Cavity-QED of a leaky planar resonator coupled to an atom and an input single-photon pulse
CCavity-QED of a leaky planar resonator coupledto an atom and an input single-photon pulse
Denis Gont¸a ∗ and Peter van Loock † Institute of Optics, Information and Photonics,Friedrich-Alexander-University Erlangen-Nuremberg, Staudtstrasse 7, 91058 Erlangen, Germany Institute of Physics, Johannes Gutenberg University Mainz, Staudingerweg 7, 55128 Mainz, Germany (Dated: November 8, 2018)In contrast to the free-space evolution of an atom governed by a multi-mode interaction withthe surrounding electromagnetic vacuum, the evolution of a cavity-QED system can be character-ized by just three parameters, (i) atom-cavity coupling strength g , (ii) cavity relaxation rate κ ,and (iii) atomic decay rate into the non-cavity modes γ . In the case of an atom inserted intoa planar resonator with an input beam coupled from the outside, it has been shown by Koshino[Phys. Rev. A , 053814 (2006)] that these three parameters are determined not only by the atomand cavity characteristics, but also by the spatial distribution of the input pulse. By an ab-initio treatment, we generalize the framework of Koshino and determine the cavity-QED parameters of acoupled system of atom, planar (leaky) resonator, and input single-photon pulse as functions of thelateral profile of the pulse and the length of resonator. We confirm that the atomic decay rate can besuppressed by tailoring appropriately the lateral profile of the pulse. Such an active suppression ofatomic decay opens an attractive route towards an efficient quantum memory for long-term storageof an atomic qubit inside a planar resonator. PACS numbers: 42.50.Ct, 42.50.Pq
I. INTRODUCTION
Cavity quantum electrodynamics (cavity-QED) is a re-search field that studies electromagnetic fields in con-fined spaces and radiative properties of atoms in suchfields. Experimentally, the simplest example of such asystem is a single atom interacting with a single mode of ahigh-finesse resonator [1]. This system bears an excellentframework for quantum communication and informationprocessing, in which atoms and photons are interpretedas bits of quantum information and their mutual inter-action provides a controllable entanglement mechanism[2].Remarkably, the evolution of a cavity-QED system canbe well characterized by just three parameters: (i) atom-cavity coupling strength g , (ii) cavity relaxation rate κ ,and (iii) atomic decay rate into the non-cavity modes γ . The cavity-QED effects become manifest clearly whenthe atom-cavity coupling g is much larger than the atomicdecay rate γ and the cavity relaxation rate κ , at the sametime. These two conditions define the (so-called) strong-coupling regime of atom-cavity interaction that ensuresthat the energy exchange between the constituents isreversible and develops faster than losses due to thecavity relaxation and the atomic decay. In the reso-nant regime, i.e., when the cavity resonant frequencymatches the atomic transition frequency, the reversibilityof energy exchange ensures that the coherent (unitary)part of atom-cavity evolution is governed by the Jaynes- ∗ [email protected] † [email protected] Cummings Hamiltonian [3] H JC = (cid:126) g (cid:0) c σ † + c † σ (cid:1) , (1)where c and c † denote the cavity mode annihilation andcreation operators, while σ and σ † are the atomic low-ering and raising operators, respectively. This Hamilto-nian describes the interaction of a two-level atom witha single-mode light field that is confined inside the res-onator.During the last decades, single-mode resonators withtypically spherical mirrors have been fabricated and uti-lized in various cavity-QED experiments. It was demon-strated that resonators with spherical mirrors can oper-ate in the strong-coupling regime, such that the coher-ent part of the atom-cavity evolution is described by theHamiltonian (1) [4]. Although a planar Fabry-Perot res-onator with a coupled atom is used to illustrate the mainfeatures of cavity-QED, there is an essential difference be-tween the resonator with spherical mirrors used in typi-cal cavity-QED experiments and a Fabry-Perot resonatorwith two coplanar mirrors. Namely, even in the caseof perfect lossless mirrors, a planar (Fabry-Perot) res-onator is intrinsically multimode with a spectrally densecontinuum of modes. Due to this essential difference,the cavity-QED parameters ( g, κ, γ ) cannot be identi-fied straightforwardly in the case of an atom coupled toa planar resonator.In recent years, however, an impressive experimentalprogress has been achieved in fabricating various planarlike resonators, i.e., two-dimensional microwave circuits(circuit-QED) [5], fiber-based (FFP) cavities [6], and di-verse micro-cavities [7]. Triggered by this experimentalprogress, the recent [8–12] and past [13–18] theoreticaldevelopments devoted to planar cavities have acquired a r X i v : . [ qu a n t - ph ] S e p an increasing attention. Although it is commonly agreedthat the Rabi oscillations cannot occur in an interact-ing system of an atom and a planar resonator becauseof a weak atom-cavity coupling, it was pointed out inRefs. [19, 20] that such system can still exhibit Rabi os-cillations, similar to those of a cavity-QED system, oncethe planar resonator is excited by a coherent externalbeam. In other words, provided that a light pulse pene-trates the resonator from outside with an appropriatelytailored spatial distribution, the coupling strength of an(otherwise weakly interacting) atom-cavity system canbe dramatically enhanced, leading to Rabi oscillations.The experimental evidences which support the exis-tence of Rabi oscillations in a coupled exciton-photonsystem confined in a planar microcavity and exposedto an external coherent beam has been presented inRefs. [21, 22]. Since the coherent part of both evolu-tions associated with confined exciton-photon and atom-photon coupled systems is governed by the Jaynes-Cummings Hamiltonian (1), these experiments providecompelling arguments that an interacting system of threeconstituents, i.e., (i) an atom weakly coupled to (ii) a pla-nar resonator, and (iii) a spatially tailored input pulse,is capable to reproduce the cavity-QED evolution. Sim-ilar to the cavity-QED system, furthermore, this (atom-cavity-pulse) system is characterized by a set of parame-ters determined not only by the atom, cavity, and reser-voir characteristics, but also by the spatial distributionof the input pulse. To our best knowledge, the problemof identifying these parameters has been addressed solelyby Koshino in Ref. [20].Using the (so-called) form-factor formalism, in this ref-erence, the author suggested three functions which corre-spond to the cavity-QED triplet ( g, κ, γ ), and he showedtheir dependence on the spatial distribution of the inputpulse. As a consequence of the developed formalism, itwas demonstrated how to suppress the atomic decay γ bytailoring appropriately the spatial distribution of this in-put pulse. However, Koshino introduced four simplifyingassumptions in his framework, namely, (i) the evolutionof the coupled atom-cavity-pulse system was describedby an ad hoc Hamiltonian, (ii) the light field had onlyone (fixed) polarization, (iii) the atom was described byan averaged (in space) dipole, while (iv) the planar res-onator accommodated only one atomic wavelength.In contrast to Koshino’s approach, in this paper, wedevelop an ab-initio theoretical framework, in which wecompletely exclude the above simplifications. In this gen-eralized framework, we derive the cavity-QED param-eters of a coupled atom-cavity-pulse system and revealthe dependence of these parameters on the atom-cavity-reservoir characteristics and spatial distribution of theinput pulse. We find explicitly the optimal lateral profilethat yields a complete vanishing of the atomic decay rateand, thus, we also find that the atomic decay can be effi-ciently suppressed by coupling of an appropriate pulse tothe resonator. Besides this optimal pulse, we consider thesituation in which a Hermite-Gaussian beam penetrates the resonator from outside. We calculate cavity-QED pa-rameters for this case and reveal their dependence on thebeam waist and the cavity length.The controllable suppression of the atomic decay opensan attractive route towards an efficient quantum memoryfor long-term storage of a single qubit that is encoded bya two-level atom coupled to the planar resonator and aninput pulse, while the atomic decay constitutes the mainsource of decoherence. Such a quantum memory poses anessential prerequisite for quantum information process-ing and quantum networking applications like quantumrepeaters [23] and quantum key distribution [24]. In thispaper, however, we address solely the physical aspectsof the suggested quantum memory, i.e., the cavity QEDbehavior of the coupled atom-cavity-pulse system, whilea quantitative characterization of the suggested quantummemory shall be addressed in our future works.The paper is organized as follows. In the next sec-tion, we analyze a leaky planar resonator and derive thequantized electromagnetic field produced inside and out-side the resonator. In Sec. II.C we discuss the limitof perfect reflectivity, which is relaxed in Sec. II.D tothe case of a high but finite reflectivity. Using the to-tal Hamiltonian of a coupled atom-cavity-pulse systemderived in Secs. III.C and III.D, we introduce the form-factor formalism and identify the cavity-QED parametersin Sec. III.C. In Sec. IV.A, we evaluate these parametersby considering an optimal lateral profile that yields a sup-pression of atomic decay, while an (experimentally feasi-ble) Hermite-Gaussian beam is considered in Sec. IV.B.A summary and outlook are given in Sec. V.
