Dirac Equation of a Free Relativistic Particle

Lucas Lamata University of the Basque Country UPV/EHU, Bilbao, Spain

BCAM, Bilbao, 8 October 2013

Laura García-Álvarez (M. Sc. student)

Urtzi Las Heras (M. Sc. student)

Julen S. Pedernales (PhD student)

Unai Alvarez-Rodriguez (PhD student)

Antonio Mezzacapo (PhD, European SOLID grant)

Simone Felicetti (PhD, CCQED Marie Curie grant)

Roberto Di Candia (PhD, CCQED Marie Curie grant)

Mikel Sanz (Postdoc, European PROMISCE grant)

Jorge Casanova (Postdoc, European PROMISCE grant)

Guillermo Romero (Postdoc, European PROMISCE grant)

Lucas Lamata (Ramón y Cajal Fellow since 2014)

Enrique Solano (Group Leader and Ikerbasque Professor)

CCQED

http://sites.google.com/site/enriquesolanogroup/

Our group develops interdisciplinary research inQuantum optics

Quantum informationRelativistic quantum mechanics

Circuit quantum electrodynamicsQuantum Biomimetics

Outline

Dirac Equation

Future Scope

Trapped-Ion Simulation

Quantum Simulations

Feynman ’82

Lloyd ’96

Simulating efficiently quantum systems

Simulating unreproducible physics

Buluta and Nori Science 326, 108 (2009)

Trapped Ions

Highly controllable Efficient 2-qubit gates

Good initialization and measurement

14-qubit entanglement and 140-gate simulator (Innsbruck), 300-ion simulator (NIST)

F>99%

F>99%

Trapped Ions in BilbaoDirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

L. Lamata,1 J. Leon,1 T. Schatz,2 and E. Solano3,4

1Instituto de Matematicas y Fısica Fundamental, CSIC, Serrano 113-bis, 28006 Madrid, Spain2Max-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany

3Physics Department, ASC, and CeNS, Ludwig-Maximilians-Universitat, Theresienstrasse 37, 80333 Munich, Germany4Seccion Fısica, Departamento de Ciencias, Pontificia Universidad Catolica 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 particlein a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position andmomentum of the Dirac particle are associated with the respective ionic variables. We show also how tosimulate the simplified 1! 1 case, requiring the manipulation of only two internal levels and one motionaldegree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung andKlein’s paradox, the transition from massless to massive fermions, and the relativistic and nonrelativisticlimits, 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 Schrodinger equationgave rise to the Klein-Gordon and Dirac equations. P. A. M.Dirac looked for a Lorentz-covariant wave equation that islinear in spatial and time derivatives, expecting that theinterpretation of the square wave function as a probabilitydensity holds. As a result, he obtained a fully covariantwave equation for a spin-1=2 massive particle, which in-corporated ab initio the spin degree of freedom. It is known[1] that the Dirac formalism describes accurately the spec-trum of the hydrogen atom and that it plays a central role inquantum field theory, where creation and annihilation ofparticles are allowed. However, the one-particle solutionsof the Dirac equation in relativistic quantum mechanicspredict some astonishing effects, like the Zitterbewegungand the Klein’s paradox.

In recent years, a growing interest has appeared regard-ing simulations of relativistic effects in controllable physi-cal systems. Some examples are the simulation of Unruheffect in trapped ions [2], the Zitterbewegung for massivefermions in solid state physics [3], and black-hole proper-ties in the realm of Bose-Einstein condensates [4].Moreover, the low-energy excitations of a nonrelativistictwo-dimensional electron system in a single layer of graph-ite (graphene) are known to follow the Dirac-Weyl equa-tions for massless relativistic particles [5,6]. On the otherhand, the fresh dialog between quantum information andspecial relativity has raised important issues concerningthe quantum information content of Dirac bispinors underLorentz transformations [7].

In this Letter, we propose the simulation of the Diracequation for a free spin-1=2 particle in a single trapped ion.We show how to implement realistic interactions on fourionic internal levels, coupled to the motional degrees offreedom, so as to reproduce this fundamental quantum-relativistic wave equation. We propose also the simulationof the Dirac equation in 1! 1 dimensions, requiring onlythe control of two internal levels and one motional degree

of freedom. We study some quantum-relativistic effects,like the Zitterbewegung and the Klein’s paradox, in termsof measurable observables. Moreover, we discuss the tran-sition from massless to massive fermions, and from therelativistic to the nonrelativistic limit. Finally, we describea possible experimental scenario.

We consider a single ion of mass M inside a Paul trapwith frequencies !x, !y, and !z, where four metastableionic internal states, jai, jbi, jci, and jdi, may be coupledpairwise to the center-of-mass (c.m.) motion in directionsx, y, and z. We will make use of three standard interactionsin trapped-ion technology, allowing for the coherent con-trol of the vibronic dynamics [8]: first, a carrier interactionconsisting of a coherent driving field acting resonantly on apair of internal levels, while leaving untouched the mo-tional degrees of freedom. It can be described effectivelyby the Hamiltonian H" " @!#"!ei# ! "$e$i#%, where"! and "$ are the raising and lowering ionic spin-1=2operators, respectively, and ! is the associated couplingstrength. The phases and frequencies of the laser fieldcould be adjusted so as to produce H"x " @!x"x, H"y "@!y"y, and H"z " @!z"z, where "x, "y, and "z areatomic Pauli operators in the conventional directions x, y,and z. Second, we consider a Jaynes-Cummings (JC) in-teraction, usually called red-sideband excitation, consist-ing of a laser field acting resonantly on two internal levelsand one of the vibrational c.m. modes. Typically, a reso-nant JC coupling induces an excitation in the internal levelswhile producing a deexcitation of the motional harmonicoscillator, and vice versa. The resonant JC Hamiltonian canbe written asHr " @$ ~!#"!aei#r ! "$aye$i#r%, where aand ay are the annihilation and creation operators associ-ated with a motional degree of freedom. $ " k

!!!!!!!!!!!!!!!!@=2M!

pis

the Lamb-Dicke parameter [8], where k is the wave num-ber of the driving field. Third, we consider an anti-JC(AJC) interaction, consisting of a JC-like coupling tuned

PRL 98, 253005 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending22 JUNE 2007

0031-9007=07=98(25)=253005(4) 253005-1 2007 The American Physical Society

LETTERS

Quantum simulation of the Dirac equationR. Gerritsma1,2, G. Kirchmair1,2, F. Zahringer1,2, E. Solano3,4, R. Blatt1,2 & C. F. Roos1,2

The Dirac equation1 successfully merges quantum mechanics withspecial relativity. It provides a natural description of the electronspin, predicts the existence of antimatter2 and is able to reproduceaccurately the spectrum of the hydrogen atom. The realm of theDirac equation—relativistic quantum mechanics—is consideredto be the natural transition to quantum field theory. However, theDirac equation also predicts some peculiar effects, such as Klein’sparadox3 and ‘Zitterbewegung’, an unexpected quivering motionof a free relativistic quantum particle4. These and other predictedphenomena are key fundamental examples for understandingrelativistic quantum effects, but are difficult to observe in realparticles. In recent years, there has been increased interest insimulations of relativistic quantum effects using different physicalset-ups5–11, in which parameter tunability allows access to differentphysical regimes. Here we perform a proof-of-principle quantumsimulation of the one-dimensional Dirac equation using a singletrapped ion7 set to behave as a free relativistic quantum particle.We measure the particle position as a function of time and studyZitterbewegung for different initial superpositions of positive-and negative-energy spinor states, as well as the crossover fromrelativistic to non-relativistic dynamics. The high level of controlof trapped-ion experimental parameters makes it possible to simu-late textbook examples of relativistic quantum physics.

