Quantum Theory of Two-Photon Wavepacket Interference in a Beam Splitter

Transcription

1 Quantum Theory of Two-Photon Wavepaket Interferene in a Beam Splitter Kaige Wang CCAST (World Laboratory), P. O. Box 8730, Beijing , and Department of Physis, Applied Optis Beijing Area Major Laboratory, Beijing Normal University, Beijing , China We study a general theory on the interferene of a two-photon wavepaket in a beam splitter. The theory is arried out in the Shrödinger piture so that the quantum nature of the two-photon interferene is expliitly presented. We find that the topologial symmetry of the probability-amplitude spetrum of the two-photon wavepaket dominates the manners of two-photon interferene whih are distinguished aording to inreasing and dereasing the oinidene probability for the absene of interferene. However, two-photon entanglement an be witnessed by the interferene manner. We demonstrate the neessary and suffiient onditions for the perfet two-photon interferene. For a two-photon entangled state with an anti-symmetri spetrum, it passes a 50/50 beam splitter with the perfet transpareny. The theory ontributes an unified understanding to a variety of the two-photon interferene effets. PACS number(s): Dv, Ta, Ar I. INTRODUCTION Photons as bosons tend to bunhing, that is, photons are more likely to appear lose together than farther apart. This happens for the photoni field with a lassial analogy. For some optial field, however, photons may behave as an opposite effet the photon anti-bunhing. It is known that photon anti-bunhing effet provides an evidene for expliitly quantum mehanial states of the optial field without lassial analogy. When two photons meet in a beamsplitter from two input ports, how do they exit from the output ports? This is an interesting topi onerning the quantum nature of photon interferene. The first experimental observations of two-photon interferene in a beam splitter were reported in 1980s. [1] [2] In the experiments, two photons to be interfered are spatially separated and degenerated with the same frequeny and polarization. This an be done by the spontaneously parametri downonversion (SPDC) of type I in a rystal, in whih a pair of photons, signal and idle, are produed. In the degenerate ase, two photons are mixed in a 50/50 beamsplitter, no oinidene ount of photons is found at two output ports. This effet was lately alled the photon oalesene interferene (CI), [3] sine two photons meeting in beamsplitter go together. The early theoretial explanation was based on indistinguishability of two single photons, that is, the interferene ours for the degenerate photons when they meet in a beamsplitter. In the further experiments, it has been found that, in addition to the degenerate photons, the two-photon interferene may our for two photons with different olors and polarizations. [4]- [7] However, for the degenerate ase when the individual signal photon and the idle photon are arranged out of their oherent range (they do not meet at the beamsplitter), the interferene is still observed. [8]- [10] Obviously, these phenomena an not be explained by the indistinguishability of two single photons. As a matter of fat, all the above experiments were performed by the SPDC, the soure emitting entangled photon pairs. A suessful theoretial explanation is to use two-photon entanglement with the help of oneptual Feynman diagrams in whih the pair of photons interfered should be seen as a whole, the two-photon or biphoton, so that the photon entanglement plays an essential role in two-photon interferene. [11] [12] Reently, the interferene of two independent photons has been studied experimentally and theoretially. [13]- [15] Santori et al [13] has demonstrated in their experiment, that two independent single-photon pulses emitted by a semiondutor quantum dot show a oalesene interferene in a beamsplitter. In this ase, it seems that one annot use the onept biphoton beause the photon entanglement is absent. As we review all these studies on two-photon interferene, it ould be onfusing why sometimes two nondegenerate photons an interfere with eah other and sometimes they annot, and sometimes the interferene of two degenerate photons ours only when they meet together and sometimes the photon meeting is not neessarily the ase. A reasonable explanation ould be attributed to photon entanglement. As mentioned above, the entangled pair for two nondegenerate photons should be seen as a biphoton and the interferene ours between biphotons. Therefore, it might be onluded that there are two kinds of interferene mehanisms: the biphoton piture for entangled photon pair and two photons piture for independent photons. If that is true, one has to fae a perplexed question, as argued in the development of quantum mehanis, how the lever photons know whether they should behave as a biphoton or a single-photon in the interferene. In this paper, we ontribute a omplete theoretial desription for two-photon interferene in a beamsplitter. Sine any realisti beam should have a finite frequeny range whih must be taken into aount, we study the two-photon 1

2 state in a wavepaket whih an be either entangled or un-entangled. The quantum desriptions of the beam splitter an be implemented in both the Shrödinger piture (S-piture) and the Heisenberg piture (H-piture). Sine the beamsplitter introdues a simple transform for field operators, the theoretial desription in the H-piture is more onvenient, and it was exploited in the most theoretial disussions. Though the two pitures should give an idential result, the desription in the H-piture averages the distint information on quantum state. Instead, we disuss this issue in the S-piture, so it an show more physial understanding for the nature of two-photon interferene. The net oinidene probability an be readily evaluated by our theory. We find that the symmetry of two-photon spetrum plays a key role in the interferene manners. We distinguish the oalesene and anti-oalesene interferenes (ACI), and find out the neessary and suffiient ondition for the perfet CI and ACI. We prove that the photon entanglement is irrelevant to CI, but neessary for ACI. Therefore, the ACI effet is the signature of two-photon entanglement, and it ould be an effetive experimental method to detet photon entanglement. However, we propose a two-photon transparent state whih an pass the beamsplitter with a full transpareny. The theory overs two ases: two-photon state in a polarization mode and in two polarization modes, whih may orrespond to the soure of SPDC of type I and type II, respetively. II. PRELIMINARY THEORETICAL DESCRIPTION A. general desription about quantum interferene Let us briefly review how the quantum interferene happens. If a quantum system onsists of more than one soure, or the interation inludes several parts, the state of the system Ψ is a oherent superposition of these soures or parts Ψ 1 = 1 α + 2 β, Ψ 2 = 3 γ + 4 δ, (1) Ψ = Ψ 1 + Ψ 2 = 1 α + 2 β + 3 γ + 4 δ. Assume that all the states α, β, γ and δ are distinguishable eah with others, there is no quantum interferene. If, however, there are some indistinguishable states generated in these oherent soures suh as Ψ 1 = 1 α + 2 β, Ψ 2 = 3 β + 4 δ, (2) Ψ = Ψ 1 + Ψ 2 = 1 α +( ) β + 4 δ, state β in two soures has to be added together. The probability of finding state β for Ψ may not be equal to the sum of those for Ψ 1 and Ψ 2, unless the interferene term =2 2 3 os[arg( 2 ) arg( 3 )] (3) is null. In other words, the quantum interferene happens if the interferene term (3) is not null. There are two reasons for the absene of interferene. One possible reason is simply that there is no indistinguishable state between two soures, that is 2 =0or 3 = 0. Another reason ould be out of phase for two oherent amplitudes, 2 and 3. In this sense, the indistinguishable state is only a neessary ondition for interferene, but not a suffiient one. The relative phase between the amplitudes 2 and 3 may settle the interferene absene, onstrutive and destrutive aording to null, positive and negative interferene terms, respetively. In the language of quantum state, in essene, the interferene originates from the oherent superposition of probability amplitudes for the indistinguishable states of different soures. We survey two-photon interferene in this piture. B. input-output transformation of quantum state in a beam splitter A beam splitter performs a linear transform for two input optial beams. In the quantum regime, the bosoni ommutation must be satisfied for all the field operators in the beamsplitter transform. So, for a lossless beamsplitter, the general transformation between input and output field operators obeys [16] ( b1 ) = S(θ, φ b τ,φ ρ ) 2 ( a1 a 2 ), S(θ, φ τ,φ ρ )= ( e iφ τ os θ e iφρ sin θ e iφρ sin θ e iφτ os θ ), (4) 2

