Resonant atom-field interaction in large-size coupled-cavity arrays
aa r X i v : . [ qu a n t - ph ] A p r Resonant atom-field interaction in large-size coupled-cavity arrays
Francesco Ciccarello
CNISM and Dipartimento di Fisica, Universita’ degli Studi di Palermo,Viale delle Scienze, Edificio 18, I-90128 Palermo, Italy (Dated: October 30, 2018)We consider an array of coupled cavities with staggered inter-cavity couplings, where each cavity modeinteracts with an atom. In contrast to large-size arrays with uniform-hopping rates where the atomic dynamicsis known to be frozen in the strong-hopping regime, we show that resonant atom-field dynamics with significantenergy exchange can occur in the case of staggered hopping rates even in the thermodynamic limit. This e ff ectarises from the joint emergence of an energy gap in the free photonic dispersion relation and a discrete frequencyat the gap’s center. The latter corresponds to a bound normal mode stemming solely from the finiteness ofthe array length. Depending on which cavity is excited, either the atomic dynamics is frozen or a Jaynes-Cummings-like energy exchange is triggered between the bound photonic mode and its atomic analogue. Asthese phenomena are e ff ective with any number of cavities, they are prone to be experimentally observed evenin small-size arrays. PACS numbers: 42.50.Pq, 05.60.Gg
I. INTRODUCTION
The tremendous interest spread over the last few years inthe dynamics of coupled-cavity arrays (CCAs) [1] has madethis emerging field a topical one within the general frameworkof quantum coherent phenomena and beyond. Many are thereasons behind such a substantial and widespread attention.Along with cold atoms in optical lattices [2] CCAs stand outas an attractive controllable test bed for many-body phenom-ena such as quantum phase transitions [3]. An appealing fea-ture of CCAs is local addressing, namely the available highcontrol of each single site in terms of performable measure-ments, quantum-state engineering and dynamical parameterstuning. From the perspective of quantum optics, CCAs openthe door to the exploration of regimes where an almost ubiq-uitous feature of so-far-investigated atom-photon dynamics,i.e. the e ff ectiveness of descriptions in terms of single-atomdynamics, does not hold any more. Importantly, the varietyof possible experimental setups prone to implement CCAs [4]and, mainly, the widespread expectation that the technologyrequired for their actual fabrication is by now at hand (at leastfor small clusters) are providing formidable motivations to ex-plore the rich physics of CCAs. In particular, one of the linesalong which current investigations are proceeding is the studyof excitation propagation in various forms such as transport ofphotons and excitons [5, 7, 8], polaritons [9, 10] and solitons[11]. In this framework, transport in CCAs with defects or im-purity atoms is also receiving considerable attention [12–15].In a typical arrangement of CCAs, the field mode of eachcavity is coupled to a two-level atom via a Jaynes-Cummings-type interaction. Energy can be thus stored in the form ofboth photons and excitons (atomic excitations) that are ableto transform into each other as well as propagate along thearray. While photons can move by direct hopping betweenneairest-neighbour cavities, excitons can travel only providedthat they are transformed into photonic excitations and even-tually converted back. The dynamics ruling excitation trans-port in CCAs is therefore non-trivial and physically attractive.While first works along this line addressed small clusters [6, 7] only more recently arbitrary-size arrays have been tackled [8].In all such scenarios, for uniform values of atom-photon inter-action strength, cavity-mode frequency and atomic detuningand whenever the decoupling of the field’s hopping Hamil-tonian in terms of normal modes is known the Hamiltoniandescribing the full many-body dynamics enjoys an attractivefeature. It can indeed be rearranged [7, 8] as the sum of de-coupled Jaynes-Cummings (JC) models [16], each coupling afield (bosonic) normal mode to its excitonic analogue. So longas the array size is small (assuming non-degenerate photonicnormal modes) resonant excitation of only one of such e ff ec-tive JC systems is possible by judicious tuning of the atomfrequency [7]. Physically, this circumstance is quite remark-able since it entails that significant amounts of energy can beexchanged between atoms and photons even in the strong-hopping regime, i.e. when the atom-photon interaction rateis much lower than the photon hopping rate. Such a pic-ture, however, can drastically change with large-size CCAs.By taking uniform-hopping rates, for instance, the passageto a large number of cavities brings about that the free fieldnormal-mode spectrum tends to a continuous band. Clearly,this rules out the possibility of any resonance and thus first-order atom-field energy exchange. As a result, in the regimeof strong hopping the atomic dynamics turns out to be frozen[8]. One may wonder whether the above picture still holdsfor large-size arrays when a non -uniform pattern of hoppingstrengths is considered. Motivated by some findings in theframework of spin chains [17] Makin et al. [8] investigateda parabolic distribution of hopping rates and found no signif-icant changes in the strong-hopping regime compared to theuniform-hopping behavior.In the present work, we show that resonant atom-photon in-teraction can take place even under strong-hopping conditionsin an arbitrary-size array with a staggered pattern of hoppingrates. In particular, we demonstrate that when certain arraysites are initially excited and each atom has negligible detun-ing from the single-cavity field a JC-like dynamics involvinga bound photonic normal mode interacting with its excitonicanalogue is triggered. Among other e ff ects, it entails a signifi-cant exchange of energy between field and atoms. This behav-ior basically stems from two features. First, the gapped natureof the normal-frequency spectrum of the hopping Hamilto-nian. Second, the occurrence under open boundary conditions(BCs) of a discrete frequency corresponding to a bound nor-mal mode at the center of the gap. Such circumstances makeresonant atom-photon dynamics possible even in the thermo-dynamic limit.The present paper is organized as follows. In Section II, weintroduce the set-up which we focus on, an array with stag-gered hopping rates, and give the Hamiltonian that describesits dynamics. We then focus on the special case where the freefield Hamiltonian reduces to that associated with a uniform-hopping array, which allows us to briefly review the dynam-ics of such arrays. In Section III, we focus on the free fieldHamiltonian in the general case, present exact solutions forits normal modes and associated frequencies and discuss itsmain properties. We then illustrate the regime which we fo-cus on and give the corresponding e ff ective representation ofthe full Hamiltonian. In Section IV, we show how an initialatomic excitation propagates along the array and discuss thesalient features of the excitation-transport dynamics. Finally,in Section V, we give some comments and draw our conclu-sions. II. HAMILTONIAN AND REVIEW OF THEUNIFORM-HOPPING SETTING
