Understanding quantum measurements within the statistical interpretation

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BSc physics and astronomy

Theoretical physics

Bachelor thesis

Understanding quantum measurements

within the statistical interpretation

by

Ernst Ippel

10437118

September 10, 2015

15 ECTS

April 2015- July 2015

Institute for theoretical physics Amsterdam

Supervisor:

Prof. dr. T.M. Nieuwenhuizen

Second assessor:

Prof. dr. J. de Boer

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Abstract

In quantum physics, the quantum measurement problem has remained a topic of discussion through-out the years. In this paper, an understanding of quantum measurements will be obtained from the solution of a dynamical model, namely, the Curie-Weiss model. In this model, the z-component of a spin-1₂ is measured through an interaction with a magnet M, consisting of N 1 spins, which are weakly coupled to a phonon bath B. The magnet initially lies in a metastable paramagnetic state, then gets triggered by the coupling with the spin-1₂ after which it relaxes towards either the up or down ferromagnetic state, resulting in a magnetization which can be read off. The magnet thus acts like a pointer, which informs us about the initial state of the tested spin. We will rely on the fact that a measurement is an interaction between the tested quantum system S and the macroscopic apparatus A=M+B. Due to the macroscopic scale of the apparatus, describing a measurement requires the use of quantum statistical mechanics, which, in turn, leads us to the statistical interpretation. Working in this quantum statistical mechanical framework restricts us to the use of ensemble theory, in which a state is represented by a density operator ˆD(t) that describes our knowledge of an ensemble of identically prepared systems S+A. The evolution of this density operator is governed by the Liouville-von Neumann equation of motion and consists of three main stages. In the first stage certain correlations are created and then destroyed, which is displayed as the decay of the off-diagonal blocks of ˆD(t). Thereafter, the correlations between the z-component of the spin and the magnet are created and registered, obtaining a final state ˆD(tf). Although ˆD(tf) has the required diagonal form, it describes an ensemble of

mea-surements, but one would like to make statements about individual runs. To make such statements, the state ˆD(tf) has to be decomposed into physical subensembles. However, in this decomposition of ˆD(tf)

lies a mathematical difficulty due to a quantum ambiguity, which is overcome by additional dynamics in the apparatus near the end of the measurement leading to relaxation of all possible subensembles towards certain stable states, satisfying the hierarchic structure. Thereafter reduction follows, in which a state can be assigned to the full system S+A in an individual run of the measurement. Altogether, quantum statistical mechanics, and therewith the statistical interpretation, appears to be sufficient to fully explain and understand quantum measurements.

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Populair wetenschappelijke samenvatting

Quantummechanica is een natuurkundige theorie die de wereld op het kleinste niveau beschrijft en al in ontwikkeling is sinds het begin van de 20ste eeuw. Hoewel de quantumtheorie door de jaren heen succesvol is gebleken in het beschrijven van deze quantumwereld, staat de interpretatie ervan nog steeds ter discussie. Een van de belangrijkste aspecten van de quantummechanica, en de interpretatie ervan, zijn quantum metingen. Dit omdat een quantum meting de enige manier is om iets te weten te komen over een quantum object. Het is echter juist in deze quantum metingen waar het zogeheten “quan-tum meetprobleem” ontstaat. Dit probleem komt voort uit een conceptuele tegenstelling tussen onze ervaring uit de macroscopische wereld, waarin elke gebeurtenis deterministisch is, en de quantumtheo-rie, welke onherleidbaar gebaseerd is op kansen. Deze tegenstelling heeft op zijn beurt geleid tot een wiskundige tegenstrijdigheid in de beschrijving van de evolutie van een quantum object tijdens een met-ing. In deze beschrijving wordt een quantum object, voordat er een meting is uitgevoerd, beschreven door een golffunctie en gedraagt zich dus als een golf. Als er dan een meting gedaan wordt, vindt er een zogeheten “ineenstorting van de golffunctie” plaats en gedraagt het quantum object zich opeens als een deeltje. De evolutie van de golffunctie wordt tijdens dit proces in eerste instantie geleid door de Schr¨odinger vergelijking, welke continu en deterministisch is. Echter, de ineenstorting van de golffunctie wordt weergeven door von Neumann’s projectie postulaat, welk discontinu en non-deterministisch is. Buiten deze wiskundige tegenstrijdigheid, wordt het meetapparaat vaak ook geheel buiten beschouwing gelaten, terwijl dit juist een essentieel onderdeel is in de beschrijving van een meting.

