SSingle photons from dissipation in coupled cavities
H. Flayac and V. Savona
Institute of Physics, ´Ecole Polytechnique F´ed´erale de Lausanne EPFL, CH-1015 Lausanne, Switzerland
We propose a single photon source based on a pair of weakly nonlinear optical cavities subject toa one-directional dissipative coupling. When both cavities are driven by mutually coherent fields,sub-poissonian light is generated in the target cavity even when the nonlinear energy per photonis much smaller than the dissipation rate. The sub-poissonian character of the field holds overa delay measured by the inverse photon lifetime, as in the conventional photon blockade, thusallowing single-photon emission under pulsed excitation. We discuss a possible implementation ofthe dissipative coupling relevant to photonic platforms.
INTRODUCTION
Single photon sources [1, 2] are fundamental buildingblocks for quantum information protocols. Current real-izations based on blockade mechanisms [3] unavoidablyrequire a strong optical nonlinearity. They are usuallyengineered with such systems as quantum dots [4–8], di-amond color centers [9], superconducting circuits [10] ortrapped atoms [11]. Although the degree of control overthese systems is steadily improving, they basically op-erate at cryogenic temperatures and (or) imply signifi-cant fabrication challenges, particularly with respect tointegration and scalability in future photonic platforms.On the other hand, nondeterministic sources relying onheralding protocols are now operating at room temper-ature in Silicon [12–16], but they require a significantinput power to trigger the four wave mixing mechanism.The unconventional photon blockade (UPB) was pro-posed as a novel paradigm to produce sub-poissonianlight in presence of a very weak nonlinearity [17]. Theseminal system consists of a pair of coherently coupledcavities embedding a Kerr nonlinear medium, where oneof the cavities is driven by a classical source [17–20]. Itwas then extended to Jaynes-Cummings [18], optome-chanical [21] or bimodal cavity [22] systems and wouldbe feasible in superconducting circuits [23], or optimizedsilicon photonic crystal platform [24, 25]. It was shownthat UPB essentially originates from a quantum inter-ference mechanism [17, 18], that arises even when thenonlinear energy per photon U (cid:28) κ , where κ is the cav-ity photon loss rate. UBP was shown to be within reachof an optimized silicon photonic crystal platform [24, 25],where it could lead to a new class of highly integrable,ultralow-power, passive single photon sources. However,the coherent mode coupling – with rate J – results adetrimental oscillation of the delayed two-photon corre-lations g (2) ( τ ), thus restricting the sub-poissonian behav-ior delays shorter than 1 /J (cid:28) /κ , under the requiredoptimal antibunching conditions [18]. As a consequence,UPB is suppressed under pulsed excitation, as the anti-bunched portion of the time-dependent field contributesminimally to the emitted pulse of duration 1 /κ [25]. Fil-tering the output pulse through a narrow time gate was shown to improve photon antibunching at the expense ofa reduced photon rate [25]. Alternative schemes [26] relyon a strong auxiliary driving field, thus departing fromthe desired low-power operation.UPB can be understood in terms of Gaussian squeezedstates [27]. For any coherent state | α (cid:105) , there exists anoptimal squeezing parameter ξ that minimizes the two-photon correlation g (2) (0), which can be made vanishingfor weak driving field. In the UPB scheme, the two cou-ple modes bring enough flexibility to tune the α and ξ values of the target mode independently [19]. A moreeffective approach would then consist in pipelining twosubsystems, one which provides the squeezing ξ and theother that induces the corresponding optimal displace-ment α . 𝐹 𝜅 Δ 𝜔 𝑈 𝜔 𝐿 𝐹 𝜅 Δ 𝜔 𝑈 𝜒 FIG. 1: (Color online) Scheme of the proposed system: Twononlinear and dissipative optical resonators are driven by mu-tually coherent fields of same frequency ω L but with differentcomplex amplitudes F , . The one-directional dissipative cou-pling from cavity 1 to cavity 2 occurs at a rate χ . In this paper, we develop such an approach by investi-gating a scheme where two optical resonators are linkedvia a dissipative – i.e. one-directional – coupling [28–30].The nature of such coupling allows independent tuning ofthe field squeezing and displacement in the second cav-ity. The most significant advance brought by the one-directional coupling however, is the absence of a normal-mode energy splitting. This removes the oscillations typ-ical of UPB, so that the condition g (2) ( τ ) < a r X i v : . [ qu a n t - ph ] J un THE MODEL