We consider a finite-length array of N low-loss cavities,where nearest-neighbour cavities are coupled together so asto allow for photon hopping. Each cavity sustains a singlefield mode of frequency ω f , which is coupled at a rate J to atwo-level atom of Bohr frequency ω a (here and throughout weuse units such that ~ = H = ˆ H f + ˆ H a + ˆ H I , (1)whereˆ H f = ω f N X x = ˆ a † x ˆ a x − κ N − X x = [1 − ( − x η ] (cid:16) ˆ a † x + ˆ a x + h . c . (cid:17) , (2)ˆ H a = ω a N X x = ˆ b † x b x , (3)ˆ H I = J N X x = (cid:16) ˆ b x ˆ a † x + ˆ b † x ˆ a x (cid:17) . (4)In Eqs. (2)-(4), ˆ a x (ˆ a † x ) is a bosonic field operator that an-nihilates (creates) a photon at cavity x , whereas ˆ b x (ˆ b † x ) isan atomic operator that annihilates (creates) an exciton onthe x th atom according to ˆ b x = h ˆ b † x i † = | g i x h e | , where | g i x ( | e i x ) is the ground (excited) state of the x th atom. ˆ H f andˆ H a are the free Hamiltonians of the field and atoms, respec-tively, while ˆ H I describes the atom-field coupling. Noticethat atomic operators associated with di ff erent sites commute, i.e. [ˆ b x , ˆ b x ′ ] = [ˆ b x , ˆ b † x ′ ] = x , x ′ since operators actingon di ff erent Hilbert spaces commute. On the other hand, it iseasily checked that ∀ x ˆ b x = [ˆ b † x ] = b x ˆ b † x + ˆ b † x ˆ b x = κ = (1 + η ) κ [ κ = (1 − η ) κ )],where κ is a hopping rate. Hence, two di ff erent hoppingstrengths are periodically interspersed along the array. Asby setting η = κ is re-trieved [8] the dimensionless staggering parameter η measuresthe distortion of such uniform-hoppings CCA. For | η | =
1, thearray reduces to a collection of independent two-cavity blocks(but one comprising a single cavity in the case of odd N ). Asketch of the whole set-up is given in Fig. 1(a). The distortedtight-binding model [18] specified by ˆ H f has recently beenharnessed (in its fermionic version) in Ref. [19] for the sakeof quantum state transfer. As we thoroughly discuss in thenext section, regardless of the value taken by η , ˆ H f in Eq. (2)can be exactly arranged in a diagonal form in terms of normal-mode field operators. In the remainder of this Section, though,we focus on the uniform-hopping case η = η =
0, the free field Hamiltonian can be expressed interms of normal modes as [8, 17]ˆ H f = X k ω k ˆ α † k ˆ α k , (5)where k = π mN + m = , ..., N ) , (6) ω k = ω f + κ cos k , (7)ˆ α k = r N + N X x = sin k x ! ˆ a x . (8)In Eqs. (5) and (8), ˆ α k and ˆ α † k are, respectively, bosonic an-nihilation and creation field operators associated with the k thphotonic normal mode, which satisfy standard commutationrules.A feature of Hamiltonian (1) is that atom-photon interac-tion strengths, cavity-mode and atomic frequencies are uni-form throughout the cavity array. This allows to arrange itin an elegant and useful form [7, 8] in terms of N decouplede ff ective JC models according toˆ H = X k h ω k ˆ α † k ˆ α k + ω a ˆ β † k ˆ β k + J (cid:16) ˆ α † k ˆ β k + ˆ β † k ˆ α k (cid:17)i , (9)where the expansion of the atomic normal-mode operatorsˆ β k ’s in terms of site operators ˆ b x ’s is apart from their di ff er-ent commutation rules fully analogous to Eq. (8). Notice that,unlike ˆ α k ’s, each ˆ β k has the same associated normal-mode fre-quency ω a .In each e ff ective JC-like Hamiltonian appearing in the de-composition (9), a field normal mode couples to its atomicanalogue through a JC-type interaction. As all of the atomicmodes have the same frequency ω a when ω a = ω k ′ for a given k ′ only the corresponding pair of field and atomic normalmodes is, in principle, resonantly excited.For instance, for N = α ± = (ˆ a ± ˆ a ) / √ ω ± = ω f ± κ , respec-tively [see Fig. 1(b)]. Analogously, the atomic normal oper-ators read ˆ β ± = (ˆ b ± ˆ b ) / √
2. When the atoms are tuned inresonance with one of these two modes by setting ∆ = ± κ (where ∆ = ω f − ω a is the atoms’ detuning from the single-cavity frequency) a JC-like dynamics occurs with a continu-ous exchange of energy between the involved photonic modedescribed by ˆ α ± and the corresponding excitonic mode [7]. Atypical behavior [7] that arises under such conditions is thatwhen one atom, say atom 1, is initially prepared in the excitedstate (with atom 2 in the ground state and no photons popu-lating the cavities) the photonic normal mode is progressivelyexcited while the excitation probability of atom 2 grows. Overthe following stage, while the field returns the received energyatom 2 keeps increasing its excitation probability until at a cer-tain time the initial excitation has fully transferred to it (withthe field and atom 1 unexcited). Afterward, the phenomenonis reversed through a further excitation-deexcitation cycle ofthe field mode until the initial state is fully retrieved. Trigger-ing such a dynamics, which is of first-order in J , through ajudicious tuning of the atomic frequency is however possibleonly in small-size arrays.Indeed, as N grows [see Fig. 1(b)] more and more photoniclevels gather in the range [ ω f − κ, ω f + κ ] as implied byEq. (7). In the thermodynamic limit N → ∞ , the photonicspectrum takes the form of a continuos band. For large arraysthis in fact suppresses any possible resonant atom-photon cou-pling due to the lack of discrete frequencies. As a result, in thestrong-hopping regime J ≪ κ a straightforward use of the in-teraction picture shows that so long as second-order processesare negligible the atomic dynamics becomes fully frozen [8].Hence, in striking contrast to the above-described process ina two-cavity array, an exciton initially localized on a givenatom is unable to move away. Also, energy exchange betweenatoms and field is suppressed. III. NORMAL MODES FOR STAGGERED HOPPINGRATES AND EFFECTIVE HAMILTONIAN
The scenario discussed above for a uniform-hopping arraysubstantially changes when η ,