In dit onderzoek gaan we uit van het feit dat een meting een interactie is tussen een macroscopisch meetapparaat A en het te testen quantum object S. Door de macroscopische schaal van het meetap-paraat is enkel het gebruik van quantum mechanica niet meer voldoende om deze interactie te kunnen beschrijven, en zal er gebruik gemaakt moeten worden van quantum statistische mechanica. Dit vereist dat we werken met ensembles van identieke systemen S+A inplaats van individuele systemen, en leid ons tot de zogehete statistische interpretatie. Deze interpretatie is een minimalistische interpretatie, wat wil zeggen dat het een interpretatie is waarbij een zo min mogelijk aantal veronderstellingen gemaakt wordt wat betreft de wiskundige beschrijving van de quantummechanica. Om de metingen the illustreren zal er gewerkt worden met het Curie-Weiss model, waarin het te testen quantum object S de z-component van een spin-1₂ is, en het meetapparaat A uit een magneet M en een phonon bad B bestaat. In eerste instantie ligt de magneet in een paramagnetische toestand, waarna deze gekoppeld wordt aan S en, afhankelijk van de toestand van de z-component van de spin-1₂, relaxeert naar een van de ferromagetis-che toestanden. Tijdens dit relaxatie proces dumpt de magneet zijn energie in het phonon bad, waarna het gehele systeem in een thermisch evenwicht eindigt. Tegen het eind van de meting vindt in het apparaat nog een relaxatie plaats die het mogelijke maakt een verband te leggen met fysische subensem-bles, waarna de magnetisatie van de magneet afgelezen kan worden en ons informeert over de toestand waarin de spin-1₂ zich in eerste instantie bevond. Al met al blijkt het gebruik van quantum statistische mechanica, en daarmee de statistische interpretatie, voldoende om quantum metingen te beschrijven en te begrijpen.

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1 Introduction

The interpretation of quantum mechanics has been a topic of interest in physics for over a century. In order to better understand the way things work at the microscopic level, and to determine which interpretation of quantum mechanics is sufficient, one must look into quantum measurements, since measurements are the only way to gain some information about a physical property of a quantum object [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. However, it is in these quantum measurements where the so-called “measurement problem” arises. This measurement problem occurs when one looks at the dynamics of the wave function, governed by the Schr¨odinger equation, and von Neumann’s first intervention often called “collapse of the wave function”, which seem to be in contradiction with one another in a mathematical way. The Schr¨odinger equation represents the evolution of a quantum system, which is a continuous process and seems to be correct when no measurements are made. If then a measurement is made, the wavefunction “collapses” and the evolution has suddenly become a discontinuous process, where in an subsequent measurement the same value will be found, meaning the wavefunction has stopped evolving. This problem originates from a conceptual contradiction between the irreducibly probabilistic theory of quantum mechanics and our everyday experience, in which each individual measurement results in a well-defined outcome. When a quantum measurement is treated as a quantum mechanical interaction between the measurement device and the tested system, the principle of the wavefunction being a lineair combination of eigenstates, also known as the superposition principle, seems to prevent this occurrence of an unique, well-defined outcome. In the following sections we first present a dynamical model which will help in elucidating quantum measurements, then an understanding of quantum measurements will be obtained through the solution of this dynamical model, and in the end some conclusions will be drawn from these results.

1.1 Interpretations of quantum mechanics

The contradictions discussed above have led to many discussions in the past about the fundamentals of quantum mechanics and have given rise to a number of accepted interpretations throughout the years. Certain ones are more widely accepted than others and the most commonly used ones will be discussed shortly below.