The Dirac equation for a spin-1/2 particle with rest mass m is givenby1

iB Ly

Lt~(ca.ppzbmc2)y

Here c is the speed of light, pp is the momentum operator, aj (j 5 1,2, 3; (a)j 5 aj) and b are the Dirac matrices (which are usually givenin terms of the Pauli matrices, sx, sy and sz), the wavefunctions y arefour-component spinors and B is Planck’s constant divided by 2p. Ageneral Dirac spinor can be decomposed into parts with positive and

negative energies E 5 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2c2zm2c4

p. Zitterbewegung is under-

stood to be an interference effect between the positive- and negative-energy parts of the spinor and does not appear for spinors that consistentirely of positive-energy (or negative-energy) parts. Furthermore, itis only present when these parts have significant overlap in positionand momentum space and is therefore not a sustained effect undermost circumstances1. For a free electron, the Dirac equation predictsthe Zitterbewegung to have an amplitude of the order of the Comptonwavelength, RZB < 10212 m, and a frequency of vZB < 1021 Hz, andthe effect has so far been experimentally inaccessible. The existence ofZitterbewegung, in relativistic quantum mechanics and in quantumfield theory, has been a recurrent subject of discussion in the pastyears12,13.

Quantum simulation aims to simulate a quantum system using acontrollable laboratory system that underlies the same mathematicalmodel. In this way, it is possible to simulate quantum systems thatcan be neither efficiently simulated on a classical computer14 nor

easily accessed experimentally, while allowing parameter tunabilityover a wide range. The difficulties in observing real quantum rela-tivistic effects have generated significant interest in the quantumsimulation of their dynamics. Examples include black holes inBose–Einstein condensates5 and Zitterbewegung for massive fermionsin solid-state physics6, neither of which have been experimentallyrealized so far. Also, graphene is studied widely in connection to theDirac equation15–17.

Trapped ions are particularly interesting for the purpose ofquantum simulation18–20, as they allow exceptional control of experi-mental parameters, and initialization and read-out can be achievedwith high fidelity. Recently, for example, a proof-of-principle simu-lation of a quantum magnet was performed21 using trapped ions. Thefull, three-dimensional, Dirac equation Hamiltonian can be simu-lated using lasers coupling to the three vibrational eigenmodes andthe internal states of a single trapped ion7. The set-up can be signifi-cantly simplified when simulating the Dirac equation in 1 1 1dimensions, yet the most unexpected features of the Dirac equation,such as Zitterbewegung and the Klein paradox, remain. In the Diracequation in 1 1 1 dimensions, that is

iB Ly

Lt~HDy~(cppsxzmc2sz)y

there is only one motional degree of freedom and the spinor isencoded in two internal levels, related to positive- and negative-energy states7. We find that the velocity of the free Dirac particle isdxx=dt~½xx, HD"=iB~csx in the Heisenberg picture. For a masslessparticle, [sx, HD] 5 0 and, hence, sx is a constant of motion. For amassive particle, [sx, HD] ? 0 and the evolution of the particle isdescribed by

xx(t)~xx(0)zppc2H1D tzijj(e2iHDt=B1)

where jj~(1=2)Bc(sxppcH1D )H1

D . The first two terms representevolution that is linear in time, as expected for a free particle, whereasthe third, oscillating, term may induce Zitterbewegung.

For the simulation, we trapped a single 40Ca1 ion in a linear Paultrap22 with axial trapping frequency vax 5 2p3 1.36 MHz and radialtrapping frequency vrad 5 2p3 3 MHz. Doppler cooling, opticalpumping and resolved sideband cooling on the S1/2 « D5/2 transitionin a magnetic field of 4 G prepare the ion in the axial motional groundstate and in the internal state jS1/2, mJ 5 1/2æ (mJ, magnetic quantumnumber). A narrow-linewidth laser at 729 nm couples the states

01

" #; jS1/2, mJ 5 1/2æ and 1

0

" #; jD5/2, mJ 5 3/2æ, which we identify

as our spinor states. A bichromatic light field resonant with the upperand lower axial motional sidebands of the 1

0

" #« 0

1

" #transition with

appropriately set phases and frequency realizes the Hamiltonian7

HD~2gD ~VVsx ppzBVsz ð1Þ

Here D 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB=2 ~mmvax

pis the size of the ground-state wavefunction,

with ~mm the ion’s mass (not to be confused with the mass, m, of the

1Institut fur Quantenoptik und Quanteninformation, Osterreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria. 2Institut furExperimentalphysik, Universitat Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria. 3Departamento de Quımica Fısica, Universidad del Paıs Vasco - Euskal HerrikoUnibertsitatea, Apartado 644, 48080 Bilbao, Spain. 4IKERBASQUE, Basque Foundation for Science, Alameda Urquijo 36, 48011 Bilbao, Spain.

Vol 463 | 7 January 2010 | doi:10.1038/nature08688

68Macmillan Publishers Limited. All rights reserved©2010

simulated particle); g 5 0.06 is the Lamb–Dicke parameter; andpp 5 iB(a 2 a)/2D is the momentum operator, with a and a the usualraising and lowering operators for the motional state along the axialdirection. The first term in equation (1) describes a state-dependent

motional excitation with coupling strength g ~VV, corresponding to adisplacement of the ion’s wave packet in the harmonic trap. The

parameter ~VV is controlled by setting the intensity of the bichromaticlight field. The second term is equivalent to an optical Stark shift andoccurs when the bichromatic light field is detuned from resonance by2V. Equation (1) reduces to the 1 1 1 dimensional Dirac Hamiltonian

if we make the identifications c ; 2g~VVD and mc2 ; BV. Themomentum and position of the Dirac particle are then mappedonto the corresponding quadratures of the trapped-ion harmonicoscillator.

To study relativistic effects such as Zitterbewegung, it is necessaryto measure Æxx(t)æ, the expectation value of the position operator ofthe harmonic oscillator. It has been noted theoretically that suchexpectation values could be measured using very short probe times,without reconstructing the full quantum state7,23,24. To measure Æxxæfor a motional state rm, we have to (1) prepare the ion’s internal statein an eigenstate of sy, (2) apply a unitary transformation, U(t), thatmaps information about rm onto the internal states and (3) recordthe changing excitation as a function of the probe time t, by mea-suring fluorescence22. In this protocol, the unitary operatorU(t) 5 exp(2igVpsxxxt/D), with xx 5 (a 1 a)D and probe Rabifrequency Vp, effectively transforms the observable sz into sin kxx,with k 5 2gVpt/D, meaning that Æxxæ can be determined by monitor-ing the rate of change of Æsin kxxæ for short probe times (Methods).Because the Dirac Hamiltonian generally entangles the motional andinternal states of the ion, we first incoherently recombine the internalstate population in 0

1

! "(Methods) before proceeding to step 1. Then

we apply the Hamiltonian generating U with the probe Rabi fre-quency set to Vp 5 2p3 13 kHz for interaction times t of up to14 ms, in 1–2-ms steps. The change of excitation was obtained by linearfits, each based on 104 to 3 3 104 measurements.

We simulate the Dirac equation by applying HD for varyingamounts of time and for different particle masses. In the experiment,

we set ~VV 5 2p3 68 kHz, corresponding to a simulated speed of lightof c 5 0.052D ms21. The measured expectation values, Æxx(t)æ, areshown in Fig. 1 for a particle initially prepared in the spinor statey(x; t 5 0) 5 (

ffiffiffiffiffi2pp

2D)21/2ex2=4D2 11

! "by sideband cooling and

application of a p/2 pulse. Zitterbewegung appears for particles withnon-zero mass, and is obtained by varying V in the range0 , V # 2p3 13 kHz by changing the detuning of the bichromaticlasers.

