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Photonic de broglie waveinterferometersStephen M. Barnett a b , Nobuyuki Imoto a & Bruno Huttner ca NTT Basic Research Laboratories , 3-1 Morinosato-Wakamiya,Atsugishi, Kanagawa, 243-01, Japanb Department of Physics and Applied Physics , University ofStrathclyde , Glasgow, G40NG, Scotlandc Group of Applied Physics Optique , Université de Genève , 20 Ruede l'Ecole de Médecine, CH1211, Genève, 4, SwitzerlandPublished online: 03 Jul 2009.

To cite this article: Stephen M. Barnett , Nobuyuki Imoto & Bruno Huttner (1998) Photonic de brogliewave interferometers, Journal of Modern Optics, 45:11, 2217-2232, DOI: 10.1080/09500349808231234

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JOURNAL OF MODERN OPTICS, 1998, VOL. 45, NO. 11, 2217-2232

Photonic de Broglie wave interferometers

STEPHEN M. BARNETT?$, NOBUYUKI IMOTOtll and BRUNO HUTTNER4 t N T T Basic Research Laboratories, 3-1 Morinosato-Wakamiya, Atsugishi, Kanagawa 243-01, Japan 1 Department of Physics and Applied Physics, University of Strathclyde, Glasgow G4 ONG, Scotland 0 Group of Applied Physics Optique, 20 Rue de 1’Ecole de Medecine, UniversitC de Genltve, CH1211 Genltve, 4, Switzerland

(Received 3 February 1998)

Abstract. In the recently proposed photonic de Broglie wave interferometer, sophisticated beam splitters are used to split the de Broglie wave of a set of photons. The photonic de Broglie or collective phase shift can, however, be found in conventional interferometry if the full photon statistics are recorded. We propose a variation of the original photonic de Broglie wave interferometer. We show that the collective phase shift can and has been found in inter- ferometry using photon pairs and propose two new photonic de Broglie wave interferometers in which normal beam splitters are used. Both of these rely on conditioning to select the cases for which all the photons behave as a single object.

1. Introduction It has long been recognized that the resolution of interferometers is ultimately

limited by quantum effects [l, 21 and that the shot noise limit to resolution can be passed by making use of non-classical states of light [3-71. All this work has been based on existing interferometers in which the optical beams are divided and recombined using beam splitters. These devices linearly superpose the two input fields to produce the two output fields [8-lo]. Jacobson et al. [ l l ] have proposed a new class of interferometer which incorporates an ‘effective’ or collective beam splitter (based on a proposal by Davidovich et al. [12]) and acts collectively on all the photons in an input mode so that all of them are transmitted or all are reflected. If the state of the two input modes a and b is l$)alq5)b, then the effective beam splitter acts to give output modes in the entangled state

By using two such effective beam splitters, it should be possible to carry out an interferemetric measurement at the so-called Heisenberg limitl. Jacobson et al. expained the enhanced resolution of their proposed interferometer in

11 e-mail: nobu@will.brl.ntt.co.jp. 7 It has also been shown to be possible to reach the Heisenberg limit using conventional

beam splitters and the same Fock state in each of the two input modes [13].

0950-0340/98 $12.00 0 1998 Taylor & Francis Ltd.

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2218 S . M . Barnett et al.

terms of the concept of photonic de Broglie waves. The idea is that, because all the photons in an input mode are behaving as a single indivisible system, their effective de Broglie wavelength is the optical wavelength divided by the number of photons in the mode ( X ~ B = X / n ) . This, in turn leads to a detected phase shift which is larger than that found in conventional interferometers by a factor of the photon number.

In this paper we describe a variation of the interferometer of Jacobson et al. which requires only one collective beam splitter. We also discuss three types of interferometer based on conventional optical beam splitters which can be made to show evidence of collective interference of photonic de Broglie waves. These are a Mach-Zehnder interferometer with equal numbers of photons in each input [13-151, a Mach-Zehnder interferometer with a two-photon absorber, and a Franson interferometer [16-181. The last two of these have already been shown to exhibit the collective effect that we seek for n = 2 [14, 17, 181. This manifests itself in a detected phase shift that is larger than 2xl /X, where I is the change in optical path length. In each case we have to provide the means either to force the incident photons to travel along the same path through the interferometer or to post-select those occasions on which they have done so. We show that, in each case, the enhanced phase shift can be understood in terms of an effective de Broglie wavelength for a number of photons behaving collectively. We begin with an introduction to the idea of a photonic de Broglie wavelength. In section 3 we present a brief account of conventional interferometry and discuss the interference in terms of photonic de Broglie phase shifts. We describe our modification of the interferometer described by Jacobson et al. and the other three interferometers in section 4, emphasizing the role of collective photonic de Broglie phase shifts. The paper concludes with a brief discussion.

