Pitfalls in the theory of carrier dynamics in semiconductor quantum dots: the single-particle basis vs. the many-particle configuration basis
PPitfalls in the theory of carrier dynamics in semiconductor quantum dots:the single-particle basis vs. the many-particle configuration basis
T. Lettau, H.A.M. Leymann,
1, 2, ∗ and J. Wiersig Institut f¨ur Theoretische Physik, Otto-von-Guericke-Universit¨at Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany Max-Planck-Institut f¨ur Physik komplexer Systeme,N¨othnitzer Strasse 38, 01187 Dresden, Germany (Dated: October 15, 2018)We analyze quantum dot models used in current research for misconceptions that arise from thechoice of basis states for the carriers. The examined models originate from semiconductor quantumoptics, but the illustrated conceptional problems are not limited to this field. We demonstrate howthe choice of basis states can imply a factorization scheme that leads to an artificial dependency be-tween two, actually independent, quantities. Furthermore, we consider an open quantum dot-cavitysystem and show how the dephasing, generated by the dissipator in the von Neumann Lindbladequation, depends on the choice of basis states that are used to construct the collapse operators.We find that the Rabi oscillations of the s-shell exciton are either dephased by the dissipative decayof the p-shell exciton or remain unaffected, depending on the choice of basis states. In a last step weresolve this discrepancy by taking the full system-reservoir interaction Hamiltonian into account.
I. INTRODUCTION
There are many well-established theories to describeopen quantum many-particle systems consisting of,e.g., quasi-free charge carriers in semiconductor het-erostructures , cavity photons , phonons , ultracoldBose-gases , polaritons , and spins in an approxi-mate way. The theories in the preceding references arerelated to mean-field theories and its successive improve-ments, like the cluster expansion (CE), in whichequations of motion (EoM) for the mean single-particleoccupations and their correlations are derived, whereashigher-order correlations are neglected.Besides the approximations that are necessary to de-scribe the interacting system itself, many of the ref-erenced models also require additional approximationsto include dissipative processes resulting from the sys-tems coupling to an external bath. The von Neu-mann Lindblad equation is a common procedure totake the influence of the exterior bath on the sys-tem into account , provided that the Born-Markov-approximation is justified .The experimental progress in the field of cavity quan-tum electrodynamics in semiconductors shows thatthere are many interesting systems in which the ba-sic assumption of mean field theories, a large Hilbertspace and weak interaction, is not valid. Many ofthese systems are sufficiently small to be described bytheir exact wave function or density matrix, formu-lated in the Hilbert space of all possible many-particleconfigurations without the need for an approximatetheory. Despite the efforts that are made to improve thetheories from both sides (exact description of relativelysmall systems and approximate description of rel-atively large systems ), there is still a gap betweenthose systems that are small enough to be described ex-actly and those that are large enough to fulfill the re-quirements of approximate theories like the CE. In this article, we describe pitfalls in the choice ofthe basis states that may occur, when applying ap-proximate theories on small systems within or close tothe mentioned gap. We use three examples to contrastapproaches that are based on a formulation in single-particle states with approaches that use many-particleconfiguration states as a basis. Although the formula-tions are equivalent, the choice of basis states can decideabout further steps. In our three examples, we show thatthe choice of basis states can suggest misleading approx-imations or determine the modeling of dissipative pro-cesses, which leads to deviations of the results, in thetwo formulations that go beyond simple approximationerrors.The remainder of this paper is organized in the follow-ing way: In Sec. II, we discuss, based on an extendedJaynes-Cummings model (JCM), the effects of a fal-lacious mean-field factorization scheme, implied by a de-scription in single-particle states.In Sec. III, we consider an open system treated in thevon Neumann Lindblad (vNL) formalism. We demon-strate that the basis states in which the collapse op-erators and with it the dissipator of the vNL equationare constructed can actually influence the modeling ofthe system. In the first example concerning the vNL(Sec. III A), the choice of basis states limits the possibil-ities to adjust the model to the experimental situation.Whereas, in the second example concerning the vNL for-malism (Sec. III B), the basis states determine whethertwo parts of the system are affected by the environmentindependently or intertwined and one system part is de-phased by the dissipative decay of the other. Finallywe recapitulate how the dissipator in the vNL can beconstructed from a system plus reservoir approach andresolve the misconception, that has led to results depend-ing on the choice of basis states (Sec. III C).The last section IV summarizes and concludes the pa-per. In the appendix, we give details of the EoM and theparameter space of the semiconductor JCM (App. A). a r X i v : . [ qu a n t - ph ] F e b Furthermore we outline the derivation of the analytic so-lution for the open system (App. B 1), and present theeffects of an additional external pump on the open system(App. B 2).
