Dynamical signatures of bound states in waveguide QED
E. Sánchez-Burillo, D. Zueco, L. Martín-Moreno, J. J. García-Ripoll
DDynamical signatures of bound states in waveguide QED
E. S´anchez-Burillo, D. Zueco,
1, 2
L. Mart´ın-Moreno, and J. J. Garc´ıa-Ripoll Instituto de Ciencia de Materiales de Arag´on and Departamento de F´ısica de la Materia Condensada,CSIC-Universidad de Zaragoza, E-50009 Zaragoza, Spain Fundaci´on ARAID, Paseo Mar´ıa Agust´ın 36, E-50004 Zaragoza, Spain Instituto de Fisica Fundamental, IFF-CSIC, Calle Serrano 113b, E-28006 Madrid, Spain
We study the spontaneous decay of an impurity coupled to a linear array of bosonic cavitiesforming a single-band photonic waveguide. The average frequency of the emitted photon is differentfrom the frequency for single-photon resonant scattering, which perfectly matches the bare frequencyof the excited state of the impurity. We study how the energy of the excited state of the impurityinfluences the spatial profile of the emitted photon. The farther the energy is from the middle ofthe photonic band, the farther the wave packet is from the causal limit. In particular, if the energylies in the middle of the band, the wave packet is localized around the causal limit. Besides, theoccupation of the excited state of the impurity presents a rich dynamics: it shows an exponentialdecay up to intermediate times, this is followed by a power-law tail in the long-time regime, and itfinally reaches an oscillatory stationary regime. Finally, we show that this phenomenology is robustunder the presence of losses, both in the impurity and the cavities.
I. INTRODUCTION
Interactions between few-level systems (or quantumimpurities) and photonic media with nonlinear disper-sion relations and band gaps give rise to a plethora of in-teresting phenomena [1]. Examples are the modificationof the level structure of the impurity [2–4], non-trivialdynamics [5–14], and charge transfer enhancement [15].A characteristic phenomenon is the appearance of boundstates [6, 16, 17] where a photonic excitation is confinedto the vicinity of the impurity. This idea has been stud-ied in various theoretical works, finding phenomena suchas suppression of decoherence [18], preservation of quan-tum correlations [19–21], or the existence of multi-photonbound states [22–24]. An instance of bound state hasbeen experimentally found [25] in a circuit QED archi-tecture [26–30], and effects of band gaps in qubit-qubitinteractions have been measured [31] in photonic crys-tals [32–34]. There are other state-of-the-art technologieswhere these states can be potentially detected, e.g. coldatoms [35, 36] and diamond structures with color centers[37, 38].In this work, we study the signatures of bound states inthe spontaneous decay in a waveguide-QED scenario. Wechoose a prototypical model where an impurity is coupledto a bosonic medium: a tight-binding model, which givesa cosine-shaped band. Due to the finite width of theband, two states appear bound to the impurity. Thisproblem has already been treated in the literature whenthe energy of excited state of the impurity is in the middleof the band [14] or when it is close to its inferior limit,so the superior limit of the band can be neglected [12].Here, we solve it for general values of the parameters,which are the energy of the excited state of the impuritywith respect to the the band and the ratio between theimpurity-photon coupling and the bandwidth.We find an energy shift of the emitted photon withrespect to the energy required to excite the impurity,provided the latter is not in the middle of the photonic band. Naively, one could argue that spectral featuresin the spontaneous emission should also appear in thescattering, since a scattering process comprises both ab-sorption and emission of the photon by the impurity.However, there is no shift in the single-photon scatter-ing [39–41]. Secondly, we study the spatial profile of theemitted photon and discuss the differences with respectto a photonic medium with a linear dispersion relation.Lastly, we find a rich dynamics in the excited state ofthe impurity. First, it decays exponentially with a decayrate different from that given by the Fermi’s golden rule,oscillating with a phase which is shifted with respect tothe bare energy of the impurity. This shift, which cor-responds to the Lamb effect, is different from the shiftfound in the energy of the emitted photon. After theinitial exponential decay, the dynamics presents an al-gebraic decay due to the presence of singularities in thedensity of photonic states. These power tails are robustunder the presence of losses, both in the impurity andthe cavities. Eventually, it reaches a stationary oscillat-ing regime.The manuscript is organized as follows. In Sect. II,we introduce the Hamiltonian and summarize both itsspectrum in the single-excitation subspace and its one-photon-scattering properties. In section III, we discussthe main results of the paper. First, we present the al-ready mentioned frequency shift of the emitted photon asa function of the coupling constant and the energy of theexcited state of the impurity. Then, we study the spatialdistribution of the emitted photon. We next characterizethe spontaneous emission when the impurity is initiallyexcited and discuss the effect of the losses. We end upwith the conclusions in Sect. IV. Some technical detailsare described in the appendices. a r X i v : . [ qu a n t - ph ] A ug II. MODELA. Hamiltonian and bound states
