De Martini et al. Vol. 19, No. 5 /May 2002/J. Opt. Soc. Am. B 1009
Stochastic quantum interferometrywith Fock states
Francesco De Martini and Paolo Mataloni
Dipartimento di Fisica and Istituto Nazionale per la Fisica della Materia, Universita di Roma ‘‘La Sapienza,’’Roma 00185, Italy
Giovanni Di Giuseppe
Dipartimento di Fisica and Istituto Nazionale per la Fisica della Materia, Universita di Roma ‘‘La Sapienza,’’Roma 00185, Italy, and Department of Electrical & Computer Engineering, Boston University, Boston,
Massachusetts 02215-2421
Fabrizio Altarelli
Department of Physics, Harvard University, Cambridge, Massachusetts 02138
Received March 14, 2001; revised manuscript received October 29, 2001
1. INTRODUCTIONInterferometry of quantum particles is rooted at the coreof modern physics, as it provides a unique tool of investi-gation and a direct demonstration of fundamental naturalproperties such as complementarity, nonlocality, andquantum nonseparability.1–4 The interferometer hasgenerally been conceived so far as a stationary device, i.e.,a measurement apparatus in which the transfer functionwi of the particle-scattering or beam-splitting elementsare independent of time (t). Only recently has the rel-evance of the t-dependent nonstationary, or stochastic,beam splitter (Sto-BS) been emphasized in connectionwith statistical properties of quantum particles. Ex-amples are provided by a neutron interferometer havingone arm regularly interrupted by a chopper,5,6 by the in-tensity second-order optical correlator,7,8 and by the first-order optical field stochastic interferometer9 realizedsome years ago by our laboratory.10,11 In the presentstudy we first analyze the properties of the first-order sto-chastic interferometer (Sto-IF), and this is compared withthe behavior of the usual stationary interferometer. As aparadigmatic example, Fig. 1 shows a Mach–Zehnderconfiguration in which the input two-way coherent scat-tering device is indeed provided by a stochastic beamsplitter (Sto-BS) in which the probabilities of photon scat-tering over the output channels, Wi(t) [ uwi(t)u2, for i5 1, 2 and S iWi(t) 5 1, are made to vary in time withintheir existence range (0–1) by means of an external non-
stationary driving signal applied to an active electro-optical device. In other words, and more generally, wedeal with the quite unusual case in which the amplitudesof the interfering particles’ Feynman paths undergo anexternally driven time perturbation: a peculiar quantumdynamical condition that, we believe, has never been in-vestigated before. In particular, the driving parametersof the Sto-BS could vary on a time scale comparable to thetransit time of the scattered particle through the interfer-ometer. As previously remarked, this peculiar dynamicalcondition may open interesting perspectives in the con-text of the welcher-weg paradigm and in connection withdelayed-choice-type experiments possibly involvingmore complicated systems, e.g., atom states in amicromaser.12,13 As said, several experiments have al-ready been devoted to the peculiar first-order Sto-IF con-dition in the optical range by adopting ‘‘stochastic’’electro-optical (EO) devices.7,10,11 The results obtained inthese works were of course found to be a direct conse-quence of the basic linearity of quantum mechanics.However, all previous experiments were performed with‘‘classical light,’’ precisely a super-Poissonian Planckianlight obtained by inducing stochastic phase disturbanceson a single-mode laser beam. In view of the general ar-guments given above, it appears relevant to investigatethe Sto-If properties under the more significant excitationof ‘‘quantum,’’ sub-Poissonian light, e.g., associated withan input Fock state generated by the modern technique of
2002 Optical Society of America
1010 J. Opt. Soc. Am. B/Vol. 19, No. 5 /May 2002 De Martini et al.
spontaneous parametric downconversion (SPDC) in asecond-order nonlinear (NL) crystal.14 In addition, in thepresent study we extend our investigation to the second-order stochastic quantum interferometer under excitationof a two-photon Fock state, again generated by SPDC.This study, never undertaken previously either theoreti-cally or experimentally to our knowledge, extends into thestochastic domain the well known condition of two-photonexcitation of a beam splitter, a condition of common inter-est in the modern field of quantum information.15–17
Precisely in this domain, the new Sto-IF is expected toprovide, for instance, a new variety of linear devices ex-hibiting new types of Hadamard unitarytransformations.18
The paper is organized as follows. In Section 2 thegeneral scattering theory of the optical lossless Sto-BS ispresented. There it is shown that if the device is excitedby an input single-mode Fock state, the particle partitionstatistics changes from the ‘‘classical,’’ Maxwell–Boltzmann to Bose–Einstein (BE) by increasing thespread of the distribution of the stocasticity parameter s.A brief analysis of the BE condition given at the end ofSection 2 emphasizes the peculiar general features of thestochastic scattering of particles over many output chan-nels, of which the Sto-BS is but a significant example.Subsection 3.A reports the general theory of the first-order Sto-If with explicit calculations of the expected in-terference visibility, and Subsection 3.B accounts for thecorresponding experimental demonstration. At last, thetwo subsections of Section 4 give a parallel, extended ac-count of the second-order interferometry, both theory andexperiment.
