Strong and weak coupling of two coupled qubits

1 MB·English

by  Elena del Valle

APS/123-QED
Strong andweakcoupling oftwo coupled qubits
Elena del Valle
∗
School ofPhysicsandAstronomy,UniversityofSouthampton, SO171BJ,Southampton, UnitedKingdom
(Dated:January13,2010)
Iinvestigatethedynamicsandpowerspectrumoftwocoupledqubits(two-levelsystems)underincoherent
continuouspumpanddissipation.Newregimesofstrongcouplingareidentified,thatareduetoadditionalpaths
ofcoherenceflowinthesystem.Dressedstatesarereconstructedevenintheregimeofstrongdecoherence.The
resultsareanalyticalandofferanexactdescriptionofstrong-couplinginpresenceofpumpinganddecayina
0
nontrivial(nonlinear)system.
1
0
2
I. INTRODUCTION

n
a
J
Quantuminformationprocessing[1] presupposesa coher-
3
ent couplingof the fundamentalbricksof quantuminforma-
1
tion,thequbits. Inrealsystems,however,decoherence,dissi-

] pationandincoherentcouplingtotheenvironmentisunavoid-
l able [2]. In the field of cavity quantumelectrodynamics[3], FIG.1:(Coloronline)Schemaofthesystemsofinterestinthistext.
l
a thenotionofcoherentcouplingisknownasstrongcoupling, Themainobjectofstudy, twocoupledqubits,issketchedin(c). It
h inthiscase,betweenlightandmatter(e.g.,theexcitedstateof willbecomparedthroughoutwithothercoupledsystems,(a),(b),(d)
- and(e).
s anatomictransition[4]oranelectron-holepairinasemicon-
e
ductor[5]). Thisleadstoaquantumsuperpositionofthebare
m
states,resultinginso-calleddressedstates[6]. Thisregimeis
. reachedinsystemsofveryhighqualityandundertightexper- An immediate extension of light-matter coupling in the
t
a imental control, so that intrinsic sources of decoherence are linear regime is the linear model, namely, that of two har-
m minimizedasmuchaspossibleandcoherentdynamicstakes monic oscillators with no restriction in the number of parti-
- over.Thesimplestdescriptionofstrongcouplingneglectsdis- cles Fig. 1(a). Its comprehensivedescription under incoher-
d
sipationaltogetherandthusreducesto thatofmerecoupling ent pumping was given in Ref. [14]. This describes for in-
n
o with strength g, introducing the notion of Rabi splitting [7]. stanceexciton-polaritonsinplanarsemiconductormicrocavi-
c Nextstep inthedescriptionincludesthe decaygi ofthebare ties[5](wherebothexcitonsandphotonsarebosons). Itwas
[ states,i=1,2[8].Thisgivesrisetoawidelyknowncriterion shown in this work how pumping calls for extended defini-

for strong-coupling: 4g> g g . This is the case of vac- tions of strong-coupling, essentially requiring absorption of
1 1 2
v uum Rabi splitting where a|t m−osto|ne excitation is involved. the pumpingrates in the decayrates. A moreimportantthe-
1 Atthislevel,whichisthemostnaturalandfundamentalsince oretical model, knownas the Jaynes-Cummingsmodel[15],
0 itdescribesoneparticle,thereisnodifferencefromtheunder- describesthecouplingofatwo-levelsystem(suchasanatom
2 lying theoretical model. Differences appear at the next step orazero-dimensionalexcitoninasmallquantumdot)witha
2 of description when the excitation scheme is taken into ac- bosonmode(typically,cavity photons),Fig. 1(b). It is more
1. count. A typical description of excitation is to consider an importantbecausemore closely related to a genuinelyquan-
0 initialcondition. Thecouplingisthen studiedasthe sponta- tumregime,thelinearmodelbeingessentiallyaclassicalde-
0 neousemissionfromthisinitialexcitedstate. Unlesstheini- scription cast in quantum-mechanical terms. Its description
1 tialconditionisrestrictedtooneexcitation(aspreviously),the under incoherent pumping was given in Ref. [16], but en-
:
v underlyingtheoreticalmodelbecomesdeterminant. Another counteredvarious difficulties to offer a complete picture. In
i importantdescriptionoftheexcitationprocessisthatofacon- particular,dressedmodesexhibitcomplexpatternsandadefi-
X
tinuouspumping,forinstanceacoherentexcitationthatdrives nitionofstrong-couplinginthissystemismuchmoredifficult
ar the system [9, 10], or an incoherentpump that feeds excita- toachieve,sincesplittingofthedressedstatesdependsonthe
tionsat a givenrate butwithoutanycoherentinput[11, 12]. excitations. Inparticular,strong-couplingcanbeenforcedby
Thelatterismoredirectlyrelatedtotheintrinsicdynamicsof pumping,leadingtosituationsofmixedweakandstrongcou-
thesystem,andistheonethatwillbeconsideredinthistext. pling,wheresomeofthestatesarebarewhilesomeothersare
With non-negligiblepumping, the underlyingtheoreticalde- dressed.
scription cannotbe ignored, and taking it into account leads In this text, I will address the case of two two-level sys-
tostrongdeviationsfromtheparadigmofstrong-couplingas tems(Fig.1(cde)),whichcompromisesbetweensimplicityof
establishedbythespontaneousemissionofoneexcitation(in thelinearmodelandrichnessoftheJaynes-Cummingsmodel.
anymodel)[13]. Inparticular,thankstothe reducedsize ofthe Hilbertspace,
I will be able to solve the problem fully analytically, as in
the linear model, a convenience not afforded by the Jaynes-
Cummingsmodel(whenincludingincoherentpumping).This
Electronicaddress:[email protected] will allow me to provide a complete picture of weak and
∗
2
strongcouplinginanontrivialsystem,andthereforeshedlight II. THEORETICALMODEL
onmorecomplicatedsystems.
TheHamiltonianfortwocoupledqubitsreads:
Mydescriptionwilladdressmoreparticularlyindependent
qubits (Fig. 1(c)), for instance superconducting (Josephson) H0=w 1s 1†s 1+w 2s 2†s 2+g(s 1†s 2+s 2†s 1), (1)
qubits[17],inthesensethattheircommutationruleswillnot
wheres aretheloweringoperatorsofthequbits,withbare
bethoseoftwofermions,thatanticommute(Fig.1(d)). Iwill 1,2
energiesw . Theyarelinearlycoupledwithstrengthg. The
alsoaddressthelattercaseforcomparisonandcompleteness, 1,2
two modes can be detuned, by a quantity D =w w , that
and obtain the elegant result that the expressions describing 1 2
issmallenough,D w , sothattherotatingwav−eapprox-
two coupled fermions are essentially identical to those de- 1,2
≪
imationisjustified. TheHilbertspaceofthecoupledsystem
scribingtwocoupledbosons,althoughthesetwosystemsare
has dimension four, with the structure 2 2. It can be de-
verydifferentin characterandbehavior(for instance bosons
⊗
composed in three subspaces (also called manifolds, rungs,
accumulate arbitrary number of particles whereas fermions
etc.) with a fixed number of excitations: the ground state,
saturateatatmostone,adistinctionrecoveredintheformal-
0,0 ,withzeroexcitation,theexcitedstateofeachqubit,
ism by merely substituting effective parameters). Also be-
{| i}
1,0 , 0,1 ,withoneexcitation,andthestate 1,1 with
causetwotwo-levelsystemscanbemappedtoonefour-level
{| i | i} {| i}
twoexcitations.
