aa r X i v : . [ qu a n t - ph ] M a r Entanglement and reduction in two-photon correlations ∗ Arthur Jabs
Alumnus, Technical University Berlin.Voßstr. 9, 10117 Berlin, [email protected](16 March 2014)
Abstract.
For a number of basic experiments on two-photon intensity correlationsit is pointed out that the results, which are usually explained in terms of the for-malism of canonical field quantization, can also be explained in terms of quantumwavepackets, where reduction and entanglement are taken into account.
Keywords:
Classical and quantum physics, quantum optics, multiphoton processes,quantum electrodynamics.
In canonical field quantization the functions that represent the fields and the conju-gate field momenta are turned into operators between which commutation relationsare postulated, for bosons modeled after the Heisenberg relations and for fermionsintroduced ad hoc. When the operator functions are expanded in Fourier series,the expansion coefficients are operators, a, a † , which satisfy commutation relationsaccording to which the product a † a has only non-negative integers as eigenvalues.Here the particle concept enters field theory: the eigenvalues are taken to be particlenumbers, a and a † create and destroy the particles in the number or Fock states.In order to avoid that the state with no particles has infinite energy normal or-dering, disregarding the commutation relations, is called on. Reduction (collapse)is not mentioned. Canonical field quantization, that here is specifically quantumelectrodynamics or quantum optics, correctly describes the experimental findings.In this note we demonstrate that in a number of experiments the findings can alsobe correctly described within quantum mechanics, without canonical quantization,when the particles (photons) are introduced in the spirit of Einstein’s photon paper[1], with some specifications. These are:(1) Photons are not point particles but special wavepackets of electromagneticwaves representing an integral number of quanta ,(2) A one-photon wavepacket, when it interacts with a measuring device, isinstantaneously contracted to a narrow place (e.g. an atom) and vanishes there: reduction [2], [3], ∗ Also paper-published in Journal of the Optical Society of America B (4), 765-770 (2014). entangled , that is, the wave function repre-senting a system of photons can no longer be written as a product of one-photonwavepackets.Compared with the machinery of canonical field quantization these conceptionsmean a more direct conceptual access to the photon-correlation experiments con-sidered in this note, and, as I think, provide an intuitively appealing picture. In anumber of articles [2 - 5] it has already been shown that the wavepacket approachalso avoids many of the conceptual difficulties of nonrelativistic quantum mechanics.In Section 2 we discuss experiments in which the two entangled signal andidler photon wavepackets (photons, for short) from spontaneous parametric down-conversion only meet and overlap in the measuring device. In Section 3 they overlapat a beam splitter before they enter the measuring device, and in Section 4 theynever overlap after they have left the source. Sections 2 and 3 are mainly concernedwith reduction, Section 4 with entanglement.In each experiment, we consider the results obtained when using classical electro-magnetic waves; then we take entanglement and reduction into account, and thencompare the results with those of quantum optics and experiment. Though basi-cally the radiation is conceived to consist of wavepackets of finite extent, in themathematical treatment the plane-wave approximation is mostly used.The most salient difference between classical and quantum predictions appears inthe visibility (relative modulation amplitude, intensity correlation function, averagecorrelation product, two-photon detection probability) V = ( C max − C min ) / ( C max + C min ) of the fringes observed in the two-photon intensity correlation as a functionof different parameter settings. Therefore our attention is focused on the formulasfor this quantity, calculated by quantum optics (qo), quantum mechanics, includingreduction and entanglement (qm), and classical optics (co). We begin