THORLABS EDU-QOP1 User manual

EDU-QOP1(/M)
Quantum Optics Kit
User Guide

Quantum Optics Kit
Table of Contents
Chapter 1 Safety........................................................................................................................... 1
1.1 Warning Symbol Definitions .................................................................................................... 1
1.2 Laser Radiation Warning.......................................................................................................... 1
1.3 Piezo Controller Warnings ....................................................................................................... 1
Chapter 2 Product Description...................................................................................................... 2
Chapter 3 Principles of Quantum Optics ....................................................................................... 4
3.1 Classical Description of Light.................................................................................................... 4
3.1.1 Behavior at a Beamsplitter.................................................................................................................................. 4
3.1.2 Interference ........................................................................................................................................................ 5
3.1.3 Polarization ......................................................................................................................................................... 6
3.2 Mathematical Formalism of Quantum Mechanics.................................................................... 8
3.2.1 Dirac Notation..................................................................................................................................................... 8
3.2.2 General Case ....................................................................................................................................................... 8
3.2.3 Application in Quantum Mechanics and Quantum Optics.................................................................................. 8
3.2.4 Linear Operators ................................................................................................................................................. 8
3.2.5 Eigenstates and Eigenvalues ............................................................................................................................... 9
3.2.6 Commutator Relations ........................................................................................................................................ 9
3.2.7 Bases of the Hilbert Space................................................................................................................................. 10
3.2.8 Observables....................................................................................................................................................... 10
3.2.9 Measurements and Superposition of Quantum States ..................................................................................... 11
3.2.10 Expectation Values............................................................................................................................................ 11
3.2.11 Mixed States...................................................................................................................................................... 11
3.2.12 Combination of Quantum Systems ................................................................................................................... 12
3.3 Quantum Description of Light.................................................................................................12
3.3.1 Fock States ........................................................................................................................................................ 12
3.3.2 Coherent States................................................................................................................................................. 13
3.3.3 Thermal States .................................................................................................................................................. 14
3.4 Proof of Quantized Light - Theory ...........................................................................................14
3.4.1 Second-Order Correlation Function .................................................................................................................. 15
3.4.2 Quantum Mechanical Treatment of a Beamsplitter.......................................................................................... 16
3.4.3 Quantum Fields ................................................................................................................................................. 17
3.5 Proof of Quantized Light - Experiment ....................................................................................18
3.5.1 Hanbury-Brown-Twiss Experiment.................................................................................................................... 18
3.5.2 Correlation Function for Single Photon Detectors ............................................................................................ 19
3.5.3 Grangier-Roger-Aspect Experiment .................................................................................................................. 20
3.5.4 Triple Coincidence Detection Scheme............................................................................................................... 21
3.5.5 Triple Coincidence Detection Scheme: Explanation 1....................................................................................... 22
3.5.6 Triple Coincidence Detection Scheme: Explanation 2....................................................................................... 22
3.5.7 Accidental Coincidences.................................................................................................................................... 23
3.6 Single Photon Interference .....................................................................................................24
3.6.1 Theoretical Description ..................................................................................................................................... 24
3.6.2 Experimental Realization................................................................................................................................... 25
3.7 Quantum Treatment of Polarization .......................................................................................26
3.7.1 Theoretical Description ..................................................................................................................................... 26
3.7.2 Experimental Realization................................................................................................................................... 27
3.8 The Quantum Eraser...............................................................................................................28
3.8.1 Theoretical Description ..................................................................................................................................... 28
3.8.2 Experimental Realization................................................................................................................................... 29
3.9 Spontaneous Parametric Down-Conversion............................................................................29

Quantum Optics Kit
Chapter 4 Experimental Concepts ...............................................................................................33
4.1 Single Photon Detectors .........................................................................................................33
4.2 Time Tagging ..........................................................................................................................34
4.2.1 Time Tagging versus Coincidence Electronics....................................................................................................34
4.2.2 Jitter and Coincidence Window .........................................................................................................................34
4.2.3 Delay Compensation..........................................................................................................................................35
Chapter 5 Kit Components ..........................................................................................................37
5.1 Pump and Alignment Laser.....................................................................................................37
5.2 Crystal and Adjustment Aids...................................................................................................38
5.3 Optics.....................................................................................................................................38
5.4 Detectors ...............................................................................................................................40
5.5 Michelson Interferometer ......................................................................................................41
5.6 Time Tagger and Software......................................................................................................42
5.7 Quantum Eraser .....................................................................................................................43
5.8 Mounting and Tools ...............................................................................................................43
5.9 Included Hardware.................................................................................................................44
5.9.1 Imperial Kit.........................................................................................................................................................44
5.9.2 Metric Kit ...........................................................................................................................................................44
Chapter 6 Quick Setup ................................................................................................................45
Chapter 7 Setup and Adjustment ................................................................................................48
7.1 Assembly of Components .......................................................................................................48
7.1.1 Pump Laser ........................................................................................................................................................48
7.1.2 Alignment Laser .................................................................................................................................................50
7.1.3 Mirrors...............................................................................................................................................................51
7.1.4 Iris Apertures .....................................................................................................................................................52
7.1.5 Half-Wave Plates................................................................................................................................................53
7.1.6 Crystal, Axicon, and Filter ..................................................................................................................................54
7.1.7 Beamsplitter ......................................................................................................................................................56
7.1.8 Detectors ...........................................................................................................................................................56
7.1.9 Detector Optics..................................................................................................................................................57
7.1.10 Beam Trap..........................................................................................................................................................58
7.1.11 Alignment Target ...............................................................................................................................................58
7.1.12 Economy Beamsplitter.......................................................................................................................................59
7.1.13 Screen ................................................................................................................................................................59
7.1.14 Mirrors for Michelson Interferometer...............................................................................................................60
7.1.15 Lens....................................................................................................................................................................61
7.1.16 Mounting the LED ..............................................................................................................................................62
7.1.17 Polarizers for Quantum Eraser...........................................................................................................................63
7.1.18 Labelling the Time Tagger..................................................................................................................................64
7.2 Preliminary Alignment............................................................................................................65
7.2.1 Software Installation..........................................................................................................................................65
7.2.2 Collimating the Pump Laser...............................................................................................................................65
7.2.3 Calibrating the Polarizer ....................................................................................................................................68
7.3 Setting Up the HBT Experiment with the Alignment Laser ......................................................70
7.4 Setting Up the Photon Pair Source..........................................................................................73
7.4.1 Setting up the Pump Laser Beam Path...............................................................................................................73
7.4.2 Aligning Both Pump and Alignment Laser on the Same Path ............................................................................75
7.4.3 Detector Positioning ..........................................................................................................................................78
7.4.4 Detector Fine Adjustment..................................................................................................................................81
7.4.5 Crystal Angle Adjustment ..................................................................................................................................83

Quantum Optics Kit
7.5 Setting Up the Grangier-Roger-Aspect Experiment .................................................................84
7.5.1 Beamsplitter Positioning ................................................................................................................................... 85
7.5.2 Third Detector Positioning ................................................................................................................................ 86
7.5.3 Third Detector Fine Adjustment........................................................................................................................ 86
7.5.4 Test Measurement ............................................................................................................................................ 87
7.6 Setting Up the Michelson Interferometer ...............................................................................88
7.6.1 Additional Alignment Path ................................................................................................................................ 89
7.6.2 Interferometer Adjustment............................................................................................................................... 90
7.6.3 Detector Positioning.......................................................................................................................................... 97
7.6.4 Detector Fine Adjustment................................................................................................................................. 97
7.6.5 Single Photon Interference Adjustment............................................................................................................ 98
7.6.6 Michelson Interferometer Test Measurement.................................................................................................. 99
7.7 Setting Up the Quantum Eraser ............................................................................................100
Chapter 8 Learning Goals and Misconceptions.......................................................................... 104
Chapter 9 Experiments ............................................................................................................. 106
9.1 HBT-Experiment with Attenuated Laser................................................................................106
9.2 Photon Pair Source...............................................................................................................107
9.3 HBT Experiment with one Arm of the Pair Source .................................................................108
9.4 Grangier-Roger-Aspect Experiment.......................................................................................108
9.5 GRA Experiment with Classical Light .....................................................................................109
9.6 Malus’ Law for Single Photons..............................................................................................110
9.7 Single Photon Michelson Interferometer ..............................................................................111
9.8 Quantum Eraser ...................................................................................................................114
Chapter 10 Additional Experiments ............................................................................................ 116
10.1 Coherence Length.................................................................................................................116
10.2 The Three-Polarizer-Paradox ................................................................................................118
10.3 Quantum Computing............................................................................................................118
10.3.1 Qubits and Gates............................................................................................................................................. 119
10.3.2 Deutsch-Jozsa Algorithm................................................................................................................................. 120
10.3.3 Optical Implementation .................................................................................................................................. 121
10.3.4 Experimental Realization................................................................................................................................. 123
10.3.5 Sample Measurements.................................................................................................................................... 126
10.3.6 Discussion of Error Sources............................................................................................................................. 127
10.3.7 Further Algorithm Ideas .................................................................................................................................. 128
10.4 Direct Event Pulse Observation.............................................................................................128
Chapter 11 Software................................................................................................................... 131
11.1 Software Installation ............................................................................................................131
11.2 General Remarks ..................................................................................................................131
11.3 Saving Results.......................................................................................................................131
11.4 Alignment Tab......................................................................................................................132
11.5 Delay Adjust Tab ..................................................................................................................132
11.6 HBT and GRA Tabs ................................................................................................................134
11.7 Malus’ Law Tab.....................................................................................................................135
11.8 Michelson Interferometer Tab..............................................................................................136

