Table Of ContentTime and spatial parity operations with trapped ions
Xiao-Hang Cheng,1,2 Unai Alvarez-Rodriguez,2 Lucas Lamata,2 Xi Chen,1 and Enrique Solano2,3
1Department of Physics, Shanghai University, 200444 Shanghai, People’s Republic of China
2Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
3IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain
(Dated: August 26, 2015)
We propose a physical implementation of time and spatial parity transformations, as well as
Galilean boosts, in a trapped-ion quantum simulator. By embedding the simulated model into an
enlargedsimulatingHilbertspace,thesefundamentalsymmetryoperationscanbefullyrealizedand
measured with ion traps. We illustrate our proposal with analytical and numerical techniques of
prototypical examples with state-of-the-art trapped-ion platforms. These results pave the way for
the realization of time and spatial parity transformations in other models and quantum platforms.
5
1
PACSnumbers: 03.67.Ac,37.10.Ty,03.30.+p,03.65.Pm
0
2
g I. INTRODUCTION
u
A
Inthelastdecade,observingquantumphenomenathat
5 aredifficultorevenimpossibletodetectinthelaboratory
2 has been possible through the concept of quantum simu-
lation [1]. Originally an idea of Richard Feynman [2], it
]
h is based on implementing a complex quantum dynamics
p on a controllable quantum system. Many proposals and
- experimentsonquantumsimulationsindifferentcontrol- |n>
t
n lableplatformssuchastrappedions[3–5],superconduct-
a ing circuits [6–8], ultracold gases [9, 10], quantum pho- |e1> ⊗ |e2> ⊗ |2>
u
tonics systems [11, 12], and optical lattices [13, 14], have |g1> |g2>
q |1>
been performed and have led to a deeper understanding
[
of a wide variety of phenomena. |0>
3
Up to now, proposed models and experimental real-
v
izations of quantum simulations with trapped ions have
6
3 been realized in spin models [15–17], quantum field the- FIG.1. (coloronline)Schemeoftheproposedexperimentfor
8 ories [18], quantum phase transitions [19], many-body theimplementationoftime,spatial,andGalileantransforma-
7 systems [20–22], fermionic and bosonic interactions [23], tions. Two ions are needed for the time parity and Galilean
0 relativistic quantum physics, including Dirac equation boost, while a single ion suffices for the spatial parity.
.
1 Zitterbewegung, [24] and its realization in the labora-
0 tory [25], Klein paradox [26, 27], and interacting Dirac
5 particles [28], among others. Recently, an implemen- properties behind the theoretical protocol for Galilean
1 tation of the Majorana equation and unphysical quan- transformations, here we propose a realistic implemen-
:
v tum operations was proposed [29], and experimentally tation in trapped-ion systems. Moreover, we develop
i realized [30, 31]. In addition, U. Alvarez-Rodriguez et a toolbox for trapped ions that can be used to imple-
X
al. [32] have developed a mathematical formulation of ment reference frame transformations for any given dy-
r
a an enlarged space or embedding space to perform linear namical equations. New interesting types of simulations
transformationsbetweenspace-timecoordinatesinagen- can emerge by adding the reference frame transforma-
eralquantumsimulator. However,theimplementationof tion to the toolbox of possible operations. This work
theseconceptsinatrapped-ionsimulator, includingpar- significantly advances the field of quantum simulations
ity P operations, has not yet been analyzed. of unphysical operations and establishes a path for im-
In this Letter, we propose the realization of time and plementing time and spatial parities in quantum optics
spatial parity operations, as well as Galilean boosts, in a systems. As a further scope, these results may allow us
trapped-ion quantum simulator. We perform analytical to enhance our capabilities when studying many-body
and numerical calculations in paradigmatic examples to interacting systems and their symmetries.
