Multiphoton interference and polarization effects

PHYSICS LETTERS A

Physics Letters A 163 (1992) 349—355 North-Holland

Multiphoton interference and polarization effects D.N. Klyshko Physics Department, Moscow State University, Moscow 119899, Russia Received 2 January 1992; accepted for publication 20 January 1992 Communicated by V.M. Agranovich

Several modifications of recently performed two-photon quantum-optical experiments are proposed together with their classical interpretation and systematization. New phenomena such as the “hidden polarization” or “higher-order Malus laws” could be observed ifthe intensity correlations are registered.

I. Introduction

7’

There seems to be a continuing interest in experiments involving the observation of intensity interference using two-photon light (TPL) in various optical schemes (see refs. [1—10]). TPL can be generated by means of two-quantum transitions in atoms or of three- or four-wave parametric processes. The parametric TPL is in some sense squeezed, so it is called squeezed vacuum (SV). It is possible also to use the classical squeezed light (SL), produced by parametric conversion ofthe usual light [11,12]. We mention another way rather unexpected and quite unpractical to squeeze the vacuum: by moving through it with constant acceleratiOn [13]. In the present communication the following topics are discussed: the classification of various effects, their unified description, classical interpretation and possible modifications using multiphoton light (MPL), SL and the usual light.

D

X X’

a

V

DA ~DXA D ~-

b

~

I

,L’ - ‘.l~~ • ‘

D

D

_____

d

—

—

2. Classification and classical interpretation The experiments in question are described by the optical scheme’s output correlation function G~1 4a,,, is the photon number <:nknl:>,They where operator. cann,,ma), be categorized into several main types (fig. 1), involving two, four or a continuum of spatial or polarization modes. In the scheme of fig.

Fig. 1. The basic schemes of intensity interference experiments, involving two (a), four (b), (c) or a continuum (d) of spatial or polarization modes.The modes are mixed by elements D (beam splitters, rotators, etc.). The output intensity correlation is measured by two detectors and a coincidence scheme.

1 a a beam splitter (BS) [4,8], an interferometer [14], or a polarization rotator can be used for mixing; it shows the correlation suppression effect [4,8,12, 14,15] and the spatial quantum beats [5,8,12]. The scheme of fig. lb embraces the usual HanburyBrown— Twiss interference (HBTI) and its TPL modifications, including the EPR-type experiments (see refs. [1,2,9,16—19]). The indexes x, y may refer to two polarization types, two [6,16] stars, two nonlinear with a common pump or two pairs crystals of conjugated modes, picked up by diaphragms from one crystal’s parametric radiation [9,17,181. The scheme offig. ic [10,12,20] is a particular case ofthe scheme

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with arbitrary frequency filters in each channel [211, which demonstrates the EPR paradox for the photon observables w, t; the same for q1, p ~ is demonstrated by the scheme of fig. 1 d [22]. The intensity interference in general has a simple classical explanation: the phase fluctuations in the input beams turn into intensity fluctuations in the output beams. Indeed, let the two in-beams of fig. 1 a have equal and constant and indepen0k (amplitudes t), k = x, y. A 50% BS (or dently diffusingorphases interferometer rotator) directs all input energy into one output mode x’ at the moments when 0 = 0 and toy’ at other moments when 0 = x. —

As a result the output intensities acquire strong fluctuations, which are perfectly anticorrelated; therefore G~~=min. This is the correlation suppression effect [4,8,15]. The spatial quantum beats [5,8,12] also have a simple classical origin: when w~, WY then there are regular beats in the output intensities, whose relative phases depend on the optical path lengths. When four or more modes are mixed, fig. 1 b, there are two possibilities depending on thewave inputinterferphases statistics: HBTI or AWl (advanced ence) [161. In classical terms they can be characterized by phase correlation or anticorrelation: Ø~± Ø~= const (the phase delays are designated according to fig. ib). Both HBTI and AWl have in essence the same classical explanation as the two-mode interference, Quantum theory predicts in all cases the same interference pattern as the classical one with a single