The photonic medium is an infinite chain of discretebosonic sites coupled to an impurity placed at site x = 0.The Hamiltonian of the combined system is ( (cid:126) = 1) H = ∆ b † b + ∞ (cid:88) x = −∞ (cid:16) (cid:15)a † x a x − J ( a † x +1 a x + a † x a x +1 ) (cid:17) + g ( b † a + a † b ) , (1)where a x and a † x annihilate and create, respectively, aphoton at position x and b and b † annihilate and createexcitations at the impurity. This impurity can be a two-level atom or qubit, another resonator, a spin, or anysystem equivalent to a qubit in the single-particle sub-space. The energy of the excited state of the impurityis ∆. From now on, we borrow the nomenclature frommolecular physics and refer to ∆ as the exciton energy;in the same way, b and b † will annihilate and create anexciton. The band of free photons is defined by a disper-sion relation which depends on both the on-site photonenergy (cid:15) and the hopping parameter J : ω k = (cid:15) − J cos k ,being k the dimensionless momentum and ω k the corre-sponding energy. In consequence, the bandwidth is 4 J .The momentum k lies in [ − π, π ). The group velocity is v k ≡ dω k /dk = 2 J sin k . The interaction Hamiltonian(the last term in Eq. (1)) is the dipole-field Hamilto-nian in the rotating-wave approximation (RWA), which isgiven by the celebrated Jaynes-Cummings model, where g is the coupling constant. A scheme of the system andthe dispersion relation ω k are shown in Figs. 1(a) and(b), respectively. This model can be realized with theinstances of quantum technologies enumerated in the In-troduction. Δ ... FIG. 1. (a) Scheme of the system.
In blue, the bosoniccoupled-cavity array. The impurity is represented as a blurredred circle. The exciton energy is ∆. (b) Dispersion rela-tion for the bosonic array . Dispersion relation ω k as afunction of the dimensionless momentum k . Due to the rotating-wave approximation, the Hamil-tonian (1) commutes with the number operator
N ≡ (cid:80) x a † x a x + b † b . Thanks to this symmetry, this modelis analytically solvable in the single-excitation subspace.A complete basis is formed by the scattering eigenstates | Ψ k (cid:105) [39] and the bound states | Ψ ± (cid:105) [42, 43]. We intro-duce now the bound states and we leave the scatteringones for the next subsection. They read | Ψ ± (cid:105) = N ± (cid:32)(cid:88) x e − κ ± | x | a † x + d ± b † (cid:33) | (cid:105) . (2)The state | (cid:105) represents the vacuum state of the sys-tem ( a x | (cid:105) = b | (cid:105) = 0). The factor N ± is a normal-ization constant, 1 / | κ ± | is the localization length, and d ± is the exciton amplitude. The energy of | Ψ ± (cid:105) is ω ± = (cid:15) − J ( e − κ ± + e κ ± ). The expressions of d ± and N ± ,as well as the computation of κ ± , are given in App. A.The quantities κ ± fix the properties of the bound states;namely, their energies ω ± , exciton amplitudes d ± , andnormalization factors N ± .We plot the bound-state energies ω ± as a function ofthe coupling constant g , as well as the band limits in Fig.2. Two cases are shown: (i) the exciton energy ∆ at themiddle of the band (∆ − (cid:15) = 0, solid lines) and (ii) ∆closer to the band bottom (∆ − (cid:15) = − J , dotted-dashedlines). The energies of the bound states lie outside ofthe band, thus they are localized (not propagating). As g → ω − (+) approaches the bottom (top) of the band.If the exciton energy coincides with the band center, theenergies of the bound states are symmetrically located.Otherwise, if the exciton energy is below the center, ∆ − (cid:15) <
0, the energy of the lower bound state ω − moves awayfrom the exciton energy ∆ faster than the energy of theupper bound state ω + does, and vice-versa. Therefore,the position of the exciton energy with respect to theband center originates an asymmetry between ω + and ω − . B. One-photon scattering
Let us now review the form of the single-particle scat-tering eigenstates of (1) and their physical implications.They read [39] | Ψ k (cid:105) = (cid:104) (cid:88) x< ( e ikx + r k e − ikx ) a † x + (cid:88) x ≥ t k e ikx a † x + d k b † (cid:105) | (cid:105) . (3)The coefficients t k and r k are the transmission and reflec-tion amplitudes for an incident plane wave, respectively.They are given by, t k = iv k ( ω k − ∆) iv k ( ω k − ∆) − g , (4) r k = t k − , (5) d k = gt k ω k − ∆ . (6) - - - / J ( ω - ϵ ) / J FIG. 2.
Bound states.
