11 November 1996

PHYSICS LETTERS A

ELSl3’IER Physics Letters A 222 (I 996) 363-369

Correlation between photocounts at beam splitter outputs for a pair of photons

John Jeffers ’

Department of Physics and Applied Physics, University of Strathclyde. John Anderson Building, 107 Rottenrow, Glasgow G4 ONG, Scotland, UK

Received 3 November 1995; revised manuscript received 10 May 1996; accepted for publication 28 August 1996 Communicated by P.R. Holland

Abstract

It is shown using conventional quantum optics theory that, contrary to recent predictions, the detection of photons in one output arm of a beam splitter does not depend upon any optical delay imposed in the other output arm before detection.

PACS: 03.65.8~: 42.50.Ar; 42.50.D~

1. Introduction

One of the predictions of quantum mechanics is that the result of a measurement made at one spacetime point can affect the results of other measurement events outside its light cone in a nonlocal manner. This

principle is the basis of experimental tests of Bell’s inequality. Such experiments use entangled states, where the entanglement is based on the conservation law for a quantity such as energy or angular momentum. It is this entanglement which is responsible for the nonlocal correlation between measurement events separated in a

spacelike way. In a similar vein Widom et al. [ l] have recently studied the photodetection of a pair of unentangled photons

with a “reasonably well-defined” frequency difference. The photons are incident upon a beam splitter from the

same arm. Output is allowed to fall upon two detectors, one in each of the possible output arms for the input photons. Light which ends up in one of the output arms undergoes a temporal delay before detection (Fig. 1) . They find that the correlation between the expected number of photons counted in each arm affects the number of photocounts in the arm with no delay. If this is correct it means that counts made in the future in the delayed arm affect those in the past in the undelayed arm. In this Letter it is shown that that conventional quantum optical theory predicts no such future-past correlations for this experimental configuration. This is in line with what one would expect on the basis of causality for the results of such an experiment.

* E-mail: john@phys.strath.ac.uk.

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364 J. Jeffers/Physics Letters A 222 (1996) 363-369

ta -td+T

Dd

Arm C

* DC -I

tc- tc+T Arm b

Fig. I. Detection of beam splitter outputs. Inputs from arms a and b are transmitted and relected at the beamsplitter and allowed to fall upon detectors in arms c and d. In arm c the light is allowed to travel further before detection.

2. Two-photon state

The state which we shall use in our paper to illustrate our argument is the continuous-mode two-photon state

of light [2,3]

12) = $it(ol)A’(a2)/0),

where the operator A^t (a1 (2) ) is

aq2,> = s dw Q.,&+~(w), (2)

where the integral runs from 0 to co, (~1 and (~2 are normalised functions of frequency, and N is the normalisation constant for the state, which is given by

IJ

2

JNj2 = 1+ do cu;(w)a2(w) . (3)

The normalisation constant therefore depends upon the mode overlap between the two photons. In order to illustrate our results we will use Lorentzian spectral functions (other normalisable functions, such as Gaussians,

will do equally well);

w(w) = &QGeimf’

0 - WI + iyl ’ (4)

(Y2(w) = meiot2

w-w2+iy2’ (5)

with t2 > tl. The frequency difference of the pair of photons is known to within the widths of the Lorentzian

functions. The Fourier transforms of the spectral functions give the possible arrival times of the photons at x = 0. In the narrow bandwidth approximation (~1~2) << 01~2)) it is easily shown that

L~I (t) = -i~e(-‘“‘-Yl)(t-tl)~( t - tl), (6)

where e(t) is the Heaviside step function. This state represents two one-photon pulses of light, of (in general) different central frequencies and widths, for which the probability for a photon to be at position x = 0 is

J. Jeffers / Physics Letters A 222 (1996) 363-369 365

an exponentially decaying function of time starting from an initial value at time tl or t2. It is instructive to calculate the normalisation constant,

i(ol -YI)T 2 ]Nl2 = 1+ “z(;, +y2) / 9 (7)

where R = 0~2 - wi and r = t2 - tl. We can see that the size of the overlap of the two photons depends upon their average frequency difference, their average arrival times and their relative widths. The normalisation constant has the usual value of & for identical spectral functions LYI = LYE.

3. Photodetection of beam splitter outputs

The system which we consider is the same as that studied by Widom et al. [ 11, a beam splitter in free space

at x = 0 whose outputs are allowed to fall upon detectors. The beam splitter is perhaps the simplest of optical

devices. It has been the study of a number of theoretical papers in quantum optics, and has been a central

component of various experiments which test the nonclassical nature of light [ 3-81. Inputs from arms a and b

are transmitted and reflected into arms c and d (Fig. 1) according to the usual beam splitter relations [ 41

Z(o) =7%(o) +Ri(@), (8) A 1

d(w) = Ib(C.0) +R&(w), (9)

where, for example, 21(w) is the continuous-mode annihilation operator in arm a [ 21. For a lossless beam splitter the reflection and transmission coefficients satisfy

1712 + (RI2 = 1. (10)