0. To illustrate this, in Fig. 1(c)we have numerically computed the normal-mode frequenciesof the free field Hamiltonian (2) for η = . N = , , , ff erences appear compared to the case η = N two continuous bands emerge, instead of a single one. Second,regardless of N a discrete normal frequency lies exactly at thecenter of the gap. These findings can be made analyticallyrigorous.To begin with, for the sake of notation compactness we de- fine the two quantities τ = η + η − , (10) ε k = κ r cos k + η sin k . (11)Note that ε k is defined positive.It can be checked that for an arbitrary odd number of cav-ities N (later on we comment on this assumption) ˆ H f can beexactly arranged in the diagonal formˆ H f = ω f ˆ α † ℓ ˆ α ℓ + X k X µ = ± ω k ,µ ˆ α † k ,µ ˆ α k ,µ , (12)where k = π m / ( N +
1) with m = , , ..., ( N − / α ℓ = η − r ητ N + − N + X x = τ x − ˆ a x − , (13) ω k ± = ω f ∓ ε k , (14)ˆ α k ± = r N + N − X x = sin ( kx ) ˆ a x ± N + X x = sin ( kx + ϑ k ) ˆ a x − . (15)The phase ϑ k appearing in Eq. (15) obeys e i ϑ k = κ (1 − η ) ε k (cid:16) e − ik − τ (cid:17) . (16)The discrete spatial functions specifying the expansion ofnormal operators Eqs. (13) and (15) in terms of site opera-tors { ˆ a x } can be shown to fulfill orthonormality conditions, aproof that we carry out in Appendix B. As such, { ˆ α ℓ , ˆ α k , ± } form a set of bosonic annihilation operators fulfilling standardbosonic commutation rules. As is to be expected, the normal-frequency spectrum { ω f , ω k ± } and corresponding set of nor-mal annihilation operators { ˆ α ℓ , ˆ α k ± } reduce to (7) and (8), re-spectively, in the special case η =
0. This check is carried outin detail in Appendix C.At variance with the problem tackled in Ref. [19] that al-lowed assumption of cyclic BCs, in deriving Eqs. (12)-(14)we have used open BCs, which are the natural ones to im-pose in a finite-length CCAs [8]. Similarly to the uniform-hopping case [17], the solutions in the continuous spectrum[see Eq. (15)] are straightforwardly obtained by superposingthose fulfilling cyclic BCs in the case of N + α ℓ is readily found by direct demonstration once oneimposes that its normal frequency be ω f (so that the hoppingterm in ˆ H f vanishes). In Appendix D, we show in detail thatupon replacement of (14), (13) and (15) on the right-hand sideof Eq. (12) the free field Hamiltonian in Eq. (2) is retrieved.The two aforementioned bands in the normal-frequencyspectrum are analytically described by Eqs. (11) and (14) [seeFig. 1(c)] with their associated annihilation operators given byEq. (15). Their energy gap ∆ ω , which coincides with the oneobtained with cyclic BCs [19], fulfills ∆ ω ≥ κ | η | , (17) FIG. 1: (Color online) (a)
Schematic sketch of an array of N coupled cavities with staggered hopping strengths κ = (1 + η ) κ and κ = (1 − η ) κ .The field mode of each cavity is coupled to an atom (red dot). (b) η = H f for N = , , ,
51. Here and in next plot ( c ) frequencies are expressed in units of κ and measured from ω = ω f . ( c ) η = . H f using open BCs for N = , , ,
51. The levels below (above) the k -axis [red (blue) dots] fallwithin the range [ ω f − κ, ω f − κ ] ([ ω f + κ, ω f + κ ]), while a single discrete frequency ω = ω f (green dot) lies at the center of the gap betweensuch intervals (its associated abscissa has been arbitrarily assigned).FIG. 2: (Color online) Amplitude of the bound mode in Eq. (13)against the cavity number for N =
51 and η = . a ), η = − .
05 ( b ), η = − . c ) and η = − . d ). As the array length N only a ff ects thenormalization factor, when N is varied the shape of these curves isuna ff ected. Notice that the amplitude vanishes at even sites. where the identity occurs in the thermodynamic limit N → ∞ .For η → | η | grows the lower bound of the gap on the right-hand sideof (17) linearly increases at a rate proportional to κ . As for thenormal mode in Eq. (13), its associated frequency ω f lies outof the two bands. Indeed, such normal mode is bound in that,di ff erently from modes (15), it does not extend over the entirearray. This is evident from the exponential functional formin Eq. (13) and Fig. 2, which shows that the mode amplitudedecreases (increases) exponentially with the cavity number x for negative (positive) values of η . Notice that while the modeamplitude strongly depends on η its frequency is fully inde-pendent of both η and κ . Using Eqs. (10) and (13), the charac-teristic length λ over which the bound mode is spread is foundas λ = | / log[(1 + η ) / (1 − η )] | . Thus the local-mode length be-comes infinite when η ≃
0, as expected, while in the range0 < | η | < | η | until it be-comes negligible when | η | ≃
1. This behavior is highlightedin Fig. 2, where the mode’s spatial profile for di ff erent η ’s isreported. A further feature which is evident in Eq. (13) andFig. 2 is that regardless of η the bound mode vanishes at evensites.We now turn our attention to the full Hamiltonian (1) inorder to highlight the implications of Eqs. (12)-(15) on theCCA’s dynamics. We first define atomic normal operators infull analogy with Eqs. (13) and (15) asˆ β ℓ = η − r ητ N + − N + X x = τ x − ˆ b x − , (18)ˆ β k ± = r N + N − X x = sin ( kx ) ˆ b x ± N + X x = sin ( kx + ϑ k ) ˆ b x − , (19)where again k = π m / ( N +
1) with m = , , ..., ( N − / ϑ k and τ still given by Eqs. (16) and (10), respectively. Itis worth pointing out that unlike { ˆ α ℓ , ˆ α k µ } the set { ˆ β ℓ , ˆ β k µ } donot obey commutation rules since, while atomic site operatorsassociated with di ff erent cavities commute, ˆ b x ˆ b † x = − ˆ b † x ˆ b x for any x . In analogy with the uniform-hopping case in theprevious Section, the full Hamiltonian in terms of field andatomic operators { ˆ α ℓ , ˆ α k ± , ˆ β ℓ , ˆ β k ± } takes the formˆ H = ˆ H ℓ + X k µ = ± h ω k ± ˆ α † k µ ˆ α κµ + ω a ˆ β † k µ ˆ β k µ + J ( ˆ β † k µ ˆ α k µ + h . c . ) i , (20)where ˆ H ℓ = ω f ˆ α † ℓ α ℓ + ω a ˆ β † ℓ β ℓ + J (cid:16) ˆ β † ℓ ˆ α ℓ + ˆ α ℓ ˆ β † ℓ (cid:17) . (21)As we show in Appendix E, the full Hamiltonian normal-mode decomposition (20) straightforwardly follows from de-composition (12) and the fact that ω a and J do not depend onthe cavity site.We take as free Hamiltonian ˆ H = ˆ H f + ˆ H a [see Eqs. (2)and (3)]. Due to [ˆ b x , ˆ b x ′ ] = x , x ′ , the commuta-tor between each site atomic operator and the free Hamilto-nian is given by [ˆ b x , ˆ H ] = ω a ˆ b x , which immediately yields[ ˆ β ξ , ˆ H ] = ω a ˆ β ξ ( ξ = ℓ, { k , µ } ). Hence, in the interaction pic-ture each normal atomic operator evolves with time accordingto ˆ β ( I ) ξ ( t ) = ˆ β ξ e − i ω a t ( ξ = ℓ, { k , µ } ). On the other hand, usingEq. (12) the normal field operators evolve in the same pictureas ˆ α ( I ) ℓ ( t ) = ˆ α ℓ e − i ω f t and ˆ α ( I ) k µ ( t ) = ˆ α k µ e − i ω k µ t . In the interactionpicture, the interaction Hamiltonian [cf. Eq. (20)] thus readsˆ H ( I ) I = J ˆ α † ℓ ˆ β ℓ e i ∆ t + J X k µ = ± (cid:16) ˆ α † k µ ˆ β k µ e i ( ω k ,µ − ω a ) t (cid:17) + h . c . (22)When ∆ =