2. Photonic de Broglie wavelength The phase shift associated with the propagation of a monochromatic, paraxial

or near plane wave through a distance 1 is Aq5 = 2xl /X, where X is the wavelength of the light. In the quantum theory of light, this phase shift can be associated with the action of a unitary operator

( 2 ) o=exp(-iAq57i) =exp(-iAq5iitci)

on the state of the field mode. Here ii, iit and 7i are the usual single-mode annihilation, creation and number operators respectively. We can associate the phase shift with the field operators or with the state vector. In the first of these, the annihilation operator becomes

ci + UtiiU = iiexp (-iAq5). ( 3 )

In the second alternative, the action of this unitary operator on a state vector I$) = En c,ln) leads to the phase shifted state

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Photonic de Broglie wave interferometers 2219

Figure 1. Schematic diagram of a Mach-Zehnder interferometer with a phase shift Ad in one arm.

so that the amplitude associated with the photon number eigenstate In) is multi- plied by the phase factor exp(-inA#J).? The majority of interferometers give a signal that is sensitive to the product of the annihilation operator in one internal path through the interferometer with the creation operator for the other path and so they are sensitive to the phase shift of the annihilation operator given by equation (3). The larger phase shifts contained in the shifted state (4) are present in conventional inteferometers but only become apparent if we record a signal that is sensitive to them. They will comprise the full phase shift only if we can find the means to suppress the signal associated with the phase shift A#J. In the next section we show where the higher phase shifts are to be found in conventional inter- ferometry before describing, in section 4, four approaches to suppressing the A4 signal so that only the higher-order photonic de Broglie interference is found.

3. Conventional interferometry The ideas discussed in this section apply to all commonly occurring inter-

ferometers exhibiting interference of classical electromagnetic waves. For brevity, however, we shall limit our discussion to the Mach-Zehnder interferometer depicted in figure 1. The interferometer consists of an input mirror M1 and an output mirror M2 and a phase shifting element in one of the internal paths which induces a phase shift A$. We choose the input and output mirrors to be symmetric, lossless beam splitters with equal transmission and reflection coeffi- cients.$ The operation of the interferometer can be adequately described by reference to only a single pair of field modes. The positive frequency part of the field is then proportional to an annihilation operator, the form of which depends on position. The six annihilation operators i?; and & are related by their interaction at the input and output 5&50 beam splitters [3-5, 8-10]:

t It is interesting to note that this simple observation is all that is required in order to

$ Losses are important in some problems and could be incorporated but would only derive unambiguously the form of the probability density for the optical phase [19, 201.

complicate our presentation. For further details see [21, 221.

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2220 S. M. Barnett et al.

a3 = - [ i 2 + i exp (-i A$)&] 2'/2

=${[I -exp(-iA$)]& + i [ l +exp(-iA$)]&}

- 1 b3 = [exp (-i A$)& + iiz]

=i{[exp(-iA$) - 1161 + i [ l +exp(-iA$)]i&}. (5 4 The phase shift A4 is found by measuring the difference between the numbers of photons detected in each of the output modes. The expectation value of this quantity is

(hi&) - (b^t,&) = ((461) - ( c i i i l ) ) cos(A$) + ((iiib^l) + (Gi i i ) ) sin(A$), ( 6 )

which clearly exhibits a simple harmonic dependence on the phase shift A$. In order to find the higher-order phase shifts depending on the photon numbers, we need to look at the photon statistics rather than just the mean numbers.

