MENG 26200: QuantumLab
Pritzker School of Molecular Engineering
University of Chicago
October 1, 2024
Contents
1 Week 1
1
1.1
Laser Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
QuED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Safety References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2 QuED Characterization: Week 2
6
2.1
Characterizing the experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
Singe Photon Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Primer on Quantum State Tomography . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.4
Bell’s Inequality
2.5
Analysis Questions for the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 QuED Experiments: Week 3
18
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2
Photon-pair Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3
Singe Photon Detection: Hanbury Brown and Twiss . . . . . . . . . . . . . . . . . . 22
3.4
Single Photon Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5
Quantum Eraser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6
QuED Applications References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7
Analysis Questions for the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 QuED Experiments: Week 4
35
4.1
Two-photon Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2
Random Number Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3
Quantum Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.4
QuED Applications References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5
Analysis Questions for the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5 MOT experiments: Week 5
44
5.1
Atomic structure of Rubidium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2
Physics Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3
Recommended Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.4
Laser characterization measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.5
Using the Spectroscopy Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.6
MOT Characterization References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.7
Analysis Questions for the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
ii
6 Trapping Atoms: Week 6
59
6.1
Create MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2
Cooling the atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.3
Analysis Questions for the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7 MOT Characterization: Week 7
64
7.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.2
Basic FI picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.3
Re-create MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.4
Detecting MOT using fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.5
Analysis Questions for the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8 NV spin experiments: Week 8
68
8.1
NV Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.2
Laser threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.3
Magnetic field calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.4
ODMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.5
Analysis Questions for the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9 NV spin experiments: Week 9
9.1
72
Analysis Questions for the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
iii
1
Week 1
1.1
Laser Safety
During this course we will be working with a three different lasers having a variety of output
wavelengths that range from the ultraviolet (UV) end of the electromagnetic spectrum to the infrared (IR), with varying output powers, beam profiles, and operating modes. It is important to always
be aware of what kind of radiation source you are working with, the related potential hazards, and
what are the appropriate safety protocols to follow.
Basic Vocabulary
A laser (an acronym from Light Amplification by Stimulated Emission of Radiation) refers to a
source of monochromatic, coherent light. There exists a great variety of types of lasers depending
mainly on their gain medium type, pumping source, operational wavelengths and their overall
structural constitution. Semiconductor laser diodes (LDs) are remarkably versatile and practical,
providing reliable, controllable, and efficient sources of laser light that are both low cost and
relatively easy to use. All of the three blocks of experiments we will be working with rely on LDs.
A review article by Wieman and Hollberg[1] provides an excellent resource for a more in depth
description of LDs.
Because a laser is directional by definition, it always consists of a primary beam that is stirred
and directed through a variety of optical elements such as lenses, cavities and crystals using mirrors
and other reflective surfaces. A specular reflection refers to the event of a laser bouncing off of
a “shinny” surface —usually a mirror— although almost any metallic surface such as those in
rings, necklaces, wrist watches and earrings can effectively act as a (undesired) mirror. A diffusive
reflection on the other hand refers to the laser beam bouncing off of “rough” surfaces such as walls,
plastics and some ceramics, although in general whether a material is “rough” or “shinny” depends
on the wavelength of the laser light.
The laser output refers to the largest energy scale associated with the device and its nominal
units depend on the operating mode of the laser. For a continuous wave (CW) mode, on which
as its name indicates the overall level of emission is constant over time, the output is reported in
Watts (W) or milliWatts (mW), and it simply refers to the amount of energy being constantly
delivered per unit time. For a pulsed laser, the light will be emitted in bursts that are spaced in
time and have some specific duration. For that case, one needs to consider the pulse duration and
the duty-cycle in order to evaluate the pulse energy (measured in Joules), and both the peak and
the average output powers.
Safety-wise, one of the most important concepts is that of maximum permissible exposure
(MPE), which simply refers to the amount of radiation to which one can be exposed without
the potential of a hazard or serious effects on your skin or eyes. The aversion response is the notion
that our eyelids will reflexively close when exposed to a bright and visible source of radiation, and
on top of this we instinctively try to move our head away. Some times this will safeguard your eyes,
or at least reduce the amount of damage. However, it is important to emphasize that we know our
eyes can only detect visible light and thus are not sensitive to UV or IR radiation. Source of light
that fall outside the visible spectrum are thus particularly more dangerous.
To avoid exposure altogether, we must make sure to keep the laser beam is well contained within
the optical path, as well as ensuring that any “unused” laser power is being dumped on a beam
trapping element. To avoid unexpected reflections, it is customary to remove any metallic objects
or jewellery around our hands before working and operating with lasers in a research lab, especially
1
when dealing with high power lasers. Lastly, it is important to wear the appropriate personal
protective equipment (PPE) on each case, such as safety goggles. These are wavelength-specific
and the most important figure of merit is usually the optical density (OD), which is a measure in
a logarithmic scale which indicates the amount of attenuation the goggles will provide for a given
color. For example, an OD of 1 reduces transmission by ten-fold, and an OD of 3 reduces the
transmission by a factor of a thousand. The required OD would depend on each laser type—the
higher the power, the higher the required OD—but an OD ≥ 5-7 is usually a good rule of thumb.
It is worth mentioning that the value of the OD is highly dependent of the wavelength, so a pair
of goggles that absorbs green light might be completely transparent to UV light for instance.
Laser classification
Because lasers are so common and they can also act as potential hazards, they are heavily
regulated by national and international norms. In the US for instance consensus standards are
recognized by organizations such as the American National Institute of Standards (ANSI) [2].
Through the last few decades the classification has been converging towards a unified international
standard that employs roman numerals going from 1 to 4 in increasing order of risk, with additional
letters specifying particular models within each class.
Class 1. These lasers are effectively incapable of producing any damage either because they
are extremely weak or strongly confined. They are safe and usually exempt from beam-hazard
measures. A typical example would be the optical component in a laser printer. More recently the
subtype “1C”, with the “C” standing in for cosmetic, has been recognized to refer to lasers that
are explicitly designed to be applied to the skin (such as that in a laser hair removal machine).
Another subtype, “1M”, with the “M” standing in for magnification, refers to lasers which are
harmless except when viewed through focusing lenses or other components such as magnifying
glasses, telescopes or binoculars. A weak, non-collimated LD is a good example of a 1M radiation
source.
Class 2. These consist of visible lasers (400- 700 nm) that are relatively weak and so the
aversion response is enough to avoid any injury. Typically the power of these lasers is less than 1
mW. A weak laser pointer would be a good example of a class 2 laser. Just like in the previous class,
the subtype “2M” specifies sources which are diverging but could otherwise become dangerous if
focused (such as the visible light of a computer mouse )
Class 3. This class refers to any laser that is significantly more powerful than the previous
class, in particular if the laser falls outside the visible range of the electromagnetic spectrum. Eye
protection should always be used when operating this class of laser. If the source is relatively
weak (1-5 mW) and the chance of injury is small, then it is classified as the subtype “3R”, with
“R” standing in for reduced controls. Any lasers falling within an output power of 5-500 mW are
otherwise classified as “3B”∗ , indicating they they are a direct eye hazard. It is a risk if reflected
from a specular reflection but relatively safe in case of a diffuse reflection event. They are relatively
safe in terms of skin exposure unless extremely focused.
Class 4. This umbrella classification refers to any high-power laser that falls outside the
previous classes. It represents a hazard to the eye and the skin after any reflecting surface, and
represent a risk and potential fire hazard even after unintended diffuse reflections. Handling this
class of laser should be done with extreme caution, and only when using adequate eye and skin
PPE and after sufficient training and experience.
∗
The “B” is remains simply due to historical reasons with respect to an older classification which labeled relatively
safer lasers as “A” subtype and “B” lasers to those that were more risky.
2
1.2
QuED
Safety Instructions
Laser safety
The setup that we will be using for generating entangled photons (quED module from quTools,
GMBH) contains a UV, 405 nm Class 3B laser. As we mentioned above, class 3B lasers are
hazardous to the eye if the beam is viewed directly, and also after specular reflections even for
short and unintentional exposures. This pumping laser is encased and interlocked such that if the
case is opened the laser will turn off by itself, and so we will not get exposed to any of the UV
light. Our pump laser is vertically polarized and emits about 20 mW when running at a current of
40 mA.
The down-conversion process is very inefficient (about 1 in 10 billion, or 0.1 PPB) and as such
even after using a relatively high-power of pumping light, we will only get a few thousand photons
of IR photons ( 810 nm) being generated. All the light that fails to get down converted is absorbed
by a series of long-pass filters. Therefore, all of the experiments we will be carrying out in this
block are eye-safe in terms of radiation and we will not need to be wearing safety goggles during
our experiments. If for any reason the interlock is bypassed and the laser is being run without
the safety enclosure, then it would be absolutely necessary to wear safety glasses at any moment.
However, remember that opening or removing the housing of the laser head might expose you and
other users to a class 3B laser. You should never operate the laser without the interlock unless you
have explicit permission and supervision of a TA/instructor.
There is a big, red push-button that engages the laser controller of the laser. In case of an
emergency, you can always click that button and immediately disable the laser. This is meant to
be an emergency measure. If you need to turn off the diode, it is better to dial down the current
to zero first.
Important considerations
Laser diodes in general can be sensitive to back-reflections, meaning that sending undesired
light back into the diode could damage it† . Then, you should be mindful of not directing any light
upstream the optical path.
The controller (quCR) is constantly maintaining the diode at a fixed temperature, regulating
the flow of current in the diode, and verifying that the interlock is engaged. Suddenly disconnecting
cables between the quCR and the down-conversion module could result in current spikes or other
transients that could damage either the Peltier cooler that regulates the temperature, or the diode
itself. More importantly, if the interlock stops functioning and/or you loose control over the current,
you might end up in a condition that is no longer eye-safe. If you need to change any of the electronic
connections to the system, you should do so while the device is off.
The pumping diode has a maximal operating current of 45 mA, and continuously running it at
or above this point will severely shorten the lifetime of the diode. As a general practice you should
always operate the diode below this level (but above threshold!). A current of 40 mA is enough to
carry out all of the experiments we will be doing for this block.
†
This is dramatically referred to as Catastrophic Optical Damage (COD))
3
Figure 1.1: Light Propagating in Fiber. Taken from Ref. [3]
Fiber Care
Fibers optics are typically composed of fused silica (glass). They are composed of a core of high
index of refracton (n) glass surrounded by a cladding of lower index glass. Light is able to travel
through fiber optic cables because it will be internally reflected on the high-to-low interface. This
relies on Snell’s law, see Fig 1.1. There are a few different types of fibers such as multimode (MM),
single mode (SM), and polarization-maintaining (PM). We will be using polarization-maintaining
fibers for the QuED experiments. To maintain polarization, stress rods are added into the cladding
to create stress within the core. This will cause the light to be polarized a particular direction.
Since these cables are literally made of glass, you should handle them gently and carefully!
Before inserting them into any port make sure they are adequately cleaned. To clean a fiber
tip you should first grab a fiber roll, and pull out a fresh cleaning wipe onto the pad. Then, gently
rub the glass end onto the pad in a figure-eight motion.
Single-photon detectors
Both because the down-conversion process is relatively inefficient, and we will also make use
of single-photon sources, we need to have detectors that are sensitive enough to distinguish single
photons. In our case, we are using Avalanche Photodetectors (APD) made from Si. APDs rely
upon the photoelectric effect to liberate an electron, and then a biasing voltage is applied so that
this generated electron will readily move. As this current moves through the APD, it will in turn
liberate more and more electrons, creating an electron avalanche. This detection scheme allows us
to map single photon detection events to measurable currents. After a single photon detection, the
APD will not be able to measure another one for a set amount of time while the material relaxes
back to its steady state. This is known as the detection window of the APD, is on the order of 30
ns for our model. Since these APDs are very sensitive to single photons, a common lamp or even a
low-power laser has the potential to destroy these devices. You should be very careful not to send
strong sources of light to the the APDs at any moment.
4
1.3
Safety References
1.
Wieman, C. E. & Hollberg, L. Using diode lasers for atomic physics. Rev. Sci. Instr. 62, 1–20
(1991).
2.
Kelechava, B. Laser Safety: Class 1, 1C, 1M, 2, 2M, 3R, 3B, and 4. American National Institute
of Standards (ANSI). https://blog.ansi.org/2018/09/laser-class-safety-1-1c-1m2-2m-3r-3b-4/#gref (2018).
3.
Fiber Optics for Sale Co. https://www.fiberoptics4sale.com/blogs/archive- posts/
95146054-optical-fiber-tutorial-optic-fiber-communication-fiber.
5
2
QuED Characterization: Week 2
Figure 2.1: Spontaneous Parametric Down Conversion Cone. Adapted from Ref. [1]
This week we will familiarize ourselves with our source of entangled photon pairs and the singlephoton detectors and optical elements we will be using through this experimental block. We will
also characterize the quantum states of single photons and entangled photons by measuring their
density matrix. Lastly, we will demonstrate the ”non-local” nature of reality with Bell’s Inequality.
2.1
Characterizing the experimental setup
The instructor/TA will first give you a quick overview of how to turn on the setup and how to use
the controller to acquire data in terms of count rates. Identify the optical components we will be
working with through this session, their purpose, and how can you mount/introduce them to the
setup:
• λ/2 Half-wave Plate (HWP): This will rotate the polarization of the light without loss.
• λ/4 Quarter-wave Plate (QWP): This will convert linear polarized light to circularly polarized
light, and vice versa. The rotation of a QWP will introduce a phase following equation 3 [2].
• Linear Polarizers (LP): This will only allow a certain polarization of light to be transmitted.
These components will follow Malus’ law.
The photons pairs created through the spontaneous parametric down conversion (810 nm) will
be emitted in random directions within a cone, (see figure 2.1). These photon pairs will be created
6
with opposite momentum so for our purposes we will collect opposite edges of the cone. This will
be achieved by using mirrors to steer these photons into our fiber ports.
Figure 2.2: Inside of Laser box
Turn on the 405 nm pumping laser diode and dial the current up to 40 mA. Use the side-mirror
in the laser box to see a a little bit of the laser light that is rejected from one of the internal
compensation crystals. We will now use the APDs to measure the threshold current for the diode.
At that point, the emission beings to be dominated by stimulated emission instead of spontaneous
emission.
As a first exercise, start dialing down the current of the diode while looking at the number pf
photons in either arm. At which point does the number of photons stop changing? By taking small
steps in the current, try to estimate the threshold current as best as you can.
Before we use the IR polarizers, take a moment to use the LED light box and use the color
filters to to make a figure similar to the shape of the PME logo. Now reproduce this same shape
using the provided polaroid films. Take a picture of both using your cellphone. Do you notice any
difference between both arrangements? Try expressing the operator that corresponds to a polarizer
explicitly as a matrix.
Adjust the HWP before the BBOs to create an output of horizontally polarized light. Now
introduce the polarizer on the manual mount and verify that its transmission follows Malus’ law.
Use the motorized rotation mount to record data through different angles.
2.2
Singe Photon Tomography
2.3
Primer on Quantum State Tomography
Given the increasing relevance and interest in quantum science, our ability to reliably create,
manipulate and characterize bigger and more complicated quantum states is an area of active
research. For systems that can be effectively isolated from their environment (a pure state), the
wavefunction ψ is sufficient to fully characterize such states, usually notated as |ψ⟩. If on the other
7
hand the system is interacting with the environment and becoming entangled with it (a mixed
state) or subject to the effects of classical noise, then it becomes necessary to employ the formalism
of the density matrix ρ̂ = |ψ⟩ ⟨ψ|.
