Few-photon transport in many-body photonic systems: A scattering approach
Changhyoup Lee, Changsuk Noh, Nikolaos Schetakis, Dimitris G. Angelakis
FFew Photon Transport in Many-Body Photonic Systems: A Scattering Approach
Changhyoup Lee, ∗ Changsuk Noh, † Nikolaos Schetakis, and Dimitris G. Angelakis
1, 2, ‡ Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 School of Electronic and Computer Engineering, Technical University of Crete, Chania, Greece 73100 (Dated: October 8, 2018)We study the quantum transport of multi-photon Fock states in one-dimensional Bose-Hubbard lattices im-plemented in QED cavity arrays (QCAs). We propose an optical scheme to probe the underlying many-bodystates of the system by analyzing the properties of the transmitted light using scattering theory. To this end,we employ the Lippmann-Schwinger formalism within which an analytical form of the scattering matrix can befound. The latter is evaluated explicitly for the two particle / photon-two site case using which we study the reso-nance properties of two-photon scattering, as well as the scattering probabilities and the second-order intensitycorrelations of the transmitted light. The results indicate that the underlying structure of the many-body statesof the model in question can be directly inferred from the physical properties of the transported photons in itsQCA realization. We find that a fully-resonant two-photon scattering scenario allows a faithful characterizationof the underlying many-body states, unlike in the coherent driving scenario usually employed in quantum Mas-ter equation treatments. The e ff ects of losses in the cavities, as well as the incoming photons’ pulse shapes andinitial correlations are studied and analyzed. Our method is general and can be applied to probe the structure ofany many-body bosonic models amenable to a QCA implementation including the Jaynes-Cummings-Hubbard,the extended Bose-Hubbard as well as a whole range of spin models. PACS numbers: 42.50.-p, 03.65.Nk
I. INTRODUCTION
Recent advances in quantum nonlinear optics and circuitQED systems [1, 2] have allowed the engineering of photon-photon interaction to the extent that strongly interacting pho-tons have started to be considered as a potential platform tosimulate many-body phenomena [3–6]. Early proposals dis-cussed the possibility to realise strongly correlated states ofphotons and polaritons in coupled QED cavity arrays (QCAs)[7–9]. Their natural advantage in local control and design, andpossibility to probe out-of-equilibrium phenomena in drivendissipative regimes, allowed QCA-based approaches to com-plement the e ff orts towards viable quantum simulators [10–18]. Experimentally, in spite of various challenges, progresshas been recently made with small scale QCAs successfullyfabricated in semiconductor and superconductor based set-ups[19–21]. Strongly interacting photons have also been createdin Rydberg media [22].A QCA, beyond its many-body character, is inherently a(quantum) optical system, thus is naturally probed by lightscattering [23]. Performing quantum measurements on theoutput (transported / scattered) light, one obtains informationabout the underlying properties of the system [24]. In thestudy of QCA simulators, the driving source has so far mostlybeen taken to be a coherent field of light described within aquantum Master equation formalism. The latter approach, al-though successfully captures the open nature of the system, isoften limited to coherent-light drives (recently a method to de-rive a master equation for the Fock-state input has been foundin [26]). This semiclassical treatment misses in our opinion animportant regime of input quantum particles being transported ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] (a)(b)
Ground state1-particle states2-particle statesN-particle states E n e r g y ω ω ... .. N photons waveguide nonlinear cavity array waveguide detector
FIG. 1: (a) Proposed method to probe the structure of bosonic many-body models as implemented QCA simulators. Photons travelingin the left waveguide are injected into the array and are transportedthrough the device to the right waveguide. In this work, the QCA isassumed to realize the Bose-Hubbard model but other models such asthe Jaynes-Cummings-Hubbard, spin models, or the extended Bose-Hubbard can also be realized [9, 17]. The injected photons scanthrough the many-body eigenstates of the simulated model and ifthey are fully resonant to the many-body states as illustrated in (b),the full information of the relevant states is mapped out faithfully inthe output spectra and correlation functions. in the system. How does a QCA many body simulator reactto general quantum input fields? Can we collect informationon the states of the many-body models simulated by studyingthe transported / scattered quantum particles (photons) from aQCA?To answer this question, we employ the Lippmann-Schwinger formalism, whose use in quantum optical systemswas pioneered by Shen and Fan [27] and led to numerous fur-ther developments [28–40]. In the context of quantum sim-ulations of many-body phenomena, using an N -photon Fockstate as the input field for an N -site cavity array seems promis-ing. As a first step towards this goal, we examine the process a r X i v : . [ qu a n t - ph ] D ec of scattering two photons on an array of coupled Kerr non-linear resonators whose dynamics are described by the Bose-Hubbard model. We first evaluate the scattering matrix ana-lytically for the case of two resonators coupled to input andoutput waveguides, and then use it to calculate the scatter-ing probabilities and the second-order correlations betweenthe scattered photons. The results indicate that the structure ofthe correlated many-body states is more clearly reflected in thescattered light fields when the individual input photon / particleenergies are fully resonant with the corresponding eigenstates(see Fig. 1). II. FEW-PHOTON TRANSPORT