1.1.1

Copenhagen interpretation

One of the first interpretations of quantum mechanics was the Copenhagen interpretation. This inter-pretation is the one mostly used in education and is taught to students at the undergraduate level. It was devised by Werner Heisenberg and Niels Bohr and tells us that quantum events should be seen as indeterminstic, meaning these events are irreducibly probabilistic. Another aspect of the Copenhagen interpretation is the wave-particle duality, in which a quantum object can be seen as a particle and a wave at the same time. This is where the so-called “collapse of the wave function” comes in when a measurement is made. This principle states that, in the absence of a measurement, the quantum object can be seen as a wave that can be represented by a wave function and is in a superpostion of eigenstates. If then a measurement is made, this wavefunction “collapses” into one of its eigenstates and loses its wavelike behaviour, thus acting like a particle. This interpretation of the wave-particle duality has evoked many discussions over the years and physicists still argue about its physical meaning.

1.1.2

Many-worlds interpretation

The many- worlds interpretation was formulated by Hugh Everett in 1957 and is an alternative to the Copenhagen interpretation. It assumes that all the possible ways in which an event can take place, do take place, but all in different “worlds”. This also implies that all the possible histories that could have taken place in our “world”, but did not, and all the possible futures that will not take place in our ”world”, will take place or have taken place in some other “world” or “universe”. Every possible quantum outcome thus occurs in some other “universe”. The many-worlds interpretation provides

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an understanding of how the deterministic equations of quantum physics and the occurence of non-deterministic events both can be correct. However, it relies on assumptions which cannot be confirmed in experiments, and is therefore not fully satisfactory.

1.1.3

Statistical interpretation

The statistical interpretation, also called the ensemble interpretation, can be seen as a minimalist in-terpretation in the sense that it has the least amount of assumptions associated with the mathematical representation of quantum mechanics. In this interpretation, a state is not represented by a wave func-tion, but instead by a density operator, which can be seen as an abstract mathematical object which describes the state of an ensemble of identical systems. Here an ensemble is defined as a being a num-ber of identically prepared systems on which the same experiment can be performed. The fundamental difference between the Copenhagen interpretation and the statistical interpretation is that in the Copen-hagen interpretation a (pure) state |ψi fully describes the state of an individual system, where in the statistical interpretation, the state, or density operator, which is an incoherent superposition of pure states as ˆρ =P

ipi|φii hφi|, describes the statistical properties of an ensemble of identical systems. Also,

the statistical interpretation is nondeterministic, meaning it doesn’t want to reduce quantum mechan-ics to a deterministic process and sees it as irreducibly probabilistic. As it turns out, working in this minimalist interpretation is sufficient to fully understand what happens during quantum measurements.

1.2 Quantum measurements

In quantum mechanics, a measurement is a dynamical process in which a system S is coupled to an apparatus A. In this process some correlations are created between the initial state of the tested system S and the final state of the apparatus A [1]. When, for example, one wants to measure an observable

b

O of a system S, where bO has eigenvalues Oi and eigenvectors |Oii, this measurement is made through

an interaction with an apparatus A. This apparatus A then gives a certain ”eigenvalue” Ai of some

physical quantity ˆA of the apparatus, where this physical quantity ˆA acts like a pointer. This value Ai

has a one-to-one correspondence with an eigenvalue Oi, from which can be inferred that the system S

initially was in the state |Oii.

As said above, performing a measurement on a system S is the only way of obtaining some information about a physical property of a quantum object. Yet it is in these measurements where the paradoxical-ity of quantum mechanics arises, namely, the question of how the superposition principle of quantum mechanics can be compatible with the fact that a single run of a measurement leads to a well-defined outcome [1]. In order to gain some information about S, one must make an observation of A. When this observation is made, one can infer some quantitative information about the system S.

1.2.1

Difference between classical and quantum measurements

One of the differences between a measurement in the classical sense and a measurement in the quantum mechanical sense lies in the non-commuting nature of observables in quantum mechanics. This means that when certain physical quantities of a system S are measured simultaneously, they cannot both take on a well-defined value, as there is always an uncertainty between them. Another difference lies in the random nature of observables in quantum mechanics; in the classical regime, the values that an observable can take are known or can be known, where in the quantum regime these values cannot be known prior to measurements. A third difference lies in the perturbation of the apparatus A to the system S. In the classical regime, this perturbation can be made negligible, leaving the initial state of S virtually unchanged. In quantum mechanics this perturbation is not negligible due to the macroscopic scale of the apparatus compared to the tested system, and one can ask if S really was in the measured state at the initial time. These differences are the reason quantum mechanics only provides us with probabilities to find a certain value of a physical quantity. However, it must be noted that these probabilities only arise through repeated measurements on an ensemble of identical systems, and not through an individual run

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of a measurement. Therefore, throughout this paper, a state of a system always refers to the state of an ensemble of identical systems, and a measurement always refers to repeated measurements performed on this large statistical ensemble of identical systems [1]. In this sense, a measurement thus provides all the possible eigenvalues Oi of some observable bO of S.