The challenge of fully characterizing and measuring a quantum state is sometimes referred to
as the Pauli problem, and is typically tackled using the technique of Quantum State Tomography
(QST). This approach consists in preparing a great number of copies of the quantum system and
performing systematic projective measurements that are usually destructive. While each of these
measurements provides only a partial amount of information about the state, if the sampling is
sufficiently spread over the phase space of the system then through some analysis it is in principle
possible to fully reconstruct the state. A simple visual metaphor for describing this process would
be to imagine measuring the shape a 3D object by taking a picture of its projected shadow along
many different directions. By putting together all of the recorded “slices” (hence the use of the
word tomography), it should be possible to create a 3D model that fully describes the shape of the
studied object.
As an example, we will now elaborate on how to perform QST for a relatively simple system
consisting on a single qubit.
Bases and notation
When describing a density matrix it is important to define a basis over which it will be expressed.
In our case, we will be simply using the |H⟩ and |V ⟩ basis. Accordingly, the density matrix can be
expressed as:
ρ̂ =
⟨H| ρ̂ |H⟩ ⟨H| ρ̂ |V ⟩
⟨V | ρ̂ |H⟩ ⟨V | ρ̂ |V ⟩
(1)
Then, we can also define other sets of orthogonal basis, starting with the “diagonal” basis:
1
|+⟩ = √ (|H⟩ + |V ⟩)
2
1
|−⟩ = √ (|H⟩ − |V ⟩) .
2
(2)
The |+⟩ and |−⟩ states are sometimes also notated as |P ⟩ and |M ⟩ and |D⟩ and |A⟩ respectively.
Lastly, we can also define the “circular”” basis:
1
|R⟩ = √ (|H⟩ + i |V ⟩)
2
1
|L⟩ = √ (|H⟩ − i |V ⟩) .
2
(3)
We can also express all of these states in their vector form:
1
|H⟩ =
0
0
|V ⟩ =
1
1
|+⟩ = √
2
1
1
1
1
|−⟩ = √
2 −1
1
|R⟩ = √
2
1
i
1
1
|L⟩ = √
2 −i
(4)
We note that the |H⟩ , |V ⟩ basis is complete, in that it can be used to create the following four
matrices:
8
µ
µ̂
θQWP
θLP
⟨ρ̂⟩µ
cH
cV
cR
cL
c+
c−
|H⟩ ⟨H|
|V ⟩ ⟨V |
|R⟩ ⟨R|
|L⟩ ⟨L|
|+⟩ ⟨+|
|−⟩ ⟨−|
0
0
+45
+45
+45
+45
0
+90
+90
0
+45
-45
⟨H| ρ̂ |H⟩
⟨V | ρ̂ |V ⟩
⟨R| ρ̂ |R⟩
⟨L| ρ̂ |L⟩
⟨+| ρ̂ |+⟩
⟨−| ρ̂ |−⟩
1
1
|H⟩ ⟨H| =
0
1
0
|H⟩ ⟨V | =
0
0
1
|V ⟩ ⟨H| =
1
0
0
|V ⟩ ⟨V | =
1
1
0 =
0
0
1 =
0
0
0 =
1
0
1 =
0
0
0
1
0
0
0
0
1
(5)
and because together they can access all the matrix elements, it is clear that any real matrix could
be written as a linear combination of these four matrices. Because ρ̂ involves complex numbers, we
will recombine these matrices into a new basis set that includes the well-known Pauli matrices:
1 0
1 = σ0 = |H⟩ ⟨H| + |V ⟩ ⟨V | =
0 1
0 1
σx = σ1 = |H⟩ ⟨V | + |V ⟩ ⟨H| =
1 0
0 −i
σy = σ2 = i(|V ⟩ ⟨H| − |H⟩ ⟨V |) =
i 0
1 0
σz = σ3 = |H⟩ ⟨H| − |V ⟩ ⟨V | =
0 −1
(6)
Projective measurements
As we mentioned above, in QST one performs projective measurements over the quantum state,
which in general we will associate with the operator µ̂ = |µ⟩ ⟨µ|. The observable associated with this
operator, µ, effectively estimates the amount overlap between the quantum state and the particular
state to which it is being projected:
µ ∝ | ⟨µ| ψ⟩|2 = ⟨ψ| µ⟩ ⟨µ| ψ⟩ = ⟨µ| ψ⟩ ⟨ψ| µ⟩ = ⟨µ| ρ̂ |µ⟩ ≡ ⟨ρ̂⟩µ .
Given the basis vectors we defined in the previous section we can then identify six different projective
measurements that we could do to the 1-qubit state using a combination of a QWP and an LP:
9
The actual measured values of µ are proportional to ⟨ρ̂⟩µ up to a variable N that depends on
factors like the intensity of the incoming light, and the efficiency of the detector. However, if the
different projective measurements are achieved only by rotations of the in-path optical elements
(in this case, the QWP and the LP), then N will be a constant for a given setup. We can then
explicitly write down the observed number of counts that we would expect to measure for each of
the given projective operators:
cH = N ⟨H| ρ̂ |H⟩
cV = N ⟨V | ρ̂ |V ⟩
cR = N ⟨R| ρ̂ |R⟩
N
=
[⟨H| ρ̂ |H⟩ + i ⟨H| ρ̂ |V ⟩ − i ⟨V | ρ̂ |H⟩ + ⟨V | ρ̂ |V ⟩]
2
cL = N ⟨L| ρ̂ |L⟩
N
=
[⟨H| ρ̂ |H⟩ − i ⟨H| ρ̂ |V ⟩ + i ⟨V | ρ̂ |H⟩ + ⟨V | ρ̂ |V ⟩]
2
c+ = N ⟨+| ρ̂ |+⟩
N
[⟨H| ρ̂ |H⟩ + ⟨H| ρ̂ |V ⟩ + ⟨V | ρ̂ |H⟩ + ⟨V | ρ̂ |V ⟩]
=
2
c− = N ⟨−| ρ̂ |−⟩
N
=
[⟨H| ρ̂ |H⟩ − ⟨H| ρ̂ |V ⟩ − ⟨V | ρ̂ |H⟩ + ⟨V | ρ̂ |V ⟩]
2
(7)
From the equation above we can see that while some of our measurements (particularly cH and
cV ) allow us to probe the diagonal terms of ρ̂ up to a proportionality constant, the off-diagonal
terms cannot be directly measured using a projective approach. Nevertheless, by constructing
linear combinations of the observables we can observe useful patterns. First,
cH + cV = N (⟨H| ρ̂ |H⟩ + ⟨V | ρ̂ |V ⟩) = N
(8)
where in the last equation we used the fact that ρ̂ should be unitary and thus Tr(ρ̂) = 1. Then, we
can use this measure to estimate the constant N , and we note as well that
cH + cV = cR + cL = c+ + c− = N ≡ S0
(9)
which corroborates the idea that the overall normalization constant should be independent of the
basis we chose. On the other hand, if we write down the contrast between the counts for each basis,
we notice that we obtain measures that we can associate with the Pauli matrices:
S1 ≡ c+ − c− = N (⟨H| ρ̂ |V ⟩ + ⟨V | ρ̂ |H⟩)
S2 ≡ cR − cL = N i (⟨H| ρ̂ |V ⟩ − ⟨V | ρ̂ |H⟩)
S3 ≡ cH − cV = N (⟨H| ρ̂ |H⟩ − ⟨V | ρ̂ |V ⟩)
(10)
The measure S2 might look particularly suspicious at first given the appearance of the imaginary
unit in the calculation. However, because ρ̂ should be Hermitian, the quantity (⟨H| ρ̂ |V ⟩ − ⟨V | ρ̂ |H⟩)
10
is a purely imaginary number, matching up with the expectation that the actual measurement cannot be a complex number. By calculating the ratio between the contrast measures and the overall
normalization factor, we can see that they actually correspond to the projection of ρ̂ over each of
the Pauli matrices:
S1
≡ (⟨H| ρ̂ |V ⟩ + ⟨V | ρ̂ |H⟩) = Tr(ρ̂ · σ1 )
S0
S2
≡ i (⟨H| ρ̂ |V ⟩ − ⟨V | ρ̂ |H⟩) = Tr(ρ̂ · σ2 )
S0
S3
≡ (⟨H| ρ̂ |H⟩ − ⟨V | ρ̂ |V ⟩) = Tr(ρ̂ · σ3 ).
S0
(11)
We then realize that the combination of these six measurements will allow us to expand ρ̂ in
terms of the Pauli basis and calculating the weight for each of the matrices. In particular,
ρ̂ =
1 X Si
σi
2
S0
i
where the factor of half is an overall normalization factor. It is clear that these measures actually
correspond to the components of the Stokes vector
 
S0
S1 
⃗ =  ,
S
S2 
S3
and we can also define a normalized vector in the Bloch sphere given by
 
S
1  1
⃗
S2 .
B=
S0
S3
11
We will set up the analyzing optics on motorized mounts and automate the data taking and analysis. Make sure you have all the necessary components for the setup. Once everything is mounted,
you will have to verify that you are capable of remotely controlling each of the components. We
will provide a working Python method that takes an IP, channel number, and the required rotation
as input. First, verify that there is an established connection between the controller and the mount
by testing a few rotation positions.
Setup:
1. Choose an arm of the QuED, and screw a LP into the breadboard, in the port closest to the
fiber launcher.
2. Screw the QWP into the breadboard on the same arm (why do you need to follow this order?).
3. Remove any polarizer/waveplate from the other arm.
4. You will use the empty arm as a trigger, so you can use the coincidences as your signal.
Experiment:
arm:
Measure the following coefficients by looking at the counts corresponding to your
Measurement Shorthand
H
V
R
L
+
-
θQWP
0°
0°
45°
45°
45°
45°
θLP
0°
90°
90°
0°
45°
-45°
It is worth noticing that these angles are measured with respect to the horizontal, but these
values will not necessarily match with the numbers that you will see in the dial of the mount. For
convenience, you can use the columns “MQWP ” and “MLP ” in the worksheet to evaluate the actual
values you will see in the mounts considering any necessary offsets.
To determine the density matrix, use the relations we derived above:
1
ρ̂ = (Iˆ + ax σ̂x + ay σ̂y + az σ̂z ).
2
ax =
c+ − c−
c+ + c−
ay =
cR − cL
cR + cL
az =
(12)
cH − cV
cH + cV
(13)
Measurements:
• Remove the HWP right before the pair of BBOs and take a measurement of ρ̂ in this case.
• Identify which element do you need to introduce in the arm to create the states ⟨+| and ⟨−|,
and then measure ρ̂ for each case.
12
• Identify which element do you need to introduce in the arm to create the states ⟨R| and ⟨L|,
and then measure ρ̂ for each case.
• Introduce the HWP right before the pair of BBOs and take a measurement of ρ̂ in this case.
2.4
Bell’s Inequality
We can use our setup to obtain an entangled quantum state by collecting at opposite positions on the
SPDC cone when the HWP is the optical path (see figure 2.1). To characterize the entanglement, we
will measure the CHSH inequality over this state. Moreover, this inequality has deep implications
about locality and the experimental violation of this inequality won the 2022 Nobel Prize in Physics.
In local realism theories, the absolute value of a particular combination of two particle correlations
is bounded by 2. α (α’) and β (β’) denote the local measurement settings of two observers. For
our purposes, α and β correspond to LP orientation.
S(α, α′ , β, β ′ ) = [E(α, β) + E(α′ , β) − E(α, β ′ ) + E(α′ , β ′ )] ≤ 2
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⊥ )
(14)
(15)
E(α, β) denotes the normalized expectation value of correlations between measurements. This
will depend on C(α, β) which is the coincidence count for a given α and β LP setting. α⊥ or β⊥
correspond to a measurement orthogonal to α or β.
C(α, β) ∝ cos2 β − α, thus E(α, β) = cos 2(β − α) so for various combinations the CHSH
√ inequality is violated. The maximum violation is found in the following table leading to S = 2 2.
Parameter
β−α
β ′ − α′
α′ − β
β′ − α
Angle
22.5°
22.5°
22.5°
67.5°
Use the worksheet to evaluate the angles at which you need to take measurements to do the
Bell inequality experiment. Then, complete the provided code and program a sequence that will
loop through all 16 configurations and process the measurements into the expectation values and
the Bell index. Start by measuring the CHSH inequality with the HWP installed in the setup, so
that you are probing the entangled state(S > 2). Rotate the mount of the waveplate so that it is
not in the way and repeat the measurement. Are the values of S consistent with your expectations?
Not only is the value of the CHSH inequality important, but so is the error. To assert a claim
for violating the CHSH inequality, one must be multiple standard deviations away from 2. The
strength of the violation is n∆ and defined by the following equations:
n∆ =
S−2
∆S
∆S(α, α′ , β, β ′ ) =
sX
a,b
13
(16)
∆E(a, b)2
(17)
∆E(a, b) = 2
[C(a, b) + C(a⊥ , b⊥ )][C(a, b⊥ ) + C(a⊥ , b)]
(C(a, b) + C(a⊥ , b⊥ ) + C(a, b⊥ ) + C(a⊥ , b))2
s
1
1
×
+
C(a, b) + C(a⊥ , b⊥ ) C(a, b⊥ ) + C(a⊥ , b)
14
(18)
2.5
Analysis Questions for the report
A) Experimental setup characterization
1. Describe the overall setup of the source of entangled photons and the purpose of each of the
components in the optical path.
2. Briefly describe why is it necessary to use a pair of crossed BBOs to generate the entangled
pairs. Prove this by writing down the state of the light before the BBOs, after passing the
first non-linear crystal, and after passing the second using Dirac notation. In the end, you
should end up with the Bell state |ψ⟩ = √12 (|HH⟩ + |V V ⟩). Hint: in your notation, keep
track of both the blue and the two potential IR photons explicitly.
3. Make a plot of the data that you took while measuring the laser diode threshold. Calculate
the current threshold from your data using the “Two-Segment Line-Fit Threshold” method
[3]. Show the fit function that you used to extract the threshold value. How does it compare
to the nominal value, 23.3 mA?
4. Attach pictures of the PME logos that you made using the color filters and the polaroid films.
Which is the darkest area in the color filters case? Which is the darkest area in the polaroid
case? If a polarizer was just a filter, these would match, but they don’t! Briefly explain why
this is so in terms of what the polarizer is actually doing.
5. Make a plot with the data you took when measuring Malus’s law. Try to fit your data to the
expected functional form and discuss how close they match and any discrepancies.
6. Extra credit [Optics, 3pts] Why do you need to use compensation crystals before and after
the pair of BBOs? Explain this in terms of some of the optical properties of BBOs.
B) Single Photon Tomography
1. In one or two sentences describe the density matrix. What is the expected value of Tr(ρ̂)?
2. Describe what is the main purpose of a HWP, QWP and LP in a laser lab. For the waveplates,
describe how these optical elements transform a polarization state both in real space (in the
laboratory frame of reference) and in the Bloch sphere.
3. Which density matrices would you expect to observe when you used the configuration with
and without the HWP in-line right before the BBOs?
4. Calculate the measured Stokes vectors and density matrices for each of the cases you measured. How do they compare with the values you were expecting? For each case, locate your
state within the Bloch sphere.
C) CHSH inequality
1. In one or two sentences, how would you briefly describe entanglement?
2. Determine S, ∆S and n∆ for your measurement sets.
3. Explain the physical meaning of measuring S > 2 for pairs of entangled photons. Were you
able to measure S > 2?