Bound-state energies ( ω ± − (cid:15) ) /J fortwo cases: (∆ − (cid:15) ) /J = 0 (solid lines) and (∆ − (cid:15) ) /J = − ω + andthe blue lower ones for ω − . As a reference, the values of(∆ − (cid:15) ) /J = 0 and (∆ − (cid:15) ) /J = − A well-known feature in this system [39–41, 44, 45] isthat it presents perfect reflection, R k ≡ | r k | = 1, if theenergy of the input photon is equal to ∆, see Eqs. (4) and(5). This is illustrated in Figs. 3(a) and (b), where R k is plotted as a function of ( ω k − (cid:15) ) /J for several valuesof ∆ and as a function of ( ω k − (cid:15) ) /J and (∆ − (cid:15) ) /J ,respectively. Considering the input as a single-photon-spectroscopy probe, we could be tempted to argue that,like in scattering, the impurity emission is also maximumat resonance. We will show that, due to the presence ofbound states, this is not the case. III. SPONTANEOUS DECAY
We discuss now the spontaneous emission of the exci-ton. For that, we consider that the impurity is excited at t = 0, | Ψ(0) (cid:105) = b † | (cid:105) , and compute the time evolution ofthe system. Spanning this state in bound and scatteringeigenstates, Eqs. (2) and (3), respectively, the state attime t is | Ψ( t ) (cid:105) = (cid:90) π − π dk π c k e − iω k t | Ψ k (cid:105) + c + e − iω + t | Ψ + (cid:105) + c − e − iω − t | Ψ − (cid:105) , (7)with c k = d ∗ k = iv k giv k ( ω k − ∆) + g , (8) c ± = (cid:18) e − κ ± − e − κ ± + g ( ω ± − ∆) (cid:19) − gω ± − ∆ . (9)In the following, we exploit these formulae to obtainour results. We first discuss the behavior of the mean Δ - ϵ = Δ - ϵ = J / Δ - ϵ = J Δ - ϵ = / - ( ω k - ϵ )/ J R k --- (a)(b) FIG. 3.
Reflection probability. (a) Reflection R k as afunction of ( ω k − (cid:15) ) /J for several values of ∆. (b) Reflection R k as a function of ( ω k − (cid:15) ) /J and (∆ − (cid:15) ) /J for g = J/ ω k , where R k = 1 (maxima in panel(a)). Notice that R k = 1 also at the band edges (Eq. (4)).Notice both graphics share horizontal axis. energy of the emitted wave packet. Then, we describe thespatial profile of that photon depending on ∆. Finally,we study the dynamics of the exciton. A. Energy shift
The state given by Eq. (7) can be used to obtain theaverage value of the Hamiltonian (1). As it is a conservedquantity, it must be equal to the value at t = 0, which is∆: (cid:104) H (cid:105) = ∆ = (cid:90) π − π dk π | c k | ω k + | c + | ω + + | c − | ω − . (10)The average energy for the propagating field is ω ph ≡ (cid:82) π − π ω k | c k | dk/ π (cid:82) π − π | c k | dk/ π = (cid:82) π − π ω k | c k | dk/ π (1 − P lig ) , (11)with P lig ≡ | c + | + | c − | . Using Eq. (10), ω ph can bewritten in a more convenient way ω ph = ∆ − | c + | ω + − | c − | ω − − P lig , (12)which shows that the energy of the emitted photon istypically different from ∆ because of the presence of thebound states. In short, the amount of energy going tothe propagating states must compensate that going tothe bound ones so that the total energy is conserved.This is the physical origin of the energy shift.This confirms the nonequivalence between scatteringand emission spectra, since the scattering resonance al-ways occurs when the input energy is ∆, see Eq. (4) andFig. 3.The energy of the emitted photon ( ω ph − (cid:15) ) /J is plottedas a function of (∆ − (cid:15) ) /J in Fig. 4(a) for several valuesof g . The closer ∆ is to the band edges, the more ω ph departs from ∆. In fact, if ∆ is close to the bottom ofthe band, where the frequency shift is larger, the effect ofthe upper bound state is negligible ( | c + | (cid:28) | c − | ), andvice-versa. In conclusion, the frequency shift survives inwaveguides without an upper cutoff. The shift increasesmonotonically with g . Eventually, as g/J → ∞ , theemitted energy coincides with the middle of the bandfor all ∆. Notice that, when the exciton energy is inthe middle of the band, i.e. when ∆ = (cid:15) , the followingrelation holds: | c + | ( ω + − ∆) = | c − | (∆ − ω − ). Insertingthis in Eq. (12), we conclude that the emitted energy isequal to the exciton one, ω ph = ∆. This is related tothe symmetry of the energy of the bound states, alreadydiscussed in Sect. II A (see Fig. 2). g = J / = J / = Jg = - - - - - - ( Δ - ϵ )/ J ( ω p h - ϵ ) / J (a)(b) (c) FIG. 4.