Quantum Optics Kit
11.9 Configuration Tab.................................................................................................................137
11.10Hidden Settings ....................................................................................................................138
Chapter 12 Technical Notes.........................................................................................................139
12.1 Different Detection Schemes................................................................................................139
12.1.1 Standard Triple Coincidences...........................................................................................................................139
12.1.2 Gated Detection Scheme .................................................................................................................................140
12.1.3 Changing the Detection Scheme in the Software ............................................................................................141
12.2 Environmental Conditions ....................................................................................................142
12.3 Avoiding Fluctuations in the Michelson Interferometer........................................................142
12.4 Accidental Coincidences .......................................................................................................144
12.5 Maximizing the Count Rate ..................................................................................................146
12.6 Choice of Polarizers ..............................................................................................................147
12.7 Polarizers vs. Quarter-Wave Plates.......................................................................................147
12.8 Axicon Design.......................................................................................................................148
12.9 Temperature Dependence of Pump Laser Wavelength .........................................................148
12.10 Adjustable Count Rate Ratio.................................................................................................148
12.11 Movable Michelson Interferometer......................................................................................149
12.12 Polarization In the SPDC Process...........................................................................................150
Chapter 13 Troubleshooting........................................................................................................151
13.1 Pump Laser not Working ......................................................................................................151
13.2 Low Count Rates with Filter..................................................................................................151
13.3 Low Count Rates with BBO crystal ........................................................................................151
13.4 Low Coincidence Count Rates...............................................................................................152
13.5 Malus’ Law Measurement is Asymmetric .............................................................................152
13.6 Michelson Interferometer Problems.....................................................................................153
13.7 Quantum Eraser Problems....................................................................................................153
Chapter 14 Laser Safety Calculation ............................................................................................154
14.1 Laser System ........................................................................................................................154
14.2 Laser Class Calculation..........................................................................................................154
14.3 Laser Safety Glasses Calculation ...........................................................................................154
Chapter 15 Acknowledgements ..................................................................................................156
Chapter 16 Warranty and RMA Information ...............................................................................157
16.1 Return of Devices .................................................................................................................157
Chapter 17 Thorlabs Worldwide Contacts...................................................................................158

Quantum Optics Kit Chapter 1: Safety
Rev. A, June 16, 2023 Page 1
Chapter 1 Safety
1.1 Warning Symbol Definitions
Below is a list of warning symbols you may encounter in this manual or on your device
Warning: Laser Radiation
General Warning
1.2 Laser Radiation Warning
Warning
The class 3B laser diode used in this kit can emit more than 50 mW of optical power, which can
cause damage to the eyes if viewed directly. The laser driver is equipped with a key switch and
safety interlock, which should be used appropriately to avoid injury. Additionally, we
recommend wearing appropriate laser safety glasses when using this kit.
1.3 Piezo Controller Warnings
General Warnings
If this equipment is used in a manner not specified by the manufacturer, the protection provided
by the equipment may be impaired. In particular, excessive moisture may impair operation.
This product generates, uses, and outputs high voltages from the SMC connector (HV Output)
that are hazardous and can cause serious injury. In any installation that uses the KPZ101 piezo
controller, it is the user’s responsibility to ensure adequate insulation and precautions are taken
to avoid shock risk. Cables for HV Out must be rated for 250 V RMS.
Spillage of fluid, such as sample solutions, should be avoided. If spillage does occur, clean up
immediately using absorbant tissue. Do not allow spilled fluid to enter the internal mechanism.

Quantum Optics Kit Chapter 2: Product Description
Page 2 MTN036012-D02
Chapter 2 Product Description
The field of quantum physics evolves quickly: quantum computers, quantum cryptography networks, and
quantum-based sensing are all nearing introduction into real world applications. Highlighting this, the 2022 Nobel
Prize in Physics was awarded to Alain Aspect, John F. Clauser, and Anton Zeilinger for their groundbreaking work
in quantum optics. In turn, teaching quantum concepts is becoming ever more important. Our Quantum Optics
Educational Kit allows students to investigate the quantum properties of light first-hand in an open and accessible
environment.
One particularly important aspect of the kit is to educate people about what constitutes a non-classical light
source. E.g., students can measure that an attenuated laser does not constitute a single photon source. The source
that is set up and analyzed in this kit is a so-called photon pair source; pairs of 810 nm photons are generated
inside a nonlinear, type-I BBO crystal from the output of a 405 nm pump laser. The photons in the pair are time-
energy entangled (though not polarization-entangled). The use of nonlinear crystals is widespread in the quantum
optics community because of their ease of alignment and their high count rates. Since all measurements are
performed with these pairs, this type of non-classical light source is also called a heralded single photon source
(i.e., one photon signaling the arrival of the other).
A look at this educational system, as displayed in Figure 1, reveals the open design that allows students to
investigate each component’s properties and function. Moreover, we do not utilize optical fibers which are often
used to guide light from the table to detectors housed somewhere else. Instead, we use free space optics and very
small single photon detectors which allow us to place the detectors in the system and perform the measurement
in that spot. This is even more important since the discussion of when and where a measurement occurs is
fundamental in quantum mechanics. The open setup requires a darkened room; see Section 12.2 for more details.
Figure 1 Overview of the Quantum Optics Setup (Breadboard not included: We recommend the
B2448FX (B60120AX) optical breadboard with the AV5(/M) damping feet.)
To analyze and correlate the signals coming from the three single photon detectors, modern time-tagging
electronics is employed. There are several experiments that students can set up and measure themselves:
•The pair source itself
•The experimental proof of a non-classical light source (Grangier-Roger-Aspect experiment)
•Experimental falsification of the misconception that attenuated lasers are single photon sources
•Behavior of single photons at a linear polarizer

