Quantum Description of Radiative Losses in Optical Cavities
Jens Oppermann, Jakob Straubel, Karolina S?owik, Carsten Rockstuhl
QQuantum Description of Radiative Losses in Optical Cavities
J. Oppermann, ∗ J. Straubel, K. S(cid:32)lowik, and C. Rockstuhl
1, 3 Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany Institute of Physics, Nicolaus Copernicus University, 87-100 Toru´n, Poland Institute of Nanotechnology, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
We present, for the first time, the quantum mechanical description of light-matter interactionin the presence of optical cavities that are characterized by radiative losses. Unique to radiativelosses is the unitary evolution and their full preservation of the coherence, in stark contrast to theusually considered dissipative losses. We elucidate the reduction of exact quantum electrodynamicequations to a form similar to the familiar Jaynes-Cummings model through the introduction andstudy of a new class of noise operators. The dynamics of this henceforth inherently dissipativemodel are then presented by formulating the resulting equations of motion. Furthermore, an input-output formalism is established, which provides a direct connection to the dynamics of output statesaccessible with detectors. The application-oriented cases of coherent and pulsed laser pumping arediscussed as inputs. Finally the single-photon dynamics in an optical cavity with significant radiativeloss - whose importance has to be contextualized in view of the prospects of light-matter interactionapplications - are reviewed according to the proposed model. The formulation is kept as generalas possible to emphasise the universal applicability to different implementations of quantum opticalsystems but from our own background we have an application in mind in the context of nanooptics.
I. INTRODUCTION
Quantum information science (QIS) employs quan-tum phenomena to elevate the processing and transferof information to levels beyond the scope of classicalphysics [1]. Consequently, in the past decades QIS hasemerged as the stepping stone towards changing the veryfoundations of the information age [2, 3] and it consti-tutes nowadays a vibrant field of research [4]. Similarly,quantum optics as one of QIS’s mainstays shares thesepromising prospects.The interaction of light and matter, that is encounteredwhenever an electromagnetic field impinges on some sortof media, is one of the most fundamental aspects ofoptics in general. As soon as discrete matter is con-sidered, such as atoms or molecules instead of a bulkmedium, it has been revealed to enable signal processingat the single-photon level [5, 6]. Cavity quantum electro-dynamics (QED) models prototypical systems to studythe quantum physical essentials of light-matter interac-tion [7, 8]. All these systems can be reduced to the pheno-type of electromagnetic field modes usually sustained bysome sort of cavity coupled to atomic transitions [9, 10].The referential case of a single-mode quantized field cou-pled to a single transition defines the Jaynes-Cummings(JC) model [11–14]. Its widespread consideration to-wards problems of both fundamental and applied scienceis a clear indication for its success.Although the notion of fundamental aspects such asnonclassical photon statistics [15] and nonclassical pho-ton properties [16] originated theoretically, experimen-tal advancement has been the driving force that putthe implementation of QED based applications in tan-gible proximity: from influencing the emission behaviorof quantum systems via microcavities [17], to controllable + QuantuminformationscienceQuantumcomputation ...circuitQEDopticalcavityQEDnanoscalephotonicsystemssingle photonscatteringleaky cavityQEDdielectriccavities
FIG. 1. Subsumption of the treatment introduced in this workinto the research activities and goals. Our work eventuallycontributes to the gear shown in the upper left corner, but ithas an impact on a multitude of further research areas. sources of single photons [18], and generation of entan-gled photons [19, 20]. More recent developments weregenerally focussed on controlling transition properties inquantum systems through tailored electromagnetic fieldmodes [21–24]. In this context, optical cavities [25–27]have proven to be a versatile and powerful tool to exer-cise control over the properties of the field modes [28–30].Consequently, a variety of investigations targeted e.g. thegeneration of squeezed states of light [31, 32], single pho-tons [33–38], entangled photons [39–44], and nonclassicallight in general [45, 46] - all relying on features of theoptical cavity.However, these results do not stand on their own asautotelic applications, but are rather motivated by thegreater goal of expanding the cavity QED tool box and to a r X i v : . [ qu a n t - ph ] J un ultimately culminate to the implementation of quantumcomputation via QIS [see Fig. 1]. This purpose imme-diately leads to the aspect of cavity mediated couplingregimes [47–51] in light-matter interaction and quan-tum coherence [52–54]. These facets have been widelydiscussed as well, including their quantum informationprospects [55–60].At this point it is crucial to highlight that anylight-matter interaction scenario utilizing optical cavitieshinges on the radiation emitted by the cavity. The emit-ted photons are the carriers of the desired features orstates and shall be used to ultimately encode informa-tion. But so far, the modeling of the emission processitself has been assumed to be equivalent to the couplingto a thermal bath [61]. This is not true. As outlinedfurther below in slightly more detail, in contrast to dis-sipation to a thermal bath, radiative losses don’t leadto the thermalization of energy and the correspondingloss of information. Instead the electromagnetic energyis transferred into the far field, where full information canbe retrieved with photodetectors. Besides the inability toactually describe the measurement process, this urgentlyprompts for an adapted description of the light-matter in-teraction in the presence of a cavity with radiation losses.To close this fundamental methodological gap we pro-pose a formulation via a set of operators introduced in thefollowing and derived from first principles. Incipiently wewould like to stress that we do not propose to replace oneassumption with another, but rather based on an analy-sis of the internal dynamics, we inferred a new structureof the emission formulation from the methodology we in-troduce here [see Fig. 2(b)]. With optical cavities univer-sally relying on radiative losses to function as potentialQIS devices, we propose a conscientious formalistic in-corporation of the radiative loss aspect of light-matterinteraction modelling in cavities into the well-establishedtheory. This entails the integration of the emission ofphotons into the measurable far field into the rigorousdescription of the unitary internal dynamics of the cav-ity [see Fig. 2(a) for the general setup we are lookingat].In lossless cavity QED, the temporal evolution of thejoint system consisting of light and matter is unitary andcan therefore be described by a Hamiltonian. The sim-plest of such systems consists of a single electromagneticmode and an atom, where the interaction between bothis described in dipole approximation. Such a system isdescribed by the Rabi Hamiltonian, which in many casescan be further simplified to yield the Jaynes-CummingsHamiltonian.However, since there exist no lossless systems in reality,the description of realistic systems requires that lossesin the system are somehow taken into account. Thisis usually achieved by including the coupling of systemoperators to a bath of other excitations, e.g. phonons.Since realistic systems contain a large number of such unwanted excitations, the energy of the system is even-tually distributed somewhat evenly across all of themand therefore effectively lost. If one thinks about opticalprocesses, then it is clear that absorption losses can bemodeled in this way [62].In this work, however, we consider the case of radiativelosses. In contrast to absorption losses, radiative lossesdo not lead to the loss of electromagnetic energy. In-stead, the energy is just displaced over time to a placefar away from the system of interest. Applying the pro-cedure outlined above would therefore require to placea perfect absorber around the cavity, effectively leadingto the far field being integrated out. Since most experi-mental setups place their detectors into the far field, thiswould mean that we lose information about the observ-ables of our theory. For this reason we introduce herean alternative formalism that splits the electromagneticfield itself up into one system and multiple bath oper-ators. This leads to dissipative dynamics and providesinsights concerning the relation between scattering andcavity modes. canonical formalism:assume thermal bath H: σ ...(a) a a a a N new formalism:infer noise operators σ , a F F F F N ...(b) l e a k y c a v i t y Q E D source detector FIG. 2. (a) System under investigation: An arbitrary opticalcavity is exposed to arbitrary incident illumination, which re-sults in emission into the far field. (b) Methodological changein formulation of emission: Coupling to thermal bath is re-placed by new coupling to chain of noise operators.