system (Fig. 1(e)), I will address this case in detail, finding
Animportantpointthroughoutthistextisthecommutation
anotherfundamentalcaseofinterest.
rulesinEq.(1),thatarethoseoftwodistinguishablesystems,
i.e.,
Themainresults, however,andthedeepestconnectionsto
be made with other models (such as the Jaynes-Cummings) s s =s †s †=0, i=1,2,, (2a)
i i i i
willbeobtainedfromthecaseoftwoqubits.Beyonditsinter-
[s ,s †] =s s †+s †s =1, i=1,2, (2b)
estforthepreviousreasons,thestudyofthecouplingbetween i i + i i i i
twoqubitsisinterestinginitsownright[18–21]: itcomesas [s ,s †]=s s † s †s =0, i= j, (2c)
i j i j − j i 6
thefundamentalsupportofentangledstates[22,23],toimple-
[s ,s ]=s s s s =0, i= j. (2d)
mentquantumgates[24,25]and,inthisquantuminformation i j i j− j i 6
processingcontext,thenaturalmodelto investigatedecoher-
Note that two operators from different systems commute.
ence[26, 27]. In thistext, the twocoupledqubitswillallow
Thesecommutationrulesfortwo-levelsystemsismostcom-
ustoinvestigatestrongcouplingunderincoherentpumping.
monly found in the literature of the Dicke model [28], that
describes a gas of two-level systems emitting in a common
Therestofthispaperisorganizedasfollows. InsectionII, radiation field. In the case of a fermion gas, this commu-
Iintroducethemodelanditsparametersanddiscussthelevel tation is an approximation, that is made for the simplicity
structure. In section III, I obtain the single-time dynamics, of the algebra and that is justified for a dilute gas by the
thatwillbeshowntobethesameindependentlyoftheunder- fact that anticommuting operators give the same final phys-
lying model. In section IV, I obtain the power spectra, that, ical results [29]. Indeed, when the wavefunctions of any
incontrastwiththeprevioussection,dependstrikinglyonthe two fermions is weakly-overlapping, symmetrized, antisym-
underlying model. Power spectra (photoluminescencespec- metrizedandnon-symmetrizedresultsarethesame[30].
trainaquantumopticalcontext)areimportantbecausethisis In ourcase, the two qubitsare stronglyinteracting, which
wherethedressedstatesmanifest. Iwillanalyzeindetailthe setsoursystemapartfromtheDickemodelanditsapproxima-
caseoftwoqubits(Sec.IVA)andcontrastitwiththatoftwo tionsinmanyrespects(seeappendixA). Forreference,Iwill
fermions(Sec. IVB) and of a four-levelsystem (Sec. IVC). analyzetheantisymmetrizedcaseoftwointeractingfermions
InsectionV,Idescribethestrongandweakcouplingregimes, (Fig. 1(d)) completely. However, the main object of interest
first in the absence (Sec. VA), and then including(VB) the inthistextisthatoftwocommutingqubits(likeintheDicke
incoherentcontinuouspump. Idefinenewregimesofstrong model), correspondingto the case of two distinguishable (in
coupling,propertothetwoqubitssystem. InsectionVI,Iil- thequantumsense)qubits(Fig.1(c)).
lustratetheresultsofprevioussectionswithexamplesofsome The level structure of Hamiltonian (1) is sketched in
interestingconfigurations: caseswherepumpingeffectisop- Fig.2(a)(atresonance).Sucha“diamond-like”configuration
timal (Sec. VIA) and detrimental (Sec. VIB) for the coher- can be mappedto a four-levelsystem (4LS),i.e., as a single
ent coupling. In section VII, based on the previous results, entity, with Hilbert space structure 1 1 1 1 (Fig. 1(e)).
⊕ ⊕ ⊕
I reconstruct the dressed states, uncovering an unexpected This description fits, for instance, the case of a multi-level
manifestation of decoherence—the emergence of additional atom[31]orofa singlequantumdotthatcanhostuptotwo
dressedstates—tobefoundonlyinthecoexistenceofpump- interactingexcitons(electron-holepairs)formingabiexciton
ing and decay along with the coherent coupling. Finally, in state [32]. On the other hand, two coupled qubits, matches
sectionVIII,Igiveasummaryofthemainresultsasmycon- the case of two nearby quantum dots directly coupled. The
clusions. Toavoiddistractioninthemaintext,mosttechnical state with doubleexcitationis thencalled an interdotbiexci-
detailsappearinappendices(B,C,D)alongwithfurtherma- tonstate[33].
terialoutsidethescopeofthisstudy(A,E). The Hamiltonian H can be diagonalized in terms of two
0
3
intermediatedressedstates, + and ,as:
| i |−i
H =w +w + + +w 1,1 1,1 (3)
0 + 11
−|−ih−| | ih | | ih |
witheigenfrequecies:
w +w D 2
w = 1 2 R, R= g2+ , (4a)
± 2 ± s 2
(cid:18) (cid:19)
w =w +w . (4b)
11 1 2
The diagonalized level structure is sketched in Fig. 2(b).
These are the same eigenfrequecies w and Rabi splitting
(givenby2R)thanforthedressedstate±softwocoupledhar-
monic oscillators up to the first manifold [14]. This equiva-
lence breaks in the manifold with two excitations where the
fermionic nature of the particles reveals and only the state
1,1 ispermitted,ascomparedtothreepossiblestatesinthe
| i
secondmanifoldofthelinearmodel: 2,0 , 1,1 , 0,2 .
{| i | i | i}
Thedynamicsofdressedstates andtheirspectralshape
|±i
dependontheamountofdecoherencethatthedissipativeand
excitationprocessesinducein the system. I willconsideran
incomingflowof excitationsthatpopulatethetwo-levelsys-
tems at rates P, P and an outgoing flow (given by the in-
1 2
verselifetime)atratesg ,g ,respectively. Thissituationcor-
1 2
respondstoanincoherentcontinuouspumporinjectioninthe
qubit that can be varied independently from the dissipation.
