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week ending PRL 108, 153602 (2012) PHYSICAL REVIEW LETTERS 13 APRIL 2012 Two-Photon Path-Entangled States in Multimode Waveguides Eilon Poem,* Yehonatan Gilead, and Yaron Silberberg Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel (Received 18 January 2012; revised manuscript received 16 February 2012; published 13 April 2012) We experimentally show that two-photon path-entangled states can be coherently manipulated by multimode interference in multimode waveguides. By measuring the output two-photon spatial correlation function versus the phase of the input state, we show that multimode waveguides perform as nearly ideal multiport beam splitters at the quantum level, creating a large variety of entangled and separable multipath two-photon states. DOI: 10.1103/PhysRevLett.108.153602 PACS numbers: 42.50.Dv, 03.67.Bg, 42.79.Gn Quantum states of photons distributed between several unlike separated photons carry relative phase information, optical paths play the central role in linear optical quantum can be coherently manipulated by MMWs. computation [1]. In particular, entangled states, where no Here we experimentally show that the answer to this decomposition into a product of either single-path or question is positive. We utilize MMI in a two-mirror, single-photon states exists, are of great importance due to tunable planar MMW [17] to implement multiport BS their nonlocal nature [2,3]. Entangled photon states are arrays of up to 5 input and 5 output ports and explore the generated either by nonlinear processes in crystals [4], by propagation of nonclassical, path-entangled two-photon cascaded emission of single quantum emitters [5], or by states through them. We measure the two-photon correla- quantum interference of nonclassical light on a beam tions at the output and ﬁnd that, for any relative phase splitter [6,7]. between the two paths, they agree very well with the Quantum interference in arrays of beam splitters (BSs) multipath, two-photon states expected at the output of ideal is used in many linear optical quantum computation multiport BSs. schemes to create and manipulate multiphoton multipath The experimental system is described in Fig. 1(a). States pﬃﬃﬃ states [8–10]. The simplest example is the Hong-Ou- of the form c ¼ ð1= 2Þðj2; 0i þ ei j0; 2iÞ are created by 2 Mandel interference [11]. In this fundamental effect, two ﬁrst splitting a 404 nm continuous-wave diode-laser beam identical photons input each on a different port of a 50:50 into two arms. The relative phase between the beams, , is BS cannot exit in two different ports due to destructive controlled and stabilized by a piezoelectrically movable interference of the two possible paths. Therefore, they mirror on one arm [see Fig. 1(a)]. The state in the two always exit bunched together on either one of the output arms at this point is j; ei i, where represents a ports, in an entangled state. A major obstacle on the way to coherent state of average photon number jj2 . Note the implementation of quantum optical circuits is the large the single-photon part of this state is c ¼ thatpﬃﬃﬃ 1 number of BSs required and the increasing complication of ð1= 2Þðj1; 0i þ ei j0; 1iÞ. The two beams then undergo their alignment. One way to overcome this problem is type I collinear degenerate spontaneous parametric down- through miniaturization of the optical circuits. Indeed, conversion in a properly oriented 2 mm long -barium- quantum interference has been recently demonstrated in borate (BBO) crystal. In this process, each 404 nm photon integrated optical circuits composed of evanescently transforms into two 808 nm photons propagating together. coupled single-mode waveguides embedded in solids Since spontaneous parametric down-conversion is much [12–14]. A conceptually different route towards robust faster than the optical period, the relative phase between implementation of quantum optical circuits may come in the two paths is kept as it was before the BBO crystal. the form of multimode interference (MMI) devices [15]. Spectral ﬁlters are used to eliminate all remaining photons These compact replacements for BS arrays, usually based outside a 3 nm band centered at 808 nm. The two-photon on planar multimode waveguides (MMWs), are already part of the state after the BBO and the spectral ﬁlters is thus used extensively in modern classical optical communica- c . There is no single-photon part, and, for low enough 2 tion networks. They have also been proposed to be useful pump powers, the content of the next