Table Of ContentPhase Selective Quantum Eraser
A. Heuer, G. Pieplow, R. Menzel
Photonik, Institut fu¨r Physik und Astronomie, Universit¨at Potsdam, D-14469 Potsdam, Germany
(Dated: January 6, 2015)
A quantum-eraser experiment is reported with photon pairs generated by two synchronously
pumpedparametricdownconverterscoupledviainducedcoherence. Thecomplementaritybetween
which-sourceinformationandtwo-photoninterferencefringevisibilityhasbeeninvestigatedbytwo
coupled interferometers.
PACSnumbers: 42.50.Ar,42.50.Dv
The complementarity principle states that performing
5 a measurement to expose both wave and particle aspect
1 of a single quantum object [1] is not possible at a any
0 given moment. The wave-like behavior often manifests
2
itself in the appearance of interference fringes, while for
n example the ’welcher Weg’ information, a particle prop-
a
erty, destroys the interference. Quantum-eraser experi-
J
ments proved to be very useful to study the complemen-
5
tarity of wave and particle aspects of photons. Since the
] first proposal by Scully and Dru¨hl [2], quantum-eraser
h protocols have been discussed extensively, especially in
p
thecontextofcomplementarityandfringevisibility[3–5].
-
t In optical interferometric studies [6], it was shown that
n FIG. 1. Schematic of the experimental setup of two coupled
interference results from the intrinsic indistinguishabil-
a
interferometers based on induced coherence.
u ity of the photon’s path. The fringe visibility vanishes,
q however , for distinguishable photon paths. Ultimately,
[
thisresultsinadirectrelationofcoherenceandindistin-
1 guishability [7]. Interference experiments that address 1. The pump light source is a frequency tripled optically
v this point using quantum eraser protocols can be found pumpedsemiconductorlasercontinuouslyoperatingata
7 in [4, 5]. wavelengthof355nm. Abeamsplitterwasusedtodivide
1
In this work we present an experiment where we the pump beam into two almost identical coherent sub-
8
0 used two coupled parametric down conversion crystals beams to pump two BBO crystals as parametric down-
0 in a cascaded arrangement. Wang, Zou and Mandel converters. Thecrystals,withlengthsof4mmeach,were
1. [8, 9] demonstrated that, in a setup such as that shown cutfordegeneratecollineartypeIphasematchingat355
0 in Fig.1, coherence can be induced without induced nm. In our case, the crystals were slightly tilted and the
5 emission,ifthephotonfluxissolowthattheprobability whole setup was arranged with filters and apertures to
1 for simultaneous generation of two photon pairs in select signal photons at a wavelength of 808 nm and the
:
v the two crystals is negligible. The coherence of the corresponding idler photons at a wavelength of 632 nm.
i separately generated signal photons depends on the To achieve induced coherence without induced emission,
X
indistinguishability of the idler photons. For perfect theidleroutputi ofthefirstBBO1crystalwasmatched
r 1
a matchingofidlermodes,thesignalfieldsfromBBO1and perfectly with the idler output i2 of the second crystal
BBO2showperfectinterference,whiletheinterferenceis BBO2 (as described in [8]). A perfect overlap of the two
lost if the idler photon’s origin becomes distinguishable. beams is essential for a high visibility in the signal pho-
With this setup we can demonstrate the effect of phase ton interference [12]. To observe the interference of the
memory with single photon interference [10]. The phase signal beams, both signal outputs of crystal BBO1 and
memory is carried by the two photon state genearted by crystal BBO2 were superimposed at the beam splitter
SPDC. Recently the induced-coherence setup was used BS2. To realize a variable phase delay, a high resolution
for a quantum imaging application [11]. delay line was introduced into the signal beam s . The
1
In contrast to the last reference an additional inter- minimum step size of the delay line was 20 nm.
ferometer for the idler photons has been added to our An additional beam splitter BS1 in between both BBO
experiment to examine the interplay of indistinguisha- crystalssplitstheidlerbeami . Itisrecombinedwiththe
1
bility and fringe visibility in two photon interference. transmitted idler beam at the beam splitter BS3, which
effectively results in a Mach-Zehnder interferometer for
AschematicofourexperimentalsetupisshowninFig. the idler beam of crystal BBO1. The path length dif-
2
FIG. 2. Measured interference signal in a standard induced
coherencesetup(thereflectedidlerbeamofBS1wasblocked).
Both, the single photon rate of detector A (a) and the coin-
cidencerateofdetectorAanddetectorB(b)wereplottedas
afunctionofthesignalpathlengthdifference∆x . Thesolid
s
curvesrepresentthebestsinusoidalfit. Wefoundaperiodof
808 nm.