The steady state reached under the pump and decay corre- FIG.2: (Coloronline)Energylevelsforthetwoqubitsdescribedby
Hamiltonian(1). Weakcoupling(a)andstrongcoupling (b)inthe
spondsto a statistical mixture of all possible quantumstates
andisdescribedbya densitymatrixr . Themasterequation absence of pump arewell definedintermsofthebare anddressed
states ,respectively. AsshowninSec.VB,theSCregimegives
ofthesystemhasthestandardLiouvillianform[34],withthe |±i
risetonewregionswhenpumpistakenintoaccount:(c)FSCwhere
correspondingLindbladterms:
dressedstatesremain ,an(d)SSCanMC,wheredressedstates
|±i
dr =Lr =i[r ,H] (5a) dforermsseadnsetwatesse,ti|sI±shio,w|On.±iM. COncloyrSreSsCpo,nwdisthtosptlhitetincagsoefwahlletrheenIew
dt have closed (both collapsing on w ). The thickness of the le|ve±lis
g 1
+ (cid:229) i(2s rs † s †s r rs †s ) (5b) representstheuncertaintyinenergydueto(a,b)thedecayand(c,d)
2 i i − i i − i i boththepumpandthedecay. Thearrowslinkingthelevelsdueto
i=1,2
pump/decayareblueforthelower(l)andredfortheupper(u)tran-
+ (cid:229) Pi(2s †rs s s †r rs s †). (5c) sitions. TransitionslabeledA,C (inivolving + or I , inigreen)
2 i i− i i − i i occuratfrequenciesdeterminedbyz ,whileB| anidD|(in±violving
i=1,2 1
|−i
or O , inorange) aredetermined byz . Thesamecolorcodeis
2
This master equation can be exactly solved given the finite use|din±tiherestofthefigurestoplotthedecompositionofthespectra.
andsmalldimensionoftheHilbertspace(whichisonlyfour).
Withinthesameformalism,wecandescribethespontaneous
emissionfromageneralinitialstatebysolvingtheequations crossLindbladtermsthatentanglel andu ,ontheonehand,
1 1
forvanishingpumping. andl andu ontheother.Ifthefourlevelsdidnotcorrespond
2 2
Iwillnotethetransitionsbetweenbarestates(cf.Fig.2(a)) totwoqubitsbuttoasingleentity(a4LS),suchasatomiclev-
as: elsorasinglequantumdotlevels,theLindbladtermswould
bewrittendirectlyintermsofthetransitionoperators,without
u1=|0,1ih1,1|=s 1s 2†s 2, (6a) thesecrossLindbladterms. Thealternativemasterequationis
l = 0,0 1,0 =s u , (6b) discussedinappendixC(cf.Eq.(C1)).
1 1 1
| ih | −
u = 1,0 1,1 =s †s s , (6c) Weintroducefermioniceffectivebroadenings:
2 | ih | 1 1 2
l = 0,0 0,1 =s u . (6d)
2 | ih | 2− 2 G 1=g1+P1, G 2=g2+P2, (7a)
G G g g
(for upper and lower transition). Note that they can all be G = 1± 2, g = 1± 2. (7b)
writtenintermsofthequbitoperators(innormalorder)since ± 4 ± 4
s =u +l (i=1,2). TheLindbladtermsofpumpanddecay
i i i
inEq.(5)areexpressedintermsoftheirs operators.Rewrit- Equation (7a) is to be compared with bosonic effective
ingthemintermsofthetransitionoperatorsu ,l leadsto broadenings,forbosonmodesaandb(suchastheharmonic
1,2 1,2
4
oscillatorsofthelinearmodel): The spectrum is composed of four peaks that I label p=
A,B,C,D, each of them with a Lorentzian (weighted by the
G˜a=ga−Pa, G˜b=gb−Pb. (8) cfoouerffirceiseonntaLntp)fraenqdueancdiiesspewrsivaendpathrte(awsesoigchiateteddbbyroKapd)e.niTnhges
p
A tilde is being used to denote the bosonic character of the (full-widthsathalfmaximum)g ,areintrinsictothesystem,
p
broadening, in the sense that the pumping strength is sub- astheycorrespondtothefourpossibletransitionsinthesys-
tractedtothedecay. Pumpingleadstobroadeningoftheline tem. ThecoefficientsK andL (derivedin appendixB) are
p p
inthefermioniccaseandnarrowinginthebosoniccase,which the parameters that are specific to the experimentalconfigu-
isaspectralmanifestationofFermiandBosestatistics. ration (such as channel of detection) or regime (steady state
Anotherconvenientnotationfor Eqs. (7) is the expression underincoherentpumpingorspontaneousemissionofanini-
ofpumpinganddecayratesintermsofG sandanewparame- tialstate). ThisformofS(w )isageneralfeatureforthepower
terrwhichrepresentsthetypeofreservoirthatthequbitisin spectraofcoupledquantumsystems[36].
contactwith[35]: Inthecaseofuncoupledqubits(g=0),thefourresonances
reducetothetwobareenergiesw andw ,broadenedbythe
g1=G 1(1−r1), g2=G 2(1−r2), (9a) effectivedecayratesG 1 and G 2, r1espectiv2ely. The peaksare,
P =G r , P =G r . (9b) in this case, pure Lorentzians. In the opposite case of very
1 1 1 2 2 2
strongcoupling(g g ,P),thespectrumisalsowellapproxi-
Allpossiblesituations(with0 ri 1),fromamediumthat matedbyLorentzia≫ns,butwiththeresonancesw p nowatthe
only absorbs excitation, ri =0≤, to≤one which only provides dressed state frequencies w , and broadened by the average
them, ri =1, are thusincludedin a transparentway. For in- rates (G 1+G 2)/2. Lorentz±ian lineshapes correspond to the
stance, a thermal bath with temperature different from zero emissionofwelldefinedisolatedmodesofthesystemHamil-
correspondstori<1/2. Theeffectofthemediumontheef- tonian,weaklyaffectedbyothermodes. Inthiscase,Kp 0.
fectivebroadeningiscontainedinG i. Thedispersivecontributionbecomesnonnegligibleinthe≈in-
Thepowerspectrumforeachqubit,i=1,2,isdefinedas: termediate situations when dissipation and decoherence (or
dephasing,cf.appendixA)areoftheorderofthedirectcou-
1 ¥ ¥
s(w )= s †(w )s (w ) = ´ G(1)(t,t+t )eiwt dtdt pling.TheHamiltonianeigenmodesarethennolongerneatly
i h i i i 2p 0 0 i leadingthedynamicsand,asaconsequence,theirbroademis-
Z Z (10)
sionlinesoverlapinenergy,producinginterferences. Thisis
where
theregimeofinterestinouranalysis,sincenewphenomenol-
ogyappearsforthedressedstatesandcouplingregimes.With
G(1)(t,t+t )= s †(t)s (t+t ) , (11)
i h i i i thisgoalinmind,Idevotethenexttwosectionstopresenting
theanalyticalexpressionsofallthequantitiesappearinginthe
isthefirstorderautocorrelationfunction.s(w )describeshow
spectrum,Eq.(13).