order—the four- for creation and detection of multiphoton states [10], as photon part—is negligible. they naturally implement Bell multiport BSs [8]. A step The beams are then inserted into an MMW made of two towards their use in quantum networks was recently made parallel metallic mirrors [17]. The distribution of one with the demonstration of Hong-Ou-Mandel interference component of the optical ﬁeld between the mirrors, of two separated photons in an MMW [16]. The question assuming perfect reﬂection and paraxial propagation, is still remains, however, whether entangled states, which given by [15] 0031-9007=12=108(15)=153602(5) 153602-1 Ó 2012 American Physical Society week ending PRL 108, 153602 (2012) PHYSICAL REVIEW LETTERS 13 APRIL 2012 MMW, calculated for a Gaussian input beam localized about x ¼ D=4, is presented. Generally, an N Â N BS would be generated by restricted interference for beams localized about x ¼ ð2p À 1 À NÞD=2N, for z ¼ qz0 =4N, where p and q are integers and p goes from 1 to N. For an even q, the BS is equal. By using Eq. (1), the transition matrix Tð1Þ ¼ ðTð1Þ Þq can be calculated for any N [18]. N;q N;1 This matrix applies for coherent beams as well as for a single photon. In order to obtain the transition matrices for states of M identical photons, one should sum up all the products of the single-photon matrix elements relevant to ~ ~ the transition between an initial and a ﬁnal M-photon conﬁguration: sﬃﬃﬃﬃﬃﬃﬃ ðMÞ N X Y ð1Þ ~ M ðTN;q Þ; ¼ ~ ~ ðT Þ ; (2) N f g j¼1 N;q j ;j ~ ~ ~ ~ ~~ where represent permutations of the initial conﬁgura- ~ tion and N (N ) is the number of all different permu- ~ ~ tations of the initial (ﬁnal) conﬁguration. The calculation of these N-port, M-photon transition matrix elements is, in general, a computationally difﬁcult task, equivalent to the calculation of the permanent of a matrix. However, it is FIG. 1 (color online). (a) The experimental setup. DL, 404 nm feasible for small enough M and N, and for N ¼ 2 there continuous-wave diode laser; SF, spatial ﬁlter; HWP, half wave even exists an analytic expression for any M [19]. Given an plate; PBS, polarizing beam splitter; RR, retroreﬂector; QWP, initial wave function (in terms of the initial amplitudes for quarter wave plate; M, mirror; BBO, -barium-borate crystal; each distribution of the M photons among the N input BPF, bandpass spectral ﬁlter; MMW, multimode waveguide; BS, ports), the ﬁnal state c out is readily found, and from it, beam splitter; MMF, multimode ﬁber; APD, avalanche photodi- any correlation function can be calculated. In particular, for ode; CE, correlation electronics. The thin purple (wide red) lines a general two-photon state, the two-photon correlation represent the pump (down-conversion) beams. Movable parts are probability between ports m and n is just marked by black double arrows. (b) Calculated intensity distri- bution in an MMW for an input beam centered about x ¼ D=4. Pð2Þ ¼ jhm; nj c out ij2 ; m;n (3) The white boxes mark propagation lengths where various 2 Â 2 BSs are implemented. (c)–(f) Schematic diagrams showing the where jm; ni is a state where one photon is on the mth conﬁgurations of input and output ports used in the experiments. output port and the other is on the nth output port. In our experimental setup, the length of the MMW is X 1 ﬁxed by the length of the mirrors, L. However, the adjust- ment of the relative length, ¼ L=z0 ¼ L=8D2 , is pos- 2 Eðx; zÞ ¼ eÀikz An sin½nðx À D=2Þ=Dei2n z=z0 ; (1) n¼1 sible through the adjustment of the separation between the mirrors. For the experiments presented below, the MMW with k ¼ 2= and z0 ¼ 8D2 =, where D is the width of was adjusted to function as 2 Â 2, 3 Â 3, 4 Â 4, and 5 Â 5 the MMW, is the wavelength of the incident light, and the BSs. Figures 1(c)–1(f) show the corresponding conﬁgura- constants An are determined by the incident ﬁeld distribu- tions of input and output ports used in these experiments. tion. From Eq. (1), it can be shown that, for any input, full We note that due to the ﬁnite conduction of the mirrors the imaging occurs at z ¼ z0 , reﬂection about the center of the MMW has a slightly different relative length for different MMW occurs at z ¼ z0 =2, and two-way, equal beam split- polarizations. To avoid any possible complications that ting with a relative phase of =2 ( À =2) occurs at z ¼ may arise, the polarization of the incident photons (in z0 =4 (z ¼ 3z0 =4). These effects are known as general both beams) was set parallel to the mirrors, such that interference [15]. Unequal BSs and BSs of more than only TE modes