Monitoring the resonantly driven Jaynes-Cummings oscillator by an external two-level emitter: A cascaded open-systems approach
MMonitoring the resonantly driven Jaynes-Cummings oscillator by an externaltwo-level emitter: A cascaded open-systems approach
Th. K. Mavrogordatos ∗ and J. Larson Department of Physics, Stockholm University, SE-106 91, Stockholm, Sweden (Dated: May 15, 2020)We address the consequences of backaction in the unidirectional coupling of two cascaded openquantum subsystems connected to the same reservoir at different spatial locations. In the spirit of[H. J. Carmichael, Phys. Rev. Lett. , 2273 (1993)], the second subsystem is a two-level atom,while the first transforms from a driven empty cavity to a perturbative QED configuration andultimately to a driven Jaynes-Cummings (JC) oscillator through a varying light-matter couplingstrength. For our purpose, we appeal at first to the properties of resonance fluorescence in thestatistical description of radiation emitted along two channels —those of forwards and sidewaysscattering —comprising the monitored output. In the simplest case of an empty cavity coupled toan external atom, we derive analytical results for the nonclassical fluctuations in the fields occupyingthe two channels, pursuing a mapping to the bad-cavity limit of the JC model to serve as a guide forthe description of the more involved dynamics. Finally, we exemplify a conditional evolution for thecomposite system of a critical JC oscillator on resonance coupled to an external monitored two-leveltarget, showing that coherent atomic oscillations of the target probe the onset of a second-orderdissipative quantum phase transition in the source. PACS numbers: 03.65.Yz, 42.50.Lc, 42.50.-p, 42.50.PqKeywords: Cascaded open systems, quantum trajectories, resonance fluorescence, dissipative quantum phasetransitions, phase bistability.
I. INTRODUCTION
The late 1980s and early 1990s witnessed the develop-ment of the formalism for describing the statistical prop-erties of light emitted from a quantum system, drivenby another nonclassical source [1–5]. While fundamen-tally interesting on its own, the theory also lends itself tothe assessment of critical behaviour in non-equilibriumquantum phase transitions. In the experiment of [6], forinstance, it was shown that the radiative decay rate ofan atom coupled to quadrature-squeezed electromagneticvacuum, generated by a Josephson parametric amplifier,can be reduced below its natural linewidth. This obser-vation corroborated Gardiner’s theoretical prediction onthe disparity of the rates at which the two polarizationquadratures are damped when an atom interacts with abroadband squeezed vacuum [7]. Concurrently with thelatter, resonance fluorescence from a driven atom whichis damped by a squeezed vacuum was studied in [8]. Fol-lowing this long path of investigation to our days, in theexplicitly cascaded setup of [9], a weakly nonlinear sys-tem comprising a superconducting resonator coupled toan artificial atom in the dispersive regime is driven bysqueezed vacuum, extending the efficient generation ofsqueezed states in a parametric amplifier comprising anarray of Josephson junctions [10]. Further along, an im-portant step in characterizing the properties of squeezingvia resonance fluorescence —and its characterstic Mollowtriplet spectrum —from artificial atoms in circuit quan-tum electrodynamics (QED) was taken in [11]. In what ∗ Email address: [email protected] concerns now the generation of time-correlated photonpairs from sources operating at the nanoscale, inelastictunneling of single photons has been very recently shownto produce highly bunched light in a process that can beconstrued as an idealized two-step cascade [12].Quantum optical systems, like those mentioned above,have come to play a crucial rˆole in the recent explo-ration of non-equilibrium phase transitions. Such dis-sipative quantum phase transitions rely fundamentallyon the balance between output and input in a back-ground of intense fluctuations, in contrast to their equi-librium counterparts. Shortly after the formulation of thecascaded-system theory, an experiment reported on theemission properties of two coupled cavities operating inthe region of optical bistability [13]. Closer to our days,the breakdown of photon blockade in zero dimensions[14], which was experimentally demonstrated in [15], isassociated with a distinct presence of quantum nonlin-earity leading to a definition of a strong-coupling “ther-modynamic limit” , where fluctuations persist, while themean-field and quantum predictions manifestly disagree.Such an out-of-equilibrium phase transition probes theparadigmatic √ n nonlinearity of the Jaynes-Cummings(JC) oscillator [16], which has been revealed in a seriesof experiments in cavity and circuit QED (see, e.g., [17]and [18]). Light-matter interaction as formulated by thedriven dissipative JC model is subject to two “thermody-namic limits” which are fundamentally different in termsof the input-output relation they dictate. One of them isa so-called weak-coupling limit , in which quantum fluctu-ations reduce to an inconsequential addition (a so-called“fuzz”) superimposed on top of the semiclassical output.The second one is a strong-coupling limit , where the sys-tem size grows together with the light-matter coupling a r X i v : . [ qu a n t - ph ] M a y strength and quantum fluctuations remain. Such a limitis associated with the occurrence of spontaneous dressed-state polarization and symmetry breaking on resonance[19] and the persistence of photon blockade off-resonance[14].Accessing the Fock states of a harmonic oscillator andassessing the statistical properties of the radiation emit-ted following excitation with a single-photon source, ei-ther coherent or incoherent, has recently revived the in-terest in the cascaded-systems formalism [20]. Subse-quently, the normalized emission spectra of a two-levelatom driven by the light emanating from another classi-cally driven two-level atom were investigated in [21] to befollowed by a detailed analysis of a regime where “wherethermal statistics and quantum coherences coexist andintertwine via quantum emitters,” as demonstrated in[22]. With regard to extended systems, a driven latticeof bidirectionally coupled cavities in the photon-blockaderegime has been assigned a quasi-thermal distributionfunction in [23], while a first-order dissipative quantumphase transition has recently been experimentally re-alized in a chain of 72 coplanar waveguide resonators[24]. Furthermore, direct correspondence between pho-ton blockade and stationary dark-state generation hasbeen recently explored in [25]. Interestingly, cascadedquantum systems have also been used for a realization ofa quantum information protocol in which spontaneouslyemitted photons from a quantum dot at a properly pre-pared state are collected and directed to a second quan-tum dot [26].How will the crucial interplay of input and output panout when information on the developed criticality is mon-itored by an external quantum system? Contextual en-tanglement for a laser oscillator illuminating a two-levelatom was studied in [27] in the frame of characterizingthe laser output-state, while a very recent recent paperreports on a superposition of macroscopically incompat-ible states, localized at the maxima and the minima ofthe dipole potential, following a detection of the electro-magnetic field [28]. An investigation of the Ising quan-tum phase transition in a quantum magnetic field [29]came after the example of [30], where an external spinis coupled to an Ising-type chain comprising a couple ofspins; such an interaction imposes a conditional evolutionon the composite system observables. In our paper, weexplore the conditional evolution of a JC oscillator aris-ing when monitoring its output by an external two-levelatom. The atom polarization together with the cavityfield form part of the forwards-scattering channel, whichoccupies a main object of our investigation. In such acascaded setup, one aims at total absorption of the inci-dent light, projecting the atom to its excited state withunit probability; a single photon representing the time-reversed wave packet would then be released by the atomin question in the course of spontaneous emission [31–33].The single-photon wave packet must then impinge fromthe full 4 π solid angle and have the appropriate temporalshape [34]. Our discussion is organized as follows. After intro-ducing the model in Sec. II, based on the cascaded-systems formalism developed in [2], we isolate the in-ternal two-level atom from the dynamics by setting itscoupling strength to the cavity equal to zero in Sec.III. We provide expressions for the incoherent spectrumand the squeezing spectrum of quantum fluctuations inSecs. III A and III B, respectively, before focusing onthe weak-excitation limit which preserves the state pu-rity. We then extract an approximate formula for thesecond-order correlation function for forwards scatteringin Sec. III C 1, which reaffirms the mapping to the bad-cavity limit. Such a correspondence gives access to amore general discussion on the second-order coherenceproperties in Sec. III C 2. In the second part of our anal-ysis, we reinstate the atom inside the cavity and assessthe implications of a light-matter coupling with growingstrength. In Sec. IV, we remain within the frameworkof the bad-cavity limit permitting the adiabatic elimina-tion of the intracavity field. In Sec. V, we abandon theperturbative analysis allowed by the distinct timescalesdefining the the bad-cavity limit, and instead move tothe strong-coupling regime, where we encounter a second-order quantum phase transition with the accompanyingspontaneous symmetry breaking for the intracavity fieldand the associated atomic polarization. We exemplifythe impact of a monitoring atom outside the cavity onthe manifestation of phase bistability for a conditionalevolution of the composite system in the course of singlequantum trajectories. In spite of the unidirectional cou-pling, we find that monitoring the bistable JC oscillatorhas an ostensible effect on how the switching events ap-pear; these events are correlated with disruptions in thecoherent-oscillation cycles of the external atom. Somebrief comments on our results and extension to futurework close out the paper. II. JC OSCILLATOR COUPLED TO A SINGLETWO-LEVEL ATOM: THE MODEL
In this work, a coherent field is driving on resonance acavity mode coupled to a two-level atom, while the out-put cavity field is directed to an external two-level atom.Both atomic transitions are as well resonant with thefrequency of the cavity mode. A traveling-wave reservoirconnects the two subsystems unidirectionally [2, 3]. Themaster equation (ME) in the Markovian approximation,for the retarded density operator ˜ ρ of the composite sys-tem at the position of the external two-level atom and inthe interaction picture, reads [2, 27, 35] d ˜ ρdt = 1 i (cid:126) [ H, ˜ ρ ] + (cid:88) k =1 , L [ C k ]˜ ρ, (1)where L [ C k ]˜ ρ ≡ C k ˜ ρC † k − (1 / C † k C k ˜ ρ − (1 / ρC † k C k isthe standard dissipation superoperator corresponding tothe collapse operator C k and taking as an argument the FIG. 1.
