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Quantum Optics
Maira Amezcua
Department of Physics and
Astronomy
Pomona College
May 2, 2012
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Contents
List of Figures
4
1 Abstract
6
2 Introduction
8
2.1 Background
8
2.1.1 Entangled States
8
2.1.2 Single Photon Interference Experiments
9
2.2 Motivation
10
2.2.1 Experiments
11
2.2.2 Future Work
14
3 Theory
14
3.1 Correlated Photon
14
3.1.1 Calcium Cascade
16
3.1.2 Spontaneous Parametric Down-conversion
17
3.2 Single Photon Interference
19
3.3 Second-order Correlation Function
22
3.4 Quantum Eraser
25
3.4.1 Complementarity
26
3.4.2 Double Slit Experiment
26
3.4.3 Two photon interference
27
3.4.4 Quantum Eraser
28
4 Materials and Methods
32
4.1 Equipment
32
4.1.1 External-cavity Diode Laser
32
4.1.2 The Laser Beam
36
4.1.3 BBO Crystal
37
4.1.4 The Detectors
38
4.1.4 (a) The A-detector
39
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4.1.4 (b) The B-detector
41
4.1.4 (c) The B’-detector
41
5 Analysis
43
6 Conclusion
50
Appendix A: Single Counting Photon Module
51
Appendix B: Coincidence Counting Module
52
7 References
53
List of Figures
Figure 1: Experimental apparatus used to demonstrate that if a down-converted photon is detected at G,
then its pair will be detected at T or R but not at both. DCC: Down-conversion crystal. PBS: polarizing
beam splitter. SPCM: single photon counting module. Adapted from Dr. Beck. ..........................................13
Figure 2: Levels of calcium. The atoms are pumped to the upper level by absorption of two photons and
emits two photons correlated in polarization [10]. .......................................................................................16
Figure 3: Spontaneous parametric down-conversion....................................................................................17
Figure 4: Type I photon pairs [15]. ..............................................................................................................18
Figure 5: Single photon interference by Grangier, Roger and Aspect [4]. ...................................................19
Figure 6: Anticorrelation parameter as function of number of cascades and trigger rate [4]. ...................20
Figure 7: Mach-Zehnder interferometer [4]. ................................................................................................21
Figure 8: Interference experiment [26]. ........................................................................................................28
Figure 9: Quantum eraser experimental setup [7]. .......................................................................................29
Figure 10: The polarization interferometer. The arrow signifies horizontal polarization and the dot
indicates vertical polarization [7]. ................................................................................................................30
Figure 11: The number of coincidence counts between detectors A and B as a function of pathlength
difference in the two arms of the interferometer. Detector B measures horizontally polarized photons [7].31
Figure 12: Measure of coincidences (a) AB, and (b) AB’ as a function of the pathlength difference between
the arms of the interferometer. The beam splitter is rotated so that detector B measures +45° polarized
photons and detector B’ measures -45° polarized photons [7]. ....................................................................31
Figure 13: Schematic of external-cavity laser...............................................................................................33
Figure 14: Spectrum of ECDL.......................................................................................................................34
Figure 15: Spectrum of ECDL.......................................................................................................................35
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Figure 16: Experimental apparatus. Components are 405nm cavity-diode laser, a pair of cylindrical lenses,
two mirror, two spherical lenses, a half-wave plate (HWP), down-conversion crystal (BBO), and a
polarizing beam-splitter (PBS).The half-wave plate before the crystal vertically polarizes the pump beam to
match the specifications of the BBO crystal. ................................................................................................36
Figure 17: The collimation lenses are on the right and the RG780 filters are on the right. The orange fiber
is the multimode optical fiber. .......................................................................................................................38
Figure 18: Detector A alignment. ..................................................................................................................40
Figure 19: The B' detector alignment setup. .................................................................................................42
Figure 20: Optical table set up with 3 detectors. ..........................................................................................43
Figure 21: Number of coincidences in AB for each 1.0s detection cycle of photon counts in detector A. This
data was taken over a window of 26.0 seconds. ............................................................................................44
Figure 22: Counts in A versus coincidences in AB, AB’ and ABB’. The values were integrated over a
period of 1.0s. The collection was made over a window of 40s.....................................................................45
Figure 23: Counts for detectors A, B, and B'. Photon counts from B and B’ are on the same order of
magnitude of counts from A. ..........................................................................................................................46
Figure 24: Coincidences for a clock rate of 0.00125s and a total integration time of 1.0s. .........................47
Figure 25: Coincidences for a clock cycle of 0.005s and an integration window of 4.0s. ............................48
Figure 26: Coincidences for a period of 0.001s and an integration time of 0.8s. .........................................48
Figure 27: g(0) values for a clock cycle of 800Hz. .......................................................................................49
Figure 28: This is the interface board after removing component C1. .........................................................51
Figure 29: Series 3 CCM which performs up to 4-fold coincidence counting. .............................................52
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1 Abstract
A pair of photons is said to be entangled if their wave function cannot be factored
into the wave functions of the individual photons. These entangled states make it possible
to test several quantum mechanical theories such as single and double interference at the
quantum level. Due to technological advances in the field of optics, we can create
entangled photon pairs through type-I spontaneous parametric down-conversion, in which
a photon of a specific frequency is converted into two photons with half the frequency
through a non-linear crystal. This thesis focuses on creating and detecting correlated
photon pairs with an
external-cavity diode laser and a beta-Barium Borate crystal.
A second experiment is carried out to find the probability of detecting a photon that
passes through a beam splitter at the two output paths.
