Dissipation-driven entanglement between qubits mediated by plasmonic nanoantennas

J. Hou,1 K. Słowik,1 F. Lederer,1 and C. Rockstuhl2, 31Institute of Condensed Matter Theory and Solid State Optics,

Abbe Center of Photonics, Friedrich-Schiller-Universität Jena, D-07743 Jena, Germany2Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany

3Institute of Nanotechnology, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany

A novel scheme is proposed to generate a maximally entangled state between two qubits by meansof a dissipation-driven process. To this end, we entangle the quantum states of qubits that aremutually coupled by a plasmonic nanoantenna. Upon enforcing a weak spectral asymmetry in theproperties of the qubits, the steady-state probability to obtain a maximally entangled, subradiantstate approaches unity. This occurs despite the high losses associated to the plasmonic nanoantennathat are usually considered as being detrimental. The entanglement scheme is shown to be quiterobust against variations in the transition frequencies of the quantum dots and deviations in theirprescribed position with respect to the nanoantenna. Our work paves the way for novel applicationsin the field of quantum computation in highly integrated optical circuits.

PACS numbers: 03.65.Ud, 73.20.Mf, 32.80.-t

I. INTRODUCTION

The remarkable performance of quantum protocols re-lies on the entanglement between quantum bits (qubits).Ideally, this entanglement should be preserved over longtime scales. Common experience predicts that dissipa-tion and the related decoherence in the system constitutea major obstacle in preserving this entanglement [1, 2].However, this intuition has been proven wrong in nu-merous recent works that focused on the generation ofentanglement by coupling qubits to a common, dissipa-tive environment [3–6]. Such dissipation driven entangle-ment schemes are extremely appealing from a practicalpoint of view since they allow to achieve entanglementindependent of the initial state of the system and wereshown to be quite robust against variations in the controlparameter.

In general, such dissipative environment can be pro-vided by metallic nanostructures. They are extremelyappealing in cavity quantum electrodynamics for devel-oping highly integrated quantum optical circuits wherequantum information is generated, processed, and de-tected at the nanoscale. At optical frequencies the prop-erties of these metallic nanostructures are dominated bythe excitation of surface plasmon polaritons, which arecarrier oscillations in the metal resonantly coupled to anexternal electromagnetic field. The excitation of surfaceplasmon polaritons permits the localization and enhance-ment of electromagnetic fields at length scales adaptedto the size of quantum emitters. Simultaneously, the ex-citation is accompanied by non-radiative dissipation ofelectromagnetic energy into heat and eventually also byradiative losses. However, this feature, usually consid-ered as a disadvantage, can be turned into a benefit indissipation driven processes, where the generation of en-tanglement ia a prime example.

So far, most of the entanglement-generation proto-

Fig. 1. Scheme of the investigated system. Two qubit carri-ers (eventually atoms, molecules or quantum dots) are posi-tioned in the vicinity of an optical nanoantenna. The systemis subject to an external driving field.

cols and entangling gates have been exploiting nanowiresas the specific metallic nanostructure. These metallicnanowires sustain propagating surface plasmon polari-tons. Such excitations are surface waves that propagatealong the nanowires over extended distances [7–9]. Insuch geometries, the quantum states of qubits encodedin quantum dots, atoms, or even molecules can be entan-gled by bringing them in close proximity to the nanowire.For identical qubits, a maximal degree of entanglementcan be obtained by choosing a suitable relative couplingstrengths to the nanowire [10, 11]. In this scheme, mul-tipartite entanglement between the qubits and the plas-mons is achieved and followed by a measurement of anumber of plasmonic excitations at the terminations ofthe nanowire. Only if no plasmons are detected, it canbe concluded that the qubits were maximally entangledwith each other.

To circumvent this probabilistic character and togenerate high degrees of entanglement in deterministicschemes, it was suggested to asymmetrically place thequbits with respect to the nanowire. For definite cou-pling strengths a maximal degree of entanglement could

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be achieved regardless of the state of the plasmonic field[1, 2]. However, large losses associated with propagat-ing surface plasmons prevent this entanglement to persistfor longer time scales. An idea of using the loss chan-nels to drive the qubits into a subradiant state that isrobust against dissipation was investigated in Ref. 12.The described scenario allows for achieving stationaryentanglement. However, it requires the system to be ini-tially driven into a predefined state and, moreover, theresulting entanglement remains significantly below themaximal possible degree. Recently, also a scheme forrobust-to-loss and high-fidelity entanglement generationwas proposed with a waveguide made from a chain ofmetallic nanoparticles [6].