The two cascaded open quantum systems connectedvia the reservoir field in the vacuum state.
The modes ofthe vacuum field that couple to the external two-level atom(TLA ) are divided between four channels. Two of them arelabeled by Γ γ and the other two by (1 − Γ) γ , with 0 ≤ Γ ≤ C . Spontaneous emission is ab-sent for the internal two-level atom (TLA ) coupled to theintracavity field with strength g . Backwards and sidewaysscattered photons are captured by the three detectors with acombined output corresponding to the collapse operator C . density matrix ˜ ρ . The coupled-system Hamiltonian inEq. (1) is H = i (cid:126) [ g ( a † σ − − aσ )+ (cid:112) Γ κγ/ a † σ − − aσ )+ ε d ( a † − a )] , (2)in which g is the coupling strength between the cavitymode (with annihilation and creation operators a and a † , respectively) and the internal atom (with polarizationand inversion operators σ − and σ z , respectively), ε d isthe amplitude of the coherent field driving the cavity and2 κ is the photon loss rate due to coupling of cavity modeto a reservoir at zero temperature. The total spontaneousemission rate due to coupling of the external atom ( σ − , σ z ) to reservoir modes is denoted by γ , while we assumethat spontaneous emission is absent for the atom insidethe cavity, unless explicitly stated otherwise (e.g., in Sec.IV). The fraction of the spontaneous emission rate intothe solid angle subtended by the source is denoted byΓ γ/
2, while we refer to Γ as the degree of focusing . Insuch a configuration, 0 ≤ Γ ≤
1, with the total sponta-neous emission rate being [2Γ + 2(1 − Γ)]( γ/
2) = γ .The two collapse operators featured in Eq. (1), reflect-ing the asymmetry of the channels coupling the externalatom to its environment due to the presence of a degreeof focusing Γ different than unity, are (see Eq. 12 of [2]) C = √ κ a + (cid:112) Γ( γ/ σ − ,C = (cid:112) [2(1 − Γ) + Γ]( γ/ σ − = (cid:112) (2 − Γ)( γ/ σ − , (3)for the single forwards-scattering ( C ) and the collectionof one backwards-scattering and two sideways-scatteringchannels (all lumped in C ). For convenience, hereinafter we omit the designation backwards-scattering when refer-ring to the field C (which is, however, the dominant con-tribution for the limiting case Γ → (cid:112) Γ κγ/
4, comprised en-tirely of coupling rates to the reservoir fields, otherwiseresponsible for dissipation. We note as well that a and σ − also couple to the reservoir modes at the same spatiallocation [2]. ME (1) is solved via exact diagonalizationfor the Liouvillian superoperators dictating the evolu-tion of the composite-system density matrix. The ME isalso unravelled into quantum trajectories via a quantumstate diffusion algorithm (see [37, 38] and the correspon-dence with a stochastic differential equation for the con-tinuous time evolution of conditioned heterodyne-currentrecords, in Sec. 18.2.3 of [39]) with adaptive stepsize. Forthe exact diagonalization, we use the exponential seriesexpansion of MATLAB’s Quantum Optics Toolbox , whilefor the generation of individual realizations we rely on anopen-source library in C ++ detailed in [40].To gain an understanding of where the unidirectionalcoupling could lead to, we begin by looking at the mean-field equations. The semiclassical equations for ˜ α ≡ (cid:104) a (cid:105) ,˜ β ≡ (cid:104) σ − (cid:105) , ζ ≡ (cid:104) σ z (cid:105) , ˜ β ≡ (cid:104) σ − (cid:105) , ζ ≡ (cid:104) σ z (cid:105) , derivedfrom the ME (1) after factorizing the coupled momentsin the equations of motion, read d ˜ αdt = − κ ˜ α + g ˜ β + ε d , (4a) d ˜ β dt = g ˜ αζ , (4b) dζ dt = − g (˜ α ∗ ˜ β + ˜ α ˜ β ∗ ) , (4c) d ˜ β dt = − ( γ/
2) ˜ β + (cid:112) κγ Γ ˜ αζ , (4d) dζ dt = − γ ( ζ + 1) − (cid:112) κγ Γ(˜ α ∗ ˜ β + ˜ α ˜ β ∗ ) . (4e)Since spontaneous emission is absent for the atom lyinginside the cavity, Eqs. (4b) and (4c) preserve the lengthof the pseudo-spin for the internal two-level atom inter-acting with the resonant cavity mode, yielding [14, 19]4 | ˜ β | + ζ = 1 . (5)Eqs. (4a) - (4c) predict the appearance of spontaneousdressed-state polarization for the JC “molecule” when ε d ≥ g/
2, producing states which become attractors inthe presence of quantum fluctuations [19]. We also notethat on the mean-field level , the equations of motion forthe atomic averages are the same as those of free-spaceresonance fluorescence, where the atom is driven by acoherent field with complex amplitude ˜ α . This is a con-sequence of the unidirectional coupling. In the steadystate, we then find˜ β , ss = − √ Y | Y | , ζ , ss = −
11 + | Y | , (6)with the dimensionless drive amplitude defined as Y ≡ (cid:112) κ Γ /γ ˜ α ss . Having now introduced the model we willbe working with, we proceed to a significant simplifica-tion by considering an empty cavity driven by coherentlight, producing an output field which is directed to theexternal two-level atom. III. COHERENTLY DRIVEN EMPTY CAVITYCOUPLED TO AN EXTERNAL TWO-LEVELATOM
Let us now consider the case where g = 0. For brevitywe drop the subscript of the atomic operators, reserving σ for the external atom only (since the internal atomhas no longer any influence on the dynamics). In thissection, we draw motivation by the analysis of the samesystem considered in [2], which we briefly summarize inthe following paragraphs.In the interaction picture, the non-Hermitian Hamil-tonian governing the evolution of the (un-normalized)conditional wave-function | ψ c ( t ) (cid:105) , as ( d/dt )( | ψ c ( t ) (cid:105) ) =[1 / ( i (cid:126) )] H | ψ c ( t ) (cid:105) has the form H = i (cid:126) [ ε d ( a † − a ) − κa † a − ( γ/ σ + σ − − (cid:112) κ ( γ/ aσ + ] . (7)Assuming an initial vacuum state for the cavity mode andgiven that the term a † σ − is absent from the Hamiltonianof Eq. (7), the conditional wavefunction can be writtenin the factorized form | ψ c ( t ) (cid:105) = | α ( t ) (cid:105) | A c ( t ) (cid:105) , where α ( t )is a coherent-state amplitude and | A c ( t ) (cid:105) is the state ofthe atom. When the field is in a coherent state, theatom does not entangle with its driving field [35]. Wefind that the amplitude α ( t ) is given by the expression α ( t ) = ( ε d /κ )(1 − e − κt ) and, upon reaching steady state,we obtain α ss = ε d /κ . Then, in the long-time limit, thewavefunction for the (external) atom alone obeys ddt | A c ( t ) (cid:105) = − (cid:32) γ σ + σ − + ε d (cid:114) Γ γκ σ + (cid:33) | A c ( t ) (cid:105) , (8)with collapse operators C = √ κ ( ε d /κ ) + (cid:112) Γ( γ/ σ − and C = (cid:112) (2 − Γ)( γ/ σ − .Now, in [2] we also read that this evolution is equiv-alent to placing the atom inside a driven cavity in the bad-cavity limit, satisfying a ME of the open driven JCoscillator on resonance (in the interaction picture) d ˜ ρdt = g [ a † σ − − aσ + , ˜ ρ ] + ε d [ a † − a, ˜ ρ ]+ κ (cid:0) a ˜ ρa † − a † a ˜ ρ − ˜ ρa † a (cid:1) + γ s σ − ˜ ρσ + − σ + σ − ˜ ρ − ˜ ρσ + σ − ) , (9)with g ≡ (cid:112) κ Γ γ/ γ s = (cid:112) (2 − γ/ κ (cid:29) ( γ/ , g ), the intracavity field operator a is then identifiable with the forwards-scattering fieldoperator (in units of the square root of photon flux) C ≡ √ κ ( ε d /κ ) + (cid:112) Γ( γ/ σ − , the statistics of whichwe wish to determine. This mapping would also leadto an enhanced emission rate of the form (1 + 2 C ) γ s ,with C ≡ g / ( κγ s ) = Γ / [2(1 − Γ)], whence (1 + 2 C ) γ s =[1 + Γ / (1 − Γ)]2(1 − Γ)( γ/
2) = γ . In this correspondence,Γ = 2 C/ (1 + 2 C ) is the proportion of the atomic reradi-ation inside the cavity seen in transmission [41]. We willdiscuss this mapping in more detail in Sec. III C 2.For the moment, we return to the solution of ME (1)in the case where the atom inside the cavity is explicitlynot involved. As in Sec. V of [35] and Sec. IIB of [5], wepropose the ansatz ˜ ρ ( t ) = | α ( t ) (cid:105)(cid:104) α ( t ) | ⊗ ρ A ( t ) , (10)leading to a reduced ME for the external atom alone, dρ A dt = 1 i (cid:126) [ H eff , ρ A ] + L [ C A ] ρ A , (11)with an effective Hamiltonian H eff = i (cid:126) (cid:112) κ Γ( γ/ α ∗ ( t ) σ − − α ( t ) σ + ]= i (cid:126) (cid:112) κ Γ γ [ α ∗ ( t ) σ − − α ( t ) σ + ] , (12)and a single collapse operator C A = √ γ σ − . (13)The coherent-state amplitude evolves again as α ( t ) =( ε d /κ )(1 − e − κt ) for a time-independent coherent drive,relaxing to the steady-state value α ss = ε d /κ . Since thecavity is in a coherent state, the neoclassical equations(4d) and (4e) are identical to the Heisenberg equationsof motion with a steady-state solution given by Eq. (6),where Y = 2 √ ε d (cid:112) Γ / ( κγ ). A. Incoherent spectrum of fluctuations for the twochannels
