Regimes of strong light-matter coupling under incoherent excitation
RRegimes of strong light-matter coupling under incoherent excitation
E. del Valle
Physikdepartment, TU M¨unchen, James-Franck-Str. 1, 85748 Garching, Germany
F. P. Laussy
Walter Schottky Institut, Technische Universit¨at M¨unchen, Am Coulombwall 3, 85748 Garching, Germany (Dated: November 7, 2018)We study a two-level system (atom, superconducting qubit or quantum dot) strongly coupled tothe single photonic mode of a cavity, in the presence of incoherent pumping and including detuningand dephasing. This system displays a striking quantum to classical transition. On the groundsof several approximations that reproduce to various degrees exact results obtained numerically, weseparate five regimes of operations, that we term “linear”, “quantum”, “lasing”, “quenching” and“thermal”. In the fully quantized picture, the lasing regime arises as a condensation of dressedstates and manifests itself as a Mollow triplet structure in the direct emitter photoluminescencespectrum, which embeds fundamental features of the full-field quantization description of light-matter interactions.
I. INTRODUCTION
The strong coupling regime is the ultimate limit oflight-matter interaction, at the level of a single quantumor a few quanta of excitations. This gave rise to the fieldof cavity Quantum Electrodynamics (cavity QED) [1],which in the recent years has blossomed in a large varietyof physical systems, from atoms [2] to semiconductors [3]passing by superconducting circuits [4] and nanomechan-ical oscillators [5]. A fascinating aspect of this funda-mental problem is how it bridges the gap between quan-tum and classical coherence. In the former case, one hasquantum superpositions of light and matter, entanglingphotons with the ground and excited states of the emit-ter. In the latter case, one has a classical photon field, acontinuous function of a continuous variable, fully spec-ified in all its attributes. The passage from one to theother can be tracked in the one-atom lasing transition.The one-atom laser is a concept first proposed and the-oretically studied by Mu and Savage [6], with the aim ofachieving lower thresholds for lasing. They encouragedexperimentalists to bring the number of atoms N in aconventional laser to unity. In a very high quality fac-tor cavity, the emitter reaches the strong coupling regimeat the single excitation level. They showed that in thisregime, a single incoherently excited emitter (a two levelsystem in the simplest case) can constitute the whole gainmedium and populate singlehandedly the cavity with avery large number of photons. If the spontaneous emis-sion rate of the atom into other modes than the cavity issmall, the growth in the population of photons exhibitsno threshold as a function of the pumping rate [7] anddevelops a classical (Poissonian) field statistics, thanksto the efficient periodic exchange of excitations with theatom.On the theoretical side, the single-atom laser has beenextensively studied for its qualities as a laser [6, 8–22], mostly solving its steady state numerically butalso through analytical techniques, such as the contin-ued fraction expansion [22] or phase-space representa- tion [12, 13, 22], and under different approximationssuch as the few-photon [9, 15, 20, 22] or semiclassi-cal [6, 11, 17, 21, 22] dynamics. On the other hand,the spectral properties, that is, atom and cavity photo-luminescence emission spectra, have been studied onlynumerically [6, 8, 9, 11, 14, 16, 17], although some ap-proximations for the linewidth of the cavity spectrum,that converges to a single peak and exhibits the stan-dard line-narrowing, provided useful analytical expres-sions [11, 17]. Given that the cavity field is coherentin the lasing regime, the back-action of the field on theemitter leads to the formation of a Mollow triplet in itsspontaneous emission spectrum [9, 14, 16], with similarproperties to that theoretically predicted by Mollow foran atom under resonant laser excitation [23]. This struc-ture is of a great fundamental interest as it represents apinnacle of nonlinear quantum optics.On the experimental side, the strong coupling regimeis now firmly established at the single and few photonlevel with atoms [24] as well as with artificial atoms , su-perconducting qubits [25, 26] or semiconductor quantumdots [27–29], and the one atom laser as described abovehas been realised in all these systems [30–32]. The Mol-low triplet under incoherent pumping has not yet beenreported in any experiment, one reason being that theytypically focus on the cavity field lasing properties. Onlythe original configuration proposed by Mollow, under res-onant coherent excitation, has been reported, also in allthe systems above, namely with a single atom [33], amolecule [34], a superconducting qubit [35] and a quan-tum dot [36–39].In this text, we consider in detail the quantum to clas-sical transition that leads to lasing in strong-coupling,starting from the fully quantized description of Jaynesand Cummings [40]. We show that quantization breaksdown as coherence is formed and a classical descriptionbecomes more appropriate. We provide several limitingcases that describe the system in its various regimes ofoperation. Varying degrees of agreement are afforded de-pending on the complexity of the approximation. We a r X i v : . [ qu a n t - ph ] M a y provide compact analytical approximations in the sim-plest cases and a straightforward numerical procedurethat leads to an excellent quantitative agreement. Weinclude pure dephasing, important in solid state sys-tems [41, 42], and arbitrary detuning between the modes.We focus more particularly on the lasing regime at res-onance, where we show that the emergence of a Mollowtriplet manifests classical nonlinearities of strong light-matter coupling in cavity-QED. As a whole, we showthat the lasing transition is a rich and complex one andwe hope to give a rather comprehensive view of its variouslimits.The remaining of this text is organized as follows. Asthe most striking and characteristic manifestation of las-ing in strong coupling is the Mollow triplet formed un-der incoherent pumping, we first revisit its coherent ex-citation counterpart, this is done in Sec. II, extending itto include pure dephasing and detuning. Starting fromSec. III, we turn to the case of incoherent excitation ex-clusively and obtain analytically the steady state, modepopulations and photon counting statistics (III A), thesystem full density matrix (III B) and the two-time cor-relators needed to compute the power (or luminescence)spectra (III C). We derive the expressions for the lasingproperties by applying the semiclassical approximation that we compare to other approximations that describethe transition into and out-of lasing. In Sec. IV, we putall these elements together and derive the spectra of emis-sion for both the cavity and the emitter. We analyse theresonances of the system (IV B) as well as the elastic scat-tering component (IV C). We apply again the semiclas-sical approximation to simplify the emitter spectra intoa compact closed-form expression for the Mollow tripletunder incoherent pumping (IV D). This expression is usedto explore the parameters where the Mollow can be ob-served experimentally. In Sec. V, we summarize our mainfindings. II. MOLLOW TRIPLET UNDER COHERENTEXCITATION
The analysis of light-scattering by a two-level system(representing an atomic transition) was first given byMollow [23], who reported the antibunching of the scat-tered light, as well as the spectral structure now knownas the
Mollow triplet [43]. It results in the case where astrong-beam of light, with frequency ω L , impinges on theemitter which has a natural frequency ω σ ( (cid:126) = 1). TheHamiltonian reads: H L ( t ) = ω σ σ † σ + Ω L ( e iω L t σ + e − iω L t σ † ) (1)where σ is the pseudo spin operator for the two-levelsystem and Ω L is its coupling strength with the opticallaser field. Note that the later, described by a complex ( c -number) wave E ( t ) ≈ Ω L ( e iω L t + e − iω L t ), is thus entirelyclassical. The explicit (and fast) time dependence in H L can be removed by going into a frame rotating with thelaser (∆ = ω L − ω σ ): H L = − ∆ σ † σ + Ω L ( σ + σ † ) . (2)The spontaneous decay and the pure dephasing sufferedby the emitter can be described by two Lindblad termsin the master equation: ∂ t ρ = i [ ρ, H L ] + (cid:104) γ σ L σ + γ φ L σ † σ (cid:105) ρ , (3)where L σ ( ρ ) = (2 σρσ † − σ † σρ − ρσ † σ ). The steady stateof this simple system ( ∂ t ρ = 0) can be solved analytically(see Appendix A) in terms of the emitter population andcoherence [23, 43]: n σ ≡ (cid:104) σ † σ (cid:105) = (Ω effL ) effL ) + γ σ γ σ + γ φ , (4a) (cid:104) σ † (cid:105) = i γ σ / L n σ (1 − i γ σ + γ φ ) . (4b)The effective coupling to the laser (intensity that effec-tively excites the emitter) is:Ω effL ≡ Ω L (cid:113) (cid:0) γ σ + γ φ (cid:1) . (5)At resonance, Ω effL → Ω L and (cid:104) σ † (cid:105) is pure imaginary. Theeffective laser intensity Ω effL is reduced with detuning byan amount that depends on the overlap in frequency be-tween the laser and the emitter lineshapes. As the laserhas no linewidth, the total emitter width γ σ + γ φ deter-mines the overlap. Therefore pure dephasing compen-sates for detuning by increasing this overlap.The normalized spectra of emission reads in the steadystate (that we set as t = 0): S σ ( ω ) = 1 πn σ (cid:60) (cid:90) ∞ (cid:104) σ † (0) σ ( τ ) (cid:105) e iωτ dτ . (6)Any two-time correlators can always be decomposedas a sum of complex damped exponentials [44]: (cid:104) σ † (0) σ ( τ ) (cid:105) = n σ (cid:88) p ( L p + iK p ) e − iω p τ e − γp τ , (7)where all the parameters, weights L p , K p , frequencies ω p and effective decay rates γ p , are real. They can beobtained by means of the quantum regression formula(see Appendix A). Eq. (6) leads to the spectrum, S σ ( ω ) = 1 π (cid:88) p L p γ p − K p ( ω − ω p ) (cid:0) γ p (cid:1) + ( ω − ω p ) . (8)In the case of Eqs. (1–3), the spectrum has four compo-nents, that we label p = coh, 0, +, − . The elastic scat-tering component ( p = coh) is a delta peak δ ( ω ) at thelaser frequency ( ω coh = γ coh = K coh = 0) with weight: L coh = |(cid:104) σ † (cid:105)| n σ = γ σ effL ) + γ σ ( γ σ + γ φ ) , (9) N excitations N+1excitations
Bare states Dressed states central (0) right (+)left (-)(0) (+)(-) (a)(b) (c) (d)