To date, however, none of these schemes could providea maximal stationary entanglement of a pair of qubitsin a deterministic manner. In our study, we report thatstrongly dissipative systems can be used to achieve thisgoal. Contrary to much of the previous work that consid-ered propagating surface plasmon polaritons in metallicwaveguides, we consider here an isolated metallic nanoan-tenna as the dissipative structure. The optical nanoan-tenna sustains localized surface plasmon polaritons thatrequire a proper quantization while studying the couplingto multiple qubits to be entangled [13, 14]. The crucialrequirement for entanglement is the existence of a cer-tain degree of asymmetry in the qubit configuration. Thisasymmetry can be easily enforced by various means. Butif the asymmetry is provided by different transition fre-quencies in the qubit carriers, the resulting entanglementturns out to be especially robust against dissipation anddecoherence, and persists even in the steady state. More-over, our protocol provides stationary entanglement, i.e.it does not depend on the initial state of the system.

Recently, a similar scheme has been proposed for adeterministic multi-qubit quantum phase gate for qubitsinteracting with suface plasmons of a nanosphere [15].Moreover, possibilities of using nanoantennas as entan-gling devices for photons [16] and nanoantennas coupledto quantum dots as sources of entangled photon pairs[17, 18] have recently been discussed.

This paper is organized as follows. In Sect. II, weintroduce the Tavis-Cummings model with damping todescribe the coupling of a pair of qubits to a lossy single-mode nanoantenna field. Next, in Sect. III) we intro-duce an effective picture for describing all observablesthat is valid in the regime of weak qubit-to-nanoantennacoupling. Such a simplified view provides us with anintuitive understanding of the entanglement-generationmechanisms. We verify the expectations based on theeffective picture by a fully numerical solution of the evo-lution of the hybrid quantum system in Sect. IV. Finally,we analyze the effect of disorder, introduced by, e.g., ex-perimental imperfections that may be present in the pro-posed scheme, on the degree of entanglement. It is shownthat the present scheme is extremely robust; rendering it

an ideal candidate for future experiments in the field ofquantum plasmonics.

II. MODEL

In this section we describe the system to be investi-gated in more detail. We briefly introduce the Lindblad-Kossakowski formalism that allows in particular to findthe steady-state density matrix of the two qubits, and toestimate the corresponding degree of entanglement.

The system consists of two qubits in the vicinity of ametallic nanoantenna, as shown in Fig. 1. The ground(|g(j)〉) and excited (|e(j)〉) states of the jth qubit areseparated by an energy ~ω(j). Each of the qubits cor-responds to a pair of flip operators σ(j)

− = |g(j)〉〈e(j)|,

σ(j)+ = σ

(j)−†and inversion operator σ(j)

z = |e(j)〉〈e(j)| −|g(j)〉〈g(j)|.

The scattering and absorption spectra of the adjacentnanoantenna are assumed to be well-characterized by sin-gle Lorentzian lines centered at frequency ωna. For theparameter range, we are interested in, this holds for theellipsoidal nanoparticles we consider here as the opticalnanoantenna [19]. It is only required that the ellipsoidsare sufficiently small such that their optical response canbe described while considering only an electric dipole mo-ment. The resonance wavelength of the ellipsoid can thenbe tuned by tailoring the axis ratio [20]. Such nanopar-ticles can readily be provided by chemical means andconstitute the base to built more complicated functionalplasmonic nanostructures [21].

With such a geometry for the optical nanoantenna inmind, we describe the field localized by the nanoantennaas a single-mode harmonic oscillator, with the annihi-lation operator a. The widths of plasmonic resonancesin the scattering and absorption spectra are given byΓsca and Γabs, respectively. For an effective coupling be-tween the nanoantenna and the qubits, the frequenciesω(j) must be close to ωna, i.e. |ω(j) − ωna| Γ, whereΓ = Γabs + Γsca is the total energy dissipation rate of thenanoantenna. A monochromatic external driving field offrequency ωdr, assumed to be almost resonant with thenanoantenna field, provides the energy to the system. Weonly take into account the coupling Ω between the driv-ing field and the nanoantenna, since in many practicalcases the direct coupling to the qubits is much smallerand can be neglected.

It is convenient to write the Tavis-Cummings Hamilto-nian of the above-described system in the frame rotatingwith the driving field frequency [22, 23]:

H =∑j=1,2.

∆ω(j)σ(j)+ σ

(j)− + ∆ωnaa

†a (1)

−∑j=1,2.

g(j)(σ

(j)+ a+ a†σ

(j)−

)− Ω

(a+ a†

),

3

where ∆ω(j) = ω(j) − ωdr and ∆ωna = ωna − ωdr are thedetunings of the driving field from the qubit transitionfrequencies and the nanoantenna resonance, respectively.In the above Hamiltonian, the first two terms correspondto the free evolution of the qubits and the nanoantennafield, and the latter two describe the qubit-nanoantennaand nanoantenna-drive coupling. In both coupling termswe have applied the rotating wave approximation, andthe coupling constants are taken real for simplicity. Wehave also set the Planck’s constant ~ = 1 in the wholemanuscript.

