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interaction-picture state vector, |Ψ (t) , for the ¬eld can be evaluated by ¬rst-order per-
turbation theory. These calculations are simpli¬ed by returning to the box-quantized
form (13.40). In this notation, the interaction Hamiltonian is
1
G(3) C (p ’ k1 ’ k2 ) e’i∆t a† 1 a† 2 + HC ,
Hem (t) = ’i
(3)
(13.49)
kk
V
k1 ,k2

where we have transformed to the interaction picture by using the rule (4.98), and in-
troduced the detuning, ∆ = ωP ’ω2 ’ω1 , for the down-conversion transition. Applying
the perturbation series (4.103) for the state vector leads to

|Ψ (t) = |0 + Ψ(1) (t) + · · · ,
(13.50)
2G(3)
1 sin [∆t/2] † †
C (p ’ k1 ’ k2 ) e’i∆t/2
Ψ(1) (t) = ’ ak1 ak2 |0 .
V ∆
k1 ,k2

According to the discussion in Chapter 6, each term in the k1 , k2 -sum (with the
exception of the degenerate case k1 = k2 ) describes an entangled state of the signal
and idler photons. Combining the limit, V ’ ∞, of in¬nite quantization volume with
the large-crystal approximation (13.22) for C yields

d3 k1 d3 k2 2G(3) 3
|Ψ (t) = |0 ’ (2π) δ (p ’ k1 ’ k2 )
3 3
(2π) (2π)
sin [∆t/2] †
— e’i∆t/2 a (k1 ) a† (k2 ) |0 . (13.51)

The limit t ’ ∞ is relevant for cw pumping, so we can use the identity
sin (∆t/2) π
lim e’i∆t/2 = δ (∆) , (13.52)
∆ 2
t’∞

which is a special case of eqn (A.102), to ¬nd

d3 k1 d3 k2 1 G(3)
|Ψ (∞) = |0 ’ 3 3 2
(2π) (2π)
3
— (2π) δ (p ’ k1 ’ k2 ) (2π) δ (ωP ’ ω1 ’ ω2 )
— a† (k1 ) a† (k2 ) |0 , (13.53)

where ω1 = ωk1 and ω2 = ωk2 .
The conclusion is that down-conversion produces a superposition of states that
are dynamically entangled in energy as well as momentum. The entanglement in en-
ergy, which is imposed by the phase-matching condition, ω1 + ω2 = ωP , provides
an explanation for the observation that the two photons are created almost simulta-
neously. A strictly correct proof would involve the second-order correlation function
¼
Three-photon interactions

G(2) (r1 , t1 , r1 , t1 ; r2 , t2 , r2 , t2 ), but the same end is served by a simple uncertainty prin-
ciple argument. If we interpret t1 and t2 as the creation times of the two photons, then
the average time, tP = (t1 + t2 ) /2, can be interpreted as the pair creation time, and
the time interval between the two individual photon creation events is „ = t1 ’ t2 . The
respective conjugate frequencies are „¦ = ω1 +ω2 and ν = (ω1 ’ ω2 ) /2. The uncertainty
in the pair creation time, ∆tP ∼ 1/∆„¦, is large by virtue of the tight phase-matching
condition, „¦ ωP . On the other hand, the individual frequencies have large spectral
bandwidths, so that ∆ν is large and „ ∼ 1/∆ν is small. Consequently, the absolute
time at which the pair is created is undetermined, but the time interval between the
creations of the two photons is small.

