Quantum gates by reverse engineering of a Hamiltonian

0.13 MB·

by  Alan C. Santos

QuantumgatesbyreverseengineeringofaHamiltonian
Alan C. Santos1,∗
1Instituto de F´ısica, Universidade Federal Fluminense,
Av. Gal. MiltonTavaresdeSouzas/n, Gragoata´, 24210-346Nitero´i, RiodeJaneiro, Brazil
(Dated:January17,2017)
ReverseengineeringofHamiltonian(REH)fromanevolutionoperatorisanusefultechniqueforprotocols
ofquantumcontrolwithpotentialapplicationstoquantuminformationprocessing. Inthispaperweintroduce
aparticularprotocoltoperformREHandweshowasthisschemecanbeusedtoperformuniversalquantum
computing by using minimal quantum resources (such as entanglement, interactions between more than two
qubitsorauxiliaryquits). Remarkable,whilepreviousprotocolrequiresthree-qubitsinteractionsandauxiliary
quitstoimplementsuchgates,ourprotocolrequiresjusttwo-qubitinteractionsandnoadditionalresource. We
applyourresultstoshowasimplementthequantumFouriertransform(QFT)motivedbyitsapplicabilityto
Shor’salgorithm. Byusingthisapproach,wecanobtainalargeclassofHamiltoniansthatallowusimplement
7
singleandtwo-quitgatesnecessaryinsubroutinesofquantumcircuitsofquantumalgorithmsas,forexample,
1
0 theShor’salgorithm.
2

n I. INTRODUCTION required[14–17].
a
In this paper we introduce a alternative way to perform
J
Currently, protocol of quantum control with time- universal QC based on REH from evolution operators. Dif-
4
1 dependent Hamiltonians, as adiabatic passage [1], Lewis- ferently from methods developed recently for REH, our ap-
Riesenfeld invariants [2, 3], transitionless quantum driving proach is independent of the initial state of the system and,

] (TQD) [4–6] and the current propose of REH from unitary therefore,ancillaqubitsisnotnecessaryinourscheme. Inthe
h
evolutions operators [7], has play an important role in quan- sectionIIwediscussthegeneralaspectsofourapproachand
p
- tuminformationprocessing(seeRef.[8]forareviewofmany weshowasobtainasetofHamiltonianwhichallowusimple-
t applicationsofthisthreefirsttechniques). Inaddition,inthe mentthequantumgatesofaquantumcircuit.Becauseourap-
n
a lastyears,manyexperimentalandtheoreticalstudieshasbeen proachisindependentofinitialstate,whilemanyprotocolsre-
u performedinordertoanalyzetherobustnessofsuchprotocols quiresauxiliaryqubitstoperformuniversalQC,ourapproach
q againstdecoherenceeffects[9–12]. need not help of auxiliary elements to implement single and
[ Remarkable, in quantum computation (QC) and informa- controlled quantum gates. In this sense, the scheme present

2 tionthesetechniquesprovidesHamiltoniansabletosolvesat- hereisanenhancedwaytoimplementquantumgates. Inthe
v isfiability problems [13], or to introduce new hybrid models section III we illustrate the results obtained here through the
8 ofQC.Infact,throughquantumcontrolprotocolswecanob- Shor’salgorithm[19]whichmakeusesofQFTassubroutines
4 tain controlable time-dependent Hamiltonians to implement ofsuchalgorithm[20,21].
8
single and controlled quantum gates. For example, quantum
1
gates of a circuit can be implemented via adiabatic Hamilto-
0
nians [14, 15] or via its superadiabatic counterpart [16–18]. II. QUANTUMGATESBYREVERSEENGINEERINGOF
.
