Exact Mapping of the 2+1 Dirac Oscillator onto the Jaynes-Cummings Model: Ion-Trap Experimental Proposal
aa r X i v : . [ qu a n t - ph ] A p r Exact Mapping of the 2+1 Dirac Oscillator onto the Jaynes-Cummings Model:Ion-Trap Experimental Proposal
A. Bermudez , M. A. Martin-Delgado and E. Solano , Departamento de F´ısica Te´orica I, Universidad Complutense, 28040 Madrid, Spain Physics Department, ASC, and CeNS, Ludwig-Maximilians-Universit¨at, Theresienstrasse 37, 80333 Munich, Germany Secci´on F´ısica, Departamento de Ciencias, Pontificia Universidad Cat´olica del Per´u, Apartado Postal 1761, Lima, Peru
We study the dynamics of the 2+1 Dirac oscillator exactly and find spin oscillations due to a
Zitterbewegung of purely relativistic origin. We find an exact mapping of this quantum-relativisticsystem onto a Jaynes-Cummings model, describing the interaction of a two-level atom with a quan-tized single-mode field. This equivalence allows us to map a series of quantum optical phenomenaonto the relativistic oscillator, and viceversa. We make a realistic experimental proposal, at reachwith current technology, for studying the equivalence of both models using a single trapped ion.
PACS numbers: 42.50.Vk, 42.50.Pq, 03.65.Pm
Current technology has allowed the implementation ofthe paradigmatic nonrelativistic quantum harmonic os-cillator in a single trapped ion [1], one of the most funda-mental toy models in any quantum mechanical textbook.However, its relativistic version, the so-called Dirac oscil-lator [2, 3], remains still far from any possible experimen-tal consideration for different fundamental and technicalreasons. We will show here that available experimen-tal tools may allow the implementation of the relativisticDirac oscillator in a single nonrelativistic trapped ion.The Dirac oscillator was introduced as an instance ofa relativistic wave equation such that its nonrelativisticlimit leads to the well-known Schr¨odinger equation forthe harmonic oscillator. This is achieved by introducingthe following coupling in the Dirac equationi ~ ∂ | Ψ i ∂t = X j =1 cα j (cid:0) p j − i mβωr j (cid:1) + βmc | Ψ i , (1)where | Ψ i is the Dirac 4-component bispinor correspond-ing to a quantum relativistic spin- particle, like the elec-tron, c is the speed of light, m is the particle rest mass,and α j , β , are the Dirac matrices in the standard repre-sentation. The interacting Hamiltonian is linear in bothmomentum p j and position r j , j = x, y, z , and ω turnsout to be the harmonic oscillator frequency. Remark thatwhen ω = 0 we recover the standard Dirac equation [4].The Dirac oscillator looks like a particular gauge trans-formation p → p − ec A that is linear in position, but thepresence of the i and the β matrix makes a crucial differ-ence. Demanding the correct energy-momentum relationfor a relativistic free particle E = p p c + m c , thesematrices are 4 × α j α k + α k α j = 2 δ jk ,α j β + βα j = 0 . (2)There has been a growing interest in simulating quan-tum relativistic effects in other physical systems, such as black hole evaporation in Bose-Einstein condensates [5]and the Unruh effect in an ion chain [6]. Another as-tonishing relativistic prediction is the Zitterbewegung [4],a helicoidal motion realized by the average position of arelativistic fermion, which has been discussed in the con-text of condensed matter systems [7] and the free-particleDirac equation in a single ion [8].Here, we shall be concerned with the Dirac oscillatormodel in 2+1 dimensions, since it is in this setting wherewe can establish a precise equivalence with the Jaynes-Cummings (JC) model [9]. In two spatial dimensions,the solution to the Clifford algebra (2) is given by the2 × α x = σ x , α y = σ y , β = σ z . Inthis case, | Ψ i can be described by a 2-component spinorwhich mixes spin up and down components with positiveand negative energies. In particular, the Dirac oscillatormodel now takes the formi ~ ∂ | Ψ i ∂t = X j =1 cσ j (cid:0) p j − i mσ z ωr j (cid:1) + σ z mc | Ψ i . (3)In this paper, we shall provide the complete (eigenstatesand energies) and exact solution of the 2D Dirac oscilla-tor in order to study its relativistic dynamics, where cer-tain collapses and revivals in the spin degree of freedomappear as a consequence of Zitterbewegung . In addition,we derive an exact mapping of the 2+1 Dirac oscilla-tor onto the JC model, an archetypical quantum opticalsystem. Furthermore, we propose the simulation of thisrelativistic dynamics in a single trapped ion, a physicalsetup possessing outstanding coherence features.Considering the spinor | Ψ i := [ | ψ i , | ψ i ] t , equa-tion (3) becomes a set of coupled equations( E − mc ) | ψ i = c [( p x + i mωx ) − i( p y + i mωy )] | ψ i , ( E + mc ) | ψ i = c [( p x − i mωx ) + i( p y − i mωy )] | ψ i . (4)In order to find the solutions, it is convenient to introducethe following chiral creation and annihilation operators a r := √ ( a x − i a y ) , a † r := √ ( a † x + i a † y ) ,a l := √ ( a x + i a y ) , a † l := √ ( a † x − i a † y ) , (5)where a x , a † x , a y , a † y , are the usual annihilation andcreation operators of the harmonic oscillator a † i = √ (cid:0) r i − i ∆ ~ p i (cid:1) , and ∆ = p ~ /mω represents theground state oscillator width. The orbital angular mo-mentum may also be expressed as L z = ~ ( a † r a r − a † l a l ) , (6)which leads to a physical interpretation of a † r and a † l .These operators create a right or left quantum of angularmomentum, respectively, and are known hence as circu-lar creation-annihilation operators. Equations (4) can berewritten in the language of these circular operators | ψ i = i mc √ ξE − mc a † l | ψ i , | ψ i = − i mc √ ξE + mc a l | ψ i , (7)where ξ := ~ ω/mc controls the nonrelativistic limit. Inorder to find the energy spectrum we shall solve the asso-ciated Klein-Gordon equation, which can be derived fromEqs. (7) as follows( E − m c ) | ψ i = 4 m c ξ a † l a l | ψ i , ( E − m c ) | ψ i = 4 m c ξ (1 + a † l a l ) | ψ i . (8)These equations can be simultaneously diagonalized writ-ing the spinor in terms of the left chiral quanta basis | n l i = 1 √ n l ! (cid:16) a † l (cid:17) n l | vac i , (9)where n l = 0 , , ... The energies can be expressed as( E n l − m c ) | n l i = 4 m c ξn l | n l i , ( E n ′ l − m c ) | n ′ l i = 4 m c ξ (1 + n ′ l ) | n ′ l i . (10)Since both components | ψ i and | ψ i belong to the samesolution, the energies must be the same E n ′ l = E n l . Thisphysical requirement sets up a constraint on the quantumnumbers n l =: n ′ l + 1. Note that, following (6), the state | n l i corresponds to a negative angular momentum. Theenergy spectrum can be described as follows E = ± E n l = ± mc p ξn l . (11)To find the corresponding eigenstates, we go back toEq. (7), and after normalization we arrive at the expres-sion for the positive and negative energy eigenstates | ± E n l i = r E nl ± mc E nl | n l i∓ i r E nl ∓ mc E nl | n l − i , (12) where the quantum number is now restricted to n l =1 , , ... In this way, we have solved the two-dimensionalDirac oscillator describing the energy spectrum and theeigenstates in terms of circular quanta. The distinctionbetween Dirac and Klein-Gordon eigenstates is an im-portant point in order to understand the dynamics of the2+1 Dirac oscillator and its realization in an ion trap.The eigenstates of the 2D Dirac oscillator