with the pioneering experiment [6] by Ghosh and Mandel. The joint proba-bility P for the detection of two photons at two points is measured as a function of theseparation ( x − x ) (Fig. 1). Signal and idler photons are produced in spontaneousparametric downconversion with no definite phase relationships. The intensity of thebeams was so low that never more than a single one-photon wavepacket at a timeappeared in each beam.According to quantum optics the probability P is { Eq. (6) in [6] } P qo ( x , x ) δx δx = 2 K K δx δx { π ( x − x ) /L ) } . (1)Here δx and δx are narrow ranges centered at x and x , respecticely. K and K are scale factors characteristic of the detectors. L ≈ λ/θ is the spacing of theclassical interference fringes when the two waves of wavelength λ come togetherat a small angle θ . Equation (1) correctly describes the experimental results. Thevisibility is the factor in front of the cosine, which here is V qo = 1. 3 -Figure 1: Geometry of the interference experiment [6].According to classical optics the probability is { Eq. (7) in [6] } : P co ( x , x ) δx δx = 2 K K δx δx " h| a A | | a B | ih ( | a A | + | a B | ) i cos 2 π ( x − x ) L . (2)Here a A and a B are the c number coefficients when the classical fields detected atsome points x , x in a distant plane are written in the form of E ( x ) = a A e i k A r A + a B e i k B r B +i δ , E ( x ) = a A e i k A r A + a B e i k B r B +i δ (3)and averaged over the relative random phase δ between signal and idler photons.Equation (2) yields the visibility V co = 2 | a A | | a B | | a A | | a A | + | a B | | a B | + 2 | a A | | a B | . (4)Here | a A | ( | a B | ) is the classical (constant) intensity of the wave arising from place A ( B ) reaching both x and x [It appears in both expressions of Eq. (3)]. Themaximum possible value is V co = , when | a A | = | a B | , in contrast to V qo = 1 fromEq. (1).Now we take into account that the radiation consists of quantum wavepackets.Conceptually, the general procedure in going from the classical to the quantum do-main is to re-interpret the classical continuous intensities as probabilities of discretequantum events. Thus in the quantum domain | a A | is proportional to the probabil-ity that wavepacket A (i.e., the one coming from place A ) will give rise to a counteither in detector D1 or in detector D2 (even if the arms from B are blocked), andthe same holds analogously for | a B | .Accordingly, | a A | | a B | is proportional to the probability that both wavepacket Aand wavepacket B will give rise to a count somewhere, these events being independentof each other. In line with this, | a A | | a A | would be proportional to the probabilitythat wavepacket A will give rise to two counts simultaneously (within an arbitrarily 4 -short time interval). Classically this would be possible, but owing to the one-quantumnature of the wavepacket it is not. In other words, coincidence counts can only bebrought about when both the one-photon packet which comes from A (through | a A | ) and that which comes from B (through | a B | ) contribute. – Note that all thisis independent of whether it was packet A or packet B that contributed to a countin a particular counter.Therefore the term | a A | | a A | as well as | a B | | a B | have to be eliminated fromEq. (4), which then yields a visibility of V qm = 1, and this coincides with V qo = 1 ofEq. (1).Thus, the situation considered leads to a simple conversion rule for convertingthe formulas of classical optics into those of quantum optics: In the formulas of classical optics eliminate all those terms that containproducts of intensities of waves coming from one and the same place inthe source.