2. STOCHASTIC BEAM SPLITTERThe optical Sto-BS is a lossless device realized by meansof a sequence of optical elements that introduce time-dependent changes on one (or several) degree of freedomof the state of the photon. In our case, the photon’s linearpolarization (p) is assumed to be a time-dependent vari-able, and the final beam-splitting process takes place in a(stationary) polarizing beam splitter, indicated by PBS inFig. 1.
Assume that a single photon in a Fock state, generatedby SPDC in a type II nonlinear crystal,19–21 with linear
polarization (p) oriented along the o axis, impinges on oneport of a Sto-BS (Fig. 1). Consider the overall transmit-tivity (T [ utu2) and the reflectivity (R [ uru2) of the de-vice. In virtue of the unitary property T 1 R 5 1, T andR can be also defined as
T 51 1 x
2, R 5
1 2 x
2,
where x is a stochastic parameter lying in the range [21,1], with distribution p(x). The general case of a Sto-BSdriven by an arbitrary noise distribution p(x) has beeninvestigated in Ref. 10. In the present paper we investi-gate the relevant case of a Gaussian distribution, withzero average value of the stochastic parameter, x 5 0,and related variance s 2:
p~xu s! [1
A2ps erfS 1
A2sD expF2S x
A2sD 2G , (1)
where the normalization condition *211 p(xu s)dx 5 1
holds. Two extreme relevant cases are considered:
s → 0: p(xu s) is equivalent to a Dirac d function.This corresponds to the case of an ordinary symmetricbeam splitter, with R 5 T 5 1/2.
s → `: p(xu s) is a constant, lims→`
p(xu s) 5 1/2.
This corresponds to a uniform distribution of R(x) 5 uru2
and T(x) 5 utu2 in the range [0–1].
We may analyze the physical conditions underlyingthis process by referring to the scheme described in Fig. 1.The input photon is transmitted through a sequence ofthree optical elements: a half-wave plate (l/2), an exter-nally driven Pockels cell (PC), and a quarter-wave plate(l/4). The optical axes of the three elements are orientedalong the directions o9–e9, o –e, and o8–e8, respectively(inset of Fig. 1). Finally, the output signal impinges on apolarizing beam splitter (PBS) where it is decomposed intwo orthogonal p components, oriented along the axes oand e.
Consider the effect induced on the photon’s polarizationstate by the various elements of Sto-BS. By assumingthe polarizations eigenstates o and e to be coincident with
Fig. 2. Poincare sphere representation of the rotation per-formed by the stochastic beam splitter on the input single-photonstate.
De Martini et al. Vol. 19, No. 5 /May 2002/J. Opt. Soc. Am. B 1011
eigenstates of sz with eigenvalues 11 and 21, uo&[ u1& and ue& [ u2&, we refer to the Poincare sphereshown in Fig. 2.22
The first half-wave plate (l/2) causes a rotation of paround its proper axes, the vector n; this action is definedby the operator
U ~1 ! 5 expS 2ip
2n • sD 5 2in • s, (2)
which, for n 5 $1/A2, 0, 1/A2%, can be expressed as
U ~1 ! 52i
A2~ sx 1 sz!. (3)
The electro-optic Pockels cell (PC) contributes with arotation d around sz , which gives
U ~2 ! 5 expS 2id
2szD 5 cos
d
2I 2 i sin
d
2sz . (4)
The quarter-wave plate (l/4) is oriented to introduce arotation of p/2 around the x axes:
U ~3 ! 5 expS ip
2sxD 5
1
A2~ I 1 isx!. (5)
By combining Eqs. (3), (4), and (5), we have
U 5 U ~3 !• U ~2 !
• U ~1 ! 5 RyS d 1p
2 D • RzS p
2 D , (6)
where
Rj~u! 5 expS 2iu
2s jD . (7)
The overall expression (6) can be finally written in thematrix form:
U
5 F cosS d
21
p
4 D expS 2ip
4 D 2sinS d
21
p
4 D expS 1ip
4 DsinS d
21
p
4 D expS 2ip
4 D cosS d
21
p
4 D expS 1ip
4 D G .
(8)The expressions of the transmission and reflection coeffi-cients of the PBS are
t 5 cosS d
21
p
4 D expS 2ip
4 D ,
r 5 sinS d
21
p
4 D expS 1ip
4 D .
This leads to the expressions of the transmissivity and re-flectivity of the Sto-BS,
T 5 utu2 51 1 sin d
2, R 5 uru2 5
1 2 sin d
2, (9)
where the stochastic parameter is sin d 5 x. The behav-ior of the Sto-BS is determined by the nonstationary char-acter of the phase angle d, which is related to the external
electric signal applied to the PC. In the experiment re-ported here, the electric signal was a periodic function oftime, and the stochasticity of x, with Gaussian distribu-tion function p(xu s), arose from the random arrival timeof the photon on the Sto-BS.