13.3.3 Experimental techniques and results
Spontaneous down-conversion in a lithium niobate crystal was ¬rst observed by Harris
et al. (1967). Shortly thereafter, it was observed in an ammonium dihydrogen phos-
phate (ADP) crystal by Magde and Mahr (1967). A sketch of the apparatus used by
Harris et al. is shown in Fig. 13.2. The beam from an argon-ion laser, operating at a
wavelength of 488 nm, impinges on a lithium niobate crystal oriented so that collinear,
type I phase matching is achieved. The laser beam enters the crystal polarized as an
extraordinary ray. Temperature tuning of the index of refraction allows the adjust-
ment of the wavelength of the down-converted, collinear signal and idler beams, which
are ordinary rays produced inside the crystal. These beams are spectrally analyzed
by means of a prism monochromator, and then detected. In the Magde and Mahr ex-
periment, a pulsed 347 nm beam is produced by means of second-harmonic generation
pumped by a pulsed ruby laser beam. The peak pulse power in the ultraviolet beam is
1 MW, with a pulse duration of 20 ns. Spontaneous down-conversion occurs when the
pulsed 347 nm beam of light enters the ADP crystal. Instead of temperature tuning,
angle tuning is used to produce collinearly phase-matched signal and idler beams of
various wavelengths.
Zel™dovich and Klyshko (1969) were the ¬rst to notice that phased-matched, down-
converted photons should be observable in coincidence detection. Burnham and Wein-
berg (1970) performed the ¬rst experiment to observe these predicted coincidences,
and in the same experiment they were also the ¬rst to produce a pair of non-collinear
signal and idler beams in SDC. Their apparatus, sketched in Fig. 13.3, uses a 9 mW,

LiNbO3
crystal
Polarizer Filter


Oven Analyzer
4880 A
Prism
argon laser
monochromator

Fig. 13.2 Apparatus used to observe spontaneous down-conversion in 1967 by Harris, Osh-
man, and Byer. (Reproduced from Harris et al. (1967).)
¼ Nonlinear quantum optics


Channel 2

2
PM
From
3520 A φ2
laser Spike
filters
φ1
ADP

Light
UV pass PM 1
trap
filter Iris
Channel 1




Monochromator PM 2
Lens

Fig. 13.3 Apparatus used by Burnham and Weinberg (1970) to observe the simultaneity
of photodetection of the photon pairs generated in spontaneous down-conversion in an am-
monium dihydrogen phosphate (ADP) crystal. Coincidence-counting electronics (not shown)
is used to register coincidences between pulses in the outputs of the two photomultipliers
PM1 and PM2. These detectors are placed at angles φ1 and φ2 such that phase matching is
satis¬ed inside the crystal for the two members (i.e. signal and idler) of a given photon pair.
(Reproduced from Burnham and Weinberg (1970).)

continuous-wave, helium“cadmium, ultraviolet laser”operating at a wavelength of
325 nm”as the pump beam to produce SDC in an ADP crystal. The crystal is cut so
as to produce conical rainbow emissions of the signal and idler photon pairs around the
pump beam direction. The ultraviolet (UV) laser beam enters an inch-long ADP crys-
tal, and pairs of phase-matched signal (»1 = 633 nm) and idler (»2 = 668 nm) photons
emerge from the crystal at the respective angles of φ1 = 52 mrad and φ2 = 55 mrad,
with respect to the pump beam. After passing through the crystal, the pump beam
enters a beam dump which eliminates any background due to scattering of the UV
photons. After passing through narrowband ¬lters”actually a combination of interfer-
ence ¬lter and monochromator in the case of the idler photon”with 4 nm and 1.5 nm
passbands centered on the signal and idler wavelengths respectively, the individual sig-
nal and idler photons are detected by photomultipliers with near-infrared-sensitive S20
photocathodes. Pinholes with e¬ective diameters of 2 mm are used to de¬ne precisely
the angles of emission of the detected photons around the phase-matching directions.
Most importantly, Burnham and Weinberg were also the ¬rst to use coincidence de-
tection to demonstrate that the phase-matched signal and idler photons are produced
¼
Three-photon interactions

essentially simultaneously inside the crystal, within a narrow coincidence window of
±20 ns, that is limited only by the response time of the electronic circuit.
In more modern versions of the Burnham“Weinberg experiment, vacuum photomul-
tipliers are replaced by solid-state silicon avalanche photodiodes (single-photon count-
ing modules), which function exactly like a Geiger counter, except that”by means
of an internal discriminator”the output consists of standardized TTL (transistor“
transistor logic), ¬ve-volt level square pulses with subnanosecond rise times for each
detected photon. This makes the coincidence detection of single photons much easier.