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2 Introduction
2.1 Background
Particle theory of light has existed since the seventeenth century but it was not
until the twentieth century that the particle nature of light was revisited and further
developed. The evidence for the particle nature of light is attributed to the photoelectric
effect, Compton scattering and Planck’s discovery of quantized energy. With Albert
Einstein’s and Max Planck’s contributions, the foundations of quantum mechanics
flourished, treating light as a particle. Later, Louis de Broglie made the bold hypothesis
that electrons behaved like waves. This wave-particle duality was at the heart of quantum
mechanics, which would support a mathematical model to explain both properties.
2.1.1 Entangled States
Quantum mechanics uses a probabilistic wave-particle model for determining the
behavior of quantum systems. It explains anomalies that cannot be resolved using
classical mechanics. With powerful computers and communication security relying on
quantum information, advances in technology make it possible test various quantum
mechanical theories using correlated photon pairs. Correlated photons are unique because
they are multi-particle systems that (a) if measured independently of each other produce
random states, and (b) the state of one depends on the state of the other. Entangled photon
pairs are used in experiments like the quantum eraser and Bell’s Inequalities.
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2.1.2 Single Photon Interference Experiments
In 1905, Albert Einstein published a paper interpreting the photoelectric effect as
light quantum striking the surface of metal and ejecting electrons [1]. His work seemed to
support the existence of a photon, but W.E. Lamb and M.O. Scully proved that Einstein’s
theory of photons was plausible but not the only one [2]. Their theory treated atoms
quantum mechanically and light classically. This semi-classical model described an
electromagnetic wave perturbing a quantized atom. The interaction exerted a force on the
atom which caused the electrons’ ejection.
The field of quantum optics was established in 1956 with the Hanbury-Brown and
Twiss experiment [3]. This was the first careful investigation of coincidences, the
occurrence of a single photon appearing in two detectors at the same time after passing
through a beam splitter. The absence of a coincidence proves photons exist. The task was
to prove that for dim light sources single photons arriving on a beam splitter would not
split, leading to anticorrelation. The wave theory of light predicts that the light arriving
on a beam splitter would be halved. In the experiment, an emission line from mercury
was isolated by a system of filters and sent to a half-silvered mirror. Two
photomultipliers with interference filters were connected to a coincidence counter.
Theoretically there would be no coincidences but the result was just the opposite. The
result of Hanbury-Brown and Twiss failed to prove the existence of photons and showed
that light traveled in bunches. The results of the experiment are complicated to
understand without applying the proper model. In this case the photomultipliers are being
treated quantum mechanically and the existence of photons is still dubious.
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It was the work of Grangier, Roger and Aspect in 1986 that made a major
breakthrough in this area of science [4]. Again they would examine the correlations
between photon detections at the outputs of a beam splitter. This simple experiment
attempted to create an improved photon source by exciting calcium atoms that would
decay to the ground state and two photons would be emitted. One photon acts as trigger
by sending a signal to the coincidence counter. This alerts the two detectors for the
second photon to expect a photon during a brief period. Grangier, Roger and Aspect
were fully successful in measuring fewer coincidences than predicted by the classical
wave theory. At last the existence of photons was confirmed in their experiment, opening
the field to other single photon experiments.
2.2 Motivation
The basis of this project is: “Do photons exist?” While we do not need to reprove
that photons do exist, this question guides students in understanding the evidence of
photons. The photoelectric effect and Compton scattering are attributed to proving
photons exist, but both phenomena can be rationalized as quantized matter interacting
with a classical electromagnetic field. The experiments presented in this paper allow for
an obvious approach to proving photons exist and make the quantum eraser and Bell’s
Inequalities easy to understand.
Whitman College and Colgate College are the front-runners in establishing
quantum mechanics laboratory coursework for undergraduates [5-9]. They have
succeeded in producing five quantum mechanics experiments which are the model for the
proposed experiments. Other colleges with quantum optic laboratories include Reed
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College, Trinity College, the University of Rochester, Dickinson College and Harvey
Mudd College. Emphasis on quantum mechanical research calls for quantum optics
training at the undergraduate level, which is why the Department of Physics and
Astronomy at Pomona College has begun to execute these experiments.
2.2.1 Experiments
The experiments delineated below are low-cost and simple to build, requiring a
standard optical table
for the setup. The proposed experiments are simple to
setup because they require few components. The nonlinear crystal and photon counter
were purchased at the beginning of this project while other equipment was gathered from
various sources. Computer software was obtained from Dr. Branning of Trinity College.
Our setup differs from other labs because we built an external-cavity diode laser as
opposed to buying a laser. We hope that by using an external-cavity diode laser, it can
increase the efficiency of pair production. During the summer, our group focused on
collimating the laser and observing the spectrum of the laser to make adjustments that
would decrease linewidth1.
Correlated Photons
The first laboratory in the series checks for correlated photons in Type I
parametric down-conversion, output beams are polarized parallel to each other. A blue
pump laser at wavelength
hits a nonlinear Barium Borate crystal, exciting
1
The linewidth is a measure of the spectral coherence of a laser. Small linewidths (~1GHz) correspond to
interference of the waves of different frequencies, in the laser, in such a way that they have a fixed relative
phase; this creates a pulse.
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two photons at lower frequencies than the input photon. While there are many photons
emitted in different directions and at different frequencies, we focus on correlated
photons that are produced in pairs at the same time. If two photons are detected in a short
time interval, we assume they are coincident and constitute a pair of correlated photons.
This experiment maximizes the coincidence counts by adjusting the vertical and
horizontal positions of the collection lenses for two detection channels.