Consider a one-dimensional array of N coupled nonlinearcavities, where the cavities at both ends are coupled to waveg-uides supporting propagating photons as shown in Fig. 1(a).The system is described by the Hamiltonian,ˆ H tot = ˆ H wg + ˆ H cc + ˆ H wc , whereˆ H wg = (cid:126) (cid:90) ∞−∞ dx (cid:16) − iv g ˆ c † L ( x ) ∂∂ x ˆ c L ( x ) (cid:17) + (cid:126) (cid:90) ∞−∞ dy (cid:16) − iv g ˆ c † R ( y ) ∂∂ y ˆ c R ( y ) (cid:17) , ˆ H cc = (cid:126) N (cid:88) j = (cid:16) ω j ˆ a † j ˆ a j + U j ˆ a † j ˆ a † j ˆ a j ˆ a j (cid:17) + (cid:126) N − (cid:88) j = J (ˆ a † j ˆ a j + + ˆ a j ˆ a † j + ) , ˆ H wc = (cid:126) (cid:90) ∞−∞ dxV δ ( x ) (cid:16) ˆ c † L ( x )ˆ a + ˆ c L ( x )ˆ a † (cid:17) + (cid:126) (cid:90) ∞−∞ dyV δ ( y ) (cid:16) ˆ c † R ( y )ˆ a N + ˆ c R ( y )ˆ a † N (cid:17) . ˆ H wg describes the propagation of photons in the waveguideswith group velocity v g , where ˆ c † L ( x < (cid:16) ˆ c † L ( x > (cid:17) andˆ c † R ( y < (cid:16) ˆ c † R ( y > (cid:17) are the creation operators for an in-coming (outgoing) photon in the left and right waveguides,respectively. ˆ H cc describes the coupled cavity system, wherethe bosonic operator ˆ a † j annihilates a photon in the j th cavitywhich has the resonant frequency ω j and nonlinearity U j . Thephoton hopping rate between the cavities is given by J . ˆ H wc describes the coupling between the waveguides to the adjacentcavities, with coupling strengths V and V . From here on, weset v g = (cid:126) =
1. To analyse the properties of the scattered pho-tons, we analytically find the two-photon scattering matrix S (2) within the Lippmann-Schwinger formalism (for a formal def-inition of the scattering matrix and a detailed derivation, see Single cavity ω + 2 U ω k k = k k Two cavities k = k − ∆ k ω δ | ǫ (1)+ !| ǫ (1) − !| ǫ (2) − !| ǫ (2)+ !| ǫ (2)0 ! FIG. 2: Energy level diagram of the two-site Bose-Hubbard QCA,where the bare-cavity energies and the coupled mode energies of thecavities are shown.One of the resonant two-photon excitation pathssatisfying Eq. (4) is illustrated by the arrows on the left, while theo ff -resonant path for identical input photons is shown on the right. Appendix A): LL (cid:104) p , p | S (2) | k , k (cid:105) = S LL δ ( k + k − p − p ) + (cid:0) r k r k δ ( k − p ) δ ( k − p ) + ( k ↔ k ) (cid:1) , (1) LR (cid:104) p , p | S (2) | k , k (cid:105) = S LR δ ( k + k − p − p ) + (cid:0) r k t k δ ( k − p ) δ ( k − p ) + ( k ↔ k ) (cid:1) , (2) RR (cid:104) p , p | S (2) | k , k (cid:105) = S RR δ ( k + k − p − p ) + (cid:0) t k t k δ ( k − p ) δ ( k − p ) + ( k ↔ k ) (cid:1) , (3)where we have used k i ( p i ) to denote the input (output) mo-menta. The subscripts LL , LR , and RR refer to which waveg-uide the two output photons have scattered to, e.g., RR meansthat two photons are in the right waveguide. r k and t k are thesingle-photon reflection and transmission coe ffi cients, respec-tively. The second lines on the right-hand side of the equa-tions describe independent single-photon scattering events,whereas the first lines describe the contributions due to thenonlinearity present in the cavity array, i.e., S LL , S LR , and S RR vanish when U = U = ω = ω = ω , U = U = U , and V = V = V . At this point it is useful to definethe total energy k + k = p + p as 2 ω + δ and the relativeenergy as ∆ k = k − k and ∆ p = p − p . The eigenenergiesof the system in the one-particle manifold are ω + (cid:15) (1) ± where (cid:15) (1) ± ≡ ± J , and the two-particle excitation subspace is com-posed of 2 ω + (cid:15) (2)0 , ± with (cid:15) (2)0 ≡ U and (cid:15) (2) ± ≡ U ± √ J + U corresponding, respectively, to the eigenstates | (cid:105) ∼ | (cid:105) − | (cid:105) , | ± (cid:105) ∼ | (cid:105) + | (cid:105) − U ∓ √ J + U √ J | (cid:105) , where | jk (cid:105) = √ j ! k ! (ˆ a † ) j (ˆ a † ) k | (cid:105) . Here, | − (cid:105) becomes theunit-filled ground state | (cid:105) in the limit of U → ∞ . InEqs. (1)-(3), the bound-terms S LL , S LR , and S RR have reso- (c) (b) ǫ (1)+ ǫ (1) − ǫ (2) − ǫ (2)+ ǫ (2)0 (d) !" ! !" ! " ! $" ! ! " " ! ! !" ! $" " ! $$ V = 0 . V = 0 . !" ! % & ’ ()* + , - , )) ./ ’ , - , ’ + δ = ǫ (2) − δ = ǫ (2)+ δ = ǫ (2)0 (a) FIG. 3: Bound state contribution in the scattering matrix element, S RR , is shown as a function of δ , for V = .
01 (blue solid) and V = . V = .
25 in (b). Panel (c) shows the resonant conditionsof ∆ k ( ∆ p ) in | S RR | for δ = , (cid:15) (1) ± , and (cid:15) (2)0 , ± for U = V = .
25. Panel (d) presents the respective eigenstate excitation amplitudes atthe corresponding two-photon energy resonances with increasing U / J when ∆ k = δ − (cid:15) (1) − (solid) and ∆ k = V = .
04. Allunits are defined with respect to J . Panel (c) clearly depicts the resonance condition written in Eq. (4) while panel (d) shows how the desiredeigenstates are more e ffi ciently excited when this condition is met (solid curves) as opposed to the o ff -resonant case (dashed curves). nances at | ∆ k | ( | ∆ p | ) = | (cid:15) (1) ± − δ | (4)for δ = , (cid:15) (1) ± , and (cid:15) (2)0 , ± , implying that the bound-term contri-butions are significant only if one of the input or output pho-tons is resonant with one of the single-photon eigenstates asillustrated in Fig. 2.First we discuss the resonance structure of the scatteringmatrix. To show an example of how the bound terms be-have, we depict S RR = (cid:82) d ∆ kd ∆ p | S RR | as a function of δ in Fig. 3(a). When the waveguide-cavity coupling strengthis weak (blue solid curve, V = .
01) we find that the reso-nant peaks at δ = , (cid:15) (1) ± , and (cid:15) (2)0 , ± are clearly distinguished,whereas for a higher coupling strength (orange dashed curve, V = .
25) resonances get broaden such that finer details arewashed out. General resonant behaviour of S RR over δ and U is also depicted in Fig. 3(b), which shows that the bound-terms have the resonances at δ = , (cid:15) (1) ± , and (cid:15) (2)0 , ± for anyvalue of U . Furthermore, | S RR | is analyzed as a function of ∆ k and ∆ p for each resonant δ in Fig. 3(c), where the reso-nant condition of Eq. (4) for ∆ k ( ∆ p ) is clearly seen. The firsttwo cases ( δ = (cid:15) (1) ± ) correspond to when each photonis resonant to a state belonging to the single excitation mani-fold, while the rest ( δ = (cid:15) (2)0 , ± ) correspond to when one photonhas either (cid:15) (1) ± , and the other has (cid:15) (2)0 , ± − (cid:15) (1) ± . Similar reso-nant mechanisms have been observed in other systems suchas a waveguide coupled to a cavity embedded in a two-levelsystem [35] or a waveguide coupled to a whispering-galleyresonator containing an atom [36].Throughout this work, we will consider two types of in-put states: 1) two photons satisfying the resonance condition(4), where for simplicity one of the input photons is assumedto have the energy (cid:15) (1) − , i.e., ∆ k = δ − (cid:15) (1) − with δ = (cid:15) (2)0 , ± (see arrows on the left side of Fig. 2); 2) two photons sat-isfying the two-photon resonance condition while having thesame energy, i.e., ∆ k = δ = (cid:15) (2)0 , ± (see arrows on theright side of Fig. 2). Later, we will show that, within the longinput pulse regime, the second-order intensity correlations inthe latter case is directly proportional to that in the coherentdriving scenario. Figure 3(d) shows the (unnormalised) two-photon eigenstate excitation amplitudes directly involved intwo-photon scattering constructed from the coe ffi cients ( e , e and e ) of the two-photon scattering eigenstate given inAppendix A 2. We see that when driven by the respectivetwo-photon eigenenergies (three circles), the fully-resonantcase (solid curves) generally excites the desired eigenstatesmore e ffi ciently than the identical-photon input case (dashedcurves). Exceptions only occur in two regimes: 1) near thelinear regime for δ = (cid:15) (2) + , where the ∆ k = U / J = δ = (cid:15) (2)0 , where the two-photon energy becomes twice the single photon eigenenergy (cid:15) (1) + . The fully-resonant photon scattering scenario thereforepromises more e ffi cient probe transmission spectroscopy ofthe multi-photon eigenstates. We will show this by explicitlycalculating the scattering probabilities. We also calculate thesecond-order intensity correlations to further characterise thescattered light and connect the observed behaviour with theunderlying states of the QCA. III. SIGNATURES OF MANY-BODY STATES INTRANSMISSION SPECTRA
In the momentum space, a general two-photon initial stateis given by | { ξ } (cid:105) = √ M ˆ c † ξ ˆ c † ξ | (cid:105) , where the normalisationfactor M = + | (cid:82) dk ξ ( k ) ξ ( k ) | is associated with the over-lap of the momentum distributions ξ i ( k ) and the continuous-mode creation operator is given by ˆ c † ξ = (cid:82) dk ξ ( k )ˆ c † L ( k ) with (cid:82) dk | ξ ( k ) | =
1. The output state is then calculated from thescattering matrix as follows: | out (2) { ξ } (cid:105) = S (2) | { ξ } (cid:105) = | out (2) { ξ } (cid:105) LL + | out (2) { ξ } (cid:105) LR + | out (2) { ξ } (cid:105) RR , where | out (2) { ξ } (cid:105) s s = (cid:82) dq dq √ M ξ ( q ) ξ ( q ) | φ (2)out (cid:105) s s , for( s , s ) ∈ { L , R } , where | φ (2)out (cid:105) LL , | φ (2)out (cid:105) LR and | φ (2)out (cid:105) RR repre-sent the two-photon wave functions associated with Eqs. (1),(2), and (3), respectively (see Appendix A 2). We assume themomentum distribution to have a narrow Gaussian profile forsimplicity, i.e., ξ j ( q ) = πσ ) / exp (cid:16) − ( q − k j ) σ (cid:17) , where ξ j isnarrowly peaked around k j . Given a narrow enough band-width with respect to the e ff ective cavity linewidth, ∝ V ,e ff ects of the pulse shape are very small as presented in Ap-pendix B 2–quantitatively similar results are obtained for boththe Lorentzian and ‘rising’ pulse profiles. We note that re-cent developments in the pulse-shaping techniques makes ourphoton scattering scenario experimentally feasible [41, 42]. A. Scattering probabilities
Using the above initial state, we first consider the scatteringprobabilities defined as, P LL = (cid:90) d p d p |(cid:104) p , p | out (2) { ξ } (cid:105) LL | , P LR = (cid:90) d p d p |(cid:104) p , p | out (2) { ξ } (cid:105) LR | , P RR = (cid:90) d p d p |(cid:104) p , p | out (2) { ξ } (cid:105) RR | , as in [33]. Figure 4 depicts them as functions of the to-tal energy δ/ J (top row), or of the photon-photon interac-tion strength U / J (lower rows). Left-hand column displaysthe fully-resonant case (see Eq. (4)) where one photon hasthe energy (cid:15) (1) − and the other has the energy δ − (cid:15) (1) − , whereasthe right-hand column displays the results when ∆ k =
0. InFigs. 4(a) and (e), we plot the two-photon transmission prob-ability ( = P RR ) for di ff erent values of interaction strengths( U / J = , , ∆ k = ff -resonant) one-photon absorption. This indicates thatthe fully-resonant Fock-state transport scheme has an ad-vantage over the identical-photon transport case in detectingtwo-photon transmission through the multi-particle correlatedstates of the QCA. In turn, this means that the two-photonscattering scenario performs better than the coherent drivingcase because: 1) the two-photons necessarily have the sameenergy in the latter and 2) the probability of finding two pho-tons in a coherent state | α (cid:105) goes as | α | (cid:28) δ = (cid:15) (2)0 , ± are further investigated as functionsof U / J . We first note that over a wide region of U / J , exceptfor the cases that coincide with the single photon resonances, P LR ≈ ∆ k = δ − (cid:15) (1) − (left hand column), while P LL ≈ ∆ k =
0. This is due to the fact that one of the two pho-tons is always resonant to the (lower) single energy state inthe fully-resonant case, while neither photon is resonant inthe ∆ k = δ = (cid:15) (2)0 and δ = (cid:15) (2) + approach the same value above U / J ∼
20. This is due to the fact that above this value of U / J ,the two states are no longer distinguishable because of theirenergy broadening ( V / J = . | (cid:105) and | + (cid:105) , induces the lit-tle shift observed in the scattering probabilities. Similarly, in δ = (cid:15) (2) − case, the energy of one of the photons approach (cid:15) (1) + within the decay bandwidth, resulting in larger two-photontransmission probability with increasing U / J . E ff ects of thiskind are absent when ∆ k = !"! ! & ’ ( )*+* ,-, . , / !"! ! ! ! " ! !"! ! % & ’ ()*) +,+ - + . / (b)(c)(a) (f)(g)(e) ! ! " ! ! ! &" ! " &" ! "’"&& ! " ( ) * +, - . / -- / ! ! " ! ! ! &" ! " &" ! "’"&& ! " ( ) * +, - . / -- / (d) (h) δ = ǫ (2)0 δ = ǫ (2)+ δ = ǫ (2) − δ = ǫ (2)0 δ = ǫ (2)+ δ = ǫ (2) − P LL P LR P RR ∆ k = δ − ǫ (1) − ∆ k = 0 !"! ! & ’ ( )*+* ,-, . , / !"! ! ! ! " !"!! ! % & ’ ()*) +,+ - + . / !"! ! ! ! " ! !"! ! % & ’ ()*) +,+ - + . / !"! ! ! !"!! ! % & ’ ()*) +,+ - + . / U/J = 1
U/J = 0
U/J = 5
FIG. 4: Left- and right-hand columns : ∆ k = δ − (cid:15) (1) − and ∆ k =
0. Two-photon transmission ( P RR ) is shown as a function of the two-photondetuning δ/ J for di ff erent photon-photon interaction strengths U / J = , , P LL , P LR , and P RR , are also shownas a function of U / J for two-photon eigenenergies δ = (cid:15) (2)0 , ± in (b)-(d) and (f)-(h). Weak waveguide-cavity coupling and the narrow bandwidthof initial photons are assumed: V / J = .