1.2.2

Ideal measurements

In this paper, the measurements that are analysed are so-called ideal measurements. For a measurement to be ideal, it has to satisfy some requirements. One of them is that the disturbance, discussed above, of the apparatus A to the system S has to be minimal. This is accomplished by setting certain constraints on the strenght of the coupling between the system S and the apparatus A. Another requirement is that the apparatus A has to be macroscopic, which means N 1, where N is the number of particles that A contains. This macroscopic property of the apparatus is essential because the apparatus should act like a pointer from which a well-defined value can be read after a measurement. Another reason is that a macroscopic apparatus can be much better controlled than a microscopic apparatus, which in turn leads to a more controllable measurement.

1.2.3

Required mechanisms for successful measurements

In a measurement process there are three main stages that have to be fulfilled in order for a measurement to be successful. The first stage is a mechanism called truncation, which sets in after the system S and the apparatus A are coupled. Due to this coupling, certain correlations between S and A will be created. Formally, truncation is the decay of the off-diagonal terms of the state, which is represented by a density operator, of S+A. This density operator describes our knowledge about a statistical ensemble

ε

of runs of a measurement performed on the compound system S+A, and can be written in matrix form when expressed in some basis. Physically, this decay means that certain correlations between S and A will vanish after some time. Which terms of the density operator of S+A are represented as the off-diagonal elements, and thus vanish, depends on the basis in which the truncation takes place. This basis is one in which both the tested observable bO of S and the pointer variable ˆA of A are diagonal [1], thus commuting. A concept closely related to truncation is reduction. The difference is that truncation is the vanishing of the off-diagonal terms of the density operator of S+A for the whole set

ε

of runs, where reduction is the transformation of the initial state of S+A into a final state for a single run of the measurement [1]. This reduction can be seen as the “collapse” of the wave function.

The second stage of a measurement is registration. After the truncation, the diagonal elements of the density operator of S+A for the whole set

ε

of runs must display certain correlations in order for a measurement to be successful. Such a measurement can only be successful when a complete correlation between an eigenstate |Oii of the tested observable bO of S and the corresponding pointer indication Ai

of A is created. The establishment of these correlations is termed as registration. These correlations ensure that, when one measures a value Ai of the pointer variable ˆA in some run, the observable bO of

S has taken, with certainty, the corresponding eigenvalue Oi [1].

The third stage is the sorting of the whole ensemble

ε

of runs, according to the pointer indication Ai, into

pure subensembles

ε

_{i after the registration. However, preceding this stage there arises a mathematical} difficulty, which lies in the decomposition of the full ensemble

ε

into physical subensembles

ε

_sub. As it turns out, finding the right decomposition requires some sort of dynamical process within the apparatus, in which the state of S+A for an arbitrary subensemble

ε

_subis led to equilibrium. This dynamical process is termed as subensemble relaxation, and is a pure dynamical process, where the sorting of runs is a mere updating of our probabilistic description [1].

1.3 Quantum statistical mechanics

At the end of a measurement, the pointer variable ˆA, wich indicates one of the values Ai, should be able

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to this macroscopic scale, quantum mechanics alone is it not sufficient to describe the dynamics of the interactions between the tested system S and the apparatus A. In order to fully describe these interac-tions between a macroscopic and a microscopic object, the use of non-equilibrium quantum statistical mechanics is needed.