were excited. In order to measure two- two ports are also possible but only for speciﬁc positions photon correlations at the output of the MMW, the light at of a localized input beam. This is known as restricted its output facet is split and imaged on two multimode interference [15]. For example, for an input beam localized ﬁbers, each connected to a single-photon detector (Si ava- about x ¼ ÆD=4, an unequal two-way BS will be realized lanche photodiode). The digital signals from the two de- at z ¼ ð2m À 1Þz0 =8, where m is an integer. This is readily tectors are input into correlation electronics that shorten seen in Fig. 1(b), where the intensity distribution inside the and multiply them, yielding the rate of coincidence events 153602-2 week ending PRL 108, 153602 (2012) PHYSICAL REVIEW LETTERS 13 APRIL 2012 in a 7 ns time window [20]. As the ﬁbers can be placed in Note that at the middle of this period (on the 50:50 BS at front of any pair of output beams [see Fig. 1(a)], both auto- ¼ 1=4) this state transforms into two separated photons, and cross correlations can be measured. Since in this exhibiting inverse Hong-Ou-Mandel interference. In con- method, the beams are split into two also when cross trast, the state c is left unchanged (up to a global phase) 2 correlations are measured, the measured rate of cross cor- for each and every application of Tð1Þ . This is because this 2;1 relations would be half that predicted by using Eq. (3). state is invariant under any unitary two-way beam splitter For the comparison of theory and experiment, we therefore with a relative phase of Æ=2 [21]. deﬁne the modiﬁed correlation probability Figure 2(b) shows the measured correlation functions ð2Þ gmn (symbols), deﬁned as the rate of coincidence events Cð2Þ ¼ Pð2Þ =ð2 À m;n Þ: m;n m;n (4) between output ports m and n, normalized by the expected accidental coincidence rate ra ¼ rm rn w, rm being the Figure 2(a) presents the calculated modiﬁed two-photon single-count rate in port m and w ¼ 7 ns the coincidence correlation probabilities [Eq. (4)] versus the phase , for measurement time window. The total integration time is 7 s three different relative propagation lengths at which 2 Â 2 for each data point. The lines are the best ﬁtted 2-period BSs are implemented [see Figs. 1(b) and 1(c)]: ¼ 1=4— sinusoidal functions. The measured background after the a 50:50 BS with a relative phase of =2; ¼ 3=8—a MMW [right panel in Fig. 2(b), empty circles] is $10 cos2 =8:sin2 =8 ( $ 85:15) BS with a relative phase of times larger than the expected accidental coincidence =2; and ¼ 1=2—a reﬂection about the center of the rate. By blocking either one of the input beams, we have MMW (a ‘‘0:100 BS’’). Note that all three cases are a part veriﬁed that indeed $90% of this background is due to a of the same series of transformations given by ðTð1Þ Þq ,2;1 small overlap between the collecting ﬁber and the neigh- ð1Þ where T2;1 is a 15:85 BS implemented for ¼ 1=8, and boring output port. This is also conﬁrmed by the increase that for q ¼ 8 ( ¼ 1) full imaging is obtained [17,18]. of the background level when the beams are brought closer The state c 0 is invariant only under reﬂection and, thus, 2 [see, e.g., Fig. 4(b) below]. The bare visibility of the returns to its initial form only when full reﬂection occurs— oscillations measured for the 50:50 BS conﬁguration once every four applications of Tð1Þ (e.g., at ¼ 1=2). 2;1 [Fig. 2(b), left panel] is 72 Æ 3%, violating the classical bound of 50% [22] by more than 7 standard deviations. This indicates that the light after the MMW is still en- tangled, even without background reduction. With the reduction of the measured background, the visibility reaches 83 Æ 3%. Figure 2(c) presents ‘‘correlation maps’’ for each of the relative lengths, showing the elements gð2Þ for two initial mn phases ¼ 0 (top) and ¼ (bottom), as extracted from the ﬁts to the measured data. For each phase, the expected periodicity with respect to the relative length of the MMW is clearly seen. We proceed to examine 3 Â 3 BSs. The input and output ports used are as illustrated in Fig. 1(d). Figure 3 shows the two possible cases. On the left we present the calculated (a) and measured (b) correlations of a 3 Â 3 equal BS ( ¼ 1=3), while on the right the correlations for an unequal BS ( ¼ 5=12) are presented. For the equal BS, the six pos- sible two-port correlation measurements divide into three pairs, oscillating with a phase difference of 2=3 between them. Each pair contains an autocorrelation of one port and the cross correlation of the other two ports. There are six FIG. 2 (color online). (a) Two-photon correlations vs the phase special phases where two such correlation pairs meet. For , calculated by using Eq. (4) for relative waveguide lengths of three of them there is an enhancement of one pair over the 1=4 (left), 3=8 (center), and 1=2 (right), where the MMW other two, while for the other three one pair is completely functions as different types of 2 Â 2 BSs, as illustrated in suppressed. This is further visualized in Fig. 3(c), where Fig. 1(c). The correlation between ports m and n is marked by ‘‘m-n.’’ (b) Measured correlation functions. The symbols are the the correlation maps for ¼ 0 and ¼ are shown. In measured values, and the lines are best ﬁtted 2 period sinusoi- contrast to the equal 3 Â 3 BS, the phase dependencies of dal functions. (c) Corresponding two-photon correlation maps the correlations in the unequal 3 Â 3 BS show only two extracted from the ﬁtted curves for ¼ 0 (top) and ¼ different phases of oscillation and only partial visibilities. (bottom). However, the two cases can be related one to the other if 153602-3 week ending PRL 108, 153602 (2012) PHYSICAL REVIEW LETTERS 13 APRIL 2012 FIG. 3 (color online). (a) Calculated two-photon correlations FIG. 4 (color online). (a) Calculated two-photon correlations vs the phase , for two cases of 3 Â 3 multiport BSs imple- vs the phase , for an equal 4 Â 4 multiport BS (left), imple- mented by setting the relative length of the MMW to 1=3 (left) mented by setting the relative length of the MMW to 1=8, and an and 5=12 (right). The locations of the input and output ports are equal 5 Â 5 multiport BS (right), for which the relative length is illustrated in Fig. 1(d). The port conﬁgurations (m-n) of all 1=5. The locations of the input and output ports are illustrated in correlations described by a certain curve are listed next to it. Figs. 1(e) and 1(f), respectively. The output port conﬁgurations (b) The measured correlations (symbols) and their sinusoidal ﬁts (m-n) of all correlations described by a certain curve are listed (lines). (c) Correlation maps for ¼ 0 and ¼ . next to it. (b) Measured correlation functions. Full black symbols represent autocorrelations, while gray-ﬁlled and empty symbols represent cross correlations. The solid lines are sinusoidal ﬁts. (c) Corresponding two-photon correlation maps for ¼ 0 and one notes that the unequal BS is composed of an equal ¼ . 2 Â 2 BS ( ¼ 1=4) on the two outer ports, followed by an equal 3 Â 3 BS of ¼ 1=6, different from that of ¼ 1=3 only in the order of the relative phases [18]. This is best In summary, we experimentally show, for the ﬁrst time, seen in the correlation maps shown in Fig. 3(c). For ¼ 0 that path-entangled quantum light can be coherently ma- the cross correlations between the two outer arms switch nipulated by multiport BSs, naturally implemented by values with the corresponding autocorrelations, while for MMI in MMWs. The manipulation is shown to maintain ¼ there is no difference between the two 3 Â 3 BSs, the information on the relative phase between the two due to the invariance of c on 2 Â 2 =2 BSs. 2 paths. This persists even for complex manipulations with To examine the effect of even more complex BSs on the a high number of output ports, allowing for the robust states c , we performed measurements for equal 4 Â 4 2 creation of a large space of two-photon multipath states, and 5 Â 5 BSs. The input and output ports used are as controlled by the initial phase. MMI is thus shown to be a illustrated in Figs. 1(e) and 1(f), respectively. Figure 4(a) robust and simple approach for the implementation of [Fig. 4(b)] presents the calculated [measured] phase depen- various quantum optical circuits of high complexity. dence of these correlations, and Fig. 4(c) shows the corre- We thank Yaron Bromberg, Yoav Lahini, and Yonatan sponding correlation maps for ¼ 0 and ¼ . In the Israel for their help. The ﬁnancial support of the Minerva case of the 4 Â 4 equal BS, all the autocorrelations and the Foundation, the European Research Council, and the cross correlations of symmetric ports oscillate together with Crown Photonics Center is gratefully acknowledged. a phase of 0, and all the cross correlations of asymmetric ports oscillate together with a phase of . The state created for ¼ 0 therefore contains no asymmetric cross correla- tions, while that created for ¼ contains only asymmet- *eilon.poem@weizmann.ac.il [1] E. Knill, R. Laﬂamme, and G. J. Milburn, Nature ric cross correlations. 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