FIG. 1: (Color online) (a) Origin of the peaks in the Mollowtriplet: the three different frequencies ( p = 0 , ± ) are found inthe four possible transitions between two Jaynes-Cummingsrungs at high intensities. Below, different Mollow tripletswhen varying detuning (b, d) or dephasing (c), with Ω L =1 . γ σ . The symmetry of the triplet is broken only under thecombined action of detuning and dephasing. The inelastic scattering part is a triplet with a centralpeak ( p = 0) and two sidebands p = ± that carry the in-formation about the light-matter interaction. The physi-cal origin of these two peaks is in the transitions betweenthe dressed states of the Jaynes-Cummings Hamiltonianat high number of excitations [45], as shown in Fig. 1(a).The two transitions between different types of dressedstates become degenerate and form the central peak (0),while transitions between the same type of dressed statesgive rise to the side bands. The full expression of thespectrum out of resonance is too lengthy to be given here.The resonant case formula is shorter. It reads: L + iK = 12 , (10a) ω = 0 , γ = γ σ + γ φ , (10b) L ± + iK ± = (10c) γ σ ( γ σ + γ φ ) (cid:2) ± γ σ − γ φ R L (cid:3) − γ σ − γ φ γ σ + γ φ (cid:2) ± i γ σ − γ φ R L (cid:3) (cid:0) γ σ ( γ σ + γ φ ) (cid:1) ,ω ± = ±(cid:60) ( R L ) , γ ± = 3 γ σ + γ φ ± (cid:61) ( R L ) , (10d)where we have defined the (half) Mollow splitting : R L = (cid:114) (2Ω L ) − (cid:0) γ σ − γ φ (cid:1) . (11) Strong coupling, where the character of the dynam-ics of the two-time correlator is oscillating rather thandamped, is defined by the appearance of this splitting( (cid:60) ( R L ) (cid:54) = 0), that is:2Ω L > | γ σ − γ φ | / . (12)We see from Eqs. (10b) and (10d) that, beyond theexpected broadening of the lines, dephasing also shiftsthe two satellite peaks. In general, this shift brings theside peaks closer to each other, inducing the transitioninto weak coupling when γ φ > γ σ + 8Ω L . However, sur-prisingly, the maximum splitting, for a fixed γ σ and Ω L ,corresponds to a nonzero dephasing, γ φ = γ σ . In fact,the splitting remains different from zero, in the presenceof dephasing, as long as γ σ − L < γ φ < γ σ +8Ω L . If thedriving field is too weak to bring by itself the system tostrong coupling (Ω L < γ σ / < γ σ − L < γ φ ≤ γ σ ), the observed lineshape always remains single peaked.The final expression for the Mollow triplet spectrumat resonance in presence of dephasing reads: S σ ( ω ) = L coh δ ( ω ) + 12 π γ σ + γ φ (cid:0) γ σ + γ φ (cid:1) + ω +1 π (cid:32) γ σ Ω − γ σ − γ φ
16 ( γ σ + ω ) (cid:33)(cid:44)(cid:32) γ σ + ω (cid:2) ( γ σ + γ φ ) + 4 ω (cid:3) + (cid:2) γ σ ( γ σ + γ φ ) − ω (cid:3) Ω + 4Ω (cid:33) . (13)The δ scattering peak and the central peak in the firstline are neatly set apart from the two side bands in therest of the expression. This decomposition is shown inFig. 1(a).The effect on the Mollow triplet of detuning the laserfrom the emitter is shown in Fig. 1(b). It spreads the sidebands apart, with asymptotes ω σ and ω σ + 2∆, while thecentral peak, pinned at the driving laser frequency ω L ,gets suppressed. The scattering peak, not shown, ulti-mately dominates the spectrum over its incoherent partwhich fades away. In any case, the lineshape remains al-ways symmetric with respect to the laser frequency (cen-tral peak), a characteristic proper to coherent excitationas we shall see later.Out of resonance, pure dephasing has a strong qual-itative effect: it breaks the above symmetry, bringingthe spectrum towards the uncoupled case with only oneLorentzian peak at ω σ with FWHM γ σ + γ φ . Even asmall dephasing enhances considerably the emitter peakrelatively to the others, as shown in Fig. 1(c). This asym-metry becomes larger the weaker the effective laser drive,that is, for lower Ω L and larger ∆ and dephasing. One asymmetricsymmetric . . . . . FIG. 2: (Color online) Degree of asymmetry V of the Mollowtriplet under coherent excitation, as a function of detuning ∆and pure dephasing γ φ for Ω L = 1 . γ σ . It is 0 (dark blue)when the side peaks have the same intensity and 1 (brightyellow) when one of them disappears completely. can quantify the visibility of this asymmetry as the dif-ference between the intensities of the two side peaks: V = | L + − L − || L + | + | L − | . (14)This is plotted in Fig. 2, where lighter (yellow) colorsrefer to smaller degrees of symmetry (minimum whenonly one peak of the two side bands survives). As apractical application, one can measure the magnitude ofpure dephasing as a function of detuning from the degreeof asymmetry. III. ONE-ATOM LASER
In the cavity QED version of this physics, the systemis described by the Jaynes-Cummings Hamiltonian [46]: H = ω a a † a + ω σ σ † σ + g ( a † σ + aσ † ) , (15)where also the light field is quantized, through the anni-hilation operator a . The detuning is now ∆ = ω a − ω σ and we consider ω a = 0 as the reference energy. TheLiouvillian, ∂ t ρ = L ρ , to describe this system in a dissi-pative context with decay ( γ c ), incoherent pumping ( P c )and pure dephasing ( γ φ ) has the form [16]: L O = i [ O, H ] + (cid:88) c = a,σ γ c cOc † − c † cO − Oc † c ) (16a)+ (cid:88) c = a,σ P c c † Oc − cc † O − Occ † ) (16b)+ γ φ σ † σOσ † σ − σ † σO − Oσ † σ ) , (16c)where ρ is the density matrix for the combined emit-ter/cavity system. The effective broadenings of the un-coupled modes are defined by Γ a = γ a − P a and Γ σ = γ σ + P σ . A. One-time correlators: populations and statistics
The light field that was previously a classical laser fieldwas fully characterised by its intensity ( | Ω L | ) and itsfrequency ( ω L ). In the fully quantized description, corre-lations between the fields should be taken into account,namely, in the steady state: N a [ n ] = (cid:104) a † n a n (cid:105) , N σ [ n ] = (cid:104) a † n − a n − σ † σ (cid:105) , (17a)˜ N aσ [ n ] = (cid:104) a † n a n − σ (cid:105) = N raσ [ n ] + iN iaσ [ n ] , (17b)with N a [ n ] and N σ [ n ] real and i ˜ N aσ [ n ] complex in generalbut real at resonance, all others being zero. The mainobservables that characterize the system are: n a = N a [1] , n σ = N σ [1] and g (2) = N a [2] /n a . (18)In the following, we provide exact implicit expressionsfor the correlators (17), that allow an efficient numericalsolution, and derive approximate analytical expressionsfor different regimes of excitation.In the case without any direct cavity pumping, P a = 0,the field correlators admit a simple expression in terms of N a [ n ] (the general equations are given in Appendix B): N σ [ n ] = P σ N a [ n − − γ a N a [ n ]Γ σ + γ a ( n − , (19a) N iaσ [ n ] = γ a g N a [ n ] , (19b) N raσ [ n ] = − ∆ γ a N a [ n ] /g Γ σ + γ φ + γ a (2 n − . (19c)This allows to obtain a single equation for N a [ n ]:0 = − (cid:104) C eff [ n ] + nγ a Γ σ + ( n − γ a − P σ Γ σ + nγ a + 1 (cid:105) N a [ n ]+ nP σ Γ σ + ( n − γ a N a [ n − − γ a Γ σ + nγ a N a [ n + 1] . (20)where we have introduced, respectively, the effective co-operativity, effective coupling and total decoherence (inthe presence of detuning and pure dephasing): C eff [ n ] = 4( g eff [ n ]) γ a Γ T [ n ] , (21a) g eff [ n ] = g (cid:112) / Γ T [ n ]) , (21b)Γ T [ n ] =Γ σ + γ φ + (2 n − γ a . (21c)Let us note that g eff [ n ] = g for all n at resonance or whendecoherence is large as compared to the detuning. As inthe case of laser excitation, the effect of detuning is toeffectively diminish the coherent coupling, which mag-nitude is linked to the spectral overlap between modes(represented by Γ T ). Here too, the decoupling caused bydetuning can be compensated by increasing decoherence (a) (b) (c) li n ea r qu a n t u m l a s i ng qu e n c h i ng t h e r m a l FIG. 3: (Color online) Exact results computed numerically for (a) γ a n a /g , (b) n σ and (c) g (2) , as a function of pumpingfor various systems, with γ a /g ∈ { . , . , . , , . , . , } from lighter to darker shades (top to bottom in (a)). Otherparameters are γ σ = 0 . g , from Ref. [47], and P a = γ φ = ∆ = 0. The different regimes of operation are designated in (a). (decay or pure dephasing) since in this case the spec-tral overlap between the cavity and the emitter increases,bringing them effectively back to resonance. Detuningand pure dephasing only appear in the cooperativity pa-rameter C eff [ n ], as noted also by Auff`eves et al. [20]. Thissituation is similar to the case of two coupled harmonicmodes [48].We reduced the whole steady-state problem of theJaynes–Cummings with emitter pumping and decay toa single equation, Eq. (20), which is however a nonlin-ear recurrence equation with non-constant coefficients,for which there is no general method leading to an exactsolution. The problem put in this form is neverthelessquite tractable numerically and we shall in the follow-ing present various limiting cases which will spell out thephysics of this problem.Since N a [0] = 1 by definition, Eq. (20) can be easilyiterated numerically to provide N a [ n ] for all n as a func-tion only of the mean number of photons in the cavity, n a . The small n equations are the most important sincethey capture the dominant few-photons correlations. Theexact expressions for n σ and g in terms of n a are, in anycase, simple enough: n σ = P σ − γ a n a Γ σ , (22a) g (2) = Γ σ + γ a γ a n a (22b) × (cid:16) P σ n a Γ σ + 2 P σ Γ σ + γ a − Γ σ + γ φ + γ a κ σ − γ a + Γ σ Γ σ (cid:17) . They are given in terms of a key parameter of the system,the Purcell rate of transfer of population from emitter tothe cavity mode: κ σ = 4( g eff [1]) γ a . (23)This parameter is large for good cavities, when cavityQED is realized at its fullest: κ σ (cid:29) g .One can obtain n a self-consistently by truncating N a [ n ] at a sufficiently high number of photons, n max , andsolving numerically the resulting finite set of equations.This gives the results plotted in Fig. 3, for (a) γ a n a /g , (b) n σ and (c) g (2) . In the best systems ( γ a , γ σ (cid:28) g ),we distinguish five regions in these plots, which we shallinvestigate in more details in the remaining of this text:1. Linear quantum regime (or simply “ linear ”), where P σ (cid:28) γ σ , keeping the emitter essentially in itsground state, very rarely excited.2. Nonlinear quantum regime (or simply “ quantum ”),where P σ ∼ γ σ is enough to probe higher ( n >
1) rungs of the Jaynes-Cummings ladder, withoutclimbing it too highly so that few photons effectsremain the dominant ones.3.
Lasing or nonlinear classical regime , when P σ (cid:29) γ σ and the emitter population ≈ .
5, the cavity canaccumulate a great number of photons and the fieldbecomes Poissonian.4.
Self-quenching regime , when P σ (cid:39) κ σ / n σ > /
2, reducingthe number of photons.5.
Thermal regime or linear classical regime , when P σ > κ σ , the emitter is always in its excited state, n σ →
1, and the dephasing induced by the pumpdisrupts the coherent coupling, so that the numberof photons is very low again and the field becomesthermal.Similar classifications have been proposed, for instanceby Poddubny et al. [17]. In the following, we addressseveral types of approximations, that perform to varyingdegrees of accuracy depending on the level of complexityinvolved and the regime under consideration. Some of ourapproximations recover known results [6, 11, 17, 21, 22],however, this comparative analysis will give us, beyondgood approximated formulas, valuable insights into theunderlying physics. It will also allow us to determinethe pumping ranges that determine each regime. As theexpression for n σ and g (2) follow straightforwardly fromthat of n a , we will not provide their explicit form in mostof the cases analysed below. -4 -4 -4 -4 (a)(b)(c)(d) FIG. 4: (Color online) Comparison between the numericalresults for n a (from Fig. 3(a)), in thick yellowish lines, withdifferent approximated solutions. In each panel, γ a decreasesexponentially from bottom to top curves, ranking from 10 g (darkest curve) to 0 . g (lightest curve). The limiting case γ a = 0 is also shown on the left upper corner, where the di-vergence is a feature of the Jaynes–Cummings model. In (a),we superimpose the linear model (dotted red) and Jaynes-Cummings truncated at one excitation (dashed blue), givenby Eq. (24)(a). In (b), the approximated semiclassical solu-tion, Eq. (32), which provides an accurate description in thelasing regime given by Eq. (61). In (c), the thermal approx-imation, given by Eq. (39) that converges to the models in(a) at low and high pumps. In (d), the cothermal approxi-mation, given by the numerical solution of Eqs. (44). In thiscase, we extend solutions to values of γ a /g till 0 . g , out ofreach numerically. A vertical guideline marks the value of γ σ = 0 . g . Other parameters are P a = ∆ = γ φ = 0.
