Quantum Optical Detector-Conditions for Local Measurements

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for Local Measurements

by

Noah D. Hegerland Oldfield

Thesis submitted for the degree of Master in Science

60 credits

Faculty of mathematics and natural sciences UNIVERSITY OF OSLO

Spring 2021

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Detector-Conditions for Local Measurements

by

Noah D. Hegerland Oldfield

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Quantum Optical Detector-Conditions for Local Measurements http://www.duo.uio.no/

Printed: Reprosentralen, University of Oslo

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Abstract

In this thesis a study of operator conditions for a photodetection system is presented. We motivate such investigations due to an ongoing discussion about noncausal effects of certain correlations functions used in photodetection experiments [5], [19], [23], [6].

Our study is presented in terms of a system consisting of a quantized electromagnetic field and an atom which acts as a source for the production of strictly localized states [15] in the system.

A specific class of strictly localized states are considered which have been shown to be close to single photon states [9].

It is argued that a good measurement operator is local and is free of vacuum expectation value contributions for times prior to the arrival of signals from a source at the detection point. We argue that the conventional normal ordering and the time-normal ordering [14], [25] satisfy the detector conditions of a good measurement operator when applied to products of free field operators at detection timest₁ andt₂ prior to the arrival of signals from a source. The study is extended to interacting fields, but an adequate description of the analytical challenges motivate the need for a numerical simulation of interacting systems.

A numerical simulation of the detection of time-normal ordered of interacting field products gives vacuum expectation value contributions fort₁, t₂<0 of the order of 10⁻⁵as measured by a L2-norm approximation given by the normalized Frobenius norm of the signal amplitude matrix for the temporal area t1, t2 <0. The norm quantifies vacuum expectation value contributions prior to the arrival of source effects for N ∼10⁴ symmetrical time values in aN×N temporal grid. Further simulations indicate that the vacuum expectation value contributions increase as the pulse duration of the source is reduced. We suggest that our numerical results could serve as instances of general verification tests of the detector conditions for interacting systems.

The modified time-normal ordering [25] is investigated in our scheme and results indicate that measurement operators employing the modified ordering contain infinite vacuum fluctuations due to the well known divergence of the vacuum expectation value of the free field product at equal times.

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Preface

The work presented in this thesis is for the degree of master in science at the university of Oslo (UiO). The background material for the topic of the thesis was researched in the fall of 2020 in conjunction with the development of the ideas for which this thesis is based upon. The writing process has taken place during the spring of 2021.

I would like to give special thanks to my supervisor Johannes Skaar at the theoretical physics section at UiO. for all the discussions and advice along the way to making this thesis possible.

Also thanks to Jan Gulla at the section of theoretical physics at UiO. for the aid with calculations of expectation values and parts of the theory section 2.3.3.

Some of the main numerical calculations performed in section 3.2 were performed with an extended version of a program written by Jan Gulla in conjunction with research of single photon states [9]. All numerical computations have been performed with programs written in python 3.

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Introduction

A

^s far as we know, any observable event is caused by at least one other event such that the event is temporally restricted to the past. The sound of an ambulance is perceived by our ears, but the cause of the sound itself can be traced backwards in time to its originating source.

However, our causal intuitions may become challenged. In the well known thought experiment of Einstein’s train, observers can disagree about the order of physical events. One might then pause and ponder whether such disagreements between observers might also allow for a complete reversal of the order of events. If so, would this not also imply that there is an ambiguity as to what forwards and backwards in time means? In Minkowski spacetime it is known that spacelike separated points represent causally disconnected events. However, such events can never influence each other, as this would require superluminal motion. Even so one perhaps one may interpret a reversal of time to take place for such events. In the case of events which are causally connected, however, one always precedes the other in time. This is a fundamental consequence of the constancy of the speed of light in all reference frames.

Many interesting paradoxes result from an apparent causality violation in physical theory. For instance Newton’s second law and Schr¨odinger’s equation allow non-restricted magnitudes for the velocities of particles. The implications of allowing superluminal motion can be investigated in Minkowski spacetime. For instance if a rock is thrown such that it travels a distance faster than the speed of light until it comes to rest at some final position. An observer in a specific reference frame might then say that the rock hit the ground before it was thrown! We could also observe the dinosaurs of earth 65 million years ago alive, in real time. This could be made possible from earth by instantaneous communication with a planet that is situated 65 million light years away. The distant planet would simply transmit the images of the 65 million year old light instantly back to earth. These types of inconsistencies due to causality are known to occur with Newton’s laws or quantum mechanics, but are barely noticeable for velocities that are much lower than the speed of light.

This introduces the main idea behind the topic of this thesis. Which is how causality might lead to computational faults in physical models. Specifically we will investigate how various forms of operator ordering affect observables of the electromagnetic field (E-field) in photodetection experiments of a photodetection system. Such a system will be formally introduced in the main section 3.1.

Following Glauber’s quantum formulation of Hanbury-Twiss-Brown type photon counting experiments [8] there have been sporadic discussions about whether certain types of correlation

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functions in the theory of photodetection might yield such causality faults [5], [19], [25]. The gist of the suggested causality violations goes along the following lines. If an observable time dependent quantityA(t) at a time t is caused at a specific time and place by the activation of some physical source. Should it then not also be unobservable at a detection point which is separated from the source by some distance due to the constancy of the speed of light? In a detector’s frame of reference the signal is emitted at the retarded timetr, while the signal would be detected at the timeto. The detection timeto must then be fundamentally restricted by the delay caused by the distance which the signal must travel from the point of origin and to the point of detection.

Consider a plane wave source emitting a signal from the position−ctr in the reference frame of a detector for the retarded timetr >0. If measurements of the signal are performed locally at the origin, then the earliest possible observation time of the observableA(t) isto= 0.

Figure 1.1: Illustration of the x-axis of a detector for a plane wave source emitting a signalA(t) at the retarded timet_r. The signal travels at the speed of lightv=ctowards the detection point atx= 0 where theA(t) is measured locally.