13.3.4 Absolute measurement of the quantum e¬ciency of detectors
In Section 13.3.2 we have seen that the process of spontaneous down-conversion pro-
vides a source of entangled pairs of photons. Burnham and Weinberg (1970) used
coincidence-counting techniques”originally developed in nuclear and elementary par-
ticle physics”to observe the extremely tight correlation between the emission times of
the two photons. As they pointed out, this correlation allows a direct measurement of
the absolute quantum e¬ciency of a photon counter. Migdall (2001) subsequently de-
veloped this suggestion into a measurement protocol. The idea behind this technique is
as follows: when a click occurs in one photon counter (the trigger detector), we are then
certain that there must have been another photon emitted in the conjugate direction,
de¬ned by momentum and energy conservation. Thus we know precisely the direction
of emission of the conjugate photon, and also its time of arrival”within a very nar-
row time window relative to the trigger photon”at any point along its direction of
propagation.
As shown in Fig. 13.4, the procedure is to place the detector under test (DUT) and
the trigger detector so that the coincidence counter can only be triggered by signals
from a single entangled pair. For a long series of measurements, the respective quantum
e¬ciencies ·1 and ·2 of the trigger detector and the DUT are de¬ned by

N1 = ·1 N (13.54)

·2
Counter
,76
ω2
Coincidence ·2
counter
ωF
Absolute
quantum
efficiency
Parametric ω1
crystal Counter
6HECCAH

Fig. 13.4 Scheme for absolute measurement of quantum e¬ciency. A pair of entangled
photons originating in the crystal head toward the ˜trigger™ detector and the ˜detector under
test™ (DUT). The parameter ·2 is the quantum e¬ciency for the entire path from the point
of emission to the DUT. (Reproduced from Migdall (2001).)
¼ Nonlinear quantum optics

and
N2 = ·2 N , (13.55)
where N is the total number of conjugate photon pairs emitted by the crystal into the
directions of the two detectors, N1 is the number of counts registered by the trigger
detector, and N2 is the number of counts registered by the DUT. We may safely
assume that the clicks at the two detectors are uncorrelated, so the probability of a
coincidence count is ·coinc = ·1 ·2 . Thus the number of coincidence counts is

Ncoinc = ·1 ·2 N , (13.56)

and combing this with eqn (13.54) shows that the absolute quantum e¬ciency ·2 of
the DUT is the ratio
Ncoinc
·2 = (13.57)
N1
of two measurable quantities. The beauty of this scheme is that this result is indepen-
dent of the quantum e¬ciency, ·1 , of the trigger detector.
Systematic errors, however, must be carefully taken into account. Any losses along
the optical path”from the point of emission of the twin photons inside the crystals
all the way to the point of detection in the DUT”will contribute to a systematic
error in the measurement. Thus the exit face of the crystal must be carefully anti-
re¬‚ection coated, and measured. Care also must be taken to use a large enough iris
in the collection optics for the conjugate photon. This will minimize absorption, by
the iris, of photons which should have impinged on the DUT. Furthermore, this iris
must be carefully aligned, so that it passes all photons propagating in the conjugate
direction determined by phase matching with the trigger photon. This ensures that no
conjugate photons are missed due to misalignment. This alignment error can, however,
be minimized by maximizing the detected signal as a function of small transverse
motions of the test detector.
However, the most serious systematic error arises in the electronic, rather than the
optical, part of the system. The electronic gate window used in the coincidence counter
is usually not a perfectly rectangular pulse shape; typically, it has small tails of lesser
counting e¬ciency, due to which some coincidence counts can be missed. These tails
can, however, be calibrated out in separate electronic measurements of the coincidence
circuitry.

13.3.5 Two-crystal source of hyperentangled photon pairs
For many applications of quantum optics, e.g. quantum cryptography, quantum dense
coding, quantum entanglement-swapping, quantum teleportation, and quantum com-
putation, it is very convenient”and often necessary”to employ an intense source of
hyperentangled pairs of photons, i.e. photons that are entangled in two or more
degrees of freedom. A particularly simple, and yet powerful, light source which yields
photon pairs entangled in polarization and momentum was demonstrated by Kwiat
et al. (1999b).
A schematic of the apparatus used for generating hyperentangled photon pairs
with high intensity is shown in Fig. 13.5. The heart of this photon-pair light source
¼
Three-photon interactions


0
8
o
45



(a)




(b)