Consider first, the case in which input mode a contains exactly N photons and input mode b is prepared in its vacuum state.? We can find the full photon statistics by using the photon number moment generating function for output mode a alone. The statistics for output mode b can then be determined from those for mode a by using the fact that the total photon number is N . The required moment generating function is (see appendix A and [ 2 3 ] )

where the colons denote normal ordering and we have used equations ( 5 c ) and (5 d). The probability that n of the N photons find their way into mode a3 is then

1 P(n) =- ( - $ ) ' M ( p ) I = 2-NCf[l - cos(Aq5)]"[1 + cos(A$)lN-". (8)

n! p=l

A Fourier expansion shows that this probability contains terms which depend on n A4 for all n between 0 and N , corresponding to photonic de Broglie phase shifts for all possible photon numbers between 0 and N . An interesting exception to this occurs for n = N / 2 for which we find that

so that the probability depends only on even multiples of A$. We can gain some insight into the origin of the appearance of these

dependences on nAq5 in the photon statistics by considering the amplitude for a given number of photons to pass through the interferometer via path b2, with the

t The photon statistics for an arbitrary but known input state in mode a can be found by weighting the results found for an N-photon input state by the probability that there are N photons in the input.

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Photonic de Broglie wave interferometers 2221

remainder passing through arm a2. The input state can be written in terms of the operators for the fields inside the interferometer as

where 10) is the two-mode vacuum state. The phase shift applies the unitary transformation (2), with ii replaced by &, to produce the state

N I$(A$)) = m- 1/22-N/2 Cy(iii)l(i@)N-l exp [-i(N - Z)A$] 10). (11 )

Each term in the summation has been subjected to a different phase shift, the value of which is proportional to the number of photons passing through the arm having the phase-shifting component. The output photon number probability distri- bution results from interference between the amplitudes for all possible distribu- tions of the photons between the two paths a2 and b2 and so depends on all the phase shifts HA$. This conclusion is also true for other input states for which the moment-generating function can be found by weighting equation (7) by the photon number probability distribution. The moment-generating functions for mode a3 given coherent and thermal input states are

I=O

- n M:yherent (p ) = exp ( -p [ 1 - cos (A$)]) (12 a)

and - - 1

M ~ ~ m a ' ( p ) = (1 +p;[1 -cos(A$)]) (12 b)

respectively, where ii is the mean photon number in the input mode a1 . These are simply the moment-generating functions for a coherent or thermal state with a mean photon number fi[l - cos (A4)]/2 (see equations (A 7) and (A 8)) so that the state of mode a3 is of the same form as that for the input state but with a phase- dependent reduction in the mean photon number. It follows that, for these 'classical' states, the photon statistics depend on the phase shift only through this change in their mean photon number. This should be contrasted with the behaviour found for non-classical states such as the photon number and squeezed vacuum states for which the form of the output state and its statistics depend on the phase shift. If mode a1 contains precisely N photons, then the moment-generating function is given by equation (7) which is not of the form associated with a photon number state (A 6). If mode a1 is prepared in a squeezed vacuum state with mean photon number ii, then the corresponding moment-generating function is

1 + p[1 - cos (A$)]. - - P2 [l - cos (A4)l'fi ) - ' I 2 , 4 ( 1 3 )

which again is not of the form associated with a squeezed vacuum state given in equation (A 9). The dependence of the output photon statistics on the phase shift in conventional interferometers is complicated by the fact that it also depends on the statistics of the input state. In order to find a signature of the purely photonic de Broglie wavelength, we must either constant an interferometer that sends all the photons along the same path through the interferometer [ l l , 14, 151 or find the

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2222 S. M . Barnett et al.

means to select those occasions on which they have all followed the same path. It is, of course, essential that we do not learn which of the two paths the photons have followed as this will destroy the interference altogether.

4. Photonic de Broglie interference 4.1. Collective beam splitters

The effective or collective beam splitter proposed by Jacobson et al. [ll] acts collectively on all the photons in each of the two input modes. This works by entangling the field modes with an atom, with ground state lg) and excited state le), in such a way that the reflection or transmission of the light depends on the atomic state. Such conditional transformations of the field have been realized in cavity quantum electrodynamics experiments with Rydberg atoms [24] and are at the heart of developments in quantum computing [25]. The collective beam splitter performs the unitary transformation associated with the unitary operator

P = lg)(gl+ le)(el exp (-(it6 - it;) exp ( ixi t i ) . 2

The action of this operator on an input state ~ $ ) a ~ ~ ) b ( l / 2 1 ~ ' ) ( ~ e ) + Ig)) produces the entangled state

1 (14) 71

(15) 1 1

' l@)a I4 )b f l ( l e ) + k)) = f l ( I $ ) a l 4 ) b k ) + 14)al$)bie)).