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3. General properties of MPL An evident (and difficult in practice) way to generalise the two-photon effects is to use N-photon light with N= 3, 4 As the TPL it also can be generated by cascade transitions in atoms or by parametric processes, described by the matter nonlinearity x (N) Practically the cascade generators using several three-wave interactions may be preferable. Belinskiiparametric recently proposed a concrete three-photon experiment [24]. In the degenerate case one has the parametric amplifier of the Nth subharmonic w 0/N [25], whose quantum noise consists of N-photon bunches; its Wigner distribution resembles an N-pointed starfish. Passing strong real noise through the amplifier we can generate a close classical analogue of MPL [12]. The N-photon nondegenerate interaction is governed by the Heisenberg equation da1 /dt= a~...a~, where t—~x~ ~ and f~is the classical pump amplitude. Multiplying it by at we find that tm all flkinvariants. are equal and therefore thealloperators (nk— n,) are In the SV case modes are equivalent, so [26] <(nk_nJ)m>=O, N, = = M In the N= 2 case M= 2N2 + N and N / 1 ‘~ (<‘~~ n 2> —N1 2), a1 a2 2 == <~~n2>). In the SV case for all is pump levels (here amoment there a nonstationary —

F= ~exp ( 1W0 t) The N-mode coincidence probability is determined by G ~ I F~2, Suppose that the radiation can get to each of the N detectors by two alternative ways (see fig. lc), then a,,, should be replaced in F by ~a,,, [exp (iOk) + exp (iØ~,,)].As a result —

difference: the lower relative level of the accidental coincidences in case of “nonclassical” input light [121. This peculiarity is a corollary of the only true enigma of quantum theory the complementarity principle (or, as one may prefer, of the nonlocality or nonpositivity of the hidden parameters distribution). Nonclassical could be not the interference phenomena itself, but the input light. This is a rather general rule in optics because any linear equilibrium optical system can be adequately described by its classical scattering matrix [23]. In other words all schemes of fig. 1 could be considered to be the classical devices (not of the simplest kind) for measuring the input light statistics [121. This approach replaces a multitude of nonclassical effects by a single one the superclassical correlations in the input light [11,12]. —

—

350

N

G (N)

fl

I 1 + exp(iOk iØ~,,) 2 —

k=

cos (0— 0’) +...,

—~

(1)

where 0 ~ O~.Thus the interference pattern depends on the sums of the optical paths, connecting each detector with the MPL source (this explains the term AWl [16]). Its high visibility V is due to the superclassical correlation 2re Fl 2/Ni...NN>> 1 (forbetween N=2 thesee modes: refs. f

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[11,12,19,26]). Note that in the classical theory l I2~N1N2 by iteration we ~ In thetocontinuous-mode free-space case convenient suppose that a pump field 1~ hasit aisfinite duration 0— T, T>> 11w 0, and that all measurements areperformedattimest,>T.Asaresultthenonstationary correlation function in the first order takes the form (cf. the N= 2 case in refs. [21,22]) FI...N_

~

> =(—ihY—’(N~l)!

T

X

J J dta

0

d3ra X (N) ~g’~-Dia ...DNa,

(2)

~

where Dap ih~[E~4), ~ xa (ra, ta), Ea E(xa) is the electric field operator and V is the nonlinear sample’s volume. The function Dap has the meaning of the field propagator, which should take into account the boundary conditions and various optical elements. The function F plays the role of the N-photon wave function or of a certain field in the 4N-dimensional space. Eq. (2) can be interpreted as follows: every point-event xa inside the sample(s) V serves as a source of this field F, which is carried by the propagators Dia toward the observation points x,. We can formally consider the effective field F(x1) for a detector at x1 to be created by stimulated scattering in the sample(s) of the advanced fields, emitted by other N— 1 detectors at the moments t of the photon’s registration [22]. This formalism describes also the MPL, emitted by excited atoms [21]. 4. Polarization properties of MPL Consider the influence of the linear optical elements on the field (see also refs. [12,14—16,21—23,27, 28]). In case of stationary experiments it can be described by the scattering matrix (SM) D [23,27]. In the normally ordered moments we can use a simple connection a’ = D*a between the output and input operators even in case of lossy (but cold) elements, when DD + ~ I. As a result the characteristic function and the P-distribution only change their scales [23] and the output moments G’ can be expressed algebraically through the input ones G. In the lossless case it is possible to use the Schrödinger representation and the corresponding photon