II. ONE-SIDED LEAKY CAVITY WITHPLANAR GEOMETRY
In order to describe a two-level atom coupled to a fieldconfined in a planar resonator, we have to consider firstan empty resonator and determine the respective quan-tized electromagnetic field. In this section, we analyzethe one-sided leaky cavity with planar geometry as shownin Fig. 1(a). This cavity consists of a perfectly reflect-ing (solid) plane mirror located at z = − (cid:96) and a leaky(semitransparent) plane mirror located at z = 0.As we mentioned in the introduction, there is an es-sential difference between a resonator with spherical mir-rors, used in typical cavity-QED experiments, and a pla-nar resonator. In the latter confinement configuration,only the normal component of wave vector (along the k z -axis) can take discrete values inside a perfectly reflecting(lossless) planar cavity, while the other two componentspropagate freely. In a leaky planar resonator, in con-trast, the semitransparent mirror at z = 0 causes thecavity relaxation, i.e., the leakage of cavity photons and,therefore, even the z -component of wave vector can neverbecome completely discrete. In contrast to the Koshino’streatment, the cavity relaxation in our approach is nota predefined function. Instead, it is determined by the FIG. 1. (Color online) (a) Multiple-reflections method [15] foran incident plane wave that penetrates the planar resonatorfrom the outside. (b) Cylindrical coordinate system in thereciprocal space. See text for details. transmissivity and reflectivity parameters of planar res-onator. This enables us to include both the intra-cavityfield and the field that leaks outside (or penetrates intothe resonator) in the same framework.
A. Semitransparent dielectric-slab mirror
Following the conventional approach (see Sec. 5.C inRef. [25]), we model the semitransparent mirror by anidealized (infinitesimally) thin layer of dielectric material,the so-called dielectric slab, with the dielectric constantaround z = 0 given by (cid:15) ( z ) = (cid:15) [1 + η δ ( z )] , (2)where (cid:15) denotes the permittivity of vacuum and η isthe positive and real parameter that encodes the trans-parency (see below). In Ref. [15], Dutra and Knight showed that the transmissivity and reflectivity of sucha thin dielectric slab are given by the expressions T ⊥ ( k ) = 2 k z k z − ˙ ι k η ; T (cid:107) ( k ) = 22 − ˙ ι k z η , (3a) R ⊥ ( k ) = ˙ ι k η k z − ˙ ι k η ; R (cid:107) ( k ) = ˙ ι k z η ˙ ι k z η − , (3b)which fulfill the equalities | R α ( k ) | + | T α ( k ) | = 1 , (4a) R α ( k ) ∗ T α ( k ) + T α ( k ) ∗ R α ( k ) = 0 , (4b)where k is the wave vector, α = ⊥ refers to the com-ponent normal to the plane of mirror (along the z axis),and α = (cid:107) refers to the component lying on the plane ofmirror ( x − y plane).Using Eqs. (3), one can readily check that the mirrorbecomes completely transparent in the limit η →
0, whileit becomes a perfect reflector in the limit η → ∞ , i.e., T α ( k ) = (cid:40) , η → , , η → ∞ , R α ( k ) = (cid:40) , η → ,m α , η → ∞ , (5)where m ⊥ = − m (cid:107) = 1. The parameter η , there-fore, determines alone the transmissivity and reflectivityof the (leaky) mirror.In our scheme, the leaky mirror at z = 0 ensures alsothat a light pulse can penetrate the resonator from out-side. The leaky mirror, therefore, is supposed to have ahigh but non-perfect reflectivity ( η (cid:29)
1) which, in thispaper, is understood as a small deviation from the perfectreflectivity limit. Since T α ( k ) and R α ( k ) are constant inthe perfect reflectivity limit [see Eqs. (5)], we can treat(to a good approximation) the transmissivity and reflec-tivity of a leaky mirror as complex valued constants T α ( k ) ∼ = T α and R α ( k ) ∼ = R α ; (6a) | R α | + | T α | = 1 , R ∗ α T α + T ∗ α R α = 0 . (6b)On top of this, moreover, the expressions (3) implyRe ( T α ) (cid:28) Im ( T α ) and Re ( R α ) (cid:29) Im ( R α ) . (7)The assumption (6a) together with relations (6b) and(7) suggest that T α and R α can be chosen in the form R α = m α (cid:112) − τ ; T α = ˙ ι τ , τ (cid:28) τ , i.e., R α starts to deviate from m α tothe second order of τ . Throughout this paper, therefore,we consider the expressions (8) to describe a leaky mir-ror, while the expressions (3) are considered to describean arbitrarily semitransparent mirror or a perfectly re-flecting mirror. B. One-sided planar resonator with asemitransparent mirror
An unconfined light propagates in free space, such thatthe positive-frequency part of its electric field E ( r , t ) isexpressed as follows [26] E (+) ( r , t ) = (cid:88) α (cid:90) d k ˆ v α ( k ) E (+) α ( k ) e ˙ ι ( k · r − k c t ) ≡ (cid:90) d k E (+) ( k , r ) e − ˙ ι k c t , (9)where k ≡ | k | denotes the modulus of wave vector,ˆ v α ( k ) denotes the unit vector specifying the direction ofa given electric-field component, while E (+) α ( k ) denote theelectric-field amplitude. We calculate how the one-sidedplanar resonator with a semitransparent mirror modifiesthe plane waves encoded by the expression E (+) ( k , r ).Having this modified expression, we then insert it backinto Eq. (9) along with the quantum counterparts ofthe field amplitudes E (+) α ( k ) and determine the quantizedelectric field in the presence of the resonator. With thehelp of quantized electric and magnetic fields, further-more, we compute the total electromagnetic energy inthe physical space that includes regions inside and out-side the cavity along with the region occupied by theleaky mirror.Similar to the theory of a Fabry-Perot resonator [27],we apply the multiple-reflection approach by summingthe reflected and transmitted plane waves as depicted inFig. 1(a). We recall that the solid mirror is a perfectlyconducting plane that implies the transformation of theamplitudes of the electric field E (+) α ( k ) → m α E (+) α ( k ); m ⊥ = − , m (cid:107) = 1 , (10)In contrast to the perfect mirror at z = − (cid:96) , the actionof a semitransparent mirror on the incident plane wavesis determined by the reflectivity and transmissivity (3),which imply the respective transformations of the ampli-tudes of the electric field E (+) α ( k ) → R α ( k ) E (+) α ( k ) ; (11a) E (+) α ( k ) → T α ( k ) E (+) α ( k ) . (11b)Using the multiple-reflection approach with relations(10) and (11), we compute the electric field inside ( − (cid:96) ≤ z <
0) and outside ( z >
0) the planar cavity region, E (+) C ( k , r ) = ∞ (cid:88) i =1 E (+) C ,i ( k , r )= 2 (cid:20) E (+) (cid:107) ( k ) L (cid:107) ( k ) cos[ k z ( z + (cid:96) )] k k ˆ z − ˙ ι E (+) (cid:107) ( k ) L (cid:107) ( k ) sin[ k z ( z + (cid:96) )] k z k ˆ s − ˙ ι E (+) ⊥ ( k ) L ⊥ ( k ) sin[ k z ( z + (cid:96) )] ˆ k ⊥ (cid:105) e ˙ ι(cid:126)k · r , (12) E (+) O ( k , r ) = E (+) ( k , r ) + ∞ (cid:88) i =1 E (+) R ,i ( k , r )= (cid:20) E (+) (cid:107) ( k ) (cid:0) e − ˙ ι k z z + P (cid:107) ( k ) e ˙ ι k z z (cid:1) k k ˆ z + E (+) (cid:107) ( k ) (cid:0) e − ˙ ι k z z − P (cid:107) ( k ) e ˙ ι k z z (cid:1) k z k ˆ s + E (+) ⊥ ( k ) (cid:0) e − ˙ ι k z z + P ⊥ ( k ) e ˙ ι k z z (cid:1) ˆ k ⊥ (cid:105) e ˙ ι(cid:126)k · r , (13)where k = { k z , k, ϑ } has been expressed in the cylindri-cal coordinate basis, while (cid:126)k = k ˆ s is the (in-plane) wavevector lying on the plane of mirror as shown in Fig. 1(b).The orthogonal unit vectors ˆ z , ˆ s , and ˆ k ⊥ ≡ ˆ s × ˆ z deter-mine the polarization of the resulting electric field, while L α ( k ) ≡ T α ( k )1 − e ι (cid:96) k z m α R α ( k ) , (14a) P α ( k ) ≡ R α ( k ) + T α ( k ) L α ( k ) m α e ι (cid:96) k z (14b)characterize the spectral response of resonator [28].At this point, we introduce the quantum counterpartof the (positive-frequency) field amplitude E (+) α ( k ) = (cid:115) (cid:126) ω k (cid:15) (2 π ) a α ( k ) , (15)where ω k ≡ c | k | = c k , while a α ( k ) is the photon annihi-lation operator that satisfies[ a α ( k ) , a † α (cid:48) ( k (cid:48) )] = δ α,α (cid:48) δ ( k − k (cid:48) ) , (16a)[ a α ( k ) , a α (cid:48) ( k (cid:48) )] = 0 . (16b)In contrast to the free-space case, the above amplitudecontains an extra factor of 2 due to the perfect mirror re-stricting the field to the half-space only [15]. By insertingEqs. (12) and (13) along with the amplitude (15) into theintegral (9), we obtain the quantized electric field inside E (+) C ( r , t ) and outside the cavity E (+) O ( r , t ) region, E (+) • ( r , t ) = (17)= (cid:88) α (cid:90) (cid:48) d k (cid:115) (cid:126) ω k (cid:15) (2 π ) U α, • ( k , z ) a α ( k ) e ˙ ι ( (cid:126)k · r − ω k t ) , where the integration (prime symbol) is restricted to thepositive k z , while the subscript • denote C or O depend-ing on the region inside or outside the cavity, respectively.In this expression, U ⊥ , C ( k , z ) = − ι L ⊥ ( k ) sin[ k z ( z + (cid:96) )] ˆ k ⊥ ; (18a) U ⊥ , O ( k , z ) = (cid:0) e − ˙ ι k z z + P ⊥ ( k ) e ˙ ι k z z (cid:1) ˆ k ⊥ ; (18b) U (cid:107) , C ( k , z ) = 2 L (cid:107) ( k ) (cid:18) cos[ k z ( z + (cid:96) )] k k ˆ z − ˙ ι sin[ k z ( z + (cid:96) )] k z k ˆ s (cid:19) ; (18c) U (cid:107) , O ( k , z ) = (cid:0) e − ˙ ι k z z + P (cid:107) ( k ) e ˙ ι k z z (cid:1) k k ˆ z + (cid:0) e − ˙ ι k z z − P (cid:107) ( k ) e ˙ ι k z z (cid:1) k z k ˆ s . (18d)Moreover, the global mode functions U α ( k , r ) ≡ e ˙ ι(cid:126)k · r [ U α, O ( k , z ) + U α, C ( k , z )] (19)form an orthonormal set (cid:90) d r U α ( k , r ) · U ∗ α (cid:48) ( k (cid:48) , r ) = (2 π ) δ α,α (cid:48) δ ( k − k (cid:48) ) , (20)where the integration over the z -axis is restricted to − (cid:96) In the previous section, we derived the quantized elec-tric and magnetic fields in the presence of a one-sidedplanar resonator as displayed in Fig. 1(a) with an arbi-trarily semitransparent mirror. We found that the energyof the electromagnetic field, localized inside and outsidethe cavity and inside the semitransparent mirror, is givenby the Hamiltonian (24). In this section, we reconsider the obtained results in the (lossless) limit of perfect re-flectivity, i.e., when the leaky mirror at z = 0 becomesa perfectly reflecting mirror ( η → ∞ ). In this case, twotypes of photon field operators, acting in the intra-cavityregion and outside region (reservoir) emerge from theglobal photon operator a α ( k ). In the next sections, weconsider these operators for the case of a leaky cavitygiven by the condition τ (cid:28) E (+) C ( r , t ) = (cid:88) α (cid:90) (cid:48) d k (cid:115) (cid:96) (cid:126) ω k π (cid:15) (2 π ) × (25) (cid:101) U α, C ( k , z ) | L α ( k ) | ˜ a α ( k ) e ˙ ι ( (cid:126)k · r − ω k t ) , where we introduced U α, C ( k , z ) ≡ L α ( k ) (cid:101) U α, C ( k , z ) and a α ( k ) ≡ (cid:112) (cid:96)/π L ∗ α ( k ) ˜ a α ( k ) . (26)In the perfect-reflectivity limit, it was showed by Dutraand Knight in Ref. [15] thatlim η →∞ | L α ( k ) | = π(cid:96) ∞ (cid:88) n = −∞ δ ( k z − k z,n ) , (27)where k z,n ≡ n π/(cid:96) . By inserting this expression intoEq. (25) and integrating over k z , we obtain the electricfield inside the cavity in the limit of perfect reflectivity, E (+) CL ( r , t ) = (cid:88) α,n (cid:90) d(cid:126)k (cid:115) (cid:126) ω n,k (cid:15) (2 π ) (cid:96) × (28) (cid:101) U α, C ( k z,n , (cid:126)k , z ) ˜ a α ( k z,n , (cid:126)k ) e ˙ ι ( (cid:126)k · r − ω n,k t ) , where ω n,k ≡ c (cid:113) k z,n + k is the (quasi-mode) frequency,and where the reduced mode functions (cid:101) U α, C form anorthogonal set (cid:90) dz (cid:101) U α, C ( k z,n , (cid:126)k , z ) · (cid:101) U ∗ α (cid:48) , C ( k z,n (cid:48) , (cid:126)k , z ) = 2 (cid:96) δ n,n (cid:48) δ α,α (cid:48) (29)with the integration being restricted to − (cid:96) < z < a α ( k z,n , (cid:126)k ) inEq. (28) depends on the discrete values of n and continu-ous values of (cid:126)k . Although the expression E (+) CL ( r , t ) is for-mally identical to the (positive-frequency parts of) elec-tric field inside a lossless planar resonator (see Ref. [15]),we remind that the operator ˜ a α ( k z,n , (cid:126)k ) has been ob-tained from a α ( k ) using the definition (26) and the dis-cretization of k z component. In order to define the cav-ity operator acting inside the cavity region only, we keepEq. (28) with replacing ˜ a α ( k z,n , (cid:126)k ) by operator c α,n ( (cid:126)k ), E (+) CP ( r , t ) = (cid:88) α,n (cid:90) d(cid:126)k (cid:115) (cid:126) ω n,k (cid:15) (2 π ) (cid:96) × (30) (cid:101) U α, C ( k z,n , (cid:126)k , z ) c α,n ( (cid:126)k ) e ˙ ι ( (cid:126)k · r − ω n,k t ) , where we interpret c α,n ( (cid:126)k ) as the cavity photon annihi-lation operator that satisfies[ c α,n ( (cid:126)k ) , c † α (cid:48) ,n (cid:48) ( (cid:126)k (cid:48) )] = δ α,α (cid:48) δ n,n (cid:48) δ (cid:16) (cid:126)k − (cid:126)k (cid:48) (cid:17) , (31a)[ c α,n ( (cid:126)k ) , c α (cid:48) ,n (cid:48) ( (cid:126)k (cid:48) )] = 0 , (31b)and is characterized by the quasi-mode frequency ω n,k .In order to derive c α,n ( (cid:126)k ) in terms of global operator a α ( k ), we multiply (scalarly) both Eqs. (17) and (30)by (cid:101) U ∗ α, C ( k z,n (cid:48) , (cid:126)k , z ) and integrate them over z from − (cid:96) to 0 using the property (29). We equate the resultingexpressions and solve them for the operator c α,n ( (cid:126)k ), c α,n ( (cid:126)k ) = 12 √ π (cid:96) (cid:90) (cid:48) d k z (cid:114) ω k ω n,k L α ( k ) a α ( k ) × (32) e ˙ ι ( ω n,k − ω k ) t (cid:90) dz (cid:101) U α, C ( k z , (cid:126)k , z ) · (cid:101) U ∗ α, C ( k z,n , (cid:126)k , z ) , which obeys the commutation relations (31).In a similar fashion, we derive the electric field validoutside the cavity region only. For this, we first express E (+) O ( r , t ) [see Eq. (17)] in the limit η → ∞ [see Eq. (5)], E (+) OL ( r , t ) = (cid:88) α (cid:90) (cid:48) d k (cid:115) (cid:126) ω k (cid:15) (2 π ) × (cid:101) U α, O ( k , z ) a α ( k ) e ˙ ι ( (cid:126)k · r − ω k t ) , (33)where the reduced mode functions (cid:101) U α, O ( k , z ) ≡ lim η → ∞ U α, O ( k , z ) = (cid:101) U α, C ( k , z − (cid:96) )form an orthonormal set (cid:90) dz (cid:101) U α, O ( k z , (cid:126)k , z ) · (cid:101) U ∗ α (cid:48) , O ( k (cid:48) z , (cid:126)k , z ) = 2 π δ α,α (cid:48) δ ( k z − k (cid:48) z ) , (34)with the integration being restricted to 0 < z < ∞ .By following the same approach as before, we keepEq. (33) with replacing a α ( k ) by operator b α ( k ) E (+) OP ( r , t ) = (cid:88) α (cid:90) (cid:48) d k (cid:115) (cid:126) ω k (cid:15) (2 π ) × (cid:101) U α, O ( k , z ) b α ( k ) e ˙ ι ( (cid:126)k · r − ω k t ) , (35)where we interpret b α ( k ) as the photon annihilation op-erator of the (reservoir) modes outside the cavity. Thisoperator satisfies the usual commutation relations[ b α ( k ) , b † α (cid:48) ( k (cid:48) )] = δ α,α (cid:48) δ ( k − k (cid:48) ) , (36a)[ b α ( k ) , b α (cid:48) ( k (cid:48) )] = 0 . (36b)In order to find the reservoir photon operator b α ( k ) interms of a α ( k ), we multiply (scalarly) both Eqs. (17) and(35) by (cid:101) U ∗ α, O ( k (cid:48) z , (cid:126)k , z ) and integrate them over z from 0 to ∞ using the property (34). We equate the resultingexpressions and solve them for the operator b α ( k ), b α ( k ) = 12 π (cid:90) (cid:48) d k (cid:48) z (cid:115) ω (cid:48) k ω k a α ( k (cid:48) z , (cid:126)k ) × (37) e ˙ ι ( ω k − ω (cid:48) k ) t (cid:90) d z U α, O ( k (cid:48) z , (cid:126)k , z ) · (cid:101) U ∗ α, O ( k z , (cid:126)k , z ) , where ω (cid:48) k ≡ c (cid:112) k (cid:48) z + k and which obeys the commuta-tion relations (36).The relations (29) and (34) along with the commuta-tion relations (31) and (36) imply that the electric fields E CP ( r , t ) and E OP ( r , t ), along with the respective mag-netic fields, enable to describe any physically achievableconfiguration of the electromagnetic field inside and out-side the planar resonator, respectively. Except for the re-gion filled by the semitransparent mirror, therefore, theoperators c α,n ( (cid:126)k ) and