A subsequent measurement which shows the atom to be in the state (1/2'/') (Ig) f 1.)) will leave the field modes in the corresponding entangled state (1/2'/') (I@),lq!~)~ f lq5)al$)b). The form of the output modes is then conditioned on the state of the atom. If we replace the two conventional beam splitters in our interferometer with such collective beam splitters, then the input state IN),1 when expressed in terms of the internal modes of the interferometer becomes

The phase shift then transforms this state into

1 1 - 21/2 ( I N ) a 2 1 0 ) b 2 k ) + Io)a21N)b21e)) f l ( lN)a210)b2bd +exp (-iNA4)10)821N)b21e)),

(17)

which clearly exhibits the collctive photonic de Broglie phase shift N A+. A n/2 pulse is applied to the atom so that its state is changed according to Ie) + (1/2'/')(le) + Ig)) and lg) -+ (1/2'/')(lg) - le)). The second collective beam splitter then transforms the state into

i[lN)a210)b2(k) - le ) ) + exp (-iNA4)10)a21N)b2(1e) + lg))] -+

1 - 23/2 [IN)a210)b2(k) + exp (-iNA4)le)> + )o)a21N)b2(exp (-iNA4)k) - l e ) ) 1 3 ( I 8 )

so that all the photons are found in one of the two output modes. A final measurement projecting the atom onto state (1/2'/') (lg) f e)) leaves the output modes in the corresponding state

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Photonic de Broglie wave interferometers 2223

so that the probability that all N photons appear in output mode a is f[l f cos (NAq5)], with the sign determined by the outcome of the measurement performed on the atom. The corresponding expectation value of the difference in the output photon numbers is

which clearly demonstrates a stronger phase dependence than that given in equation (6) for a conventional interferometer.

It is interesting to note that it is possible to see the photonic de Broglie phase shift in an interferometer containing a collective beam splitter only at the input and a conventional beam splitter at the output if the detailed photon statistics are recorded. This might be an easier experiment to realize than one requiring two collective beam splitters. Consider the state (16) after the light has passed through the first collective beam splitter. As we have already noted, a measurement which projects the atom onto the state (1/21/2(lg) f le)) will leave the internal modes in the state

= (1 - $ ) N * ( $ ) N ~ ~ ~ [N(Aq5+;)], ( 2 3 )

where the expectation value has been obtained using the phase-shifted state given by the right-hand side of equation ( 2 2 ) . The moment-generating function contains all the information required to find the full photon statistics of the ouput mode a. In particular, the probability that the number of photons found in the mode is even is given by

which clearly displays the full photonic de Broglie phase shift. If the initial input state is not a photon number eigenstate, then this expression will have to be averaged over the initial photon number probability distribution. For a general state input into mode a, the probability that an even number of photons will be found in ouput mode a is

P(even) = f (1 f cos ( i ~ i ~ 8 ) ) = f (1 f Re {Ma, (1 - exp (ie)]}), ( 2 5 )

where we have set 8 = Aq5 + n / 2 . For a coherent state ( I C Y ) ) this is

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2224 S. M . Barnett et al.

P(even) = 411 f exp [1a12 (cos e - 111 cos (la12 sin e l ) , (26)

which, apart from a phase shift, is proportional to the expectation value of the output photon number found if a second collective beam splitter is used [l 13. This quantity is very sensitive to small phase shifts such that 8 is small. In this limit the probability (26) reduces to

P(even) M f [I f exp (- f )aI2e2) cos (la12e)], (27)

which clearly depends on the mean photonic de Broglie phase shift laI2 A+.

4.2. Equal intensity inputs The photonic de Broglie phase shift described above arises because all the

photons traverse the same path through the interferometer. A closely related effect can be obtained with a conventional interferometer if both of the input modes a and b contain precisely the same number of photons [13-151 so that the total photon number is even. Simple symmetry arguments then show that the number of photons passing along each arm of the interferometer is always even [9]. It then follows that all the interfering amplitudes contributing to give the output photon probabilities will be subject to phase shifts that are even multiples of A+.

