Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity
aa r X i v : . [ qu a n t - ph ] D ec Correlated two-photon transport in a one-dimensional waveguide side-coupled to anonlinear cavity
Jie-Qiao Liao and C. K. Law Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong,Shatin, Hong Kong Special Administrative Region, People’s Republic of China
Ø We investigate the transport properties of two photons inside a one-dimensional waveguideside-coupled to a single-mode nonlinear cavity. The cavity is filled with a nonlinear Kerr medium.Based on the Laplace transform method, we present an analytic solution for the quantum states ofthe two transmitted and reflected photons, which are initially prepared in a Lorentzian wave packet.The solution reveals how quantum correlation between the two photons emerge after the scatteringby the nonlinear cavity. In particular, we show that the output wave function of the two photons inposition space can be localized in relative coordinates, which is a feature that might be interpretedas a two-photon bound state in this waveguide-cavity system.
PACS numbers:
I. INTRODUCTION
Creating quantum correlations among photons hasbeen a subject of major interest for studying the founda-tions of quantum theory as well as applications in quan-tum information science. As direct interactions betweenphotons in free space are extremely weak, generationof correlated photons generally requires nonlinear me-dia. Electromagnetically induced transparency and pho-ton blockade are mechanisms that have been exploitedto achieve strongly interacting photons [1–4]. Recently,studies of two-photon scattering from a two-level systeminside a one-dimensional (1D) waveguide have also re-ported various features of photon correlation [5–7]. Forexample, Shen and Fan [6] have discovered the existenceof two-photon bound states, and Roy [7] has indicatedan interesting application of the system as a few-photonoptical diode. We also note that Shi and Sun [8] have em-ployed a formal scattering theory to study multi-photontransport in a 1D waveguide.In this paper, we investigate the correlation propertiesof two photons in a 1D waveguide that is side-coupledto a nonlinear cavity filled with a Kerr medium (Fig. 1).The nonlinear cavity plays the role of a scatterer. It isworth noting that such a Kerr nonlinearity has also beenemployed in coupled cavity array systems for studyingquantum phase transition [9–14] and nonclassical photonstatistics [15–18]. Here we will focus on the transportproperties of two photons determined by the long timesolution of the Schr¨odinger equation, assuming the initialphotons are in wave packet forms. We will present an an-alytic solution based on the Laplace transform method,which has been applied to related photon-atom scatter-ing problems [19]. From the two-photon transmissionand reflection amplitudes, we show how the two scat-tered photons can be correlated in frequency and posi-tion variables, with the latter revealing photon bunchingand anti-bunching effects. Our solution also reveals atwo-photon resonance condition when the incident pho-ton energies match the cavity frequency shifted by the
Single-mode cavity U Kerr medium J c w Waveguide
FIG. 1: (Color online). Schematic diagram of the physicalsetup. A 1D waveguide is coupled to a cavity filled with aKerr-type nonlinear medium. Photons injected from the left-hand side of the waveguide are scattered by the nonlinearcavity. As a result, photons are reflected or transmitted inthe waveguide.
Kerr interaction. The behavior of transmission and re-flection near the resonance will be discussed.