We may write, for example, the action of the operator Uon the o polarization of the input single-photon purestate, uC& in 5 uo&:
uC& int 5 UuC& in
5 expS 2ip
4 D F cosS d
21
p
4 D uo& 1 sinS d
21
p
4 D ue&G ,
(10)
where uC& int is the intermediate polarization state just af-ter the Sto-BS.
A. Bose–Einstein Correlations by Stochastic Two-WayScatteringIt can be enlightening to analyze in more detail the sto-chastic partition over the two (M 5 2) output modes i5 1, 2 of the lossless Sto-BS of n particles associatedwith a Fock state un& and injected into one of the inputmodes of the device, the other input mode being in thevacuum state u0&.7–11 Consider the partition probabilityPnh of the n input particles into h and (n 2 h) outputparticles. In the case of stationary BS splitting (i.e., by aSta-BS) with elementary probabilities W1 and W2 5 12 W1 , this quantity is expressed by the well known Ber-
noulli distribution Pnh 5 W1(n2h)W2
hChn , Ch
n [ @n!/h!(n2 h)!# expressing ‘‘classical’’ absence of photon correla-
tions in the BS process. When the splitting is carried outby the Sto-BS, the above expression is replaced more gen-erally by
Pnh 5 22nChnE
21
11
@~1 1 x !~n2h !~1 2 x !h#p~xu s!dx.
(11)
The Sta-BS condition is easily recovered by setting s5 0 in Eq. (1): p(xu s) 5 d (x). The opposite conditions → ` implies the uniform distribution lims→` p(xu s)5 1/2, which in turn implies a driven, deterministiccorrelation of the state on the output particles. The cor-responding value of the integral is found to be gener-ally expressed in terms of gamma functions by thebeta function23: B@(n 2 h 1 1), (h 1 1)# [ G(n 2 h1 1)G(h 1 1)/G(n 1 2). This leads in our case to theresult Pnh 5 (n 1 1)21 expressing the Bose–Einstein(BE) partition law of n particles over M 5 2 boxes, i.e.,output modes. This result has been testedexperimentally.7 Interestingly enough, the realization ofthe general BE partition law (Pnh)BE 5 n!(M 2 1)!/(n1 M 2 1)! of n photons over a very large number of out-put scattering channels M @ 2 has also been successfullytested in a set of experiments in which a single laserbeam was stochastically (elastically) scattered by an en-semble of small polystyrene particles in Brownian motionat various temperatures T. There the transition fromBE to classical partition statistics was recovered by freez-ing of the motion of the scattering particles at low T.
1012 J. Opt. Soc. Am. B/Vol. 19, No. 5 /May 2002 De Martini et al.
Equation (11) was first introduced by Laplace and inves-tigated in detail by De Finetti in the context of exchang-able sequences in probability theory.24, 25, 26 In this con-text it might be found at first sight surprising that thevisibility V of the output patterns of the first- and second-order interferometers is lowered in the case of Sto-BS inrespect to the maximum value obtained with a Sta-BS.We may easily clarify this point as follows. We haveshown10,11 that the attainment in the former case of atrue photon Bose–Einstein condensation over the two in-terferometer modes implies a reduction, 2DH loc , of theparticle-localization information entropy (i.e., accountingfor the particlelike photon’s character) from the maximumvalue H loc 5 ln 2 5 0.693 to the value H loc 5 0.50. Thisimplies a corresponding increase by DH loc of the informa-tion entropy related to the complementary photon’s wave-like property, and consequently it leads to a loss of fringevisibility. More physically, the deterministic routing ef-fect induced by the Sto-BS on the input photon beamspoils the complete aleatory character of the coherentscattering property of the Sta-BS. The price paid for thein-principle information gained on the path taken by theparticle emerging from the Sto-BS is precisely a loss offringe visibility. Of course, this is a well known featureof all quantum phenomena ascribable to Bohr’s comple-mentarity. If we keep analyzing the interference phe-nomena reported here in the context of quantum informa-tion, we may note that the perturbation of the aleatoryscattering character of the input BS of the interferometer,and then the 2DH loc decrease of the information contentof the system, may be viewed as a kind of decoherence onthe state of the photon. In fact, in virtue of this decoher-ence process the pure qubit state implied by the normal(i.e., Sta-BS) two-way interferometer is transformed bythe effect of the Sto-BS into a mixed state. The detectedloss of fringe visibility may be indeed related formally tothe complex parameter of this mixing process.