Existence of Photons
The basic idea behind this experiment is that light displays particle properties at
the quantum level. Classically light on a beam splitter can be detected in two directions of
polarization but the same does not hold for a photon; thus, a single photon incident on a
beam splitter can be transmitted or reflected but not both. Fig 2. illustrates the setup for
this experiment which aims to demonstrate that when a down-converted photon is
counted at detector G, its pair will be detected at either detector T or R. Since a photon
cannot be split, it cannot be detected in two channels at the same time. This experiment
also measures the degree of second-order coherence parameter,
the probability
of measuring a photon at detectors T and R in the same time interval. The purpose of this
experiment is to show that quantum mechanically
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Figure 1: Experimental apparatus used to demonstrate that if a down-converted photon is detected at G, then its pair
will be detected at T or R but not at both. DCC: Down-conversion crystal. PBS: polarizing beam splitter. SPCM: single
photon counting module. Adapted from Dr. Beck.
Quantum Eraser
Effectively, if two correlated photons are produced from the same source and the
polarization is measured for one of the photons then they will not interfere. However, if
the path that was taken by the photon can be erased then the photons should interfere.
Here the interference is calculated by looking at the coincidence counts based on the path
each photon took.
Bell’s Inequalities
This experiment tests local realism for two correlated photons. Local realism says
that for two photons from the same source have defined polarizations once emitted.
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However, quantum mechanics is not bound by local realism and one must be abandoned
to explain the results in the experiment. Based on coincidence counts for both photons the
probability that the photons had a particular set of polarizations can be determined.
2.2.2 Future Work
Due to several unforeseen setbacks, only the first two experiments of the
sequence are underway. There is still much work that needs to be done to improve the
laser, coincidence counts, and begin employing the second set of laboratories. More
optical equipment must be purchased, including a second beta-Barium Borate crystal.
This work will be continued through the summer of 2012 as part of the Summer
Undergraduate Research Program (SURP).
3 Theory
3.1 Correlated Photon
Quantum entanglement has made its way into applications such as quantum
imaging, quantum information processing, and quantum lithography. Correlated twophoton systems are a statistical mixture of states in which the state of one particle can be
determined by the state of the other particle. Another characteristic of entangled photon
pairs is that if measured independently of each other the states are random. In this
particular project we are concerned with type I correlated photons, down-converted pairs
with the same polarization.
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A two-particle system is described as being in an entangled state if its wave
function cannot be factored into a product of individual photon polarization states. One
example that displays mutual dependence of polarization orientations with an equal
superposition of two states is:
(3.1)
Where
is the horizontally polarized state and
is the vertically polarized state, and the
subscripts refer to the individual photons. For the wave function given above, the
probability of obtaining the result
or
is
. This implies that half the
time the photons are horizontally polarized and the other half they are vertically polarized.
When the polarization is measured, the photons collapse into one state. Entangled downconverted photons are used to test single-photon interference, the quantum eraser and
Bell’s Inequalities.
In order to conduct experiments observing single photons it is important to have
an adequate single-photon source. The Hanbury-Brown and Twiss experiment [3]
attempted to create a stream of photons by focusing on the 435.8-nm light from an
emission line of a mercury arc. Mercury atoms collided with electrons; the interactions
excited atoms to various states, including one that rapidly decayed to the ground state.
This excitation would emit a photon of wavelength 435.8-nm. This cycle produced light
that then traveled to the detectors, which had a 435.8-nm interference filter to ensure that
only photons of the correct wavelength entered the photomultiplier. The Hanbury-Brown
and Twiss correlated photon method had several discrepancies and the mercury arc had
several imperfections. The glass bulb lamp introduced differential phase-shifts that
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reduced correlations. With newer technology available more techniques were presented
to create correlated photons. Two methods will be described in the following sub-sections.
3.1.1 Calcium Cascade
The work of Grangier, Roger and Aspect [4] used a well-established technique by
exciting a stream of collimated calcium atoms by means of two-photon laser excitation.
Calcium atoms were excited to a high s-state that decayed to the ground state thru an
intermediate p-state:
photons of frequency
[10]. This cascade in calcium gave two
and
, correlated in polarization.
Figure 2: Levels of calcium. The atoms are pumped to the upper level by absorption of two photons and emits two
photons correlated in polarization [10].
An atomic beam of calcium is produced by a tantalum oven, operating at temperature of
atoms/cm3. Two lasers with
1000°K [11]. This induces an interaction region of
parallel polarizations, a single-mode krypton ion laser (
single-mode Rhodamine 6G dye laser (
Running each laser at 40mW, the cascade rate is
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) and a tunable cw
), illuminate the calcium beam at 90°.
per second. The calcium
cascade proved to be sufficient enough; however, with advancements in technology the
amount of equipment to make correlated photons was reduced.
3.1.2 Spontaneous Parametric Down-conversion
Parametric down-conversion, or “frequency splitting of light” [12], splits a single
pump photon (
) into two photons of lower frequency (
) in a nonlinear crystal
[13-17].
Figure 3: Spontaneous parametric down-conversion.
The light is a monochromatic plane wave with wave vector
and frequency
.The
nonlinear crystal is fabricated to have a similar index of refraction to the linear medium
surrounding the crystal. The daughter photons, often named the “signal” and “idler”
photons have definitive properties because of the invariance of the medium which
conserves energy and momentum. These are the relations of the wave vectors and
frequencies of the parent and daughter photons:
(3.1.1)
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(3.1.2)
Emission of the products is almost simultaneous. There are two types of parametric
down-conversion: type I and type II. Type I down-conversion produces photon pairs with
parallel polarization, while type II produces oppositely polarized photon pairs. Any
crystal can only support one type of down-conversion and the polarization of the beam
must be adjusted accordingly to match the crystal. However, for one crystal there is not a
unique solution to the wavelength and direction of the daughter photons. The photons are
always produced in pairs but our project focuses on degenerate signal and idler photons,
those with the same wavelength.