We will now carry on with the ME produced for theexternal atom alone which, after the field amplitude hasrelaxed to its final value, can be written in the standardform of free-space resonance fluorescence, as dρ A dt = 1 i (cid:126) [ H eff , (t (cid:29) κ − ) , ρ A ] + L ( C A ) ρ A , (14)in which the effective Hamiltonian has relaxed to H eff , (t (cid:29) κ − ) = (1 / (cid:126) ω A σ z + i (cid:126) ε d (cid:112) Γ γ/κ ( σ − e iω A t − σ + e − iω A t ) , (15)where ω A is the atomic frequency (coinciding with thefrequency of the drive and the resonance frequency of theintracavity mode). In the Appendix, we derive the corre-lation functions needed to calculate the incoherent spec-trum of the forwards-emitted field from the steady-statefirst-order correlation function [42]. These are the sameas in ordinary resonance fluorescence since the fluctua-tions ∆ C , and ∆ C † , (where ∆ C , ≡ C , − (cid:104) C , (cid:105) ss )are proportional to ∆ σ − and ∆ σ + , respectively, and thequantum regression formula relies on the Bloch equa-tions (in which appropriately modified coefficients fea-ture). For all the steady-state averages, (cid:104)·(cid:105) ss , the limit t → ∞ has already been attained. Therefore, one onlyrequires the Hamiltonian of Eq. (15) when assessing thecoherence properties of the source field radiated by theatom outside the cavity as a stationary process.Adopting the scaling of [41] and following the standardprocedure (see, e.g., Sec. 2.3.4 of [42]), we write the in-coherent optical spectrum of the forwards and sidewaysscattered fields as the Fourier transform of the first-orderfluctuation correlation function for the slowly varying op-erators ˜ C , ˜ C , at a scaled angular frequency displaced bythe atomic resonance, as S C , , inc ( ω ) = 12 π ( (cid:104) ∆ ˜ C † , ∆ ˜ C , (cid:105) ss ) − × (cid:90) ∞−∞ dτ e i ( ω − ω A ) τ (cid:104) ∆ ˜ C † , (0)∆ ˜ C , ( τ ) (cid:105) ss = 1 π (cid:18) Y (1 + Y ) (cid:19) − × Re (cid:18)(cid:90) ∞ dτ e i ( ω − ω A ) τ (cid:104) ∆˜ σ + (0)∆˜ σ − ( τ ) (cid:105) ss (cid:19) , (16)where the (slowly varying) operators ˜ C , ( t ) and ˜ C † , ( t )are defined in a frame rotating with ω A . The incoherentspectrum evaluates to (see Eq. 1 of [41] and Eq. 22 of[21]) S C , , inc ( ω ) = 12 π (cid:40) (cid:18) Y Y (cid:19) −
11 + ( ω − ω A ) − (cid:20) /Y − /Y −
5) 12 δ (cid:21) / − δ/ / − δ ) + ( ω − ω A ) − (cid:20) /Y − − (1 /Y −
5) 12 δ (cid:21) / δ/ / δ ) + ( ω − ω A ) (cid:41) , (17) where τ ≡ γτ / ω ≡ ω/γ , ω A ≡ ω A /γ and δ ≡ δ/γ ,with δ = ( γ/ √ − Y . The above expression is nor-malized (to unit area) with respect to the dimensionlessangular frequency ω . At the exceptional point, δ = 0(see the Appendix), the incoherent spectrum is given bythe expression [see Eq. (5) of [41]] S C , , inc , cr ( ω ) = 92 π (cid:40)
11 + ( ω − ω A ) − ω − ω A ) [(3 / + ( ω − ω A ) ] (cid:41) , (18)yielding a narrower distribution than the free-spaceLorentzian spectrum, before the Rabi doublet emerges. B. Squeezing of quantum fluctuations and thespectrum of squeezing
The incoherent spectrum of the quantum fields occu-pying the two channels, corresponding to the operators C , C , is intimately tied to the spectrum of squeezingwhich, unlike the former, can assume negative values.Essentially, the spectrum of squeezing for both channelsassumes the same form as in ordinary resonance fluores-cence, since the source operators ∆ σ ± obey the same op-tical Bloch equations as we have already pointed out (fora discussion on the self-homodyning of squeezed flores-cence and its contribution to antibunching see, e.g., theform Eq. 37 and the ensuing discussion in [43], and com-pare to Sec. 2.3.6 of [42]). Defining the field quadratures∆ ˜ X , ≡ ˜ X , − (cid:104) ˜ X , (cid:105) ss with √ κ ˜ X ≡ (1 / C + ˜ C † )and √ κ ˜ X ≡ − i (1 / C − ˜ C † ), we calculate the nor-mally ordered quadrature variances in the steady state(see also Eq. 32 of [43]), (cid:104) : (∆ ˜ X , ) : (cid:105) ss = 14 [2 (cid:104) ∆ ˜ C † ∆ ˜ C (cid:105) ss ± (cid:104) (∆ ˜ C ) (cid:105) ss ± (cid:104) (∆ ˜ C † ) (cid:105) ss ]= 14 (cid:18) Γ γ κ (cid:19) (cid:104) (cid:104) σ z (cid:105) ss − (2 ± (cid:104) ˜ σ + (cid:105) (cid:105) = 14 (cid:18) Γ γ κ (cid:19) Y ( Y ∓ Y ) . (19)Squeezing of the steady-state quantum fluctuations oc-curs only for the field quadrature ˜ X , which is in phasewith the steady-state polarization (cid:104) ˜ σ − (cid:105) ss , for Y <
1; thisvariance is an explicit function of the degree of focusing.The spectrum of squeezing for the outward-fieldquadrature, as measured via a homodyne detectionscheme employing a local oscillator with phase θ , is [41] S C , sq ( ω, θ ) = 8 ηπ (cid:90) ∞ dτ cos( ω τ ) × Re (cid:16) (cid:104) ∆ ˜ C † (0)∆ ˜ C ( τ ) (cid:105) ss + e iθ (cid:104) ∆ ˜ C † (0)∆ ˜ C † ( τ ) (cid:105) ss (cid:17) , (20)where η stands for the product of the collection and de-tection coefficients. Once more, we need to sequester for-mulas of ordinary resonance fluorescence from the Ap-pendix. For θ = 0, the quantity in the integral of Eq.(20) is the normally ordered correlation function of thein-phase quadrature X , which, as we anticipate when τ = 0, is explicitly negative for Y (cid:28) (cid:104) : ∆ ˜ X (0)∆ ˜ X ( τ ) : (cid:105) ss = 12 (cid:18) Γ γ κ (cid:19) [ (cid:104) ∆˜ σ + (0)∆˜ σ + ( τ ) (cid:105) ss + (cid:104) ∆˜ σ + (0)∆˜ σ − ( τ ) (cid:105) ss ]= − (cid:18) Γ γ κ (cid:19) Y (1 + Y ) × (cid:40) (cid:104) − Y + (cid:16) γ δ (cid:17) (1 − Y ) (cid:105) e − (3 γ/ − δ ) τ + (cid:104) − Y − (cid:16) γ δ (cid:17) (1 − Y ) (cid:105) e − (3 γ/ δ ) τ (cid:41) . (21)From this function, one computes the squeezing spec-trum for the field quadrature of forwards-scattered lightwhich is in phase with the induced atomic polarizationas S C , sq ( ω,
0) = − ηπ (cid:18) Γ γ (cid:19) Y (1 + Y ) × (cid:40) (cid:20) − Y + (cid:18) δ (cid:19) (1 − Y ) (cid:21) / − δ (3 / − δ ) + ω + (cid:20) − Y − (cid:18) δ (cid:19) (1 − Y ) (cid:21) / δ (3 / δ ) + ω (cid:41) . (22)On the other hand, for the fluctuations in quadrature to (cid:104) ˜ σ − (cid:105) ss ( θ = π/ S C , sq ( ω, π/
2) = (2 κ ) 8 ηπ × (cid:90) ∞ dτ cos( ω τ ) (cid:104) : ∆ ˜ X (0)∆ ˜ X ( τ ) : (cid:105) ss = 8 ηπ (cid:18) Γ γ (cid:19) (cid:90) ∞ dτ cos( ω τ ) × ( (cid:104) ∆˜ σ + (0)∆˜ σ − ( τ ) (cid:105) ss − (cid:104) ∆˜ σ + (0)∆˜ σ + ( τ ) (cid:105) ss )= 4 ηπ (cid:18) Γ γ (cid:19) Y Y
11 + ω . (23)Hence, we recover the general result linking the incoher-ent spectrum to the spectrum of squeezing [41] S C , inc ( ω + ω A ) = (16 η (cid:104) ∆ ˜ C † ∆ ˜ C (cid:105) ss ) − × [ S C , sq ( ω, θ ) + S C , sq ( ω, θ + π/ , (24)applying in our case for θ = 0 (see also Fig. 2 of [21] foran explicit formation of the incoherent spectrum as a bal-ance of Lorentzians with positive and negative weights). For weak excitation strengths, Y (cid:28)
1, the spectrum S C , sq ( ω,
0) takes negative values due to squeezing offluctuations in phase with the mean induced polariza-tion; this lies at the root of the squared Lorentzian pro-file whose origins date back to Mollow as reported in1969 [see [41] as well as Eq. (4.21) of [44] and discussionbelow]. The incoherent spectrum of Eq. (17) and thenormalized squeezing spectra as given by S C , sq ( ω ) ≡ (cid:16) η (cid:104) ∆ ˜ C † ∆ ˜ C (cid:105) ss (cid:17) − S C , sq ( ω ) (25)for the forwards emission are depicted in Fig. 2. In thisfigure, where Y > / (2 √
2) for all frames, we witness thedevelopment of the characteristic Mollow triplet for anincreasing degree of focusing, a sign of dominant inco-herent scattering. The Rabi sidebands of Figs. 2(c) and2(d) are perfectly captured by the spectrum of squeezingfor strong focusing, as predicted by the dominant contri-bution of the second term in the sum of Eq. (24) to theincoherent spectrum, for large values of Y . C. Second-order coherence for the two channels
To get a deeper insight for the statistics of the for-wards and sideways-scattered light we now consider thesecond-order correlators. The second-order correlationfunction for the forwards-scattering field, correspondingto the operator C , is: g (2) C ( τ ) = (cid:104) ˜ C † (0) ˜ C † ( τ ) ˜ C ( τ ) ˜ C (0) (cid:105) ss (cid:104) ˜ C † ˜ C (cid:105) = tr { ˜ C † (0) ˜ C (0) e ˜ L τ [ ˜ C (0)˜ ρ ss ˜ C † (0)] }(cid:104) ˜ C † ˜ C (cid:105) = (cid:104) ( ˜ C † ˜ C )( τ ) (cid:105) ˜ ρ (0)=˜ ρ (cid:48) ss (cid:104) ˜ C † ˜ C (cid:105) ss , (26)where in passing from the first to the second line wehave once more employed the quantum regression for-mula. The (normalized) initial state of the atomic system˜ ρ (0) = ˜ ρ (cid:48) ss , for which the above averages are evaluated,is given by ˜ ρ (cid:48) ss ≡ ˜ C ˜ ρ ss ˜ C † tr( ˜ C ˜ ρ ss ˜ C † ) . (27)We note here that the steady-state mean photon flux inthe forwards direction, featuring in the above expression,is given by the expression (cid:104) ˜ C † ˜ C (cid:105) ss = (2 κ ) 18Γ γκ Y Y [(1 − Γ) + Y ]= γ Y Y [(1 − Γ) + Y ] . (28) -20 -10 0 10 2000.10.2 S -20 -10 0 10 2000.050.10.150.2-20 -10 0 10 2000.10.2 S -20 -10 0 10 2000.10.2 (i)(i) (i)(ii) (ii)(ii) (ii)(i) (b)(c) (d)(a) FIG. 2.