It then begs the question of what sorts of detections we should expect prior to the signal arriving at the detection point. Is it sufficient to expect experimental measurements of the local observable A(t) to give the value zero before the signal actually reaches our point of observation? Might it not also be sufficient to measure any existing background noise or reference values as equivalent to zero? These ideas motivate the notion of detector conditions for which is one of the main topics of this thesis. However, we must also take into account that quantum fields are prone to contain infinite vacuum fluctuations. A field property which is usually dealt with by an appropriate operator ordering, which in turn acts to eliminate such divergences.

This thesis will be structured into four main parts. The first part consists of this introduction which will go on to explain the origin of the causality issues while giving a brief summary of its developments to the present time. The introduction will also include a brief motivation for why this problem warrants an investigation and why it might affect what we’ll call ultrafast optics.

The second part, theoretical background, will give a brief introduction to the relevant elements of field theory, quantum optics and the specific correlation functions and operator orderings which will be subjected to investigation in the main part. Following this the necessary elements of the pulse mode formalism, strictly localized close-to-single photon states and local operators will be introduced.

In the main section 3 we will investigate a system consisting of a free E-field and a source undergoing a transition from an initial state with all empty field modes to a final state where the source has created a strictly localized close-to-single photon state. The entire transition will be taken into account by identifying three events which are associated with individual time intervals.

First there is a time for which nothing happens in the system, second a source activates and affects

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the E-field, lastly the source deactivates and leaves the effects in the field and source system.

From there we shall investigate how the various operator orderings of products of both free and interacting field operators satisfy or fail to satisfy detector conditions. Operators satisfying the conditions will be dubbed good measurement operators. The time-normal ordering will be further examined numerically in a simulated photodetection experiment where we shall argue that nonlocal effects might occur and in turn would indicate noncausal effects due to the choice of operators. We conclude the main part by also computing the vacuum expectation value of field operators using a recently suggested modified ordering [25].

The findings and investigations of the main part will be discussed in section 4 and the concluding remarks of the thesis will be given in section 5.

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1.1 The Short Pulse Causality Problem

Here we will give a reiteration of the historical origin of what we shall call theshort pulse causality problem of quantum optics.

1.1.1 Roy J. Glauber and the Theory of Photodetection

Observables measured in optical experiments typically involve the amplitudes of the classical electromagnetic fieldEc(r, t), such as the electromagnetic field intensity atr= 0,I(t)∝E^∗(t)E(t) [17]. The electromagnetic field intensity is thus defined in terms of quadratic products of field amplitudes.

It is well known that coherent sources produce interference patterns. Such interference properties of the electromagnetic field are naturally quantified by correlation functions. Which in turn quantify what we could consider to be better described as a global property of the E-field, namely coherence.

For a classical electromagnetic field amplitude Ec(t), the correlation function for a Michelson interferometer [17] can be written in terms of the field amplitudes as

g⁽¹⁾(t₁, t₂) = D

E^∗_c(t1)Ec(t2)E rD

I(t1)ED I(t2)E

. (1.1)

Here the superscript (1) of g refers to the exponential degree of the intensity quantity in the numerator of (1.1), which is seen to be one, as there are only two field amplitudes. Hence, (1.1) is referred to as afirst order correlation function.

Any introduction to quantum optics should involve a brief introduction to what some might call the founding effect of quantum optics, which is the Hanbury Brown and Twiss (HBT) effect.

This section will be somewhat based upon a lecture given by Alain Aspect [1].

In the 1950’s paper by Hanbury et al. the authors sought to demonstrate that their interferometry method, based upon correlation measurements, could be used to determine the size of distant stars. One main advantage of their method was that it was much less prone to being affected by atmospheric noise. The experimental method consisted of directing monochromatic light from a source through a beam splitter, before measurements of the photons was performed by a photomultiplier. The photon measurements were finally sent through to a correlator¹.

1A correlator is essentially just the physical apparatus which computes the correlation function (1.1).

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11 1.1. The Short Pulse Causality Problem

Figure 1.2: A visualization of a simplified optical circuit for the Hanbury Brown and Twiss effect.

The correlation function for two detections at the same spatial position, but for different times t1, t2 was shown to be given by [4], [17]

g⁽²⁾(t1, t2) = D

I(t1)I(t2)E D

I(t1)ED

I(t2)E = D

E^∗(t1)E^∗(t2)E(t1)E(t2)E D

E^∗(t1)E(t1)ED

E^∗(t2)E(t2)E. (1.2) Since there is now a quadratic dependency of the intensities in the numerator we call (1.2) a second order correlation function.

An informal prescription for the HBT method [4], [1] can be described as follows. The HBT experiment essentially measures the coherence lengthL_c=λ/αbetween a set of detection points P₁ andP₂. Hereλis the wavelength of the measured light from some distant source, and αis the angular diameter of the object to be measured.

Figure 1.3: Illustration of the angular diameter and the distance between the points of correlation.

When the detectors are close, the correlation function gives a factor of 2, while at certain separated distances, it gives the value 1.

An important note about the first and second order correlation functions (1.1),(1.2) is that they are statistically related by the central limit theorem [1], [17]

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g⁽²⁾(t1, t2) = 1 +

g⁽¹⁾(t1, t2)

2

. (1.3)

On a historical note, the HBT method would actually cause a controversy at the time, which was due to the lack of a consistent quantum mechanical interpretation of their method. This came to spark the creation of the field of quantum optics!

In quantum mechanics, the electromagnetic field is understood by the definition of a Hermitian operator, which represent the physical observables of nature. In accords with the wave-particle duality, we should then also be able to state an interpretation of the second order correlation function (1.2) in terms of a scheme which is based on the statistical counting of photon particles.