can beexpressed more transparently in terms of 2-componentPauli spinors | χ ↑ i and | χ ↓ i| + E n l i = α n l | n l i| χ ↑ i − i β n l | n l − i| χ ↓ i , | − E n l i = β n l | n l i| χ ↑ i + i α n l | n l − i| χ ↓ i , (13)where α n l := r E nl + mc E nl and β n l := r E nl − mc E nl are real.From this expression we observe that the energy eigen-states present entanglement between the orbital and spindegrees of freedom. This property is extremely importantsince the following initial state | Ψ(0) i := | n l − i| χ ↓ i = i β n l | + E n l i − i α n l | − E n l i (14)superposes states with positive and negative energies,and this is the fundamental ingredient that leads to Zit-terbewegung in relativistic quantum dynamics. This phe-nomenon, due to the interference of positive and negativeenergies, has never been observed experimentally. Thereason is that the amplitude of these rapid oscillationslies below the Compton wavelength, where pair creationis allowed, and the one-particle interpretation falls down.Now, the evolution of this initial state can be expressedin the energy basis as | Ψ( t ) i = i β n l | + E n l i e − i ω nl t − i α n l | − E n l i e i ω nl t , (15)where ω n l := E n l ~ = mc ~ p ξn l (16)describes the frequency of oscillations. Writing thisevolved state in the language of Pauli spinors, | Ψ( t ) i = (cid:18) cos ω n l t + i √ ξn l sin ω n l t (cid:19) | n l − i| χ ↓ i + s ξn l ξn l sin ω n l t ! | n l i| χ ↑ i , (17)we observe an oscillatory dynamics between | n l i| χ ↑ i and | n l − i| χ ↓ i . The initial state, | n l − i| χ ↓ i , which hasspin-down and n l − Zitterbewegung [4].To clarify this issue further, we calculate the time evo-lution of the following physical observables, that catchthe full essence of the system dynamics, h L z i t = − ( n l − ~ − ξn l ξn l ~ sin ω n l t, h S z i t = − ~ + ξn l ξn l ~ sin ω n l t, h J z i t = ~ ( − n l ) , (18)where J z = L z + S z stands for the z -component of thetotal angular momentum. The latter relations describea certain oscillation in the spin and orbital angular mo-mentum, while the total angular momentum is conserveddue to the existent invariance under rotations around thez-axis. It is important to highlight that these oscillationshave a pure relativistic nature. In the nonrelativisticlimit ξ ≪
1, these oscillations become vanishingly small h L z i t = − ( n l − ~ − ξn l ~ sin Ω n l t + O ( ξ ) , h S z i t = − ~ + 4 ξn l ~ sin Ω n l t + O ( ξ ) , (19)where Ω n l := mc (1 + 2 ξn l ) / ~ stands for the oscillationfrequency in the nonrelativistic limit. In this limit thenegative energy components are negligible and thereforethe Zitterbewung disappears.The results discussed so far allow a precise mappingbetween two seemingly unrelated models: the Jaynes-Cummings model of Quantum Optics and the 2D Diracoscillator. Starting from Eq. (7), we may write the Diracoscillator Hamiltonian as H = 2i mc p ξ (cid:16) a † l | ψ ih ψ | − a l | ψ ih ψ | (cid:17) + mc σ z = ~ ( gσ − a † l + g ∗ σ + a l ) + mc σ z , (20)where σ + , σ − , are the spin raising and lowering opera-tors, and g := 2i mc √ ξ/ ~ is the coupling strength be-tween orbital and spin degrees of freedom. In QuantumOptics, this Hamiltonian describes a Jaynes-Cummingsinteraction, that has been studied in cavity QED andtrapped ions [1, 10], among others. Within this novelperspective, the electron spin can be associated with atwo-level atom, and the orbital circular quanta with theion quanta of vibration, i.e., phonons. As we will see be-low, the central result of Eq. (20) allows both physicalsystems, the JC model and the 2D Dirac oscillator, toexchange a wide range of important applications.We will show now how