This work is organized as follows. In Sec. II the elec-tromagnetic cavity and bath operators are defined andtheir equations of motion derived. The bath operatorsare then integrated out to yield closed equations contain-ing only the operators of interest. Section III establishesthe connection between cavity and far field dynamics,allowing for the formulation of an input-output scheme.In Sec. IV the formalism is used to retrieve a general-ized form of the experimentally well-established Jaynes-Cummings model, which can also describe the far fielddynamics. We summarize our findings in Sec. V. TheAppendices contain some of the more involved calcula-tions needed throughout this work.
II. THE CAVITY
We consider the problem of a quantized electromag-netic field coupled to a single two-level system (TLS) inelectric dipole approximation. We further assume thepresence of a localized and lossless dielectric structuregiving rise to classical light scattering. The Hamiltonianof the system can be found in many textbooks on quan-tum optics and reads [63] H = (cid:88) λ (cid:90) d k (cid:126) ω k a † k ,λ a k ,λ + (cid:126) ω a σ z (cid:88) λ (cid:90) d k (cid:16) E k ,λ ( r a ) · d a † k ,λ σ − + h.c. (cid:17) , (1)where ω k = c | k | , c is the speed of light in vacuum, d and r a are the transition dipole moment and spatialposition of the TLS, σ z is a Pauli matrix, σ − is the cor-responding Pauli lowering operator and a k ,λ are photonicannihilation operators. For the rest of this work we dropthe mode index k from ω k , making the dispersion rela-tion implicit. The electromagnetic field modes are of theform [64] E k ,λ ( r ) = (cid:115) (cid:126) ω (2 π ) (cid:15) (cid:15) b ˆ e k ,λ e i kr + E ( s ) k ,λ ( r ) , (2)where (cid:15) b is the permittivity of the background mediumand E ( s ) k ,λ ( r ) is the scattered field, which arises due to thespatially inhomogeneous dielectric function (cid:15) ( r , ω ). Thephotonic operators satisfy the usual harmonic oscillatorcommutation relations (cid:104) a k ,λ , a † k (cid:48) ,λ (cid:48) (cid:105) = δ ( k − k (cid:48) ) δ λλ (cid:48) , (3)with all other commutators vanishing. The descriptionoffered by the Hamiltonian in Eq. (1) is exact and general,but usually not tractable. Luckily, many problems ofpractical interest feature electromagnetic resonances, i.e.the electromagnetic field at the position of the TLS onlytakes on appreciable values over a set of comparativelynarrow frequency ranges. In the following we will assumethe existence of a single resonant electromagnetic modein the vicinity of the transition frequency ω a of the TLS.All other modes will be assumed to be too far detunedto yield any significant contribution and therefore canbe ignored. By further assuming a Lorentzian frequencydependence, we write E k ,λ ( r a ) = E δ λ,λ (cid:114) Γ2 π g (ˆ k ) ω − ω − iΓ / , (4) where ω is the central frequency and Γ is the linewidthof the mode and g (ˆ k ) describes the angular dependence.The Kronecker delta δ λ,λ signifies that there is indeedonly one electromagnetic mode and not two degenerateones of different polarization. This constitutes no limita-tion since in practice one can describe the polarizationsin the coupled-uncoupled basis. Since only one polariza-tion couples to the TLS we shall drop polarization indices λ from here on. In the following, we will show that theabove assumptions allow us to reduce the exact Hamil-tonian to an approximate but tractable form. TakingEq. (4) into account, we can derive evolution equationsin the Heisenberg picture for the operators from Eq. (1)that read as˙ a k = − i ωa k − i E · d (cid:126) (cid:114) Γ2 π g (ˆ k ) ω − ω − iΓ / σ − , (5)˙ σ − = − i ω a σ − + i E ∗ · d ∗ (cid:126) (cid:90) d k (cid:114) Γ2 π g ∗ (ˆ k ) ω − ω + iΓ / σ z a k . (6)Inspection of Eq. (6) motivates the following definitionof a resonant mode annihilation operator: a := (cid:90) c / d k √ Gω (cid:114) Γ2 π g ∗ (ˆ k ) ω − ω + iΓ / a k , (7) G := (cid:90) d Ω k (cid:12)(cid:12)(cid:12) g (ˆ k ) (cid:12)(cid:12)(cid:12) , (8)where d Ω k = sin( θ k ) dθ k dϕ k denotes integration oversolid angles. Please note that the normalization constant G in Eq. (8) is chosen in such a way that the harmonicoscillator commutation relations are satisfied: (cid:2) a, a † (cid:3) = 1 , (9)and all other commutators vanishing. The originalHeisenberg Eqs. (5) and (6) can now be rewritten in termsof the newly defined resonant mode operators. The pro-cess is detailed in Appendix A and the results read˙ a = ( − iω − Γ / a − i κσ − − i F , (10)˙ σ − = − i ω a σ − + i κ ∗ (cid:18) − i Γ2 ω (cid:19) σ z a + i κ ∗ ω σ z F , (11)where the effective light-matter coupling constant κ isdefined as κ = √ Gc / ω E · d (cid:126) = (cid:114) G π πc / ω √ E max · d (cid:126) , (12)with E max being the field strength at resonance. Tothe best of our knowledge this is the first time that thecoupling strength between an open cavity mode and aquantum emitter has been calculated from first princi-ples rather than from phenomenological considerations.The new operator F appearing in Eqs. (10) and (11)belongs to a family of operators defined as F n := (cid:90) d kω c / √ G (cid:114) Γ2 π g ∗ (ˆ k ) ( ω − ω ) n a k . (13)Comparing Eqs. (7) and (13), we notice that the opera-tors F n are not associated with the resonance mode at ω ,but rather with a broad range of frequencies. For this rea-son we will call F n noise operators from here on. Pleasenote that these noise operators were retrieved withouthaving introduced a thermal bath.In order to construct a closed system of equations, weneed equations of motion for F n . The derivation of theseis detailed in Appendix B. The results read˙ F n = − i ω F n − i F n +1 . (14)The set of equations (10), (11), and (14) is closed andcan therefore be used in their current form to describethe system dynamics. In the spirit of the theory of openquantum systems, however, we wish to find a set of equa-tions that only contains the system operators a and σ − as dynamic quantities.As is detailed in Appendix C, the equations of motionof the noise operators (14) can