3 where a i and b i are the field annihilation operators for the input and output ports, respetively. The subsript i (i =1, 2) symbolizes the ports in the same propagation diretion. θ haraterizes the refletion and the transmission rates, for instane, θ = π/4 for a 50/50 beamsplitter. φ τ and φ ρ are two phases allowed in the unitary transformation (4). In transformation (4), two input beams, a 1 and a 2, should be in the same mode. In other words, two input photons are indistinguishable as soon as they are mixed in the beamsplitter. Transformation (4) is arried out in the H-piture. In priniple, it an solve all the problems onerning the beamsplitter transform by evaluating expetation values of field operators. Nevertheless, the method wipes out the information of what happens for quantum state. This shortoming an be avoided in the S-piture. The transform of wavevetor in the S-piture, orresponding to the operator transformation (4), Ψ out = U Ψ in, has been disussed in Ref. [16], in whih an expliit expression of U has been given. Alternatively, we use a simpler method to evaluate the output wavevetor. It is a similar method as for a dynami quantum system when the evolution of operators has been known in the H-piture, one may obtain the state evolution in the S-piture without solving the Shrödinger equation. [17] The method requires two onditions: (i) the initial state is known as Ψ in = f(a 1,a 1,a 2,a 2 ) Θ ; (ii) the inverse evolution of state Θ is known as Θ = U 1 Θ. In the beamsplitter ase, we set Θ = 0 and, obviously, a vauum state 0 is always onserved as U 0 = U 1 0 = 0. So that the output state is obtained as Ψ out = U Ψ in = Uf(a 1,a 1,a 2,a 2 ) 0 = Uf(a 1,a 1,a 2,a 2 )U 1 U 0 = Uf(a 1,a 1,a 2,a 2 )U 1 0 (5) = f(ua 1 U 1,Ua 1 U 1,Ua 2 U 1,Ua 2 U 1 ) 0 = f(b 1, b 1, b 2, b 2) 0, where b i Ua i U 1 and b i Ua i U 1 (i =1, 2). Beause b i = U 1 a i U is known due to Eq. (4), one may obtain its inverse transform as ( ) ( ) ( ) b1 = S 1 a1 a1 (θ, φ τ,φ ρ ) = S( θ, φ a τ,φ ρ ). (6) 2 a 2 b 2 It is not diffiult to hek S(θ, φ τ,φ ρ )S( θ, φ τ,φ ρ )=I. For the onveniene in use, we write b 1(ω) =osθe iφτ a 1 (ω) sin a θe iφρ 2 (ω), (7) b 2 (ω) =sinθeiφρ a 1 (ω)+osθe iφτ a 2 (ω). C. quantum states of single-photon and two-photon wavepakets A single-photon state of the monohromati beam with the frequeny ω, travelling in a given propagation diretion, is written as a α (ω) 0. The index α denotes a partiular polarization or a spatial mode. Any pratial beam has a finite bandwidth, so a general form of the single photon state an be expressed as Φ s = ω C α (ω)a α (ω) 0, (8) where C α (ω) is the spetrum of the probability amplitude. The single-photon wavepaket orresponding to the above state is obtained as 0 E α (+) (z,t) Φ s = ω E α (ω)c α (ω)e iω(z/ t) C α (z/ t), (9) where the field operator for the polarization mode α is desribed by E α (+) (z,t) ω E α (ω)a α (ω)exp[iω(z/ t)], (10) and E α (ω) is the field amplitude per photon. C α (z/ t) 2 shows an expetation for the field intensity at (z,t). A single-photon state an be in a multimode superposition. The onept mode onerned here refers to polarization or spatial distribution, but not radiation frequeny and propagation diretion, beause the diretion has been given and the frequeny dependene has been inorporated into state (8). A single-photon state of two modes is written as 3

4 Φ s = ω [C α (ω)a α (ω)+c β(ω)a β (ω)] 0, (11) where α and β are mode index. Then we onsider a quantum state ontaining two photons separated spatially. Eah photon has a given propagation diretion, designated by the subsripts 1 and 2, so that the separated two photons are ready to be inident upon two input ports of a beamsplitter. In the desription of a two-photon wavepaket, we disuss two ases. Case I : two-photon wavepaket in the same polarization (or spatial) mode If we assume two photons to be in the same polarization mode, the two-photon state in a general form an be expressed as Φ two = C(ω 1,ω 2 )a 1 (ω 1)a 2 (ω 2) 0. (12) Note that two photons in the state a 1 (ω 1)a 2 (ω 2) 0 are distinguishable even if ω 1 = ω 2 sine they are separated physially. C(ω 1,ω 2 ) denotes a spetrum of two-photon wavepaket. The orresponding two-photon wavepaket for state (12) is given by 0 E (+) 1 (z 1,t 1 )E (+) 2 (z 2,t 2 ) Φ two = E(ω 1 )E(ω 2 )C(ω 1,ω 2 )e iω1(z1/ t1) e iω2(z2/ t2) C(z 1 / t 1,z 2 / t 2 ). (13) Equation (12) an desribe both entangled and un-entangled two-photon states. fatorized as If two-photon spetrum an be C(ω 1,ω 2 )=C 1 (ω 1 )C 2 (ω 2 ), (14) the two-photon state Φ two is un-entangled. That is, the two-photon state onsists of two independent single-photon wavepakets suh as 0 E (+) 1 (z 1,t 1 )E (+) 2 (z 2,t 2 ) Φ t = E(ω 1 )E(ω 2 )C 1 (ω 1 )C 2 (ω 2 )e iω1(z1/ t1) e iω2(z2/ t2) C 1 (z 1 / t 1 ) C 2 (z 2 / t 2 ), where C j (z/ t) is the single-photon wavepaket designated by Eq. (9). Otherwise, if the fatorization (14) is impossible, Eq. (12) defines a frequeny-entangled two-photon state. The orresponding two-photon wavepaket (13) an not be fatorized into two single-photon wavepakets as Eq. (15) does. This kind of two-photon state an be generated in the SPDC of type I. For example, in the degenerate ase, the two-photon spetrum of a pair of entangled photons an be expressed in a symmetri form [11] C(ω 1,ω 2 )=g(ω 1 + ω 2 Ω p )e [(ω1 Ω)2 +(ω 2 Ω) 2 ]/(2σ 2), (16) where Ω and σ are the entral frequeny and the bandwidth, respetively, for both the signal and the idle beams. Ω p = 2Ω is the entral frequeny for the pump beam. Funtion g(x) desribes the phase mathing. For simpliity, it an be assumed as a Gaussian (15) g(ω 1 + ω 2 Ω p )=Ae (ω1+ω2 Ωp)2 /(2σ 2 p ), (17) where σ p is the bandwidth of the pump beam. In the ase of σ p 0, Eq. (17) tends to a δ funtion g(ω 1 + ω 2 Ω p )=Aδ(ω 1 + ω 2 Ω p ). (18) As an important example, a set of Bell states, whih onsist of two monohromati photons with frequenies Ω 1 and Ω 2 being in the same polarization, are written as Φ ± =(1/ 2) [δ(ω 1 Ω 1 )δ(ω 2 Ω 1 ) ± δ(ω 1 Ω 2 )δ(ω 2 Ω 2 )]a 1 (ω 1)a 2 (ω 2) 0, (19a) Ψ ± =(1/ 2) [δ(ω 1 Ω 1 )δ(ω 2 Ω 2 ) ± δ(ω 1 Ω 2 )δ(ω 2 Ω 1 )]a 1 (ω 1)a 2 (ω 2) 0. (19b) 4