1 However, these schemes requires auxiliary qubits in order to AHAMILTONIAN
0
canimplementuniversalQC.Wesayuniversalinsensethat,
7
1 given an unknown input state, we should be able to imple- Let us start by considering the Schro¨dinger equation (we
: ment any single- and two-qubit quantum gate on this qubit. set(cid:126)=1throughoutthemanuscript)
v
Ingeneral,theHamiltoniansthatimplementsanadiabaticdy-
i
X namics,aswellassuperadiabaticHamiltonianandthosepro- H(t)|ψ(t)(cid:105)=i|ψ˙(t)(cid:105), (1)
videdbyREH,mustevolvesthesystemfrominitialstatethat
r
a iseigenstateoftheadiabaticHamailtonian. Thus,sometimes, suchthatthedynamicsisunitaryand, therefore, therearean
suchHamiltoniansmustbehighlydegenerate[16,17]andso operator U(t) where we can write |ψ(t)(cid:105) = U(t)|ψ(0)(cid:105). From
we can encode any unknown state in the initial state of the equation above (valid for any |ψ(0)(cid:105)), the Hamiltonian reads
system. Consequently, ancillaqubitsarenecessaries. Onthe as
otherhand,evenwhenwehaveancillaqubitsavailable,when
wewantimplementtwo-qubitquantumgatestheproblemof H(t)=iU˙(t)U†(t), (2)
manybodiesinteractionsmustbehighlight.Infact,ingeneral
weneedinteractionsbetweenthequbitsofourtargetsystem whichisawayso-knowntoobtaintheHamiltonianassociate
as well as between the target system and the ancilla qubits, with evolution operator U(t). In general, this equation is the
such that sometimes interactions of three (or more) qubits is starting point in protocols of reverse engineering in closed
quantum systems [7, 22, 23], as well as transitionless quan-
tum driving [4–6, 8]. The operator U(t) can be defined of
manydifferentway,buttherearepropertiesthatsuchoperator
∗[email protected]ff.br must satisfy, namely, the unitary condition (U(t)U†(t) = 1)
2
andU(0) = 1. Letusnowusetheseconditionstodefinethe Notice that the evolution operator is not ψ -independent,
inp
operatorU(t)as in sense that we do not need to know any information about
(cid:88) thestate|ψinp(cid:105)tousethemethod. Therefore,theHamiltonian
U(t)= eiϕn(t)|n(t)(cid:105)(cid:104)n(t)|, (3) H(t) associate with operator U(t) is also ψ -independent.
n inp
Thus, single-qubit universal operations can be performed by
where |n(t)(cid:105) constitutes an orthonormal bases for the Hilbert using this approach, where no additional quantum resource
space associated with the system and ϕn(t) are real free pa- (suchasentanglementorauxiliaryqubits,forexample)isre-
rameters.ItiseasyshowthatU(t),definedabove,satisfiesthe quired.
unitarity condition U(t)U†(t) = 1 for any set of parameters
ϕ (t), but the imposition U(0) = 1 restricts the initial values
n
of ϕn(t) and it must comply with the condition ϕn(0) = 2nπ B. Two-qubitquantumgates
forn∈Z.
Differently from others protocols [1–7], the definition of
To show that our protocol can be used to implement uni-
the operator U(t) can not be view as an operator that drives
versal QC, we must show as implement two-qubit quantum
the system from initial state |n(0)(cid:105) to |n(T)(cid:105). This subtle dif-
gatesthroughit. Bywritingatwo-qubitinputstateinitsmost
ferencecharacterizesthepresentprotocolasanalternativeto generalformas|ψ (cid:105)=a|00(cid:105)+b|01(cid:105)+c|10(cid:105)+d|11(cid:105)andby
inp,2
eachothersschemes.Moreover,asweshallsee,thisapproach
definingtheoperator
canbeusefulinsomeprotocolsofquantuminformationpro-
(cid:88)
creessosiunrgce,,sfuocrhexaasmimplpel.ementing universal QC using minimal U2(t)= k=1,2|nk,+(t)(cid:105)(cid:104)nk,+(t)|+eiϕk(t)|nk,−(t)(cid:105)(cid:104)nk,−(t)|,
(10)
whereweset(withk¯ =k−1)
A. Single-qubitgates |nk,+(t)(cid:105)=cosθk(t)|k¯0(cid:105)+eiφk(t)sinθk(t)|k¯1(cid:105), (11)
|nk,−(t)(cid:105)=−sinθk(t)|k¯0(cid:105)+eiφk(t)cosθk(t)|k¯1(cid:105), (12)
Inthissectionwewillshow, withoutadditionalresources,
whichsinglequantumgatescanbeimplementedbyusingthe with the initial conditions ϕ (0) = ϕ (0) = 2nπ (due the re-
1 2
scheme presented here. To this end, we start by considering quirement U (0) = 1). Now we have six free parameters
2
thatasingle-qubitgatecanbeviewasalineartransformation and we need set them in order to obtain the expected result
on an arbitrary quantum state |ψ (cid:105) = a|0(cid:105)+b|1(cid:105), and so let at the end of the dynamics. The operator U (t) is a gen-
inp 2
usconsiderthetransformation|ψ(t)(cid:105) = U (t)|ψ (cid:105),wherethe eraltransformoftwo-qubitstateand,byadjustingadequately
1 inp
operatorU (t)isgivenby our free parameters, U (t) can be an entangled gate or only
1 2
a composition of two independent single-qubits gates (i.e.,
U1(t)=|n+(t)(cid:105)(cid:104)n+(t)|+eiϕ(t)|n−(t)(cid:105)(cid:104)n−(t)|, (4) U2(t) = A1(t)⊗ A2(t)). In order to analyze some results by
using its most general form, we keep our discussion without
where
considersomeparticularcaseforU (t),butparticularizations
2
|n+(t)(cid:105)=cosθ(t)|0(cid:105)+eiφ(t)sinθ(t)|1(cid:105), (5) forU2(t)willbedoneinthesectionIII.
