Table Of ContentSubradiance and entanglement in atoms with several independent decay channels
Martin Hebenstreit, Barbara Kraus, Laurin Ostermann and Helmut Ritsch
∗
Institut fu¨r Theoretische Physik, Universit¨at Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria
(Dated: January 9, 2017)
Spontaneous emission of a two–level atom in free space is modified by other atoms in its vicinity
leading to super- and subradiance. In particular, for atomic distances closer than the transition
wavelength the maximally entangled antisymmetric superposition state of two individually excited
atomic dipole moments possesses no total dipole moment and will not decay spontaneously at
all. Such a two-atom dark state does not exist, if the atoms feature additional decay channels
towards other lower energy states. However, we show here that for any atomic state with N −1
independent spontaneous decay channels one can always find a N-particle highly entangled state,
which completely decouples from the free radiation field and does not decay. Moreover, we show
that this state is the unique state orthogonal to the subspace spanned by the lower energy states
7 with this property. Its subradiant behavior largely survives also at finite atomic distances.
1
0 PACSnumbers: 42.50.Pq,42.50.Ct,42.50.Wk,07.60.Ly
2
n
Thedecayofanexcitedatomicstatetowardslowerly- multi-partite entangled states, where all N 1 indepen-
a −
ing states via spontaneous emission is one of the most dent decay channels are suppressed. For these states the
J
striking consequences of the quantum nature of the free totaldipolemomentsonalloftheseN 1transitionssi-
5 −
radiation field [1]. Heuristically introduced even be- multaneously vanish and at least in principle the optical
] fore e.g. by Einstein, the spontaneous emission rate excitation in this state will be stored indefinitely.
h Γ = ω3µ2 , called A-coefficient, is proportional to the After introducing our model atom system and the
-p squar3e(cid:15)d0πt(cid:126)rca3nsition dipole moment µ2 between the upper generalized, unique multi-atom dark states, we will dis-
nt and lower atomic state and the third power of the tran- cuss their special entanglement properties and possi-
ble quantum information processing based schemes to
a sition frequency ω [2].
u Interestingly, it turns out that for several particles prepare them. In the final part of the paper we nu-
q the emission process is not independent but can be col- merically study some more realistic geometries using
[ Lambda type atoms, where the effect of dipole-dipole
lectively enhanced or reduced depending on the atomic −
coupling leads to a reduced but still finite lifetime as
1 arrangement [3]. It was already noted some time ago
populationofthesingleexciteddarkstateviadecayfrom
v that these superradiant and subradiant collective states,
1 where a single excitation is distributed over many parti- multiply excited states will accumulate.
7 Interestingly,arelatedphenomenonappearsinV-type
cles,areentangledatomicstates[4,5]. Althougharecent
4 atoms with two excited and one ground state, where a
classical coupled dipole model also leads to subradiance-
1 singlegroundstateatomcanpreventthedecayofseveral
likephenomena[6],themostsuperradiantandtheperfect
0
excitations. This is only shortly exhibited in Appendix
. dark states for two two-level atoms with states (g , e )
1 | (cid:105) | (cid:105) II to avoid confusion with the Λ-geometry.
arethemaximallyentangledsymmetricandantisymmet-
0
ric dipole moment superposition states
7
e
1
ψ =(eg ge )/√2. (1) | i
:
v | ±(cid:105) | (cid:105)±| (cid:105)
i While superradiance on a chosen transition persists in
arX tcmhauyeltccilahesavene,nlwealth,oetmnhsetrwheeitihsatnsoeomvecropamolspdsleeecstsaeeylsycmhdaaonrrkneetslshtaaftrneomofonraestdwineo-- |g1i ��2�1� �3� �i��N��1 |gN�1i
gle excited state e to several lower lying states g as
| (cid:105) | i(cid:105) g
schematically depicted in Fig. 1. Hence, in practise the 2 g g
| i 3 i
| i | i
observation of subradiant states is much more difficult
than seeing superradiance, as all other decay channels
need to be excluded [7]. FIG. 1. Level scheme of an atom with several independent
decay channels of different polarization or frequency.
