Gaussian modeling and Schmidt modes of SPDS biphoton states
aa r X i v : . [ qu a n t - ph ] J un Gaussian modeling and Schmidt modes of SPDS biphoton states
M V Fedorov ∗ , Yu M Mikhailova , and P A Volkov A.M. Prokhorov General Physics Institute,Russian Academy of Sciences,38 Vavilov st., Moscow, 119991, Russia Max-Planck-Institut f¨ur Quantenoptik,Hans-Kopfermann-Strasse 1,D-85748, Garching, Germany ∗ E-mail: [email protected]
A double-Gaussian model and the Schmidt modes are found for the biphoton wavefunction characterizing spontaneous parametric down-conversion with the degeneratecollinear phase-matching of the type I and with a pulsed pump. The obtained resultsare valid for all durations of the pump pulses, short, long and intermediately long.
PACS numbers: 03.67.Bg, 03.67.Mn, 42.65.Lm
I. INTRODUCTION
Biphoton states generated in Spontaneous Parametric Down-Conversion (SPDC) can behighly entangled in continuous variables characterizing signal and idler photons, such asangular variables or frequencies [1, 2]. For SPDC processes with the type-I phase matchingthe degree of entanglement is characterized by values of the Schmidt number K up to300 in the case of short pump pulses. Such high entanglement reflects a high degree ofcorrelations accumulated in the biphoton state and, in principle such highly-correlated statescan be useful for goals of quantum information and quantum cryptography. But, of course,there is a big gap between finding that the degree of entanglement is very high and itsapplication in practice. We think that an important step in the direction of closing this gapis finding the Schmidt modes and eigenvalues of the reduced density matrix rather than onlythe Schmidt number K . This task is not quite trivial because Schmidt modes are knownonly for double-Gaussian wave functions [3, 4], whereas in some cases even a possibility ofmodeling the biphoton wave function by a double-Gaussian function is problematic. Wewill show below that in such cases a reasonable way of derivation consists in calculatingfirst the reduced density matrix (even if approximately) and, then, modeling the latter by adouble-Gaussian function. Eigenfunctions of such a model double-Gaussian density matrixare just the Schmidt modes. By continuing this procedure, and using general connectionsbetween the density matrix and wave function, we show that the double-Gaussian model ofthe spectral wave function exists, and it’s found explicitly.The paper is structured in the following way. In the next Section we give a brief overviewof the relation between the double-Gaussian wave functions and density matrices and deriva-tion of their Schmidt modes. In Section 3 we summarize the main features of the spectralbiphoton wave function for the degenerate collinear SPDC process with a pulsed pump. InSection 4 we describe how the Gaussian modeling can be applied for finding Schmidt modesin the case of long pump pulses. In Section 5 we describe Gaussian modeling of the re-duced density matrix and derivation of the Schmidt modes in the case of short pump pulses.In Section 6 we formulate some generalizations of the derived specific results and suggesta single double-Gaussian model of the biphoton spectral wave function valid for arbitrarypump-pulse durations. II. REDUCED DENSITY MATRIX AND SCHMIDT MODES FORDOUBLE-GAUSSIAN BIPARTITE WAVE FUNCTIONS
Let x and x be some continuous variables of particles “1” and “2” and Ψ( x , x ) bea bipartite wave function. Lets us consider here only the case of symmetric bipartite wavefunctions for which Ψ( x , x ) = Ψ( x , x ). The most general form of the double-Gaussianwave function satisfying this condition is given byΨ( x , x ) = r πab exp (cid:20) − ( x + x ) a (cid:21) exp (cid:20) − ( x − x ) b (cid:21) , (1)where a and b are parameters related to widths of distributions in x , determined by eachGaussian factor in Equation (1), a > b >
0. Alternatively, the same wave function canbe presented in a different form, convenient for determining the double-Gaussian Schmidtmodes [4] Ψ( x , x ) = α √ π exp (cid:26) α (cid:20) − µ − µ ) ( x + x ) + 2 µ (1 − µ ) x x (cid:21)(cid:27) . (2)The parameters a and b are related to µ and α by equations µ = a − ba + b , α = r ab and a = √ α r µ − µ , b = √ α r − µ µ , (3) − < µ < x , x ), are given by ρ ( x , x ; x ′ , x ′ ) = Ψ( x , x )Ψ( x ′ , x ′ ) and ρ r ( x , x ′ ) = Z dx ρ ( x , x ; x ′ , x ) . (4)After integration, the reduced density matrix can be presented in the same forms as describedabove for the wave function, (1) and (2) ρ r ( x , x ′ ) = 1 e a r π exp (cid:20) − ( x + x ′ ) e a (cid:21) exp (cid:20) − ( x − x ′ ) e b (cid:21) = e α s π e µ − e µ exp (cid:26)e α (cid:20) − e µ − e µ ) ( x + x ′ ) + 2 e µ (1 − e µ ) x x ′ (cid:21)(cid:27) . (5)The pairs of parameters e a, e b and e α, e µ are related to each other by the same formulas as a, b and α, µ (3): e µ = e a − e b e a + e b , e α = r e a e b and e a = √ e α s e µ − e µ , e b = √ e α s − e µ e µ . (6)Moreover, with a simple algebra we find a series