Parity-relevant Zitterbewegung and quantum simulation by a single trapped ion

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by  Kunling Wang

Parity-relevant Zitterbewegung and quantum simulation by a single trapped ion
Kunling Wang1,2, Tao Liu3, Mang Feng1∗, Wanli Yang1 and Kelin Wang4
1 State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, China
2 Graduate School of the Chinese Academy of Sciences, Beijing 100049, China
3 The School of Science, Southwest University of Science and Technology, Mianyang 621010, China
4 The Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
Zitterbewegung (ZB), the trembling of free relativistic electrons in a vacuum could be simulated
byasingletrappedion. WefocusonthevariationsofZBunderdifferentparityconditionsandfind
no ZB in the case of odd or even parity. ZB occurs only for admixture of the odd and even parity
states. WealsoshowthesimilarroleplayedbytheparityoperatorforthetrappedioninFock-state
representationandthespaceinversionoperatorforarealisticrelativisticelectron. AlthoughtheZB
1
effectisinvisibleinarelativisticelectron,preparationofthetrappedionindifferentparitystatesis
1
asophisticated job,whichmakesitpossible toobservetheparityrelevantZBeffectswithcurrently
0
available techniques.
2

n PACSnumbers: 31.30.J-,42.50.Dv,03.65.Pm
a
J

4 SincethediscoverybySchro¨dinger[1],Zitterbewegung eration of the origin of the ZB, i.e., the interference be-
(ZB), i.e., the trembling of free relativistic electrons, has tweenthe positive-andnegative-energycomponents,the
]
h drawn more and more attention and interests over past ZBisalsorelevanttoparityofthestates. Tounderstand
p years[2–8]. Ithasbeengenerallybelievedthatthe trem- therelevantphysics,wewillcomparetheparityoperator
- bling of a relativistic electron is resulted by the interfer- of the trapped ion with the space inversion operator of
t
n ence between negative and positive energy components, the realistic relativistic electron. The experimental fea-
a a typical relativistic feature of the Dirac electron. Up sibility to observe the parity-relevant ZB effects will be
u
to now, however, no direct observation of ZB has been justified.
q
achieved due to inaccessibility with current experimen- Under the radiation of three laser lights with red-
[
tal techniques, which led to some questioning on the ZB detuning, blue-detuning, and carrier transition, respec-

2 [6,9]. Moreover,therehavebeenalternativeexplanations tively, the interacting Hamiltonian of a single trapped
v for the origin of the ZB, such as the continuously vir- ion reads [10, 16]
4
tual transition process between different internal states
1 inviewofquantumfieldtheory[8]ortherelevancetothe H =i~ηΩ(a+ a)σˆx+~Ωσz, (1)
6 −
complexphasefactorincontextofspace-timealgebra[3].
5 where η is the Lamb-Dicke parameter, Ω and Ω are, re-
e
1. By quantum simulation, some relativistic effects have spectively, the effective Rabi frequency and the effec-
1 recently been demonstrated in some controllable physi- tive Larmor frequency of the ion. a+e(a) is the cre-
0 cal systems, such as graphene, semiconductor,supercon- ation (annihilation) operator of the quantized motion of
1 ductor and trapped ion [10–18]. It was recently consid- the ion. σ and σ are usual Pauli operators. Defining
x z
v: ered that the ZB occurs not only under the relativistic p = i~(a+ a)/2∆ with ∆ the size of the ground state
i condition, but also extensively in the dynamics of a sys- wavefuncti−on,wemayrewritetheHamiltonianasaform
X
tem with more than one degree of freedom [14, 19]. The analogous to a 1+1 dimensional Dirac equation
r demonstrationof the ZB beyond the relativistic electron
a helps us to further understand the Dirac equation and HD =2η∆Ωpσx+~Ωσz (2)
the relevant relativistic phenomena.
provided c:=2η∆Ω and mc2 :=~Ω.
Inthiswork,wefocusontheroleofparityplayedinthe e
Introducing the parity operator
simulationofDiracequationforthe ZBeffectbyasingle
e
trappedion. Likein[10,16],wealsoemploythemotional Πˆ =eiπ(a+a−21+21σz) (3)
degreesoffreedomoftheiontosimulatethepositionand
momentum of the relativistic electron, and the internal commuting with the Hamiltonian in Eq. (1), which
degrees of freedom to refer to the energy states. Since means Πˆ is a conserved quantity under the Hamiltonian
themotionalstateoftheioncouldbequantized,wemay Hˆ,westudy belowthe dynamicsofthe trappedionwith
discusstheprobleminnumber-staterepresentation. The different parity states, i.e., of the odd or even parity, or
keypointofourworkistointroduceaparityoperatorΠˆ, of admixture of the both.
by which we show that, besides the conventionalconsid- The ZB is relatedto the averageposition x(t) of the
h i
trappedion. Sinceourinterestisinthestatesundersome
parityconditions,wehavetofindthecommoneigenfunc-
tionsofHˆ andΠˆ. Tothisend,wedefine p astheeigen-
∗E-mail: [email protected] function of (a+ a) by assuming (a+ a|)ip = ip p .