We investigate the particle dynamics in the crossover from relat-ivistic to non-relativistic dynamics. The data in Fig. 1 well matchnumerical simulations based on equation (1), which are shown assolid lines. The error bars are obtained from a linear fit assumingquantum projection noise, which dominates noise caused by fluctua-tions of control parameters. In addition, the data were fitted with aheuristic model function of the form Æxx(t)æ 5 at 1 RZBsin vZBt toextract the effective amplitude, RZB, and frequency, vZB, of theZitterbewegung shown in the inset. As the particle’s initialmomentum is not dispersion free, the amplitude and frequency areonly approximate concepts. From these data, it can be seen that thefrequency, vZB < 2V, grows linearly with increasing mass, whereasthe amplitude decreases as the mass is increased. Because the mass ofthe particle increases but the momentum and the simulated speed oflight remain constant, the data in Fig. 1 show the crossover from thefar relativistic to non-relativistic limits. Hence, the data confirm thatZitterbewegung decreases in both limits, as theoretically expected. Inthe far-relativistic case, this is because vZB vanishes; in the non-relativistic case, it is because RZB vanishes.

The tools with which we simulate the Dirac equation can also beused to set the initial state of the simulated particle precisely. Theparticle in Fig. 2a was given an average initial momentumÆpp(t 5 0)æ 5 B/D by means of a displacement operation using theHamiltonian H~Bg ~VVsx xx=D. The initial state of this particle consists

6

5

4

3

2

1

1 2 300

50

100

0

0 50 100 150

⟨x⟩ (

)

^∆

t (µs)

0.00

0.50

∆R

ZB (

)

ωZB (kH

z)

Ω Ω~/η

Figure 1 | Expectation values, Æxx(t)æ, for particles with different masses.The linear curve (squares) represents a massless particle (V 5 0) moving atthe speed of light, which is given by c 5 2g ~VVD5 0.052Dms21 for all curves.From the top, the other curves represent particles of increasing masses. TheirCompton wavelengths are given by lC ; 2g ~VVD/V 5 5.4D (down triangles),2.5D (diamonds), 1.2D (circles) and 0.6D (up triangles), respectively. Thesolid curves represent numerical simulations. The figure showsZitterbewegung for the crossover from the relativistic limit, 2g ~VV?V, to thenon-relativistic limit, 2g ~VV=V. Inset, fitted Zitterbewegung amplitude, RZB

(squares), and frequency, vZB (circles), versus the parameter V/g ~VV (which isproportional to the mass). Error bars, 1s.

x ( )∆

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0 20 40 60 80 100 120 140 160t (µs)

a

b

0.0

–0.2 –5 5

01( )

10( )

0 –5 50 –5 50

0.2

|ψ(x

)|2

t = 0 µs t = 75 µs t = 150 µs

∆λC = 0.62

⟨x⟩ (

)

^∆

Figure 2 | Zitterbewegung for a state with non-zero average momentum.a, Initially, Zitterbewegung appears owing to interference of positive- andnegative-energy parts of the state,y(x; t 5 0) 5 eix=Dex2=4D2

(ffiffiffiffiffi2pp

2D)1=2 11

! ". As these parts separate, the

oscillatory motion fades away. The solid curve represents a numericalsimulation. Error bars, 1s. b, Measured (filled areas) and numericallycalculated (solid lines) probability distributions, | y(x) | 2, at times t 5 0, 75and 150ms (as indicated by the arrows in a). The probability distributioncorresponding to the state 0

1

! "is inverted for clarity. The vertical solid line

represents Æxxæ as plotted in a. The two dashed lines indicate the respectiveexpectation values for the positive- and negative-energy parts of the spinor.

NATURE | Vol 463 | 7 January 2010 LETTERS

69Macmillan Publishers Limited. All rights reserved©2010

Trapped Ions in BilbaoDirac Equation and Quantum Relativistic Effects in a Single Trapped Ion

L. Lamata,1 J. Leon,1 T. Schatz,2 and E. Solano3,4

1Instituto de Matematicas y Fısica Fundamental, CSIC, Serrano 113-bis, 28006 Madrid, Spain2Max-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany

3Physics Department, ASC, and CeNS, Ludwig-Maximilians-Universitat, Theresienstrasse 37, 80333 Munich, Germany4Seccion Fısica, Departamento de Ciencias, Pontificia Universidad Catolica 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 particlein a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position andmomentum of the Dirac particle are associated with the respective ionic variables. We show also how tosimulate the simplified 1! 1 case, requiring the manipulation of only two internal levels and one motionaldegree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung andKlein’s paradox, the transition from massless to massive fermions, and the relativistic and nonrelativisticlimits, 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 Schrodinger equationgave rise to the Klein-Gordon and Dirac equations. P. A. M.Dirac looked for a Lorentz-covariant wave equation that islinear in spatial and time derivatives, expecting that theinterpretation of the square wave function as a probabilitydensity holds. As a result, he obtained a fully covariantwave equation for a spin-1=2 massive particle, which in-corporated ab initio the spin degree of freedom. It is known[1] that the Dirac formalism describes accurately the spec-trum of the hydrogen atom and that it plays a central role inquantum field theory, where creation and annihilation ofparticles are allowed. However, the one-particle solutionsof the Dirac equation in relativistic quantum mechanicspredict some astonishing effects, like the Zitterbewegungand the Klein’s paradox.

In recent years, a growing interest has appeared regard-ing simulations of relativistic effects in controllable physi-cal systems. Some examples are the simulation of Unruheffect in trapped ions [2], the Zitterbewegung for massivefermions in solid state physics [3], and black-hole proper-ties in the realm of Bose-Einstein condensates [4].Moreover, the low-energy excitations of a nonrelativistictwo-dimensional electron system in a single layer of graph-ite (graphene) are known to follow the Dirac-Weyl equa-tions for massless relativistic particles [5,6]. On the otherhand, the fresh dialog between quantum information andspecial relativity has raised important issues concerningthe quantum information content of Dirac bispinors underLorentz transformations [7].

In this Letter, we propose the simulation of the Diracequation for a free spin-1=2 particle in a single trapped ion.We show how to implement realistic interactions on fourionic internal levels, coupled to the motional degrees offreedom, so as to reproduce this fundamental quantum-relativistic wave equation. We propose also the simulationof the Dirac equation in 1! 1 dimensions, requiring onlythe control of two internal levels and one motional degree

of freedom. We study some quantum-relativistic effects,like the Zitterbewegung and the Klein’s paradox, in termsof measurable observables. Moreover, we discuss the tran-sition from massless to massive fermions, and from therelativistic to the nonrelativistic limit. Finally, we describea possible experimental scenario.

We consider a single ion of mass M inside a Paul trapwith frequencies !x, !y, and !z, where four metastableionic internal states, jai, jbi, jci, and jdi, may be coupledpairwise to the center-of-mass (c.m.) motion in directionsx, y, and z. We will make use of three standard interactionsin trapped-ion technology, allowing for the coherent con-trol of the vibronic dynamics [8]: first, a carrier interactionconsisting of a coherent driving field acting resonantly on apair of internal levels, while leaving untouched the mo-tional degrees of freedom. It can be described effectivelyby the Hamiltonian H" " @!#"!ei# ! "$e$i#%, where"! and "$ are the raising and lowering ionic spin-1=2operators, respectively, and ! is the associated couplingstrength. The phases and frequencies of the laser fieldcould be adjusted so as to produce H"x " @!x"x, H"y "@!y"y, and H"z " @!z"z, where "x, "y, and "z areatomic Pauli operators in the conventional directions x, y,and z. Second, we consider a Jaynes-Cummings (JC) in-teraction, usually called red-sideband excitation, consist-ing of a laser field acting resonantly on two internal levelsand one of the vibrational c.m. modes. Typically, a reso-nant JC coupling induces an excitation in the internal levelswhile producing a deexcitation of the motional harmonicoscillator, and vice versa. The resonant JC Hamiltonian canbe written asHr " @$ ~!#"!aei#r ! "$aye$i#r%, where aand ay are the annihilation and creation operators associ-ated with a motional degree of freedom. $ " k

!!!!!!!!!!!!!!!!@=2M!

pis

the Lamb-Dicke parameter [8], where k is the wave num-ber of the driving field. Third, we consider an anti-JC(AJC) interaction, consisting of a JC-like coupling tuned

PRL 98, 253005 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending22 JUNE 2007