Fig. 13.5 (a) High-intensity spontaneous down-conversion light source: two identical, thin,
highly nonlinear crystals are stacked in a ˜crossed™ con¬guration, i.e. the crystal axes lie in
perpendicular planes, as indicated by the diagonal markings on the sides. The crystals are
so thin that it is not possible to tell if a given photon pair emitted by the stack comes from
the ¬rst or from the second crystal. Hence the crossed stack produces polarization-entangled
pairs of photons. (b) Schematic of apparatus to produce and to characterize this photon-pair
light source. (Reproduced from Kwiat et al. (1999b).)

consists of two identically cut, thin (0.59 mm), type I down-conversion crystals”β
barium borate (BBO)”that are stacked in a crossed con¬guration, i.e. with their
optic axes lying in perpendicular planes. What we will call the vertical plane is de¬ned
by the optic axis of the ¬rst crystal and the direction of the pump beam, while the
horizontal plane is de¬ned by the optic axis of the second crystal and the pump beam.
The crystals are su¬ciently thin so that the waist of the pump beam”a continuous-
wave, ultraviolet (wavelength 351 nm), argon-ion laser”overlaps both. Since these
are birefringent (type I) crystals, the ultraviolet pump enters as an extraordinary
ray, and the pair of red, down-converted photon beams leave as ordinary rays. The
two crystals are identically cut with their optic axes oriented at 33.9—¦ with respect
to the normal to the input face. The phase-matching conditions guarantee that two
degenerate-frequency photons at 702 nm wavelength are emitted into a cone with a
half-opening angle of 3.0—¦ .
Under certain conditions, this arrangement allows one to determine the crystal of
origin of the twin photons. For example, if the pump laser is V -polarized (i.e. linearly-
polarized in the vertical plane), then type I down-conversion would only occur in the
¬rst crystal, which would produce H-polarized (i.e. linearly-polarized in the horizontal
plane) twin photons. Similarly, if the pump laser were H-polarized, then type I down-
conversion would only occur in the second crystal, which would produce V -polarized
twin photons. However, suppose that the pump laser polarization is neither horizontal
nor vertical, but instead makes an angle of 45—¦ with respect to the vertical axis. This
state is a coherent superposition, with equal amplitudes, of horizontal and vertical
polarizations. Thus when this 45—¦ -polarized pump beam is incident on the two-crystal
½¼ Nonlinear quantum optics

stack, a down-conversion event can occur, with equal probability, either in the ¬rst or
in the second crystal. If the photon pair originates in the ¬rst crystal, both photons
would be H-polarized, whereas if the photon pair originates in the second crystal, both
photons would be V -polarized.
The thickness of each crystal is much smaller than the Rayleigh range (a few
centimeters) of the pump beam, and di¬raction ensures that the spatial modes”i.e.
the cones of emission in Fig. 13.5(a)”from the two crystals overlap in the far ¬eld,
where the photons are detected. This situation provides the guiding principle behind
this light source: for a 45—¦ -polarized pump beam, it is impossible”even in principle”
to know whether a given photon pair originated in the ¬rst or in the second crystal. We
must therefore apply Feynman™s superposition rule to obtain the state at the output
of the pair of tandem crystals. If the crystals are identical in thickness and the pump
is normally incident on the crystal face, the result is the entangled state
1 1
¦+ = √ |1k1 H , 1k2 H + √ |1k1 V , 1k2 V . (13.58)
2 2
The notation 1k1 H denotes the horizontal polarization state of one member of the pho-
ton pair”originating in the ¬rst crystal”and 1k2 H denotes the horizontal polarization
state of the conjugate member, also originating in the ¬rst crystal. Similarly, 1k1 V de-
notes the vertical polarization state of one member of the photon pair”originating in
the second crystal”and 1k2 V denotes the vertical polarization state of the conjugate
member, also originating in the second crystal. The phase-matching conditions ensure
that the down-converted photon pairs are emitted into azimuthally conjugate direc-
tions along rainbow-like cones, so that they are entangled both in momentum and in
polarization. Hence this light source produces hyperentangled photon pairs.
The entangled state |¦+ is one of the four Bell states de¬ned by
1 1
¦+ ≡ √ |1k1 H , 1k2 H + √ |1k1 V , 1k2 V , (13.59)
2 2
1 1
¦’ ≡ √ |1k1 H , 1k2 H ’ √ |1k1 V , 1k2 V , (13.60)
2 2
1 1
Ψ+ ≡ √ |1k1 H , 1k2 V + √ |1k1 V , 1k2 H , (13.61)
2 2
1 1
Ψ’ ≡ √ |1k1 H , 1k2 V ’ √ |1k1 V , 1k2 H . (13.62)
2 2
These are maximally entangled states that form a basis set for the polarization states
of pairs of entangled photons with wavevectors k1 and k2 . The states |¦+ and |¦’
can be generated by two crossed type I crystals, and the states |Ψ+ and |Ψ’ can be
generated by a pair of crossed type II crystals.
More generally, the two crystals could be tilted away from normal incidence around
an axis perpendicular to the direction of the pump laser beam. This would result in
phase changes which lead to the output entangled state
eiξ
1
¦ ; ξ = √ |1k1 H , 1k2 H + √ |1k1 V , 1k2 V ,
+
(13.63)
2 2
½½
Three-photon interactions