II. FALLACIOUS FACTORIZATION
To illustrate a conceptual problem that can arise froma Hartree-Fock-like factorization of expectation values,we consider a model with Jaynes-Cummings interaction,introduced in , with the Hamiltonian H = ωb † b + ε e e † e + ε h h † h − ( gheb † + h . c . ) , (1)where b ( † ) annihilates (creates) a cavity mode photonwith frequency ω and e ( † ) /h ( † ) annihilates (creates) anelectron/hole, with energy ε e /ε h , respectively. Thedipole matrix element g can be chosen real, and all pa-rameters are specified in units of (cid:126) . This Hamiltoniandescribes a two-level quantum dot (QD) embedded ina semiconductor environment, coupled to a single cav-ity mode. It would be identical to the JCM, if one re-stricts the electronic states to fully correlated electronsand holes, i.e., restricting the electronic states to a sin-gle exciton (electron-hole pair). However, in order todescribe the semiconductor properties of a QD, an in-dependent occupation of the electron and hole states isallowed in this model, which we term the semiconduc-tor JCM. Both systems perform a coherent exchange be-tween the cavity photons and the exciton, called Rabioscillations . The calculation of the time evolution us-ing EoM for the expectation values produces a hierarchyof coupled equations. We will show that a factorizationof many-particle expectation values into single-particleexpectation values, often used to truncate hierarchies ofEoM, is not only unnecessary in this exemplary model,but also leads to conceptually wrong conclusions.To derive the EoM, we follow the approach of , inwhich the photons are not described by creation and an-nihilation operators, but by the photon probability distri-bution. This allows for a variant of the CE in which thephotonic part is treated exactly, and is termed the pho-ton probability CE by the authors of . The expectationvalues of interest are the hole f h = (cid:10) h † h (cid:11) and the electron f e = (cid:10) e † e (cid:11) occupation, the occupation of the Fock-stateswith n photons p n = (cid:104)| n (cid:105)(cid:104) n |(cid:105) , and the imaginary part ofthe photon-assisted polarization ψ n = Im (cid:104)| n + 1 (cid:105)(cid:104) n | he (cid:105) .From the Heisenberg equation for the generalized occu-pations f en = (cid:10) | n (cid:105)(cid:104) n | e † e (cid:11) and f hn = (cid:10) | n (cid:105)(cid:104) n | h † h (cid:11) , with f e/h = (cid:80) n f e/hn , we obtain the time derivativesd t f e/hn =2 g √ n + 1 ψ n , (2)d t p n =2 g √ n + 1 ψ n − g √ n ψ n − , (3)d t ψ n = − g √ n + 1 ( p n +1 − f hn +1 − f en +1 ) − g √ n + 1 (cid:0) C Xn +1 − C Xn (cid:1) , (4) . . . . N (a) exactHartree-Fock t/ g . . . . g ( ) ( ) (b) FIG. 1: Time evolution of the semiconductor JCM with ex-act and factorized EoM (Hartree-Fock), for the mean photonnumber N (a) and the photon autocorrelation g (2) (0) (b); Ini-tial conditions: p = 1, f e , f h = 0 . , . where the diagonal terms are zero since the cavity is cho-sen to be in resonance with the QD. The EoM for thephoton-assisted polarization couples to the higher-orderterm C Xn = (cid:10) | n (cid:105)(cid:104) n | e † eh † h (cid:11) (5)that describes the electron-hole correlation. In the JCM,in which electrons and holes are perfectly correlated, themany-particle term C Xn can be expressed exactly by thealready known single electron expectation values f en , thusclosing the hierarchy. In a semiconductor environment,the assumption of perfectly correlated electrons and holesis not valid . Therefore, the term C Xn can no longer beexpressed by a single single-particle term. Guided by the single-particle basis, one can proceedwith an approximate treatment of C Xn . The first orderof the photon probability CE results in the factorization C Xn = (cid:10) | n (cid:105)(cid:104) n | e † eh † h (cid:11) ≈ (cid:10) | n (cid:105)(cid:104) n | e † e (cid:11) (cid:10) | n (cid:105)(cid:104) n | h † h (cid:11) (cid:104)| n (cid:105)(cid:104) n |(cid:105) = f en f hn p n , (6)which is related to a neglect of the electron-hole correla-tion δ = (cid:10) e † eh † h (cid:11) − (cid:10) e † e (cid:11) (cid:10) h † h (cid:11) , (7)and corresponds to the Hartree-Fock approximation.With the applied factorization one obtains a closed setof EoM and is able to calculate the dynamics of the elec-tronic and photonic occupations N = (cid:10) b † b (cid:11) = (cid:80) np n ,and the photon autocorrelation function at zero delaytime τ = 0 g (2) (0) = (cid:10) b † b † bb (cid:11) (cid:104) b † b (cid:105) = (cid:80) ( n − n ) p n ( (cid:80) np n ) . (8)Figure 1 shows the dynamics of the semiconductorJCM for a cavity with an initially prepared single-photonFock state and f e , f h = 0 . , .
1. The dimensionlesscharge C = f h − f e of the QD is preserved by theHamiltonian. It is proposed in that, with a fixed FIG. 2: The dependence of the maximum amplitude of g (2) (0)on the charge C and the oscillation ability O . (a): g (2)max (0) independence of C for the exact and factorized (Hartree-Fock)EoM. (b): g (2)max (0) (green area) in dependence of C and O , andthe electron-hole correlation δ (blue/red contour plot in the C - O -plane) which increases with O from δ = − / to δ = / .The special case of δ = 0 is marked by the black curves. Theinitial conditions are δ = 0, f e + f h = 1 and p = 1. Tofix the additional free parameter we have chosen the initialprobabilities for | G (cid:105) and | X s (cid:105) to be equal, which is equivalentto the restriction to a fixed number of total excitations. number of total initial excitations (i.e., p n = const and f e + f h = const ), the charge C determines the maxi-mum amplitude of the photon autocorrelation function g (2)max (0). Figure 2(a) shows the dependence of g (2)max (0)on C , when the system is initially prepared in a single-photon Fock state p = 1 and f e + f h = 1 for the exact(see next paragraph) and the factorized system. Thecurves have their maximum at C = 0, which suits thenotion that the probability that an electron can recom-bine with a matching hole, i.e, the ability of the systemto oscillate, is directly connected to C . The exact andthe approximate curve deviate, since the electron-holecorrelation δ is forced to be zero for all times in the fac-torized version of the EoM, whereas δ = 0 is only aninitial condition in the exact EoM (see next paragraph). When the system is described in many-particle config-urations, it becomes apparent that the previous con-clusion is only an artifact of the focus on single-particleproperties, which manifests in the factorization of theterm C Xn . Factorizing the term C Xn forces the electron-hole correlation δ to be