In this paper, we introduce a new class of subradiant
or dark states appearing for atoms with several indepen-
dent transitions. As a key result of this work we find Model: Let us assume a collection of N identical N-
that for systems of N particles one can construct highly level emitters with a set of N 1 low energy eigenstates
−
g , where i 1,...,N , which are dipole coupled to
i
| (cid:105) ∈ { }
a higher energy state e separated by the excitation
energies (cid:126)ω (see Fig. 1|)(cid:105). The atomic center of mass
i
∗ [email protected] motion is treated classically with fixed positions ri for
2
i 1,...,N , which we will assume to be close-by, g ,occupationstriviallywillbestationaryunder with
i
∈ { } | (cid:105) L
i.e. within a cubic wavelength. For each atom i and L[ρ ] = 0. The much more interesting task is to find
g
transition j we define individual Pauli ladder operators states, ρ , featuring atomic excitations, which will not
e
σji± describing transitions between the i-th atom’s ex- decay under L and are stationary with eigenvalue zero,
citedstate|e(cid:105)i andeachofitsj lowerenergystates|gj(cid:105)i, i.e. L[ρe]=0.
respectively. For the case of two two-level atoms such dark states
The coupling of each atomic transition i,j to the elec- are well known and have been confirmed experimentally
tromagneticvacuumleadstoanindividualfreespacede- decades ago [11]. A generic dark state can then be writ-
cayrateΓi. Asallatomsarecoupledtothesamevacuum ten as in Eq. (5), i.e., ψ2 = ψ . As a central claim of
modes, thjese decay rates are modified by pairwise inter- this work we show tha|tdt(cid:105)his f|or−m(cid:105)ula can be generalized
actions with neighboring atomic transitions k,l, which to the case of N atoms with N 1 independent optical
−
upon elimination of the field modes can be described by transitions between the upper state 0 = s0 = e and
| (cid:105) | (cid:105) | (cid:105)
mutualdecayratesΓijkl,withΓiji =Γj [3,8]. Note,thatin N −1 lower states |i(cid:105)=|si(cid:105)=|gi(cid:105) in the form
additiontomodifyingthedecayproperties,thecollective
1 (cid:88) (cid:79)
coupling of the atoms to the vacuum modes also induces ψN = sgn(π) s , (5)
energy level shifts Ωik as presented in [8]. For simplicity | d (cid:105) √N! | π(i)(cid:105)
we will first assumejal highly symmetric arrangement of π∈SN i
the particles, so that all particles acquire equal energy where the sum runs over all permutations π of N ele-
shifts Ωik = Ω , which can be incorporated in effective ments. Using the criterion for pure states to be station-
jj j
transition frequencies [9]. Our central interest targets ary under given in [12], we show in Appendix I that
L
the modifications of the emission rates via the collective thisN N-levelstateoftotalspin0istheuniquestation-
decay mechanism. ary state which is orthogonal to the subspace where all
In terms of the operators defined above with the ex- particles are in g for some i. A symmetric variant of
i
| (cid:105)
cited state energy set to zero, the dipole coupled atomic this state, denoted by ψN , with all positive signs will
| sr(cid:105)
Hamiltonian is given by [9] be its super-radiant analogue.
(cid:88) (cid:88)(cid:88)
H = −ω¯jiσji−σji++ Ωijkσji+σjk−. (2) Decayofsingleexcitationsinatriangulararray
i,j i=k j 1.0
(cid:54) Decayof ψ3
| di
The full dynamics of the coupled open system includ- Indep. decay
0.8
ing decay is then governed by a master equation for the Decayof ψ3
| si
density matrix ρ of the whole system of N multilevel
0.6
emitters, reading
∂ρ 0.4
=i[ρ,H]+ [ρ]. (3)
∂t L
0.2
Following standard quantum optical assumptions and
methods,theeffectiveLiouvillianforthecollectivedecay 0.0
summed over all transitions and atom pairs reads [9, 10] 0 1 2 3 4 5 6 7 8 9
Γt
L[ρ]=12(cid:88)i,k (cid:88)j Γijk(cid:2)2σji−ρσjk+ (4) FIG. 2. Upper state population decay of three close by in-
teracting Λ-type atoms in a triangular configuration of size
−σji+σjk−ρ−ρσji+σjk−(cid:3). d (cid:28) λ with Γ1j2 = 0.95Γ starting from the ideal dark state
(blue line). For comparison the dashed black line gives the
While this can be a rather complex and complicated caseofindependentatomdecay,whiletheredlinecorresponds
expression for a general atomic arrangement [9], in the to a fully symmetric state with a superradiant decay on both
case of atoms much closer to each other than the tran- transitions.