There are several energy dissipation channels in theconsidered system. Besides the dominating mechanismswhere energy is scattered and absorbed by the nanoan-tenna at a total rate of Γ, we take into account dissipationeffects induced in the qubit-carriers by photonic vacuum,i.e., the spontaneous emission rate γ(j) of each qubit.For simplicity, we do not include additional dephasingchannels for qubits, which is an approximation commonlymade in related literature [1, 2, 6, 10, 12, 24, 25].

The Lindblad operator reads thus [26]:

L (ρ) = DΓ

(ρ, a, a†

)+∑j=1,2.

Dγ(j)

(ρ, σ

(j)− , σ

(j)+

), (2)

where ρ is the density matrix of the full system (thequbits and the nanoantenna field) and

Dγ (ρ,A,B) =γ

2(2AρB −ABρ− ρAB) . (3)

Typically, the dissipation rates in atoms or quantum dots(in the range of 1 ∼ 100 MHz) are several orders of mag-nitude smaller than those associated with nanoantennas(∼ 100 THz) [27]. Nevertheless, they might influenceboth the results at long time-scales and the steady-stateof the system, which will be our main subject of interest.

In principle, another dissipation channel should betaken into account for short interqubit distances, belowthe wavelength of the qubit transition 2πc/ω(j), where cis the vacuum speed of light. The channel arises due tothe interaction of the two qubits through the surroundingphotonic vacuum. This leads to collective effects such asvacuum-induced sub- and superradiance [3, 28–30]. Asit will turn out later, these effects can only support thepreservation of entanglement in our scheme. We thus donot include them here and consider instead somethinglike the worst-case scenario for the entanglement genera-tion.

The dynamics of the system is described by theLindblad-Kossakowski equation:

ρ = −i[H, ρ] + L (ρ) . (4)

To find the steady-state of the system ρ(t→∞), we solvethe above equation with the left-hand side set to zero,using a freely-available quantum optics toolbox [31]. For

this purpose we truncate the field’s Hilbert space suf-ficiently large, depending on the driving field strengthΩ. The steady-state density matrix of the qubit subsys-tem ρqb(t → ∞) can be found by performing a partialtrace over the field degrees of freedom: ρqb(t → ∞) =Trfield [ρ(t→∞)].

Coupling of the qubits to a common electromagneticmode may lead to their significant entanglement, whichcan be quantified in terms of concurrence, defined as [32]

C(ρqb) = max0, λ1 − λ2 − λ3 − λ4, (5)

where λk are square roots of eigenvalues of ρqbρqb

in descending order, ρqb = (σy ⊗ σy) ρqb∗ (σy ⊗ σy),

σy =

(0 −ii 0

)stands for the Pauli matrix and the as-

terisk denotes complex conjugation. The concurrenceranges between 0 (for product states) and 1 (for maxi-mally entangled states such as Bell states [33]). We men-tionthat the concurrence and other entanglement mea-sures have been analyzed in the context of plasmonics,e.g. in Ref. 25.

Before we proceed with the results obtained within themodel presented in this section, we will introduce an ef-fective description of the qubit-qubit system, which canbe obtained by an adiabatic elimination of the field. Suchprocedure is strictly valid in the weak qubit-nanoantennacoupling regime, defined as |g(j)| Γ. As we willsee, however, this approximative approach provides uswith intuitions that hold also beyond the weak-couplingregime.

III. EFFECTIVE DYNAMICS IN THE WEAKQUBIT-NANOANTENNA COUPLING REGIME

To find the effective description of the qubit-qubit sub-system we are going to start by analyzing the Heisenbergequations of motion of the qubit and field operators. Ifthe coupling of the field to the qubits is weak comparedto the dissipation rate of the nanoantenna, it is possibleto eliminate the field degree of freedom by substitutingthe adiabatic solution to the field. As the result, equa-tions of motion for the qubit operators only are obtained.These three steps will be described in subsection IIIA.An analysis of the resulting equations will allow to findan effective Hamiltonian and effective Lindblad terms insubsection III B. Such procedure leads to a drastic sim-plification of the Lindblad-Kossakowski formalism. Thevalidity of the resulting simplified approach will be veri-fied in subsection III C. To understand the entanglement-generation mechanisms, we will transform the effectiveHamiltonian and Lindblad term to the Dicke basis, moresuitable to describe an effectively hybridized system oftwo qubits (subsection IIID). Additionally, an expressionfor concurrence in terms of only a few density matrix el-ements will be given. This will allow for a more intuitive

4

analysis of entanglement generation in our scheme in asummarizing subsection III E.