15
4. Read a little about what Bell states are and explicitly write down the four Bell states in the
|H⟩ , |V ⟩ basis using Dirac notation.
5. Say we measure one of the entangled photons as |H⟩. If it is in a Bell State (|Φ+ ), do we
know what state the other photon is in? Why?
6. Extra credit [Quantum Optics, 6 pts]: Follow the tutorial “Bell inequality with photons”
available on canvas under Files/Tutorials to show that the angles that we used for the measurements are the ones that result in the maximal possible violation of Bell’s inequality.
7. Extra credit [Statistics, 3 pts]: derive the relation for ∆E using error propagation. (Hint:
define two variables γ ≡ C(a, b) + C(a⊥ , b⊥ ) and δ ≡ C(a, b⊥ ) + C(a⊥ , b) and propagate errors
over these two variables)
16
QuED Characterization References
1.
QuED Manual. QuTools. https://www.qutools.com/files/quED/quED_manual.pdf.
2.
James, D. F. V., Kwiat, P. G., Munro, W. J. & White, A. G. Measurement of qubits. Phys.
Rev. A 64, 052312 (2001).
3. The differences between threshold current calcualtion methods LINK. MKS Instruments (2020).
17
3
QuED Experiments: Week 3
3.1
Introduction
Last week, we performed single-photon tomography and measured the CHSH inequality using our
photon pairs. In this module, we will now perform photon pair tomography, confirm the singlephoton nature of our source, and turn to interferometry to further study the quantum nature of
the photon pairs.
3.2
Photon-pair Tomography
In this experiment we will characterize the photon pair generation of the QuED using tomography.
Last week we determined that 6 measurements were necessary to map the single photon density
matrix with a LP and QWP. To map the photon pair density matrix, 36 measurements are required,
as we now need to mount a QWP-LP on each arm, and perform 6 measurements for each of the
6 configurations of the other arm. Needless to say, this mapping is more complex and doing all
measurements by hand could be time consuming, so we will set up the analyzing optics on motorized
mounts and automate the data taking and analysis.
Primer on the two-photon analysis
Because we will now be characterizing a two-qubit system, the Hilbert space for this system will
be the result of combining the Hilbert spaces associated with each qubit via a tensor product:
|ψ⟩ ∼ |µ1 ⟩ ⊗ |µ2 ⟩ ≡ |µ1 µ2 ⟩ .
We will implicitly assume the 1, 2 order and omit those subindices to simplify the notation. The
corresponding density matrix ρ̂ is now a 4 × 4 matrix, which can be expressed in the combined
|H1 ⟩ , |H2 ⟩ , |V1 ⟩ , |V2 ⟩ basis as:


⟨HH| ρ̂ |HH⟩ ⟨HH| ρ̂ |V H⟩ ⟨HH| ρ̂ |HV ⟩ ⟨HH| ρ̂ |V V ⟩
 ⟨HV | ρ̂ |HH⟩ ⟨HV | ρ̂ |V H⟩ ⟨HV | ρ̂ |HV ⟩ ⟨HV | ρ̂ |V V ⟩ 

ρ̂ = 
 ⟨V H| ρ̂ |HH⟩ ⟨V H| ρ̂ |V H⟩ ⟨V H| ρ̂ |HV ⟩ ⟨V H| ρ̂ |V V ⟩ 
⟨V V | ρ̂ |HH⟩ ⟨V V | ρ̂ |V H⟩ ⟨V V | ρ̂ |HV ⟩ ⟨V V | ρ̂ |V V ⟩
(19)
We can expand this matrix as a weighted sum of the equivalent 2-qubit Pauli matrices, that is:
ρ̂ =
1X
ri,j σij
4
i,j
where σij = σi ⊗ σj , for example:
 1
1 0
1 0
1 0
σ00 = σ0 ⊗ σ0 =
⊗
=
 1
0 1
0 1
0
0
and
18
 
0
1 0
1
0
 0
1
0
1
 = 
0
1 0  0
1
1
0 1
0
0
1
0
0
0
0
1
0

0
0
,
0
1
 0
1 1
1 0
0 1
σ0x = σ0 ⊗ σx =
⊗
=
 0
0 1
1 0
0
1
 
1
0 1
0
0
 1
0
1
0
 = 
1
0 1  0
1
0
1 0
0
1
0
0
0
0
0
0
1

0
0

1
0
and so on for the rest of the 16 matrices. Just like last week we will rely on performing a set of
projective measurements to evaluate the density matrix, except now there will be a projection on
each arm. A few examples of these conditions are shown in the table below:
µ
µ̂
θQWP
(1)
θLP
(1)
θQWP
(2)
θLP
(2)
⟨ρ̂⟩µ
cHH
cHV
cHR
cHL
cH+
cH−
cV H
..
.
|HH⟩ ⟨HH|
|HV ⟩ ⟨V H|
|HR⟩ ⟨RH|
|HL⟩ ⟨LH|
|H+⟩ ⟨+H|
|H−⟩ ⟨−H|
|V H⟩ ⟨HV |
..
.
0°
0°
0°
0°
0°
0°
0°
..
.
0°
0°
0°
0°
0°
0°
+90°
..
.
0°
0°
+45°
+45°
+45°
+45°
0°
..
.
0°
+90°
+90°
0°
+45°
-45°
0°
..
.
⟨HH| ρ̂ |HH⟩
⟨V H| ρ̂ |HV ⟩
⟨RH| ρ̂ |HR⟩
⟨LH| ρ̂ |HL⟩
⟨+H| ρ̂ |H+⟩
⟨−H| ρ̂ |H−⟩
⟨HV | ρ̂ |V H⟩
..
.
Processing the obtained counts values is slightly more complicated than the 1-qubit case (see for
example Ref. [1]). One way of doing the calculations is to first define, for convinience, normalized
coincidences counts fij defined as:
fij =
cij
cij + ci,j⊥ + ci⊥ ,j + ci⊥ j⊥
where i⊥ and j⊥ correspond to the orthogonal eigenvectors to i and j for each basis. For example:
fH+ =
cH+
cH+ + cH− + cV + + cV −
After that, we can define a Tmn matrix that will tell us the weight of each of the two-qubit
Pauli matrices. The evaluation of the matrix elements is given by:
Tmn =
X
(−1)pi +qj fij
i,j
where pi and qj are parity indices that depend on the specific basis. They equal zero for the H, +, R
cases, and they equal 1 for the V, −, L cases. However, in the cases when σ0 is involved, the values
of these indices are overwritten as zero (mathematically, they are multiplied by Kronecker deltas
δm0 and δn0 ). It is easier to provide a few examples to clarify these rules. For instance:
Tzx = (−1)0+0 fH+ + (−1)0+1 fH− + (−1)1+0 fV + + (−1)1+1 fV −
= fH+ − fH− − fV + + fV −
19
(20)
In the cases when one of the matrices is σ0 , we can average over the null index through the
other potential bases:
1 (0)
(0)
(0)
Tz0 = (Tzx
+ Tzy
+ Tzz
).
3
(0)
where Tij refers to the evaluation of the matrix element with the corresponding overwritten values
of the indices, for instance:
(0)
Tzx
= (−1)0 fH+ + (−1)0 fH− + (−1)1 fV + + (−1)1 fV −
= fH+ + fH− − fV + − fV −
(21)
(0)
Tzy
= (−1)0 fHR + (−1)0 fHL + (−1)1 fV R + (−1)1 fV L
= fHR + fHL − fV R − fV L
(22)
(0)
Tzz
= (−1)0 fHH + (−1)0 fHV + (−1)1 fV H + (−1)1 fV V
= fHH + fHV − fV H − fV V
(23)
In this formulation T00 = 1 always. Then, the density matrix can be written as:

Tx0 + Txz−iTy0−iTyz
Txx−iTxy−iTyx−Tyy
T00 + T0z + Tz0 + Tzz
T0x−iT0y + Tzx−iTzy
1 T0x + iT0y + Tzx + iTzy
Txx + iTxy−iTyx + Tyy Tx0−Txz−iTy0 + iTyz 
T00−T0z + Tz0−Tz

ρ̂ = 

T00 + T0z−Tz0−Tzz
T0x−iT0y−Tzx + iTzy 
4 Tx0 + Txz + iTy0 + iTyz Txx−iTxy + iTyx + Tyy
Tx0−Txz + iTy0−iTyz
T0x + iT0y−Tzx−iTzy
T00−T0z−Tz0 + Tzz
Txx + iTxy + iTyx−Tyy

It is clear that this numerical recipe is too cumbersome to be carried out by hand in a reasonable
time, so we will rely on a python code which we will provide you to do this evaluation that takes all
36 projective measurements as an input. It is also worth mentioning that error propagation would
be significantly more complicated than in the 1-qubit case. Moreover, assuring that the evaluated
density matrix satisfies all of the requirements for it to be physical in the presence of noise in the
data also requires special care[2].
Setting up the motorized mounts
We will need a total of 4 motorized mounts for our measurement, two for the LPs and two for the
QWPs. As of now each quED module can only control two mounts, so we will need to remotely
control two modules to carry our our measurement.
As a first task, make sure you have all the necessary components for the setup, turn on both
quED modules, verify they are online and take note of the IP addresses of each module.
Use the check-list in your worksheet to go over all of the required hardware and make sure
they are all working as expected. One of the LPs is already mounted. Your instructor/TA will
demonstrate how to mount the other LP and you will have to mount both of the QWPs.
20
Once everything is mounted, you will have to verify that you are capable of remotely controlling
each of the components. We will provide a working Python method that takes an IP, channel #,
and the required rotation as input. First, verify that there is an established connection between the
controller and the mount by testing a few rotation positions. After that, carry out the calibration
procedure for each mount.
Calibration procedure:
1. ”Home” the mount using the quED module, wait for the double click.
2. Disconnect the ethernet connection to release the mount.
3. Rotate to desired position, θ = 0 as measured from the horizontal (adjusting by eye is enough).
4. Reconnect to relock the mount
Once all of the four mounts are being controlled remotely, we will now mount them on the
setup. Fig. 3.1 below shows how the mount position should look like. Notice that the arrangement
is quite tight and there is not much leeway between the mounts and either the fiber couplers or
the mirror mounts. You will have to be careful to avoid bumping any of the other elements in the
arrangement. The center of gravity of the mounts makes it so that they tumble by themselves, so
you need to hold them steadily with your hand until they are properly bolted down.
Figure 3.1: Layout of the motorized mounts for the two-photon tomography measurements
Once everything is set, you will need to finish up a method in Python that will take the
measurement. We will provide a working code that takes a measurement already, so you only need
to create two nested loops that parse through the 36 possible configurations.
We will also rely on python code to take care of the analysis of the data. You will measure 36
combinations, consisting of HH, HV, RR, RH, +H, +−, V+, etc. HH corresponds to measuring
both arms in the horizontal and horizontal basis, RL corresponds to measuring an arm in the right
basis and the other in the left basis, −− corresponds to measuring both arms in the minus diagonal
basis.
21
Shorthand
HH
VR
R−
++
V−
...
...
Coincidences
5000
4923
1100
...
...
...
...
Measurements:
• Find the density matrix for the photon pairs without the HWP in the laser cavity.
• Find the density matrix for the photon pairs with the HWP in the laser cavity.
• Find the density matrix for the photon pairs with the HWP in the laser cavity, but flipped
around.
• Determine what additional optical element and orientation you would need to use to get the
other Bell states. Ask your TA/instructor for it and take a two-photon tomography to verify
you got the remaining Bell states.
• Now try that same element at a different orientation
• If you want, try any other optical element you would like to put in the path to check what
its effect is on ρ̂
3.3
Singe Photon Detection: Hanbury Brown and Twiss
The Hanbury Brown and Twiss (HBT) experiment in 1956 [3] demonstrated single photon detection
by measuring intensity correlations between two detectors. This correlation function is commonly
referred to as the second order correlation function— notated as g (2) (τ )— and it is commonly
used to quantify the performance of single photon emitters. For a classical field (one that is fully
described by Maxwell’s equations) the correlation function is defined as:
g (2) (τ ) =
⟨I(t)I(t + τ )⟩
⟨I(t)⟩ ⟨I(t + τ )⟩
(24)
where I(t) refers to the intensity of the incoming beam and ⟨·⟩ refers to performing a time average.
If we further assume that the light source is stationary—meaning that although it might have some
fluctuations, its overall statistics do not change as a function of time—then we can re-interpret the
time averaging as an ensemble averaging over many repetitions. For these kinds of experiments
using single-photon sources, however, one is usually looking at the output of a detector counter
instead of a power- or an intensity-meter. Given that the number of photons hitting a detector
should be proportional to the intensity of the beam, we could also rewrite the expression for the
correlation as:
g (2) (τ ) =
⟨n(t)n(t + τ )⟩
⟨n(t)⟩ ⟨n(t + τ )⟩
22
(25)
where n(t) is the number of registered counts as a function of time. In our experiments, we will
only look at g (2) (τ = 0). The physical interpretation of g (2) is that it provides us with a number
that characterizes the probability p of detecting multiple photons within an specific time window
(or, for the τ = 0 case, on two events separated by an infinitesimal time). It is worth mentioning,
however, that it is not just a probability in itself, given that it can be larger than 1. Rather,
g (2) (τ ) evaluates that probability p(τ ) when normalized (think compared) with the probability of
observing coincidences by randomly choosing a different time window, pr , that is:
g (2) (τ ) ∼
p(τ )
pr
(26)
Making sense of g (2) (τ ) heuristically
From a classical perspective, we should strictly have g (2) (τ = 0) ≥ 1. The field can either have
some fluctuations (positive correlation, g (2) > 1), or no fluctuations at all (no correlation, g (2) = 1).
The case of observing anti-correlations, that is g (2) < 1, can only occur in a quantum picture. An
example of a non-classical field is one containing exactly one photon only. In this case, g (2) = 0
exactly. This is a reflection of the statement that if we measure a photon at some time, the chance
that we will observe any other photon related to the same emission process is simply zero, because
there was only one photon to begin with, and they are indivisible. For those cases, the light is
described as being “anti-bunched”, or following sub-Poissonian statistics. If we plot a time trace
of the output signal of a single photon detector for these different cases, it would look like Fig. 3.2
below
Figure 3.2: Schematic of the different cases for g (2) (τ )
23
For the g (2) (τ ) < 1 case, given a detection, the probability of having one right after such event
is negligible, p ∼ 0. For a given integration time, there will be Nc counts out of a total of Nt
windows, such that pr ∼ Nt /Nc . Regardless of the value of pr , the fact that p ∼ 0 “quenches” the
value of g (2) to something close to zero and thus g (2) (τ ) < 1.
For the g (2) (τ ) = 1 case, given a detection, the probability of having one right after such event
is just p. Without calculating such value, given that there are no correlations at all between photon
counting events, we should have that pr = p, that is, the likelihood of hitting a coincidence right after
is just as good as that of trying at any other random time. In this case, g (2) (τ ) = p/pr ∼ p/p = 1
necessarily.
For the g (2) (τ ) > 1 case, given a detection, the probability of having one right after such event
is very high, p ∼ 1. Regardless of the value of pr , we know that it should be less than one, and
thus g (2) (τ ) ∼ 1/pr > 1.
An analogy for g (2) (τ ) using dice
It is possible to draw a nice picture to motivate the physical meaning behind g (2) (τ ) using dice.