to implement the dynamics ofEq. (3) in a single ion inside a Paul trap, which was shownto follow the dynamics of Eq. (20). The Dirac spinor willbe described by two metastable internal states, | g i and | e i , as follows | Ψ i := | ψ i| e i + | ψ i| g i (21) while the circular angular momentum modes will be rep-resented by two ionic vibrational modes, a x and a y . Cur-rent technology allows an overwhelming coherent con-trol of ionic internal and external degrees of freedom [1].There, three paradigmatic interactions, the carrier, red-, and blue-sideband excitations, can be implemented atwill, independently or simultaneously [11]. For example,using appropriately tuned lasers, it is possible to producethe following interactions H JC i = ~ η i ˜Ω i h σ + a i e i φ + σ − a † i e − i φ i + ~ δ i σ z ,H AJC i = ~ η i ˜Ω i h σ + a † i e i ϕ + σ − a i e − i ϕ i , (22)where { a i , a † i } , with i = x, y , are the phonon annihila-tion and creation operators in directions x and y , ν i arethe natural trap frequencies, η i := k i p ~ / M ν i are theassociated Lamb-Dicke parameters depending on the ionmass M and the wave vector k , δ i and ˜Ω i are the ex-citation coupling strengths and φ, ϕ , the red and bluesideband phases. Remark that the term ~ δ i σ z , in H JC i ofEq. (22), stems from a detuned JC excitation.A suitable combination of the above introduced exci-tations (22), with proper couplings and relative phases,can reproduce the following Hamiltonian H = c (cid:2) σ gex p x + σ gey p y (cid:3) + mωc (cid:2) σ gex y − σ gey x (cid:3) + mc σ gez (23)with σ gex := | g ih e | + | g ih e | , σ gey := − i( | e ih g | − | e ih g | ), σ gez := | e ih e | − | g ih g | , and the following parameter corre-spondence c = √ η ˜Ω ˜∆ ,mc = ~ δ,mωc = ~ √ η ˜Ω ˜∆ − , (24)where ˜∆ := ˜∆ i is the width of the motional ground state,˜Ω := ˜Ω i , η := η i , ∀ i = x, y . The remarkable equivalenceof the Dirac oscillator Hamiltonian (3) and the interac-tion in Eq. (23) shows that it is possible to reproducethe 2D Dirac oscillator, with all its quantum relativis-tic effects, in a controllable quantum system as a singletrapped ion.For the sake of illustration, note that the effectiveterms appearing in Eq. (23) can be achieved by suitablelinear combinations of H JC i and H AJC i in (22), i = x, δ x = δ, φ = π , ϕ = π → √ ~ η ˜Ω ˜∆ σ gex p x + ~ δσ gez ,i = y, δ y = 0 , φ = 0 , ϕ = π → √ ~ η ˜Ω ˜∆ σ gey p y ,i = x, δ x = 0 , φ = π , ϕ = π → √ ~ η ˜Ω ˜∆ − σ gey x,i = y, δ y = 0 , φ = 0 , ϕ = 0 → √ ~ η ˜Ω ˜∆ − σ gex y. (25)Note that in the trapped ion picture, the important pa-rameter ξ = 2( η ˜Ω /δ ) can take on all positive values,assuming available experimental parameters: η ∼ . ∼ − Hz, and δ ∼ − Hz [1]. The ability to ex-perimentally tune these parameters will allow the exper-imenter to study otherwise inaccessible physical regimesthat entail relativistic and nonrelativistic phenomena.For example, the
Zitterbewegung is encoded in the spindegree of freedom, and we can associate Rabi oscillationsto the interference of positive and negative energy so-lutions. Setting the initial state | i| χ ↓ i ↔ | i| g i , theinternal degree of freedom evolves according to Eq. (18) h S z i t = − ~ ξ ξ ~ sin ω t, (26)where ω = δ √ ξ , see Eq. (16), stands for the fre-quency of the Zitterbewegung oscillations and can takeon a wide variety of measurable values.In order to simulate this dynamics in an ion-trap table-top experiment, the ion must be cooled down to its vi-brational ground state | i , with a current efficiency above99% [1]. To estimate the observable (26), one can makeuse of the powerful tool called electron shelving, where h S z i t = ~ P e ( t ) −