Provided that modes 6 and 6 are independent then the independence of modes E and d^ is ensured by the

criterion

Z?* + 7’R = 0. (11)

Light which emerges in arms c and d is allowed to fall upon identical photodetectors D, and Dd. We will calculate three quantities which can be measured by these detectors, the expectation values of the number of

photocounts at each detector, and the expectation value of the product of the number of photocounts. In the narrow bandwidth limit the expectation value of the number of photocounts in arm c is

tc+T

= 2 1 dt 1 dw / dw’ ei(o-w’)t(Ct (w)E(w’)), (12)

where the second line is obtained by substitution of the Fourier relation between the frequency and temporal

operators [ 21

2(t) = & J do P(w)e-‘“‘. (13)

Here 77 is the detector quantum efficiency and tc is the start of the detection window of length T. Strictly

speaking we ought to replace the time t in the exponent by t - x/c, where x is the propagation distance from

366 J. Jeffers/ Physics Letters A 222 (1996) 363-369

the beam splitter to the detector. In fact this is unnecessary because in free space light travels at speed c no matter what its frequency, and so all propagation delays can be accounted for simply by allowing arbitrary detector start times and an arbitrary integration time. This is borne out by the occurrence of a conjugate pair

of operators in ( 12). In their paper Widom et al. [ 1] assume that there is a delay in arm c which allows the

photons which reach detector Dd to be detected before those which fall upon D,. Here all propagation-induced

time delays are accounted for by adjustment of tc and t d, where td is the detection window start time for detector Dd. Any dependence of measured quantities at either detector on an optical delay introduced in arm c

or d should therefore show itself as a functional dependence on the difference in detection times tc - td. The beam splitter relations can be used to write the number of counts as

(m,) = g rTdt/ dw / dw’ eit”-“‘)‘(81(w)a(w’))

k

for vacuum input from arm b. If the input from arm a is a two-photon state then

then be evaluated as

The first term is the squared modulus of the probability amplitude for photon 1 to be detected at DC, and the

(14)

this expectation value can

+c.c. . >I (15)

second term is the corresponding quantity for photon 2. The other terms are the interferences between the two amplitudes. This result simplifies in the limit where tc is a time much earlier than either tt or t2, and T is long

enough for the photons to reach the detectors with absolute certainty. Then the Fourier relation for the spectral

functions and the expression for the overlap of the pair of photons gives

(m) = 2T4712 (16)

as expected, because in the limit where the detector efficiency and the transmission coefficient tend to unity

the detector ought to see both photons. The long detector integration time result has previously been derived by Fearn and Loudon [ 31. Similarly we can write for the detector in arm d

(md) = $$f ‘yTdf / do / dw’ ei(o-w’)f(Gt (w)a(w’))

td

dw a;(w)q(o) +c.c. (17)

It is clear from the general expressions that neither (m,) nor (md) depend upon the time difference t, - td, and so they are independent of any delays in the output arms of the beam splitter. These results can be illustrated using the Lorentzian spectral functions defined in the previous section. For example, the number of counts at

detector D, is

J. Jeffers / Physics Letters A 222 (19%) 363-369 367

(m,) = q (e-“““ql - e-*y’=)e(r, - r,) + 1 --t 2

+ (1 -e-2y1(rc+T-rl))O(rt -t,)O(t, +T-tl) + 1 -+2

+ 4~1~2

a2 + (r1 + Y2)2

[e(in-y2)(tc-t2)-yl(tc+r2-2fl) (1 _ ,W~-Y~--Y~V)~(~~ _ r2j

+ (e -2Y17 _ e(iR-Y2)(r~+T-f~)-Yl(fc+~+f2-2fl))~(~2 _ t,)~(l, + T _ r2) + c.c.3

>

, (18)

where the shorthand 1 -+ 2 means replace r(y) 1 by r(y)2. Different terms in Eq. ( 18) can be related to different orderings of the various detection times and photon arrival times. For example, the term proportional

to 8( r, -rl ) is the square of the probability amplitude for the situation in which there is some chance that photon

1 arrived at detector c before the detector start time. Similarly, the term proportional to O( rl - rc) f?( rc + T - rl ) is the square of the amplitude for the situation where the possible photon arrival times span the latter part of

the detector integration time. In this case there is also some chance that the photon will arrive at the detector after the end of the detector integration time. Similar interpretations apply to other terms.