Quantum Optics Kit Chapter 2: Product Description
Rev. A, June 16, 2023 Page 3
•Single photon interference in a Michelson interferometer
•Quantum eraser experiment
All these experiments are described in detail in this manual and are made experimentally accessible by clear
alignment procedures and experimental alignment tools that greatly help in reliably finding the signals of interest.

Quantum Optics Kit Chapter 3: Principles of Quantum Optics
Page 4 MTN036012-D02
Chapter 3 Principles of Quantum Optics
3.1 Classical Description of Light
In this section, a summary of the classical description of light is given, focused on the properties that play a role in
the experiments in this kit. It is often useful to compare the quantum mechanical results derived in later sections
to the classical results of this section.
Classically, light is described as an electromagnetic field governed by the Maxwell equations.

󰇍



(1)

󰇍


(2)

󰇍


󰇍


(3)

󰇍

󰇧
󰇍

󰇨
(4)
Here, 
󰇍

and 
󰇍

are the electric and magnetic fields, is the charge density, is the electric current density, is
the vacuum permittivity constant, and is the vacuum permeability constant.
By combining the Maxwell equations, the electromagnetic wave in vacuum can be described by a single equation:
󰇧

󰇨
󰇍

󰇛󰇜
(5)
One solution of this equation is a plane wave with:

󰇍

󰇛󰇜
󰇍


󰇍


(6)
Here, 
󰇍

is the wave vector and is the angular frequency.
A Cartesian coordinate system can be chosen in such a way that the propagation of the wave is along the z-
direction, i.e., the wave vector is 
󰇍


󰇍

, with 
󰇍

being the unit vector in the z-direction. The following equation
therefore applies:

  
(7)
Here, is the wavelength, is the frequency, and is the speed of light.
The electric field emitted by a laser can be approximated as a plane wave and the intensity is the square of the
electric field 
󰇍

.
3.1.1 Behavior at a Beamsplitter
If a plane wave with an amplitude 
󰇍

is incident on the input port of a beamsplitter with a transmission coefficient
and a reflection coefficient , the output fields are still plane waves with amplitudes of 
󰇍


󰇍

and 
󰇍



󰇍

, respectively. The output intensities are then:

󰇍


(8)

󰇍


(9)
Under the assumption of an ideal lossless beamsplitter, the squares of the coefficients add up to 1:

(10)

Quantum Optics Kit Chapter 3: Principles of Quantum Optics
Rev. A, June 16, 2023 Page 5
3.1.2 Interference
Assume the experimental setup
1
shown in Figure 2: An incoming light wave, 
󰇍

(green arrow), is split at a
beamsplitter. The light from each output port, 
󰇍

and 
󰇍

(blue dashed arrows), then hits a mirror at normal
incidence and is reflected backwards into the beamsplitter. The waves from the two arms recombine at the
beamsplitter and leave in the direction of the detector, 
󰇍

(such as a screen, camera, or eye) as well as back
towards the light source, 
󰇍

(red dashed arrows).
Figure 2 Schematic of a Michelson Interferometer
If the input field has an amplitude and the beamsplitter ratio is 50:50, then the amplitudes and in the
arms are:


(11)
Due to the path length difference of arms and , the fields in the arms become out of phase with a phase
difference :


(12)
The amplitude of the field at the output port is then:

󰇛󰇜    
󰇛󰇜
(13)
The intensity measured at the detector on the output path is then:

 󰇛󰇜 󰇛󰇜
(14)
Where is the intensity of the initial input. Hence, depending on the path length difference, the output intensity
can reach from 0 (destructive interference) to (constructive interference).
When combining two beams at a beamsplitter, energy conservation dictates that in one of the outgoing beams,
the incoming beams are phase-shifted by an additional to each other
2
. In which outgoing beam this occurs
depends on the actual design of the beamsplitter. Here, we assign the additional phase-shift to the output beam
going back towards the laser, 
󰇍

. The field and intensity in this path are then:

󰇛󰇜
󰇛󰇜
(15)

󰇛󰇜󰇛󰇜 󰇛󰇛󰇜󰇜 󰇛󰇜
(16)
1
Thorlabs offers the EDU-MINT2(/M) Educational Kit for Classical Interferometry. The manual is freely available on the product webpage and introduces the
Michelson interferometer in more detail.
2
M. Fox, Quantum Optics: An Introduction. (Oxford University Press, Oxford, 2006).

