1. Experimental Measurement of Bell’s Inequality in Type I Spontaneous Parametric
Downconversion
Deanna Zapata
University of California Los Angeles
(Dated: March 18, 2016)
We prepare polarization entangled photons through Type I spontaneous parametric
down-conversion to demonstrate quantum nonlocality. A test of Clauser, Horne, Shi-
mony and Holt version of Bell’s inequality determines S=2.525±0.027 in clear contra-
diction of hidden variable theories.
Keywords: Local Hidden Variable Theory, Entanglement,Non-Linear Optics
I. INTRODUCTION
In 1935, Einstein, Podolsky, and Rosen published
their now famous EPR Paradox paper, which set out to
theoretically demonstrate that quantum theory did not
fully describe the nature of particles. Quantum mechan-
ics states that a pair of particles in an entangled state,
for instance the singlet spin configuration of a positron
and electron: |ψ = 1√
2
(↑−↓+ + ↓−↑+) are correlated in
such a way that if particle 1 is measured spin up, then
particle 2 must have a spin down when measured.[1]
Whether these correlated particles are measured in the
same lab, or several light-years away, the outcome is
the same. This gives rise to the ”Copenhagen” inter-
pretation, or that of non-locality, which suggests that
a measurement on one particle immediately collapses
the state of the other, even if they are nowhere near
each other. Einstein, Podolsky, and Rosen considered
this absurd based on the principle of locality, meaning
information cannot travel faster than the speed of light;
therefore suggesting the particles were created with
well-defined spins all along, and the theory of quantum
mechanics was incomplete. In order to reconcile this
with reality, Einstein, Podolsky, and Rosen suggested
perhaps there was some local hidden variable, λ, which
provided a correction to quantum theory. [1]
CHSH Bell Inequality
After investigating many theories of locality, Bell proved
that any local hidden variable theory is incompatible
with quantum mechanics. Here we present an experi-
ment which demonstrates photon entanglement to test
the CHSH Bell Inequality as follows [2]:
The distribution of the hidden variable λ is described as
ρ(λ) ≥ 0 where
ρ(λ)dλ = 1 (1)
We assume locality and realism that for the signal pho-
ton, the measurement is completely determined by λ
when measured at the angle α, specified by the function
A(λ, α). The values of A(λ, α) may be +1 for Vα (vertical
polarization) detection or -1 for Hα (horizontal polariza-
tion) detection. Similarly, a function B(λ, β) describes
the outcome of the idler photon as +1 for Vβ detection
or -1 for Hβ detection. The probabilities of finding VαVβ,
VαHβ, HαVβ, HαHβ is given by integrals, respectively
PV V (α, β) =
1 + A(λ, α)
2
1 + B(λ, β)
2
ρ(λ)dλ (2)
PV H(α, β) =
1 + A(λ, α)
2
1 − B(λ, β)
2
ρ(λ)dλ (3)
PHV (α, β) =
1 − A(λ, α)
2
1 + B(λ, β)
2
ρ(λ)dλ (4)
PHH(α, β) =
1 − A(λ, α)
2
1 − B(λ, β)
2
ρ(λ)dλ (5)
The first measure of correlation is
E(α, β) = PV V (α, β)+PHH(α, β)−PV H(α, β)−PHV (α, β)
(6)
which is the expectation value of all measurements and
is equal to +1 when all polarizations agree, and -1 when
all polarizations disagree. The other measure is
S(α, α , β, β ) = [E(α, β)+E(α , β)−E(α, β )+E(α , β )] ≤ 2
(7)
where α(α ), β(β ) are different polarizer angles that rep-
resent two observers measuring the photons in each arm.
Although S itself does not have any physical meaning,
it was proved by Clauser, Horne, Shimony, and Holt to
be less than or equal to 2 for any hidden variable theory
and arbitrary polarizer angle values. In this experiment,
we choose these values to be α = −45◦
, β = −22.5◦
, α =
0◦
, β = 22.5◦
.
II. EXPERIMENTAL SETUP
Type I Spontaneous Parametric Down-
conversion
We prepare correlated photons in the state
|Φ+
=
1
√
2
(|V 1 |V 2 + |H 1 |H 2) (8)

2. 2
FIG. 1. Quantum Entanglement Demonstrator (quED) unit
that is the source of entangled photons for this experiment.
