Bounds for the Distance Dependence of Correlation Functions of Entangled Photons in Waveguides
Andrei Khrennikov, Borje Nilsson, Sven Nordebo, Igor Volovich
aa r X i v : . [ qu a n t - ph ] M a y Bounds for the Distance Dependence ofCorrelation Functions of Entangled Photons inWaveguides
A. Khrennikov, B. Nilsson, S. NordeboInternational Centre for Mathematical Modellingin Physics and Cognitive SciencesSchool of Computer Science, Physics and MathematicsLinnæus University, SE-3159 V¨axj¨o, SwedenI.V. VolovichSteklov Mathematical InstituteRussian Academy of SciencesGubkin St. 8, 117966, GSP-1, Moscow, RussiaDecember 7, 2018
Abstract
The distance dependence of the probability of observing two pho-tons in a waveguide is investigated. The Glauber correlation functionsof the entangled photons in waveguides are considered and the spatialand temporal dependence of the correlation functions is evaluated.We derive upper bounds to the distance dependence of the probabil-ity of observing two photons. These inequalities should be possible toobserve in experiments.
The transmission of light in waveguides, in particular its quantum properties,is a topic of great interest in optics. Investigations of quantum correlations1nd entanglement among photons have been in focus of in the foundationsof quantum theory and its applications to quantum information science andmetrology.With the emergence of quantum communication links over long distances[1, 2, 3, 4], there is a need for a detailed study of the dependence of correla-tions and entanglement among photons on distance.Since direct interactions between photons in free space are extremelyweak, generation of correlated photons generally requires nonlinear mediasuch as the parametric down conversion. Recently, studies of two-photonscattering from a two-level system inside a one-dimensional waveguide havereported various features of photon correlation [5, 6, ? , 7]. In particular aformal scattering theory to study multi-photon transport in a waveguide wasemployed [8]. We also mention a general model describing nonlinear effectsin propagation to and from the system in the quantum state, based on one-dimensinal model of field-atom interaction [9], see even [10]. It describesspatiotemporal quantum coherence for the case of spontaneous emission froma single excited atom. In [5] this model was applied to the two-photon inputwave packets. This field of research, spatiotemporal behavior of correlationsof two photons propagating in nonlinear media, is closely related to studieson nonlinear response of a single atom to an input of two photons, e.g., froma single photon source [11]. This response can be observed in the correlationsbetween the two output photons. Here it is also very important to understandspatiotemporal dependence of correlations.There are thus sound reasons to study how correlation functions of entan-gled photons in hollow waveguides behave in space and time. The physicalmechanism to model is dispersion that spreads pulses in space and time caus-ing attenuation with distance. The spreading limits also the bit rate for agiven waveguide length because of mixing of pulses. Repeaters can be usedat some length intervals, which cause higher costs and problems with pre-serving the quantum state through the repeater. Information on space andtime properties of the correlation functions is thus of engineering interest.Such information is provided in the current paper by modelling the effect ofmodal dispersion.The study is also a preparation for more elaborate models, including ma-terial dispersion in hollow waveguides and fibres, cf. e.g. [12], [13]. Todescribe the situation, a brief background is presented [14] on optic fibres.For a fibre, it is possible to purify the material to the extent that losses fromscattering from the impurities can be neglected in some wavelength bands.2xamples of such bands are the 1.3 µ m and 1.55 µ m bands. The dispersioncan be reduced to a negligible level for a certain wavelength within such aband, and a low dispersion can then be reached for a narrow wavelengthband. Such a reduction of dispersion can be reached [14] with a dispersioncompensator which is a special fibre with tuned length, after the transmis-sion fibre, having the opposite dispersion to the transmission fibre. Anothermethod [14] is dispersion shift, i.e., a choice of the transversal fibre dimen-sions. The chromatic dispersion, defined as the combination of modal andmaterial dispersion, can be made to vanish for both methods at the designwavelength.It is worth mentioning the existence of repeaters for