The scenario is to imagine that you are organizing a series of particles in an empty lattice and are
deciding what kind of spacing to have between each of them. For this, you will be rolling a die
(1-6) to determine the amount of empty cells that you will be leaving between each particle. For
simplicity, let’s assume that there are six times as many lattice sites (Nt ) as there are particles Np .
In analogy to the definition of g (2) (τ ), we could calculate a g (2) (x) correlation index that tells you
how likely these atoms are to be right next to one another, a kind of normalized nearest-neighbor
correlation function:
g (2) (x) ∼
p(one particle right after another)
p(choosing a spot at random and finding a particle)
Figure 3.3: DIfferent probability distribution for dice analogies with g (2) (τ )
24
(27)
For the first case, let’s imagine that you are using a loaded dice such that it always lands on a 6.
Because of this, the probability density function (PDF) of the particle-particle distance f (d) = δd,6
as shown in Fig. 3.3a. Because of this, p = f (1) = 0. Given the description, pr = Np /Nt = 1/6.
Regardless of this, g (2) (x) = 0 and thus we can identify this with the ”anti-bunched” case. Notice
that in this case the distribution looks like a lattice in space, while in the case of photons the
uniform distribution occurs over detection time intervals.
As a second case, let’s now assume that we are using a regular, fair dice. Accordingly, the
corresponding PDF is uniform with f (d) = 1/6 (see Fig. 3.3b) and thus p = f (1) = 1/6. In this
case g (2) (x) = 1 and so we associate this with the laser-like source that is completely uncorrelated.
As the third and final case, let’s assume that the dice is unfair and it is loaded towards the lower
numbers. There are a few distributions that we could use but for simplicity let’s assume a linearly
decreasing distribution (see Fig. 3.3c). In this case, f (d) ∼ (6 − d)/15, and thus p ∼ f (1) = 1/3.
Given this, g (2) (x) = (1/3)/(1/6) = 2 and thus we have a one-to-one correspondence to thermal
light. Other distributions of the unfair dice could in principle be associated with more unusual
correlation states such as n-photon states.
g (2) (τ ) for different sources of light
First, let’s look at the value of g (2) (τ = 0) for a coherent source of light like a laser. In that case,
the number of detected photons is proportional to the intensity. Even if there are some overall
fluctuations in the intensity, in principle they should be mean-zero variations and be independent
from one another, and thus on average the value of n(t) will be constant. In the language of
correlations, this uniformity corresponds to the idea that even if we fully characterize the intensity
profile of the source at time t, this would not give us any useful information to predict when will
the next photon arrive. In that case, g (2) (τ = 0) = 1, the light source is coherent, and it follows
Poissonian statistics.
In particular, we can relate the operating principle of a laser with g (2) (τ = 0) = 1, in that the
process of constantly sustaining stimulated emission in a medium will effectively act as a feedback
mechanism. In the case on which some positive time correlations develop, this will momentarily
“deplete” the medium, leading to a lower rate of population inversion and thus a weaker field right
after this event. The event of some negative time correlations developing will be a mirror case of
this scenario, and thus overall the laser will maintain its g (2) (τ = 0) = 1 character.
For other classical sources of light like the incandescent filament of a light bulb, sometimes
referred to as “chaotic” or “thermal” light, it is possible to have correlated intensity fluctuations
on the source. In those cases, there is a tendency for detection events to be sometimes closer to
each other: given that a detection occurred at t, there is a higher probability of another detection
occurring close to it. In this case, g (2) (τ = 0) ≥ 1, the light source is incoherent, the photons
are said to be “bunched”, and it follows super-Poissonian statistics. In fact, for thermal light,
g (2) (τ = 0) = 2.
While we can draw a nice mental picture for the values of g (2) for a laser and a single-photon
source, explaining why g (2) (τ = 0) = 2 for a thermal source is a little subtle. In fact, if one
simply imagines photons as bullets being emitted from the source, it might even seem a little
counterintuitive that a thermal, “chaotic” source consisting of a great number of emitters that are
uncorrelated both in time and phase would actually result in photons bunching with one another.
Should not the thermal, random case be the one where we are not able to discern any pattern
in the arrival of simultaneous photons? This apparent paradox is arising because the picture of
25
photons as simple hard-core particles is simply not correct. Despite their individualized particlelike properties, photons also display wave-like properties in that they can have relative phases and
interfere with one another. The cases of fields with only one photon or with fields where all the
photons are in-phase allow us to draw a simple picture from the point of view of superposition, but
the thermal case is the hardest one to deal with from that perspective given that all the involved
phases are random, and because of that all of them need to be considered in the calculation without
much room for simplifying assumptions.
Despite this, a useful picture that leads to the intuition behind g (2) (τ = 0) = 2 for a thermal
source consists in remembering that after a laser is passed through a strongly scattering medium
such as frosted glass, a “speckle” pattern will emerge. This granulated image will consist of a
distribution of patches of light of different extents, but with a characteristic length scale that is
Fourier-coupled with the roughness length scale of the scattering surface. Then, we can imagine a
photocount measurement as a small cross-section that is “plowing” through this speckle pattern.
Whenever a photon is detected, we are passing through one of the patches, as so the likelihood
of finding another photon “close-by” is high— hence g (2) (τ = 0) > 1. Although this picture is a
little hand-wavy, some experiments[4] have actually used elements such as a rotating frosted-glass
to create a “pseudo-thermal” light source out of a laser for which g (2) (τ = 0) = 2.
g (2) (0) as a function of photocounts
In our experiments we will be calcualting the second-order correlation function as ratio between
counts or coincidences counts. Above we saw how at least on the classical formulation g (2) (0) was
a ratio between the average of the intensity at two points in time and a “normalization” that takes
into account the intensity for each of those events. Quantum-mechanically we would express the
correlation function as averages over creation and annhiliation operators, although they too will
result in photon-number operators. Given that we can relate intensity with the number of photons,
and the number of photons with the probability of them being transmitted or reflected through our
beam splitter, we can also express g (2) as a ratio of probabilities.
First, we can write down a relationship between the number of counts Ni for a particular event
(say, a reflection or a transmission) and the probability of that event occurring, pi , by noting
that the photodetector will be making a collection over an integration time τi , and that each
count/coincidence measurement occurs within a detection time ∆T W . Then, the total number of
potential detection windows is given by Nw = τi /∆T W , and we can then write:
Ni = pi · Nw = pi
τi
∆T W
and thus we can use the measured number of counts/coincidences to back-calculate the event
probability given that the integration time is user-defined and the detection window is a constant
for the photon-counter (in our case, ∆T W = 30 ns). In particular:
pi = Ni ·
∆T W
τi
For the unheralded case, where we are just monitoring the coincidences between those photons
that are transmitted (NT ) by the BS and those that are reflected (NR ), we can write:
26
g (2) (0) =
pRT
NRT
τi
=
·
.
pR · pT
NR NT ∆T W
The heralded case is slightly more complicated in that we now also need to consider the event
of a triple coincidence between reflection, transmission and heralding, NHRT . Then, we can write:
g (2) (0) =
pHRT
pH · pR · pT
the trick to obtain a simple expression is to re-express the product the probabilities of the three
events as:
pH · pR · pT =
pHR · pHT
pH
and thus we can then obtain:
g (2) (0) =
NHRT NH
.
NHR · NHT
Note that the heralding in this case makes it so that the measurement is independent of both τi
and ∆T W , as the gating effectively takes those variables into account.
HBT Experiment:
1. We will use one arm of the QuED for the HBT; the other can be used to herald. Make sure
all the fibers are clean. Using the fibers, carefully connect one arm to HBT module and both
HBT outputs to the QuCR. Plug the other arm directly into the QuCR.
2. Now we will need to look at the counts. What channels do you need to look at? What values
do these correspond to in g 2 (τ )?
3. Record these channels and determine g 2 (τ )
4. Perform a heralded g 2 (τ = 0)h measurement. Hint: This requires a second photon and a
third APD, see figure 2.3.
3.4
Single Photon Interference
Background
Wave-Particle Duality: In classical physics there are the concepts of waves and particles.
Waves, for example electromagnetic fields, can interfere. The field of two waves add up and the
intensity is the square of the sum. In a Michelson interferometer the optical path difference between
two arms define if the waves interfere constructively or destructively in the output arm (see Fig.
3.5).
Classical particles are well localized at any time and the path of the particle is known (Fig. 3.5).
A particle can only take one of the two possible paths through the interferometer. Therefore the
27
Figure 3.4: Schematic for a Heralded HBT Measurement
Figure 3.5: Michelson Interferometer
particle intensity in the output arm of the interferometer can not depend on the length difference
of the two interferometer arms a and b.
A photon is the smallest energy packet of the electromagnetic field. Many experiments demonstrate interference of electromagnetic fields and therefore even weak electromagnetic fields should
interfere. But photon anti-bunching and the anti-correlation of photon pairs anticipate a particle like behavior of photons. Grangier et al [5] showed that single photons can interfere. They
demonstrated photon counting statistics which are ”‘in contradiction with any classical wave model
of light”’ and interference of these photons in a Mach-Zehnder interferometer. Their experiment
proved that a photon can not be described by the classical ”‘wave”’ or ”‘particle”’ concept.
Interference:
by [6]:
The electric field of a plane wave propagating in the z-direction can be represented
Ek (z, t) = Re{Aei(kz−ωt) }
(28)
Consider two monochromatic waves:
E1 (z, t) = Re{Aei(kz−ωt) }
i(kz−ωt+∆ϕ)
E2 (z, t) = Re{Ae
28
(29)
}
(30)
It can be shown that the time-averaged intensity of the summed fields is:
Isum = ⟨E1 + E2 ⟩2t = I0 · (1 + cos(∆ϕ))
(31)
The visibility of the interference fringes is defined as [7]:
V =
I max − I min
I max + I min
(32)
Michelson Interferometer: The beam splitter in a Michelson interferometer divides an incoming wave into two arms. The two waves are reflected by mirrors and overlap on the beam splitter.
The phase difference ∆ϕ = k · d of the two waves depends on the optical path difference between
the arms d and the wave number k. The optical path difference can be controlled effectively by
varying the refractive index in one arm of the interferometer.
Coherence Length: The photons emitted from the quED are not plane waves, but wave packets.
The spatial extent of the photon wave packet is characterized by the coherence length Lc . If the
optical path difference is varied within this coherence length, interference fringes will be observed.
However, outside of the coherence length, no fringes will be observed. The intensity pattern is
given by [6, 8]:
I(d) = I0 · (1 + cos(2πd/λ)) · exp{−|d/Lc |}
(33)
where the exponential decay in the visibility V (d) = exp{−|d/Lc |} measures the degree of
coherence, so-called g (1) . Additionally, the coherence length gives a coherence time τc = Lc /c,
which characterizes the uncertainty in the photon energy [7]:
∆E ≈ h/τc
(34)
Setup
Figure 3.6: Experimental Layout
A schematic view of the setup can be seen in Fig. 3.6a (note that we are using a manual
version). To connect the Michelson interferometer (Fig. 3.6b) with the quED, one of the output
fibers of the quED is connected via a fiber adapter to the input fiber of the interferometer. The
second output fiber of the quED has to be extended as well to balance out the path lengths (the
29
quCR will not register any coincidences otherwise). The extended fiber and the output fiber of the
interferometer is connected to the APD unit of the quCR. The polarizers in the interferometer are
not necessary for this experiment and have to be swung out of the beam path.
Alignment: Before the experiment is performed, the alignment of the mirrors in the interferometer should be checked. For that, block one path of the interferometer with a piece of paper. Adjust
the set-screws of the mirror mount in the open path to reach the count rate maximum. Repeat for
the other path to reach approximately the same count rate in both arms.
Path Length Control: With the quED-MI, the optical path lengths are controlled via a glass
wedge and not via the mirror position (Fig. 3.7). The optical path length of light in the glass
wedge is extended because of its higher refraction index in comparison with the distance in air.
The thicker the wedge at position x of the beam, the bigger the delay δ:
δ = 2 · x tan(α) · (nbr − 1)
(35)
where α is the wedge angle and nbr is the refractive index of the glass (Fig. 3.7). It is important
that the effect is doubled, because the beam passes through the wedge twice (thus the factor of
2). There is a slight change of beam angle due to the wedge, but it is constant with respect to
the effective thickness and therefore does not disturb the coupling. The retardation that stems
from the minimal thickness of the wedge is constant and is neglected. Constant retardations are
compensated by the mirror positions and the second, fixed wedge in the other arm of the setup.
Figure 3.7: Interferometer Wedge
Procedure
The interference fringes will be observed when the two arms have nearly equal path lengths. Begin
by coarsely sweeping the wedge through the beam path, looking for oscillations in the coincidence
counts. Once found, use the micrometer to precisely map out the visibility of the fringes.
30
3.5
Quantum Eraser
From last week’s tomography measurements, we know that the photons have a polarization state.
So far, the Michelson interferometer has been unaware of this information. Next, we will study
what happens to the interference of photons when the setup ”looks” at their polarization.
Setup
The experimental configuration begins the same as in the Single Photon Interference measurements.
The quED-MI is designed in such a way that the polarisation of the photons at the input of the
interferometer is diagonal (i.e. 45 degrees between horizontal and vertical). We will place polarizers
(shown in Fig. 3.6b) into the beam paths of both arms, as well as into the output path.
Procedure
Search for interference fringes under three different configurations:
1. No polarizers in any path (just reconfirm the results from your previous measurements).
2. Introduce the horizontal polarizer in path a and the vertical polarizer in path b (Fig. 3.8a).
3. With these polarizers in place, add the diagonal polarizer to the output path (Fig. 3.8b).
Figure 3.8: Quantum Eraser
3.6
QuED Applications References
1.
James, D. F. V., Kwiat, P. G., Munro, W. J. & White, A. G. Measurement of qubits. Phys.
Rev. A 64, 052312 (2001).
2.
Smolin, J. A., Gambetta, J. M. & Smith, G. Efficient method for computing the maximumlikelihood quantum state from measurements with additive gaussian noise. Physical review
letters 108, 070502 (2012).
3.
Hanbury Brown and Twiss. Correlation between photons in two coherent beams of light.
Nature 177, 27 (1956).
31
4.
Arecchi, F., Gatti, E. & Sona, A. Time distribution of photons from coherent and Gaussian
sources. Physics Letters 20, 27–29 (1966).
5.
Grangier, P., Roger, G. & Aspect, A. Experimental Evidence for a Photon Anticorrelation
Effect on a Beam Splitter: A New Light on Single-Photon Interferences. Europhysics Letters
1, 173. https://dx.doi.org/10.1209/0295-5075/1/4/004 (1986).
6.
Loudon, R. The quantum theory of light (OUP Oxford, 2000).
7.
Jelezko, F., Volkmer, A., Popa, I., Rebane, K. K. & Wrachtrup, J. Coherence length of photons
from a single quantum system. Phys. Rev. A 67, 041802. https://link.aps.org/doi/10.
1103/PhysRevA.67.041802 (4 2003).
8.
Langangen, Ø., Vaskinn, A., Skagerstam, B.-S., et al. Interference of light in a MichelsonMorley interferometer: A quantum optical approach. International Journal of Optics 2012.
Also found on arXiv:1102.4324 (2012).
9.
Wootters, W. K. Entanglement of formation of an arbitrary state of two qubits. Physical
Review Letters 80, 2245 (1998).
10.
Coffman, V., Kundu, J. & Wootters, W. K. Distributed entanglement. Physical Review A 61,
052306 (2000).
32
3.7
Analysis Questions for the report
A) Photon-pair Tomography
For the following questions, you are free to use python packages such as Qutip.