The correlation of the number of photocounts at each detector is found in the same way,

r,+T t,j+T

(mcp) = r12 J J dr dr’ (2+(r)t(r)i+(r’)~(r’>). rr fd

(19)

Again we use the Fourier transform and beam splitter relations to write this correlation as

(m md) = 7j+y2pq2 “+T fd+T c 4,rr2 J dr J &J do, . . dw”’ ei(w-o~)rei(o”-o”‘)I’(at(W)ii+(Wn)~(W’)~(O1,l)).

fc td

(20)

This expectation value can also be evaluated for general spectral functions; it is

(m c md) = ‘2’7’2’R’2 'rlTdr 'd+Tdr' lq (r)a*(r’) + aI (r’)a2(r) 12. INI2 J J

(21)

The first term inside the modulus is the amplitude for photon 1 to be detected at DC and photon 2 to be detected

at Dd, and the second term is the amplitude for the converse to occur. In the long integration time limit this formula simplifies to

(m,md) = 2,72]7(2]R12. (22)

This result also agrees with earlier authors’ calculations [ 31. In their paper Widom et al. [ 1] point out that the correlation depends upon the delay imposed in arm c. It is not completely clear from the above expression that this is so and so we illustrate the formula using Lorentzian spectral functions,

368 J. Jeffers / Physics Letters A 222 (1996) 363-369

(m md) = 71*1~l*1~1* c

INI2

x {[em ( 2Yl(kll) (1 _ e- 2Y’T)B(tc - tl) + (1 -ee- 2y’(rc+T-‘wt, - t,)f3(t, +T - t1)]

x [e -*Yl(cJ-rz) (, _ e- 2y2T)O( td - t2) + (1 - e- 2y’(rd+T-‘2))o(t2 - td)o(td + T - t2)] + 1 +, 2)

+ 4y,y2e*(~~t~+~2~2)

a*+ (r1 +Y*)*

{[e(-iR-yI-y2)h(l _ e(-ifl-yl-y2)T)e(t c

_ t2)

+ (,(-ifl-yl-y2)r2 - e(-in-Yl-Y2)(fc+T))e(t2 - t,)e(t, + T _ t2)~

x [e(iR-yl-y2)fd 1 _ e( ( ifbJ~-Y2)T)e(td _ t2)

+ (e(ifi-yl-~2)r2 - e(iR-y’-Y2)(‘d+T))e( t2 - td)e( td + T - t2)] + c.c.} , (23)

where the shorthand 1 * 2 means replace t(~)1(2) by t(~)*(l). This formula is a little opaque, but it is clear that the interference terms contain factors which, when multiplied together, do indeed depend upon the

measurement time difference tc - &j, and therefore on any delay introduced. Such a dependence is to be expected

in a coherence measurement of this kind. As has previously been pointed out this delay time has no effect

on the individual expected numbers of photocounts at either detector. Also, the dependence on delay clearly vanishes in the long detector integration time limit,

This leads us to the question of how such results were obtained by Widom et al. [ 11. They derive their

results using a different formalism to that described here. Firstly they write down a “properly symmetrised

two-photon wavefunction” (their Eq. (4) > ,

(W4)

in which, for example, yi is the amplitude for one of the incident photons to arrive at detector i, and Si denotes

the corresponding amplitude for the other photon. The expected number of photons counted at detector Dd, and the correlation can be written in terms of this,

(md) = 21+dd1* + kdl* + I$dcl*~ W5)

(mcmd) = j$cdl* + ~~dc/dc~*~ W6)

These last two equations allow the number of counts at detector d to be written in terms of the correlation. It

is the denominator in (W4) which gives rise to the dependence of the expected number of photocounts upon delay. This is an artifact of the fact that the authors have chosen to write the wavefunction in terms of detected amplitudes. In fact the wavefunction would be better expressed in terms of photon amplitudes in the input arms of the beam splitter. This form of wavefunction contains no dependence on delays imposed after the beam

splitter, and a calculation of the expected number of photocounts at either detector from this starting point is in agreement with the long integration time limits of Eqs. (1.5) and (17).

There would also appear to be another problem with the analysis in Ref. [ 11. Any valid quantum mechanical

analysis of this system ought to take account of detection times. In order for the frequency difference between the two photons to be resolvable in principle then the detectors must have an integration time of greater than the reciprocal of the frequency difference. This is a long time, and contains many optical cycles at the individual photon frequencies. The cosinusoidal modulation of the number of photocounts depends upon an unexplained phase fatcor 4, which, when detection times are accounted for, contains temporal modulations at the individual photon frequencies, and when these are integrated over the detection time the modulation will average to zero.

J. Jeffem / Physics Letters A 222 (1996) 363-369 369

4. Conclusions

In this Letter we have studied the photodetection of a pair of unentangled photons which fall upon a beam splitter from the same arm. The probability of detecting photons at an earlier time in one of the beam splitter

output arms does not depend on the probability of detecting photons later in the other output arm. This is in line

with intuitive expectations based on causality. If, however, one considers the correlation between photocounts in different output arms, then, as expected, it depends upon the measurement time difference, and thus on

any delay introduced after the beam splitter. This is only true, however, if there is at least some chance for

the photons to arrive before the beginning or after the end of the detection window. Otherwise the detector integration time effectively runs from ---co to 00. However, the fact that the correlation can in some cases

depend upon the delay is irrelevant to the number of counts obtained in either arm as these quantities do not depend upon the correlation. Thus it has been shown here that conventional quantum optical theory predicts that the number of photocounts in one output arm does nor depend upon any delay introduced before detection

in the other output arm.

Acknowledgement

The author would like to thank Steve Barnett and Norbert Ltitkenhaus for useful discussions and the U.K. Engineering and Physical Science Research Council for financial assistance.

References

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