energyisdistributedandisthusoffundamentalinterest. Ina
quantumopticalcontext, thiscan be observeddirectlyin the
opticalemission, butforgenerality,Iwill keeptheterminol-
III. SINGLE-TIMEDYNAMICS
ogy of power spectrum. For the steady state spectrum, the
running time t is taken at infinite values (thereby removing
one integral). In appendix B, we make use of the quantum We start by analyzing the relevant average quantities
regression formula[34] in its most generalform to compute needed to compute the weights Lp and Kp in the spec-
(1) trum(13),thatis,populationsandcoherences:
G as well as other two- and one-time correlators (second
i
ordercorrelationfunctionsaregiveninappendixE).Weshall
n = s †s , n = s †s R, (14a)
focusoni=1inthefollowing,withoutlossofgenerality: 1 h 1 1i 2 h 2 2i ∈
n = s †s C, (14b)
G(11)(t,t+t )=n1p=A(cid:229),B,C,D[Lp(t)+iKp(t)]e−iw pt e−g2pt , n1co1rr=hsh1†1s 1s2i2†s∈2i ∈R. (14c)
(12)
where n = s (w )dw , the population of qubit 1, is ni is theprobabilitythatqubiti isexcited. The sum n1+n2,
1 1 that can go up to two, is the total excitation in the system.
used to normalize the expression for the spectrum (so that
S (w )dw =1R): ncorr isthe effectivecoherencebetweenthequbitsdueto the
1 directcoupling.n isthejointprobabilitythatbothqubitsare
11
R excited. Itisalsothepopulationofstate 1,1 . Ifthequbits
S (w )= 1 (cid:229) L g2p wereuncoupled,wewouldhaven11=n1n|2. i
1 p p∈{A,B,C,D}" p g2p 2+(w −w p)2 latIitonisoimfapomrotadnet(hsearye,ntoo=utlsin†esthefdoirffaeqreunbcietabnedtwnee=npoap†au-
(cid:0) (cid:1) w w p foranharmonicoscillat1or,chf.1Fig1i.1(a)and(c)),anda pohpulai-
K − , (13)
− p g2p 2+(w −w p)2# tai†on0o,f0a satnadtes(†r 100a,0nd,rr˜e1s0pfeocrtitvheelys)t.atTeh|e1,p0oipuwliatthio|n1o,0fith=e
fromEqs.(10)and(12). (cid:0) (cid:1) inte|rmediiate sta1te| 1,i0 (resp. 0,1 ) is given by n n
1 11
| i | i −
5
(resp. n n ). This is also the probability of having only
2 11
−
oneofthequbitsexcited. Thepopulationofthegroundstate
isgivenby1 n n +2n .
1 2 11
− −
Inthespontaneousemissioncase,n ,n andn havethe
1 2 corr
same solutions (depending only on the initial condition n0,
1
n0 andn0 )thantheircounterpartforthetwocoupledlinear
2 corr
oscillators, n , n and n . However, the two models differ
a b ab
for n : in the case of coupledqubits, it decaysfrom its ini-
11
tialvalue,n11(t)=e−4g+tn011,whereasinthelinearmodel,the
populationr˜ oscillatesasa resultoftheexchangewiththe
11
other states that are available in the second manifold ( 2,0
| i
and 0,2 ).Althoughpopulationsofthemodeshavethesame
| i
dynamics both in the two harmonic oscillators and the two
coupled qubits, the underlyingpopulationsof their states do
not. For instance, the decay fromthe initial condition 1,1
lheaardmsotoninc1o=scinl1la0t+orns,11on=erh1a0s+,nr 1=1foar†tahe=qur˜bit,+wrh˜ile+f|o2rr˜thei. FsoIlGid.b3l:ac(Ckolilnoer)oinnltihnee)SPCorwegeirmsepe(cg1tr=umg,aSn1d(wg2)=, fgro/m2)afoqrutbhiets(ttehaidcyk
a h i 10 11 20 stateundervanishingpump(P1=0.02gandP2=0.01g).Thespectra
Sincen1=na(andalson2=nb)thestatesaredifferentlypop- iscomposedoffourpeaksarisingfromtheloweranduppertransi-
ulated(r 10=r˜10,Etc.). tions (blue and red thin lines, respectively). In dashed purple, the
6
Inthesteadystate, allthesemeanvaluescanbewrittenin linearmodelspectrumforcomparison.
termsof effectivepumpand decayparameters, asin the two
harmonic oscillators, but now following fermionic statistics
(i=1,2): of a boson or fermion character, respectively. This provides
a neat picture of bunching/antibunching from excitations of
Peff
nSS= i , (15a) differentmodesthatareotherwiseofthesamecharacter.
i g eff+Peff Inthemostgeneralcase,withpumpanddecay,beforethe
i i
g +g P +P steady state is reached, also the transient dynamics of the
g eff=g + 1 2 Q , Peff=P+ 1 2Q , (15b)
i i G +G i i i G +G i mean values n1, n2 and ncorr for the coupled qubits maps to
1 2 1 2
nScoSrr= D g2iG (n1−n2), (15c) tmheoncicororesscpilolantdoirnsg, wavitehraognelsynG˜a, nbG .anWdencabanocfocnoculupdleedthheanr-,
+ →
− thatthesingle-timedynamicsofthequbitisruledbya(half)
withthecorrespondinggeneralizedPurcellrates Rabifrequencyofthesameformthaninthebosoncase[14]:
4(geff)2 4(geff)2
Q1= G , Q2= G , (16) R1TD= g2 (G +iD /2)2, (20)
2 1 − −
q
andtheeffectivecouplingstrength onlywithFermion-likeeffectivebroadenings,Eqs.(7).Since,
incontrasttothebosoncasewherethereisonlyoneRabipa-
g
geff= . (17) rameter,anotherexpressionwillariseinthetwo-timedynam-
1+ D /2 2 icsforcoupledqubits,IwillrefertoEq.(20)asthesingle-time
G
+ dynamics(half)Rabifrequency.
r
(cid:16) (cid:17)
We conclude this section by noting that the magnitudes
Finally,n takesasimpleintuitiveforminthesteadystate,
11
studieduptonowareindependentonhavingtwoqubits,two
nSSP +nSSP identicalfermionsora4LS.Thecrosstermsappearinginthe
nS11S= 1 G 2+G 2 1 ≤nS1SnS2S. (18) master equation for the first case, due to the correlationsin-
1 2
duced by the incoherent processes, do not affect the steady
ThisshouldbecontrastedwiththecounterpartofnSS fortwo statepopulations,asisshowninappendixC.
11
bosonicmodes(aandb),forwhich:
nSSP +nSSP
a†ab†b SS=2nSSnSS a b b a nSSnSS. (19) IV. POWERSPECTRA
h i a b − G a+G b ≥ a b
This is an interesting manifestation of the sym- A. Twoqubits
metry/antisymmetry of the wavefunction for two
bosons/fermions, that is known to produce such an at- Thegeneralexpressionsforthespectrumandtwo-timecor-
tractive/repulsive character for the correlators. Here we see relatorsoftwocoupledqubitsadmitanalyticsolutionsatres-
that quantum (or correlated) averages nˆ nˆ (with nˆ the onanceinthesteadystateofanincoherentcontinuouspump.