Emitted energy. (a)
Average energy of theemitted photon ( ω ph − (cid:15) ) /J as a function of (∆ − (cid:15) ) /J for g = J/ , J/ , J, J . For reference, the straight line rendersthe diagonal ω ph = ∆. In (b) and (c) , we plot | c k | as afunction ( ω k − (cid:15) ) /J and (∆ − (cid:15) ) /J for g = J/ g = J/ ω k . For each ∆, wenormalize c k such that max k ( | c k | ) = 1. We also study the energy distribution of the emitted photon | c k | . We plot it as a function of ( ω k − (cid:15) ) /J and(∆ − (cid:15) ) /J for the representative cases of g = J/ g = J/ ω k = ∆. However, as g increases(right panel), | c k | reaches its maximum for ω k (cid:54) = ∆,being the difference larger the closer ∆ is to one of theband edges. This deviation of the maximum away from∆ implies a frequency shift of the emitted photon, asalready seen in Eq. (12) and Fig. 4(a). The reasonis simple. In the spontaneous emission some energy isreleased into the bound states, with a mean energy thatdoes not generally match the exciton energy. Therefore,the coupling into flying photons must compensate for thisimbalance. However, due to the fact that bound andscattering states are orthogonal, the former do not playany role in the latter. It is worthy to emphasize that thismechanism is rather general. In any photonic systemsupporting single-particle bound states, the frequency ofthe flying photon arising from spontaneous emission willpresent a shift with respect to that of the exciton.We also characterize the emission probability intopropagating modes, P emission ≡ − P lig = 1 −| c + | −| c − | ,in Fig. 5. Two effects are observed. First, the emissioninto bound states is negligible ( P emission (cid:39)
1) in the range g/J (cid:28)
1. Increasing this ratio, P emission decreases. Be-sides, the closer ∆ is to the band gap, the smaller P emission is. Anyway, the emission probability is appreciable for re-ally large values of the ratio g/J : for instance g/J (cid:39) . P emission (cid:39) .
25 for the values of ∆ considered inFig. 5.
FIG. 5.
Probability of photon emission.
Probability ofemitting a flying photon, P emission = 1 − P lig = 1 − | c + | −| c − | , as a function of g/J for ∆ − (cid:15) = − J/ , − J, − J/ , B. Emitted field
We now study the spatial profile of the emitted field.We compute the amplitudes in position space, φ x ( t ) ≡ (cid:104) | a x | Ψ( t ) (cid:105) [Cf. App. B]. The photon probability distri-bution | φ x ( t ) | is shown in Fig. 6, at time t = 75 /J and g = J/
5, for two values of the detuning: ∆ − (cid:15) = 0 (bluesolid) and ∆ − (cid:15) = − J (red dashed). The vertical solidblack lines represent | x | = x max ≡ v max t , defined in termsof the maximum group velocity v max = v k = π/ = 2 J .The probability | φ x | is mostly confined within thecausal cone. For | x | > x max , it is not zero but it de-cays exponentially, as expected for the free-field scalar propagator [46, Sect. 4.5], [47, Sect. 2]. If ∆ is in themiddle of the band, the emitted photon has a momentumdistribution peaked around k = π/
2, where v k = v max . If∆ (cid:54) = (cid:15) , the velocity of the emitted photon is not peakedaround v max so the maximum of | φ x | is below x max (seethe dashed red curve of Fig. 6, where ∆ − (cid:15) = − J ).Lastly, notice that the emitted photon would be wellpeaked around | x max | in position space, independentlyof the value of ∆, if the dispersion relation were linear. Δ - ϵ = Δ -J - -
75 0 75 15000.0050.01 x | ϕ x - ϵ = FIG. 6.
Field distribution. | φ x | as a function of x attime t = 75 /J for ∆ − (cid:15) = 0 (solid blue) and ∆ − (cid:15) = − J (dashed red) for coupling g = J/
5. The black solid verticallines render the propagation limit | x | = x max = v max t , with v max = v k = π/ = 2 J . C. Impurity dynamics
We finish with a detailed study of the exciton dynam-ics. From Eq. (7), we extract the time dependence of theamplitude of the exciton b † | (cid:105) c e ( t ) ≡ (cid:104) | b | Ψ( t ) (cid:105) = c se ( t ) + c be ( t ) , (13)with c be ( t ) = (cid:80) α = ± | c α | e − iω α t and c se ( t ) = (cid:82) π − π dk | c k | e − iω k t / π the contributions from the boundand scattering states repectively, see Eqs. (2) and (3).First, we focus on c se ( t ): c se ( t ) = e − i(cid:15)t g πJ (cid:90) − dy F ( y ) e i yJt , (14) with F ( y ) = (cid:112) − y − y ) ((∆ − (cid:15) ) /J + 2 y ) + ( g/J ) . (15)The behavior of c se ( t ) is determined by the kernel F ( y ), y-* Δ y- F ( y ) L ( y ) F ( y ) g = G ( y ) - (a)(b) - - F ( y ) FIG. 7.