5 Case II : two-photon wavepaket in two orthogonal polarization (or spatial) modes We assume that there are two un-entangled single-photon wavepakets traveling in different diretions, and eah of them is desribed by a two-mode superposition state (11). The ombined state for the two photons is written as Φ ss = [C 1α (ω 1 )a 1α (ω 1)+C 1β (ω 1 )a 1β (ω 1)] [C 2α (ω 2 )a 2α (ω 2)+C 2β (ω 2 )a 2β (ω 2)] 0 (20) ω 1 ω 2 = [C 1α (ω 1 )C 2α (ω 2 )a 1α (ω 1)a 2α (ω 2)+C 1β (ω 1 )C 2β (ω 2 )a 1β (ω 1)a 2β (ω 2) +C 1α (ω 1 )C 2β (ω 2 )a 1α (ω 1)a 2β (ω 2)+C 1β (ω 1 )C 2α (ω 2 )a 1β (ω 1)a 2α (ω 2)] 0. In the general ase, a two-mode two-photon wavepaket an be expressed as Φ two = Φ αα + Φ ββ + Φ αβ + Φ βα, (21) where Φ mm = C mm (ω 1,ω 2 )a 1m (ω 1)a 2m (ω 2) 0, m = α, β (22a) Φ αβ = C αβ (ω 1,ω 2 )a 1α (ω 1)a 2β (ω 2) 0. α β (22b) There are four two-photon spetra, C αα (ω 1,ω 2 ),C ββ (ω 1,ω 2 ),C αβ (ω 1,ω 2 ),C βα (ω 1,ω 2 ), whih desribe a two-mode two-photon wavepaket. If the fatorization of Eq. (21) into Eq. (20) is impossible, the two-mode wavepaket is entangled. This kind of two photon states an be generated in the SPDC of type II, in whih a pair of down-onverted photons, o-ray and e-ray, are polarization-orthogonal. A very famous example is the set of Bell states onsisting of two photons: one with frequeny Ω α and polarization α and the other one with frequeny Ω β and polarization β, Φ ± =(1/ 2) [δ(ω 1 Ω α )δ(ω 2 Ω α )a 1α (ω 1)a 2α (ω 2) ± δ(ω 1 Ω β )δ(ω 2 Ω β )a 1β (ω 1)a 2β (ω 2)] 0, (23a) Ψ ± =(1/ 2) [δ(ω 1 Ω α )δ(ω 2 Ω β )a 1α (ω 1)a 2β (ω 2) ± δ(ω 1 Ω β )δ(ω 2 Ω α )a 1β (ω 1)a 2α (ω 2)] 0. (23b) The Bell states Ψ ± an be generated in the SPDC proess of type-ii, in whih two down-onverted photons ome from the overlap of the o-ray and e-ray ones. [19] However, a half-wave-plate an hange the polarization between horizontal and vertial, so that by using two orthogonal half-wave-plates in two paths, the Bell states Φ ± an be obtained from Ψ ± (if one sets Ω α =Ω β ). By taking into aount the bandwidths of the beams, the polarization entanglement state generated in SPDC of type II an be desribed as Ψ w (θ) = g(ω 1 + ω 2 Ω p )[e (ω1 Ωα) 2 /(2σ 2 α ) (ω2 Ω β) 2 /(2σ 2 β ) a 1α (ω 1)a 2β (ω 2) (24) +e iθ e (ω1 Ω β) 2 /(2σ 2 β ) (ω2 Ωα)2 /(2σ 2 α ) a 1β (ω 1)a 2α (ω 2)] 0, where Ω p =Ω α +Ω β, and we have assumed that o-ray and e-ray have different entral frequenies and bandwidths. The phase θ depends on the rystal birefringene. But if one puts a wave-plate in path 1, it is possible to introdue an additional relative phase to the polarization β, so that the phase θ an be set as desired. [19] III. TWO PHOTON INTERFERENCE IN A BEAM SPLITTER A. output quantum states and oinidene probability Equation (5) an be used to alulate the output quantum state for any input state inident upon a beamsplitter. We fous on the input states of a two-photon wavepaket, as has been shown in the last setion. Case I : For the input state (12), the orresponding output state after the beamsplitter transform is obtained as 5

6 Φ two out = C(ω 1,ω 2 )b 1 (ω 1)b 2 (ω 2) 0 (25) = C(ω 1,ω 2 )[a 1 (ω 1)e iφτ os θ a 2 (ω 1)e iφρ sin θ][a 1 (ω 2)e iφρ sin θ + a 2 (ω 2)e iφτ os θ] 0 = C(ω 1,ω 2 ){[a 1 (ω 1)a 1 (ω 2)e iφ a 2 (ω 1)a 2 (ω 2)e iφ ]osθsin θ +[a 1 (ω 1)a 2 (ω 2)os 2 θ a 2 (ω 1)a 1 (ω 2)sin 2 θ]} 0, where φ = φ τ + φ ρ. In the summation taken in the whole frequeny spae, the states orresponding to (ω 1,ω 2 )and(ω 2,ω 1 ) are indistinguishable and should be added together. For doing it, we may take = ω 1<ω 2 + ω 1=ω 2 + ω 1>ω 2, and then exhange the variables ω 1 and ω 2 in the last summation. In result, Eq. (25) is written as Φ two out = {[C(ω 1,ω 2 )+C(ω 2,ω 1 )][a 1 (ω 1)a 1 (ω 2)e iφ a 2 (ω 1)a 2 (ω 2)e iφ ]osθsin θ (26) ω 1<ω 2 +[C(ω 1,ω 2 )os 2 θ C(ω 2,ω 1 )sin 2 θ]a 1 (ω 1)a 2 (ω 2) +[C(ω 2,ω 1 )os 2 θ C(ω 1,ω 2 )sin 2 θ]a 1 (ω 2)a 2 (ω 1)} 0 + C(ω, ω){[(a 1 (ω))2 e iφ (a 2 (ω))2 e iφ ]osθsin θ +(os 2 θ sin 2 θ)a 1 (ω)a 2 (ω)} 0. ω For a 50/50 beamsplitter (In the following text, we onsider only the ase of the 50/50 beam splitter.), Eq. (26) is redued to ( Φ two out =(1/2) {[C(ω 1,ω 2 )+C(ω 2,ω 1 )][a 1 (ω 1)a 1 (ω 2)e iφ a 2 (ω 1)a 2 (ω 2)e iφ ] (27) ω 1<ω 2 +[C(ω 1,ω 2 ) C(ω 2,ω 1 )][a 1 (ω 1)a 2 (ω 2) a 1 (ω 2)a 2 (ω 1)]} + ) C(ω, ω)[(a 1 (ω))2 e iφ (a 2 (ω))2 e iφ ] 0. ω In Eq. (27), the first and last terms desribe two photons exiting from the same output port, whereas the seond term desribes two photons exiting from two different ports resulting in a lik-lik in oinidene measurement. The oinidene probability for a lik-lik detetion at two output ports is evaluated by P =(1/4) 2 C(ω 1,ω 2 ) C(ω 2,ω 1 ) 2 =(1/4) C(ω 1,ω 2 ) C(ω 2,ω 1 ) 2, (28) ω 1<ω 2 where we have onsidered that C(ω 1,ω 2 ) C(ω 2,ω 1 ) 2 is symmetri with respet to the diagonal ω 1 = ω 2 in the frequeny spae (ω 1,ω 2 ) and vanishes at ω 1 = ω 2. It an be expressed in the integration form as P =(1/4) dω 1 dω 2 C(ω 1,ω 2 ) C(ω 2,ω 1 ) 2 (29) = 1 2 ( dω 1 ) dω 2 [C(ω 1,ω 2 )C (ω 2,ω 1 ) +..], where the normalization dω 1 dω 2 C(ω 1,ω 2 ) 2 = 1 has been applied. What we have done from Eq. (25) to Eq. (26) is in fat the same operation as Eq. (2). Physially, in the input two-photon wavepaket, it may ontain two soures: soure 1 a photon of frequeny ω 1 at port 1 and the other photon of frequeny ω 2 at port 2 with the amplitude C(ω 1,ω 2 ); soure 2 a photon of frequeny ω 2 at port 1 and the other photon of frequeny ω 1 at port 2 with the amplitude C(ω 2,ω 1 ). The above pair of soures, a 1 (ω 1)a 2 (ω 2) 0 and a 1 (ω 2)a 2 (ω 1) 0, in the input state an generate indistinguishable output states. In the spetral plane for the input state in whih eah point orresponds to a biphoton sub-state with 6