1 2 i
h i
number operator) are higher/smaller than classical (uncor- Fromnowon,wewillreferalwaystothissituationand,there-
related) averages nˆ nˆ , depending on whether they are fore,Iwilldropthesteadystatelabelinthenotation.
1 2
h ih i
6
The fourcoefficientsL +iK appearingin Eq. (12), now cooperativebehavior of two coupled modes, similarly to the
p p
definedinthesteadystate,read:(cf.appendixB) superradianceoftwo atomsintheDicke modelortherenor-
malization with the mean number of photons in the Jaynes-
1 n
L +iK = 2(2z +iG )(R iG )+a +a 2 CummingsModel.
A A 1 2 1 2
16Rz1 − − n1 Thelastandmostimportantparameterappearinginthepre-
+2g n P1 (2z +iG )+2(R+z +iG ) ncorr , vious expressions is a Rabi frequency for the two-time dy-
−G 1 2 1 + n namics,thatforcoupledqubitsdiffersfromitscounterpartfor
+ 1
h i (21ao) single-timedynamics(cf.Eq.(20)):
1 n
L +iK = 2(2z +iG )(R+iG ) a a 2
B B 16Rz2 2 2 − − 1− 2n1 R= g2 (Dsg)2 G 2 . (27)
nP n − − −
+2g 1 (2z +iG )+2(R z iG ) corr , q
G 2 2 − 2− + n This is the true analog of the (half) Rabi frequency of the
+ 1
h i (2o1b) linearmodelsincethisvalue, notitssingle-timecounterpart,
determines strong or weak coupling (emergence of dressed
1 n
LC+iKC= 2(2z1 iG 2)(R iG ) a1 a2 2 states). Atvanishingpump,therenormalizedcouplingGcon-
16Rz1 − − − − − n1 verges to g, and both R and R1TD converge to the standard
n P n
+2g 1 (2z iG ) 2(R z +iG ) corr , expressionforthe(half)Rabisplitting[14]:
−G 1− 2 − − 1 + n
+ 1
h i (21co)
R = g2 g 2. (28)
LD+iKD= 1 2(2z2 iG 2)(R+iG )+a1+a2n2 0 q − −
16Rz2 − − n1 The normalized power spectrum of qubit 1 follows from
nP n
+2g 1 (2z iG ) 2(R+z iG ) corr . Eq.(13)withthecoefficientswehaveobtained.Thepositions
G + 2− 2 − 2− + n1 and broadeningsof the four peaks are given respectively by
h i (2o1d) the real and imaginary parts of z and z . Their expressions
1 2
remainvalidin thespontaneousemission case bysetting the
Theyaredefinedintermsoftheparameters
pumpingratestozero.
a1=Gg22 [4G 2++2P1(P2−2G +)−P2G 1], a2=Gg22 P1(P1−g1), blaFcikg)uarned3itsisdeacnoemxpaomspitlieonoifntfhoeursppeecatkrsum(thiSn1(bwlu)ea(inndsroedli)d.
+ + Thesplitpositionsofthefourpeaks, whichindicatethesys-
(22)
tem is in the SC regime, are marked with two vertical blue
and the correspondingfrequenciesand decay rates, that also
lines. The two peaks that correspond to the lower manifold
appearexplicitlyinEq.(12):
transitions,AandDinFig.2(b),appearwithathinblueline.
g g
A +iw =2G +iz , B +iw =2G +iz , (23a) Upper transitions, B and C in Fig. 2(b), appear with a thin
A + 1 B + 2
2 2 red line. All resonances are at the same positions but the
g g
C +iw =2G iz , D +iw =2G iz . (23b) stronger dispersive part of upper transitions leads to a shift
C + 1 D + 2
2 − 2 − of their maximum. The upper transitions are much weaker
in intensity (magnified 30 to be visible) due to the small
Theyalldependontwocomplexparameters,z andz :
1 2 ×
pump. The double excitation of the system is very unlikely
z = (Dsg)2+(iG R)2. (24) (n11 = 0.0004). The lineshape is therefore close to that of
1,2 +± twocoupledharmonicoscillators,plottedwithadashedpur-
Thedegreeofsymmetrqy,Ds,isarealdimensionlessquantity, ple line for comparison. The system is in the linear regime
whereallmodelsofFig.1fortwocoupledmodesconverge.In
between0and√2,givenby
thefollowingsections,wewillseehowthelineshapeschange
(g P +g P)/2 whenenteringthenonlinearregime.
Ds= 1 2 2 1 . (25)
G
p +
This quantity is proper to the coupled qubits case and its B. Twofermions
physical meaning will be clarified later. Its value is linked
to the symmetry between the different parameters. For in-
As noted before, although the expressions for the single-
stance, Ds =1 when all parameters are equal to each other,
timedynamics(populations,coherence,Etc.)fortwocoupled
g =g =P =P. Ontheotherhand,Ds=0ifoneofthepa-
1 2 1 2 fermions (anticommuting operators for modes 1 and 2) are
rameters(anyofthem)ismuchlargerthantheothers.Itleads
thesamethanforthetwocoupledqubits,theirpowerspectra
toarenormalizedcouplingstrength
aredifferent. Ascomparedtothecoupledqubits,thecoupled
G=Dsg, (26) fermionsspectrum assumes a simple and fundamentalform,
closelyrelatedtothatofthetwoharmonicoscillators:thefor-
that reaches a maximum when the parameters are such that mal expression is the same, differing only in the parameters
Ds = √2. Such an enhancement, by √2, is related to the (effectivebroadenings,populations,Etc.). Inparticular,only
7
oneRabiparameter,thesingle-timeRabifrequency,R1TD,de- V. STRONGANDWEAKCOUPLINGREGIMES
termines both the single- and two-time dynamics. The two-
fermionsspectraarethusobtainedbysimplysubstitutingthe
Thestandardcriterionforstrongcoupling(SC)isbasedon
fermionic parameters (Eq. 7) in the expression of the linear
thesplittingatresonanceofthebarestatesintodressedstates.
model[14]. Thismanifestsintheappearanceoft -oscillationsinthetwo-
This simplicity and likeliness to the linear model stems timecorrelatorsandasplittingofthepeaksthatcomposetheir
fromthefundamentalnatureoftheproblem:twoidentical(in- spectrum.