where the phase ξ depends on the tilt angle. Instead of tilting the two tandem crystals,
it is more convenient to tilt a quarter-wave plate placed in front of them, so that an
elliptically-polarized pump beam emerges from the quarter-wave plate with the major
axis of the ellipse oriented at 45—¦ with respect to the vertical. Then the down-converted
photon pair emerges from the tandem crystals in the entangled state |¦+ ; ξ , with
a nonvanishing phase di¬erence ξ between the H“H and V “V polarization-product
states. The phase of the entanglement parameter ξ can be easily adjusted by changing
the relative phase between the horizontal and vertical polarization components of the
pump light, i.e. by changing the ellipticity of the ultraviolet laser beam polarization.
In the actual experiment, schematically shown Fig. 13.5(b), a combination of a
prism and an iris acts as a ¬lter to separate out the ultraviolet laser pump beam from
the unwanted ¬‚uorescence of the argon-ion discharge tube. A polarizing beam splitter
(PBS) acts as a pre¬lter to select a linear polarization of the laser beam. Following
this, a half-wave plate (HWP) allows the selected linear polarization to be rotated
around the laser beam axis. The beam then enters a quarter-wave plate (QWP)”
whose tilt angle allows the adjustment of the relative phase ξ of the entangled state
in eqn (13.63)”placed in front of the tandem crystals (BBO).
Separate half-wave plates (HWP) and polarizing beam splitters (PBS) provide
polarization analyzers, placed in front of detectors 1 and 2, that allow independent
variations of the two angles of linear polarization, θ1 and θ2 , of the photons detected
by Geiger counters 1 and 2, respectively. The irises in front of these detectors were
around 2 mm in size, and the interference ¬lters (IF) had typical bandwidths of 5 nm
in wavelength. The iris sizes and interference-¬lter bandwidths were determined by
the criterion that the detection should occur in the far ¬eld of the crystals, and by
phase-matching considerations.
Under these conditions, with a 150 mW incident pump beam and a 10% solid-angle
collection e¬ciency”arising from the ¬nite sizes of the irises placed in front of the
detectors”the hyperentangled pair production rate was around 20, 000 coincidences
per second. Standard coincidence detection of the correlated photon pairs in this ex-
periment was accomplished by means of solid-state Geiger counters (silicon avalanche
photodiodes with around 70% quantum e¬ciency, operated in the Geiger mode), in
conjunction with a time-to-amplitude converter and a single-channel analyzer, with a
coincidence time window of 7 ns. The polarization states of the individual photons were
analyzed by means of rotatable linear polarizers, with the analyzer angle for detector
2 being rotated relative to that of detector 1 (whose analyzer angle was kept ¬xed at
’45—¦ ).
Typical data are shown in Fig. 13.6. The singles rate (the output of an individ-
ual Geiger counter) shows no dependence on the relative angle of the two analyzers,
indicating that the photons were individually unpolarized. On the other hand, coin-
cidence measurements showed that the relative polarization of one photon in a given
entangled pair with respect to the conjugate photon was very high (with a visibility
of 99.6 ± 0.3%). This means that an extremely pure two-photon entangled state has
been produced with a high degree of polarization entanglement. Such a high visibility
in the two-photon coincidence fringes indicates a violation of Bell™s inequalities”see
eqn (19.38)”by over 200 standard deviations, for data collected in about 3 minutes.
½¾ Nonlinear quantum optics

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