7 an amplitude C(ω 1,ω 2 ), the diagonal ω 1 = ω 2 divides the plane into two parts. The interferene ours between a pair of symmetri points with respet to the diagonal, as shown in Fig. 1. Equation (29) for the oinidene probability shows an interferene term C(ω 1,ω 2 )C (ω 2,ω 1 )+... As pointed out in Se. II-A, the neessary and suffiient ondition for the absene of interferene is dω 1 dω 2 [C(ω 1,ω 2 )C (ω 2,ω 1 ) +..] = 0. (30) It results in a 50% oinidene probability, that is, the probability of that two photons go together is equal to that they exit from different ports. Obviously, if the spetrum of the input state satisfies C(ω 1,ω 2 )C(ω 2,ω 1 ) =0, (31) the interferene disappears. In this ase, there is no pair of two-photon states to be interfered, as shown in Fig. 1a. But ondition (31) is not neessary for the absene of interferene. The other ondition for the absene of interferene is out of phase between amplitudes C(ω 1,ω 2 )andc(ω 2,ω 1 ), i.e. arg C(ω 1,ω 2 ) arg C(ω 2,ω 1 )=(n +1/2)π. Otherwise, the phase differene of two biphoton states determines inrease and derease of the oinidene probability. Now we an answer the question raised in Introdution. In the language of quantum state, it is lear to show the nature of interferene based on the biphoton, but not two photons. What we emphasize is that this interferene mehanism does not ask for any preondition for the input two-photon state, either entangled or un-entangled. We will show in the following that entangled two-photon wavepaket behaves in a distint interferene manner different from un-entangled one. Case II : If we assume that the beamsplitter does not hange the polarization of the input beam, Eq. (7) an be still used to the polarization modes α and β individually. In a two-photon state with two polarizations, the biphoton state with the same polarization αα is distinguishable from the states with the same polarization ββ and the ross polarizations αβ and βα. Therefore, the interferene an not our among them. For the input state shown in Eq. (21), one may alulate the output state in suh a way Φ two out = Φ αα out + Φ ββ out + Φ αβ+βα out, (32a) Φ mm out = U Φ mm, m = α, β (32b) Φ αβ+βα out = U( Φ αβ + Φ βα ). (32) Equation (32b) has already been alulated in Eqs. (25)-(27). Equation (32) an be alulated as Φ αβ+βα out = [C αβ (ω 1,ω 2 )b 1α(ω 1 )b 2β(ω 2 )+C βα (ω 1,ω 2 )b 1β(ω 1 )b 2α(ω 2 )] 0 (33) = [C αβ (ω 1,ω 2 )b 1α (ω 1)b 2β (ω 2)+C βα (ω 2,ω 1 )b 1β (ω 2)b 2α (ω 1)] 0 = {[C αβ (ω 1,ω 2 )+C βα (ω 2,ω 1 )][a 1α (ω 1)a 1β (ω 2)e iφ a 2α (ω 1)a 2β (ω 2)e iφ ]sinθos θ +[C αβ (ω 1,ω 2 )os 2 θ C βα (ω 2,ω 1 )sin 2 θ]a 1α (ω 1)a 2β (ω 2) [C αβ (ω 1,ω 2 )sin 2 θ C βα (ω 2,ω 1 )os 2 θ]a 2α (ω 1)a 1β (ω 2)} 0. For a 50/50 beamsplitter, it is written as Φ αβ+βα out =(1/2) {[C αβ (ω 1,ω 2 )+C βα (ω 2,ω 1 )][a 1α (ω 1)a 1β (ω 2)e iφ a 2α (ω 1)a 2β (ω 2)e iφ ] (34) +[C αβ (ω 1,ω 2 ) C βα (ω 2,ω 1 )][a 1α (ω 1)a 2β (ω 2) a 2α (ω 1)a 1β (ω 2)]} 0. In the summation, the first term shows α and β photons traveling together, and the seond term shows α and β photons exiting separately from two output ports, ausing a lik-lik ounting. For the input state of Eq. (21), the normalization is desribed as 1=n αα + n ββ + n αβ + n βα, (35a) n mm = dω 1 dω 2 C mm (ω 1,ω 2 ) 2, m = α, β, (35b) n αβ = dω 1 dω 2 C αβ (ω 1,ω 2 ) 2, α β, (35) 7

8 where n ij indiates the probability proportion of the input state Φ ij of Eq. (21). For ase II, we onsider polarization-sensitivity of detetion system whih an distinguish the output oinidene probability for two photons with a partiular onfiguration of polarizations. Similar to Eq. (29), the oinidene probability for the same polarized photons is obtained as P mm =(1/4) dω 1 dω 2 C mm (ω 1,ω 2 ) C mm (ω 2,ω 1 ) 2 (m = α, β) (36) = 1 2 n mm ( 1 1 2n mm dω 1 ) dω 2 [C mm (ω 1,ω 2 )Cmm(ω 2,ω 1 ) +..], where the normalization (35b) has been used. The oinidene probabilities for two output photons with the ross polarizations an be obtained by Eq. (34). Two oinidene probabilities, α(β) photon at port 1 and β(α) photon at port 2, are the same as P αβ = P βα =(1/4) dω 1 dω 2 C αβ (ω 1,ω 2 ) C βα (ω 2,ω 1 ) 2 (37) = 1 4 (n αβ + n βα ) ( 1 1 n αβ + n βα dω 1 ) dω 2 [C αβ (ω 1,ω 2 )Cβα (ω 2,ω 1 ) +..], where the normalization (35) has been used. If the detetion does not distinguish polarization, the total oinidene probability is deteted as P = P αα + P ββ = P αβ dω 1 dω 2 [2C αβ (ω 1,ω 2 )Cβα (ω 2,ω 1 )+ m=α,β C mm (ω 1,ω 2 )Cmm (ω 2,ω 1 ) +..]. As mentioned above, beause of the distinguishability of the polarization onfiguration, the interferene between two photons with the same polarization is independent of that for the ross polarizations. The ondition of the absene of interferene for the same polarized photons is the same as that in ase I. (see Eq. (30)) As for the states Φ αβ and Φ βα shown in Eq. (22b), the ondition for the absene of two-photon interferene is a null interferene term dω 1 dω 2 [C αβ (ω 1,ω 2 )Cβα(ω 2,ω 1 ) +..] = 0. (39) It results in the oinidene probability of the ross-polarized photons P αβ (38) = P βα = 1 4 (n αβ + n βα ), (40) so that the oinidene probability P αβ + P βα =2P αβ is one half of the probability proportion n αβ + n βα for the pairs of ross-polarized photons. Again, the probability that two photons go together is the same as that they exit separately. Equation (37) shows that the interferene ours between the input states Φ αβ and Φ βα. If only one state, either Φ αβ or Φ βα, exists, the interferene never happens beause of C αβ (ω 1,ω 2 )C βα (ω 2,ω 1 ) =0. (41) Similarly, the phase differene between two amplitudes, C αβ (ω 1,ω 2 )andc βα (ω 2,ω 1 ), dominates the ourrene of interferene. When both the input states Φ αβ and Φ βα oexist, the interferene ours between two soures: soure 1 a photon (ω 1,α) at port 1 and the other photon (ω 2,β) at port 2 with the amplitude C αβ (ω 1,ω 2 ); soure 2 a photon (ω 2,β) at port 1 and the other photon (ω 1,α) at port 2 with the amplitude C βα (ω 2,ω 1 ). Soures 1 and 2 ome from the input states Φ αβ and Φ βα, respetively. Note that in this ase two photons at two input ports are always orthogonal in polarization and there is no degenerate photons. Undoubtedly, the effet annot be understood in the two photons piture. To understand the interferene mehanism in ase II, we give a simple explanation. The pair of soures, a 1α (ω 1)a 2β (ω 2) 0 and a 1β (ω 2)a 2α (ω 1) 0, beome indistinguishable when they are mixed in the beamsplitter. This an be seen by omitting the subsripts 1 and 2 of the reating operators. However, the pair of soures, a 1α (ω 1)a 2β (ω 2) 0 and a 1α (ω 2)a 2β (ω 1) 0, are still distinguishable as the subsripts 1 and 2 have been omitted. So that the interferene an not our when the state Φ αβ (or Φ βα )existsbyitselfevenifithasa symmetri spetrum C αβ (ω 1,ω 2 )=C αβ (ω 2,ω 1 ). 8

