. Remark that theterm ~δiσz, in HJC

i of Eq. (26), stems from a detuned JC excitation.

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3.2 Dirac Oscillator Hamiltonian

A suitable combination of the above excitations (26), with proper couplingsand relative phases, can reproduce the following Hamiltonian

H = c[σge

x px + σgey py

]+ mωc

[σge

x y − σgey x

]+ mc2σge

z (27)

with the ionic Pauli matrices

σgex :=|g〉〈e|+ |g〉〈e|,

σgey :=− i(|e〉〈g| − |e〉〈g|),

σgez :=|e〉〈e| − |g〉〈g|,

(28)

and the following parameter correspondence

c =√

2ηΩ∆,mc2 = ~δ,mωc = ~

√2ηΩ∆−1,

(29)

where ∆ := ∆i is the width of the motional ground state, Ω := Ωi, η :=ηi,∀i = x, y. The remarkable equivalence of the Dirac oscillator Hamilto-nian (7) and the interaction in Eq. (27) shows that it is possible to reproducethe 2D Dirac oscillator with all its quantum relativistic effects, in a control-lable quantum system as a single trapped ion.

For the sake of illustration, note that the effective terms appearing inEq. (27) can be achieved by suitable linear combinations of HJC

i and HAJCi

in (26),

i = x, δx = δ, φ = 3π2

, ϕ = π2→ √

2~ηΩ∆σgex px + ~δσge

z ,

i = y, δy = 0, φ = 0 , ϕ = π → √2~ηΩ∆σge

y py,

i = x, δx = 0, φ = π2, ϕ = π

2→ √

2~ηΩ∆−1σgey x,

i = y, δy = 0, φ = 0 , ϕ = 0 → √2~ηΩ∆−1σge

x y.

(30)

Note that in the trapped ion picture, the important parameter ξ = 2(ηΩ/δ)2

can take on all positive values, considering available experimental parameters:η ∼ 0.1, Ω ∼ 0 − 106Hz, and δ ∼ 0 − 106Hz [16]. The ability to experimen-tally tune these parameters will allow the experimenter to study otherwiseinaccessible physical regimes that entail relativistic and non-relativistic phe-nomena.

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3.3 Zitterbewegung Measurement

For example, the Zitterbewegung is encoded in the spin degree of freedom,and we can associate Rabi oscillations with the interference of positive andnegative energy solutions. Once the initial state is set to |0〉|χ↓〉 ↔ |0〉|g〉,the internal degree of freedom evolves according to Eq. (22)

〈Sz〉t = −~2

+4ξ

1 + 4ξ~ sin2 ω1t, (31)

where ω1 = δ√

1 + 4ξ, see Eq. (20), stands for the frequency of the Zitterbe-wegung oscillations and can take on a wide variety of measurable values.

In order to simulate this dynamics in an ion-trap tabletop experiment, theion must be cooled down to its vibrational ground state |0〉, with a currentefficiency above 99% [16]. To estimate the observable (31), one can make useof the powerful tool known as electron shelving, where

〈Sz〉t =~2

[2Pe(t)− 1] (32)

can be calculated through the measurement of the probability of finding theion in the excited state Pe(t).

Another fundamental result of the JC model which can be directly mappedto the Dirac oscillator is the existence of collapses and revivals in the atomicpopulation, which is claimed to be a clear evidence of the quantization ofthe electromagnetic field. To produce this effect, an initial state |z〉|g〉 isrequired, where |z〉 is an initial circular coherent state,

|Ψ(0)〉 = e−|z|2/2

∞∑nl=0

znl

√nl!|nl〉|g〉, (33)

with z ∈ C. After an interaction time t,

〈Sz〉t = −~2

+ ~∞∑

nl=0

4ξ(nl + 1)|z|2nle−|z|2

[1 + 4ξ(nl + 1)]nl!sin2(ωnl+1t). (34)

This expression can be understood as an interference effect of terms withdifferent frequencies ωnl+1 leading to collapses and revivals. A novel feature

12

of the Dirac oscillator is the appearance of these collapses and revivals in theorbital circular motion of the particle, shown in