A. Adiabatic elimination of the field operators

The Heisenberg equations of motion of the qubits andthe field read:

σ(j)z = −

(σ(j)z + 11

)γ(j) (6)

−2ig(j)(a†σ

(j)− − σ

(j)+ a

)+ f (j)

z ,

σ(j)− = −

(i∆ω(j) +

γ(j)

2

)σ

(j)− − ig(j)σ(j)

z a+ f(j)− , (7)

a = − (i∆ωna + Γ/2) a (8)

+i

Ω +∑j=1,2.

g(j)σ(j)−

+ fa,

where f (j)z , f (j)

− and fa are fluctuation operators includedto preserve the commutation relations [34].

As we have mentioned in the previous section, the dis-sipation channel of the nanoantenna field dominates overthe one of the qubits by orders of magnitude: Γ γ(j).In this case, we can adiabatically eliminate the field oper-ators [35]. Such elimination consists in substituting theadiabatic expression for the annihilation operator (seealso Refs. 36 and 37)

a =i(

Ω +∑j=1,2. g

(j)σ(j)−

)+ fa

i∆ωna + Γ/2(9)

into equations (6) and (7). We arrive at effective evolu-tion equations:

σ(j)z (t) = −

(σ(j)z + 11

)γ

(j)eff + 2i

(Ω

(j)eff σ

(j)+ − Ω

(j)eff

∗σ

(j)−

)+2iξeff

(σ

(j)+ σ

(k)− − σ

(k)+ σ

(j)−

)−γ′eff

(σ

(j)+ σ

(k)− + σ

(k)+ σ

(j)−

)+ F (j)

z , (10)

σ(j)− (t) = −

(i∆ω

(j)eff +

γ(j)eff

2

)σ

(j)− − iΩ

(j)eff σ

(j)z

+

(iξeff +

γ′eff

2

)σ(j)z σ

(k)− + F

(j)− , (11)

where k 6= j, ∆ω(j)eff = ω

(j)eff −ωdr and we have introduced

the single-qubit parameters

γ(j)eff = γ(j) + Γg(j)2

/Z, (12)

ω(j)eff = ω(j) −∆ωnag

(j)2/Z,

Ω(j)eff = g(j)Ω (∆ωna + iΓ/2) /Z,

Z = (Γ/2)2 + ∆ω2na,

the parameters describing an effective collective be-haviour

γ′eff = Γg(1)g(2)/Z, (13)ξeff = −g(1)g(2)∆ωna/Z,

and modified fluctuation operators

F (j)z = f (j)

z +2ig(j)

Γ/2 + i∆ωnaσ

(j)+ fa + H.c.,

F(j)− = f

(j)− −

ig(j)

Γ/2 + i∆ωnaσ(j)z fa,

where H.c. stands for a Hermitian conjugate.

B. Effective Hamiltonian and Lindblad term

Equations equivalent to (10) and (11) can be ob-tained in the density-matrix approach from an effectiveLindblad-Kossakowski equation

ρqb = −i[Heff , ρqb] + Leff(ρqb), (14)

with the effective Hamiltonian (see also [38])

Heff =∑j=1,2.

[∆ω(j)

effσ(j)+ σ

(j)− −

(Ω

(j)eff σ

(j)+ + Ω

(j)eff

∗σ

(j)−

)]+ξeff

(σ

(1)+ σ

(2)− + σ

(2)+ σ

(1)−

), (15)

and the effective Lindblad term

Leff

(ρqb)

=∑j=1,2.

Dγ(j)eff

(ρqb, σ

(j)− , σ

(j)+

)(16)

+∑

k 6=j=1,2.

Dγ′eff

(ρqb, σ

(k)− , σ

(j)+

).

The obtained Hamiltonian and Lindblad term describeeffectively the dynamics of two qubits. The effects in-duced by the nanoantenna mode are now described byeffective parameters, that can be clearly interpreted:

• The interaction to the nanoantenna mode shifts thetransition frequency of the jth qubit to ω(j)

eff .

• Each of the qubits is coupled to an external field,whose Rabi frequency has been rescaled by thepresence of the nanoantenna to an effective valueΩ

(j)eff .

• Now the nanoantenna mediates an effective dipole-dipole interaction ξeff between the qubits.

• The lifetime of the excited state of the jth qubit issignificantly shortened, such that the correspond-ing decay rate is enhanced to the value of γ(j)

eff .

• Each of the qubits is capable of modifying the elec-tromagnetic environment of the other. This leadsto a collective decay rate γ′eff and correspondingsub- and superradiance effects [28–30].

5

0

5

50

5

5

0

0.25

0.5

0 5x103 10x103

0

0.25

0.5

0 50 100

0

0.1

0.2

0 5 10Γt

(a)

(b)

(c)

ρSS

ρSS

ρSS

0

2

Fig. 2. Evolution of the symmetric-state occupationprobability in the full- (red solid lines) and the effective-Hamiltonian picture (blue dashed lines). The qubits, thenanoantenna and the driving field are assumed to be onresonance. The system is initially set to its ground state.The following normalized parameters were chosen: (a) weak-coupling regime: g = 0.01Γ, Ω = 0.05Γ; (b) intermediatecase: g = 0.1Γ, Ω = 0.5Γ; (c) strong-coupling regime: g = Γ,Ω = 0.5Γ. In each case γ = 10−8Γ.