sition wavelength, all Γik = Γ become approximately
j j
independent of the atomic indexes (i,k), essentially re- For a 3-level Λ-atom (N =3) one explicitly gets
ducingtoasingleconstantΓ . Forsimplicity,wewillfirst
j
1
also assume equal decay rates on all transitions Γj = Γ, ψ3 = eg g + g g e + g eg
i.e. equal dipole moments and Clebsch-Gordon coeffi- | d(cid:105) √6{| 1 2(cid:105) | 1 2 (cid:105) | 2 1(cid:105) (6)
cients. This will hardly be exactly true for any real eg g g g e g eg .
2 1 2 1 1 2
atomic configuration (besides a J = 0 to J = 1 tran- −| (cid:105)−| (cid:105)−| (cid:105)}
sition), but it will not change the essential conclusions which is a state within the set of maximally entangled
below. tripartite states of qutrits [13].
Collective atomic dark states: Obviously any This state has zero total dipole moment µ =
j
atomic density matrix ρ involving only ground states, (cid:10)(cid:80) σi(cid:11) = 0 on both transitions. As shown in Fig. 2,
g i j
3
for a sub-wavelength triangular atomic configuration it operators S, ψN S N ψN , which implies several
| d (cid:105) ∝ ⊗ | d (cid:105)
exhibitssubradiantdecaysubjecttothemasterequation important properties. First, if one particle is measured
Eq.(3). Astheatomswillnotonlyundergocollectivede- inanybasisandthemeasurementoutcomeaswellasthe
cay but also experience energy shifts from the resonant chosenbasisareannounced,thentheotherN 1particles
dipole-dipole coupling Ωijk in Eq. (2), the dark states in canbetransformeddeterministicallytothes−tate|ψdN−1(cid:105)
Enaqm. (ic5)minixignegnewriatlhaorethneortsteaigteenssitnadtuesceosfaHfi.niHteenlcifee,tidmye- bnyotpdearrfokrmfoirnNg LU1saotnolmy.sNwoitthe,Nhowe1vedre,ctahyacth|aψndNn−el1s(cid:105). is
as it is the case for conventional dark states [14]. Note Second, the g−eometric measure−of entanglement [19]
thatthesubradiantstatesdiscussedherearenotthedark canbecomputedeasilyandoneobtainsthatE (ψN )=
stetamtse.sTaphpereea,rianpgairntictuwloarlasuseprerepxocsititaitoinonofogfrΛou-tnydpestasytess- 1pe−ndmixaxI|)a1[2(cid:105),0..]..,|aDNu(cid:105)e(cid:12)(cid:12)(cid:104)tao1,t.h.a.t,,aWN|ψ=dN(cid:105)1(cid:12)(cid:12)21= 1ψ−NN1g!ψ(|Nsedei(cid:105)sAapn-
decouples from the laser excitation for each atom sepa- N! −| d (cid:105)(cid:104) d |
entanglement witness (see e.g. [21]), i.e. tr(Wρ ) 0
rately [15] and leads to a coherent population trapping sep ≥
for any separable state ρ , and there exists a state ρ
without any excitation. sep
s.t. tr(Wρ)<0.