1. Choose any two of the two-qubits Pauli matrices σij (other than σ00 and σ0x ) and write them
down explicitly as 4 × 4 matrices.
2. Using your data, determine ρ̂ for each of the measurements you performed. Try displaying ρ̂
in a 3D bar plot.
3. Calculate Tr(ρ̂) for all cases.
4. Calculate the Von Neumann Entropy for each case: S = −Tr(ρ̂ log ρ̂). Note that if your
measured matrix has negative eigenvalues, you will need to do some data processing. See for
example this paper by Smolin et al. [2].
5. Do a partial trace for the measured density matrix corresponding to |Φ+ ⟩ on each qubit. What
is the reduced density matrix for each case? Do these differ depending on which photon you
trace out?
6. Where you able to observe all Bell states? Describe the hardware that you had to add/change
on each case to go from |Φ+ ⟩ to the other states, and explain to which quantum gates
(X,Y,Z,H, etc) do each of these changes correspond.
7. Calculate the fidelity of your measured ρ̂ for each case with respect to all the four Bell States
and express it as a percentage. In our case, F (ρ̂, σ̂) = Tr(ρ̂σ̂), where σ̂ is the density matrix
for one of the Bell states.
8. Extra credit [Linear algebra, 6 pts]. For each of your measured density matrices, calculate
their concurrence, their tangle, and their entanglement entropy[9, 10].
B) Singe Photon Detection: Hanbury Brown and Twiss
1. What value of g 2 (τ = 0) would you expect to measure for the UV pumping beam? What
value of g 2 (τ = 0) did you get for the non-heralded down-converted photons? Explain the
relationship between these two in terms of SPDC.
2. What value of g 2 (τ = 0) did you measure for the heralded case? Explain whether this
measurement is sufficient to confirm or deny if you are working with a single-photon source.
3. Do your results allow you to state whether we have a single-photon source? What kind of
protocol do you need to follow to use the SPDC source as a single-photon source?
C) Single photon interference
1. During the measurements, the HWP before the BBOs was removed, such that the photons
you were using in the interferometer were not entangled. Explain why you don’t need the
photons to be entangled to carry out this experiment. Would you get the same results if they
were entangled?
33
2. Attach a plot of the single-photon interference signal that you observed showing at least one
full period of constructive and destructive interference, along with the fit that you are using
to analyze your data.
3. What is the maximum visibility you measured? Express it as a percentage.
4. What is the photon wavelength that you can extract from this data? How does it compare
to what you were expecting. Try to estimate an error in the wavelength calculation.
5. Attach a plot showing the decaying envelope in the interference. Describe how did you analyze
the data to extract the coherence length of the photons.
6. What is the coherence length and coherence time that you measured?
7. What is the corresponding photon bandwidth (∆λ)? Note: You will need to use the equation
E = hc/λ to evaluate ∆E as a function of ∆λ. It is a common mistake to think that ∆E =
hc/∆λ, although that is not the case! What is the right expression for ∆E?
8. Extra credit [Quantum Optics, 5 pts] Explain what would happen to the interference pattern
you observed if you introduce a polarizer on each arm, but crossed with each other (H vs V)?
What would happen if you introduce a polarizer at 45 o right at the output of the BS?
34
4
QuED Experiments: Week 4
4.1
Two-photon Interference
Background
For the single photon impinging on a balanced non-polarizing beam splitter there is an equal
chance of being reflected or transmitted. If two photon-counting detectors are positioned at the
output ports of the beam splitter, the photon is registered at one or the other detector with equal
probability, but never in coincidence at both detectors. The observed perfect anti-correlations
between the detection events provide a simple experimental test of the photon indivisibility (i.e.,
the HBT measurement we performed in week 2).
Considering the experimental scenario where two photons simultaneously enter the input ports
of the beam splitter, one would expect four possibilities for the output: both are transmitted (Fig.
4.1a), both are reflected (Fig. 4.1b), and one is transmitted while the other reflected (Fig. 4.1c,d).
The detectors at the beam splitter outputs should thus register half of the photons in coincidence
(Fig. 4.1a,b), while the other half is not detected in coincidence (Fig. 4.1c,d).
Figure 4.1: Hong-Ou-Mandel Effect
Photon Indistinguishability This intuitive explanation however fails to account for two-photon
interference [1], which occurs if the two photons are indistinguishable in all degrees of freedom, i.e.
the photons have the same wavelength, polarization and spatio-temporal mode. In such a case the
first two possibilities - when both photons are transmitted or reflected - cannot be distinguished
from one another because there is always one photon in either output mode. As a result, the two
possibilities are coherently superposed. Due to unitarity of the beam splitter transformation there
is always an overall π-phase shift between the two possibilities, canceling each other completely.
The involved π-phase shift is universal and independent of a specific beam splitter, and therefore
the described destructive interference appears for any practical realization.
For the remaining possibilities, the photons always exit the same output port of the beam
splitter. This manifests in the absence of the coincidence counts. By scanning the relative time
delay between the photon arrivals at the beam splitter (Fig. 4.1e), the degree of temporal distinguishability of the two photons is effectively changed, and therefore a dip in the coincidence
count rate can be observed (i.e., the so-called Hong-Ou-Mandel dip [2]). For zero delay, the perfect
photon overlap in the time domain is ensured, and the coincidence rate should theoretically drop
to zero. This is however never the case in practice because of experimental imperfections such
as the the deviation from the ideal 50:50 beam splitter splitting ratio, imperfect spatial-mode, or
polarization-mode overlap. Therefore the dip with a limited visibility is always recorded experimentally. The measured visibility corrected for the imperfect splitting ratio gives a direct measure
of indistinguishability of the input photons. A dip below 50% confirms photon indistinguishability.
35
Unlike the interference effects in conventional Mach-Zehnder or Michelson interferometers, the
Hong-Ou-Mandel effect does not require the phase stability of the interferometer arms. The path
differences of the arms need not be kept constant to within a fraction of wavelength, but only to
within a fraction of the photon coherence length. From a practical point of view the Hong-OuMandel effect can be viewed as a method to gauge the femtosecond time intervals (corresponding
to micrometer length scales) between the photons and by implication the length of the photon wave
packets.
Setup
As in Fig. 4.2, the output fibers of the quED are connected to the inputs of the quED-HOM. The
outputs of the Add-On are connected to the detectors of the control and readout unit quCR. In the
motorized version, the motor driver qu3MD has to be connected via USB to the quCR, the motor
itself plugged into site 3 of the qu3MD.
Figure 4.2: Experimental Layout
Alignment
To realign the fiber-to-fiber coupling in both free-space lines for maximum coupling:
1. Make sure that the quED setup is as shown in Fig. 4.2a with the source wave plate removed
and that the count rates are high.
2. Block one of the free space lines and check the measured single count rates. Optimize the
coupling by carefully tuning both adjustment screws of the mirror holder on one side of the
unblocked free space line. The single counts rates should be approximately at 25000-30000
counts per second in both detectors.
3. Repeat with the other free space line.
4. With both lines unblocked, you should measure between 1000 and 2000 coincidences per
second.
Procedure
The search of the interference dip is performed by changing the length of the free-space line using
the linear translation stage and observing changes in the measured coincidence count rate. As soon
as a significant drop in the coincidence count rate is observed, the two interferometer arms are
adjusted to have the same length and the interference dip is found. With the motorized version,
36
the linear translation stage is controlled by the quCR. For this, the ”‘linear scanning”’ Tab is used.
For an initial coarse search, a step size of a few microns can be used.
4.2
Random Number Generation
A single photon at the push of a button remains a subject still being researched heavily. However,
weak coherent pulses are an easy approximation, sufficient for our application. Weak coherent
pulses are just weak laser pulses. As such, the photon numbers in one pulse obey the Poisson
distribution, where the probability of k photons in one pulse is given by:
Pk = e−λ
λk
k!
(36)
with the average photon number λ.
To generate and gauge weak coherent pulses, switch to the pulsed laser mode in the laser tab
and activate the Picture-in-Picture (PiP) Overlay, such that you can modify the pulse settings from
anywhere in the quCR software. A single pulse is specified by its amplitude and its duration, and
you can modify the frequency with which the pulses are generated.
Switch to the count rate panel to see the signals from the APDs. Please note that (while the
sync checkbox is active) only counts happening during a laser pulse are displayed, but they are still
integrated over the integration interval. So, if you set the frequency to 1000 Hz and the Integration
time to 100ms, you can see how many photons are detected during 100 pulses. You can also display
the average number of photons per pulse if you activate the per pulse checkbox. The average
number of photons per pulse can then be adjusted by changing the pulse duration and the pulse
amplitude. You can also try what happens when the laser is switched off.
The randomness of photon detection events can be used to generate random sequences of bits.
The setup is the same as in the HBT measurement from week 2. The photon detection events,
however, are analysed in a different way. Usually, we count rates of detected events: singles and
n-fold coincidences. Here, we only look for coincidences between detector 0 and detector 1 or 2.
When detector 0 and 1 register a photon at the same time (within the coincidence time window),
we attribute a binary 0 to it, when detector 0 and 2 click simultaneously, a binary 1 is produced.
Following this protocol, the RNG module will create a stream of zeros and ones. You can choose
portions of the sequence to serve as Alice’s and Bob’s respective random lists.
4.3
Quantum Cryptography
Quantum cryptography has emerged as a promising discipline for applying quantum physics to
the secure communication and processing of information [3–5]. Here, we will use single photons
to create shared quantum keys, which cannot be secretly stolen by an eavesdropper, according to
the Bennett and Brassard BB84 protocol [6]. Using these keys, messages - such as, “Quantum Lab
rocks!” - can be securely communicated.
Background
Key Distribution with Photons Conventional cryptography schemes that are founded on a
secret key are only then completely secure when each key is as long as the message itself and used
37
only once. Thus, the encryption problem shifted to the exchange of a secret key. At the moment,
mostly asymmetric schemes (e.g. RSA) using a public key (for encryption) and a private key (for
decryption) are being employed. The secureness relies essentially on the impossibility to factorize
a large number in its prime factors. With faster computers, the possibility of a quantum computer
or newly found algorithms, the security of these conventional systems is diminishing.
With the BB84 protocol, named after its inventors and the year of publication (Bennett and
Brassard, 1984 [6]), it is possible to use the quantum physical properties of photons to transfer a
secret key between two parties, tap-proof. When that has happened, the message can be encrypted
and sent via an open classical channel. Because of that, the term quantum cryptography is actually
misleading, it should be called quantum key distribution (QKD) instead.
Quantum Key Distribution
technique in two ways:
The quantum nature of photons can be leveraged to improve this
1. Photon detection events can generate the random bases/bits sequences initially held by Alice
and Bob.
2. Transmission of single photons makes eavesdropping impossible.
The BB84 Protocol The two parties involved in the secure communication are called Alice and
Bob by convention. Alice operates a source of single photons that she can individually prepare
in a linear polarization state known to her and sends them to Bob. Prior to sending each photon
to Bob, Alice has two choices to make: first, which polarization basis to use - rectilinear (i.e.,
horizontal/vertical) or diagonal (i.e., plus/minus); second, which bit value to send (i.e., horizontal
(0) or vertical (1); plus (0) or minus (1)). Ideally, these decisions should be made randomly. Bob
receives the photons and can choose one basis per photon for a polarization measurement. In doing
so, a meaningful bit is only received when Bob and Alice chose the same basis. Thus, they have
to communicate their basis choice. If Bob detects no photon because of losses in the quantum
channel, the corresponding bit is discarded. The following steps are then done using a classical
(public) channel. Bob sends a list with his measurement bases to Alice. She compares the list to
her own and tells Bob which bases match. All bits with non-matching bases are discarded by both
parties. The remaining bits make up the secret key.
Detection of Eavesdroppers In general, it is assumed that Eve (from eavesdropper) can intercept both the classical communication channel and the photons sent by Alice. So how is it
still possible to exchange a secret key, or rather, to detect the attempt? The BB84 protocol takes
advantage of two important properties of single photons:
• By a single measurement, the polarization state of a single photon cannot be determined fully
(remember tomography!).
• The polarization state of a single photon may not be copied (the so-called No-CloningTheorem [7]).
Eve could, e.g., intercept part of the photons from Alice and perform a polarization measurement,
just as Bob would. But, if Bob receives no photons at the proper time, the bits are just discarded
and Eve gains no information about the key. Therefore, Eve sends one photon onward to Bob in
38
each case, with the polarization measured by Eve. This kind of attack is called an intercept-resend
attack.
Bob and Alice can detect such an attempt by comparing part of the key. That is to say, if Eve
chooses a basis that does not conform with Alice’s and Bob’s, the Bit received by Bob is random.
Thus, it can happen that Bob’s Bit does not match with Alice’s. If some of those ”‘errors”’ are
found, they can assume that the photon channel is compromised and their key is not secure. There
are of course more sophisticated ways of comparing the key than just publishing part of it, called
error correction schemes. Some popular choices are the cascade protocol or the low density parity
check.
The complete BB84 protocol is summarized step-by-step in Fig. 4.3.
Figure 4.3: BB84 Protocol
Setup
Source We will implement this protocol using streams of photons, instead of single photons. The
source should be set up in CW mode without the laser half waveplate inserted, such that only
horizontally polarized photons will be produced.
Polarizers Figure 4.4 displays the layout of the quED. The essential components are the half
waveplate and linear polarizer, which are inserted in one arm of the setup (the other arm will not
be used). Alice will use only the half waveplate to send photons with polarizations chosen from her
basis and bits lists. Bob will use only the linear polarizer, which is controlled separately from the
quCR unit on his side of the lab, to set his detection bases and bits.
Detection For each measurement, Bob must determine the bit value of the detected photons
from the observed count rate. Considering the minimum and maximum count rates, an appropriate
threshold for sorting can be identified - for a count rate below (above) the threshold, the bit value
is 0 (1).
39
Figure 4.4: Two-table Layout
Procedure
Carry out the BB84 protocol as detailed above to send messages between the two tables. The main
steps include:
1. Set the half waveplate and polarizer angles for each transmission. Both Alice and Bob must
keep track of their basis and bits lists.
2. Bob determines the detected bit value from the measured count rate.
3. Share classically. First, Bob sends his bases. Alice then confirms which subset of the bases
list agrees with hers. Lastly, Bob shares a few remaining bits, which Alice confirms to
establish no interference from an eavesdropper (this step may be skipped if you do not fear
eavesdropping). The remaining sequence of bits is the quantum-secure key, which Alice and
Bob hold independently; ideally it is identical, but you cannot compare.
4. Once you have created a key, use it to encrypt, send, and decode a message. The included
python script converts a text string to a sequence of zeros and ones. Depending on Alice’s
key, the sequence is scrambled. This ciphered message is then given to Bob, who can use his
key to decode the message.
4.4
QuED Applications References
1.
Mandel, L. Quantum effects in one-photon and two-photon interference. Rev. Mod. Phys. 71,
S274–S282. https://link.aps.org/doi/10.1103/RevModPhys.71.S274 (2 1999).
40
2.
Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecond time intervals between
two photons by interference. Phys. Rev. Lett. 59, 2044–2046. https://link.aps.org/doi/
10.1103/PhysRevLett.59.2044 (18 1987).
3.
Gisin, N., Ribordy, G., Tittel, W. & Zbinden, H. Quantum cryptography. Rev. Mod. Phys.
74, 145–195. https://link.aps.org/doi/10.1103/RevModPhys.74.145 (1 2002).
4.