This module comprises a laser diode (1), beam steering mir-
rors (2,7), beam shaping optics (3a,3b,3c), half-wave plate
(4), pre-compensation birefringent crystal (5), non-linear
down-conversion crystals (6), irises (8), rotating polarizers
(9), longpass filters (10), collimators (11), and single mode
fibers (12).
where |V and |H are vertically and horizontally polar-
ized photons, respectively. This is done using the princi-
ple of spontaneous parametric down-conversion (SPDC)
in a quantum entanglement demonstrator (quED). Pho-
tons in an intense laser beam pass through a non-linear
beta-barium borate (BBO) crystal and spontaneously
convert into pairs of photons, called the signal and idler,
each with half of the frequency of the incident photons,
in agreement with energy and momentum conservation.
ω3 = ω2 + ω1 (9)
k3 = k2 + k1 (10)
where ω3, ω2, ω1; k3, k2, k1 represent the frequen-
cies and wave vectors of the pump laser beam (3), sig-
nal (2), and idler (1). Figure 1 shows the setup of the
experiment. Label 1 shows the laser diode head which
emits photons of wavelength 403nm that pass through
a series of optics (2,3b,4,5,3c,6), including a collimating
lens, a mirror to control pointing, a negative spherical
lens, a cylindrical lens which compensates for the ellipti-
cal polarization of the laser beam, a half-wave plate, and
a pre-compensation birefringent crystal which is used to
set the relative phase in the set of photon pairs, and cor-
rect for the time delay which arises between the photon
pairs born in the second down-conversion crystal relative
to the first.[3] Each photon from the pair is emitted out
of the crystals and fiber coupled through a guiding mirror
(7) and a polarizing filter in each arm (9). This output
goes to photodiodes built into the quED control unit and
is read from the front panel screen.[3]
Polarization
Because the BBO crystal is Type I phase matched, the
signal and idler photons emerge with the same polar-
ization, either horizontal or vertical. These crystals and
cross polarized so that either polarization from the pump
beam can down-convert.[4] To create an entangled state,
we linearly polarize the pump diode laser (using a half-
wave plate) at some angle θ from the vertical, then shift
the phase (using the pre-compensation crystal) by φ. The
photons enter the down-conversion crystal in state
|ψ = cos θ |V + exp[iφ] sin |H (11)
and the down-converted photons (signal and idler)
emerge as
|ψ DC = cos θ |H s |H i + exp[iφ + δ] sin |V s |V i (12)
We put rotating polarizers in the signal and idler
photon paths and adjust them to angles α and β to mea-
sure the coincidence rates for each polarization of the
photons. We use the rotated polarization basis
|V α = cos α |V − sin α |H (13)
|H α = sin α |V + cos α |H (14)
where |V α describes a state with polarization rotated α
from the vertical, and |H α a state with polarization α
from the horizontal. We find the probability of coinci-
dence detection, for example VαV β, to be
PV V (α, β) =| Vα|s Vβ|i |ψ DC |2
(15)
and more generally also include VαHβ, HαVβ, HαHβ for
any pair of polarizing angles. Using the rotated polariza-
tion basis in Equations 13 and 14, we find Equation 15
reduces to
PV V (α, β) =
1
2
cos2
(β − α) (16)
for our state |Φ+
when θ = 45◦
and φ = 0◦
.
III. RESULTS
CHSH Bell Inequality Violation
To find the correlations of photons for the
value E, four values of coincidence rate measure-
ments are needed for each combination of polarizer
angles: (α, β),(α , β),(α, β ), and (α , β ). This is
accomplished by taking coincidence measurements at
(α, β),(α⊥, β),(α, β⊥), and (α⊥, β⊥) for each combination
of α, α , β, β . The values measured in Table 1 correspond
to different angle orientations that represent local mea-
surements of two observers, are used to determine the
correlation between photon measurements. These angles
were chosen to be α = −45◦
, β = −22.5◦
, α = 0◦
, β =
22.5◦
. Also shown is the average number of accidental
coincidences which may occur during the coincidence in-
terval τ of each other. This is used to show that such
background is relatively small and constant, and when
subtracted from the coincidence rates, decreases the over-
all |S|. Therefore, a finding of |S|> 2 is not an artifact
of accidental background. [2]

3. 3
TABLE I. Measured values for single (Nα, Nβ) and coinci-
dence (Ncoin) rates of entangled photons per second corre-
sponding to angle orientations for the two polarizers (α, β).