As an important consequence of the uniqueness of the
Usingthiswitness, itcanbeshown, thattheentangle-
dark state, no such state can exist for a smaller num-
ment of the states ψN is persistent under particle loss.
ber of atoms. Hence, when considering M atoms with | d (cid:105)
In otherwords, if thepartial traceof thestate ψN over
N 1 independent optical transitions between the up- | d (cid:105)
− any particle is performed, the resulting state is still en-
per state e and their N 1 lower states g , where the
| (cid:105) − | i(cid:105) tangled. ForN =3thiscanbeverifiedbycalculatingthe
emitted photons on each transition are distinguishable,
negativity of the reduced density matrix. For higher N,
the following picture emerges: for M < N only ground
however, the statement can be proven using the witness
states are stationary under . In case M > N, how-
L mentioned before and even holds under loss of a large
ever, extra stationary states involving excitations can be
portion of the particles.
found. They are given by tensor products of states that
Preparing collective dark states: As these totally
arestationaryforpartsofthesystemandsuperpositions
antisymmetric states have intricate entanglement prop-
of these states. To give a simple example for the case
erties they cannot be prepared from a product ground
M =6 and N =3 the states
state with simple local operations. In the following we
|ψ(cid:105)=(α|ψd3(cid:105)⊗|ψd3(cid:105)+β|gigj(cid:105)⊗|ψd3(cid:105)⊗|gk(cid:105))/√2 (7) bpreogpeonseeratwlizoedwatoysptroepparreepaψrNe thfoersNtat>e 3|ψ.d3(cid:105), which can
are dark for any α,β C. In both methods we init|iadlly(cid:105)prepare the state ψ =
Entanglement pro∈perties of the dark states: The (01 10 )/√2 for two of the particles denoted|as−p(cid:105)ar-
dark states ψN given in Eq. (5) are complex entangled t|icle(cid:105)s−1|an(cid:105)d 2, which can be achieved by applying a
| d (cid:105)
states, whose mathematical properties havebeen consid- CNOT to the two particles in the initial product state
ered before and will be shortly recapitulated here. For (0 1 )/√2 1 , respectively.
instance, it has been shown that ψN can be used to | I(cid:105)n−|th(cid:105)e firs⊗t |m(cid:105)ethod we then prepare particle
| d (cid:105)
solve the Byzantine agreement problem, the N strangers 3 in the state 2 and apply the 3-qutrit gate
problem, the secret sharing problem, and the liar de- e i2π/9(X X X+h.c|.),(cid:105)whereX = 1 0 + 2 1 + 0 2,
− ⊗ ⊗
tection problem [16, 17]. Moreover, it has been shown in order to obtain the state ψ3 u|p(cid:105)t(cid:104)o|loca|l(cid:105)p(cid:104)ha|se|ga(cid:105)t(cid:104)es|.
that there exists no local hidden variable model describ- This preparation procedure|cadn(cid:105)be easily verified noting
ing quantum predictions for the state ψN . To this end, that X3 =1.
| d (cid:105)
generalized Bell inequalities that are violated by ψN Alternatively,weprepareparticle3in + =(0 + 1 +
have been constructed for any N [17]. | d (cid:105) 2 )/√3 and apply the two-qutrit unita|ry(cid:105)U =| 0(cid:105) |0(cid:105)
What makes these states so useful for the above men- X| (cid:105)+ 1 1 X2+ 2 2 1 on the particle pai|rs(cid:105)((cid:104)3,|1⊗)
tioned tasks are their rather special entanglement prop- and (|3,(cid:105)2(cid:104)) i|n⊗order t|o(cid:105)o(cid:104)bt|a⊗in ψ3 .