9 B. oalesene interferene In the last subsetion, we show the interferene term in the representation of quantum state. In the presene of interferene, the interferene term inreases or dereases the oinidene probability with respet to that of the absene of interferene. It is neessary to define two manners of two-photon interferenes, the oalesene interferene (CI) and the anti-oalesene interferenes (ACI), aording to the oinidene probabilities less and more than that for the absene of interferene, respetively. For the CI effet, the probability of the fat that two photons travel together is more than the probability of the fat that they exit apart. In the extreme ase, two photons always go together and the oinidene probability is null, one alls it the perfet oalesene interferene. Case I : Aording to Eqs. (27) or (29), the suffiient and neessary ondition for the perfet CI is that the two-photon wavepaket has a symmetri spetrum in the whole frequeny range C(ω 1,ω 2 ) C(ω 2,ω 1 ). (42) A symmetri spetrum an be aquired for both entangled and un-entangled two-photon wavepakets. For example, a pair of degenerate photons generated in SPDC of type I, suh as shown in Eq. (16), has a symmetri spetrum. The three Bell states Φ ± and Ψ + desribed in Eqs. (19) are also symmetri. These examples of two-photon entangled states show the perfet CI. For two independent single-photon wavepakets, the two-photon spetrum is the produt of two single-photon spetra shown in Eq. (14). If two single-photon spetra are idential, C 1 (ω) =C 2 (ω) =C(ω), the symmetri ondition (42) is fulfilled. This means two idential single-photon wavepakets perform the perfet CI. On the ontrary, if the symmetri spetrum (42) is satisfied for two independent wavepakets, i.e. C 1 (ω 1 )C 2 (ω 2 ) C 2 (ω 1 )C 1 (ω 2 ), it has C 1 (ω 1 ) C 2 (ω 1 ) C 1(ω 2 ) = s, (43) C 2 (ω 2 ) where s is a onstant independent of frequeny. By taking into aount the normalization 1 = C 1 (ω) 2 dω = C2 (ω) 2 dω, one obtains s = e iθ and hene C 1 (ω) =e iθ C 2 (ω). (44) The spetrum of two un-entangled single-photon wavepakets is then C(ω 1,ω 2 )=e iθ C 2 (ω 1 )C 2 (ω 2 ). The phase fator independent of frequeny is atually trivial. In this sense, we an onlude that the neessary and suffiient ondition for the perfet CI of two independent single-photon wavepakets is that two single-photon wavepakets are idential. Furthermore, we prove that, for two independent single-photon wavepakets, the oinidene probability is not greater than one half. Sine one has dω 1 dω 2 C(ω 1,ω 2 )C (ω 2,ω 1 ) (45) = dω 1 dω 2 C 1 (ω 1 )C 2 (ω 2 )C1 (ω 2 )C2 (ω 1 )= dωc 1 (ω)c2 (ω) 2, Eq. (29) is written as [ P = ] dωc 1 (ω)c2 (ω) 1 2. (46) This is the right reason why we distinguish CI and ACI effets. In ase I, the ACI effet never happens for two un-entangled single-photon wavepakets. Equation (46) shows that if two independent single-photon spetra never overlap in the whole frequeny range, i.e. C 1 (ω)c 2 (ω) 0, there is no two-photon interferene. In the two photons piture, it would be explained by the distinguishability of two input photons, i.e. two photons have different frequenies. But this explanation is inonsistent with the fat that two non-degenerate photons an interfere in some other ases. However, in the biphoton piture, we an find the orret understanding. To be expliit, we assume that the spetrum of eah single-photon wavepaket is a Gaussian type C i (ω) exp[ (ω Ω i ) 2 /(2σ 2 )] (i =1, 2). If the two single-photon spetra have the same entral frequeny Ω 1 =Ω 2 = Ω, i.e. they are idential, the orresponding two-photon spetrum C(ω 1,ω 2 ) exp[ ((ω 1 Ω) 2 +(ω 2 Ω) 2 )/(2σ 2 )] is symmetri with respet to the diagonal ω 1 = ω 2, as shown by the ontour plot of the spetrum in Fig. 2a. If, however, the differene of the two entral frequenies is larger than the 9

10 bandwidth, Ω 2 Ω 1 >σ, the two single-photon spetra do not overlap. In the spetral spae for the two-photon states, the entre of the two-photon spetrum deviates from the diagonal, as shown in Fig. 2b. Similar to Fig. 1a, there are few pairs of photons to be interfered. In this theory, the net oinidene probability an be alulated for showing the manners of two-photon interferene. Experimentally, it would be diffiult to detet the net oinidene probability beause of the lower quantum detetion effiieny. A simple way is to ompare the relative value of the oinidene probability with respet to a referene, for example, the one for the absene of interferene. In experiment, this an be done by introduing different paths for two inident beams. Let us assume a two-photon spetrum C s (ω 1,ω 2 ) to be defined at an optial soure emitting two beams separately. These two beams, traveling different paths z 1 and z 2, are ready to input into two ports of a beamsplitter. Aording to Eq. (13), the new spetrum after the paths is written as C(ω 1,ω 2 )=C s (ω 1,ω 2 )e i(ω1z1/+ω2z2/). (47) If C s (ω 1,ω 2 ) is symmetri, C(ω 1,ω 2 ) beomes asymmetri at the unbalaned position z 1 z 2. As an example, we onsider a soure emitting a two-photon wavepaket with the spetrum as C s (ω 1,ω 2 )=g(ω 1 + ω 2 Ω p )e [(ω1 Ω1)2 +(ω 2 Ω 2) 2 ]/(2σ 2), (48) where Ω p =Ω 1 +Ω 2 desribes the phase mathing in whih Ω p and Ω i (i =1, 2) are the entral frequenies for the pump beam and two onverted beams, respetively. σ defines the bandwidth for two onverted beams. This is the typial form of two-photon wavepaket generated in SPDC of type I by taking into aount the two down-onverted beams non-degenerate. If g(ω 1 + ω 2 Ω p ) an not be fatorized as g 1 (ω 1 )g 2 (ω 2 ), it desribes an entangled two-photon wavepaket. However, Eq. (48) an desribe an un-entangled two-photon wavepaket too, as long as g(x) =1. For the spetrum (48), we alulate the oinidene probability by Eq. (29) and obtain (see Appendix A) P =(1/2)[1 e 1 2 z2 (σ/) 2 e 1 2 ( Ω/σ)2 ], (49) where the path differene z = z 2 z 1 and the frequeny deferene Ω = Ω 2 Ω 1. The equation displays a well known interferene dip at the balaned position z 1 = z 2, observed in the previous experiments, for example, in Refs. [1] and [2] for the entangled two-photon state, and in the reent experiment reported in Ref. [13] for two independent single photons. The width of the dip is defined by the oherent length of the single-photon beam /σ. When the path differene z exeeds far the oherent length z >>/σ,the CI disappears, showing the referene (P =1/2) for the absene of interferene. In the degenerate ase Ω = 0, the oinidene probability is null at the balaned position and the perfet CI ours due to the symmetry of the spetrum (48). The level of dip rises as the differene of the entral frequenies Ω is inreased. However, it is interesting that the oinidene probability does not depend on the form of funtion g(x), so that the present theory ontributes an uniform desription for both entangled and un-entangled two-photon wavepakets. In a general ase when the bandwidths of two single-photon beams are not equal, in order to evaluate the oinidene probability, funtion g(x) must be given. Let the spetrum of two-photon wavepaket C s (ω 1,ω 2 )=Ae (ω1+ω2 Ωp)2 /(2σ 2 p ) e (ω1 Ω1)2 /(2σ 2 1 ) (ω2 Ω2)2 /(2σ 2 2 ), (50) where σ p is the bandwidth for the pump beam. The oinidene probability is alulated as P =(1/2)[1 (σ s /σ f )e 1 2 z2 (σ s/) 2 e 1 2 ( Ω/σ f ) 2 ], (51) where two effetive bandwidths are defined as 2σ1 2 σ s = σ2 2 σ1 2 +, (52a) σ2 2 σp(σ σ f = + σ2 2 )+4σ2 1 σ2 2 2(σp 2 + σ2 1 + σ2 2 ). (52b) The effetive bandwidths σ s and σ f determine the spatial oherent range and the frequeny range of two-photon interferene, respetively. In the extreme ase σ p 0 whih desribes the maximum two-photon entanglement, two effetive bandwidths are equal: σ f = σ s. However, when σ p, the two onverted beams are not entangled, the oinidene probability is written as 10