|n (t)(cid:105)=−sinθ(t)|0(cid:105)+eiφ(t)cosθ(t)|1(cid:105), (6) FromEqs. (10-12)wecanwritetheevolvedstateas
−
|ψ (cid:105)=U (t)|ψ (cid:105)
with θ(t), ϕ(t) and φ(t) being real free parameters. It is easy inp,2 2 inp,2
show that the conditions U (t)U†(t) = 1 and U (0) = 1 are =α(t)|00(cid:105)+β(t)|10(cid:105)+γ(t)|01(cid:105)+δ(t)|11(cid:105) (13)
1 1 1
satisfiedifwechooseϕ(t)suchthatϕ(0) = 2nπ, forintegers
withthefollowingcoefficients
n. The parameters associated with the quantum gate to be
iTmopslheomwenttheadtawreeecnacnordeeadllyintiompplaermamenettesrisnθg(let)-,qϕu(bti)tagnadteφs(bt)y. α(t)= aσ1,+(t)−σ1,−(t)[acosθ1(t)+beiφ1(t)sinθ1(t)], (14)
2
ugsivinegntbhyeoperatorU1(t),letusconsidertheevolvedstate|ψ(t)(cid:105) β(t)= bσ1,+(t)+σ1,−(t)[bcosθ1(t)−ae−iφ1(t)sinθ1(t)],(15)
2
|ψ(t)(cid:105)=U1(t)|ψinp(cid:105)=α(t)|0(cid:105)+β(t)|1(cid:105), (7) γ(t)= cσ2,+(t)−σ2,−(t)[ccosθ2(t)+deiφ2(t)sinθ2(t)], (16)
wherethecoefficientsα(t)andβ(t)aregiven,respectivelyby 2
α(t)= aσ+(t) − σ−(t)[acosθ(t)+be−iφ(t)sinθ(t)] (8) δ(t)= dσ2,+(t)+σ2,−(t)[dcos2θ2(t)−ce−iφ2(t)sinθ2(t)],(17)
2 2
β(t)= bσ+(t) + σ−(t)[bcosθ(t)−aeiφ(t)sinθ(t)] (9) wσkh,±er(e0)ag=ai2nδw+∓e.hAavseindetfihneecdasσek,o±f(ts)in=gl(ee-iϕqku(tb)i±t g1a)t,es,utchhetohpa-t
2 2
eratorU (t),aswellastheHamiltonianassociatetoU (t),is
2 2
withσ (t)=(eiϕ(t)±1). Byusingtheinitialconditionϕ(0)= ψ -independent.