zero and introduces a constraint JCM | + s i | G i | X s i |− s i FIG. 3: Illustration of the electronic configurations | i (cid:105) of thesemiconductor JCM. The original JCM consists of the stateswithin the dashed box, which are the only ones appearing inthe interaction part of the Hamiltonian in Eq. (9). to the system that eventually results in the artificial con-nection between g (2)max (0) and C . A reformulation in termsof many-particle configurations, depicted in Fig. 3, re-veals which electronic states of the QD take part in theRabi oscillations. The state of the system is determinedby four coefficients c i = (cid:104)| i (cid:105)(cid:104) i |(cid:105) , corresponding to themany-particle configuration states | i (cid:105) . Examining theHamiltonian formulated in this basis H = ωb † b + (cid:88) ε i | i (cid:105)(cid:104) i | − (cid:0) g | G (cid:105)(cid:104) X s | b † + h . c . (cid:1) (9)reveals that only | G (cid:105) and | X s (cid:105) take part in the Rabioscillations. This finding suggests the definition of anew quantity, the oscillation ability O = (cid:104)| G (cid:105)(cid:104) G |(cid:105) + (cid:104)| X s (cid:105)(cid:104) X s |(cid:105) , which actually determines the amplitudeof the Rabi oscillations, rather than the charge C = (cid:104)| + s (cid:105)(cid:104) + s |(cid:105) − (cid:104)|− s (cid:105)(cid:104)− s |(cid:105) . Even in the case of C = 0and fixed excitations, one could have no Rabi oscilla-tions at all, if the initial electronic state of the systemis equally distributed between the configurations | + s (cid:105) and |− s (cid:105) . Figure 2(b) shows g (2)max (0) in dependence of O and C for a constant amount of total excitations (seeApp. A 2). The amplitude of g (2) (0) increases with O , independent of C . The correlation between electronsand holes δ is depicted as a contour plot at the bottomof Fig. 2 (b), varying from anticorrelated to fully corre-lated electrons and holes with increasing O . The specialcase for the Hartree-Fock factorization δ = 0 is markedby the three black curves. Following this path in param-eter space, one regains the artificial dependence of themaximum amplitude of g (2) (0) on the charge C . Thisdependence, projected in the C - g (2) (0)-plane, is identicalto the black curve in Fig 2 (a). In conclusion, we have demonstrated that in thesemiconductor JCM, not the charge of the QD C butthe oscillation ability O determines the maximum of g (2) . The constraint in configuration space introducedby the factorization scheme can lead to a misconcep-tion about the systems dynamic, in our case it is theconnection between the charge of the QD and its abil-ity to perform Rabi oscillations. The effect of this con-straint in configuration space is especially drastic in ourcase, since a connection between observables of the sys-tem is derived, g (2) max (0) = f ( O, C ), in contrast to a casewhere the dependence of an observable on an external pa-rameter is derived, e.g. the input-output characteristics (cid:10) b † b (cid:11) = f ( P ump ext ) of a laser.One can avoid problems and misconceptions like thisby describing the finite states of the carriers localizedin the QD in the basis of its many-particle configura-tions as we have demonstrated here. When a formula-tion in single-particle states is desirable one should in-clude all correlations between the localized single-particlestates [App. A] since correlations between single-particlestates are strong in finite systems . There are manyapproaches on QD-(cavity) systems described in the lit-erature, that either find a formulation that includes allpossible many-particle configurations of the system or if this is not possible use hybrid factorization schemesrelated to the cluster expansion. These factorizationschemes are hybrid approaches in the sense that thecorrelations between the carriers localized in a QD arefully included, while correlations between other systemparts are treated approximately by factorization e.g. cor-relations between, different QDs , QD and delocal-ized wetting-layer states , and QD states and continuumstates of the light-field in free space . III. OPEN SYSTEMS AND THECONSTRUCTION OF THE DISSIPATOR
In the previous section, we have discussed misleadingresults that arise from an approximation scheme thattruncates the hierarchy of EoM. In this section we demon-strate that when open quantum system are described inBorn-Markov approximation using the vNL equationd t ρ = i [ ρ, H ] + (cid:88) i γ i (cid:18) L i ρL † i − L † i L i ρ − ρL † i L i (cid:19) = i [ ρ, H ] + D ( ρ ) (10)it can make a significant difference whether the dissipator D is constructed in a single-particle or in a many-particleconfiguration basis. We demonstrate that a misleadingassumption can already be incorporated in the construc-tion of the EoM, thus producing questionable results evenif the basic EoM for ρ is then solved without further ap-proximations. A. Hole capture
As a first introductory example, we consider the holecapture of a semiconductor QD. To model the hole cap-ture from delocalized wetting layer states, the modelillustrated in Fig. 3 is augmented by further localizedstates. For cylindrical QDs these states are the p-shellstates, which are energetically higher than the s-shellstates. Restricting the model to one spin direction andone state in the p-shell results in four single-particlestates that can be occupied by up to four carriers. Thismodel, consisting of 16 possible many-particle configura- tions (see Ref. for details), is the basis for many modelsused to describe semiconductor QDs .The excitation of the QDs is facilitated by electronand hole capture from the quasi-continuous wetting layerstates into the p-shell. To describe the hole capture inthe single-particle basis, one uses a single collapse opera-tor, L = h † p in Eq. (10), that creates a hole in the p-shell.Assigned to this process is a hole capture rate Γ h . Thisformulation treats the hole capture in the p-shell inde-pendently of the occupation of the other states. However,the carriers are captured due to phonon and Coulombscattering of the delocalized wetting layer carriers intothe localized QD states and the single-particle-energiesof the QD states are renormalized by the Coulomb inter-action. Since the scattering rates depend on the energiesof the final state, the hole capture rate of a positivelycharged QD is lower than the one of a negatively chargedQD , as illustrated in Fig. 4. To model the hole capturein a way that takes different capture rates into account,one needs to construct a collapse operator for each tran-sition between two many-particle configurations in whicha hole