It is worth to stress that the direct interplay between thequbits given by ξeff , as well as the collective behaviour de-scribed by γ′eff , are qualitatively new features, that havebeen missing in the generic Hamiltonian (1) and Lindbladterm (4).

The above-described effective formalism can be easilygeneralized to a greater number of qubits. Note also thata similar result for qubits coupled to nanowires was re-cently derived within the Green’s function formalism inRef. 38.

C. Validity of the effective description

Now the validity of the simplified description will beverified by rigorous full-Hamiltonian calculations.

In Fig. 2 the evolution of the symmetric-state occu-

pation probability, calculated by the full- and effective-Hamiltonian approaches, i.e. by solving Eq. (4) andEq. (14. respectively, are compared. The qubits werechosen to be identical (γ(1) = γ(2) ≡ γ = 10−8Γ),resonant with both the nanoantenna and the drivingfield (ω(1) = ω(2) = ωna = ωdr), and symmetricallypositioned with respect to the nanoantenna (implyingg(1) = g(2) ≡ g). The evolution in the full and the effec-tive picture is compared for three values of the normal-ized coupling strength: g/Γ = 0.01, 0.1, 1, correspond-ing to the so-called weak qubit-to-nanoantenna couplingregime, an intermediate case, and the strong-couplingregime [14, 39, 40]. In each case, a relatively strong driv-ing field was chosen: Ω/Γ = 0.05 for the weak-couplingcase, and 0.5 otherwise.

As expected, the effective approach is in perfect agree-ment with the exact one in the first case [Fig. 2(a)], wherethe loss rate of the field exceeds the coupling to otherevolution channels: Γ |g|, |Ω|. Only in such param-eter regime the evolution of the field is truly adiabatic,i.e. the annihilation operator can be represented by thequasi-stationary solution given by Eq. (9). As the con-ditions deviate from the genuine weak-coupling regime,the effective approach becomes approximative [Fig. 2(b)]and eventually gives wrong results [Fig. 2(c)]. However,its capacity lies not only in the fact that it simplifies cal-culations under the weak-coupling conditions. The mainadvantage of the effective Hamiltonian approach is thatit provides us with an intuitive insight into the processesthat are responsible for the evolution of the system, inparticular in entanglement-generation mechanisms.

This will become more obvious if the effective Hamil-tonian and Lindblad term are transformed to a basismore suitable to describe an effectively hybridized sys-tem of two qubits. Such transition will be performedin the following subsection. It permits to analyze theentanglement-generation mechanisms, i.e. to use directlyan analytical expression for the definition of conditions,where large degrees of stationary entanglement are ex-pected. Later in Sect. IV, we will show that the resultscome up to the expectations beyond the weak-couplingregime too.

D. Transition to the Dicke basis

The effective Hamiltonian acquires an in-teresting form in the so-called Dicke basis:|E〉 = |e(1)e(2)〉 , |S〉 =

(|e(1)g(2)〉+ |g(1)e(2)〉

)/√

2,|A〉 =

(|e(1)g(2)〉 − |g(1)e(2)〉

)/√

2, |G〉 = |g(1)g(2)〉,

where the symbols E and G stand for the bi-excited andtotal ground states, respectively, and S and A - for thesymmetric and antisymmetric superpositions of stateswith a single excitation shared between the two qubits

6

E

S

A

G

ξeff δωξeff

E

S

A

G

ΩA

ΩS

ΩS

ΩA

γsuper

(a) (b)

γsuper

γsub

γsub

Fig. 3. Scheme of the hybridized two-qubit system rep-resented in the Dicke basis: (a) energy shifts and coherentcouplings described by the effective Hamiltonian (17) and (b)superradiant and subradiant dissipation channels: γsuper =(γ

(1)eff + 2γ′eff + γ

(2)eff

)/2, γsub =

(γ

(1)eff − 2γ′eff + γ

(2)eff

)/2, as

described in Eqs. (18).

[28]. In the Dicke basis the effective Hamiltonian reads

Heff =∑

m=E,S,A,G

Em|m〉〈m| (17)

+δω (|A〉〈S|+ |S〉〈A|)−ΩS (|S〉〈G|+ |E〉〈S|)−H.c.