Further experiments where this rule can easily be seen to apply are: [7] { Eq. (8) (30) } , [8] { Eq. (67) (72) } , [9] { Eq. (4) (4 . } , [10] { Eq. (4) (41) } . The first(classical) formula goes over to the second (quantum) formula by the conversionrule: eliminate I , I or h I i or I I , I II or M , N in the respective papers. Next we consider the paper [11] by Kwiat et al., which is concerned with the co-incidence counting rate P of two detectors measured under various polarizationsand phase differences of the signal and idler photons. The simplified setup is out-lined in Fig. 2. A signal photon s and an idler photon i , both with horizontal linearFigure 2: Geometry of the interference experiment [11].polarization, are emitted from the crystal KDP in different directions, reflected attwo mirrors, unified at the 50:50 beam splitter (BS) and led into two detectors D1and D2, which are connected with a coincidence counter (CC). The setup is sym-metrical with respect to the beam splitter, mirrors, and detectors. In the way ofthe signal photon, a halfway plate is placed, which rotates the plane of polarization 5 -by φ . In front of the detectors, linear polarizers can be placed at angles θ and θ ,respectively, to the horizontal. Each arm contains only one photon at a time.We consider the coincidence rate P between the two detectors as a function of φ , θ , and θ . P is proportional to the joint probability considered in Section 2. Theexperimental results can be described by the quantum-optics formulas in Eqs. (8),(A4), (13), (A5) in [11]:(a) P qoa = 12 sin φ without polarizers , (5)(b) P qob = 14 sin φ with one polarizer , (6)(c) P qoc = 14 sin φ sin ( θ − θ ) with two polarizers . (7)We now want to derive the formulas to which classical optics would lead us in thesesituations. Let the classical field amplitudes be A s and A i , corresponding to a A and a B in the notation of Section 2. At the detectors we consider separately thehorizontal H and vertical V components. In classical treatment, the probability ofa detector click is proportional to the intensity of the radiation at the place of thedetector. Coincidence counts are then proportional to the product of the intensitiesat detector D1 ( I H1 + I V1 ) and detector D2 ( I H2 + I V2 ) after averaging over the relativerandom phase δ between signal and idler pulses: P c ∝ ( I H1 + I V1 )( I H2 + I V2 ) δ . The intensity at detector D1, say, is just the sum ( I H1 + I V1 ) because there are nointerference terms between the horizontal and vertical components.Case (a):Only a half-way plate and no polarizers.The horizontal component of the wave at detector D1 is then A H1 = 1 √ A s e i( ω s t + k s r s + δ ) cos φ + 1 √ A i e i( ω i t + k i r i + π/ . (8)The factor exp(i π/
2) = i comes from the reflection of the idler wave at the sym-metric beam splitter, the factor 1 / √ φ from the half-way plate in the way of the signal wave. We may omit theterms i( ωt + kr ) in the exponentials because they can be absorbed in the relativerandom phase δ . Thus Eq. (8) simplifies to A H1 = 1 √ A s cos φ e i δ + i √ A i . Analogously, the vertical component of the wave at detector D1 can be written as A V1 = 1 √ A s sin φ e i δ . A H2 = i √ A s cos φ e i δ + 1 √ A i , A V2 = i √ A s sin φ e i δ . Thus the total intensity at detector D1 becomes I = I H1 + I V1 = | A H1 | + | A V1 | = 12 (cid:16) A + A + 2 A s A i cos φ sin δ (cid:17) and at detector D2, I = I H2 + I V2 = | A H2 | + | A V2 | = 12 (cid:16) A + A − A s A i cos φ sin δ (cid:17) . Then I I = 14 (cid:16) ( A + A ) − A A cos φ sin δ (cid:17) , and after averaging over δ (sin δ = ), and with A = I s etc., we arrive at P co ∝
14 ( I + I ) + 12 I s I i sin φ. (9)Applying the conversion rule results in P qm ∝ I s I i sin φ , which, except for theunessential proportionality factor I s I i , coincides with the quantum optical value inEq. (5).Case (b):Half-way plate plus one polarizer.With polarizer angle θ in front of detector D1 classical optics yields P co ∝ I θ ( I H2 + I V2 ) δ I θ = | A θ | A θ = i √ A s cos( θ − φ ) + √ A i cos θ e i δ A H2 = √ A s cos φ + i √ A i e i δ A V = √ A s sin φ ,and after some calculation, one arrives at P co ∝ I cos ( θ − φ ) + 14 I cos θ + 14 I s I i sin φ. (10)Application of the conversion rule results in P qm = I s I i sin φ , which is essentiallyEq. (6) and gives one half the value in Eq. (5).Case (c):Half-way plate plus two polarizers.With the polarizer angles θ and θ classical optics yields 7 - P co ∝ | A | | A | δ ,where A = √ A s cos( θ − φ ) + i √ A i cos θ e i δ A = i √ A s cos( θ − φ ) + √ A i cos θ e i δ and this finally results in P co ∝ I cos ( θ − φ ) cos ( θ − φ ) + 14 I cos θ cos θ + 14 I s I i sin φ sin ( θ − θ ) . (11)Applying the conversion rule leaves only the second line of Eq. (11), which essentiallycoincides with Eq. (7).Another, more complicated experiment, where the conversion rule by eliminating h I i , h I i , converts the classical formulas into those of quantum optics is [12] { Eq.