of formulas relating the parameters char-acterizing the density matrix with the parameters characterizing the wave function e α = α, e µ = µ , e a = r a + b , e b = ab r a + b , (7)and a = r e a (cid:16)pe a + e b − pe a − e b (cid:17) b = r e a (cid:16)pe a + e b + pe a − e b (cid:17) . (8)The Schmidt modes { ψ n ( x ) , χ n ( x ) } are defined as eigenfunctions of the reduced densitymatrices ρ r ( x , x ′ ) and ρ r ( x , x ′ ). For symmetric wave functions Ψ( x , x ) two functionsin the Schmidt-mode pairs are identical, χ n ( x ) = ψ n ( x ). In a general case, to find Schmidtmodes one has to solve the integral equation Z dx ′ ρ r ( x , x ′ ) ψ n ( x ′ ) = λ n ψ n ( x ) , (9)where λ n are eigenvalues of the reduced density matrix. The Schmidt modes diagonalize thereduced density matrix and, hence, the latter can be presented as the sum of products ofSchmidt modes ρ r ( x , x ′ ) = ∞ X n =0 λ n ψ n ( x ) ψ n ( x ′ ) . (10)According to the Schmidt theorem, the wave function Ψ( x , x ) also can be expanded in aseries of products of the Schmidt modesΨ( x , x ) = ∞ X n =0 p λ n ψ n ( x ) ψ n ( x ) . (11)For the double-Gaussian wave functions analytical expressions for the Schmidt modes areknown and they follow from the formula [4]exp (cid:26) α (cid:20) − µ − µ ) ( x + x ) + 2 µ (1 − µ ) x x (cid:21)(cid:27) = √ π p − µ ∞ X n =0 µ n ϕ n ( αx ) ϕ n ( αx ) , (12)where ϕ n are the oscillator wave functions ϕ n ( x ) = (2 n n ! √ π ) − / e − x / H n ( x ), H n ( x ) are theHermite polynomials. The Schmidt modes and eigenvalues of the reduced density matrixare easily found now from Equations (1), (2), (3), and (12) ψ n ( x , ) = √ α ϕ n ( αx , )= (cid:18) ab (cid:19) / ϕ n √ x , √ ab ! = (cid:18) e a e b (cid:19) / ϕ n √ x , pe a e b ! (13)and p λ n = p − µ µ n = 2 √ aba + b (cid:18) a − ba + b (cid:19) n = ( − n s e b e a + e b e a − e b e a + e b ! n/ ,λ n = 4 ab ( a − b ) n ( a + b ) n +1) = 2 e b e a + e b e a − e b e a + e b ! n . (14)The Schmidt number, characterizing the degree of entanglement in the state Ψ( x , x ) isdefined as K = ∞ X n =0 λ n ! − = 1 + µ − µ = 1 + e µ − e µ = a + b ab = e a e b , (15)which agrees with more general expressions of Refs. [4, 6]. III. SPECTRAL BIPHOTON WAVE FUNCTION
Let us remind now the main features of the biphoton spectral wave function for SPDCwith the type-I degenerate collinear phase matching and with a pulsed pump. The maingeneral expression has the form [2, 5]Ψ( ν , ν ) ∝ exp (cid:18) − ( ν + ν ) τ (cid:19) × sinc (cid:26) L c (cid:20) A ( ν + ν ) − B ( ν − ν ) ω (cid:21)(cid:27) , (16)where τ is the pump-pulse duration, L is the length of a crystal, ν and ν are deviations offrequencies of the signal and idler photons ω , from the central frequencies ω (0)1 = ω (0)2 = ω / | ν , | ≪ ω , ω is the central frequency of the pump spectrum, A and B are thetemporal walk-off and dispersion constants A = c (cid:16) k ′ p ( ω ) (cid:12)(cid:12) ω = ω − k ′ ( ω ) | ω = ω / (cid:17) = c ( p ) g − ( o ) g ! ,B = c ω k ′′ ( ω ) | ω = ω / , (17)v ( p ) g and v ( o ) g are the group velocities of the pump and ordinary waves, and k and k p are thewave vectors of signal and pump photons.The first term on the right-hand side of Equation (16) is the pump spectral field strengthtaken in the Gaussian form. The second term is the sinc-function characterizing the crystaland propagation in it of the pump and signle/idler photons. The argument of the sinc-function contains both linear and quadratic terms in ν , . As we assume that | ν , | ≪ ω ,in principle, the quadratic term is much smaller than the linear one. Nevertheless, thequadratic term can never be dropped because it is crucially important for determining thefinite-width single particle spectrum [without the quadratic term both factors on the right-hand side of Equation (16) depend only on ν + ν , which makes the integral R dν | Ψ( ν , ν ) | independent of ν and single-particle spectrum infinitely wide]. As for the linear term in theargument of the sinc-function it can be efficiently eliminated in the case of long pump pulsesbut cannot be dropped if pulses are short. The control parameter separating the regions ofshort and long pulses is given by [1] η = ∆ ν ∆ ν ≈ c τAL = 2 τL/ v ( p ) g − L/ v ( o ) g , (18)i.e., it’s equal to the ratio of the double pump-pulse duration to the difference of timesrequired for the the pump and idler/signal photons for traversing all the crystal. Pumppulses are short if η ≪ η ≫ η ∼ τ ∼ K and the parameter R defined as the ratio of the single- to coincidencespectral widths of the corresponding photon distributions, R = ∆ ν ( s )1 / ∆ ν ( c )1 [2]. Both pa- FIG. 1:
Entanglement parameters: R , calculated analytically [2], and K , calculated numerically [7], for acrystal LiIO of the length L = 0 . rameters were calculated analytically as function of the pump-pulse duration τ and werefound to be rather close to each other. Also, for the same wave function (16), the Schmidtnumber K ( τ ) was calculated numerically [7], and the result was found to be in a very goodagreement with the analytical function R ( τ ) of Ref. [2] (see Fig. 1). The degree of spectralentanglement was found to be rather high at all values of the pump-pulse duration andespecially high in the cases of sufficiently short and long pulses (far from the minimumwhich was found to occur at η ∼ τ ∼ η ≪