FIG. 3. Single rate (a,c) of detector B and coincidence rate
ference of the two beams was controlled using a variable
(b,d)betweendetectorAandBasafunctionoftheidlerpath
trombonedelayinthetransmittedbeampath. Thepho-
lengthdifference∆x . Thesolidcurvesrepresentthebestfitt
i
tons were detected with fiber coupled avalanche photo resulting in a period of 633 nm. During the measurement of
diodes(SPCM-AQRH-13,PerkinElmer)atthepositions (a,b) only crystal BBO1 was pumped. For the measurement
shown in Fig. 1. The measurement scenario can be al- of (c,d) both crystals were pumped and signal beam s1 was
tered with two beam blocks. To measure the classical blocked.
induced coherence [8] the reflected idler beam of beam
splitter BS1 can be blocked. For the coherence measure-
depends linearly on the transmissivity [9], thus without
ment of the idler beams a beam block can be inserted
the beam splitter a visibility of more than 80% would
into the signal beam s .
1 havebeenrecorded(amplitudetransmissivityt=0.67of
In our first experimental run we examined the coher- beam splitter BS1).
ence properties of the signal beams. A beam block is ToadjustthealignmentoftheMach-Zehnderinterfer-
used to block the reflected idler beam of beam splitter ometer for the idler photons, only the BBO1 crystal was
BS1. Thus the detector B will only see a fraction of the pumped. A high contrast interference signal is recorded
idlerphotonsi1fromcrystalBBO1whicharetransmitted by detector B when the delay line of the upper idler
through the beam splitter BS1 and superimposed with beam is varied (see Fig. 3(a,b)). The visibilities of the
theidlerphotonsi2fromcrystalBBO2. Thepumppower fringes of the single count rate and the coincidence rate
forthetwocrystalswas35mWeach. Theresultingcount between detector B and A were both above 95%. If the
ratesatthedetectorswere22,000photons/secand14,000 second crystal is also pumped, the single photon count
photons/sec for detector A and B, respectively. Under rate increases but the visibility is reduced significantly
these conditions the probability for simultaneous gener- (V ≈ 50% ) as seen in Fig. 3(c). This reduction cannot
ation of two biphotons during the measuring interval of be attributed to an unbalanced interferometer. It seems
2 ns is less than 10−4. This insured induced coherence that all the photons generated in the BBO2 crystal are
instead of stimulated emission as the mechanism for the added incoherently to the photons from the first crystal.
generation of the second biphoton and thus, stimulated This becomes even more apparent when the coincidence
emission can be neglected. measurements (see Fig. 3(d)) are analyzed: Inserting a
The interference contrast is measured by varying the beam block in the signal arm s and recording the co-
1
delay line in the signal beam s and recording the count incidence rate between detector B and A does not show
1
rates of signal detector A and the coincidence counts of any variation, when modulating the path length differ-
detectors Aand B.The resultsof themeasurementscor- enceoftheidlerbeams. Fromthequantumopticalpoint
responding to first-order interference are shown in Fig. of view it is clear that there is no coherence between the
2(a) and those corresponding to second-order interfer- idler photons generated in BBO1 and BBO2 because of
ence are showen in Fig. 2(b). The data was fitted by the non overlapping signal beams. The idler photon’s
a least squares fitting routine. We found visibilities of pathways s or s are therefore totally distinguishable.