(14) (5a) } . Finally, we consider several aspects of an experiment proposed by Franson [13]:(A) Its experimental realization by Kwiat et al. in the case of narrow coincidencetime windows [14].(B) Experimental realizations in the case of wide coincidence windows.(C) A classical calculation for case (A).Case (A):Narrow coincidence window.The principal scheme of the experimental setup [14] is shown in Fig. 3. The signaland idler photon wavepackets are entangled because they satisfy the conditions t s = t i (12) k s + k i = k p (13) ω s + ω i = ω p , (14)where the indices s, i , and p refer to the signal photon, the idler photon, and thepump-beam photon, respectively. Each downconverted photon has a substantialbandwidth, but the sum of their frequencies is fixed to within the pump band-width, which is negligible in the experiment. The pairs were selected such that ω s and ω i are centered at ω p /
2. Thus, with λ p = 351 nm, λ s and λ i were each centeredat λ = 702 nm.As shown in Fig. 3, each photon packet passes a Mach-Zehnder-like interferom-eter, where it is split in two equal parts. One part goes the long way and the otherthe short way, and then the two are recombined. The difference ∆ L between the 8 -Figure 3: Outline of the experimental setup in [14].long and the short way was about 63 cm, the (coherence) length σ x of the signaland idler packets was less than about 36 µ m, and the coincidence time window was τ = 1 . δl = cτ = 44 cm. Thus δl < ∆ L ,and this means a short coincidence window. The arrangement is the same for bothphotons. They are then registered in the detectors, which are assumed to have 100%efficiency. Filters F of ∆ λ = 10 nm bandwidth around λ = 702 nm were placed infront of the detectors.As already stated, there are no definite phase relations between signal and idlerphotons, and, after leaving the crystal, the two photons never meet again. Yet, aspredicted by Franson, based on quantum optics, interference fringes with visibility1 are observed in the coincidence rates between detectors Ds and Di, when ∆ L isslowly changed.The explanation based on quantum wavepackets is as follows. In order to con-struct the wave function of the system of the entangled photons, we start with aproduct ψ s ψ i = ( ψ Ss + ψ Ls )( ψ Si + ψ Li ) = ψ Ss ψ Si + ψ Ss ψ Li + ψ Ls ψ Si + ψ Ls ψ Li , (15)where ψ Ss is the short-way part, and ψ Ls the long-way part of the signal wavepacket, 9 - ψ s , and the same for the idler packet, ψ i . Explicitly we write: ψ Ss ( x s , t ) = Z k s e ψ ( k s ) e i( k s x s − ω s t ) d k s . (16)The normalized function e ψ ( k ) determines the shape (including the width σ x ) of theunsplit signal and idler packets as well as of their short-way and the long-way parts.To simplify writing, we write k instead of k , and x instead of x . x is then a pointon the (optical) path counted from the source (the crystal), and k is the componentof k in the direction of the path.Similarly, we then write: ψ Ls ( x s , t ) = Z k s e ψ ( k s ) e i( k s ( x s − ∆ L ) − ω s t ) d k s (17)because we can assume that ψ Ls has the same shape as ψ Ss , but is displaced by ∆ L .Replacing the indices s in Eqs. (16) and (17) by i gives us ψ Li ( x i , t ) and ψ Si ( x i , t ),and the first product on the right-hand side of Eq. (15) becomes ψ Ss ψ Si = Z k s Z k i e ψ ( k s ) e ψ ( k i ) e i( k s x s − ω s t + k i x i − ω i t ) d k s d k i . However, the condition in Eq. (13) requires the insertion of δ ( k i − ( k p − k s )) [or δ ( k s − ( k p − k i ))] under the integral, which then can no longer be written as aproduct of a signal and an idler packet, but turns into ψ SSsi = e i( k p x i − ω p t ) Z e ψ ( k s ) e ψ ( k p − k s ) e i k s ( x s − x i ) d k s (18)for any values of x i and x s . Applying the same procedure to the product ψ Ls ψ Li andcomparing the result with Eq. (18) we obtain: ψ LLsi = e − i k p ∆ L ψ SSsi . (19)In the same way, we obtain: ψ SLsi = e i( k p ( x i − ∆ L ) − ω p t ) Z e ψ ( k s ) e ψ ( k p − k s ) e i k s ( x s − x i +∆ L ) d k s , (20)and ψ LSsi is equal to ψ SLsi of Eq. (20), except that the indices i and s are interchanged.Now we apply these formulas to the coincidence probability measured in theexperiment [13]. Both detectors were to click at the same time within a small regis-tering time window τ ≪ ∆ T = ∆ L/c , and both detectors lie at the same distancefrom the source, i.e., x s = x i . In this case it is ψ SLsi = ψ LSsi = 0 (21) 10 -because the integrand in Eq. (20) is a product of two functions, one of which,exp(i k ∆ L ), oscillates rapidly as a function of k , while the other, e ψ ( k ) e ψ ( k p − k ),is a relatively smooth function of k . The integral is the closer to zero the larger∆ k ∆ L , where ∆ k is the width of the smooth function. In fact ∆ k ∆ L ≫ k ∆ x ≈ / k ∆ L ≫ L/ ∆ x ≫
1. As signal and idler packet overlap in k space thewidth ∆ k of the product function is comparable to the width σ k of the single parts ψ Ss , ψ Ls etc. Thus ∆ x ≈ σ x and ∆ L/ ∆ x ≫ L/σ x ≫
1. This means thatthe widths σ x of the parts ψ Ss , ψ Ls etc. are small compared with ∆ L , which is anotherway to see that no coincidences between the short-way part of one packet and thelong-way part of the other are possible.Then Eqs. (19) and (21) yield the coincidence probability C (A)qm = | ψ si | = | ψ SSsi + ψ LLsi | = | ψ SSsi | | − e − i k p ∆ L | = | ψ SSsi | k p ∆ L ) , (22)and this means visibility V qm = 1, the same as predicted by quantum optics. Themeasured value of only 0 . < τ > ∆ T . Such a casewas studied in the experiments [15-17]. In this case, the coincidence probability isthe sum of three probabilities, which refer to three different physical situations:(B1) Coincidences between the short-way parts of the signal and the idler packetand between the long-way parts of the two packets.(B2) Coincidences between the short-way part of the signal packet and the long-way part of the idler packet.(B3) Coincidences between the short-way part of the idler packet and the long-way part of the signal packet.Case (B1) is equivalent with detectors which have the same narrow coincidencewindow and lie at the same distance from the source, x s − x i = 0. This is the casealready treated in case (A), where C (A)qm is given by Eq. (22), which with Eq. (18)reads C (B1)qm = (cid:12)(cid:12)(cid:12)(cid:12)Z e ψ ( k s ) e ψ ( k p − k s )d k s (cid:12)(cid:12)(cid:12)(cid:12) k p ∆ L ) . (23)Case (B2) is equivalent with detectors which have the same narrow coincidencewindow, and detector Ds lies at x s = x i + ∆ L while Di lies at x i ; that is, x s − x i =+∆ L . In this case, the integral for ψ SSsi [Eq. (18)] contains the factor exp(+i k s ∆ L ),which is now the rapidly oscillating factor. By a reasoning like that after Eq. (21)the integrals in Eqs. (18) and (19) now result in ψ SSsi = ψ LLsi = 0 .
11 -The integral for ψ SLsi [Eq. (20)] contains the factor exp(i k s ( x s − x i + ∆ L )), whichwith x s − x i = +∆ L becomes the rapidly oscillating factor and leads to ψ SLsi = 0 . The integral for ψ LSsi contains the factor exp(i k s ( x s − x i − ∆ L )), which with x s − x i =+∆ L becomes 1 and leads to | ψ LSsi | = (cid:12)(cid:12)(cid:12)(cid:12)Z e ψ ( k i ) e ψ ( k p − k i )d k i (cid:12)(cid:12)(cid:12)(cid:12) ≡ (cid:12)(cid:12)(cid:12)(cid:12)Z e ψ ( k s ) e ψ ( k p − k s )d k s (cid:12)(cid:12)(cid:12)(cid:12) . Thus C (B2)qm = | ψ LSsi | = (cid:12)(cid:12)(cid:12)(cid:12)Z e ψ ( k s ) e ψ ( k p − k s )d k s (cid:12)(cid:12)(cid:12)(cid:12) . (24)Case (B3) means that x s − x i = − ∆ L , and repeating the steps that led us in(B2) to Eq. (24) results in C (B3)qm = C (B2)qm , (25)which is as it should, due to the signal/idler symmetry. With Eqs. (23), (24), and(25) the total coincidence probabilitiy in case (B) is C (B)qm = C (B1)qm + C (B2)qm + C (B3)qm = (cid:12)(cid:12)(cid:12) ψ LSsi (cid:12)(cid:12)(cid:12) k p ∆ L ) . (26)This means visibility V qm = , which is confirmed in the above-mentioned experi-ments.Cases (A) and (B) were realized in the experiment [18] and can be treated inthe same way as the experiment in [14], though the paths of the signal and idlerphotons in [18] were not spatially separated. The visibility obtained was V = 0 . V = 0 .