illustrateourprotocol, whichisbasedonstate-of-the-art The formalism introduced in Ref. [32] allows us to
trapped-iontechnologies. Weshowthatthisproposal,in- implement the quantum simulation of reference frame
cluding state initialization, dynamics, and measurement, transformations in the lab. This symmetry transforma-
can be efficiently implemented in current experiments. tion is described by a linear relation between the initial
While Ref. [32] focuses on the underlying mathematical (t,x) and the final (t(cid:48),x(cid:48)) coordinates, x(cid:48) = (cid:80) α x ,
i ij ij j
2
i,j = 0,1. The spinor in the enlarged space is de- The Hamiltonian in the enlarged space, H = σx⊗H =
1
fined as Ψ(x,t) = (ψe,ψo)T, where the even and odd σxcpˆ, where σx is the Pauli operator acting on ion 1,
1 1
part of any wave function can be expressed as ψe,o = which can be realized by implementing a blue- and red-
1[ψ(x,t)±ψ(x(cid:48),t(cid:48))]. Therefore, the dynamical informa- sideband simultaneously [3, 24],
2
tionofψ(x,t)andψ(x(cid:48),t(cid:48))isencodedintheevolutionof
Ψ. Moreover,throughajudiciouschoiceofmeasurement H=ηΩ˜(σ+a†eiφb+σ−ae−iφb)+ηΩ˜(σ+aeiφr+σ−a†e−iφr),
1 1 1 1
observables, one can perform a reference frame transfor- (3)
mationviaalocalσz operator,orevenobservespacetime with proper phases for blue- and red-sideband φ =π/2,
b
correlation functions between different reference frames. φ = −π/2. Here, ηΩ˜ = c and i(a†−a)/2 = pˆ∆ with
r 2∆
Throughout this paper we consider the evolution in the (cid:112)
∆ = 1/2mν. We depict in Fig. 1, a scheme of the
simulatedHilbertspaceasgivenbytheSchr¨odingerequa-
experimentalsetupwithtwoionsinteractingwithlasers.
tioni∂ ψ =−ic∂ ψ,wherecisasimulatedspeedoflight
t x The initial state in the enlarged space is given as,
and we fix (cid:126) = 1. The corresponding equation for Ψ in
the embedding space, for arbitrary Galilean transforma- (cid:18)1(cid:19)
Ψ(x,t=0)= ⊗ψ(x,t=0), (4)
tions, may be written as 0
i∂ Ψ=−i(α˜ 1+α˜ σx)∂ Ψ, (1)
t 1 2 x where ψ(x,t = 0) can be described as a Gaussian
where α˜1,2 = [c(α11 ±α00)∓α10]/(2α11). We explain w(cid:112)av√e packet, ψ(x,t = 0) = ψ0(x,t = 0)eip0x =
now how to use this representation for the implementa- ( 2π∆)−1e−x2/4∆2eip0x. In a trapped-ion setup, this
tioninatrapped-ionsystem. IntheLamb-Dickeregime, can be achieved by cooling the motional mode to the
(cid:112)
η (cid:104)(a+a†)2(cid:105) (cid:28) 1, the Hamiltonian describing the in- ground state, which is a Gaussian, and displacing it by
teraction between an ion and a laser driving is [3] simultaneous red and blue sidebands with the Hamilto-
nian p xˆσx, where xˆ = ∆(a+a†), with different phases
H(t)=Ω0σ+[1+iη(ae−iνt+a†eiνt)]ei(φ−δt)+H.c., in the0a an2d a† operators as compared to the simulating
(cid:112) HamiltonianH. Thiswillallowonetoachieveanaverage
where δ =ω−ω is the laser detuning, η =k 1/2mν is
0 p momentum, using the auxiliary second ion initialized
the Lamb-Dicke parameter [3], k is the wave number of 0
in an eigenstate of the σx operator. After applying the
theexternalfield,misthemassofion,ν isthefrequency 2
evolutionpropagatorexp(−iHt), wecanevolvethestate
of a static potential harmonic oscillator, Ω is the cou-
0 for any time. The solution reads
plingstrength,aanda† aretheannihilationandcreation
operators of any suitable vibrational mode of the ion 1
string, that we choose to be the center of mass motional Ψ(x,t)= (cid:112)√ (5)
2 2π∆
mode, and φ is the field phase. In spin-1/2 language,
Hσ+aWm=hile|teon(cid:105)n(cid:104)δiga|=n=0H,(σax=c+aΩrir(σiσeyr+)e/rei2φs,o+σn−aσn−=cee|−gci(cid:105)aφ(cid:104)n)e.|b=Ae o(rbσedtxa-−sinideieσdbywa)/nit2dh., ×−ee−−(c(4tc+∆4t+∆x2x)22)2eeipip00(c(tc+t+xx))++ee−−(c(t4c−∆t4−∆x2x)22)2ee−−ipip00(c(tc−t−xx)) .