04 and σ/ J = . ff erence in the behaviour of scattering probabilities as a function of U / J when two-particle states are probed fully resonantly with di ff erent energy photons via the one-particle manifold (left column) comparedto the case where a virtual (o ff -resonant) one-photon absorption is required (right column). In addition, in the former case, transmission isgenerally significantly larger which makes this approach experimentally more e ffi cient (see text for more details). B. Intensity-intensity correlations
The scattering probabilities reveal the presence of the multi-photon correlated states, but no information about the actualcorrelations is given. For the latter, one may employ the second-order correlation function between positions z and z : g (2) s s ( z , z ) = (cid:104) out (2) { ξ } | ˆ c † s ( z )ˆ c † s ( z )ˆ c s ( z )ˆ c s ( z ) | out (2) { ξ } (cid:105)(cid:104) out (2) { ξ } | ˆ c † s ( z )ˆ c s ( z ) | out (2) { ξ } (cid:105)(cid:104) out (2) { ξ } | ˆ c † s ( z )ˆ c s ( z ) | out (2) { ξ } (cid:105) where ( s , s ) ∈ { R , L } . Here, we focus on the transmittedlight, whose correlation function can be written as g (2) RR ( z , z ) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:82) { ξ ( k ) } φ RR ( z , z ) (cid:12)(cid:12)(cid:12)(cid:12) M (cid:82) dx (cid:16) | (cid:82) { ξ ( k ) } φ LR ( x , z ) | + | (cid:82) { ξ ( k ) } φ RR ( x , z ) | (cid:17) (cid:82) dx (cid:16) | (cid:82) { ξ ( k ) } φ LR ( x , z ) | + | (cid:82) { ξ ( k ) } φ RR ( x , z ) | (cid:17) , (5)where (cid:82) { ξ ( k ) } ≡ (cid:82) dk dk ξ ( k ) ξ ( k ), and φ (2) LR and φ (2) RR repre-sent the two-photon wave functions associated with Eqs. (2),and (3), respectively (see Appendix A 2). In this work, we willconcentrate on the zero-delay case, i.e., z = z = ff er- ent correlations for di ff erent values of ∆ k , since the distin-guishability of the photons a ff ects the intensity-intensity cor-relations. Specifically, g (2)initial increases from g (2)initial = when k = k to g (2)initial = | k − k | (cid:29) σ (see Appendix B 1). !"! ! ! !"! ! " $ ! % ! " $ %% !"! ! ! !"!! ! % ! " " ! ! " ! ! ! "’""&"’&""&"&"""&" ! ! " ( ! " " ! ! " ! ! ! &" ! " "’""&"’"&""’&""& ! " ( ! " $$ (b)(a) (d)(c) ∆ k = δ − ǫ (1) − ∆ k = 0∆ k = δ − ǫ (1) − ∆ k = 0 δ = ǫ (2)0 δ = ǫ (2)+ δ = ǫ (2) − U/J = 1
U/J = 0
U/J = 5
FIG. 5: Left- and right-hand columns : ∆ k = δ − (cid:15) (1) − and ∆ k =
0. Second-order intensity correlation function, g (2) RR , is shown as a functionof two-photon detuning δ/ J for di ff erent photon-photon interaction strengths U / J = , , U / J for two-photon eigenenergies δ = (cid:15) (2)0 , ± in (b) and (d). The same parameters are chosen as used in Fig. 4. We highlight herethe direct mapping of the correlations of the many-body state (cid:15) (2) − onto the transmitted light g (2) RR (the green dotted line in (b)), as the formerapproaches monotonically the Mott-like state | , (cid:105) with increasing U / J . This does not hold in the identical-photons case however, where g (2) RR first increases before it dips down to follow the correlations of the state. The same behaviour is also found in the coherent-driving scenario[16]. (See the detailed discussion in Section III. B regarding the rest of the states and regimes, and di ff erences between the two approaches.) Figures 5(a) and (c) plot the zero-delay second-order cor-relations against the total energy δ/ J . In the absence of non-linearity, the ∆ k = g (2) RR = /
2: being linear,the system does not change the statistics of the (identical) in-put photons. On the other hand, in the fully-resonant case,there are peaks when the photons have the energies ( (cid:15) (1) − , (cid:15) (1) − )and ( (cid:15) (1) − , (cid:15) (1) + ), resulting in g (2) RR ≈ / g (2) RR ≈
1, respec-tively. Away from these points, g (2) RR ∼ U = , (cid:15) (2)0 / ∆ k =
0. This behaviour is not asso-ciated with any multi-photon correlated state, but arises dueto a quantum interference between di ff erent path ways to thetwo-photon excitation in the second cavity.To see in detail how the second-order intensity correlationschange with the interaction strength, we plot g (2) RR as a func-tion of U / J at two-photon energies δ = (cid:15) (2)0 , ± for the cases of ∆ k = δ − (cid:15) (1) − in (b) and ∆ k = U / J , the transmitted light attwo-photon eigenenergies are anti-bunched (bunched) when ∆ k = δ − (cid:15) (1) − ( ∆ k = g (2) RR in the fully-resonant case provides a more faithful char-acterisation of the underlying multi-photon correlated states.Perhaps this is best illustrated by the δ = (cid:15) (2)0 (blue solid)curves. This state is proportional to | , (cid:105) − | , (cid:105) regardlessof the value of U / J , and therefore has a constant g (2) RR . This isexactly what is observed in the fully-resonant case in contrastto the identical-photons case, as long as the state is resolvedfrom the state at (cid:15) (2) + (i.e., below U / J ∼ g (2) RR at (cid:15) (2) − shows the expected monotonic behaviour in the fullyresonant case, due to the increase in | , (cid:105) component with in- creasing U / J . In the identical-photons case, large bunchingis observed before g (2) RR decreases and dips below 1 only when U / J >
20. Similar behaviour is also found in the coherent-driving scenario [16].We attribute the qualitative di ff erences between the twocases to the presence or absence of the resonant single-photontransmission. In the fully-resonant case, this is guaranteed bydefault and moreover the single photon transmission probabil-ity is robust at ≈ U / J . This pro-vides a nice constant background against which the second-order correlations can be measured. Such a background fieldis absent when ∆ k = g (2) RR = /
2, the same as the background correla-tion) at U / J = (cid:15) (1) + ) cominginto resonance with (cid:15) (2)0 / C. Comparison with the coherent-driving scenario
Somewhat surprisingly, the intensity-intensity correlationsfor the identical-photons case are quantitatively very similarto those obtained from the coherent driving scenario. Thiscan be seen by writing down the expressions for the correla-tion function in both cases. In the scattering formalism thecoherent input field is incorporated by writing the input wavepacket as | α (cid:105) = e ˆ c † α − ˆ c α | (cid:105) where ˆ c † α = (cid:82) dk α ( k )ˆ c † ( k ) with themean photon number ¯ n = (cid:82) dk | α ( k ) | , and choose a Gaussianwave packet α ( k ) = √ ¯ n (2 πσ ) / exp (cid:16) − ( k − k c ) σ (cid:17) , where α ( k ) is narrowly peaked around k c . We here assume thatthe coherent-field is weak such that the mean photon number¯ n (cid:28)
1. In this case, the output state | out α (cid:105) = (cid:80) n S ( n ) | α (cid:105) canbe approximated as | out α (cid:105) ≈ e − ¯ n / ( | (cid:105) + S (1) ˆ c † α | (cid:105) + S (2) (ˆ c † α ) | (cid:105) ) , where S (1) and S (2) are given as Eqs. (A8) and (A62), respec-tively. For the output state, the second-order intensity correla-tions can be calculated from g (2) RR , coherent ( z , z ) = (cid:104) out α | ˆ c † R ( z )ˆ c † R ( z )ˆ c R ( z )ˆ c R ( z ) | out α (cid:105)(cid:104) out α | ˆ c † R ( z )ˆ c R ( z ) | out α (cid:105)(cid:104) out α | ˆ c † R ( z )ˆ c R ( z ) | out α (cid:105) (6) ≈ | (cid:82) { α ( k ) } φ RR ( z , z ) | e − ¯ n | (cid:82) dk α ( k ) φ R ( z ) | | (cid:82) dk α ( k ) φ R ( z ) | , where (cid:82) { α ( k ) } ≡ (cid:82) dk dk α ( k ) α ( k ), and φ R ( x ) represents thesingle-photon wave function (see Appendix A 1). On the otherhand, the correlations in Eq. (5) can be approximated for theidentical two-photon input ( ∆ k =
0) as g (2) RR , two − photon ( z , z ) ≈ | (cid:82) { ξ ( k ) } φ RR ( z , z ) | | (cid:82) dk ξ ( k ) φ R ( z ) | | (cid:82) dk ξ ( k ) φ R ( z ) | . ! ! " ! ! ! &" ! " "’"&&&""&" " &" ! &" ! " ( ! " $ %% ! ! ! " ! ! ! $&$ ! ! " ’ ! " " (a)(b) U/J = 1
U/J = 5 g (2) RR, two − photon g (2) RR, coherent g (2)ss FIG. 6: Second-order intensity correlations for the transmitted lightas a function of the two-photon detuning δ/ J for U / J = ,
5, ob-tained from the scattering approach with the identical two-photon in-put (blue solid) and a coherent-state input (orange dashed), and alsofrom the master-equation formailsm (green dashed). For the latter,
Ω = V ¯ n = . γ = V = .
04 are used.
One easily finds that the two cases only di ff er by a factor of1 /
2, identical to the di ff erence in the initial correlations, i.e., g (2) RR , two − photon ≈ g (2) RR , coherent . (7)This is numerically demonstrated in Fig. 6, where we plot thezero-delay correlations in Eqs. (5) (multiplied by 2) and (6)as a function of two-photon detuning δ/ J for U / J = U / J = V = .