3. THEORY OF FIRST-ORDER STOCHASTICINTERFEROMETRYWe refer to our model first-order stochastic interferom-eter, the Mach–Zehnder Sto-If shown in Fig. 1. The in-put beam is split by the Sto-BS in two different paths cor-responding to the orthogonal p polarizations o and e,before being recombined at the output by a stationary po-larizing beam splitter PBS. If f represents the phasedifference between the two arms of the interferometer,the output state can be expressed as
uC&out 5 Rz~f !uC& int
5 Rz~f !UuC& in 5 U~R !uC& in . (12)
In this expression, f 5 4pdz/l, where dz represents thelength difference of the two arms.
We can also write
rout~p! 5 U~R !r in~n!U†~R ! 5I 1 p • s
2, (13)
where r in(n) is defined as
r in~n! 5 uo&^ou 5I 1 n • s
2, (14)
with n 5 $0, 0, 1%. By means of the isomorphism be-tween the two-dimensional polarization Hilbert space andthe three-dimensional Poincare sphere space,27, 28 we ob-tain
U~R !n • s U†~R ! 5 n • Rs 5 RTn • s. (15)
In this way we obtain p 5 RTn. The rotation matrix R
associated with U(R) is
R 5 RzS p
2 DRyS d 1p
2 DRz~f !
5 F 2sin f cos f 0
2cos f sin d 2sin f sin d cos d
cos f cos d sin f cos d sin dG , (16)
where Rj(u) is the matrix for the u angle rotation aroundthe j axes.
The last element of the interferometer shown in Fig. 1consists of the polarization analyzer A(m), inserted afterthe output BS and in front of the detector D. A(m) se-lects the polarization component along the direction mand creates the conditions to observe the interferencefringe pattern.
A convenient expression of the fringe visibility is ob-tained by evaluating the photon-detection probability:
P~m, 6! 5 TrF1 6 m • s
2rout~p!G
5 TrF1 6 U †~R !m • sU~R !
2r in~n!G . (17)
Since a rotation R of the state is equivalent to a rotationby R21 of the measurement apparatus, we get
P~m, 6! 51
2@1 6 m • RTn# 5
1
2@1 6 m • p#
51
2@1 6 Rm • n#. (18)
For a 45° orientation of m in respect to o, m 5 $1, 0, 0%(o8 axis) is obtained:
P~d, w! 51
2~1 6 cos d cos w!. (19)
The instantaneous value of the first-order fringe visibil-ity, defined as V (1) 5 Imax 2 Imin /Imax 1 Imin , becomes
V ~1 !~d! 5 ucos du, (20)
with 2p/2 < d < p/2. The average value of the visibilitywith respect to x, V (1)(x) 5 (1 2 x2)1/2, is given by the ex-pression
^V ~1 !~x !&x 5 E21
11
p~x !V ~1 !~x !dx. (21)
De Martini et al. Vol. 19, No. 5 /May 2002/J. Opt. Soc. Am. B 1013
The evaluation of the integral may be given by use of theexpression (1) for p(x).
By adopting the integral representation of the hyper-geometric function 1F1(a, b; z) the expression of thefirst-order visibility is23
^V ~1 !&~ s! 5p
4
1F1F3
2, 2;
1
2s 2G1F1F1,
3
2;
1
2s 2G. (22)
Exactly the same result is obtained by the explicit evalu-ation of the integrals within the following general expres-sion for the first-order visibility evaluated10 by a differenttheoretical approach:
^V ~1 !&~ s! 5p
2
E2`
1`
P~x!J1~2px!
xdx
E2`
1`
P~x!sin~2px!
xdx
. (23)
Here J1(2px) is a Bessel function and P(x)[ @erf(A2s)21#21 exp@2(A2psx)2# is the Fourier trans-form of the Gaussian p(xu s). The function ^V (1)&( s) isreported in Fig. 3 (solid curve) as a function of the noiseparameter s in the range 0–100. For s → `, i.e., for afully white-noise spectrum, the asymptotic expansion1F1(a, b; z) leads to the limit value: ^V (1)&(`) 5 p/4' 0.785.7,10 The other limit s 5 0, corresponding to thecommon stationary first-order interferometry, leads to thewell known value ^V (1)&(0) 5 1 independent of the aver-age number of particles ^n& associated with the inputstate.10
A. ExperimentIn view of the configuration of our experiment we firstfind it convenient to express the average first-order vis-ibility as a function of d:
^Vm~1 !& 5
1
DTE
DTV ~1 !~d!dt. (24)
Fig. 3. First-order fringe visibility ^V (1)&, Eq. (22), as a functionof the stochastic parameter s : the solid curve represents thetheoretical prediction; the dotted curve corresponds to the best-fit analysis of the experimental data.
In the above expression, DT represents the characteristicperiod of variation of d. We need to consider the simul-taneous realization of the two conditions: (a) DT@ DT8, where DT8 is the transit time of the photonthrough the Sto-BS, and (b) DT ! DT9, where DT9 corre-sponds to the time necessary to collect the statistical en-semble of photocounts for each value of the phase shift f.Both conditions are easily satisfied in the experiment be-cause DT8 is in the picosecond time domain, and DT9 canbe of the order of a second. Since x 5 sin d, we make useof the following condition:
1
DTdt 5 p~xu s!dx. (25)
As a consequence,
dt
dx5 DTp~xu s! 5
DT
A2ps erfS 1
A2sD expF2S sin d
A2sD 2G .
