# Photonic de broglie wave interferometers

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<ul><li><p>This article was downloaded by: [University of California Davis]On: 04 November 2014, At: 20:39Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK</p><p>Journal of Modern OpticsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tmop20</p><p>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 , Universit de Genve , 20 Ruede l'Ecole de Mdecine, CH1211, Genve, 4, SwitzerlandPublished online: 03 Jul 2009.</p><p>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</p><p>To link to this article: http://dx.doi.org/10.1080/09500349808231234</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor & Francis makes every effort to ensure the accuracy of all the information (theContent) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. 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Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions</p><p>http://www.tandfonline.com/loi/tmop20http://www.tandfonline.com/action/showCitFormats?doi=10.1080/09500349808231234http://dx.doi.org/10.1080/09500349808231234http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditions</p></li><li><p>JOURNAL OF MODERN OPTICS, 1998, VOL. 45, NO. 11, 2217-2232 </p><p>Photonic de Broglie wave interferometers </p><p>STEPHEN M. BARNETT?$, NOBUYUKI IMOTOtll and BRUNO HUTTNER4 t N T T Basic Research Laboratories, 3-1 Morinosato-Wakamiya, Atsugishi, Kanagawa 243-01, Japan 1 Department of Physics and Applied Physics, University of Strathclyde, Glasgow G4 ONG, Scotland 0 Group of Applied Physics Optique, 20 Rue de 1Ecole de Medecine, UniversitC de Genltve, CH1211 Genltve, 4, Switzerland </p><p>(Received 3 February 1998) </p><p>Abstract. In the recently proposed photonic de Broglie wave interferometer, sophisticated beam splitters are used to split the de Broglie wave of a set of photons. The photonic de Broglie or collective phase shift can, however, be found in conventional interferometry if the full photon statistics are recorded. We propose a variation of the original photonic de Broglie wave interferometer. We show that the collective phase shift can and has been found in inter- ferometry using photon pairs and propose two new photonic de Broglie wave interferometers in which normal beam splitters are used. Both of these rely on conditioning to select the cases for which all the photons behave as a single object. </p><p>1. Introduction It has long been recognized that the resolution of interferometers is ultimately </p><p>limited by quantum effects [l, 21 and that the shot noise limit to resolution can be passed by making use of non-classical states of light [3-71. All this work has been based on existing interferometers in which the optical beams are divided and recombined using beam splitters. These devices linearly superpose the two input fields to produce the two output fields [8-lo]. Jacobson et al. [ l l ] have proposed a new class of interferometer which incorporates an effective or collective beam splitter (based on a proposal by Davidovich et al. [12]) and acts collectively on all the photons in an input mode so that all of them are transmitted or all are reflected. If the state of the two input modes a and b is l$)alq5)b, then the effective beam splitter acts to give output modes in the entangled state </p><p>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 </p><p>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 </p><p>beam splitters and the same Fock state in each of the two input modes [13]. </p><p>0950-0340/98 $12.00 0 1998 Taylor & Francis Ltd. </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f C</p><p>alif</p><p>orni</p><p>a D</p><p>avis</p><p>] at</p><p> 20:</p><p>39 0</p><p>4 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>2218 S . M . Barnett et al. </p><p>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. </p><p>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. </p><p>2. Photonic de Broglie wavelength The phase shift associated with the propagation of a monochromatic, paraxial </p><p>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 </p><p>( 2 ) o=exp(-iAq57i) =exp(-iAq5iitci) </p><p>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 </p><p>ci + UtiiU = iiexp (-iAq5). ( 3 ) </p><p>In the second alternative, the action of this unitary operator on a state vector I$) = En c,ln) leads to the phase shifted state </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f C</p><p>alif</p><p>orni</p><p>a D</p><p>avis</p><p>] at</p><p> 20:</p><p>39 0</p><p>4 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>Photonic de Broglie wave interferometers 2219 </p><p>Figure 1. Schematic diagram of a Mach-Zehnder interferometer with a phase shift Ad in one arm. </p><p>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. </p><p>3. Conventional interferometry The ideas discussed in this section apply to all commonly occurring inter- </p><p>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]: </p><p>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 </p><p>derive unambiguously the form of the probability density for the optical phase [19, 201. </p><p>complicate our presentation. For further details see [21, 221. </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f C</p><p>alif</p><p>orni</p><p>a D</p><p>avis</p><p>] at</p><p> 20:</p><p>39 0</p><p>4 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>2220 S. M. Barnett et al. </p><p>a3 = - [ i 2 + i exp (-i A$)&] 2'/2 =${[I -exp(-iA$)]& + i [ l +exp(-iA$)]&} </p><p>- 1 b3 = [exp (-i A$)& + iiz] </p><p>=i{[exp(-iA$) - 1161 + i [ l +exp(-iA$)]i&}. (5 4 The phase shift A4 is found by measuring the difference between the numbers of photons detected in each of the output modes. The expectation value of this quantity is </p><p>(hi&) - (b^t,&) = ((461) - ( c i i i l ) ) cos(A$) + ((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. </p><p>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 ] ) </p><p>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 </p><p>1 P(n) =- ( - $ ) ' M ( p ) I = 2-NCf[l - cos(Aq5)]"[1 + cos(A$)lN-". (8) </p><p>n! p=l </p><p>A Fourier expansion shows that this probability contains terms which depend on n A4 for all n between 0 and N , corresponding to photonic de Broglie phase shifts for all possible photon numbers between 0 and N . An interesting exception to this occurs for n = N / 2 for which we find that </p><p>so that the probability depends only on even multiples of A$. We can gain some insight into the origin of the appearance of these </p><p>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 </p><p>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. </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f C</p><p>alif</p><p>orni</p><p>a D</p><p>avis</p><p>] at</p><p> 20:</p><p>39 0</p><p>4 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>Photonic de Broglie wave interferometers 2221 </p><p>remainder passing through arm a2. The input state can be written in terms of the operators for the fields inside the interferometer as </p><p>where 10) is the two-mode vacuum state. The phase shift applies the unitary transformation (2), with ii replaced by &, to produce the state </p><p>N I$(A$)) = m- 1/22-N/2 Cy(iii)l(i@)N-l exp [-i(N - Z)A$] 10). (11 ) </p><p>Each term in the summation has been subjected to a different phase shift, the value of which is proportional to the number of photons passing through the arm having the phase-shifting component. The output photon number probability distri- bution results from interference between the amplitudes for all possible distribu- tions of the photons between the two paths a2 and b2 and so depends on all the phase shifts HA$. 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 </p><p>I=O </p><p>- n M:yherent (p ) = exp ( -p [ 1 - cos (A$)]) (12 a) </p><p>and - - 1 </p><p>M ~ ~ m a ' ( p ) = (1 +p;[1 -cos(A$)]) (12 b) </p><p>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 m...</p></li></ul>