− − | i − | i
2
Straightforwarddeduction yields SubstitutingEqs. (7),(8),(11)and(12)intoEq. (5),we
may write down the co-eigenstate of E and odd parity
p = 1 e−p42 ∞ inHn(p) n , (4) to be, +
| i √42π √n! | i
n=0
X 1
whereHn(x)isaHermitepolynomialdefinedbyHn(x)= ψEo+ =W(p)( SE+(p) + SE+(−p) )⊗ √2(|pi−|−pi)
t(h−e1)snteaxte2/s2idndxnn{e|p−ix}2/a2re[2o0r]t.honIotrmcoaulldandbecopmropvleetne,tahnadt (cid:12)(cid:12)(cid:12) +EQW′(p)( S(cid:12)(cid:12)E+(p) (cid:11)− S(cid:12)(cid:12)E+(−p) (cid:11))⊗(√12(|pi+|−pi)
each state p corresponds to the momentum p =
~p/(2∆). | i mom (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11)
Substituting Eq.(4) into Eq.(1) and diagonalizing Hˆ where Q is a coefficient to be determined. We only keep
yields the eigenvalues some reasonable terms by setting Q = W(p)/W′(p),
i.e., elimination of the terms of S (p) p and
E± =± (~Ω)2+(η~Ωp)2 SE+(−p) |pi. Then we have, (cid:12) E+ (cid:11)|− i
q (cid:12)
and the eigenstates e (cid:12)(cid:12) ψ(cid:11)o = 1 ψ (p) ψ ( p) . (13)
η~Ωp T E+ √2 E+ − E+ −
(cid:12)SE+(p)(cid:11)=N(p)"1,|E+|+e ~Ω# , Similarly,(cid:12)(cid:12)(cid:12) we Ehave oth(cid:2)e(cid:12)(cid:12)r co-eig(cid:11)enst(cid:12)(cid:12)ates of E+(cid:11)(cid:3)and even
(cid:12) η~Ωp T parity, E− and different parities [21],
S (p) =N(p) − ,1 ,
E− "|E−|+~Ω # ψe = 1 [ψ (p) + ψ ( p) ], (14)
with N(p(cid:12)(cid:12)) = (cid:11)(E +~Ω)/2 Ee the normalization E+ √2 | E+ i | E+ − i
| ±| | ±| (cid:12) (cid:11) 1
factor. So the tpotal eigenfunctions of Hˆ are ψE+(p) = (cid:12)ψEo− = √2[|ψE−(p)i+|ψE−(−p)i], (15)
S (p) p and ψ (p) = S (p) p .
E+ ⊗| i E− E− ⊗|(cid:12)i (cid:11) (cid:12) (cid:11) 1
(cid:12) Under(cid:11)theparityo(cid:12)perator(cid:11)Πˆ,w(cid:12) eassum(cid:11)efol(cid:12)lowingpar- (cid:12)ψEe− = √2[|ψE−(p)i−|ψE−(−p)i]. (16)
i(cid:12)ty states in the num(cid:12)ber-state re(cid:12)presentation,
(cid:12) (cid:11)
(cid:12)
Based on these eigenstates, the average position of
o = + f 2m+1 + f 2m , (5)
| i | i 2m+1| i |−i 2m| i the ion x(t) = (aˆ++a)∆ can be calculated by the
m m h i h i
X X evolved states with odd or even parity, where
e = + f 2m + f 2m+1 , (6)
2m 2m+1
| i | i | i |−i | i
where f (kXm= 2m, 2m + 1, mXm= 0,1,2, ) are nor- |ψo(t)i= ap ψEo+(p) e−i|E~+|t + bp ψEo−(p) ei|E~−|t
malizedkcoefficients. o stands for odd·p··arity state Xp (cid:12)(cid:12) E Xp (cid:12)(cid:12) E
due to Πˆ|oi = −|oi an|di|ei for even parity state with |ψe(t)i= ap (cid:12)ψEe+(p) e−i|E~+|t + bp (cid:12)ψEe−(p) ei|E~−|t,
Πˆ e = e . are the eigenstates of σz with eigenval- Xp (cid:12) E Xp (cid:12) E
| i | i |±i (cid:12) (cid:12)
ues 1, respectively. (cid:12) (cid:12)
± with a and b the coefficients determined by the initial
For our purpose,we may construct the spin states p p
|±i condition.
by positive energy eigenstates as
Tocalculate x(t) ,wefirstconsider dx/dt , theaver-
+ =W(SE+(p) + SE+( p) ), (7) age velocity, h i h i
| i | i | − i
=W′(S (p) S ( p) ), (8)
E+ E+
|−i | i−| − i dx ∆
or by negative energy eigenstates as, dt = i~ aˆ++aˆ,HˆD =2η∆Ωhσˆxi.