1.3.1

Irreversibility of the measurement process

In a measurement, both the truncation and the registration must be irreversible processes in order to provide a well-defined, observable outcome. After truncation, S+A ends up in a mixed state which is represented by a diagonalized density matrix, where its diagonal elements entail the required correlations between the tested observable and the apparatus. Since only these correlations are desired, this density matrix has to remain diagonal. From this it follows that the truncation has to be an irreversible process so that recurrences of the off-diagonal terms, which entail the un-desired correlations, of the density matrix of S+A are prevented. The registration has to be an irreversible process because of the requirement that the apparatus A has to register the obtained pointer indication Aiin a robust and permanent manner [1].

This permanent registration is crucial in the measurement process, since it makes an observation of the registered result possible. For robust registration to take place, an amplification, within the apparatus A, of the signal of S is needed. To ensure this amplification, and the irreversibility of the truncation and registration, a macroscopic apparatus A is thus needed, which asks for the use of quantum statistical mechanics. Due to this macroscopic scale, the initial state of this apparatus cannot be expressed as some pure state |ψi, since preparing A in a pure state requires a controlling of a complete set of commuting observables [1], which, in practice, is impossible. Rather than in a pure state, A lies initially in a mixed state, which is represented by some density operator. Equally, the final state of the apparatus is also a mixed state and represented by some density operator. These mixed states arise from the fact that the number of state vectors for a given pointer indication Ai is huge for a macroscopic object [1], in the

sense that some macrostate may have many microstates in statistical physics. This results in not being able to represent such states with a single state vector.

1.3.2

Representation of states in quantum statistical physics

In a way, a measurement process resembles a relaxation process as seen in statistical physics, where the apparatus tends to an equilibrium state with highest macroscopic entropy S. However, there are some fundamental differences between a measurement in quantum physics and a relaxation in statistical physics. The first is that A, initially in a metastable state, can have multiple final equilibrium states with corresponding pointer indication Ai. The second difference is that, in a quantum measurement, A

needs some sort of triggering from the measured system S to evolve to an equilibrium state, where this final state of A has to be correlated to the final state of S. In statistical physics, this evolution to the state with highest entropy S, occurs spontaneously.

Since the states of S and A are represented by density operators, the evolution of the full joint state of S+A is not governed by the schr¨odinger equation, but instead by the Liouville-von Neumann equation of motion i~dˆρ/dt = [ ˆH, ˆρ]. At some time t < 0, before the measurement, the density operator of S is denoted as

r

ˆ

(0), and the density operator of A as ˆR(0). The density operator of the joint state space of S+A,which we denote by ˆD(0), is then written as the tensor product of the two individual state spaces as [1]

ˆ

D(0) =

r

ˆ

(0) ⊗ ˆR(0). (1) At the end of the measurement, at some time t = tf, the final, normalized state ˆriof S can be are written

as [1] ˆ ri= 1 pi Πir(0)Πˆ i, (2)

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where Πi is the projection operator which projects the initial state of S onto one of its final states ˆri,

and pi is defined as [1]

pi≡

Tr

s(ˆr(0)Πi), (3)

which functions as a normalizing factor. When we denote the final states of A as ˆRi, we can define the

final density operator of the compound system S+A, for the full ensemble

ε

of runs, as [1] ˆ D(tf) = X i (Πi

r

ˆ

(0)Πi) ⊗ ˆRi= X i

p

i

r

ˆ

i⊗ ˆRi. (4)

Eventually, we want to obtain a final state of S+A in the form of ˆDi=

r

ˆ

i⊗ ˆRi [1] for any subensemble

ε

i of runs, implying that the state for the whole set

ε

of runs can be written asP

i

piDˆi when looking

at (4). Such a subset

ε

_{i is obtained by unambiguously splitting the whole ensemble}

ε

of runs into these subensembles with corresponding fraction pi of runs, and corresponding pointer indiciation Ai.

If this final state of S+A can be produced, reduction can be realized, which means that the state ˆDi

can be assigned to S+A in an individual run of a measurement. However, it is in this splitting of the whole ensemble

ε

into subensembles

ε

_{i where difficulties arise and the quantum ambiguity comes in. To} overcome these difficulties, a so-called hierarchic structure is needed. This hierarchic structure requires that, at the final time t = tf, any subensemble

ε

subis represented by a density operator

ˆ

Dsub(tf) =

X

i

q

iDˆi, (5)

where the qiterms represent arbitrary weights. When this hierarchic structure is fulfilled, reduction can

be justified.