1. Linear model approximations
In the linear regime, where the emitter is excited withvery low probability, it can be well approximated byanother harmonic oscillator [48]. This allows to find aclosed-form analytical solution: n a ≈ κ σ κ σ (Γ σ + γ a ) + Γ σ (Γ σ + γ a + γ φ ) P σ , (24a) n σ ≈ κ σ + γ a + γ σ + γ φ κ σ (Γ σ + γ a ) + Γ σ (Γ σ + γ a + γ φ ) P σ . (24b)Equivalent expressions for n a and n σ are obtained inthe first order truncation of a continuous fraction ex-pansion of these quantities, as recently shown by Gart-ner [22]. They are also formally identical to those ob-tained by truncating the Jaynes-Cummings model at thefirst rung of excitation [20, 47, 49][65]. The only differ-ence is in the effective broadening Γ σ , that appears witha − sign in the case of coupled bosons, Γ σ → γ σ − P σ [48],and a + sign in the truncated Jaynes–Cummings model,Γ σ → γ σ + P σ . At P σ (cid:28) γ σ , the sign becomes irrelevantwith Γ σ ≈ γ σ and the population grows linearly withpumping, n a ≈ C P σ , with the slope: C = κ σ κ σ ( γ σ + γ a ) + γ σ ( γ σ + γ a + γ φ ) . (25)This agrees with the numerical results for P σ (cid:47) γ σ asshown in Fig. 4(a). In the case where γ σ = 0, we simplyhave n a ≈ P σ /γ a . Interestingly, the two models alsoprovide the right formula in the high pumping regimewhere the number of photons is low again,lim P σ →∞ n a = κ σ P σ , (26)but the emitter is completely saturated.The bosonic populations diverge at two values ofpumping (where the denominators vanish): P ± = γ σ + κ σ + γ a + γ φ (cid:32) ± (cid:115) − κ σ γ a κ σ + γ a + γ φ (cid:33) . (27)For good systems with small cavity decay rates, we have P − ≈ γ σ and P + ≈ κ σ + γ σ + γ φ . At P − , the twopopulations diverge but remain positive, a manifestationin this model that the system enters the lasing oscil-lations where both populations are “inverted”. For in-termediate pumpings, where the system is in the lasingregime P − < P σ < P + , both populations are negative,meaning that the physics in this regime is out of reachof this model. For large enough pumping, P σ > P + ,the system exits the lasing regime as n a in the bosonicmodel converges again to the exact numerical result fol-lowing Eq. (26). However, n σ remains negative for all P σ > P − , meaning that its population remains there-after inverted. The bosonic model thus provides an ac-curate description of the transition in and out of lasingthough the appearance of divergences and negative pop-ulations, as we will confirm later when linking it to thefull Jaynes–Cummings system. The truncated Jaynes-Cummings formulas remain always positive and insteadof a divergence, the two-level system becomes inverted,with n σ → T ( n ) = n na / ( n a + 1) n +1 and g (2) = 2. On the other hand, truncating the Jaynes-Cummings model at one excitation means that one ex-cludes the possibility to have two photons at a time inthe cavity, which results in perfect photon antibunching, g (2) = 0. Both models provide the exact n a solution ofthe Jaynes–Cummings model in two and opposite limit-ing cases, namely, γ a → n a = P σ γ σ − P σ , (28)with g (2) = 2, and γ a → ∞ for the truncated Jaynes–Cummings model, where n a = P σ γ σ + P σ κ σ γ a , (29)with g (2) = 0. This is seen in Fig. 4(a), where γ a /g = 10(lowest curve) is already a good approximation for ∞ ,while γ a /g = 0 (uppermost curve) is exact. In the firstlimiting case, recovered by the bosonic model, the systemaccumulates photons in the most effective way possible,which leads to a divergence in the number of photons,also in the exact Jaynes-Cummings model. The Rabidelivery of photons is so efficient in the one-atom laserthat, unless there is a leakage of photons from anotherchannel, the accumulation of photons is unbounded. Itis unlimited by the strong-coupling feed.In the opposite limit of weak and inefficient coupling,the emitter undergoes population inversion unaffectedby the cavity while the cavity gets an effective pump-ing of photons from n σ through a very weak Purcell rate κ σ [20, 22]. For γ a > g the truncated Jaynes-Cummingsmodel provides a quantitatively good agreement for theentire range of pumping, as shown in Fig. 4(a), if oneexcludes the behaviour of g (2) . This limit is also ac-counted for exactly by a series expansion in P σ , given inAppendix C, since the coupling is perturbative. In thiscase, the system goes directly from the quantum linearto the classical linear regime.One can obtain an exact expression for g (2) in the linearregime solving Eqs. (20) truncated, not at one, but at twophotons, that is, for n = 1 , N a [3] = 0). Notethat at P σ = 0, N a [ n ] = 0 for all n , but we are interestedin the limit: g (2) P σ → = lim P σ → N a [2] n a , (30) that remains different from zero, since N a [ n ] ∝ P nσ . Thismethod leads, to first order in P σ , to the populations ofEq. (24) and to: g (2) P σ → = 2 κ σ ( γ a + γ σ ) + γ σ ( γ a + γ σ + γ φ ) κ σ (3 γ a + γ σ ) + ( γ a + γ σ )(3 γ a + γ σ + γ φ ) . (31)This result is exact and valid for any set of parameters,as one can check by simply solving the equations to thenext order of truncation ( n = 1 , , N a [4] = 0). Ingeneral, g (2) ∈ [0 ,
2] in the linear regime of the Jaynes–Cummings dynamics, although in Fig. 3(c) we only show0 ≤ g (2) ≤ since we have chosen γ σ ≈
2. Semiclassical approximation
Given that, as shown in Fig. 3(c), the cavity field be-comes Poissonian in the lasing regime, we can find goodapproximated solutions for the steady state under the as-sumption T[ n ] = e − n a n na /n !, which leads to N a [ n ] = n na .To establish the lasing in strong-coupling regime, one alsoneeds a good cavity, so we shall assume γ a (cid:28) g , and highenough pumping, P σ (cid:29) γ σ , γ a . Plugging N a [ n ] = n na intoEq. (20) and solving the resulting equation for n = 1, orequivalently, imposing g (2) = 1 in Eq. (22b), gives n a ≈ Γ σ γ a (cid:16) − γ σ Γ σ − Γ σ + γ φ κ σ (cid:17) . (32)This is a very good approximation for the region wherethe cavity field behaves classically [6, 12], equivalent tosolving the n = 0 equation in Eqs. (20) [21, 22]. Theprobability of finding the emitter in its excited statereads: n σ ≈ (cid:16) σ + γ φ κ σ (cid:17) . (33)The approximated n a is plotted with blue dashed linesin Fig. 4(b) for comparison with the numerical resultsand exhibits a remarkable agreement for a large pump-ing range of practical interest (the axis is in log scale).Similar agreement is found for n σ [21].The two expressions for the populations have astraightforward interpretation. Two parameters deter-mine the populations (neglecting for the sake of sim-plicity the small correction brought by the emitter de-cay, 2 γ σ / Γ σ (cid:28) cavity feeding ” and the “ emit-ter feeding ” efficiencies, defined as F a = Γ σ / (2 γ a ) and F σ = (Γ σ + γ φ ) /κ σ , respectively. They follow as: n a ≈ F a (1 − F σ ) and n σ ≈ (1 + F σ ) / . (34)The cavity population increases linearly with pumping( n a ≈ F a ) while the emitter is half occupied ( n σ ≈ / n a and P σ in the—aptly denominated—linear regime, as givenby Eq. (25). A similar linear relationship n a ≈ C P σ also holds in the lasing regime, when P σ (cid:28) κ σ but be-yond the quantum regime, P σ > γ a , γ σ , this time with aslope C defined as: C = 12 γ a . (35)The transition between the two types of linear behaviours n a ≈ C i P σ , i = 1 →
2, and the question of the thresholdin this process is an interesting subject [14] that wouldbring us too far astray and that we postpone to anotherwork [50]. We will only comment in the present text thatthis intermediate region is the less liable to the types ofapproximation that we derive here, since it lies at thefrontier between the very few and the very large num-ber of excitations, and no good approximation can re-produce it to a high degree of accuracy other than bykeeping track of all correlations between the particles,which, being a N body type of problem, implies a nu-merical procedure. We call this intermediate region the“quantum nonlinear” or simply the “quantum” regime.When the pumping is sizable as compared to κ σ , theemitter occupation starts to show signs of saturation, in-creasing linearly with pumping, and quenching the linearincrease of the cavity population. F σ represents thereforethe degree to which the pumping succeeds in populat-ing the emitter itself, against the coherent exchange ofpopulation that feeds the cavity with efficiency F a . Themaximum population of the cavitymax( n a ) ≈ κ σ γ a (cid:0) − γ σ + 2 γ φ κ σ (cid:1) , (36)is reached at the intermediate rate P σ | max( n a ) ≈ κ σ (cid:0) − γ σ + γ φ κ σ (cid:1) . (37)The present approximated expressions are valid until n σ approaches 1, then, the self-quenching dominates the dy-namics and n a →
0, at around: P max ≈ κ σ − γ σ − γ φ , (38)that is, when the pump reaches the effective transfer rateof excitation towards the cavity mode. At this point thestatistics changes to thermal due to pump-induced deco-herence. Note that P max ≈ P + , found in the previoussubsection when analysing the bosonic model.
3. Thermal approximation
Assuming a thermal state for the cavity field, withstatistics T ( n ) = n na / ( n a + 1) n +1 and thus with N a [ n ] = n ! n na , satisfies in good approximation Eqs. (20) when the system undergoes self-quenching and gets driven into thethermal regime ( P σ > κ σ ). The n a obtained from theequation ( n = 1), or equivalently, imposing g (2) = 2 inEq. (22b), reads: n a ≈ γ a (cid:110) (Γ σ + γ a ) × (cid:115) P σ γ a / Γ σ Γ σ + γ a + (cid:2) σ + γ a + γ φ κ σ − P σ Γ σ + γ a + γ a Γ σ (cid:3) − Γ σ (cid:0) Γ σ + γ φ κ σ + 2 γ σ Γ σ − (cid:1) − γ a (cid:0) σ + γ φ κ σ + 2 (cid:1) − γ a Γ σ (cid:0) Γ σ κ σ + 1 (cid:1)(cid:111) . (39)This solution converges to the linear result of Eq. (24)to first order in P σ when P σ < γ σ . The opposite limit,well into the thermal region, is that already found withEq. (26) to first order in 1 /P σ , which provides the de-creasing tail at large pumpings, as shown in Fig. 4. Thelines converge to the same universal curve when plotting γ a n a /g in Fig. 3(a). In the intermediate region, it pro-vides a good approximation when the system is not goodenough to lase, at large dissipation rates, similarly to thetruncated Jaynes-Cummings formula. Of course, having g (2) = 2 throughout, does not, in general, represent wellthe statistics.Fig. 4(c) may seem to shown that Eq. (39) capturesthe qualitative behaviour of n a even in the lasing regime.It does not, however, provide the change into the linearlasing slope from n a ∼ C P σ to C P σ , discussed in theprevious subsection, which is a serious conceptual short-coming.In two particular cases, the thermal solution becomesexact for Eq. (20), namely, γ c = P c = 0 with c = a (pre-viously discussed) on the one hand and its counterpart c = σ on the other hand. In both cases N aσ = 0 and N σ [ n ] = ¯ n σ N a [ n −
1] with N a [ n ] = n !¯ n na , where:¯ n a = P c γ c − P c and ¯ n σ = P c γ c + P c . (40)The steady state in this case is two uncorrelated ther-mal fields at the same temperature. This is independentof the coupling strength g , which only determines thespeed at which this steady state is achieved. Althoughthe discontinuity at g = 0 might seem counterintuitive, itis physically clear that the thermal equilibrium does nototherwise depend on details of the microscopic couplings.The case P σ = γ σ = 0, that corresponds to thermal ex-citation of the cavity mode only, presents interesting as-pects which we also postpone to a future work.The case P a = 0 = γ a = 0, appearing in Fig. 4(c) asthe uppermost exact curve, diverges, as we noted pre-viously, at P σ ≥ γ σ , meaning that the system does nothave a steady state but rather its intensity grows with-out bounds. This is to be interpreted as the instabilityaccompanying the transition into lasing [12]. FIG. 5: (Color online) Comparison between the numerical g (2) (in thick lines) and the cothermal approximation (dashedlines), from Eq. (43) with n a of Fig. 4. It interpolates neatlybetween the lasing and the thermal regions (1 →
2) but alsoprovides a very good agreement in the linear regime.
4. Cothermal approximation
A very good quantitative approximation for the wholepumping range is obtained assuming that the field is in aso-called cothermal state [51], which is the field with, inaverage, n coh a coherent photons and n tha thermal photons(with a total cavity population n a = n tha + n coh a ). In thiscase, the photonic statistics reads:T[ n ] = e − n coh a / (1+ n tha ) ( n tha ) n (1 + n tha ) n +1 L n [ − n coh a n tha (1 + n tha ) ] , (41)and moments of the distribution are: N a [ n ] = n !( n tha ) n L n [ − n coh a n tha ] , (42)with L n the n -th Laguerre polynomial. The second ordercoherence function is g (2) = 2 − (cid:0) n coh a n a (cid:1) . (43)One can write the set of Eqs. (20) for n =1, 2 using theparametrization of Eq. (42):0 = (cid:104) C eff [1] + γ a Γ σ − P σ Γ σ + γ a + 1 (cid:105) n a − P σ Γ σ + 2 γ a Γ σ + γ a [2 n a − ( n coh a ) ] , (44a)0 = (cid:104) C eff [2] + 2 γ a Γ σ + 2 γ a − P σ Γ σ + 2 γ a + 1 (cid:105) [2 n a − ( n coh a ) ] − P σ Γ σ + γ a n a + 2 γ a Γ σ + 2 γ a [6 n a − n a ( n coh a ) + 4( n coh a ) ] . (44b)Equations (44) can be easily solved numerically (twocoupled nonlinear equations), extracting the physical so-lution n a , n coh a ≥
0. They yield an excellent agree-ment with the complete numerical solution as shown in Fig. 4(d) for the cavity population and Fig. 5 for g (2) . Wetook advantage of this to extend the results to cases outof reach of our numerical procedure, namely, γ a /g downto 0 .