Quantum Optics Kit Chapter 3: Principles of Quantum Optics
Page 6 MTN036012-D02
It is apparent that the sum of the intensities in the two output arms is always equal to the input intensity ,
meaning that the energy is conserved.
3.1.3 Polarization
The Maxwell equations show that the electromagnetic field is a traverse field, i.e., the field vectors are
perpendicular to the propagation direction2. Figure 3 shows an example of such a wave.
Figure 3 An electromagnetic wave propagating along the z-direction. The electric field is marked as a
solid blue line, and the magnetic field is marked as a red dashed line.
The polarization is defined by the direction of the electric field; hence the magnetic field is omitted in the following
equations. For a plane wave propagating in z-direction, the electric field vector is in the xy-plane and can be
described as:

󰇍


󰇍


󰇍


(17)
For this plane wave, the following equations apply:
󰇛󰇜
(18)
󰇛󰇜
(19)
Here, and are the amplitudes of the fields in x- and y-direction, respectively, and is the phase factor,
describing the fact that the field can oscillate in separate phases in the x- and y-direction. The overall amplitude
of the electric field is then:

(20)
The polarization vector describes the direction of the electric field oscillation and is defined as:


󰇍



󰇍


(21)
and the electric field 
󰇍

can be described in terms of the polarization vector (see the literature
3
for a more detailed
derivation) as:

󰇍

󰇛󰇜
(22)
To understand the meaning of the polarization vector, it is helpful to investigate several special cases.
Linear Polarization
If , i.e., the x- and y-components of the electric field are oscillating in phase, then is a real vector, describing
a line which is offset by an angle from the x-axis:


(23)
3
M. Beck, Quantum Mechanics, Theory and Experiment. (Oxford University Press, Oxford, 2012).

Quantum Optics Kit Chapter 3: Principles of Quantum Optics
Rev. A, June 16, 2023 Page 7
The electric field oscillates along this line, and the light is called linearly polarized. Figure 4 displays an example of
a linearly polarized electric field.
Figure 4 Electric Field of a Linearly Polarized Wave. The perspective in the right graph is along the
negative z-direction (towards the source).
Circular Polarization
If 
and  , the polarization vector becomes 

󰇍


󰇍

. The subscript L denotes left-
circular polarization. Calculating the electric fields in this case and setting z = 0 to only view the oscillation in time
at a fixed point, one obtains:

󰇛󰇜
(24)

󰇛󰇜
(25)
This means that the direction of the electric field is constantly changing and tracing a circle in the x-y-plane, hence
the name circular polarization. The rotation is counter-clockwise (when looking towards the source) and called
left-circularly polarized.
Figure 5 Electric Field of a Left-Circularly Polarized Wave. The perspective in the right graph is along the
negative z-direction (towards the source).
A phase difference of 
results in clockwise rotation (again, when looking towards the source) and the light
is then called right-circularly polarized. In this case, the polarization vector becomes:
 

󰇍


󰇍


(26)
Elliptical Polarization
Elliptical polarization is the general case when none of the special cases above apply. The phase shift and the ratio
of the amplitudes in the x- and y-directions determine the orientation and eccentricity of the ellipse.
Linear Polarizers
Linear polarizers are optical elements that only transmit the component of incoming light that is linearly polarized
along the polarizer axis. The light after the polarizer is then linearly polarized along this axis.

Quantum Optics Kit Chapter 3: Principles of Quantum Optics
Page 8 MTN036012-D02
The polarization of linearly polarized light passing a linear polarizer will be projected onto the polarizer axis, and
the intensity will be attenuated according to Malus’ Law:
 