II. PHYSICAL MODEL
The physical model under investigation consists of aninfinitely long 1D waveguide and a nonlinear cavity lo-cated at the origin (Fig. 1). We consider a single-modefield in the cavity, which couples to right- and left-propagating fields of the waveguide via the side cou-pling [20, 21] so that photons can tunnel between thewaveguide and the nonlinear cavity. The Hamiltonian(with ~ = 1) of the system is given byˆ H = ω c ˆ a † ˆ a + U a † ˆ a † ˆ a ˆ a + Z ∞ dkω k (ˆ r † k ˆ r k + ˆ l † k ˆ l k )+ J Z ∞ dk h ˆ a † (ˆ r k + ˆ l k ) + (ˆ r † k + ˆ l † k )ˆ a i . (1)Here ˆ a and ˆ a † are annihilation and creation operatorsassociated with the cavity mode with the resonance fre-quency ω c . The second term in Eq. (1) describes the Kerrnonlinear interaction with strength U . The Hamiltonianof free fields propagating in the waveguide is described bythe third term, where ˆ l k (ˆ l † k ) and ˆ r k (ˆ r † k ) are, respectively,the annihilation (creation) operators for left- and right-propagating waves with wave number k and frequency ω k . These operators satisfy the commutation relations[ˆ l k , ˆ l † k ′ ] = [ˆ r k , ˆ r † k ′ ] = δ ( k − k ′ ) , [ˆ l k , ˆ r † k ′ ] = 0 . (2)Finally, the last term in the Hamiltonian (1) repre-sents the coupling between the cavity and the waveguide,where J is the tunneling strength.For convenience, we introduce even- and odd-paritymodes operators of the waveguide,ˆ b k ≡ √ r k + ˆ l k ) , ˆ c k ≡ √ r k − ˆ l k ) , (3)so that Hamiltonian (1) can be rewritten asˆ H = ˆ H ( o ) + ˆ H ( e ) (4)with ˆ H ( o ) = Z ∞ dkω k ˆ c † k ˆ c k , (5a)ˆ H ( e ) = ω c ˆ a † ˆ a + U a † ˆ a † ˆ a ˆ a + Z ∞ dkω k ˆ b † k ˆ b k + g Z ∞ dk (ˆ a † ˆ b k + ˆ b † k ˆ a ) . (5b)Here g ≡ √ J is introduced. We see that the interac-tion involves only even modes, and photons in the oddmodes evolve freely in the waveguide. Therefore we shallfocus on the calculation of the transport properties of thephotons in even modes.In the rotating frame with respect to ˆ H ( e )0 = ω c ˆ a † ˆ a + ω c R ∞ dk ˆ b † k ˆ b k , the Hamiltonian ˆ H ( e ) can be simplified toˆ H ( e ) I = U a † ˆ a † ˆ a ˆ a + Z ∞ dk ∆ k ˆ b † k ˆ b k + g Z ∞ dk (ˆ a † ˆ b k + ˆ b † k ˆ a ) , (6)where ∆ k = ω k − ω c is the detuning. In this paper thedispersion relation for the modes in the waveguide is as-sumed to be linear, i.e., ω k = v g k , and we will set thespeed of light in the waveguide as v g = 1. III. SINGLE-PHOTON TRANSPORT
As a preparation for finding the solution for two-photon scattering, we first consider the single-photonproblem [22–26]. Note that the Kerr nonlinearity haszero effect for single photon states. The main purposein this section is to present the single-photon transmis-sion and reflection coefficients, which will appear in thetwo-photon solution later in the paper.In the single-excitation subspace, an arbitrary statecan be written as | ϕ ( t ) i = α ( t ) | i c |∅i + Z ∞ dkβ k ( t ) | i c | k i , (7) where | i c |∅i stands for the state with one photon in thecavity and no photon in the waveguide, and | i c | k i de-notes the state with a vacuum cavity field and one photonin the k th (even) mode of the waveguide. The time de-pendent variables α ( t ) and β k ( t ) are the respective prob-ability amplitudes.By the Schr¨odinger equation i | ˙ ϕ ( t ) i = ˆ H ( e ) I | ϕ ( t ) i , wehave ˙ α ( t ) = − ig Z ∞ dkβ k ( t ) , (8a)˙ β k ( t ) = − i ∆ k β k ( t ) − igα ( t ) . (8b)By performing the Laplace transform defined by ˜ f ( s ) ≡ R ∞ f ( t ) e − st dt , Eq. (8) becomes s ˜ α ( s ) − α (0) = − ig Z ∞ dk ˜ β k ( s ) , (9a) s ˜ β k ( s ) − β k (0) = − i ∆ k ˜ β k ( s ) − ig ˜ α ( s ) , (9b)where α (0) and β k (0) are the initial values of the proba-bility amplitudes.Assuming that initially the cavity is in the vacuumstate and an incident single photon in the waveguide isprepared in a wave packet with a Lorentzian spectrum,the initial condition reads α (0) = 0 , β k (0) = G ∆ k − δ + iǫ , (10)where δ and ǫ are the detuning and spectral width ofthe photon, and G = p ǫ/π is a normalization constant.The choice of β k (0) in Eq. (10) has the advantage thatanalytic solutions can be