Strongly driven nonlinear quantum optics in microring resonators
SStrongly driven nonlinear quantum optics in microring resonators
Z. Vernon ∗ and J.E. Sipe Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada, M5S 1A7 (Dated: August 16, 2018)We present a detailed analysis of strongly driven spontaneous four-wave mixing in a lossy in-tegrated microring resonator side-coupled to a channel waveguide. A nonperturbative, analyticsolution within the undepleted pump approximation is developed for a cw pump input of arbitraryintensity. In the strongly driven regime self- and cross-phase modulation, as well as multi-pair gen-eration, lead to a rich variety of power-dependent effects; the results are markedly different than inthe low power limit. The photon pair generation rate, single photon spectrum, and joint spectralintensity (JSI) distribution are calculated. Splitting of the generated single photon spectrum intoa doublet structure associated with both pump detuning and cross-phase modulation is predicted,as well as substantial narrowing of the generated signal and idler bandwidths associated with theonset of optical parametric oscillation at intermediate powers. Both the correlated and uncorrelatedcontributions to the JSI are calculated, and for sufficient powers the uncorrelated part of the JSIis found to form a quadruplet structure. The pump detuning is found to play a crucial role in allof these phenomena, and a critical detuning is identified which divides the system behaviour intodistinct regimes, as well as an optimal detuning strategy which preserves many of the low-powercharacteristics of the generated photons for arbitrary input power.
PACS numbers: 42.65.Lm, 42.65.Wi, 42.50.Dv
I. INTRODUCTION
Integrated optical microresonators continue to developas a promising platform for generating, controlling, andmeasuring quantum states of light [1–10]. Advances infabricating such chip-based structures is enabling theconstruction of micron-scale optical ring resonators withreported quality factors Q of over one million [4, 11].By exploiting the nonlinear optical response of the ringmedium, combined with the massive enhancement of in-traring pump intensity made possible by the large Q val-ues of the resonator, a wide variety of nonlinear opticalphenomena can be realized using relatively modest inputpowers. Entangled photon pair generation in silicon mi-crorings has been demonstrated using mere µ W of pumppower [5, 8], and optical parametric oscillation in a sil-icon nitride microring has been observed using a pumppower of only 50 mW [12]. Arrays of coupled silicon mi-crorings have also been investigated as a potential sourceof heralded single photons [13].Such high-Q microrings are ideal for investigations ofstrongly driven nonlinear optical effects. Depending onthe application, these effects can be undesirable or highlysought after: multi-pair production from an entangledphoton pair source contaminates the sought after en-ergy correlation, whereas optical parametric oscillation(OPO) only arises in the strongly driven regime. Theo-retical studies of quantum nonlinear optics in integratedmicroresonators have typically treated the nonlinearityperturbatively [14–21], which limits calculations to quan-tities relating to a single generated photon pair.Recently we presented a general theoretical treatment ∗ [email protected] of photon pair generation arising from spontaneous four-wave mixing (SFWM) in microring resonators, fully ac-counting for the quantum effects of scattering losseswithin the resonator [14]. As our focus was on the ef-fects of such losses, we limited ourselves to the low powerregime in which a perturbative solution of the relevantequations of motion provides an adequate description ofthe pair generation process. In this work we extend ouranalysis to a more strongly driven regime, where pertur-bative strategies are inadequate and competing nonlin-ear effects, including self-phase modulation (SPM) andcross-phase modulation (XPM), become important. Werestrict ourselves to one of the most common pump statesused in experiment, that of a coherent, narrowband con-tinuous wave pump beam, for which a nonperturbative,analytic solution to the semiclassical equations of motionis achievable within the undepleted pump approximation.This approximation limits us to pump intensities belowthe onset of OPO, the threshold for which is clearly indi-cated by our equations; OPO in such structures will bethe subject of a later communication. Even below theOPO threshold, the subtle interplay between the variousnonlinear terms that couple the ring modes, as well asthe effects of multiple photon pair generation, give riseto a rich variety of nonlinear optical phenomena that areaccessible by varying only two input parameters, namelythe pump intensity and detuning.In Sec. II we begin by assembling the relevant Hamil-tonian and field operators for the ring-channel system.In Sec. III a brief review and summary of our earlier [14]theoretical framework is presented, wherein the system’sdynamics are reduced to a set of coupled ordinary dif-ferential equations for the ring operators alone. Steadystate solutions for the pump mode, incorporating the ef-fects of SPM and scattering losses, are developed in Sec.IV, and the stability of those solutions is studied. The a r X i v : . [ qu a n t - ph ] A ug equations of motion for the signal and idler modes arethen solved in Sec. V, enabling the calculation of physi-cal quantities including the photon generation rate, singlephoton power spectrum, and joint spectral intensity dis-tribution. For each of these measurable quantities thecorresponding predictions at low and high pump powersare compared, and we identify a set of experimental fea-tures, or “smoking guns,” that distinguish the qualitativebehaviour at high pump powers from that at low pumppowers. II. HAMILTONIAN AND FIELDS
We consider an integrated microring resonator side-coupled to a channel waveguide, as illustrated in Fig. 1.We assume the ring size and quality factor Q have beenchosen such that the ring accommodates individual res-onant modes which are well separated in frequency; thatis, we are in the high finesse limit. While a simple gener-alization of our framework can be used to treat arbitrarilymany ring resonances, we restrict our model for the timebeing to contain only three ring modes of interest. Thefull system Hamiltonian can then be written [14] as H = H channel + H ring + H coupling + H bath , (1)wherein H channel refers to the channel fields, H ring to thering modes, H coupling to the coupling between the chan-nel and ring, and H bath to any modes into which ringphotons may be lost, as well the coupling of those modesto the ring modes. Introducing channel fields ψ J ( z ), thechannel Hamiltonian is H channel = (cid:88) J (cid:34) ¯ hω J (cid:90) dz ψ † J ( z ) ψ J ( z )+ i ¯ hv J (cid:90) dz (cid:32) dψ † J ( z ) dz ψ J ( z ) − H . c . (cid:33) (cid:35) , (2)where the fields satisfy the usual commutation relations[ ψ J ( z ) , ψ J (cid:48) ( z (cid:48) )] = 0 , (cid:104) ψ J ( z ) , ψ † J (cid:48) ( z (cid:48) ) (cid:105) = δ ( z − z (cid:48) ) δ JJ (cid:48) . (3)The index J ∈ { P, S, I } runs over three fields of interest,respectively labelled P , S and I for pump, signal andidler, with corresponding reference frequencies ω J andpropagation speeds v J . Each field ψ J contains frequencycomponents centred at ω J , taken to be the resonant fre-quency of the corresponding ring mode, and ranges overa bandwidth that does not overlap with those of otherfields ψ J (cid:48) , but involving excitation over sufficiently longdistances that the Dirac δ function in (3) is a good ap-proximation [14]. By allowing the fields to have differentpropagation speeds we include the possibility of group ve-locity dispersion between the different channel fields. TheHamiltonian (2) does assume group velocity dispersion within each channel field is negligible, but it is straight-forward to include arbitrary dispersion. The spatial co-ordinate z ranges from z = −∞ to z = + ∞ with thecoupling to the ring assumed to take place at a singlepoint z = 0. Within this point coupling approximationthe coupling Hamiltonian becomes H coupling = (cid:88) J (cid:16) ¯ hγ J b † J ψ J (0) + H . c . (cid:17) , (4)in which we have introduced ring-channel coupling co-efficients γ J , as well as discrete ring mode annihilationoperators b J . In addition to the physical channel, tosimulate scattering losses in the ring we include an ex-tra “phantom channel” into which ring photons can belost. The phantom channel similarly accommodates threefields φ J ( z ) with respective propagation speeds u J andcoupling coefficients µ J , and is represented as H bath by achannel and coupling Hamiltonian identical to those forthe physical channel [14].The Hamiltonian for the ring modes can be written as H ring = (cid:88) J ¯ hω J b † J b J + H NL , (5)where H NL includes all the nonlinearity in the system.Since the fields will be most intense within the ring res-onator, we neglect channel nonlinearities and take H NL tocontain only ring mode operators. In this work we con-sider effects arising from the third-order nonlinear sus-ceptibility in the ring, taking H NL = (cid:16) ¯ h Λ b P b P b † S b † I + H . c . (cid:17) + ¯ hηb † P b † P b P b P + ¯ hζ (cid:16) b † S b † P b S b P + b † I b † P b I b P (cid:17) . (6)The first term is responsible for SFWM, in which twopump photons are converted to a signal and idler pho-ton pair. The second leads to SPM of the pump, whilethe latter two are responsible for XPM between the pumpand signal and idler modes. It is safe to neglect SPM andXPM terms that involve only the signal and idler modes,since the power in those modes will be small comparedto that in the pump mode. While we focus in this workon SFWM involving a single pump mode, it is straight-forward to incorporate multiple pump modes into ourmodel. The nonlinear coupling coefficients Λ, η and ζ are not independent, as they arise from the same nonlin-ear susceptibility, but we formally leave them arbitraryfor the time being so that the effects of each term in H NL can more easily be identified. Obtaining expressions forthese constants depends on the approximations used toderive the nonlinear sector of the ring Hamiltonian. Wepresent our derivation of this Hamiltonian and the asso-ciated constants Λ, η and ζ in Appendix A, arriving atan estimate of Λ ≈ ¯ hω P cn n V ring , (7)with η = Λ / ζ = 2Λ. In this expression n refersto the nonlinear refractive index of the ring material, n FIG. 1. Integrated ring-channel system geometry with la-belled ring modes and incoming and outgoing outgoing chan-nel fields. Photons generated in the ring may exit to thephysical channel or be lost to the upper effective “phantomchannel”. to the linear refractive index, and V ring to the volumeof the ring mode. For the silicon nitride rings used intypical experiments [12], with n ≈ . × − m / W[22] this yields Λ ∼