The truncation, registration and subensemble relaxation are going to be exemplified by the Curie-Weiss model, which will be introduced in the following section.

2 The Curie-Weiss model

The model which is used to help in elucidating the quantum measurement problem in this paper is the Curie-Weiss model [1]. In this model the tested object is a microscopic system S which is coupled to a macroscopic apparatus A.

2.1 The whole system S+A

In the whole compound system S+A, the tested system S will be taken as a spin-1₂, and the apparatus A will consist of a magnet M and a phonon bath B. The observable ˆO of the spin-1₂ that will be measured, is the z-component

s

ˆ

z of its spin. This observable

ˆ

s

z is represented by its conventional pauli operator,

in the form of the matrix

ˆ

s

z= ~₂

σ

ˆ

z =~₂ 1 0 0 −1 ! , (6)

and has eigenvalues

s

i equal to ±1, leaving the ~/2 out for simplicity. To prevent the statistics of

ˆ

s

z from being disturbed too much, it is required that it nearly commutes with the hamiltionan of the

compound system S+A. This hamiltonian of the whole system S+A can be written in terms of individual hamiltonians as [1]

ˆ

H = ˆHS+ ˆHSA+ ˆHA, (7)

where ˆHSA is the hamiltonian of the coupling between S and A.

The apparatus A= M+B should initially lie in some metastable state, from which it can undergo a phase transition into one of several equilibrium states by interacting with S. This phase transition can be seen

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as a symmetry breaking, in which the initial symmetric state ˆR(0) of A has unbroken invariance, but the final equilibrium states ˆRiare asymmetrical, with broken invariance. Since

s

ˆ

z has two eigenstates with

eigenvalues

s

i= ±1, it is logical to choose an apparatus with two stable final states, which then must

have a one-to-one correspondence with these eigenstates of

s

ˆ

z. To allow for this feature, an apparatus

in the form of A = M+B is chosen, as issued above. In this model, also known as the Ising model [251], the magnet M consists of N 1 spins where each spin n has corresponding Pauli operators

σ

ˆ

_a(n), with a = x, y, z and n = 1, 2, ..., N . At some temperature T ≤ Tc, where Tc is the Curie temperature, the

magnet M can transit to two possible ferromagnetic states due to interaction with S. The magnetization ˆ

m of the magnet in these ferromagnetic states has the values m ' ±1, and will act as the pointer indication Ai. This magnetization is denoted as [1]

ˆ m = 1 N N X n=1

ˆ

σ

(n)_z , (8)

where N has a finite value. For the metastable initial state ˆR(0) of A, the paramagnetic state of M is chosen. The magnet is prepared at a temperature T0 > Tc to ensure this paramagnetic state, after

which it can relax towards one of its final ferromagnetic states ˆRi, with i =↑, ↓, at T0≤ Tc. The phonon

bath B is initially prepared at a temperature T < Tc, as to make sure the magnet can dispose its energy

during its relaxation. At the end of the measurement, the pointer variable Ai will have taken the value

+mF when A is in the state ˆR↑, and −mF if A is in the state ˆR↓, from which some information about

S can be inferred.

2.1.1

The tested system S

As mentioned above, the tested observable

ˆ

s

z of S must commute, or atleast nearly, with the the

hamiltonian ˆH of S+A in order for a measurement to be ideal. The easiest way to satsify this feature is setting the hamiltonian of S equal to zero, as ˆHS = 0, which can be justified by the fact that the measured

observable

s

ˆ

zis almost constant during an ideal measurement. The hamiltonian of the coupling between

S and A is given by the spin-spin coupling in the z-direction, denoted as [1] ˆ HSA=

−gˆ

s

z N X n=1

ˆ

σ

(n)_z = N gˆszm,ˆ (9)

where g > 0 is the strength of this coupling. The coupling will be turned on at the beginning of the measurement at t = 0, and turned off at some time t < tf [1]. The number N of spins in M in (9) should

be taken sufficiently large, but finite, as to guarantee the occurence of a clear phase transition form ˆR(0) to some final state ˆRi.