obtained conveniently. In addi-tion, by noting that ǫ → α ( s ) = 1 s + γ πigG δ − i ( s + ǫ ) , (11a)˜ β k ( s ) = G s + i ∆ k (cid:18) k − δ + iǫ + 1 s + γ γδ − i ( s + ǫ ) (cid:19) . (11b)Note that in obtaining Eq. (11), we have made the ap-proximation: R ∞ g s + i ∆ k dk ≈ R ∞−∞ g s + i ∆ k d ∆ k = γ/ γ = 2 πg .Taking the inverse Laplace transform of Eq. (11), inthe long time limit, γt/ → ∞ and ǫt → ∞ , we have α ( t → ∞ ) = 0 , β k ( t → ∞ ) = ¯ t k β k (0) e − i ∆ k t , (12)where ¯ t k = ∆ k − iγ/ k + iγ/ . (13)Equation (12) shows that the scattering process resultsin a phase shift θ k for a single photon with wave vector k , where the phase shift is defined by exp( iθ k ) = ¯ t k .In terms of the left- and right-propagation modes, ifwe assume a photon packet is incident onto the cavityfrom the left, then the initial state can be written as | ϕ (0) i = Z ∞ dkβ k (0)ˆ r † k |∅i = 1 √ Z ∞ dkβ k (0)(ˆ b † k + ˆ c † k ) |∅i . (14)In the long-time limit, the wave function becomes, | ϕ ( t → ∞ ) i = Z ∞ dkβ k (0) e − i ∆ k t ( t k ˆ r † k + r k ˆ l † k ) |∅i , (15)where the transmission and reflection amplitudes are de-fined as t k = ∆ k ∆ k + iγ/ , r k = − iγ/ k + iγ/ . (16)A similar result has been obtained for the case that asingle photon is scattered by a two-level system in a 1Dwaveguide [22], namely, the transmission amplitude t k iszero at the exact resonance. This effect was also reportedin Ref. [20] for side coupling with a classical field. IV. CORRELATED TWO-PHOTONTRANSPORTA. Equations of motion and solution
We now turn to the two-photon scattering problem.Since the total excitation number operator of the systemis a conserved quantity, we can restrict the calculation tothe two-excitation subspace. An arbitrary state in thissubspace has the form: | Φ( t ) i = A ( t ) | i c |∅i + Z ∞ dkB k ( t ) | i c | k i + Z ∞ dp Z p dqC p,q ( t ) | i c | p , q i , (17)where | i c |∅i is the state of two photons in the nonlinearcavity and no photon in the waveguide, and | i c | k i isthe state with one photon in the cavity and one photonwith wave number k in the waveguide. The last termrepresents the state with no photon in the cavity andtwo photons with wave numbers p and q in the waveguide. A ( t ), B k ( t ), and C p,q ( t ) denote the respective probabilityamplitudes.By the Schr¨odinger equation, the probability ampli- tudes are governed by:˙ A ( t ) = − iU A ( t ) − i √ g Z ∞ dkB k ( t ) , (18a)˙ B k ( t ) = − i ∆ k B k ( t ) − i √ gA ( t ) − ig Z ∞ dpC p,k ( t ) , (18b)˙ C p,q ( t ) = − i (∆ p + ∆ q ) C p,q ( t ) − ig ( B p ( t ) + B q ( t )) . (18c)We assume that the two injected photons are initiallyprepared in a Lorentzian wave packet. The initial condi-tion of the system reads, A (0) =0 , B k (0) = 0 , (19a) C p,q (0) = G (cid:18) p − δ + iǫ q − δ + iǫ + 1∆ q − δ + iǫ p − δ + iǫ (cid:19) , (19b)with the normalization constant G = ǫ √ π (cid:18) ǫ ( δ − δ ) + 4 ǫ (cid:19) − / . (20)Here δ j and ǫ j ( j = 1 ,
2) are parameters defining thedetunings and spectral widths of the two photons. Notethat C p,q has been symmetrized in Eq. (19b) because ofthe bosonic character of photons.We are interested in the asymptotic solution of C p,q ( t )in the long time limit. After a lengthy calculation (seeAppendix A), we obtain for t ≫ γ − and ǫ − , C p,q ( t ) = (¯ t p ¯ t q C p,q (0) + B p,q ) e − i (∆ p +∆ q ) t , (21)where ¯ t p and ¯ t q are defined in Eq. (13). The expressionof B p,q is given by B p,q = − U G γ (cid:0) ∆ p + i γ (cid:1) (cid:0) ∆ q + i γ (cid:1) (∆ p + ∆ q − U + iγ ) × p + ∆ q − δ − δ + 2 iǫ ) × " (cid:0) ∆ p + ∆ q − δ + iǫ + i γ (cid:1) + 1 (cid:0) ∆ p + ∆ q − δ + iǫ + i γ (cid:1) . (22)From Eqs. (21) and (22), we notice that the term B p,q isa non-factorizable function of p and q , implying a correla-tion between the two output photons. B p,q has a numer-ator proportional to the strength of the Kerr nonlinearity U in the cavity. In the case U = 0, Eq. (21) reduces toa simple expression C p,q ( ∞ ) = ¯ t p ¯ t q C p,q (0) exp[ − i (∆ p +∆ q ) t ], describing two independent scattered photons. B. Two-photon correlation in frequency variables