This would mean that a replacement of the classical intensitiesI(t) in (1.2) with probabilities p(t) for detecting photon particles should be possible. The expectation values of the intensities are thus mapped onto the joint probabilities p(t1, t2) for the detection of two photons at times t₁, t₂ [1]

D

I(t1)I(t2)E

↔D

p(t1, t2)E

. (1.4)

This means one obtains the second order correlation function in terms of the photon probabilities

g⁽²⁾(t₁, t₂) = D

p(t1, t2)E D

p(t1)ED

p(t2)E. (1.5)

According to probability theory [2], we should expect (1.5) to giveg⁽²⁾(t1, t2) = 1 for independent events. Which would imply that the photons measured att1 andt2 are independent. However, the measurement resultg⁽²⁾(t₁, t₂) = 2 in the HBT method would then only make physical sense if the photons where in fact not independent. At the time this seemed to be a major inconsistency within the HBT method, as it seemed absurd to suggest that photons which were emitted from distant stars could be dependent.

In light of this, the authors HBT gave an explanation of this phenomena in a series of papers [3]

by interpreting the statistical relation (1.3). The first term was explained to be the contribution due to having true independent particles, essentially standard random noise. The second due to the randomness of the particle wave function phases, as beat notes of random waves. However, the formal quantum interpretation, coherent with quantum mechanics, is by most recognized to be stated first by Roy J. Glauber.

Glauber’s toy model, as explained by Alain Aspect [1], gives the crux of his main idea. Given a system of two photon emittersE₁, E₂ and two detectors D₁, D₂. All of which may be in either an excited state |ei or in a ground state |gi. The transition from both emitters being in the ground state, to both detectors becoming excited, indicates that a measurement of two photons has taken place. There are two paths that the photons could have taken, either each photon is measured by the corresponding indexed or the reverse indexed detector. We can illustrate this with some simple diagrams.

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13 1.1. The Short Pulse Causality Problem

Figure 1.4: Diagram representation of the possible photon paths in a detection experiment in Glauber’s toy model.

To each diagram there is associated a probability amplitude and Glauber argued that the amplitudes must be added before taking the squared modulus

Figure 1.5: Diagram representation of taking the squared modulus of the sum of the amplitudes.

which he argued meant that the probability amplitudes actually interfere.

In order to acquire the correlation function for photon counting, Glauber considered an interaction picture Hamiltonian of an atom source interacting with an electromagnetic field [19], [20]

H(t) =H0+V(t). (1.6)

Here H0 is the Hamiltonian for a quantum harmonic oscillator and V(t) is the interaction part of the Hamiltonian , given by

V(t) =−dE(t) =−d0

h

E⁺(t)L++E⁺L₋+E⁻(t)L++E⁻(t)L₋i

(1.7) where dis the electric dipole operator for electric dipole transitions between the atomic source states andE(t) =E⁺(t) +E⁻(t) is the sum of the positive and negative frequency parts of the electric field operator [19], [20].

Performing the rotating wave approximation (RWA) amounts to disregarding terms which both excite the field and atomic states [20]. In this example these are the terms E⁺L₋ and E⁻L₊. Which means the interaction Hamiltonian could effectively be approximated to

V(t)≈ −d0

h

E⁺(t)L++E⁻(t)L₋i

(1.8) Further details of Glauber’s procedure will not be required for our purposes. However, such detailed treatments can be found elsewhere[19], [20].

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For our purposes we shall find it sufficient to state Glauber’s result.

For what he defines as an ideal photon detector,²the probability per unit time that a photon is detected is [8], [17], [18]

dP(t)

dt ∝ hE⁻(t)E⁺(t)i. (1.9)

Which is the quantity that we shall refer to as the photon count rate. The photon count rate (1.9) can be generalized toN-fold delayed coincidence experiments for timest₁, t₂,· · ·, t_N, which effectively multiplies additional positive and negative frequency field operators in a specific order, according to [8], [14]

d

dtP(t₁, t₂,· · ·, t_N)∝

E⁻(t₁)E⁻(t₂),· · ·E⁻(t_N)E⁺(t_N)E⁺(t_N₋₁)· · ·E⁺(t₁)

. (1.10) The successes of Glauber’s photodetection theory can hardly be overstated in the current era of quantum optics [11].

From this point on we will not directly study the photon count rate. However, we shall bear in mind that it is proportional to the correlation functions, which will be studied in great detail.

1.1.2 Bykov and Tatarski

In a paper published in 1988, the authors Bykov and Tatarskii point out how one might identify a causality violation within the scheme of Glauber’s photon count rate correlation function (1.10).

Specifically they go on to demonstrate, by an example, that such correlation functions 1.10 do not vanish for a causal input signal [5]. They provide a reasoning for how a general causality violation can be identified due to the Glauber correlation functions. After which they go on to demonstrate such violations with a specific example. A brief account of their example will be given in this subsection.

Their general reasoning states that due to the integral form of input signals V(t), it can be shown that conjugate integrals relate the real and imaginary parts ofV by the Hilbert [21], [5]

transform

ImV(t) = 1 π

Z ∞

−∞

dt⁰ ReV(t⁰)

t−t⁰ . (1.11)

Such that for a signal that can be made explicitlycausal by the multiplication with a Heaviside function Θ (A.4)V(t)∝Θ(t), the signal is zero for t <0. However, due to the induced Hilbert transforms (1.11) there are nonlocal effects even for explicitly causal signalsV (1.11).

To demonstrate, informally, along the lines of the example given [5], we could define

V(t) = Θ(t) sinωt (1.12)

Issues of causality could then arise, if for instance (1.9), depends on both the imaginary and real parts of (1.12). We could then see directly that nonlocal effects would be induced by the Hilbert transform (1.11).

2Which is taken to bea system of negligible size ( atomic) which has frequency dependent photoabsorption probability[8]

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15 1.1. The Short Pulse Causality Problem The authors go on to suggest that it might be natural to replace the measurement operator in (1.9) by

hE⁻(t₁)E⁺(t₂)i → h:E(t₁)E(t₂) :i (1.13) where :: indicates the normal ordering of the operators such that all annihilation operators are placed to the right of all creation operators [22]. Such replacements are argued to remove the types of causality issues, as pointed out by the authors [5].

Other papers prior to Bykov and Tatarskii have also suggested causal issues with such correlation functions [6], by applying an expansion method with the Liouville operator [14].