b α ( k ) cover the entire continuumof Fock spaces spanned by the global operator a α ( k ). Inother words, this operator can be expanded as a α ( k ) = (cid:88) n A α,n ( k ) c α,n ( (cid:126)k ) + (cid:90) (cid:48) d k (cid:48) z B α ( k , k (cid:48) z ) b α ( k (cid:48) z , (cid:126)k ) , (38)where A α,n ( k ) and B α ( k , k (cid:48) z ) are defined by the meansof commutators A α,n ( k ) = (cid:20) a α ( k ) , (cid:90) d(cid:126)k (cid:48) c † α,n ( (cid:126)k (cid:48) ) (cid:21) , (39a) B α ( k , k (cid:48) z ) = (cid:20) a α ( k ) , (cid:90) d(cid:126)k (cid:48) b † α ( k (cid:48) ) (cid:21) . (39b)Since the global photon operator obeys the eigenop-erator equation [ a α ( k ) , H F ] = (cid:126) ω k a α ( k ) [see Eq. (24)],the expansion (38) can be traced back to Fano’s diag-onalization technique utilized in Ref. [29] to analyze acoupled bound-continuum system. In the framework ofcavity QED, this technique has been exhaustively studiedin Ref. [30], where a α ( k ), c α,n ( (cid:126)k ), and b α ( k ) were identi-fied as the dressed, bare (or quasi-cavity), and reservoirphoton operators, respectively (see also Ref. [31]). D. The case of a high but finite reflectivity We found above that a α ( k ) can be expressed with thehelp of operators c α,n ( (cid:126)k ), b α ( k ) and functions (39). Thisexpected result, obtained for a lossless resonator, is basedon the ability to describe any attainable electromagneticfield configuration inside or outside the resonator usingthe expression (30) or (35), respectively.In this section, we show that the expansion (38) holdstrue also in the case of a high but finite reflectivity, i.e., aleaky cavity. In order to proceed, we replace the functions R α ( k ) and T α ( k ) in (30) and (35) by the expressions (8).The structure of E (+) CP ( r , t ) and E (+) OP ( r , t ) implies that thereduced mode functions (cid:101) U α, C ( k z,n , (cid:126)k , z ), (cid:101) U α, O ( k , z ) andthe orthogonality relations (29), (34) remain unchanged.The operators c α,n ( (cid:126)k ) and b α ( k ), in contrast, includeimplicitly the reflectivity and transmissivity by meansof the spectral response function L α ( k ) [see Eqs. (14)].In the case of a leaky cavity with τ (cid:28) 1, to a goodapproximation, this response function takes the form L ( ω, k ) ∼ = c (cid:96) ∞ (cid:88) n =0 − τω − ω n,k + ˙ ι c τ / (4 (cid:96) ) , (40)where, without loss of generality, we replaced k z by the(frequency valued) parameter ω divided by c . In the de-nominator of this expression, moreover, we have imposedthe k dependence by means of the quasi-mode frequency ω n,k . In the limit of vanishing k , the resulting functionreduces to Eq. (9.49) derived in Ref. [28] for the case ofan one-dimensional leaky cavity, where the contributionof continuous and unconfined modes has been omitted.By inserting (40) in Eqs. (32) and (37) with replace-ment k z → ω/c , we compute explicitly c α,n ( (cid:126)k ) and b α ( k ), c α,n ( (cid:126)k ) = (cid:90) (cid:48) dω − τ / (2 √ π (cid:96) ) ω − ω n,k + ˙ ι c τ / (4 (cid:96) ) a α ( ω, (cid:126)k ) , (41) b α ( k ) = (cid:90) (cid:48) dω a α ( ω, (cid:126)k ) [ δ ( c k z − ω ) (42)+ lim ξ → + c k z − ω − ˙ ι ξ ∞ (cid:88) n =0 − τ / (4 π (cid:96) ) ω − ω n,k + ˙ ι c τ / (4 (cid:96) ) (cid:35) , where the global photon operator fulfills the relations[ a α ( ω, (cid:126)k ) , a † α (cid:48) ( ω (cid:48) , (cid:126)k (cid:48) )] = c δ α,α (cid:48) δ ( ω − ω (cid:48) ) δ (cid:16) (cid:126)k − (cid:126)k (cid:48) (cid:17) , [ a α ( ω, (cid:126)k ) , a α (cid:48) ( ω (cid:48) , (cid:126)k (cid:48) )] = 0 . We show below that these operators fulfill the com-mutation relations (31) and (36), respectively. Using thesame arguments as in the previous section, i.e., the possi-bility to describe any configuration of the electromagneticfield using (30) and (35), we conclude that the expansion a α ( ω, (cid:126)k ) = (cid:88) n A n ( ω, k ) c α,n ( (cid:126)k ) + (cid:90) (cid:48) d k z B ( ω, k z ) b α ( k )(43)replaces Eq. (38) in the case of a leaky cavity, where A n ( ω, k ) = − τ / (2 √ π (cid:96) ) ω − ω n,k − ˙ ι c τ / (4 (cid:96) ) , (44a) B ( ω, k z ) = δ ( c k z − ω )+ lim ξ → + τ / (2 √ π (cid:96) ) c k z − ω + ˙ ι ξ ∞ (cid:88) n =0 A n ( ω, k ) . (44b)In order to show that c α,n ( (cid:126)k ) and b α ( k ) satisfy thecommutation relations, we first use Eqs. (41) and (44a),for which the commutator (31a) reduces to the expression δ α,α (cid:48) δ n,n (cid:48) δ (cid:16) (cid:126)k − (cid:126)k (cid:48) (cid:17) (cid:90) (cid:48) dω c | A n ( ω, k ) | . (45) FIG. 2. (Color online) (a) Schematic view of the experi-mental setup that realizes the proposed scenario. (b) Se-ries of branches which characterize the dispersion relation ω n,k = c (cid:112) k z,n + k associated with the frequency of cavityquasi-modes. (c) The total form-factor (63) for N = 1 and N = 19 being summed over the polarizations. See text fordetails. Now we use Eqs. (42) and (44b), for which the commu-tator (36a) reduces to the expression δ α,α (cid:48) δ (cid:16) (cid:126)k − (cid:126)k (cid:48) (cid:17) (cid:90) (cid:48) dω c B ( ω, k (cid:48) z ) B ∗ ( ω, k z ) . (46)It can be readily checked that the integral in (45) is equalto one, while the integral in (46) is equal to δ ( k z − k (cid:48) z )up to the contribution O ( τ ), which is negligibly smalldue to the (leaky cavity) condition τ (cid:28) III. FORM-FACTORS AND THE CAVITY-QEDPARAMETERS In the introduction, we explained that an interactingsystem of three constituents: (i) an atom coupled to (ii) aplanar resonator, and (iii) an input pulse that penetratesthe resonator from outside, can exhibit Rabi oscillationsreproducing the cavity-QED evolution. In Fig. 2(a) wedisplay the experimental setup that could realize this sce-nario. In this setup, an atom at rest is located inside theplanar resonator, while the single-photon wave packet | in (cid:105) = (cid:88) α (cid:90) d r ψ ( z ) ϕ α ( x, y ) ˜ b † α ( r ) | vac (cid:105) (47)characterized by a non-trivial spatial distribution pen-etrates the resonator at normal incidence. Here ˜ b α ( r )denotes the Fourier transform of b α ( k ), while | vac (cid:105) is thephoton field vacuum state, such that a α ( k ) | vac (cid:105) = 0.The functions ψ ( z ) and ϕ α ( x, y ) describing the spatialdistribution of the light pulse are normalized, such that (cid:90) dz | ψ ( z ) | = (cid:88) α (cid:90) dx dy | ϕ α ( x, y ) | = 1 , (48)and where | ψ ( z ) | is localized in the z > g, κ, γ ). This require-ment would establish a one-to-one correspondence to theconventional cavity-QED framework. Secondly, these pa-rameters have to depend not only on the atom and cavitycharacteristics (as in cavity-QED) but also on the lateralprofile ϕ α ( x, y ) of the single-photon pulse (47).The identification of these parameters, framed by theabove two criteria, was proposed by Koshino in Ref. [20].In this reference, the author observed that the atom-fieldcoupling extracted from the total Hamiltonian includingan atom, the light field, and the atom-cavity interactionencodes the entire triplet of parameters. Using the so-called form-factor formalism that gives a proper frame-work to isolate and study various couplings of a givenHamiltonian, Koshino calculated the cavity-QED param-eters in question and demonstrated that the atomic decayrate γ becomes considerably suppressed once the lateralprofile ϕ α ( x, y ) of the input single-photon pulse is appro-priately tailored.However, Koshino introduced four simplifying assump-tions in his framework, namely, (i) the evolution of thecoupled atom-cavity-pulse system was described by an adhoc Hamiltonian, (ii) the light field had only one (fixed)polarization, (iii) the atom was described by an aver-aged (in space) dipole, while (iv) the planar resonator accommodated only one atomic wavelength. In the gen-eralized framework we derived in the previous section,the first two simplifications have been already excluded.In this