be formally solved withoutfurther approximations to yield F n ( t ) = e − i ω t ∞ (cid:88) m =0 ( − i t ) m m ! F m + n (0) , (15)where the form of F m (0) defines the type of illumina-tion, as can be seen from Eq. (13) and the examples ofSection III. Please note that the operators F n ( t ) can beinterpreted as input parameters, as Eq. (15) tells us thatthere exists no backaction from the system. Insertion ofEq. (15) into Eqs. (10) and (11) now yields a closed setof equations of motion for the operators a and σ − , wherethe lowest order input operator serves as a pump term˙ a ( t ) =( − iω − Γ / a ( t ) − i κσ − ( t ) − i F ( t ) , (16)˙ σ − ( t ) = − i ω a σ − ( t ) + i κ ∗ (cid:18) − i Γ2 ω (cid:19) σ z ( t ) a ( t )+ i κ ∗ ω σ z ( t ) F ( t ) . (17)Using Eqs. (16) and (17) allows one to find equations ofmotion for all observables of the system, i.e. expectationvalues of arbitrary operators. The inital values of opera-tors containing a and F n can be inferred from the initialstate of the quantized electrodynamic field using the def-initions (7) and (13). The structure of these equationsresembles the one presented conceptually in Fig. 2(b).Instead of the simultaneous coupling of the system to a ? ExcitationSystemDynamicsMeasurement ? FIG. 3. Schematic depiction of an experimental setup. Thesystem is excited by a beam of light, the interaction betweencavity mode and matter takes place and finally the outgoingradiation is measured in the far field. larger number of bath operators, the operators that de-scribe the evolution of our actual system is only coupledto one noise operator. In a sequential type of processeach noise operator then couples to the next.We have now succeeded not only at describing theinternal quantum dynamics of a cavity with radiativelosses, but also at linking them to the external field viathe noise operator F ( t ). In the following section thislink will be used to formulate an input-output schemecapable of describing real experimental setups. III. INPUT-OUTPUT FORMALISM
At this point we succeeded in describing the internaldynamics of a cavity-matter system in a tractable andrigorous manner. However, this is not yet sufficient tomake predictions about actual experiments. A typicalquantum optical experiment consists of probing an opti-cal system with a beam of light and measuring the out-going radiation. This process is schematically shown inFig. 3. In the following, we will establish a quantitativerelation between the different parts of the systems.
A. Input Operators
Most experimental illumination schemes in optics usea light beam with a waist diameter much larger than thesystem under consideration. For all practical purposessuch a light beam can be considered as a plane wavewith the wave vector pointing along the beam axis. Forthis reason, the following subsections describe how anillumination with coherent plane waves can be describedby means of the noise operator F ( t ).
1. Continuous pumping
One of the most common pumping schemes is excita-tion by a continuous laser beam, i.e. the incident light ismonochromatic, coherent, and polarized. In terms of thescattering eigenmode operators this means (cid:104) a k ( t = 0) (cid:105) = α P δ ( k − k P ) , (18)where α P is the amplitude and k P is the wave vector ofthe laser beam. Using Eq. (13) one finds (cid:104) F n (0) (cid:105) = C ( ω P − ω ) n , (19) C := c / ω P (cid:114) Γ2 πG g ( ˆk P ) α P . Substitution of Eq. (19) into the expectation value ofEq. (15) now yields (cid:104) F ( t ) (cid:105) = C e − i ω t ∞ (cid:88) n =0 ( − i ( ω P − ω ) t ) n n != C e − i ω t e − i( ω P − ω ) t = C e − i ω P t . (20)Equation (20) implies that the equation of motion for a contains a pump term of constant amplitude, which os-cillates at the pump laser frequency. Terms of this form have been routinely employed when discussing drivenquantum systems, but now we actually have the meansto quantify the relation between laser intensity and pumpstrength.
2. Pulsed pumping
Assume now that the laser used to pump the system isnot continuous, but pulsed. This means that a range offrequencies centered around the laser frequency is excitedaccording to (cid:104) a k (cid:105) = α P δ ( ˆk − ˆk P )e − ∆ ( ω − ω ) . (21)Inserting this into the expectation value of Eq. (13) leadsto (cid:104) F n (0) (cid:105) = C (cid:90) ∞ dωω ( ω − ω ) n e − ∆ ( ω − ω ) , (22) C = c / (cid:114) Γ2 πG g ∗ ( ˆk P ) α P . The frequency integral can be evaluated after extendingthe lower integration boundary to −∞ to yield [65] (cid:104) F n (0) (cid:105) = Cω √ π (2 n − n ∆ n +1 (23) (cid:104) F n +1 (0) (cid:105) = C (cid:112) pi (2 n + 1)!!2 n +1 ∆ n +3 , (24)where the double factorial is defined as n !! = (cid:100) n/ (cid:101)− (cid:89) k =0 ( n − k ) . (25)Insertion of Eqs. (23) and (24) into the expectation valueof Eq. (15) now leads to (cid:104) F ( t ) (cid:105) = e − i ω t ∞ (cid:88) n =0 ( − i t ) n (2 n )! (cid:104) F n (0) (cid:105) + e − i ω t ∞ (cid:88) n =0 ( − i t ) n +1 (2 n + 1)! (cid:104) F n +1 (0) (cid:105) = e − i ω t √ πCω ∆ ∞ (cid:88) n =0 (cid:18) − t (cid:19) (2 n − n )! + e − i ω t √ πC (cid:18) − i t (cid:19) ∞ (cid:88) n =0 (cid:18) − t (cid:19) (2 n + 1)!!(2 n + 1)!= e − i ω t √ πCω ∆ ∞ (cid:88) n =0 n ! (cid:18) − t (2∆) (cid:19) + e − i ω t √ πC (cid:18) − i t (cid:19) ∞ (cid:88) n =0 n ! (cid:32) − t (2∆) (cid:33) = e − i ω t √ πCω ∆ (cid:20) − i t ω ∆ (cid:21) e (cid:16) − t (cid:17) ≈ √ πCω ∆ e (cid:16) − t (cid:17) e − i (cid:16) ω − ω (cid:17) t , (26)where in the last step ω ∆ << F n ExcitationSystemDynamicsMeasurement
FIG. 4. The noise operators are defined in terms of scatter-ing modes and can therefore bridge the gap between far fieldexcitation and internal dynamics. the pump term gains a Gaussian envelope with a tempo-ral spread equal to the inverse of the frequency spread.The above examples demonstrate, that the noise op-erators are fully capable of describing arbitrary inputschemes in a quantitative manner. They form a link be-tween the excitation and internal dynamics of the opticalsystem, as is illustrated in Fig. 4.