1.1.3 Milonni, James and Fearn

The article titledPhotodetection and causality in quantum optics by Milonni, James and Fearn [19] can be considered a response to the notion of Bykov and Tatarski types of causality issues within Glauber’s correlation functions. Their response argues by the use of a simple method that exploits explicit causality, as in (1.12). They’re argument can be summed up as the following.

In essence, they follow Glauber’s original approach [8], except they explicitly keep the Heaviside function for a causal signal, such as (1.12), and they demonstrate that it makes a difference at which point one performs the rotating wave approximation (RWA). They show that if RWA is performed directly on the Hamiltonian, as Glauber does, it does indeed lead to a causality violation. Perhaps trivially since that simply gives Glauber’s correlation functions. However, if the RWA is performed at a later step in the computation [19], this gives explicitly causal correlation functions. The Glauber correlation function (1.10) would then become multiplied by a Heaviside function

d

dtP(t)∝

E⁻(t)E⁺(t)

Θ(t). (1.14)

And they go on to say that it is only when the RWA is performed by neglecting counter rotating terms in the Hamiltonian, that is becomes necessary to extend the typical frequency integrals to also include the lower negative infinity in the integration bounds of the spectral integral [19].

This notion of performing RWA at different stages has been studied in a recent paper [7]. The procedure of applying RWA to the Hamiltonian at the outset is referred to as the pre-trace rotating wave approximation.

1.1.4 Plimak and Stenholm

Following the paper by Milonni et al. in 1995 [19], a paper by Plimak and Stenholm [25] presents a continuation of the causality examinations of Bykov and Tatarskii where they point out that it is still an issue to identity universal causal quantities measured by a macroscopic detector. They present an example where the proposed causal correlation function (1.13) is shown by an example to exhibit non-causal effects. This example will be somewhat reproduced in section 3.2.1. They instead propose an alternative method of operator ordering based on a series of papers [23], [24].

The authors Plimak and Stenholm suggest to have identified a general operator ordering which is applied when computing Glauber-like correlation functions. More about operator ordering in section 2.2. They go on to replace this general ordering by another, which then leads to their supposedly general causal correlation function. This specific modified ordering will be examined

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in this thesis in section 3.3. The works of Milonni et al. can be clearly distinguished from those of Plimak and Stenholm, as their work does not seem to be concerned with the application of approximations, such as RWA. Rather their work is viewed as a continuation of the papers by Bykov and Tatarski [5] and D. Haan [6].

1.2 Motivations for the Investigations

The motivations behind the studies in this thesis can be summarized from three main viewpoints:

a philosophical, a phenomenological and a technological.

Due to Einstein’s special relativity, and perhaps from our own physical intuition about cause and effect, we have all reason to expect that nature obeys the causality principle. The fact that the velocity of light is finite leads to the manifestation of causality as a computational factor in physical theories. If so, then theories which do not obey causality, might lead to incorrect predictions that do not correspond to our experimental values.

The issues pointed out by Bykov and Tatarski seem to be practically related to detection experiments where the source duration is taken to be an experimental parameter. Thus, it is natural to ask whether a hypothetical experiment or simulation could be proposed in order to test such noncausal effects for various operators? If so, which observables would be relevant for such an experiment? Note that breaches in causality should perhaps not be called phenomenological in the usual sense, but instead in the sense where we view them as theoretical artefacts.

The technological motivation behind the studies in this thesis could be related to the practical use of such phenomena where short source duration times become important. Such as in the field of ultrafast quantum optics, where such considerations might be crucial in order to generate photons at ultrashort source durations on demand. In order to achieve maximum precision within such experiments, one must then also make an account for the possibility of Bykov and Tatarski type nonlocal effects.

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Chapter 2

Theoretical Background

In this chapter, we shall cover the necessary theoretical background material for the investigations performed in the main part.

First we shall give a sufficient description of a quantized electromagnetic field for our purposes.

Then a sufficient account will be given for the properties of the E-field and its separation into positive and negative frequency parts. A brief overview of how the notions of causality and locality are related will also be given.

Furthermore we will state the general elements of operator ordering before moving on to relevant topics from quantum optics, such as the analytic signal transformations, which will become important for our analysis in the main chapter.

Finally the topics of the pulse mode formalism, local operators and the specific quantum states known as strictly localized states will be introduced.

2.1 Elements of Quantum Field Theory

In the time of Newton, the gravitational force was thought to influence objects apparently instantly over any distance. After the experimental works of Faraday, among others, the conventional idea of a field was supposedly introduced to physics by Maxwell in 1849 [26].

Classical vector fields are usually considered to be the gravitational field and the non quantized electromagnetic field. However, classical fields and particles were long viewed as distinct concepts of nature. The mental picture is typically that fields serve as the mediums for which particles would move through, like a branch following a stream’s forced trajectory through the watercourse.

Quantum field theory is thus viewed by many as the unification of these ideas, providing a mathematical framework for describing the universe in terms of fields which obey the postulates of quantum mechanics and special relativity.

2.1.1 The Principle of Locality

The principle of locality in physics can be stated heuristically as the notion that any physical object is only directly influenced by its immediate surroundings. Thereby excluding the possibility of instantaneous influences by a signal sent from a far away source, limited by the finite velocity

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of lightc. In quantum field theory, it is common nomenclature to refer tolocal functions orlocal theories. The latter is simply a theory which satisfies the principle of locality.

Alocal function, however, is typically in the physics literature [22, p. 271], [10, p. 39] a function f of the form of

f(x, ∂φ(x),· · ·, ∂ⁿφ(x)) (2.1)

for finite nand a function φ. In layman’s terms the function f(x) can be understood as local at the coordinate x in the following sense. If f can be expressed by simple functions φ and derivatives ofφ. Then by analogy, a Taylor expansion off about some point x₀

f(x) =f(x0) + ∂

∂xf(x0)x+1 2

∂²

∂x²f(x0)(x−x0)²+· · · (2.2) could be understood as local in the sense that the farther away fromx0we wish to have precision about the function f, the more global is the required information about the functionf. This intuition is commonly used in introductions to local Lie algebras as well.