Incoherent scattering and squeezing of fluctuations in the forwards-scattering channel for increasing degree of focusing.
Incoherent spectra S C , inc ( ω ) [curves (i)] obtained from Eq. (17), showing the development of the Mollow triplet, and normalizedsqueezing spectra S C , sq ( ω − ω A ) [curves (ii)] obtained from Eq. (22) (centered at ω A ) of the forwards-propagating field for anincreasing degree of focusing: Γ = 0 . , . , . , . (a) - (d) , respectively. The remaining parameters read: κ/γ = 200and ε d /γ = 50. For weak driving, Y (cid:28)
1, this quantity reduces to (cid:104) ˜ C † ˜ C (cid:105) ss ≈ (2 κ ) (cid:16) ε d κ (cid:17) (1 − Γ) , (29)provided that Γ is not too close to unity.
1. The weak-excitation limit
The calculation is significantly simplified in the low-excitation limit. We follow closely the treatment in Sec.13.2.3 of [39] regarding the statistics of a weak intracavityfield in the bad-cavity limit. Within the Hilbert space ofthe external atom, the density matrix in the steady state,following Eq. (6) with (the real) Y = 2 √ ε d (cid:112) Γ / ( κγ ), is(˜ ρ A ) ss = 12 2 + Y Y | (cid:105)(cid:104) | + 12 Y Y | (cid:105)(cid:104) |− √ Y Y ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) . (30)In the limit Y (cid:28)
1, the steady state may then be ap-proximated by a pure state as(˜ ρ A ) ss ≈ | ˜ A ss (cid:105)(cid:104) ˜ A ss | , (31)with | ˜ A ss (cid:105) = | (cid:105) − √ Y | (cid:105) . (32)The reduced state given by Eq. (27), prepared under thecondition that a photodetection occurs in the forward direction at τ = 0, is then also approximately pure andcan be written in the factorized form(˜ ρ A ) (cid:48) ss ≈ | ˜ A (cid:48) ss (cid:105)(cid:104) ˜ A (cid:48) ss | , (33)with | ˜ A (cid:48) ss (cid:105) = ˜ C | ˜ A ss (cid:105) (cid:113) (cid:104) ˜ A ss | ˜ C † ˜ C | ˜ A ss (cid:105) =[ ε d /κ + (cid:112) Γ γ/ (4 κ ) ˜ σ − ] | ˜ A ss (cid:105) (cid:113) (cid:104) ˜ A ss | [ ε d /κ + (cid:112) Γ γ/ (4 κ ) ˜ σ + ][ ε d /κ + (cid:112) Γ γ/ (4 κ ) ˜ σ − ] | ˜ A ss (cid:105) , (34)For the un-normalized state in the numerator of Eq. (34)we write | ˜ A ss (cid:105) = (cid:16) ε d /κ + (cid:112) Γ γ/ (4 κ ) ˜ σ − (cid:17) [ | (cid:105) − ( Y / √ | (cid:105) ]= (cid:16) ε d /κ − (cid:112) Γ γ/ (8 κ ) (cid:17) | (cid:105) − ( ε d /κ )( Y / √ | (cid:105) . To dominant order in the driving-field amplitude, thestate norm, equal to the square of the denominator inEq. (34), is12 κ tr( ˜ C ˜ ρ ss ˜ C † ) ≈ (cid:32) ε d κ − ε d κ (cid:114) Γ γ κ (cid:115) κ Γ γ (cid:33) = ( ε d /κ ) (1 − Γ) , (35)an expression we have already met in Eq. (29). Bringingthe different pieces together, we write the reduced statein the same order of magnitude with respect to Y as | ˜ A (cid:48) ss (cid:105) ≈ | (cid:105) − − Γ Y √ | (cid:105) = | (cid:105) − ε d κ (cid:115) κ Γ γ (1 − Γ) | (cid:105) . (36)The relaxation of the conditional state | ˜ A (cid:48) ss (cid:105) to thesteady state | ˜ A ss (cid:105) occurs via the action of the propagator e ˜ L A τ , where the Liouvillian super-operator ˜ L A is definedthrough the ME of Eq. (11):˜ L A ≡ − ε d (cid:112) Γ γ/κ [ σ + − σ − , · ]+ γ σ − · σ + − σ + σ − · − · σ + σ − ) ≈ − ε d (cid:112) Γ γ/κ [ σ + , · ] − γ { σ + σ − , ·} . (37)This approximation, preserving the purity of the state,is justified only for a weak excitation which guarantees anegligible photon emission probability during the relax-ation of the atom back to the steady state. The driveterm is accounted for by a non-Hermitian Hamiltonian(retaining only the term proportional to σ + ) preserv-ing the norm of the steady-state reduced density ma-trix as unity plus a first-order correction in the drivestrength. Under this assumption, the second-order cor-relation function in the forwards direction can be recastin the form g (2) C ( τ ) ≈ ( ε d /κ ) (1 − Γ) ×(cid:104) ˜ A ( τ ) | [ ε d /κ + (cid:112) Γ γ/ (4 κ ) ˜ σ + ][ ε d /κ + (cid:112) Γ γ/ (4 κ ) ˜ σ − ] | ˜ A ( τ ) (cid:105) , (38)with initial condition | ˜ A (0) (cid:105) = | ˜ A (cid:48) ss (cid:105) . Having eliminatedthe quantum jumps due to spontaneous emission fromthe Liouvillian of Eq. (37), the conditional wavefunctionevolves under a non-Hermitian Hamiltonian, obeying theSchr¨odinger equation ddτ | ˜ A ( τ ) (cid:105) = (cid:32) − ε d (cid:114) Γ γκ σ + − γ σ + σ − (cid:33) | ˜ A ( τ ) (cid:105) . (39)We expand the conditional state as | ˜ A ( τ ) (cid:105) = x ( τ ) | (cid:105) + y ( τ ) | (cid:105) , where the complex expansion coefficients x ( τ ) , y ( τ ), satisfy the set of coupled linear differentialequations dxdτ = 0 , dydτ = − γ y − ε d (cid:114) Γ γκ x. (40)The initial conditions should match the conditional stateof Eq. (36) following the measurement of a photon inthe forwards direction, yielding x (0) = 1 and y (0) = − ε d /κ ) (cid:112) κ Γ / [ γ (1 − Γ)] .The solution to the set of coupled equations (40) andtheir associated initial conditions produces the state | ˜ A ( τ ) (cid:105) = | (cid:105)− ε d κ (cid:115) κ Γ γ (cid:18) − Γ e − ( γ/ τ (cid:19) | (cid:105) . (41) g C ( ) () (iii)(iv)(ii)(i) (a) g C ( ) () (ii)(i) (iv)(iii) (b) FIG. 3.
Second-order coherence of forwards scattering in theweak-excitation limit.
Second-order correlation function offorwards photon scattering g (2) C ( τ ) vs the dimensionless delay γτ , calculated using Eq. (43). In (a) we depict photon an-tibunching for Γ = 0 . , . , . , . (b) the approach to extreme pho-ton bunching for Γ = 0 . , . , . , .
82 as depicted by thecurves (i)-(iv), respectively.
Finally, consistent with the initial approximation in Eq.(32), which amounts to keeping terms linear in thedriving-field amplitude, we write (cid:16) ε d /κ + (cid:112) Γ γ/ (4 κ ) ˜ σ − (cid:17) | ˜ A ( τ ) (cid:105)≈ ( ε d /κ ) | (cid:105) − Γ( ε d /κ )[1 + Γ / (1 − Γ) e − ( γ/ τ ] | (cid:105) = { ε d / [ κ (1 − Γ)] } { − [Γ / (1 − Γ) ] e − ( γ/ τ } | (cid:105) , (42)whence, substituting in Eq. (38), we finally obtain the second-order correlation function for the forwards emis- g C ( ) () (ii) (a) (iii)(i) g C ( ) () (b) (iii) (ii)(i) FIG. 4.