0031-9007=07=98(25)=253005(4) 253005-1 2007 The American Physical Society

LETTERS

Quantum simulation of the Dirac equationR. Gerritsma1,2, G. Kirchmair1,2, F. Zahringer1,2, E. Solano3,4, R. Blatt1,2 & C. F. Roos1,2

The Dirac equation1 successfully merges quantum mechanics withspecial relativity. It provides a natural description of the electronspin, predicts the existence of antimatter2 and is able to reproduceaccurately the spectrum of the hydrogen atom. The realm of theDirac equation—relativistic quantum mechanics—is consideredto be the natural transition to quantum field theory. However, theDirac equation also predicts some peculiar effects, such as Klein’sparadox3 and ‘Zitterbewegung’, an unexpected quivering motionof a free relativistic quantum particle4. These and other predictedphenomena are key fundamental examples for understandingrelativistic quantum effects, but are difficult to observe in realparticles. In recent years, there has been increased interest insimulations of relativistic quantum effects using different physicalset-ups5–11, in which parameter tunability allows access to differentphysical regimes. Here we perform a proof-of-principle quantumsimulation of the one-dimensional Dirac equation using a singletrapped ion7 set to behave as a free relativistic quantum particle.We measure the particle position as a function of time and studyZitterbewegung for different initial superpositions of positive-and negative-energy spinor states, as well as the crossover fromrelativistic to non-relativistic dynamics. The high level of controlof trapped-ion experimental parameters makes it possible to simu-late textbook examples of relativistic quantum physics.

The Dirac equation for a spin-1/2 particle with rest mass m is givenby1

iB Ly

Lt~(ca.ppzbmc2)y

Here c is the speed of light, pp is the momentum operator, aj (j 5 1,2, 3; (a)j 5 aj) and b are the Dirac matrices (which are usually givenin terms of the Pauli matrices, sx, sy and sz), the wavefunctions y arefour-component spinors and B is Planck’s constant divided by 2p. Ageneral Dirac spinor can be decomposed into parts with positive and

negative energies E 5 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2c2zm2c4

p. Zitterbewegung is under-

stood to be an interference effect between the positive- and negative-energy parts of the spinor and does not appear for spinors that consistentirely of positive-energy (or negative-energy) parts. Furthermore, itis only present when these parts have significant overlap in positionand momentum space and is therefore not a sustained effect undermost circumstances1. For a free electron, the Dirac equation predictsthe Zitterbewegung to have an amplitude of the order of the Comptonwavelength, RZB < 10212 m, and a frequency of vZB < 1021 Hz, andthe effect has so far been experimentally inaccessible. The existence ofZitterbewegung, in relativistic quantum mechanics and in quantumfield theory, has been a recurrent subject of discussion in the pastyears12,13.

Quantum simulation aims to simulate a quantum system using acontrollable laboratory system that underlies the same mathematicalmodel. In this way, it is possible to simulate quantum systems thatcan be neither efficiently simulated on a classical computer14 nor

easily accessed experimentally, while allowing parameter tunabilityover a wide range. The difficulties in observing real quantum rela-tivistic effects have generated significant interest in the quantumsimulation of their dynamics. Examples include black holes inBose–Einstein condensates5 and Zitterbewegung for massive fermionsin solid-state physics6, neither of which have been experimentallyrealized so far. Also, graphene is studied widely in connection to theDirac equation15–17.

Trapped ions are particularly interesting for the purpose ofquantum simulation18–20, as they allow exceptional control of experi-mental parameters, and initialization and read-out can be achievedwith high fidelity. Recently, for example, a proof-of-principle simu-lation of a quantum magnet was performed21 using trapped ions. Thefull, three-dimensional, Dirac equation Hamiltonian can be simu-lated using lasers coupling to the three vibrational eigenmodes andthe internal states of a single trapped ion7. The set-up can be signifi-cantly simplified when simulating the Dirac equation in 1 1 1dimensions, yet the most unexpected features of the Dirac equation,such as Zitterbewegung and the Klein paradox, remain. In the Diracequation in 1 1 1 dimensions, that is

iB Ly

Lt~HDy~(cppsxzmc2sz)y

there is only one motional degree of freedom and the spinor isencoded in two internal levels, related to positive- and negative-energy states7. We find that the velocity of the free Dirac particle isdxx=dt~½xx, HD"=iB~csx in the Heisenberg picture. For a masslessparticle, [sx, HD] 5 0 and, hence, sx is a constant of motion. For amassive particle, [sx, HD] ? 0 and the evolution of the particle isdescribed by

xx(t)~xx(0)zppc2H1D tzijj(e2iHDt=B1)

where jj~(1=2)Bc(sxppcH1D )H1

D . The first two terms representevolution that is linear in time, as expected for a free particle, whereasthe third, oscillating, term may induce Zitterbewegung.

For the simulation, we trapped a single 40Ca1 ion in a linear Paultrap22 with axial trapping frequency vax 5 2p3 1.36 MHz and radialtrapping frequency vrad 5 2p3 3 MHz. Doppler cooling, opticalpumping and resolved sideband cooling on the S1/2 « D5/2 transitionin a magnetic field of 4 G prepare the ion in the axial motional groundstate and in the internal state jS1/2, mJ 5 1/2æ (mJ, magnetic quantumnumber). A narrow-linewidth laser at 729 nm couples the states

01

" #; jS1/2, mJ 5 1/2æ and 1

0

" #; jD5/2, mJ 5 3/2æ, which we identify

as our spinor states. A bichromatic light field resonant with the upperand lower axial motional sidebands of the 1

0

" #« 0

1

" #transition with

appropriately set phases and frequency realizes the Hamiltonian7

HD~2gD ~VVsx ppzBVsz ð1Þ

Here D 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB=2 ~mmvax

pis the size of the ground-state wavefunction,

with ~mm the ion’s mass (not to be confused with the mass, m, of the

1Institut fur Quantenoptik und Quanteninformation, Osterreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria. 2Institut furExperimentalphysik, Universitat Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria. 3Departamento de Quımica Fısica, Universidad del Paıs Vasco - Euskal HerrikoUnibertsitatea, Apartado 644, 48080 Bilbao, Spain. 4IKERBASQUE, Basque Foundation for Science, Alameda Urquijo 36, 48011 Bilbao, Spain.

Vol 463 | 7 January 2010 | doi:10.1038/nature08688

68Macmillan Publishers Limited. All rights reserved©2010

•Klein (theory)•Klein (experiment)•Majorana equation•Unphysical operations

simulated particle); g 5 0.06 is the Lamb–Dicke parameter; andpp 5 iB(a 2 a)/2D is the momentum operator, with a and a the usualraising and lowering operators for the motional state along the axialdirection. The first term in equation (1) describes a state-dependent

motional excitation with coupling strength g ~VV, corresponding to adisplacement of the ion’s wave packet in the harmonic trap. The

parameter ~VV is controlled by setting the intensity of the bichromaticlight field. The second term is equivalent to an optical Stark shift andoccurs when the bichromatic light field is detuned from resonance by2V. Equation (1) reduces to the 1 1 1 dimensional Dirac Hamiltonian

if we make the identifications c ; 2g~VVD and mc2 ; BV. Themomentum and position of the Dirac particle are then mappedonto the corresponding quadratures of the trapped-ion harmonicoscillator.

To study relativistic effects such as Zitterbewegung, it is necessaryto measure Æxx(t)æ, the expectation value of the position operator ofthe harmonic oscillator. It has been noted theoretically that suchexpectation values could be measured using very short probe times,without reconstructing the full quantum state7,23,24. To measure Æxxæfor a motional state rm, we have to (1) prepare the ion’s internal statein an eigenstate of sy, (2) apply a unitary transformation, U(t), thatmaps information about rm onto the internal states and (3) recordthe changing excitation as a function of the probe time t, by mea-suring fluorescence22. In this protocol, the unitary operatorU(t) 5 exp(2igVpsxxxt/D), with xx 5 (a 1 a)D and probe Rabifrequency Vp, effectively transforms the observable sz into sin kxx,with k 5 2gVpt/D, meaning that Æxxæ can be determined by monitor-ing the rate of change of Æsin kxxæ for short probe times (Methods).Because the Dirac Hamiltonian generally entangles the motional andinternal states of the ion, we first incoherently recombine the internalstate population in 0

1

! "(Methods) before proceeding to step 1. Then

we apply the Hamiltonian generating U with the probe Rabi fre-quency set to Vp 5 2p3 13 kHz for interaction times t of up to14 ms, in 1–2-ms steps. The change of excitation was obtained by linearfits, each based on 104 to 3 3 104 measurements.