enrottieesth[1a7t],thwehbiciphawrteiteweilnltdainsgculesmseinnttshheafroeldlobweitnwge.enFiarsnty, Let us point out, that giv|end(cid:105)|ψdN−1(cid:105) for N −1 parti-
cles, the state ψN can be prepared by preparing par-
oIconlffe|toψρhtdjNeh,e(cid:105)pwraohwrvtieocirrchdlaeislssl,patohnaberdttaritiecnhdleeeusdcrbeebusdyttdpopefeanrrtsfthoiitercymleemnijnsa,getimrstihbxpelrefooppirasorarmtntiaiyaolxnptiaamrlraattciloe-. (cid:80)tUiciNl=e=−0N2(cid:80)|ii+Nin=−0111(cid:105)/|(cid:104)√ii(cid:105)||N,(cid:104)itd|o(cid:80)(cid:105)⊗aiNl=lX−0p1ia(+r−1t,i1c)lw(eNhp−ea1rie)r(s1X+(Ni),|=ij(cid:105)).a|0nH(cid:105)de(cid:104)nNacpe−p,l|1yψ|idNn+g(cid:105)
identity. This also implies that the state is contained in can be prepared recursively. In a similar manner the
the maximally entangled set [13] as it cannot be reached state ψN can be prepared. This can be achieved by us-
| sr(cid:105)
from any other (not Local Unitary (LU)-equivalent [18]) ing ψ instead of ψ as the initial state of particles 1
+
state by Local Operations and Classical Communication and|2 a(cid:105)nd omitting|th−e(cid:105)minus sign in the initial state of
(LOCC). Furthermore, it can be shown easily that these particleN. However, thepropertiesof ψN areverydif-
| sr(cid:105)
states can be transformed via LOCC into another pure, ferentfrom ψN ,ase.g. ψN hasmuchlesssymmetries.
| d (cid:105) | sr(cid:105)
entangled state. Another difference can be found in the geometric mea-
Animportantpropertyof ψN isthatforallinvertible sure of entanglement E (ψN ) = 1 N!/NN [20, 22],
| d (cid:105) g | sr(cid:105) −
4
which is smaller than Eg for the dark state. Decayfromthefullyinvertedstateinachain
Dissipative dynamics of dipole-dipole coupled 100
Decayof ψ
atom arrays: In the following we will exhibit the col- | 1ei
Dickefor ψ
lectivedynamicsofthesystemforvariousconfigurations, 10−1 | 1ei
where we restrict ourselves to the case of three atoms Decayof|ψ1gi
with two decay channels, i.e. three Λ-systems, in a equi- Decayof|ψd3i
lateraltriangleor,alternatively,anequidistantchain. As 10−2
shown in Fig. 2 at a small but finite distance the dark
statewilldecaymuchslowerthanindependentatoms. In
10−3
addition, changing all the minus signs to plus signs leads
to superradiance on both transitions.
10−4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Decayofdifferentstates Γt
1.0
FIG. 4. Decay of three Λ-type atoms in an equidistant
0.8 chain of distance ≈ λ/4 with nonequal Γik starting from the
totallyinvertedstate|eee(cid:105). Theredlinegivestheexcitedstate
Decayof ψ3
0.6 | di population per atom, the blue line√shows the population of the
Decayof|ψunpi two ground states (|g1(cid:105)+|g2(cid:105))/ 2 and the black line gives
0.4 Decayof ψ the dark state |ψ3(cid:105) fraction during the decay. For compari-
| pumpi d
son the red dashed line exhibits ideal collective atomic decay
0.2 with all equal Γik (Dicke case). Note, that during the evolu-
tionthedark(gray)state,whichdecaysmuchslower,becomes
populated partially.
0.0
0 1 2 3 4 5 6 7 8 9
Γt
tive emission [8], which has been proposed and used for
FIG. 3. Upper state population decay for three close spaced tailored deterministic entanglement generation between
lambda-type atoms for three different single excited initial the ground states of interacting Λ-atoms [24, 25].
states. The blue line corresponds to the pure dark state |ψ3(cid:105),
d Recently we also became aware of another, even
the red line gives the case of the two atom dark state for opti-
√
more complicated scheme based on three M-shaped 5-
cally pumped atoms 1/ 2(|eg1(cid:105)−|g1e(cid:105))|√g1(cid:105) and green corre-
spondstoanunpolarizedproductstate1/ 2(|eg (cid:105)−|g e(cid:105))|g (cid:105) level atoms interacting via three coupled cavities and
1 1 2
involving all three atomic states lasers [26]. Here a dark tripatite entangled state involv-
ing only atomic ground states of the three atoms with
As a second example we demonstrate the surprising very similar properties to our case N =3 is found to be
fact that a nearby atom in the final state of the transi- efficientlypopulatedbyadeterministicdissipativeprepa-
tioncanbeusedtosuppressthedecayofanatomexcita- ration scheme.