We consider two driven Kerr resonators with dissipa-tive (one-directional) coupling from the first to the sec-ond cavity as sketched in Fig.1. Our goal is to find aregime of parameters for which cavity 2 – from now ondenoted target cavity – displays sub-poissonian photonstatistics. In the frame rotating at the frequency ω L ofthe driving fields, the system Hamiltonian readsˆ H = (cid:88) j =1 , (cid:104) − ∆ j ˆ a † j ˆ a j + U j ˆ a † j ˆ a † j ˆ a j ˆ a j + F ∗ j ˆ a j + F j ˆ a † j (cid:105) , (1)where F j ( j = 1 ,
2) are the complex driving field ampli-tudes for each cavity, ∆ j = ω j − ω L are the cavity modedetunings, and U j are the strengths of the Kerr nonlin-earities. The open system dynamics obeys the quantummaster equation i ∂ ˆ ρ∂t = (cid:104) ˆ H , ˆ ρ (cid:105) − i (cid:88) j =1 , κ j ˆ D [ˆ a j ] ˆ ρ + iχ ˆ D [ˆ a , ˆ a ] ˆ ρ , (2)where ˆ D [ˆ a j ] ˆ ρ = { ˆ a † j ˆ a j , ˆ ρ } − a j ˆ ρ ˆ a † j describe the dissipa-tion into the environment at rates κ j , and ˆ D [ˆ a , ˆ a ] ˆ ρ =[ˆ a ˆ ρ, ˆ a † ] + [ˆ a , ˆ ρ ˆ a † ] models the dissipative coupling at arate χ = √ ηκ κ . Here, we have defined a transfer effi-ciency η ∈ [0 ,
1] [31], relating the coupling to the dissipa-tion rates.Before directly solving Eq. (2), it is useful to study thesystem in the limit of weak driving fields F , →
0. Inthis limit, analytical expressions for the various expec-tation values can be obtained by assuming pure statesand restricting to the n ≤ g (2)2 (0) = (cid:104) ˆ a † ˆ a † ˆ a ˆ a (cid:105)(cid:104) ˆ a † ˆ a (cid:105) (cid:39) | c | | c | , (3)where | c | and | c | are the probabilities of having zerophotons in cavity 1 and, respectively, 1 and 2 photonsin the target cavity [see Eqs.(A.18,A.21)]. By requiring c = 0, one obtains a condition for an optimal sub-poissonian behavior, g (2)2 (0) (cid:39)
0. This optimal conditioncan be met, provided that the driving fields fulfill F | opt = iF ˜∆ ˜ U ± (cid:112) − U ˜ U ˜∆ ˜∆ (cid:16) ˜∆ + U (cid:17) χ , (4)where we defined ˜∆ , = − ∆ , − iκ , /
2, ˜∆ = ˜∆ + ˜∆ ,˜ U = ˜∆ + U , and assuming F ∈ R + without loss ofgenerality. Eq.(4) reveals several interesting features: (i) F | opt needs to carry the proper magnitude and phase. (ii) F | opt depends linearly on F , which cannot there-fore be set to zero. Indeed, an undriven target cavitywould simply act as a spectral filter [32], thus essentiallyrecovering the single mode statistics. (iii) The optimalfield amplitude doesn’t depend on U , which can there-fore be set arbitrarily small. If under the assumption thatthe sub-poissonian character originates from the optimalsqueezing mechanism described in Ref.[27], then this fea-ture hints at the fact that cavity 1 is here the main sourceof squeezing. (iv) In the case where U = 0, an optimalvalue of the driving field is still well defined but resultsin a vanishing occupation of the target cavity. (v) Wemade no assumptions on the value of U j , which may beset arbitrarily smaller or larger than the loss rates κ j . RESULTS AND DISCUSSION
From now on, we will consider the case of cavities withequal loss rates κ = κ = κ and and nonlinearities U = U = U . We solve numerically Eq. (2) in thestationary limit, on a Hilbert space truncated to include N max quanta per mode. As a figure of merit, in Fig.2(a),we show the two-photon correlation g (2)2 (0) for the targetcavity, as a function of its average occupation n (blueline). We assumed for this calculation the most interest-ing regime of weak nonlinearity compatible with Siliconphotonic crystal cavities where U = 10 − κ [25]. We ad-ditionally assumed ∆ = ∆ = 0 for simplicity. Since U (cid:28) κ , the optimal condition (4) approximately reducesto F | opt (cid:39) iF κ χ (5) -3 -2 -1 U/ κ g ( ) ( ) × -4 g (0)Opt. Ther.Opt. Pure Non-Gaussian -8 -7 -6 -5 -4 -3 -2 -1 n -8 -7 -6 -5 -4 -3 -2 -1 g ( ) ( ) Master Eq.Lin.Lin. Opt. (a) (b)
FIG. 2: (Color online) (a) Target cavity two-photon corre-lations g (2)2 (0) versus its occupation n . Here U = 10 − κ , F = F | opt and χ = κ . Blue line: exact master equation.Red line: linearized model. Yellow line: linearized modelwhere F was obtained for each value of n from numericalminimization of g (2)2 (0). (b) g (2)2 (0) versus U at fixed occu-pation n = 10 − . The dashed black and red line denote thethermal and pure state Gaussian boundaries set in Ref. 27. For increasing driving field amplitudes, the valueof N max required for convergence becomes exceedinglylarge. To extend the range of accessible n values, welinearize with respect to the mean-field solution (see Ap-pendix B). The result is plotted in Fig.2(a) (red line) andmatches perfectly with the full quantum treatment from n > − where the mean field dominates over fluc-tuations. The optimal two-photon correlation behaveslinearly as a function of n , with the exception of largeroccupancies where a nonlinear increase in g (2)2 (0) is dis-played. This behavior resides in the limited range ofvalidity of Eq.(4) which loses accuracy as the 3 photonprobability rises. For the largest values of n we thereforesearched for the optimal parameters numerically, usingthe amplitude and phase of F as free parameters. Thevalue g (2)2 (0) (cid:54) . n (cid:39) .