Our use of the termlocal, which will be defined in the subsection 2.4, will at least be strongly related to this notion.

2.1.2 Quantum Electrodynamics

Afield φ(x) can in a broad sense be defined as any quantity continuously distributed throughout some space by mapping the set of pointsxonto some field valueφ(x). Aquantum field is a field that also obeys the postulates of quantum mechanics and special relativity. The coordinates xare defined as vectors in Minkowski spacetime x^µ = (ct,r) [22], while the fieldsφ(x) act as Hermitian operators on the system Hilbert space. In the Heisenberg picture, such time dependent field operators, in conjunction with some quantum state|ψi, describe quantum systems.

Our study in this thesis will be conducted for the single component electromagnetic field operator E(t,r). Where in section 3.1 the operator will be considered in the interaction picture by the free field representation introduced in this subsection. While in section 3.2, we shall consider an E-field operator in a specific representation in the Heisenberg picture. The measurements will be performed at the originr= 0 in the frame of our detector.

In the interaction picture, the field operator can be written [17] in terms of its Fourier expansion as

E(t) = Z ∞

−∞

dωE(ω)a(ω)e^−iωt (2.3)

whereE(ω) are the electromagnetic field mode functions, which we will take to be E(ω)∝√ ω, [9]. We will abusively refer toE(ω) as simply, field modes.

1

The field modes are multiplied by the annihilation and creation operators{a, a^†}for a quantum harmonic oscillator field mode. Furthermore, we can separate (2.3) into a sum of integrals taken over the integration limits (−∞,0) and (0,∞)

1A fieldmoderefers to natural frequencies that the field will tend to vibrate with.

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19 2.1. Elements of Quantum Field Theory

E(t) = Z ∞

0

dωE(ω)a(ω)e^−iωt+ Z ∞

0

dωE^∗(ω)a^†(ω)e^iωt. (2.4)

The terms in (2.4) thus completely separate the positive e^−iωt and negativee^+iωt frequencies, such that the field operator to be written simply as the sum of its positive and negative frequency parts

E(t) =E⁺(t) +E⁻(t). (2.5)

Here the positive and negative frequency parts are conjugates, in the sense

E⁺(t)†

=E⁻(t). (2.6)

The field operator (2.5) is what we shall refer to as thefree field operator.

However, it is easy to get used to the intuition that the positive and negative frequency parts of the fieldE(t) arealwaysassociated with the annihilation and creation operators{a, a^†}. This is not always the case and depends on the representation of the E-field. The positive and negative frequency parts are generally functions of both creation and annihilation operators. Even so we can show that for a general representation of a field operator, the conjugate property (2.6) still holds. Take a general separation of an E-field operator

Z ∞ 0

dω A⁺(ω)e^−iωt+ Z ∞

0

dω A⁻(ω)e^iωt (2.7)

where the field modesA^±(ω) are both assumed to contain a mix ofaanda^†. By the hermiticity of (2.7) it follows that the positive and negative frequency parts still satisfy the conjugate relation (2.6). This holds as long the frequency integral can be separated into integrals over purely positive and negative frequency parts as in (2.4).

2.1.3 Causality

Causality is the relationship between cause and effect of events, while a causal event is often defined to be an event such that its cause precedes its event in time. Another common term for the notion of a causal event isEinstein causality orrelativistic causality. These terms define the notion of causality in terms of its contingency upon the finiteness of the velocity of light.

Perhaps one could say specialized in a more appropriate way and made concrete.

In the language of special relativity provides a Minkowski diagram, can be taken to represent an event S by the origin of the diagram. Since the velocity of light is finite, the cause of the event at the origin must then be another event which must be situated in the past light cone.

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Figure 2.1: Minkowski diagram illustrating the eventS and its past light cone.

Thus, we might takeS to not be causal if the cause of S is outside the past light-cone. Since the speed of light c is the supremum velocity of the universe, it follows from special relativity that all communication between causal events at different spacetime points, must do so by a communication velocity that is less than or equal toc.

What types of physical objects are then to obey causality? Would it make sense to say that an observable quantity, like the momentum of a particle should respect causality? The answer to such questions will not specifically be tackled in this thesis. However, we could say that since causality is a property of events in spacetime, we might seek to view causal properties of observables through the guise of events in Minkowski spacetime.

Since causality must always be understood in terms of relationships between at least two points in spacetime, we could call timelike four-vectors causal, in the sense that they lie within the future or past light cone of a Minkowski diagram. Noncausal effects might be useful to view as transformations of such vectors. For example, if we performed a transformationA of a timelike vectorv^µ and the resulting transformation gave a spacelike vectorA(v^µ)

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21 2.2. Operator Ordering

Figure 2.2: Minkowski digram illustrating a rotation A of a timelike vector v^µ resulting in a spacelike vectorA(v^µ). The 45^◦ tangents in dotted lines indicate the speed of lightc.

For the purposes of this thesis, this has simply been a practical intuition for a view of causality violations in terms of Minkowski diagrams.

We should also note that there is also a usage of the term causality within the theory of Fourier transformations. The termcausal in this context [21] is used for so calledcausal time functions , whereas a functionf(t) iscausal if fort <0, we havef(t) = 0.

In this thesis we will usually refer to Einstein causality, if nothing else is specified.

2.1.4 Relationship between Locality and Causality

In order to discuss causality, signals between events require a means to physically travel from A to B. One might say that locality, as discussed in 2.1.1, is one way of ensuring this. In this sense, we could state that causality requires locality. However, locality alone might not be sufficient as it does not restrict the velocity at which information travels. Which means perhaps locality in addition to demanding the finiteness of the velocity of light is sufficient. In this thesis, however, we shall define a local operator in subsection 2.4 by a related, but more specific notion of the locality discussed in these subsections.

2.2 Operator Ordering

The idea of operator ordering refers to the concept of assigning a specific conditional order to how a set of operators should act on some space. Often the topic is introduced in introductory courses in quantum field theory. For instance one encounters the common normal ordering scheme of operator ordering usually when introduced to computing vacuum expectation values over classical fields.