1. They were found in the work [2] to be given by∆ ω ( c )shotrt = 5 . cAL , ∆ ω ( s )shotrt = r A ln(2) ω Bτ . (19)Note finally that a very small difference between the Schmidt number K ( τ ) (exact andnumerical, [7]) and the parameter R ( τ ) (analytical, [2]) raises often a question why are theyso close to each other whereas the wave function (16) is rather significantly non-double-Gaussian? In this work we give an answer: actually, the wave function (16) is “hiddenlydouble-Gaussian”. And we make this hidden double-Gaussian character of the wave function(16) explicit by finding the double-Gaussian models for the regions of short and long pumppulses and by generalizing these findings for all values of the pump pulse duration (Section6). IV. GAUSSIAN MODELING AND SCHMIDT MODES IN THE CASE OFLONG PUMP PULSES
In the case of long pump pulses, η ≫
1, the Gaussian function on the right-hand sideof Equation (16) is much narrower the sinc-function. For this reason ν ≈ − ν , and thiseliminates efficiently the term linear in ν and ν in the argument of the sinc-function andreduces Ψ( ν , ν ) to the formΨ( ν , ν ) ∝ exp (cid:20) − ( ν + ν ) τ (cid:21) sinc (cid:20) BL cω ( ν − ν ) (cid:21) . (20)The wave function of a similar form was considered earlier in the analysis of angular en-tanglement of SPDC biphoton states [8], and then some first Schmidt modes were foundnumerically. On the other hand, the form of the sinc-function in Equation (20) is alreadyvery convenient for Gaussian modeling. Parameters of the corresponding Gaussian functioncan be chosen from the coincidence condition of the Full Widths at Half Maxima (FWHM) ofthe squared functions sinc( u ) and exp( − γu ) which gives γ = 0 . (cid:20) BL cω ( ν − ν ) (cid:21) → exp (cid:20) − . B L c ω ( ν − ν ) (cid:21) , (21)reduces the wave function of Equation (20) to the double-Gaussian form (1)Ψ( ν , ν ) = r πab exp (cid:20) − ( ν + ν ) τ a (cid:21) exp (cid:20) − ( ν − ν ) b (cid:21) , (22)with the parameters a and b given by a = 2 √ ln 2 τ and b = r c ω . BL = c r π . BLλ , (23)where λ is the central wavelength of the pump. Note, that owing to the assumption η ≫ L ≫ λ , the ratio of the parameters b and a is large: b/a ∼ η p L/λ ≫
1. By using Equations (3) we find also other parameters of the double-Gaussian wave function, µ and α : µ long = − b − ab + a ≈ − ab = − cτ r .
249 ln 2
BLλ π (24)and α long = r ab = r τc (cid:18) . BLλ π ln 2 (cid:19) / . (25)The Schmidt number (15) for long pump pulses appears to be given by K long = a + b ab ≈ b a = √ π √ .
249 ln 2 cτ √ λ L B ≈ . cτ √ λ L B . (26)This result coincides perfectly with the earlier derived expression for the parameter R in theapproximation of long pump pulses [2] R long = √ . π cτ √ λ L B ≈ . cτ √ λ L B . (27)Though the coefficients in Equations (26) and (27) look different, numerically they are seento be amazingly close to each other. Note that the parameter R long was found in the paper[2] without any Gaussian modeling and in the frame of a procedure absolutely different fromthat used above for the derivation of the Schmidt number K long .In terms of K long , Equations (24) and (25) for the parameters µ and α can be rewrittenas µ long = ≈ − K long and α long ≈ a p K long = τ p K long ln 2 . (28)Again in terms of K long , the Schmidt modes (13) and eigenvalues of the reduced densitymatrix (14) take the form ψ n ( ν , ) = p τ / K long ln 2) / ϕ n " τ ν , p K long ln 2 (29)and p λ n ≈ ( − n s K long (cid:18) − K long (cid:19) n , λ n ≈ K long (cid:18) − K long (cid:19) n . (30)The derivations of this section were based on a direct Gaussian modeling of the wavefunction, without explicit consideration of the density matrix. As we will see in the followingsection, in the case of short pump pulses such simple procedure does not work, and to applyGaussian modeling, one has to calculate first the reduced density matrix. This indicates arather important qualitative difference between the cases of long and short pump pulses. V. SCHMIDT MODES IN THE CASE OF SHORT PUMP PULSES