B. Output Operators
Up to this point we have only been concerned withthe internal dynamics of the system under external ir-radiation. But in order to describe actual experiments,we also need to consider the dynamics of output states,that are experimentally accessible via detectors. If all de-tectors are placed in the far field, have sufficiently smallapertures, and are sensitive to only a narrow frequencyrange, the output modes we are interested in are of theform E (out) k ,λ ( r ) = C k ,λ ˆ e k ,λ e i kr + E ( i ) k ,λ ( r ) , (27)where E ( i ) k ,λ ( r ) only contains incoming field components.A solution to Maxwell’s equations of the form describesa complicated scenario, in which the scattering responsesof all incident fields interfere destructively in all but onespacial direction. This leads to a single plane wave as anoutgoing field.One should of course ask, whether solutions of the form(27) exist and how one can find them. In order to an-swer these questions, we first recall that the macroscopic Maxwell equations are invariant under time-reversal, ifno absorption losses are present [66]. This means thatthe system Hamiltonian is invariant under the antiuni-tary time reversal operator T [67]: T HT = H. (28)We can therefore exchange the electric field for its time-reversed counterpart, without changing the structure ofthe Hamiltonian. Therefore, the Heisenberg equationsof motion also keep their form under time-reversal. Thetime-reversed electric field operator reads T E ( r ) T = (cid:90) d k [ T E k ,λ ( r ) a k T + h.c. ]= (cid:90) d k (cid:2) E ∗ k ,λ ( r ) T a k T + h.c. (cid:3) = (cid:90) d k (cid:2) E ∗− k ,λ ( r ) T a − k T + h.c. (cid:3) . (29)We now define the time-reversed photon operators a (out) k = T a − k T, (cid:16) a (out) k (cid:17) † = T a †− k T, (30)which satisfy harmonic oscillator commutation relations.Inserting the scattering eigenmodes in Eq. (2) intoEq. (29) yields the time-reversed field modes E ∗− k ,λ ( r ) = C ∗− k ,λ ˆ e ∗− k ,λ e i kr + E ( s ) ∗− k ,λ ( r ) . (31)We now see that the time-reversed scattering modes (31)are indeed of the form (27), since the complex conjugateof an incoming multipole field is an outgoing multipolefield. The relevant output modes are therefore the onesdescribed by the operators a (out) k , which obey the Heisen-berg equations˙ a (out) k = − i ωa (out) k − i E ∗ · d (cid:126) (cid:114) Γ2 π g ∗ ( − ˆ k ) ω − ω + iΓ / σ − . (32)The important thing to notice now is that the dynamicsof σ − can be calculated without refering to any outputmodes, so that σ − can be treated as an external parame-ter in Eq. (32). This enables us to make output calcula-tions without having to keep track of an infinite numberof operators. This is schematically depicted in Fig. 5.To summarize this section, we have established a wayto treat the internal dynamics of an open cavity underarbitrary illumination with only a small number of oper-ators. We have furthermore established a way to relatethe results of the internal calculations to the temporal dy-namics of individual Fourier components of the far field.This enables us to completely describe optical systemswith radiative losses in terms of experimentally accessiblequantities. Examples of this procedure are demonstratedin the following Section. SystemDynamicsMeasurement a k(out) FIG. 5. The time reversed scattering modes have only a singleoutgoing Fourier component and therefore describe the fieldradiated away into a specific direction.
IV. RELATION TO JAYNES-CUMMINGSMODEL
In order to offer a verification of the theory developedabove, we now turn to the task of retrieving the well-established Jaynes-Cummings model from our formalism.Consider an input state of the form | Φ (cid:105) := (cid:90) d k c / ω √ G (cid:48) g (cid:48) (ˆ k ) (cid:114) Γ2 π ω − ω − iΓ (cid:48) / a † k | (cid:105) , (33)where the normalization factor G (cid:48) is defined as G (cid:48) := (cid:90) d Ω k (cid:12)(cid:12)(cid:12) g (cid:48) (ˆ k ) (cid:12)(cid:12)(cid:12) . (34)It is easy to check that the initial state | Φ (cid:105) is properlynormalized, i.e. (cid:104) Φ | Φ (cid:105) = 1. We will now proceed toderive, from Eqs. (16) and (17), the Heisenberg equa-tions for the number operators a † a and σ + σ − , as well asthe appropriate initial conditions arising from the initialstate | Φ (cid:105) .We first want to determine the effect of the zero-timenoise operators F n (0) on the initial state | Φ (cid:105) : F n (0) | Φ (cid:105) = (cid:90) d k c ω √ GG (cid:48) g (cid:48) (ˆ k ) g ∗ (ˆ k ) √ ΓΓ (cid:48) π ( ω − ω ) n ω − ω − iΓ (cid:48) / | (cid:105) . (35)Comparing Eq. (35) with Eq. (B1), we see that the fre-quency integrals are formally identical. Since the fre-quency integral in Eq. (B1) vanishes, as is demonstrated in Appendix B, we conclude that F n (0) | Φ (cid:105) = 0 . (36)But since the zero order noise operator F ( t ) is of theform (15) at all times, we conclude that F ( t ) | Φ (cid:105) = 0 . (37)Next we consider the action of the cavity operator at zerotime on the initial state: a (0) | Φ (cid:105) = (cid:90) d Ω k g (cid:48) (ˆ k ) √ G (cid:48) g ∗ (ˆ k ) √ G √ ΓΓ (cid:48) π (cid:90) dω ω − ω − iΓ (cid:48) / ω − ω + iΓ / | (cid:105) = (cid:18) g √ G ∗ g (cid:48) √ G (cid:48) (cid:19) √ ΓΓ (cid:48) (Γ + Γ (cid:48) ) / | (cid:105) , (38)where the scalar product between two angular functionsis defined as a ∗ b := (cid:90) d Ω k (cid:104) a (ˆ k ) (cid:105) ∗ b (ˆ k ) . (39)From Eq. (38) it is now easy to obtain the initial photonnumber in the cavity (cid:104) Φ | a † (0) a (0) | Φ (cid:105) = (cid:20) g √ G ∗ g (cid:48) √ G (cid:48) (cid:21) (cid:34) √ ΓΓ (cid:48) (Γ + Γ (cid:48) ) / (cid:35) . (40)Please note that due to the Cauchy-Schwarz inequality[68] (cid:20) g √ G ∗ g (cid:48) √ G (cid:48) (cid:21) ≤ (cid:18) g √ G ∗ g √ G (cid:19) (cid:18) g (cid:48) √ G (cid:48) ∗ g (cid:48) √ G (cid:48) (cid:19) = 1 , (41)and that the geometric mean of two numbers is alwayssmaller than the algebraic mean [68] √ ΓΓ (cid:48) (Γ + Γ (cid:48) ) / ≤ . (42)It therefore follows that the initial number of cavity pho-tons in Eq. (40) is smaller than 1, as is required from thefact that only a single photon is incident.We now turn to the problem of deriving equations ofmotion for the number operator expectation values a † a and σ + σ − . To this end we first use Eqs. (16), (17) and(37) to derivedd t (cid:104) a † a (cid:105) = − Γ (cid:104) a † a (cid:105) + 2Im (cid:2) κ (cid:104) a † σ − (cid:105) (cid:3) , (43)dd t (cid:104) σ + σ − (cid:105) =2Im (cid:20) κ (cid:18) ω (cid:19) (cid:104) a † σ z σ − (cid:105) (cid:21) , (44)dd t (cid:104) a † σ − (cid:105) = (cid:20) − i( ω a − ω ) − Γ2 (cid:21) (cid:104) a † σ − (cid:105) +i κ ∗ (cid:104) σ + σ − (cid:105) + i κ ∗ (cid:18) − i Γ2 ω (cid:19) (cid:104) σ z a † a (cid:105) , (45)which does not yet form a closed system of equations.In order to change this, we first use the Pauli operatoridentity σ z σ − = − σ − , (46)in order to change (cid:104) a † σ − σ z (cid:105) in Eq. (44) to − (cid:104) a † σ − (cid:105) .Next we rewrite the remaining problematic operator as (cid:104) σ z a † a (cid:105) = 2 (cid:104) a † σ + σ − a (cid:105) − (cid:104) a † a (cid:105) , (47)and use the fact that the total excitation number of theoriginal Hamiltonian (1) (cid:90) d ka † k a k + σ + σ − , (48)is conserved. This means that, for the given initial state,the solution to the Schr¨odinger equation is of the form | Φ( t ) (cid:105) = (cid:90) d kA ( k , t ) | k , g (cid:105) + B ( t ) | , e (cid:105) , (49)where g and e refer to the ground and excited state ofthe atom, respectively. Combining Eqs. (47) and (49),we notice that (cid:104) σ z a † a (cid:105) = − (cid:104) a † a (cid:105) . (50)Inserting this result into Eq. (45), we see that the equa-tions of motion are now indeed closed.The above equations of motion are very similar to theones derived from the Jaynes-Cummings (JC) Hamilto-nian in the single photon case, except for the dampingterms and the small asymmetry in the light-matter cou-pling terms. We therefore expect to find the well-knownphenomena of the JC model when solving them. In or-der to see if these expectations are accurate, we presenta number of numerical simulations for different systemparameters. We assume without loss of generality that g (cid:48) = g and Γ (cid:48) = Γ, so that the initial photon number inthe cavity is exactly 1. Figure 6(a) shows the result for asystem in the strong coupling regime, i.e. κ > Γ /