classical optic fibresusing optical rather than previously used electrical methods [15] and fibreswitches that preserve the quantum state of the photon [16].We shall study the asymptotic behaviour of the Glauber correlation func-tions for the entangled states of two photons in waveguides and show theirvanishing for large distances. We estimate the rate of decrease of correlationsand present upper bounds for the correlation functions.To estimate the correlation function we shall use some results on theproperties of solutions of the (1+1)-dimensional Klein-Gordon equation anal-ogously used in the Haag-Ruelle scattering theory.We prove that the probability density P ( z , t , z , t ) observing one pho-ton at point z along the waveguide at time t and another photon at point z at time t satisfies the following inequality P ( z , t , z , t ) ≤ C ( t + | t | )( t + | t | ) (1)for some t and all z , t , z , t .The bound (1) and the bounds (26), (27) (see below) should be possibleto observe in experiments.Note that the decreasing of the correlations for entangled states in emptyspace was found in [ ? ], see also discussion in [ ? ], [ ? ]. In this paper we haveconsidered the waveguides and found the universal bound for the correlations,see also [ ? ] for preliminary considerations. Let E j ( r , t ) be the j -th component ( j = 1 , ,
3) of the electric field operatorat the space time point r , t . The operator can be written as the sum of the3ositive and negative frequency parts: E j ( r , t ) = E (+) j ( r , t ) + E ( − ) j ( r , t ) . Theprobabilities of photo detection are given by Glauber‘s formulas, [17]. In par-ticular, the probability that a state ψ of the radiation field will lead to thedetection at time t of a photon with the polarization along the direction j bya detector atom placed at point r is proportional to the first-order correlationfunction P ψ ( r , t, j ) = h ψ | E ( − ) j ( r , t ) E (+) j ( r , t ) | ψ i . The joint probability of ob-serving one photo ionization with polarization j at point r at time between t and t + dt and another one with polarization j at point r between t and t + dt with t ≤ t is proportional to P ψ ( r , t , j ; r , t , j ) dt dt , wherethe second-order correlation function is P ψ ( r , t , j ; r , t , j ) = h ψ | E ( − ) j ( r , t ) E ( − ) j ( r , t ) E (+) j ( r , t ) E (+) j ( r , t ) | ψ i (2)The waveguide is a hollow conducting cylindrical tube Γ ⊂ R alongthe z axis with boundary surface S and with a cross section Ω with thebounding curve ∂ Ω in the xy -plane. It will be assumed that the walls haveinfinite conductivity. Appropriate boundary conditions are posed: E t | S =0 , H n | S = 0 , where E t is the component of electric field E tangential to theboundary of the waveguide and H n is the component of magnetic field H normal to the boundary.It is well known that in the interior of the waveguide the solutions of theMaxwell equations without sources can be divided into two sets of solutions,the so called T M modes with H z = 0 and the T E modes with E z = 0 [19, 20].A general solution is a linear combination of the T M and
T E modes.The general solution of the Maxwell equations in the waveguide can bewritten as follows [18]. Let ϕ nν ( z, t ) be any function satisfying the Klein-Gordon equation ( ∂ ∂t − ∂ ∂z + m nν ) ϕ nν ( z, t ) = 0 . (3)Here n = 1 , , ... and ν = T M or T E . We define for ν = T M E nν ( r , t ) = e nν ( x, y ) m − nν ∂∂z ϕ nν ( z, t ) (4)+ n z m nν v n ( x, y ) ϕ nν ( z, t ) , H nν ( r , t ) = − h nν ( x, y ) m − nν ∂∂t ϕ nν ( z, t )where v n is the solution of the eigenvalue problem4 ∂ ∂x + ∂ ∂y + m n ) v n ( x, y ) = 0 , ( x, y ) ∈ Ω , (5) v n | ∂ Ω = 0with the properties Z Ω v n ( x, y ) v n ′ ( x, y ) dxdy = δ nn ′ , (6) X n v n ( x, y ) v n ( x ′ , y ′ ) = δ ( x − x ′ ) δ ( y − y ′ ) , (7)where m n > ν = T M e nν ( x, y ) = ∇ T v n ( x, y ) , (8) h nν ( x, y ) = n z × ∇ T v n ( x, y ) = n z × e nν ( x, y ) . E nν ( r , t ) and H nν ( r , t ) are defined in an analogous manner [ ? ] for ν = T E .The general solution of the Maxwell equations in the waveguide can nowbe written in the form E ( r , t ) = X nν E nν ( r , t ) , H ( r , t ) = r ǫ µ X nν H nν ( r , t ) . (9)We write the solution of the Klein-Gordon Eq. (3) in the form ϕ n ( z, t ) = Z dk p πω n ( k ) ( a + n ( k ) e iω n ( k ) t − ikz (10)+ a n ( k ) e − iω n ( k ) t + ikz ) , where ω n ( k ) = p k + m n , and quantize it by taking a n ( k ) , a + n ( k ) as theannihilation and creation operators.Now the quantum electromagnetic field in the waveguide is reduced to aset of massive (1+1)-dimensional Klein-Gordon