A common example where normal ordering is applied can be given by the computation of the expectation value of the total field momentum Pof a classical field φ(x) [22]

hPi=h0|1 2

Z ∞

−∞

d³p p (2π)³

apa^†_p+a^†_pap

|0i. (2.8)

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Applying the commutation relation [ap, a^†_p0] = (2π)³δ(p−p⁰) for the field modes gives hPi=h0|1

2 Z ∞

−∞

d³p p (2π)³

2apa^†_p+δ³(0)

|0i (2.9)

which evaluates to infinity due to the delta function contributionδ³(0).

These sort of infinite vacuum contributions are expected for quantum fields, since the fields are defined in terms of integrals over an infinite number of field modes. However, they pose as a common nuisance in calculations. We could then make the argument that only differences in momentum, is what can be measured. Hence we should in practice correct for thislargenumber bysubtracting it from the actual value. Giving the remaining observable difference.

The normal ordering is thus a means of computationally implementing this physical condition by ensuring that the annihilation operators are all ordered to the right of all creation operators. Since the vacuum state|0idefines the lower cut-off bound of the energy eigenstates of photon frequency modes, this means the annihilation operators evaluate the vacuum state to zeroap|0i= 0. The delta function in (2.9) would then be eliminated.

So the actual measurable quantity would then be taken to be what is known as the normal ordered vacuum expectation value of the total momentum

h0|:P:|0i (2.10)

Generally we could define anordering as the operationOrd(R, A), where Ais an operator and Ris an ordering condition.

The notation :P: indicates that all annihilation operators contained withinPare to be ordered to the right of all creation operators.

In the same way we shall go on to define various orderings in terms of an ordering ruleR. The rule tells the ordering how to act on an operatorA(a, a^†) which is given by arbitrary products and sums of annihilation and creation operators which may act on respective Hilbert spaces, for instance

A(a, a^†) =a₁a^†₂a₂a₃a^†₄ (2.11) which will be used as our defining example.

Definition 1 (Normal Ordering) LetR=N be the condition of moving all annihilation operators to the right of all creation operators. Then

Ordn

N, A(a, a^†)o

=:

a1a^†₂a2a3a^†₄

:=a^†₂a^†₄a1a2a3

denotes the normal ordering of A(ai, a^†_j).

Another useful ordering is introduced typically when computing the general solution to the time evolution operator of the Schr¨odinger equation. For a time dependent Hamiltonian noncommut- ing at the timest, t₀, the Taylor expansion of

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U(t, t₀) =1+

∞

X

n=1

(−i

~ )ⁿ

Z t t₀

dt₁ Z t1

t₀

dt₂ · · · Z tn−1

t₀

dt_nH(t₁)H(t₂)· · ·H(t_n) (2.12) is called theDyson series [22].

By ordering all the operators in (2.12) such that all the later time Hamiltonians are to the left of all earlier time Hamiltonians in the product of the series, we obtain [22]

U(t, t0) =Texpn

− i

~ Z t

t0

dt H(t)o

. (2.13)

Note that there are no physical assumptions which are made going from (2.12) to (2.13). It is a true equality. In this case, the ordering enabled the Dyson series to be written in a nice and compact form.

In the case of time orderings the operatorAwill be taken to be a function of time variables such that it is separable into products of other operatorsAi(ti) evaluated for the timesti∈ {t1,· · ·tN} forN ∈N.

Definition 2 (Time Ordering) LetR=Tbe the condition of moving all operators which are evaluated at the latest times to the leftof operators evaluated at earlier times. Then

Ordn

T, A(t1,· · ·, tN)o

=Tn

A(t1,· · ·tN)o

=A1(t1)· · ·AN(tN) denotes the time ordering of A(t1,· · · , tN)fort1> t2>· · ·> tN.

A note on the time ordering. The operators A(t1,· · · , tn) =A1(t1)· · ·AN(tN) are assumed to have time dependencies which are fixed to the operators. This means that ifA1(t1) =ae^it¹ and A2(t2) =a^†e^it², then the time ordering would also order the annihilation and creation operators, even if they are not explicitly time dependent. The time dependence is thus defined for the operator overall.

We shall also define the reverse of the time ordering

Definition 3 (Anti Time Ordering) Let R= ¯Tbe the condition of moving all operators which are evaluated at the latest times to therightof all operators evaluated at earlier times.

Then

Ordn

T, A(t¯ ₁,· · · , t_N)o

= ¯Tn

A(t₁,· · · , t_N)o

=A(t_N)· · ·A(t₁) denotes the anti time ordering of A(t1,· · · , tN)fort1> t2>· · ·> tN.

Now, as was stated in (2.7). It is not generally the case that positive and negative frequencies e^∓iωt always come paired with annihilation and creation operators as E-fields in the interaction

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picture. For general E-field operators in the Heisenberg picture, certain operator orderings become ambiguous, such as the normal order of positive and negative frequency parts. One way of dealing with such ambiguities is to apply the general property which was pointed out in 2.1.

Since a general E-field can always be written as a sum of its positive and negative frequency parts, we can define an ordering which instead orders the frequency parts directly. For this we will assume the operatorA to be expressed as a general sum and product of positive and negative frequency parts, our defining example is taken to be the simple productA⁺A⁻. However, the ordering is defined for arbitrary products of different positive and negative frequency part operators such asA⁺₁A⁻₂A⁺₃ and so on.

Definition 4 (Frequency Ordering) Let R=F be the condition of moving all operators containing only positive frequenciese^−iωtto theleftof all operators containing only negative frequenciese^+iωt. Then

Ordn

F, A⁺A⁻o

=:A⁺A⁻ :_F=A⁻A⁺ denotes the frequency ordering ofA⁺A⁻.

We define the frequency ordering in this way as it equates to the normal ordering when the operatorA⁺A⁻ corresponds to positive and negative frequency parts of the free fieldE(t), as in the interaction picture.