Let us express the results in terms of the left- andright-propagating modes. Assuming the two photons areinjected from the left-hand side of the waveguide, thenthe initial wave function can be written as | ψ (0) i = Z ∞ Z ∞ dpdqC p,q (0) ˆ r † p ˆ r † q |∅i (23)According to Eq. (3) and the solution (21), we obtain thelong-time wave function, up to an overall phase factorexp[ − i (∆ p + ∆ q ) t ], as | ψ ( t → ∞ ) i = Z ∞ Z ∞ dpdq ( C rrp,q ˆ r † p ˆ r † q + C llp,q ˆ l † p ˆ l † q ) |∅i + Z ∞ Z ∞ dpdq ( C rlp,q ˆ r † p ˆ l † q + C lrp,q ˆ l † p ˆ r † q ) |∅i , (24)where C rrp,q = t p t q C p,q (0) + 14 B p,q , (25a) C llp,q = r p r q C p,q (0) + 14 B p,q , (25b) C rlp,q = t p r q C p,q (0) + 14 B p,q , (25c) C lrp,q = t q r p C p,q (0) + 14 B p,q . (25d)Here C rrp,q and C llp,q are, respectively, the two-photontransmission and two-photon reflection amplitudes,which correspond to the processes in which two photonswith wave numbers p and q are transmitted into the right-propagation mode or reflected into the left-propagationmode. In addition, C rlp,q ( C lrp,q ) relates to the processwhere the photon with wave number p ( q ) is transmit-ted into the right-propagation mode and the photon withwave number q ( p ) is reflected into the left-propagationmode.We point out two interesting situations revealing thestrong correlation of output photons in the frequency do-main. The first situation is achieved by injecting twoidentical photons with δ = δ = 0 and a narrow spectralwidth ǫ ≪ γ . This corresponds to the case when the peakfrequency of the photons coincides with the resonant cav-ity. In this case the two photons are mainly reflected anduncorrelated [Fig. 2(a)], but if they are transmitted, theyare strongly correlated [Fig. 2(b)]. This can be seen bythe fact that t p = t q = 0 at zero detuning, and hencethe transmission of both photons is dominated by the B p,q term. In other words, the two-photon transmissionnear δ = δ = 0 is almost entirely due to the nonlinear-ity in the cavity. Such a pair of transmitted photons isfrequency correlated with the two-photon transmissionprobability concentrated along the line ∆ p + ∆ q = 0[Fig. 2(b)]. The uncertainty in the frequencies of indi-vidual transmitted photons is of the order of γ , whereas FIG. 2: (Color online). (a) and (b) are plots of | γC llp,q | and | γC rrp,q | , respectively, when δ = δ = 0. (c) and (d) are plotsof | γC rrp,q | and | γC llp,q | , respectively, when δ = δ = U/ U/γ = 10 and ǫ/γ = 0 . the uncertainty in the sum of the frequencies of both pho-tons is of the order of ǫ . The smaller ǫ , the narrower isthe distribution.The second situation of interest is two-photon reso-nance occurring when the sum of energies of the two in-cident photons equals to the energy of a cavity containingtwo photons, i.e., δ + δ = U . In this case the photonscan jointly enter the cavity. We show in Figs. 2(c) and2(d) an example with δ = δ = U/ ≫ γ , where thefrequency correlation appears more effectively in the re-flected amplitude C rrp,q , since r j ≈ j = p, q ). This isshown in the narrow distribution in Fig. 2(d). The trans-mission part [Fig. 2(c)], although they carry most of theprobabilities, are almost uncorrelated. C. Two-photon correlation in position variables
We now discuss the spatial features of the output pho-tons. For simplicity, but without loss of generality, weconsider the monochromatic limit ǫ → h x , x | ψ rr i ≈ − π M G e iE ( x c − t ) θ ( t − x c ) φ rr ( x ) , (26)with φ rr ( x ) = t δ t δ cos( δx ) − UE − U + iγ × γ ( E + iγ ) − δ e ( iE − γ )2 | x | . (27)Here we have defined x c = ( x + x ) / x = x − x forthe center-of-mass and relative coordinates respectively,and E = δ + δ and δ = ( δ − δ ) /