c
also known as Jaynes-Cummings (JC) interaction, is re-
Furthermore,thequantumstatesinthesimulatedspaces
alisedinthecaseofδ =−ν. ThisHamiltonianiswritten
as Hr = Ω˜η(aσ+eiφr +a†σ−e−iφr). Respectively, when are obtained reversing the initial mapping,
iδnt=eraνc,tioanbcluane-sbideeabcahniedv,edan,tain-JdayitnsesH-Camumiltmoniniagns c(aAnJCbe) ψ(x,t)=(1,1)Ψ(x,t)= (cid:112)√1 e−(ct4−∆x2)2e−ip0(ct−x),
2π∆
etrxaptreesnsoewd hasowHtbo=apΩ˜pηly(at†hσe+seeitφebc+hnaiqσu−ees−tioφbg)e.nWeraeteilltuhse- ψ(x,−t)=(1,1)σzΨ(x,t)= (cid:112)√1 e−(c4t+∆x2)2eip0(ct+x).
time and spatial parity transformations in trapped ions. 2π∆
(6)
II. TIME PARITY TRANSFORMATION WeplotinFig.2(a)theinitialwavepacketinthesimu-
latedspace,andinFig.2(b)theevolvedandtime-parity-
As a first example, we show how to use two trapped transformed wavepackets in Eq. (6). We calculate now
ions to simulate a time parity transformation, (t,x) → thepositionaveragevaluesinthesimulatedspaceforthe
(−t,x), (α ,α ,α ,α ) = (−1,0,0,1). Here, we different reference frames and their correlation,
00 01 10 11
choosetheHamiltonianinthesimulatedspaceasatime-
independent one, H = He = cp and the momentum (cid:104)xˆ(cid:105)ψ(x,t) =ct,(cid:104)xˆ(cid:105)ψ(x,−t) =−ct,(cid:104)xˆ(cid:105)ψ(x,t),ψ(x,−t) =0.
(7)
p > 0, which describes a massless Dirac Hamiltonian
The spacetime correlations may be measured in dif-
without the internal degree of freedom. The correspond-
ferent ways with current ion technology. The one we
ingone-dimensionalSchr¨odingerequationintheenlarged
introduce here extends the physical principle employed
space can be expressed as
in [25]. The measurement is performed upon the σz
1
i∂ Ψ=σxcpˆΨ. (2) observable of the first ion, associated with the enlarged
t 1
3
space degree of freedom, via fluorescence detection. To 0.4 0.4
achieve this, a state-dependent displacement operator (a) (b)
0.3 t = 0 Μs0.3 t = 2 Μs
U = exp(−ikxˆσx/2) is applied to the internal state 2 2
coofst(hkixˆs)σiozn+asnind(kt1xˆh)eσyjoi[n2t5].mIondeo,rdwerithtoAdet=ectU,†eσ.g1z.U, th=e #!"xΨ00..12 #!"xΨ00..12
1 1
spacetime correlation (cid:104)xˆ(cid:105)ψ(x,t),ψ(x,−t) in the simulated 0.0 0.0
space, one should measure (cid:104)xˆ(σ1x + 1)σ1z(cid:105)Ψ(x,t) follow- 0."410 "5 0 5 100."410 "5 0 5 10
ing Eq. (6). Here, the operator (σ1x +1)σ1z acts on the 0.3 (c) x!#" t = 2 Μs0.3 (d) x!#"
qubit degree of freedom of the enlarged space, and the 2 2
measurement can be decomposed into two parts, one for "x 0.2 x" 0.2
! !