01, shown on Figs. 4 and 5 as the dashed lines only,without the corresponding numerical calculation. Thetransitions from the linear regime to lasing and from las-ing to the thermal regime are well accounted for withinthis approximation. Although they are not exact, theyprovide a fairly good quantitative description of the en-tire pumping range for any set of parameters, particularlyof the quantum regime which eludes most approximationschemes.The transition into a thermal field can be induced byincreasing the incoherent pumping but also other sourcesof decoherence such as dissipation, pure dephasing or de-tuning. In Fig. 6, we plot the extracted n a and g (2) ,respectively, under the cothermal approximation, as afunction of pumping, when increasing detuning and puredephasing. The lasing to thermal transition is evenmore apparent in the Mandel- Q parameter, defined as Q = n a ( g (2) − g eff [7]. Adding dephas-ing, γ φ , may compensate this reduction and smoothenthe threshold up to a certain point (as it also increasesto the total decoherence rate Γ T ). This observation wasmade by Auff`eves et al. [20]. In general, the maximumintensity, max( n a ), and coherence achieved, always de-crease with both detuning and dephasing. However, de-tuning makes it occur at higher pumpings while dephas-ing does at lower ones. Pure dephasing is thus indeedmaking both transitions, from quantum to lasing andfrom lasing to self-quenching, occur at smaller pumping. B. Density matrix
We have obtained in the previous section good approx-imations for the statistics of the cavity field, T[ n ], as wellas the main quantities of interest such as n a . To computethe photoluminescence spectrum, what we will do in nextsection, we also need the full density matrix in the steadystate. In the very strong-coupling regime, we can relateit analytically to T[ n ], as we show in this section.The full statistics is most conveniently obtained fromthe master equation with elements ρ m,i ; n,j for m , n photons and i , j excitation of the emitter ( m, n ∈ N , i, j ∈ { , } ). Rather than to consider the equations ofmotion for the matrix elements directly, it is clearer andmore efficient to consider only elements that are nonzeroin the steady state. These are:p [ n ] = ρ n, n, , p [ n ] = ρ n, n, , q[ n ] = ρ n, n − , , (45)0 FIG. 6: (Color online) (a) n a , (b) g (2) and (c) Q -Mandel parameter, as a function of the pumping rate, extracted with thecothermal approximation. Detuning is fixed in each column to the value while pure dephasing increases from 0 (thick curves)to 12 g (dashed curves) in steps of 2 g . Detuning brings a threshold for lasing and while dephasing can help to reduce it, it alsomakes the system less robust to quenching. Parameters are γ a = 0 . g , P a = 0 = γ σ = 0. and correspond to, respectively, the probability to have n photons with (p ) or without (p ) excitation of the emit-ter, and the coherence element between the states | n, (cid:105) and | n − , (cid:105) , linked by the Hamiltonian Eq. (15).Both p and p are real. It is convenient to separate q into its real and imaginary parts, q[ n ] = q r [ n ] + i q i [ n ],as they play different roles in the dynamics. The equa-tions for these quantities are derived from the Liouvillianequation (16). They read (their full form is given in Ap-pendix B): ∂ t p [ n + 1] = D phot { p [ n + 1] } (46a) − P σ p [ n + 1] + γ σ p [ n + 1] − g √ n + 1q i [ n + 1] ,∂ t p [ n ] = D phot { p [ n ] } (46b) − γ σ p [ n ] + P σ p [ n ] + 2 g √ n + 1q i [ n + 1] ,∂ t q i [ n + 1] = D phot { q i [ n + 1] }− (cid:104) Γ σ + γ φ Γ σ + γ φ (cid:105) q i [ n + 1]+ g √ n + 1(p [ n + 1] − p [ n ]) , (46c)where we have separated the cavity dynamics into a su-peroperator D phot , which exact expression is given in theAppendix. If this photonic dynamics is much slower thanthe rest, that is γ a , P a (cid:28) Γ σ , g , one can solve the emit-ter dynamics separately, ignoring D phot [6, 52]. This as-sumes that the photon distribution T[ n ] = p [ n ] + p [ n ] does not change during the excitation and interactionwith the emitter that happens at a timescale, of order1 /P σ or 1 /g , much faster than the photon one, of order1 /γ a . This approximation becomes exact for the per-fect cavity, when γ a = 0 (and P a = 0, which we alreadyassumed), which is why the system admits an exact so-lution (cf. Eq. (40)). Neglecting the photon dynamicsallows one to link all the density matrix elements withthe photon statistics T[ n ]:p [ n + 1] ≈ (47a) κ a ( n + 1) (cid:16) P σ Γ σ T[ n ] + γ σ Γ σ T[ n + 1] (cid:17) + γ σ T[ n + 1]2 κ a ( n + 1) + Γ σ , p [ n ] ≈ (47b) κ a ( n + 1) (cid:16) P σ Γ σ T[ n ] + γ σ Γ σ T[ n + 1] (cid:17) + P σ T[ n ]2 κ a ( n + 1) + Γ σ , q i [ n + 1] ≈ κ a √ n + 12 g (p [ n + 1] − p [ n ]) (47c)= − κ a √ n + 12 g P σ T[ n ] − γ σ T[ n + 1]2 κ a ( n + 1) + Γ σ . where κ a = 4( g eff [1]) Γ σ + γ φ (48)is the Purcell rate of transfer of population from the cav-ity mode to the emitter (cf. Eq. (23)) and g eff follows1the definition in Eq. (21) for negligible γ a . Note thatp [ n ] + p [ n ] is not strictly equal to T[ n ], due to our ap-proximations, but the numerical discrepancy is small inthe regime of interest where the number of photons ishigh and n ≈ n + 1. In particular, the equality holds ex-actly in the aforementioned case of γ a = P a = 0, thanksto some nontrivial simplifications of the expressions whenT[ n ] is thermal. C. Two-time correlators
We now turn to the problem of the steady state opticalemission spectrum, that consists in computing two-timecorrelators of the type (cid:104) c † (0) c ( τ ) (cid:105) with c = a , σ . We canlink the two time correlators to the quantities derived inthe previous sections following an implementation of thequantum regression theorem that relies explicitly on thedensity matrix ρ : (cid:104) c † (0) c ( τ ) (cid:105) = (cid:88) k,l ρ c [ k ; l ] ( τ ) (cid:104) l | c | k (cid:105) , (49)where ρ c [ k ; l ] ( τ ) = (cid:104) c † ( | l (cid:105) (cid:104) k | )( τ ) (cid:105) is in the Schr¨odinger pic-ture, where the states evolve and operators have theirsteady state values [53]. The indices k , l go through allthe states in the system Hilbert space ( k = ( k , k ) with k = 0 , , . . . , n max for the photons and k = 0 , ρ c [ k ; l ] follow the same masterequation as the density matrix elements ρ [ k ; l ] . Since sim-ilar approximations can also be naturally implemented,this will allow us to provide closed-form solutions for thetwo-time correlators, as is detailed in Appendix D. Wegive here the main lines of the derivations and introducethe key quantities that lead to the final result. We sin-gle out, again, only the nonzero elements. For two-timecorrelators, they can be gathered in four functions of n : S i [ n ] ≡ ρ c [ n,i ; n − ,i ] , i = 0 , , n ≥ , (50a) Q [ n ] ≡ ρ c [ n, n, , n ≥ , (50b) V [ n ] ≡ ρ c [ n, n − , , n ≥ . (50c)The two-times correlators follow from these quantities(with c = σ , a respectively) as: (cid:104) σ † (0) σ ( τ ) (cid:105) = ∞ (cid:88) n =0 Q [ n ] , (51a) (cid:104) a † (0) a ( τ ) (cid:105) = ∞ (cid:88) n =0 ( √ n + 1 S [ n + 1] + √ nS [ n ]) . (51b)Each term n of these sums accounts for the transitionsbetween the rungs n + 1 and n , as in the case of sponta-neous emission [16], the first rung, being given by n = 0.The equations of motion for the quantities in Eqs. (50)are extracted from the master equation (cf. Eq. D5), andcan also be, like for the single time dynamics, separated into a slow photonic dynamics embedded in a superop-erator D phot on the one hand, and a fast emitter andcoupling dynamics on the other hand: ∂ τ S [ n + 1] = D phot { S [ n + 1] } (52a) − P σ S [ n + 1] + γ σ S [ n + 1]+ ig ( √ nV [ n + 1] − √ n + 1 Q [ n ]) ,∂ τ S [ n ] = D phot { S [ n ] } (52b)+ P σ S [ n ] − γ σ S [ n ] − ig ( √ n + 1 V [ n + 1] − √ nQ [ n ]) ,∂ τ Q [ n ] = D phot { Q [ n ] } (52c) − (cid:16) Γ σ + γ φ − i ∆ (cid:17) Q [ n ]+ ig ( √ nS [ n ] − √ n + 1 S [ n + 1]) ,∂ τ V [ n + 1] = D phot { V [ n + 1] } (52d) − (cid:16) Γ σ + γ φ i ∆ (cid:17) V [ n + 1]+ ig ( √ nS [ n + 1] − √ n + 1 S [ n ]) , for n ≥
1. After some long, but straightforward alge-bra, we can express S , [ n ] and Q [ n ] in terms of p , [ n ]and q i [ n ], which, in turn, are expressed in terms of thestatistics T[ n ]. This can be done for arbitrary parame-ters, including detuning. However, simple expressions arepossible only at resonance, where we can write Eq. (49)as: (cid:104) c † (0) c ( τ ) (cid:105) = E c + n c ∞ (cid:88) n =0 [ C c I [ n ] e − iR I [ n ] τ e − (3Γ σ + γ φ ) τ/ + R.s.i. C c O [ n ] e − iR O [ n ] τ e − (3Γ σ + γ φ ) τ/ + R.s.i.] . (53)“R.s.i” stands for “Rabi sign inversion” and is the oper-ation that consists in changing the sign of R I , O keepingall other quantities the same. The first term E c thatfactors out of the sum, is independent of τ , due to theapproximation of very large photonic lifetime. It is dis-cussed separately in section IV C. The most fundamentalquantities that arise in the above treatment are the n thmanifold inner and outer (half ) Rabi frequencies : R O , I [ n ] = (cid:114) g ( √ n + 1 ± √ n ) − (cid:16) Γ σ − γ φ (cid:17) . (54)The (half) Rabi frequency of the linear regime, R isrecovered as the particular case n = 0 with R = R O [0] = R I [0] = (cid:114) g − (cid:16) Γ σ − γ φ (cid:17) . (55)The coefficient C c I [ n ] (id. for C c O [ n ]) are complex quan-tities in general that we decompose into their real andimaginary part as C c I [ n ] = L c I [ n ] + iK c I [ n ] (56)2They are a function of R I [ n ] (and C O of R O [ n ]).As stated previously, the coefficients C c I , O [ n ] are writ-ten in terms of the system parameters and the steadystate photon distribution T[ n ] only. Their general fullexpression are too long to be given. Only for the simplestcase of γ σ , γ φ = 0 and c = σ , the expressions simplifysufficiently to be reproduced here: C σ I , O [ n ] = α σ I , O [ n ] n σ T[ n ] + β σ I , O [ n ] n σ T[ n −
1] (57)where: α σ I , O [ n ] = (cid:0) P σ (cid:1) + g (1 + n ) P σ + 8 g (1 + n ) + iP σ R I , O [ n ] × (58a) (cid:0) P σ (cid:1) − g (1 + n ∓ (cid:112) n (1 + n )) P σ + 8 g (1 + n ) ,β σ I , O [ n ] = ± g P σ (4 + 3 iP σ R I , O [ n ] ) × (58b)2 g ( (cid:112) n (1 + n ) ± n ) + P σ ( (cid:112) n (1 + n ) ∓ n )4(8 g n + P σ )(4 g + 4 g P σ (1 + 2 n ) + P σ ) , where the notations ± and ∓ associate I to the uppersign and O to the lower one. IV. MOLLOW TRIPLET UNDERINCOHERENT EXCITATION
The luminescence spectra can, like all other quanti-ties, be obtained “exactly” through numerical compu-tations [9, 16]. In our case, computing Eqs. (52) andapplying the definition of Eq. (8) for the power spec-tra, one arrives to results such as those shown in Fig. 7,where PL spectra are computed both for the cavity andthe emitter, as a function of pumping. In insets (a–f),we select various snapshots plotted in log-scale that il-lustrate the regimes discussed previously, namely: (a)shows a case from the linear regime, where only the firstrung of the Jaynes–Cummings ladder is occupied, thesystem being otherwise in vacuum. This yields the vac-uum Rabi splitting. The differences in lineshapes in thisregime have been amply discussed elsewhere [54]. In (b),one is in the quantum nonlinear regime, where the spec-trum has a complex structure featuring the many peaksthat arise from transitions between the lower rungs of theJaynes–Cummings ladder. In (c), one leaves the quantumregime, with a collapse of the quantum nonlinear peaksthat “melt” to form an emerging structure of much re-duced complexity, namely, a triplet, as seen in (d) whereone enters the lasing regime that develops in (e) and isfully formed in (f). The triplet is particularly evidentin the emitter power spectrum, where it is neatly visiblealso in a linear scale. Quenching is not shown explicitlyin this plot but is elsewhere [16].A remarkable feature of lasing in strong-coupling arisesin the form of a scattering peak, well known from Mol-low’s results (cf. Eq. (13)) where it is due to the driving -5-10 5 10 (a) (d) (b)(e) (c)(f) -5-10 5 10 -5-10 5 10 - - - - - (a)(b)(c)(d)(e)(f ) FIG. 7: (Color online) Cavity (left blue) and emitter (rightpink) spectra of emission computed numerically. As a func-tion of pumping, a transition from (a) the quantum linearregime to (f) lasing can be followed passing through (b)the quantum nonlinear regime that (c) melts into (d) and(e) structures of much reduced complexity, namely, triplets.A Mollow triplet is neatly visible in the