25 (yellow line) for such a weakly non-linear system. We note that in the presence of a thermalbackground, e.g. if microwave photons [23] are envis-aged, the system would display a value of g (2)2 (0) = 2 inthe limit of vanishing driving fields. The function wouldtherefore present a minimum at finite driving amplitude.By assuming a linewidth κ = 1 µ eV of state-of-the-art photonic crystal cavities, we can extract a maximumemission rate as high as R = n κ/ (cid:126) = 380 MHz. The cor-responding intracavity power at zero detuning for cavityresonances (cid:126) ω , = 0 . P in = ω c ( F + F ) / (cid:126) = 15 . F (cid:39) √ n κ = 2 F . The real input power canbe estimated to 50 × P in = 778 pW when taking into ac-count a conservative value for the in-coupling efficiency[33]. This value is about 30 times smaller than the typi-cal input power required for single photon operation withquantum dots [4].It was shown [27] that, under the assumption that thestate is Gaussian, a lower bound on g (2) (0) exists. Inparticular, for mean occupancies n (cid:28)
1, this bound isgiven by g (2) (0) | p (cid:39) |(cid:104) ˆ a (cid:105)| for a pure displaced-squeezedstate, and by g (2) (0) | th (cid:39) √ ¯ n eff for a correspondingthermal (i.e. mixed) state with mean occupation ¯ n eff .For a general mixed state, we can define an effective ther-mal occupation ¯ n eff = (1 / Trˆ ρ − / (cid:28)
1, which thenroughly measures the degree of mixedness. We show inFig.2(b) the computed g (2)2 (0) as a function of U (blueline) at constant occupation n = 10 − (where Eq.(4)holds), and compare it to the pure and thermal limits(dashed lines) that are independent of U for given val-ues of n and κ . Photons in the target cavity achieve avalue of g (2) (0) lying below the thermal limit and, from U/κ > − , crossing the pure state boundary. In thiscase, the state departs from a Gaussian state, which waschecked by identifying negative Wigner distribution areas(not shown).We have studied the impact of variable detunings ∆ , when the other parameters are fixed. The results for n FIG. 3: (Color online) Color maps of (a) the target cavityoccupation n and (b) two-photon correlation g (2)2 (0), com-puted as a function of the cavity detunings. Here U = 10 − κ , χ = κ and we set the optimal condition (4) at ∆ , = 0 and n = 10 − . and g (2)2 (0) are presented in Fig.3. The panel (a) showsthat the occupation vanishes for small detunings. Thisis due to destructive interference between the input fromcavity 1 and the field driving the target cavity. In par-ticular, under the condition F = − iχF / ˜∆ , the coef-ficient c is suppressed therefore favoring photon pairs(see Appendix A). As shown in Fig.3(b), in the region∆ < | κ/ | , we observe both a strong bunching up to g (2)2 (0) = 30 (red areas) or strong antibunching (blueareas) where the optimal antibunching condition holds.As already discussed, antibunching results from the in-terplay between the squeezing brought by cavity 1 andthe field displacement induced by the driving field on thetarget cavity [27]. The results are essentially unchangedwhen U = 0 as dictated by Eq.(4).As discussed above, the UPB scheme displays anti-bunching only for values of the time delay smaller than1 /J (cid:28) /κ [17, 18, 25], thus preventing simple oper-ation under pulsed input. This is ultimately due to thenormal-mode energy splitting in the spectrum of the two-resonators, of the order of 2 J . The dissipative couplingovercomes this difficulty, as the normal-mode splitting isabsent and the emitted photons are characterized by thespectrum of the target cavity (see Fig.A5). We show inFig.4 the g (2)2 ( τ ) function computed at steady state forthe optimal parameter values (red line), and compare itto the UPB result (blue line) for the same value of U .In the dissipative case, the antibunching survives over τ > /κ and oscillations are absent. The single photonregime, defined by g (2)2 ( τ ) < .
5, is preserved over theshaded time frame and behaves similarly to conventionalsources [3].We studied the pulsed regime in more detail by a di-rect time integration of Eq.(2) where we assumed inputGaussian pulses F j exp[ − ( t − t j ) /σ t ]. A key quantityin assessing the single-photon emission under pulsed ex-citation is the two-photon correlation averaged over two g ( ) () UPBDiss.