Figure 4: Type I photon pairs [15].
Parametric down-conversion is used because it is easier and cheaper to implement than
other methods such as calcium excitation2.
2
Calcium atoms are excited to a high s-state that decayed to the ground state thru an intermediate p-state,
by giving off two photons.
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3.2 Single Photon Interference
The issue being addressed is whether a photon can interfere with itself [18-21].
Quantum mechanically a photon can interfere with itself and is the very basis of the
theory, but the outcome fails for a single system. To understand the wave-particle duality
and the quantum behavior of a photon it is necessary to analyze both single photon
experiments executed by Grangier, Roger and Aspect (GRA) [4].
The first experiment is based around the particle nature of light, violating the
classical wave model. Fig. 5 shows the experiment setup.
Figure 5: Single photon interference by Grangier, Roger and Aspect [4].
The detection of the first photon
and
for a time duration
at photomultiplier
3
triggers photomultipliers
in view of the second photon
photomultipliers feed into single and coincidence counters for
count rate for the gate,
and
. These two
. GRA denote
as the
as the single rates for the signal photomultipliers, and
the coincidence rate. The probabilities of these measurements during the time interval
are as follows,
3
Where
is the lifetime of the calcium cascade.
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(3.2.1)
A classical approach to this experiment generates the inequality
.
(3.2.2)
This inequality means that the classical coincidence probability
product of the single counts
is greater than the
the probability of accidental coincidences. Any
violation of this inequality is a characterization of non-classical behavior.
Figure 6: Anticorrelation parameter as function of number of cascades and trigger rate [4].
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The experimental values reinforce the particle nature and emphasize that a photon cannot
be split at a beam splitter. The second experiment by GRA builds a Mach-Zehnder
interferometer around the first beam splitter; this is the actual single photon interference
experiment.
Figure 7: Mach-Zehnder interferometer [4].
If an intense light source was the input to the Mach-Zehnder interferometer then
interference fringes could be constructed based on the outputs of
and
. The
particle aspect would tell us that a single photon in the system could be detected at either
output but not both. However, due to the wave-particle duality of quantum mechanics by
altering the path difference between the arms of the interferometer and observing over a
large time interval one can reconstruct interference fringes. Changing the path difference
between the photomultipliers changes the probability of the path the photon traveled and
we observe fluctuations in counts of each photomultiplier.
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3.3 Second-order Correlation Function
The results of quantum optics experiments relating to photon interference can be
explained classically in terms of coherence. In a classical field, an electromagnetic wave
is described perfectly by Maxwell’s equations. The first-order coherence function
determines the visibility of interference fringes based on electric field fluctuations.
However, Maxwell’s equations cannot be applied in the realm of quantum mechanics so
the degree of second-order coherence
is applied to quantify the results of
Hanbury-Brown and Twiss experiments. The second-order (temporal) correlation
function measures coherence of intensity fluctuations with time delay τ [22-23].
(3.3.1)
Where
and
are the electric field and intensity, respectfully at time . The
brackets indicate ensemble averages as opposed to time averages. The degree of secondorder coherence can be written as the correlations between the intensities of the
transmitted
and reflected
beams. It is a function of the time delay between
measurements of intensity:
(3.3.2)
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The present work is simultaneous
intensity measurements. Assuming a 50/50
beam splitter in which the transmitted, reflected and incident
intensities are related
by,
(3.3.3)
If we apply this to classical fields, we find that
(3.3.4)
In our experiments do not measure intensity directly so we use other quantities we can
measure experimentally. The function is interpreted as the probability that a coincidence
occurs:
(3.3.5)
where
interval
and
, and
are the probabilities of a photodetection at detectors T and R in a time
is the joint probability of making detections at both detectors. These
probabilities are calculated by finding the average rate of detections multiplied by the
time interval
. The average rate of detections is just the total number of detections
divided by the total counting time
. The probabilities for detections and coincidences
are as follows:
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(3.3.6)
Substituting equations in (2.6) into (2.5) yields
(3.3.7)
These functions are used to make two-detector measurements of
, but usually
three detectors are used. A second source beam incident on a third detector is used as a
gate (labeled G). The expression then becomes
(3.3.8)
The probabilities are calculated a bit differently because they can be normalized.
(3.3.9)
The expression for (2.8) in terms of number of coincidences is
(3.3.10)
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As Grangier, Roger and Aspect noted, “a photon can only be detected once,” thus the
probability of finding any coincidences in detectors G, T and R are zero (
), and
(3.3.11)
This means that the photocount at the transmitted and reflected detectors must be
anticorrelated because the photon must choose a path. Since the electronics cannot make
high speed measurements, often times the time interval
coincidences are measured in all three detectors yielding
is long enough that
. For all classical light,
coincidences are measured in all three detectors such that
3.4 Quantum Eraser
In quantum mechanics interference is the superposition of states for several
indistinguishable paths. Understanding the wave-particle duality requires investigation of
mixed states and their impact on interference. The single photon interference experiments
demonstrate that photons can interfere with themselves and changing the system destroys
interference. The visibility of the fringe pattern is a result of “which-way” information
known by the observer. Similar experiments can be performed using correlated photon
pairs to show the relationship between interference and entanglement. The development
of two photon interference experiments has led to the quantum eraser, which recovers
interference by “erasing” the information about the source of the detected photon.