−ΩA (|A〉〈G| − |E〉〈A|)−H.c.,

In these expressions the physically most important fea-tures are (see also Fig. 3a):

• energy shifts by the effective dipole-dipole term ξeff :EE = ∆ω

(1)eff +∆ω

(2)eff , ES = 1

2 (∆ω(1)eff +∆ω

(2)eff )+ξeff ,

EA = 12 (∆ω

(1)eff + ∆ω

(2)eff )− ξeff , EG = 0;

• the coherent coupling between the symmetric andantisymmetric states, present if only the effectivetransition frequencies of the two qubits are shiftedwith respect to each other: δω = 1

2 (∆ω(1)eff −∆ω

(2)eff );

as follows from the second of Eqs. (12), such coher-ent coupling is possible either for non-symmetricpositions of the qubits with respect to the nanoan-tenna, that lead to different coupling strengthsg(1) 6= g(2), or for different transition frequenciesω(1) 6= ω(2);

• the possibility of a direct access from the ground(and excited) state to the symmetric and antisym-metric states by a coherent coupling to the ex-ternal field Ω; this coupling is described by thethird of Eqs. (12) and effective parameters ΩS =1√2(Ω

(1)eff + Ω

(2)eff ), ΩA = 1√

2(Ω

(1)eff − Ω

(2)eff ).

The effective Lindblad term (16) rewritten in the Dickebasis is somewhat cumbersome, and, therefore, we willnow only analyze its contributions to the diagonal terms

of the density matrix. They read as:[Leff

(ρqb)]

EE= −ρqb

EE (γsuper + γsub) ,[Leff

(ρqb)]

SS= −

(ρqb

SS − ρqbEE

)γsuper

+1

2<(ρqb

AS

)(γ

(1)eff − γ

(2)eff

),[

Leff

(ρqb)]

AA= −

(ρqb

AA − ρqbEE

)γsub

+1

2<(ρqb

AS

)(γ

(1)eff − γ

(2)eff

), (18)[

Leff

(ρqb)]

GG= ρqb

SSγsuper + ρqbAAγsub

−<(ρqb

AS

)(γ

(1)eff − γ

(2)eff

).

Thus there are two decay channels in the system (seeFig. 3b): a superradiant one from the bi-excited, viathe symmetric, to the ground state, with the rateγsuper ≡

(γ

(1)eff + 2γ′eff + γ

(2)eff

)/2; and a subradiant one,

via the antisymmetric state, with the rate γsub ≡(γ

(1)eff − 2γ′eff + γ

(2)eff

)/2. The population dynamics is ad-

ditionally complicated by a coupling to the coherenceρqb

AS.The interpretation becomes more straight while con-

sidering identical qubits placed in highly symmetric po-sitions with respect to the antenna, i.e. for ω(1) = ω(2),g(1) = g(2) and γ(1) = γ(2). Then, without drivingfield Ω, the Hamiltonian is diagonal in the Dicke ba-sis, as can be clearly seen from Eqs. (12,13) and (17).Thus, the effective evolution of the Dicke-states occupa-tion probabilities is only given by the Lindblad term (18):

˙ρqbmm =

[Leff

(ρqb)]mm

. Moreover, γ(1)eff = γ

(2)eff ≡ γeff ,

and therefore the superradiant rate simplifies to γeff+γ′eff ,and the subradiant rate is equal to the vacuum-inducedspontaneous decay rate γ(1) = γ(2) ≡ γ [28, 38]. To agood approximation the antisymmetric state is decoupledfrom the rest of the system at time scales comparable to1/γeff , but its population decays exponentially at timescales of the free-space spontaneous emission rate 1/γ.

It should be noted, that if the qubits are separatedby distances less than 2πc/ω(j), coupling to the pho-tonic vacuum itself is a source of sub- and superradiance[3, 28]. Then, the lifetime of the antisymmetric state iseven longer, which naturally promotes the preservationof entanglement (see also Ref. 41). In this paper we dis-regard this simple effect and prove that even when theantisymmetric state is not entirely decay-free, large val-ues of stationary entanglement can be reached.

E. Evaluation of the effective formalism in thecontext of entanglement generation

Writing down the expression for concurrence one canobtain in the effective picture for a weak driving field Ω.In this case, the coherence parameters ρqb

Ep, p ∈ A,S,G

7

are small and the concurrence reads [24]

Ceff(ρqb) ≈ max 0, CM , (19)

CM =

√(ρqb

SS − ρqbAA

)2

+ 4=(ρqb

SA

)2

− 2

√ρqb

GGρqbEE.

Having Eqs. (17-19) at hand, we can summarizethe effective picture obtained in this section from theentanglement-generation point of view:

• Eq. (19) suggests that in the investigated config-uration, the entanglement can attain large valueswhen the qubit-qubit system is driven into eitherthe symmetric or the antisymmetric state with ahigh probability, and at the same time the proba-bility of the bi-excited state |E〉 remains low.