fields. Let us consider oneof the modes. We define a one particle state | ψ i = Z g ( k ) a † k | i , (11)5here | i is the Fock vacuum. The probability density to detect the photonat the point z along the waveguide at time t is proportional to P ( z, t ) = h ψ | ϕ ( − ) ( z, t ) ϕ (+) ( z, t ) | ψ i , (12)where ϕ (+) ( z, t ) = 1(2 π ) / Z R dk √ ω k a k e − iω k t + ikz , ϕ ( − ) ( z, t ) = h.c., (13)and ω k = √ k + m , m > P ( z, t ) = | A ( z, t ) | , (14)where A ( z, t ) = h | ϕ (+) ( z, t ) | ψ i = Z dkg ( k ) e − iω k t + ikz √ πω k . (15)The function A ( z, t ) is a solution of the Klein-Gordon equation. Let usconsider the question of how the solution behaves in the limit of large t in aframe of reference moving with a constant velocity v ≤ V <
1. In this frame,corresponding to the time x (1 − v ) /v after the wave front at t = x , the fieldis given by A ( vt, t ) = Z dkg ( k ) e − it ( ω k − kv ) √ πω k . (16)For sufficiently smooth function g ( k ) one can use the stationary phase methodto get A ( vt, t ) = g ( k ) e − it ( ω k − k v ) − iπ/ √ πω k s πtω ′′ ( k ) + O ( 1 t ) . (17)Here k is the solution of the equation ω ′ ( k ) = v , i.e. k = mv/ √ − v andone has ω ′′ ( k ) >
0. For the probability P ( z, t ) we obtain P ( vt, t ) = 1 t | g ( k ) | ω k ω ′′ ( k ) + O ( 1 t / ) . (18)More elaborated results on asymptotic expansions of the solution to theKlein-Gordon equation are given by H¨ormander [22].6 The entangled photon correlation functionwith distance dependence
Now we define a two particle entangled state (biphoton) | ψ i = Z f ( k , k ) a † k a † k | i , (19)where | i is the Fock vacuum and f ( k , k ) is the two-photon wave functionwhich is a symmetric function, f ( k , k ) = f ( k , k ) because we deal withbosons.The probability to detect one particle at the point z along the waveguideat time t and another particle at the space point z at time t is proportionalto P ( z , t , z , t ) = h ψ | ϕ ( − ) ( z , t ) ϕ ( − ) ( z , t ) ϕ (+) ( z , t ) ϕ (+) ( z , t ) | ψ i . (20)The expression for P ( z , t , z , t ) (20) can be written as P ( z , t , z , t ) = | A ( z , t , z , t ) | , (21)where A ( z , t , z , t ) = h | ϕ (+) ( z , t ) ϕ (+) ( z , t ) | ψ i = Z dk dk (22) { e − iω k t + ik z √ πω k e − iω k t + ik z √ πω k f ( k , k ) + ( k ↔ k ) } . Note that the function A ( z , t , z , t ) satisfies the Klein-Gordon equationwith respect to z , t and z , t .Let us suppose that one of the photons is observed in a frame of referencemoving with velocity v and another photon is observed in a frame of referencemoving with velocity v . By using the stationary phase method we obtainfor large t and t : A ( z , t , z , t ) = s πt ω ′′ ( k ) s πt ω ′′ ( k ) f ( k , k ) 12 √ πω k √ πω k e − iπ/ (23) { e − it ( ω k − ik v ) e − it ( ω k − ik v ) + ( k ↔ k ) } . k = mv / p − v , k = mv / p − v , z = v t and z = v t .Therefore P ( z , t , z , t ) = | f ( k , k ) | t t ω ′′ ( k ) ω ′′ ( k ) ω k ω k . (24) |{ e − it ( ω k − ik v ) e − it ( ω k − ik v ) + ( k ↔ k ) }| It is interesting to see the difference between the entangled wave function f ( k , k ) and the separable one by looking to it with an explicitly indicateddependence on the spacetime coordinates: f ( k , k ) = f ( m z t / s − z t , m z t / s − z t ) (25)Let the wave function of two photons f ( k , k ) in a waveguide be a smoothfast decreasing function. Then the probability of observing two photonsin the waveguide should satisfy the following bounds. For any n , n =0 , , , , ... there exist constants C n n such that for | z | ≥ | t | , | z | ≥ | t | , (26)one has P ( z , t , z , t ) ≤ C n n (1 + | z | ) n (1 + | z | ) n . (27)Furthermore there exists a constant C such that P ( z , t , z , t ) ≤ C | t || t | , (28)for all z , t , z , t . An asymptotic estimate for C , valid for large t or t , isprovided by (24).An explicit expression for the wave function of the biphotons in a specialcase is given in [21]: f ( k , k ) = ik f P ( k + k ) p k k ( k + k ) , (29)where f P ( k + k ) is a Gaussian function describing the pumping photons.By using this form of the wave function we obtain the asymptotic formulafor the probability in this special case.8o conclude, the main result of this paper is the bounds to the probabilitydensity (1) and (26), (27) which, in principle, should be possible to test inexperiments. Acknowledgements . One of the authors (I. Volovich) would like tothank the International Centre for Mathematical Modelling in Physics andCognitive Sciences for the support during his visit to Linnæus University.
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