30 March 1992

language. The output state

I

>‘

can be found through

the input one, I > =—F(a~)l0>,a1a nd~)the by the 10>SM[27]. following algorithm: I >‘(fig. =F(D The two-mode mixing la) is described by an —

SUYY(2) matrix Dyx~ ( t, r) with components D~,, = 2+IrI2~T+R=l. The XY— D* =t x,Dy can __D* modes differ—r, by ltI the directions, then D describes a BS of an interferometer. In another case the modes differ only by polarizations, then D is the Jones matrix and the indexesx, ymay referto some linear (es, e~)or circular (e~,e) basis vectors. We will use mainly the polarization language. The rotation of the linear basis by an angle (0) is realized by a rotator having t = cos (0) and r= sin (0). The compensator, introducing the relative phase de‘

lay O~—0~E2O,has t=exp(iØ) and r=0. The Mach—Zehnder interferometer has DMZ = D 2D (exp (iØ), 0)D1, where D1 describe the BSs. If they have real parameters with D2 = D j-’ and T= R=0.5, then tMz=cos(O) and r~=i sin(Ø). Thus the interferometer and the rotator are mathematically equivalent (if only the absolute values of t, rare of importance). An arbitrary two-mode N-photon state forms a multiplet with (N+ 1)-fold degeneracy:

I

=

~N c IN—p, p> p=O

Im,n>_=In>~Im>~,

(3)

where ~ I c~ 12 = 1. Thus the state is defined by its “polarization” vector c= (cr), which has the spinor transformation properties [27]. In the N= 2 case one can prepare an arbitrary state by means ofthree 2~-crystalswith types I andspeII cifically orientedand ~~a common pump. According to phase matching (3) (flm N—p) p!fl! I cj,, 12 (4) G’~’~ <:n~n~,: > = ~ (p—l)!(~—k)!~ p

From (3) follows [27] that the output polarization is c, = ~ ~ cq, where S= ~N and D~ is the irreducible SU (2) matrix of dimension (2S+ 1)2. It can be put in the following form: k2 tkt*Irm ( r’~) D(v) — (p!~!q!~!)~2 ~j, k!flm!n! (5) pq — —

‘

where p, q=0, 1,

...,

N,

k 1 =max(0, ~—q), k2=

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min(~7,~), l=q—fi+k, m=ji—k, n—_q~—k.Note the symmetry:

2t~~(_r*)P and from (4) follows the “resolu(~‘)u1’ tion enhancement” [7] and the “multiphoton Malus

D~(r, t) =D~~(t, _r*) =D~(t*,r)

law” [27]: G’°’°’—T’~—cos2”(Ø). (b) Consider the two-photon input state

_r~i(s)(t*, —

_r*)

(6)

.

I ~

ForN=l wehaveD~”2~_—D; forN=2 and 3 we give as examples only the second lines: ~

~

\/~t*r)

(7)

DW2~= (~~,/it2r~,t(T— 2R), r( 2T— R), ~ t*r2) .

(8)

=

tr’~,T—R,

It should be mentioned that the analogous SU (2) description of the BS was given in ref. [28] and of the MPL polarization in ref. [27]. We consider below several examples of the MPL polarization properties. The one-photon polarization vector c= (c 0,2 just c1) as can projected onto the thebeclassical polarization PoincaréInsphere S vector. the N-photon case we have the 52N 2)

projective sphere anda two-dimensional the transformation C_4D(N/on x (I, r)c spans only subspace it. In the usual interferometry the visibility V equals the degree of polarization (or coherence) p, which is an SU(2) invariant. All pure one-photon states havep= 1, but for N~2p can be zero. Nevertheless the field in such states has some transverse structure, revealed by the registration of the higher moments, see fig. 2. This property may be called the “hidden” or “higher-order” polarization (or coherence). (a)LetI >=IN,0>,i.e.co=l.From(5)wefind the well-known binomial distribution: c=D~=

=Il>+Il>~.