(26)
We may express the first-order visibility in the form
^Vm~1 !& 5
1
DTE
2p/2
p/2
V~d!dt
dx
dx
dddd, (27)
where dx/dd 5 cos d. It is obtained
^Vm~1 !& 5
1
A2ps erf~1/A2s!E
2p/2
p/2
expF S sin d
A2sD 2Gcos2 ddd.
(28)
This expression leads to the integral representation givenin Section 3, Eq. (22).
Consider the experimental configuration adopted forthe investigation of first-order interference (Fig. 4). Thephoton source was a mode-locked Nd:YAG laser (modelCoherent-Antares), operating at a repetition rate of 76-MHz frequency, upconverted by third-harmonic genera-tion (THG) and emitting 50-ps coherent light pulses withwavelength l 5 354.7 nm. These pulses excited a collin-ear SPDC process in a 1-mm-thick b-barium borate NLcrystal cut for type II collinear phase matching at an
Fig. 4. Experimental apparatus for first-order stochastic inter-ferometry.
1014 J. Opt. Soc. Am. B/Vol. 19, No. 5 /May 2002 De Martini et al.
angle 49.4° with respect to the optical axis. The twoemitted correlated photons, polarized along the orthogo-nal p directions (o, e), corresponding to the optical axesof the crystal, were selected at the common wavelengthl 5 709.4 nm. One of the photons was transmittedthrough the polarizing beam splitter (PBS1) located infront of Sto-BS in Fig. 4 and detected by D1 .
The corresponding output signal was adopted as thetrigger for the conditional measurement performed by de-tector D2 placed at the output of the Sto-IF apparatus.The radiation detected by D1 and D2 was filtered by twocorresponding equal interference filters (IFs) with band-width Dl 5 6 nm. The detectors were Si avalanchephoton-counting modules EGG-SPCM200 with quantumefficiencies .65%. The output signals of the D’s fed a co-incidence counter within time windows of 5 ns.
The photon emerging from the Sto-BS device (ex-pressed by the gray area in Fig. 4), followed suitable op-tical paths within the single polarizing beam splitterPBS2 to realize, with two couples of external mirrors, thestandard Mach–Zehnder interferometer (MZ-If )configuration.29 The arms of the MZ-If were associatedwith the orthogonal states of the p polarization (o, e).In order to restore the indistinguishability between theoptical paths of MZ-If, a p analyzer A, oriented at 45°with respect to (o, e), was inserted in front of D2 .Fringe-visibility patterns were obtained and registered byvarying the optical-path difference dz between the twoarms of the MZ-If by a piezoelectric translation drive.
The main element of the Sto-BS was a transverselymodulated LiNbO3 Pockels cell, with a characteristicphase-shift voltage Vp/2 5 (282 6 8)V at l 5 709.4 nm.The Pockels cell was driven by an adjustable generatorthat supplied the high-voltage waveform expressed byv(t) 5 A(t) u( ft) with u( ft) being a cw zero-meansquare-wave carrier with a first-harmonic frequency f5 100 kHz, amplitude modulated by a waveformA(t).10,11 The overall signal consisted of a periodic se-quence of equal positive quasi-triangular pulses with du-ration Dt 5 1.4 ms, each starting at a different time tj5 (tj21 1 Dt) and expressed by Aj(t) 5 sin21@ g(t)#within 0 < t < 1, t [ 2(t 2 tj)/Dt. Each triangularpulse of the sequence was completed in the interval 1< t < 2 by a pulse equal to Aj(t) but symmetrical intime, i.e., having in common with Aj(t) the maximumvoltage value and decreasing toward zero with negativeslope. In these expressions the function g(t) [ ux(t)u isthe solution of the equation 2p(xu s) 5 dt (uxu)/duxu. Thesin21 dependence of Aj(t) accounts for the Malus law ofpolarization, and g(t) expresses the effect of the distribu-tion function p(xu s) of the driving signal. The functionAj(t) was synthesized, for different values of s, by electri-cally programmable read-only memory devices. The in-set of Fig. 4 shows the electrical waveform v(t), normal-ized to its maximum amplitude Vp/2 and related to theGaussian distribution p(xu s) given in Eq. (1) with s5 0.5. Alternatively, a truly stochastic excitation wasprovided by a carrier u( ft) supplied by a narrow-bandGaussian noise source having a Lorentzian spectrum withdf/f ' 0.1 and a center frequency f ' 100 kHz. Furtherdetails on the Pockels cell driving are found in Refs. 10and 11.