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3.4.1 Complementarity
A general concept in quantum mechanics, complementarity is interpreted as a
consequence of the uncertainty principle. Two observables are complementary if
knowing the outcome of one observable destroys knowledge of the second. Two
examples of complementary will be illustrated [24].
Example 1: For a single particle with position and momentum , if the position is
measured at any moment in time it prevents measuring the momentum. The
momentum then is in a large range of equally probable states.
Example 2: Consider a spin-½ particle with two orthogonal spin components. If the
horizontal spin component as a definite value (left or right) then both values of the
vertical component (up or down) each have a probability of 50%.
For entangled photon pairs produced by type-I spontaneous parametric down-conversion,
the signal and idler photons have the same polarization. The two photon interference
experiment treats entanglement and interference as complementary observables.
Determining the which-way path information of the signal photon by its polarization
yields full which-way information for the idler photon and annihilates any interference.
However, negating which-way information from both of these photons restores
interference. We now turn to the double slit experiment to understand the quantum eraser.
3.4.2 Double Slit Experiment
The archetypal double slit experiment [25] demonstrates the wave nature of light
and probabilistic nature of quantum mechanics. A photon incident on a thin plate with
two parallel slits and is observed on a screen some distance behind the plate. The
probability that the photon arrives at any given point on the screen is calculated by the
probability of the photon travelling through the top slit or the bottom slit. The
interference pattern arises from indistinguishability of which slit the photon crossed.
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Using the double slit experiment we can derive the relation between the complementary
observables: which-way path information and interference. First assume that the incident
photon is vertically polarized, or in the vertical polarization state
. This does not
change the path the photon took nor does it determine which slit it traversed. By adding a
half-wave plate (HWP) to the upper slit, we can observe which slit the photon passed
through because it rotates the polarization by 90° to
Classical mechanics relates the electric fields
, the horizontal polarization state.
and
to the
corresponding wave passing through the top and bottom slits, respectively. The
interference term
is equal to zero because the fields are orthogonal. Interference
fringes can be regained by erasing the which-way path information of the photon, thus
restoring the photons to the vertical polarization state
.A
polarizer is placed in
front of the screen to mix up the polarization states and the origin of the photon cannot be
known; thus, the interference pattern is recovered. The double slit experiment led to two
photon interference tests and the next section will discuss the experiment carried out by
Zou et al [26].
3.4.3 Two photon interference
Consider the interference experiment by Zou et al. indicated in Fig. 8, in which
two mutually coherent pump waves are pumped into two similar nonlinear crystals NL1
and NL2.
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Figure 8: Interference experiment [26].
Parametric down-conversion occurs at NL1 and NL2, with the emission of a signal
photon and an idler photon at each. We observe interference between the signal photons
when the paths of the idler photons
photodetectors
are aligned. The rate of coincidences at
display a fourth order interference because the detectors cannot
distinguish between the photon pairs. If the connection between
is broken by
misalignment or a beam-block, it is possible to determine whether the signal photons
come from NL1 or NL2. In this case the alignment of the two idlers serves as the
quantum eraser.
3.4.4 Quantum Eraser
The quantum eraser [27-31] model employed in this project is taken from Gogo et
al. experimental arrangement [7]. There are two parts to the setup: the polarization
interferometer and the quantum eraser. Fig. 9 indicates the entire experimental layout.
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Figure 9: Quantum eraser experimental setup [7].
The source used produces entangled photon pairs in the polarization state,
(3.4.1)
Any photon has a 50% chance that it is horizontally or vertically polarized. The signal
photon then goes into a polarization interferometer illustrated in Fig. 10.
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Figure 10: The polarization interferometer. The arrow signifies horizontal polarization and the dot indicates vertical
polarization [7].
The beam displacing prism (BDP) splits the signal beam into vertically and horizontally
polarized components and the half-wave plate flips the polarizations. The second BDP
realigns the beams, which go through a half-wave plate to interfere on the polarizing
beam splitter (PBS). The pathlength difference between the two polarization beams is
adjusted by moving the second BDP. Theoretically the photon takes both paths and
interference will occur. However, if we can determine the polarization of the photon we
can determine which path was taken. The quantum eraser exists in the idler signal; the
half-wave plate in front of B,B’ detectors constitutes the eraser. This plate is oriented
such that the idler beam horizontally polarized are detected at B and vertically polarized
idler photons are detected at B’. If a vertically polarized idler photon is detected at B’
then we know that the signal photon must also be vertically polarized.
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Figure 11: The number of coincidence counts between detectors A and B as a function of pathlength difference in the
two arms of the interferometer. Detector B measures horizontally polarized photons [7].
Rotating the half-wave plate erases any which-way information and interference is
renewed. Any knowledge of the original state of polarization is removed, yielding new
polarizations.
Figure 12: Measure of coincidences (a) AB, and (b) AB’ as a function of the pathlength difference between the arms of
the interferometer. The beam splitter is rotated so that detector B measures +45° polarized photons and detector B’
measures -45° polarized photons [7].
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4 Materials and Methods
4.1 Equipment
4.1.1 External-cavity Diode Laser
Diode lasers are widely used in optical experiments because they are compact,
simple and inexpensive [32]. We used an external-cavity diode laser (ECDL) because it is
capable of reasonable continuous wave output powers with high efficiency, very stable,
and tunable [33]. For our experiments we wanted to tune our laser to a wavelength of
because our crystal is cut to produce down-converted photons at that input
wavelength. We constructed our own blue diode laser using the essentials for an
external-cavity: laser diode, collimating lens and a diffraction grating. All the
components are fastened unto a base plate, with adjustable positions for the lens and
diffraction grating. The lens is mounted into a threaded tube and its position can be
adjusted by rotating the lens cover. This is done by inserting a needle into the pinhole on
the rim of the lens cover. The diffraction grating moves by turning either of the two
adjustment knobs; one knob translates the grating horizontally and the other changes the
vertical dimension. Fig. 11 illustrates the position of the basic units.