Second-order quantum correlations for a weakeningcoupling to the external atom.
Second-order correlation func-tion of forwards photon scattering g (2) C ( τ ) in (a) and sidewaysscattering g (2) C ( τ ) in (b) vs the dimensionless delay γτ , for γ/κ = 0 . , . , . ε d /κ = 0 . . sion in the weak-excitation approximation g (2) C ( τ ) ≈ (cid:34) − (cid:18) Γ1 − Γ (cid:19) e − ( γ/ τ (cid:35) . (43)This expression, which is explicitly independent of thedriving-field amplitude, agrees with Eq. (41) of [43] with C ≡ g / ( κγ (cid:48) ) = Γ / [2(1 − Γ)], in the correspondence wehave introduced in Sec. III. The transition from pho-ton antibunching to bunching for varying degrees of fo-cusing Γ is depicted in Fig. 3. The function g (2) C ( τ )in the weak-coupling limit has a minimum at the delay γτ m = 4 ln[Γ / (1 − Γ)] (see also Eq. 42 of [43]), which g ( ) C () (ii)(iii)(i) (iii)(ii)(i) FIG. 5.
Probing the intracavity light-matter coupling strengthin the bad-cavity limit.
Second-order intensity correlationfunction of the sideways scattered light field, g (2) C ( τ ) vs γτ ,extracted from the solution of ME (1) with γ s = 0 for in-creasing g/ε d assuming values 0 .
05, 1, and 2 . ε d /κ = 0 . . γ/ε d = 0 . γ s = 0. is relevant for Γ ≥ .
5. In particular, for Γ = 0 . τ m = 0 and g (2) C (0) = 0, as we can see for curve (iv)in Fig. 3(a). The occurrence of the minimum value of g (2) C ( τ m ) = 0 for τ m > →
1. The excited atom then letsa photon pass through while it is dealing with the one itis about to emit [2, 35]; as a result, closely-spaced pho-ton pairs are detected, in stark contrast with the scat-tered field of ordinary resonance fluorescence when theamplitude of the coherent drive is very small. We alsonote that, owing to the fact that terms proportional to σ − · σ + —destroying the state purity —are omitted fromEq. (37) in this approximation, g (2) C ( τ ) is initially pro-portional to the waiting-time distribution of photon emis-sions in the forwards direction, w C ( τ ), a probability dis-tribution over waiting times to the next jump associatedwith a photon emission, integrated to unity. This pro-portionality relation holds true for delay times γτ muchsmaller than the mean time between jumps (scaled by theatomic lifetime) which, in the limit of a vanishingly weakexcitation, extends to infinity (see [45] and Sec. 13.2.4of [39]). The waiting-time distribution w C ( τ ), however,ultimately decays to zero at long times (as a probabil-0ity distribution function), while g (2) C ( τ ) is asymptotic tounity.
2. More on the mapping to a two-level atom inside acoherently driven bad cavity
Following on with our mapping for larger drivingstrengths and comparing with Eqs. (17) and (23) of [43],we identify the second-order correlation function of Eq.(26) with the more involved expression g (2) C ( τ ) − − C [1 + Y (1 + 2 C ) ] × e − (3 γ/ τ (cid:110) [1 − C − Y (1 + 2 C ) ] cosh( δτ )+ γ δ [1 + 2 C − Y (1 + 2 C )(5 + 2 C )] sinh( δτ ) (cid:111) , (44)where the parameters Y , δ and C have been defined inSec. III, all involving the effective coupling strength g = (cid:112) κ Γ γ/ beyond thebad-cavity limit when mapped to the parameters of thecascaded-system setup . There is no entanglement be-tween the atom and the cavity field due to their direct coupling [35]. This means that a condition of the kind κ/γ (cid:29) not imposed. Here, the presence of C addresses adifferent physical mechanism: instead of being the Pur-cell factor enhancing the spontaneous emission rate toarbitrary values set by the intracavity coupling strength,it promotes the sideways-emission rate to its full 4 π value, γ s → γ . As for the sideways emission —for the field op-erator C —one employs directly the familiar result fromfree-space resonance fluorescence, namely (see Appendixand [42]), g (2) C ( τ ) = (cid:104) ˜ C † (0) ˜ C † ( τ ) ˜ C ( τ ) ˜ C (0) (cid:105) ss (cid:104) ˜ C † ˜ C (cid:105) = 1 − e − (3 γ/ τ (cid:18) cosh δτ + 3 γ δ sinh δτ (cid:19) , (45)with g (2) C (0) = 0 for all values of Γ. Further evi-dence on the distinct character of the forwards-emissionfield alongside its difference from the ordinary reso-nance fluorescence of sideways emission, is given by thenumerically-computed correlation functions of Fig. 4 be-yond the weak-excitation limit. The results depicted herecoincide with the analytical predictions of Eqs. (44) and(45). In Fig. 4(a), the decay of g (2) C ( τ ) to a minimumbelow unity [curve (i)] (approaching arbitrarily low val-ues for ε d /κ → Y (as we keep ε d /κ
100 105 110 115 120 t p e ( t ) + p e ( t ) (ii) (i) (a)
40 42 44 46 48 50 t p e ( t ) + p e ( t ) (ii)(i) (b) FIG. 6.
Rabi oscillations in single quantum trajectories. (a)
Time-dependent probability p e ( t ) ≡ . (cid:104) σ z ( t ) (cid:105) ) of findingthe atom outside the cavity in the excited state, extractedfrom a single realization. Curve (i) depicts p e ( t ) from theunravelling of the full ME (1) in the presence of sidewaysspontaneous emission by the internal two-level atom, with g/ε d = 0 . γ s /ε d = 0 . γ/ε d = 0 . .
9. Curve(ii), depicting 1+ p e ( t ) (for visual clarity), originates from thereduced ME of ordinary resonance fluorescence. (b) Same asin frame (a), but for γ/ε d = 0 . | n = 1 (cid:105) for the generation ofcurves (i) in both frames. constant and increase κ/γ ) [curve (iii)], in the approachto a coherent-state output. At the same time, g (2) C ( τ )shows the expected onset of the free-space resonance flu-orescence ringing associated with the pronounced Rabidoublet in the optical and squeezing spectra we have metin Fig. 2. For the computation of higher-order correla-tion functions for the forwards-scattering channel, one1employs directly Eqs. 22 and 23 of [43]. IV. ADIABATIC ELIMINATION OF THEINTRACAVITY FIELD: FREE-SPACERESONANCE FLUORESCENCE REVISITED
We now bring the atom inside the cavity into play ina perturbative fashion, assuming a coupling to the cav-ity mode with a small but finite strength g (cid:28) κ . Thisinteraction is considered together with emission from thesides of the cavity at a rate γ s , adding another dissipa-tion channel to the ME (1) via the term L [ C ]˜ ρ , with C = √ γ s σ − . We focus on a hierarchy of timescales setby the condition κ (cid:29) ( ε d , g, γ s / L [ C ]˜ ρ ), we obtain thefollowing equations of motion for the time-varying aver-ages of the coupled cavity mode field, atomic polarizationand inversion of the internal two-level atom (where thetilde on top of the operators signifies the equivalence toa frame rotating with the atomic frequency ω A ): d (cid:104) ˜ a (cid:105) dt = − κ (cid:104) ˜ a (cid:105) + g (cid:104) ˜ σ − (cid:105) + ε d , (46a) d (cid:104) ˜ σ − (cid:105) dt = − γ s (cid:104) ˜ σ − (cid:105) + g (cid:104) σ z ˜ a (cid:105) , (46b) d (cid:104) ˜ σ (cid:105) dt = − γ s (cid:104) ˜ σ (cid:105) + g (cid:104) ˜ a † σ z (cid:105) , (46c) d (cid:104) σ z (cid:105) dt = − γ s ( (cid:104) σ z (cid:105) + 1) − g ( (cid:104) ˜ σ ˜ a (cid:105) + (cid:104) ˜ a † ˜ σ − (cid:105) ) , (46d)and for the external atom, d (cid:104) ˜ σ − (cid:105) dt = − γ (cid:104) ˜ σ − (cid:105) + (cid:112) κγ Γ (cid:104) σ z ˜ a (cid:105) , (47a) d (cid:104) ˜ σ (cid:105) dt = − γ (cid:104) ˜ σ (cid:105) + (cid:112) κγ Γ (cid:104) ˜ a † σ z (cid:105) , (47b) d (cid:104) σ z (cid:105) dt = − γ ( (cid:104) σ z (cid:105) + 1) − (cid:112) κγ Γ ( (cid:104) ˜ σ ˜ a (cid:105) + (cid:104) ˜ a † ˜ σ − (cid:105) ) . (47c)From the Heisenberg-Langevin equation (12) of [3], byvirtue of the unidirectional coupling between the two cas-caded systems and in a frame rotating with ω A , we obtain[43] ˜ a ( t ) = ε d κ + gκ ˜ σ − ( t ) + 1 κ ˆ ξ ( t ) , (48)where ˆ ξ ( t ) is the quantum-noise term arising from theinteraction of the cavity mode with the field modes ofa reservoir (and contains the sum of the corresponding annihilation operators over those modes). Here, we take (cid:104) ˆ ξ ( t ) (cid:105) = 0, assuming that the reservoir is in the vacuumstate. Substituting the expression of Eq. (48) for the adi-abatically eliminated intracavity field into the equationsof motion (46) and (47), in which the system operatorshave been pre-ordered in such a fashion as to make clearthat every term involving the reservoir field coming from˜ a ( t ) is zero, yields the Bloch equations for the two-levelatom inside the cavity, d (cid:104) ˜ σ − (cid:105) dt = − γ s C ) (cid:104) ˜ σ − (cid:105) + Y √ (cid:104) σ z (cid:105) , (49a) d (cid:104) ˜ σ (cid:105) dt = − γ s C ) (cid:104) ˜ σ − (cid:105) + Y √ (cid:104) σ z (cid:105) , (49b) d (cid:104) σ z (cid:105) dt = − γ s (1 + 2 C )( (cid:104) σ z (cid:105) + 1) − Y √ (cid:104) ˜ σ (cid:105) + (cid:104) ˜ σ − (cid:105) ) , (49c)and for the external atom, d (cid:104) ˜ σ − (cid:105) dt = − γ (cid:104) ˜ σ − (cid:105) + ε d (cid:114) γ Γ κ (cid:104) σ z (cid:105) + g (cid:114) γ Γ κ (cid:104) ˜ σ − σ z (cid:105) , (50a) d (cid:104) ˜ σ (cid:105) dt = − γ (cid:104) ˜ σ (cid:105) + ε d (cid:114) γ Γ κ (cid:104) σ z (cid:105) + g (cid:114) γ Γ κ (cid:104) ˜ σ σ z (cid:105) , (50b) d (cid:104) σ z (cid:105) dt = − γ ( (cid:104) σ z (cid:105) + 1) − ε d (cid:114) γ Γ κ ( (cid:104) ˜ σ (cid:105) + (cid:104) ˜ σ − (cid:105) ) − g (cid:114) γ Γ κ ( (cid:104) ˜ σ ˜ σ − (cid:105) + (cid:104) ˜ σ − ˜ σ (cid:105) ) , (50c)where in this case there is explicit spontaneous-emissionenhancement for the internal two-level system by (1 +2 C ), with the Purcell factor C = g / ( κγ s ) dependingon the strength of the coherent intracavity light-matterinteraction. Equations (49a)-(49c) can be solved inde-pendently of the quantities pertaining to the externalatom, and the steady-state solution follows from the stan-dard treatment of free-space resonance fluorescence (foran atom placed outside the cavity) as [43] (cid:104) ˜ σ ± (cid:105) ss = − √ Y (1 + 2 C )(1 + 2 C ) + Y , (cid:104) σ z (cid:105) ss = − (1 + 2 C ) (1 + 2 C ) + Y , (51)where Y ≡ √ gε d / ( κγ s ) is the scaled dimensionlessdrive amplitude. When γ s →
0, we can instead write (cid:104) ˜ σ ± (cid:105) ss = − √ Y (cid:48) Y (cid:48) , (cid:104) σ z (cid:105) ss = −
11 + Y (cid:48) , (52)2 FIG. 7.