0, i.e. each atom is on resonance with its own cavitymode, Eq. (22) shows that while the contribution to ˆ H I fromthe bound mode is constant in time, those arising from thenormal modes in the two energy bands rotate at frequenciesthat are at least equal to half the energy-gap lower bound 2 κ | η | [see (17)]. Hence, provided that J /κ ≪ | η | , (23)such contributions are rapidly rotating and thus do not a ff ectthe system’s dynamics. When ∆ ,
0, this conclusion still holdsprovided that the detuning is at most of the same order of mag-nitude of J and obeys | ∆ | ≪ κ | η | . Therefore, in the aboveregime the e ff ective Hamiltonian in the Schr¨odinger picturereduces toˆ H e ff = ˆ H ℓ + X k µ = ± (cid:16) ω k ± ˆ α † k µ ˆ α κµ + ω a ˆ β † k µ ˆ β k µ (cid:17) . (24)Eq. (24) embodies a central finding of this work, namelythe possibility that the complex many-body atom-photon in-teraction reduces to an e ff ective JC-like coupling described byˆ H ℓ between a bound photon mode and its excitonic analoguewith all of the remaining field and atoms’ modes freely evolv-ing. According to Eq. (23), when | η | ≤ J ≪ κ . In the thermodynamic limit, i.e. inpractice for large-size CCAs, this marks a major di ff erencefrom the uniform-hopping array [8] discussed in the previoussection. IV. DYNAMICS OF EXCITATION TRANSPORT
Our next goal is to shed light on the features of the sys-tem’s time evolution in the one-excitation subspace in linewith Refs. [7, 8]. We recall that, despite normal atomic oper-ators associated with di ff erent modes in general do not com-mute, in the one-excitation subspace they e ff ectively do [20].We denote by | i the system’s state with zero excitations,either photonic or atomic, and define states | Ψ ± i as | Ψ ± i = A ± ˆ α † ℓ | i + B ± ˆ β † ℓ | i , (25)with A ± = J p ( ∆ ± Ω ) + J , (26) B ± = ∆ ± Ω p ( ∆ ± Ω ) + J . (27)Here, Ω = √ ∆ + J is the usual Rabi frequency associatedwith the JC-like Hamiltonian (21). | Ψ ± i are eigenstates of ˆ H ℓ ,and hence of ˆ H e f f , with eigenvalues ( ω a + ω f ) / ± Ω /
2. Usingthis and taking as paradigmatic initial state | Ψ (0) i one suchthat a given atom x is excited, i.e. | Ψ (0) i = ˆ b † x | i , at a latertime t the system has evolved according to | Ψ ( t ) i = h | ˆ β ℓ ˆ b † x | i (cid:16) B + e − i Ω / t | Ψ + i + B − e i Ω / t | Ψ − i (cid:17) + X k X µ = ± h | ˆ β k ,µ ˆ b † x | i e i ∆ / t β † k ,µ | i , (28)up to an irrelevant phase factor. Eq. (28) was obtained byexpanding | Ψ (0) i in the basis of stationary states of ˆ H e f f {| Ψ ± i , ˆ α k , ± | i , ˆ β k , ± | i} . In the following analysis of the im-plications of Eq. (28), we shall make use of the completenessof the basis of single-photon states { ˆ α † ℓ | i , ˆ α † k , ± | i} , i.e.ˆ α † ℓ | ih | ˆ α ℓ + X k X µ = ± ˆ α † k ,µ | i h | ˆ α k ,µ = , (29)where is the identity operator in the one-photon Hilbertspace of the field. Eq. (29) straightforwardly follows fromorthonormality identities (B7)-(B9).Let us first consider the case that x is even. As the boundmode (13) does not overlap even sites (see previous section)we trivially have h | ˆ β ℓ ˆ b † x | i = β † k ,µ | i do contribute to | Ψ ( t ) i . This immediatelyyields that h | ˆ a x | Ψ ( t ) i = x , i.e. no field excitationis developed. As for h | ˆ b x | Ψ ( t ) i , namely the probability am-plitude to find the x th atom excited, use of Eq. (29), which isclearly valid for operators ˆ β k ± ’s as well, along with Eq. (15)entails h | ˆ b x | Ψ ( t ) i = δ x , x . In other words, when x is even | Ψ ( t ) i = | Ψ (0) i , i.e. the atomic excitation is frozen analogouslyto the uniform-hopping case [8]. The freezing behavior how-ever may not occur when x is odd . Indeed, through a rea-soning similar to the one carried out above it is immediateto prove that for odd x if x is even then both h | ˆ a x | Ψ ( t ) i and h | ˆ b x | Ψ ( t ) i vanish, namely the initial excitation can onlyspread over odd cavities. On the other hand, when x is oddprojection of Eq. (28) onto ˆ a † x | i and ˆ b † x | i respectively yield h | ˆ a x | Ψ ( t ) i = − i N τ x + x − J Ω sin Ω t e − i ∆ t , (30) h | ˆ b x | Ψ ( t ) i = δ x , x + N τ x + x − " cos Ω t − i ∆Ω sin Ω t ! e − i ∆ t − , (31)where N is the square of the factor out of the sum in Eq. (13).Eqs. (30) and (31) fully describe the time evolution of the ar-ray in the regime such that the system Hamiltonian is wellapproximated by ˆ H e f f as given in Eq. (24). To illustrate theessential features of the dynamics, in Fig. 3 we address thecase of x = |h | ˆ b x | Ψ ( t ) i| and |h | ˆ a x | Ψ ( t ) i| , in the time interval t ∈ [0 , π/ Ω ] (afterwardsthe same behavior is cyclically re-exhibited). The exciton ini-tially present at cavity 1 (so that at t = t = π/ Ω when the fieldbound mode attains its maximum amplitude. Next, the fieldmode returns energy while the exciton probability spreadingcontinues. At t = π/ Ω , the field is again fully unexcitedbut, remarkably, the exciton is no more localized on the firstatom. Rather, for each atom in an odd cavity (except cav-ity 1) the excitation probability reaches its maximum value.Finally, between t = π/ Ω and t = π/ Ω while the field under-goes a further excitation-deexcitation cycle analogous to theprevious one the excitonic distribution progressively localizearound the first cavity until at t = π/ Ω the initial state is fullyretrieved. All of such features can be easily and accuratelypredicted once the moduli of Eqs. (30) and (31) are squared,which when ∆ = p f , x ( t ) = N J Ω ! τ x + x − sin Ω t , (32) p a , x ( t ) = " δ x , x + N τ x + x − cos Ω t − ! , (33)where p f , x ( p a , x ) stands for the photonic (atomic) probabilityexcitation.It is worth mentioning that one can give a pictorial descrip-tion of the above dynamics in terms of breathing-mode be-haviors. While the field bound mode exhibits pure “transversebreathing” (with respect to the array axis), the excitonic modein addition to this also shows “longitudinal breathing” sincethe atomic excitation somehow cyclically propagates alongthe array axis and localizes again on the starting atom.The setting η = − .