04 and ¯ n = . Ω (ˆ a + ˆ a † ) is added in the ˆ H cc without considering ˆ H wg and ˆ H wc , and then the second-orderintensity correlations g (2)ss is calculated for the steady state ρ ss obtained from a quantum optical master equation with a dis-sipation rate of γ = V . The numerical calculations of themaster equation formalism show the consistent results as com-pared to the scattering approach for the coherent-state input,i.e., g (2)ss ≈ g (2) RR , coherent .From these results, we conclude that the fully-resonantscattering scenario has advantages over the conventionalcoherent-driving scenario in characterising the correlations ofthe underlying many-body QCA states. IV. EFFECTS OF PHOTON LOSSES
Lastly, we address the issue of dissipation into non-guidedmodes. Within the scattering formalism used in this work,Markovian photon losses with the rate γ bath can be accountedfor by either introducing a waveguide for each cavity [43],or equivalently using a combination of the scattering theoryand the input-output formalism [32, 44]. In calculating thetwo-photon scattering matrices, it has been found that the ef-fects of losses can be treated exactly by replacing the cav-ity frequency ω j with ω j − i γ bath / H cc [43]. Using this method, we have calculated the two-photontransmission probability ( P RR ) in the presence of extra pho-ton losses in the cavities, as shown in Fig. 7(a). As expected,the transmission probability decreases and broadens as γ bath ! ! " ! ! ! &" ! " &" ! "’""&"’"&""’&""& ! " ( ) * +, - . / -- / (b)(a) γ bath /J = 0 γ bath /J = 0 . γ bath /J = 0 . ! ! " ! ! ! "’""&"’"&""’&""& ! " ( ! " " FIG. 7: For the fully-resonant case, two-photon transmissions ( P RR )and second-order correlations as a function of the total energy δ/ J with γ bath / J = , .
02, and 0 .
04 when V / J = .
04 and U / J = increases while V / J remains fixed. Things are a little morecomplicated for the second-order intensity correlation func-tion g (2) RR . We must add extra contributions–in which one pho-ton is in one of the extra loss channels–to the denominator ofEq. (5). However, as a first consideration, one can ignore thee ff ects of ‘quantum jumps’ on these terms and calculate g (2) RR using the non-Hermitian Hamiltonian. The results are plottedin Fig. 7(b), showing the e ff ect of losses for γ bath / J = , . .
04 when V / J = .
04 and U / J = V. SUMMARY AND DISCUSSION
To summarise, we have proposed a few-photon transportscenario to probe the many body structure of strongly corre-lated models simulated in QCAs. We have demonstrated thefeasibility of our proposal by analytically calculating the scat-tering matrix of the two-photon, two-site Bose-Hubbard QCAand studying the scattering probabilities and correlation func-tions. Signatures of strongly correlated multi-particle stateswere found in scattering probabilities and the second-orderintensity correlations. We have compared two cases: 1) thefully-resonant case in which two input photons have tailoredenergies to match the single-particle and two-particle eigenen-ergies of the model in question; 2) the identical-photons casein which two input photons have identical energies and aretwo-photon resonant with one of the two-particle states. Wefind that the multi-photon fully-resonant excitation scenariois advantageous over the alternative, in that it allows highertransmission probabilities and a more faithful mapping of theintensity-intensity correlations. Finally, we noted a corre-spondence between the identical-photon scattering case andthe coherent-driving case, illustrating that the fully-resonantFock-state scattering method has advantages over the latter.The e ff ects of losses in the cavities, as well as the incomingphotons’ pulse shapes and initial correlations are studied andanalyzed.A generalisation to larger arrays or number of photons isstraightforward but the calculation is involved. To this end,field theoretic methods such as LSZ reduction formula [45],or a general connection between the scattering matrix andGreen’s functions of the local system [46] might prove help-ful in deducing the properties of higher N -photon scatteringmatrices which provides an interesting avenue for future re-search. Another interesting topic is to see whether a multi-coloured coherent driving fields can be used to obtain similarphysics as studied in this work. We also note that our resultsare general and can be applied to probing the structure of anymany-body bosonic models amenable to a QCA implementa-tion including the Jaynes-Cummings-Hubbard, the extendedBose-Hubbard and a whole range of spin models.Finally, we note that the scheme presented in this work canbe experimentally demonstrated in a variety of systems, suchas semiconductor microcavities [19], photonic crystal coupledcavities [47], coupled optical waveguides [48, 49], and super-conducting circuits [1, 20, 21]. In the latter, a dimer arraysimilar to the one we have described here has been fabricatedand measured with high e ffi ciency [20, 21]. Acknowledgments
We thank D. E. Chang and M. Hartmann for helpful discus-sions, and C. Lee thanks P. N. Ma for useful comments aboutnumerical calculations. We would like to acknowledge thefinancial support provided by the National Research Founda-tion and Ministry of Education Singapore (partly through theTier 3 Grant “Random numbers from quantum processes”),and travel support by the EU IP-SIQS.
Appendix A: Scattering eigenstate
In this section, we provide a detailed derivation of the scat-tering matrices for the single- and two-photons cases.
1. Single-photon scattering
Single-photon scattering eigenstates are written as | E (1) (cid:105) = (cid:90) ∞−∞ dx φ L ( x )ˆ c † L ( x ) | (cid:105) + (cid:90) ∞−∞ dy φ R ( y )ˆ c † R ( y ) | (cid:105) + e ˆ a † | (cid:105) + e ˆ a † | (cid:105) . (A1)The time independent Schr¨odinger equation ˆ H tot | E (1) (cid:105) = E (1) | E (1) (cid:105) with E (1) = k leads to the following set of equa-tions, − i ∂∂ x φ L ( x ) + V δ ( x ) e = E (1) φ L ( x ) , (A2) − i ∂∂ y φ R ( y ) + V δ ( y ) e = E (1) φ R ( y ) , (A3) ω e + Je + V φ L (0) = E (1) e , (A4) ω e + Je + V φ R (0) = E (1) e . (A5)From Eqs. (A2) and (A3), the discontinuity relations are givenby φ L (0 + ) = φ L (0 − ) − iV e = √ π − iV e and φ R (0 + ) = φ R (0 − ) − iV e = − iV e , provided that the initial regionsare considered as φ L ( x < = √ π e ikx and φ R ( y < = φ L (0) = ( φ L (0 + ) + φ L (0 − )) and φ R (0) = ( φ R (0 + ) + φ R (0 − )). Now, solving Eqs. (A2) and (A3) in theregion x > y > φ L ( x ) = √ π ( θ ( − x ) + r k θ ( x )) e ikx φ R ( y ) = √ π t k θ ( y ) e iky . The transmission and reflection coe ffi cients are found fromthe relations r k = − √ π ie V + t k = − √ π ie V , where e and e are calculated from Eqs. (A4) and (A5): e = (cid:113) π V ( − iV − E (1) + ω )4 J + ( V − i ( E (1) − ω ))( V − i ( E (1) − ω )) , e = − J (cid:113) π V J + ( V − i ( E (1) − ω ))( V − i ( E (1) − ω )) . Thus, the explicit expressions of transmission and reflectioncoe ffi cients are written as r k = J − ( V + i ( E (1) − ω ))( V − i ( E (1) − ω ))4 J + ( V − i ( E (1) − ω ))( V − i ( E (1) − ω )) , (A6) t k = iJV V J + ( V − i ( E (1) − ω ))( V − i ( E (1) − ω )) . (A7)As expected, nonlinear e ff ects do not appear in this single pho-ton case, and hence the transmission and reflection of singlephotons are equivalent to the case of two two-level atoms [50],or two linear resonators [51]. Using these results, φ L ( x ) and φ R ( y ) construct the single photon scattering matrix [28, 33] S (1) = (cid:90) dk | φ (1)out (cid:105) k (cid:104) φ (1)in | , (A8)where the input and output states of single photon are writ-ten as | φ (1)in (cid:105) k = (cid:82) dx φ L ( x < c † L ( x ) | (cid:105) and | φ (1)out (cid:105) k = | φ (1)out (cid:105) L + | φ (1)out (cid:105) R , with | φ (1)out (cid:105) L = (cid:82) dx φ L ( x > c † L ( x ) | (cid:105) and | φ (1)out (cid:105) R = (cid:82) dy φ R ( y > c † R ( y ) | (cid:105) .