The experiment was performed by collecting severalstatistical samples of the measured visibility correspond-ing to different values of s (varying in the range 0.3–100)and by comparing in each case the visibility obtained instochastic conditions (s Þ 0) with that measured in sta-tionary conditions (s 5 0). In this way, we could obtaindirectly the average reduction of visibility for differentvalues of the noise parameter s.
The experimental results of the average visibility Vm(1)
versus s are shown in Fig. 3. The best fit gives a visibil-ity reduction for increasing s of ;20%, in agreement withthe theoretical predictions and with the experimental re-sults previously obtained for coherent light (see Fig. 2 ofRef. 10). The best-fit curve representing the results iscalculated by assuming that the stochasticity parameter xvaries within the reduced interval [20.9, 0.9] instead ofthe integration interval [21, 1] appearing in the theoret-ical result expressed by Eq. (21). This reduction of do-main of stochastic variation is due to a noncomplete ran-domization of the R and T optical parameters of theSto-BS obtainable with our experimental EO driving sys-tem. Precisely, the EO response of the adopted Pockelscell could not follow the 100-kHz modulation frequency ofthe driving high-voltage electric noise generator. An-other possible source of disturbance is the presence ofacoustic-wave resonances in the LiNbO3 crystal of our EOdevice caused by piezoelectric ringing for driving frequen-cies larger than 100 kHz.30
4. SECOND-ORDER STOCHASTICINTERFEROMETRY
A. TheoryThe process of stochastic interferometry involving a two-photon state may be modeled by consideration of two en-tirely different optical schemes corresponding to experi-ments carried out so far in the stationary regime. Astochastic extension of these two different schemes isshown by the two drawings in Fig. 5.
Consider first the scheme shown in Fig. 5(a), whereeach incoming photon particle with equal p polarizationexcites one of the two input modes kj ( j 5 1, 2) of a sym-metric beam splitter (BS), having the respective statespreviously stochastically phase shifted by (p/2 1 d1) and(p/2 1 d2) by two equal (quarter-wave plate plus Pockelscell) EO devices. This is the mixed-state injected versionof the well known stationary secondary-order interferom-eter, first introduced by Mandel and coworkers15 and byAlley and coworkers.16
The effect of the overall input device is represented bythe unitary operator:
Vh~Sh! 5 expF iS dh
21
p
4 D sxG5 cosS dh
21
p
4 D I 1 i sinS dh
21
p
4 D sx , (29)
where Sh 5 Rx(dh 1 p/2) and h 5 1, 2. In the case ofequal input (o) polarization for the two photons the finalstate lies on the y, z plane of the Poincare sphere; i.e., it iselliptically polarized.
De Martini et al. Vol. 19, No. 5 /May 2002/J. Opt. Soc. Am. B 1015
Assume the input product state:
uC& in 5 uc1&1 ^ uc2&2 , (30)
where u c j&a is the u c j& polarization state for the a mode.The state after transformation and before injection in theBS is
uC& int 5 V1uc1&1 ^ V2uc2&2 . (31)
The state after the linear superposition in the BS, char-acterized by parameters t, r, and utu2 1 uru2 5 1, is
with uc j&a 5 Vju c j&a . Prime and unprimed indexes ac-count for the detector excitation at the different times, tand t 1 t. In order to refer more precisely to the experi-mental conditions, the time t accounts for the temporalposition of the center of mass of the photon wave packet(wp) assumed Gaussian with a full width expressed bythe photon’s coherence time tcoh .
where the two square brackets account, respectively, forsingle-photon coincidences involving detection on bothoutput modes and for two-photon detection on either oneof the output modes.
The intensity correlation measurement, performed bytwo equal single-photon detectors Dj coupled to both out-put modes of a single BS, is evaluated by the average:
where G is an appropriate proportionality factor. For nowp overlap the interference term disappears and
R` 5 G@ utu4 1 uru4#. (36)
The second-order interference visibility V (2) is defined inthe same way as for the first-order case. For wp overlapit is found
V ~2 ! 5uR` 2 R0u
R`
52uru2utu2
utu4 1 uru4 u^c2uc1&u2. (37)
and, for no wp overlap, V (2) 5 0. In the case of a losslesssymmetric beam splitter,31 V (2) 5 u^c2uc1&u2, which de-pends on the mutual projection of the two-photon polar-ization states transformed by the operator (29). For d1
5 d2 , i.e., V1 5 V2 , i.e., for a fully in-phase p transfor-mation, the visibility attains its maximum value: V (2)
5 1. In general, for an input product state uC& in5 uo&1 ^ uo&2 , it is found
V ~2 !~d1 , d2! 51
2@1 1 cos~d1 2 d2!#, (38)
which shows the effect of the phase shift d1 2 d2 in thestochastic regime. For two statistically independent d1, 2Gaussian phase distributions, with the same s, we obtain
^V ~2 !&~ s! 51
2$@1 1 @^V ~1 !&~ s!#2%. (39)
The dashed curve of Fig. 6 shows the behavior of
Fig. 6. Second-order fringe visibility ^V (2)& as a function of thestochastic parameter s. The dashed curve corresponds to theexpression of the theoretical ^V (2)& for the noncollinear case, Eq.(39). The solid curve represents the curve for ^V (2)& in the col-linear case, Eq. (57). It can be compared with the dotted curve,which corresponds to the best-fit representation of the experi-mental data.