Bimodal field driving a two-level atom . Time-dependent averages (cid:104) σ y ( t ) (cid:105) of the slowly varying y -polarization componentfor the atom inside (in solid purple line) and outside (in green dots) the cavity, extracted from a single quantum trajectory. Forthe trajectory in the top panel, Γ = 0 .
1, and for the trajectory in the bottom panel, Γ = 0 .
9. The remaining parameters are: g/κ = 100, γ/κ = 40 and ε d /g = 0 . | n = 1 (cid:105) . with Y (cid:48) = 2 √ { gε d / [ κγ (1 + 2 C )] } → √ ε d /g .We will now decouple the moments of Eqs. (50a)-(50c)in the mean-field approximation, and seek the steady-state solutions of the modified equations of motion forthe external two-level system, − γ (cid:104) ˜ σ − (cid:105) ss + ε d (cid:114) γ Γ κ (cid:20) gε d (cid:104) ˜ σ − (cid:105) ss (cid:21) (cid:104) σ z (cid:105) ss = 0 , (53a) − γ (cid:104) ˜ σ (cid:105) ss + ε d (cid:114) γ Γ κ (cid:20) gε d (cid:104) ˜ σ (cid:105) ss (cid:21) (cid:104) σ z (cid:105) ss = 0 , (53b) − γ ( (cid:104) σ z (cid:105) ss + 1) − ε d (cid:114) γ Γ κ (cid:20) gε d (cid:104) ˜ σ (cid:105) ss (cid:21) × ( (cid:104) ˜ σ (cid:105) ss + (cid:104) ˜ σ − (cid:105) ss ) = 0 . (53c)The solution to these equations —equivalent to free-space3resonance fluorescence —is, as usual, (cid:104) ˜ σ ± (cid:105) ss = − √ Y (cid:48)(cid:48) Y (cid:48)(cid:48) , (cid:104) σ z (cid:105) ss = −
11 + Y (cid:48)(cid:48) , (54)but now with Y (cid:48)(cid:48) ≡ ε d (cid:112) / ( κγ )[1 + ( g/ε d ) (cid:104) ˜ σ (cid:105) ss ].For g/ε d (cid:28) ε d /γ s (cid:28)
1, one recovers the semiclas-sical dynamics predicted by Eqs. (4). When γ s →
0, weobtain Y (cid:48)(cid:48) γ s → ≡ ε d (cid:115) κγ (cid:20) −
11 + 2( ε d /g ) (cid:21) , (55)tending to zero for small ratios ε d /g . The effect of re-ducing the dimensionless drive amplitude Y (cid:48)(cid:48) γ s → whenincreasing the ratio g/ε d is reflected in the intensity cor-relation functions for sideways scattering of Fig. 5, wherenumerical results from the solution of the ME (1) arecompared to the analytical expression of Eq. (45), withthe appropriate scaled amplitude, taken from Eq. (55).The long-time approach of the second-order coherencefunction for sideways scattering to unity, as depicted incurve (i) of the main plot (for g/ε d (cid:28) g/γ ∼ ε d /κ ) , which is a sign of departure from the va-lidity of the adiabatic elimination of the intracavity fieldand the mean-field dynamics of free-space resonance flu-orescence. Otherwise, the two sets of curves are in goodagreement.Prompted by this semiclassical argument, we will nowcompare the probability to find the external two-levelatom in the excited state in single quantum trajecto-ries unravelling the full ME (1), to the resonance fluo-rescence corresponding to the equations of motion (50a)-(50c). We assume a factorization of moments wherebythe atomic polarization and inversion of the internal two-level atom are kept equal to their steady-state values atall times. In other words, we compare to the solution ofa reduced ME where a coherent field drives the externalatom, with an amplitude set by the mean-field steady-state operator averages of the atom inside the cavity. Theresults are depicted in Fig. 6, where we can observe thata decreasing spontaneous emission rate γ brings the in-creasingly coherent Rabi oscillations in phase with themonitored output of resonance fluorescence. The oscilla-tions depicted in the curves (ii), following from unravel-ling the ME of the free-space resonance fluorescence, cor-respond to the steady-state solution of the optical Blochequations —as given in Eq. (54) —for the scaled driveamplitude Y (cid:48)(cid:48) . This amplitude is in turn defined from theparameters employed for the solution of the full ME (1),unravelled when producing the curves (i). In the mean-while, the inversion for the atom inside the cavity (inthe cascaded configuration) remains virtually fixed at itssteady-state value (cid:104) σ z (cid:105) ss ≈ − .
95, yielding a very small probability of finding the atom in the excited state —inline with the weak-excitation limit of Sec. III C 1 —as weexpect in the bad-cavity limit of the JC interaction we arehere considering. By lowering substantially the photonloss rate 2 κ with respect to the coupling strength g andconsidering the limit of zero spontaneous emission, weaccess the strong-coupling regime which is not amenableto perturbation theory: there, modifications occur at thelevel of coherent quantum dynamics due to the significantJC nonlinearity (Sec. 13.3 of [39]) transmitted throughto the monitored output. V. QUANTUM-FLUCTUATION BIMODALSWITCHING DRIVING AN EXTERNALTWO-LEVEL ATOM
The strong-coupling regime is defined by the condi-tion g/κ (cid:29)
1; here, g/ ε d are of the same order ofmagnitude, while we also assume that the internal atomis not radiatively coupled to the modes of the vacuumfield ( γ s = 0), carrying on from Sec. II. By doing sowe reach the limit of “zero system size” γ / (8 g ) → ε d = g/ ε d ≥ g/
2, the neoclassical (see Sec. IIC of [14])steady-state intracavity field is bimodal according to theexpression [19, 46]˜ α ss = ε d κ (cid:34) − (cid:18) g ε d (cid:19) (cid:35) ± i g κ (cid:115) − (cid:18) g ε d (cid:19) , (56)identifying a complex-conjugate pair of state amplitudes.In the strong-coupling limit, where g/κ (cid:29)
1, and suffi-ciently away from the critical point, the mean-field so-lutions (6) for the two-level atom outside the cavity are(for Γ κ/γ ∼
1, guaranteeing | Y | (cid:29) β , ss ≈ − √ Y | Y | = − (cid:114) γ Γ κ α ss | α ss | , (57)yielding˜ β , ss ≈ − (cid:114) γ Γ κ (cid:115) − (cid:18) g ε d (cid:19) ± i g ε d × (cid:40)(cid:16) ε d κ (cid:17) (cid:34) − (cid:18) g ε d (cid:19) (cid:35)(cid:41) − / , (58)while ζ , ss ≈ − | Y | = − γ κ (cid:40)(cid:16) ε d κ (cid:17) (cid:34) − (cid:18) g ε d (cid:19) (cid:35)(cid:41) − . (59)4 FIG. 8.
Driving the external atom above and below the critical point of symmetry breaking.
Time-dependent averages (cid:104) σ y ( t ) (cid:105) of the slowly varying y -polarization component for the atom inside (in solid purple line) and outside (in green dots) the cavity,extracted from a single quantum trajectory. For the quantum trajectory depicted in the top panel, ε d /g = 0 . bottom panel, ε d /g = 0 . Q ( x + iy ) of the intracavityfield, with the corresponding steady-state photon number given underneath. The remaining parameters are: g/κ = 100, γ/κ = 0 .