each summand, (cid:104)xˆσ1z(cid:105)Ψ(x,t), and −i(cid:104)xˆσ1y(cid:105)Ψ(x,t). These #Ψ0.1 #Ψ0.1
twomeasurementscanbeobtainedfromthederivativeof 0.0 0.0
theAobservablewithrespecttok inthelimitk(cid:104)xˆ(cid:105)(cid:28)1, "-1100 "5 0 5 1100!-1100 !5 0 5 10
inwhich∂ (cid:104)A(cid:105)≈(cid:104)xˆσy(cid:105), withalocalrotationinthefirst x!#" x!""
k 1
case to change σy into σz. Moreover, computing (cid:104)A(cid:105) for
1 1
sizable k, for the cases of initial σ1z and σ1y eigenstates FIG. 2. (color online) (a) Probability distribution |ψ(x)|2
in the internal state, allows one to obtain cos(kxˆ) and at time t = 0. Probability distribution |ψ(x)|2 at time t =
sin(kxˆ). Via Fourier transform, we can obtain the posi- 2×103∆ with c = 10−3 and v/c = 0.8 for time parity (b),
tion wavepacket probability distribution. The previous spatial parity (c), and Galilean Boost (d). Dashed blue lines
procedure enables us, among other things, to compute are time evolutions and solid red ones are the corresponding
spacetimecorrelationfunctionswithoutfulltomography, Galilean group transformations.
which can reduce significantly the required resources.
the even and odd components of the spinor Ψ.
III. SPATIAL PARITY TRANSFORMATION ψ(x,t)= (cid:112)√1 e−(c4t∆−2x)2e−ip0(ct−x), (10)
2π∆
The second case we consider is the simulation of
a spatial parity transformation, (t,x) → (t,−x), ψ(−x,t)= (cid:112)√1 e−(c4t∆+2x)2e−ip0(ct+x). (11)
(α ,α ,α ,α ) = (1,0,0,−1). The initial state 2π∆
00 01 10 11
in the simulated space coincides with the one of time
(cid:112)√ We plot in Fig. 2(c) the evolved and spatial-parity-
parity, ψ(x,t = 0) = ( 2π∆)−1e−x2/4∆2eip0x. So transformed wavepackets in Eqs. (10) and (11). We also
does the Hamiltonian in the simulated space, H = obtain the expectation values in each of the correlations.
He = cpˆ. It is obvious that ψ(−x,t = 0) =
((cid:112)√2π∆)−1e−x2/4∆2e−ip0x. TheHamiltonianintheen- (cid:104)xˆ(cid:105)ψ(x,t) =ct, (cid:104)xˆ(cid:105)ψ(−x,t) =−ct,
larged space H goes to σx⊗He, and the initial spinor in
1 (cid:18) (cid:19)
the enlarged space can be expressed as 1 −1
(cid:104)xˆ(cid:105) =(cid:104)Ψ(x,t)| xˆ|Ψ(x,t)(cid:105)
ψ(x,t),ψ(−x,t) 1 −1
Ψ(x,0)= 2(cid:112)√12π∆e−4x∆22 (cid:18)eeiipp00xx+−ee−−iipp00xx (cid:19). (8) ==(cid:104)(cid:104)ΨΨ((xx,,tt))||σσ1zzxxˆˆ||ΨΨ((xx,,tt))(cid:105)(cid:105)−−ii(cid:104)(cid:104)ΨΦ((xx,,tt))||σσz1yxˆxˆ||ΦΨ((xx,,tt))(cid:105)(cid:105)
1 1
This can be achieved by initializing the internal state =−2ip0∆2e−(2c∆t)22e−2p20∆2, (12)
associated with the enlarged space in the (1,0)T state,
cooling the motional mode to the ground state, which is where Φ(x,t)=e−iπσ1x/4Ψ(x,t). These can be measured
a Gaussian, and performing a conditional displacement asinthepreviousexample. Wepointoutthatthespatial
of the motional state with the Hamiltonian p xˆσx. We paritycasecanbedistinguishedfromthetimeparitycase
0 1
point out that for spatial parity one ion suffices for both through the spacetime correlation, which is different in
state initialization and simulation. both cases. The computation of these correlations with
Accordingly, the state in the enlarged space consider- our method makes full state tomography unnecessary.
ing time-evolution results,
1
IV. GALILEAN BOOST
Ψ(x,t)= (cid:112)√ (9)
2 2π∆
×ee−−((cctt44−−∆∆xx22))22ee−−iipp00((cctt−−xx))+−ee−−((cc44tt++∆∆xx22))22ee−−iipp00((cctt++xx)) . cvht)aH,ne(grαee0,0aw,sαseo0c1pi,raαotpe10do,sαwe1itt1hh)e=asGi(m1a,ul0ill,ae−tainovn,b1oo),ofswath,re(ertfe,exrve)ni→scet(hfter,axrme−le-
ativevelocity[33]. Here,weconsiderthepreviousHamil-
Asinthepreviouscase,inordertorecoverthewavefunc- tonian and initial state in the simulated space. The cor-
tion in each of the simulated spaces we add or subtract respondingHamiltonianinthesimulatingenlargedspace
4
reads
111
(cid:16) v(cid:17) v
H(t)= c+ 1pˆ− σxpˆ. (13)
2 2
(cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125)
000...999666
H1(t) H2(t)
Moreover, we can calculate the initial spinor state and FFF
computethetimeevolution. Theexpressionforthequan- 000...999222
tum states in the simulated spaces reads,
ψ(x,t)= (cid:112)√1 e−(ct4−∆x2)2e−ip0(ct−x), (14) 000...888888 (a)
2π∆
000 000...238033833333333333
ψ(x−vt,t)= (cid:112)√1 e−(ct−4x∆+2vt)2e−ip0(ct−x+vt),(15) 111 ηηηΩΩΩttt
2π∆
000...999666
We plot in Fig. 2(d) the evolution of the wavepackets
with and without Galilean boost in Eqs. (14) and (15).
Moreover, the expectation values for position xˆ are FFF
000...999222
(cid:104)xˆ(cid:105) =ct, (cid:104)xˆ(cid:105) =(c+v)t, (16)
ψ(x,t) ψ(x−vt,t)
(cid:104)xˆ(cid:105)ψ(x,t),ψ(x−vt,t) = 12t(2c+v)e−t82∆v22e−ip0tv. (17) 000...888888 (b)
Forthetrapped-ionsimulationtheinitializationofthe 111000 000...238033833333333333
spinor can be done similarly to the time parity case. For ηηηΩΩΩttt
the subsequent dynamics we divide the Hamiltonian in
000...999666
Eq. (13) into two parts to implement its evolution in the
trapped-ion system. To realize H in the laboratory, we
1
FFF
propose to use a second auxiliary ion initialized in an
000...999222
eigenstate of σx,
2
(cid:16) v(cid:17) (cid:16) v(cid:17)
H |Ψ(cid:105)|+(cid:105)= c+ 1pˆ|Ψ(cid:105)|+(cid:105)≡ c+ σxpˆ|Ψ(cid:105)|+(cid:105). 000...888888 (c)
1 2 2 2
(18)
000 000...2380330833.5333333333 1
Then, the Hamiltonian can be implemented as ~
ηηηΩΩΩttt
(cid:16) v(cid:17)
H(cid:48) =σxpˆ c+ =iηΩ˜ σx(a†−a), (19)
1 2 2 1 2
with ηΩ˜ = (c+ v)/2∆. Moreover, the second term in FIG. 3. (color online) Fidelity F =Tr[ρ|ψI(cid:105)(cid:104)ψI|] of (a) time
1 2 parity, (b) spatial parity, and (c) Galilean boost with Ω˜ =
Eq. (13) can be realized as
0.01ν(red,upper),Ω˜ =0.025ν(black,middle)andΩ˜ =0.04ν
v (purple, lower). ρ denotes the final state after initialization
H =− σxpˆ=iηΩ˜ σx(a†−a), (20)
2 2 1 2 1 and dynamics, evolved with Eq. (21), and |ψI(cid:105) denotes the
idealevolvedstateinabsenceofimperfections. Wepointout
andwithηΩ˜ =− v ,throughsimultaneousredandblue that Ω˜ in the Galilean boost case (c) equals Ω˜1 in Eq. (19).
2 4∆ WeconsidertheparametersoftrappedCa+ioninsomeofthe
sideband excitations upon the first ion.