(27)
Here,  and  are the intensities before and after the polarizer, respectively.  is the angle between the
polarization of the incoming light and the x-axis, while is the angle between the polarizer axis and the x-axis.
The intensity of circularly-polarized light will always be attenuated by a factor of 
when passing through a linear
polarizer, irrespective of the polarizer axis.
3.2 Mathematical Formalism of Quantum Mechanics
Before the quantum mechanical treatment of the electromagnetic field is introduced in the next section, a very
brief introduction to quantum mechanics is given in this section, only as far as will be required for the calculations
in the remainder of the chapter. A more comprehensive introduction can be found in the literature
4
,
5
.
3.2.1 Dirac Notation
In this chapter, the Bra-Ket notation introduced by Dirac will be used. In the following subsection, a brief overview
of the components and their mathematical and physical meanings are given.
3.2.2 General Case
A ket is written in the form . Mathematically it is a vector in a vector space V.
A bra is written in the form . Mathematically it is a linear form f: V→, mapping a vector into the plane of
complex numbers.
Letting the linear formact on a ket is denoted as . The result of this operation is a complex number.
3.2.3 Application in Quantum Mechanics and Quantum Optics
In quantum physics, quantum states can be represented by vectors in a complex Hilbert space (the exact nature
of that space depends on the problem at hand), which facilitates calculation in most cases. These vectors are
denoted by kets. The symbol(s) inside the ket are a label denoting the state, they are not a mathematical entity
on which operations can be performed. For example, in general and 
.
As an inner product 󰇛󰇜is defined in every Hilbert space (the being a placeholder for any possible element of
the Hilbert space), the bra can be understood as positioning the vector in the first position of this inner
product 󰇛󰇜. A bra acting on a ket is then understood as the inner product 󰇛󰇜. This is
typically interpreted as a projection of state onto the state , i.e., the probability amplitude for the state 
to collapse into the state . The inner product is antilinear, i.e., exchanging the components gives the complex
conjugate:

(28)
3.2.4 Linear Operators
Linear operators are denoted 󰆹and when acting on a ket, output a ket, e.g., 󰆹. A typical way to construct
linear operators is the outer product. If is a ket and is a bra, then the outer product is defined as 
and acts on a ket in the following way:
󰇛󰇜
(29)
Linear operators are distributive, i.e.:
4
M. Beck, Quantum Mechanics, Theory and Experiment. (Oxford University Press, Oxford, 2012).
5
R. Loudon, The Quantum Theory of Light. (Oxford University Press, Oxford, 1965).

Quantum Optics Kit Chapter 3: Principles of Quantum Optics
Rev. A, June 16, 2023 Page 9
󰆹
󰆹 
(30)
󰆹󰇛󰇜󰆹󰆹
(31)
In the following, we will list some special types of operators which are of importance later.
Adjoint Operator
For every linear operator 󰆹there exists an operator 󰆹, for which the following holds true:
If 󰆹, then 󰆹for all and 
(32)
The operator 󰆹(pronounced “A-dagger”) is called the adjoint operator to 󰆹.
Identity Operator
The identity operator 
has no effect on any state, i.e.:

for all 
(33)
Inverse Operator
The inverse operator to 󰆹is denoted as 󰆹and defined as:
󰆹󰆹 󰆹󰆹

(34)
Unitary Operators
A unitary operator is an operator for which the following equation holds true:

   

(35)
Projection Operators
An operator of the form is called a projection operator because it projects any state in the Hilbert
space onto the state . Projection operators are idempotent, i.e.:

(36)
This means that after projecting once onto the state any subsequent projection onto the same state does
not change anything.
Hermitian Operators
An operator is called Hermitian or self-adjoint, if it is identical to its own adjoint operator, i.e.:

(37)
Hermitian operators play a key role in quantum mechanics as they represent measurable physical quantities,
such as energy or momentum.
3.2.5 Eigenstates and Eigenvalues
For a linear operator 󰆹, if there exist one or more state vectors for which:
󰆹
(38)
these vectors are called eigenstates of 󰆹and the (in general complex numbers) are called eigenvalues.
If is an eigenstate of 󰆹, then the following relation applies to the corresponding bra vector and the adjoint
operator󰆹:
󰆹
(39)
Here, is the complex conjugate value of .
3.2.6 Commutator Relations
In general, operators are not commutative, i.e.: 󰆹
 󰆹. The commutator of two operators is defined as:

Quantum Optics Kit Chapter 3: Principles of Quantum Optics
Page 10 MTN036012-D02
󰆹
󰆹  󰆹
(40)
If 󰆹
, the operators are said to be commuting.
3.2.7 Bases of the Hilbert Space
In any Hilbert space, there exists at least one orthonormal basis, i.e., a set of vectors for which the following
relation holds:

(41)
With being the Kronecker delta function defined as:
 

(42)
An orthonormal basis contains exactly as many elements as the dimension of the Hilbert space and a linear
combination of these bases vectors can be used to describe any vector in the Hilbert space. Let be the basis
vectors in an N-dimensional Hilbert space, then any vector-normalized vector (meaning ) can be
expressed as:


(43)
Here, the are complex coefficients for which the following holds:


(44)
If a ket is expressed via a basis of the Hilbert space, the corresponding bra is:


(45)
Here, the basis vectors are replaced by their corresponding bra and the coefficients are the complex conjugates
of the ones used to express the ket.
Additionally, for an orthonormal basis, the following relation applies:


 
(46)
Here, is called a projection operator as explained in Section 3.2.4. The eigenstates of a Hermitian
operator in a finite dimensional Hilbert space always form an orthonormal basis, and the eigenvalues are real
numbers.
3.2.8 Observables
An observable is any physically measurable quantity, such as momentum, energy, polarization, and many more.
Every observable corresponds to a Hermitian operator, and the possible results of a measurement of the
observable are the eigenvalues of the corresponding operator.
For a system in any normalized state , the probability of measuring the value is:
󰇛󰐝󰇜
(47)
Here, is the eigenvalue corresponding to the eigenstate .
Using Equation (46) and the fact that the eigenvectors of a Hermitian operator form a basis of the Hilbert space,
we can rewrite any Hermitian operator with the eigenvectors as


 
(48)

Quantum Optics Kit Chapter 3: Principles of Quantum Optics
Rev. A, June 16, 2023 Page 11
3.2.9 Measurements and Superposition of Quantum States
Hidden in the prior subsection is an integral property of quantum physics, the superposition of states. As stated
previously, measuring an observable of a particle in the state can in general result in any eigenvalue of
the operator . This is because, prior to the measurement, the particle is in a superposition of all states .


The result of the measurement is not defined beforehand! The superposition coefficients give the
probabilities of different measurement results (and define the state ).
When considering multiple measurements, there are two cases to distinguish:
Multiple measurements on the same particle:
After a measurement that yielded as a result, the state is changed to . This means that every consecutive
measurement of the same observable will yield with 100% probability (unless the state was changed in the
meantime, e.g., by measuring a different observable). This process of changing the state due to a measurement
is often called a collapse of the state (superposition of states), as in: “The measurement resulted in so the
state collapsed to .”
Multiple measurements on identical particles:
Here, a critical difference between classical and quantum physics becomes visible: in classical physics, a quantity
like momentum is always well defined (even if unknown). When performing measurements of this quantity on
an ensemble of identical particles in the same state, one will always get the same result. This is not the case in
quantum physics: measurements on an ensemble of particles in the identical state can have different results.
The result before the measurement is not only unknown, but unknowable!
3.2.10 Expectation Values
If many measurements of an observable are performed on a state (not on the same system, but on an
ensemble of systems in the identical state), then the expected average of the measurement results is called the
expectation value of the measurement. It is calculated as follows:

(49)
The probability of measuring a specific eigenvalue of is:
󰇛󰐝󰇜
(50)
For the third equality we used the antilinear property of the inner product; see Equation (28). We can see that the
probability of measuring an eigenvalue is the expectation value of the projection operator onto the
corresponding eigenstate .
The variance of the measurement is:

(51)
3.2.11 Mixed States
States that can be expressed by a single ket vector are called pure states. However, there are states of the
electromagnetic field that cannot be represented by a pure state. Instead, they are statistical mixtures of pure
states, i.e., one can only give the probabilities for the system to behave as being in one of two or more pure states.
These states are called mixed states
6
and are expressed by the so-called density operator:


 

 
(52)
6
R. Loudon, The Quantum Theory of Light. (Oxford University Press, Oxford, 1965).

Quantum Optics Kit Chapter 3: Principles of Quantum Optics
Page 12 MTN036012-D02
The are the weights for the pure state and can be interpreted as the probabilities of the mixed state to behave
as if being in the corresponding pure state when an observable is being measured. Note that they are not the
probability of the mixed state being the pure state. The number N here can be any number.
3.2.12 Combination of Quantum Systems
Assume you have two quantum systems (such as particles), called A and B. Each system has a corresponding
Hilbert space, and , and a set of orthonormal basis vectors and . If the two quantum systems
interact, then the state of the combined system is an element of the enlarged Hilbert space  . The
symbol denotes the tensor product. The dimension of is the product of the dimensions of and and
is an orthonormal basis of .
Let be a normalized vector in and be a normalized vector in . Then is also
normalized and an element of . To simplify the notation, we define:

(53)
Expressed in the orthonormal basis , the state is:

  
(54)
The inner product between two vectors and in is defined as:

(55)
An observable of a subsystem is still represented by an operator. Let be an observable in subsystem A. Then
the operator is 
, acting only on the part of the combined state that describes subsystem A and leaving
subsystem B unchanged. Often, this is only written as .
The expectation value of for a combined state is:

(56)
3.3 Quantum Description of Light
In this section, we will give a brief overview of the quantum description of light, focusing on the light states that
are most important for the experiments in this kit. A more comprehensive introduction can be found in the
literature
7
.
To treat the electromagnetic field quantum mechanically, it is assumed that every mode of the field corresponds
to a harmonic oscillator
8
. The Hamiltonian of a mode of the electromagnetic field can then be written as:



(57)
Here,  
 with being the Planck constant, is the resonance angular frequency, and is the number
operator. The number operator can be expressed in terms of the annihilation operator and the creation operator
:

(58)
Quantum mechanics thus treats the electromagnetic field as quantized excitation in (energy) packages of .
These packages are called photons.
3.3.1 Fock States
A single photon is the fundamental excitation quantum of the electromagnetic field, i.e., the smallest possible
excitation of the field. It is described by the so-called Fock state . Fock states in general describe the
7
R. Loudon, The Quantum Theory of Light. (Oxford University Press, Oxford, 1965).
8
M. Beck, Quantum Mechanics, Theory and Experiment. (Oxford University Press, Oxford, 2012).