Integrand for c se ( t ) . (a) Kernel F ( y ) in logarithmicscale for g = J/ F ( y ) for g = 0 (black, dotted), and G ( y ) (Eq. (C8),black, dotted-dashed). We fix ∆ = (cid:15) . In (b), we zoom in F ( y ) around y (cid:39) −
1, with the same parameters as those usedin (a). The kernel F ( y ) reaches a maximum ay y = y ∗− and∆ y − = y ∗− + 1. Notice that the scale is not logarithmic in thiscase. which is related to the density of photonic states as afunction of the dimensionless energy y = cos k . This ker-nel is plotted in Fig. 7(a). At sufficiently long times, theoscillating term in the integral (14), e i yJt , cancels outany smooth contribution of F ( y ). Therefore, the asymp-totic relaxation dynamics is governed by the sharpestpeaks and the singularities of F ( y ). There are three maincontributions: (i) a Lorentzian peak, associated to a poleof F ( y ) in the complex plane, (ii) two peaks appearing at y ∗± , with y ∗± close to ± F ( y ) around y = − y = ±
1, where the first derivative of F ( y ) is discontin-uous. All these features are clearly seen Figs. 7(a) and(b).The Lorentzian peak gives an exponential decay c se ( t ) ∼ e − ( iϕ +1 / τ ) t . This is equivalent to an excitedatom emitting photons into the free space. This contri-bution is the fastest and main one for short-enough times, t < τ , since it comes from the widest peak in F ( y ), seeFig. 7(a). We compare the (numerical) exact results for τ and δϕ ≡ ϕ − ∆, computed by integrating Eq. (14),with those obtained with the Lorentzian approximationof F ( y ) in Fig. 8. We also compare the results to thoseobtained with Fermi’s Golden Rule: τ FGR0 = J sin k ∆ /g ,with k ∆ such that ω k ∆ = ∆, and ϕ FGR = ∆. Fermi’sGolden Rule describes accurately the exact results when∆ is around the middle of the band, but correctionsare necessary when ∆ gets closer to the band edges andwhen the coupling g increases. The exciton energy ap-pears in the phase of the exponential up to a correction: ϕ = ∆ + δϕ . Thus, δϕ is the Lamb shift due to the cou-pling to the photonic bath. Notice that this Lamb shiftis different from the energy shift of the emitted photon(compare Fig. 4 to Figs. 8 (c) and (d)), even thoughboth converge to ∆ in the limit g/J →
0. In fact, assaid, there is another characteristic energy of the systemwith a different behaviour: the single-photon reflectionresonance, which occurs exactly at the bare excitationenergy ∆ (see Fig. 3 and Eq. (5)).At later times, t (cid:29) τ , the singular parts of F ( y )are relevant. Singularities give non-exponential decays[5, 11]. In particular, the contribution of the peaks of F ( y ) at y ∗± , with y ∗± (cid:39) ±
1, starts to dominate. Let usdefine the widths of these peaks at y ∗± as ∆ y ± ≡ | y ∗± ∓ | (see Fig. 7(b)). For short-enough times, when e i Jyt canbe considered to be constant for y ∈ ( − , − y − )and y ∈ (1 − ∆ y + , F ( y ) can be approx-imated by setting g = 0 (black dotted curve in Fig.7(a)). At g = 0, the kernel diverges as 1 / (cid:112) − y when y → ±
1. This kind of singularity gives an al-gebraic decay t − / for c se ( t ). For long-enough times,when e i Jty cannot be taken as a constant, we haveto consider the full kernel, with the actual value of g .Therefore, the mentioned divergences are rounded offand the algebraic decay is modified by exponential fac-tors. In other words, these peaks provide a contribution c se ( t ) = t − / ( a − e − i Jt e − t/ τ , − + a + e i Jt e − t/ τ , + ), with τ , ± = (4 J ∆ y ± ) − . The values of the constants a ± , aswell as the details on the computation, are shown in App.C.Eventually, these exponential contributions vanish.The only surviving contribution comes from the singular-ities at the band edges. There, F ( y ) is not differentiableand gives a non-exponential (power-law) contribution forall times to c se ( t ), which dominates for t (cid:29) τ , τ , ± .We show in App. C that this contribution goes as c se ( t ) ∼ t − / cos(2 Jt − π/ t − / and t − / decay was already discussed in [12], butthey did not see the oscillating factors, since they tookthe exciton energy really close to the lower part of theband, neglecting the contribution of the upper boundstate. As mentioned, this decay with t − / originates from a discontinuity in the derivative of the density ofphotonic states and is quite common in impurity decayproblems [48], both for continuous systems [49, 50] andfor discrete ones [51–53].The contribution of the bound states c be ( t ) is muchsimpler: it gives an oscillatory term which persists forinfinitely long times: P be ( t ) ≡ | c be ( t ) | = | c + | + | c − | +2 | c + c − | cos(( ω + − ω − ) t ), [11, 14]. (a)(c) (b)(d)g=J/5 g=3J/10 - ( Δ - ϵ )/ J δ φ / J - ( Δ - ϵ )/ J δ φ / J - ( Δ - ϵ )/ J τ / τ F G R ( Δ = ϵ ) - ( Δ - ϵ )/ J τ / τ F G R ( Δ = ϵ ) FIG. 8.