We simulate the Dirac equation by applying HD for varyingamounts of time and for different particle masses. In the experiment,

we set ~VV 5 2p3 68 kHz, corresponding to a simulated speed of lightof c 5 0.052D ms21. The measured expectation values, Æxx(t)æ, areshown in Fig. 1 for a particle initially prepared in the spinor statey(x; t 5 0) 5 (

ffiffiffiffiffi2pp

2D)21/2ex2=4D2 11

! "by sideband cooling and

application of a p/2 pulse. Zitterbewegung appears for particles withnon-zero mass, and is obtained by varying V in the range0 , V # 2p3 13 kHz by changing the detuning of the bichromaticlasers.

We investigate the particle dynamics in the crossover from relat-ivistic to non-relativistic dynamics. The data in Fig. 1 well matchnumerical simulations based on equation (1), which are shown assolid lines. The error bars are obtained from a linear fit assumingquantum projection noise, which dominates noise caused by fluctua-tions of control parameters. In addition, the data were fitted with aheuristic model function of the form Æxx(t)æ 5 at 1 RZBsin vZBt toextract the effective amplitude, RZB, and frequency, vZB, of theZitterbewegung shown in the inset. As the particle’s initialmomentum is not dispersion free, the amplitude and frequency areonly approximate concepts. From these data, it can be seen that thefrequency, vZB < 2V, grows linearly with increasing mass, whereasthe amplitude decreases as the mass is increased. Because the mass ofthe particle increases but the momentum and the simulated speed oflight remain constant, the data in Fig. 1 show the crossover from thefar relativistic to non-relativistic limits. Hence, the data confirm thatZitterbewegung decreases in both limits, as theoretically expected. Inthe far-relativistic case, this is because vZB vanishes; in the non-relativistic case, it is because RZB vanishes.

The tools with which we simulate the Dirac equation can also beused to set the initial state of the simulated particle precisely. Theparticle in Fig. 2a was given an average initial momentumÆpp(t 5 0)æ 5 B/D by means of a displacement operation using theHamiltonian H~Bg ~VVsx xx=D. The initial state of this particle consists

6

5

4

3

2

1

1 2 300

50

100

0

0 50 100 150

⟨x⟩ (

)

^∆

t (µs)

0.00

0.50

∆R

ZB (

)

ωZB (kH

z)

Ω Ω~/η

Figure 1 | Expectation values, Æxx(t)æ, for particles with different masses.The linear curve (squares) represents a massless particle (V 5 0) moving atthe speed of light, which is given by c 5 2g ~VVD5 0.052Dms21 for all curves.From the top, the other curves represent particles of increasing masses. TheirCompton wavelengths are given by lC ; 2g ~VVD/V 5 5.4D (down triangles),2.5D (diamonds), 1.2D (circles) and 0.6D (up triangles), respectively. Thesolid curves represent numerical simulations. The figure showsZitterbewegung for the crossover from the relativistic limit, 2g ~VV?V, to thenon-relativistic limit, 2g ~VV=V. Inset, fitted Zitterbewegung amplitude, RZB

(squares), and frequency, vZB (circles), versus the parameter V/g ~VV (which isproportional to the mass). Error bars, 1s.

x ( )∆

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0 20 40 60 80 100 120 140 160t (µs)

a

b

0.0

–0.2 –5 5

01( )

10( )

0 –5 50 –5 50

0.2

|ψ(x

)|2

t = 0 µs t = 75 µs t = 150 µs

∆λC = 0.62

⟨x⟩ (

)

^∆

Figure 2 | Zitterbewegung for a state with non-zero average momentum.a, Initially, Zitterbewegung appears owing to interference of positive- andnegative-energy parts of the state,y(x; t 5 0) 5 eix=Dex2=4D2

(ffiffiffiffiffi2pp

2D)1=2 11

! ". As these parts separate, the

oscillatory motion fades away. The solid curve represents a numericalsimulation. Error bars, 1s. b, Measured (filled areas) and numericallycalculated (solid lines) probability distributions, | y(x) | 2, at times t 5 0, 75and 150ms (as indicated by the arrows in a). The probability distributioncorresponding to the state 0

1

! "is inverted for clarity. The vertical solid line

represents Æxxæ as plotted in a. The two dashed lines indicate the respectiveexpectation values for the positive- and negative-energy parts of the spinor.

NATURE | Vol 463 | 7 January 2010 LETTERS

69Macmillan Publishers Limited. All rights reserved©2010

Dirac equation in a trapped ion

Brief prehistory of the Dirac equation

a) The energy of a free classical particle is

b) The quantum version is the Schrödinger equation, a nonrelativistic wave equation:

E = p2

2m

i ∂∂tφ = −

2

2m∇2φ

d) A natural quantum relativistic version would be the Klein-Gordon equation:

−2

∂2

∂t 2φ = −2c2∇2φ +m2c4φ

c) The energy of a free relativistic particle, following special relativity, can be obtained from

E2 = p2c2 +m2c4

where we have replaced and E→ i ∂

∂t p→−i∇

By then, nature and physical consistence said YES to Schrödinger and NO to Klein-Gordon!

Dirac equation in a trapped ion Brief history of the Dirac equation

a) Dirac wanted linear spatial and time derivatives so as to keep similarities with special relativity, and to have the future of a particle defined by the wavefunction at a given time. He linearized the relativistic energy operator and was led to propose E = p2c2 +m2c4

i ∂∂tφ = (c α.p + βmc2 )φ

where , , the matrices satisfy , p = −i

∇ = px , py , pz( )

α = αx ,αy ,αz( ) αk ,β 4 × 4 αk

2 = β2 = I4

αk =0 σ k

σ k 0

#

$

%%

&

'

(( , βk =

I2 00 −I2

#

$

%%

&

'

((

and Pauli matrices σ x =0 11 0

"

#$

%

&' , σ y =

0 −ii 0

"

#$

%

&' , σ z =

1 00 −1

"

#$

%

&'

φThe high price to pay was to have a four-component wavefunction: the Dirac bispinor .

i ∂∂tφ =

mc2 c σ .p

c σ .p −mc2

%

&

''

(

)

**φ

In this Dirac representation, we can write the Dirac equation for a free spin-1/2 charged particle as

φ =

φaφbφcφd

"

#

$$$$$

%

&

'''''

with Dirac bispinor

Dirac equation in a trapped ion

c) Other two components are related to positive and negative eigenvalues of the Dirac Hamiltonian: this can be seen by choosing zero-momentum (p=0) and diagonalizing

i ∂∂t

φ+φ−

$

%

&&

'

(

)) =

mc2 00 −mc2

$

%&&

'

())

φ+φ−

$

%

&&

'

(

)) with eigenvalues E = ±mc2

The negative-energy solutions of the Dirac equation, though mathematically consistent, were conflictive. In fact, the linear superposition of positive and negative energy solutions leads to a jittering motion of a free relativistic particle called Zitterbewegung.

Dirac tried to solve the problem proposing the vacuum to be a sea filled of negative-energy electrons, and a hole in the Dirac sea would behave as a positively charged particle, he thought it was the proton, history considers this as the prediction of the existence of the positron and the universe of antimatter.

b) Two bispinor components were meant to predict ab initio the spin-½ of the electron.