〈Lz〉t = −~|z|2 − ~∞∑

nl=0

4ξ(nl + 1)|z|2nle−|z|2

[1 + 4ξ(nl + 1)]nl!sin2(ωnl+1t). (35)

The generation of an initial circular coherent state will require two sequen-tial applications of the technique described in Ref. [16] in an initial motionalground state. These two operations should be applied with a relative phase

such that Dl(z) = Dx(z)Dy(−iz), where Dj(z) = eza†j−z∗aj , j = x, y. Theobservable of Eq. (34) can be measured via a similar electron-shelving tech-nique, while the observable of Eq. (35) can be measured via the mapping ofthe collective motional state onto the internal degree of freedom [16].

3.4 Dirac Oscillator Chiral Partner

It is worth mentioning that the chiral partner of the 2D Dirac oscillatorHamiltonian (7) can be obtained through the substitution ω → −ω, andconsists on right-handed quanta. This Hamiltonian presents similar featuresas those discussed above, can also be simulated with trapped ions, and canbe exactly mapped onto an anti-Jaynes-Cummings interaction

H = ~(garσ− + g∗a†rσ

+) + mc2σz, (36)

with similar parameters. It is precisely this chirality which allows an exactmapping between the JC, AJC, and the left-handed and right-handed 2DDirac oscillator. This essential property, which is missing in the 3D case,prevents an exact mapping of Eq. (6) onto a JC-like Hamiltonian.

4 Conclusions and Future Directions

In conclusion, we have demonstrated the exact mapping of the 2+1 Diracoscillator onto a Jaynes-Cummings model, allowing an interplay betweenrelativistic quantum mechanics and quantum optics. We gave two relevantexamples: the Zitterbewegung and the collapse-revival dynamics. In addi-tion, we showed that the implementation of a 2D Dirac oscillator in a single

13

trapped ion, with all analogies and measured observables, is at reach withcurrent technology.

The main importance of this work has been the development of a mappingbetween two apparently disconnected models. This fact allows an interplaybetween the Quantum Optics and Relativistic Quantum communities, whichcan be translated in to a deeper understanding of different phenomena whicharise in both fields. Although Eq.(24) may seem a happy coincidence, it turnsout that this analogy can be pushed further.

We have found that the 3+1 Dirac oscillator can also be mapped ontoa Quantum Optics model, which in this higher-dimensional case gets moreinvolved. The relativistic Hamiltonian can be described as a three-modefour-level Jaynes-Cummings-like model. We are currently working on howthis analogy can serve to explain the striking properties of the relativisticmodel, and to hopefully predict some novel phenomena.

We have also found that a relativistic electron subjected to a constantmagnetic field can be mapped onto a combination of Jaynes-Cummings andAnti-Jaynes-Cummings coupling. This allows us to obtain the relativisticLandau levels with the corresponding eigenstates in a straight manner. Fromthe quantum optics perspective, we have also predicted some novel dynam-ical effects in the relativistic electron. Carefully selecting an appropriateinitial state, we obtain similar Zitterbewegung properties in the spin degreeof freedom as we have described in this paper. Furthermore, we have alsopredicted the generation of Dirac cats ( i.e. Relativistic Schrodinger cats )in the orbital degree of freedom under the ultra-relativistic regime.

All these results cannot be discussed here, and will be detailed in a forth-coming publication. Nevertheless, they point towards a deeper relationshipbetween Quantum Optics and Relativistic Quantum Mechanics. Further-more, since current ion trap technology allows the simulation of such rela-tivistic system as discussed above, some of the predicted phenomena maybe tested experimentally in a near future. Hopefully, some other unrealizedrelativistic effects may also emerge from the experimental results, which willpuzzle our scientific minds.

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5 Publications

The work described in this text is based on the following text submitted forpublication :

A. Bermudez, M. A. Martin-Delgado, E. Solano. arXiv:0704.2315 (2007),

15

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[24] A carrier interaction consists on a resonant classical driving onto a two-level system, while the red-sideband and the blue-sideband correspondto a JC and anti-JC (AJC) interactions, respectively.

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