10 Hz. For typical silicon rings [5],with n ≈ . × − m / W [23] this calculation predictsΛ ∼ Hz.
III. EQUATIONS OF MOTION
The Heisenberg equations of motion for the field op-erators ψ J ( z, t ) and φ J ( z, t ) and the ring operators b J ( t )follow from the Hamiltonian (1), and can be simplifiedby the introduction of auxiliary quantities[14]; here wesummarize the results.The equations of motion for the channel fields are (cid:18) ∂∂t + v J ∂∂z + iω J (cid:19) ψ J ( z, t ) = − iγ J b J ( t ) δ ( z ) , (8)with similar expressions obeyed by the phantom channelfields φ J ( z, t ). Note that the solutions to these equationscontain a discontinuity at z = 0, which is a consequenceof our point-coupling assumption. To avoid explicitlydealing with this discontinuity, it is helpful to introduceformal channel fields which we identify as those fields which are incoming and outgoing with respect to thecoupling point. We define the incoming field ψ J< ( z, t )by ψ J< ( z, t ) = ψ J ( z, t ) for z < , (9)and extend it to z ≥ (cid:18) ∂∂t + v J ∂∂z + iω J (cid:19) ψ J< ( z, t ) = 0 . (10)This confers a false future on ψ J< ( z, t ), correspondingto the free evolution of the incoming field without anycoupling to the ring. We similarly define the outgoingfield ψ J> ( z, t ) by taking ψ J> (0 , t ) = ψ J ( z, t ) for z > , (11)and demanding that for all z (cid:18) ∂∂t + v J ∂∂z + iω J (cid:19) ψ J> ( z, t ) = 0 , (12)giving ψ J> ( z, t ) a false past to the left of the couplingpoint. By an identical procedure we may define the in-coming and outgoing phantom channel fields φ J< ( z, t )and φ J> ( z, t ). Since we will primarily be concerned withthe properties of the photons generated in the ring, whichexit to one of the channels and propagate to positive z ,all calculations involving the ring’s output will be carriedout on the outgoing fields ψ J> ( z, t ). Our goal is thereforeto construct an explicit solution for these fields in termsof the incoming fields ψ J< ( z, t ). Indeed, since these fieldsfreely propagate, the field at large positive z (where anymeasurements on the generated photons would occur) isentirely determined by the outgoing field at z = 0, ψ J ( z, t ) = e − iω J z/v J ψ J> (0 , t − z / v J ) for z > . (13)It therefore suffices to construct a solution for ψ J> (0 , t ),which can be very simply related to the incoming field ψ J< (0 , t ) and the corresponding ring operator b J ( t ) [14]via ψ J> (0 , t ) = ψ J< (0 , t ) − iγ J v J b J ( t ) . (14)For each operator O J ( t ) it will be convenient to definethe corresponding slowly-varying barred operator O J ( t ), O J ( t ) = e iω J t O J ( t ) . (15)In terms of these quantities and the incoming and out-going fields, the equations for the ring mode annihilationoperators b J ( t ) are found to satisfy (cid:18) ddt + Γ P + 2 iηb † P ( t ) b P ( t ) (cid:19) b P ( t ) = − iγ ∗ P ψ P < (0 , t ) − iµ ∗ P φ P < (0 , t ) − i Λ ∗ b † P ( t ) b S ( t ) b I ( t ) e − i ∆ ring t , (16a) (cid:18) ddt + Γ S + iζb † P ( t ) b P ( t ) (cid:19) b S ( t ) = − iγ ∗ S ψ S< (0 , t ) − iµ ∗ S φ S< (0 , t ) − i Λ b P ( t ) b P ( t ) b † I ( t ) e i ∆ ring t , (16b) (cid:18) ddt + Γ I + iζb † P ( t ) b P ( t ) (cid:19) b I ( t ) = − iγ ∗ I ψ I< (0 , t ) − iµ ∗ I φ I< (0 , t ) − i Λ b P ( t ) b P ( t ) b † S ( t ) e i ∆ ring t , (16c)where we have introduced the ring mode detuning∆ ring = ω S + ω I − ω P , (17)as well as the total effective linewidths Γ J ,Γ J = Γ J + M J , (18)where Γ J and M J denote the damping rates associatedwith the physical channel and phantom channel cou-plings, respectively: Γ J = | γ J | v J ,M J = | µ J | u J . (19)These total damping rates can be simply related to thequality factors Q J of the resonator modes; for example,for the pump resonance Q P = ω P Γ P , (20)which yields Q P ∼ for a ring with Γ P = 1 GHzgiven a pump with wavelength λ = 1550 nm. The cou-pled set of driven, damped ordinary differential equations(16) fully describes the nonlinear dynamics of the ring-channel system. Combined with the channel transforma-tion (14), a solution to this system of equations permitsthe calculation of any measurable quantities on the out-going photons in the channel.It is important to note at this stage that our treat-ment neglects the effect of ring heating due to the largecirculating pump power present in the ring. Such ther-mal effects are routinely observed in experimental inves-tigations of microring systems, and typically manifest asan effective power-dependent drift in the resonant fre-quencies of the ring as it undergoes thermal expansion[12]. For slowly varying and cw pumps a simple wayto account for this is through the addition of a pumpphoton number-dependent correction to each resonance.Our model already incorporates a similar effect: SPMand XPM of each mode are represented by preciselysuch terms. The inclusion of thermal resonance drift cantherefore be modelled by altering the coefficients η and ζ in Eqs. (16), which would be replaced by effective con-stants η eff and ζ eff , η eff = η + η thermal ,ζ eff = ζ + ζ thermal . (21) While η and ζ are both positive, η thermal and ζ thermal would be negative, since as the ring expands the res-onant frequencies are typically lowered [24]. Dependingon the relative magnitude of the thermal drift coefficientscompared to the SPM and XPM strengths, in some cir-cumstances η eff and ζ eff may become negative. While forthe remainder of this work we neglect thermal drift ofthe ring resonances, so that η thermal = ζ thermal = 0, weemphasize that our conclusions do not depend sensitivelyon this assumption unless otherwise stated. IV. STEADY STATE PUMP SOLUTION
The set of coupled equations (16) treats both the pumpand signal and idler modes quantum mechanically, retain-ing the operator nature of b J ( t ) for each J . While this isnecessary if one wishes to fully account for the nonclassi-cal properties of the pump mode, in typical experiments[5, 6] the system is pumped by a coherent laser beam orpulse. In such situations the initial pump state is de-scribed by setting each incoming pump mode to a coher-ent state. The pump field can then be well approximatedby its expectation value, which is a classical function oftime. To implement this semiclassical approximation wetake b P ( t ) → β P ( t ) = (cid:104) b P ( t ) (cid:105) . (22)In addition to treating the pump classically, we also im-plement the undepleted pump approximation. In theequation for the ring pump mode (16a) the term involv-ing b † P b S b I accounts for the effect on the pump modewhen a signal-idler photon pair is produced. Neglect-ing such effects, we drop this term and instead take thesemiclassical pump amplitude β P ( t ) to satisfy (cid:18) ddt + Γ P + 2 iη | β P ( t ) | (cid:19) β P ( t )= − iγ ∗ P (cid:104) ψ P < (0 , t ) (cid:105) , (23)in which we have assumed (cid:104) φ P < (0 , t ) (cid:105) = 0, so that thereis no incoming pump energy in the phantom channel.Note that while this approximation amounts to neglect-ing pump depletion due to photon pair generation, linearpump losses are still accounted for in our model, as ev-idenced by the presence of the damping term Γ P in Eq.