+ =W′(S ( p) S (p) ), (9) (cid:28) (cid:29) Dh iE
| i | E− − i−| E− i Taking the odd parity as an example, we heave
=W(S (p) + S ( p) ), (10)
E− E−
|−i | i | − i
with normalization factors W = |E±|/(2|E±|+2~Ω) hψo(t)|σˆx|ψo(t)i= a∗pap ψEo+(t) σˆx ψEo+(t)
and W′ = [|E±|(|E±|+~Ω)]/p2(η~ΩP)2. Moreover, Xp D (cid:12)(cid:12) (cid:12)(cid:12) E
theoddandqevenmotionalstatesareassociatedwiththe + b∗pbp ψEo−(t)(cid:12)σˆx(cid:12)ψEo−(t)
momentum eigenstates |pi, i.e., e Xp D (cid:12)(cid:12) (cid:12)(cid:12) E
f2m+1 2m+1 1 (p p ), (11) + a∗pbpe2i|E~±|t ψEo+(t)(cid:12)σˆx(cid:12)ψEo−(t)
m | i∝ √2 | i−|− i Xp D (cid:12) (cid:12) E
X (cid:12) (cid:12)
f2m|2mi∝ √12(|pi+|−pi). (12) + b∗pape−2i|~E±|t ψEo−(t)(cid:12)σˆx (cid:12)ψEo+(t) .(17)
Xm Xp D (cid:12)(cid:12) (cid:12)(cid:12) E
(cid:12) (cid:12)
3
It is easy to check in Eq. (17) that ψo(t) σˆ ψo(t) =0
since every term is zero. As a resulht, x(t|) xre|mainis un- 1.2 β=0,π/2 (a) 9 β=0,π/2 (b)
h i β=π/12 8 β=π/12
changed. Similarly we have ψe(t) σˆx ψe(t) =0. These 1 β=π/6 β=π/6
h | | i 7
results imply that a trapped ion in an eigenstate of Πˆ β=π/4 β=π/4
0.8
6
would be on average static.
Because Eq. (17) involves both the positive and nega- ∆) 0.6 ∆) 5
tive energy components, the ZB should occur, according 〈〉x ( 0.4 〈〉x ( 4
to the conventional viewpoint, due to their interference. 3
Our result, however, presents that the ZB depends not 0.2 2
only on the interference between the positive and nega-
0 1
tive energy components, but also on parity of the states. λ=0.6∆ λc=5.4∆
c 0
Tobemoreclarified,wehavenumericallycalculatedin −0.2
0 50 100 150 200 250 0 50 100 150 200 250
Fig. 1 the average position of the trapped ion with the t (µs) t (µs)
initial state (cosβ + +sinβ ) 0 . Since both + n
| i |−i | i | i| i
and n are states with definite parity, the case with
FIG.1: (Coloronline)Numericalsimulationofhxˆ(t)iinunits
|−i| i
α = β is of the most mixed parity. By changing the of ∆ for initial state (cosβ|+i+sinβ|−i)|0i, where |±i are
values of β, we may see clearly from the figure that the eigenstates of σz and |0i is the ground motional state. λc =
ZB occurs only in the admixture of the odd and even 2ηΩ˜∆/Ω is the Compton wavelength. We consider (a) λc =
parity eigenstates. 0.6∆, near the non-relativistic limit and (b) λc =5.4∆ close
Ourresultcouldbeunderstoodbytheviewpointin[19] to the relativistic limit. From the bottom to top, the curves
that the ZB could appear in any system, besides the rel- correspond toβ =0orπ/2, π/12, π/6 andπ/4, respectively.
ativistic system, with more than one degree of freedom.