(9)

Bothrotator the angular andp aresozero, the leavesmomentum the vectorN÷ (9)—N_ unchanged, the field seems to be completely unpolarized. An elegant group-theoretical classification of such states for the multimode case was given by Karassiov [29], who called them “scalar”. Butifwepassthestate (9) througha ~A-platewith an orientation angle U (which has t= ~ [1 + i cos(20)] and r ~i sin(20)), then we get G,,k (:n~’:> = cos2(20) and ~ =sin2(20). This an example of the hidden polarization. that aiscompletely scalar field, invariant to any SUNote (m) transformation, dent modes [23].is the thermal field with indepen(c) The state Ii >.~ II >~ also has hidden polarization. According to (4), (7) after a rotator (or interferometer) G’kk = sin2 (20) and ~ = cos2 (20); thus if 0=45°,then ~ This is the correlation suppression effect [4,8,12,14,15]. Let now N= 25 be even and I > = IS> IS> ~, i.e. C 5:= 1. According to (5) the out-amplitude of the same state is s

s

2

c~=D~=~(k) Tk(_R)S_kPs(X),

(10)

where x=cos(20) = T—R and P~(x) are the Legendre polynomials. Thus the out-moment ~ (5’.c~)~goes to zero S times as T changes from 0 to 1, see fig. 2. This is the generalisation of the correlation suppression effect.

5. Mixing of arbitrary two-mode states a

b

Fig. 2. The higher-order Malus laws: the dependence of basisx’, (a) and y’ of relative <:n’,~n~:> to the(b) initial on the basis orientation x, ‘. The solid of thelines detectors’ conespond to the input states Ik>~Ik>~ with k=l (a) and 2 (b), see eq. (10); the dotted lines to two independent coherent states of equal intensities, eq. (13). The input light in all cases is not p0larized in the usual sense, i.e. = = const.

352

In the Heisenberg approach the optical scheme 12~a, where performs amthe (ar,binary a~) (a, transformation b). Analogously a’=D~’ A~’= D (N/2)*A (N) where A~,N) =a~’b~/(~!p!)”2 and p 0, 1 N, ~ = N—p. Thus we find the transformation rule for the fourth moments:

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PHYSICS LETFERS A

~ 2R

,

2S

~ ~R) p=O q~0

~

(11)

where 2R=—k+l, 2Smm+n, p~=2R—p,ã~m2S—q and ~S)mD~S)[j!(2S_i)!/j!(2S_j)!]l/2 This transformation holds in the classical theory as well, where a and a + ma * are c-numbers and A (N) is a classical spinor. The SU (2) matrixes D ~ reflect the symmetry ofthe problem, not its quantum character, so the photon language is not obligatory, If the in-modes are independent, stationary and equivalent, then eq. (11) gives Gkj~=N2s~ I ~ l2g~g~, (12) where g = G”°/N”and 2S= k+ l_—p +~. For coherent states g ~ = 1, 50 G”=~N2[3+cos(4Ø)], G2’2’= ~¼N4[4l +20 cos(40)+3 cos(80)]

,

(13)

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ref. [12]). The inertial detection diminishes V, see ref.Consider [12]. next two symmetrical in-beams with independent phases and correlated intensities, K~ 0. For SV ,c=l, for a pure two-photon state from (3), (4) follows tC=K’= — 1. For thermal beams a2=N2+N, gxy=l—iclKmin, —NI (N+ 1), and from (14) follows ?Cmin

V’ (1 — 2Kmin/K) g~, 1 = (g~, — 1) cos2 (20), a’2 ic’ cos2 (20) (15) —i = 1 + K sin2 (20), = a K l+Ksin2(2Ø) —

