Fermi's golden rule for spontaneous emission in absorptive and amplifying media
Sebastian Franke, Juanjuan Ren, Marten Richter, Andreas Knorr, Stephen Hughes
.. Fermi’s golden rule for spontaneous emission in absorptive and amplifying media
Sebastian Franke,
1, 2, ∗ Juanjuan Ren, Marten Richter, Andreas Knorr, and Stephen Hughes Technische Universität Berlin, Institut für Theoretische Physik,Nichtlineare Optik und Quantenelektronik, Hardenbergstraße 36, 10623 Berlin, Germany Department of Physics, Engineering Physics, and Astronomy,Queen’s University, Kingston, Ontario K7L 3N6, Canada (Dated: February 26, 2021)We demonstrate a fundamental breakdown of the photonic spontaneous emission (SE) formuladerived using Fermi’s golden rule, in absorptive and amplifying media, where one assumes theSE rate scales with the local photon density of states, an approach often used in more complex,semiclassical nanophotonics simulations. Using a rigorous quantization of the macroscopic Maxwellequations in the presence of arbitrary linear media, we derive a corrected Fermi’s golden rule andmaster equation for a quantum two-level system (TLS) yielding the direct appearance of a quantumpumping term and a modified decay rate that is net positive. We show rigorous numerical resultsof the temporal dynamics of the TLS for an example of two coupled microdisk resonators, forminga gain-loss medium, and demonstrate the clear failure of the semiclassical theory.
Spontaneous emission (SE) of a two-level system (TLS)has been a fundamental topic since the birth of quantumelectrodynamics and is one of the standard metricsfor current applications in nanophotonic and nanoplas-monic systems interacting with quantum emitters, suchas molecules and quantum dots. Photonic engineeringallows one to modify the local density of states (LDOS)which in turn leads to an enhancement or a suppressionof the SE decay rate, e.g., by using photonic crystals .The SE rate, Γ , is associated with the transition prob-abilities with respect to a pertubation ˆ V , formalized intoFermi’s golden rule (FGR). In terms of a TLS quantumdipole interaction and the photonic LDOS, the estab-lished FGR in the photonics community is Γ LDOS ( r a ) = 2 (cid:126) (cid:15) d · Im[ G ( r a , r a , ω a )] · d ∗ , (1)where G is the photonic Green function at the posi-tion r a of the quantum emitter, with a dipole moment d = (cid:104) e | ˆ d | g (cid:105) = d n d and frequency ω a . From a quantumfield theory, the SE rate is also connected to the vacuumfluctuations of the quantized electric field ( ˆ E ), through Γ VF ( r a ) = 2 π (cid:126) d · (cid:104) | [ ˆ E ( r a , ω a ) , ˆ E † ( r a , ω a )] | (cid:105) · d ∗ , (2)where | (cid:105) is the vacuum state. The interpretation of SEdecay can thus be thought of as a consequence of radia-tion reaction or from vacuum fluctuations , which areidentical in the case of lossy media, as both simply de-pend on the LDOS in this case. Thus, the theory of SEdecay can often approximate the TLS as a harmonic os-cillator (bosonic), which yields the same result as a weakexcitation approximation.Consequently, in the literature, Eqs. (1) and (2) arecommonly used as a basis for calculating important op-tical figures of merit, such as the Purcell factor and theradiative β -factor, even with gain media . Indeed, Purcell’s formula is simply a special case of Eqs. (1)-(2)for a single electromagnetic mode, aligned to the dipoleand frequency of the TLS. Figure 1. Schematic of a quantum emitter (red dot) at po-sition r a , treated as a TLS and weakly interacting with anamplifying ( (cid:15) I ( r , ω ) < ) and lossy ( (cid:15) I ( r , ω ) > ) spatially in-homogeneous electromagnetic environment (grey areas). Theemitter