25 in Fig. 3 was chosen in order to bet-ter highlight these phenomena. Higher values of η reduce thenumber of involved cavity sites [we recall that the bound modecharacteristic length is given by λ = / log[(1 + η ) / (1 − η )], seeEq. (13) and Fig. 2]. For lower values of η more cavities getinvolved in the dynamics but the overlap between the exci-tonic bound mode and the initial state shrinks. As the compo-nent of the initial state that is orthogonal to the bound mode remains frozen in the light of Eq. (24), such overlap clearlya ff ects the maximum amount of energy that can be exchangedbetween atoms and photons. This can be seen in Fig. 4, wherefor di ff erent η ’s we plot the overall atomic (photonic) excita-tion probability P x p a , x ( P x p f , x ) against time. Remarkably,in full analogy with a standard JC model [16] the atom-fieldexchange of energy occurs so that the initially excited atomfully retrieves the energy released to the array regardless ofthe array size. Notice that this takes place at a rate given bythe very same Rabi frequency associated with a single isolatedcavity. As anticipated, the higher η the larger is the exchangedamount of energy. In particular, in the limit η → − degenerate single-cavity JCmodel, which can describe the intermediate situation betweena standard single-cavity JC and a uniform-hopping array.When the initial localized excitation is purely photonic, en-ergy exchange proceeds analogously. However, while the ex-citonic distribution evolves with time exactly as the photonicone in the above case (thus exhibiting pure transverse breath-ing) an analogous argument does not hold for the photonic dis-tribution. This exhibits a behavior more complex than thosein Fig. 3 and, remarkably, it spreads with time over the entirearray. This can be seen with the help of Eq. (24): while the un-bound atomic normal modes all have the same frequencies, sodo not the photonic ones. Hence, unlike the process in Fig. 3where the part of the initial wavefunction not overlapping theatomic local mode remains frozen, when the initial excitationis photonic the non-overlapping part freely propagates alongthe array in the form of photonic excitation. V. CONCLUSIONS
In this work, we have considered an array of coupled cav-ities and atoms with interspersed hopping strengths. Wehave first presented analytical solutions for the normal eigen-modes of the free field Hamiltonian under open BCs to high-light the emergence of two continuous bands with a boundmode occurring at the center of their energy gap. In con-trast to uniform- and parabolic-couplings arrays [8] where inthe strong-hopping regime the atomic dynamics is frozen, wehave shown that depending on which cavity is initially exciteda significant exchange of energy between atoms and photonscan arise. The associated dynamics is basically the one oc-curring with a standard JC model, where the aforementionedphotonic bound mode and its excitonic analogue play the rolesof the cavity mode and two-level atom, respectively. Remark-ably, the Rabi frequency associated with such e ff ective JC-type dynamics is the same as the one associated with a singleisolated cavity. In real space, an excitation initially localizedwithin one cavity periodically spreads over nearby cavity sitesof the same parity, in the form of both photonic and atomicexcitations, and then localizes back on the starting site so thatthe initial conditions are retrieved. Interestingly, there is an FIG. 3: (Color online) Snapshots of the atomic (red dots) and photonic (blue dots) excitation probabilities against the cavity number at timeinstants within the range t ∈ [0 , π/ Ω ] and for the initial state where atom 1 is fully excited. We have set η = − . N =
101 and (in units of J ) κ = ω f = ∆ =
0. Although the plots were obtained from numerical solutions of the exact Hamiltonian ˆ H , in practice they remainidentical when Eqs. (30) and (31) are employed. Only cavity numbers such that both excitation probabilities significantly di ff er from zero areshown.FIG. 4: Overall atomic (solid line) and photonic (dashed line) exci-tation probabilities against time for η = − . , − . , − . ,
1. Theinitial state is the one where atom 1 is fully excited. We have set N =
101 and (in units of J ) κ = ω f = ∆= intermediate time instant at which while the field is fully un-excited and the initially localized exciton is spread over thecharacteristic range of the bound mode.In this work we have restricted to the case of odd N . Indeed,in our staggered tight-binding model with open BCs the evenand odd cases cannot be treated on the same footing like withcyclic conditions [19]. In the even case, while major featuressuch as the presence of a band gap with a discrete level atits center still hold the discrete level, when present, becomestwofold [21]. We have thus focused on the odd case merely forthe sake of argument in order to better highlight the physicale ff ects that we have presented. A comprehensive treatment ofboth cases will be the subject of a future publication [22].Localized (bound) normal modes often occur in solid-statephysics [23] typically in the vicinity of localized defects orimpurities that break the translational invariance of the hostlattice. In a similar vein, they also appear in various CCAs scenarios such as arrays with one [12] or two [13] impurityatoms, T-type arrays with a single impurity atom [14] and inCCAs with one or two detuned cavities [15]. Here, the boundmode responsible for the phenomena that we have presentedarises in a somewhat di ff erent way since no impurities or de-fects are present. Rather, its emergence is essentially a pureboundary e ff ect stemming solely from the finiteness of the ar-ray length (we recall that under cyclic BCs this mode is absent[19]). As such, aside from the specific context here addressedthe present work provides a paradigmatic example of bound-ary e ff ects in a CCAs scenario.Concerning an experimental test of the phenomena pre-sented in this paper, arguments analogous to those recentlydiscussed elsewhere for CCAs with controllable hoppingstrengths [8, 24] hold here as well. It is important to pointout that even though we have often assumed large-size arraysin our numerical examples all the discussed e ff ects in fact donot depend on the number of cavities. This makes their obser-vation feasible even with a small-size array, a setting that iswidely expected to become accessible in the imminent future.Finally, it is worth mentioning that in the one-excitationHilbert space of the field the bound mode in Eq. (13) in factdefines an invariant state of the free field Hamiltonian ˆ H f . Inparticular, notice that it makes the hopping part of the freefield Hamiltonian (2) e ff ectively vanish (this is the reason whythe Rabi frequency associated with the e ff ective JC-type dy-namics is the same as the one associated with a single isolatedcavity). Invariant subspaces are states able to inhibit the trans-port of excitations through a quantum network. As such, theyplay a major role, although indirect, in some models recentlyproposed to explain the observed highly e ffi cient excitationtransfer in light-harvesting complexes [25]. Our findings pro-vide a novel mechanism and context (of course well di ff erentfrom the aforementioned ones) where invariant subspaces canalso play a significant role in excitation transport. Acknowledgments
Fruitful discussions with D. Burgarth, G. Falci,G. M. Palma, M. Paternostro and M. Zarcone are grate-fully acknowledged.