distinguishable)particlescoupledlinearly,obeyingfullytheir In a naive approach to the problem of defining strong-
quantum statistics. The two coupled qubits (or the Jaynes- coupling in a system other than the linear model, one could
Cummingsmodel[16]),bymixingdifferenttypesofparticles thinkthattheconditionforSCis´ (R1TD)=0(atresonance),
(distinguishablemodes)andthereforebreakingcommutation leadingto thefamiliarinequality,g> G 6 . However,thisis
rules, result in the more complex description and richer dy- notthecasewheneverpumpanddecay|ar−e|bothtakenintoac-
namicspresentedintheprevioussection. count. Instead,onemustfindtheconditionforasplittingbe-
tweentheneweigenstates,thatis,thetwopairsofpeaksform-
ingthespectrum. Thepeaksarepositionedsymmetricallyin
twopairsabouttheoriginatw = ´ (z )and,therefore,
C. four-levelsystem(4LS) p ± 1,2
´ (z )=0 or ´ (z )=0 (32)
1 2
Also in the four-level system (with no cross Lindblad 6 6
terms),theexpressionsforthesingle-timedynamics(popula- is the mathematical condition for SC in this system. Given
tions,coherence,Etc.) arethesamethanforthelinearmodel, thattherearetwodifferentparametersz andz onwhichthe
1 2
and herealso theirpower spectraare different. Theparame- conditionrelies, the SC/WC distinction must be extended to
tersforthe4LSspectraareofabosoniccharacter,cf.Eq.(8): cover new possibilities. Thus, instead of only one relevant
parameter,G /g,aswasthecaseinthelinearmodel,SCbe-
tweentwoqu−bitsisdeterminedbythreeparameters:
G˜ =g P , G˜ =g P , (29a)
1 1 1 2 2 2
G˜− G˜ − G /g, G +/g and Ds. (33)
G˜ = 1± 2, (29b) −
± 4 This gives rise to the situations listed in Table I, that are
R˜= g2 G˜2 . (29c) discussedinthefollowingsections.
− −
q
The relevant parameters that characterize the coupling sim- R ´ (z1) ´ (z2) Acronym Typeofcoupling
plifyto: R =0 =0 FSC FirstorderStrongCoupling
| | 6 6
iR =0 =0 SSC SecondorderStrongCoupling
| | 6 6
iR 0 =0 MC MixedCoupling
G˜
R= +R˜, (30a) i|R| 0 6 0 WC WeakCoupling
G | |
+
z =R˜ iG˜ , (30b) TABLE I: Type and nomenclature of coupling for two coupled
1,2 +
± qubits. Beyondtheusualweakcoupling(WC)andstrongcoupling
(heredenotedFSC)encounteredinthelinearmodel,thesystemex-
recoveringtheconventionalstrongcouplingcriterionbasedon
hibitstwonew regions: MixedCoupling (coexistenceof weakand
oneparameteronly,thebosonic(half)RabifrequencyR˜. The
strongcoupling)andSecondorderStrongCoupling(withtwodiffer-
resultingspectralstructurethenconsistsoftwopairsofpeaks entsplittingsoftwopairsofdressedstates).
sittingat ´ (R˜)with:
±
g 3(P +P)+g +g
A +iw = 1 2 1 2 +iR˜, (31a)
A
2 4 A. Vanishingpumpandspontaneousemission
g 3(g +g )+P +P
B +iw = 1 2 1 2+iR˜, (31b)
B
2 4 In the case of vanishing pump, that corresponds as well
gC +iw = 3(g1+g2)+P1+P2 iR˜, (31c) to spontaneousemission, the standard SC and WC hold. In
C
2 4 − this limit, we recover the familiar expression for the half
g2D +iw D= 3(P1+P24)+g1+g2−iR˜. (31d) Rz1a,2bi freq(uRe0ncyigR+,)R21=TDR→0 Ri0g.+[T3h9e].parameters simplify to
→ ± ±
Thepositionsandbroadeningsofthefourpeaksare:
Notethatg arealwayspositiveforanycombinationofthepa- p
p
g g
rameters,incontrastwiththoseoftwobosonicmodes,where A +iw =g +iR , B +iw =3g +iR , (34a)
the system can diverge. Therefore, the values of pump and 2 A + 0 2 B + 0
g g
decayratesherearenotlimited,alwaysleadingtoaphysical C +iw =3g iR , D +iw =g iR . (34b)
C + 0 D + 0
steadystate. 2 − 2 −
8
Fromhere,theassociatedconditionforSCreducestoR be-
0
ingreal,ormoreexplicitlyg> g ,asinthelinearmodelat
vanishingpump[seeFig.2(b)].|In−S|C,thetwopairsofpeaks
p=A,D and p=B,C sit on the same frequencies although
they have different broadenings. Excited states have shorter
lifetime,sinceeachexcitationcandecay. FromEqs.(34),the
two dressed states undergothe transitioninto weak coupling
(WC)simultaneously[asinFig.2(a)].InWC,R iR and
0 0
→ | |
bothparametersz becomeimaginary,giving
1,2
w =0, p=A,B,C,D, (35a)
p
g g
A =g R , B =3g R , (35b)
+ 0 + 0
2 −| | 2 −| |
g g
C =3g + R , D =g + R , (35c)
+ 0 + 0
2 | | 2 | |
with g 0, since g R in this regime. The four peaks
p + 0
≥ ≥| |
collapse into four Lorentzians at the origin, all differing in
theirbroadenings.
As a resultof the two pairs of peakssitting always on the
same two (or one) frequencies, the final spectra can only be
either a single peak or a doublet, both shapes being possi-
FIG.4: (Coloronline)PhasespaceofthesteadystateStrong/Weak
ble in SC or WC regimes (as in the linear model and for
the same reasons[14]). An intuitive derivationand interpre- Couplingregimesasafunctionofpumpforg1=gandg2=g/2. In
StrongCoupling (SC,blue), one can distinguish tworegions, First
tation of these results is given in appendix D, based on the
order (FSC, light blue) and Second order (SSC, dark blue) Strong
so-calledmanifoldpicture[13],whichconsistsinconsidering Coupling. Weak Coupling (WC) isin purple and Mixed Coupling
transitions between eigenstates of a non-hermitian Hamilto- (MC)ingreen.Thedashedbluelinesenclosethetworegionswhere
nian,withenergiesbroadenedbytheimaginarypart. twopeakscanberesolvedinthepowerspectrumofthefirstqubit,
Inthe steadystate case butin thelimitofvanishingpump S1(w ).OnefallsinSCandtheotherinWC.
(the linear regime), only the vacuum and first manifold are
populated. The spectra in this limit converge with the lin-
ear modeland also it can be analyzed in terms of manifolds theinterplayofpumpanddecayisdetrimental(sinceEq(36)
by straightforwardextension. The spectrumin Fig. (3) is an implies that G<g). It is not possible to reach the optimum
example of SC for vanishing pump as we can see from the effectivecouplingandmaximumsplittingofthespectrallines
factthatthelowertransitionpeaks,inblue,dominateoverthe givenby2√2g. Eq.(36)leadstothe followingexplicitcon-
broaderandweakupperpeaks,inred. Inthiscase, thesplit- ditionforg:
tingofthedressedmodesgivesrise toa splittinginthe final
G
spectrum(inblack). Inwhatfollows,wetakethisSCconfig- g> | −| . (37)
uration(g =gandg =g/2)asastartingpointtoexplorethe 1 (Ds)2
1 2
| − |
effectofanon-negligibleincoherentcontinuouspump.