Dirac equation in a trapped ion

The 1+1 Dirac equation

a) The 1+1 Dirac equation requires a wavefunction with only two components, and we write it here, after a suitable rotation around x-axis, in the Majorana representation:

i ∂∂tφ = cσ x px +mc

2σ z( )φ =mc2 cpxc px −mc2

%

&

''

(

)

**φ φ =

φaφb

"

#

$$

%

&

''

with Dirac spinor

The spin interpretation disappears and the positive and negative energy solutions survive.

b) The eigenstates are E± =

mc2 + E±

mc2 + E±( )2 + c2 px2

cpxmc2 + E±( )2 + c2 px2

!

"

######

$

%

&&&&&&

with eigenvalues E± = ± c2 px2 +m2c4

Dirac equation in a trapped ionIII) Simulating free Dirac particles and Zitterbewegung

a) We consider a two-level ion in a harmonic Paul trap. We assume that cooling techniques allow the preparation of the ionic motional ground state. The associated Hamiltonian is

8 trapped ions at Innsbruck

L. Lamata, J. León, T. Schätz, and E. Solano, Phys. Rev. Lett. 98, 253005 (2007)

b) We will show how to implement the quantum simulation of the 1+1 Dirac equation in such a way that the Dirac spinor will be represented by the ionic internal degrees of freedom, while the position and momentum of the free Dirac particle will be represented by the ion position and momentum.

Trapped ion model

H0 = ν(a

†a + 12)

Challenge: how would you simulate a free Schrödinger particle with a single trapped ion? HSch =p2

2m

R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. F. Roos, Nature 463, 68 (2010)

Dirac equation in a trapped ionBasic interactions in trapped ions

a) The carrier excitation:

Hσφ= Ωσφ = Ω σ +eiφ +σ −e−iφ( )

φ = 0→ Hσ x= Ωσ x

φ = −π2→ Hσ y

= Ωσ y

'

()

*)

b) The red sideband excitation:

Hr = η Ωr σ

+aeiφ r +σ −a†e−iφ r( )

Hb = η Ωb σ

+a†eiφ b +σ −ae−iφ b( )c) The blue sideband excitation:

x =

2Mνa† + a( ) = Δ a† + a( )

d) The linear superposition of red and blue sideband excitations:

Hr+b = η Ωσφ αx + βpx( ) px = i

Mν2

a† − a( ) = i2Δ

a† − a( )with

Dirac equation in a trapped ionQuantum simulation of the 1+1 Dirac equation

a) The previous linear superposition plus qubit detuning yields an effective Hamiltonian corresponding to the Majorana representation of the 1+1 Dirac Hamiltonian for a free particle:

i ∂∂tφ = HD

ionφ = 2ηΔ Ωσ x px + Ωσ z( )φ =Ω 2ηΔ Ω px

2ηΔ Ω px −Ω

(

)

**

+

,

--φ,

i ∂∂tφ = HDφ = cσ x px +mc

2σ z( )φ =mc2 cpxc px −mc2

%

&

''

(

)

**φ

to be compared with the original:

Ω = mc2

2ηΔ Ω = c

$%&

'&producing the parameter correspondence:

This shows that the simulated speed of light is tunable via the laser amplitude, and the particle rest mass via the laser frequency!

b) Similar steps produce the quantum simulation of higher dimensional Dirac equations

Dirac equation in a trapped ionc) If we consider the relativistic limit, , the Dirac dynamics produces constantly growing Schrödinger cats as in quantum optical systems: mc

2 cpx m→ 0( )

HDion = 2ηΔ Ωσ x px + Ωσ z → HD

rel= 2ηΔ Ωσ x pxSee, for example, Solano et al., PRL (2001), Solano et al., PRL (2003), Haljan et al., PRL (2005),

and Zähringer et al., PRL (2010).

d) If we consider now the nonrelativistic limit, , the Dirac dynamics would be happy to have a quantum optician calculating the second-order effective Hamiltonian:

mc2 cpx

HDI = 2ηΔ Ω σ +e2iΩt +σ −e−2iΩt( )pxz → H eff=σ z

px2

Ω2η2Δ2 Ω2

'

()

*

+,

=σ zpx2

2m

We have solved our previous challenge: this result represents the quantum simulation in a trapped ion of a free Schrödinger particle via the simulation of the nonrelativistic limit of the Dirac equation!

withsimulated mass m =

νΩ2η2 Ω2 M

Quantum simulation of the Zitterbewegung

a) The Zitterbewegung (ZB) is a jittering motion of the expectation value of the position operator . It appears as a consequence of the superposition of positive and negative energy components.

In the Heisenberg picture, we can write the evolution of the Dirac position operator

x(t) = x(0)+ c

2 pxHD

t + ic2HD

e2iHDt / −1( ) σ x −cpxHD

#

$%

&

'(

x(t)

b) The prediction of ZB is considered controversial, see several papers appeared in the last few years questioning existence/absence. The predicted ZB frequency/amplitude for our “relativistic” ion are

ωZB 0 −106Hz

xZB 0 −103 A

ω

ZB~ 2 E

D/ = 2 p

0

2c2 + m2

c4/ ≡ 2 2ηΔ Ωp

0( )2

/ +Ω2

xZB~

2mc

mc2

ED

2

≡η2 ΩΩΔ

4η2 Ω2Δ2 p0

2+

2Ω2~ Δ

From the experimental point of view, the ZB frequency looked cool!

However, the ZB amplitude was disappointing: how can one measure in the lab the ion position as a function of the interaction time with a resolution beyond the width of the motional ground state?

Dirac equation in a trapped ion

Dirac equation in a trapped ion

ρat−m (0) = + + ρ

m where + =

1

2g + e( )

x(t) =dP

e(t)

dtt=0

where Pe(t) = Tr ρ

at−m (t) e e

Lougovski et al., Eur. Phys. J. D (2006); Bastin et al., J. Phys. B: At. Mol. Opt. Phys. (2006); Franca Santos et al., PRL (2006); Gerritsma et al., Nature (2010), Zähringer et al., PRL (2010);

Casanova et al., PRA 81, 062126 (2010).

Theory of “instantaneous” measurements

a)  The answer to the previous question is certainly: with an innovative and powerful method! We have created such a method but we will just present introductory guidelines.

If the initial state of the probe qubit and the unknown motional system is

It can be proved that after a red-sideband excitation during an interaction time “t”

It is possible to encode relevant motional system observables in the short-time dynamics of the probe qubit, in fact we can get all in the first and second derivatives at t=0!

b) There are a number of papers studying different results for the “instantaneous” measurements. Some of them are theoretical and some of them have already seen the light of experiments.

Dirac equation in a trapped ion

Our theoretical proposals work, by their fruits you will know them!

“Instantaneous” measurements of ZB with sub-Δ resolution in Nature 2010. Reconstruction of absolute square wavefunction

of quantum walks in PRL 2010.

Trapped Ions in BilbaoQuantum Field Theories,

Casanova, Lamata, ..., Solano, PRL ’11Fermion Lattice Models,

Casanova, Mezzacapo, Lamata, Solano, PRL ’123

and in the time- and space-dependent phases that areassociated to energy and momentum conservation.

The bosonic field will, in addition, be written as

A(t, x) = a0ei!0teik0x + a†

0ei!0teik0x. (10)

Consequently, the resulting interaction Hamiltonian is

H = gX

i,j=f,f

F i,j(pi

, pj

, k0, t) i†pi

jpj

a0 + H.c., (11)

where ipii=f,f

= bin, d†in, and

Ff,f (pf

, pf

, k0, t) =

Zdx|G

f

(pf

, x, t)|2eik0x

ei!0t,

F f ,f (pf

, pf

, k0, t) =

Zdx|G

f

(pf

, x, t)|2eik0x

ei!0t,

Ff,f (pf

, pf

, k0, t) =

ZdxG

f

(pf

, x, t)Gf

(pf

, x, t)

ei(pf+pfk0)x

ei(!f+!f!0)t. (12)

This Hamiltonian contains the self-interacting dynam-ics given by |f, f , ni $ |f, f , n ± 1i (|f, f , ni denotesthe state with one fermion, one antifermion, and nbosons), mediated by b†

inbinak0 , b†

inbina†k0

, dind†ina

k0 and

dind†ina†

k0. It also includes pair creation and annihilation

processes given by |f, f , ni $ |0, 0, n ± 1i, mediated bydinbina†

k0and b†

ind†ina

k0 in the quasi-resonant case, as well

as dinbinak0 and b†

ind†ina†

k0in the far o↵-resonant case.