Exponential decay. (a), (b) τ /τ FGR0 (∆ − (cid:15) = 0)and (c), (d) δϕ/J as a function of the position of the excitonenergy with respect to the band for (cid:15) = ∆. The coupling is g = J/ g = 3 J/
10 (right panels). We divide τ by the decay time given by the Fermi’s Golden Rule at themiddle of the band, τ FGR0 (∆ − (cid:15) = 0). The red solid curve andthe black dashed one correspond to the Fermi’s Golden Ruleand to the single-pole approximation, respectively. The bluepoints are computed numerically; we fit the exact dynamicscomputed with (14) to an exponential for t < τ . We sum up all this information in Fig. 9, where we plotthe impurity dynamics for ∆ − (cid:15) = 0 and g = J/ a − e − i Jt + a + e i Jt (arisingfrom the peaks around y ∗± ), cos(2 Jt − π/
4) (from the sin-gularities at y = ± ω + − ω − ) t ) (from c be ( t )).The population P e ( t ) ≡ | c e ( t ) | is drawn as a black, dot-ted curve. It first decays as e − t/τ . In addition, thebound-state term dominates over the remaining contribu-tions from the scattering states. Therefore, after a tran-sient period, P e ( t ) achieves the stationary regime of P be ( t )(purple, dashed curve; remind that we are not showingthe oscillations). We also show P se ( t ) ≡ | c se ( t ) | in thered solid curve. After the initial exponential decay with e − t/τ , where P se ( t ) (cid:39) P e ( t ), it decays sub-exponentially.To see the different contributions to this sub-exponentialdecay more clearly, we plot it in the inset in log-log scale.After the mentioned exponential decay with e − t/τ , it fol-lows a decay with t − e − τ for τ (cid:28) t (cid:39) τ (as ∆ − (cid:15) = 0, τ ≡ τ , + = τ , − ; in particular τ (cid:39) τ for the chosenparameters). Eventually, as t (cid:29) τ , P se ( t ) goes with t − .The agreement between the analytical predictions (blue P e P es P eb - - - - / τ P es e - t τ t - e - t τ t - - - - - - - / τ / τ τ FIG. 9.
Impurity dynamics. P e ( t ) (black, dotted), P se ( t )(red, solid), and P be ( t ) (purple, dashed) for ∆ − (cid:15) = 0 and g = J/ P se ( t )in log-log scale with the three contributions: the exponentialdecay (blue, dashed), the power-law with t − (green, dotted),and the decay with t − (orange, dotted-dashed). For the sakeof clarity, we average the oscillations. dashed curve for e − t/τ , green dotted curve for t − e − t/τ and orange dotted-dashed curve for t − ) and the exact(numerical) integration is clear in the figure.Finally, even though we have focused on the case with∆ in the middle of the band, the mathematical analysisshown in App. C is general, so another choice of param-eters will give the same qualitative behavior.
1. Losses
Here we incorporate losses to the model. We add animaginary part both to the exciton energy and the cavityenergy, ˜∆ = ∆ − iγ e / (cid:15) = (cid:15) − iγ c / (cid:15) by ˜∆ and ˜ (cid:15) , respectively. We take γ e / c /g ∼ − .
15. Considering losses in the exciton, theintegrand F ( y ) resembles to the lossless case (see Figs.7 and 10), apart from the fact that now it is a complexfunction; the same happens if we instead add losses tothe cavities. Therefore, we can repeat the analysis of thelossless case.We illustrate the modifications with γ e (cid:54) = 0 in Fig. 11.Initially, it still decays exponentially, but the decay rateis a sum of the previous one, 1 /τ , and γ e : the amplitudereads c sc ( t ) ∝ e − ( iϕ +1 / τ + γ e / t (see Fig. 11(a)). Thepower law with t − , c se ( t ) = t − / ( a − e − i Jt e − t/ τ , − + a + e i Jt e − t/ τ , + ), is preserved. The coefficients a ± ,whose expressions are shown in App. C, get modified10 − % at most for the chosen values of γ e . Lastly, theasymptotic decay with t − does not depend on ∆ (seeApp. C). The robustness of the power-law tails is seenin Fig. 11(b).If we instead consider lossy cavities, γ c (cid:54) = 0, there is F ( y )| Re ( F ( y )) γ e | Im ( F ( y )) γ e - FIG. 10.