—

—

Note, that the second scheme with D’=D’ would recover the initial statistics. If in eqs. (14a), (14b) we put G~~=G~~=0 and consider T and R to be stochastically modulated in time, then we get a trivial classical description of the experiment [301: the observed output moments G’kl are linearly connected with the input moment ~ through T2, R2 and TR, averaged in time.

where we put T= cos2 (0) (see fig. 2). For thermal fields g(P) =p! and G’= G, i.e., there is no interference. Thus a plane wave with two orthogonally polarized components, which have independently diffusing phases and equal mean intensities with nonthermal fluctuations, has the hidden polarization. Let now ==..._—0, then from (11) follows

Suppose now that the in-beams have different frequencies w~w, and that the scheme has some dispersion: arg ( t) = a (w) and arg ( r) = fl( w). From a’ = D*a we find (T=R = 2G~~ G~Y 1— Vcos(y), V= ~ + ~ + 2G~~’

G’~~ = T2GXX + R 2GYY + 4TRGXY,

(1 4a)

Let 1A, l~be the distances between the BS and the detectors, then the displacement of the BS by z= ~(lA—lB) gives a(w)=/3(w)=zw/c and y=

(1 4b)

2z( wi,, — WY) /c. Now (16) describes the spatial quantum beats, observed in refs. [5,8] using SV. According to (16) it is possible also to use classical light (~iass~ ~). For uncorrelated identical beams, correlated Poissonian and Gaussian beams, and for SV we find from (16) that V equals respectively

~

TR(Gxx+GYY)+(T—R)2Gxy

1 + Vcos (40) ~ — 4G~~

,

— G~.

—

~

+~

+

G~~’

(1 4c)

In classical theory GXY ~ ~(G~+ ~ so Vciass ~ ~. In the SV case ~ 1 /N= 2 (here g~=Gk// NkNI), therefore V=(N+l)/(3N+ 1) and ifN<< 1 then V=l (thisfollowsfrom(l4c)atonceifweput there G,~=0).This limit is achieved with low intensity TPL, which was actually used in refs. [2—101. The condition N>> 1, which gives V= ~, corresponds to strongly squeezed light (classical or quantum, see

y=a(Wx)+/3(Wx)—a(Wy)—fl(wy)

1 N+K 1 +ic(1+ 1/N) l+g~~’2N+,c’ 3+ic(l+l/N)’

(16)

.

2N+ 1 4N+l (17)

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6. Four-mode mixing ~

Using the notations of fig. lb, we find = TA TBGXX +RARBGYY + TARBGXY +RA

TBGY~

+tAtBrArBexp(iO_)Gj

(18)

,

=

=



(19)

±—

x

y,

t12exp(iO2), Fk=(N~+Nk) where Ø~are the pump phases. Let N~=N 3, and tM=rM=2’, then ~ 1+ Vcos(Ø~—Ø~+O~), where again V= (N+ 1) / (3N+ 1). (b) The J= 0—*J’ = 0 two-quantum transitions in atoms give

G=0,

>=2~/2(I

(20)). Thus it is possible to repeat the polari-

zation EPR experiments using the usual light, which

are possible delays and otherfourth moments are supposed to be zero. There are two possible ways to modulate the coincidences: bythe changing 2(0M) or delays ~the mixing parameters TM=cos (a) In case of two SV sources with a common pump [6,9,16—18] we have Gkk=2N~+Nk, G~~=N~N 3,, G _F*F

I

(22)

for thermal beams g= 2 and

(cf.

A+ B+ G~=

0±= 0~— 0~ ± (0~ — 0~), G

,

G~,,~=l+~ cos(20A—20B)

where Gk,

(d) In the polarization experiment 0— = 0, so (21) takes the form 2= 1 +(TATB+RARB) (g— 1) G~,,~/N +2(TATBRAR B)\i/2.