46 in case (B).Case (C):A classical calculation.It is interesting to compare the quantum case (B1), where x s = x i , V qm = 1, withthe following classical calculation. A possible classical analog would be a situationwhere the electromagnetic signal and idler pulses are independent of each other.This might be expressed explicitly by multiplying the signal pulse, say, by exp(i δ )with random δ , but such a factor would drop out in the calculations below.In the classical case, the signal and idler pulses even before measurement lie inrather narrow intervals about ω s , k s , and ω i , k i , respectively, albeit still permittingsufficiently short packets in x space. Then a statistical ensemble of different runs isconsidered, where different runs have different values of ω and k , satisfying, however,the conditions in Eqs. (12), (13), and (14) in each single run.In the quantum case, the coincidence rate was obtained by first averaging (inte-grating over k s in a rather wide interval) the product of the signal and idler ampli-tudes , meaning entanglement, and then taking the absolute square. In the classical 12 -case, waves may be added but never multiplied. Classically only intensities may bemultiplied (cf. Section 3). Thus we calculate the classical intensities at detectorsDs and Di, multiply the two, and then average the product over k s and k i . Such aprocedure is like that applied to the case of two spin-1/2 particles in a spin-singletstate in [5, Appendix C].Thus we again start with a product of a signal and an idler pulse, E s E i = ( E Ss + E Ls )( E Si + E Li ) = E Ss E Si + E Ss E Li + E Ls E Si + E Ls E Li (27)as in Eq. (15). Here we write E Ss etc. instead of ψ Ss etc. in order to emphasize that ourwavepackets are now pulses of classical electromagnetic waves. As k and ω now liein narrow intervals, we approximate the pulses by integrals like those for ψ Ss ( x s , t )in Eq. (16), where, however, the range of the integration is so small that we canapproximate the integrals by their integrands multipied by ∆ k . For example [cf.Eq. (16)], E Ss ( x s , t ) = e ψ ( k s ) e i(k s x s − ω s t ) ∆ k s E Ls ( x s , t ) = e ψ ( k s ) e i(k s (x s − ∆L) − ω s t ) ∆ k s , and analogous formulas for E Si and E Li . Then the product in Eq. (27), with k i = k p − k s from Eq. (13), turns into E s E i = E Ss E Si (cid:16) e − i( k p − k s )∆ L + e − i k s ∆ L + e − i k p ∆ L (cid:17) ∆ k s ∆ k i . In averaging | E s E i | (∆ k → d k and integrating over a wide interval) all those ofthe 16 terms which contain factors like exp( ± i k s ∆ L ) average to zero, following thereasoning after Eq. (21). The surviving terms are | E s E i | = (cid:12)(cid:12) E Ss E Si (cid:12)(cid:12) (cid:16) e +i k p ∆ L + e − i k p ∆ L (cid:17) = | E s E i | k p ∆ L ) , (28)which means visibility V co = , that is half the quantum value in Eq. (22) in thesituation considered.In essence, the term k p ∆ L = ( k s + k i )∆ L arises from the product of the signaland idler waves and also from the product of the classical intensities. The particularvalue V qm = 1 arises from averaging over the product of the amplitudes, rather thanover the product of the intensities, that is, from entanglement. References and notes [1] A. Einstein, “ ¨Uber einen die Erzeugung und Verwandlung des Lichtes be-treffenden heuristischen Gesichtspunkt,” Ann. Physik (Leipzig) , 132-148(1905). English translation by A.B. Arons and M.B. Peppard, “Einstein’s pro-posal of the photon concept - a translation of the Annalen der Physik paperof 1905,” Am. J. Phys. , 367-374 (1965). 13 -[2] A. 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