Scarani, V. et al. The security of practical quantum key distribution. Rev. Mod. Phys. 81,
1301–1350. https://link.aps.org/doi/10.1103/RevModPhys.81.1301 (3 2009).
5.
Portmann, C. & Renner, R. Security in quantum cryptography. Rev. Mod. Phys. 94, 025008.
https://link.aps.org/doi/10.1103/RevModPhys.94.025008 (2 2022).
6.
Bennett, C. H. & Brassard, G. Quantum cryptography: Public key distribution and coin
tossing. Theoretical Computer Science 560, 7–11. https://doi.org/10.1016%5C%2Fj.tcs.
2014.05.025 (2014).
7.
Wootters, W. K. & Zurek, W. H. A single quantum cannot be cloned. Nature 299, 802–803
(1982).
8.
Bista, A., Sharma, B. & Galvez, E. J. A demonstration of quantum key distribution with
entangled photons for the undergraduate laboratory. American Journal of Physics 89, 111–
120 (2021).
41
4.5
Analysis Questions for the report
A) Two-photon interference
1. During the measurements, the HWP before the BBOs was removed, such that the photons
you were using in the interferometer were not entangled. Explain why you don’t need the
photons to be entangled to carry out this experiment. Would you get the same results if they
were entangled?
2. Attach a plot of the HOM interference pattern that you measured as well as the fit that you
are using to analyze your data.
3. Calculate the average noise that you observed in the baseline. How does this uncertainty
compare to what you would expect assuming Poissonian statistics?
4. Is the dip well described by a Gaussian line profile? Try to evaluate a figure of merit for your
fit such as the coefficient of determination or a chi-square goodness of fit test.
5. What is the relative depth of the dip that you observed? Relate this to the distinguishability
of the photons. Identify a couple reasons that could be reducing the visibility and what
hardware changes would you implement to fix address them.
6. What is the width of the dip? What does this tell you about the photons? Relate this with
to the coherence length of the photons, Lc , and compare it to the estimation of it you did
last week.
B) Random Number Generation
1. Briefly describe the scheme that you used to distinguish the “0” and “1” cases to generate
your own set of random numbers.
2. What is the bias that you measured in your data set?
3. Briefly consult reference NIST’s description of their test suite for random number generators†,
and perform at least 3 test(s) with the python test suite*. Describe what each of the method
is doing and whether your data set passed the test.
† Bassham, L. et al. A Statistical Test Suite for Random and Pseudorandom Number Generators
for Cryptographic Applications. LINK .
* Ang, S. K. & Churchill, S. NIST Randomness Testsuite 2017. LINK
C) Quantum cryptography
1. Describe the setup that you used to simulate Alice and Bob. What is the effective distance
over which the signal was being transmitted?
2. Do Alice’s and Bob’s keys agree? What message did Alice (Bob) send (receive)?
3. Try decoding the message after introducing an error to the key. How different is it?
4. Why is this protocol not ”quantum-secure?”? If you were Eve, how would you attack this
system?
42
5. Describe why this setup is not compatible if we wanted to work in the single-photon case.
What hardware changes would you have to implement to make it compatible with the singlephoton case?
6. Extra credit [Quantum Optics, 6pts]. Read the methods section of a recent paper by Galvez et.
al [8] and explain how do they use a Quartz crystal to simulate the effects of an Evesdropper.
Based on the coherence length and times that you measured using the HOM interferometer,
what would be the minimal thickness of the quartz crystal that you would need for our SPDC
source?
43
5
MOT experiments: Week 5
In this first week of the Atomic Physics block we will spend some time getting familiar with the
laser and spectroscopy components that we will be working with. We will also explore the atomic
level structure of Rubidium, and observe many of its electronic sublevels by doing spectroscopy on
a Rubidium vapor cell.
Laser Safety
The laser equipment that we will be working with will be quite different from the ones we were using
in the QO block. The QuED source was a fully interlocked laser system and its overall output was
just a few thousands of photons. Moreover, all of that light was fiber-coupled and well contained,
so all of the experiments were quite eye-safe and we didn’t even need to wear laser safety goggles.
In this new module, however, we will be using an near IR, class 3B 780nm laser with a linearly
polarized output and has a power 60mW at the outlet. Like the quED pump, this laser is hazardous
even in the case of short specular exposures! Further, because it is near IR, the light is barely
detected by the eye, making perceived light intensities (if any) far lower than their real power. As
a result, if the laser is fully powered, we will be wearing laser safety goggles at all times when the
laser is on. Moreover, all of the light is running free-space through our setup instead of being fiber
coupled, so the laser is much more accessible and prone to being redirected. This is particularly
relevant for us because unlike in many optical setups, the MOT has vertical beam paths in sections
of the setup (in the beam-separator and at the vacuum tube) which make accidental exposure to
the beam more likely. When performing alignment, use white sheets of paper or business cards to
observe the beam. You can also use the IR camera to view the beam and faint reflections the eye
cannot detect.
Because all of this setup is arranged in free-space, you should always be mindful of where the
beam is and make sure your eyes are never in the beam path (even if you are wearing safety goggles).
In addition, you should make sure that you do not introduce your hands along the beam paths of
the MOT. Avoid wearing anything that has metallic surfaces while working with optics, especially
on your hands, such as wrist watches, chains, necklaces, rings and piercings (among others). They
can cause stray reflections of the laser beam into an unsuspecting eye.
Additional considerations
The glass cell where we will be carrying out the laser cooling is being constantly kept at a
pressure of less than a few nTorr (10−9 Torr) to ensure ultra-high vacuum conditions and avoid
background-gas collisions. An ion pump is constantly maintaining this conditions and should remain
ON at all times. The switch for the pump is similar and right next to the coils switch, and you
should be careful to never turn the pump off when turning on the coils.
The mirrors that we will use for stirring the MOT beams are gold-plated to enhance their reflectivity. However, that translates into them being very fragile if directly touched. While you are
in the lab, be mindful of where your arms and hands are at every moment to avoid unintentionally
touching the optical components. Whenever you are handling and mounting bare optical components, you should always be wearing gloves to avoid leaving fingerprints on them. If optics appear
dirty, ask the TA to come in and evaluate if cleaning is necessary. The gold plated mirrors that are
part of the miniMOT should only be cleaned using compressed air.
For similar conditions as those explained in the quED section, you should avoid suddenly disconnecting cables or wires from the laser controllers. In addition, because we will be using a PID
loop, it is important to avoid making very dramatic changes on the laser operating conditions if
44
the feedback loop is engaged. The Vescent laser diode and spectroscopy module are specifically
designed for wavelength tunable excitation of specific Rb transitions. The laser diode is driven
using current flow from the laser diode controller. While changes in the driving current change
the output laser power, they also change the output laser wavelength, allowing us to tune the laser
to relevant transitions. Like the quED setup, reflection of the beam back into the laser diode can
damage the diode and should be avoided. Further, insuring that the temperature controllers are
stable before powering on the laser protects them from potential damage.
Because we will be eventually using an RF source for modulating the laser diode, you should
be careful of always having a load attached to the RF generator. Make sure to verify with your
TA/instructor that the conditions for enabling the generator are appropriate when it is time to use
it on weeks 6 and 7.
The maximum current at which you should operate the laser diode is 110 mA, and running it
above this condition can potentially damage the diode. The controller will limit your range but if
the warning red LED turns on on the laser controller then you should dial the current back down
below the limit.
The source of Rb atoms is a sputtering dispenser. In general operating it at a current of around
3.5 A is sufficient and you should not need to change this amount significantly. If for any reason
you feel like you need to change this numbers, please discuss it first with your TA/instructor.
The Cold-Quanta MiniMOT has a large glass vacuum cell, it is important that nothing ever
come in contact with the glass cell, as if the vacuum were to break, the entire instrument would
need to be replaced. Be incredibly careful when adjusting optics in the proximity of the glass vapor
cell.
5.1
Atomic structure of Rubidium
Rubidium (Rb) is a light-gray metallic element corresponding to atomic number 37. Because
it belongs to the Alkali metals family, it shares most of they general properties like having a
hydrogen-like electronic structure, being very soft, having a relatively low melting point, and being
quite reactive with water∗ . On Earth, naturally abundant Rubidium has two isotopes, 85Rb and
87
Rb, with a relative abundance of ∼75% and ∼25%, respectively† . In its neutral form Rb has 37
protons, 37 electrons, and 48 and 50 neutrons for 85Rb and 87Rb, respectively, making both atomic
species being composite bosons. Its electronic configuration is given by [Kr]5 s1, and so electronic
transitions in Rubidium occur by raising and lowering its outer, single valence electron through the
different atomic levels.
In order to describe the atomic structure of Rubidium, it is important to briefly recall all of
the quantum numbers that we will be using, summarized in Table 1 below. In general, atomic
levels are noted using what is referred to as “spectroscopic notation”, and more or less quantum
numbers are specified depending on the level of detail with which the structure is being described,
as summarized in the Table below.
where n is the principal quantum number, S labels the spin of the valance electron, L corresponds
to the orbital angular momentum, and J is the total electronic angular momentum, which accounts
for the coupling between both the intrinsic and the orbital degrees of freedom: J = L + S. As we
∗
Precisely,2 Rb + 2 H2O
H2 + 2 RbOH. This is an exothermic reaction that generates hydrogen! Therefore,
a Rubidium fire would be really bad and cannot be put out with water.
†87
Rb is technically radioactive, although its half-life is longer than the age of the universe, so it is stable for
human purposes
45
Term Symbol
example
Coarse
Fine
Hyperfine
nL
5S
nLJ
5S1/2
nLJ , F = ...
5S1/2 , F = 2
can see from Rb being on the fifth row of the periodic table and from its electronic configuration,
n=5. At its ground state, this electron will be in an S (L=0), spherically symmetric orbital, and
correspondingly J=1/2. Thus, we label the ground state as 5S1/2 . Full notation is given by:
n2S+1 LJ
(37)
although the 2S + 1 term becomes more useful once we depart from the alkali metal family
and more than once valance electron is involved in the calculations. The natural transition for
Rb corresponds to exciting its valence electron from its ground state to the nearest excited state,
which in this case would be the 5P orbital from a coarse structure point of view. The wavelength
of this transition is near-IR and corresponds to λ = 780 nm. However, because of the coupling
between the spin and the angular momentum that the electron has in the P orbital, 5P splits into
two fine structure levels corresponding to J = 1/2 and J = 3/2. The transition 5S1/2 → 5P1/2 is
traditionally referred to as a D1 line and the transition 5S1/2 → 5P3/2 corresponds to the D2 line,
with a corresponding transition wavelength of 795 nm and 780 nm, respectively. Because the D1
line is too far detuned beyond the tuning range of our diode, in the lab we will only be able to
observe excitation to the D2 line. Both the ground state 5S1/2 and the excited state 5P3/2 all split
into multiple hyperfine levels according to F = I + J ∈ I + J, ...|I − J|. In particular, we note that
for 85Rb, I = 5/2, and for 87Rb, I = 3/2. In the ground state, 85Rb exhibits a splitting of ∼ 3GHz
between the F = 3 and F = 2 sublevels while its 87Rb has a larger splitting of ∼6.8 GHz between
F = 2 and F = 1. The excited state manifolds on the other hand contain the F ∈ 4, 3, 2, 1 and
F ∈ 3, 2, 1, 0 levels and show a much smaller splitting on the order of 100’s of MHz, as detailed in
Fig. below.
5.2
Physics Background
Light and matter can interact in any one of 3 ways - absorption, spontaneous emission, and stimulated emission. T. Hansch and A. Schawlow first suggested that absorption of lasers light could
be used to cool atoms back in 1975[1]. The intuition behind this is quite elegant: imagine a laser
beam as a coherent stream of photons of a single frequency. Upon hitting an atom, an electronic
transition occurs with a finite probability and the photon momentum is imparted to the atom. Now
the decay of the electronic excited state results in the spontaneous emission of a photon as well,
but the momentum recoil from this process occurs in a random direction. Over many absorptionemission cycles then, the average momentum kick that the atom accrues will be along the direction
of the laser. Now, if we detune the laser to a frequency lower than that of the relevant electronic
transition, those atoms moving towards the laser with a certain velocity will see a resonant laser
beam (those atoms will see the laser with an increased frequency due to the Doppler shift) and
thus will absorb photons. Over time those atoms will get slowed down in that direction. If we
then imagine subjecting an atom to 3 orthogonal laser beams, it will be slowed down along those
3 directions. Adding 3 more orthogonal laser beams, each of which is counter-propagating to one
from the previous 3, the atom’s velocity is reduced from all 6 directions. This shrinks the velocity
46
Table 1: Summary of relevant quantum numbers for the atomic structure of Rubidium.
Symbol
Name
Physical meaning
Formula
n
Principal
distance of electron
from nucleus
n=5
S
Spin
intrinsic angular momentum
of the electron
S = 1/2
L
Orbital
shape of electron orbital
orbital angular momentum
L = 0,1,2,3,... labeled
as S,P,D,F,...
J
Electronic
net angular momentum
of the electron
I
Nuclear spin
net angular momentum of
protons and neutrons in nucleus
F
Total
net angular momentum
of electron + nucleus
J=L+S
Notes
Sometimes labeled as
“total” (unfortunately)
85
Rb, I = 5/2
For 87Rb, I = 3/2
F=I+J
F is always an integer for Rb
space that the atom can explore. As we know from thermodynamics and statistical physics, the
temperature of an atomic ensemble scales as the mean of the velocity square. Thus, reduction of
this quantity implies that the atom is cooled.
This does not give the full story. The atoms are still free to leave the trapping region in this
setup. To trap the atoms in position space as well however we need to add DC magnetic fields to
Zeeman shift the atomic energy levels. For this purpose, there are 2 requirements: a homogeneous
magnetic field to cancel out the natural background magnetic field due to the Earth, and a magnetic
field gradient to have a spatially-dependent Zeeman shift.
The set up of a successful magneto-optical trap is thus the culmination and intersection of
several different fields of physics - optics, atomic physics, quantum mechanics, etc. A magnetooptical trap was first demonstrated in 1987 when Raab et. al. [2] in the group of David Pritchard
and Steven Chu trapped sodium atoms. Steven Chu went on to win the Nobel Prize in 1997 for
this experiment. Let us see how the added magnetic field gradient can help us trap the atoms in
physical space.
Assuming that there is zero magnetic field at the intersection point of the 6 MOT beams, let
us then imagine that there exists a spatially homogeneous gradient in that region. The geometry
of the MOT laser beams and magnetic field generating coils is shown in Fig. 5.1b. The current
flows in opposite directions in each of the 2 coils. The arrows in the center of the diagram point
toward the direction of the field at that point. You can see the 6 laser beams and their respective
polarizations pointed towards the center of the MOT. Thus, the magnetic field near the zero point
will be linear as:
B = a⃗x + b⃗y + c⃗z
(38)
The magnetic sublevels of the ground states of the atom are then shifted by:
∆E = gF µB BmF , mF = 0, ±1, .., ±F
47
(39)
(a) Atomic energy levels in the MOT magnetic field gradi-(b) Geometry of the MOT lasers and direction of magnetic
ent
field
Figure 5.1: Picture taken from page no. 192, Atomic Physics, C. J. Foot. [3]
The magnetic sublevels of the excited states of the atom are shifted by:
∆E ′ = gF µB BmF ′ , mF ′ = 0, ±1, .., ±F ′
(40)
The total energy shift is then given by:
∆(E − E ′ ) = gF µB B(mF ′ − mF )
(41)
Since the nuclear spin for both states is the same, ∆mF = ∆mJ . The spatial dependence of
the energy levels along one dimension is shown in Fig.5.1a (the energy diagram is not to scale, the
optical transition is much larger than the Zeeman shifts). Thus, σ − transitions are favoured as they
bring the energy levels closer to resonance with the laser beams. The two forces due to radiation
pressure that are then felt by any atomic state are given by:
S
h̄⃗kγ
F⃗+,− = ±
2 1 + S + [2(δ ∓ ωD ∓ µ′ B/h)/γ]2
(42)
Here F⃗+,− corresponds to the forces due to σ +,− light, respectively, S = I/IS is the saturation
parameter, ωD is the Doppler shift, δ = ωLaser − ωT ransition is the overall laser detuning, and
µ′ = gF µB (mF ′ − mF ) is the effective magnetic moment of the atomic transition.