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Figure 13: Schematic of external-cavity laser.
The grating is mounted in a Littrow configuration so the first order incident light on the
grating is reflected back to the laser diode. This second weaker beam collapses into the
main beam, increasing the output power as the feedback is above or reaches the threshold.
Because the size of the cavity changes the frequency of the light, mechanical movement
and thermal expansion must be avoided. Our laser is mounted on a larger base that is
bolted to the optical table. A cooler has been attached to the laser but we have not used it.
We use a lens to coarsely collimate the bare diode laser without any grating
feedback. We use mirrors to steer the beam across the lab and adjust the position of the
lens along the axis of the beam propagation. If the lens is too far from the diode, the beam
focuses at 5 feet from the laser diode. To facilitate beam collimation, we use an index
card with varying dot sizes so that we can compare the size of the beam up close and far
away. Diode lasers are single-mode and the linewidth of a bare diode is
[34].
Once the bare laser diode is coarsely collimated, the grating is attached. To maximize the
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feedback from the grating, we operate the laser at just above threshold and change the
length of the cavity by adjusting the horizontal and vertical positions of the grating. At
low currents, the laser diode is sensitive to feedback because it cannot lase by itself; the
grating is adjusted so a second beam appears near the main output beam. This is the fine
adjustment to optimize the feedback and reduce linewidth.
Figure 14: Spectrum of ECDL.
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Figure 15: Spectrum of ECDL.
We used a spectrograph to verify that our laser was at the desired wavelength
. The first spectrum was taken in November 2011 with our spectrograph and
has a linewidth of
linewidth of
Fig. 13 is a recent spectrum of our ECDL with a
Within this four month interval, the linewidth increased and
several other wavelengths are present because of the thermal expansion of the cavity and
losing feedback of the laser. Both spectra were taken when the laser power output was
which was measured with a power meter.
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4.1.2 The Laser Beam
Figure 16: Experimental apparatus. Components are 405nm cavity-diode laser, a pair of cylindrical lenses, two mirror,
two spherical lenses, a half-wave plate (HWP), down-conversion crystal (BBO), and a polarizing beam-splitter
(PBS).The half-wave plate before the crystal vertically polarizes the pump beam to match the specifications of the
BBO crystal.
Due to the inherent astigmatism of laser diodes, a laser diode beam is elliptical;
several techniques can be used to circularize the beam. We initially placed an anamorphic
prism between the ECLD and the first mirror. Anamorphic prisms consist of two wedgeshaped prisms. Anamorphic prisms correct the beam shape by expanding or contracting
the beam in one direction without changing the other direction by adjusting the angle of
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the prisms and the incident light. Even when we used a 3.1x anamorphic prism, the beam
was still elliptical when it reached the crystal. Instead we used a pair of cylindrical lenses
to circularize the beam. In the case of our laser, the beam was diverging horizontally,
while the vertical dimension did not. To minimize aberrations, we placed two cylindrical
convex-lenses L1 and L2 with focal lengths
respectively, in
such a way that the convex surfaces face the collimated beam. We placed L1 a distance
from the ECLD module and positioned L2
from the first
lens. This resulted in a circularized beam.
To increase the intensity of the beam, we used a pair of spherical lenses to form a
telescope; thus, increasing the intensity of the beam. A pair of plano-convex spherical
lenses L3 and L4 with focal lengths
respectively, was placed
before the half-wave plate. These lenses were chosen such that the magnification
reduced the beam size by a fifth. Based on preliminary tests, reducing the beam increased
the number of counts in each channel.
4.1.3 BBO Crystal
We used a nonlinear crystal, 5x5 mm aperture and 1-mm-long beta-barium borate
(BBO) crystal, cut for type-I spontaneous parametric down-conversion. The direction of
the down-converted photons is determined by the angle formed by the optic axis of the
crystal and the propagation of the beam, the phase-matching angle .4 Our crystal was cut
with a phase-matching angle of
so that incoming light at 405nm
produces two entangled photons of wavelength 810nm at
4
from the pump beam axis.
For an explanation of phase-matching refer to Appendix B of Galvez et al. [6]
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The crystal was mounted on a stage that would allow us to adjust the orientation of the
crystal. Once the collection lenses were set up, we adjusted the horizontal and vertical tilt
of the crystal so that it produced maximum counts in both channels.
4.1.4 The Detectors
We used a single photon counting module [34] with four channels (SPCM-AQ4C)
as our photodetector. The signal and idler beams were collected with fiber optic
collimation lenses and directed at different input channels of the counting module. Each
channel consists of a fiber optic collimation lens, two optical fibers and RG780 filter
between the fibers. The photons passed from a fiber optic collimation lens to a multimode fiber to a RG780 filter and redirected from a second fiber optic collimation lens
into a single mode fiber to the photon detector. The RG780 filter acts as a high pass filter
to block blue photons from the laser beam, while transmitting down-converted light. The
SPCM-AQ4C is highly sensitive to light and all connections must be made before turning
on the system.
Figure 17: The collimation lenses are on the right and the RG780 filters are on the right. The orange fiber is the
multimode optical fiber.