Appendix A: Useful formulas
In this Appendix, we prove the three identities (to be usedin the following Appendixes)sin ( kx + ϑ k ) = κ (1 − η ) ε k { sin[ k ( x − − τ sin( kx ) } , (A1) N ± X x = sin( kx ) sin( k ′ x ) = N + δ kk ′ , (A2) N + X x = (cid:8) sin[ k ( x − k ′ x ) + sin[ k ′ ( x − kx ) (cid:9) = N +
12 cos k δ kk ′ , (A3)where k = π m / ( N +
1) with m = , ..., ( N − / k ′ with m ′ being the associated integer). Eq. (A1)is straightforwardly checked by expressing the sine functionin terms of complex exponentials and then replacing e ± i ϑ k through Eq. (16) assin ( kx + ϑ k ) = κ (1 − η ) ε k e ikx ( e − ik − τ ) − e − ikx ( e ik − τ )2 i = κ (1 − η ) ε k { sin[ k ( x − − τ sin( kx ) } . (A4)As for the remaining identities, using the well-known sum for-mula for geometric series it turns out that N + X x = e ikx = N + X x = e π imxN + = N + m = [ − + ( − m ] (1 − e ik )2(1 − cos k ) m , . (A5)By taking the real part of Eq. (A5) the corresponding formulafor the cosine sum is obtained as N + X x = cos ( kx ) = ( N + m = − + ( − m m , . (A6)Upon application of prosthaphaeresis formulas each productof sines appearing in the sum (A2) can be decomposed as thesum of two cosines so as to yield that N ± X x = sin( kx ) sin( k ′ x ) = N ± X x = cos[( k − k ′ ) x ] − cos[( k + k ′ ) x ]2 = ( n N + − h − + ( − m ′ io = N + m = m ′ m , m ′ = N + δ kk ′ , (A7) where we have used (A6) to replace the cosine sums.Eq. (A2) is thus proven (note that the sum remains un-changed if the upper bound ( N + / N − / m = ( N + / m , N + X x = cos [( k ± k ′ ) x − k ] = Re e − ik N + X x = e i ( k ± k ′ ) x = h − + ( − m + m ′ i Re " e ik − e ∓ ik ′ = h − + ( − m + m ′ i (cos k − cos k ′ )(cos k − cos k ′ ) + (sin k ± sin k ′ ) , (A8)which holds for k , k ′ . For k = k ′ we obtain that N + X x = cos [( k ± k ′ ) x − k ] | k = k ′ = P N + x = cos [ k (2 x − = N + cos k . (A9)In the upper case (corresponding to the case k + k ′ ) the sumvanishes since clearly so does the sum over both sin(2 kx ) andcos(2 kx ). Use of prosthaphaeresis formulas allows to decom-pose each product of sines in Eq. (A3) in terms of a cosinesum. Thereby, the first sum in the left-hand side of (A3) canbe decomposed as N + X x = sin[ k ( x − k ′ x = cos[( k − k ′ ) x − k ] − cos[( k + k ′ ) x − k ]2 , (A10)while the decomposition of the second sum in (A3) is obtainedfrom (A10) by exchanging k with k ′ . It is now clear that forany k , k ′ the left-hand side of (A3) vanishes given that anexchange of k with k ′ evidently transforms (A8) into its oppo-site. On the other hand, it is immediate to see that when k = k ′ Eqs. (A9) and (A10) entail that the left-hand side of (A3) re-duces to ( N + / k δ k , k ′ . The proof of identity (A3) is thuscomplete. Appendix B: Orthonormality conditions
Here, we demonstrate that the set of discrete functions spec-ifying the normal operators in Eqs. (13) and (15) fulfill or-thonormality conditions. To this aim, we first set a suitable no-tation (to be used in the next appendixes as well). Firstly, werelabel each continuous-band normal mode associated with { k , ±} as { k , ± } , i.e. we replace ± with the numerical index µ = ±
1. The expansion of each normal annihilation operator isrewritten asˆ α ℓ = N X x = ϕ ℓ, x ˆ a x = N + X x = ϕ ℓ, x − ˆ a x − + N − X x = ϕ ℓ, x ˆ a x , (B1)ˆ α k µ = N X x = ϕ k µ, x ˆ a x = N + X x = ϕ k µ, x − ˆ a x − + N − X x = ϕ k µ, x ˆ a x , (B2)where by virtue of Eqs. (13) and (15) the discrete real func-tions ϕ ’s are defined as ϕ ℓ, x − = A τ x − , ϕ ℓ, x = , (B3) ϕ k µ, x − = µ B sin ( kx + ϑ k ) , ϕ k µ, x = B sin ( kx ) , (B4)where for compactness of notation we have set A = η − r ητ N + − , (B5) B = r N + . (B6)We recall that τ is defined according to Eq. (10). Our goal isto prove that the set of N discrete functions { ϕ ℓ, x , ϕ k µ, x } satisfythe orthonormality conditions N X x = ϕ ℓ, x ϕ ℓ, x = , (B7) N X x = ϕ ℓ, x ϕ k µ, x = , (B8) N X x = ϕ k µ, x ϕ k ′ µ ′ , x = δ kk ′ δ µµ ′ . (B9)Upon use of Eqs. (B3) and (B5) along with the well-knownsum formula for geometric series the following identity holds N + X x = τ x − = τ N + − τ − = ( η − ( τ N + − η = A − . (B10)Hence, it turns out that N X x = ϕ ℓ, x ϕ ℓ, x = N + X x = ϕ ℓ, x − = A N + X x = τ x − = , (B11)which proves Eq. (B7).Using now identity (A1) along with Eqs. (B3) and (B4), theleft-hand side of