The last kind of transitions, as well as self-interactions,are o↵-resonant and would be neglected in the weak cou-pling regime, but would be allowed in our formalism forUSC/DSC regimes [18]. In our proposed setup, all per-turbative series terms are included, as shown in Fig. 1.

For practical purposes, we consider |k0| !0, i.e.,a slow massive boson. We may then approximateFf,f (p

f

, pf

, k0, t) = F f ,f (pf

, pf

, k0, t) = g1 exp(i!0t),

and Ff,f (pf

, pf

, k0, t) = g2 exp[(t T/2)2/(22t

) + it],where g2/g1 gives the relative strength of the pair cre-ation with respect to the self-interaction, = !

f

+!f

!0

and T is the total time of the process, being t

the tem-poral interval of the interaction region. Thus, the self-interactions are always on, while the pair creation andannihilation take place only when the fermion and an-tifermion wavepackets overlap, as they should. Accord-ingly, the Hamiltonian we aim to simulate is

H = g1ei!0t

b†inbina0 + dind†

ina0

(13)

+ g2e (tT/2)2

22t

heitb†

ind†ina0 + ei(2!0+)tdinbina0

i+ H.c.

We propose to implement this Hamiltonian dynamics ina system of two trapped ions, see Fig. 2a. The bosonicmode will be encoded in the center-of-mass (COM) vi-bronic mode of the two-ion system. We envision to

FIG. 2. (a) Setup for the trapped-ion simulation. (b)|hf, 0, 0| (t)1i|2 as a function of t in units of !0 (red/uppercurves), where | (t)1i is the evolved state from | (0)1i =|f, 0, 0i, and average number of virtual bosons (blue/lower

curves), ha†0a0i, for g1 = 0.15!0, 0.1!0, 0.05!0, 0.01!0, g2 = 0.

The largest amplitudes correspond to the largest couplings.(c) |hf, f , 0| (t)2i|2 as a function of t in units of !0 (red/upperleft curve), where | (t)2i is the evolved state from | (0)2i =|f, f , 0i, and average number of virtual bosons (blue/lower

left curve), ha†0a0i, for g1 = 0.01!0, g2 = 0.21!0, t = 3/!0,

T = 30/!0, = 0. (d) The same as (c) for g1 = 0.1!0,g2 = !0, t = 4/!0, T = 30/!0, = 0.

map the 4-dimensional Hilbert space associated to thefermionic/antifermionic operators onto 4 internal levelsof the first ion. For this, we consider a Jordan-Wignermapping, b†

in = I +, bin = I , d†in = +

z

,din =

z

, and encode it in four internal levels ofthe first ion, |1i, |2i, |3i, |4i, e.g., b†

in = |4ih3| + |2ih1|,d†in = |4ih2| |3ih1|, the vacuum state is state |1i, and

|fi = |2i, |fi = |3i, |f, fi = |4i. Accordingly, Hamil-tonian (13) results in

H = g(t)

|4ih1| a0e

it + |1ih4| a†0e

it

g(t)

|1ih4| a0e

i(2!0+)t + |4ih1| a†0e

i(2!0+)t

g1

|3ih3| |2ih2|

a0e

i!0t + a†0e

i!0t

+g1 I

a0e

i!0t + a†0e

i!0t

. (14)

Here, the first line corresponds to a detuned red sidebandinteraction between |4i and |1i with time-dependentRabi frequency g(t) = g2 exp[(t T/2)2/22

t

]. Thesecond line is a detuned blue sideband interaction, be-tween the same levels and with the same Rabi frequency.The third line can be developed applying detuned redand blue sideband interactions to |3i and |2i to pro-duce (|3ih2| |2ih3|)[a0 exp(i!0t)+a†

0 exp(i!0t)]/i, anda rotation of |3i and |2i with a classical field to pro-

3

1

2

3

4 5 6

7 8 9

10 11

1 ... n

...

12

4

10

2 3

9

3a) b)

n

n

FIG. 1. (a) Mapping of a fermionic Hamiltonian onto anion string. The couplings between fermions 1 and 4 (resp., 1and 10) are nonlocal when applying the Jordan-Wigner trans-formation. (b) Ecient mapping of the tunneling couplingb†1b10 + b†10b1 in trapped ions. This highly nonlocal couplingcan be implemented with Mølmer-Sørensen gates (dark blueand green), local exp(iz) gates (red), exp[±i(/4)

Pi

y

i

](yellow) and exp[±i(/4)

Pi

x

i

] (cyan) gates.

One of the main appeals of our method is that theencoding of fermionic models in a lattice that has cou-plings beyond nearest neighbors, in principle to arbitrarynumbers of neighbors, is feasible. This also means thatwe can apply the Jordan-Wigner transformation for twoand three spatial dimensions, not just for one, withoutemploying additional virtual Majorana fermions [34]. Allthis is due to the fact that the fermionic operators areencoded in nonlocal spin operators that are eciently im-plementable. Thus, the Hamiltonians we are proposingto simulate are highly nonlocal, but eciently realizable.In order to show this, we plot in Fig. 1 the mappingof a solid-state 3D fermionic system onto an ion string.As opposite to the nearest-neighbor tunneling couplingbetween fermions 1 and 2, which is local, the couplingsbetween 1 and 4, and between 1 and 10, are nonlocalin terms of the Jordan-Wigner transformation (see yel-low lines in Fig. 1a). Nevertheless, we can implementthem in an ecient way. In Fig. 1b we show the im-plementation of the tunneling coupling b†1b10 + b†10b1 =x

1 z

2 z

3 ...z

9 x

10 +y

1 z

2 z

3 ...z

9 y

10

in trapped ions. This highly nonlocal coupling is a globalunitary of 210 210 dimensions. In the general case, itwould require a number of elementary gates of 220 '106 [2], but with our mapping it can be decomposed easilythrough Mølmer-Sørensen gates (dark blue and green),local exp(iz) gates (red), exp[±i(/4)

Pi

y

i

] (yellow)and exp[±i(/4)

Pi

x

i

] (cyan) gates (see Fig. 1b).Numerical simulations.– In order to compare the e-

ciency of the Trotter decomposition with the exact case,we have realized numerical simulations of the Fermi-Hubbard Hamiltonian, Eq. (4), for di↵erent levels ofTrotter expansion and for the exact diagonalization case.We have considered the case of three lattice sites, with

2 6 100

0.02

0.04

0.06a)

Ut 5 15 250.94

0.96

0.98

1b)

nT

5 15 250.86

0.9

0.94

0.98

1d)

nT

0.5 1.5 2.50

0.4

0.8

1c)

wt nTUt

FIG. 2. (a) hb†2#b2#i(t) (dashed, blue), and hb†3"b3"i(t) (solid,red) as a function of Ut, for a number of Trotter steps n

T

=15, and (b) fidelity |h (t

F

)| (tF

)T

i|2 as a function of nT

,for Ut

F

= 10, where | (t)i is the state evolved with exactdiagonalization, and | (t)

T

i is the Trotter-evolved state, for| (0)i = |

T

(0)i = b†1"b†1#|0i, for |w|/U = 0.1. (c) hb†2#b2#i(t)

(dashed, blue), and hb†3"b3"i(t) (solid, red), for nT

= 15, and

(d) fidelity |h (tF

)| (tF

)T

i|2, for UtF

= 2.5, where | (t)i isthe state evolved with exact diagonalization, and | (t)

T

i isthe Trotter-evolved state, for | (0)i = |

T

(0)i = b†1"b†1#|0i,

for |w|/U = 4. In (a) and (c), the lines are obtained withexact diagonalization and the dots with Trotter expansion.

six modes (two spins per site), to be simulated with sixtwo-level trapped ions. The resulting Hamiltonian is

H = w(b†1"b2" + b†1#b2# + b†2"b3" + b†2#b3# + H.c.)