2. Thetemporal oscillations of energy between electromagneticfield and atom, known as Rabi oscillations, are clearly vis-ible. Due to the dissipative nature of the cavity mode theoscillations are enveloped by a monotonically decreasingfunction, which represents the total number of (electro-magnetic and atomic) excitations in the system. Thestep-like shape of the total excitation can be explainedby recalling that only the electromagnetic mode is dis-sipative. Hence, whenever all of the energy is stored inthe atom, no dissipation occurs and the total excitationnumber has a plateau. The result for a weakly coupledsystem is shown in Fig. 6(b). The cavity photon is seento decay rapidly, exciting the atom only weakly. Theatomic excitation then also starts decaying, although theatom itself is non-dissipative. This is, of course, due tothe atom being coupled to the dissipative cavity. In order to further investigate the decay of the atomvia the cavity, we turn to the case where the atom isinitially excited, while the electromagnetic field is in itsground state: | Φ (cid:105) = | , e (cid:105) . (51)One can easily convince oneself that the equations ofmotion derived above still hold for the initial state inEq. (51). The initial conditions, however, have to bechanged to (cid:104) σ + σ − (cid:105) = 1 and (cid:104) a † a (cid:105) = 0. If we restrictourselves to the case κ (cid:28) Γ, we can obtain an approxi-mate solution by adiabatic elimination (AE) of the cavitymode [69]. The AE result reads (cid:104) σ + ( t ) σ − ( t ) (cid:105) = exp (cid:32) − | κ | Γ t (cid:33) . (52)The numerical result for the complete set of equations ofmotion is shown in Fig. 6(c) and compared to the AEresult. We see an excellent agreement between the ap-proximate and the complete solution. The deexcitationof the atom to due to its coupling to the electromagneticvacuum is, of course, just the well-known phenomenon ofspontaneous emission. The dependence of the emissiontime on the coupling strength is a manifestation of thePurcell effect [17].If the decay of an initially excited atom is indeed due tospontaneous emission, then the energy should be trans-ferred to the output modes described in Sec. III. In orderto verify this, we first derive from Eqs. (16), (17) and(32) the single-photon equations of motion ddt (cid:104) a (out) † k a (out) k (cid:105) =2Im (cid:104) ζ k (cid:104) a (out) † σ − (cid:105) (cid:105) , (53) ddt (cid:104) a (out) † k σ − (cid:105) = − i∆ k (cid:104) a (out) † k σ − (cid:105) + i ζ ∗ k (cid:104) σ + σ − (cid:105) − i κ ∗ (cid:104) a (out) † k a (cid:105) , (54) ddt (cid:104) a (out) † k a (cid:105) = (cid:18) − i∆ k − Γ2 (cid:19) (cid:104) a (out) † k a (cid:105) + i ζ ∗ k (cid:104) σ + a (cid:105) − i κ (cid:104) a (out) † k σ − (cid:105) , (55)where the definitions ζ k := E ∗ · d (cid:126) (cid:114) Γ2 π g ∗ ( − k ) ω k − ω + iΓ / , (56)∆ k := ω − ω k , (57)were used. Since we assume κ (cid:28) Γ, Eq. (55) can beadiabatically eliminated. Equations (54) then become ddt (cid:104) a (out) † k σ − (cid:105) = (cid:34) − i∆ k − | κ | i∆ k + Γ / (cid:35) (cid:104) a (out) † k σ − (cid:105) + i ζ ∗ k (cid:104) σ + σ − (cid:105) + ζ ∗ k κ ∗ i∆ k + Γ / (cid:104) σ + a (cid:105) . (58) o cc up a ti on nu m b e r (c) 1.00.80.60.40.20.0 5 10 Γ t photonatomtotal o cc up a ti on nu m b e r (d) 1.00.80.60.40.20.00 20 40 Γ t10 30 photonatomtotal o cc up a ti on nu m b e r (e) 1.00.80.60.40.20.00 2 4(4| κ | / Γ )t1 3atom (exact)atom (AE)(a) (b) o cc up a ti on nu m b e r o cc up a ti on nu m b e r Γ t log ( κ / Γ ) Γ t log ( κ / Γ )0.50.01.0 2.0 4.0 0.0-0.5 0.5 1.00.50.0 2.0 4.0 -0.5 0.0 0.5 FIG. 6. Results of numerical simulations of Eqs. (44) with a single-excitation input state: (a) Temporal system dynamics fordifferent values of the coupling-to-loss ratio κ/ Γ when a single photon is injected into the system. Red colors correspond to thephotonic and blue colors to the atomic excited state. (b) Same as (a) but with the atom initially excited. (c) Strong couplingcase κ = Γ = 10 − ω . The characteristic Rabi oscillations are clearly visible. (d) Weaker coupling κ = 10 − ω and Γ = 10 − ω .Please note the different timescales of photonic and atomic decay. (e) Atomic decay for κ = 10 − ω and Γ = 10 − ω with theatom initially excited. The solid line refers to the numerical solution of the exact equations of motion and the dashed line isthe approximate result obtained from adiabatic elimination (AE). This equation can be further simplified, if one uses theAE result (cid:104) σ + a (cid:105) ≈ − i κ Γ / (cid:104) σ + σ − (cid:105) (cid:28) (cid:104) σ + σ − (cid:105) . (59)Equation (58) can then be solved to yield (cid:104) a (out) † k σ − (cid:105) t = i ζ ∗ k (cid:90) t dt (cid:48) exp (cid:32)(cid:34) − i∆ k − | κ | i∆ k + Γ / (cid:35) [ t − t (cid:48) ] (cid:33) (cid:104) σ + σ − (cid:105) t (cid:48) , (60) where the indices of the expectation values denote thetime of evaluation. Inserting Eq. (60) into Eq. (53) andintegrating the resulting equation leads to (cid:104) a (out) † k a (out) k (cid:105) = 2 | ζ k | Re (cid:40)(cid:90) t dt (cid:48) (cid:90) t (cid:48) dt (cid:48)(cid:48) exp (cid:34)(cid:32) − i∆ k − | κ | i∆ k + Γ / (cid:33) ( t (cid:48) − t (cid:48)(cid:48) ) (cid:35) (cid:104) σ + σ − (cid:105) t (cid:48)(cid:48) (cid:41) . (61)Using the AE result Eq. (52), the integrations in Eq. (61)can be easily performed. The result reads (cid:104) a (out) † k a (out) k (cid:105) = | ζ k | Re (cid:40) Γ / | κ | k + | κ | / (i∆ k + Γ / − | κ | / (Γ / (cid:34) − exp (cid:32) − | κ | Γ / t (cid:33)(cid:35) +2 1 − i∆ k − | κ | / (i∆ k + Γ /
2) 1i∆ k + | κ | / (i∆ k + Γ / − | κ | / (Γ / (cid:34) − exp (cid:32) − i∆ k − | κ | i∆ k + Γ / t (cid:33)(cid:35)(cid:41) . (62)0Considering the denominators in Eq. (62), it becomesclear that the number of photons with frequency detun-ing ∆ k is small, unless ∆ k (cid:46) | κ | / (Γ / k (cid:28) Γ /