is pumped by the amplifying medium with a rate Γ gain and its free spontaneous decay Γ is enhanced by Γ loss / Γ .The environment also induces a gain and loss related pho-tonic Lamb shift ∆ LS of the TLS frequency ω a . In this Letter, we show that this LDOS view ofSE is in general incorrect , and introduce a revi-sion to the usual FGR for the SE decay of a point dipole(Eq. (1)), for the case of dielectric media with amplifyingregions. From a macroscopic quantum theory of loss andgain, we derive the master equation of a TLS interactingwith its electromagnetic environment in the weak cou-pling limit through a second-order Born-Markov approx-imation, which gives rise to terms associated with thepump (originating from the gain region) and the decay(originating from the loss region) of the emitter. Using acoupled gain-loss resonator structure, we show that whilethe common Maxwell treatment of SE leads to negative a r X i v : . [ phy s i c s . op ti c s ] F e b Purcell factors and nonphysical temporal evolutions ofthe TLS, our scheme leads to a positive Purcell factorwith non-vanishing steady states . We explain why thecommon assumption, that the SE rate scales with theLDOS fails , and present a resolution to this breakdown.We show that the SE in such media cannot be used infurther semiclassical calculations, but must be treatedquantum mechanically, and that the usual limit of a har-monic oscillator model fails. We then present the correctFGR for describing SE (which also includes absorptionprocesses from the gain medium) in such media.
Theory.
We consider the general Hamiltonian H = H a + H B + H I of a TLS interacting with the electromag-netic field in a lossy and amplifying media , H a = (cid:126) ω a σ + σ − , (3a) H B = (cid:126) (cid:90) d r sgn( (cid:15) I ) (cid:90) ∞ d ω ω b † ( r , ω ) · b ( r , ω ) , (3b) H I = − (cid:20) σ + (cid:90) ∞ d ω d · ˆ E ( r a , ω ) + H . a . (cid:21) , (3c)where the spatial integral is over all (real) space, sgn isthe sign function, (cid:15) I is the imaginary part of the per-mittivity, ω a is the frequency of the TLS, and σ ± arethe Pauli operators acting as lowering and raising oper-ators of the TLS; b ( † ) ( r , ω ) are the bosonic annihilation(creation) operators of the medium and the electromag-netic degrees of freedom. The medium-assisted electricfield operator ˆ E ( r a , ω ) fulfills the Helmholtz equation, (cid:2) ∇ × ∇ × − (cid:15) ( r , ω ) ω /c (cid:3) ˆ E ( r , ω ) = iωµ ˆ j N ( r , ω ) , where ˆ j N ( r , ω ) is the current noise operator, which preservesthe fundamental QED commutation relation for arbitrarymedia . In the presence of lossy and amplifying media, itis given as ˆ j N ( r , ω )= ω (cid:112) (cid:126) (cid:15) | (cid:15) I ( r , ω ) | /π [Θ( (cid:15) I )ˆ b ( r , ω ) +Θ( − (cid:15) I )ˆ b † ( r , ω )] , where Θ[ (cid:15) I ] ( Θ[ − (cid:15) I ] ) is the Heavi-side function with respect to the spatial region, R − V gain ( V gain ), with passive (active) dielectric permittivity (cid:15) I ( r , ω ) > ( (cid:15) I ( r , ω ) < ), cf. Fig. 1.The Helmholtz equation has the formal source-field solution ˆ E ( r , ω )= i (cid:82) d sG ( r , s , ω ) · ˆ j N ( s , ω ) / ( ω(cid:15) ) ,where G ( r , s , ω ) is the photon Greenfunction of the medium, which satisfies [ ∇ × ∇ ×− (cid:15) ( r , ω ) ω /c ] G ( r , s , ω )= ω δ ( r − s ) /c , withsuitable radiation boundary conditions. The macroscopicGreen function quantization is valid under strict linearresponse of the electromagnetic field in the dielectricmedium (cf. Supplemental Material, Ref. 20). Breakdown of the SE rate from vacuum fluctuations .To derive the correct SE for amplifying media, we look athe time derivative of the density operator (cid:104) f, | ρ | f, (cid:105) ,with a final emitter state | f (cid:105) at time t with respect to ainitial emitter state | i, (cid:105) ( | f (cid:105) (cid:54) = | i (cid:105) ) at time t . Treat-ing the interaction between the emitter and the pho-tons pertubatively up to the second order leads to FGR,and we derive the transition rate from | e, (cid:105) to | g, (cid:105) as Γ ( e, → ( g, ) ≡ Γ loss , with Γ loss = 2 (cid:126) (cid:15) d · K ( r a , r a , ω a ) · d ∗ , (4)where K ( r , r (cid:48) ) = (cid:82) R − V gain d s (cid:15) I ( s ) G ( r , s ) · G ∗ ( s , r (cid:48) ) , andin the purely lossy case ( V gain → ∅ ) leads to theLDOS-SE formula, Eq. (1), through the Green iden-tity (cid:82) R d s (cid:15) I ( s ) G ( r , s ) · G ∗ ( s , r (cid:48) ) = Im[ G ( r , r (cid:48) )] . Sim-ilarly, the transition rate from | g, (cid:105) to | e, (cid:105) is obtained, Γ ( g, → ( e, ) ≡ Γ gain , with Γ gain ( r a ) = 2 (cid:126) (cid:15) d · I ( r a , r a , ω a ) · d ∗ , (5)where I ( r , r (cid:48) ) = (cid:82) V gain d s | (cid:15) I ( s ) | G ( r , s ) · G ∗ ( s , r (cid:48) ) . Notethat Γ gain ( r a ) ≥ , since | (cid:15) I | ≥ and the integral canbe recast into a form (cid:82) | f | by using properties of theouter product and the reciprocity theorem, G ( r , r (cid:48) ) = G T ( r (cid:48) , r ) . This same holds true for Γ loss .First, we recognize that the form of Γ gain ( r a ) and Γ loss ( r a ) is fundamentally different from the usually as-sumed LDOS-SE formula in the photonics community;however, we can relate them to Eq. (1), so that Γ loss ( r a ) = Γ LDOS ( r a ) + Γ gain ( r a ) , (6)and from Eq. (1) and (4), we deduce that Γ LDOS =Γ loss − Γ gain , which yields a result that is clearly incon-sistent with the rigorous treatment of FGR. Second, wecan investigate the contributions associated with vacuumfluctuations. Inserting the source field expansion of theelectric field into Eq. (2), then Γ VF = Γ LDOS , identicalto the results obtained from the commonly used FGR. Wenote that this is only the case, because the more generalcommutator instead of the anti-normal ordered operatorexpression is used as the basis for the quantum (vacuum)fluctuations, Eq. (2). This is in contrast to some resultsin Ref. 15, where Eq. (2) (replacing [ ˆ E , ˆ E † ] by ˆ E ˆ E † ) wasdetermined to equal Eq. (1), as a suppression of SE. Thisis clearly not correct. Thus, the semi-classical LDOS-SEmodel is related to quantum (vacuum) fluctuations in thepresence of gain, but it is no longer related to any physi-cally relevant rate. Next, the impact of the rigorous FGRtreatment on the dynamics of the TLS is demonstrated. Dynamical equations of TLS densities and dephasing .To obtain a more general description of the TLS dynam-ics in the presence of gain, we use the density matrixpicture, starting with the Liouville-von Neumann equa-tion ∂ t ρ = − i [ H, ρ ] / (cid:126) , where ρ is the total density oper-ator and H is Hamiltonian from Eq. (3). Applying thesecond-order Born Markov approximation, we obtain thereduced TLS master equation, whose projections on thebasis elements of the TLS subspace are ∂ t ρ ee a = − Γ loss ρ ee a + Γ gain ρ gg a , ∂ t ρ gg a = − ∂ t ρ ee a , (7a) ∂ t ρ eg a = − i [ ω a + ∆ LS ] ρ eg a − [Γ loss + Γ gain ] ρ eg a , (7b)where ρ a = tr B ρ is the trace with respect to the elec-tromagnetic degrees of freedom, ρ ee a = (cid:104) e | ρ a | e (cid:105) ( ρ gg a = ( ω a − ω + ) /γ + P u r ce ll f a c t o r F P ×10 ω − ω I ω II ω III F iso p × F loss P F LDOS P Γ gain / Γ F dip P Figure 2. Left: Schematic of a z -polarized quantum dipole with dipole moment d placed between a lossy and an amplifyingmicrodisk resonator, as well as the two hybridized QNMs (2D model, areas A