1] (27)can be obtained through the measurement of the proba-bility of obtaining the ionic excited state P e ( t ) with ex-traordinary precision.Another fundamental result of the JC model which canbe mapped straightforward to the Dirac oscillator is theexistence of collapses and revivals in the atomic popu-lation, which is claimed to be a direct evidence of thequantization of the electromagnetic field. To producethis effect an initial state | z i| g i is required, where | z i isan initial circular coherent state, | Ψ(0) i = e −| z | / ∞ X n l =0 z n l √ n l ! | n l i| g i , (28)with z ∈ C . After an interaction time t , h S z i t = − ~ ~ ∞ X n l =0 ξ ( n l + 1) | z | n l e −| z | [1 + 4 ξ ( n l + 1)] n l ! sin ( ω n l +1 t ) . (29)This expression can be understood as an interference ef-fect of terms with different frequencies ω n l +1 leading tocollapses and revivals. A novel feature of the Dirac oscil-lator is the appearance of these collapses and revivals inthe orbital circular motion of the particle, reflected in h L z i t = − ~ | z | − ~ ∞ X n l =0 ξ ( n l + 1) | z | n l e −| z | [1 + 4 ξ ( n l + 1)] n l ! sin ( ω n l +1 t ) . (30)The generation of an initial circular coherent state willrequire two sequential applications of the technique de-scribed in Ref. [1] on an initial motional ground state.These two operations should be applied with a relativephase such that D l ( z ) = D x ( z ) D y ( − i z ), where D j ( z ) =e za † j − z ∗ a j , j = x, y . The observable of Eq. (29) can be measured via a similar electron-shelving technique, whilethe observable of Eq. (30) can be measured via the map-ping of the collective motional state onto the internaldegree of freedom [1].It is worth mentioning that the chiral partner of the 2DDirac oscillator Hamiltonian (3) can be obtained throughthe substitution ω → − ω , and consists on right-handedquanta. This Hamiltonian presents similar features asthose discussed above, and can be exactly mapped ontoan anti-Jaynes-Cummings interaction H = ~ ( ga r σ − + g ∗ a † r σ + ) + mc σ z , (31)with similar parameters. It is precisely this chiralitywhich allows an exact mapping between the JC, AJC,and the lefthanded and righthanded 2D Dirac oscillator.This essential property, missing in the 3D case, forbidsan exact mapping of Eq. (1) onto a JC-like Hamiltonian.In conclusion, we have demonstrated the exact map-ping of the 2+1 Dirac oscillator onto a Jaynes-Cummingsmodel, allowing an interplay between relativistic quan-tum mechanics and quantum optics. We gave two rel-evant examples: the Zitterbewegung and collapse-revivaldynamics. In addition, we showed that the implementa-tion of a 2D Dirac oscillator in a single trapped ion, withall analogies and measured observables, is at reach withcurrent technology.A.B. and M.A.MD. aknowledge DGS grant under con-tract BFM2003-05316-C02-01 , and CAM-UCM grantunder ref. 910758. E.S. acknowledges finnancial supportof EuroSQIP and DFG SFB 631 projects. [1] D. Leibfried, R. Blatt, C. Monroe, D. Wineland, Rev.Mod. Phys. , 281 (2003).[2] D. Ito, K. Mori, E. Carrieri, N. Cimento
51 A , 1119,(1967).[3] M. Moshinsky, A. Szcepaniak, J. Phys. A , L817,(1989).[4] W. Greiner, “Relativistic Quantum Mechanics: WaveEquations” , (Springer, Berlin, 2000).[5] L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys.Rev. Lett. , 4643 (2000).[6] P. M. Alsing, J. P. Dowling, and G. J. Milburn, Phys.Rev. Lett. , 220401 (2005).[7] J. Schliemann, D. Loss, and R. M. Westervelt, Phys. Rev.Lett. , 206801 (2005).[8] L. Lamata, J. Le´on, T. Sch¨atz, and E. Solano,quant-ph/0701208.[9] E. T. Jaynes, F. W. Cummings, Proc. IEEE , 89(1963).[10] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod.Phys.73