In the case of short pump pulses we cannot use anymore the above-described procedureof a direct Gaussian modeling of the wave function (16). Indeed, in this case the pumpspectral function is much wider than the sinc-function and, hence, the linear term in thesinc-function’s argument cannot be eliminated. As for the quadratic term, it cannot bedropped too because it is crucially important for appropriate determination of the single-particle spectrum. As for the sinc-function of Equation (16) with its full argument, it’srather difficult to see directly any possibility of its replacement by any Gaussian function.For example, a simple substitution of sinc( u ) by e − u gives a super-Gaussian function withquadratic and both third- and forth-power terms in ν , under the symbol of exponent.This is not a simplification at all and does not provide a way for analytical derivation ofthe Schmidt modes. So, we will apply here a somewhat subtler method. At first, we willcalculate approximately the reduced density matrix corresponding to the wave function (16).The reduced density matrix found in such a way appears to be much more appropriate fordouble-Gaussian modeling than the wave function. For the Gaussian model of the reduceddensity matrix, its eigenfunctions (Schmidt modes), eigenvalues and the Schmidt numberare easily found. After this, with the help of equations of Section II, we will find parametersof the effective double-Gaussian model for the original wave function (16), which appears tobe very non-trivial and hardly guessable in advance.0 A. Density matrix
In a general form the density matrix corresponding to the wave function (16) is given by ρ r ( ν , ν ′ ) ∝ Z dν exp (cid:26) − [( ν + ν ) + ( ν ′ + ν ) ] τ (cid:27) × sinc (cid:26) L c (cid:20) A ( ν + ν ) − B ( ν − ν ) ω (cid:21)(cid:27) × sinc (cid:26) L c (cid:20) A ( ν ′ + ν ) − B ( ν ′ − ν ) ω (cid:21)(cid:27) . (31)Let us make in this equation a series of approximations similar to those made in Section V A of the work [2] and based, actually, on a single physical reason: under the assumptionabout short duration of pump pulses, η ≪
1, the Gaussian factor in Equation (31), as afunction of either ν , ν ′ , or ν , is much smoother than both sinc-functions. Compared to thelocalization region of the Gaussian function ∼ /τ both the difference | ν − ν ′ | and sums | ν + ν | , | ν + ν ′ | are very small: they are on the order of c/AL , as they are determined bythe linear terms in the arguments of the sinc-functions. Owing to this, we can approximate ν by − ν and by − ν ′ in small quadratic terms in the arguments of the first and secondsinc-functions in Equation (31), thus linearizing their arguments with respect to ν : ρ r ( ν , ν ′ ) ∝ Z dν exp (cid:26) − [( ν + ν ) + ( ν ′ + ν ) ] τ (cid:27) × sinc (cid:26) AL c [ ν − ¯ ν ( ν )] (cid:27) sinc (cid:26) AL c [ ν − ¯ ν ( ν ′ )] (cid:27) , (32)where ¯ ν ( ν ) = − ν + 4 Bν Aω and ¯ ν ( ν ′ ) = − ν ′ + 4 Bν ′ Aω . (33)The slowly changing exponential function in Equation (32) can be taken out of the integral,for example, at ν = [ ¯ ν ( ν ) + ¯ ν ( ν ′ )] to giveexp (cid:26) − τ ln 2 (cid:20) ( ν − ν ′ )
16 + B ( ν + ν ′ ) A ω (cid:21)(cid:27) ≈ exp (cid:20) − τ B ( ν + ν ′ ) A ω ln 2 (cid:21) , (34)where in the last approximate expression we have dropped all small terms proportional to( ν − ν ′ ) . In this approximation, instead of ν = [ ¯ ν ( ν ) + ¯ ν ( ν ′ )], we could choose forevaluation of the exponential factor either ν = ¯ ν ( ν ) or ν = ¯ ν ( ν ′ ). In all cases the finalresult would be the same as given by the last expression in Equation (34).1The remaining integral of the product of two sinc-functions is calculated with the help ofEquation (37) of the paper [2] Z dν sinc (cid:26) AL c [ ν − ¯ ν ( ν )] (cid:27) sinc (cid:26) AL c [ ν − ¯ ν ( ν ′ )] (cid:27) = π sinc (cid:26) AL c [ ¯ ν ( ν ) − ¯ ν ( ν ′ )] (cid:27) ≈ π sinc (cid:20) AL c ( ν − ν ′ ) (cid:21) . (35)By combining Equations (34) and (35) together, we find the following representation for thereduced density matrix ρ r ( ν , ν ′ ) ∝ exp (cid:20) − τ B ( ν + ν ′ ) A ω ln 2 (cid:21) sinc (cid:20) AL c ( ν − ν ′ ) (cid:21) . (36)Note that in the last approximate expression of Equation (35) all quadratic terms in theargument of the sinc-function are dropped. Though such approximation could not be usedfor the wave function (16), it appears to be well applicable for the density matrix (35) becausethe quadratic terms in the argument of the sinc function are small and, most important,because the linear term is proportional to the difference of frequencies ν − ν ′ rather thantheir sum, whereas the exponential factor in Equation (36) depends on ν + ν ′ . This featureof the density matrix contrasts to that of the wave function (16) in which both the pumpspectrum and the linear term in the argument of the sinc-function depend on the sum offrequencies ν + ν . B. Gaussian modeling and Schmidt modes for short pump pulses
Both factors in the reduced density matrix (36) can be modeled by Gaussian functionsby means of substitutionsexp (cid:20) − τ B ( ν + ν ′ ) A ω ln 2 (cid:21) → exp (cid:20) − γ τ B ( ν + ν ′ ) Aω √ ln 2 (cid:21) (37)and sinc (cid:20) AL c ( ν − ν ′ ) (cid:21) → exp (cid:20) − γ A L ( ν − ν ′ ) c (cid:21) , (38)where γ and γ are fitting factors to be defined below.With these substitutions the reduced density matrix (36) takes the double-Gaussian form(5) ρ r ( ν , ν ′ ) = 1 e a r π exp (cid:20) − ( ν + ν ′ ) e a (cid:21) exp (cid:20) − ( ν − ν ′ ) e b (cid:21) (39)2with e a = (ln 2) / s Aω γ τ B , e b = cAL r γ . (40)In terms of the control parameter η (18), roughly, e a ∼ e b s Lλ η ≫ e b, (41)because for short pulses η ≪ L ≫ λ .With the help of Equations (13), (14), and (15) we can find now other parameters of thedouble-Gaussian reduced density matrix ( e α and e µ ), as well as the Schmidt number K . Thelatter is given by K short = e a e b = (ln 2) / ALc s γ Aω γ τ B = A / √ B L √ λ cτ (ln 2) / r πγ γ . (42) C. Gaussian modeling of the original wave function