It is clear from equation (6) that the mean values of the photon numbers found in the two output modes will be equal and that the interference is not reflected in this quantity. The photon statistics, however, are sensitive to twice the phase shift [14,15]. We can show this by calculating the moment-generating function [23] for the photon number found in output mode a . For precisely N photons in each of the input modes we find that

x [l - cos (A@)]"[l + cos (A+)]'-", (28)

where we have used equations (5 c) and (5 d) to express 63 in terms of 61 and 61. The odd powers of cos (A+) cancel between terms involving m and 1 - m, so that the moment-generating function depends only on cos2(A$) = f [l + cos (2 A+)] and hence only on twice the phase shift A$. The experimental demonstration of this double phase shift [14] can therefore be viewed as an observation of photonic de Broglie phase shift for two photons.

4.3. Selection by two-photon absorption The previous two sections have shown how collective phase shifts can be

observed by forcing either all the photons or an even number of them along one of the interfering paths. It is also possible to find the same phenomenon if we can determine those occasions on which all the photons travelled along the same path and examine the interference conditional on the photons having travelled along the same path. Naturally, this approach does not produce an enhanced phase resolu- tion as those occasions on which all the photons do not pass along the same path are discarded and do not contribute to the observed interference.

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Photonic de Broglie wave interferometers 2225

bl

Figure 2. Mach-Zehnder interferometer with a two-photon absorber designed to absorb only if there are photons present in both arms of the interferometer. The polarization at any point is represented as horizontal (h), vertical (w) or circular (.*I.

Consider the arrangement depicted in figure 2 which consists of a Mach- Zehnder interferometer with normal 50-50 beam splitters as the input and output mirrors and a set of polarization-selective elements. Input mode a contains N horizontally polarized photons and mode b is prepared in its vacuum state. A series of polarization-sensitive elements are inserted into the interferometer and these are designed to deplete the light level by two-photon absorption but in such a way that the absorption can occur only if photons are present in both arms. The absence of any two-photon absorption will then indicate that all the photons passed along one of the interferometer arms but will not tell us which path they took. The first half- wave and polarizing beam splitter ensure that photons reaching the first quarter- wave plate along path a2 are horizontally polarized but those travelling along path b2 are vertically polarized. The first quarter wave plate then transforms these into 0- and u+ circular polarizations respectively. The two-photon absorber is designed to absorb only pairs of photons having opposite spin and so will register absorp- tions only if there are photons in both arms of the interferometer. The remaining polarization-sensitive elements undo the transformations of the polarization.

The two-photon absorber is designed to absorb photons in pairs, with each pair consisting of one 0- and one 0' photon. It is specifically designed not to be able to absorb two 0- or 0' photons. Such a two-photon absorber can be realized using a two-photon resonant transition, but far from a one-photon resonant transition, between two J = 0 atomic levels (figure 3 ) . t Resonant transitions can only occur if the atomic and hence the field angular momentum is unchanged, that is if the two absorbed photons have opposite spins: one u- and one IT+. The occurrence of a two-photon absorption can be detected by fluorescence from the excited atom. With a long enough absorber, all the photons in one arm of the interferometer will be absorbed.

Such an absorber assures us that detecting all N photons in the two output modes and no fluorescence from the two-photon absorber is only possible when all N photons are either transmitted or reflected by the first beam splitter because some photons must be absorbed if the N photons are distributed between the two

t A suitable example might be the 4s2 'So and 4p2 'So levels in calcium used as a source of photon pairs in testing Bell's inequalities and single-photon interfe.rence [26,27].

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2226 S . M. Barnett et al.

= 1

Figure 3. Possible level scheme for a suitable atomic two-photon absorber. Only the angular-momentum-conserving transition for which A m = 0 should be allowed.

internal paths. On those occasions when the photons have all taken the same path, the state of modes a2 and b2 resembles that prepared by a collective beam splitter:?

The phase shift multiplies the second term in this state by the photonic de Broglie phase shift factor exp (-iN A4) and the output beam splitter recombines the two modes to give the output state

N

n=O

It then follows that the probability for counting an even number of photons in mode a3 is

P(even) = [l + cos (N A+)]. (31)

This probability clearly exhibits the dependence on the collective phase shift NAq5. It also agrees with the result found in section 4.1 for the interference at a beam splitter of a two-mode state prepared by a collective beam splitter (once allowance is made for the differing phase shifts associated with reflection at the input beam splitter).