IfR=0, then, also´ (pz )=´ (z )=0. FSC includesthe
1 2
standar6dSCregimeintheabsenceofp6 ump(g> G thatis
B. Non-negligiblepump impliedbyEq.(37)). Itisthemostextendedregio|n−in|Fig.4,
colored in light blue. In this case, the spectrum of emission
Whenpumpistakenintoaccount,allthetypesofcoupling follows the expected pattern: two pairs of peaks, A, D and
B,C, are placed one on top of each other, althoughthey are
listedinTableIareaccessible. TheseareplottedinFig.4as
afunctionofthepumpingrates. ThestartingpointistheRabi differentlybroadened[seethespectrainFig.3].
InFig.5(a)and(b)wetrackthebroadeningsandpositions
frequencyR,Eq.(27),thatiseitherrealorpureimaginary.
First,letusconsiderthecasewhere: of the four peaks (A and D in blue and B andC in red) as a
functionofpump,throughtheSCregionofFig4,onadiago-
R= R G2<g2 G 2 , (36) nallinedefinedbyP2=P1/2.Followingthemfromvanishing
| | ⇔ − − pump,wherethemanifoldpictureisexact,thefourpeakscan
fromwhichfollowsthatz =z ,andtherefore´ (z )=´ (z ). be easily associated with the lower and upper transitions of
1 ∗2 1 2
This is the most standard situation that we already found in Fig. 2(b), and that is why we keep the same color code and
the absence of pumping. It is sketched in Fig. 2(c). To dis- notation. Dressedstates , closetotheHamiltonianones,
|±i
tinguish it from the other types of coupling to be discussed can still be defined in the system, but with the modifiedfre-
shortly,IwillfromnowoncallitFirstorderStrongCoupling quenciesw ´ (z ),bothaffectedequallybydecoherence.
1 1
(FSC).Notethatcondition(36)canonlybesatisfiedifDs<1, By constr±uction, the resulting spectra in this regime can
therefore, when the renormalization of the coupling through only be a doublet or a single peak, depending on the mag-
9
nitude of the broadeningof the peaks (that always increases
with pump and decay) against the splitting of the lines (that
always decreases). As in the limit of vanishing pump, ob-
servingadoubletinthespectradoesnotimplysplittingofthe
dressedstates(andthus,SC)[37],butherethetendencyisal-
waysthesame: thelowerthepumpandthedecay,thebetter
theresolutionofthesplitting.
Second,letusconsiderthesituationoftheRabifrequency
beingimaginary:
R=iR G2>g2 G 2 . (38) FIG.5: (Coloronline)Broadenings(a),(c),andpositions(b),(d)of
| | ⇔ − − thelinesthatcomposethespectraasafunctionofpumpforthedecay
Thisresultsinthreepossibilities, listedinTableI(WC, SSC parameters g1=g and g2 =g/2. In the plots of the first column,
andMC),thatconstitutethethreeremainingregionsdelimited the pump P1 varies with P2=P1/2, moving upwards in the phase
inFig.4. Inwhatfollowswefindthespecificitiesofeachof spaceofFig4. TheverticalguidelineshowsthecrossingfromFSC
thesethreeregimes. toWC.Intheplotsofthesecondcolumn, thepumpP2 varieswith
TheWeakCouplingregime(WC,inpurple)ischaracterized P1 =0.2g, moving in diagonal in the phase space of Fig 4. The
vertical guidelinesshow thecrossing fromFSCtoSSCand finally
by
toMC.Thedashedbluelinerepresentsthesplittingasitisresolved
z =iz , z =iz , z =z G< G R (39) inthefinalspectrum S1(w ). Thecolor code(red-blue andorange-
1 | 1| 2 | 2| 16 2 ⇔ | +−| || green)correspondstothatofthetransitionsinFig.2.
and therefore ´ (z )=´ (z )=0. Note that condition (39)
1 2
is not analyticalin terms of the relevantparameters(33). In
markstheborderbetweenthetwokindsofSC,withtheopen-
WC,thefourpeaksareplacedattheoriginwithfourdifferent
ingofa“bubble”forthepositionsw andw (thatwereequal
broadenings. The dressed states have collapsed in energy to A B
w . in the FSC region), and the convergenceof all the broaden-
1
ings.Inprinciple,onecanexpectthatquadrupletsandtriplets
Up to here, we have remained within the SC and WC re-
may form out of the four peaks. However, the broadenings
gionsalreadyknownfromthelinearmodel.Wenowconsider
andcontributionsofthedispersiveparts(givenbyK )aretoo
thetwonewregionsofSC,propertothecoupledqubits,that p
large to let any fine splitting emerge clearly. The spectra in
IcallSSCandMC,respectively:
thisregionreducetosingletsanddoublets.However,weshow
SSC: Whenbothparametersz arereal,then:
1,2
inSec.VIthroughsomeexamplesthattheymaybedistorted,
z = z , z = z , z <z G> G + R . (40) doubtlesslyreflectingthemultipletstructure.
1 1 2 2 1 2 +
| | | | ⇔ | | ||
MC: Whenz isimaginaryandz real,orequivalently,
1 2
WerefertoitasSecondorderStrongCouplingregime(SSC,
colored in dark blue in the phase space). Here, the broad- z =iz , z = z G R <G< G + R ,
1 1 2 2 + +
enings of the four peaks are equal, g /2=2G , but the po- | | | | ⇔ | −| || | | (|4|1)
p +
sitions of the pairs of peaks are different, w = z and we enterthe last newregionin Fig.4. Thisis a Mixed Cou-
A,C 1
w = z . The reason is that the bare energi±es| o|f the plingregime(MC,coloredingreeninthephasespace)where
B,D 2
±| |
modes undergo a second order anticrossing induced by the the two inner peaks, A and C—as well as the reconstructed
interplaybetweencoupling,pumpanddecay.Theenergiesof eigenstate I —have collapsed at the origin, like in WC.
thedressedstatesareaffecteddifferentlybydecoherence,up However,th|e±twio outerpeaks, BandD—aswellas O —
to the point where we may picture the physicsin terms of a are still split. As in SSC, the broadening of the pea|ks±dioes
newtypeofeigenstates. TheassociationoftheAandD(with notallowforadistinctionbetweenupperoflowerresonances.
w ) as the peaks correspondingto lower transitions and B The collapsed resonance is in this case at the bare energy
A,D
andC(withw )touppertransitionsiscompletelyarbitrary w =0becausethatisthetotalaveragebareresonanceinthe
B,C 1
inthisregion,giventhatthebroadeningsofthepeaks,which system. In Sec. VIA we will see that when the qubits are
ledustosuchassociationinFSC,arenowequal.Thisimplies detuned,thisresonancehappensat(w +w )/2= D /2.