004 and Γ = 0 .
95. As in Fig. 7, for both quantum trajectories the same seed to the random-number generator andinitial conditions were used; both atoms were initialized in their ground states, and the cavity field in the Fock state | n = 1 (cid:105) . For the two-level atom inside the cavity, the correspond-ing quantities read [see Eqs. (4b)-(4c) and (5) of Sec.II]˜ β , ss = ± i ˜ α ss | ˜ α ss | = − g ε d ± i (cid:115) − (cid:18) g ε d (cid:19) , ζ , ss = 0 . (60)Defining λ ≡ ( g/ ε d ) , we observe that at the criticalpoint, λ c = 1, both the field amplitudes ˜ α ss and the atomic polarization states ˜ β (1 , , ss display a pitchfork-like bifurcation. The complex order parameter ˜ α ss of Eq.(56) points to a scaling of the form ( λ − λ c ) / , iden-tifying a critical exponent equal to 1 /
2. However, themoduli of the external polarization, | ˜ β , ss | , and inver-sion, ζ , ss , scale instead as ( λ − λ c ) − / and ( λ − λ c ) − ,respectively, for large excitation amplitudes Y away fromthe critical point, while the modulus of the internal po-larization, | ˜ β , ss | is equal to 1 / ζ , ss remains fixed at zero.Quantum-fluctuation bistable switching above thresh-old, stabilizing the mean-field states as attractors [19], isdepicted in Fig. 7 for low and high degree of focusing tothe external atom, and a steady-state average intracavityphoton number (cid:104) n (cid:105) ss ≡ (cid:104) a † a (cid:105) ss ≈
43. The imaginary partof the polarization-operator average has opposite signsfor the two atoms, 1 and 2, as correctly predicted by thesemiclassical Eqs. (57) and (60). The two metastablestates with conjugate polarization have a lifetime thatsignificantly exceeds the κ − -timescale set by dissipation.Since γ/κ (cid:29)
1, one cannot distinguish individual Rabioscillations (for the external atom) during the lifetimeof each of the two states of polarization with oppositeimaginary parts. Increasing the degree of focusing fromΓ = 0 . . ≤ κt ≤
80) in Fig.7(a) when focusing to the external atom is stronger [Fig.7(b)]; in the latter case, the Bloch vector explores largerregions of the unit sphere.In Fig. 8, we observe the distinction between driv-ing the external two-level atom by a state that fluctuatesand by a state with nonzero mean-field amplitude ˜ α ss slightly below and above the critical point, respectively.Quantum-fluctuation switching between the two semi-classical solutions of the external atom, as seen in bothframes, is simultaneously accompanied by phase jumpsof the cavity-field amplitude. Hence, the two subsys-tems become phase correlated via radiative coupling tothe same reservoir. Here, γ/κ (cid:28) Y than the one used in Fig. 7); there-fore, individual Rabi oscillations are visible, with a pe-riod which is comparable to the lifetime of the metastablestates. Switching events to a different metastable state—and accordingly to a different excitation ladder of theJC spectrum [14, 19] —in the top panel of Fig. 8, arecorrelated with a clear disruption of the Rabi-oscillationcycles. This is a disruption of the phase arising from theswitching of the drive-field phase, in contrast to a dis-ruption due to spontaneous emission —which resets theatomic oscillation to the ground state —seen in trajec-tories of regular resonance fluorescence. For the bottompanel of Fig. 8, the scaled amplitude of the field driv-ing external atom is Y = 0, since the cavity is drivenby a field with an amplitude below its threshold value(whence ˜ α ss = 0). Nevertheless, coherent oscillations inthe imaginary part of the atomic-polarization average,though visibly more distorted, can still be discerned, to-gether with their disruptions, correlated with transitionsto a different metastable state. We need to emphasizehere that this is a regime of pronounced fluctuations, be- ing around the critical point of a second-order quantumphase transition in zero dimensions, where a departurefrom the mean-field predictions is to be expected whenone monitors directly the output of the bistable oscillator(see Sec. 16.3.6 of [39]). In fact, the Q -function for thecavity-field distribution in Fig. 8 evidences two maximafor complex-conjugate amplitudes, before the appearanceof the expected mean-field bifurcation; this suggests aconditional evolution roughly of the type described inEqs. (50-52) of [19], affected and monitored by the ex-ternal two-level atom, in spite of driving below threshold. VI. CONCLUDING DISCUSSION ANDFUTURE WORK
In this work, we have derived analytical results for thestatistics of the forwards and sideways emission channelby means of a mapping to an atom inside the coherentlydriven cavity coupled to the supported resonant modewith a strength determined by the dissipation rates ofthe initial cascaded-system configuration. For this pur-pose, we at first set to zero the coupling strength be-tween the cavity and the two-level atom comprising theJC oscillator, developing a treatment which relied on sev-eral well-known results from ordinary resonance fluores-cence and the bad-cavity limit of QED. We then broughtprogressively the JC dynamics into play, reflected in thenonlinearity of the nonclassical light emanating from thefirst subsystem coupled to an atomic scatterer. Througha succession of coupled equations of motion for the twocascaded subsystems, we employed a semiclassical andnumerical investigation to compare the solution of the fullME with free-space resonance fluorescence for the atom-scatterer lying outside the cavity. The latter is driven byan effective field whose amplitude is determined by thesteady-state polarization and inversion averages of theatom inside the cavity.By promoting the intracavity coupling strength be-tween the two constituents of the JC oscillator we haveeventually moved to a regime where the quantum natureof the field driving the external atom cannot be reducedto a mean-field or perturbative description. The out-put of the bistable oscillator in the region of the criticalpoint, forming part of the forwards-scattering channel,involves actively both coupled quantum degrees of free-dom in the JC interaction (which is not the case when g/κ (cid:28) , γ s /κ (cid:28) γ = 0), wehave also found that the conditional evolution of phasebistability in the coupled degrees of freedom is differentfrom the trajectories depicted in Fig. 7 for the same seed6to the random-number generator. Therefore, our analy-sis has been largely based upon the coherence propertiesof resonance fluorescence and the associated Rabi oscilla-tions, with reference to the different statistics of forwardsand sideways scattering for an external atom coupled tothe coherently driven cavity; such a disparity arises dueto the interference with the coherent cavity output field,and varies with the degree of focusing. Disruptions inthese coherent oscillations, emerging on approaching thecritical point, are correlated with the switching eventsrealizing phase bistability in the individual trajectoriesunravelling the evolution of the full system density ma-trix (source plus target).Let us pause here to comment a little bit further onthe mapping to the bad-cavity limit, which, as we havealready pointed out, effectively amounts to placing theexternal atom inside the cavity and dealing with JC dy-namics, even if perturbatively. In the cascaded-systemconfiguration, the forwards dipole scattering is made sig-nificant by strongly focusing the cavity output onto theexternal emitter so that a significant fraction of the 4 π solid angle seen by the atom is occupied by the incom-ing mode. In this case, the drive field must be mode-matched to the dipole mode that naturally arises in thecoupling of a dipole transition to the electromagnetic fieldin free space —this remains a considerable challenge onthe experimental front. There is no Purcell enhance-ment involved in generating the mode overlap and, inthat sense, our mapping is rather formal. Nevertheless,the effective “enhancement” of the sideways spontaneousemission rate to its full 4 π value, points us to the ME (11) in which a classical field is driving the target atom(a central message of [35]) where the trace over the cavityfield has worked out to yield exactly the total emissionrate γ .Following the resurgence of interest in the emissionproperties of a target driven by an explicitly quantumsource, an immediate extension of our work could touchupon driving the external atom with the output of aJC oscillator in the regime of photon blockade, probingits persistence in the “thermodynamic limit” of strongcoupling, where the scale parameter grows with the cou-pling strength. Here, the output is a stream of distinctphoton assemblies, corresponding to the multiphotonresonances responsible for the blockade, and comprises asource of manifestly nonclassical light (depending on thestrength of the driving, both bunching and antibunchingmay occur —see Sec. 3.3 of [47]). The experiment wouldthen focus on the radiation properties of such a sourcein an environment containing the external atom-target—another quantum nonlinear oscillator —monitoringthe composite conditional evolution via the two channelsof the distributed forwards and sideways emission. ACKNOWLEDGMENTS