gain / loss with disk radius R = 5 µ m and gap distance d gap = 1155 nm). The dielectric constants of the resonators are assumed to be (cid:15) loss = (2 + 10 − i ) and (cid:15) gain = (2 − · − i ) ,giving rise to nearly degenerate QNM eigenfrequencies (cid:126) ˜ ω + [eV] = 0 . − . · − i and (cid:126) ˜ ω − [eV] = 0 . − . · − i (cf. Ref. 21 for numerical details on the QNM calculations). Right: Purcell factor of the emitter-cavity systemas function of TLS frequency shifted with respect to real part of the QNM hybrid frequencies ˜ ω + , showing results obtainedfrom LDOS-SE model for the isolated lossy disk (grey dashed) and hybrid structure (black), Eq. (1), the full dipole solution,as well as the rigorous FGR (magenta, Eq. (4)). Additionally, the contribution of the pump from FGR is shown (blue). (cid:104) g | ρ a | g (cid:105) ) is the excited (ground) state density and ρ eg a = (cid:104) g | ρ a | e (cid:105) is the dephasing with Lamb shift ∆ LS = d · Re[ G ( r a , r a , ω a )] · d / ( (cid:126) (cid:15) ) . Compared to the LDOS-SEmodel, the ground state and excited state densities arecoupled through the loss and gain rate. Moreover, the de-phasing decays with Γ loss +Γ gain rather then Γ loss − Γ gain .Interestingly, the effective Lamb shift ∆ LS of the dephas-ing oscillation is identical to the purely lossy case.Thus, we find three fundamental corrections to com-mon formulas in the current literature : (i) the SE rateobtained from the heuristic FGR and vacuum fluctua-tions is corrected by an additional term − Γ gain ; (ii)a non-vanishing transition process from the equilibriumground state | g, (cid:105) to the excited states with one gain me-dia excitations missing | e, gain (cid:105) occurs; (iii) the gain andloss rate appear as coupling constants in the dynamicalequations of the TLS densities, which is clearly missing inthe usual models , which associate all interactionprocesses as decay terms. Numerical results for coupled microdisk resonators . Toexplicitly demonstrate the impact and consequences ofthe corrected formula for SE, we now investigate the Pur-cell enhancement and dynamics of a TLS in an examplarycavity-QED setup, where the quantum dipole is placed inthe gap between a lossy and an amplifying resonator, asillustrated in Fig. 2. Such resonators are commonly usedto explore the enhanced sensing capabilities, lasing, andunidirectional transmission near exceptional points .For calculating the quantum parameters appearing inEq. (7), the classical photon Green function must bedetermined, which can be a tedious task for arbitraryshaped scattering structures. However, it has been shownthat for dipole positions r a and a reference position r near or inside the cavity regions, then the scattering part ofthe Green function can be accurately represented by aquasinormal mode (QNM) expansion of the form G ff ( r a , r , ω ) = (cid:88) µ A µ ( ω )˜ f µ ( r a )˜ f µ ( r ) , (8)where A µ ( ω )= ω/ (2(˜ ω µ − ω )) is the particular choice ofthe spectral coefficient, ˜ ω µ are the QNM eigenfre-quencies, and ˜ f µ ( r ) is the QNM eigenfunction, solving (cid:2) ∇ × ∇ × − (cid:15) ( r , ˜ ω µ )˜ ω µ /c (cid:3) ˜ f µ ( r )=0 together with openboundary conditions. Due to the choice of outgoing radi-ation conditions, the QNM eigenfrequencies are complexwith ˜ ω µ = ω µ − iγ µ , where γ µ is the HWHM of the QNMresonance with center frequency ω µ . For γ µ > , theQNM eigenfunctions diverge as a further consequence ofthe radiation conditions; however, since r = r a appears inthe first part of Γ loss , while r ∈ V gain appears in Γ gain , theQNM expansion is well-defined for calculating the decayand pump rates of quantum emitters near cavity regions.Taking into account the