2. We note thatEq. (26) is a product of the center-of-mass wave function -10 -5 0 5 100.00.20.40.60.81.00 5 10 15 200.00.20.40.60.81.0 | rr ( x ) | x U / U / U / U /(a) | rr ( ) | U / (b) FIG. 3: (Color online). Spatial features of two-photon trans-mission at δ = δ = 0. (a) | φ rr ( x ) | as a function of scaledrelative coordinate γx , for various values of the scaled Kerrparameter U/γ . (b) | φ rr (0) | as a function of the scaled Kerrparameter U/γ . and the relative wave function φ rr ( x ), with exp[ iE ( x c − t )] θ ( t − x c ) describing the center-of-mass motion of thetwo transmitted photons. The second term of φ rr ( x ) isa function localized around x = 0 with a width γ − . Weremark that a similar feature was reported in Ref. [6] in aphoton-atom scattering problem, where the exponentialdecaying function is connected to the existence of photonbound states.To reveal spatial correlations, we take δ = δ = 0 sothat the first term of φ rr ( x ) can be suppressed. This isshown in Fig. 3(a) for various values of U . Note that | φ rr ( x ) | is proportional to the joint probability of pho-tons with a separation x , therefore the decaying featurecorresponds to photon bunching. In particular, the jointprobability of having both transmitted photons at thesame position increases with increasing U , but it satu-rates when U ≫ γ [Fig. 3(b)]. For the two-photon reflec-tion amplitude in position space, we carry out a similarcalculation and obtain, h x , x | ψ ll i ≈ − π N G e − iE ( x c + t ) θ ( t + x c ) φ ll ( x ) , (28)with φ ll ( x ) = r δ r δ cos( δx ) − UE − U + iγ × γ ( E + iγ ) − δ e ( iE − γ )2 | x | . (29)At δ = δ = δ = 0, the second term causes a dip in | φ ll ( x ) | at x = 0 [Fig. 4(a)], which is a signature ofphoton antibunching as the reflected photons repel each -10 -5 0 5 100.00.20.40.60.81.0 x | ll ( x ) | U / =0 U / =0.5 U / =1 U / =10(a) | ll ( ) | U / (b) FIG. 4: (Color online). Spatial features of two-photon re-flection at δ = δ = 0. (a) | φ ll ( x ) | as a function of scaledrelative coordinate γx , for various values of the scaled Kerrparameter U/γ . (b) | φ ll (0) | as a function of the scaled Kerrparameter U/γ . other. As U increases, the joint probability of havingboth reflected photons at the same position decreases[Fig. 4(b)], which is in contrast to the transmitted part.Finally we describe the effects of two-photon resonancearound δ + δ = U discussed in the previous section. Forsimplicity we again consider the case δ = δ here. InFig. 5, we illustrate the dependence of the relative two-photon wave function on E = δ + δ . The effect of two-photon resonance is most apparent in Fig. 5(a), where thereflected two-photon wave function is strongly localizedaround x = 0 when E = U . Away from the resonance,the reflected two-photon wave function exhibits an oscil-latory pattern in x [Fig. 5(c)], which is controlled by thetwo-photon detuning E − U . We also plot the transmit-ted two-photon wave function in Figs. 5(b) and 5(d), inwhich similar oscillatory patterns are observed. V. CONCLUSIONS
In conclusion, we have presented an analytic solutionof two-photon scattering inside a one-dimensional waveg-uide that is side-coupled to a Kerr-type nonlinear cavity.The system provides a scheme to realize correlated two-photon transport. The Kerr nonlinearity is found to cor-relate photons in frequency variables such that ∆ p + ∆ q is a constant, which is a constraint because of the energyconservation. In position space, we have shown that theKerr nonlinearity can cause the two photons ‘stick’ to-gether with an average separation distance of the orderof v g γ − . We may interpret the result as a two-photon FIG. 5: (Color online). Dependence of spatial features on E = δ + δ . (a) | φ ll ( x ) | and (b) | φ rr ( x ) | . Examples atparticular values of E are shown in (c) and (d). In thesefigures, we use U = 10 γ and δ = 0. bound state, because of the exponential decaying shapeof relative wave function. However, because of the in-terference with single photon processes described by thefirst term in Eq. (25), features of photon correlation mayonly be observed efficiently in certain directions. Finally,we note that recent studies of a related topic have consid-ered using a single atom as a scatterer [6, 27]. However,in view of recent progress in achieving a giant Kerr non-linearity [1–3], our work suggests that a nonlinear cavitymay be an alternative way of regarding the correlatedtwo-photon transport problem. Acknowledgments
This work is supported by the Research Grants Councilof Hong Kong, Special Administrative Region of China(Project No. 401408).