In accordance with the discussion of Section 2, there is a one-to-one correspondence be-tween the double-Gaussian reduced density matrix and the double-Gaussian wave function.This means that, as we have received the double-Gaussian representation (39) for the re-duced density matrix of the biphoton state arising in the degenerate collinear SPDC processwith the type-I phase matching in the limit of short pump pulses, the original wave function(16) also must be representable in the double-Gaussian form, even though it’s not evident inadvance. Parameters of the double-Gaussian model of the wave function can be found fromEquations (40) and (8), the latter of which can be simplified in the approximation e a ≫ e b togive a ≈ e b √ cAL √ γ , b ≈ √ e a = (ln 2) / s Aω γ τ B , (43)with the relation between a and b opposite to that occurring between e a and e b (41): a ∼ b r λ ηL ≪ b. (44)With a and b given by Equations (43), the double-Gaussian spectral wave function modelingthat of Equation (16) is given by Equation (1) (with x , substituted by ν , ). In theapproximation a ≪ b (44) the coincidence and single-particle spectra corresponding to this3wave function are determined by the curves dw ( c ) ( ν ) dν ∝ exp (cid:20) − ( ν + ν ) a (cid:21) ν =const (45)and dw ( s ) ( ν ) dν ∝ exp (cid:18) − ν b (cid:19) . (46)The coincidence and single-particle spectral widths are defined as FWHM of the curves (45)and (46) ∆ ν ( c )1 = 2 a √ ln 2 = 2 cAL s ln 2 γ , ∆ ν ( s )1 = b √ ln 2 = (ln 2) / s Aω γ τ B . (47)By comparing these formulas with the earlier derived analytical expressions for spectralwidths (19), we find that all functional dependences in these two sets of formulas are identi-cal. As for numerical coefficients, they can be made identical too if we choose appropriatelythe fitting parameters γ and γ in the Gaussian model substitutes (37) and (38) for twofactors in the reduced density matrix (36), (39). Found from these conditions parameters γ , are given by γ = √ ln 2 ≈ . , γ = 4 ln 2(5 . = ln 2(2 . ≈ . . (48)With these fitting parameters defined, we can write down now all final expressions for themodel double-Gaussian wave function, Schmidt number, Schmidt modes and eigenvalues ofthe reduced density matrix for the case of short pump pulses. So, Equations (43) for theparameters a and b become equal to a = 2 . √ ln 2 cAL = 3 . cAL , b = r Aω τ B , (49)and the double-Gaussian model (1) of the wave function (16) itself takes the formΨ ( G )short = ALπc r γ K short × exp (cid:20) − γ A L ( ν + ν ) c (cid:21) exp (cid:20) − γ τ B ( ν − ν ) Aω √ ln 2 (cid:21) ≈ . ALπc √ K short × exp (cid:20) − . A L ( ν + ν ) c (cid:21) exp (cid:20) − . τ B ( ν − ν ) Aω √ ln 2 (cid:21) . (50)Comparison of this expression with that of Equation (16) shows that they ar not alike at all.But they do describe a very similar behavior of the wave function Ψ( ν , ν ). Similarity and4differences between the expressions (16) and (50) are illustrated by two pictures of Figure2. In these pictures localization regions of the wave functions (16) and (50) are shown bythick solid lines. “Centers of mass” of these lines correspond to maxima (equal to unit) ofthe narrowest factors in the formulas for these wave functions. For the exact wave function(16) the narrowest factor is the sinc-function, which is maximal when its argument equalszero. This condition yields ν c . m . ( ν ) = ν + Aω B (cid:16) A − p A + 8 BAν /ω (cid:17) . (51)∆ ν ( c )shotrt = 5 . c/AL . For the model double-Gaussian wave function the narrowest part is FIG. 2:
Thick solid lines determine localization regions of the wave functions ( a ) (16) and ( b ) (50), dashedlines determine limitations of these regions along the thick solid lines; calculations for LiIO , pump wave-length λ = 400 nm and pump-pulse duration τ = 50 fs. the first exponential factor on the right-hand side of Equation (50) and, hence, ν c . m . ( ν ) = − ν . (52)Widths of the thick lines in Figure 2 are determined by the coincidence spectral widthfor short pump pulses (19), ∆ ν ( c )1 = 5 . c/AL , and, thus, the wave-function localizationregions are regions under the thick solid curves in the ( ν , ν )-map. Note, that in thepictures of Figure 2 widths of the thick solid curves are slightly increased compared to ∆ ν ( c )1 to distinguish clearer these curves from other lines in figures.Dashed lines in the pictures of Figure 2 determine limitations of the wave-function lo-calization regions in the longitudinal direction imposed by the remaining slowly varyingfactors in Equations (16) and (50) [the Gaussian pump spectral function in (16) and thesecond Gaussian factor on the right-hand side of Equation (50)]. Mathematically these lim-itations can be found from the condition that