1016 J. Opt. Soc. Am. B/Vol. 19, No. 5 /May 2002 De Martini et al.
^V (2)( s)& as a function of s. In the limit of a uniform dis-tribution, s → `, the second-order visibility is ^V (2)&5
12@1 1 (p/4)2# > 0.808.
Assume now an input p-polarization entangled state:
uC& in 51
A2@ uca&1 ^ ucb&2 1 exp~if !ucb&1 ^ uca&2].
(40)
The transformation in terms of the operator (29) is
u c& int 51
A2@ uca
1&1 ^ ucb2&2 1 exp~if !ucb
1&1 ^ uca2&2],
(41)
where ucak& j 5 Vku ca& j . After the BS superposition, we
have
uC&out 51
A2@t2uca
1&3 ^ ucb2&48 1 r2ucb
2&38 ^ uca1&4
1 exp~if !~t2ucb1&3 ^ uca
2&48 1 r2uca2&38 ^ ucb
1&4)]
1 uNoCoinc&. (42)
The uNoCoinc& state accounts for two-photon detection onone output mode and does not lead to coincidences.
The coincidence rate detected on both output modes isthe average of expression (34) over the input state. Incase of full photon wp’s overlapping, t ! tcoh , it is found,
R0 5 G$@ utu4 1 uru4#@1 1 u^caucb&u2cos f #
1 t2r* 2Re@^cb2ucb
1&^ca1uca
2&exp~if !# 1 c.c.%,
(43)
which reduces, in the absence of overlap, to
R` 5 G@ utu4 1 uru4#@1 1 u^cau cb&u2cos f #. (44)
The second-order interference visibility is
V ~2 ! 52uru2utu2
utu4 1 uru4 uRe@^cb2ucb
1&^ca1uca
2&exp~if !#u, (45)
which depends on the entanglement phase f. For the in-put state uC& in 5 1/A2@ uo&1 ^ ue&2 1 exp(if )ue&1 ^ uo&2] anda lossless symmetric BS we obtain the expression
V ~2 !~d1 , d2 ; f ! 51
2@1 1 cos~d1 2 d2!#ucos~f !u,
(46)
which coincides with that obtained for the case of an in-put product state but with a new factor accounting for theentanglement phase.
If we consider the same experimental configurationshown by Fig. 5(a) with the normal BS replaced by a po-larizing beam splitter, we have the stochastic counterpartof the second-order interferometer. In this case, we ob-tain the expression of the second-order visibility for theinput product state,
V ~2 !~d1 , d2! 5cos d1 cos d2
1 1 sin d1 sin d2, (47)
and for the input entangled state,
V ~2 !~d1 , d2 ; f !
5cos d1 cos d2 1 ~1 2 sin d1 sin d2!cos f
1 2 sin d1 sin d2 1 cos d1 cos d2 cos f. (48)
Consider now the second-order Sto-IF shown by theconfiguration given in Fig. 5(b), which coincides with thatadopted in our experiment. The two-photon state, ex-pressed as uC& in 5 uo& ^ ue& and generated in a type II NLcrystal by collinear SPDC, propagates through a quarter-wave plate (l/4) and a Pockels cell (PC) before being pro-cessed by a polarizing beam splitter (PBS). This second-order polarization-interference method, to our knowledge,has been first adopted in our laboratory.32 Consider theeffect of the transformation expressed by the Sto-BS op-erator (29) on the input p-polarization state of two pho-tons generated with mutual delay t : uC& in 5 u c1&1^ u c2&18 . By adoption of the same notation as for thenoncollinear case, we have
uC& int 5 uc1&1 ^ uc2&18 . (49)
The state emerging from the PBS, which transmits andreflects the o and the e polarizations, respectively, is theoutput state and is written as
where cjk,lq 5 ^ juck&^lucq&, with j, l 5 o, e and k, q 5 1,2, corresponds to the product of the projections of the pho-ton polarization state on the PBS basis. We expect inter-ference when the two photons wp’s overlap: t ! tcoh .However, when this is not the case, the interference dis-appears, as we shall see, and the two particles are split bythe PBS as classical particles.