• In this context, the configuration with equal qubit-to-nanoantenna coupling strengths g(1) = g(2) isespecially interesting. In this case, the existenceof an approximately nondecaying antisymmetricstate, which is at the same time a maximally en-tangled state, follows from Eqs. (18). This opensan opportunity to produce high degrees of entan-glement between the qubits being robust againstdissipation. The other maximally entangled state,the symmetric one, decays quickly due to superra-diance.

• It turns out, however, as can be seen from Eq. (17),that it is the symmetric state that can be directlyaccessed by a coherent drive (for g(1) = g(2) theeffective coupling to the antisymmetric state dis-appears naturally: ΩA = 0). The antisymmetricstate is decoupled from the rest of the Hilbert space,unless a certain degree of asymmetry between thequbits is introduced, resulting in δω 6= 0. Becausewe have already set g(1) = g(2), the asymmetrymust be evoked by different transition frequenciesω(1) 6= ω(2). Then, a high occupation probabilityof the (approximately) nondecaying antisymmetricstate can be reached through a coherent couplingδω to the symmetric one.

We conclude that two qubits symmetrically coupled to ananoantenna mode should be efficiently driven into theantisymmetric state if their transition frequencies are dif-ferent from each other. Then, the population can bedriven from the ground to the symmetric state with acoherent drive ΩS, and from the symmetric state to theantisymmetric one, by the coupling induced by the dif-ference in the transition frequencies δω. The small de-cay rate of the antisymmetric state allows it to remainhighly-occupied, which leads to a large concurrence. Thisscheme is at the heart of our proposal.

In this section we have developed a simplified approachto describe the qubit-qubit dynamics for qubits weakly

coupled to a lossy nanoantenna field mode. By an adi-abatic elimination of the operators associated with thefield, we arrived at an effective Lindblad-Kossakowski for-malism. Even though the effective picture is only valid inthe weak qubit-to-nanoantenna coupling regime, it is sim-ple and instructive. Moreover, as we will show in Sect.IV, its predictions for entanglement-generation mecha-nisms hold beyond this regime. As the conclusion of thissection we find that two qubits symmetrically coupled toa nanoantenna mode should be efficiently driven into theantisymmetric state if only their transition frequenciesdiffer from each other.

IV. ASYMMETRY-INDUCED INTERQUBITENTANGLEMENT

In this section we verify the insights gained above.First, we solve the Lindblad-Kossakowski equation (4)with the full Hamiltonian (1) for the perfect case oftwo qubits characterized by equal coupling strengths tothe nanoantenna mode: g(1) = g(2) ≡ g. The transi-tion frequencies of the qubits are anti-symmetrically de-tuned from the nanoantenna resonance: ω(1) = ωna +δω,ω(2) = ωna − δω.

A. Perfect case of equal coupling strengths andanti-symmetric detuning

In Fig. 4, steady-state values of the antisymmetric-state probability ρAA(t → ∞) and the correspondingconcurrence C, are plotted as functions of the normalizedcoupling strength g/Γ and driving-field Rabi frequencyΩ/Γ, for the detuning δω = 10−3Γ and the vacuum-induced spontaneous-emission rate γ(1) = γ(2) ≡ γ =10−8Γ. The parameter range analyzed in Fig. 4 corre-sponds to nanoantennas investigated in Ref. 19: silvernanospheres and nanospheroids of both the characteris-tic size and the qubit-to-nanoantenna distance up to 100nm.

As expected, it follows from Fig. 4 that the obtainedstationary entanglement can be extremely large. Nat-urally, both ρAA(t → ∞) and C are equal to zero for avanishing coupling to the nanoantenna resonance (g = 0)or without driving field (Ω = 0). The results quickly growwith g and small values of Ω and reach unity or almostunity for coupling constants g > 0.05Γ, for a considerablerange of Ωs. As follows from Eq. (17), for strong drivingfields the role of the processes that distribute the popula-tion between the |G〉, |S〉 and |E〉 states (the terms pro-portional to ΩS) is increased with respect to the coherenttransfer between the symmetric and antisymmetric states(the term ∼ δω). This is why ρAA(t→∞) drops for largevalues of Ω. The concurrence is further decreased due tothe increased probability of the bi-excited state occupa-

8

0 0.1 0.2 0.30

0.3

0.5

0

1

Ω/Γ

g/Γ

(a)

0.1

0.20.5

0

1

0 0.1 0.2 0.3Ω/Γ

0

0.3

g/Γ 0.1

0.2

(b)

Fig. 4. (a) The steady-state probability of the antisymmetricstate ρAA(t→∞) and (b) the steady-state concurrence C asfunctions of the normalized coupling constant g/Γ and drivingfield Ω/Γ, for two qubits anti-symmetrically detuned from thenanoantenna resonance: δω = ω(1) − ωna = ωna − ω(2) =10−3Γ, and vacuum-induced spontaneous emission rate γ =10−8Γ.

tion ρEE(t→∞) [see the second term in Eq. (19)]. How-ever, it is important to stress that a concurrence close tounity can be obtained in a large parameter space.