(23).In this work we consider the case of a continuous wave(cw) pump beam injected in to the channel, so that (cid:104) ψ P < (0 , t ) (cid:105) = pγ ∗ P e − i ∆ P t , (24)where ∆ P is the detuning of the injected pump from thering pump resonance, and p is a constant related to theinput pump power P in in the channel at the couplingpoint via p = (cid:114) P P in ¯ hω P . (25)In steady state, after the ring pump mode has come toequilibrium with the channels, we expect there to be aconstant average number of pump photons N P in thering, where N P = lim t →∞ | β P ( t ) | . (26)Defining (cid:101) β P ( t ) = e i ∆ P t β P ( t ), from Eq. (23) we have (cid:18) ddt + Γ P + i (2 η | β P ( t ) | − ∆ P ) (cid:19) (cid:101) β P ( t ) = − ip. (27)It is not difficult to show that N P will be constant onlywhen (cid:101) β P ( t ) has both constant amplitude and constantphase, so that d (cid:101) β P ( t ) /dt = 0. Setting this time derivativeto zero in the above equation and taking the modulussquared of the result, we find that in steady state N P must be a root of the cubic equation C P ( N P ) = 0 , (28)where C P ( N P ) ≡ η N P − η ∆ P N P + (Γ P + ∆ P ) N P − | p | . (29)In the absence of SPM (when η →
0, or when the inputpower is very small), N P is related to the incoming powerby a simple linear function, N P = | p | Γ P + ∆ P . (30)The presence of SPM, however, complicates the task ofdetermining N P as a function of | p | for a given detuning∆ P and nonlinearity η . The cubic equation (28) has ingeneral as many as three real, positive roots. Further-more, only some of these may correspond to stable solu-tions of (23). Before solving for the roots of C P ( N P ), wefirst derive a set of criteria to assess the stability of anysuch solution.To determine whether or not a given root of (29) isstable, we conduct an analysis similar to that of Hoff,Nielsen and Andersen [25]. For a given constant solution (cid:101) β (0) P to (27), we define the fluctuation amplitude δβ P ( t )via (cid:101) β P ( t ) = (cid:101) β (0) P + δβ P ( t ) . (31) Keeping terms up to first order in δβ P , the equations ofmotion satisfied by δβ P ( t ) and δβ ∗ P ( t ) can be written as ddt (cid:18) δβ P ( t ) δβ ∗ P ( t ) (cid:19) = F (cid:18) δβ P ( t ) δβ ∗ P ( t ) (cid:19) , (32)where F is the 2 × F = − Γ P − i (4 ηN P − ∆ P ) − iη (cid:104) (cid:101) β (0) P (cid:105) iη (cid:104) (cid:101) β (0) ∗ P (cid:105) − Γ P + i (4 ηN P − ∆ P ) . (33)For a given solution to be stable, we require the realpart of both eigenvalues of F to be negative, so thatthe fluctuation term δβ P ( t ) will decay with time. Theseeigenvalues are f ± = − Γ P ± (cid:113) η N P − (4 ηN P − ∆ P ) . (34)Now, Re( f − ) < f + ) < η N P − (4 ηN P − ∆ P ) < Γ P . (35)Solving this inequality, we find that any solution N P forEq. (28) corresponds to a stable solution of Eq. (27)if | ∆ P | is below a “critical detuning”, | ∆ P | < ∆ critical ,where ∆ critical = √ P . (36)When | ∆ P | > ∆ critical , a solution N P of (28) correspondsto a stable solution of (27) if and only if N P lies outsidea certain interval, N P / ∈ ( N − , N + ), where N ± = 13 η (cid:18) ∆ P ± (cid:113) ∆ P − ∆ (cid:19) . (37)Having established the stability criteria for a given N P ,we return to the task of finding real, positive roots of (28).While analytic expressions for the roots exist, it is moreinstructive to use indirect arguments to study their na-ture. Taking the derivative of C P with respect to N P ,we find that dC P /dN P = 0 at N P = N ± . Thus, for sub-critical detunings ( | ∆ P | < ∆ critical ), where the N ± arenot purely real, there are no local extrema – it is easy toshow that a graph of C P ( N P ) is monotonically increasingand intersects the N P axis only once, leading to a sin-gle real, positive root N P which corresponds to a stablesolution. On the other hand, for supercritical detunings( | ∆ P | > ∆ critical ), the function C P ( N P ) goes through alocal maximum at N − and minimum at N + . The numberof times C P ( N P ) intersects the N P axis is then deter-mined by the power parameter | p | ; varying it translatesthe graph of C P ( N P ) vertically. If C P ( N − ) > C P ( N + ) <
0, the graph of the function intersects the N P axis three times, indicating the existence of three real,positive values of N P . The outer two correspond to stablesolutions, while the inner root is unstable. These multi-ple cases are illustrated in Fig. 2, in which N P is plotted FIG. 2. (Colour online) Steady state average photon numberin the ring pump mode as a function of channel input powerwith Γ P = 1 GHz, η = 1 Hz for (a) zero detuning, (b) sub-critical detuning, (c) supercritical detuning. Red and bluecurves indicate stable solutions, green unstable. The dashedline respresents choice of optimal detuning to maximize N P at each input power (∆ P = ∆ opt P ( N P ) = 2 ηN P ). as a function of input power for various detunings. Forvalues of | ∆ P | above the critical detuning of √ P thereexists a region of optical bistability, in which two stableequilibrium average pump photon numbers for a giveninput power are permitted, a phenomenon that has beenobserved experimentally in microring systems [24]. Inthis region the two stable solutions are separated by anunstable (and therefore physically inaccessible) range of N P . Also plotted in this figure is the case of “optimaldetuning”, ∆ P = ∆ opt P ( N P ), in which ∆ P is not taken tobe fixed, but chosen to exactly cancel the effect of SPMas P in is increased,∆ P = ∆ opt P ( N P ) = 2 ηN P , (38)which restores the simple linear relationship between N P and | p | , N P = | p | Γ P . (39)This behaviour is indicated by the dashed line in Fig.2, which corresponds to a stable pump solution for allinput powers, always lies on or above the fixed-detuningcurves, and at each input power corresponds to the choiceof detuning that maximizes N P . V. SIGNAL AND IDLER DYNAMICS
Having developed the steady state pump solution, wereturn to the signal and idler equations of motion. Wefirst develop an exact solution to these equations, validfor a cw pump of arbitrary intensity, and then use thissolution to calculate the photon pair generation rate, aswell as the one- and two-photon spectra of the generatedphotons.