is created, with rates depending on the configura-tions. Two exemplary transitions are illustrated in Fig. 4,which create a hole in the p-shell, and correspond to theoperators L = | ++ (cid:105)(cid:104) + s | and L = | X p (cid:105)(cid:104)− p | , with therates Γ + h < Γ − h , respectively.This example illustrates two possible ways to constructa transition of a carrier, triggered by the environment,within the dissipator D : (i) using single-particle creationand annihilation operators, resulting in a single collapseoperator (e.g., L = h † p for the hole capture). (ii) using aset of different collapse operators formulated as transitionoperators between many-particle configurations (e.g., L and L ). This formulation allows for a direct distinctionbetween different many-particle configurations.Note that the dissipator in (i) can be also obtainedusing configuration operators, L = h † p = (cid:80) ij | i (cid:105)(cid:104) j | (with i, j chosen so that L creates a p-shell hole). Accordingly,a combination of creation and annihilation operators canregain a distinction between the configurations as in (ii).However, these alternative ways would result in a ratherclumsy notation. The conclusion of this introductory example is thatthe dissipator D can be constructed in two diffrent waysand that it appears necessary to formulate the dissipatorin the basis of many-particle configurations. B. Non-local dephasing
In this example, we show that the two ways to con-struct the dissipator D , described in Sec. III A, lead todifferent results even when the rates for the different col-lapse operators formulated in the many-particle basis, areequal. We emphasize that the same set of operators isused in both constructions of D and that the only differ-ence is how the operators enter the dissipator.Such a situation is the vacuum Rabi oscillation of an Γ + h | + s i | ++ i Γ − h |− p i | X p i FIG. 4: Illustration of the transition from the many-particleconfiguration | + s (cid:105) to | ++ (cid:105) and |− p (cid:105) to | X p (cid:105) , correspondingto the collapse operators L and L , respectively. Both tran-sitions result in the capture of a hole in p-shell of the QD.The capture rate of a positively charged hole depends on theQD’s charge (Γ + h < Γ − h ). β Γ g XXX p g ψ X g ψ X β X s G g ψ s g ψ s β ΓΓ(a) (b)
FIG. 5: (a): Illustration of the system dynamics in the single-particle basis. (b): Illustration of the transitions between themany-particle configurations revealing that there are two Rabicycles, connected by the decay of the p-exciton. electron-hole pair in the s-shell in resonance with a highquality cavity mode, in presence of the spontaneous de-cay of an electron-hole pair in the p-shell, as illustratedin Fig. 5 (a). The basis states of the Hilbert space forthis system are | n, i (cid:105) , where n is the number of cavityphotons and i denotes the electronic configuration of theQD. With an initially empty cavity and a QD preparedin the biexciton state, four electronic configurations arecoupled by the vNL equation: The ground state config-uration | G (cid:105) , the s-exciton | X s (cid:105) , the p-exciton | X p (cid:105) andthe biexciton | XX (cid:105) configuration, as illustrated in Fig. 6.The Jaynes-Cummings interaction Hamiltonian reads H JC = − (cid:0) gh s e s b † + h . c . (cid:1) = − (cid:0) g ( | G (cid:105)(cid:104) X s | + | X p (cid:105)(cid:104) XX | ) b † + h . c . (cid:1) (11)in the single-particle and the configuration basis respec-tively. The dissipator generates the spontaneous loss ofexcitons in the p- and s-shell, with the rates Γ and β , re-spectively. In contrast to the hole capture in Sec. III A,the decay rates in the p-shell are independent of the os-cillatory state of the s-exciton. Unlike the Hamilton op-erator, which is independent of the formulation, the ef-fect of the dissipator D depends on its formulation sincethe collapse operators enter nonlinearly. In the single-particle basis, the loss of the p-shell exciton is generatedby L sp = h p e p (formulation (i)). The same operator canbe constructed by a sum of configuration operators L sp = | G (cid:105)(cid:104) X p | + | X s (cid:105)(cid:104) XX | , (12) | G i | X s i | X p i | XX i FIG. 6: Illustration of the many-particle configurations | G (cid:105) , | X s (cid:105) , | X p (cid:105) , and | XX (cid:105) , which are the basis states for the QD-model exhibiting non-local dephasing. which is still formulation (i). In the many-particle for-mulation, the spontaneous loss of p-shell excitons is gen-erated by two collapse operators L G = | G (cid:105)(cid:104) X p | and L X = | X s (cid:105)(cid:104) XX | , (13)with equal rates γ G = γ X = Γ (formulation (ii)). Thesame holds for the spontaneous exciton loss in the s-shell,with the loss rate β and the collapse operators chosenaccordingly. In the single-particle basis, the dynamics of the s- andthe p-shell are decoupled, which can be seen in the EoMfor the single-particle operator expectation valuesd t (cid:10) e † p e p (cid:11) = − Γ (cid:10) e † p e p (cid:11) , d t (cid:10) e † s e s (cid:11) = − β (cid:10) e † s e s (cid:11) + 2 gψ (cid:124)(cid:123)(cid:122)(cid:125) Rabi + , d t ψ = − βψ + g (cid:0)(cid:10) b † b (cid:11) − (cid:10) e † s e s (cid:11)(cid:1)(cid:124) (cid:123)(cid:122) (cid:125) Rabi − , d t (cid:10) b † b (cid:11) = − gψ (cid:124)(cid:123)(cid:122)(cid:125) Rabi + , (14)with ψ being the imaginary part of photon-assisted polar-ization ( ψ = Im (cid:10) h s e s b † (cid:11) ) and Rabi ± marking the termsresponsible for the Rabi-oscillations. The p-shell occu-pation decays exponentially with rate Γ, the s-shell oc-cupation oscillates with the vacuum Rabi-frequency 2 g and decays with rate β , and the polarization is subjectto the dephasing introduced by the spontaneous losses β in the s-shell. Fig. 5 (a) illustrates the dynamics of thesingle-particle occupations of the system. In the configuration basis, the required quantitiesto formulate the EoM are the occupations of the ba-sis states ( XX n , X np , X ns , G n ) with photon num-ber n , e.g., G n = (cid:104)| n, G (cid:105)(cid:104) n, G |(cid:105) and the photon-assisted polarizations between bi- and p-exciton ψ nX =Im( (cid:104)| n, XX (cid:105)(cid:104) n + 1 , X p |(cid:105) ) and between s-exciton andground state ψ ns = Im( (cid:104)| n, X s (cid:105)(cid:104) n + 1 , G |(cid:105) ). Since westart with an empty cavity, the EoM are restricted tothe first photon block ( n = 0 ,
1) and readd t XX = − (Γ + β ) XX + 2 gψ X , d t ψ X = − gXX − (Γ + β / ) ψ X + gX p , d t X p = − gψ X − Γ X p , d t X s = Γ XX − βX s + 2 gψ s , d t ψ s = { Γ , } ψ X − gX s − β / ψ s + gG , d t G = Γ X p − gψ s , d t X p = βXX − Γ X p , d t G = βX s + Γ X p . (15)The curled brackets { Γ , } in the fifth line of Eqs. (15)mark the difference between the single-particle (i: Γ) andthe configuration basis (ii: 0) in the EoM, which wewill discuss in more detail below. For further discus-sion it is convenient to formulate the EoM in matrix formd t r = M r , where the column vector r = (cid:0) XX , . . . , G (cid:1) T contains the dynamical quantities as listed in Eqs. (15)(for initial state r = (1 , , . . . , T ) and the parametermatrix M reads M = − Γ − β g − g − Γ − β / g − g − ΓΓ − β g { Γ , } − g − β / g Γ − g β − Γ
00 0 0 β Γ . (16)The matrix M can be separated into eight blocks, indi-cated by the lines in Eq. (16). We refer to these blocksrow-by-row. Block I describes the Rabi oscillations withfrequency 2 g on its off-diagonal elements and the decayof excitation and the dephasing of the polarization onits diagonal elements. The same holds for block IV andthe off-diagonal elements of these blocks correspond tothe terms Rabi ± in Eq. (14). Since there is no pump-ing in this system, block II, which transports occupa-tion from lower to higher electronic states, is zero. BlockIII together with the diagonal elements of block I de-scribes the transfer of population from the part of thesystem with a p-shell exciton to the one without. Theoccupation that is lost due to the negative sign of theΓs in block I is transferred to occupations without a p-shell exciton by the positive Γs of block III. The entryin curled brackets M ψ s ,ψ X = { Γ , } reflects the differ-ence between the construction of the dissipator in single-particle basis (i) ( M ψ s ,ψ X = Γ) and in configuration ba-sis (ii) ( M ψ s ,ψ X = 0). In the single-particle basis, thephoton-assisted polarization is transferred from the XX - X p oscillation to the X s - G oscillation. Therefore, onlythe s-exciton decay with rate β and not the p-excitondecay with rate Γ contributes to the dephasing of ψ in Eq. (14). The loss of polarization ψ X with rate Γ is trans-ferred to ψ s with exactly the same rate. On the contraryin the many-particle basis, the polarization ψ X , which islost by the decay of the p-shell exciton, is not picked upby the polarization ψ s , thus the element M ψ s ,ψ X is zero.The remaining blocks can be interpreted analogously, byassociating pairs of positive and negative entries in thesame column with a transfer of occupation. The transi-tions between the states are illustrated in Fig. 5 (b). The solutions of the numerical integration of Eq. (15)are shown in Fig. 7. In panel (a) and (b), a generic casefor the time evolution of the system, for the configura-tion probabilities in (a) and the single-particle occupa-tions in (b), is depicted. Here the deviations of the twoapproaches are visible but one might overlook or dismissthem as irrelevant. The results for the occupation of thes-exciton state | X s (cid:105) and s-shell electron (cid:10) e † s e s (cid:11) depend onthe construction of the dissipator. The results obtainedin the single-particle basis are labeled by the subscript’sp’. The initially prepared biexciton (panel (a), shadedarea) oscillates with the Rabi frequency 2 g and decayswith the rate Γ + β . The s-exciton occupation increaseswith rate Γ, oscillates with the Rabi frequency, and de-cays with the rate β . This behavior holds for both, theconstruction of the dissipator in the many-particle config-urations and in the single-particle basis. The two curvesdeviate in the fact that in the single-particle basis, theoscillations have a larger amplitude than in the many-particle basis, and that in the single-particle basis, theground state is fully occupied within each Rabi cycle.An alternative representation of the dynamics is given inpanel (b), in which the single-particle expectation val-ues for the electrons (cid:104) e † i e i (cid:105) are shown. The p-electron isdecaying with rate Γ in both formulations of the dissipa-tor, whereas the oscillation of the s-shell electron dependson the formulation of the dissipator. To emphasize thecharacteristic difference between the two constructionsof the dissipator we consider the limiting case of vanish-ing s-shell decay ( β = 0), where the deviations are notblurred by a circumstantial dephasing mechanism, with (cid:104) e † i e i (cid:105) shown in panel (c). In the single-particle basis, thes-shell performs Rabi oscillations with a constant ampli-tude of / , while the p-shell exciton decays exponentially.In the many-particle basis, the p-shell electron decay de-phases the Rabi oscillations in the s-shell, resulting in adiminished amplitude in the long-term behavior. Due tothe construction of the dissipator in the non-local basisof the many-particle configurations the dissipation in onesystem part induces non-local dephasing in an otherwiseindependent system part.In many cases, e.g. in cw-lasers, the long term behav-ior or the steady state of the system are of interest. Thesimple form of the EoM in the case of vanishing β al-lows to derive analytic expressions for the dependenceof the amplitude of the Rabi oscillations on the rate Γ(see App. B 1 for details). In the long term behavior the . . . P r o b a b ili t y (a) X s sp X s XX . . P r o b a b ili t y (b) (cid:10) e † s e s (cid:11) sp (cid:10) e † s e s (cid:11)(cid:10) e † p e p (cid:11) t/ g . . . P r o b a b ili t y (c) FIG. 7: Dynamics of the s-shell Rabi oscillations and sponta-neous p- and s-shell decay obtained with a the dissipator con-structed in the single-particle basis (i) (’sp’, red dashed line)and in the many-particle basis (ii) (black line). (a): Occupa-tion probability of the s-exciton X s , the shaded area marksthe biexciton occupation XX . (b, c): Occupation probabilityof s-shell electron occupation (cid:10) e † s e s (cid:11) , the shaded area marksthe p-shell electron occupation (cid:10) e † p e p (cid:11) . The decay rates are β = 0 .