1 2
54%forfirst-orderinterferenceand91%forsecond-order To erase the ’welcher Weg’ information of the idler
interference. These are remarkable values and indicate photons, the beam block was removed. The signal pho-
a high degree of induced coherence and good alignment ton detector A can can now measure photons from both
of optical elements in the setup. In an induced coher- crystals. This results in two coupled interferometers for
ence experiment, the visibility of first-order interference the coincidence measurement of detectors A and B. The
3
a =t a +r a , (2)
i02 1 i01 1 i0
where t and r are the transmissivity and reflectivity of
1 1
the beam splitter BS1 and a is the annihilation oper-
i0
ator of the vacuum field mode in the empty port of the
beam splitter. In our subsequent calculations we assume
that the idler field i that interacts with the pump p
02 2
in BBO2 is perfectly aligned with the transmitted idler
FIG. 4. Coincidence rate of detector A and B depending on
field i and that the conversion efficiency is so low that
the signal path length difference ∆x for two different phase 01
s
delays ∆ϕi of the idler interferometer. a) high interference the generated idler field i1 of BBO1 contributes negligi-
contrast at a phase delay ∆ϕ ≈π/2 and b) low interference bly (compared to the vacuum idler) to the generation of
i
contrastataphasedelay∆ϕi ≈0. Thesolidcurvesrepresent the signal s2 in the second crystal. After propagation
the best sinusoidal fit. We found a period of 808 nm. throughthesetup(seeFig. 1)thetotalelectricfieldam-
plitudes of the signal field at detector A and of the idler
field at detector B are:
coincidenceratewillbeafunctionofbothpathlengthdif-
ferences∆xs and∆xi,respectively. Inthisrunoftheex- AA = r2 ei∆ϕs(cid:0)as01+iC1a†i01(cid:1)
periment we kept the phase difference of the idler beams +t (cid:0)a +iC (cid:0)t a† +r a† (cid:1)(cid:1) ,
∆ϕ = 2π/λ ∆x fixed. Fig. 4(a) shows the measured 2 s02 2 1 i01 1 i0
i i
coincidence rate as a function of the signal path length A = r r (cid:0)a +iC a† (cid:1)+r t a
B 3 1 i01 1 s01 3 1 i0
difference∆xs withanidlerphasedelay∆ϕi ≈π/2. The +t ei∆ϕi(cid:0)t a +r a +it C a† +iC a† (cid:1) ,
phasewasdeterminedbytheamplitudeofthesinglepho- 3 1 i01 1 i0 1 1 s01 2 s02
(3)
toncountrateofdetectorB.Weexpectthelowestcount
where r and t are the reflectivity and transmissivity of
x x
rate when there is no phase delay. This is due to the de-
the respective beam splitter. The phase factors ∆ϕ and
i
structive interference of photons from the crystal BBO1.
∆ϕ are caused by the idler interferometer and the sig-
s
The count rate peaks at ∆ϕ = π. The visibility of the
i nalinterferometer,respectively. FromA andA wecan
A B
interference signal is relatively high and reaches a value
compute the normally ordered correlation functions that
of 97%. For ∆ϕ ≈0 the single count rate of detector B
i provide the counting rates measured by the ideal pho-
reaches a minimum and the visibility is reduced down to
ton detectors A and B. Because signal and idler modes
40% as can be seen in Fig.4(b).
areinitiallyunoccupied,weevaluatethecorrelationfunc-
In the following section we discuss results using a sim-
tionsintheirrespectivevacuumstates. Wecalculatethe
plified theory [13–15]. It captures the essence of what
correlation functions to the first order of the C’s, which
wemeasuredwithoutinvokingacumbersomeformalism.
is consistent with our assumption that there is at most
We can assume that the down-conversion efficiency γ2 is
onlyonesinglebiphotonintheapparatusduringanyde-
so small that there is no more than a single photon pair
tection interval. The signal photon counting rate at B is
presentatanytime. Duetothisverylowpairgeneration
proportional to
rate,weassumethatthepumpfieldsp andp donotde-
1 2
plete and can hence be described with classical complex R ∝ (cid:104)A† A (cid:105)
amplitudes A and A . We work in the limit of per- B B B
fect phase mapt1ching forp2all involved modes. We assume = |C1(cid:0)r1r3+t1t3ei∆ϕi(cid:1)|2(cid:104)as01a†s01(cid:105) (4)
all fields to be monochromatic and that the interaction +|C t ei∆ϕi|2(cid:104)a a† (cid:105)
2 3 s02 s02
with the crystals is pointlike. With these assumptions, = |C (cid:0)r r +t t ei∆ϕi(cid:1)|2+|C t |2 . (5)
thefieldamplitudesforthesignal/idlerbehindBBO1are 1 1 3 1 3 2 3
[13]:
In accordance with the experimental findings (see Fig.