Usually ωD and µ′ B/h are much smaller than δ, so we can expand the denominators in Eq. 42
and keep only linear terms. Then, the total trapping force is given by:
F⃗ = −α⃗r − β⃗v
(43)
Therefore, as shown in Fig. 5.1a, the atoms moving away from the trap are closer to resonance. We
can thus see that the atoms are trapped in both physical and velocity space.
48
5.3
Recommended Reading
These are a couple of references that is not required of you to complete this experiment, but that
will serve as useful references if you’re ever in a research environment.
1. Foot, Christopher J. Atomic Physics. 1st edition, Oxford University Press, 2005.
2. Laser Cooling and Trapping, Harold J. Metcalf, Peter van der Straten Pages 149-164
5.4
Laser characterization measurements
First we must turn on the laser. The Vescent Laser Diode, like many other laser diodes, requires
temperature stabilization. First, confirm that the laser temperature controller is on and that the
laser diode is at the correct temperature. You can see this on the Vescent Laser Controller (D2105-200) front panel which has a ’Temp Status’ light (Fig. 5.2). This LED should be lit green, as
it is in the figure. With the temperature properly stabilized, flick the switch labelled ”Laser” at
the bottom left of the front panel to the ”ON” setting to turn the laser on. The ”Laser On” LED,
also on the bottom left of the front panel, should be lit green.
Once the laser is powered on and current is being applied, adjust the ”Coarse Current” knob.
Install a power meter in the beam path, and use it to determine what is the current threshold of
the laser. If you take small enough steps, you should be able to observe an exponential increase
right before the threshold point, right at the point where the current is sufficient to sustain the
stimulated emission regime.
Now, we will move on to measuring the waist of the laser. For this, we will be using the
knife-edge method, were we use a razor blade to gradually clip the laser beam until no light makes
through. If you trace the light power as a function of blade displacement, you will map out a
Gaussian error function. By fitting your data to the equation:
1
x − x0
I ∼ A 1 − erf
2
w
(44)
where A and x0 are an overall fitting parameters, x refers to the blade displacement, and w
corresponds to the 1/e radius of the laser assuming the beam follows a Gaussian profile.
5.5
Using the Spectroscopy Module
Set up an opto-mechanical arrangementment that will let you fix the bullet cam right above the
spectroscopy module. Then, take some time to look through the components of the spectroscopy
module, and identify the paths that the pump and the probe beam follow. Spend some time
familiarizing yourself with the laser current controller and the laser PID loop. In your worksheet,
write down Which knobs and input/output channels will be necessary for measuring the saturated
absorption spectra.
Although a laser diode is often labeled as producing a single frequency, that frequency can be
tuned by adjusting the current flowing through the laser. In order to make a MOT next week, we
need to precisely control the frequency of the laser onto resonance with particular transitions in
the Rb spectrum. This is done with the assistance of the spectroscopy module, the first component
the laser passes through.
49
Figure 5.2: Laser Controller front panel
50
Figure 5.3: Rubidium 85 and 87 F = 3 to F ′ spectrum
The spectroscopy module measures the absorption of a room temperature vapor cell of Rb. By
tuning the current through the laser diode we can tune the laser frequency, and then see the changes
in absorption as a function of frequency from the changes in signal measured by the spectroscopy
module.
Install the vapor cell in the beam path such that it passes through it right before the light
gets coupled into the MOT setup. Adjust its height and orientation until the beam roughly passes
through the center of the cell.
With the Oscilloscope ON, flip the ’Laser State’ switch down to the ’Ramp’ setting, and rotate
the Ramp Amp clockwise till it’s maximum. On the oscilloscope, take some time to identify which
signals are being monitored on each channel, and what is their interpretation.
Take some time to think about the electronic structure of rubidium, and how many resonances
and cross-over signals would you expect to observe as you sweep through the current.
Start increasing the current in the laser diode. Identify the currents at which you observe
resonances in the vapor cell, and correlate that with what you are observing in the scope. Save
spectra of the set of transitions from for both Rb 85 and 87 (both a picture and a text file) and
label the peaks by their F’ value, as well as the observed cross-over features.
Use the set of peaks corresponding to the 87 Rb, F = 2 to F ′ transitions to deduce a calibration
51
Figure 5.4: Rubidium 85 Doppler-broadened D2 spectrum
52
between the time axis measured in the scope and the frequency of the laser. Overlap a trace of the
change in current using your scope. Use this data to try to estimate the change in the output laser
frequency and wavelength that a shift of 1mA in the driving current of the diode produces.
Lastly, move the bullet-cam so that you can observe the vapor cell. Take some pictures of the
cell as you sweep through the absorption resonances.
5.6
MOT Characterization References
1.
Hänsch, T. W. & Schawlow, A. L. Cooling of gases by laser radiation. Optics Communications
13, 68–69 (1975).
2.
Raab, E. L., Prentiss, M., Cable, A., Chu, S. & Pritchard, D. E. Trapping of neutral sodium
atoms with radiation pressure. Physical review letters 59, 2631 (1987).
3.
Foot, C. J. Atomic physics (OUP Oxford, 2004).
4. The differences between threshold current calcualtion methods LINK. MKS Instruments (2020).
53
Figure 5.5: Laser Servo front panel
54
Figure 5.6: Mini-MOT control panel
55
Figure 5.7: Laser Controller front panel
56
Figure 5.8: Schematic of the Mini-MOT
57
5.7
Analysis Questions for the report
A) Laser characterization
1. Make a plot of the data that you took while measuring the laser diode threshold. Calculate the
current threshold from your data using both the “Two-Segment Line-Fit Threshold” method
and the “First Derivative (dL/dl) Threshold” methods [4]. Show the fit function that you
used to extract the threshold value. How does each compare to the nominal value, 44 mA?
2. Based on your measurement, determine the laser safety classification for the laser at your
operating conditions.
3. Attach a plot of the waist measurement that you did using the knife-edge method. What is
the extracted 1/e diameter for the laser beam?
4. Use the waist information to calculate the intensity of the beam when running with a current
of 90 mA.
B) Setup devices and controllers
1. Briefly describe the operating principle of a PID loop.
2. Explain how can we use the output of the spectroscopy module as the error signal for the
servo.
C) Saturated Absorption Spectroscopy
1. Provide a detailed plot of each the absorption spectra you measured. Process your data using
the calibrations you measured to make sure that the horizontal axis corresponds to frequency
units. Identify as many of the peaks and features as possible, including the Doppler-free
resonances and the cross-over signals.
2. How would the recorded spectra look like if you tried blocking the pump bean?
3. In the 87 Rb, F = 2 → F ′ spectrum, calculate the ratio between the separation between the
2 → 3 transition and the 2 − 3 cross-over peak, and the separation between the 2 − 3 and
the 1 − 3 cross-over peaks. Does it match what you would expect by analysing the reported
splittings for the hyperfine states of 87 Rb?
4. Use your extracted data to estimate the change in the output laser frequency of the diode
when the current is changed by 1 mA.
5. Use the previous result to evaluate the change in wavelength of the laser when the current
is changed by 1 mA. Note: You will need to use the equation c = λν to evaluate ∆λ as a
function of ∆ν. It is a common mistake to think that ∆λ = c/∆ν, although that is not the
case! What is the right expression for ∆λ?
58
6
Trapping Atoms: Week 6
For this week, we will focus on trapping and cooling atoms, that is, we will achieve a MagnetoOptical Trap (MOT). Following the successful creation of a MOT, we will explore the behavior of
the MOT using mostly qualitative approaches.
6.1
Create MOT
The goal here is to create a stable MOT. When you have successfully captured atoms in the MOT,
you will see a bright blob appear on the IR camera in the center of the glass cell.
Aligning the Laser into the mini-MOT kit
To do this, hang the alignment card on extensions to the cage from the mini-MOT kit. Placing the
card at the end of the extensions will allow you to align the position of the beam, while placing the
card close to the mini-MOT entrance will allow you to align the angle of the beam. While both
mirrors adjust both angle and position, the further mirror (M1) from the entrance, generally adjusts
the position more, while the closer mirror (M2) generally adjusts the angle more. As a result, we
can iteratively align the lasers position and angle by moving the corresponding mirror to center the
laser on the alignment card at the corresponding position. You may wish to work with the lights
off to more clearly see the laser spot. Once you are comfortable with the iterative approach, you
can try simultaneously tuning the two mirrors counter to one another, holding the position on the
alignment card constant while adjusting the incoming angle to the card. This approach is called
’walking’ the beam. When the laser goes through the center of the alignment card at the close and
far positions, the beam is aligned into the mini-MOT setup. M1 and M2 are mounted in ”kinematic
mounts” which each have 2 adjustable knobs - the one higher up aligns the beam vertically and
the lower knob aligns the beam horizontally. The general alignment procedure is as follows:
1. Hang the alignment card on the end of the cage rod extension.
2. Use M1 to align the beam onto the center of the alignment card.
3. Now hang the alignment card on the beginning of the cage rod extensions.
4. Use M2 to adjust the beam until it is centered on the alignment card.
5. You will now find that the beam is misaligned on the alignment card if hung at the end of
the cage extension rods. Therefore, repeat steps 1-4 until you find that the beam path has
been aligned to be centered at both positions of the alignment card.
Balancing the Power in Each Arm
Now that the beam is aligned into the mini-MOT, we want to balance the powers on all 3 axes of
the MOT. The power in each of the arms should be within a tenth of the saturation intensity of
the atomic transition we’re interested in. Balancing the powers can be done via rotation of the 2
half-waveplates in the telescope module of the Mini-MOT. To balance the power in the different
arms, begin by measuring each of the powers using the provided laser power meter. Ensure that
the power meter is at the correct measurement wavelength (780 nm).
59
1. Using the first half-wave plate, adjust the power in the bottom (first) arm to be 1/3 the total
power.
2. Then, adjust the second half wave plate to balance the powers between the two upper arms.
3. Repeat steps 1 and 2 until the powers are approximately equal (to within 10-15% of each
other).
From our alignment, we were able to achieve 0.6 mW/cm2 in each arm. If you are having trouble
achieving that power, use a white piece of paper to look at the beam shape and intensity. If
the beam is noticeably off center, you may need to adjust your alignment to achieve the required
powers. It is also worth noting that a homogeneous field results in a good atom trap.
Polarization Next, we need to make sure all the MOT beams have the correct polarization. The
polarization of each beam decides how much orbital angular momentum is imparted to the atoms.
Thus, to cycle on a transition, the beams must all be circularly polarized. Use the polarimeter to
make sure that the beams are circularly polarized with the same handedness at when exiting the
cage. If this is not the case, you can adjust the quarter wave plates (labelled L/4 in the cage) to
achieve maximum circular polarization in each arm.
Another aspect of the alignment which you are not expected to correct is the collimation of the
beam. If the beam does not appear collimated at the output of the cage, call a TA before attempting
to correct the divergence yourself.
Retroreflectors Finally, check that the 3 beams overlap with their retroreflections and intersect
at one point in the vacuum chamber using the alignment card. You can do this by attaching cage
extensions (provided to you) onto the exit window of each beam from the cage. Then place the
alignment card on the cage extension and make sure that the retroreflection is centered. Note:
You may need to hold the IR camera and view the card through the camera screen to see the
retroreflections.
Congrats! You have fully aligned the Mini-MOT laser into the kit, we will now focus on getting
the laser ready for trapping.
Repumping The first task is to set up a repump laser. While most Rb85 atoms are pumped by
the laser from F=3 to F’=4, they may with a small finite probability get excited to F’=3 instead.
They then have a decay pathway to F=2 in which case they would be dark (insensitive) to the
pumping beams. To overcome this, we can excite the atoms from F=2 repeatedly to F’=3 until
they decay down back to F=3 and they can start cycling again. To set up repumping, there are 3
ways: firstly, one can use a different laser, red detuned from the F=2 to F’=3 transition. Secondly,
one could modulate the laser diode current amplitude at a frequency red-detuned from the repump
transition to generate co-propagating pumping and repump beams. Thirdly, we can phase modulate
the pump beam with an electro-optic modulator at the repump frequency. We will be going the
route of modulating the current intensity with the help of a microwave frequency source (SRS).
Turning on the repump will change the Rb spectrum on the oscilloscope quite a bit as at any
time there are 2 transition probabilities adding up to generate a voltage signal. Set the strength of
the frequency source to be 15 dBm. Adjust the frequency of the RF from around 3.2 GHz down
to 2.945 GHz in small steps. Initially, the spectrum will look something like the one shown in
60
Figure 6.1: The separated spectra generated by the repump and cycler.
61
Figure 6.2: Rb spectrum with both cycler and repump on.
Fig. 6.1. At each adjustment, make sure that you keep track of the transition dip of interest, and
observe the repump and pumping spectra merge into one. It should look like the spectrum shown
in Fig. 6.2. Lock the laser to the positive slope of the F=3 to F’=4 transition (the encircled slope
in the oscilloscope trace, Fig.6.2), far enough away from the minima that you are sufficiently reddetuned for a MOT, but not so far that your lock is weak to perturbations. This is the leftmost dip
on the full Doppler-broadened transition peak - you can see that the positive slope of the dip has
a higher peak to allow for larger red-detunings. Note: You may end up locking onto the incorrect
slope if you Zoom into the wrong side of the ramp. To keep things simple, always work with the
region with low Servo-TTL (as shown in the blue trace in Fig. 6.2).
6.2
Cooling the atoms
Now that the repump is working, simply flick the display setting of the Mini-MOT to the magnetic
field coils. Now flip the polarity of the magnetic field coils and observe the cloud of atoms fluoresce.
Congratulations, you have created your first MOT!
62
6.3
Analysis Questions for the report
1. Briefly explain how does the miniMOT setup creates the counter-propagating beams that
have the opposite polarization handedness from the ”input” beams (which optic(s) in the
setup produce them)?
2. Which polarity did you have to use for the coils? What change in the setup would you have
to do if you wanted to create a MOT using the opposite polarity?
3. In the setup, the modulation of the LD creates two main sidebands at ν +Ω and ν −Ω. Which
of these sidebands is the one that ends up acting as repumping light?
4. Attach a picture of your MOT.
5. Discuss why you can displace the MOT using a permanent magnet.
6. Extra credit [Mathematical Physics, 5 pts] prove that the modulation of the laser current
results in frequency sidebands (Hint: you might need to use the Jacobi-Anger expansion).
63
7
MOT Characterization: Week 7
This week, the focus is on characterizing the MOT. This includes direct detection of the MOT
fluorescence, the loading curve, and measuring the number of atoms that are trapped.
7.1
Background
The number of atoms in a sample, N , is one of the most useful parameters for AMO experiments.