11 P = 1 [ 1 2σ 1σ 2 2 σ1 2 + exp( σ2 1σ 2 ] 2 σ2 2 σ1 2 + z2 σ2 2 2 Ω2 σ1 2 + ). (53) σ2 2 It verifies P < 1/2 for two independent single-photon wavepakets. Case II : In the general form of ase II, the input state inludes four parts shown in Eq. (21). We have already indiated in Se. III-A that there is no interferene among the states Φ αα, Φ ββ and Φ αβ + Φ βα, so that they an be disussed independently. For state Φ mm, disussion is the same as ase I. For Φ αβ + Φ βα,onemayuse Eq. (34), or Eq. (37) to study the CI effet. Therefore, for ase II, the suffiient and neessary onditions for the perfet CI are obtained as C mm (ω 1,ω 2 ) C mm (ω 2,ω 1 ), m = α, β (54a) C αβ (ω 1,ω 2 ) C βα (ω 2,ω 1 ). (54b) We note that the onditions (54a) and (54b) are for the perfet CI of two photons with the same polarization (state Φ mm ) and with orthogonal polarizations (state Φ αβ + Φ βα ), respetively. Similar to ase I, the perfet CI an be aquired for both entangled and un-entangled two-photon wavepaket. For the entangled two-photon states, for instane, the Bell states Φ ± and Ψ + desribed by Eqs. (23a) and (23b) satisfy the symmetri onditions (54a) and (54b), respetively. However, the polarization-entangled two-photon wavepaket Ψ w (θ =0), defined by Eq. (24), satisfies ondition (54b). Then we onsider two independent single-photon wavepakets, desribed by Eq. (20). The normalization of eah single-photon wavepaket requires n jm = dω C jm (ω) 2, n jα + n jβ =1, (j =1, 2, m = α, β). (55) If two single-photon wavepakets are idential C 1m (ω) =C 2m (ω), m = α, β (56) it is readily to verify that the symmetri onditions (54) have been satisfied. As the same as ase I, two idential single-photon wavepakets show the perfet CI. On the other hand, if the symmetri ondition Eq. (54b) has been satisfied for two independent single-photon wavepakets, one obtains C 1α (ω 1 ) C 2α (ω 1 ) = C 1β(ω 2 ) = s, (57) C 2β (ω 2 ) where the onstant s is independent of frequeny. By taking into aount the normalization (55), one obtains s = e iθ, and hene C 1α (ω) =e iθ C 2α (ω), C 1β (ω) =e iθ C 2β (ω). (58) This means that, for two independent single-photon wavepakets, if only ondition (54b) has been satisfied, ondition (54a) must be satisfied, too, and the two wavepakets are idential in addition to a phase. In other words, for two independent single-photon wavepakets, if the perfet CI for photons with the ross polarizations has been observed, one an predit the perfet CI for photons with the same polarization. We alulate the oinidene probability for two independent single-photon wavepakets. Using Eqs. (36) and (45), we obtain P mm = 1 2 [n 1mn 2m dωc 1m (ω)c2m (ω) 2 ], m = α, β, (59) where n 1m n 2m is the probability of two m-polarized photons entering the beamsplitter. Equation (59) shows that (1/2)n 1m n 2m is the referene oinidene probability for the absene of interferene of the m-polarized photons. This means that the ACI effet annot be observed in detetion of the oinidene probability of the same polarized photons. By taking into aount the integral dω 1 dω 2 C αβ (ω 1,ω 2 )Cβα (ω 2,ω 1 )= dω 1 dω 2 C 1α (ω 1 )C 2β (ω 2 )C1β (ω 2)C2α (ω 1) (60) = dωc 1α (ω)c 2α(ω) 11 dωc 2β (ω)c 1β(ω),

12 and Eq. (37), the oinidene probability for the pairs of ross-polarized photons is obtained as P αβ = P βα = 1 4 (n 1αn 2β + n 1β n 2α ) 1 4 [ dωc 1α (ω)c2α(ω) dωc 2β (ω)c1β(ω) +..], (61) where n 1α n 2β + n 1β n 2α is the probability of two ross-polarized photons entering the beamsplitter. Again, (1/2)(n 1α n 2β + n 1β n 2α ) is the referene oinidene probability for the absene of interferene for two ross-polarized photons. Different from P mm, P αβ + P βα an be higher or lower than this referene. Finally, the total oinidene probability (38) is written as [ P = 1 2] 1 2 dω[c 1α (ω)c2α(ω)+c 1β (ω)c2β(ω)] 1 2, (62) where the normalization (55) has been applied. This result tells us that, for two independent single-photon wavepakets, the ACI effet never ours for the polarization-insensitive detetion. Of ourse, eah sub-oinidene probability, suh as P mm (m = α, β) orp αβ + P βα, is no more than one half, too. In onsequene, two-photon oalesene interferene in a beamsplitter an inspet the identity for two independent single-photon wavepakets or, substantially, the symmetry of the spetrum of two-photon wavepaket. Obviously, CI is not a riterion for two-photon entanglement. We have proved in Eqs. (46) and (62) that ACI effets annot our for two independent single-photon wavepakets. Therefore, ACI is the signature of two-photon entanglement. We will disuss ACI in details in the next two sub-setions. C. anti-oalesene interferene and two-photon transparent state The other manner of two-photon interferene is just opposite of the oalesene interferene: the oinidene probability at the output of beamsplitter is greater than that of the absene of interferene. In the extreme ase two photons never go together, we all it the perfet ACI. Case I : Aording to Eq. (27), the neessary and suffiient ondition for the perfet ACI is C(ω 1,ω 2 )= C(ω 2,ω 1 ) (63) in the whole frequeny spae. We all Eq. (63) the anti-symmetri two-photon spetrum. Obviously, it satisfies C(ω, ω) = 0. One an see immediately from Eq. (27) that only the output states for two photons traveling in different ports remain so that the oinidene probability is unity. Furthermore, when the anti-symmetri ondition (63) is satisfied, the output state (27) is redued to Φ two out = C(ω 1,ω 2 )[a 1 (ω 1)a 2 (ω 2) a 1 (ω 2)a 2 (ω 1)] 0 (64) ω 1<ω 2 = C(ω 1,ω 2 )a 1 (ω 1)a 2 (ω 2) 0 C(ω 2,ω 1 )a 1 (ω 1)a 2 (ω 2)] 0 ω 1<ω 2 ω 2<ω 1 = C(ω 1,ω 2 )a 1 (ω 1)a 2 (ω 2) 0 = Φ two. It means that when the perfet ACI ours, the output state is idential to the input. In other words, a two-photon wavepaket with the anti-symmetri spetrum (63) is invariant under the 50/50 beamsplitter transform, or it is the eigenstate with the unity eigenvalue. Note that the eigenstate is not unique, and it an be any two-photon wavepaket satisfying ondition (63). Physially, the two-photon wavepaket is perfetly transparent passing beamsplitter, so we all it two-photon transparent state. Of ourse, the two-photon transparent state must be in entanglement, sine two independent single-photon wavepakets never show ACI effet. A well-known example of two-photon transparent state is the Bell state Ψ whih is defined by (19b) and satisfies the anti-symmetri ondition (63). So the Bell state Ψ is the eigenstate in a 50/50 beamsplitter transform. This is the reason why the Bell state Ψ an be measured by a oinidene ounting in the teleportation sheme. [18] Case II : Similarly as disussed above, the neessary and suffiient onditions for the perfet ACI are C mm (ω 1,ω 2 ) C mm (ω 2,ω 1 ), m = α, β, (65a) C αβ (ω 1,ω 2 ) C βα (ω 2,ω 1 ). (65b) 12