B. Analyzing the robustness of the scheme

It is important to double-check the sensitivity of the re-sults presented in Fig. 4 against variations of nominal pa-rameters of the system which might be eventually causedby experimental imperfections. The steady-state concur-rence as a function of the normalized qubit-transition-frequency detunings from the nanoantenna resonanceδω(j)/Γ ≡

(ω(j) − ωna

)/Γ is shown in Fig. 5(a) for

g = Ω = 0.2Γ, γ = 10−8Γ. As expected, the concur-rence vanishes for identical qubits, i.e. for δω(1) = δω(2),and grows fast for very small transition-frequency differ-ences (the apparent discontinuity is a result of imperfectresolution of the figure). The result proved to be robustagainst small deviations from the optimal conditions: theconcurrence remains high even if the absolute values ofthe detunings δω(j) are not equal, as long as they have op-posite signs. (For the nanoantenna parameters describedin Ref. 19, this corresponds to a tolerance of a few THzfor the concurrence to exceed 0.9.) When the detuningsare too large, however, the qubits are no longer in exactresonance with the nanoantenna and the driving field,which leads to smaller concurrence.

We proceed in analyzing the steady-state concurrencefor unequal coupling constants g(1) 6= g(2). The result isshown in Fig. 5(b) for δω = 10−3Γ, Ω = 0.1Γ, γ = 10−8Γ.Interestingly, even though such imperfect configurationleads to additional asymmetry in the system, it resultsin smaller degrees of steady-state entanglement. Thisis because for unequal coupling constants the antisym-metric state undergoes a decay with a rate proportionalto(g(1) − g(2)

)2[see the third of Eqs. (18)]. Conse-

quently, its steady-state probability is smaller. This isin agreement with the results obtained in Refs. 1 and

−0.05 0 0.05−0.05

0.05

0 0.5

0

1

δω(1)

δω(2)

eff

eff

(a)

1

0.9

0.8

0.70.199 0.200 0.2010.199

0.201

0.2

g(1)

g(2)

(b)

Fig. 5. (a) The steady-state concurrence C as a function ofnormalized detunings of the qubits δω(1)/Γ and δω(2)/Γ, forthe coupling constant g = 0.2Γ. (b) Concurrence as a functionof different qubit-nanoantenna coupling constants g(1) andg(2), for a symmetric detuning δω = 10−3Γ. In both cases thedriving field Ω = 0.1Γ, vacuum-induced spontaneous emissionrate γ = 10−8Γ.

12, where asymmetric placements of the qubits with re-spect to a nanowire or a microtoroid were considered forthe purpose of entanglement generation at short time-scales. The resulting entanglement did not, however,persist in the steady-state. Such sensitivity of the con-currence to the coupling strengths implies in particulara need for precisely placing the qubits with respect tothe nanoantenna surface. This seems to be within thereach of the state-of-the-art technology, which allows fora control over the qubit-to-nanoantenna distance with asub-nanometer resolution [42–47].

In the last section we have numerically solved theLindblad-Kossakowski equation to confirm that a verylarge degree of stationary interqubit entanglement, withconcurrence reaching one, can be obtained with twoqubits of different transition frequencies, symmetricallycoupled to a nanoantenna. We have found that thescheme is robust against perturbations in the detuningof the qubits from the nanoantenna resonance. For thesteady-state concurrence to be large it is however crucialthat the antisymmetric state remains strongly subradi-ant, i.e. that the effective loss rates of all qubits areequal. This requires a precise positioning of the qubitswith respect to the nanoantenna.

V. CONCLUSIONS

A cavity-QED formulation of the problem of two qubitscoupled to a single nanoantenna resonance has been con-sidered in the context of interqubit entanglement gener-ation.

The underlying physical mechanisms were analyzedby a simplistic effective description, strictly valid in theweak-coupling regime only. The effective description di-rectly leads to the conclusion that the qubits may bedriven into a maximally entangled steady state for sym-metric coupling to the mode of the nanoantenna, but an

9

antisymmetry in the transition frequencies of the qubits.Great advantages of the proposed scheme are that (1)

it is independent of the initial state of the system; and(2) it is robust against dissipation through channels asso-ciated with both the nanoantenna and the qubits. How-ever sensitive to deviations in the coupling strength be-tween the qubits and the nanoantenna field, the proposedscheme has been proven stable with respect to perturba-tions in their spectral properties.

Implementation of entanglement generation and morecomplicated quantum-information protocols with plas-monic structures promises faster quantum computingwith miniaturized devices.

ACKNOWLEDGEMENTS

This work was partially supported by the German Fed-eral Ministry of Education and Research (PhoNa) and bythe Thuringian State Government (MeMa).

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