3(c)) equation (5) contains the incoherent contribution
A = a +i C a† ,
s1 s01 1 i01 (1) to the detector signal: The second term on the right
A = a +i C a† of equation (??) (|C |2) corresponds to the rate of idler
i1 i01 1 s01 2
photongenerationintheBBO2crystal. Althoughthein-
where a and a are the free, unperturbed operators distinguishableidlerphotonsinducethecoherenceofthe
s01 i02
satisfying the relation a |Ψ(cid:105) = a |Ψ(cid:105) = 0 for the signalphotonsthereisnovisibleinterferencebetweenthe
s01 i01
initial state |Ψ(cid:105). C = γA is a constant that incorpo- idlerphotonsgeneratedinBBO1andinBBO2. However,
i pi
rates the crystal properties and the pump intesity. Due thejointdetectionrateofsignalandidlerphotonsshows
to the beam splitter BS1 (see Fig. 1) the unperturbed high contrast second-order interference. The coincidence
idler operator a is subject to the usual transformation countingrateR fordetectioneventsatdetectorAand
i02 AB
4
fringe visibility as a function of the idler phase delay is:
√
| 8cos(∆ϕ /2)|
V(∆ϕ )= i . (8)
i 2−cos(∆ϕ )
i
The visibility is 0 for ∆ϕ =n·2π and is 1 for odd mul-
i
tiples of π/2 as can be seen in Fig 5(b). The high vis-
ibility close to 1 that we observed for ∆ϕ =π/2 and
i
a significantly reduced visibility for ∆ϕ =0 reproduces
i
the theoretical findings [see Fig. (4)]. Due to the im-
FIG. 5. a) Coincidence rate as a function of time: The path
perfect balance of the beam splitters BS1 and BS3, the
length differences of the signal and the idler interferometer
contrast in Fig. (4) b) does not vanish. Instabilities in
∆x and ∆x are varied at constant speed of 20 nm/s. b)
s i
Visibiltiy as a function of the idler phase delay ∆ϕi as given the setup which lead to fluctuations in ∆ϕi are another
by expression 7 reason for not completely extinguishing the contrast. In
combination with the very sharp resonance at ∆ϕ =0,
i
measuring a visibility of 0 is very challenging.
detector B is In the ideal setup of perfectly balanced beam splitters
onecanexplainthelossandrecoveryofinterferencecon-
R ∝(cid:104)A†A† A A (cid:105)
AB A B B A trast in a simple way: The idler photons originating
= |C r ei∆ϕs(cid:0)r r +t t ei∆ϕi(cid:1)+C t t ei∆ϕi|2 . fromBBO1willinterferedestructivelyatthebeamsplit-
1 2 1 3 1 3 2 2 3
(6) ter BS3 for a phase delay ∆ϕi =n·2π. Thus, only the
The three terms in equation (6) represent the three in- idler photons originating from the BBO2 crystal will be
terferingpathsthatasignalandidlerphotoncantakein counted at detector B. This tags the signal photon’s ori-
order to be counted at A and at B simultaneously. The gin, which prevents interference in the signal interferom-
first probability amplitude (i) ir2r1r3C1ei∆ϕs represents eter. When balancing the rate of idler photons originat-
thefollowingevents: Signalphotons1acquiresthephase ing from BBO1 and BBO2 by adjusting the idler phase
delay ∆ϕs and is reflected at BS1, while the idler i1 is ∆ϕi =π/2the’whichcrystal’tagofthesignalphotonis
reflected at BS1 and BS3. (ii) ir2t1t3C1ei(∆ϕs+∆ϕi) rep- erased and interference in the signal/signal interferome-
resents: Signal photon s1 acquires the phase delay ∆ϕ ter is restored. Our experiment is thus a realization of
s
andisreflectedatBS2whiletheidlerphotoni1istrans- a quantum eraser. Intermediate states of the idler phase
mitted trough BS1, aquires a phase delay ∆ϕ and is only make partial ’which crystal’ information available
i
transmitted through BS3. (iii) it2t3C2ei∆ϕi represents: and therefore they reduce the interference contrast. In
Signal photons are transmitted through BS2, the idler summary, an experiment with two coupled interferome-
photon i2 acquires a phase delay ∆ϕ , and is transmit- ters for correlated signal and idler photons generated by
i
ted through BS2. spontaneous parametric down conversion has been pre-
We now assume, for the sake of simplicity, equal con- sented. Thecouplingoftheinterferometerswasachieved
version rates (C = C ) and perfectly balanced beam byinducedcoherence. Thedetuningoftheidlerinterfer-
1 2
splitters. The coincidence rate R is: ometer allows for the selection of the origin of the pho-
AB
tons. We thus demonstrated a quantum eraser protocol
√
R =2−cos(∆ϕ )− 8cos(∆ϕ /2)× forthecomplementarityofwave-likeandparticle-likebe-
AB i i
(7)
havior of photons .
sin(∆ϕ −∆ϕ /2) .
s i
Sinceourexperimentisbasedontwocoupledinterferom-
eters,thecoincidenceratevarieswithbothphasedelays. ACKNOWLEDGMENT
Whenbothdelaysarevariedsimultaneouslyabeatstruc-
turemanifestsitself. Fig. 5(a)showsthecalculatedcoin- We are grateful to P. W. Milonni for valuable and il-
cidencerateasafunctionoftime. Thepathlengthdiffer- luminating discussions.
encesofthesignalandtheidlerinterferometers∆x and
s
∆x arevariedataconstantspeedof20nm/s. Although
i
the path length differences are changed with the same
velocity, the phase delay is different due to the different
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