+U(b†1"b1"b†1#b1# + b†2"b2"b

†2#b2# + b†3"b3"b

†3#b3#). (7)

Notice that the number of terms to be implemented scaleslinearly with the number of modes, 5N/2 4 (11 in thiscase, for N = 6). At the same time, the nonlocal gatesupon several ions are eciently implementable with fewlasers, such that the number of gates in each term of theHamiltonian is, in the worst case, linear in the numberof modes, and in many cases just constant.

In Fig. 2, we plot (a) hb†2#b2#i(t) (dashed, blue),

and hb†3"b3"i(t) (solid, red) as a function of Ut, fora number of Trotter steps n

T

= 15, and (b) fidelity|h (t

F

)| (tF

)T

i|2 as a function of nT

, for UtF

= 10,where | (t)i is the state evolved with exact diagonal-ization, and | (t)

T

i is the Trotter-evolved state, for| (0)i = |

T

(0)i = b†1"b†1#|0i, and for |w|/U = 0.1. (c)

hb†2#b2#i(t) (dashed, blue), and hb†3"b3"i(t) (solid, red) asa function of Ut, for a number of Trotter steps n

T

= 15,

Majorana fermions and protected qubits

Mezzacapo, Casanova, Lamata, Solano, NJP ’13

0 20 400.5

1MFQ(a) (b)

Jt

δh =0δh =-0.01Jδh =-0.005Jδh =0.005J

δh =0.01J

z

z

z

z

z

F

Holstein Model and unbounded Hs

Mezzacapo, Casanova, Lamata, Solano, PRL ’12 3

0 500 1000 1500 2000−1

−0.5

0

0.5

1

1

h2ziIh2

ziEhn2iIhn1iIh1ziE h3

ziEh3ziIh1

ziIt

1

h2ziIh2

ziEhn2iIhn1iIh1ziE h3

ziEh3ziIh1

ziIt1

h2ziIh2

ziEhn2iIhn1iIh1ziE h3

ziEh3ziIh1

ziIt

1

h2ziIh2

ziEhn2iIhn1iIh1ziE h3

ziEh3ziIh1

ziIt1

h2ziIh2

ziEhn2iIhn1iIh1ziE h3

ziEh3ziIh1

ziIt1

h2ziIh2

ziEhn2iIhn1iIh1ziE h3

ziEh3ziIh1

ziIt

1

h2ziIh2

ziEhn2iIhn1iIh1ziE h3

ziEh3ziIh1

ziIt

1

h2ziIh2

ziEhn2iIhn1iIh1ziE h3

ziEh3ziIh1

ziIt

1

h2ziIh2

ziEhn2iIhn1iIh1ziE h3

ziEh3ziIh1

ziI1thn1iE

Fidelity Loss

1

h2ziIh2

ziEhn2iIhn1iIh1ziE h3

ziEh3ziIh1

ziI1thn1iE

Fidelity Loss F (t)

FIG. 2. (color online). Dynamics for the 3 + 1 ions configu-ration of the NN XX Hamiltonian. Dotted curves stand forhi

ziE for the exact dynamics, and solid curves stand for hiziI

for realistic ion interactions (i = 1, 2, 3 for the first, secondand third ion). The parameters are chosen in order to havemaxima in the fidelity F (t) = |h E(t)| I(t)i|2 of 0.995 (topblack curve) at time steps of 333 1t. These time steps canbe chosen as Trotter steps.

possibility of obtaining an Ising field in linear chains oftrapped ions has been proposed and realized [32, 36].However, in implementing NN interactions between morethan two ions, one must be careful in designing an ap-propriate set of lasers and detunings in order to minimizethe spurious non-nearest-neighbor (NNN) e↵ects. To thisextent, we have realized numerical simulations for a 3+1ions setup [37], using one set of two pairs of counter-propagating lasers detuned close to the shifted center ofmass (COM) shifted mode of frequency 1 = 1 !0/3to drive the first two ions (detunings ±1), and anotherset of lasers detuned close to a second mode of frequency2, that in the case of 3+1 ions can be chosen as thebreathing mode, addressing the second and the third ion(detunings ±2). For a generic number of ions, Rabi fre-quencies i of the lasers driving the i-th and the i+1-thions are chosen to achieve the desired strength in theIsing coupling, according to [36],

HNN =N1X

i=1

2i

" NX

m=1

i,mi+1,mm

2i 2

m

!

+i,N+1i+1,N+1N+1

2i 2

N+1

x

i xi+1. (7)

In Fig. 2, the first and second ion are driven with twopairs of counterpropagating lasers with detuning close tothe shifted COM mode (1 = 1.0187 1 for !0 = h/4).The Rabi frequencies are chosen properly in order toreach a NN interaction of h/2 = 0.001 1. Lasers drivingthe second and the third ions are detuned close to theshifted breathing mode at 2 = 1.731 1 [35], with pa-rameters 2 = 1.71196 1. Detunings are chosen to have adynamics decoupled with respect to the phonons at time

steps 333 t and a negligible NNN interaction [37]. Atthese times, the ion spins match the exact value, phononsare detached from spins and the fidelity oscillation (topblack curve) F (t) = |h E(t)| I(t)i|2 reaches maxima,with peaks of 0.995.

The initial state, as in all our numerical simulations,except where specified, is chosen to mimic a configura-tion in which one electron is injected at the center of aone dimensional lattice provided with Holstein interac-tions. To this extent, all the spins are initialized in theopposite Z direction, except the one in site N/2, in caseof even N , or (N + 1)/2 in case of odd N . The spin ofthe last ion has to be initialized along the Z direction inorder to be a passive ion with respect to the dynamics,according to the protocol for the implementation of H3

given below. The vibrational modes are assumed to beinitially cooled down to the ground state with resolvedsideband cooling [33].

Notice that one can always implement a perfect NNcoupling by using more stroboscopic steps. A possibil-ity is to decompose the global NN into nearest-neighborpairwise interactions. Another possibility is to design acounter, non-nearest-neighbor interaction step betweenpairs of non-nearest neighbor ions in order to eliminatethe spurious NNN imperfections. Given that one has anunwanted hi,j

ixj

x, one can add more Trotter steps tothe protocol of the form hi,j

ixj

x in order to have anHamiltonian free of NNN couplings. The dynamics as-sociated to the step with H2 is implemented similarly tothe one of H1, with a di↵erent choice of the initial phasesof the lasers, in order to achieve a YY interaction.

The Hamiltonian H3 is realized as a combination of2N red and blue detuned lasers with appropriate ini-tial phases in order to recover a coupling of the i-thion (i = 1, ...N) with the mi-th normal (shifted) modei,mi

iix(b†mi

+ bmi). The i-th ion is driven with red

and blue detuned lasers to the mi-th mode, establishinga one-to-one correspondence between the first N ions andthe first N normal modes. Moreover, the last ion of thechain is driven by 2N lasers detuned in order to be cou-pled with the same modes of the ions in the chain. Twoadditional rotations of the spins of all ions around theY axis are applied before and after coupling the spins tothe phonons. They can be obtained by acting two timeswith a global beam upon all the N + 1 ions at the sametime. The Hamiltonian describing this process is,

Hep =NX

i=1

(ii,miiz+N+1,iN+1,mi

N+1z )(bmi+b†mi

).

(8)The Rabi frequencies of the lasers must be chosen accord-ing to i = g/2i,mi

, N+1,i = g/2N+1,mi. If the last

ion is initialized with the spin aligned along the Z axisand not addressed by spin flip gates during the simula-tion, the previous described gates result in the e↵ective

Conclusions

• Dirac equation: landmark QM/SR unification

• Trapped ions:

first quantum simulation of Dirac physics

• First steps quantum simulation of QFTs

Thank you for your attention!