A. Exact solution
We begin by writing the equations (16b–16c) for thesignal and idler ring operators in the presence of a classi-cally described cw pump that leads to a ring pump am-plitude of the form β P ( t ) = β P e − i ∆ P t , where β P is aconstant. Letting (cid:101) b x ( t ) = e i ∆ P t b x ( t ) for x = S, I weobtain ddt (cid:18)(cid:101) b S ( t ) (cid:101) b † I ( t ) (cid:19) = M (cid:18)(cid:101) b S ( t ) (cid:101) b † I ( t ) (cid:19) + D ( t ) , (40)where M is the 2 × M = (cid:32) − Γ S − i ( ζ | β P | − ∆ P ) − i Λ β P i Λ β ∗ P − Γ I + i ( ζ | β P | − ∆ P ) (cid:33) , (41)and D ( t ) the driving term responsible for quantum fluc-tuations from the physical and phantom channels, D ( t ) = (cid:32) − ie i ∆ P t ( γ ∗ S ψ S< (0 , t ) + µ ∗ S φ S< (0 , t )) ie − i ∆ P t ( γ I ψ † I< (0 , t ) + µ I φ † I< (0 , t )) (cid:33) . (42)In obtaining M we have assumed the ring resonances areequally spaced, so that ∆ ring = ω S + ω I − ω P = 0;the pump detuning, however, is left arbitrary. Previ-ously [14] we employed a perturbative approach in thefrequency domain to solve these equations, while ignor-ing the effects of SPM and XPM. While this provides anadequate description of the pair generation process forlow pump powers, a nonperturbative strategy is neededto treat the strongly driven case. In the cw regime, where M is time-independent, this coupled set of linear ordinarydifferential equations can be solved exactly in the timedomain for arbitrary pump intensities by taking (cid:18)(cid:101) b S ( t ) (cid:101) b † I ( t ) (cid:19) = t (cid:90) −∞ dt (cid:48) G ( t, t (cid:48) ) D ( t (cid:48) ) , (43)where the 2 × G ( t, t (cid:48) ) is givenby G ( t, t (cid:48) ) = e (cid:82) tt (cid:48) Mdt (cid:48)(cid:48) = e M · ( t − t (cid:48) ) = (cid:18) g D ( t, t (cid:48) ) g A ( t, t (cid:48) ) g ∗ A ( t, t (cid:48) ) g ∗ D ( t, t (cid:48) ) (cid:19) . (44)For simplicity we henceforth assume the ring-channelcoupling constants and propagation speeds for each modeare the same, γ J = γ , v J = u J = v and µ J = µ so Γ J = Γfor each J . The matrix elements g D and g A are then givenby g D ( t, t (cid:48) ) = e − Γ( t − t (cid:48) ) (45) × (cid:18) cosh[ ρ ( t − t (cid:48) )] − i ζN P − ∆ P ρ sinh[ ρ ( t − t (cid:48) )] (cid:19) and g A ( t, t (cid:48) ) = − i Λ β P ρ sinh[ ρ ( t − t (cid:48) )] , (46)in which we have introduced the dynamical parameter ρ , ρ = (cid:113) Λ N P − ( ζN P − ∆ P ) . (47)Depending on the pump photon number N P and detun-ing ∆ P , ρ may be either purely real, purely imaginary,or exactly zero. Indeed, as will become clear in the fol-lowing sections, ρ serves as an important parameter incharacterizing the system’s behaviour.With explicit solutions written down for the ring op-erators (cid:101) b J ( t ), we can make use of the incoming-outgoingchannel field relation (14) to determine ψ J> (0 , t ). Wefind for the signal ψ S> (0 , t ) = (48) (cid:90) dt (cid:48) (cid:20) q SS ( t, t (cid:48) ) ψ S< (0 , t (cid:48) ) + p SS ( t, t (cid:48) ) φ S< (0 , t (cid:48) )+ q SI ( t, t (cid:48) ) ψ † I< (0 , t (cid:48) ) + p SI ( t, t (cid:48) ) φ † I< (0 , t (cid:48) ) (cid:21) , where we have introduced the temporal response func-tions q xx (cid:48) ( t, t (cid:48) ) for the physical channel and p xx (cid:48) ( t, t (cid:48) ) forthe phantom channel: q SS ( t, t (cid:48) ) = δ ( t − t (cid:48) ) − | γ | v θ ( t − t (cid:48) ) e − (Γ+ i ∆ P )( t − t (cid:48) ) (49) × [cosh[ ρ ( t − t (cid:48) )] − i ζ | β P | − ∆ P ρ sinh[ ρ ( t − t (cid:48) )] , and q SI ( t, t (cid:48) ) = (50) − γ Λ β P vρ θ ( t − t (cid:48) ) e − i ∆ P ( t + t (cid:48) ) e − Γ( t − t (cid:48) ) sinh[ ρ ( t − t (cid:48) )] . The phantom channel response functions are related tothese via p SS ( t, t (cid:48) ) = µ ∗ γ ∗ ( q SS ( t, t (cid:48) ) − δ ( t − t (cid:48) )) (51)and p SI ( t, t (cid:48) ) = µγ q SI ( t, t (cid:48) ) . (52)Similar response functions p Ix ( t, t (cid:48) ) and q Ix ( t, t (cid:48) ) can beintroduced for the idler fields, which, due to our assump-tion of equal coupling coefficients and propagation speedsfor the signal and idler fields, are identical to those for thesignal: p IS = p SI , p II = p SS , q IS = q SI and q II = q SS . B. Photon generation rate
Armed with explicit expressions for the outgoing fields ψ S> and ψ I> , we can calculate any measurable quantityrelated to the generated signal and idler photon pairs.Of particular interest is the photon pair generation rate,one of the primary figures of merit used in assessing thepractical utility of the ring-channel system. The steadystate outgoing flux of signal photons J S into the physicalchannel can be calculated via J S = lim t →∞ v (cid:104) ψ † S> (0 , t ) ψ S> (0 , t ) (cid:105) = lim t →∞ (cid:90) dt (cid:48) | q SI ( t, t (cid:48) ) | . (53)Computing the integral, we find J S = 2ΓΛ N P Γ − ρ . (54)The nature of the scaling of J S with pump photon num-ber N P depends intimately on the character of ρ , thebehaviour of which as a function of N P for various de-tunings is illustrated in Fig. 3. Recalling (47), we findthat ρ is real when N P ∈ [∆ P / , ∆ P / Λ], with ρ = 0 atthe endpoints of this interval, and imaginary otherwise.For low enough N P , when ρ ≈ i | ∆ P | , J S scales quadrat-ically with the number of pump photons N P . Since inthis regime N P is directly proportional to the channelinput power P in , the overall scaling of J S with P in re-mains quadratic, in agreement with experiment [5]. Asthe pump power increases, however, the scaling of J S isaffected by several separate power-dependent processes.First, for a fixed detuning ∆ P , the SPM-induced driftof the pump resonance slows the scaling of N P with chan-nel input power P in , as demonstrated in Fig. 2. Second, FIG. 3. (Colour online) Real and imaginary parts of ρ as afunction of N P for zero, subcritical and supercritical detun-ings, indicating that ρ is always either purely real or purelyimaginary. The red line indicates ρ = Γ. The transitionbetween ρ being purely imaginary and purely real occurs atthe points N P = ∆ P /
3Λ and N P = ∆ P / Λ. For supercrit-ical detunings, there exist points where ρ = Γ (representedon the plot by red diamonds), indicating the onset of OPObehaviour. The nonlinear parameter Λ taken as Λ = 10 Hz.FIG. 4. (Colour online) Photon pair generation rate as afunction of channel input power for various detunings. Thedashed curve indicates the optimal detuning case. Systemparameters for this plot are η = 1 Hz, Γ = 1 GHz. XPM between the pump, signal and idler modes effec-tively shifts the resonance lines of the signal and idlermodes, compromising the resonance enhancement of thepair generation process. Finally, for supercritical detun-ings | ∆ P | > ∆ critical , ρ → Γ when N P → N ± , where N ± are the same two photon numbers that define thestability of the pump solution (37). In that limit thephoton flux J S formally diverges. This unphysical pre-diction corresponds to the onset of optical parametric oscillation[11, 12]. As this threshold is approached, stim-ulated emission leads to photon pairs being generatedfaster than the rate at which they are removed from thering, preventing the system from reaching a steady statewithin our model. We are prevented from treating thiscase by our assumption of an undepleted pump. Ourresults are expected to be valid when the intraring con-version efficiency E ring is much less than unity; this ef-ficiency, defined as the ratio between the steady statesignal (or idler) and pump photon numbers, can be ex-pressed as E ring = J S N P = Λ N P Γ − ρ . (55)While in future work we intend to investigate the OPOregime and the associated effects of pump depletion, forthe time being we restrict ourselves to regimes where E ring (cid:28)
1; in all examples presented below this inequal-ity is satisfied.Perhaps most remarkable is the special regime of opti-mal detuning, wherein ∆ P is chosen to maximize N P ateach channel input power, ∆ P = ∆ opt P ( N P ) as defined inEq. (38). For this choice ρ = 0 identically for all N P , ρ = (cid:113) Λ N P − ( ζN P − ∆ P ) = (cid:115) Λ N P − (cid:18) N P − N P (cid:19) = 0 . (56)The photon pair flux then maintains its quadratic scalingwith both N P and channel input power P in over its entiredomain: J S = 8Γ Λ (¯ hω P ) Γ P . (57)This cancellation between the effects arising from pho-ton pair generation, XPM, and the SPM-dependent de-tuning strategy ∆ opt P ( N P ) arises from the simple fixedrelationship between the associated nonlinear couplingstrengths Λ, η and ζ . Crucial for this phenomenon isthat the strength of the photon pair generation processscale quadratically with the pump photon number N P .This cancellation effect would therefore not be possibleusing, for example, spontaneous parametric downcon-version, the strength of which would scale linearly with N P . The presence of thermal resonance drift would notcompromise the existence of an N P -dependent detuningstrategy that yields ρ = 0 over all N P , though such astrategy would no longer correspond to that which alsolinearizes and maximizes the relationship between N P and channel input power.The photon pair generation rate as a function of chan-nel input power is plotted in Fig. 4 for various values of∆ P alongside this optimal detuning case. For lower pow-ers, when SPM and XPM are negligible, a pump beamwith ∆ P = 0 gives the best scaling of J S . For inter-mediate powers the detuning may be tweaked to combatSPM and XPM in order to maximize J S , while for highpowers the optimal detuning strategy of ∆ P = ∆ opt P ( N P )beats any fixed subcritical detuning. The behaviour ofthese curves suggests a simple experiment to identify thepresence of nonperturbative, strongly driven effects: onecould simply measure the outgoing signal or idler poweras a function of pump input power for a set of fixed,subcritical pump detunings. For fixed nonzero detun-ings ∆ P < ∆ critical , strongly driven effects are indicatedby the presence of a global maximum of generated sig-nal power at intermediate pump input power, followed bydecreasing signal power approaching an asymptotic valueof ¯ hω S lim N P →∞ J S ( N P ) = ¯ hω S . (58)For critically coupled ring systems Γ ≈ Γ / hω S Γ /
3. If thermaldetuning of the ring resonances is included this asymp-totic power will be different; however, the qualitative be-haviour of the signal power as a function of pump powerwill be unchanged.