Quantum Optics Kit Chapter 3: Principles of Quantum Optics
Rev. A, June 16, 2023 Page 13
fundamental excitations of the electromagnetic field. They are the eigenstates of the Hamiltonian and of the
number operator:

(59)
The following equations hold for Fock states:

(60)

(61)
Here, the names of the operators and becomes clear. The annihilation operator acting on a Fock state results
in a Fock state with one less photon, while the creation operator acting on a Fock state results in Fock state with
one additional photon. The ground state of the electromagnetic field is and is called the vacuum state. It
describes the complete absence of photons.
The operators and do not commute. We can calculate the commutator by letting it act on a state and
apply the definitions above:
󰇟󰇠
󰇛󰇜
(62)
As this holds for any state , we obtained:
󰇟󰇠
(63)
The expectation value and the variance of the photon number in a Fock state are:

(64)

(65)
This means that a field mode in a Fock state contains exactly n photons; see the left graph in Figure 6. Generating
Fock states experimentally has long been a challenge. Nowadays, reliable methods exist to prepare light in the
state n = 1, but Fock states with higher numbers require increasingly complex techniques.
3.3.2 Coherent States
A plane electromagnetic wave with constant amplitude, such as one emitted by a (strongly attenuated) laser, is
different from a Fock state, with a number of photons that matches the energy contained in the wave. Instead,
such a plane wave is described by a coherent state.
Such a coherent state can be expressed as a linear combination of Fock states because Fock states form an
orthonormal basis of the Fock space. The coherent state is:





(66)
Coherent states are the eigenstates of the annihilation operator , i.e., . The annihilation operator
is not Hermitian, so in general, the eigenvalues are complex numbers. Coherent states are the closest quantum
mechanical analog to classical oscillators.
The mean photon number (expectation value of the number operator) of a coherent state is:

(67)
The variance of such a state is:

󰇛󰇜
(68)
Here, we used the commutator relation of the annihilation and creation operators:
(see Section 3.3.1). The variance is equal to the expectation value, which hints at a Poisson

Quantum Optics Kit Chapter 3: Principles of Quantum Optics
Page 14 MTN036012-D02
distribution of the photon numbers. Indeed, it can be proven that the probabilities of a coherent state measured
in different Fock states is Poissonian
9
, as depicted in the center graph of Figure 6.
3.3.3 Thermal States
Besides Fock states and coherent states, there are other, more chaotic states of light. This includes thermal
radiation emitted by a black body, fluorescent light sources, and LEDs. Chaotic light cannot be described by a pure
quantum state. Instead, it is a statistical mixture of pure states; for more information, see Section 3.2.11.
All the chaotic light sources above share the same photon number distribution for a fixed mean photon number
:
󰇛󰇜
󰇛󰇜
(69)
It can be shown that the variance for the probability distribution in Equation (69) is always:

(70)
As a black body is the most prominent example of this kind of light source, all states with this probability
distribution are called thermal states in the following even if the explicit density matrix depends on the exact type
of the source. For example, the state of a black body with a temperature can be expressed by the following
density matrix
10
:
󰇧
󰇨 
(71)
Here, is the angular frequency, is the Boltzmann constant, and the exponential of the number operator is
defined by a power series expansion. One can see from Equation (70) that the photon number variance of thermal
states is always larger than for a Poissonian statistic such as displayed by coherent states. Figure 6 shows a graph
of the photon number probabilities for a Fock state, a coherent state, and a thermal state with the same mean
photon number 5.
Figure 6 Photon number probabilities for a Fock state (left), a coherent state (center), and a thermal
state (tight), all with the same mean photon number <n> = 5. Be aware that the scaling for the first plot is
different than for the second and third to provide better visibility.
11
3.4 Proof of Quantized Light - Theory
In educational literature, the quantization of light is often motivated by the photoelectric effect
12
. However, this
phenomenon can be completely described by semiclassical theories that do not require a quantization of the
electromagnetic field. Only the energy levels of the atoms in the matter are quantized in these theories
13
, while
9
M. Fox, Quantum Optics: An Introduction. (Oxford University Press, Oxford, 2006).
10
R. Loudon, The Quantum Theory of Light. (Oxford University Press, Oxford, 1965).
11
The figure was composed with the Qutip Quantum Toolbox for Python.
12
B. J. Pearson and D. P. Jackson, American Journal of Physics, 78(5), 471 (2010).
13
R. Loudon, The Quantum Theory of Light. (Oxford University Press, Oxford, 1965).