Integrand for c se ( t ) with an imaginary part in ∆ . Kernel F ( y ) in logarithmic scale for γ e = 0 (red, solid), aswell as its real and imaginary part for γ e = g/
10. The otherparameters are those of Fig. 7. a global factor e − γ c t/ multiplying c sc ( t ) (see Eq. (14)).When integrating F ( y ), the imaginary part in (cid:15) adds anincreasing exponential e γ c t/ to c sc ( t ), contrarily to ∆(see the denominator of F ( y ), Eq. (15); ∆ and (cid:15) haveopposite signs). This increasing exponential cancels outwith the global factor e − γ c t/ . Therefore, no modifica-tions are seen in the initial exponential regime (see Fig.12(a)). The global factor e − γ c t/ suppresses the powerlaws in the long-time limit. If the characteristic time ofthe losses 1 /γ c is larger than τ , ± , we can see the power-law tails for intermediate times (see Fig. 12(b)). IV. CONCLUSIONS
We have discussed the differences betweenspontaneous-decay and scattering spectra. As weargued in the text, naively we could expect thatthe scattering resonance should coincide with thespontaneous-emission energy. However, whereas thescattering resonance is always equal to the excitonenergy, we have shown that the emission frequency isshifted. In particular, this shift is more clear as thecoupling increases and/or the exciton energy is closer tothe band edges. We have also seen that the profile of theemitted photon strongly depends on the exciton energywith respect to the photonic band. Lastly, the presenceof bound states and a nontrivial density of states makesthe impurity dynamics nontrivial, with three dynamicalregimes: exponential decay, power-law with a transitionfrom t − to t − , and oscillatory asymptotic regime. Thisdynamics has proven to be robust under the presenceof losses, both in the atom and in the cavities. Eventhough the population at the power-law regime is verysmall, it could be measured. In fact, such power lawshave already been measured in a context of dissolvedorganic materials, where the fluorescence follows an
10 10010 - - - - t / τ P e s (a)(b) γ e = γ e = γ e = γ e = - - - - - / τ P e s FIG. 11.
Impurity dynamics for γ e (cid:54) = 0 . (a) P se ( t ) inlogarithmic scale for several values of γ e . The thicker lines arethe exact results, whereas the thinner ones are the analyticalprediction for the exponential regime: P se ( t ) ∝ e − (1 /τ + γ e ) t .(b) The same in log-log scale and in the long-time regime.The values of γ e are those of panel (a). algebraic decay at long times [54].Some features, such as the spectroscopic shifts in thespontaneously emitted photons, can be detectable bytuning up and down the frequency of the exciton withrespect to the band edge. For probing the dynamics,we suggest using a more sophisticated protocol that (i)places the exciton energy at the right frequency, (ii) thenexcites it and after a finite time t (iii) detunes the excitonand probes dispersively its excited state population. Allthese ideas can be implemented in state-of-the-art setupswith superconducting cavities and transmon qubits [25]and also with quantum dots in photonic crystals [32–34]. ACKNOWLEDGMENTS
We acknowledge support by the Spanish Ministe-rio de Economia y Competitividad within projectsMAT2014-53432-C5-1-R, FIS2015-70856-P (Cofunded byFEDER), and No. FIS2014-55867-P, the Gobiernode Aragon (FENOL group), CAM Research NetworkQUITEMAD+, and the European project PROMISCE. (a)(b) γ c = γ c = - g γ c = - g γ c = - g - - - - - / τ P e s
10 10010 - - - - t / τ P e s FIG. 12.