the corresponding squared Gaussian function5in Equations (16) and (50) are larger or equal to, e.g., 1/2, which gives ν c . m . ( ν ) ≤ ν dashed1 exact ( ν ) , ν dashed − ( ν ) ≤ ν c . m . ( ν ) ≤ ν dashed +1 Gauss ( ν ) , (53)where ν dashed1 exact ( ν ) and ν dashed ± ( ν ) are the functions determining location of the dashed lines,correspondingly, in Figures 2( a ) and 2( b ): ν dashed1 exact ( ν ) = − ν + 2 ln 2 τ , ν dashed ± ( ν ) = ν ± (ln 2) / r Aω . Bτ . (54)These formulas, as well as the pictures of Figure 2, show clearly that mechanisms, limitingthe localization regions of the exact (16) and model (50) wave functions are absolutelydifferent. But results are seen to be almost identical. Moreover, the only difference in theresulting localization regions in the pictures ( a ) and ( b ) of Figure 2 is a very small curvatureof the thick solid line in Figure 2( a ) missing in Figure 2( b ). This curvature arises owing tothe quadratic term in the argument of the sinc-function in wave function (16). As discussedabove, this term is crucially important for determining the single-particle spectrum andlimitation of the localization region of the exact wave function (16). In the model function(50) quadratic terms are missing. Moreover, even if taken into account in some way, suchterms are not expected to be important at all. All integral characteristics of the SPDCspectra are identical if calculated with the exact and model double-Gaussian wave functions.In particular, this is true for for the Schmidt number K and single-particle and coincidencespectral widths. As for the Schmidt modes and eigenvalues of the reduced density matrix, itwould be very interesting to calculate them numerically to compare with the above derivedformulas (58) and (60). We hope to be able making such comparison later.With the fitting parameters γ , determined by Equation (48) and the parameters a and b given by Equations (49), Equation (42) for the Schmidt number K short takes the form K short = b a = √ π ln 22 . A / √ B L √ λ cτ = 0 . A / √ B L √ λ cτ . (55)This expression coincides exactly with that of the work [2] for the parameter R in the regimeof short pulses.The reduced density matrix (39), after all substitutions, takes its final form ρ r, short ( ν , ν ′ ) = r BτπAω × exp (cid:20) − Bτ Aω ( ν + ν ′ ) (cid:21) exp (cid:20) − A L ln 2(5 . c ) ( ν − ν ′ ) (cid:21) . (56)6For the parameter µ , determining eigenvalues of the reduced matrix, we get formally thesame expression as in the case of long pump pulses, µ ≈ − ab , though with differentconstants a and b , which gives finally µ ≈ − K short = − . √ π ln 2 √ BA / √ λ cτL , (57)and eigenvalues of the reduced density matrix are given by λ n ≈ K short (cid:18) − K short (cid:19) n (58)with K short determined by Equation (55).The scaling factor α of the Schmidt modes (13) in the case of short pump pulses has theform α short = 1 a √ K short = ALc √ ln 22 . √ K short = 0 . ALc √ K short , (59)and the Schmidt modes are given by ψ n ( ν , ) = r ALc (ln 2) / √ .
78 ( K short ) / ϕ n ν , ALc √ ln 22 . √ K short ! ≈ . s ALc √ K short ϕ n (cid:18) . ν , ALc √ K short (cid:19) . (60) VI. GENERALIZATIONS
By comparing Equations (24) and (30) with (57) and (58), we find that they look identicalwith the only substitution: K long ⇋ K short . So, we can suggest the following simplestinterpolation rule for a transition between the regions of short and long pulses consisting inthe assumption that even in the intermediate region ( η ∼
1) the dependence of µ and λ n ofthe Schmidt number K remains the same as in the asymptotic regions η ≪ η ≫ µ ( τ ) = − K ( τ ) ,λ n ( τ ) = 2 K ( τ ) (cid:18) − K ( τ ) (cid:19) n ≈ K ( τ ) exp (cid:18) − nK ( τ ) (cid:19) . (61)Of course,these relations are proved rigorously only in the limits of short ( η ≪
1) andlong ( η ≫
1) pump pulses. But as the Schmidt number K ( τ ) is large also in the case ofintermediate pulse durations, the assumption that Equations (61) remain valid also in the7case η ∼ K ( τ ) in the region ofintermediate pulse durations either has to be taken from numerical calculations [7] or in itsturn has to be defined by means of a more or less reasonable interpolation. The simplestinterpolation K ( τ ) = q K ( τ ) + K ( τ ) was suggested and used in our earlier work [2].Below we discuss somewhat more elaborate ways of interpolating K ( τ ) into the region η ∼ η ∼
1. In other words, we assume that for any values of the control parameter η , small,intermediate, and large, the biphoton wave function (16) can be satisfactorily modeled byΨ( ν , ν ; τ ) = s πa ( τ ) b ( τ ) exp (cid:20) − ( ν + ν ) a ( τ ) (cid:21) exp (cid:20) − ( ν − ν ) b ( τ ) (cid:21) , (62)where the widths a ( τ ) and b ( τ ) are determined by Equations (23) and (49) in the cases oflong and short pump pulses and have to be defined yet in the case of intermediately longpulses. By comparing expressions in Equations (23) and (49) we find easily that they arerelated to each other by the following simple formulas a short a long = 1 .