Finally we can define what can perhaps be aptly described as an exotic ordering. It has been used in recent studies of causality [25] and is motivated as a general ordering based on the natural orderings performed for correlation functions in photodetection theory [8], [14], [18].

The ordering can be defined in terms of the already defined orderings.

For this we shall need an operator defined by terms of purely positive and negative frequency parts in addition to terms which mix positive and negative frequency parts. Taking the notation in definition 4, we shall add a time dependence such that mixed products are writtenA^±(t₁,· · ·, t_N), while pure positive and negative frequency parts are simply denoted by A⁺(t₁,· · ·, t_N) and A⁻(t₁,· · ·, t_N). We shall thus define the ordering over operators of the form

A(t1,· · ·, tN) =A⁺(t1,· · · , tN) +A⁻(t1,· · ·, tN) +A^±(t1,· · ·, tN). (2.14) The first and second terms can be written as products of positive and negative frequency part operators, for instance

A⁺(t1,· · ·, tN) =A⁺(t1)· · ·A⁺(tN). (2.15) While the last termA^± indicates that it is a mixed product ofA⁺andA⁻ which in turn can be written as products such as (2.15).

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Definition 5 (Time-Normal Ordering) Let R =T N be the condition of time ordering positive frequency operators and anti time ordering negative frequency operators, in products of such operators. The frequency ordering will be performed on the result. Then

Ordn

T N, A(t1,· · · , tN)o

=Tn

A⁺(t1,· · · , tN)o + ¯Tn

A⁻(t1,· · ·, tN)o + : ¯Tn

A⁻(t1,· · · , tN)o Tn

A⁺(t1,· · · , tN)o :F

is the time-normal ordering ofA(t1,· · ·, tN).

Finally we shall give an effective definition of the ordering suggested by Plimak and Stenholm [25] to address the short pulse causality problem. It is a modified time-normal ordering in the sense that for terms of an operator A, which are products of purely positive or negative frequency parts, they should have the time or anti time ordering performed before the frequency parts are extracted. We shall define the analytic signal transforms which extract frequency parts in the following subsection 2.3.2. We shall state more clearly how positive and negative frequency parts are extracted in section 2.3.2. For now a positive frequency part can be viewed as the result of transformationsA^±_t such that for a product of positive frequency part operators A⁺(t1,· · ·, tN) =A⁺(t1)· · ·A⁺(tN), we use the following compact notation for sequences of the transformsA⁺_t

A⁺(t1)· · ·A⁺(tN) =A⁺_t₁· · · A⁺_t_NA(t1)· · ·A(tN)≡ A⁺_t₁_,···_,t_NA(t1)· · ·A(tN). (2.16)

Definition 6 (Modified Time-Normal Ordering) Let R = M T N be the condition of time-normal ordering, but reversing the order of which the positive and negative frequency parts are extracted in the purely time or purely anti time ordered terms. Then

Ordn

M T N, A(t1,· · ·, tN)o

=A⁺_t₁_,···_,t_NTn

A(t1,· · · , tN)o

+A⁻_t₁_,···_,t_NT¯n

A(t1,· · ·, tN)o + : ¯Tn

A⁻(t1,· · · , tN)o Tn

A⁺(t1,· · · , tN)o :F

is the modified time-normal ordering of A(t₁,· · ·, t_N).

A general word of warning about orderings, inserting commutators before an ordering is performed might create some mathematical paradoxes which best be avoided. A simple demonstra- tion can be made by the following example.

LetV be some operator with positive frequency partV⁺and negative frequency partV⁻which together satisfyV⁻V⁺=1, but not the reverseV⁺V⁻6=1. If we were to compute the frequency ordering ::F on the productV⁺V⁻, this would yield the result

:V⁺V⁻:_F=V⁻V⁺=1. (2.17)

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From the definition ofV we might then also perform computations before taking the ordering, obtaining

:V⁻V⁺:F=:1:F=1. (2.18)

Alas, the error made in (2.18) has resulted in the faulty result16=1. So one should then avoid performing such computations within the ordering operations and instead view the argument of an ordering as more akin to a label.

2.3 Quantum Optics

Quantum optics is the aspect of optics that deals with the behaviour of light as broken up into individual quanta, and with these quanta behaving, uh I would have to say, rather strangely.

Roy J. Glauber We have already introduced the historical development of quantum optics in chapter 1, resulting in a fully quantum explanation and interpretation of the Hanbury-Brown-Twiss experiment. In this section we shall define some variants of correlation functions which are the main subjects of interest for our investigations in the main section. Also necessary topics from quantum optics will be introduced or stated, such as the analytic signal transformations and the pulse mode formalism.

2.3.1 Photon Counting

How many photons are registered at a photon detector within a given time? This is quantified by thephoton count rateP_i→f for transitions of an initial stateito a final statef. The graduate student of physics will first have encountered such probability measures of quantum observables when derivingFermi’s golden rule in a quantum mechanics course.

For our purposes of photon counting we shall be interested in three different variants of the count rate. Which one will depend on the operator ordering and the choice of positive and negative frequency parts of the operator in question. However, in general, the count rateP(t) for a given number of photon detections is proportional to anth degree correlation function [17], such that

ρ⁽ⁿ⁾(t)∝g⁽ⁿ⁾(t). (2.19)

We shall be concerned with second order correlation functions, such that n = 2. Three such correlation functions will be defined where the two first will be studied primarly in the first two section of the main chapter 3.1-3.2, while the third will be studied in the third section of the main chapter 3.3. First there is Glauber’s originally derived correlation function using the rotating wave approximation as described in section 1. The correlation functions will be denoted bygas in (2.19), but we shall assign subscript letters to distinguish the correlation functions. As usual

|0idenotes the vacuum ground state of a quantized electromagnetic field.

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27 2.3. Quantum Optics

Definition 7 (Glauber Correlation Functions) The second order correlation function corresponding to Glauber’s original derivation is given by

g⁽²⁾_G (t₁, t₂)∝ h0|E⁻(t₁)E⁺(t₂)|0i whereE denotes the free field operator.