FIG. 4: (Color online) Delayed two-photon correlation func-tion g (2)2 ( τ ) (red line) computed in the steady state regimeunder continuous wave driving, for the target cavity at n =10 − , U = 10 − κ and χ = κ . The gray area highlights thesingle photon regime. The blue line shows the oscillatingUPB counterpart obtained for the same value of U requir-ing J = 19 . κ , ∆ j = 0 . κ and F = 0. times [25] g (2)pulse = (cid:82) G (2)2 ( t , t ) dt dt (cid:82) n ( t ) n ( t ) dt dt , (6)where G (2)2 ( t , t ) = (cid:104) ˆ a † ( t )ˆ a † ( t )ˆ a ( t )ˆ a ( t ) (cid:105) and n ( t ) = (cid:104) ˆ a † ( t )ˆ a ( t ) (cid:105) . For optimal single-photon opera-tion, the duration of the excitation pulses should be op-timized so to be shorter than the sub-poissonian timewindow (see Fig.4), while allowing enough time for thebuildup of the squeezing (see Fig.A1). A suitable delay∆ t = t − t = 1 . /κ between the two pulses (see Ap-pendix C) has also been introduced here to circumventthe onset of strong bunching in the earliest part of theoutput pulse.We show in Fig.5(a) the computed cavity occupations n , ( t ). Here, the target cavity has an average occupa-tion of n pulse = (cid:82) n ( t ) dt (cid:39) × − . Fig.5(b) shows thecorresponding two-time correlation function g (2)2 ( t , t ).The contour plot highlights the occupation of the tar-get cavity n ( t , t ) = (cid:112) n ( t ) n ( t ), which peaks wellinside the sub-poissonian portion of the plot. For thepresent case, we obtained g (2)pulse (cid:39) .
3. Single photon op-eration may be enhanced via an optimal pulse shaping (atask beyond the scope of this study). Further enhance-ment may be obtained through time-gating the outputpulse, as already suggested for the UPB [25]. If one ap-plies a time gate of duration ∆ T = 5 /κ , highlighted bythe dashed lines in Fig.5(b), the two-photon correlationis reduced to g (2)pulse < . n pulse (cid:39) − . In line with the steady-statediscussion and for the parameters we chose here, whichfit in the requirements of condition (4), we can estimatea single photon rate of R = 2 . /κ = 13 ns. Note that this rate could easily be increased by one order of magnitude by considering anumerically optimized pump amplitudes as in Fig.2(a)(yellow curve). κ t κ t g (2)2 (t ,t ) t n j × -3 Cavity 1Cavity 2 κ (a) (b) FIG. 5: (Color online) Pulsed regime: (a) Time dependentcavity occupation. (b) Two-time two-photon correlation func-tion g (2)2 ( t , t ). The quantity n ( t , t ) is displayed as a con-tour plot. The dashed-white lines denote a time-gate windowresulting in g (2)pulse < .
1. The parameters are U = 10 − κ , χ = κ , σ t = 5 /κ , ∆ , = 0, F = 0 . κ , F = F | opt and∆ t = 1 . /κ . The dissipative coupling considered so far can be im-plemented through an intermediate coupling element,which may be a waveguide or a third optical resonator,as investigated in Ref.[34]. The coupling element acts asan engineered reservoir, effectively generating the quan-tum interference required for the one-directional trans-mission. In the case of a third cavity, the correspondingHamiltonian readsˆ H = (cid:88) j =1 (cid:104) − ∆ j ˆ a † j ˆ a j + U j ˆ a † j ˆ a † j ˆ a j ˆ a j + F ∗ j ˆ a j + F j ˆ a † j (cid:105) + (cid:88) j (cid:54) = k =1 (cid:104) J jk ˆ a † j ˆ a k + J ∗ jk ˆ a † k ˆ a j (cid:105) (7)where the auxiliary mode is not driven, i.e. F = 0, andis ideally characterized by a large dissipation rate κ (cid:29) κ , . J jk describe coherent photon hopping amplitudes.This system is well approximated by Eq. (2) under theconditions J = iχ/ J = J = (cid:112) − iJ κ / J , currently feasiblee.g. using waveguide delay lines [35]. The directionalityof the coupling can be tested numerically (see AppendixD). In presence of the auxiliary resonator, the optimal an-tibunching condition is displaced in the parameter space.We have identified a new condition by running a steadystate optimization with | F | and φ as free parameters,and setting F = 0 . κ , U j = 10 − κ , ∆ j = 0, and κ = 10 κ . We obtained a value of g (2)2 (0) = 3 . × − for | F | = 6 . × − κ and φ (cid:39)
0, proving the single-photon operation.An efficient single-photon source should be bench-marked against the current state-of-the-art, representedby quantum emitters in resonant cavities [4–11], and her-alded sources[12–16]. Recent advances have led to close-to-ideal single-photon operation for both schemes. How-ever, quantum emitters and heralded source respectivelyrequire cryogenic temperature and high input power.The present proposal brings a significant advantage inthat it naturally operates at ultra-low power, in the nWrange as estimated above for a photonic crystal cavity.This must be compared to the 24 nW of Ref.[4] and tothe mW range of heralded sources [13]. Expected single-photon rates are in the MHz range for the three schemes[6, 13, 25]. Photon purity (i.e. the value of g (2) (0)) canbe made here arbitrarily large, as seen in Fig.2(a). More-over here, photons are emitted within the narrow spec-trum of the target cavity (see Fig.A5(a)). Within theassumptions of our model, the indistinguishability de-gree amounts to 99 .