Innsbruckexperiments[25]η=0.06andrealisticdecoherence
rates [4] Γ =Γ =Γ =3.7×10−7ν and Γ =6.2×10−7ν.
h c − φ
V. DISCUSSION
Toanalyzetherobustnessofthesimulatingsystem,we where the Lindblad superoperators are L(Xˆ)ρ =
computedthedynamicswithamasterequationincluding (2XˆρXˆ†−Xˆ†Xˆρ−ρXˆ†Xˆ)/2. Here, H is the trapped-
T
differentdecoherencesources. Weconsideredunintended ion Hamiltonian corresponding to each of the three
carrier transitions due to off-resonant coupling, heating cases analyzed, namely, time parity, spatial parity, and
Γ ,phononlossΓ ,dephasingΓ ,andspontaneousemis- Galilean boost. We include carrier and counterrotating
h c φ
sion Γ , sideband terms in the dynamics, i.e., without perform-
−
ing vibrational rotating-wave approximation. Therefore,
ρ˙ =−i[HT,ρ]+ΓhL(a†)ρ+ΓcL(a)ρ thismasterequationaccountsforallthesignificantdeco-
+Γ L(σz)ρ+Γ L(σ−)ρ, (21) herence and error sources present in current trapped-ion
φ −
5
experiments.We plot in Figs. 3(a)-(c) and Figs. 4(a)-(c)
111 the fidelities of trapped calcium and beryllium ions af-
ter state initialization and evolution with the dynamics
in Eq. (21) for the cases of time parity, spatial parity,
000...999666 and Galilean boost. For the state initialization part, we
compute the dynamics with an equivalent master equa-
FFF tion for the corrresponding initialization Hamiltonian as
000...999222
described for each case (a)-(c) in the text. We consid-
ered for the initialization time p ∆ = ηΩ˜t = 1 in all
0
000...888888 (a) cases. We point out that because of the relatively small
atomic mass of beryllium, large trap frequency, sizable
Lamb-Dicke factor and the corresponding large decoher-
000 000..00.000164661666666666777
111 ence rates are introduced in the realistic experiments of
ηηηΩΩΩttt
NIST [4]. Our work shows a significant feasibility in dif-
ferent trapped-ion setups.
000...999666
FFF
000...999222 VI. CONCLUSIONS
To summarize, we have proposed the physical imple-
000...888888 (b) mentation of fundamental symmetry transformations in-
cluding time and spatial parity with trapped ions. The
000 000..00.000164661666666666777 formalism permits as well to perform Galilean boosts
111
ηηηΩΩΩttt with the same technology. By embedding the simulated
physical system into an enlarged Hilbert space living in
the trapped-ion system, the proposed formalism can be
000...999666
carried out with current ion-trap setups. Furthermore,
our work establishes a path for the realization of parity
FFF
000...999222 transformations in other quantum platforms and many-
body interacting models.
000...888888 (c)
VII. ACKNOWLEDGEMENTS
000 000..00.00010646.615666666666777 1
~
ηηηΩΩΩttt
We acknowledge funding from National Natural Sci-
ence Foundation of China (61176118, 11474193), Shang-
haiPujiangProgram(13PJ1403000),ShuguangProgram
FIG. 4. (color online) Fidelity of (a) time parity, (b) spatial (14SG35), Program for Eastern Scholar, Specialized Re-
parity, and (c) Galilean boost with Ω˜ = 0.01ν (red, upper), search Fund for the Doctoral Program of Higher Edu-
Ω˜ =0.025ν (black, middle) and Ω˜ =0.04ν (purple, lower),by cation (2013310811003), Basque Government IT472-10
considering trapped Be+ ion as an example. We use the pa- and BFI-2012-322, Spanish MINECO FIS2012-36673-
rameters in some of the NIST experiments [4]: Lamb-Dicke C03-02, Ram´on y Cajal RYC-2012-11391, UPV/EHU
factorη=0.3,decoherenceratesΓh =Γc =Γ− =3.7×10−7ν EHUA14/04, UPV/EHU Grant No. UFI 272 11/55,
and Γ =6.2×10−7ν.
φ PROMISCE, and SCALEQIT EU projects.
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form Galilean group transformations on nonrelativistic
dynamics of massive particles.