25 and Γ = 0 . g for panel (a) and (b), in panel (c) the rate β = 0. − − Γ / g . . A m p li t ud e FIG. 8: Asymptotic effect of the non-local dephasing on theamplitude of the oscillation of (cid:104) X s (cid:105) in dependence of thescaled decay rate ˜Γ. amplitude of the Rabi oscillations A can be expressed by A | t (cid:29) = 12 (cid:113) (˜Γ + 2) + ˜Γ (˜Γ + 4) , ˜Γ = Γ2 g . (17)Note the peculiar result that the long term effect of thenon-local dephasing is strongest, when its rate is thesmallest since in this case the Rabi oscillations are ex-posed to the dephasing for the longest time. As it canbe seen in Fig. 8, the minimal amplitude is / for almostvanishing but nonzero decay rates ˜Γ. In the opposite caseof an immediate p-exciton decay, the amplitude remainsat its maximum value of / since no polarization couldbuild up to be dephased.Going beyond this minimal example one can furtherincrease the non-local dephasing effect by exploiting thesame mechanism discussed above. Adding an additionalpump process to the dissipator D , compensating the p-shell loss, ties the s-exciton permanently to the dephasinginfluence of the p-shell. In this case the Rabi oscillationsin the s-shell would completely vanish, when the dissipa-tor is constructed in the configuration basis (ii). Whereas when the dissipator is constructed of in the single-particlebasis (i), the Rabi oscillations in the s-shell would againnot be effected at all by the p-shell (see App. B 2).The problematic conclusion to this section is that theoutcome of the EoM depends crucially on the choice ofbasis states for constructing the dissipator. In the nextsection we will see how this problem can be resolved andthat, in contrast to our first example in Sec. II, the non-local dephasing effect is not an artifact of an approxima-tion error. C. System plus reservoir approach
The discrepancies between the results, when the dissi-pator D is constructed in either single-particle (i) or theconfiguration basis (ii) originate from deviating approxi-mations and assumptions about the system-reservoir in-teraction, already build into the construction of the dis-sipator D itself. To see where the crucial assumptionsdeviate we discuss in this section how the dissipator de-scribing the decay of a p-shell exciton in Sec. III B canbe derived from a system plus reservoir approach.Starting from the von Neumann equation d t χ = i [ χ, H ]for the full density operator χ describing the QD-cavity-mode system and a reservoir of non-confined modes,we derive the EoM for the reduced density operator ρ = Tr R ( χ ) in Born-Markov approximation . To thisend we divide the Hilbert space H into a reservoir part H R consisting of the non-confined modes and a systempart H S = H QD ⊗ H C consisting of the QD and the con-fined cavity mode. The QD Hilbert space itself consistsof the s- and p-shell subspace H QD = H s ⊗ H p . Afterrecapitulating how one can derive the general EoM for ρ , where we essentially follow the approach from Ref. ,we compare the obtained EoM (formulated in the single-particle and in the configuration basis) to the EoM for ρ used in the previous section.Assuming a reservoir of harmonic modes with fre-quency ω k that are annihilated(created) by r ( † ) k , we canformulate the reservoir Hamiltonian H R and the system-reservoir interaction Hamiltonian H S⇔R as H R = (cid:88) k ω k r † k r k , (18) H S⇔R = (cid:88) j (cid:32)(cid:88) k κ jk r † k L j + (cid:88) k κ j ∗ k r k L † j (cid:33) = (cid:88) j (cid:16) R † j L j + R j L † j (cid:17) (19)respectively. In H S⇔R the sum over all reservoirmodes is summarized in the reservoir operators R j cou-pling to the system operators L j in full rotating waveapproximation . The operators L j , will be chosen as L sp according to Eq. (12) in the single-particle (i) andas L G and L X according to Eq. (13) in the configurationbasis formulation (ii).In Born approximation the full density operator χ ( t )factorizes to χ ( t ) = ρ ( t ) ⊗ ρ T R , where ρ T R is the reservoirdensity operator in thermal equilibrium. We trace overthe reservoir R and reformulate the von Neumann equa-tion in the interaction picture for χ ( t ) = ρ ( t ) ⊗ ρ T R as anintegro-differential equationd t ρ ( t ) = (cid:90) t dt (cid:48) Tr R (cid:0) [ H S⇔R ( t ) , [ ρ ( t (cid:48) ) ρ T R , H S⇔R ( t (cid:48) )]] (cid:1) , describing the dissipative influence of the reservoir R onthe reduced density operator ρ . Now we insert the gen-eral Hamiltonian from Eq. (19) and execute the commu-tators and collect all reservoir operators in the reservoircorrelations Tr R (cid:0) • ρ T R (cid:1) = (cid:104)•(cid:105) R . When the reservoir occu-pations can be neglected the only contributing reservoircorrelations are (cid:104) R j ( t (cid:48) ) R † i ( t ) (cid:105) R and the EoM for the re-duced density operator readsd t ρ ( t ) = (cid:88) i,j (cid:90) t dt (cid:48) (cid:68) R j ( t (cid:48) ) R † i ( t ) (cid:69) R (cid:110) L i ( t ) ρ ( t (cid:48) ) L † j ( t (cid:48) ) − L † j ( t (cid:48) ) L i ( t ) ρ ( t (cid:48) ) + L i ( t ) ρ ( t (cid:48) ) L † j ( t (cid:48) ) − ρ ( t (cid:48) ) L † j ( t (cid:48) ) L i ( t ) (cid:111) . When the time scales of the reservoir and the system canbe separated we can apply the Markov approximation,which corresponds to (cid:68) R j ( t (cid:48) ) R † i ( t ) (cid:69) R = (cid:88) kl δ kl κ i ∗ l κ jk e iω k t (cid:48) e − iω l t (20)= (cid:88) k κ i ∗ k κ jk e − iω k ( t − t (cid:48) ) ≈ γ ji δ ( t − t (cid:48) ) , and we obtaind t ρ = (cid:101) D ( ρ ) = (cid:88) i,j γ ji (cid:110) L i ρL † j − L † j L i ρ − ρL † j L i (cid:111) . (21)Here the dissipator (cid:101) D has a more general non-diagonalform in the collapse operators L i and rates γ ij , in con-trast to the dissipator D in Eq. (10) used in Sec. III B.This non-diagonal dissipator appears in many systems,e.g., in open resonators the non-diagonal form of thedissipator induces correlations between different photonmodes . We now use the non-diagonal dissipator (cid:101) D from Eq. (21) and insert the system operators L j fromthe system-reservoir interaction Hamiltonian formulatedin the single-particle basis and in the configuration basis. In the single-particle basis the system-reservoir inter-action Hamiltonian reads H sp S⇔R = (cid:88) k κ k r † k h p e p + (cid:88) k κ k r k e † p h † p = R † sp L sp + R sp L † sp = (cid:88) j = sp (cid:16) R † j L j + R j L † j (cid:17) with κ k being the coupling strength of reservoir mode k to the p-exciton. This Hamiltonian leads to the dissipator (cid:101) D sp ( ρ ) =Γ (cid:110) L sp ρL † sp − L † sp L sp ρ − ρL † sp L sp (cid:111) , (22)where we have identified the only appearing rate γ spsp with the rate Γ from the previous section. Equation (22)is identical to the dissipative part of the EoM (10) usedin Sec. III B in single-particle formulation (i) with L j = L sp . Using Eq. (12) for L sp and L ( † ) G L ( † ) X = 0 we canreformulate Eq. (22) to (cid:101) D sp ( ρ ) = Γ (cid:110) L G ρL † G − L † G L G ρ − ρL † G L G (cid:111) + Γ (cid:110) L X ρL † X − L † X L X ρ − ρL † X L X (cid:111) + 2Γ L G ρL † X + 2Γ L X ρL † G , (23)which corresponds to the Γ dependent part of Eqs. (15)and (16) with M ψ s ,ψ X = Γ. In the configuration basis the system-reservoir inter-action Hamiltonian reads H C S⇔R = (cid:88) k κ Gk r † k L G + (cid:88) k κ Xk r † k L X + (cid:88) k κ G ∗ k r k L † G + (cid:88) k κ X ∗ k r k L † X = R † G L G + R † X L X + R G L † G + R X L † X = (cid:88) j = G,X (cid:16) R † j L j + R j L † j (cid:17) where we have allowed the dipole-matrix elements κ jk todepend on s-exciton state j = G, X , which would notbe possible in the single-particle basis. By inserting thesystem operators operators L j into Eq. (21) we obtain (cid:101) D C ( ρ ) = γ CGG (cid:110) L G ρL † G − L † G L G ρ − ρL † G L G (cid:111) + γ CXX (cid:110) L X ρL † X − L † X L X ρ − ρL † X L X (cid:111) + 2 γ CXG L G ρL † X + 2 γ CGX L X ρL † G . This dissipator (cid:101) D C is in general not in agreement with thediagonal dissipator D from Eq. (10). For rates γ Cij = Γthe dissipator (cid:101) D C agrees with the dissipator constructedin the single-particle basis (cid:101) D sp in Eq. (23). If we as-sume the system-reservoir coupling strength to be inde-pendent of the s-shell exciton κ Gk = κ Xk = κ k , we ob-tain γ Cij = Γ and thus (cid:101) D C = (cid:101) D sp . In fact in this casethe system-reservoir interaction Hamiltonians are identi-cal with H C S⇔R = s ⊗ H sp S⇔R , where s is the identityoperator in H s . This resolves the problematic conclu-sion from Sec. III B and we see that starting from thesystem-reservoir interaction Hamiltonian leads to a dis-sipator that is in general non-diagonal and independentfrom the choice of basis states . H R H p = ⇒ κ G/Xk H s H R H p = ⇒ κ k H s (a) (b) FIG. 9: Illustration of the different reservoir couplings. In theleft figure (a) the reservoir coupling elements κ are indepen-dent of the state of s-exciton. The interaction Hamiltonian H sp S⇔R operates in H p ⊗ H R , thus the reservoir interacts onlywith a single localized state. In the right figure (b) the reser-voir coupling elements κ j depend on the state of s-excitonthus H C S⇔R operates in H s ⊗ H p ⊗ H R and the p-excitonloss is connected to a non-local measurement of the s- andp-exciton state corresponding to L G/X . When we use the diagonal form of the dissipator ad-hoc as done in Eq (10), we implicitly make strong as-sumptions about the reservoir, namely that the reservoircorrelations result in rates γ CGG = γ CXX = Γ and γ CXG = γ CGX = 0 . (24)Nevertheless, from a formal point of view it is possible toconstruct a reservoir Hamiltonian that leads to the ratesin Eq. (24) and thus the described non-local dephasing ef-fect. To this end it is however necessary that the couplingstrengths κ jk depend on the s-exciton state and thus thesystem-reservoir Hamiltonian interacts non-locally withthe QD as illustrated in Fig. 9. IV. CONCLUSION
We have shown how the choice of basis states canchange the dynamics of a system, if an approximation isinvolved in the calculation. In our first example, the ap-pearance of the equations, formulated in a single-particlebasis, suggested a factorization scheme, which created anartificial dependence between two actually independentquantities. We have analyzed this dependence in termsof the systems many-particle basis states, in which therelations between the quantities can be seen directly.In the second part, we have investigated an open sys-tem treated in Born-Markov-approximation, where thereservoir influence is modeled by a dissipator in Lindbladform. We have shown that the way, in which an equal setof collapse operators enter the dissipator, has a profoundinfluence on the systems dynamics. The construction ofthe dissipator determines if the Rabi oscillations of thes-shell exciton are non-locally dephased by the decay ofthe p-shell exciton.The problem of formulation dependent dynamics, hasbeen resolved, by taking the system-reservoir interactionHamiltonian into account. Starting from the full Hamil-tonian and evaluating the reservoir correlation functions, we have shown that in both formulations, the s-shell Rabioscillations are independent of the p-shell decay. How-ever we also shown that the non-locally dephased s-shelloscillations can actually occur when the system-reservoirinteraction Hamiltonian depends on the whole QD state.In contrast to the first example, the misconception insecond part arises not from an inappropriate approxi-mation scheme, but from the notion that two differentlyconstructed dissipators would describe the same physicalsituation.
V. ACKNOWLEDGMENTS
We thank T. Pistorius for a stimulating discussion,that has led to our last example. We would also liketo thank the two unknown referees who provided us withvery helpful hints for improvements and constructive crit-icism. T. Lettau and H.A.M. Leymann have contributedequally to this work.0
Appendix A: Semiconductor JCM1. Equation of Motion
The EoM for the photon-assisted polarization can beobtained byd t (cid:104) he | n + 1 (cid:105)(cid:104) n | ] (cid:105) = (cid:104) i [ H, ghe | n + 1 (cid:105)(cid:104) n | ] (cid:105) = − ig (cid:42) (cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40) heb † he | n + 1 (cid:105)(cid:104) n | (cid:124) (cid:123)(cid:122) (cid:125) ∝ hh + e † h † bhe | n + 1 (cid:105)(cid:104) n | − he | n + 1 (cid:105)(cid:104) n | (cid:8)(cid:8)(cid:8) heb † (cid:124) (cid:123)(cid:122) (cid:125) ∝ hh + | n + 1 (cid:105)(cid:104) n | e † h † b (cid:43) = − ig √ n + 1 (cid:10) e † eh † h | n (cid:105)(cid:104) n | − ee † hh † | n + 1 (cid:105)(cid:104) n + 1 | (cid:11) = − ig √ n + 1 (cid:104) e † eh † h | n (cid:105)(cid:104) n | − (1 − e † e )(1 − h † h ) | n + 1 (cid:105)(cid:104) n + 1 |(cid:105) = ig √ n + 1( p n +1 − f en +1 − f hn +1 ) + ig √ n + 1 (cid:104) e † eh † h | n + 1 (cid:105)(cid:104) n + 1 | − e † eh † h | n (cid:105)(cid:104) n | (cid:124) (cid:123)(cid:122) (cid:125) C Xn (cid:105) . (A1)The EoM for the semiconductor JCM can be closed by calculating the derivative of C Xn , which readsd t ( e † eh † h | n (cid:105)(cid:104) n | ) = (cid:10) i [ H, e † eh † h | n (cid:105)(cid:104) n | ] (cid:11) − ig (cid:42) hee † eh † hb † | n (cid:105)(cid:104) n | + (cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40) e † h † e † eh † hb | n (cid:105)(cid:104) n | (cid:124) (cid:123)(cid:122) (cid:125) ∝ h † h † − (cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40)(cid:40) e † eh † hhc | n (cid:105)(cid:104) n | b † (cid:124) (cid:123)(cid:122) (cid:125) ∝ hh − e † eh † he † h † | n (cid:105)(cid:104) n | b (cid:43) = ig (cid:10) ee † ehh † hb † | n (cid:105)(cid:104) n | + e † ee † h † hh † | n (cid:105)(cid:104) n | b (cid:11) = ig √ n + 1 (cid:10) eh | n + 1 (cid:105)(cid:104) n | + e † h † | n (cid:105)(cid:104) n + 1 | (cid:11) = − ig √ n + 1 (cid:104) he | n + 1 (cid:105)(cid:104) n | − h . c . (cid:105) =2 g √ n + 1 ψ n , (A2)and couples only to the already known polarization ψ n .