The intensity correlation at the output of the PBS is ex-pressed by the average of the operator (34) over the inputstate that postselects in this way the first contribution ofEq. (50).33,34
In the present interferometry experiment the measure-ments are carried out by both detectors Dj , and the mea-sured quantity is the number of coincidence detectionscounted within a suitable time window. The coincidencerates for photon wp’s overlap and nonoverlap are given re-spectively by
R0 5 Gu^ouc1&^euc2& 1 ^euc1&^ouc2&u2u, (52)
R` 5 G@ u^ouc1&^euc2&u2 1 u^euc1&^ouc2&u2#. (53)
The second-order visibility (37) is then generally ex-pressed by
V ~2 ! 5 2u^ouc1&^euc2&^euc1&^ouc2&u
u^ouc1&^euc2&u2 1 u^euc1&^ouc2&u2. (54)
For an input state uC& in 5 uo&1 ^ ue&18 , the above formulareduces to
V ~2 !~d! 5cos2 d
1 1 sin2 d. (55)
De Martini et al. Vol. 19, No. 5 /May 2002/J. Opt. Soc. Am. B 1017
Note that Eq. (47) reduces to Eq. (55) for d1 5 d2 . Thismeans that the collinear case is equivalent to the noncol-linear case, operating with a polarizing beam splitter andwith injection of a product state or of a singlet entangledstate (f 5 p/2).
As in the case of first-order interferometry, the averagevalue of the visibility with respect to the stochasticity pa-rameter x 5 sin d is calculated by means of the expres-sion
^V ~2 !~x !&x 5 E21
11
p~x !V ~2 !~x !dx. (56)
However, experimentally we can measure the mean val-ues ^R0&x and ^R`&x , and hence we can obtain the experi-mental mean visibility:
^Vm~2 !~ s!& 5 u^R`&x 2 ^R0&xu/^R0&x . (57)
The theoretical behavior of ^V (2)&( s) as a function of s isreported in Fig. 6 (solid curve). In the limit s → ` wehave ^V (2)& 5 1/2.
B. ExperimentA second-order stochastic interferometry experiment wasperformed by adopting the optical configuration shown inFig. 5(b). A detailed experimental layout is given in Fig.7. The correlated photons of each pair were SPDC gen-erated at the same l 5 709.4 nm with a coherence timetcoh 5 100 fs by a type II (b-barium borate) crystal cut forcollinear emission with the same third-harmonic-generated mode-locked laser apparatus described in Sec-tion 3. In the condition of collinear emission, both emit-ted photons belong to the same spatial mode k and havemutually orthogonal linear p polarizations, directed alongthe ordinary (o) and extraordinary (e) axes of thecrystal.14 Generally, at the exit of the NL crystal the twophoton wp’s do not overlap because of the crystal birefrin-gence and the different group velocities. In the experi-ment this effect was fully compensated by two couples ofmicrometrically adjustable mirrors placed at the exitmodes of a polarizing beam splitter PBS1 shown in Fig. 7.The two-photon product state uC& 5 uoe& [ uo& ^ ue& wastransmitted through the Sto-BS, corresponding to the se-quence of optical elements located within the gray area ofFig. 7.
Fig. 7. Experimental apparatus for second-order stochastic in-terferometry.
The photons emerging from the two output sides of thepolarizing beam splitter PBS2 were detected by D1 and D2after filtering by two equal IF’s having a bandwidth Dl5 6 nm. The Sto-BS and the related d-driving deviceswere identical to the one described in Section 3.10,11
As in the case of the first-order interference measure-ments, the experiment was performed by collecting sev-eral statistical samples of the second-order visibility fordifferent values of the variance of the stochastic param-eter s. Figure 6 shows the experimental results corre-sponding to the measurements of ^Vm
(2)& as a function of s.Note there the loss of the visibility for increasing values ofs starting from the stationary condition value s 5 0.The experimental results and the corresponding best fit(see the dotted curve of Fig. 6) give a ;20% variation ofthe second-order visibility, which is lower than the theo-retical prediction of 50% given by the solid curve of thesame Fig. 6. We believe that this effect can be attributedto the same limitations of the frequency response of theEO modulator, already discussed in connection with theresults of the first-order Sto-IF experiment.30 This prob-lem is caused by the incomplete randomization of thebeam-splitter parameters. As a consequence, the shapeof the distribution of the stochasticity parameter s andthe amount of the visibility variation are modified.
5. CONCLUSIONIn conclusion, we have reported, for the first time to ourknowledge, a full theoretical account of the first- andsecond-order interferometry in the stochastic regime by aFock-state photon excitation. The theoretical resultswere substantiated by two experiments involving themodern technique of nonlinear spontaneous parametricdownconversion. We believe that the present results arerelevant for a deeper understanding of the statisticalproperties of quantum particles within the fundamentalframework of field interferometry operating in exotic con-ditions.
This study was supported by the Progetto di RicercaAvanzata of the Istituto Nazionale per la Fisica della Ma-teria (PRA-Cat 97), by the Ministero per l’Universita’ eper la Ricerca Scientifica e Tecnologica, and by the Field-Effect-Transistor European Network IST-2000-29681 (Ac-tive Teleportation and Entangled States InformationTechnology) on Quantum Information.
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