13 The same as ase I, ondition (65a) gives the invariant state However, under ondition (65b), one obtains Φ αβ out = Φ mm out = Φ mm, m = α, β. (66) C αβ (ω 1,ω 2 )[a 1α (ω 1)a 2β (ω 2) a 2α (ω 1)a 1β (ω 2)] 0 (67) = [C αβ (ω 1,ω 2 )a 1α (ω 1)a 2β (ω 2)+C βα (ω 2,ω 1 )a 2α (ω 1)a 1β (ω 2)] 0 = Φ αβ, α β. Again, a two-photon wavepaket with the anti-symmetri spetra (65) is transparent passing the beamsplitter. It is readily to hek that the polarization-entangled anti-symmetri Bell state Ψ and state Ψ w (θ = π), defined by Eqs. (23b) and (24), respetively, fulfill onditions (65) and hene are the two-photon transparent states. We have already indiated that, there is no interferene among two-photon pairs, αα, ββ and αβ, so that onditions (65a) and (65b) are for the perfet ACIs of two photons with the same polarization and the orthogonal polarizations, respetively. For ase II, it is possible that one of the two-photon spetra is symmetri and the other one is antisymmetri. For example, we onsider two independent single-photon wavepakets whih are idential. Then a phase shift θ for β polarized beam is introdued by inserting a wave-plate in path 1. The ombined two-photon state is written as Ψ ss (θ) = [C α (ω 1 )a 1α (ω 1)+e iθ C β (ω 1 )a 1β (ω 1)] [C α (ω 2 )a 2α (ω 2)+C β (ω 2 )a 2β (ω 2)] 0 (68) ω 1 ω 2 = [C α (ω 1 )C α (ω 2 )a 1α (ω 1)a 2α (ω 2)+e iθ C β (ω 1 )C β (ω 2 )a 1β (ω 1)a 2β (ω 2) +C α (ω 1 )C β (ω 2 )a 1α (ω 1)a 2β (ω 2)+e iθ C β (ω 1 )C α (ω 2 )a 1β (ω 1)a 2α (ω 2)] 0. The two-photon spetra of state Ψ ss (θ = π) satisfy the symmetri ondition (54a) for the photon pairs αα and ββ, and the anti-symmetri ondition (65b) for the photon pair αβ. In result, the photon pairs with the same polarization travel together while the photon pairs with the orthogonal polarizations exit from different ports. Nevertheless, the total oinidene probability in polarization-insensitive detetion must satisfy Eq. (62). D. observation of anti-oalesene interferene effet The detetion of entanglement is the one of the important tasks in quantum information. We have proved in the previous subsetions that the ACI effet is the signature of two-photon entanglement so that it an be an useful and simple method to demonstrate entanglement. The photon entanglement state generated in the soure may have a symmetri spetrum showing the CI effet. Due to the fat that the manners of interferene depend on the relative phase of the interferene term whih inreases or dereases oinidene probability, we introdue an additional phase in two-photon wavepaket to hange the manners of the interferene from CI to ACI. Case I : We onsider a two-photon spetrum in the form of Q(ω 1,ω 2 )=g(ω 1 + ω 2 2Ω)f(ω 1 Ω)f(ω 2 Ω), (69) in whih f(x) desribes a spetral profile idential for two single-photon beams and g(x) desribes possible entanglement. Obviously, spetrum (69) is symmetri. If f(x) is a Gaussian, we have already alulated the oinidene probability as shown in Eq. (49) whih is in fat irrelevant to photon entanglement. In order to introdue an additional phase in the spetrum, one an set an unbalaned Mah-Zehnder interferometer in one path of the beam. This method was proposed in the previous experiments. [8] [9] We explain this method again in the S-piture by the spetra feature for two-photon state. Let beam 1 be split into two parts, and one travels a short path L s,andthe other a long path L l. Then these two sub-beams inorporate a beam again whih interferes with beam 2 traveling a path z 2. The new two-photon spetrum at the input ports of beamsplitter is obtained as C(ω 1,ω 2 )=Q(ω 1,ω 2 )[e iω1l l/ + e iω1ls/ ]e iω2z2/ =2Q(ω 1,ω 2 )e i(ω1z1+ω2z2)/ os(ω 1 L/), (70) where z 1 =(L l + L s )/2 and L =(L l L s )/2. We set ν i = ω i Ω, (i =1, 2), Eq. (70) is written as 13