Appendix A: Solution of Eq. (18) by the Laplace transform method
In this appendix, we give a detailed derivation for the solution of Eq. (18) which governs the transport of two photonsin a waveguide. We use the Laplace transform method to solve these equations. Under the initial condition (19),Eq. (18) becomes ( s + iU ) ˜ A ( s ) = − i √ g Z ∞ dk ˜ B k ( s ) , (A1a)( s + i ∆ k ) ˜ B k ( s ) = − i √ g ˜ A ( s ) − ig Z ∞ dp ˜ C p,k ( s ) , (A1b)[ s + i (∆ p + ∆ q )] ˜ C p,q ( s ) = C p,q (0) − ig ( ˜ B p ( s ) + ˜ B q ( s )) . (A1c)Substituting Eqs. (A1a) and (A1c) into Eq. (A1b), and making use of the initial condition, we obtain the equationfor the variable ˜ B k ( s ) as[∆ k − i ( s + γ/ B k ( s ) = Z ∞−∞ (cid:18) g U − is + g ∆ p + ∆ k − is (cid:19) ˜ B p ( s ) d ∆ p +2 πgG (cid:18) k + δ − i ( s + ǫ ) 1∆ k − δ + iǫ + 1∆ k + δ − i ( s + ǫ ) 1∆ k − δ + iǫ (cid:19) . (A2)where we have made the approximation R ∞ dp g s + i (∆ p +∆ k ) ≈ γ/ B k ( s ) Eq. (A2), by inspection, takes the form:˜ B k ( s ) = 2 πgG ∆ k − i ( s + γ/ (cid:18) k + δ − i ( s + ǫ ) 1∆ k − δ + iǫ + 1∆ k + δ − i ( s + ǫ ) 1∆ k − δ + iǫ (cid:19) (1 + ˜ F k ( s )) , (A3)with ˜ F k ( s ) = − iγ (cid:18) k + δ − i ( s + ǫ ) 1∆ k − δ + iǫ + 1∆ k + δ − i ( s + ǫ ) 1∆ k − δ + iǫ (cid:19) − × "(cid:18) U − is − iγ + 1 δ + ∆ k − i ( s + ǫ ) (cid:19) δ − i (cid:0) s + ǫ + γ (cid:1) δ + δ − i ( s + 2 ǫ )+ (cid:18) U − is − iγ + 1 δ + ∆ k − i ( s + ǫ ) (cid:19) δ − i (cid:0) s + ǫ + γ (cid:1) δ + δ − i ( s + 2 ǫ ) . (A4)Then from Eq. (A1c) we obtain the following expression˜ C p,q ( s ) = G s + i (∆ p + ∆ q ) (cid:20) iγs + γ + i ∆ p (cid:18) s + ǫ + i (∆ p + δ ) 1∆ p − δ + iǫ + 1 s + ǫ + i (∆ p + δ ) 1∆ p − δ + iǫ (cid:19) + iγs + γ + i ∆ q (cid:18) s + ǫ + i (∆ q + δ ) 1∆ q − δ + iǫ + 1 s + ǫ + i (∆ q + δ ) 1∆ q − δ + iǫ (cid:19) − γ s + γ + iU s + 2 ǫ + i ( δ + δ ) (cid:18) s + ǫ + γ + iδ + 1 s + ǫ + γ + iδ (cid:19) (cid:18) s + γ + i ∆ p + 1 s + γ + i ∆ q (cid:19) − γ s + 2 ǫ + i ( δ + δ ) 1 s + ǫ + γ + iδ (cid:18) s + ǫ + i ( δ + ∆ p ) 1 s + γ + i ∆ p + 1 s + ǫ + i ( δ + ∆ q ) 1 s + γ + i ∆ q (cid:19) − γ s + 2 ǫ + i ( δ + δ ) 1 s + ǫ + γ + iδ (cid:18) s + ǫ + i ( δ + ∆ p ) 1 s + γ + i ∆ p + 1 s + ǫ + i ( δ + ∆ q ) 1 s + γ + i ∆ q (cid:19) + (cid:18) p − δ + iǫ q − δ + iǫ + 1∆ q − δ + iǫ p − δ + iǫ (cid:19)(cid:21) . (A5)Until now, we have obtained the expression for ˜ C p,q ( s ). Then we can get the expression for the probability amplitude C p,q ( t ) by performing the inverse Laplace transform of ˜ C p,q ( s ). In particular, since we are interested in the outputstate of the two photons, here we present only the long-time solution of C p,q ( t → ∞ ) as C p,q ( t → ∞ ) = (¯ t p ¯ t q C p,q (0) + B p,q ) e − i (∆ p +∆ q ) t , (A6)where ¯ t p and ¯ t q have been defined in Eq. (13), and the expression for the correlation term is B p,q = − U G γ (cid:0) ∆ p + i γ (cid:1) (cid:0) ∆ q + i γ (cid:1) (∆ p + ∆ q − U + iγ ) 1(∆ p + ∆ q − δ − δ + 2 iǫ ) × " (cid:0) ∆ p + ∆ q − δ + iǫ + i γ (cid:1) + 1 (cid:0) ∆ p + ∆ q − δ + iǫ + i γ (cid:1) . (A7) Appendix B: Derivation of two-photon output state in position space
In this appendix, we derive the wave function of the two-photon output state (24) in position space. For thetwo-photon transmission process, the corresponding wave function in position space can be written as h x , x | ψ rr i = Z ∞ Z ∞ dpdqC rrp,q h x , x | ˆ r † p ˆ r † q |∅i≈ M Z ∞−∞ Z ∞−∞ ( t p t q C p,q (0) + B p,q / e − i (∆ p +∆ q ) t e i ∆ p x e i ∆ q x d ∆ p d ∆ q + x ↔ x . (B1)In Eq. (B1), symmetrization of the two photons has been taken into account by introducing h x , x | ˆ r † p ˆ r † q |∅i = M ( e i ∆ p x e i ∆ q x + e i ∆ p x e i ∆ q x ). According to the initial condition given in Eq. (19b), we can get the expressionfor the independent transport part as M Z ∞−∞ Z ∞−∞ t p t q C p,q (0) e − i (∆ p +∆ q ) t e i ∆ p x e i ∆ q x d ∆ p d ∆ q = − π M G t δ − iǫ t δ − iǫ e ( iE +2 ǫ )( x c − t ) cos( δx ) θ ( t − x c ) , (B2)where we introduce the center-of-mass coordinator x c = ( x + x ) /
2, the relative coordinator x = x − x , the totalmomentum E = δ + δ , and the relative momentum δ = ( δ − δ ) / θ ( x ) is the Heaviside step function and t δ − iǫ is defined in Eq. (16). Note that here we have taken the approximation exp[ γ ( x − t ) / → γ/ ≫ ǫ .According to Eq. (A7), the Fourier transform of the correlation part B pq can be written as M Z ∞−∞ Z ∞−∞ B p,q e − i (∆ p +∆ q ) t e i ∆ p x e i ∆ q x d ∆ p d ∆ q = A + A , (B3)with A l = − M U G γ Z ∞−∞ Z ∞−∞ q + iγ/
2) 1(∆ p + iγ/
2) 1(∆ p + ∆ q − U + iγ ) 1(∆ p + ∆ q − δ − δ + 2 iǫ ) × p + ∆ q − δ l + iǫ + iγ/ e i ∆ p ( x − t ) d ∆ p e i ∆ q ( x − t ) d ∆ q , (B4)for l = 1 ,
2. The Fourier transform of the correlation part can be obtained as A + A = 8 π M G U ( E − U − iǫ + iγ ) γ ( E + iγ − i ǫ ) − δ e ( iE +2 ǫ )( x c − t ) e ( iE +2 ǫ − γ ) | x | θ ( t − x c ) . (B5)According to Eqs. (B2) and (B5), the second term in Eq. (B1) can be obtained by making the replacement x c → x c and x → − x . Then h x , x | ψ rr i = − π M G e ( iE +2 ǫ )( x c − t ) θ ( t − x c ) φ rr ( x ) , (B6)with φ rr ( x ) = t δ − iǫ t δ − iǫ cos( δx ) − UE − U − iǫ + iγ γ ( E + iγ − i ǫ ) − δ e ( iE +2 ǫ − γ )2 | x | . (B7)Using the same method, we can obtain the wave function for the two-photon reflection state, h x , x | ψ ll i = Z ∞ Z ∞ dpdqC llp,q h x , x | ˆ l † p ˆ l † q |∅i ≈ − π N G e − ( iE +2 ǫ )( x c + t ) θ ( t + x c ) φ ll ( x ) , (B8)with φ ll ( x ) = r δ − iǫ r δ − iǫ cos( δx ) − UE − U − iǫ + iγ γ ( E + iγ − i ǫ ) − δ e ( iE +2 ǫ − γ )2 | x | , (B9)where N is defined by h x , x | ˆ l † p ˆ l † q |∅i = N ( e − i ∆ p x e − i ∆ q x + e − i ∆ p x e − i ∆ q x ). [1] H. Schmidt and A. Imamoˇglu, Opt. Lett. , 1936 (1996).