2, whichallows for the following simplification of Eq. (62): (cid:104) a (out) † k a (out) k (cid:105) ≈ | ζ k | k + (cid:104) | κ | / (Γ / (cid:105) (cid:34) − k t ) exp (cid:32) − | κ | Γ / t (cid:33) + exp (cid:32) − | κ | Γ / t (cid:33)(cid:35) . (63) Likewise, the expression (56) for ζ k can be simplified toyield | ζ k | ≈ (cid:12)(cid:12)(cid:12)(cid:12) E · d (cid:126) (cid:12)(cid:12)(cid:12)(cid:12) Γ2 π (cid:12)(cid:12)(cid:12) g ( − ˆ k ) (cid:12)(cid:12)(cid:12) (Γ / = c ω (cid:12)(cid:12)(cid:12) g ( − ˆ k ) (cid:12)(cid:12)(cid:12) G | κ | π (Γ / . (64)To acquire the total output photon number, the result inEq. (63) has to be integrated over the wave vector. UsingEq. (64) and writing the cosine in terms of exponentialfunctions, one finds (cid:90) d k (cid:104) a (out) † k a (out) k (cid:105) ≈ (cid:90) ∞−∞ dω ωω | κ | π Γ / k + (cid:104) | κ | / (Γ / (cid:105)(cid:34) (cid:32) − | κ | Γ / t (cid:33) − exp (cid:32) − | κ | Γ / t + i∆ k t (cid:33) − exp (cid:32) − | κ | Γ / t − i∆ k t (cid:33)(cid:35) = (cid:34) − exp (cid:32) − | κ | Γ / t (cid:33)(cid:35) = 1 − (cid:104) σ + σ − (cid:105) t , (65)where the integrations are performed with standard con-tour integral techniques. We therefore find that the to-tal excitation number is a constant of motion and thatthe number of photons asymptotically reaches 1 for largetimes. This is in perfect agreement with the requirementsof the original Hamiltonian (1), as well as the physical in-tuition regarding spontaneous emission. V. CONCLUSION
We demonstrated the extenstion of the unitary internalcavity dynamics in QED by a rigorous quantum descrip-tion of radiative losses. Unlike the canonical formula-tion based on a phenomenological coupling to a thermalbath, we have derived a description employing a chain ofnoise operators. Furthermore, we added input and out-put channels to the formalism that allow for a completedescription of the dynamics: starting from incident farfield illumination, incorporating all unitary cavity relatedprocesses, and culminating in far field emission. The pro-cedure suggested here consists of the following steps: • Characterize the cavity classically by determiningthe resonance frequency and linewidth as well asthe dependence on the illumination direction. • Calculate the light-matter coupling constant fromthe properly normalized field strength (according to [64]), the cavity parameters and the emitter’stransition dipole moment. • Evaluate F ( t ) by means of the zero-time noise op-erators F n (0) and the initial photonic state. • Calculate the internal dynamics of the cavity mode. • Solve the equations of motion for the output modesof interest.We discussed single-photon dynamics in a leaky cav-ity coupled to a single atom and retrieved the familiarJaynes-Cummings model, but with the added possibil-ity of calculating the far field dynamics. However, theformalism presented here can be employed to describea multitude of different scenarios of light-matter inter-action, which go beyond the simple Jaynes-Cummingsmodel. We hope that this work will pave the way towardsa more rigorous description of open optical cavities andtheir interaction with the far field.
ACKNOWLEDGMENTS
The study was supported by the Karlsruhe School ofOptics and Photonics (KSOP). The authors also wishto thank the Deutscher Akademischer Austauschdienst(PPP Poland) and the Ministry of Science and HigherEducation in Poland.1
Appendix A: Derivation of System OperatorHeisenberg Equations
Using the definition (7) together with the Heisenbergequation of motion (5), one derives:˙ a = (cid:90) d k c / √ Gω (cid:114) Γ2 π g ∗ (ˆ k ) ω − ω + iΓ / a k = (cid:90) d k c / √ Gω (cid:114) Γ2 π g ∗ (ˆ k ) ω − ω + iΓ / − i ω ) a k − i E · d (cid:126) σ − (cid:90) d k c / √ Gω Γ2 π (cid:12)(cid:12)(cid:12) g (ˆ k ) (cid:12)(cid:12)(cid:12) ( ω − ω ) + (Γ / = − i (cid:90) d k c / √ Gω (cid:114) Γ2 π g ∗ (ˆ k ) a k ω − ω + iΓ / ω − iΓ / ω − ω + iΓ / − i √ Gc / E · d (cid:126) σ − (cid:90) ∞ dω Γ2 π ω − ω + ω ( ω − ω ) + (Γ / . (A1)Since we assume Γ (cid:28) ω , the lower integration boundaryin the second term can be approximately shifted to −∞ .Noticing that the part of the second integral antisymmet-ric in ω − ω vanishes and splitting up the fracture underthe first integral, one arrives at˙ a = − i (cid:90) d k c / √ Gω (cid:114) Γ2 π g ∗ (ˆ k ) a k + ( − i ω − Γ2 ) (cid:90) d k c / √ Gω (cid:114) Γ2 π g ∗ (ˆ k ) ω − ω + iΓ / a k − i √ Gc / E · d (cid:126) σ − . (A2)Employing the definitions in Eqs. (7), (12) and (13) thisbecomes ˙ a ≈ ( − iω − Γ / a − i κσ − − i F . (A3)Turning now to the atom dynamics, the equation of motion (6) can be written˙ σ − = − i ω a σ − + i E ∗ · d ∗ (cid:126) (cid:90) d kω (cid:114) Γ2 π g ∗ (ˆ k ) ωω − ω + iΓ / σ z a k = − i ω a σ − + i E ∗ · d ∗ (cid:126) (cid:90) d kω (cid:114) Γ2 π g ∗ (ˆ k ) σ z a k ω − ω + iΓ / ω − iΓ / ω − ω + iΓ / − i ω a σ − + i E ∗ · d ∗ (cid:126) σ z (cid:90) d kω (cid:114) Γ2 π g ∗ (ˆ k ) a k + i (cid:18) ω − i Γ2 (cid:19) E ∗ · d ∗ (cid:126) σ z (cid:90) d kω (cid:114) Γ2 π g (ˆ k ) ω − ω + iΓ / a k . (A4)Using once again the definitions in Eqs. (7), (12) and (13)this can be written˙ σ − = − i ω a σ − + i E ∗ · d ∗ (cid:126) (cid:90) d k (cid:114) Γ2 π g ∗ (ˆ k ) ω − ω + iΓ / σ z a k . (A5) Appendix B: Derivation of Noise OperatorHeisenberg Equations