free-space contribution, one canthen formulate the first term of Γ loss by approximating G ( r a , r a ) ≈ G ( r a , r a ) + G ff ( r a , r a ) , and Γ gain by ap-proximating G ( r a , s ) ≈ G ff ( r a , s ) , where G is the Greenfunction of the homogeneous background medium .For the coupled disk resonator system, two hybridizedQNMs µ = + , − appear in the optical frequency regime,which stem from the coupling of the fundamental modesof the isolated resonators . Their respective field distri-butions is shown in Fig. 2(left), and they are used as thebasis for the expansion of Eq. (8).As a first example, we derive the Purcell enhancementfactors with respect to the free-space emission rate Γ ,i.e., F P = Γ / Γ for the different models. Besides theusual SE formulas, where Γ LDOS is the relevant decayrate (Eq. (1)), and the rigorous FGR model with rates Γ loss , Γ gain (Eq. (4), (5)), one can additionally define aPurcell factor from classical dipole-Maxwell calculationsthrough Poynting vector terms , which serves as a fur-ther calculation, that is independent from any mode ex-pansion of the photon Green function. Note that theQNM and dipole (classical) Purcell factors for an iso-lated gain resonator are negative, but gain on its owndoes not constitute linear media.The different model results as a function of TLS fre-quency are shown in Fig. 2(right). First, we note thatthe Purcell factor of the isolated lossy resonator disk (in-dicated by the grey dashed line) is roughly one order ofmagnitude smaller compared to the enhancements of thehybrid system, regardless of the model used. Second, andmost significantly, the full dipole-Maxwell solution as wellas the result obtained from the LDOS-SE model show ahighly non-Lorentzian lineshape, that produces negativePurcell factors which are clearly unphysical . Here, thegain-induced terms are taken into account as negativecontributions, that compensate the loss. This concep-tional problem is commonly ignored or never discussedin the literature, e.g., with exceptional points connectedto PT -symmetric systems . Note that although weshow negative LDOS values, the two-QNM expansionis in excellent agreement with the full classical solution(without any approximations). In contrast to the semi-classical models, the results obtained from the quantumdynamical model are strictly positive. ρ ee a ρ ee a (0) = 0 ω a = ω I ω a = ω II ω a = ω III t Γ loss ρ ee a ρ ee a (0) = 1 Figure 3. Temporal dynamics of the TLS excited state occu-pation ρ ee a with initial states ρ ee a (0) = 0 (top) and ρ ee a (0) = 1 (bottom) for the TLS frequencies indicated by vertical linesin Fig. 2 using the full non-linear master equation (solid) andthe classical model (dashed). Next, we discuss the consequences of these modelson the TLS population dynamics. We concentrate on three critical TLS frequency points that are indicatedby vertical dashed lines in Fig. 2: (i) ω a = ω I , where Γ LDOS reaches its maximum; (ii) ω a = ω II , where Γ loss reaches its maximum (and Γ LDOS its minimum), and(iii) ω a = ω III , where Γ LDOS = 0 . In the quantummodel, the temporal evolution of the densities and thedephasing is derived via Eq. (7), while in the LDOS-SEmodel, Γ loss → Γ LDOS and Γ gain → . The results areshown in Fig. 3. In the case of ρ a (0) = | g (cid:105)(cid:104) g | (TLSin the ground state initially), the quantum model pre-dicts a non-trivial time evolution, where the TLS occu-pation reaches a non-zero steady state with occupation ρ ee a ( t → ∞ ) = Γ gain / [Γ loss + Γ gain ] after few decay pro-cesses ( ∼ t Γ loss ), which is independent on the initialstate. In contrast, the LDOS-SE model obviously pre-dicts a trivial time evolution, since both the dephasingand