Inourcase,theZBeffectisoriginatedfromtheinterplay
between the internal and motional degrees of freedom We have PˆϕE¯±(p¯;x¯,t) = ±ϕE¯±(−p¯;x¯,t). Under such
a consideration,we may easily find the odd parity states
of the ion. With respect to the conventional viewpoint
of interference between the positive and negative energy
1
components,wemaycheckEq. (17)againwhichinvolves ϕoE¯+ = √2[ϕE¯+(p¯)−ϕE¯+(−p¯)],
both positive and negative energy states. The internal-
1
mdiffoteiroennatl-esntaertgeyinctoermpplaoynleenatdstetromtsh.eOinntecrefetrheencioenbeitswienena ϕoE¯− = √2[ϕE¯−(p¯)+ϕE¯−(−p¯)],
certainparity state, however,the interference is destruc-
and the even parity states
tive,yielding x =0. SotheZBappearsonlyinthecase
h i
of admixture of different parity states, which allows the 1
constructive interference between different energy com- ϕeE¯+ = √2[ϕE¯+(p¯)+ϕE¯+(−p¯)],
ponents. Simply speaking, whether the ZB appears or
1
not, is decided by both the interference and symmetry, ϕeE¯− = √2[ϕE¯−(p¯)−ϕE¯−(−p¯)].
the latter of which is reflected by parity.
For a realistic relativistic electron, there is no demand
Theyareformallyconsistentwiththeco-eigenstatesEqs.
to quantize the motional freedom, but the parity opera-
(13)-(16) for the trapped ion. In this sense, we consider
tor Πˆ discussed above reminds us of the space inversion that Πˆ and Pˆ play the same role in the respective sys-
operator Pˆ defined as [22], tems. In other words, Πˆ is something like an inversion
parityoperatorinthenumberstaterepresentation,which
Pˆϕ(x¯,t)=σˆ ϕ( x¯,t), (18)
z − moves population back and forth between the internal
and motional states.
where ϕ(x¯,t) is the wavefunction of the relativistic elec-
Followingthesamestepsasforthetrappedion,wecan
tron, and we have used the overline to represent the pa-
immediatelyfind x¯ =0oftherelativisticelectronunder
rameters of the relative electron in order to distinguish h i
definite parity states. So the ZB of the relativistic elec-
from the ones of the trapped ion. For a relativistic elec-
tron, the eigenstates of Hˆ are, tron occurs when the electron is in both the admixture
D of different parity states and the admixture of different
energycomponents. Aparticle,suchasarelativisticelec-
ϕE¯+(p¯;x¯,t)=N(p¯)(cid:18)|E¯+|c1+p¯m¯c2(cid:19)eip~¯x¯−i|E¯~+|t tbreonstoartiactorraipnphedarimono,nsitcaoysicnigllaintiaonc.erOtatihneprwarisitey,astmatoevwinilgl
ϕE¯−(p¯;x¯,t)=N(p¯) |E¯−−|1+cp¯m¯c2 eip~¯x¯+i|E¯~−|t, psuarretliycleexwpietrhiednicffeertehnetZeBne.rgycomponentscoexisting will
(cid:18) (cid:19)
Since the ZB in a real relativistic electron is inacces-
where m¯ is the mass of the electron, p¯andx¯ are,respec- sible with current technique, we have to resort to other
tively,themomentumandtheposition,E¯ standforthe systems, such as the trapped ion, to observe the parity-
±
energies and N(p¯) the normalization factor. relevantZBeffect. Withdifferenttrappedion’smotional
4
states, the eigenstates of the parity operator have abun- thaninanymixedparitystates. Thisis advantageousto
dantforms,suchas n and + n n quantum gate operation.
|±i⊗| i | i⊗| iA±|−i⊗| iB
where n is the displaced coherent state [23]. With In summary, we have investigated the dynamics of a
| iA(B)
currentlyavailableion-traptechnique,ithasalreadybeen single trapped ion under some certain parity conditions.
achieved the cooling of the single ion to the motional Our study has shown that the ZB in the trapped ion is
ground state [24]. Preparation of the ion to a certain relevantnotonlytotheinterferencebetweendifferenten-
motional Fock state or a certain coherent state has also ergy components, but also to parity. To understand the
been a sophisticated job [25]. As a result, using the par- physics related to the realistic Dirac particle, we have
ity states mentioned above, we may check the variation discussedthecorrespondenceoftheparitybetweenarel-
of the ZB with respect to different parity conditions fol- ativistic electron and a trapped ion. Experimental feasi-
lowing the operations in [16]. bility of observing our results by a trapped ion has been
Besides the simulation of Dirac equation, the parity justified with currently available techniques. We argue
operator Πˆ also has application in quantum computing. thatourstudyisnotonlyhelpfultoexploretheZBeffect
Πˆ commutes not only with Hˆ, but also with other laser- itself, but alsousefulto further understandthe quantum
ioninteractionHamiltonians,suchasHˆ =σ a+σ a+, characteristic of the ultracold trapped ion.
r + −
a usually used Hamiltonian for logic gate operation [26]. TheworkisfundedbyNationalNaturalScienceFoun-
If we prepare the ion in the state + 0 or other parity dation of China under Grants No. 10974225 and No.
| i| i
states, the ion will be more steady during the operation 11004226,and by Chinese Academy of Sciences.
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