section, we avoid the remaining two assumptionsand generalize, in this way, the paper of Koshino. A. Input light-pulse coupled to the resonator The light pulse (47) can be expressed in k -space as | in (cid:105) = (cid:88) α (cid:90) d k (cid:101) ψ ( k z ) (cid:101) ϕ α ( (cid:126)k ) b † α ( k ) | vac (cid:105) , (49)where (cid:101) ψ ( k z ) and (cid:101) ϕ α ( (cid:126)k ) are the Fourier transforms of ψ ( z )and ϕ α ( x, y ), respectively, and describe the frequencydistribution of the input light pulse in k -space. Similarto Eq. (48), these two functions are normalized (cid:90) d k z | (cid:101) ψ ( k z ) | = (cid:88) α (cid:90) d(cid:126)k | (cid:101) ϕ α ( (cid:126)k ) | = 1 . (50)In the conventional approach [30, 32], a single-photonstate with a non-trivial frequency distribution is typicallygiven by the expression | (cid:105) = (cid:82) dω k G ( ω k ) b † ( ω k ) | vac (cid:105) ,where ω k is proportional to the modulus of k . In our case,however, the frequency distribution is given by (cid:101) ψ ( k z ) and (cid:101) ϕ α ( (cid:126)k ) depending on different wave-vector components,while the integration is performed over the k -space.Using (42) and (44), we express (49) in the form | in (cid:105) = 1 c (cid:90) dω (cid:101) ψ ( ω ) d † ( ω, (cid:101) ϕ ) | vac (cid:105) , (51)that is similar to the conventional single-photon state | (cid:105) shown above, and where we introduced the pulse operator d ( ω, (cid:101) ϕ ) ≡ (cid:88) α (cid:90) d(cid:126)k (cid:101) ϕ ∗ α ( (cid:126)k ) × (cid:34) ι τ (cid:114) π(cid:96) ∞ (cid:88) n =0 A ∗ n ( ω, k ) (cid:35) a α ( ω, (cid:126)k ) , (52)that depends on the lateral profile (cid:101) ϕ α . This operator,moreover, fulfills the commutation relations[ d ( ω, (cid:101) ϕ ) , d † ( ω (cid:48) , (cid:101) ϕ )] = c δ ( ω − ω (cid:48) ) ; (53a)[ d ( ω, (cid:101) ϕ ) , d ( ω (cid:48) , (cid:101) ϕ )] = 0 , (53b)and exhibits the properties d † ( ω, (cid:101) ϕ ) | vac (cid:105) = (cid:101) ψ ∗ ( ω ) | in (cid:105) , (54a) d ( ω, (cid:101) ϕ ) | in (cid:105) = (cid:101) ψ ( ω ) | vac (cid:105) . (54b)These properties suggest that d † ( ω, (cid:101) ϕ ) creates a single-photon state (49) weighted by (cid:101) ψ ∗ ( k z ), while d ( ω, (cid:101) ϕ ) an-nihilates the respective state resulting into the vacuumstate weighted by (cid:101) ψ ( k z ). In order to complete the deriva-tions in this subsection, we invert the relation (52) a α ( ω, (cid:126)k ) = (cid:101) ϕ α ( (cid:126)k ) (cid:34) − ˙ ι τ (cid:114) π(cid:96) ∞ (cid:88) n =0 A n ( ω, k ) (cid:35) d ( ω, (cid:101) ϕ ) , (55)where we used Eq. (50) along with the relation˙ ι τ (cid:114) π(cid:96) | A n ( ω, k ) | = A ∗ n ( ω, k ) − A n ( ω, k ) . (56) B. Atom coupled to the resonator In this section, we derive the Hamiltonian that governsthe evolution of the intra-cavity field coupled to a two-level atom by using the electric field (30) and the photonfield operator (41) of a leaky cavity. We consider an atomat rest inside the resonator as displayed in Fig. 2(a). Theinternal structure of the atom is completely characterizedby the states | g (cid:105) (ground) and | e (cid:105) (excited), which ful-fill the usual orthogonality and completeness relations.We recall that the cavity quasi-modes are characterizedby the frequency ω n,k that has both discrete and contin-uous contributions. These quasi-modes are grouped bybranches indexed by n as can be seen in Fig. 2(b), where ω c ≡ ω , = c k z, defines the lower cut-off frequency.We assume that the atomic transition frequency ω a isequal or above this lower cut-off frequency, such that theatom couples at least to one quasi-mode of the resonator.We choose the position of atomic center-of-mass r (cid:48) = { , , − (cid:96)/ } and switch to the Schr¨odinger picture, inwhich the electric field (30) is time-independent. In thedipole approximation, the Hamiltonian ( n = 1 , . . . , N ) H AC = −Q r · (cid:2) E (+) CP ( r (cid:48) ) + E (–) CP ( r (cid:48) ) (cid:3) = −D (cid:0) σ † + σ (cid:1) N (cid:88) α,n (cid:90) d(cid:126)k (cid:115) (cid:126) ω n,k (cid:15) (2 π ) (cid:96) × (cid:20) ˆ ρ · (cid:101) U α, C (cid:18) k z,n , (cid:126)k , − (cid:96) (cid:19) c α,n ( (cid:126)k ) + H.c. (cid:21) (57)describes the electric-dipole coupling between a two-levelatom and N cavity quasi-modes, where N is the num-ber of intersection points between ω a and the branchesof ω n,k [see Fig. 2(b)]. In the above Hamiltonian, Q is the electric charge, σ = | g (cid:105)(cid:104) e | is the atomic (excita-tion) lowering operator. We also introduced the notation (cid:104) g |Q r | e (cid:105) ≡ D ˆ ρ with D being the (real) dipole matrix el-ement of the atomic transition and ˆ ρ being the unit realvector that determines the polarization of transition.In the rotating-wave approximation, we express theHamiltonian (57) in the form H AC = (cid:126) N (cid:88) α,n (cid:90) d(cid:126)k (cid:104) λ α,n ( (cid:126)k ) σ † c α,n ( (cid:126)k ) + H.c. (cid:105) , (58) where we introduced the atom-field coupling λ α,n ( (cid:126)k ) ≡ −D (cid:114) ω n,k (cid:15) (2 π ) (cid:96) (cid:126) ˆ ρ · (cid:101) U α, C (cid:18) k z,n , (cid:126)k , − (cid:96) (cid:19) . (59)Using (24) and (58), we compose the total Hamiltonian H T = H F + H A + (cid:126) N (cid:88) α,n (cid:90) d(cid:126)k (cid:104) λ α,n ( (cid:126)k ) σ † c α,n ( (cid:126)k ) + H.c. (cid:105) (60)that governs the evolution of a coupled atom-field system,and where H A = (cid:126) ω a σ z / H T = H F + H A + (cid:126) N (cid:88) α,n (cid:90) (cid:48) dω d(cid:126)k (cid:104) λ α,n ( (cid:126)k ) × A ∗ n ( ω, k ) σ † a α ( ω, (cid:126)k ) + H.c. (cid:105) . (61)Using Eqs. (55) and (56), finally, we express the aboveHamiltonian in the (second) equivalent form H T = H F + H A + (cid:126) N (cid:88) α,n (cid:90) (cid:48) dω d(cid:126)k (cid:104) λ α,n ( (cid:126)k ) (cid:101) ϕ α ( (cid:126)k ) × A n ( ω, k ) σ † d ( ω, (cid:101) ϕ ) + H.c. (cid:3) . (62) C. Form-factors and the cavity-QED parameters Following the approach of Koshino, we define the total form-factor F T ( ω, N ) ≡ c (cid:126) (cid:88) α (cid:90) d(cid:126)k (cid:12)(cid:12)(cid:12) (cid:104) vac , e | H T a † α ( ω, (cid:126)k ) | vac , g (cid:105) (cid:12)(cid:12)(cid:12) = 1 c (cid:88) α (cid:90) d(cid:126)k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n λ α,n ( (cid:126)k ) A ∗ n ( ω, k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (63)that isolates the coupling between the atom and globalphoton field a α ( ω, (cid:126)k ).Assuming that the atomic dipole ˆ ρ lies in the planeparallel to the mirrors, i.e., ˆ ρ · ˆ z = 0, we calculate ana-lytically the total form-factor (63) F T ( ω, N ) = D ω c (cid:15) π (cid:126) c N (cid:88) n, odd (cid:18) n + ω ω c (cid:19) × (cid:18) 12 + 1 π arctan (cid:20) πτ (cid:18) ωω c − n (cid:19)(cid:21)(cid:19) (64a) ≡ D ω c (cid:15) π (cid:126) c F ◦ T (cid:18) ωω c , N (cid:19) , (64b)that has the units of frequency, while the summation over n is only over the odd and positive integer values. Wedisplay in Fig. 2(c) the total form-factor F T ( ω, N ), for0 N = 1, N = 19 and τ = 10 − . It is clearly seen that thetotal form-factor depends on the square of ω and has asteplike behavior at the points ω = n ω c ( n = 1 , , , . . . ).This figure displays the spectral mode density of a lossless(1D confined) planar resonator [7, 28, 33] and, therefore,it reveals the physical meaning of the total form-factor.The chosen value of τ = 10 − is compatible with theassumptions of Sec. II.A, by which a leaky mirror devi-ates only slightly from a perfect one. Throughout thispaper, therefore, we consider this specific value to eval-uate various expressions involving τ . We remark, more-over, that the total form-factor derived by Koshino (seeEq. (15) in Ref. [20]) is proportional to F T ( ω, ω /ω c since the (cid:107) -component of light polarization was disregarded.Although we assumed that an input light-pulse pene-trates the cavity as shown in Fig. 2(a), the total form-factor (64a) is independent of the lateral profile ϕ α ( x, y ).We observe, however, that the total Hamiltonian ex-pressed in the form (62) contains two operator pairs σ d † ( ω, (cid:101) ϕ ) and σ † d ( ω, (cid:101) ϕ ), such that σ d † ( ω, (cid:101) ϕ ) | vac , e (cid:105) = (cid:101) ψ ∗ ( ω ) | in , g (cid:105) , (65a) σ † d ( ω, (cid:101) ϕ ) | in , g (cid:105) = (cid:101) ψ ( ω ) | vac , e (cid:105) , (65b)These properties along with the atom-field coupling λ α,n ( (cid:126)k ) (cid:101) ϕ α ( (cid:126)k ) A n ( ω, k ) suggest that, for an appropri-ately tailored (cid:101) ϕ α ( (cid:126)k ) [equivalently ϕ α ( x, y )], the atom-field evolution governed by the Hamiltonian (62) can re-semble the evolution of a cavity-QED system, that is e − ˙ ι (cid:126) H T t | in , g (cid:105) = | vac , e (cid:105) , (66)where the photon field (52) plays the role of cavity pho-ton field in cavity-QED [see (1)]. In other words, if thesingle-photon pulse (47) penetrates a planar resonatorwith an atom in the ground state, then (after a certaintime interval t ) this pulse can be completely absorbed bythe atom.Furthermore, we define the second form-factor F C ( ω, (cid:101) ϕ, N ) ≡ c (cid:126) (cid:12)(cid:12) (cid:104) vac , e | H T d † ( ω, (cid:101) ϕ ) | vac , g (cid:105) (cid:12)(cid:12) , (67)to which we refer below as the cavity form-factor, sinceit contains the coupling between an atom and the ( (cid:101) ϕ α -dependent) photon field (52) that might reproduce thecavity-QED evolution (66). By inserting the total Hamil-tonian (62) into the expression (67), we readily obtain F C ( ω, (cid:101) ϕ, N ) = 1 c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n,α (cid:90) d(cid:126)k λ α,n ( (cid:126)k ) (cid:101) ϕ α ( (cid:126)k ) A n ( ω, k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (68)The cavity-QED like behavior (66), exhibited by theatom-cavity-pulse system with an appropriate inputpulse, suggests that the cavity form-factor (68) plays therole of the spectral mode density corresponding to thecompletely (3D) confined light. In a cavity-QED system with reasonable small losses, in turn, this density pro-duces a resonance peak centered around ω ◦ and describedby the Lorentzian L ( ω ) = κ π g ( ω − ω ◦ ) + κ / , (69)where its area and the half-width are identified with g and κ , respectively [7, 28, 33]. Using this analogy andprovided that the cavity form-factor resembles a sharplypeaked resonance, we identify the atom-field couplingstrength with the expression g ( (cid:101) ϕ, N ) ≡ (cid:18)(cid:90) dω F C ( ω, (cid:101) ϕ, N ) (cid:19) , (70)while the cavity relaxation ratio κ ( (cid:101) ϕ, N ) is identified withthe half-width of F C ( ω, (cid:101) ϕ, N ).We notice that in contrast to the total form-factor (63),the modulus in (68) is moved outside the integral. Withthe help of Cauchy-Schwarz inequality, this feature leadsto the relation F C ( ω, (cid:101) ϕ, N ) ≤ F T ( ω, N ) . (71)Since the resonance peak (69) describing the spectralmode density of a cavity-QED system is below the(quadratically growing) curve given by Eq. (64a) and cor-responding to the spectral mode density of a planar res-onator, both above form-factors are in perfect agreementwith the relation (71) and the present discussion.The above relation suggests the third form-factor, F N ( ω, (cid:101) ϕ, N ) ≡ F T ( ω, N ) − F C ( ω, (cid:101) ϕ, N ) , (72)to which we refer below as the non-cavity form-factor,since it gives the difference between the spectral modedensities of (i) the (1D confined) atom-cavity system and(ii) the atom-cavity-pulse system that behaves as a (3Dconfined) cavity-QED system with losses. Since the cav-ity form-factor (68) satisfies the relation (71) and encodes g ( (cid:101) ϕ, N ) and κ ( (cid:101) ϕ, N ), we identify the expression (72) withthe atomic decay rate, γ ( ω, (cid:101) ϕ, N ) ≡ F N ( ω, (cid:101) ϕ, N ) . (73)In accordance with the two criteria we formulated inthe beginning of this section, we defined three form-factors characterizing the coherent and incoherent partsof evolution of the coupled atom-cavity-pulse system. Be-ing defined in a similar fashion as in the conventionalcavity-QED, these form-factors enable us to compare theoverall performance of our setup to an arbitrary cavity-QED system. We recall that the input state (51) is asingle-photon state with a non-trivial frequency distribu-tion (cid:101) ψ ( ω ), where the parameter ω contributes to all theform-factors, while the lateral profile (cid:101) ϕ α ( (cid:126)k ) contributesonly to the cavity and non-cavity form-factors. In or-der to enhance the atom-field interaction, in the nextsection, we determine the optimal frequency distribution (cid:101) ψ opt and the optimal lateral profile (cid:101) ϕ opt α , for which theatomic decay rate vanishes.1 FIG. 3. (Color online) (a) The cavity (solid curve) and thetotal (dashed curve) form-factors for N = 1 and f = ω a .The cavity form-factor resembles a nice Lorentzian boundedby the total form-factor. (b) The non-cavity form-factor for N = 1 and f = ω a . See text for details. IV. ANALYSIS OF LATERAL PROFILES In the previous sections, we identified the cavity-QEDparameters which characterize both coherent and inco-herent parts of the evolution of a coupled atom-cavity-pulse system with losses. We also suggested that thesetup displayed in Fig. (2)(a) can reproduce the cavity-QED evolution once an appropriately tailored single-photon pulse and a proper frequency distribution are pro-vided at the input. In this section, we demonstrate thata coupled atom-cavity-pulse system behaves as a cavity-QED system and we evaluate the cavity-QED parametersby considering several predefined pulses. A. Optimal spatial distribution Before we evaluate cavity-QED parameters for a pre-defined single-photon pulse, we determine first (cid:101) ϕ opt α and (cid:101) ψ opt , for which the atomic decay rate vanishes. Accord-ing to the definition (73), we require the vanishing of theleft part of (72). This leads to the equation F T ( ω, N ) = F C ( ω, (cid:101) ϕ opt α , N ) , (74) which is an extreme case of Eq. (71). Using Eq. (50) andthe cavity form-factor (68), we solve the above equationfor the lateral profile. The obtained solution (cid:101) ϕ opt α ( (cid:126)k , f, N ) = N (cid:88) n λ ∗ α,n ( (cid:126)k ) A ∗ n ( f, k ) (cid:112) c F T ( f, N ) , (75)fulfills Eq. (50) and depends on the parameters f and N .We insert the above optimal profile back into Eq. (68)and obtain the optimal cavity form-factor F opt C ( ω, f, N ) = 1 c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n,α (cid:90) d(cid:126)k λ α,n ( (cid:126)k ) × (cid:101) ϕ opt α ( (cid:126)k , f, N ) A n ( ω, k ) (cid:12)(cid:12)(cid:12) . (76)Considering, as before, that the atomic dipole ˆ ρ lies inthe plane parallel to the mirrors, we calculate analyticallythe optimal cavity form-factor (76), which takes the form F opt C ( ω, f, N ) = D ω c (cid:15) π (cid:126) c (cid:16) τ π (cid:17) (cid:12)(cid:12)(cid:12) F ◦ C ( ωω c , fω c , N ) (cid:12)(cid:12)(cid:12) F ◦ T (cid:16) fω c , N (cid:17) , (77)where F ◦ T ( u, N ) has been defined in (64b), while F ◦ C ( u, v, N ) = N (cid:88) n, odd n ˙ ι π + 4 π arctanh (cid:20) π ( n − u + v ) π ( v − u )+˙ ι τ (cid:21) π ( v − u ) + ˙ ι τ +˙ ι π v + π ( u + v ) arctanh (cid:20) π ( n − u + v ) π ( v − u )+˙ ι τ (cid:21) π ( v − u ) + ˙ ι τ . (78)In Fig. 3(a), we display F opt C ( ω, ω a , 1) by a solid curvethat corresponds to the situation, in which the resonatoraccommodates just one wavelength associated with theatomic transition frequency, that is (cid:96) = c π/ω a . This im-plies that only one quasi-mode couples to the atom, thatis ω a = ω c and N = 1. It is clearly seen that the solidcurve resembles a nice Lorentzian, while the peak of thisLorentzian is bounded by the total form-factor F T ( ω, f in Eq. (77), and namely, this parameter setsthe central frequency of the resulting Lorentzian (solidcurve). In Fig. 3(b), furthermore, we display