the densities are proportional to the initial state.In the case of ρ a (0) = | e (cid:105)(cid:104) e | (TLS in the excited stateinitially), the difference is even more pronounced: whilethe result from the quantum model again reveal non-trivial but physical meaningful steady states, the LDOS-SE model predicts unphysical solutions, that can eitherlead to infinite, constant or vanishing excited state popu-lations for t → ∞ . This a consequence of the wrong inter-pretation of loss and gain-induced rates. We note that,although the dynamical equations (Eq. (7)) are formallysimilar to typically laser rate equations with thermal ex-citation, the steady states and interpretation of the gainsources is fundamentally different; while in the here pre-sented case, two different reservoirs (gain and loss) arecoupled to the emitter, the pumping in the typical mod-els comes through a finite thermal photon number froma lossy environment.We stress the appearance of unphysical solutions areconnected to the classical limit of the quantum equa-tions. This is achieved by assuming a weak excitationapproximation. In this limit, the excited state densityis ρ ee a ( t ) = exp( − Γ LDOS t ) ρ ee a (0) + P ( t ) , where P ( t ) =Γ gain (cid:2) − exp( − Γ LDOS t ) (cid:3) / Γ LDOS , which is very similarto the behaviour of the solutions from the LDOS-SE.However, this assumption is only valid for Γ gain (cid:28) Γ loss ,which is no longer the case for the example above (where Γ gain ∼ Γ loss ). Indeed, for any case where amplificationconstitutes a significant part of the dielectric medium,the classical limit of the TLS is violated, which in turnleads to a contradiction that the SE rate of the TLS isproportional to the LDOS . Thus, the common wisdomof SE scaling with the LDOS is no longer correct.
To conclude, we have introduced a quantum dynam-ical approach to rigorously account for gain and decayof a TLS in linear dielectric media. In this formula-tion, the amplification enters the model through a re-versed Lindblad term in a master equation, which onlyinvolves positive quantum rates. This is in complete con-trast to the usual applied SE models, where the LDOScan exhibit negative values, explicitly shown for a cou-pled gain-loss resonator. This not only leads to a fix ofthe classical (LDOS) SE of a TLS in an amplifying andabsorptive environment, but also implies a TLS pumpingprocess through the gain region; this is a consequence ofthe operator ordering, manifesting in clear (and unique)quantum phenomena from vacuum QED. These resultsalso open up new avenues into currently proposed quan-tum theories for exceptional points and PT -symmetriclike system , and future extensions of the theory to-wards strong light-matter coupling. At a fundamentallevel, they also show that SE is indeed a uniquely quan-tum mechanical process that cannot always be describedclassically or in the limit of a harmonic oscillator model. Acknowledgements . We acknowledge funding fromQueen’s University, the Canadian Foundation for Inno-vation, the Natural Sciences and Engineering ResearchCouncil of Canada, and CMC Microsystems for the pro-vision of COMSOL Multiphysics. We also acknowl-edge support from the Deutsche Forschungsgemeinschaft(DFG) through SFB 951 Project B12 (Project number182087777) and the Alexander von Humboldt Foundationthrough a Humboldt Research Award. This project hasalso received funding from the European Unions Hori-zon 2020 research and innovation program under GrantAgreement No. 734690 (SONAR). ∗ [email protected] Paul Adrien Maurice Dirac, “The quantum theory of theemission and absorption of radiation,” Proceedings of theRoyal Society of London. 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