95% (see Appendix E). Such a highvalue would obviously be reduced in the presence to puredephasing or fluctuations in the driving fields. This inci-dentally represents a second advantage – in addition topulsed operation – of the present scheme on the originalUPB, where instead photons are emitted over the spec-trum of the normal modes of the two cavities. The prob-abilistic emission character remains a limitation of bothUPB and cascaded schemes. We are confident howeverthat this difficulty may be overcome soon by devisingnew schemes based on the present dissipative couplingparadigm.
CONCLUSION
We have proposed a scheme for a single-photon sourceoperating under weak nonlinearity and relying on dissi-pative, one-directional coupling between two optical cav-ities. Such approach enables single-photon generationover pulsed excitation, thus overcoming the main limi-tation of the unconventional photon blockade. We haveproposed a three-cavity configuration that enables theone-directional coupling and may be realized on severalplatforms, including weakly nonlinear photonic crystalcavities, coupled ring resonators. The scheme could begeneralized to several cascaded optical cavities, aimingat suppressing the n -photon probabilities to enhance thesingle-photon operation or pair production.The authors acknowledge fruitful discussions with D.Gerace and M. Minkov. Appendix A: Weak pump limit
Given the system Hamiltonianˆ H = (cid:88) j =1 , (cid:104) − ∆ j ˆ a † j ˆ a j + U j ˆ a † j ˆ a † j ˆ a j ˆ a j + F ∗ j ˆ a j + F j ˆ a † j (cid:105) , (A.8)we can express the time-dependent quantum state as anexpansion on the basis of occupation number eigenstates.In the limit of weak driving fields F , →
0, it is then pos-sible to retain only terms in this expansion, whose coeffi-cients depend to leading order in the driving field ampli-tudes as O ( F j F k ) with j + k ≤
2. From the Schr¨odingerequation, it can be easily inferred that the coefficient c jk depends exactly as O ( F j F k ) to leading order. Hence,this weak driving limit coincides with approximating thetime-dependent state as | ψ (cid:105) (cid:39) c | (cid:105) + c | (cid:105) + c | (cid:105) (A.9)+ c | (cid:105) + c | (cid:105) + c | (cid:105) where, | jk (cid:105) denotes a Fock state with j photons in thefirst cavity and k photons in the second one. The equa-tions governing the time-dependence of the coefficientsare found from the solution of the Schr¨odinger equation˜ H | ψ (cid:105) = i (cid:126) ∂ t | ψ (cid:105) written for the non-hermitian Hamilto-nian ˜ H = ˆ H − i (cid:88) j =1 , κ j ˆ a † j ˆ a j − iχ ˆ a ˆ a † (A.10)where the first term stands for the cavity losses and thesecond one for the one-directional transmission. We ob-tain the following coupled set of equations for the coeffi-cients c jk ( t ) i ˙ c = F ∗ c + F ∗ c (A.11) i ˙ c = F c + ˜∆ c + F ∗ √ c + F ∗ c (A.12) i ˙ c = F c + ˜∆ c + F ∗ √ c + F ∗ c − iχc (A.13) i ˙ c = F √ c + 2 (cid:16) U + ˜∆ (cid:17) c (A.14) i ˙ c = F √ c + 2 (cid:16) U + ˜∆ (cid:17) c − iχ √ c (A.15) i ˙ c = F c + F c + (cid:16) ˜∆ + ˜∆ (cid:17) c − iχ √ c (A.16)Note that the underlined terms in Eqs.(A.12,A.13) areof third order in F , , according to the criterion derivedabove, and are therefore negligible within the present ap-proximation. In the steady state where ˙ c jk ( t ) = 0, im-posing the normalization condition c = 1 and solvingthe resulting equations for the c jk coefficients iteratively,we obtain the explicit expressions c = − F ˜∆ (A.17) c = − F ˜∆ − iχ F ˜∆ ˜∆ (A.18) c = F F ˜∆ ˜∆ + iF (cid:16) ˜∆ + ˜∆ + U (cid:17) χ ˜∆ ˜∆ (cid:16) ˜∆ + ˜∆ (cid:17) (cid:16) ˜∆ + U (cid:17) (A.19) c = F ˜∆ (cid:16) ˜∆ + U (cid:17) √ c = F ˜∆ (cid:16) ˜∆ + U (cid:17) √ iF F √ (cid:16) ˜∆ + ˜∆ (cid:17) (cid:16) ˜∆ + U (cid:17) χ ˜∆ ˜∆ (cid:16) ˜∆ + ˜∆ (cid:17) (cid:16) ˜∆ + U (cid:17) (cid:16) ˜∆ + U (cid:17) − F (cid:16) ˜∆ + ˜∆ + U (cid:17) χ ˜∆ ˜∆ (cid:16) ˜∆ + ˜∆ (cid:17) (cid:16) ˜∆ + U (cid:17) (cid:16) ˜∆ + U (cid:17) with the definition ˜∆ j = − ∆ j − iκ j /