On top of being a basic figure of merit to diagnose the well-functioning of the lasers and quantifying
signals, it also allows us to estimate densities and collision rates in the sample, which become crucial
when using a MOT as a starting point for processes such as evaporative cooling.
In general there are three main imaging methods for probing atomic samples:
1. Fluorescence imaging (FI): measuring the amount of spontaneously emitted light (fluorescence) from a sample directly after it has absorbed a pulse of near-resonant light. A larger
signal is indicative of a larger amount of atoms in the sample.
2. Absorption imaging (ABS): measuring the amount of attenuation in the light profile
(absorption) after the sample was illuminated with a pulse of resonant light. The darker the
”shadow” that the sample casts, the more atoms it should be composed of.
3. Polarization phase-contrast imaging (PHC): analyzing the intensity , phase and polarization profiles of a far-detuned light that has gone through the atomic sample (dispersion).
The density profile of the gas can then be extracted by modeling the interaction between the
light and the cloud.
Both ABS and PHC require a more involved analysis and modeling of the interaction between the
probing beam and the cloud of atoms. Thus, they usually require the sample to be cold, dense and
small. Although our MOT has a remarkably low temperature, it is still not a feasible sample for
either of these methods. Thus, we will rely on FI for characterizing our MOT.
7.2
Basic FI picture
We will model our atomic sample as a uniform, spherical sample composed of Na atoms in total.
Then, we would like to answer the question: how many photons, Nγ will this sample spontaneously
emit after it has been pumped with a brief pulse of near-resonant light?
If we assume a simplified picture of a two-level system undergoing spontaneous emission (associated
with the Einstein coefficient Aeg ), then Nγ will be given by:
Nγ =
tp
ρee Na
τ
(45)
where tp is the pulse duration, τ = 1/Γ is the excited state lifetime, and ρee is the fraction of the
atoms in the excited state. For our case, 85 Rb, τ = 26.2348 ns and Γ = 2π × 6.066 MHz. The
quantity Ne = ρee Na corresponds to the total amount of atoms that are in the excited state, and
ρee can be evaluated by solving the optical Bloch equations for an ensemble of atoms (see Foot):
64
ρee =
1
2 I/Is
4
∆ 2
Γ
(46)
+ I/Is + 1
Here ∆ is the detuning of the pulse with respect to the resonant frequency, I is the intensity of the
beam, and Is is the saturation intensity, which is given by:
Is =
π hc
Γ
3 λ3
t
where for our case λ = 780 nm and thus Is = 1.671 mW/cm2 . The factor τp estimates the number
of emission events that will occur during the probing pulse. Note that it would be one if the pulse
duration exactly matched the lifetime of the excited state, but in general the pulse will be much
longer.
Eq. (45) takes care of all the physics of the system, and now we need to take into account
experimental conditions and constraints. These photons will be emitted uniformly in 3D space, but
we will only be able to collect a small fraction η of these photons with our camera/photodetector.
We estimate this ”collection efficiency” by η ∼ Ω/4π, where Ω is the collection solid angle subtended
by our measuring device, compared to the total 4π steradians available for a spherically-symmetric
emission. Ω can be evaluated exactly using calculus, but for a sensor that is sufficiently far from
the sample, and is also sufficiently small such that the small-angle approximation is valid, it can
be approximated as
Ω∼
πr2
d2
where r is the radius of the light-collecting optic (most probably a lens), and d the distance between
the sample and the optic.
Then, the number of detected photons, Np will be given by:
Np = η
tp
ρee Na
τ
(47)
The signal that we will measure, assuming we use either a CCD or a CMOS sensor, will not be
in terms of number of photons but in the number ”counts” of photoelectrons (times whatever
multiplicative gain the device has). This is characterized by the quantum efficiency, ξ of the
sensor, which measures the number of counts per every incoming photon. ξ is usually reported
as a percentage. For example, if ξ = 33%, the sensor will report 1 count per every 3 photons.
Conversely, for every ”count” event, we would expect to have received 3 photons.
We can measure ξ for a sensor with area A by exposing it to a pulse with a fixed duration te and
a known intensity I. Then, assuming every photon was detected, we can write the total received
energy in different forms:
ε = I A te = Nc Q = Np
hc
λ
(48)
where Nc is the number of measured counts after such pulse and Q is the device quantum efficiency
relating energy (Joules) with respect to counts. Q is experimentally useful in terms of units, but
for our purposes we can evaluate a ”per photon” equivalent of Q:
65
Q=
Qλ
hc
such that Q = 1/ξ. Finally, we are ready to write down a relationship between the number of
measured counts/photons after a FI pulse and the number of atoms in the sample:
Np = Nc Q = η
tp
ρee Na
τ
(49)
and we can solve for Na in terms of the experimentally relevant quantities:
Na =
7.3
Nc Q
t
η τp ρee
(50)
Re-create MOT
In the first week, you aligned the beam into the MiniMOT kit, balanced the powers in the different
arms, checked the polarization and retro-reflection before turning on the coils and getting your cold
atom gas. This week, the setup should already aligned, balanced and properly retro-reflected. As
a result, putting you should be able to quickly create your MOT using the frequency and power on
the RF generator that you used last week. Once you have a MOT, we can move on to characterizing
the MOT properties.
7.4
Detecting MOT using fluorescence
Before we setup the CMOS camera we will be using to image the MOT, we can measure its quantum
efficiency using the procedure described in section 7.2. Introduce the camera in one of the arms of
the MOT and see what kind of image is recorded. If the sensor is saturated, you will need to make
use of a ND filter to reduce the overall power of the beam, and take this attenuation factor into
account. Take a picture and use it to estimate ξ.
We now should set the lens with which we’ll be collecting light for the sensor. Remove the
bullet cam and replace it with a provided mini-flashlight. You will most likely need to use the ND
filter again to prevent the sensor from saturating. Install the lens right in front of the camera at
a distance that roughly corresponds to the focal length and then proceed to adjust it until you
observe a sharp image of the flashlight. The LED should look like a square with sharp edges. Once
you finish this step, we will no longer be adjusting the separation between the lens and the CMOS,
and we’ll be moving them together using the provided breadboard.
Once this step is done, reset the bullet cam and the MOT. Make sure you can observe the
atoms in the bullet them, and then proceed to adjust the CMOS + lens assembly until you are
able to clearly observe the MOT in the camera. Verify that you are observing the atoms by
slightly displacing them with a magnet and seeing them move in the CMOS. The signal should also
disappear when the light is blocked or the coils are turned off.
Adjust the snapshot duration such that the image of the MOT in the CMOS is not saturated.
Then try to optimize the MOT by adjusting the parameters of the lock such as the overall offset
of the lock and the amplitude of the current modulation.
66
After this, use the camera to record a background image of the field of view when there is no
IR light passing through the setup, and another background image when the IR is passing through,
but the coils are turned off. Let the MOT load until it reaches a steady state and take another
picture. We will use those pictures to estimate the number of atoms in the sample and its size.
After this step, use the recording feature of the camera to try to take a video of the MOT
loading until it reaches a steady state. We will analyze this video to characterize the loading of the
MOT in terms of its overall size and the number of atoms as a function of time.
Record a video of the atoms leaking from the MOT after you block each of the arms of the
MOT.
Finally, record a video of the MOT as you turn off the coils, and try to see if you can observe the
explosion of the MOT. For this step, use as many frames per second (FPS) as the camera allows.
7.5
Analysis Questions for the report
1. What is the quantum efficiency ξ you measured? How does it compare to the nominal value
of ξ = 33.7%?
2. What was the distance between the light-collecting lens and the CMOS? How does it compare
to the reported focal length of the lens?
3. Use your recorded pictures of the MOT to isolate the MOT fluorescence by subtracting the
background image you recorded (with light, no magnetic field gradient). Then, estimate the
number of atoms in your sample. From your picture, measure the radius of the cloud and
estimate its overall volume assuming a spherical/ellipsoidal shape. What is the density of the
cloud? How does it compare to the density of rubidium at STP conditions?
4. Briefly discuss how do the atoms leak from the MOT after blocking each of the beams. Is
there any difference between any of the arms?
5. Analyze the frames of the recorded video of the MOT loading, and prepare a plot of atom
number and cloud radius as a function of loading time.
6. Were you able to observe the MOT exploding? What change in the setup could you do to
try to better observe the explosion?
67
8
NV spin experiments: Week 8
In this week we are going to experiment with nitrogen vacancy (NV) center spins in diamond.
These spins can serve as qubits and have shown very long coherence times. At the same time, they
are sensitive probes with atomic scale resolution and therefore a popular platform for quantum
sensing applications.
8.1
NV Background
Carbon forms in one of the allotropes a cubic lattice, called diamond. Within the diamond lattice
numerous types of defects can exist. One of these defects is the nitrogen-vacancy (NV) center. It
is formed by a substitutional nitrogen atom associated with a neighboring vacancy in the diamond
crystal structure. The NV center can be excited by light in the green spectrum. The excited
state decays back to the ground state either directly or via an intermediate shelving state with
different fluorescence intensity. The decay path depends on the electron spin of the NV center.
This allows optical readout and initialization at room temperature. Further the electron spin can
be manipulated by microwave radiation. Applying magnetic and electric fields the energy levels of
the spins can be shifted. Hence, a vast amount of experiments and measurement applications are
possible.
Figure 8.1: The NV center consists of a nitrogen atom adjacent to a vacancy in the carbon lattice of diamond.
Therefore, the dynamics of the NV center allow applications like spin initialization and state
readout. Therefore, the center is suitable for quantum sensing applications like magnetic field
sensing, measuring spin relaxation time and optically detected magnetic resonance - ODMR. Due
to their scalability, long coherence times and ability for interaction with photons, NV centers are of
high interest for research in quantum information processing. Qubits can be defined as spin states
of single electron or nuclear spins.
68
Figure 8.2: Energy-level scheme of the NV center. A green laser excites the NV center to the 3E state, from
where it either decays to 3A or undergoes intersystem crossing to the metastable state 1A, before decaying
to the ms = 0 ground state.
Applying microwave radiation with 2.87 GHz to the NV center drives the transition between
ms= 0 and ±1 in its electronic ground state. Additional excitation with laser light in the green
spectrum populates ms= ±1 in the electronic exited state. This state has a significant possibility of
intersystem crossing from 3E to the shelving state 1A with a long lifetime. From there it decays with
high probability back to the ms= 0 ground state. Due to the longer lifetime, the fluorescence gets
quenched by applying microwave radiation at the frequency D, resonant to the transition between
the spin levels. Hence, the fluorescence of a NV center is brighter when it is in the ms= 0 state.
Appling a magnetic field B, the ms= ±1 levels of the ground state split up linearly according to the
Zeeman effect. Thus, the microwave resonance frequency also shifts. This allows measurements of
magnetic fields with high sensitivity and high spatial resolution.
Figure 8.3: The grey curve shows the decrease of fluorescence at the resonance frequency. The ms= ±1
ground level of the NV center splits up linearly by applying a magnetic field (blue) and indicate the shift in
the resonance frequencies.
8.2
Laser threshold
For this set of experiments we will be using a 50 mW green diode laser (520 nm, model L520P50
from Thorlabs) to excite the NV centers in the diamond sample. As a first step, characterize the
69
laser threshold of this laser using a power-meter head and also determine the maximum output
power we can achieve. In this case the control only allows us to set a ”percentage” for driving the
diode, so we’ll measure the threshold as a function of this internal variable.
8.3
Magnetic field calibration
In order to perform the ODMR measurements, we first need to calibrate the magnetic field of the
permanent magnet we will be using. Use a caliper to determine the length and the diameter of the
cylindrical permanent magnet. The magnetic field at a distance z from its circular face along its
symmetry axis is given by:
B0
B(z) =
2
z+H
z
p
−√
R2 + z 2
R2 + (z + H)2
!
(51)
where B0 is a fitting constant, R is the diameter of the cylinder, and H is its height. We will use
a magnetic field sensor (Yoctopuce, model Yocto-3D-V2) to do the calibration. Mount the sensor
so that the facet of the magnet is parallel to the sensor, and use the dovetail rail to carefully vary
the distance between the sensor and the magnet. Use Eq. (51) to fit your data.
8.4
ODMR
Given that we now have an estimate of the field of the magnet as a function of distance, mount
the magnet on the rotating breadboard, and place the magnet as close as possible to the diamond.
Take ODMR spectra while varying the angle between the magnetic field and the diamond, and
try to observe features corresponding to the magnetic field being aligned to the [100] and the [110]
crystallographic directions in the diamond.
Once you have found these features, use the dovetail rail to vary the distance between the
diamond and the magnet for both of these directions, and trace the position of the peaks as a
function of distance between the magnet and the diamond. By using Eq. (51), plot the splitting
of the ODMR features as a function of the magnitude of the magnetic field. Use those plots to
extract the gyromagnetic ratio of the electron.
70
8.5
Analysis Questions for the report
1. Make a plot of the output power of the laser as a function of the driving ”percentage”. What
is the maximum output power you were able to measure
2. What is the color of the light the diamond fluoresces with after being excited with the green
light? Why are they not the same?
3. Make a plot of the magnetic field of the cylindrical magnet as a function of distance from its
facet. How does the fit to Eq. (51) compare to your data?
4. Make a plot of that shows the different ODMR spectra you measured as a function of the
angle between the diamond and the input magnetic field. At which angle can you distinguish
more peaks? At which angle can you distinguish the least amount of peaks? Identify which
of the spectra correspond to B beign aligned with the [100] and the [110] crystallographic
directions in the diamond
5. Make a plot of the ODMR spectra, and of the splitting between the ODMR features as
a function of magnetic field. From this plot, try to extract the gyromagnetic ratio of the
electron. What is the dominant source of error in your estimation?
71
9
NV spin experiments: Week 9
In this last week we will continue our experiments with nitrogen vacancy (NV) center spins in
diamond. Last week we explored the dependence of the position of the ODMR peaks as a function
of magnetic field strength and orientation. This week we will try to observe the effects of the
nuclear magnetic spin of nitrogen.
Because 14 N has I = 1, we need to include additional terms in the Hamiltionan of our system.
That is, we go from:
⃗ · S⃗z
H ∼ DSz2 + γe B
where D = 2.88GHz is the zero-field splitting and γe is the gyromagnetic ratio of the electron to
⃗ · S⃗z − QIz2 − An I⃗ · S⃗z
H ∼ DSz2 + γe B
where Q = 4.946 MHz is the quadrupole splitting and AN = 2.19 MHz corresponds to the hyperfine
interaction. This translates into the features being further split into 3 contributions corresponding
to mI = 0, ±1. The observed frequencies would be given by:
νmI = D − γe Bz + mI AN
where mI = 0, ±1. Thus, by measuring these features we can estimate AN .
Because these features are narrow, it is necessary to make sure that sources of broadening such
as power broadening are kept to a minimum in our measurement.
For the measurements, place the permanent magnet sufficiently close to the diamond such that
you can identify one of the contributions corresponding to some NV− orientation. Reduce the
frequency span to try to observe the triple feature within the resonance. The separation within
each peak corresponds to AN .
Once you observe these features, vary the laser power and the overall driving RF power to see
if you can find an optimal condition for observing these features.
9.1
Analysis Questions for the report
1. Make a plot of one of the isolated ODMR features. Were you able to observe the triple feature
due to the nuclear spin coupling?
2. Based on this plot, provide an estimate for AN
3. Briefly describe the effects of increasing the laser power and the driving RF power in the
spectra. Were you able to identify optimal conditions for the measurement?
72