tion. For this we start from an anti-symmetric two-atom A central question in our configuration concerns the
state ψ2 = (g ,e e,g )/√2, which will not decay extend to which a dissipative preparation works as well
| d(cid:105) | 1 (cid:105)−| 1(cid:105)
onthefirsttransitionto g . Thisstate,however,decays for the case of several independent decay channels. Let
1
| (cid:105)
on the second transition towards (g ,g g ,g )/√2. us point out that, by construction, it is obvious that the
1 2 2 1
Now, let us add a third atom in eit|her of(cid:105)t−he|two g(cid:105)round darkstate, ψN ,isalsodecoupledfromanyfurthersym-
| d (cid:105)
states. As shown in Fig. 3, a third atom prepared in g metric laser excitation, i.e. an excitation by a laser on
1
| (cid:105)
will not prevent decay (red line), while a third nearby anytransitionsusinganequalphaseoneachatom. Hence
atom in the state g partially prevents total decay and the state is also dark in absorption measurements simi-
2
| (cid:105)
results in a finite excited state population probability at lar to coherent population trapping in the ground state
long times(green line). Hence, after some time the sys- manifold [15].
tem has either decayed to (g ,g ,g g ,g ,g )/√2 In a first approach to populating the dark state one
1 2 2 2 1 2
| (cid:105)−| (cid:105)
or ends up in the dark state ψ3 . Thus preparing two can simply start from the totally inverted state eee for
| d(cid:105) | (cid:105)
atom dark states in the vicinity of an unpolarized state threeatomsplacedatasuitablefinitedistance,wherethe
ofindependentgroundstateatomsprovidesamethodfor off diagonal elements of the matrix Γik acquire negative
probabilistic preparation of the dark states. values. In Fig. 4 we demonstrate this mechanism for
It is known that in spatially extended systems with a three qutrit chain with a distance of about λ/4. A
non-uniform radiative coupling coefficients Γik, there is comparison of the excited state fraction decay for the
no perfect dark state but only long lived subradiant finitesizedchain(redline)withtheidealcollectivedecay
states[23]. Inthiscasethefreespacespontaneousdecay (dashed red line) shows a slowdown of the decay at later
from a multiply excited state can also sometimes end up times,whereasmallfractionofthepopulationendsupin
insuchagraystate,thuscreatingentanglementbycollec- the dark state (black line). As the dark state has only a
5
verysmalloverlapwithanyproductstate,thisfractionis partite states with a plethora of quantum information
small but can become relatively important at late times. applications. They can be prepared by a sequence of a
Since the ideal dark state ψ3 acquires a finite lifetime bipartite or tripartite gates or via tailored spontaneous
| d(cid:105)
at finite distances, this fraction decays as well but at a emission from multiply excited states in optical lattices.
much slower rate. A generalization to multiple excitations and several ex-
cited states can be envisaged.
Conclusions: Asourkeyresultweshowthatthecon- Acknowledgements WethankM.Moreno-Cardoner,
cept of dark or subradiant states can be generalized to D. Chang and R. Kaiser for helpful discussions and ac-
multiple decay channels, if one includes one more parti- knowledge support from DARPA (LO and HR) and the
cle than decay channels. The corresponding dark states Austrian Science Fund (BK and MH) (FWF) grants
are completely anti-symmetric, highly entangled multi- Y535-N16 and DK-ALM: W1259-N27.
[1] P. A. Dirac, in Proceedings of the Royal Society of Lon- [25] A. A. Svidzinsky, X. Zhang, and M. O. Scully, Physical
donA:Mathematical,PhysicalandEngineeringSciences, Review A 92, 013801 (2015).
Vol. 114 (The Royal Society, 1927) pp. 243–265. [26] X. Shao, Z. Wang, H. Liu, and X. Yi, Physical Review
[2] V. Weisskopf, Naturwissenschaften 23, 631 (1935). A 94, 032307 (2016).
[3] R. Lehmberg, Physical Review A 2, 883 (1970). [27] Notethatstatescontainingnoexcitationsmayalwaysbe
[4] Z.FicekandR.Tana´s,PhysicsReports372,369(2002). added, as they are stationary.
[5] B.Bellomo,G.Giorgi,G.Palma, andR.Zambrini,arXiv
preprint arXiv:1612.07134 (2016).