± inp
2nπwecanseethatα(0) = aandβ(0) = b,becauseσ (0) = Inthisdiscussionwehavenotlabeledatargetandcontrol
±
2δ+±. Therefore, from Eqs. (7-9), an arbitrary single-qubit qubit and this choice can be done through the definition of
rotationcanbeperformed. the free parameters. Thus, the operator U (t) encompasses a
2
3
largeclassoftwo-qubitgates, likeentanglingquantumgates Finally, let us discuss about implementation of the CNOT
(as CNOT or some controlled single-qubit unitary rotations) gate. Without loss of generality, we consider a bipartite sys-
ornon-entanglinggates(suchastheSWAPgate). teminitiallyinthestate
|ψ (cid:105)=a|0(cid:105) |0(cid:105) +b|0(cid:105) |1(cid:105) +c|1(cid:105) |0(cid:105) +d|1(cid:105) |1(cid:105) (20)
inp,2 c t c t c t c t
III. UNIVERSALSETOFQUANTUMGATESAND where the subscript “c” and “t” labels the control and target
QUANTUMFOURIERTRANSFORM qubit,respectively. Thus,underthisencodingthestateofthe
systematendoftheevolutionmustbewrittenas
Inordertoshowaswecantoimplementasetofuniversal |ψ (cid:105)=a|0(cid:105) |0(cid:105) +b|0(cid:105) |1(cid:105) +γ(T)|1(cid:105) |0(cid:105) +δ(T)|1(cid:105) |1(cid:105) (21)
out,2 c t c t c t c t
quantumgatesbyusingtheresultsdevelopedhere,inthissec-
tionweconsider, withoutlossesgenerality, somechoicesfor whereγ(t)andδ(t)aregivenbytheEqs.(16)and(17),respec-
the free parameters previously discussed. As application of tively. Noticethatthecoefficientsaandbwerenotchanged,
thismethod,wediscussaboutthesetofquantumgatesneces- therefore, from Eqs. (14) and (15) we must set ϕ1(t) = 0.
sarytoimplementthequantumFouriertransform[20],which Withthischoice,theparametersθ1(t)andφ1(t)becomesaddi-
hasanimmediateapplicationtotheShor’salgorithm[19]. tionalfreeparametersthatmaybeusedtosimplifytheHamil-
tonian. Inaddition,toobtaintheCNOToperationrightly,we
needchooseourparametersuchthatγ(T) = d andδ(T) = c.
A. Quantumgatesfor(approximately)universalQC Therefore,fromEqs. (16)and(17),thisresultcanbeachieve
fromthechoicesθ (T)=π/2,ϕ (T)=πandφ (T)=0.
2 2 2
In thissection we provides aset of choices of thefree pa-
rameters of our approach which allow us implement an uni-
B. ApplicationtoquantumFouriertransform
versal set of quantum gates composed by phase shift gates,
HadamardgateandCNOTgatethatmaybeusedtoperform
universalQCwitharbitraryefficiency[20,24]. Now,letusdiscusstheapplicabilityofourapproachtoim-
Forphaseshiftgates,givenanyentry|ψ (cid:105)=a|0(cid:105)+b|1(cid:105)we plement the discrete quantum Fourier transform (QFT). This
inp
havethecorrespondingoutput|ψ (cid:105) = a|0(cid:105)+eiξb|1(cid:105),forany exampleisconsideredagoodexampleduetheconnectionof
out
value 0 < ξ < 2π. Thus, from Eqs. (8) and (9), such a gate theQFTwiththeShor’salgorithm[19,20]. Toimplementthe
may be implemented if we choose ϕ(T) = ξ and θ(T) = 0. quantum circuit for QFT we need of Hadamard, controlled
Notice that we have boundary conditions for the parameter phase shift gates in addition with the SWAP gate [20]. The
ϕ(t)givenbyϕ(0) = 0andϕ(T) = ξ,soϕ(t)cannotassume Hadamard gate was discussed in the previous section, there-
anyform. Ontheotherhand, theparameterθ(t)hasasingle fore we will discuss about controlled phase shift and SWAP
condition that must be satisfied, where such a condition can gate.
beobtainedsettingθ(t)=0. Moreover,wecanuseitasafree Let us start from Eq. (21) to discuss the controlled phase
parameter in order to simplify the Hamiltonian that evolves shiftgate. AsinthecaseoftheCNOTgate,herewewillcon-
thesystem. Infact,puttingθ(t)=0theHamiltonianforphase sider our bipartite system as given by the Eq. (20) and the
shift gates is given by (in order to obtain simple Hamiltoni- final state given by Eq. (21), however, here we need satisfy
ans, throughout the manuscript we consider that the system γ(T) = c and δ(T) = eiξd. Now, the choice of the free pa-
evolvesuptoaglobalphase) rametersnecessarytoachievethisresultmightbeθ2(T) = 0,
ϕ (T) = ξ, ϕ (T) = 0 and φ (T) = 0. Under this considera-
2 1 2
ϕ˙(t) tions,wecansetφ (t)=θ (t)=0andtheHamiltoniancanbe
H (t)= σ , (18) 2 2
ph 2 z writtenas
whereσz = diag[1, −1]isthePaulimatrixandthesymbol H(t)= ϕ˙2(t)(cid:2)1c⊗σzt+σzc⊗1t−σzc⊗σzt(cid:3) , (22)
4
“˙” denotes temporal derivatives. This choice is not unique
andwecanhavemanyotherspossibilitiesifwesetθ(t) (cid:44) 0, where the parameter θ1(t) has not any relevance because we
butifwepickθ(t)(cid:44)0thecorrespondingHamiltonianmaybe put ϕ1(t) = 0. The Hamiltonian above is the more simple
notassimpleastheHamiltonianobtainedabove. Hamiltonian that we can obtain with this method, but is not
The Hadamard gate is an exclusive gate of quantum com- uniqueandcanassumeothersformfromdifferentchoicesof
puters due its ability of generating quantum superpositions theparametersφ2(t)andθ2(t).