14 C(ν 1,ν 2 )=2e iω(z1+z2)/ Q(ν 1,ν 2 )e i(ν1z1+ν2z2)/ os(ν 1 L/ + θ), (71) where the additional phase θ = Ω L/ = 2π L/λ. λ = 2π/Ω is the wavelength for eah single-photon beam. In the ase of the perfet phase mathing in SPDC, g(x) δ(x), one obtains C(ν 1,ν 2 )=±2e iω(z1+z2)/ δ(ν 1 + ν 2 )f(ν 1 )f(ν 2 )e i(ν1z1+ν2z2)/ os(ν 1 L/), θ = nπ, (72a) C(ν 1,ν 2 )=±2e iω(z1+z2)/ δ(ν 1 + ν 2 )f(ν 1 )f(ν 2 )e i(ν1z1+ν2z2)/ sin(ν 1 L/), θ =(n + 1 )π. (72b) 2 At the balaned position z 1 = z 2, spetrum (72a) is symmetri and spetrum (72b) is anti-symmetri so that the phase θ dominates the interferene manners hanged between the perfet CI and ACI. Now, we onsider the two-photon spetrum Q(ω 1,ω 2 ) at the soure defined by Eqs. (16) and (17). Using Eq. (29), we alulate the oinidene probability for the two-photon spetrum (71) (see Appendix B) β2 P = 1 2 {1 1 B [os(2θ) 1 e 2 ( 2+β 2 L2 + z 2 )( σ ) e 1 2 ( L+ z)2 ( σ ) e 1 2 ( L z)2 ( σ )2 ]}, (73a) B =1+os(2θ)exp[ 1+β2 2+β 2 L2 ( σ )2 ], (73b) where β σ p /σ and z = z 2 z 1. The three terms in the square brakets of Eq. (73a) ontribute to the interferene ourring mainly at the three positions of beamsplitter: the first term for z = 0 and the last two terms for z = ± L. Similar to Eq. (49), the oherent length of the single-photon beam /σ defines the width of the interferene dip (or peak) so that only when L is larger than /σ the dips an be apart in spae. In the first term, the phase 2θ may affet the interferene manners, CI, ACI or the absene of interferene, whereas in the last two terms it shows only the CI effet. However, to show a signifiant interferene effet at the balaned position, it should satisfy the ondition 2+β L < 2 β 2 σ = 2+β 2, (74) σ p where /σ p is the oherent length for the pump beam. Sine σ p is related to two-photon entanglement, /σ p is also alled the two-photon oherent length. For σ p << σ (β <<1), that is the two-photon oherent length is muh larger than the single-photon oherent length, it is possible that the optial path differene of two beams L exeeds the single-photon oherent length /σ, but ondition (74) is satisfied. This fat has been demonstrated experimentally in Ref. [9]. For the perfet phase mathing in SPDC, g(x) = δ(x) is set in Eq. (16), the oinidene probability an be alulated by Eq. (29) P = 1 2 {1 1 1+os2θ e [os 2θ 1 2 L2 ( σ e 1 2 z2 ( σ )2 + 1 )2 2 e 1 2 ( L+ z)2 ( σ ) e 1 2 ( L z)2 ( σ )2 ]}. (75) If L >>/σ, the above equation is approximately written as P 1 2 {1 os(2θ)e 1 2 z2 ( σ )2 1 2 e 1 2 ( L+ z)2 ( σ )2 1 2 e 1 2 ( L z)2 ( σ )2 ]}. (76) This result was obtained in the previous study [9]. (Note that Eq. (76) is approximately valid sine it gives P < 0 at z =0forθ = nπ.) At the balaned position, Equation (75) is simplified as P = (1 os 2θ)[1 e 1 2 L2 ( σ )2 ]. (77) 2[1 + os 2θ e 1 2 L2 ( σ )2 ] It shows that, for an ideal two-photon entanglement in frequeny, the perfet CI and ACI our by setting the phase θ = nπ and θ =(n + 1/2)π, respetively. This is onsistent with the symmetry of two-photon spetrum indiated by Eq. (72). To show the feature of the entanglement, we also apply this method to two independent single-photon wavepakets for omparison. In this ase, we set g(x) onstant in Eq. (16), and the two single-photon spetra are separable as C 1 (ν) =A 1 e ν2 /(2σ 2) e iνz1/ os(ν L/ + θ), (78) C 2 (ν) =A 2 e ν2 /(2σ 2) e iνz2/, 14

15 in whih the phase fator independent of frequeny is negleted. By using Eq. (46), we alulate the oinidene probability for the above two independent single-photon spetra (see Appendix C) P = 1 2 {1 1 2(1 + os 2θ e L2 ( σ )2 ) e iθ e 1 4 ( L+ z)2 ( σ )2 + e iθ e 1 4 ( L z)2 ( σ )2 2 } (79) = 1 2 {1 1 [os 2θ e 1 1+os2θ e L2 ( σ 2 ( L2 + z 2 )( σ )2 + 1 )2 2 e 1 2 ( L+ z)2 ( σ ) e 1 2 ( L z)2 ( σ )2 ]}. The first line of Eq. (79) shows learly P < 1/2. We note that the same results as Eqs. (75) and (79) an also be obtained from Eq. (73) by setting β 0andβ, respetively. In Figs. 3-5, we plot the oinidene probabilities for the three examples of two-photon spetra: the maximum two-photon entanglement desribed by g(x) =δ(x), the arbitrary entanglement desribed by Eq. (17) with σ p = σ, and the two independent single-photon wavepakets (78), whih are indiated by solid, dashed and dotted lines, respetively. Figures 3a and 3b show the oinidene probability versus the phase 2θ at the balaned position z 1 = z 2 for L(σ/) =1 and 3, respetively. It shows that the phase θ dominates the manners of interferene. For the maximum entanglement (solid line), the perfet CI and ACI our at the phase 2θ = 2nπ and (2n + 1)π, respetively, and it is independent of the normalized optial path differene L(σ/). For an arbitrary entanglement, however, both CI and ACI an our, but not perfet (dashed line). As for two independent single-photon wavepakets in the oherent range for single-photon ( L(σ/) = 1), the CI ours, but there is no ACI effet (dotted line). In experiments, it would be diffiult to measure the net oinidene probability due to a lower quantum effiieny. The urves in Fig. 3 are unable to witness ACI effet if there is not a referene for oinidene probability. Alternatively, one may san the position of beamsplitter to show the two-photon interferene. In Fig. 4, we plot the oinidene probability versus the normalized position of the beamsplitter z(σ/) for L(σ/) = 1. The referene of oinidene probability has been shown at the large z(σ/). In Fig. 4a, by hoosing the phase 2θ =(2n +1)π, the ACI effet has been shown at the balaned position, and it witnesses the two-photon entanglement of the input state. In Fig. 4b, 2θ =2nπ, the CI ours, and there is no signifiant differene for the three ases. In Fig. 4 (also in Fig. 5) for 2θ =(2n +1/2)π, however, the three urves oinide exatly, showing the CI. As a matter of fat, for 2θ =(2n +1/2)π, Eqs. (73), (75) and (79) beome idential P = 1 2 {1 1 2 [e 1 2 ( L+ z)2 ( σ )2 + e 1 2 ( L z)2 ( σ )2 ]}. (80) In this ase, the interferene is independent of photon entanglement evaluated by the bandwidth σ p of the pump beam. In Fig. 5, we set a larger L(σ/) =3, for whih the traveling path differene L of two photons is larger than the oherent length /σ of the single-photon beam. The two side-dips emerge approximately at the position z = L. Different from Fig. 4, for two independent single-photon wavepakets (shown by the dotted lines), the interferenes disappear at the balaned position for any value of phase θ. But the CI and ACI still our for the entangled two-photon wavepaket by hoosing proper phases. The interferene effet shown in Fig. 5 has been reported in Ref. [9], in whih the authors demonstrate that the single-photon wavepaket onept is not always appropriate for two-photon interferene measurements. Due to the fat that in the experiment two photons to be interfered are in entanglement the Feynman-type diagrams of biphoton amplitudes were applied in their theoretial analysis. In the present theory we show an uniform desription for a general wavepaket ontaining two photons, whether in entanglement or not. It has shown that the entangled twophoton wavepaket may behave in interferene manners similar to or different from the un-entangled one. But only ACI effet is the signature of two-photon entanglement. The various effets an be understood by the two-photon spetra whih are typially desribed by Eqs. (71), (69), (16) and (17). We plot the ontours for the envelope of the spetra by omitting the osillatory phase fator (i.e. setting z 1 = z 2 = 0 in Eq. (71)) for simpliity. Eah point in the spetral plane orresponds to a two-photon state with the amplitude C(ν 1,ν 2 ). In the ontour, pairs of points symmetri with respet to the diagonal ν 1 = ν 2 ontribute to interferene so that the topologial harateristi of ontour may illustrate the interferene manners. Figures 6 and 7 show the ontours of the spetra for the un-entangled (by setting g(x) = 1) and entangled (by setting σ p =(1/3)σ) wavepakets, respetively, in whih the bright and the dark with respet to the bakground indiate respetively the positive and negative values of amplitudes. In Figs. 6 and 7, the parameters are hosen as (a) L(σ/) =1andθ = nπ; (b) L(σ/) =1andθ =(n +1/2)π; () L(σ/) = 3andθ = nπ; (d) L(σ/) = 3andθ =(n + 1/2)π. For un-entangled two-photon wavepaket, in Fig. 6, the ontours of spetra are symmetri with respet to the Cartesian axis, but not to the diagonal. As for entangled two-photon wavepaket, the ontours in Fig. 7 show approximate symmetry with respet to the diagonal: the symmetri for Figs. 7a and 7 and the anti-symmetri for Figs. 7b and 7d. Case II : We disuss two examples: one is the polarization entangled two-photon wavepaket desribed by Eq. (24), and the other one onsists of two independent single-photon wavepakets being in two orthogonally polarized 15