C. Single photon spectrum
Another physical quantity of interest is the spectrallineshape of signal and idler photons that are emittedfrom the ring. For low power cw pumps these singlephoton spectra typically exhibit a Lorentzian lineshape[5, 6, 14] with a characteristic width determined by thetotal effective linewidths of the microring cavity reso-nances. As we now demonstrate, these spectral charac-teristics are significantly different in the strongly pumpedregime. We develop results for the signal field spectrum;the idler field will have identical properties.We define the power spectrum [27] for the signal chan-nel field as ν S ( ω s ; t ) = (59)lim T →∞ T t + T/ (cid:90) t − T/ dt (cid:90) dτ √ π g (1) ( t, t + τ ) e iω s τ where the first-order temporal coherence function g (1) ( t , t ) is defined by g (1) ( t , t ) = v (cid:104) ψ † S> (0 , t ) ψ S> (0 , t ) (cid:105) = ΓΓ (cid:90) dt (cid:48) q ∗ SI ( t , t (cid:48) ) q SI ( t , t (cid:48) ) . (60)In writing (59) we have introduced the relative frequencyco-ordinate ω s , which corresponds to a frequency offsetfrom the ring reference ω S . The physical frequency ω s associated with ω s is therefore ω s = ω S + ω s . (61) FIG. 5. (Colour online) Origin of splitting in the signal andidler lineshapes. Green, blue, and red curves indicate pump,signal and idler resonances, respectively, and could respresentthe enhancement factor [26] that would characterize the ratioof the intensity in the ring to the incident channel intensityin a linear experiment. The dashed green line indicates apump detuned by ∆ P , which leads to a generated pair havingeither its signal or idler photon detuned by ∼ P from thecorresponding resonance, as indicated by the signal and idlerpairs connected by dashed black lines. The presence of pairsfrom both cases leads to a doublet structure for both thesignal and idler lineshapes. In the remaining sections we adopt this notation of low-ercase subscripts for frequency offsets: ω s for the signal, ω p for the pump and ω i for the idler.Evaluating (60) and setting t = t , t = t + τ we obtain g (1) ( t, t + τ ) = (62)ΓΛ N P ρ e i ∆ τ e − Γ | τ | ρ cosh[ ρ | τ | ] + Γ sinh[ ρ | τ | ]Γ − ρ , which is independent of t , depending only on the relativetime difference τ , as would be expected for a cw pump.Taking the Fourier transform, we arrive at an expressionfor the lineshape, ν S ( ω s ) = (63)4ΓΓΛ N P √ π | Γ − ρ + i ( ω s − ∆ P ) | | Γ + ρ + i ( ω s − ∆ P ) | , with an identical equation for the idler lineshape ν I ( ω i ).This expression takes the form of a product of twoLorentzians. We consider first subcritical detunings.When ρ is imaginary, these Lorentzians have identi-cal characteristic widths δω = Γ and are centred on ω s = ∆ P ± | ρ | . For low powers, when ρ ≈ i | ∆ P | thespectrum is therefore peaked at ω s = 0 and ω s = 2∆ P ,in agreement with the perturbative calculation [14]. Thissplitting is easily understood as a consequence of thetradeoff between energy conservation and resonance en-hancement of the pair generation process. As illustratedin Fig. 5, when a photon pair is produced with a de-tuned pump, either the signal photon or idler photon ina pair, but not both, can be generated within a ring reso-nance; energy conservation then requires the other to be0 FIG. 6. (Colour online) Spectral lineshape ν S ( ω s ) scaled to unit maximum vs. ring pump photon number N P . For each plotwe take Γ = 1 GHz and Λ = 10 Hz. (a) ∆ P = 0, (b) ∆ P = 0 . critical ≈ . P = 1 . critical ≈ . P = ∆ opt P ( N P ) = 2 ηN P . The origin of the frequency axis corresponds to the ring resonance at ω S . generated with a frequency that lies away from its corre-sponding resonance. This is seen in the N P → N P a sim-ilar splitting can arise from the effective XPM-induceddetuning of the signal and idler ring resonances even fora pump with ∆ P = 0, as seen in Fig. 6(a) for large N P .When ∆ P = 0 the lineshape begins as a singly-peakedLorentzian, eventually splitting to a doublet structurewhen ρ becomes imaginary as a consequence of XPM.For nonzero ∆ P , as N P increases, XPM effectivelycounters the pump detuning and the extent of this split-ting is reduced as | ρ | decreases, eventually vanishingwhen N P = ∆ P / N P is increased further, ρ be-comes real and ceases to contribute to spectral split-ting, resulting instead in an effective correction to thelinewidth. The lineshape then takes the form of a prod-uct of two Lorentzians, both centred on ω s = ∆ P , withrespective widths δω ± = Γ ± ρ . As ρ becomes compara-ble to Γ the smaller of these two widths becomes domi-nant, leading to a lineshape with overall effective width δω ≈ Γ − ρ . For subcritical detunings, as demonstratedat large N P in Fig. 6(b), the spectral splitting is thenresumed as ρ once again becomes imaginary.For supercritical detunings, as the threshold for opti-cal parametric oscillation is approached ρ → Γ and thebandwidth of the emitted signal and idler photons be-comes arbitrarily narrow, as seen in Fig. 6(c). This fol-lows from our idealization of the pump as an indefinitelycoherent cw beam; in actual experiments the bandwidthof the generated photons will become comparable to thatof the pump, a phenomenon that has been observed instrongly pumped experiments on silicon nitride micror-ings [12].Finally, as shown in Fig. 6(d), in the special case ofoptimal detuning when ∆ P = ∆ opt P ( N P ), so that ρ = 0,the lineshape remains peaked at a single N P -dependentfrequency for each N P , with unchanging characteristicwidth δω = Γ, precisely mimicking the low-power resultat zero detuning.Experimentally, measuring the signal or idler lineshape1as a function of input power for a nonzero, subcriticaldetuning as in Fig. 6(b) would reveal the richness of thestrongly driven regime, and illustrate the behaviour ofthe ρ paramater, which incorporates the effects of bothXPM and pair generation. D. Joint spectral intensity
To assess the degree of spectral correlation between thesignal and idler modes, it is instructive to study the jointspectral intensity distribution of the generated photonpairs. While it is straightforward to define this quantityfor a system driven by a train of weak pump pulses, inwhich multi-pair generation can be neglected, it is a moresubtle task to craft a sensible measure of spectral correla-tion in the strongly driven cw regime. In particular, thereis no single function that characterizes a joint probabil-ity amplitude of signal and idler photons, since in generalthere will be far more than two photons in the quantumstate of the signal and idler modes. Furthermore, evenfor weak cw pumps, if one introduces outgoing channelannihilation operators c J ( ω j ) via ψ J> (0 , t ) = (cid:90) dω j √ π c J ( ω j ) e − iω j t (64)and naively calculates expectation values of the form (cid:104) c † S ( ω s ) c † I ( ω i ) c S ( ω s ) c I ( ω i ) (cid:105) , the idealization of a zero-bandwidth cw pump leads to ill-defined expressions in-volving the square of Dirac δ distributions.To resolve these difficulties, in Appendix B we developa model of a typical experiment used to characterize theJSI for weakly driven systems, in which the coincidencecount rate of signal and idler photons at respective fre-quencies ω s and ω i is measured. We the extend the def-inition of the JSI to strongly driven systems by definingthe JSI to equal the calculated outcome of such an ex-periment for arbitrary input power. This definition re-duces to the usual result for single-pair output states,and serves as a sensible measure of spectral correlationbetween the signal and idler fields. This coincidence ratecan be written as I corr ( ω s , ω i ) = v δt (2 π ) (cid:90) dt ... (cid:90) dt (65) (cid:20) e iω s ( t − t ) e iω i ( t − t ) T ( t ) T ( t ) T ( t ) T ( t ) × (cid:104) ψ † S> ( t ) ψ † I> ( t ) ψ S> ( t ) ψ I> ( t ) (cid:105) (cid:21) , where T ( t ) is the Fourier transform of a transmissionfunction ˆ T ( ω ) that resolves the frequencies of the signaland idler photons prior to detection, T ( t ) = (cid:90) dω √ π ˆ T ( ω ) e − iωt , (66)and δt is the temporal resolution of the coincidencecounter. In this expression the spatial dependence of the field operators ψ J> ( z, t ) has been suppressed; the signaland idler arms of the JSI measurement are assumed tooccur at balanced distances from the ring-channel cou-pling point.The four-time expectation value (cid:104) ψ † S> ( t ) ψ † I> ( t ) ψ S> ( t ) ψ I> ( t ) (cid:105) is found to natu-rally split into two parts, v (cid:104) ψ † S> ( t ) ψ † I> ( t ) ψ S> ( t ) ψ I> ( t ) (cid:105) = (67) A ∗ ( t , t ) A ( t , t ) + g (1) ( t , t ) g (1) ( t , t ) , where A ( t , t ) = (cid:90) dt (cid:48) (cid:20) q SI ( t , t (cid:48) ) q II ( t , t (cid:48) ) (68)+ p SI ( t , t (cid:48) ) p II ( t , t (cid:48) ) (cid:21) . The function g (1) is precisely the first-order coherencefunction defined in Eq. (60) used to calculate the singlephoton spectrum, g (1) ( t , t ) = ΓΛ N P ρ e i ∆ P ( t − t ) e − Γ | t − t | (69) × ρ cosh[ ρ | t − t | ] + Γ sinh[ ρ | t − t | ]Γ − ρ . The A ( t , t ) term, after computing the integrals, is givenby A ( t , t ) = γ Λ β P v e − Γ | t − t | e − i ∆ P ( t + t ) (70) × [ a sinh[ ρ | t − t | ] + a cosh[ ρ | t − t | Γ − ρ , where the constants a and a are defined by a = ρ − i ζN P − ∆ P ρ Γ ,a = Γ − i ( ζN P − ∆ P ) . (71)The JSI can therefore be expressed as the sum of corre-lated and uncorrelated terms, I ( ω s , ω i ) = I corr ( ω s , ω i ) + I uncorr ( ω s , ω i ) , (72)where I corr ( ω s , ω i ) = (73) δt (2 π ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) dν (cid:90) dν ˆ A ( ν , ν ) ˆ T ( ω s − ν ) ˆ T ( ω i − ν ) (cid:12)(cid:12)(cid:12)(cid:12) and I uncorr ( ω s , ω i ) = δt (2 π ) (74) × (cid:90) dν (cid:90) dν (cid:104) ˆ g (1) ( ν , − ν ) ˆ T ( ω s − ν ) ˆ T ( ω s − ν ) (cid:105) × (cid:90) dν (cid:48) (cid:90) dν (cid:48) (cid:104) ˆ g (1) ( ν (cid:48) , − ν (cid:48) ) ˆ T ( ω i − ν (cid:48) ) ˆ T ( ω i − ν (cid:48) ) (cid:105) . I uncorr can be expressed asa separable product of functions of ω s and ω i , while I corr cannot. Each is expressed as a convolution of the Fouriertransforms ˆ A ( ν , ν ) and ˆ g (1) ( ν , ν ) of the A ( t , t ) and g (1) ( t , t ) functions,ˆ A ( t , t ) = (cid:90) dt √ π (cid:90) dt √ π A ( t , t ) e iν t e iν t , (75) and similarly for g (1) ( ν , ν ), with the transmission filterfunction ˆ T ( ν ) ˆ T ( ν ). The A and g (1) functions are deter-mined by the dynamics of the signal and idler modes inthe ring, while their convolution with the T functions re-flects the frequency averaging that arises from the finiteresolution of a realistic JSI measurement.Computing the Fourier transform, we find for ˆ A ˆ A ( ν , ν ) = Γ Λ N P δ ( ν + ν − P ) (cid:32) − i ζN P − ∆ P ρ ( i ∆ ν − Γ + ρ )( − i ∆ ν − Γ + ρ ) + 1 + i ζN P − ∆ P ρ ( i ∆ ν − Γ − ρ )( − i ∆ ν − Γ − ρ ) (cid:33) , (76)with ∆ ν = ( ν − ν ) /
2. The term in parentheses multiplying the δ function varies on the scale of Γ. Assuming thatthe measurement frequency resolution δω trans is much narrower than this, the slowly varying term can be pulled outof the integrals in (73), leaving I corr ( ω s , ω i ) ≈ δt Γ Λ N P [ D ( ω s − ∆ P , ω i − ∆ P )] (77) × (cid:12)(cid:12)(cid:12)(cid:12) − i ζN P − ∆ P ρ ( i ( ω i − ∆ P ) − Γ + ρ )( − i ( ω i − ∆ P ) − Γ + ρ ) + 1 + i ζN P − ∆ P ρ ( i ( ω i − ∆ P ) − Γ − ρ )( − i ( ω i − ∆ P ) − Γ − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) where D ( ω s , ω i ) = (78)12 π (cid:90) dν (cid:90) dν δ ( ν + ν ) ˆ T ( ω s − ν ) ˆ T ( ω i − ν ) . The function D ( ω s , ω i ) can be interpreted as the“smoothed” version of the Dirac δ ( ω s + ω i ) distribution,and arises from the finite bandwidth of the JSI measure-ment scheme; D ( ω s − ∆ P , ω i − ∆ P ) is sharply peakedand uniform along the energy-conserving antidiagonalline ω s + ω i − P = 0 with characteristic width δω trans (the measurement resolution) in the direction orthogonalto that line.Finally, taking the Fourier transform of g (1) ( t , t ), wefind ˆ g (1) ( ν , − ν ) = δ ( ν − ν ) (79) × N P | Γ − ρ + i ( ν − ∆ P ) | | Γ + ρ + i ( ν − ∆ P ) | . As with ˆ A , apart from the δ function this is slowly vary-ing compared to the measurement resolution; the termmultiplying the δ function can be pulled out of the inte-gral in Eq. (74). The uncorrelated contribution to theJSI I uncorr is therefore well approximated by I uncorr ( ω s , ω i ) ≈ δt π (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) dω | ˆ T ( ω ) | (cid:12)(cid:12)(cid:12)(cid:12) ν S ( ω s ) ν I ( ω i ) , (80)where ν S ( ω s ) and ν I ( ω i ) are precisely the single-photonlineshape functions given by Eq. (63) as derived in theprevious section. The uncorrelated part of the JSI istherefore proportional to the simple product the signaland idler lineshapes. For low power cw pumps, wherein multi-pair gener-ation is insignificant, the uncorrelated part of the JSI I uncorr is negligible and I corr dominates. The JSI thentakes the form of a narrow antidiagonal line correspond-ing to the energy-conserving condition ω s + ω i − P = 0.For ∆ P = 0, the line is singly peaked, as illustrated inFig. 7(a). For nonzero ∆ P the line is distributed amongtwo peaks separated by 2∆ P , as evident in Fig. 7(b),consistent with the single photon spectrum derived in theprevious section. At higher powers, such as in Fig. 7(c),this splitting can also arise from XPM-induced signal andidler detuning even for a pump with ∆ P = 0. When thesplitting is due to XPM-induced signal and idler detun-ing, the JSI remains centred on the unperturbed ringresonances. On the other hand, when pump detuning isresponsible for the splitting, the JSI is translated by ∆ P along both frequency axes.In Fig. 8 the uncorrelated contribution I uncorr to theJSI is plotted for the same pump parameters as in Fig.7. The weight of the uncorrelated contribution is ex-tremely small compared to the correlated contributionat low powers, as indicated by the scales in Figs. 7 and8, but grows to an appreciable level at high powers. For∆ P = 0, as in Fig. 8(a), at low N P the uncorrelatedpart of the JSI displays a single peak centred at the ori-gin. In the regimes that give rise to split lineshapes, asillustrated in Fig. 8(b) and 8(c), the uncorrelated contri-bution takes the form of four distinct peaks, symmetri-cally placed about the centre of the overall distribution.Two of these peaks lie on the antidiagonal, overlappingwith the correlated contribution. The remaining two lieon the diagonal, and would therefore appear to violateenergy conservation if assumed to correspond to signaland idler photons that originated from the same pair.It is therefore natural to interpret these peaks as corre-3sponding to signal and idler photons that are detectedfrom separate pairs. As these uncorrelated, “non-energyconserving” peaks are well separated from the correlatedpart of the JSI, they are uncontaminated by the corre-lated contribution to the JSI. The properties of photonpairs detected in these peaks would therefore be expectedto differ from those detected in the antidiagonal peaks.We intend to investigate such properties in future work.The form of the JSI depends qualitatively on whether ρ is imaginary or real, a behaviour we saw earlier in thesingle photon spectrum. When ρ is imaginary, and thuscontributes to the frequency terms in the denominators ofEqs. () and (), a splitting in the JSI appears. When ρ isreal, and acts as an effective correction to the linewidth Γ,the JSI is localized to a single line arising from the over-lap of I corr with a single peak in I uncorr . For sufficientlydetuned pumps, in the regime of real ρ the uncorrelatedcontribution can be large enough to be visible on the JSIplot without exaggeration or scaling. Indeed, for super-critical detunings | ∆ P | > ∆ critical , as the OPO thresholdis approached and ρ → Γ the uncorrelated contributionvastly dominates over the correlated contribution, as seenin Fig. 9. This is an expected consequence of the rapidgrowth in the photon pair generation