2. Parameter space and Correlation
To capture the full range of possible initial electronicconfigurations of the QD in the semiconductor JCM weneed not only the charge C , and the oscillation ability O , but also the QDs inversion I in terms of the expan-sion coefficients of the density matrix c i . To this end weconsider the following transformation: O = c + c c = 12 ( O − I ) C = 1 − c − c − c c = 12 ( O + I ) I = c − c c = 1 − O − C . (A3)The coordinates are bounded by the values C ∈ [ − , O ∈ [0 , | C | ] I ∈ [ − O, O ] , (A4)which is reflected by the triangular shape of the g (2) max (0) = f ( O, C ) plot in Fig. 2(b). To specify the num- ber of excitations in the QD, we have chosen the initialcondition I = 0, i.e., c = c . Appendix B: Dephasing1. Analytical solution
Since we are interested in the long term effect of thenon-local dephasing without s-shell decay, β = 0, onlysix of the eight equations of Eq. (15) have to be takeninto account. The matrix M simplifies to M = (cid:32) W (Γ) G W (0) (cid:33) , (B1)where W and G are defined by W (Γ) = − Γ ω − ω / − Γ ω / − ω − Γ , G = Γ { Γ , }
00 0 Γ , ω = 2 g .In the many-particle basis (formulation (ii)) , the so-lution of the system with the initial condition r =(1 , , , , , T reads XX ψ X X p X s ψ G = T / − (cid:16) ΓΓ+2 iω + 1 (cid:17) e iωt − (cid:16) ΓΓ − iω + 1 (cid:17) e − iωt / e − Γ t / e − Γ t e − iωt / e − Γ t e − iωt , (B2)where T is the transformation matrix, consisting of alleigenvectors of M . For Γ (cid:54) = 0, the asymptotic behavioris determined by the first three rows of Eq. (B2). For X s ( t + τ ), with τ → ∞ , we obtain X s ( t ) | t (cid:29) = (2Γ + 4 ω ) cos ωt + 2Γ ω sin ωt + 4 ω ) + 12= (cid:112) (2Γ + 4 ω ) + (2Γ ω ) + 4 ω ) sin ( ωt + ϕ ) + 12 , where ϕ is an irrelevant phase. The oscillation of X s iscentered around / , and its amplitude varies from / forsmall, but nonzero dephasing rates Γ, to / , for greatΓ (cid:29) ω .In the single-particle perspective (formulation (i)), thesolution reads XX ψ X X p X s ψ G = T / − / e iωt − / e − iωt / e − Γ t / e − Γ t e iωt / e − Γ t e iωt (B3)and the coefficients do not depend on ω or Γ. Therefore,the amplitude of X s | t (cid:29) = 12 (sin ( ωt + δ ) + 1) (B4) stays / for all values of Γ and ω .
2. Pumped p-exciton
In Sec. III B in the main text we have presented a min-imal example that induces the non-local dephasing effect.To demonstrate that the dephasing can become signifi-cantly stronger we present a further exploitation of thenon-local dephasing mechanism.When we add a pumping process to the p-exciton withrate P to our model in Sec. III B we obtain a case wheredifferent constructions of the dissipator (i) and (ii) resultin an entirely different long term behaviour of the system.The pumping process is induced by the adjunct collapseoperators for the p-exciton decay. In the single-particlebasis (i) this is L P sp = e † p h † p = | X p (cid:105)(cid:104) G | + | XX (cid:105)(cid:104) X s | (ad-joint of Eq. (12), and in the configuration basis (ii) theseoperators are L P G = | X p (cid:105)(cid:104) G | and L P X = | XX (cid:105)(cid:104) X s | (ad-joint of Eq. (13)). As shown in Fig. 10, the s-shell elec-tron occupation (cid:104) e † s e s (cid:105) reaches a steady state for a dissi-pator constructed in the configuration basis, whereas inthe single-particle basis the occupation (cid:104) e † s e s (cid:105) sp performsRabi-oscillations for all times, independent of the p-shelldecay and pumping. t/ g . . . P r o b a b ili t y (cid:10) e † s e s (cid:11) sp (cid:10) e † s e s (cid:11)(cid:10) e † p e p (cid:11) FIG. 10: Dynamics of (cid:10) e † s e s (cid:11) in the single-particle basis (reddashed line) and in the many-particle basis (black line), and of (cid:10) e † p e p (cid:11) (shaded area), for the same parameters as in Fig. 7(c)with additional p-shell pump P = 0 . ∗ Electronic address: [email protected] U. Hohenester and W. P¨otz, Phys. Rev. B , 13177(1997). W. Hoyer, M. Kira, and S. W. Koch, Phys. Rev. B ,155113 (2003). M. Kira and S. W. Koch, Phys. Rev. A , 022102 (2008). M. Mootz, M. Kira, and S. W. Koch, J. Opt. Soc. Am. B , A17 (2012). W. W. Chow, F. Jahnke, and C. Gies, Light Sci Appl ,e201 (2014), URL . M. Lorke, T. R. Nielsen, J. Seebeck, P. Gartner, and F. Jahnke, Phys. Rev. B , 085324 (2006), URL http://link.aps.org/doi/10.1103/PhysRevB.73.085324 . J. Kabuss, A. Carmele, and A. Knorr, Phys. Rev. B ,064305 (2013). D. Witthaut, F. Trimborn, H. Hennig, G. Kordas,T. Geisel, and S. Wimberger, Phys. Rev. A , 063608(2011). F. Trimborn, D. Witthaut, H. Hennig, G. Kordas,T. Geisel, and S. Wimberger, Eur. Phys. J. D , 63(2011). J. Tignon, T. Hasche, D. S. Chemla, H. C. Schnei-der, F. Jahnke, and S. W. Koch, Phys. Rev. Lett. , http://link.aps.org/doi/10.1103/PhysRevLett.84.3382 . M. D. Kapetanakis and I. E. Perakis, Phys. Rev. Lett. ,097201 (2008). J. Fricke, Annals of Physics , 479 (1996). J. Fricke, V. Meden, C. W¨ohler, and K. Sch¨onhammer,Annals of Physics , 177 (1997). W. Hoyer, M. Kira, and S. W. Koch, in
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