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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
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
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
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
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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
                                                                                    ðMÞ        N X Y ð1Þ
                                                                                 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
                  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-
Eðx; zÞ ¼ eÀikz         An sin½nðx À D=2Þ=DŠei2n 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

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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)
correlations are measured, the measured rate of cross cor-         for each and every application of Tð1Þ . This is because this
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
                                                                   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-
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).
                                                                   [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
                                                                   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

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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.
                                                                    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
                                                                    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-              *
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PRL 108, 153602 (2012)                   PHYSICAL REVIEW LETTERS                                                     13 APRIL 2012

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