[6] G.Facchinetti,S.Jenkins, andJ.Ruostekoski,Physical I. APPENDIX
Review Letters 117, 243601 (2016).
[7] W. Guerin, M. O. Arau´jo, and R. Kaiser, Physical Re-
view Letters 116, 083601 (2016). Proofs of some properties of |ψdN(cid:105): In this ap-
[8] L.Ostermann,H.Zoubi, andH.Ritsch,OpticsExpress pendix, we prove some of the statements about the state
20, 29634 (2012). ψN , given in Eq. (5), made in the main text.
| d (cid:105)
[9] S.Kra¨merandH.Ritsch,TheEuropeanPhysicalJournal First, we show, that the state ψN is the unique state
| d (cid:105)
D 69, 1 (2015). (up to superpositions with states containing no excita-
[10] H. S. Freedhoff, Physical Review A 19, 1132 (1979). tions) that is stationary for a system consisting of N
[11] P. Grangier and J. Vigu´e, Journal de Physique 48, 781 atoms with N 1 independent optical transitions be-
(1987). −
tweentheupperstate e andN 1lowerstates g . To
[12] B.Kraus,H.P.Bu¨chler,S.Diehl,A.Kantian,A.Micheli, | (cid:105) − | i(cid:105)
this end, we make use of the criterion for pure states to
and P. Zoller, Physical Review A 78, 042307 (2008).
be stationary under derived in [12]. These conditions
[13] J.I.deVicente,C.Spee, andB.Kraus,PhysicalReview L
read as follows.
Letters 111, 110502 (2013).
[14] D. Plankensteiner, L. Ostermann, H. Ritsch, and 1. Q Φ =λ Φ for some λ C,
C. Genes, Scientific Reports 5 (2015). †| (cid:105) | (cid:105) ∈
[15] B.Dalton,R.McDuff, andP.Knight,JournalofModern 2. c = λ Φ for some λ C with (cid:80) g λ 2 =
Optics 32, 61 (1985). l l| (cid:105) l ∈ l l| l|
Re(λ),
[16] M.Fitzi,N.Gisin, andU.Maurer,PhysicalReviewLet-
ters 87, 217901 (2001). (cid:80)
[17] A. Cabello, Physical Review Letters 89, 100402 (2002). where Q = P − iH, P = lglc†lcl, and cl are the
quantum jump operators with dissipative rates g [12].
[18] Note that LUs do not alter the entanglement properties l
inthestateastheycanbeappliedreversibly.Hence,only Here, we have t(cid:80)hat {cl}l = {Sl−}l, with the jump op-
non-LU transformations are relevant in this context. erators Sl− = jσl−,j. As the Sl− are nilpotent we
[19] H. Barnum and N. Linden, Journal of Physics A: Math- get λ = 0 l. Making a general ansatz for Φ , Φ =
l
ematical and General 34, 6787 (2001). (cid:80) c ∀ i ,...,i ,wegetthenecess|ar(cid:105)yc|on(cid:105)di-
[20] M. Hayashi, D. Markham, M. Murao, M. Owari, and tioin1,.c..,iN i1,..=.,iN0| 1i ,...,Ni(cid:105), where at least one i = e
S. Virmani, Physical Review A 77, 012104 (2008). i1,...,iN ∀ 1 N k
[21] B. M. Terhal, Linear Algebra and its Applications 323, [27]andnoik =gl,for|Φ(cid:105)tobeaneigenstateofSl−. As,
according to the conditions, this must hold for all l, we
61 (2001); M. Lewenstein, B. Kraus, J. I. Cirac, and
get c = 0, unless i ,...,i = e,g ,...,g .
P. Horodecki, Physical Review A 62, 052310 (2000). Hencie1,,...Φ,iN = (cid:80) c˜{π1(1),...N,}π(N{) . F1urtherNm−o1r}e,
[22] M.Aulbach,D.Markham, andM.Murao,NewJournal | (cid:105) π∈Sn π| (cid:105)
of Physics 12, 073025 (2010). by applying Sl− onto |Φ(cid:105) one can derive the necessary
[23] H. Zoubi and H. Ritsch, EPL (Europhysics Letters) 82, conditions
14001 (2008).