Accidental coincidences (Nacc = τNαNβ/T) are also included
assuming the coincidence window is τ = 25ns and T = 1s. [2]
α β Nα Nβ Ncoin Nacc
-45◦
-22.5◦
29164 27940 1075 20.4
-45◦
22.5◦
31071 28003 440 21.7
-45◦
67.5◦
36655 27948 180 25.6
-45◦
112.5◦
34782 28285 1250 24.6
0◦
-22.5◦
29006 27360 1570 19.8
0◦
22.5◦
31944 27909 1750 22.3
0◦
67.5◦
36543 27326 330 24.9
0◦
112.5◦
35139 27512 420 24.2
+45◦
-22.5◦
29146 28691 560 20.9
+45◦
22.5◦
31442 28693 1600 22.5
+45◦
67.5◦
36585 28592 1250 26.1
+45◦
112.5◦
34874 28711 200 25.0
90◦
-22.5◦
29220 31071 180 22.7
90◦
22.5◦
31187 31486 250 24.5
90◦
67.5◦
36403 31077 1410 28.3
90◦
112.5◦
35116 31039 1600 27.2
The normalized expectation value for the correla-
tions between measurements is given by Equation 16,
where each term is a coincidence rate measurement with
the two polarizers set at the corresponding angle values
[3]
E(α, β) =
C(α, β) − C(α, β⊥) − C(α⊥, β) + C(α⊥, β⊥)
C(α, β) + C(α, β⊥) + C(α⊥, β) + C(α⊥, β⊥)
(17)
and the error in E is
∆E(α, β)
= 2
[C(α, β) + C(α⊥, β⊥)][C(α, β⊥) + C(α⊥, β)]
[C(α, β) + C(α, β⊥) + C(α⊥, β) + C(α⊥, β⊥)]2
×
1
C(α, β) + C(α⊥, β⊥)
+
1
C(α, β⊥) + C(α⊥, β)
(18)
Using Equations 6, 7, and 16 for our chosen polar-
ization angles we find our expected value for S and its
error [3]
S(α, α , β, β ) = 2
√
2 (19)
∆S(α, α , β, β ) =
a=α,α b=β,β
∆E(a, b)2 (20)
which violates the CHSH inequality we derive in Equa-
tion 7.
Table 2 shows our calculated values for each E and
S as well as their associated errors.
TABLE II. Calculated normalized expectation values from our
data set and our value of S.
E(α, β) 0.517±0.016
E(α , β) 0.708±0.012
E(α, β ) -0.633±0.013
E(α , β ) 0.667±0.012
S(α, α , β, β ) 2.525±0.027
IV. CONCLUSION
It has been demonstrated that the entangled pho-
tons can be produced relatively easily in the lab by means
of spontaneous parametric down-conversion using non-
linear crystals which may be tuned to produce different
states. These polarization entangled photons have been
shown to violate the CHSH Bell inequality and local re-
alistic hidden variable theories. From Table 2 we find
our value of S = 2.525 ± 0.027 is in direct violation of
the CHSH Bell inequality by over 19 standard deviations,
thus unambiguously eliminating the possibility of a hid-
den variable and is consistent with quantum mechanics.
V. SUMMARY
Based on the reported results we found experimen-
tal measurements in direct contradiction of local hidden
variable theories which attempt to reconcile quantum me-
chanics with locality. We used a diode laser with a two
crystal configuration to create Type I entangled photons
in the |Φ+
state as our source for measurement. Coinci-
dence rate measurements sensitive to polarization angles
show the correlation to polarization states. We report
a value for S which is in clear violation of local realis-
tic hidden variable theories by more than 19 standard
deviations.
REFERENCES
[1] Griffiths, David Introduction to Quantum Mechan-
ics, Pearson, 2nd Edition (2005)
[2] Dehlinger,Dietrich and Mitchell,M.W. Entangled
photons, nonlocality and Bell inequalities in the un-
dergraduate laboratory arxiv.org, 27 May 2002.
[3] quED Tools Entanglement Demonstrator - Opera-
tion Manual Version 1.4, 7 Nov. 2013
[4] Catalano, Jesse L. Spontaneous Parametric Down-
Conversion and Quantum Entanglement Univer-
sity Honors Theses. PDXScholar (2014)