392 ln 2 η = 1 . η ≈ η and b long b short = r cτ × . LA = 1 . √ η ≈ √ η. (63)By using the idea of interpolation we assume that in a general case we can define thefunctions a ( τ ) and a ( τ ) as being given by a ( τ ) = a short f a ( η ) and b ( τ ) = b long f b ( η ) , (64)where the smoothing functions obey the conditions f a ( η ) | η ≪ = 1 , f a ( η ) | η ≫ = 1 η ; f b ( η ) | η ≪ = 1 √ η , f b ( η ) | η ≫ = 1 . (65)Of course, there are infinitely many ways of choosing the smoothing functions f a ( η ) and f b ( η ). To restrict somehow this manifold of possibilities, let us take these functions in theform f a ( η ) = (1 + η s ) − /s and f b ( η ) = (1 + η s ) / s √ η , (66)8where s > f a ( η ) and f b ( η ) Equations (67) arereduced to a ( τ ) = a short (1 + η s ) − /s ; a ( τ ) ω = 1 . πA √ ln 2 λ L (1 + η s ) − /s ,b ( τ ) = b long (1 + η s ) / s √ η ; b ( τ ) ω = r λ π . BL (1 + η s ) / s √ η . (67)The dependences a ( τ ) and b ( τ ) are characterized by two curves in Figure 3. The dashed partsof these curves characterize smooth transitions from the short-pulse to long-pulse regions.One of the main features of the widths a ( τ ) and b ( τ ) seen clearly in Figure 3 is that at all FIG. 3:
Parameters a ( τ ) and b ( τ ) (67) of the general double-Gaussian spectral wave function (62), calcu-lations for LiIO crystal, L = 0 . s = 2 . values of the pump-pulse duration τ , or of the control parameter η , always b ( τ ) ≫ a ( τ ).This feature of the model wave function (62) differs significantly from that of the exactwave function (16). In the latter the pump spectral function and sinc-function change theirroles at η ∼
1: in the region η ≪ η ≫ τ . Thissimplifies, for example, the definition of the coincidence and single-particle spectral widths.In terms of a ( τ ) and b ( τ ) they are given by∆ ν ( c )1 ( τ ) = 2 a ( τ ) √ ln 2 , ∆ ν ( s )1 ( τ ) = b ( τ ) √ ln 2 , (68)for all τ .9With the definitions of Equations (67) we get the following generalized expression for theSchmidt number K ( τ ) = b ( τ )2 a ( τ ) = √ π ln 22 . √ × . A r LBλ [1 + η s ( τ )] / s p η ( τ ) . (69)The dependence K ( τ ) (or K ( η )) is similar to that characterized by the curves in Figure 1.The minimum of the function K ( τ ) is achieved at η = 2 − /s and K ( τ ) | min = √ π ln 22 . √ × . A r LBλ / s /s . (70)The case s = 3 corresponds to the earlier used interpolation [2]. If we want to increase K ( τ ) | min by 12% to make the curve K ( τ ) closer to the numerically calculated one [7],we have to take s = 2 .
21 in our formulas (66)-(70). This gives K ( τ ) | min = 83 (insteadof K ( τ ) | min = 73 of the work [2]), and the curve K ( η ) appears in this case practicallyindistinguishable from the numerically calculated one. But of course, the main noveltyof the described here method of interpolation is not only in a possibility of getting this12% improvement. The main message is in the assumption about a possibility of modelingthe wave function (16) at arbitrary values of the pump-pulse duration τ by the double-Gaussian wave function (62) with the parameters a ( τ ) and b ( τ ) given by Equations (67).This assumption is proved to be correct in the cases of long and short pulses. At the“integral level”, correctness of the suggested model (62) is also proved for all values of τ because the Schmidt number K ( τ ) is found to be practically identical in the exact numericalcalculations [7] and in the given above analytical derivation for the model double-Gaussianwave function (62) with s = 2 .
21. Localization regions of the exact (16) and model (62)wave functions in the case of intermediate pump-pulse durations are also very close to eachother and, qualitatively, similar to that shown in Figure 2( b ). A more detailed comparison ofthe exact (16) and model (62) wave functions in the case of intermediately long pump pulsescan involve comparison of eigenvalues of the reduced density matrix and of a structure ofthe Schmidt modes. But for the exact wave function (16) at η ∼ λ n ( τ ) of the reduced densitymatrix, as said above, we assume that this expression is valid for not only for small andlarge η but also also in all the region between these two asymptotic limits. In accordancewith the Schmidt theorem, the weights with which the Schmidt modes are represented in the0expansion (11) are given by (cid:12)(cid:12)(cid:12)p λ n ( τ ) (cid:12)(cid:12)(cid:12) . As an example, let us consider the case η = 1, when K ( τ ) = 87. In this case the values of (cid:12)(cid:12)(cid:12)p λ n ( τ ) (cid:12)(cid:12)(cid:12) η =1 are located along the line shown in Figure4. At n = K = 87, we get (cid:12)(cid:12)(cid:12)p λ ( τ ) (cid:12)(cid:12)(cid:12) η =1 = 0 . (cid:12)(cid:12) √ λ n (cid:12)(cid:12) , (cid:12)(cid:12) √ λ (cid:12)(cid:12) = 0 .