The idea presented in this section is based on our ability to establish the magnitude of the difference in the number of photons in the two internal arms of the interferometer without also determining its sign. This might also be achieved if we can perform a quantum non-demolition measurement of the square of the difference in the photon numbers in the two internal modes. The output statistics would then depend on Aq5 multiplied by the magnitude of this photon number difference.

t The phase factor multiplying the second term in the superposition has a different phase from that in equation (21) because of different input-output relations for the collective and conventional beam splitters. T h e iN in equation (29) arises from the Nth power of the reflection coefficient i/2'I2.

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Photonic de Broglie wave interferometers 2227

photon UV-vL

(b) Figure 4. Generalized Franson interferometer with N entangled photons: ( a ) each

photon in a different interferometer; ( b ) all N photons in the same interferometer.

4.4. N-photon Franson interferometer Our final example of photonic de Broglie interference is a generalization of the

Franson interferometer [16-181. Our discussion of this system differs from the previous systems in that it requires the full continuum of field modes represented by continuum creation and annihilation operators [28-311. The idea employs a high-order parametric process to produce N correlated photons (Figure 4). The photons have a set of frequencies w1 w2,. . . WN (not necessarily equal) which are constrained to satisfy the relation

where wp is the frequency of the pump photon. The N photons are different directions satisfying the phase matching condition and sent

emitted in to N arms

each having a delay line with an optical phase shift 6, and the same delay I-. The delay lines are constructed with symmetrical 5&50 beam splitters. If we know the times at which the photons were emitted, then it is possible to determine which path was taken by the photons by noting when they arrive.

We can use the beam-splitter input-output relationships to relate the output operators b, and d , to the input operators Ci, and 2,. Each of the input modes c, is

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2228 S. M. Barnett et al.

assumed to be the vacuum and so the incident N-photon entangled state can be expressed as?

where the &(urn) are the continuum creation operators [25-271 satisfying the commutation relations

(&(u), ai(w’)] = 6kl6(w - w’ ) , (34)

and 10) is the continuum vacuum state. It is convenient to change the mode index from w to t by means of the Fourier transform [28,29]

& ( t ) = - dwexp (-iwt)&(w). (35)

It then follows that these time-domain operators satisfy the commutation relation

so that we can express the input state in terms of localized modes as

where the normalization has been omitted for simplicity (see footnote to text before equation (33)). Equation (37) clearly shows the simultaneity of the N photons; they all appear at the same time, but this time is not determined by the state. An analysis incorporating finite coherence times would show that the photons are all emitted within a time comparable with the inverse of the bandwidths of the photons or their coherence time ?-photon. This time is much shorter than the pump coherence time T~”, , ,~ . The N-photon interference de- scribed below takes place under the condition Tphoton < T < T~” , , ,~ . The lower bound ensures that no single-photon interference occurs while the upper bound ensures that it is impossible to determine whether the N photons have taken the longer path (through the delay line) or the shorter path. If all the photons arrive at the detectors simultaneously (to within a time of about Tphoton), then it follows that they must all have passed through the long or short paths through their respective interferometers. It is not possible, however, to determine which path was taken as we do not know the time at which the photons were emitted (to better than about T~,,,,,~). It follows that on those occasions when all the photons arrive simul- taneously we can expect to find a collective phase shift. This has already been observed for a pair of photons generated by spontaneous parametric down- conversion [ 17,181.

t Note that this state is not normalized and is, therefore, not a true field state [30]. We can provide a more rigorous discussion by taking account of the finite bandwidths of the pump and of the down-converted fields or equivalently by incorporating their coherence times [31]. To do so here would unnecessarily complicate our presentation. The expressions given in the text should be thought of as representing the limiting forms in which the pump bandwidth tends to zero (or its coherence time tends to infinity) and the bandwidth of each down-converted photon tends to infinity (or its coherence time tends to zero).