2. The cavities oc-cupations and their second order coherence functions arethen computed as n = (cid:104) ˆ a † ˆ a (cid:105) = | c | + | c | + 2 | c | (cid:39) | c | (A.22) n = (cid:104) ˆ a † ˆ a (cid:105) = | c | + | c | + 2 | c | (cid:39) | c | (A.23) g (2)1 (0) = (cid:104) ˆ a † ˆ a † ˆ a ˆ a (cid:105) n (cid:39) | c | | c | (A.24) g (2)2 (0) = (cid:104) ˆ a † ˆ a † ˆ a ˆ a (cid:105) n (cid:39) | c | | c | (A.25)In Eqs.(A.22,A.23), we resort to the fact that c (cid:29) c , c (cid:29) c , c , c (see Fig.A1(a)). Interestingly the c coefficient vanishes under the condition F = − i χ ˜∆ F (A.26)It corresponds to the condition where the laser drivingof cavity 2 interferes destructively with the input iχc from cavity 1. In that case the | (cid:105) state and thereforephoton pairs are favored in the target cavity.For single photon operation, we can now look morespecifically at the conditions required for g (2)2 (0) to vanishnamely when c = 0. In such a case we obtain theoptimal value for the cavity 1 pump amplitude F | opt = i F ˜∆ ˜ U ± (cid:113) − F U ˜ U ˜∆ ˜∆ (cid:16) ˜∆ + U (cid:17) χ (A.27) where we defined ˜∆ = ˜∆ + ˜∆ , ˜ U = ˜∆ + U . Notethat we kept the pump amplitude complex all along thederivation. The optimal antibunching condition there-fore requires a proper phase relation between the drivingfields. Considering as above ∆ , = 0, U , = U , κ , = κ and F ∈ R + the target cavity occupation in the optimalcondition reduces to n (cid:39) F Uκ ( κ − U ) (A.28)and therefore strongly increases with U especially whenit approaches κ .The validity of the above assumptions can be checkedfrom Fig.A1(a) showing the analytical solutions toEqs.(A.17-A.21) and specifically the | c jk ( t ) | coefficientevolution under condition (A.27) within the 2 photonsphoton subspace. It which reveals the dynamical sup-pression of the c coefficient and the clear hierarchy be-tween the coefficients of different manifolds separated byat least 5 orders of magnitude in such weak pump limit.In Fig.A1(b) we show the corresponding g (2)2 ( t, t ) vanish-ing using the full n expression and accounting for theEqs.(A.17,A.18) underlined terms (blue line), or neglect-ing the c and c contribution and the underlined terms(dashed red line). One can see that we obtain a perfectmatch between the curves which insures the legitimacyof Eq.(A.25). - - - - - - κ t | c j k | κ t g ( ) ( t,t ) FullSimplified (b) (a) | c | | c | | c | | c | | c | | c | Fig.A 1: (Color online) (a) Log scale wavefunction coeffi-cients evolution in the 2 photons subspace. (b) Target cav-ity two photon correlation evolution using the full n expres-sion (blue line) or neglecting the c , c contribution andthe Eqs.(A.17,A.18) underlined terms (dashed red line). Here F = 10 − κ and F = F | opt . The other parameters are thoseof Fig.2(a) of the main text. Appendix B: Strong pump limitAnalytical linearized approach
In the opposed limit of strong pump amplitude wherethe cavity fields are dominantly classical, it is convenientto resort to standard linearization techniques allowingto check the convergence of the master equation results.The operators are expanded as ˆ a j = α j + δ ˆ a j given α j = (cid:104) ˆ a j (cid:105) the classical fields amplitudes and δ ˆ a j the quantumfluctuations operators. Dropping terms of orders largerthan 2 in δ ˆ a j in Eq.(A.8) we can derive a set of linearizedquantum Langevin equations i ˙ˆ a = (cid:104) − ∆ + 4 U | α | − i κ (cid:105) ˆ a (A.29)+ 2 U α ˆ a † + i √ κ ˆ a in1 i ˙ˆ a = (cid:104) − ∆ + 4 U | α | − i κ (cid:105) ˆ a (A.30)+ 2 U α ˆ a † + i √ κ ˜ a in2 for the fluctuations where we have dropped the δ no-tation. The operators ˆ a in1 and ˜ a in2 account for the in-put noise in each cavities Here the cavity 1 outputˆ a out1 = √ κ ˆ a + ˆ a in1 is driving the cavity 2 input ˜ a in2 = √ η ˆ a out1 + (cid:112) (1 − η )ˆ a in2 when an undelayed coupling is as-sumed and where ˆ a in2 is the local noise acting on cavity2. The parameter η ∈ [0 ,