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The SPCM outputs electrical pulses that went into a coincidence counting module
(CCM) designed by Dr. Branning of Trinity College [35]. The CCM takes up to fourinput channels, allowing the user control over the logic and interfaces with a computer
over USB. The software attached to the system is a LabVIEW vi that allows the user to
control the clock cycle, sample size, bin size, and integration time of the system. The
clock cycle is the smallest time interval for a single sample of counts. Most of our
experiments have a clock cycle of 800Hz. The sample size is the minimal interval for
data collection, written to the screen or file, for an arbitrary number of clock cycles. Each
of our sample sizes is only one clock cycle. The bin size is the number of samples the
software accumulates before updating the on-screen data. We chose a bin size of 800
samples because it provides a higher count and coincidence rate. Finally, the integration
time is based on the time it takes to acquire the 800 samples, which in our case is 1.0s.
4.1.4 (a) The A-detector5
The angle between the signal (and idler) beam at the input pump beam is
collection lens for channel A is placed on a translating stage so that it makes a
. The
angle
with the pump beam and faces the crystal. Next, the alignment laser was connected to the
single mode fiber for channel A instead of the multi-mode fiber to avoid alignment issues
between the multi- and single mode fibers. If the alignment beam is not visible, the tip
and tilt of the fiber optic collimation lenses between fibers needs to be adjusted such that
the beam from the collection lens is a point source. A coarse alignment is made by
shining a laser beam backward through the fiber and onto the crystal. The vertical and
5
The A-detector or detector A refers to the collection lens for channel A.
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horizontal tilt of the collection lens mount is adjusted so the backward beam hits the
center of the crystal.
3°
Figure 18: Detector A alignment.
This coarse alignment points detector A in the general direction of the down-converted
photons. Next, a fine alignment is done by reconnecting the single mode fiber to channel
A of the SPCM. Using the LabVIEW program, counts in channel A were measured. The
horizontal tilt of the collection lens mount is adjusted until the photon count rate is
maximized and then, the vertical tilt of the mount is tweaked until the counts are
optimized. The knob of the translation stage for the collection lens is turned to change the
angle the detector makes with the crystal. Using the translation stage, detector A was slid
left and right on the optical table until the count rate was maximized. Again, the tilt of the
collection lens mount was readjusted. This process was repeated several times, always
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adjusting the tilt before sliding the detector. Once the count rate is maximized the
detection optics for channel B can be aligned.
4.1.4 (b) The B-detector6
We used a similar procedure for the coarse alignment of the collection lens for
channel B. For the fine alignment, instead of observing the count rates, we maximized
the coincidences between channels A and B to optimize the alignment for the collection
lens for channel B. We maximize AB coincidences in order to detect as many entangled
photon pairs as possible. Next, the horizontal and vertical tilt of detector B was adjusted
to maximize AB coincidence counts. Then, detector B was translated to increase AB
coincidences. These two steps were repeated until the collection lens for channel B was
in the optimal position for AB coincidences.
4.1.4 (c) The B’-detector7
Before setting up the collection lens for channel B’, two irises, a half-wave plate
(HWP), and a polarizing beam splitter (PBS) are aligned using a backward beam from the
collection lens for channel B. These components are centered such that they are in the
path of the down-converted photons traveling to detector B. Since down-converted light
is vertically polarized, we can use a half-wave plate and a PBS to fine tune the orientation
of polarization so that it splits equally between the two collection lenses. By rotating the
HWP, we adjusted the input polarization to be
with respect to the axis of the PBS.
6
The B-detector or detector B refers to the collection lens for channel B.
7
The B’-detector or detector B’ refers to the collection lens for channel B’.
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We can easily transmit or reflect the beam by rotating the HWP, which is useful during
the alignment of the third collection lens.
Figure 19: The B' detector alignment setup.
We placed the B’ collection optics on the reflection side of the beam-splitter and
used the coarse alignment method described in the sections above. The alignment laser
must shine through the beam-splitter and irises onto the down-conversion crystal. Next,
to fine-align detector B’, we observed AB’ coincidences. The half-wave plate was rotated
to maximize AB’ counts because the orientation of the wave-plate changes AB and AB’
coincidence counts. We adjusted the horizontal and vertical tilt of the collection lens for
channel B’ to increase AB’ coincidences. We slid detector B’ until the coincidence rate
was at its highest and repeat the process several times until the collection lens for channel
B’ was in the optimal position for AB’ count rates. Finally, we rotated the half-wave
plate to equalize AB and AB’ coincidences.
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5 Analysis
Figure 20: Optical table set up with 3 detectors.
After aligning detectors A and B using the method described in section 4.1.4, we
observed coincidences between channels A and B. These measurements were aimed at
verifying that counted photons were indeed down-converted photons, and to compare the
number of detected photons against the coincidence rate. First, detection channels were
set to a clock period of
, but because counts were exceedingly
low to detect coincidences the data was integrated over 1.0 second cycles. The graph
Page 43 of 55
below displays the number of correlations between channels A and B for each 1.0s cycle
of channel A counts when the laser was outputting 1.49W of power.
Figure 21: Number of coincidences in AB for each 1.0s detection cycle of photon counts in detector A. This data was
taken over a window of 26.0 seconds.
The data demonstrates that there are not many coincidences for the amount of photons
incident on the detectors. It is likely that we are not completely blocking out the blue
photons from the laser. However, comparing the collection data, counts in channel A
were higher by a factor of 10. Although correlations were optimized before taking any
data, it is possible that the beam block was in the way of path B and caused lower count
rates. Several adjustments were made to the position of the beam block and detector B
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but counts were not improved. In order to measure the second-order correlation function,
coincidences AB’ are counted by the addition of the third detector. Once detector B’ is in
the optimal position for AB’ coincidences, correlations AB, AB’ and ABB’ were
calculated as shown in Fig. 22 below.