2. Two-photon scattering
For the two-photon scattering problem, a general form oftwo-photon eigenstates | E (2) (cid:105) = | E (2)1 (cid:105) + | E (2)2 (cid:105) + | E (2)3 (cid:105) is given as | E (2)1 (cid:105) = (cid:90) ∞−∞ dx dx φ LL ( x , x ) 1 √ c † L ( x )ˆ c † L ( x ) | (cid:105) + (cid:90) ∞−∞ dy dy φ RR ( y , y ) 1 √ c † R ( y )ˆ c † R ( y ) | (cid:105) + (cid:90) ∞−∞ dx dy φ LR ( x , y )ˆ c † L ( x )ˆ c † R ( y ) | (cid:105) , | E (2)2 (cid:105) = e √ a † ˆ a † | (cid:105) + e ˆ a † ˆ a † | (cid:105) + e √ a † ˆ a † | (cid:105) , | E (2)3 (cid:105) = (cid:90) ∞−∞ dx (cid:16) φ L ( x )ˆ c † L ( x )ˆ a † + φ L ( x )ˆ c † L ( x )ˆ a † (cid:17) | (cid:105) + (cid:90) ∞−∞ dy (cid:16) φ R ( y )ˆ c † R ( y )ˆ a † + φ R ( y )ˆ c † R ( y )ˆ a † (cid:17) | (cid:105) . | E (2)1 (cid:105) represents two photons in either the left or right waveg-uide, | E (2)2 (cid:105) represents two photons in the coupled cavities, and | E (2)3 (cid:105) describes one photon in one of the waveguides and theother in one of the cavities. We here obtain the two-photonscattering eigenstates by imposing the open boundary condi-tion. The Schr¨odinger equation ˆ H tot | E (2) (cid:105) = E (2) | E (2) (cid:105) gives − i ∂∂ x φ LL ( x , x ) − i ∂∂ x φ LL ( x , x ) + V √ δ ( x ) φ L ( x ) + δ ( x ) φ L ( x )) = E (2) φ LL ( x , x ) , (A9) − i ∂∂ x φ LR ( x , y ) − i ∂∂ y φ LR ( x , y ) + V δ ( x ) φ R ( y ) + V δ ( y ) φ L ( x ) = E (2) φ LR ( x , y ) , (A10) − i ∂∂ y φ RR ( y , y ) − i ∂∂ y φ RR ( y , y ) + V √ δ ( y ) φ R ( y ) + δ ( y ) φ R ( y )) = E (2) φ RR ( y , y ) , (A11) − i ∂∂ x φ L ( x ) + φ L ( x ) ω + φ L ( x ) J + V δ ( x ) e √ + V √ φ LL ( x , + φ LL (0 , x )) = E (2) φ L ( x ) , (A12) − i ∂∂ x φ L ( x ) + φ L ( x ) ω + φ L ( x ) J + V δ ( x ) e + V φ LR ( x , = E (2) φ L ( x ) , (A13) − i ∂∂ y φ R ( y ) + φ R ( y ) ω + φ R ( y ) J + V δ ( y ) e + V φ LR (0 , y ) = E (2) φ R ( y ) , (A14) − i ∂∂ y φ R ( y ) + φ R ( y ) ω + φ R ( y ) J + V δ ( y ) e √ + V √ φ RR ( y , + φ RR (0 , y )) = E (2) φ R ( y ) , (A15) ω e √ + Je + U e √ + V φ L (0) = E (2) e √ , (A16) ω e + ω e + Je √ + Je √ + V φ L (0) + V φ R (0) = E (2) e , (A17) ω e √ + Je + U e √ + V φ R (0) = E (2) e √ . (A18)Let us first solve these equations in the half spaces, x < x , x < y , and y < y . In this case, there are three quadrants: (cid:192) x < x < x < y < y < y < (cid:193) x < < x ,0 x < < y , y < < y , and (cid:194) < x < x , 0 < x < y ,0 < y < y . The Initial conditions for the amplitudes in theregion (cid:192) are given as φ LL ( x < , x < = √ π ( e ik x + ik x + e ik x + ik x ) , (A19) φ LR ( x < , y < = , (A20) φ RR ( y < , y < = . (A21)The discontinuity relations of the two-photon amplitudesacross x , x , y , y = φ LL (0 + , x ) = φ LL (0 − , x ) − i V √ φ L ( x ) , (A22) φ LL ( x , + ) = φ LL ( x , − ) − i V √ φ L ( x ) , (A23) φ LR (0 + , y ) = φ LR (0 − , y ) − iV φ R ( y ) , (A24) φ LR ( x , + ) = φ LR ( x , − ) − iV φ L ( x ) , (A25) φ RR (0 + , y ) = φ RR (0 − , y ) − i V √ φ R ( y ) , (A26) φ RR ( y , + ) = φ RR ( y , − ) − i V √ φ R ( y ) . (A27)Similarly, the discontinuity relations of the cavity-photon am-plitudes across the origin are given from Eqs. (A12)-(A15): φ L (0 + ) = φ L (0 − ) − iV e √ , (A28) φ L (0 + ) = φ L (0 − ) − iV e , (A29) φ R (0 + ) = φ R (0 − ) − iV e , (A30) φ R (0 + ) = φ R (0 − ) − iV e √ . (A31)Two-photon and cavity-photon amplitudes are also discontin-uous at x , x , y , y = φ LL (0 , x ) = φ LL ( x , =
12 ( φ LL (0 + , x ) + φ LL (0 − , x )) , (A32) φ LR (0 , y ) =
12 ( φ LR (0 + , y ) + φ LR (0 − , y )) , (A33) φ LR ( x , =
12 ( φ LR ( x , + ) + φ LR ( x , − )) , (A34) φ RR (0 , y ) = φ RR ( y , =
12 ( φ RR (0 + , y ) + φ RR (0 − , y )) . (A35)From these, the coupled linear inhomogeneous first-orderdi ff erential equations (A12), (A13), (A14), and (A15) in re-gion (cid:192) can be rewritten as i ∂∂ x (cid:32) φ L ( x < φ L ( x < (cid:33) = ω − E (2) − iV JJ ω − E (2) − iV (cid:32) φ L ( x < φ L ( x < (cid:33) + (cid:32) √ V φ LL ( x < , − ) V φ LR ( x < , − ) (cid:33) + (cid:32) V e √ V e (cid:33) δ ( x ) , (A36) i ∂∂ y (cid:32) φ R ( y < φ R ( y < (cid:33) = ω − E (2) − iV JJ ω − E (2) − iV (cid:32) φ R ( y < φ R ( y < (cid:33) + (cid:32) V φ LR (0 − , y < √ V φ RR (0 − , y < (cid:33) + (cid:32) V e V e √ (cid:33) δ ( y ) . (A37)We solve these with the discontinuity relations and the initialconditions in Eq. (A19)-(A21) to find φ L ( x < = √ π ( χ L k e ik x + χ L k e ik x ) , (A38) φ L ( x < = √ π ( χ L k e ik x + χ L k e ik x ) , (A39) φ R ( y < = , (A40) φ R ( y < = , (A41)where χ L k = A (cid:16) M − ( k + λ − ) − M + ( k + λ + ) (cid:17) ,χ L k = A (cid:16) M − ( k + λ − ) − M + ( k + λ + ) (cid:17) , χ L k = A (cid:16) k + λ − ) − k + λ + ) (cid:17) ,χ L k = A (cid:16) k + λ − ) − k + λ + ) (cid:17) , A = √ V J (cid:113) J − ( V − V + i ( ω − ω )) , M ∓ = − iV + iV + ω − ω ) ∓ √ V J / A J ,λ ∓ = (cid:16) − iV − iV − E (2) + ω + ω ) ∓ √ V J / A (cid:17) . (A42)Substituting Eqs. (A28)-(A31) into Eqs. (A16)-(A18), weobtain e , e and e as follows1 e = √ V (cid:16) J φ L (0 − ) + φ L (0 − )( V + V − i ( E (2) − ω − ω )) × (2 iU + V − iE (2) + i ω ) + J φ L (0 − )(2 U − iV − E (2) + ω ) (cid:17) /η, (A43) e = V (cid:16) J φ L (0 − ) + φ L (0 − )( − U + iV + E (2) − ω ) (cid:17) ( − U + iV + E (2) − ω ) /η, (A44) e = √ JV (cid:16) J φ L (0 − ) + φ L (0 − )( − U + iV + E (2) − ω ) (cid:17) /η, , (A45) η = (cid:16) (2 iU + V − iE (2) + i ω )( V + V − i ( E (2) − ω − ω ))(2 U − iV − E (2) + ω ) + J (8 U + U − iV − iV + − E (2) + ω + ω )) (cid:17) . Here, we note that the amplitudes of two-photon excitations tobe in the same cavity, e and e , approach zero in the limitof U and U → ∞ as these two-photon excitations requirean infinite amount of energy.Substituting the initial conditions in region (cid:192) andEqs. (A38), (A39), (A40), and (A41) into the discontinuityrelations, we obtain φ LL ( x < , + ) = √ π ( r k e ik x + r k e ik x ) , (A46) φ LR ( x < , + ) = π ( t k e ik x + t k e ik x ) , (A47) φ RR ( y < , + ) = = , (A48)where the single-photon transmission and reflection coe ffi -cients for E (2) = k + k are defined as r k j = (1 − i V √ χ L k j ) , t k j = − i V √ χ L k j , where j = ,
2. These are same as Eqs. (A6) and (A7). Wenow solve Eqs. (A9)-(A11) in region (cid:193) with the initial condi-tions in Eqs. (A46) - (A48) to find φ LL ( x < , x > = √ π ( r k e ik x + ik x + r k e ik x + ik x ) , (A49) φ LR ( x < , y > = π ( t k e ik x + ik y + t k e ik x + ik y ) , (A50) φ RR ( y < , y > = . (A51)Then solving eqs. (A12), (A13), (A14), and (A15) in region (cid:194) with the boundary conditions for φ L (0 + ), φ L (0 + ), φ R (0 + ), φ R (0 + ), φ LL ( x > , − ), φ LR ( x > , − ), φ LR (0 − , y > φ RR (0 − , y > φ L ( x > = √ π (cid:16) r k χ L k e ik x + r k χ L k e ik x + M − c L − e − i λ − x + M + c L + e − i λ + x (cid:17) , (A52) φ L ( x > = √ π (cid:16) r k χ L k e ik x + r k χ L k e ik x + c L − e − i λ − x + c L + e − i λ + x (cid:17) , (A53) φ R ( y > = √ π (cid:16) t k χ L k e ik y + t k χ L k e ik y + M − c R − e − i λ − y + M + c R + e − i λ + y (cid:17) , (A54) φ R ( y > = √ π (cid:16) t k χ L k e ik y + t k χ L k e ik y + c R − e − i λ − y + c R + e − i λ + y (cid:17) , (A55)where c L ∓ = ± A (cid:16) π V (cid:0) M ± φ L (0 + ) − φ L (0 + ) (cid:1) − r k ( k + λ ∓ ) − r k ( k + λ ∓ ) (cid:17) , c R ∓ = ± A (cid:16) π V (cid:0) M ± φ R (0 + ) − φ R (0 + ) (cid:1) − t k ( k + λ ∓ ) − t k ( k + λ ∓ ) (cid:17) . Here, c L ∓ =
0, and c R ∓ = U = U =
0, sothat φ L ( x > , φ L ( x > , φ R ( y > , φ R ( y >
0) have onlysingle-photon behaviours.Equations (A22)-(A27) can be rewritten as φ LL (0 + , x > = √ π (cid:16) r k r k e ik x + r k r k e ik x + B LL − e − i λ − x + B LL + e − i λ + x (cid:17) , (A56) φ LR (0 + , y > = π (cid:16) t k r k e ik y + t k r k e ik y + B LR − e − i λ − y + B LR + e − i λ + y (cid:17) , (A57) φ RR (0 + , y > = √ π (cid:16) t k t k e ik y + t k t k e ik y + B RR − e − i λ − y + B RR + e − i λ + y (cid:17) , (A58)with B LL − = − i V √ M − c L − , B LL + = − i V √ M + c L + , B LR − = − i V √ M − c R − , B LR + = − i V √ M + c R + , B RR − = − i V √ c R − , and B RR + = − i V √ c R + .Finally, substituting Eqs. (A52), (A53), (A54), and (A55)and then applying the initial conditions Eqs. (A56), (A57),2and (A58), we solve Eqs. (A9)-(A11) in region (cid:194) φ LL (0 < x < x ) = √ π (cid:16) r k r k e ik x + ik x + r k r k e ik x + ik x + B LL − e i ( k + k + λ − ) x − i λ − x + B LL + e i ( k + k + λ + ) x − i λ + x (cid:17) ,φ LR (0 < x < y ) = π (cid:16) t k r k e ik x + ik y + t k r k e ik x + ik y + B LR − e i ( k + k + λ − ) x − i λ − y + B LR + e i ( k + k + λ + ) x − i λ + y (cid:17) ,φ RR (0 < y < y ) = √ π (cid:16) t k t k e ik y + ik y + t k t k e ik y + ik y + B RR − e i ( k + k + λ − ) y − i λ − y + B RR + e i ( k + k + λ + ) y − i λ + y (cid:17) . One can repeat the above calculations for the other half-spaces to obtain φ LL (0 < x < x ) = φ LL (0 < x < x ) | x ↔ x , φ LR (0 < y < x ) = π (cid:16) r k t k e ik x + ik y + r k t k e ik x + ik y + B LR − e i ( k + k + λ − ) y − i λ − x + B LR + e i ( k + k + λ + ) y − i λ + x (cid:17) ,φ RR (0 < y < y ) = φ RR (0 < y < y ) | y ↔ y , where B LR − = − i V √ c L − and B LR + = − i V √ c L + .From the above results, the full solution of the two-photoneigenstates are given by the amplitudes φ LL ( x , x ) = √ π (cid:16) (cid:88) P (cid:0) θ ( − x ) θ ( − x ) + θ ( x ) θ ( x ) r k P r k P (cid:1) e ik P x + ik P x + (cid:88) Q e i ( k + k ) x Q (cid:16) B LL − e i λ − ( x Q − x Q ) + B LL + e i λ + ( x Q − x Q ) (cid:17) θ ( x Q − x Q ) θ ( x Q ) (cid:17) , (A59) φ LR ( x , y ) = π (cid:16) (cid:88) P θ ( x ) θ ( y ) t k P r k P e ik P x + ik P y + e i ( k + k ) x (cid:16) B LR − e i λ − ( x − y ) + B LR + e i λ + ( x − y ) (cid:17) θ ( y − x ) θ ( x ) + e i ( k + k ) y (cid:16) B LR − e i λ − ( y − x ) + B LR + e i λ + ( y − x ) (cid:17) θ ( x − y ) θ ( y ) (cid:17) , (A60) φ RR ( y , y ) = √ π (cid:16) (cid:88) P θ ( y ) θ ( y ) t k P t k P e ik P y + ik P y + (cid:88) Q e i ( k + k ) y Q (cid:16) B RR − e i λ − ( y Q − y Q ) + B RR + e i λ + ( y Q − y Q ) (cid:17) θ ( y Q − y Q ) θ ( y Q ) (cid:17) , (A61)where E (2) = k + k . P = ( P , P ) and Q = ( Q , Q ) arepermutations of (1 ,
2) needed to account for the bosonic sym-metry of the wave function.Here, all the B ’s become zero if the cavities are linear, i.e., U = U =