1] stands for the fraction ofcavity 1 field that is transmitted to the cavity 2. Hencewe obtain i ˙ˆ a = (cid:104) − ∆ + 4 U | α | − i κ (cid:105) ˆ a (A.31)+ 2 U α ˆ a † + i √ κ ˆ a in1 i ˙ˆ a = (cid:104) − ∆ + 4 U | α | − i κ (cid:105) ˆ a (A.32)+ 2 U α ˆ a † + i √ ηκ κ ˆ a + i √ ηκ ˆ a in1 + i (cid:112) (1 − η ) κ ˆ a in2 which reveal squeezing terms of strength U j α j governedby the steady state classical fields that fulfill0 = (cid:104) − ∆ + 2 U | α | − i κ (cid:105) α + F (A.33)0 = (cid:104) − ∆ + 2 U | α | − i κ (cid:105) α + F − iχα (A.34)Eqs.(A.43,A.44) admit cumbersome but analytical solu-tions that can be plugged into Eq.(A.31,A.32) which canbe recast in the form ˙ u = ˆ A u + n where u = (ˆ a , ˆ a † , ˆ a , ˆ a † ) T (A.35) n = ( √ κ ˆ a in1 , √ κ ˆ a † in1 , √ ηκ ˆ a in1 + (cid:112) (1 − η ) κ ˆ a in2 , √ ηκ ˆ a † in1 + (cid:112) (1 − η ) κ ˆ a † in2 ) T (A.36)andˆ A = − i ˜∆ U α − U α ∗ − ˜∆ ∗ − iχ U α iχ − U α ∗ − ˜∆ ∗ (A.37)where we used the definition ˜∆ j = − ∆ j + 4 U j | α j | − iκ j /
2. We can finally write the following Lyapunov equationˆ A ˆ V + ˆ V ˆ A T = − ˆ D (A.38)for the steady state correlation matrixˆ V = (cid:104) ˆ a ˆ a (cid:105) (cid:104) ˆ a ˆ a † (cid:105) (cid:104) ˆ a ˆ a (cid:105) (cid:104) ˆ a ˆ a † (cid:105)(cid:104) ˆ a † ˆ a (cid:105) (cid:104) ˆ a † ˆ a † (cid:105) (cid:104) ˆ a † ˆ a (cid:105) (cid:104) ˆ a † ˆ a † (cid:105)(cid:104) ˆ a ˆ a (cid:105) (cid:104) ˆ a ˆ a † (cid:105) (cid:104) ˆ a ˆ a (cid:105) (cid:104) ˆ a ˆ a † (cid:105)(cid:104) ˆ a † ˆ a (cid:105) (cid:104) ˆ a † ˆ a † (cid:105) (cid:104) ˆ a † ˆ a (cid:105) (cid:104) ˆ a † ˆ a † (cid:105) (A.39)Given that in the absence of thermal photons the onlynon-zero noise correlations are (cid:104) ˆ a in j ˆ a † in j (cid:105) the correspond-ing matrix readsˆ D = 12 κ χ χ − η ) κ (A.40)Eq.(A.38) can be vectorized in the form (cid:16) ˆ I ⊗ ˆ A + ˆ A ⊗ ˆ I (cid:17) vec( ˆ V ) = − vec( ˆ D ) (A.41)which is nothing but a set of linear equation solvableanalytically. Then, from the knowledge of ˆ V , we cancompute the second order coherence functions as g (2) j (0) (cid:39) N j + 4 n j N j + 2 n j + 2 Re (cid:2) α ∗ j (cid:104) ˆ a j (cid:105) (cid:3) + |(cid:104) ˆ a j (cid:105)| [ n j + N j ] (A.42)with the definitions N j = | α j | and n j = (cid:104) ˆ a † j ˆ a j (cid:105) for theclassical and fluctuation populations respectively. Fi-nally we note that while such linearized approach onlyallows for gaussian states, it remains suitable for the de-scription of squeezed states. Semiclassical Approach
While the above linearized approach allows to effi-ciently compute the steady state solutions and is espe-cially suitable for optimization, it cannot directly addressthe system dynamics, disregards non-Gaussian states anddoesn’t give access to the system density matrix. Ifneeded one can therefore deploy a semiclassical approachto overcome these limitations where the classical field dy-namics is governed by i ˙ α = ˜∆ α + F ( t ) (A.43) i ˙ α = ˜∆ α + F ( t ) − iχα (A.44)and the fluctuations are evolving according to the masterequation i ∂ ˆ ρ f ∂t = (cid:104) ˆ H f , ˆ ρ f (cid:105) − i (cid:88) j =1 , κ j ˆ D [ δ ˆ a j ] ˆ ρ f (A.45)+ iχ ˆ D [ δ ˆ a , δ ˆ a ] ˆ ρ f where the associated Hamiltonian readsˆ H f = (cid:88) j =1 , (cid:104) ˜∆ j ˆ a † j ˆ a j + U j (cid:16) α ∗ j ˆ a j + α j ˆ a † j (cid:17)(cid:105) (A.46)+ (cid:88) j =1 , U j (cid:104) ˆ a † j ˆ a † j ˆ a j ˆ a j + 2 α ∗ j ˆ a † j ˆ a j ˆ a j + 2 α j ˆ a † j ˆ a † j ˆ a j (cid:105) when dropping the δ notations. Note that we keep herenonlinear terms of all orders for an exact description ofthe system. The advantage of this method with respectto the direct master equation treatment stems from thefact that the fluctuation occupation is very small in thedisplaced reference frame set by the classical field whichallows to work with a much smaller truncated Hilbertspace and this at an arbitrary pump power. It turnsinto a great advantage for the global optimization rou-tine that converges much faster and small memory costin the semiclassical case. If needed, the full system den-sity matrix can be reconstructed applying a multimodedisplacement operator according toˆ D ( α , α ) = e α ˆ a † − α ∗ ˆ a e α ˆ a † − α ∗ ˆ a (A.47)ˆ ρ = ˆ D ( α , α )ˆ ρ f ˆ D † ( α , α ) (A.48)where ˆ ρ f as been preliminary inflated with zeros to asuitable size imposed by the classical amplitudes. Anycorrelation can be accessed by reconstructing the globaloperators ˆ a j ( t ) = α j ( t ) I + δ ˆ a j evaluated on ˆ ρ f ( t ). Weshow in Fig.A2 results corresponding to Fig.2(a) of themain text using the above semiclassical approach (seecaptions). Appendix C: Optimal pulse delay