with elements of the computational basis. More specifically, Byconsideringthestate|ψinp,2(cid:105),theSWAPgateimplement
given a quantum state |ψ (cid:105) = a|0(cid:105)+b|1(cid:105) we have its corre- theexchangeb→candc→b.However,suchachangeisnot
inp √ √
spondingoutput|ψ (cid:105) = (a+b)/ 2|0(cid:105)+(a−b)/ 2|1(cid:105). We possiblefromEqs. (14-17)andweneedredefineourscheme.
out
canimplementsuchoperationbysettingθ(T)=π/4,φ(T)=0 For this particular case of the SWAP gate, we redefine the
andϕ(T) = π. Inthiscase,wehaveonlyonefreeparameter, states|nk,±(t)(cid:105)ofthefollowingway
namely,φ(t)andsoweconsiderφ(t) = 0inordertosimplify |n1,+(t)(cid:105)=cosθk(t)|00(cid:105)+eiφk(t)sinθk(t)|11(cid:105), (23)
theHamiltoniangivenby |n1,−(t)(cid:105)=−sinθk(t)|00(cid:105)+eiφk(t)cosθk(t)|11(cid:105), (24)
H (t)= ϕ˙√(t)(σ +σ ). (19) |n2,+(t)(cid:105)=cosθk(t)|01(cid:105)+eiφk(t)sinθk(t)|10(cid:105), (25)
Had 2 2 z x |n2,−(t)(cid:105)=−sinθk(t)|01(cid:105)+eiφk(t)cosθk(t)|10(cid:105), (26)
4
wherethecoefficientsgivenintheEqs. (14-17)becomes the initial state of the system. We illustrate the results ob-
tained here through the Shor’s algorithm that makes uses of
α(t)= aσ1,+(t)−σ1,−(t)[acosθ1(t)+deiφ1(t)sinθ1(t)], (27) QFTassubroutinesofsuchalgorithm.Whilethefirstpurpose
2 toimplementQFT,fromadiabaticdynamicsusingcontrolled
β(t)= bσ2,+(t)+σ2,−(t)[bcosθ2(t)−ce−iφ2(t)sinθ1(t)],(28) quantumevolution,requiresthree-qubitsinteractions[15,21],
2 ourprotocolrequiresjusttwo-qubitinteractions.Byusingthis
γ(t)= cσ2,+(t)−σ2,−(t)[ccosθ2(t)+beiφ2(t)sinθ2(t)], (29) applepmroeancths,inwgelecaanndobtwtaoin-qauiltarggaetecsl,atshseoreffHoraemtihlitsonmiaenthsotdoicman-
2
beausefultechniquetoperformuniversalQCinmanyphys-
δ(t)= dσ1,+(t)+σ2,−(t)[dcosθ1(t)−ae−iφ1(t)sinθ1(t)].(30) icalsystems.