1 2
−
that, ratherthantwo dressed states, and + (Fig.2(b)) Again,althoughonemayexpectatripletinMC, onlydis-
|−i | i
asintheconventionalstrongcoupling(FSC),thesystemnow torted singlets are observed in the best of cases due to the
exhibitsfourdressedstates: I , I , O and O . They broadeninganddispersiveparts. InFig. 5(c) and(d) we can
+ +
areplottedinFig.2(d): I |(−reisp|. Oi | )−hiaveen|ergiiessplit see the transition from SSC into MC, at the second vertical
at z ,givingrisetoth|e±ininerpea|ks,±iingreen(resp. z , line.
1 2
±| | ±| |
giving rise to the outer peaks, in orange). The physicalori- Notethat,inthissystem,thepumpingmechanismisequiv-
ginofthisremarkabledeparturefromtheconventionalstrong alent to an upward decay, due to the ultimate saturation of
couplingpicturewillbediscussedinsectionVII. thequbitandthe symmetryinthe schemaoflevelsthatthey
We can see how peak broadenings and positions change form. Themasterequationissymmetricalunderexchangeof
when going from FSC to SSC in Fig. 5(c) and (d). In this thepumpandthedecay(g P)whenthetwo-levelsofboth
i i
↔
case, we track the peaks by varying P for a fixed P, mov- qubit are inverted ( 0,0 1,1 and 1,0 0,1 ) [40].
2 1
| i↔| i | i↔| i
ing upwards in the phase space. The first vertical guideline Consequently,theparametersz ,z andR,andalsothepopu-
1 2
10
lationsofallthelevels, aresymmetricinthesame way,asit
happenswithjustonequbit. Inothersystems, likethelinear
model, the Jayne-Cummings model or simply a single har-
monicoscillator,theeffectofthepumpextendsupwardstoan
infinitenumberofmanifoldswhilethedecaycannotbringthe
systemlowerthanthegroundstate. Thereisnonaturaltrun-
cationforthepump(thatultimatelyleadstoadivergence),as
there is for the decay. But with coupled qubit, state 1,1
| i
is the upper counterpart of 0,0 , undergoing a saturation.
| i
Thisimplies,forinstance,thatthe(transient)dynamicsinthe
limitofvanishingdecayisexactlythesameasthatofvanish-
ing pumpand that in such case we can also apply the mani- FIG.6: Factorscontributing toDs: (a) √G 1G 2/[(G 1+G 2)/2] as a
foldmethodtoobtaintherightpositionsandbroadeningsasa function of G 1,G 2 and (b) √r1+r2 2r1r2 as a function of r1,r2.
−
functionofpump,inthesamewaythatwedidasafunctionof The values corresponding to the contour lines are marked on the
decayonly. Weonlyhavetotakeintoaccountthementioned plots.Bothfunctionstakevaluesfrom0(darkblue)to1(light).
symmetry consistently. As long as the dynamics moves up-
wardsordownwardsonly,evenwhenintermediatestatesare
coupled, the manifold picture is suitable. The manifold di- A. Optimallypumpedcases:g<G √2g
≤
agonalization breaks, however, in the presence of both non-
negligiblepumpanddecay.ThisisdiscussedinappendixD. Let us explore the optimally pumped cases by consider-
ingparametersonthediagonalG =G inFig. 6(a)together
1 2
withtheantidiagonalr +r =1inFig.6(b). Thereservoirs
1 2
VI. PARTICULARCASES haveoppositenaturebutinteractwithequalstrengthwiththe
qubits. Thiscorrespondstothesituationwherethedecayand
In this section, we illustrate the rather abstract previous pumpingparametersareequalinacrossedway:
discussions with examples. The symmetry in the decay and
pumping rates determines the effective coupling, emission P1=g2, and P2=g1. (45)
propertiesand dressed states. Let us start by expressing Ds,
the magnitude quantifying such symmetry, in terms of the The system has a totalinputthatis equalto the totaloutput,
reservoirparameters(G iandri), PTOT=P1+P2=gTOT=g1+g2,andalsoequalPurcellrates,
Q =Q . Theexcitedandgroundstatesareformallyequiva-
1 2
√G G lentinthedynamics.
Ds=√2 1 2 r +r 2r r . (42)
(G +G )/2 1 2− 1 2 Figure 7 shows the different coupling regimes accessible
1 2
p with this configuration, as a function of P1 and P2 with the
Inthisform,itsphysicalmeaningismoreclear.Therearetwo samecolorcodethaninFig.4. ThisconfigurationisinFSC
separate factors to discuss: the symmetry in the strength of only when all parameters are equal, Ds =1, (blue line) and
thecouplingstothereservoirs,givenby√G 1G 2/[(G 1+G 2)/2] there is totalsymmetry in the system. Otherwise, one of the
andplottedinFig.6(a),andthesymmetryinthenatureofthe newtypeofcoupling(SSCinblueorMCingreen)isrealized
reservoirsgivenby√r1+r2 2r1r2,plottedinFig.6(b). asthecouplingiseffectivelyimproved,G>g.Intheinsetthe
−
Thecouplinggisenhancedwhen typeofspectralshapesthatresultsisshown.
g<G≤√2g, thatis, 1<Ds≤√2, (43) Lineshape Lsl Lcon
singlet 1 2
whichhappenswhenr >1/2andr <1/2(ortheotherway
1 2 distortedsinglet 1 6
around). This corresponds to the two squared regions with
doublet 3 4
lighter colors in Fig. 6(b). Then, the two reservoirs are of distorteddoublet 3 8
oppositenatures: the reservoirof the first qubitprovidesex- triplet 5 6
citations(P >g )whiletheotherabsorbsthem(P <g ). If quadruplet 7 8
1 1 2 2
thisisaccompaniedbysimilarinteractionstrengths,G G ,
1 2
enhancementoccurs.Werefertothesesituationsasopti∼mally TABLEII:ThelineshapesS1(w )aredefinedbytwoquantities: Lsl
pumpedandstudytheminSec.VIA. isthenumberoftimesthatS1(w )changesslope,thatis,thenumber
On the other hand, if the reservoirs are of the similar na- ofrealsolutionstotheequationdS1(w )/dw =0;Lconisthenumber
tures,bothr ,r 1/2or 1/2,bothprovidingorabsorbing of times that S1(w ) changes concavity, that is, the number of real
particles,the1nt2he≥systemi≤sdetrimentallypumped: solutionstotheequationd2S1(w )/dw 2=0.
0 G g, thatis, 0 Ds 1. (44) TheverticalaxisinFig.7,withP1=g2=g andP2=g1=0,
≤ ≤ ≤ ≤ is illustrative of all the possible coupling regions and line-
It correspondsto the two squaredregionswith darkercolors shapes. This is the extreme situation of optimal pumping
inFig.6(b).WestudythisinSec.VIB. where the two reservoirs interact equally strongly with the