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Photonic de Broglie wave interferometers 2229

The state of the output modes for the N interferometers can be found by using the continuum-mode generalizations of the beam splitter relations to obtain

Note that the time shift arises because of the different lengths of the two paths through the interferometer. On substituting equations (38) into the form for the input state (37), we find that the output state is

N

= 2-N /dtn{[b^l( t ) + iJi(t)] - exp (i&)[i;(t + T) + idi(t + lo). (39) fl= 1

We assume that the response times of the detectors are much shorter than the delay r so that coincidence counting of the N photons is possible. The reduced state corresponding to the N photons being detected simultaneously is then

where the variable of integration has been changed from t + r to t in the second product. This is valid as the range of the integral is over T~~~~ (taken to be infinite in the above equation) which is much larger than r. Note that the normalization factor has again been omitted as it is not essential. It is clear that the probability for the photons appearing simultaneously at the N detectors is

This clearly shows that the interference occurs with the sinusoidal dependence on the sum of all the phase shifts. This type of interference has been successfully demonstrated for the case N = 2 [17,18].

For the degenerate case, the N photons are emitted in the same direction and enter a single interferometer at the same (unknown) time. As shown in figure 4(b), the N paths in figure 4(a) are superposed into one. Hence there is only one delay line having delay time r and phase shift 9. We assume that detectors a and b can record the photon numbers with a time resolution smaller than r. If all the photons are detected at the same time, then they must have all taken the same path and the numbers of photons found in the two detectors will be determined by the same probabilities as found in section 4.3. In particular, the probability that detector a records an even number of photons is

P(even) = 3 [ 1 + cos (NO)], (42)

which clearly depends on the photonic de Broglie phase shift N A$.

5. Discussion The propagation or time evolution of n photons shifts the phase of the state b y

n A$, where A$ is the change in the phase of the field. This simple observation is

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2230 S. M . Barnett et al.

the origin of the idea of photonic de Broglie interference. We have shown that integer multiples of the elementary phase shift A4 can be found in conventional interferometry in the output photon statistics. True photonic de Broglie inter- ference can be found if we can suppress the individuality of the photons in the interferometer. The most extreme example of this is the original proposal by Jacobson et al. [ l l ] in which all the photons are forced to travel along the same route through the interferometer by the action of an effective or collective beam splitter. We have shown that the photonic de Broglie phase shift can also be found in the output photon statistics for an interferometer in which only the input mirror is a collective beam splitter.

Photonic de Broglie interference can also be found in conventional interfero- metry if the photons can be forced to behave collectively. The simplest example exploits the property that coincident photon pairs incident on a 50-50 beam splitter will both leave in the same direction [32] to ensure that the numbers of photons travelling along each path of the interferometer is always even. It then follows that the observed photon statistics will depend on twice A4 [14]. An interferometer built on this principle should be able to reach the so-called Heisenberg limit for phase-shift resolution.

It is also possible to find the photonic de Broglie phase shift if we can perform measurements conditional on all the photons having travelled along the same (unknown) path through the interferometer. We have described two possible methods for achieving this: one based on two-photon absorption and the other based on coincidence counting with an interferometer having very different lengths for the two optical paths. The latter of these two has already been successfully demonstrated for photon pairs [17,18]. It should be emphasized that neither of these two schemes presents an enhanced phase resolution over the shot noise limit. This is simply because only those cases where all the photons took the same path are recorded and this comprises only 2-N of the total number of experimental runs.

We conclude by noting that photonic de Broglie interference is extremely sensitive to decoherence due to coupling with the environment. In particular, the loss of only one photon from the internal paths to the environment will destroy the interference. This is because it would then, in principle, be possible to determine from which mode the photon was lost and hence would constitute a Welcher-Weg measurement on all the remaining photons.

Acknowledgments

of N T T Basic Research Laboratories is greatly appreciated. Discussion on polarization-dependent two-photon absorption with Dr Koashi

Appendix. The moment-generating function

mode having annihilation and creation operators ii and iit is The moment-generating function for photon number associated with a single

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Photonic de Broglie wave interferometers 2231

A number of statistical measures can be calculated once the moment-generating function is known. In particular probability that there are n photons in the mode and the mth factorial moment are

It follows from the definition (A 1) that

M ( 2 ) = P(even) - P(odd),

so that the probability that the photon number is even is

P( even) = 4 [ 1 + M ( 2)]. (A 5)

The moment-generating functions for the photon number state In) and for coherent, thermal and squeezed vacuum states with mean photon number ii are

(PI = ( 1 - PYl

(PI = exp (-P47

(PI = ( 1 + Pfi)-ll

M y y p ) = ( 1 + 2pn - p2n)-”2.

(A 6)

(A 7)

(A 8)

(A 9)

M n u m b e r

Mcoherent

Mtherrnal

For a more complete discussion see [23].

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