2
Whentheinevitableinteractionofoursystemwiththeex-
Therefore,nowthenewcoefficientsallowusmaketheex- ternal medium (reservoir) is considered, decoherence effects
change b → c and c → b. By considering ϕ (t) = 0 maybeabigproblemandthisproblemsincreaseswithnum-
1
we guarantee that α(t) = a and δ(t) = d that is a require- berofparticlesinoursystem. Ontheotherhand,large-scale
ment of the SWAP gate. Finally, if we put θ (T) = π/2, quantumcomputingrequiresthemanipulationofmanyqubits
2
ϕ (T) = π and φ(T) = 0, we drive the system from the that remaining with a good robustness against decoherence
2
|ψ (cid:105) = a|0(cid:105) |0(cid:105) +b|0(cid:105) |1(cid:105) +c|1(cid:105) |0(cid:105) +d|1(cid:105) |1(cid:105) tothefinal effects. In this scenario, the scheme presented here may be
in,2 c t c t c t c t
state|ψ (cid:105)=a|0(cid:105) |0(cid:105) +c|0(cid:105) |1(cid:105) +b|1(cid:105) |0(cid:105) +d|1(cid:105) |1(cid:105). more robust to decoherence than the protocols that requires
in,2 c t c t c t c t
auxiliaryqubits,alreadywedonotneedauxiliaryqubitsand,
consequently, the error produced by such qubits is removed
IV. CONCLUSION here. Itwouldbeinterestingtomakethisanalysisinorderto
comparativethisschemewithothersknownprotocols. Other
In summary, in this paper we have introduced a scheme interesting point is the energetic resource to implement this
toperformuniversalQCviareverseengineeringofaHamil- protocol. Can we optimize it, for example, from additional
tonian from the evolution operator. We discuss the general free parameters? If we can, how can we? We leave these as
aspects of our approach and show as obtain a set of Hamil- openproblemsforfutureresearch.
tonian that allow us implement an universal set of quantum ACKNOWLEDGMENTS
gates. Our method is an economic scheme that can be view
asanalternativetoothersmethodpresentintheliterature. In WeacknowledgefinancialsupportfromtheBrazilianagen-
fact, while many protocols requires auxiliary qubits to per- ciesCNPqandtheBrazilianNationalInstituteofScienceand
form universal QC, our approach does not need help of aux- Technology for Quantum Information (INCT-IQ). Specially,
iliary elements to implement single and controlled quantum wewouldliketothanktoNatanaelMourafromUniversidade
gates, because the method presented here is independent of RegionaldoCaririforusefuldiscussions.
[1] U. Gaubatz, P. Rudecki, S. Schiemann and K. Bergmann, J. [13] E.Farhietal,Science292,472(2001).
Chem.Phys.92,5363.(1990). [14] D.Bacon,S.T.Flammia,PhysRevLett.103,120504(2009).
[2] H.R.Lewis,Phys.Rev.Lett.18,510(1967). [15] I.Hen,Phys.Rev.A91,022309(2015).
[3] H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 [16] A.C.SantosandM.S.Sarandy,Sci.Rep.5,15775(2015).
(1969). [17] A.C.Santos,R.D.SilvaandM.S.Sarandy,Phys.Rev.A93,
[4] M. Demirplak and S. A. Rice, J. Phys. Chem. A 107, 9937 012311(2016).
(2003). [18] I.B.Coulamy,A.C.Santos,I.HenandM.S.Sarandy,Front.
[5] M. Demirplak and S. A. Rice, J. Phys. Chem. B 109, 6838 ICT3,19(2016).
(2005). [19] P.W.Shor,Proceedingsofthe35thSymposiumonFoundations
[6] M.V.Berry,J.Phys.A:Math.Theor.42,365303(2009). ofComputerScience,124(1994).
[7] Yi-HaoKangetal,Sci.Rep.6,30151(2016). [20] M. A. Nielsen and I. L. Chuang, Quantum Computation and
[8] E.Torronteguietal,Adv.At.Mol.Opt.Phys.62,117(2013). QuantumInformation.CambridgeUniversityPress,Cambridge
[9] A. M. Childs, E. Farhi, J. Preskill, Phys Rev A. 65, 012322 (2000).
(2001). [21] IHen,Front.Phys.2,44(2014).
[10] M.H.S.Amin,D.V.AverinandJ.A.NesteroffPhysRevA. [22] M. Herrera, M. S. Sarandy, E. I. Duzzioni and R. M. Serra,
79,022107(2009). Phys.Rev.A89,022323(2014).
[11] Y.-X.Duetal,Nat.Commun.7,12479(2016). [23] J.Jing,L.-A.Wu,M.S.SarandyandJ.G.Muga,Phys.Rev.A
[12] S. An, D. Lv, A. del Campo and K. Kim, Nat. Commun. 7, 88,053422(2013).
12999(2016). [24] A.Barencoetal,Phys.Rev.A52,3457(1995).