[2] A. Imamoˇglu, H. Schmidt, G. Woods, and M. Deutsch,Phys. Rev. Lett. , 1467 (1997).[3] S. E. Harris and Y. Yamamoto, Phys. Rev. Lett. , 3611(1998).[4] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T.E. Northup, and H. J. Kimble, Nature (London) , 87(2005).[5] K. Kojima, H. F. Hofmann, S. Takeuchi, and K. Sasaki,Phys. Rev. A , 013803 (2003)[6] J. T. Shen and S. Fan, Phys. Rev. Lett. , 153003(2007); Phys. Rev. A , 062709 (2007).[7] D. Roy, Phys. Rev. B , 155117 (2010).[8] T. Shi and C. P. Sun, Phys. Rev. B , 205111 (2009).[9] M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio,Nat. Phys. , 849 (2006); Phys. Rev. Lett. , 160501(2007); Laser and Photon Rev. , No. 6, 527 (2008).[10] A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L.Hollenberg, Nat. Phys. , 856 (2006).[11] D. G. Angelakis, M. F. Santos, and S. Bose, Phys. Rev.A , 031805(R) (2007). [12] M. J. Hartmann, Phys. Rev. Lett., , 113601 (2010).[13] N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, Phys.Rev. A , 031803(R) (2008).[14] A. Tomadin, V. Giovannetti, R. Fazio, D. Gerace, I.Carusotto, H. E. T¨ureci, and A. Imamoˇglu, Phys. Rev.A , 061801(R) (2010).[15] D. Gerace, H. E. T¨ureci, A. Imamoˇglu, V. Giovannetti,and R. Fazio, Nat. Phys. , 281 (2009).[16] T. C. H. Liew and V. Savona, Phys. Rev. Lett. ,183601 (2010).[17] S. Ferretti, L. C. Andreani, H. E. T¨ureci, and D. Gerace,Phys. Rev. A , 013841 (2010).[18] M. Bamba, A. Imamoˇglu, I. Carusotto, and C. Ciuti,arXiv:1007.1605.[19] T. S. Tsoi, M.Phil. thesis, The Chinese University ofHong Kong, 2009.[20] Y. Xu, Y. Li, R. K. Lee, and A. Yariv, Phys. Rev. E ,7389 (2000).[21] M. F. Yanik, W. Suh, Z. Wang, and S. Fan, Phys. Rev.Lett. , 233903 (2004).[22] J. T. Shen and S. Fan, Opt. Lett. , 2001 (2005); Phys. Rev. Lett. , 213001 (2005); Phys. Rev. A ,023837(2009); , 023838 (2009).[23] D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D.Lukin, Nat. Phys. , 807 (2007).[24] T. S. Tsoi and C. K. Law, Phys. Rev. A , 063832(2008); Phys. Rev. A , 033823 (2009).[25] L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori,Phys. Rev. Lett. , 100501 (2008); Z. R. Gong, H. Ian, L. Zhou, and C. P. Sun, Phys. Rev. A , 053806 (2008);Y. Chang, Z. R. Gong, and C. P. Sun, arXiv:1005.2274.[26] J. Q. Liao, J. F. Huang, Y. X. Liu, L. M. Kuang, andC. P. Sun, Phys. Rev. A , 014301 (2009); J. Q. Liao,Z. R. Gong, L. Zhou, Y. X. Liu, C. P. Sun, and F. Nori,Phys. Rev. A81