emitter spectrum,whereas the cavity gives rise to a single narrowing line. Notethat in the cuts (a–f), spectra are displayed in log-scale. Pa-rameters are γ a = 0 . g , γ σ = 0 . g , P a = γ φ = ∆ = 0. laser scattering photons off the atom. This δ peak alsoforms in the cavity QED version of this problem, as seenin Fig. 7(f) on the emitter spectrum as the very sharppeak sitting on top of the incoherent Mollow triplet. Notethat this peak arises from a numerical computation. Itis a close counterpart of the Rayleigh scattering peak ofresonance fluorescence, although this is the coherent fieldgrown by the emitter itself in the lasing process that isthe source of the scattered photons, on the very emitterthat created them in the first place. More interestingly,this peak which, in the lasing regime, is maximum andnarrower (with the same linewidth as the cavity), formssmoothly from a similar depletion when approaching thelasing regime from below, as seen in Fig. 7(d), where anequally narrow absorption peak is carved in the emerg-ing triplet. As opposed to scattering, in this case, the3 FIG. 8: (Color online) Lasing as a condensation of dressed states: transitions energies between the dressed states when solvingnumerically the dynamics exactly are shown. The color code (online) has blue shades corresponding to a positive weight ofthe transition in the cavity emission and red shades corresponding to a negative weight. In (a) all regimes are shown over allenergies, in (b) a close-up of (a) is given for frequencies | ω p | < g lying between the vacuum Rabi splitting, in (c) a close-up of (b)is given in the lasing regime, showing an extremely complex structure of the exact solution obtained in the full-quantizationpicture, although the final result washes out completely most of this underlying information, to provide only the single narrowline of a lasing system. cavity is coherently and resonantly absorbing excitationsfrom the emitter (this depletion is pinned at the cavityenergy). As the field is building its coherence, it sucks en-ergy very efficiently from its source, until a point, shownin Fig. 7(e), where the cavity does not require such acoherent absorption to keep building in intensity. Thismarks the transition between the point where the systemis building its coherent field to the one where it is fullyformed and acting back on the emitter.These results are certainly beautiful (an animationof this transition is available in the supplementary ma-terial of Ref. [55]) but obtaining them numerically isan intensive computer task since the Hilbert space be-comes very large, whereas the final output certainly looksamenable to a simple analytical description. Solving thesystem numerically nevertheless gives access to every sin- gle transition that occur in the many rungs of the Jaynes–Cummings ladder through the dressed state decomposi-tion of Eq. (8). This yields a surprisingly complex struc-ture, shown in Fig. 8, where we plot the positions ω p where the system emits, weighted by the intensity L p of the transition such that resonances disappear withvanishing intensities. Fig. 8(a) gives an overall picturewhile a zoom of the transitions lying between the Rabidoublet is given in Fig. 8(b), showing an emerging andintricate structure, further zoomed in (c) in the lasingregion. The inner peaks form “bubbles” that open andcollapse around the origin, where lies the lasing mode.Such behaviours of the dressed states also appear in sim-pler systems such as two coupled two-level emitters in-coherently pumped, which can be solved fully analyti-cally [56]. The bubbles formation result from a complex4interplay between pumping and decay, which open newchannels of coherence flow in the system. Figure 8(a),shows clearly the satellite peaks of the Mollow triplet.Although the lines are neatly splitted the one from theother, their increasing broadening allows the formationof a smooth spectral shape in the lasing regime, that onecan follows with the naked eye from the “melting” of thequantized structure, as shown in Fig. 7. Fig. 8(a) alsoshows how the inner peaks ultimately all converge at theorigin, thereby forming the lasing mode. In the lasingin strong coupling scenario, lasing can thus be seen as aBose condensate of the dressed states [57–59]. It is fasci-nating to follow the formation of a coherent and classicalfield from a fully quantized picture, but this brings littleinsights into the actual phenomenon. Beside hinting atits underlying richness and complexity, Fig. 8 essentiallyshows that a complete and fully quantized description ofa system that is behaving basically classically is hope-lessly complicated, keeping track of a huge amount ofirrelevant details, while the behaviour is well accountedfor by a few macroscopic degrees of freedom, such as anintensity n a and a off-diagonal coherence element (cid:104) a (cid:105) .Fig. 7 thus illustrates, in one of the most fundamentalsystem of quantum optics, the breakdown of the quan-tum picture in a quantum to classical transition. Evenin the simplest and exactly solvable system, it is diffi-cult to read much, and we surmise that the condensationof dressed states in the lasing process is out of reach ofthe present understanding of dissipative quantum optics,calling for a framework such as that developed for con-servative systems [60–62].Since the intricate patterns of Fig. 8 occur at a dif-ferent energy scale than that of the observables and donot show up in the spectra, one can hope in the wakeof the excellent approximations derived previously to getsimilar analytical results also in the spectral domain. Inthe following, we derive such an approximate descriptionof the exact picture presented above [21], allowing us toread the essential physics of this transition in the lasingregime. A. Spectral decomposition
The Fourier transform in Eq. (6) of the two-time corre-lators (53) provides an approximated expression for thespectrum of emission: S c ( ω ) = (cid:60) ( E c ) n c δ ( ω ) + 1 π ∞ (cid:88) n =0 [+ L c I [ n ] γ I [ n ]2 − K c I [ n ]( ω − ω I [ n ]) (cid:0) γ I [ n ]2 (cid:1) + ( ω − ω I [ n ]) + R.s.i.+ L c O [ n ] γ O [ n ]2 − K c O [ n ]( ω − ω O [ n ]) (cid:0) γ O [ n ]2 (cid:1) + ( ω − ω O [ n ]) + R.s.i. ] , (59) where we have introduced the positions and broadenings: ω I , O [ n ] = (cid:60) ( R I , O [ n ]) , (60a) γ I , O [ n ] = 3Γ σ + γ φ − (cid:61) ( R I , O [ n ]) , (60b)so that the optical spectrum is composed of a series ofLorentzian lines at frequencies ω p with linewidths γ p andweight L p plus interference terms weighted by K p . Theselines arise from transitions between rungs n + 1 and n ofthe Jaynes-Cummings ladder. The spectra are normal-ized to unity, therefore, the incoherent part of the spectra(second and third lines) is normalized to 1 − (cid:60) ( E c ) /n c .Each transition can exhibit weak or strong coupling, thatis, the rungs being split or not into dressed states, sim-ilarly to the case with no pumping [16]. When split, allpeaks broadenings are the same, regardless of the rung: γ I [ n ] = γ O [ n ] = (3Γ σ + γ φ ) / n ] taken as Poissonian. The first case,with γ a = 0 . g , corresponds to a very good cavity well inthe strong coupling regime with the emitter, such as isrealized in circuit QED. The second case, with γ a = 0 . g ,corresponds to a less favourable situation representativeof the state of the art of systems such as quantum dotsin microcavities [63]. In both plots, the pump increasesfrom the top to the bottom curves and features the quan-tum to classical crossover. The transitions between theJaynes-Cummings rungs are first resolved individually, atlow pumping, and then merge into a Mollow triplet. Theapproximated spectrum of emission differs from the nu-merical exact result at low pumpings, since the assump-tion of much larger pumping than decay does not applyhere. The regime of validity for our approximated spec-trum and that for the observation of the Mollow tripletwith incoherent excitation is: γ a , γ σ , γ φ (cid:28) g < P σ (cid:28) κ σ . (61)Note that out of resonance, one must consider g eff [1] in-stead of g in Eq. (61). As the position of the peaks is stillwell approximated, however, this approximation providesan instructive and physically transparent picture of theMollow triplet formation. We analyse these peak posi-tions in more details in the next subsection. B. Peak positions
Let us recall the expression of the Jaynes-Cummingsquadruplets positions in the spontaneous regime, in orderto appreciate better the features brought by the incoher-ent pump [16]:Ω O , I [ n ] = (cid:60) (cid:104)(cid:115) g ( n + 1) − (cid:18) γ a − γ σ (cid:19) ± (cid:115) g n − (cid:18) γ a − γ σ (cid:19) (cid:105) . (62)5 (cid:45) (cid:45) (cid:45) Γ a (cid:144) g (cid:61) P Σ (cid:144) g (cid:61) (cid:174) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Ω (cid:144) g Γ a (cid:144) g (cid:61) P Σ (cid:144) g (cid:61) (cid:174) (cid:45) (cid:45) FIG. 9: (Color online) Approximated spectra of emission,Eq. (59), as pumping is increased and the system undergoesthe quantum-to-classical transition. The plots correspond totwo different cavity lifetimes ( γ a = 0 . g and 0 . g ). Parame-ters, not specified in the plots, are P a = γ σ = γ φ = ∆ = 0.In inset, the comparison between the approximated-analytical(thin blue) and exact-numerical (thick black) spectra for thelowest pump P σ = 0 . g . Although the approximation breaksin the quantum regime, it shows how the Mollow triplet isformed. These branches are plotted as thin dashed lines in Fig. 10(only the positive frequency ones) as a function of theirdecoherence rate Γ = γ a − γ σ . They can be exactlymapped with transitions between rungs in the dissipativeJaynes-Cummings ladder, exhibiting three regimes de-pending on their splittings: (a) at low decoherence, bothrungs—initial and final—are split (in the strong couplingregime) and we have four peaks (two shown in the fig-ure). (b) When the lower rung enters weak coupling andrecovers the bare modes, only two transitions (one in thefigure) can be seen from the upper, still split, rung. (c)At large decoherence, none of the rungs are split and alltransitions are at the origin.In the same Fig. 10, we have plotted with thick linesthe positions in the presence of pumping, Eq. (60a), È G È(cid:144)H L Ω I , O @ n D (cid:144) g FIG. 10: Transition energies (positions of the spectral peaks)as a function of decoherence | Γ | / (4 g ) from rungs n = 1 , , , γ a − γ σ , Eq. (62), thin dashed lines) and in the pres-ence of pumping (Γ = Γ σ − γ φ , Eq. (60a) thick lines). Theline starting at ω = g corresponds to the linear regime orfirst-rung-to-vacuum transition ( (cid:60) ( R ), thick black line) andis common to both cases. The curves starting above (resp.below) 1 correspond to the outer ( inner ) transitions. as a function of the corresponding decoherence rate,Γ = P σ + γ σ . The positions, paired around the centre(still only the positive half is shown), also depend onthe rung. In the linear regime, n = 0, these are simplythe two Rabi transitions ±(cid:60) ( R ), with the same struc-ture as two coupled harmonic oscillators or the sponta-neous regime that we have just described. However, for n > n a (cid:29) ω O [ n ] ≈ (cid:60) (cid:114) ng − (cid:16) Γ σ / (cid:17) (see Fig. 9).In the presence of pumping, it is no longer possibleto map exactly the position of the peaks composing thespectra with transitions from the Jaynes-Cummings lad-der. The effect of the incoherent pump, before being sostrong to close the dressed state splitting of the Jaynes-Cummings rungs, is to make it homogeneous throughoutthe ladder. Then, the inner peaks do not close becausethe rungs enter the weak coupling regime but rather be-cause the transitions coincide with the cavity frequency.In any case, the Jaynes-Cummings structure is very muchdistorted and the transitions between its rungs very muchmixed, in such a way that it is no longer possible to re-construct the ladder. Furthermore, in our derivation ofEq. (54) we have neglected the interplay between the cav-ity and emitter dynamics (by assuming them to have verydifferent time scales) and, therefore, Fig. 9 shows the iso-lated effect of the incoherent pump in contrast with the6complex features we observed in Fig. 8. C. Elastic scattering E c It is possible to give the general expression (with de-tuning, dephasing and decay) for the elastic scatteringcontribution to the spectra, for both emitter and cavityemission: (cid:60) ( E σ ) = 4 g Γ σ + γ φ ∞ (cid:88) n =0 (63a) γ σ ( κ a + 2Γ σ ) √ n q i [ n + 1] + P σ ( κ a − σ ) √ n q i [ n ] κ a (cid:104) σ + γ φ ) (cid:105) + 4Γ σ + 4Γ σ κ a (2 n + 1) , (cid:60) ( E a ) = ∞ (cid:88) n =0 κ a (cid:104) σ + γ φ ) (cid:105) + 4Γ σ + 4Γ σ κ a (2 n + 1) × (cid:110) γ σ ( κ a (4 n + 1) + 2Γ σ )(1 + n )T[ n + 1]+ P σ ( κ a (4 n + 3) + 2Γ σ ) n T[ n ] (cid:111) . (63b)Note that (cid:60) ( E a ) > (cid:60) ( E σ ) can benegative, as we pointed out in the discussion of Fig. 7.The expressions (63) support our qualitative discussions.In the cavity emission, the scattering peak is simply thedirect emission of photons through the cavity mode andtherefore, is always positive. For the emitter, however,these coherent cavity photons must “convert” into mate-rial excitations before being emitted, which implies thepossibility of interferences that can be positive or nega-tive. This change of sign is even more clear when γ σ = 0where E σ < P σ < √ g . At low pumping, this carvesa “hole” in the spectra pinned at the cavity frequency.For P σ > √ g , the delta peak is positive. In the caseof the cavity, most of the emission comes from it, andnot the de-excitation of the dressed states, that is, fromthe photons that undergo an efficient interaction with theemitter.When the two-level system is not sufficiently popu-lated/inverted (at low P σ ), the cavity has a smaller elas-tic scattering contribution. The emitter sees an inter- ference hole being carved in its spectrum, reminiscentof a Fano resonance. The emitter represents an efficientpumping for the cavity in a linear way. As we alreadymentioned, the cavity is sucking the cavity photons (at ω a ) out of the emitter. Although at very low pumping( P σ ≈ γ a ) these approximations are not valid (a deltafunction weighted negatively means a negative spectrum)and do not provide a quantitative agreement with thenumerical results, they are interesting to understand thequalitative features that are obtained numerically.When the two-level system is saturated in the self-quenching regime (at high P σ ), it is mainly in its excitedstate. The cavity spectrum tends towards the bare cav-ity emission (thermal regime). In the exact solution, thisis a Lorentzian with the cavity linewidth γ a . In our ap-proximated solution, the cavity bare emission is a deltafunction and the two-level system emission loses com-pletely the elastic scattering component because there isweak interaction with the cavity which provides it.In the lasing region, 1 < P σ < g , the triplet appearsin both channels of emission: cavity and emitter. Theemitter grows some small positive elastic scattering com-ponent on top of the triplet, as a collateral effect fromthe strong interaction with the cavity (proportional toq i [ n ]). D. Semiclassical approximation
In the lasing regime, where the n and n + 1 rungs thatare close to n a (having n a (cid:29)
1) have similar splittingsand the cavity field becomes Poissonian (coherent), sim-ple expressions—in regard to the complexity of the un-derlying machinery to obtain them—arise for the spectraof emission [21]. The variable n becomes continuous ascompared to the large intensity n a and, given that thedistribution is Poissonian (peaked at the mean value),we can consider only the case n = n a in Eq. (59). Theintegral over the distribution T[ n ] simplifies to 1. In thisregime, the inner peaks have collapsed into the centrewhile the outer remain splitted. Substituting n a (and n σ for the normalization) from Eq. (59), we obtain the finalexpression for the emitter spectrum: S σ ( ω ) = (cid:16) P σ Γ σ + γ φ + κ σ − Γ σ κ σ (cid:17) δ ( ω ) + 12 π Γ σ + γ φ (cid:0) Γ σ + γ φ (cid:1) + ω + 1(Γ σ + γ φ + κ σ )[( − P σ + Γ σ ) κ σ + ((3Γ σ + γ φ ) + 4(Γ σ − P σ ) κ σ ) ω + 4 ω ] (cid:110) − P σ κ σ (3Γ σ + γ φ + κ σ )+2 P σ Γ σ [3Γ σ +4Γ σ ( γ φ +2 κ σ )+( γ φ + κ σ )( γ φ +3 κ σ )]+4 P σ κ σ ω − (Γ σ + γ φ + κ σ )[Γ σ (3Γ σ + γ φ +2 κ σ )+(Γ σ − γ φ ) ω ] (cid:111) . (64)7 Mollow tripletobservable Mollow triplet not observable N o M o ll o w t r i p l e t FIG. 11: (Color online) Comparison between the position ofthe side peaks ( (cid:60) ( R O ), in solid blue) and those observed inthe emitter spectrum ( ω obs , in solid purple, above the filling),as a function of pumping. In dashed is shown the neck of theside peak so that the filled area delimits approximately itshalf-width. Similarly to the coherent excitation case, it is composedof an elastic scattering term (delta peak), a central peak(a Lorentzian peak with FWHM Γ σ + γ φ ) and two sidebands. When splitted, these have a FWHM (3Γ σ + γ φ ) / ω O [ n a ] ≈ (cid:60) ( R O ) where R O = (cid:114) (2 P σ − Γ σ ) κ σ − (cid:0) Γ σ + γ φ (cid:1) . (65)This is the Mollow splitting in the case of incoherentexcitation, analogous to R L in Eq. (11). It is plottedin Fig. 11. Contrary to the laser excitation, this split-ting can now close due to the pumping intensity P σ . InFig. 11, we show the domains where the Mollow triplet isclearly resolved. This naturally requires that the systemis able to enter the regime of lasing in strong-coupling,which starts at figures of about γ a /g ≈ .
1. This is thecase shown in the Figure, where we compare the Mol-low splitting, Eq. (11), with the observed splitting, rep-resented as the shaded area which is delimited by themaximum (upper boundary, solid) and the neck (lowerboundary, dashed) of a side peak. When the Mollowsplitting is maximum, decoherence has however broad-ened so much the satellite peaks that no triplet is ob-servable anymore.Applying the same procedure, for the cavity emissionwe have S a ( ω ) = δ ( ω ), that is, a purely elastic spectrumwith negligible linewidth. A more accurate approxima-tion to the FWHM of the elastic peaks in this regime,which reproduces the typical lasing cavity line narrow-ing, is given by the broadening γ L : γ L = 2 g γ a /P σ (66)as derived by Poddubny et al. [17]. (b) (c)(a) numericalanalyticalcoherent FIG. 12: (Colour online) Mollow triplet in the direct PLspectra from the emitter, S σ ( ω ) for γ a = 0 . g , P a = 0, γ σ = 0 . P σ = 7 g ( n a ≈ . n σ ≈ . L = √ n a g and also an equiv-alent emitter broadening, γ σ → γ σ + P σ . The two types ofMollow lineshapes are clearly different even though their peakpositions and broadening are equal. Also, as compared to thecoherent excitation case, the Mollow triplet under incoher-ent pumping is a resonant structure that becomes stronglyasymmetric with detuning. In Fig. 12, we show the excellent agreement betweenthe analytical approximation (in dashed lines) and theexact numerical computation (solid lines). In Fig. 12(a),the case of resonance—the one of most interest—is wherethe Mollow triplet is best observed. Its analytical ex-pression is given by Eq. (64), where we also include thescattering peak, (both as a result of the numerical pro-cedure and approximated by Eq. (66)) seen as a verysharp line, which we have truncated in the plot, as it ex-tends more than one hundred times higher than is shown.In Fig. 12(a), we also provide further evidence that theMollow triplet formed under incoherent pumping is ofa different nature than that formed under coherent ex-citation [21], by superimposing the coherent excitation8 (a) incoherent excitation (b) coherent excitation W e i gh t o f t h e p ea k s centralsidebands FIG. 13: Contributions to the Mollow triplet under (a) in-coherent and (b) coherent excitation for the three types ofpeaks: the central Lorentzian peak (constant at 1 /
2, solidline), the side bands (dashed) and the elastic scattering peak(dotted). The parameters are chosen so that the two Mollowtriplets can be compared on equal grounds, as explained inthe text, for γ a = 0 . g , P a = 0, γ σ = γ φ = 0 and ∆ = 0. Mollow triplet (dotted line). We take Ω L → √ n a g and γ σ → γ σ + P σ to compare both expressions on equalgrounds (both lineshapes remain dissimilar even whenparameters are left completely free). This substitutionmakes both types of triplet share the same position andbroadening for their three peaks. However their relativeweight is different, as shown in Fig. 13. These strongqualitative departures result in the striking differencesbetween the final spectra, although the peaks have iden-tical characteristic if taken in isolation.When the system is not at resonance, in sharp con-trast with the conventional Mollow triplet that retainedits qualitative features (cf. Fig. 1), the Mollow tripletunder incoherent pumping becomes strongly asymmet-ric, as it recovers the scenario of an anticrossing of twomodes. The out-of resonance case is studied in detailsin Fig. 14. Whereas both detuning and dephasing wereneeded to break the symmetry of the conventional Mol-low triplet, the one formed under incoherent pumping islost by detuning alone, pumping playing already the roleof dephasing. Dephasing has otherwise the expected ef-fect of smearing out and broadening the spectral features.Analytical results can also be given for the non-resonantcase when ∆ (cid:54) = 0, but, as for the conventional Mollow,they are too long to be reasonably written down. We plotit on top of the numerical solution in Figs. 12(b) and (c)where one can see the semiclassical approximation is ex-cellent there as well. V. SUMMARY AND OUTLOOKS
We investigated the steady states of the Jaynes–Cummings Hamiltonian established under the interplayof decay and incoherent pumping (of the emitter), in-cluding pure dephasing and detuning for wider generality.This is the simplest and most fundamental realization ofa fully quantized system, realized with atoms, quantumdots or superconducting qubits in a cavity. We identifyfive regimes through semi-analytical and approximated (a) (b) (c)
FIG. 14: Counterpart of Fig. 1 for the Mollow triplet underincoherent excitation. In contrast to the coherent excitationcase, detuning alone breaks the Mollow triplet. Parametersare γ a /g = 0 . γ σ /g = 0 . P a = 0 and P σ /g = 7. solutions, all confronted to exact numerical solutions.These are: i the linear quantum regime, ii the nonlin-ear quantum regime, iii the nonlinear classical regime,or lasing regime, iv the self-quenching regime and v thethermal regime. We provided closed-form analytical ex-pressions that account for most of theses regimes and asimple numerical procedure (solving a set of two couplednonlinear equations) that afford an excellent descriptionover the entire range of excitation and all the five regimesthat we have outlined. This also allows a transparentreading of the physics involved, namely, the first regimeinvolves the lowest rung of the Jaynes–Cummings ladderonly and corresponds to spontaneous emission. A linear-model (two coupled oscillators) and a truncated Jaynes–Cummings model offer two complementary views of thisregime. The quantum regime is the one where the systemstarts to climb the ladder, requiring a full record of all thequantum correlators involved. This is therefore the mostcomplicated regime from the point of view of the amountof information required to describe it, since no good ap-proximation can synthesize the dynamics of a few quanta.In very good systems, this manifests spectrally in a com-plex structure of peaks at anharmonic frequencies. Aspumping is further increased, lasing ensues which bringsback the system to a simple level of description in a semi-classical approximation. A single narrowing line in thecavity or a variation of the Mollow triplet for the emitterdescribe accurately the system. In the fully quantizedpicture, lasing appears as a condensate of dressed states,with a complex pattern that however does not manifestin the observable, showing a breakdown of the quantizedpicture in favour of a classical description.The Jaynes–Cummings model that arose as a challengefor full-field quantization [64] remains to this day a pro-ficient source of theoretical and experimental investiga-tions into the quantum realm. The realization in thelaboratory of the nontrivial quantum physics that it cov-ers will shed light on foremost issues such as the quan-9tum to classical crossover, emergence of coherence, las-ing and quantum nonlinearities. A solid understandingof the various regimes it realizes may also lead to usefuldevices and applications, from single-photon sources tolow-threshold lasers. Acknowledgements
We thank Paul Gartner for comments and discussions.We acknowledge support from the Newton InternationalFellowship scheme, Emmy Noether project HA 5593/1-1funded by the German Research Foundation (DFG) andEU FP-7 Marie Curie Initiative ‘SQOD’.