2.1.2

Features of the apparatus A

Since the apparatus A consists of two individual systems, namely, the magnet M and the phonon bath B, its hamiltonian can also be written in terms of individual hamiltonians as [1]

ˆ HA= ˆHM+ ˆHM B+ ˆHB, (10) where we denote ˆHM as [12] ˆ HM = −N X q=2,4 Jq ˆ mq q + ˆVM = −N J2 ˆ m2 2 − N J4 ˆ m4 4 + ˆVM, (11) where ˆm is the magnetization, Jq is the coupling constant for each q-plet of spins σ(n) of M and ˆVM

(99) is an additional interaction term to which we’ll come back in section 5. In this paper, only the q = 4 interactions, also known as super-exchange interactions will be considered, for which the lower bound 3J4 > J2 must be satisfied. This is because, for these super-exchange interactions, the magnet

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and as shown in [1], first-order interactions are better able to satisfy the requirements for a successful registration. When only considering the pure quartic case q = 4, the coupling constant J2 of second

order transitions can be set to zero, which results in the hamiltonian for the magnet M being written as [1] ˆ HM = −N J4 ˆ m4 4 + ˆVM = − J4 4N3 N X i,j,k,l=1

ˆ

σ

_z(i)

σ

ˆ

_z(j)

σ

ˆ

_z(k)

ˆ

σ

(l)_z + ˆVM (12)

The main reason for choosing quartic interactions over quadratic interactions is that, for q = 4, the paramagnetic equilibrium state is stable, even at temperatures T < Tc, and the lifetime of this state is

much longer than the duration of a measurement.

Since phonons are bosons, the hamiltonian of the interaction between M and B can be choosen in the standard spin-boson form [13,14,15], denoted as

ˆ HM B= √

γ

N X n=1 (ˆσ(n)_x Bˆ_x(n)+ ˆσ(n)_y Bˆ_y(n)+ ˆσ_z(n)Bˆ(n)_z ) =√

γ

N X n=1 X a=x,y,z ˆ σ(n)_a Bˆ_a(n) (13) where ˆBa(n)is some hermitean linear combination of phonon operators [1] and γ represents the strength

of the coupling between M and B, which must be weak, meaning γ 1 [1]. These interactions between ˆ

σ_a(n)and ˆB(n)a lead the apparatus A to equilibrium. The hamiltonian for the bath alone is given by [1]

ˆ HB= N X n=1 X a=x,y,z X k

~ω

k

ˆ

b

†(n) k,a

ˆ

b

(n) k,a, (14)

where each set of phonon modes, associated with each component a of each spin σ(n)_{, is labelled by}

k, n, a, with for every k a corresponding eigenfequency

ω

k, and ˆb

(n)

k,ais the ladder operator.

2.2 States and interactions

It must be kept in mind that, in this paper, a state refers to the state of a large statistical ensemble of identically prepared systems, characterized by a density operator, and a measurement refers to repeated measurements, or an ensemble of measurements, on this statistical ensemble of systems.

The full density operator ˆD of S+A will be expressed in the eigenbasis of the tested observable

s

ˆ

z, with

eigenstates |↑i and |↓i with corresponding eigenvalues

s

i= 1 for i =↑ and

s

i= −1 for i =↓, as [1]

ˆ D = ˆ_R_↑↑ _Rˆ_↑↓ ˆ R↓↑ Rˆ↓↓ ! , (15)

where each block ˆRij, with i =↑, ↓ and j =↑, ↓, is an operator that entails the correlations between S

and A and lives in the state space of A. By tracing out the bath B form these blocks ˆRij, we can obtain

the operators in the state space of the magnet as [1] ˆ

Rij= TrBRˆij, (16)

where again i =↑, ↓ and j =↑, ↓.