Using the definition in Eq. (13) together with theHeisenberg equations (5) and (6) we arrive at˙ F n = − i ω F n − i F n +1 − i κω Γ2 π σ − (cid:90) ∞ dωω ( ω − ω ) n ω − ω − iΓ / . (B1)As can be easily seen, the above frequency integral ishighly divergent. This is due to the fact that we as-sumed a perfect Lorentzian frequency dependence of theelectromagnetic field at the emitter position. In a realsystem, however, one would not expect this assumptionto hold for frequencies far off-resonance. Especially forvery high frequencies one expects rapid oscillations of thefield strength, so that the high frequency contributionsaverage out to zero. Equation (4) therefore has to bemodified to take the off-resonance contributions into ac-count. We do this by adding an Gaussian envelope thatdecays on time scales large compared to the Lorentzianlinewidth Γ, but small compared to ω : E k ,λ ( r a ) = E δ λ,λ (cid:114) Γ2 π g (ˆ k )e − ( ω − ω ) /β ω − ω − iΓ / , (B2)Γ (cid:28) β (cid:28) ω . (cid:90) ∞ dωω ( ω − ω ) n e − ( ω − ω ) /β ω − ω + iΓ / ≈ (cid:90) ∞−∞ dωω ( ω − ω ) n e − ( ω − ω ) /β ω − ω + iΓ / , (B3)where the lower integration boundary has been approxi-mately extended to −∞ , since the exponential functiondecays much faster than any polynomial can grow.The integral in Eq. (B3) can now be solved by contourintegration techniques, if one introduces an auxiliary fac-tor of exp( ± i (cid:15)ω ). But while (cid:15) can just be chosen to beinfinitesimally small, the choice of sign in the exponentleads to very different results. This is due to the factthat the integrand only possesses a pole in the upper half-plane. Hence, in order to find a meaningful result we needto eliminate one of the two possibilities by physical rea-soning. This is similar to choosing the retarded insteadof the advanced Green’s function, since the later violatescausality. However, in the current case it is not immedi-ately obvious which solution is the unphysical one. Onlyonce we obtain the solutions for both possible equationswill it be obvious which one to choose. For this reasonwe consider, for the moment, both possible solutions:˙ F n = − i ω F n − i F n +1 , (lower half-plane) (B4)˙ F n = − i ω F n − i F n +1 − i κ (cid:20) ω (cid:21) (cid:18) i Γ2 (cid:19) n +1 . (upper half-plane)(B5)The details of solving both of these equations will be pre-sented in Appendix C, where we show that the solutionfor integration over the lower half-plane is the physicalone. Appendix C: Solution of Noise Operator HeisenbergEquations
We start by considering the equation of motion (B5)in order to demonstrate its unphysical nature. Formallysolving and then iterating the equation leads to F n ( t ) =e − i ω t ∞ (cid:88) n =0 ( − i t ) n n ! F n (0)+ Γ κ (cid:20) ω (cid:21) ∞ (cid:88) n =0 ( Γ2 ) n J n ( t ; t ) , (C1)where the operator valued terms J n ( t ) read J n ( t ; t ) = (cid:90) t dt . . . (cid:90) t n dt n +1 e i ω ( t n +1 − t ) σ − ( t n +1 ) . (C2) We can now use induction to calculate the values of J n ( t ).Since J ( t ) is of the form J ( t ; t ) = (cid:90) t dt e i ω ( t − t ) σ − ( t ) , (C3)the following induction hypothesis is consistent with thebase case: J n ( t ; t ) = (cid:90) t dt (cid:48) e i ω ( t (cid:48) − t ) σ − ( t (cid:48) ) ( t − t (cid:48) ) n n ! . (C4)Performing the induction step is now straightforward J n +1 ( t ; t ) = (cid:90) t dt (cid:48)(cid:48) J n ( t (cid:48)(cid:48) ; t )= (cid:90) t dt (cid:48)(cid:48) (cid:90) t (cid:48)(cid:48) dt (cid:48) e i ω ( t (cid:48) − t ) σ − ( t (cid:48) ) ( t (cid:48)(cid:48) − t (cid:48) ) n n != (cid:90) t dt (cid:48) e i ω ( t (cid:48) − t ) σ − ( t (cid:48) ) (cid:90) t dt (cid:48)(cid:48) ( t (cid:48)(cid:48) − t (cid:48) ) n n ! Θ( t (cid:48)(cid:48) − t (cid:48) )= (cid:90) t dt (cid:48) e i ω ( t (cid:48) − t ) σ − ( t (cid:48) ) (cid:90) tt (cid:48) dt (cid:48)(cid:48) ( t (cid:48)(cid:48) − t (cid:48) ) n n != (cid:90) t dt (cid:48) e i ω ( t (cid:48) − t ) σ − ( t (cid:48) ) ( t − t (cid:48) ) n +1 ( n + 1)! , (C5)which is of the required form. Inserting Eq. (C5) intoEq. (C1) now yields F n ( t ) =e − i ω t ∞ (cid:88) n =0 ( − i t ) n n ! F n (0)+ κ (cid:20) ω (cid:21) Γ (cid:90) t dt (cid:48) e ( − i ω +Γ / t − t (cid:48) ) σ − ( t (cid:48) ) , (C6)where the infinite sum was performed to yield an ex-ponential function. Close inspection of Eq. (C6) revealsthat the second term is divergent in time due to the factorexp[(Γ / t ], which can be pulled in front of the integral.But this would mean that the noise operators grow with-out limit, driving the temperature of the system towardsinfinity. The equations of motion (B5) is therefore clearlyunphysical.Turning to (B4) we find the formal solution F n ( t ) = e − i ω t F n (0) − i (cid:90) t dt (cid:48) e − i ω ( t − t (cid:48) ) F n +1 ( t (cid:48) ) . (C7)Iteration of Eq. (C7) then leads to the form F n ( t ) = e − i ω t ∞ (cid:88) m =0 F n + m ( t )( − i) m I m ( t ) , (C8) I m ( t ) = (cid:90) t dt . . . (cid:90) t m − dt m . (C9)3The elements of the series I m ( t ) can be easily calculatedby induction. First we notice that the base case I ( t ) = 1 , (C10)is in agreement with the assumption I m ( t ) = t m m ! . (C11)We now proceed with the inductive step I m +1 ( t ) = (cid:90) t dt I m ( t ) = (cid:90) t dt t m m ! = t m +1 ( m + 1)! , (C12)hence proving our assumption. Substitution of Eq. (C12)into Eq. (C8) gives the final result F n ( t ) = e − i ω t ∞ (cid:88) m =0 ( − i t ) m m ! F n + m ( t ) . (C13) ∗ [email protected][1] J. I. Cirac and H. J. Kimble, Nat. Photon. , 18 (2017).[2] R. J. Schoelkopf and S. M. Girvin, Nature , 664(2008).[3] H. J. Kimble, Nature , 1023 (2008).[4] P. Zoller et al. , Eur. Phys. J. D , 203 (2005).[5] A. Imamo˘glu, D. D. Awschalom, G. Burkard, D. P. Di-Vincenzo, D. Loss, M. Sherwin, A. Small, et al. , Phys.Rev. Lett. , 4204 (1999).[6] K. Hennessy, A. Badolato, M. Winger, D. Gerace,M. Atat¨ure, S. Gulde, S. F¨alt, E. L. Hu, andA. Imamo˘glu, Nature , 896 (2007).[7] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atoms and Photons, Introduction to Quantum Electrody-namics (Wiley, New York, 1989).[8] L. Allen and J. H. Eberly,
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