Appendix A: Quantum regression formula forcoherent excitation
Two-time correlators of the type (cid:104) σ † (0) σ ( τ ) (cid:105) canbe computed by means of the quantum regressionformula [16]. Once we find the set of opera-tors C { m,n } = σ † m σ n (with m , n ∈ { , } ) and theregression matrix M mnm (cid:48) n (cid:48) that satisfy Tr[ C { m,n } L O ] = (cid:80) { m (cid:48) n (cid:48) } M mnm (cid:48) n (cid:48) Tr[ C { m (cid:48) ,n (cid:48) } O ] for a general operator O ,then the equations of motion for the two-time correla-tors ( τ ≥
0) in the steady state ( t = 0) read: ∂ τ (cid:104) O (0) C { m,n } ( τ ) (cid:105) = (cid:88) { m (cid:48) n (cid:48) } M mnm (cid:48) n (cid:48) (cid:104) O (0) C { m (cid:48) ,n (cid:48) } ( τ ) (cid:105) . (A1)The corresponding regression matrix M is given, in thecase of a coherent and classical excitation of the emitter,as explained in section II, by: M mnmn = i ∆ L ( m − n ) − γ σ m + n ) − γ φ m − n ) , (A2a) M mn − m,n = i Ω L [ m + 2 n (1 − m )] , (A2b) M mnm, − n = − i Ω L [ n + 2 m (1 − n )] . (A2c)and zero everywhere else. We concentrate on comput-ing g (1)1 , which corresponds to setting O = σ † and hav-ing { m, n } = { , } in Eq. (A1). We obtain the followingdifferential equation: ∂ τ v L ( τ ) = − M L v L ( τ ) + A L (cid:104) σ † (cid:105) , (A3)where v L ( τ ) = (cid:104) σ † (0) σ ( τ ) (cid:105)(cid:104) σ † (0) σ † ( τ ) (cid:105)(cid:104) σ † (0) σ † σ ( τ ) (cid:105) , A L = i Ω L − (A4)and M L = − i ∆ L + γ σ + γ φ − i Ω L i ∆ L + γ σ + γ φ i Ω L − i Ω L i Ω L γ σ . (A5) The solution is v L ( τ ) = e − M L τ [ v L (0) − M − L A L (cid:104) σ † (cid:105) ] + M − L A L (cid:104) σ † (cid:105) , (A6)in terms of the steady steady values (cid:104) σ † (cid:105) and v L (0) =[ (cid:104) σ † σ (cid:105) , ,
0] (initial condition of Eq. (A4)). We also need,therefore, to compute the steady state of the system, u L = (cid:104) σ (cid:105)(cid:104) σ † (cid:105)(cid:104) σ † σ (cid:105) , (A7)which can be done, again, by means of the general for-mula in Eq. (A1). This time we take O = 1 and { m, n } = { , } , { , } , { , } and find the equation: ∂ t u L = 0 = − M L u L + A L . (A8)The solution is u L = M − L A L . It allows us to simplifyEq. (A6) further as v L ( τ ) = e − M L τ [ v L (0) − u L (cid:104) σ † (cid:105) ] + u L (cid:104) σ † (cid:105) , (A9)The final explicit solutions for the mean values and cor-relators of interest are presented in the main text. Appendix B: Derivation of the field correlators anddensity matrix
The equations of motion of the correlators in Eq. (17)can be derived from the master equation (16) by simplyapplying the general formula (cid:104) O (cid:105) = Tr( Oρ ) or from therules given by the quantum regression formula [16]: ∂ t N a [ n ] = − n Γ a N a [ n ] + n P a N a [ n −
1] (B1a)+ 2 gnN iaσ [ n ] ,∂ t N σ [ n ] = − (cid:2) Γ a ( n −
1) + Γ σ (cid:3) N σ [ n ] (B1b)+ P a ( n − N σ [ n −
1] + P σ N a [ n − − gN iaσ [ n ] ,∂ t N iaσ [ n ] = − (cid:104) Γ a n −
1) + Γ σ + γ φ (cid:105) N iaσ [ n ] (B1c)+ P a n ( n − N iaσ [ n −
1] + ∆ N raσ [ n ]+ g (2 N σ [ n + 1] + nN σ [ n ] − N a [ n ]) ,∂ t N raσ [ n ] = − (cid:104) Γ a n −
1) + Γ σ + γ φ (cid:105) N raσ [ n ] (B1d)+ P a n ( n − N raσ [ n − − ∆ N iaσ [ n ] , with n ≥ N a [0] = 1. In the steady state,with P a = 0, one can further simplify these equationsand write N aσ [ n ] and N σ [ n ] in terms of N a [ n ] to obtainEqs. (19).The master equation (16) can also be rewritten in0terms of the density matrix elements in Eq. (45) as: ∂ t p [ n ] = − (cid:2) ( γ a + P a ) n + P a (cid:3) p [ n ] (B2a)+ γ a ( n + 1)p [ n + 1] + P a n p [ n − − P σ p [ n ] + γ σ p [ n ] − g √ n q i [ n ] ,∂ t p [ n ] = − (cid:2) ( γ a + P a ) n + P a (cid:3) p [ n ] (B2b)+ γ a ( n + 1)p [ n + 1] + P a n p [ n − − γ σ p [ n ] + P σ p [ n ] + 2 g √ n + 1q i [ n + 1] ,∂ t q i [ n ] = − (cid:104) ( γ a + P a ) n − Γ a (cid:105) q i [ n ] (B2c)+ γ a (cid:112) ( n + 1) n q i [ n + 1] + P a (cid:112) ( n − n q i [ n − − Γ σ + γ φ i [ n ] + g √ n (p [ n ] − p [ n − − ∆ q r [ n ] ,∂ t q r [ n ] = − (cid:104) ( γ a + P a ) n − Γ a (cid:105) q r [ n ] (B2d)+ γ a (cid:112) ( n + 1) n q r [ n + 1] + P a (cid:112) ( n − n q r [ n − − Γ σ + γ φ r [ n ] + ∆ q i [ n ] . At resonance, the real part of the coherence distribution,q r , gets decoupled and vanishes in the steady state. As aresult, only Eqs. (B2b)–(B2d) need to be solved. When g vanishes, q i does not couple the two modes anymore, andtheir statistics become thermal, like in the boson case.Through the off-diagonal elements q i , the photon densitymatrix can vary between Poissonian, thermal (superpois-sonian) and subpoissonian distributions.In order to solve these equations in the steady state, weneglect the photonic dynamics (as explained in the maintext) and further substitute q r [ n ] = [2∆ / (Γ σ + γ φ )]q i [ n ] inthe equation for q i , and write everywhere p [ n ] as T[ n ] − p [ n ] and p [ n + 1] as T[ n + 1] − p [ n + 1], so that q r [ n ],p [ n ], p [ n + 1] do not appear explicitly in the remainingthree equations. Then, the equations read in matricialform: ∂ t u [ n ] = − M [ n ] u [ n ] + A [ n ] (B3)with: u [ n ] = p [ n + 1]p [ n ]q i [ n + 1] , A [ n ] = γ σ T[ n + 1] P σ T[ n ]0 , M [ n ] = Γ σ g √ n + 10 Γ σ − g √ n + 1 − g √ n + 1 g √ n + 1 Γ σ + γ φ + Γ σ + γ φ . (B4)The solution in the steady state is u [ n ] =( M [ n ]) − A [ n ] which gives the result of the text,Eq. (47). Note that this solution is only exact in thecase γ a = P a = 0. Appendix C: Perturbative regime of interaction inthe limit of weak coupling
Eq. (20) can be solved exactly as a series Taylor expan-sion on the pumping P σ . For this, we rewrite Eq. (20)in terms of the fraction F [ n ] = N a [ n + 1] /N a [ n ] = n a g ( n +1) /g ( n ) (for n > F [ n − F [ n ] + B n A n ) = C n A n (C1)where A n = 2 γ a Γ σ + nγ a , (C2a) B n = 1 C eff [ n ] + nγ a Γ σ + ( n − γ a − P σ Γ σ + nγ a + 1 , (C2b) C n = nP σ Γ σ + ( n − γ a . (C2c)All quantities can be expanded, or assumed to have asolution in the case of F [ n ], in power series of P σ : B n A n = ∞ (cid:88) k =0 α k [ n ] P kσ , C n A n = ∞ (cid:88) k =0 β k [ n ] P kσ , (C3a) F [ n ] = ∞ (cid:88) k =0 f k [ n ] P kσ . (C3b)A key feature of this expansion is that β [ n ] = 0 and f [ n ] = 0 for all n (we recall the linear behaviour of n a at low pumping). With this considerations, Eq. (C1) nowreads ∞ (cid:88) q =1 ∞ (cid:88) k =1 f q [ n − f k [ n ] + α k [ n ]) P k + qσ = ∞ (cid:88) t =1 β t [ n ] P tσ . (C4)We further change the sum index q , on the left hand sideof the equation, for t = q + k , so that we can get rid ofit and of the pump dependence: t (cid:88) k =1 f t − k [ n − f k [ n ] + α k [ n ]) P k + qσ = β t [ n ] . (C5)The exact solution can be found exactly and recurrentlyas: f [ n ] = β [ n + 1] α [ n + 1] , (C6a) f [ n ] = β [ n + 1] − f [ n ]( f [ n + 1] + α [ n + 1]) α [ n + 1] , (C6b) . . .f t [ n ] = β t [ n + 1] − f t − [ n ] (cid:80) t − k =1 ( f k [ n + 1] + α k [ n + 1]) α [ n + 1] . (C6c)1This method is only useful in practical terms when theeffect of the coupling is perturbative, that is, at very lowpumping or in the weak coupling regime, where only afew terms of the expansion are needed. Otherwise, inorder to reproduce non perturbative effects such as thetransition into lasing, the sum should be performed to allorders of P σ , which might not be practical numerically. Appendix D: Derivation of the two-time correlators
We can obtain the elements ρ ck ; l ( τ ) needed to computethe two-time correlators in Eq. (49), in an equivalent wayas ρ [ k ; l ] , as they follow the same master equation [53]: ∂ t ρ [ k ; l ] = (cid:80) i,j M (cid:104) k ; li ; j (cid:105) ρ [ i ; j ] . That is, we can solve ∂ τ ρ c [ k ; l ] ( t + τ ) = (cid:88) i,j M (cid:104) k ; li ; j (cid:105) ρ c [ i ; j ] ( t + τ ) . (D1)The initial conditions are given in terms of the steadystate density matrix: ρ c [ k ; l ] (0) = (cid:104) c † ( | l (cid:105) (cid:104) k | )(0) (cid:105) = (cid:88) α ρ [ k ; α ] (cid:104) α | c † | l (cid:105) . (D2)Let us be more specific by writing the formulas for thetwo correlators of interest. For the emitter spectra, wehave (cid:104) l , l | σ | k , k (cid:105) = δ l ,k δ l , δ k , , which gives: (cid:104) σ † (0) σ ( τ ) (cid:105) = n max (cid:88) n =0 ρ σ [ n, n, ( τ ) . (D3)For the cavity spectra, we have (cid:104) l , l | a | k , k (cid:105) = √ k δ l ,k δ l ,k − , which gives: (cid:104) a † (0) a ( τ ) (cid:105) = n max (cid:88) n =0 (cid:88) i =0 , √ nρ a [ n,i ; n − ,i ] ( τ ) . (D4) ρ c is obtained by solving the master Eq. (D1) in bothcases, but the initial conditions that follow in each casefrom Eq. (D2), are different. For the emitter correlator,they are ρ σ [ n,i ; m,j ] (0) = ρ [ n,i ; m, δ j, , while for the photon, ρ a [ n,i ; m,j ] (0) = √ m + 1 ρ [ n,i ; m +1 ,j ] . In the same way aswhen solving the steady state distributions, we write themaster equation for ρ c [ n,i ; m,j ] only for the elements thatwill be different from zero during the evolution with τ .We have to include all the elements that are nonzero inthe initial condition plus those that are linked to them.One can check that the nonzero elements are the same forthe initial conditions of both the cavity and the emittercorrelators, those defined in Eq. (50). They follow the equations: ∂ τ S [ n ] = − (cid:2) γ a + P a n −
1) + P a (cid:3) S [ n ] (D5a)+ γ a (cid:112) n ( n + 1) S [ n + 1] + P a (cid:112) n ( n − S [ n − − P σ S [ n ] + γ σ S [ n ] + ig ( √ n − V [ n ] − √ nQ [ n − ,∂ τ S [ n ] = − (cid:2) γ a + P a n −
1) + P a (cid:3) S [ n ] (D5b)+ γ a (cid:112) n ( n + 1) S [ n + 1] + P a (cid:112) n ( n − S [ n − P σ S [ n ] − γ σ S [ n ] − ig ( √ n + 1 V [ n + 1] − √ nQ [ n ]) ,∂ τ Q [ n ] = − (cid:104) ( γ a + P a ) n + P a (cid:105) Q [ n ] (D5c)+ γ a ( n + 1) Q [ n + 1] + P a nQ [ n − − (cid:16) Γ σ + γ φ − i ∆ (cid:17) Q [ n ]+ ig ( √ nS [ n ] − √ n + 1 S [ n + 1]) ,∂ τ V [ n ] = − (cid:2) ( γ a + P a ) n − γ a (cid:3) V [ n ] (D5d)+ γ a (cid:112) ( n − n + 1) V [ n + 1] + P a (cid:112) n ( n − V [ n − − (cid:16) Γ σ + γ φ i ∆ (cid:17) V [ n ]+ ig ( √ n − S [ n ] − √ nS [ n − . As explained in the main text, we solve these equationsby neglecting the very slow photonic dynamics. We thendefine the steady state and slow varying function X [ n ] ≡ S [ n ](0) + S [ n ](0) , (D6)and substitute S [ n ] = X [ n ] − S [ n ] and S [ n + 1] = X [ n + 1] − S [ n + 1], so that we can rewrite the equationsin a matricial form (B3): ∂ τ u [ n ]( τ ) = − M [ n ] u [ n ]( τ ) + A [ n ] (D7)with: u [ n ] = S [ n + 1] S [ n ] Q [ n ] V [ n + 1] , A [ n ] = γ σ X [ n + 1] P σ X [ n ]00 , and M [ n ] = Γ σ ig √ n + 1 − ig √ n σ − ig √ n ig √ n + 1 ig √ n + 1 − ig √ n Γ σ + γ φ − i ∆ 0 − ig √ n ig √ n + 1 0 Γ σ + γ φ + i ∆ . (D8)The solution in the steady state is u [ n ]( τ ) = e − M [ n ] τ (cid:16) u [ n ](0) − ( M [ n ]) − A [ n ] (cid:17) + ( M [ n ]) − A [ n ] . (D9)2For the emitter correlator, the initial condition derivedfrom Eq. (D2) reads: u [ n ](0) = (cid:16) σ + γ φ + i (cid:17) q i [ n + 1]0p [ n ]0 . (D10)which implies substituting X [ n ] = (cid:16) σ + γ φ + i (cid:17) q i [ n ] in A [ n ]. The final correlator is found by taking the thirdelement of the vectorial solution and summing contribu-tions from all rungs: (cid:104) σ † (0) σ ( τ ) (cid:105) = (cid:80) ∞ n =0 ( u [ n ]) , whichcorresponds to Eq. (51a).For the photonic correlator, the initial condition is: u [ n ](0) = √ n + 1p [ n + 1] √ n p [ n ] (cid:16) σ + γ φ − i (cid:17) √ n + 1q i [ n + 1] (cid:16) σ + γ φ + i (cid:17) √ n q i [ n + 1] . (D11)which implies substituting X [ n ] = √ n T[ n ] in A [ n ].The final correlator is found as: (cid:104) a † (0) a ( τ ) (cid:105) = (cid:80) ∞ n =0 (cid:104) √ n + 1( u [ n ]) + √ n ( u [ n ]) (cid:105) , which correspondsto Eq. (51b).