rate in this regime– multiple photon pairs are generated in sufficiently largequantities that joint detection of a signal and idler photonoriginating from the same energy-conserving pair is un-likely relative to the probability of detecting a signal andidler photon which originated from separate pairs andthus obey no relationship in energy. Another effect seenas ρ → Γ is the narrowing of the entire JSI distribution toa small point-like peak centred on ( ω s , ω i ) = (∆ P , ∆ P ).Within our idealization of a zero-bandwidth cw pump thearea of this point would be limited only by the frequencyresolution of the JSI measurement scheme, though in ac-tual experiments the finite pump bandwidth would serveas a fundamental lower bound for the overall extent ofthe JSI.Perhaps the most definitive experimental indication ofstrongly driven effects lies in the top-right and bottom-left uncorrelated peaks of the JSI distribution for a de-tuned pump as in Fig. 8(b). For sufficient detuningsthese peaks are well separated from the antidiagonal andthus easily distinguished from the correlated part of theJSI. For low powers, wherein only one photon pair isgenerated in the ring at any given time, they would beentirely absent from the measured JSI. As the power in-creases, any non-spurious coincidence detection of pho-tons in these regions indicates multi-pair generation, asphotons generated in those peaks do not conserve energyand therefore must be associated with separate, indepen-dently produced pairs. VI. CONCLUSION
We have investigated the strongly driven regime ofspontaneous four-wave mixing in microring resonators for
FIG. 7. (Colour online) Correlated part I corr of joint spectralintensity distribution, scaled to unit maximum, of signal andidler photon pairs for a pump with (a) ∆ P = 0 , N P = 10,(b) ∆ P = 3Γ , N P = 10, and (c) ∆ P = 0 , N P = 2 × .Ring parameters are taken as Γ = 1 GHz and Λ = 10 Hz.The splitting evident in (b) arises from the pump detuning,whereas in (c) the XPM-induced detuning of the signal andidler ring modes is responsible. FIG. 8. (Colour online) Uncorrelated part I uncorr of jointspectral intensity distribution, scaled to unit maximum, ofsignal and idler photon pairs. Pump parameters are (a) ∆ P =0 , N P = 10, (b) ∆ P = 3Γ , N P = 10, and (c) ∆ P = 0 , N P =2 × . Ring parameters are taken as Γ = 1 GHz and Λ = 10Hz. FIG. 9. (Colour online) Joint spectral intensity distribution,scaled to unit maximum, of signal and idler photon pairs for(a) ∆ P = 1 . critical with N P = 9 . × (90% of OPOthreshold) and (b) ∆ P = 1 . critical with N P = 1 × (95%of OPO threshold). Ring parameters are taken as Γ = 1 GHzand Λ = 10 Hz. a cw pump input. A nonperturbative, exact analytic so-lution to the semiclassical equations of motion within theundepleted pump approximation was developed, whichpermits the calculation of any physical quantity relatedto the outgoing signal and idler fields while fully takinginto account intraring scattering losses. The effects ofself- and cross- phase modulation, as well as multi-pairgeneration, were found to drastically alter the nature ofthe photon pair generation process at high powers. Acritical pump detuning of ∆ critical = √ Appendix A: Calculation of nonlinear couplingconstants
To estimate the nonlinear coupling constants Λ, η and ζ , we present in this section a derivation of the nonlinearsector H NL of the ring Hamiltonian. For the moment we imagine the ring has been decoupled from both thephysical and phantom channels, so that it is idealized asa perfect, isolated cavity. We expand the electric field E ( r ) and electric displacement field D ( r ) in the ring interms of discrete ring modes E α ( r ) and D α ( r ) as E ( r ) = (cid:88) α (cid:114) ¯ hω α b α E α ( r ) + H . c ., D ( r ) = (cid:88) α (cid:114) ¯ hω α b α D α ( r ) + H . c ., (A1)where ω α are the mode frequencies and b α the associ-ated annihilation operators. The contribution to the ringHamiltonian arising from the third-order nonlinear sus-ceptibility can be written [28] as H NL = (A2) − (cid:15) (cid:90) d r Γ ijkl (3) ( r ) D i ( r ) D j ( r ) D k ( r ) D l ( r )with implied summation over repeated lowercase Ro-man indices, where (cid:15) is the permittivity of vacuum andΓ ijkl (3) ( r ) represents the nonlinear response coefficients.Within the rotating wave approximation, only keepingrelevant terms for the pump, signal and idler modes, weobtain H NL = (A3) − (cid:15) (cid:18) (cid:19) ¯ hω P (cid:114) ¯ hω S hω I Q SIP P b † S b † I b P b P − (cid:15) (cid:18) (cid:19) ¯ hω P (cid:114) ¯ hω S hω I Q P P IS b † P b † P b I b S − (cid:15) (cid:18) (cid:19) (cid:18) ¯ hω P (cid:19) Q P P P P b † P b † P b P b P − (cid:15) (cid:18) (cid:19) (cid:18) ¯ hω P hω S (cid:19) Q SP SP b † S b † P b S b P − (cid:15) (cid:18) (cid:19) (cid:18) ¯ hω P hω I (cid:19) Q IP IP b † I b † P b I b P , where the constants Q IJKL are given by Q IJKL = (A4) (cid:90) d r (cid:16) Γ ijkl (3) ( r )( D iI ( r )) ∗ ( D jJ ( r )) ∗ D kK ( r ) D lL ( r ) (cid:17) . As is typically done for dispersive media we take [29]Γ ijkl (3) ( r ) (A5)= χ ijkl (3) ( r ) (cid:15) n ( r ; ω ) n ( r ; ω ) n ( r ; ω ) n ( r ; ω ) , where n ( r ; ω ) is the linear refractive index of the ringmedium at frequency ω , and χ ijkl (3) ( r ) is the frequency-dependent nonlinear susceptibility. To evaluate the coef-ficients Q IJKL we introduce co-ordinates for the ring r ⊥ and l φ , such that the volume element d r = ρdρdφdz (A6)6can be written as d r = ρdρdzR dl φ = d r ⊥ dl φ , (A7)where R is the nominal ring radius and l φ = Rφ , whichvaries from 0 to 2 πR ≡ L , the nominal ring circumfer-ence. The co-ordinate r ⊥ is understood as shorthand forthe pair ( ρ, z ). Writing the mode fields E α ( r ) as E α ( r ) = e α ( r ⊥ ) e ik α l φ √ L , (A8)where k α = 2 πn α /L for integer n α , we can simplify H NL to H NL = (A9) − L ¯ hω P (cid:114) ¯ hω S hω I Q (cid:48) SIP P b † S b † I b P b P − L ¯ hω P (cid:114) ¯ hω S hω I Q (cid:48) P P IS b † P b † P b S b S − L (cid:18) ¯ hω P (cid:19) Q (cid:48) P P P P b † P b † P b P b P − L (cid:18) ¯ hω P hω S (cid:19) Q (cid:48) SP SP b † S b † P b S b P − L (cid:18) ¯ hω P hω I (cid:19) Q (cid:48) IP IP b † I b † P b I b P , in which the reduced constants Q (cid:48) IJKL are given by Q (cid:48) IJKL = 1 √ Z I Z J Z K Z L (A10) × (cid:20) (cid:90) d r ⊥ dl φ (cid:15) χ ijkl (3) ( r ⊥ , l φ )( e iI ( r ⊥ )) ∗ ( e jJ ( r ⊥ )) ∗ × e kK ( r ⊥ ) e lL ( r ⊥ ) e i ( k K + k L − k I − k J ) l φ (cid:21) . (A11)The modes (A1) are normalized [29] such that Z α = (A12)1 L (cid:90) d r ⊥ dl φ (cid:15) n ( r ⊥ ; ω α ) e ∗ α ( r ⊥ ) · e α ( r ⊥ ) γ gp ( r ⊥ ; ω α ) , = 1 , (A13)where γ gp ( r ⊥ ; ω α ) is the ratio of the group and phase ve-locities of the ring medium at each frequency and spatialpoint. However, we display the Z α explicitly in (A10) sothat the expression can be used regardless of whether ornot the e α ( r ⊥ ) are normalized such that Z α = 1.To estimate the constants Q (cid:48) IJKL we approximate theratio γ gp ≈ e α ( r ⊥ ) to be ofuniform magnitude within the ring and vanish elsewhere.We take this uniform magnitude to be unity and assumethat for the modes of interest (2 k P − k I − k S ) L (cid:28)
1. Formodes with polarization mainly perpendicular to the ringplane, the relevant susceptibility will be χ zzzz (3) ≡ χ (3) , in-dependent of position in the ring; this can be immediately generalized to treat other mode polarizations. We thenhave Z α = (cid:15) n A, (A14)where A is the cross-sectional area of the ring. Taking ω S ≈ ω I ≈ ω P in the prefactors of (A9), we finally obtain H NL ≈ (cid:16) ¯ h Λ b P b P b † S b † I + H . c . (cid:17) + ¯ hηb † P b † P b P b P + ¯ hζ (cid:16) b † S b † P b S b P + b † I b † P b I b P (cid:17) , (A15)where Λ = 3¯ hω P χ (3) (cid:15) n LA (A16)and η = Λ / ζ = 2Λ. In terms of the more ex-perimentally accessible nonlinear refractive index n =3 χ (3) / (cid:15) cn this becomesΛ = ¯ hω P cn n LA , (A17)which is in line with the results of similar derivations [25].