Eq. (B8) can be arranged as N X x = ϕ ℓ, x ϕ k µ, x = µ A B N + X x = τ x − sin ( kx + ϑ k ) ∝ N + X x = n τ x − sin[ k ( x − − τ x sin( kx ) o ∝ N − X x = τ x sin( kx ) − N + X x = τ x sin( kx ) = . (B12)where we have used the fact that in the second sum on thelast line either of the terms corresponding to x = x = ( N + / k = π m / ( N + N X x = ϕ k µ, x ϕ k ′ µ ′ , x = B µµ ′ N + X x = sin ( kx + ϑ k ) sin ( k ′ x + ϑ k ′ ) + N − X x = sin ( kx ) sin ( k ′ x ) . (B13)Upon use of Eqs. (16), (A2) and (A3), the first sum withinsquare brackets on the right-hand side is given by N + X x = sin ( kx + ϑ k ) sin ( k ′ x + ϑ k ′ ) = " κ (1 − η ) ε k (1 + τ − τ cos k ) × N + δ kk ′ = N + δ kk ′ , (B14)where we have used the identity1 + τ − τ cos k = ε k κ (1 − η ) , (B15)which can be checked straightforwardly through definitions(10) and (11) and application of half-angle formulae. UsingEq. (A2) along with Eq. (B14), Eq. (B13) takes the form N X x = ϕ k µ, x ϕ k ′ µ ′ , x = µµ ′ δ kk ′ + δ kk ′ = δ kk ′ δ µµ ′ , (B16)which shows that identity (B9) holds.All of the three orthonormality identities (B7)-(B9) are thusproved. Appendix C: Normal modes in the special case η = In this Appendix, we prove that the set of normal frequen-cies and corresponding annihilation operators appearing in thedecomposition of ˆ H f (12) and defined in Eqs. (13)-(15) re-duce to (7) and (8), respectively, in the special case η =
0, asexpected.Using Eqs. (11) and (14), in the case that η = α k + and ˆ α k − become, respectively ω k + | η = = ω f + κ cos( k / , (C1) ω k − | η = = ω f − κ cos( k / = ω f + κ cos k − π ! . (C2)According to identity (A1), proven in Appendix A, for η = kx + ϑ k ) becomes proportional to the sum oftwo sines [ τ → − kx + ϑ k ) | η = = (cid:16) kx − k + kx (cid:17) cos k k = sin " k x − = − sin " k − π ! (2 x − . (C3)0Also, sin( kx ) can be arranged in either of the equivalent formssin( kx ) = sin " k x ) = sin " k − π ! (2 x ) . (C4)In the light of Eqs. (15), (C3) and (C4) ˆ α k ± can be arranged asˆ α k + | η = = r N + N X x = sin " k x ˆ a x , (C5)ˆ α k − | η = = r N + N X x = sin " k − π ! x ˆ a x . (C6)Given that k = π m / ( N +
1) with m = , ..., ( N − / N − / − k − π ! = ( π ( N + − m ) N + ) N − m = = π NN + ,..., π [( N + / + N + . (C7)Eq. (C7) shows that the normal modes corresponding toEqs. (C2) and (C6) coincide with the last ( N − / η = α ℓ | η = = lim η → " η − r ητ N + − N + X x = ( − x − ˆ a x − = − r N + N X x = sin " π [( N + / N + x ˆ a x , (C8)thereby coinciding with (8) for m = ( N + / m = ( N + / ω f . Hence, in the special case η = Appendix D: Free-field normal-mode decomposition
In this Appendix, we will explicitly check the validity ofEq. (12), i.e. the normal-mode decomposition of the free fieldHamiltonian ˆ H f . Due to the orthonormality conditions (B7)-(B9), the (real) matrix of the coe ffi cients that define the normaloperators [cf. Eqs. (B3)-(B4)] is orthogonal. Its inverse thuscoincides with its transpose, which yields thatˆ a x = ϕ ℓ, x ˆ α ℓ + X k ,µ ϕ k µ, x ˆ α k µ . (D1)By using this equation to express operators { ˆ a x } in terms ofnormal operators, the free field Hamiltonian (2) becomesˆ H f = ω f N X x = ϕ ℓ x ˆ α † ℓ ˆ α ℓ + X k µ X k ′ µ ′ N X x = ϕ k µ, x ϕ k ′ µ ′ , x ˆ α † k µ ˆ α k ′ µ ′ + X k µ N X x = ϕ ℓ, x ϕ k µ, x (cid:16) ˆ α † ℓ ˆ α k µ + h . c . (cid:17) + N − X x = ρ x ϕ ℓ, x + ϕ ℓ, x ˆ α † ℓ ˆ α ℓ + X k µ X k ′ µ ′ N − X x = ρ x f k ′ µ ′ k µ, x ˆ α † k µ ˆ α k ′ µ ′ + X k µ N − X x = ρ x g k µ, x (cid:16) ˆ α † ℓ ˆ α k µ + h . c . (cid:17) , (D2)where we have set ρ x = − κ (cid:2) − ( − x η (cid:3) , (D3) f k ′ µ ′ k µ, x = ϕ k µ, x + ϕ k ′ µ ′ , x + ϕ k µ, x ϕ k ′ µ ′ , x + , (D4) g k µ, x = ϕ ℓ, x + ϕ k µ, x + ϕ ℓ, x ϕ k µ, x + . (D5)The three sums over x between brackets on the first line ofEq. (D2) coincide with the left-hand sides of the orthonormal-ity identities (B7)-(B9). Moreover, due to Eq. (B3) the coe ffi -cient of ˆ α † ℓ ˆ α ℓ on the second line clearly vanishes. Using these facts, Eq. (D2) considerably simplifies asˆ H f = ω f ˆ α † ℓ ˆ α ℓ + X k µ ˆ α † k µ ˆ α k µ + X k µ X k ′ µ ′ N − X x = ρ x f k ′ µ ′ k µ, x ˆ α † k µ ˆ α k ′ µ ′ + X k µ N − X x = ρ x g k µ, x (cid:16) ˆ α † ℓ ˆ α k µ + h . c . (cid:17) . (D6)A comparison between Eqs. (D6) and (12) with the help ofEqs. (11) and (14) shows that the proof of (12) is now