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In this Appendix, we calculate first and second-order correlation functions for the source field of the atom outsidethe cavity, which we then use to derive expressions for the incoherent spectra and spectra of squeezing of the twochannels in Secs. III A and III B, respectively, as well as their second-order coherence properties in Sec. III C 2.The expectation value of the fluctuations in s (cid:62) ≡ (˜ σ − , ˜ σ + , σ z ) —defined in a frame rotating with ω A —populatingthe vector of fluctuation operators ∆ s ≡ ∆˜ σ − ∆˜ σ + ∆ σ z ≡ ˜ σ − ˜ σ + σ z − (cid:104) ˜ σ − (cid:105) ss (cid:104) ˜ σ + (cid:105) ss (cid:104) σ z (cid:105) ss , (A.1)evolves in time according to ddt (cid:104) ∆ s (cid:105) = M (cid:104) ∆ s (cid:105) . (A.2)The matrix M is provided by the Bloch equations, extracted from the ME of resonance fluorescence, dρdt = − i ω A [ σ z , ρ ] − γY √ σ + e − iω A t − σ − e iω A t , ρ ] + γ σ − ρσ + − σ + σ − ρ − ρσ + σ − ) , (A.3)as M ≡ − γ/ √ κγ Γ α ss − γ/ √ κγ Γ α ss − √ κγ Γ α ss − √ κγ Γ α ss − γ = − γ − Y / √
20 1 − Y / √ √ Y √ Y . (A.4)The quantum regression formula, then, dictates the evolution of the first-order correlation function, (cid:104) ∆˜ σ + (0)∆ s ( τ ) (cid:105) ss , ddτ (cid:104) ∆˜ σ + (0)∆ s ( τ ) (cid:105) ss = M (cid:104) ∆˜ σ + (0)∆ s ( τ ) (cid:105) ss . (A.5)The initial conditions are given by the steady-state values (cid:104) ∆˜ σ + ∆ s (cid:105) ss = (1 + (cid:104) σ z (cid:105) ss ) − (cid:104) ˜ σ + (cid:105) ss (cid:104) ˜ σ − (cid:105) ss − (cid:104) ˜ σ + (cid:105) − (cid:104) ˜ σ + (cid:105) ss (1 + (cid:104) σ z (cid:105) ss ) = 12 Y (1 + Y ) Y − √ Y (A.6)8The formal solution to Eq. (A.5) is given by (cid:104) ∆˜ σ + (0)∆ s ( τ ) (cid:105) ss = S − exp( λ τ ) S (cid:104) ∆˜ σ + ∆ s (cid:105) ss , (A.7)where λ ≡ SM S − = diag( λ , λ , λ ), with λ = − γ/ λ , = − γ/ ± δ , is a diagonal matrix formed bythe eigenvalues of M . Here, the shift δ captures the dependence of the eigenvalues on the driving strength, and isdefined as δ ≡ ( γ/ √ − Y . There is one special point where M becomes non-diagonalizable, namely when δ = 0[or Y = 1 / (2 √ exceptional point , at which two of the eigenvalues, λ and λ , coalesce. Atthat point, these two eigenvalues switch from purely real (relaxing response) to complex (decaying and oscillatoryresponse), which coincides with the formation of the Mollow triplet depicted in Fig. 2. Since M is a non-Hermitianmatrix, its left and right eigenvectors are in principle not equivalent; the rows of S are then populated by the lefteigenvectors of M , while the columns of S − are populated by the right eigenvectors of M . The right eigenvectorcorresponding to the eigenvalue λ is e = (1 / √ , − , T (which is also equal to the transpose of the correspondingleft eigenvector). The remaining right and left eigenvectors corresponding to the eigenvalues λ , assume the form e = c (1 , , A ) T , e = c (1 , , A ) T and e (cid:48) = c (cid:48) (1 , , A (cid:48) ), e (cid:48) = c (cid:48) (1 , , A (cid:48) ), respectively. The coefficients featuringin the third components of the eigenvectors read: A = (cid:16) δ − γ (cid:17) √ Y γ , A = − (cid:16) δ + γ (cid:17) √ Y γ , A (cid:48) = (cid:16) γ − δ (cid:17) √ Y γ , A (cid:48) = (cid:16) γ δ (cid:17) √ Y γ . (A.8)Following then the prescription, we write Eq. (A.7) in the form (cid:104) ∆˜ σ + (0)∆ s ( τ ) (cid:105) ss = / √ c c − / √ c c A c A c exp( λ τ ) / √ − / √ c (cid:48) c (cid:48) A (cid:48) c (cid:48) c (cid:48) c (cid:48) A (cid:48) c (cid:48) (cid:104) ∆˜ σ + ∆ s (cid:105) ss , where exp( λ τ ) = e − ( γ/ τ e − (3 γ/ − δ ) τ
00 0 e − (3 γ/ δ ) τ . The orthonormality of right and left eigenvectors produces the system of equations A A (cid:48) c c (cid:48) + A A (cid:48) c c (cid:48) = 1 , (A.9a) A (cid:48) c c (cid:48) + A (cid:48) c c (cid:48) = 0 . (A.9b)Solving the above system of equations yields c c (cid:48) = [ A (cid:48) ( A − A )] − = (1 / γ/ (4 δ )] , A (cid:48) c c (cid:48) = − A (cid:48) c c (cid:48) = Y γ/ (4 √ δ ) , c c (cid:48) = (1 / − γ/ (4 δ )] . Then, for the various first-order correlation functions we obtain (see also [5, 44]) (cid:104) ∆˜ σ + (0)∆˜ σ − ( τ ) (cid:105) ss = 12 Y (1 + Y ) (cid:40) (cid:20) Y −
12 ( − (cid:21) e − ( γ/ τ + (cid:104) c c (cid:48) Y + c c (cid:48) ( −
1) + A (cid:48) c c (cid:48) ( √ Y ) (cid:105) e − (3 γ/ − δ ) τ + (cid:104) c c (cid:48) Y + c c (cid:48) ( −
1) + A (cid:48) c c (cid:48) ( √ Y ) (cid:105) e − (3 γ/ δ ) τ (cid:41) = 14 Y Y e − ( γ/ τ − Y (1 + Y ) (cid:104) − Y + (cid:16) γ δ (cid:17) (1 − Y ) (cid:105) e − (3 γ/ − δ ) τ − Y (1 + Y ) (cid:104) − Y − (cid:16) γ δ (cid:17) (1 − Y ) (cid:105) e − (3 γ/ δ ) τ , (A.10)9 (cid:104) ∆˜ σ + (0)∆˜ σ + ( τ ) (cid:105) ss = 12 Y (1 + Y ) (cid:40) (cid:20) − Y + 12 ( − (cid:21) e − ( γ/ τ + (cid:104) c c (cid:48) Y + c c (cid:48) ( −
1) + A (cid:48) c c (cid:48) ( √ Y ) (cid:105) e − (3 γ/ − δ ) τ + (cid:104) c c (cid:48) Y + c c (cid:48) ( −
1) + A (cid:48) c c (cid:48) ( √ Y ) (cid:105) e − (3 γ/ δ ) τ (cid:41) = − Y Y e − ( γ/ τ − Y (1 + Y ) (cid:104) − Y + (cid:16) γ δ (cid:17) (1 − Y ) (cid:105) e − (3 γ/ − δ ) τ − Y (1 + Y ) (cid:104) − Y − (cid:16) γ δ (cid:17) (1 − Y ) (cid:105) e − (3 γ/ δ ) τ , (A.11) (cid:104) ∆˜ σ + (0)∆ σ z ( τ ) (cid:105) ss = 12 Y (1 + Y ) (cid:40) (cid:104) A c c (cid:48) Y + A c c (cid:48) ( −
1) + A A (cid:48) c c (cid:48) ( √ Y ) (cid:105) e − (3 γ/ − δ ) τ + (cid:104) A c c (cid:48) Y + A c c (cid:48) ( −
1) + A A (cid:48) c c (cid:48) ( √ Y ) (cid:105) e − (3 γ/ δ ) τ (cid:41) = 12 √ Y (1 + Y ) (cid:40) (cid:104) − (cid:16) γ δ (cid:17) (2 − Y ) (cid:105) e − (3 γ/ − δ ) τ + (cid:104) (cid:16) γ δ (cid:17) (2 − Y ) (cid:105) e − (3 γ/ δ ) τ (cid:41) . (A.12)We also note that (cid:104) ∆˜ σ + (0)∆ s ( τ ) (cid:105) ss = (cid:104) ˜ σ + (0)˜ σ − ( τ ) (cid:105) ss (cid:104) ˜ σ + (0)˜ σ + ( τ ) (cid:105) ss (cid:104) ˜ σ + (0) σ z ( τ ) (cid:105) ss − (cid:104) ˜ σ + (cid:105) ss (cid:104) ˜ σ − (cid:105) ss (cid:104) ˜ σ + (cid:105) (cid:104) ˜ σ + (cid:105) ss (cid:104) σ z (cid:105) ss = (cid:104) ˜ σ + (0)˜ σ − ( τ ) (cid:105) ss (cid:104) ˜ σ + (0)˜ σ + ( τ ) (cid:105) ss (cid:104) ˜ σ + (0) σ z ( τ ) (cid:105) ss − Y (1 + Y ) Y Y / √ . We now calculate the (normalized) second-order correlation function, g (2)ss ( τ ) = (cid:104) (cid:104) σ + σ − (cid:105) ss + lim τ →∞ (cid:104) σ + (0) σ z ( τ ) σ − (0) (cid:105) ss (cid:105) − [ (cid:104) σ + σ − (cid:105) ss + (cid:104) σ + (0) σ z ( τ ) σ − (0) (cid:105) ss ] (A.13)requiring the third component of the vector (cid:104) σ + (0) s ( τ ) σ − (0) (cid:105) ss . Using once more the quantum regression theorem,this vector evaluates to (cid:104) σ + (0) s ( τ ) σ − (0) (cid:105) ss = (cid:104) σ + σ − (cid:105) ss (cid:104) s ( τ ) (cid:105) ρ (0)= | (cid:105)(cid:104) | = 12 Y Y (cid:104) s ( τ ) (cid:105) ρ (0)= | (cid:105)(cid:104) | , giving (cid:104) σ + (0) σ z ( τ ) σ − (0) (cid:105) ss = − Y (1 + Y ) (cid:20) Y e − (3 γ/ τ (cid:18) cosh δτ + 3 γ δ sinh δτ (cid:19)(cid:21) , (A.14)and (cid:104) σ + (0)˜ σ ± ( τ ) σ − (0) (cid:105) ss = − √ Y (1 + Y ) (cid:20) − e − (3 γ/ τ (cid:18) cosh δτ + 3 γ δ sinh δτ (cid:19)(cid:21) − √ Y Y e − (3 γ/ τ γ δ sinh δτ. (A.15)Finally, g (2)ss ( τ ) = (2 (cid:104) σ + σ − (cid:105) ss ) − (cid:104) (cid:104) σ z ( τ ) (cid:105) ρ (0)= | (cid:105)(cid:104) | (cid:105) = 1 − e − (3 γ/ τ (cid:18) cosh δτ + 3 γ δ sinh δτ (cid:19) ..