Definition 8 (Bykov-Tatarskii Correlation Functions) The second order correlation function corresponding to Bykov and Tatarskii’s suggestion [5], is defined as

g_BT⁽²⁾(t₁, t₂)∝ h0|T :E(t₁)E(t₂) :|0i

where T :: is the time-normal ordering (5) T :: andEdenotes the general field operator in the Heisenberg picture.

Finally we shall mention the correlation function suggested by Plimak and Stenholm to be a candidate for a solution to the short pulse causality problem. They suggest a correlation function applying a modified time-normal ordering [25]denotedT ::_{P S}, which can be stated as

Definition 9 (Plimak-Stenholm Correlation Functions) The second order correlation function corresponding to Plimak and Stenholm’s suggestion [25], is defined as

g⁽²⁾_{P S}(t₁, t₂)∝ h0|T :E(t₁)E(t₂) :_{P S}|0i

whereT ::_{P S} is the modified time-normal ordering (6) andEdenotes the general field operator in the Heisenberg picture.

2.3.2 Analytic Signal

As we have stated in section 2.1, the electric field operatorE(t) in the interaction picture representation can be separated into positive and negative frequency parts

E(t) = Z ∞

0

dωE(ω)a(ω)e^−iωt+h.c=E⁺(t) +E⁻(t) (2.20) this means that we shall associate positive frequencies with negative exponentials e^−iωt and negative frequencies with positive exponentialse^iωt. The reason for the reverse definition is that usually positive frequencies refer to the positional part exponentialse^±ik·r[17], but as was noted in chapter 1, we have set r= 0 for our detection frame.

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The positive and negative parts of a field operator can generally be defined in a variety of different ways. Perhaps the most intuitive and straightforward is by simply taking the Fourier transformation of a given signal, then truncating all negative or positive frequencies, before finally taking the reverse Fourier transformation back again. This should then leave only positive or negative frequencies in the field operator, denoted by a positive or negative superscriptf^±

f^±(t) =A^±{f(t)}=F⁻¹n Fn

f(t)o

Θ(±ω)o

(2.21) whereF is the Fourier transformation [18] and Θ denotes the Heaviside function (A.4).

From the alternative definition of the Heaviside function Θ(x) = ¹₂(1 +sgn(x)) and the convo- lution theorem [18], another form of (2.21) can be derived as

A^±n f(t)o

=f(t) +iHn f(t)o

(2.22) whereHdenotes the Hilbert transform [18].

There is also a third formulation which follows from theSokhotski-Plemej theorem, see for instance [13, p. 336], [27, p. 113]

A^±n f(t)o

= Z ∞

−∞

dt⁰δ^±(t−t⁰)f(t⁰) (2.23) where

δ^±(t) = lim

ε→0

1

t∓iε (2.24)

are the positive and negative frequency parts of the Dirac-delta function.

2.3.3 Pulse Mode Formalism

The pulse mode formalism is essentially a different choice of basis states for which the appropriate photon states can be represented.

We can define a photon state |ηi in a Fock space with the well known frequency mode basis

|n₁, n₂,· · ·i. Here the single photon states |n_ii are defined in terms of the number of photons that occupy theith frequency mode with frequencyω_i. We can thus perform the change of basis

|n1, n2,· · ·i → |nξ1, nξ2,· · ·i (2.25) such that the photon states are instead given in terms of the number of photons in specific pulse modes ξn(ω) [9]. The pulse modes form a complete set {ξn(ω)}_n∈_N which span theL2(0,∞) function space.

Furthermore, the pulse modes satisfy

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29 2.3. Quantum Optics

X

n

ξ_n^∗(ω)ξn(ω⁰) =δ(ω−ω⁰), (completeness) Z ∞

0

dω ξ_n^∗(ω)ξm(ω) =δnm, (orthonormality).

Such that an occupied pulse mode state can be defined according to the well known definition for the frequency mode basis

|nii= a^†_n_i

√n_i |0ii. (2.26)

Here|1iiwould indicate that a single photon occupies theithpulse mode. We will only explicitly denote the photon states with subscripts such as 1i when necessary, otherwise conventional frequency basis notation will be used.

The annihilation and creation operators are redefined in this formalism, given in terms of the pulse modes ξand of the frequency basis ladder operators

a^†_n= Z ∞

0

dω ξn(ω)a^†(ω). (2.27)

We can use the complete set of pulse modes {ξn(ω)}n to write the free field operators E(t) in terms of the pulse mode basis as

E(t) =X

n

En(t)an+X

n

E_n^∗(t)a^†_n (2.28)

where the free field amplitudes are defined as En(t) =

Z ∞ 0

dωE(ω)ξn(ω)e^−iωt. (2.29)

From these definitions, we can see from (2.28) that the free field retains the form of a sum of positive and negative frequency parts E(t) =E⁺(t) +E⁻(t) where the positive frequency part is easily identified as

E⁺(t) =X

n

En(t)an. (2.30)

Quantum Optical Detector-Conditions for Local Measurements

for Local Measurements

by

Noah D. Hegerland Oldfield

Thesis submitted for the degree of Master in Science

60 credits

Faculty of mathematics and natural sciences UNIVERSITY OF OSLO

Spring 2021

Detector-Conditions for Local Measurements

by

Noah D. Hegerland Oldfield

Abstract

Preface

Contents

Chapter 1

Introduction

A

1.1 The Short Pulse Causality Problem

1.1.1 Roy J. Glauber and the Theory of Photodetection

1.1.2 Bykov and Tatarski

1.1.3 Milonni, James and Fearn

1.1.4 Plimak and Stenholm

1.2 Motivations for the Investigations

Chapter 2

Theoretical Background

2.1 Elements of Quantum Field Theory

2.1.1 The Principle of Locality

2.1.2 Quantum Electrodynamics

2.1.3 Causality

2.1.4 Relationship between Locality and Causality

2.2 Operator Ordering

2.3 Quantum Optics

2.3.1 Photon Counting

2.3.2 Analytic Signal

2.3.3 Pulse Mode Formalism