Figure 22: Counts in A versus coincidences in AB, AB’ and ABB’. The values were integrated over a period of 1.0s.
The collection was made over a window of 40s.
This demonstrates that the photon must choose path B or path B’ because there are no
ABB’ coincidences. The coincidences were significantly higher than our first experiment
but not comparable to the total photon count. One salient feature of adding the second
half-wave plate and detector B’ is that the B counts, as well as AB coincidences,
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increased. AB or AB’ coincidences are favored based on the orientation of the half-wave
plate after the crystal. First, the wave-plate is oriented to maximize the counts in channel
B and detector B is adjusted to amplify AB coincidences. Then, the wave-plate is rotated
to favor B’ counts and detector B’ is adjusted to improve AB’ coincidences. Before
taking data, the wave-plate was oriented such that AB and AB’ coincidences were even.
The photon counts in channel A were comparable to the sum of photon counts in B and
B’. We are unsure why the number of counts in channel B increased.
Figure 23: Counts for detectors A, B, and B'. Photon counts from B and B’ are on the same order of magnitude of
counts from A.
The number of coincidences is used to calculate the second-order correlation function for
three detectors because there are no correlations between AB and AB’ coincidences. This
Page 46 of 55
means that from observing coincidence counts we cannot tell which path the photon took.
Instead our observations validate that light is a superposition of two waves can be
deconstructed when incident on a beam-splitter. Increasing and decreasing the integration
window does not suggest any correlations but increases and decreases the coincidence
rates respectively.
Figure 24: Coincidences for a clock rate of 0.00125s and a total integration time of 1.0s.
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Figure 25: Coincidences for a clock cycle of 0.005s and an integration window of 4.0s.
Figure 26: Coincidences for a period of 0.001s and an integration time of 0.8s.
Page 48 of 55
Once we measured the number of counts and coincidences we calculated the secondorder correlation function for 3-detectors based on equation 3.3.10.
Where
is the number of photon counts and
coincidences. Below is a table of
,
and
are the number of
values.
NA
NA,B
NA,B'
NA,B,B'
g(2)A,B,B'(0)
95876.06
167
163
0
0
96058.06
173
183
0
0
96480.06
155
159
0
0
96340.06
163
155
0
0
96523.06
138
170
0
0
96928.06
166
181
0
0
96308.06
185
170
2
6.124519
96919.06
170
172
0
0
96533.06
170
167
0
0
96362.06
165
193
0
0
96921.06
188
165
1
3.12447
96254.06
146
170
0
0
Figure 27: g(0) values for a clock cycle of 800Hz.
Page 49 of 55
Overall,
for our data points. The instances in which
occur
because the sampling rate is not small enough to detect single photons. Thus, within the
given sampling rate photons may be detected at B and B’, generating
.
6 Conclusion
The quantum optics experiments described above are well on their way. While the
first two experiments have been executed, there is still much work that must be done
before testing Bell’s Inequalities and building the quantum eraser. At the moment efforts
are being made to increase coincidence counts and count rates. It is crucial that
coincidences are increased to display entanglement and confirm the existence of photons.
There are still other issues that must be dealt with. More spectra must be taken and the
laser should be tuned to 405nm. A critical issue is automating the data analysis process.
LabVIEW saves the data in a text file but getting to large amounts of data takes quite a
bit of time. The next step is to create a program that retrieves and analyzes the data. The
future endeavor is to produce more calculations of the second order correlation parameter.
Under quantum mechanics,
means that there are no correlations between
all three channels because it would be impossible to detect a photon in channels B and B’
at the same time. Until then the latter experiments cannot progress.
Page 50 of 55
Appendix A: Single Counting Photon Module
For our apparatus we used Perkin Elmer’s single photon counting module
(SPCM-AQ4C) with 760-1-interface board (SPCM-AQ4C-IO). This unit consists of four
independently fiber-coupled photon-counting channels, which contain fiber optical
receptacles pre-aligned to the detector. The SPCM-AQ4C uses silicon avalanche
photodiodes with a peak photon detection efficiency of 60% at 650nm. This module has a
dark count rate of 500 cps, but it has little effect on our experiments. The interface board
provides power connectors to the power supplies and BNC connectors that output TTL
(transistor-transistor logic) signals. The module requires +2Volts, +5 Volt, and +30 Volt
power supplies. We used a B&K Precision 1672 triple output DC regulated power supply
that provides one 5V fixed output and two variable outputs. During one of the
experiments, the wires for the +5V power supply shorted and burned out the capacitor C1
on the interface board. Removing this component from the interface board did not affect
the function of our counting module.
Figure 28: This is the interface board after removing component C1.
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Appendix B: Coincidence Counting Module
We used a coincidence counting module (CCM) designed by Dr. Branning of
Trinity College [36], which uses logical AND gates to count coincidences. The TTL
outputs of the SPCM-AQ4C are directed at the CCM via BNC connectors. The detector
pulses enter a pulse-shaping circuit that reduces the width between 10-18 ns. This allows
the pulses to overlap and reduce the number of accidental coincidences. The pulses are
sent to the inputs of an AND gate where the output will read true if both detector pulses
arrive at the same time. The output signals are sent over to a LabVIEW program via
FTDI USB. The CCM software provides the drivers for FTDI USB interface device. The
LabVIEW software displays the counts in real time and/or saves them to a disk.
Figure 29: Series 3 CCM which performs up to 4-fold coincidence counting.
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