[24] A. Gonza´lez-Tudela, V. Paulisch, D. Chang, H. Kimble,
and J. I. Cirac, Physical Review Letters 115, 163603 c + (8)
i1,...,ir−1,e,ir+1,...is−1,gl,is+1
(2015).
+c =0 i .
i1,...,ir−1,gl,ir+1,...is−1,e,is+1 ∀ j
6
Using this argument consecutively three times, it fol- obtain E (ψN )=1 1 .
g | d (cid:105) − N!
lows that c˜ = c˜ , for arbitrary transpositions σ.
π σ π
This implies Φ − ψ◦N and hence completes the proof.
| (cid:105)∝| d (cid:105)
II. APPENDIX
Generalization to V-systems: two excitation
dark state A second generic configuration of a three
Let us now show that the geometric measure of
entanglement of ψN is given by E (ψN ) = 1 level system with decay is a V type system, where two
Tmoax|tah1i(cid:105)s,...,e|anNd(cid:105),(cid:12)(cid:12)(cid:104)aw1e|,.d.e.x(cid:105),palNici|tψldyN(cid:105)(cid:12)(cid:12)fi2n=d 1o−nNge1!|o(sfdee(cid:105)thalesop[2r0o]d−)-. u|gp(cid:105)p.erTshtiasteiss (t|yep1i(cid:105)c,a|lel2y(cid:105))thdeeccaaysetofotrheasazemroe garnoguunldarstmaote-
mentumgroundstate. Interestinglyitturnsoutthatthe
uct states a ,...,a that maximize the overlap
(cid:12)(cid:12)(cid:104)a1,...,aN|ψ|dN1(cid:105)(cid:12)(cid:12). InoNr(cid:105)dertofindsuchastate,notethat rdeaprlkacsitnagteth|ψed3set(cid:105)actoenssgtrauncdteedisinatlshoeasanmonewdeacyaaysinagbsotvaetebsy.
duetothesymmetryS N of ψN ,wecanchoose a ar-
⊗ | d (cid:105) (cid:12) | N(cid:105) (cid:12) In contrast to the case investigated in this manuscript
bitrarily and still get maximal overlap (cid:12)(cid:104)a1,...,aN|ψdN(cid:105)(cid:12) it, however, stores two excitation quanta in the three
by choosing a ,..., a optimal. Let us choose
1 N 1 atoms. This is shown in Fig. 5, where we compare its
w|oareNde(cid:105)rsu=tbos|emNqau−xe{inm1|t(cid:105)li.yz(cid:105)eWhthaeevoeo|bvtetaori−lnapc(cid:105)(cid:104)h}aNo1No(cid:12)(cid:12)|sψ(cid:104)eadN1,(cid:105).a=..,Na1N|−ψ1dN|ψ−dN1(cid:105)−.1(cid:105)Ii(cid:12)(cid:12)nn, sainlnotdwersedustepicenargy-rqa(udreeisadtnitolinnd,ee)wcahwyeitt(hhliegrihnttdhebisplueceno)ud.ledInttbieastgaoenmneasrda(dlyiizetelilodonwtao)l
i i 1,...,N 1
span 0 ,..., N 2 . The procedu{r|e (cid:105)i}s ∈repeat−ed re- storing N 1 excitations in N atoms creating an almost
{| (cid:105) | − (cid:105)} −
cursivelyinordertofix a ,...,a = 0,...,N 1 and inverted collective dark state in a realistic setup.
1 N
| (cid:105) | − (cid:105)
7
doubly excited three V-atoms chain decay
1.0
ground state: dark
ground state: super
ex. states dark
0.8
ex. states superradiant
ground state indep
ex. states indep
0.6
populations0.4
0.2
0.0
0 1 2 3 4 5
Γt
FIG. 5. Decay of three V-type atoms in small chain d (cid:28) λ
withinitiallytwoexcitations. Bluegivesthegroundstateand
red and the excited state populations. The other lines show
the dynamics for a super-radiant initial state for comparison.