15. This results show that falling of (cid:12)(cid:12) √ λ n (cid:12)(cid:12) with a growing n is ratherslow, and in reality there is a rather large interval of n > K , giving not too small contributioninto the expansion of the wave function in the sum of the Schmidt-mode products (11). FIG. 4:
Square roots of the eigenvalues of the reduced density matrix (61) at η = 1 and K ( τ ) given byEquation (69). The Schmidt modes of the model double-Gaussian wave function (62) have the sameform as in the asymptotic cases of long (29) and short (60) pump pulses, but now with thegeneralized parameters a ( τ ) and b ( τ ) (67): ψ n ( ν , ; τ ) = 1 p α ( τ ) ϕ n (cid:20) ω α ( τ ) ν , ω (cid:21) , (71)where ω α ( τ ) is the dimensionless generalized scaling factor characterizing the dependenceof ψ n on the dimensionless variables ν , /ω : ω α ( τ ) = ω s a ( τ ) b ( τ ) = 2 . √ AB / (cid:18) Lλ (cid:19) / η / (1 + η s ) / s . As a function of the control parameter η , the dimensionless scaling factor of the Schmidtmodes is plotted in Figure 5. The dimensionless scaling factor is large practically at allvalues of η . With a growing η the scaling factor monotonously grows, which means thatlocalization regions of all Schmidt modes shrink. At η = 1, Eq. (VI) gives ω α ( τ ) ≈ . n th Schmidt mode (71) can be estimated as δν n ( τ ) ∼ √ nα ( τ ) = r n a ( τ ) b ( τ )2 . (72)1 FIG. 5:
Dimensionless scaling factor ω α ( τ ) (VI) vs. the control parameter η , for LiIO crystal, L = 0 . λ = 400 nm, s = 2 . This width grows with the increasing mode number n as √ n . For n = K ( τ ) Equation (72)gives δν K ∼ b ( τ ) / ν ( s ) / √ ln 2 = 0 . ν ( s ) ; δν n ( τ ) reaches ∆ ν ( s ) at n ≈ . K ( τ ). It’strue, however, that the contribution of such high-number Schmidt modes to the expansion(11) is very small: for the same parameters as used above for all estimates, we get √ λ . K ∼
6% of √ λ (to be compared with √ λ K ∼
37% of √ λ ). This means that practically for allimportant Schmidt modes their spectral width is smaller than the single-particle width ofthe biphoton spectrum as a whole.As an example, some spectral Schmidt modes are shown in Figure 6 for the same param-eters as in Figure 5 and η = 1. FIG. 6:
Schmidt modes ψ , ψ and ψ . VII. CONCLUSION
Summarizing, we found double-Gaussian models of the SPDC type-I wave function (16)in both regions of short and long pump pulses. In the case of short pump pulses even2a possibility of Gaussian modeling was not evident in advance. Its derivation required atransition to the reduced density matrix where Gaussian modeling appears to be much morenatural and simple than in the case of the wave function. Then, after finding a double-Gaussian model for the reduced density matrix and using general relations between itsparameters and those of the wave function, we were able to find the double-Gaussian model(50) of the original wave function (16) in the case of short pump pulses. Though the formsof these too functions are significantly different, the double-Gaussian model reproduces astructure of the exact wave function rather well and almost in all details, which is seenclearly in two pictures of Figure 2. Having the double-Gaussian models of the exact wavefunction in the regions of short and long pulses we found also in these two cases the Schmidtmodes (29) and (60). Finally, by analyzing features of all parameters in the asymptoticcases of long and short pump pulses and using the most appropriate interpolation to theregion of intermediately long pump pulses, we found a possibility of suggesting the model ofa double-Gaussian wave function [Equations (62) and (67)], which we assume to be valid forarbitrary values of the pump-pulse duration τ . For this model wave function we found alsothe Schmidt number K ( τ ) (69), Schmidt modes [Equations (71) and (VI)], and eigenvalues ofthe reduced density matrix (61). The Schmidt number found in such a way agrees perfectlywell with the results of its numerical calculation [7]. We assume that the functions ψ n ( ν , )of Equation (71) represent equally well the spectral Schmidt modes in all range of the pump-pulse durations, short, long and intermediate. Comparison with exact numerical calculationsof the Schmidt modes is expected to be interesting and fruitful. Also we hope that the derivedresults can be tested experimentally. For example, an interesting scheme of an experimentcan involve splitting of the biphoton beam for two channels by a nonselective beam-splitter,installing in one of two channels a spectral mask gating through it only a single spectralmode and measuring the coincidence spectrum in the second channel. According to theSchmidt theorem the coincidence spectrum measured in such a way is identical to that ofthe mode gated through in the first channel, and this coincidence of two spectra deservesits experimental verification. We hope also that in future the data about spectral Schmidtmodes can be used for practical applications in the problems of quantum information andquantum cryptography.3 References [1] Fedorov M V, Efermov M A, Volkov P A, Moreva E V, Straupe S S and Kulik S P 2008
Phys.Rev. A Phys. Rev. A J. Opt. Soc. Am. ′ Ren A B, Banaszek K and Walmsley I A 2003
Quantum Information and Computation Phys. Rev. A J. Phys. B S467[7] Mauerer W and Silberhorn C 2008 to be published in QCMC’08 Proceedings AIP[8] Law C K and Eberly J H 2000
Phys. Rev. Lett.84