Appendix B: An operational definition of the jointspectral intensity
When characterizing a source of entangled photonpairs, the joint spectral intensity (JSI) distribution is of-ten introduced in the low power limit, when the state ofthe signal and idler modes is well approximated by | ψ SI (cid:105) = p vac | vac (cid:105) (B1)+ (cid:90) dω s (cid:90) dω i f ( ω s , ω i ) a † S ( ω s ) a † I ( ω i ) | vac (cid:105) , where | p vac | < a † J ( ω ) refer tothe creation operators at frequency ω for the signal andidler modes[30]. The unsymmetrized and unnormalizedJSI for the such a state is defined as | f ( ω s , ω i ) | , andis proportional to the probability density per unit timeof jointly detecting a signal and idler photon pair withrespective frequencies ω s and ω i . For strongly pumpedsources, when multiple photon pairs are generated in sig-nificant quantities so that higher order terms involvingmore than two creation operators appear in the state,it is less straightforward to define a single function thatcharacterizes the energy relationship between simultane-ously detected signal and idler photons. Instead, onecan operationally extend the definition of the JSI tostrongly pumped sources by calculating for arbitrary in-put power the outcome of experiments designed to mea-sure | f ( ω s , ω i ) | in the low power limit. In this sectionwe develop such a calculation for a typical measurementscheme employed to measure the JSI of photon pairs pro-duced in a microring resonator.We consider a standard experimental setup [31] tomeasure coincidence rates between signal and idler pho-tons of particular frequencies as illustrated in Fig. 10.7 FIG. 10. (Colour online) Schematic of experimental setup formeasuring the joint spectral intensity distribution. Pump,signal and idler { P, S, I } outputs from the ring resonatorare incident on a filter F that removes the pump compo-nent from the beam. Signal and idler fields with modes c S and c I are then separated by dichroic beamsplitter DBS, andindependently filtered by monochromators, which are imple-mented by frequency-dependent beamsplitters BS1 and BS2.Each monochromator-beamsplitter transmits in a small win-dow δω trans about ω i and ω s , respectively. Broadband pho-todetectors D1 and D2 measure the detector modes c S, det and c I, det , and are connected to coincidence counter CC to regis-ter joint detection events within a temporal resolution of δt .Vacuum is input to the empty ports of BS1, BS2 and DBS. The signal and idler fields are separated, and each fieldis sent through a separate monochromator set to trans-mit photons in some small range δω trans about a centrefrequency ω s for the signal and ω i for the idler. Placedafter each monochromator are broadband photodetectorsconnected to a coincidence counter to identify simultane-ously detected signal and idler photons. The transmis-sion frequencies ω s and ω i are independently controllable,and correspond to a single point (or, more accurately, sin-gle bin) on the JSI plot, which is produced by scanningthrough ω s and ω i and measuring the corresponding co-incidence rate. The transmission width is chosen to bemuch smaller than the linewidth of the measured pho-tons, δω trans (cid:28) Γ, so that the full 2D spectrum can beresolved.The monochromators can be simply modelled asfrequency-dependent beamsplitters. Provided both thesignal and idler arms of the experiment are balanced,the spatial dependence of the fields after the ring canbe suppressed; all fields in this section are understood tobe evaluated immediately after the ring-channel couplingpoint. We introduce annihilation operators c S ( ω s ) and c I ( ω i ) for the ring output fields ψ S> ( t ) and ψ I> ( t ) as inEq. (64). We can then apply the appropriate transforma-tion to obtain the annihilation operators c J, det ( ω j ) for thefields seen by the detectors placed after the monochro-mators. For the signal, we have c S, det ( ω ) = ˆ T ( ω − ω s ) c S ( ω ) + ˆ R ( ω − ω s ) c S, vac ( ω ) , (B2)in which c S, vac refer to the modes on the other input portof the monochromator-beamsplitter, into which only vac-uum is present. The transmission and reflection functions ˆ T ( ω ) and ˆ R ( ω ) determine which frequencies are transmit-ted by the the monochromator. For example, a simplefilter may be modelled by a transmission function with arectangular frequency profile,ˆ T ( ω ) = (cid:40) , − δω trans < ω < δω trans , otherwise . (B3)The exact choice of ˆ T ( ω ) is not important for our pur-poses; for simplicity we only assume ˆ T ( ω ) is a real, suffi-ciently narrow, symmetric function of ω . The reflectionfunction will be irrelevant, though it will satisfy the usualrestrictions to correctly model a beamsplitter.In exactly the same manner, modes c I, det ( ω ) seen bythe detectors of the idler arm can be introduced. We canthen write down the fields measured by each detector inthe usual way, ψ J, det ( t ) = (cid:90) dω j √ π c J, det ( ω j ) e − iω j t . (B4)In a typical coincidence measurement the signal detectoris continuously activated, and detection of a signal pho-ton at time t is used to trigger the activation of the idlerdetector (which is placed at a small delay relative to thesignal detector) for a very short time δt , so that the idlerdetector samples the idler field during the time interval[ t − δt/ , t + δt/ I ( ω s , ω i ; t ) at time t of coincident detection events at ω s and ω i of the sig-nal and idler detectors is given by the standard Glauberformula involving the fields at each detector [32], I ( ω s , ω i ; t ) = (B5)lim T →∞ (cid:34) T t + T/ (cid:90) t − T/ dt (cid:48) t (cid:48) + δt/ (cid:90) t (cid:48) − δt/ dt (cid:48)(cid:48) v (cid:104) ψ † S, det ( t (cid:48) ) ψ † I, det ( t (cid:48)(cid:48) ) ψ S, det ( t (cid:48) ) ψ I, det ( t (cid:48)(cid:48) ) (cid:105) (cid:35) . (B6)In steady state the expectation value depends only ontime difference | t (cid:48)(cid:48) − t (cid:48) | , which in the integrand is at most δt . Provided the coincidence resolution time δt is muchsmaller than the timescale on which the expectation valuevaries (in our case Γ − ), this expression for I ( ω s , ω i ; t ) isthen well approximated by I ( ω s , ω i ; t ) ≈ (B7) v δt (cid:104) ψ † S, det ( t ) ψ † I, det ( t ) ψ S, det ( t ) ψ I, det ( t ) (cid:105) . Proceeding to expand the detector fields in terms oftheir constituent modes, we find I ( ω s , ω i ; t ) = (B8) v δt (2 π ) (cid:90) dν ... (cid:90) dν (cid:20) e i ( ν + ν − ν − ν ) × ˆ T ∗ ( ν − ω s ) ˆ T ∗ ( ν − ω i ) ˆ T ( ν − ω s ) ˆ T ( ν − ω i ) × (cid:104) c † S ( ν ) c † I ( ν ) c S ( ν ) c I ( ν ) (cid:105) (cid:21) . c J ( ν i ) in terms of their respec-tive parent fields and then carrying out the integrationover each ν i , we arrive at Eq. 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