Impurity dynamics for γ c (cid:54) = 0 . (a) P se ( t ) inlogarithmic scale for several values of γ c . (b) The same inlog-log scale and in the long-time regime. The values of γ c arethose of panel (a). The power laws survive for intermediatetimes for moderate values of γ c , but they disappear if γ c istoo large (black curve). Appendix A: Bound States
We provide the explicit expressions for d ± , κ ± , and N ± appearing in the main text (Eq. (2)). The excited-stateamplitude of the impurity d ± is d ± = gω ± − ∆ . (A1)In order to compute κ ± , we define η ± ≡ e − κ ± and usethe eigenvalue equation H | Ψ ± (cid:105) = ω ± | Ψ ± (cid:105) [43] η ± + ∆ − (cid:15)J η ± + g J η ± − ∆ − (cid:15)J η ± − . (A2)This equation has four solutions. However, we have twoconstrains: (i) Re( κ ± ) >
0, because the photonic cloudmust be localized around the impurity and cannot ex-plode at x → ±∞ , and (ii) Im( κ ± ) = 0 , π , since theenergies ω ± = (cid:15) − J ( e − κ ± + e κ ± ) are real. With theserestrictions, there are only two solutions for η ± , whichcan be found numerically.If we take the limit J → ∞ , where the dispersiontends to be linear, the valid solutions for η ± are ±
1, soRe( κ ± ) = 0. Therefore, | Ψ ± (cid:105) are not bound anymore.In fact, they converge to the scattering states | Ψ k (cid:105) with k = 0 and k = π , that is, those at the band edges.The normalization factor is N ± = (cid:18) e − κ ± − e − κ ± + | d ± | (cid:19) − / . (A3)Finally, c ± = (cid:104) | σ − | Ψ ± (cid:105) = ( N ± d ± ) ∗ can be obtained(Eq. (9)), since we know both d ± , Eq. (A1), and N ± ,Eq. (A3). Appendix B: Emitted field
The profile of the emitted field φ x ( t ) = (cid:104) | a x | Ψ( t ) (cid:105) isgiven by φ x ( t ) = 12 π (cid:90) π − π dkc k e − iω k t (cid:104) | a x | Ψ k (cid:105) (B1)+ c + e − iω + t (cid:104) | a x | Ψ + (cid:105) + c − e − iω − t (cid:104) | a x | Ψ − (cid:105) , where we have used Eq. (7). In order to compute theamplitude (cid:104) | a x | Ψ k (cid:105) we take the expression of | Ψ k (cid:105) , Eq.(3), for k > (cid:104) | a x | Ψ k (cid:105) = (cid:26) e ikx + r k e − ikx x < ,t k e ikx x ≥ . (B2)If k < (cid:104) | a x | Ψ k (cid:105) = (cid:26) t k e ikx x < ,e ikx + r k e − ikx x ≥ . (B3)The amplitudes (cid:104) | a x | Ψ ± (cid:105) are computed by projectingon | Ψ ± (cid:105) (Eq. (2)): (cid:104) | a x | Ψ ± (cid:105) = N ± e − κ ± | x | . (B4) Appendix C: Impurity dynamics: analyzing theintegrand1. Exponential decay
In order to extract the first exponential decay, wecan approximate F ( y ) by L ( y ) = a p / ( y − y p ), being y p the pole corresponding to the peak of F ( y ), with − < Re( y p ) < y p ) >
0, and a p the residueof F ( y ) at y = y p . The value of y p is found numerically,equating the denominator of F ( y ) to 0 (see Eq. (15)).The residue a p is computed by definition. We extend theintegration domain to ±∞ . Then, applying the residuetheorem c se ( t ) = i a p ( g/J ) e − i(cid:15)t e i y p Jt , (C1)By computing this numerically, we obtain the decay rate τ = (4 J Im( y p )) − and the phase ϕ = (cid:15) − J Re( y p ), asshown in Fig. 8 in the main text.
2. Sub-exponential regime: t − / The kernel F ( y ) has a sharp behavior around y ∗± . Infact, it diverges when y → ± g = 0. In order to takeinto account this contribution, we can approximate F ( y )by F ( y ) | g =0 (see blue, dashed curve of Fig. 7(a)) c se ( t ) (cid:39) g e − i(cid:15)t πJ (cid:90) − dy e i yJt (cid:112) − y ((∆ − (cid:15) ) /J + 2 y ) . (C2)If 2∆ y ± Jt (cid:28)
1, with ∆ y ± = | y ∗± ∓ | , the oscillatoryterm e i yJt will not be sensitive to the difference between F ( y ) and F ( y ) | g =0 when y is close to the edges. As weare concerned in the contribution around ±
1, we can ap-proximate the integral as: c se ( t ) (cid:39) g e − i(cid:15)t √ πJ (cid:18) J (∆ − (cid:15) − J ) (cid:90) ∞− dy e i yJt √ y (C3)+ J (∆ − (cid:15) + 2 J ) (cid:90) −∞ dy e i yJt √ − y (cid:19) . These integrals are analytical c se ( t ) (cid:39) g e − i(cid:15)t √ πJt (cid:18) e − i Jt (∆ − (cid:15) − J ) + e i Jt (∆ − (cid:15) + 2 J ) (cid:19) . (C4)In consequence, P se ( t ) decays with ( Jt ) − after the ini-tial exponential decay if τ (cid:28) t (cid:28) τ , ± , with τ , ± =(4 J ∆ y ± ) − . We can rewrite the last expression byadding the decaying exponentials with τ , ± : c se ( t ) (cid:39) g e − i(cid:15)t √ πJt (cid:18) e − i Jt (∆ − (cid:15) − J ) e − t/ τ , − + e i Jt (∆ − (cid:15) + 2 J ) e − t/ τ , + (cid:19) . (C5)The constants a ± introduced in the main text can beidentified as a − = g √ πJ (∆ − (cid:15) − J ) , (C6) a + = g √ πJ (∆ − (cid:15) + 2 J ) . (C7)
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