1. Elastic scattering term
The second line in Eq. (D9), B [ n ] = ( M [ n ]) − A [ n ] (D12)is independent of τ , due to the approximation of infi-nite lifetime γ a ≈
0. If γ a was of the order of g, Γ σ , wecould not have assumed X [ n ] to be τ -independent and,therefore, nor B [ n ]. In the regime where our approxi-mation is valid, this term gives rise to the first element in Eq. (53), E c , which turns into a delta function at thecavity frequency in the frequency domain, Eq. (59).For the emitter, the intensity of this contribution isfound from summing the third element of B [ n ], for allrungs n ≥ E σ = (cid:80) ∞ n =0 ( B [ n ]) . For the cavity case, itis found as E σ = (cid:80) ∞ n =0 (cid:104) √ n + 1( B [ n ]) + √ n ( B [ n ]) (cid:105) .
2. First rung
In principle, we must solve separately the Rabi dynam-ics of the first rung with the ground state, n = 0, havinga 2 × S [0] = 0 and V [0] = 0): u [0] = (cid:18) S [1] Q [0] (cid:19) , A [0] = (cid:18) γ σ X [1]0 (cid:19) , M [0] = (cid:18) Γ σ igig Γ σ + γ φ − i ∆ (cid:19) . (D13)For the emitter, the initial conditions are: u [0](0) = (cid:32)(cid:16) σ + γ φ + i (cid:17) q i [1]p [0] (cid:33) . (D14)In the photonic case, the initial conditions are: u [0](0) = (cid:32) p [1] (cid:16) σ + γ φ − i (cid:17) q i [1] (cid:33) . (D15)One can check that the solution for n = 0 is finally thesame as taking the limit n → n >
1. Therefore, there is no further needof separating the Rabi from the rest of rungs in our ex-pressions, although it will find simpler expressions for thequantities of interest and we may point them out. Forinstance, if γ σ = 0, in both cases, A [0] = 0 and thesolution simplifies to: u [0]( τ ) = e − M [0] τ u [0](0). [1] S. Haroche and J.-M. Raimond, Exploring the Quantum:Atoms, Cavities, and Photons (Oxford University Press,2006).[2] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod.Phys. , 565 (2001).[3] G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, andA. Scherer, Nat. Phys. , 81 (2006).[4] Y. Makhlin, G. Sch¨on, and A. Shnirman, Rev. Mod.Phys. , 357 (2001).[5] T. Kippenberg, Nature , 458 (2008).[6] Y. Mu and C. M. Savage, Phys. Rev. A , 5944 (1992).[7] H. J. C. P. R. Rice, Phys. Rev. A , 4318 (1994).[8] C. Ginzel, H.-J. Briegel, U. Martini, B.-G. Englert, andA. Schenzle, Phys. Rev. A , 732 (1993).[9] M. L¨offler, G. M. Meyer, and H. Walther, Phys. Rev. A , 3923 (1997). [10] B. Jones, S. Ghose, J. P. Clemens, P. R. Rice, and L. M.Pedrotti, Phys. Rev. A , 3267 (1999).[11] O. Benson and Y. Yamamoto, Phys. Rev. A , 4756(1999).[12] T. B. Karlovich and S. Y. Kilin, Opt. Spectrosc. , 343(2001).[13] S. Y. Kilin and T. B. Karlovich, Sov. Phys. JETP ,805 (2002).[14] J. P. Clemens, P. R. Rice, and L. M. Pedrotti, J. Opt.Soc. Am. B , 2025 (2004).[15] T. B. Karlovich and S. Y. Kilin, Laser Phys. , 783(2008).[16] E. del Valle, F. P. Laussy, and C. Tejedor, Phys. Rev. B , 235326 (2009).[17] A. N. Poddubny, M. M. Glazov, and N. S. Averkiev,Phys. Rev. B , 205330 (2010). [18] A. Gonzalez-Tudela, E. del Valle, E. Cancellieri, C. Teje-dor, D. Sanvitto, and F. P. Laussy, Opt. Express , 7002(2010).[19] S. Ritter, P. Gartner, C. Gies, and F. Jahnke, Opt. Ex-press , 9909 (2010).[20] A. Auff`eves, D. Gerace, J.-M. G´erard, M. F. Santos, L. C.Andreani, and J.-P. Poizat, Phys. Rev. B , 245419(2010).[21] E. del Valle and F. P. Laussy, Phys. Rev. Lett. ,233601 (2010).[22] P. Gartner, arXiv:1105.2189 (2011).[23] B. R. Mollow, Phys. Rev. , 1969 (1969).[24] A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer,J. McKeever, and H. J. Kimble, Phys. Rev. Lett. ,233603 (2004).[25] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S.Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J.Schoelkopf, Nature , 162 (2004).[26] J. M. Fink, M. G¨oppl, M. Baur, R. Bianchetti, P. J. Leek,A. Blais, and A. Wallraff, Nature , 315 (2008).[27] J. P. Reithmaier, G. Sek, A. L¨offler, C. Hofmann,S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Ku-lakovskii, T. L. Reinecker, and A. Forchel, Nature ,197 (2004).[28] T. Yoshie, A. Scherer, J. Heindrickson, G. Khitrova,H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, andD. G. Deppe, Nature , 200 (2004).[29] E. Peter, P. Senellart, D. Martrou, A. Lemaˆıtre, J. Hours,J. M. G´erard, and J. Bloch, Phys. Rev. Lett. , 067401(2005).[30] J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, andH. J. Kimble, Nature , 268 (2003).[31] O. Astafiev, K. Inomata, A. O. Niskanen, T. Yamamoto,Y. A. Pashkin, Y. Nakamura, and J. S. Tsai, Nature ,588 (2007).[32] M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, andY. Arakawa, Nat. Phys. , 279 (2010).[33] F. Y. Wu, R. E. Grove, and S. Ezekiel, Phys. Rev. Lett. , 1426 (1975).[34] G. Wrigge, I. Gerhardt, J. Hwang, G. Zumofen, andV. Sandoghdar, Nat. Phys. , 60 (2008).[35] O. Astafiev, A. M. Zagoskin, A. A. A. Jr., Y. A. Pashkin,T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai,Science , 840 (2010).[36] A. Muller, E. B. Flagg, P. Bianucci, X. Y. Wang, D. G.Deppe, W. Ma, J. Zhang, G. J. Salamo, M. Xiao, andC. K. Shih, Phys. Rev. Lett. , 187402 (2007).[37] A. N. Vamivakas, Y. Zhao, C.-Y. Lu, and M. Atat¨ure,Nat. Phys. , 198 (2009).[38] E. B. Flagg, A. Muller, J. W. Robertson, S. Founta, D. G.Deppe, M. Xiao, W. Ma, G. J. Salamo, and C. K. Shih,Nat. Phys. , 203 (2009).[39] S. Ates, S. M. Ulrich, A. Ulhaq, S. Reitzenstein,A. L¨ooffler, S. H¨ofling, A. Forchel, and P. Michler, Nat. Photon. , 724 (2009).[40] E. Jaynes and F. Cummings, Proc. IEEE , 89 (1963).[41] S. M. Ulrich, S. Ates, S. Reitzenstein, A. L¨offler,A. Forchel, and P. Michler, arXiv:1103.1594 (2011).[42] C. Roy and S. Hughes, arXiv:1102.0254 (2011).[43] R. Loudon, The quantum theory of light (Oxford SciencePublications, 2000), 3rd ed.[44] E. del Valle,
Microcavity Quantum Electrodynamics (VDM Verlag, 2009).[45] C. Cohen-Tannoudji and S. Reynaud, J. phys. B.: At.Mol. Phys. , 345 (1977).[46] B. W. Shore and P. L. Knight, J. Mod. Opt. , 1195(1993).[47] A. Laucht, N. Hauke, J. M. Villas-Bˆoas, F. Hofbauer,G. B¨ohm, M. Kaniber, and J. J. Finley, Phys. Rev. Lett. , 087405 (2009).[48] F. P. Laussy, E. del Valle, and C. Tejedor, Phys. Rev. B , 235325 (2009).[49] A. Auff`eves, J.-M. G´erard, and J.-P. Poizat, Phys. Rev.A , 053838 (2009).[50] E. del Valle, F. Laussy, and J. Finley, Unpublished.(2011).[51] F. P. Laussy, I. A. Shelykh, G. Malpuech, and A. Ka-vokin, Phys. Rev. B , 035315 (2006).[52] M. O. Scully and M. S. Zubairy, Quantum optics (Cam-bridge University Press, 2002).[53] K. Mølmer (1996), URL .[54] F. P. Laussy, E. del Valle, and C. Tejedor, Phys. Rev.Lett. , 083601 (2008).[55] F. P. Laussy and E. del Valle, J. Phys.: Conf. Ser. ,012018 (2010).[56] E. del Valle, Phys. Rev. A , 053811 (2010).[57] A. ˘Imamo¯glu, R. J. Ram, S. Pau, and Y. Yamamoto,Phys. Rev. A , 4250 (1996).[58] A. ˘Imamo¯glu and R. J. Ram, Physics Letter A , 193(1996).[59] F. P. Laussy, G. Malpuech, and A. Kavokin, Phys. Stat.Sol. C , 1339 (2004).[60] M. Berry, New Scientist , 47 (1987).[61] M. C. Gutzwiller, Sci. Am. , 78 (1992).[62] F. Haake, Quantum Signatures of Chaos , vol. 2nd ed.(Springer-Verlag, New York, 2001).[63] R. Ohta, Y. Ota, M. Nomura, N. Kumagai, S. Ishida,S. Iwamoto, and Y. Arakawa, Appl. Phys. Lett. ,173104 (2011).[64] C. Stroud, A jewel in the crown (Institute of Optics,2004), chap. 30.[65] The truncation cannot be done at the level of Eq. (20),since this one related to the number of photons nn