reduced1to demonstrating the identities N − X x = ρ x g k µ, x = , (D7) N − X x = ρ x f k ′ µ ′ k µ, x = − µ ε k δ kk ′ δ µµ ′ . (D8)In addition, we need to prove that ε k is given by Eq. (11).As for the proof of (D7), we use Eqs. (B3) and (B4) toderive the following identities N − X x = ( ± x ϕ k µ, x + ϕ ℓ, x = ± N − X x = ϕ k µ, x ϕ ℓ, x − = ± γτ , (D9) N − X x = ( ± x ϕ k µ, x ϕ ℓ, x + = ± N − X x = ϕ k µ, x − ϕ ℓ, x − = γ , , (D10)where γ = AB N − X x = sin( kx ) τ x . (D11)These identities along with Eqs. (D3) and (D5) allow tostraightforwardly arrange the left-hand side of (D7) in theform N − X x = ρ x g k µ, x = τ + + ητ − η ! γ = " η + τ − ( η − γ = , (D12)where we have used definition (10). Eq. (D7) is thus demon-strated.To prove Eq. (D8), we consider the following identitiesholding for any x = , ..., ( N − / ρ x − ϕ k µ, x ϕ k ′ µ ′ , x − = − µ ′ B κ sin( kx ) (1 + η ) sin( k ′ x + ϑ k ′ ) , (D13) ρ x ϕ k µ, x ϕ k ′ µ ′ , x + = − µ ′ B κ sin( kx ) (1 − η ) sin[ k ′ ( x + + ϑ k ′ ] , (D14)where we have used Eqs. (B4) and (D3). Eqs. (D13) and(D14) sum to ρ x − ϕ k µ, x ϕ k ′ µ ′ , x − + ρ x ϕ k µ, x ϕ k ′ µ ′ , x + = − µ ′ B κ sin( kx ) S , (D15)where S = (1 + η ) sin( k ′ x + ϑ k ′ ) + (1 − η ) sin[ k ′ ( x + + ϑ k ′ ] . (D16)Using now identities (A1) and (B15) along with definition(10), S can be arranged in the form S = κ (1 − η ) ε k ′ (cid:8)(cid:2) (1 − η ) − τ (1 + η ) (cid:3) sin( k ′ x ) + ( η + (cid:0) sin[ k ′ ( x − + sin[ k ′ ( x + (cid:1)(cid:9) = κ (1 − η ) ε k ′ h (1 − τ − τ cos k ′ ) sin( k ′ x ) i = ε k κε k ′ sin( k ′ x ) . (D17)In deriving Eq. (D17) we have made use of prosthaphaeresisformulas to replace the sum of sines on the second line of (D17) with the product 2cos k ′ sin( k ′ x ). Using this result, uponsum of (D15) over x = , ..., ( N − / N − X x = ρ x − ϕ k µ, x ϕ k ′ µ ′ , x − + ρ x ϕ k µ, x ϕ k ′ µ ′ , x + = − µ ′ ε k δ kk ′ , (D18)where we have used identity (A2) to carry out the sum ofsin( kx ) sin( k ′ x ).Consider now the left-hand side of Eq. (D8), where the sumcan be split into the sums over even and odd terms. A com-parison of this with the left-hand side of Eq. (D18) shouldmake clear that (D8) can be obtained by adding (D18) to thequantity obtained from (D18) through exchange of { k , µ } with { k ′ , µ ′ } . This argument leads to N − X x = ρ x f k ′ µ ′ k µ, x = − ( µ + µ ′ ) ε k δ kk ′ = − µ ε k δ kk ′ δ µµ ′ , (D19)which proves that Eq. (D8) holds.Although we have proven both the identities (D7) and (D8),to finalize the explicit demonstration of Eq. (12) we need toprove that ε k is given by Eq. (11) (note that in this Appendixup to this stage we have left ε k unspecified). To accomplishthis task, we point out that Eq. (16) entails the constraint1 = κ (1 − η ) ε k | e − ik − τ | , (D20)i.e. the squared modulus of the complex exponential on theleft-hand side must be unitary. By bringing ε k to the left-handside and calculating the squared modulus on the right-handside Eq. (D20) takes the form ε k = κ n(cid:2) ( η −
1) cos k − ( η + (cid:3) + ( η − sin k o = κ cos k + η sin k ! , (D21)where in the last step we have made use of half-angle formu-lae. By taking the square root of (D21) Eq. (11) is retrieved.The proof of decomposition (12) is therefore complete. Appendix E: Full Hamiltonian normal-mode decomposition
Here, we prove that the full Hamiltonian (1) can be de-composed in terms of normal operators according to Eq. (20).Having proven in Appendix D that (12) holds, it su ffi ces todemonstrate that N X x = ˆ b † x ˆ b x = ˆ β † ℓ ˆ β ℓ + X k µ ˆ β † k µ ˆ β k µ , (E1) N X x = (cid:16) ˆ a † x ˆ b x + ˆ b † x ˆ a x (cid:17) = (cid:16) ˆ α † ℓ ˆ β ℓ + ˆ β † ℓ ˆ α ℓ (cid:17) + X k µ (cid:16) ˆ α † k µ ˆ β k µ + ˆ β † k µ ˆ α k µ (cid:17) . (E2)We first note that due to Eqs. (18) and (19) the orthonormalityconditions (B7)-(B9) entail that Eq. (D1) holds for the atomic2operators as well, i.e.ˆ b x = ϕ ℓ, x ˆ β ℓ + X k ,µ ϕ k µ, x ˆ β k µ . (E3)Using this equation to express each atomic site operator inEq. (E1) in terms of normal operators along with the or-thornormality identities (B7)-(B9), the proof of Eq. (E1) isin fact identical to the one carried out in the case of field op-erators [cf. first line of Eq. (D2) and the quantity between thefirst brackets on the right-hand side of Eq. (D6)]. 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