the non-cavity form-factor F T ( ω, − F opt C ( ω, ω a , 1) that is iden-tified with the (optimal) atomic decay rate γ ◦ ( ω ). It canbe clearly seen that the atomic decay is efficiently sup-pressed in the region ω ≤ ω c . This restriction, in turn,suggests the profile of the frequency distribution (cid:101) ψ opt ( ω ).2We conclude that our system in Fig. 2(a) behaves as acavity-QED system once the optimal single-photon pulse | opt (cid:105) = (cid:88) α (cid:90) d k (cid:101) ψ opt ω ◦ ( k z ) (cid:101) ϕ opt α ( (cid:126)k , ω a , b † α ( k ) | vac (cid:105) is provided at the input, where (cid:101) ψ opt ω ◦ ( ω ) can be modeledby a narrow-band Gaussian distribution with the centralfrequency ω ◦ , such that ω ◦ ≤ ω c . Without this inputpulse, the spectral mode density describes an atom beingweakly coupled to the photon field confined in a planarresonator as seen Fig. 2(c) [18, 19].We assume now that the central frequency ω ◦ matchesthe atomic transition frequency ω a (= ω c ) and we calcu-late g ◦ , κ ◦ , and γ ◦ ( ω a ) using the atomic data λ a = 852 nm ; D = 4 . Q a , (79)which correspond to the D -transition of a Cesium atom[34], and where a is the Bohr radius. This atomic data,along with the Lorentzian (69) plotted in Fig. 3(a), yieldsthe cavity-QED parameters( g ◦ , κ ◦ , γ ◦ ( ω a )) = 2 π (cid:0) , , . − (cid:1) MHz . (80)We see that the cavity relaxation rate oversteps notablythe atom-field coupling strength, while the atomic decayrate is negligibly small if compared to both g ◦ and κ ◦ .The values (80) are obtained in the case when the res-onator accommodates just one atomic wavelength, suchthat the atom is coupled to one single cavity quasi-mode,or equivalently, ω a = ω c and N = 1. We stress that thesuppression of atomic decay for ω ◦ < ω c in a resonator ac-commodating just one atomic wavelength was expected,since there are no available quasi-modes below the cavitycut-off frequency to which an input pulse can couple inorder to facilitate the atomic emission. We show below,however, that the inhibition of atomic emission occurs inour setup even for ω ◦ > ω c in a resonator that accom-modates more than one atomic wavelength. This resultcannot be explained by the lack of available cavity quasi-modes and constitutes a peculiar feature of the coupledatom-cavity-pulse system shown in Fig. 2(a).In order to proceed, we consider the resonator thataccommodates three atomic wavelengths, such that theatom is coupled to three quasi-modes, or equivalently, ω a = 3 ω c and N = 3. In Figs. 4(a) and (b), we displaythe cavity (solid curve) and non-cavity form-factors, re-spectively. As in the previous case, the cavity form-factorresembles a nice Lorentzian bounded by the total form-factor (dashed curve), while the atomic decay rate is sup-pressed inside a small window centered at ω = 3 ω c . Us-ing the atomic data (79) along with the Lorentzian (69)plotted in Fig. 4(a), we calculate g ◦ , κ ◦ , and γ ◦ ( ω a ),which take the values( g ◦ , κ ◦ , γ ◦ ( ω a )) = 2 π (cid:0) , , . − (cid:1) MHz . (81)If we compare these values to (80), we conclude thatthe cavity relaxation rate oversteps slightly the coupling FIG. 4. (Color online) (a) The cavity (solid curve) and thetotal (dashed curve) form-factors for N = 3 and f = 3 ω c .The cavity form-factor resembles a nice Lorentzian boundedby the total form-factor. (b) The non-cavity form-factor for N = 1 and f = 3 ω c . (c) Cavity-QED parameters g ◦ N and κ ◦ N as functions of N . See text for details. strength g ◦ , while the atomic decay rate γ ◦ ( ω a ) remainsnegligibly small.To reveal the dependence of g ◦ and κ ◦ on N , weconsider the case when the resonator accommodates N atomic wavelengths, that is (cid:96) = N c π/ω a . This impliesthat the atom is coupled to N cavity quasi-modes, suchthat ω a = N ω c . With this in mind, the optimal pulse | opt N (cid:105) = (cid:88) α (cid:90) d k (cid:101) ψ opt ω a ( k z ) (cid:101) ϕ opt α ( (cid:126)k , ω a , N ) b † α ( k ) | vac (cid:105) penetrates the resonator, where the central frequency ω ◦ ω = ω a . In Fig. 4(c), we display g ◦ N / π (dashed curve)and κ ◦ N / π (solid curve) as functions of N , where thecavity length is bounded by N = 100. This restric-tion corresponds to the length of typical macroscopicresonators used in cavity-QED experiments in the op-tical domain. It is clearly seen that in the region N > 10, the cavity relaxation rate becomes slightly smallerthan the respective g ◦ N parameter leading, therefore, tothe atom-field evolution characterized by the inequality g ◦ N > κ ◦ N (cid:29) γ ◦ N ( ω a ). This regime ensures that the en-ergy exchange in the coupled atom-field system developsfaster than the losses due to the cavity relaxation andthe atomic decay. We remark that one reason, why thecurves in Fig. 4(c) drop with growing N , is the fact thatthe cut-off frequency that appears in Eqs. (63) and (77)drops with growing (cid:96) , which itself is proportional to N .To summarize this section, we determined the opti-mal lateral profile of the input pulse that ensures van-ishing of the non-cavity form-factor identified with theatomic decay rate. Using the cavity form-factor asso-ciated with the optimal input pulse | opt N (cid:105) , we studiedthe dependence of cavity-QED parameters on the cav-ity length that is proportional to the number of cavityquasi-modes coupled to the atom. We confirmed thatthe atomic decay rate becomes dramatically suppressedonce the central frequency of pulse matches the atomictransition frequency. In contrast to the atomic decay for N = 1, which is suppressed for a rather large window as-sociated with frequency distribution (cid:101) ψ opt ( ω ), the respec-tive window for N > B. Hermite-Gaussian beam In the previous section, we exploited the vanishing ofthe atomic decay rate in order to determine the optimalinput pulse | opt N (cid:105) that implies a dramatic suppressionof atomic decay rate. For an appropriately large cavitylength, moreover, this optimal pulse leads to an atom-field evolution with g ◦ N > κ ◦ N (cid:29) γ ◦ N ( ω a ). Although theparameter window for N > 1, in which γ ◦ N ( ω a ) becomesefficiently suppressed, is rather small to be accessible inpractice, the results we obtained provide us with the rel-evant insights about the cavity-QED like behavior of thecoupled atom-cavity-pulse system shown in Fig. 2(a).Apparently, the lateral profile (75) has a complicatedshape that makes the experimental generation of the re-spective spatial profile ϕ opt α ( x, y, f, N ) very challenging.This conclusion along with a small parameter windowfor N > 1, in which the atomic decay becomes sup-pressed, suggest us to consider a specific input pulse thatcan be easily tailored in an experiment. In this section,we consider the Hermite-Gaussian beams TEM , andTEM , of the waist w , which we identify with the (cid:107) and ⊥ polarization-components of ϕ Gα ( x, y, w ), respectively. FIG. 5. (Color online) The cavity form-factor (solid curve)and the associated Lorentzian (dashed curve) for N = 1 and(a) w = 100 µ m, (b) w = 500 µ m. See text for details. Using these beams, we analyze the cavity-QED param-eters as functions of w and the number of quasi-modescoupled to the atom, N .We observe that the dependence in (75) on the radialpart of (cid:126)k [see Fig. 1(b)] poses the main difficulty con-cerning the generation of this lateral profile in practice.The dependence on the angular part of (cid:126)k , in contrast, issimple and is encoded in the atom-field coupling (59) λ (cid:107) ,n ( (cid:126)k ) = λ ◦(cid:107) ,n ( k ) cos ϑ ; λ ⊥ ,n ( (cid:126)k ) = λ ◦⊥ ,n ( k ) sin ϑ , where the dipole orientation assumption ˆ ρ · ˆ z = 0 andthe explicit form of mode functions (18) have been used.Motivated by this simple angular dependence (preservedby the Fourier transform), we suggest the identification ϕ G (cid:107) ( x, y, w ) = 1 √ F ( x, w ) F ( y, w ) [TEM , ] ; (82a) ϕ G ⊥ ( x, y, w ) = 1 √ F ( x, w ) F ( y, w ) [TEM , ] , (82b)where F n ( z, w ) = (cid:18) π (cid:19) (cid:114) n + n ! w H n (cid:16) zw (cid:17) e − z w . (83)The lateral profiles (82) are the simplest and experi-mentally most feasible beams which exhibit the same an-gular dependence in physical space as the Fourier trans-form of (cid:101) ϕ opt α ( (cid:126)k , f, N ). These beams depend on the waist4 FIG. 6. (Color online) (a) The cavity form-factor (solid curve) and the associated Lorentzian (dashed curve) for N = 5 and w = 500 µ m. The last peak resembles an almost ideal Lorentzian. (b), (c), and (d) Cavity-QED parameters g Gw (dashed curve), κ Gw (solid curve), and γ Gw (dotted curve) as functions of w for N = 1, 15, and 29, respectively. See text for details. w and fulfill the normalization condition (48). The cor-responding input pulse, therefore, takes the form | HG (cid:105) = (cid:88) α (cid:90) d k (cid:101) ψ opt ω ◦ ( k z ) (cid:101) ϕ Gα ( (cid:126)k , w ) b † α ( k ) | vac (cid:105) , (84)where (cid:101) ψ opt ω ◦ was defined in the previous subsection, while (cid:101) ϕ Gα ( (cid:126)k , w ) = w k ˙ ι √ π e − w k (cid:0) cos [ ϑ ] δ α, (cid:107) + sin [ ϑ ] δ α, ⊥ (cid:1) (85)satisfies the normalization condition (50). We insert thislateral profile into Eq. (68) and obtain F GC ( ω, w, N ) = 1 c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n,α (cid:90) d (cid:126)k λ α,n ( (cid:126)k ) (cid:101) ϕ Gα ( (cid:126)k , w ) A n ( ω, k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (86)which becomes after the evaluation F GC ( ω, w, N ) = D ω c (cid:15) π (cid:126) c (cid:18) τ w ω c π c (cid:19) × (cid:12)(cid:12)(cid:12)(cid:12) F • C (cid:18) ωω c , w ω c c , N (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (87)with the notation F • C ( u, v, N ) = N (cid:88) n sin (cid:20) πn (cid:21) × (cid:90) ∞ n (cid:112) s ( s − n )( n + s ) e − v ( s − n ) dsu − s − ˙ ι τ / (4 π ) . (88) We recall that only in the case when the cavity form-factor resembles a sharply peaked function, it plays therole of spectral mode density in a cavity-QED systemwith losses. We have checked that, in contrast to theoptimal cavity form-factor (77), the form-factor (87)yields only deformed Lorentzians for small beam waists w . However, the larger the waist we consider, the lessdeformed the peaks we obtain. Considering the res-onator that accommodates only one atomic wavelength,for instance, in Figs. 5(a) and (b) we display F GC ( ω, w, w = 100 µ m and 500 µ m, respectively,where the condition ω c = ω a along with the atomicdata (79) have been used. The dashed curves depict theLorentzians obtained as the best fit to the respective solidcurves. It is seen that the solid curve in Fig. 5(a) is no-tably deformed with regard to the dashed one, while bothcurves in Fig. 5(b) almost coincide.We have checked, furthermore, that the total form-factor gives the major contribution to the atomic decayrate γ Gw ( ω ) = F T ( ω, − F GC ( ω, w, ω < ω c [see Fig. 2(c)]. Using the Lorentzians (dashedcurves) from Figs. 5(a) and (b), we calculate the cavity-QED parameters (cid:0) g Gw , κ Gw , γ Gw (0 . ω a ) (cid:1) = 2 π (cid:0) , , . − (cid:1) MHz , (89a) (cid:0) g Gw , κ Gw , γ Gw (0 . ω a ) (cid:1) = 2 π (cid:0) , , . − (cid:1) MHz , (89b)for the beam waists w = 100 µ m and 500 µ m, respectively,where the central frequency of input pulse is slightly5detuned from the atomic transition frequency, that is ω ◦ = 0 . ω a . These parameters suggest that the cavityrelaxation rate drops for a larger waist of the beam, whilethe atomic decay rate remains negligible if compared toother parameters. We stress, however, that a large waist w of the input beam requires a large surface size of theresonator which, from an experimental point of view, islikely incompatible with a small-volume resonator thataccommodates just one atomic wavelength.Before we turn to a big-volume resonator that accom-modates N atomic wavelengths, we remark that, in con-trast to the optimal form-factor (77) that produces a sin-gle peak at ω = f , the form-factor (87) produces a seriesof peaks at ω = n ω c ( n = 1 , , , . . . ), which resemble niceLorentzians only for large waists w . To illustrate this fea-ture, we consider the resonator that accommodates fiveatomic wavelengths, that is ω a = 5 ω c and N = 5. InFig. 6(a), we display F GC ( ω, µ m , 5) (solid curves) us-ing the atomic data (79). As in the previous Figure, thedashed curves depict the Lorentzians obtained as the bestfit to the respective solid curves.It can be seen that the cavity form-factor producesthree different peaks at ω/ω c = 1 , , and 5, while only thepeak at ω = ω a resembles an almost perfect Lorentzian.The three Lorentzians (dashed curves) in Fig. 6(a) yield (cid:0) g Gw , κ Gw , γ Gw (0 . ω c ) (cid:1) = 2 π (cid:0) , , . − (cid:1) MHz , (90a) (cid:0) g Gw , κ Gw , γ Gw (3 . ω c ) (cid:1) = 2 π (6 , , . 8) MHz , (90b) (cid:0) g Gw , κ Gw , γ Gw (4 . ω c ) (cid:1) = 2 π (10 , , . 8) MHz , (90c)corresponding to central frequencies of the input pulsewhich are slightly detuned from ω c , 3 ω c , and 5 ω c , re-spectively. We see that the last peak resembles not onlyan almost perfect Lorentzian, but also implies a higher g Gw and a smaller cavity relaxation rate κ Gw than thoseparameters associated with the other two peaks. Theatomic decay rate γ Gw , in contrast, increases due to themajor contribution of the total form-factor that vanishesin the region ω < ω c .After we pointed out the main features of the cavityform-factor (87) for N = 1 and N = 5, let us consider aresonator that accommodates N = 1, 15, and 29 atomicwavelengths and attempt to reveal the dependence ofcavity-QED parameters on the beam waist w . Althoughwe noticed that a large waist of the input beam is likelyincompatible with a small cavity size, for completeness,we include the case N = 1 in our considerations. Fromthe case N = 5 analyzed above, we learned that theform-factor produces ( N + 1) / g Gw / π (dashed curve), κ Gw / π (solid curve), and γ Gw / π (dotted curve) as func-tions of w for the above mentioned three values of N . Although in all three figures the cavity relaxation rateis efficiently suppressed for large w , it still remains no-tably higher than the atom-field coupling strength g Gw .The atomic decay rate that is negligibly small for N = 1oversteps slightly g Gw for N > 1, which is in agreementwith the observations we already made [see (90)]. It isclearly seen, furthermore, that the atom-field evolutionfor N = 1 implies κ Gw > g Gw (cid:29) γ Gw (0 . ω a ). For N > w , in contrast, the atom-field evolution implies κ Gw > γ Gw (0 . ω a ) > g Gw , while for a reasonably high w the same evolution implies γ Gw (0 . ω a ) > κ Gw > g Gw .To summarize this section, we considered the Hermite-Gaussian input pulse (84) instead of the optimal pulse | opt N (cid:105) . Using the cavity form-factor (87) associated withthis (experimentally feasible) input pulse, we studied thedependence of cavity-QED parameters on the beam waistand the number of cavity quasi-modes coupled to anatom. In contrast to the results we obtained in the previ-ous section, the atomic decay rate becomes dramaticallysuppressed only for N = 1 and the central frequencythat is slightly detuned from the atomic transition fre-quency. For N > 1, however, the atomic decay ratebecomes non-negligible and it oversteps the atom-fieldcoupling strength, while for a reasonably large waist ofthe beam, the atomic decay rate oversteps both the atom-field coupling strength and the cavity relaxation rate. Weconclude, therefore, that an input beam that reproducesonly the angular part of the optimal lateral profile (cid:101) ϕ opt α ,is insufficient to achieve the cavity-QED evolution, suchthat the atom-field energy exchange develops faster thanthe losses due to the cavity relaxation and the atomicdecay. V. SUMMARY AND OUTLOOK In this paper, we generalized the framework of Ref. 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