+2 Re[lAtBr~r~exp(i0+)G+

30 March 1992

l>~Il>~+Il>~l1>~) .

would give V~~. Actually such an experiment has been already performed in ref. [321. As the last example consider the Franson scheme, see fig. 1 c and refs. [10,12,201. Each beam is transformed by the interferometer: ak=~ak[l+exp(i0k)I where Ok = Wklk/C (ik are the path differences, B). Simple calculation gives

k=

A,

G~B=~GAB[l+~cos(OA+0$)+~cos(0A—0B) +cos(0A)+cos(OB)].

(23)

1k are much larger than the coherence lengths, If thethe field at each detector is a superposition of then two independent beams — the short-path one and the long-path one. Suppose that the frequencies Wk are slowly varying in time, then the last two terms in (23) would disappear after averaging and we again have two options: 0A±OB=const.

Now Gkk=G+=~,G~ 3,=G.=0, and if 0+=0, then eq. (18) takes the familiar form [1]

G~= I tA tB + rA rB 12 20A — 20B) ~ . (20) ~[1 +cos( (c) Let G+=0, and G_~A0.In case of two independent beams with amplitudes ~ a,,,, each beam having been split into two, we have G~~=G= N~N 3,.In the symmetrical case with real t, r 2= 1 +(T~T~—R~RB)(g—2) G~,,~/N + ItAIB exp(iØ ) +rArB 12, (21) =

where g= GXX/N 2, In the interferometer case T= ~ so ~ 1 + Vcos(0), where V= (1 +g) ~. Thus two one-photon, coherent and thermal beams give V= 1, ~ and ~.

354

Actually the scheme of fig. lc can be reduced to the four-mode scheme of fig. lb considered above if we take the interferometers’ output mirrors in fig. 1 c as the mixing mirrors in fig. lb. Two correlated shortpath beams correspond to the x-source and two longpath beams to the y-source. In the SV case the interference phase is 0A+OB=k(IA+IB) [10]. If the in-modes are excited coherently by an usual source, then the phase is k(IA—IB). If we take into account the finite radiation bandthen for width 2y QmyT<< and the coincidence 1 we get V=(N+ time window l)/(3N+T 1) [12], for SV and ~ for Gaussian light. But if Q>> 1, then V=(N+l)/(2QN+2)forSVand V=1/2Q<
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nomena, based on the scattering matrix (input— output) formalism, was demonstrated. This formalism reduces many problems to standard algorithms and facilitates direct comparison with the classical theory. Three basic types of effects have been singled out, depending on the presence of phase correlation, anticorrelation, or on the absence of correlation in the input light beams. The phase fluctuations are converted by the optical scheme into the observed intensity (anti)correlations. All effects, observed with two-photon light sources could be observed also with the classical light if the radiation bandwidthis smaller than the reciprocal detectors’ averaging time. Perhaps some of the modifications considered here may be of interest for experimental investigation.

[7] E. Mohler, J. Brendel, R. Lange and W. Martienssen, Europhys. Lett. 8 (1989) 51. [8] J.G. Rarity and P.R. Tapster, Phys. Rev. A 41(1990) 5139. [9] J.G. 2495.Rarity and P.R. Tapster, Phys. Rev. Lett. 64 (1990) [10] P.G. Kwiat, W.A. Vareka, C.H. Hong, H. Nathel and R.Y. Chiao, Phys. Rev. A 41(1990) 2910. [11] B.Ya. Zeldovich and D.N. Klyshko, JETP Lett. 9 (1969)

Acknowledgement

[21] D.N. Klyshko, Phys. Lett. A 128 (1988) 133; Soy. Phys. Usp.32 (1989) 555.

I am grateful to A.V. Belinskii for fruitful discussions.

[22] D.N. Klyshko, Soy. Phys. Usp. 31(1988) 74; Soy. Phys. JETP67 (1988) 1131. [23] D.N. Klyshko, Phys. Lett. A 137 (1989) 334. [24]A.V. Belinskii, JETP Lett. 54 (1991) 11. [25]P.V. Elyutin and D.N. Klyshko, Phys. Lett. A 149 (1990)

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