As discussed in the main text, the g (2)pulse can be min-imized by imposing the proper delay ∆ t = t − t be-tween the cavity pulses. We have therefore computed inFig.A3 the g (2)pulse dependance versus ∆ t for the param-eters of Fig.5 which demonstrates the occurrence of anoptimal value. Appendix D: Unidirectional Coupling
In the main text we proposed the three cavity configu-ration as a candidate system for the required dissipativecoupling. In that case, the master equation takes thestandard form i ∂ ˆ ρ∂t = (cid:104) ˆ H , ˆ ρ (cid:105) − i (cid:88) j =1 κ j ˆ D [ˆ a j ] ˆ ρ (A.49)where ˆ H is defined in (7). We set the conditions J = iχ/ J = J = (cid:112) − iJ κ / F = 0, U = U = U =10 − κ , κ = κ = κ and κ = 10 κ . To demonstrate the -3 -2 -1 F / κ -8 -6 -4 -2 N j Classical N N -3 -2 -1 κ -16 -14 -12 -10 -8 -6 -4 n j Fluctuations n n -3 -2 -1 κ -8 -6 -4 -2 N j + n j Total N +n N +n -8 -7 -6 -5 -4 -3 + n -7 -6 -5 -4 -3 -2 -1 g ( ) ( ) (a) (b)(c) (d) F /F / N Fig.A 2: (Color online) Semiclassical steady state solutions forthe parameters of Fig.2 of the main text under the optimalcondition. (a) Classical, (b) fluctuations and (c) total cavitiesoccupations versus the pump amplitude F . (d) Cavity 2second order coherence function versus its total occupation N + n . κ ∆ t0.40.50.6 g ( ) pu l s e Fig.A 3: (Color online) Integrated cavity 2 emission statistics g (2)pulse over one pulse at variable delays ∆ t between the 2 cavitypulses. unidirectional transmission we solve the system dynamicsthrough Eq.(A.49), in the cases where only one cavity isdriven as shown in Fig.A4 and setting the nonzero pumpamplitude to 0 . κ . As one can see, while driving thecavity 1 solely results in a nonzero field in the cavity2, driving only the latter results in a vanishing cavity1 occupation as expected. It demonstrates the efficientunidirectional transmission from cavity 1 to cavity 2 inthe parameter range we consider. -13 -11 -9 -7 -5 -3 -7 -6 -5 -4 -3 n j Cavity 1Cavity 2Cavity 3 F F (a) (b) κt κt Fig.A 4: (Color online) Log scale cavity occupations in thecase where we drive (a) only the cavity 1: F = 0 . κ and F = 0 (b) only the cavity 2: F = 0 and F = 0 . κ . Appendix E: Indistinguishability
As discussed above, a strong asset of the dissipa-tive coupling is the absence of normal mode splittingthat characterizes the UPB. The spectrum S ( ω ) = (cid:82) (cid:104) ˆ a † (0)ˆ a ( τ ) (cid:105) dτ of the target cavity is shown in Fig.5(a)and is nothing but a Lorenzian spectrum of width κ .This strongly favors the indinstinguishability of the sin-gle photon emission which can be quantified by simulat-ing a Hong-Ou-Mandel experiment. Let us imagine thatthe output of the target cavities of two identical cascadedsystems a and b are mixed in a 50 /
50 beamsplitter. Onecan show [36] that the coincidence probability delayed by τ between the beamsplitter outputs reads P a,b ( τ ) = 12 (cid:16) G (2)2 ( τ ) − | G (1)2 ( τ ) | + G (1)2 (0) (cid:17) (A.50)where G (1)2 ( τ ) = (cid:104) ˆ a † (0)ˆ a † ( τ ) (cid:105) and G (2)2 ( τ ) = (cid:104) ˆ a † (0)ˆ a † ( τ )ˆ a ( τ )ˆ a (0) (cid:105) . Interestingly in the steady stateregime, G (1)2 ( τ ) is constant as one can see in Fig.5(b)(blue line). Therefore P a,b ( τ ), shown by the red lineof Fig.5(b) in its normalized form, is essentially definedby the two-photon correlation G (2)2 ( τ ) and in particu-lar P a,b (0) = G (2)2 (0). The indistinguishability degree isquantified by the visibility V = P a,b (+ ∞ ) − P a,b (0) P a,b (+ ∞ ) + P a,b (0) = 1 − g (2)2 (0) (A.51)which amounts to V = 99 .
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