0, so that each photon undergoes the individualscattering process and the energy of each photon is preserved.If the system, on the other hand, is nonlinear, the bound-statecontributions become important, modifying the photon statis-tics of the output light as shown in the main text. In the limitof U → ∞ , B ’s become exactly the same as those of the cou-pled two-level atoms [50]. Finally, we can find the two-photonscattering matrix from, as in [33], S (2) = (cid:90) dk dk | φ (2)out (cid:105) k , k (cid:104) φ (2)in | , (A62)where the input and output states are written as | φ (2)in (cid:105) k , k = (cid:90) dx dx φ LL ( x < , x <
0) 1 √ c † L ( x )ˆ c † L ( x ) | (cid:105) | φ (2)out (cid:105) k , k = | φ (2)out (cid:105) LL + | φ (2)out (cid:105) LR + | φ (2)out (cid:105) RR , where | φ (2)out (cid:105) LL = (cid:90) dx dx φ LL ( x > , x >
0) 1 √ c † L ( x )ˆ c † L ( x ) | (cid:105) , | φ (2)out (cid:105) LR = (cid:90) dx dy φ LR ( x > , y > c † L ( x )ˆ c † R ( y ) | (cid:105) , | φ (2)out (cid:105) RR = (cid:90) dy dy φ RR ( y > , y >
0) 1 √ c † R ( y )ˆ c † R ( y ) | (cid:105) . The scattering matrix elements between the input ( k , k )and output ( p , p ) momentums are given as3 LL (cid:104) p , p | S (2) | k , k (cid:105) = r k r k δ ( k − p ) δ ( k − p ) + r k r k δ ( k − p ) δ ( k − p ) + S LL δ ( k + k − p − p ) , (A63) LR (cid:104) p , p | S (2) | k , k (cid:105) = r k t k δ ( k − p ) δ ( k − p ) + r k t k δ ( k − p ) δ ( k − p ) + S LR δ ( k + k − p − p ) , (A64) RR (cid:104) p , p | S (2) | k , k (cid:105) = t k t k δ ( k − p ) δ ( k − p ) + t k t k δ ( k − p ) δ ( k − p ) + S RR δ ( k + k − p − p ) , (A65)where S LL = π (cid:16) B k , k LL − ( − i λ k , k − + p + − i λ k , k − + p ) + B k , k LL + ( − i λ k , k + + p + − i λ k , k + + p ) (cid:17) , S LR = π (cid:16) B k , k LR − − i λ k , k − + p + B k , k LR + − i λ k , k + + p + B k , k LR − − i λ k , k − + p + B k , k LR + − i λ k , k + + p (cid:17) , S RR = π (cid:16) B k , k RR − ( − i λ k , k − + p + − i λ k , k − + p ) + B k , k RR + ( − i λ k , k + + p + − i λ k , k + + p ) (cid:17) . Appendix B: Intensity-intensity correlation
In this section, we discuss the equal-time second-order in-tensity correlations of the initial two-photon wavepacket andstudy the e ff ects of pulse-shape on the correlations of thetransmitted light.
1. Correlations between the two initial photons
Here, we analyse the initial correlations for two photonsgiven in the main text: | { ξ } (cid:105) = √ M ˆ c † ξ ˆ c † ξ | (cid:105) . The correlationfunction g (2)initial ( x , x ) can be written as g (2)initial ( x , x ) = | g ( x ) g ( x ) + g ( x ) g ( x ) | M g ( x ) g ( x ) , where g ( x ) = √ π (cid:90) dk ξ ( x ) e ikx , ! !" ! !"! ! !"$ ! %"& ! %"’$"(%"$%"(!"$ $"(%"$%"(!"$ ! ! ! ! " " " $ " % & " ! !" ! ! " ! ! " " " $ " % & (a)(b) FIG. 8: (a) g (2)initial (0) as a function of M , constructed from di ff erentvalues of momentums k and k for initial two photons. (b) g (2)initial asa function of δ when ∆ k = δ − (cid:15) (1) − . g ( x ) = √ π (cid:90) dk ξ ( x ) e ikx , g ( x ) = | g ( x ) | + | g ( x ) | + (cid:112) M − (cid:0) g ( x ) g ∗ ( x ) + g ( x ) g ∗ ( x ) (cid:1) . In Fig. 8(a), we depict a monotonic relation between the auto-correlation, g (2)initial (0), and the overlap of initial wave packets, M , constructed from di ff erent values of momenta k and k .The auto-correlation g (2)initial (0) has a maximum at M = | k − k | (cid:29) σ ) and a minimum at M = k = k ). Figure 8(b) shows that g (2)initial has a minimum of 0 . δ = (cid:15) (1) − when ∆ k = δ − (cid:15) (1) − (corre-sponding to the case of ∆ k = g (2)initial depends on δ when ∆ k = δ − (cid:15) (1) − , while g (2)initial = . ∆ k = δ .
2. E ff ects of pulse shape in narrow-band regime In this section, we show that the e ff ects of pulse shapein photon scattering is negligible given a narrow enoughbandwidth. For this purpose, we examine equal-time auto-correlations in the transmitted light, g (2) RR , for three di ff erenttemporal envelopes, Gaussian, Lorentzian, and Rising distri-butions, respectively given as Ξ G ( t ) = √ σ exp[ − σ t − ik t ](2 /π ) / , Ξ L ( t ) = √ σ exp[ − σ | t | − ik t ] , Ξ R ( t ) = √ σ exp[ σ t / − ik t ] θ ( − t ) , where σ is the inverse temporal pulse width and k is the cen-tral momentum. In the momentum space, they read ξ G ( k ) = exp[ − ( k − k ) / σ ](2 πσ ) − / ,ξ L ( k ) = (cid:112) /πσ / (( k − k ) + σ ) − ,ξ R ( k ) = (cid:112) /π √ σ (2 i ( k − k ) + σ ) − , where σ can be seen as the bandwidth of each profile.Figure 9 plots g (2) RR as a function of the probe detuning forthree di ff erent pulse profiles. The continuous lines are the4results for the Gaussian profile, whereas the results for theLorentzian and Rising profiles are marked by the red and bluedots, respectively. These results clearly demonstrate the in-sensitivity of the intensity correlations to the pulse profile, asexpected in the narrow-band regime. We have also checkedthat the probabilities are similarly insensitive to the pulse pro-file. ! ! ! " ! ! ! $&$ ! ! " " " ’ ! !" " ’ ! $" " ’ "" ’ ! " $$ ! ! ! " ! ! ! $&$ ! " " " ’ ! !" " ’ ! $" " ’ "" ’ ! " $$ (a)(b) ∆ k = δ − ǫ (1) − ∆ k = 0 GaussianLorentzianRising
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