To obtain the marginal density operators of S+A at any time t, we can trace either the system S, the magnet M or the bath B out of the full density operator ˆD as follows [1]

ˆ

r

= TrAD,ˆ R = Trˆ SD,ˆ RˆM = TrS,BD,ˆ RˆB = TrMR = Trˆ S,MD,ˆ D = Trˆ BD,ˆ (17)

where

r

ˆ

belongs to the tested system S, ˆR to the apparatus A, ˆRM to the magnet M, ˆRB to the phonon

bath B and ˆD to S+M. Since the full density operator ˆD is written as a density matrix in the eigenbasis of

ˆ

s

z, the marginal density operator ˆD of S+M can also be expressed in matrix form as [1]

ˆ D = TrBD =ˆ ˆ_R ↑↑ Rˆ↑↓ ˆ R↓↑ Rˆ↓↓ ! , (18)

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where the operators ˆRij, given in (16), entail the full statistics of S+M. Now by tracing M out of (18),

the density operator of S can also be obtained in matrix form as [1]

ˆ

r

= TrMD =ˆ

r↑↑ r↑↓

r↓↑ r↓↓

!

=

r

↑↑|↑i h↑| +

r

↑↓|↑i h↓| +

r

↓↑|↓i h↑| +

r

↓↓|↓i h↓| , (19)

where the terms

r

ij = TrMRˆij, with i =↑, ↓ and j =↑, ↓, describe the system S alone. Since the only

correlations of interest are the ones between the tested observable

ˆ

s

z of S and the apparatus A, and not

the correlations between the transverse components

ˆ

s

x,

s

ˆ

y and A, we note that (15) should end up at

some time t = tf as [1] ˆ D(tf) = p↑Rˆ⇑ 0 0 p↓Rˆ⇓ ! = p↑|↑i h↑| ⊗ ˆR⇑+ p↓|↓i h↓| ⊗ ˆR⇓= X i piDˆi, (20)

where the off-diagonal blocks, which represent these correlations between A and the transverse com-ponents of the spin, have vanished. The ˆR⇑ and ˆR⇓ terms represent the density operators of the final

equilibrium states of M+B, in which M either has a magnetization m⇑ in the positive z-direction, or a

magnetization m⇓ in the negative z-direction at the end of a measurement. The pi terms, with i =↑, ↓,

represent he probabilities of these events occuring, where, according to born’s rule, these terms are given by p_i = TrSr(0)Πˆ i = r↑↑(0) with i = j =↑, ↓. Since the state of the bath doesn’t change significantly

during the measurement, it can be traced out of (20) as in (18), giving rise to the final state of S+M alone as [1] ˆ D(tf) = p↑RˆM ⇑ 0 0 p↓RˆM ⇓ ! = p↑|↑i h↑| ⊗ ˆRM ⇑+ p↓|↓i h↓| ⊗ ˆRM ⇓, (21)

which describes the full statistics of the correlated states of the tested spin ˆs of S and the magnet M at the end of a measurement. Looking at (17), we can see that ˆR = TrSD = ˆˆ R↑↑+ ˆR↓↓ which gives

rise to the form ˆRM = ˆR↑↑+ ˆR↓↓ for the densitiy operator of M. From this operator the probability

distribution Pdis

M (m, t), which gives the probabilities of the observable ˆm taking one of its eigenvalues

m, where −1 ≤ m ≤ 1, can be derived as [1]

P_Mdis(m, t) = TrMRˆM(t)δm,mˆ with

X

m

P_Mdis(m, t) = 1, (22) where dis stands for discontinuous and δm,mˆ is the projection operator which projects the operator

ˆ

RM(t) on the subspace m of ˆm. Since M is macroscopic as N 1, and the level spacing is given by

δm = 2/N , the magnetization m can be treated as a continuous variable and (22) becomes PM(m, t) = N 2P dis M (m, t) with Z 1 −1 PM(m, t)dm = 1. (23)

In the next section we will see that the density operator of M only depends on the spin observables ˆσz(n)

through its hamiltonian ˆHM as in (12). When this is the case, the operator ˆRM(t) can be obtained, by

inverting equation (22), as [1] ˆ RM(t) = 1 G( ˆm)P dis M ( ˆm, t), (24)

where G( ˆm) is the multiplicity of the observable ˆm of M, where this multiplicity gives the number of microstates for which ˆm takes on a certain eigenvalue m. Since the operators of S+M, given in (18), only depend on the observable ˆm, they can also be written in the form [1]

ˆ Rij(t) = 1 G( ˆm)P dis ij ( ˆm, t), (25)

where then the correlations of components between the spins of S and M are given by [1] C_xdis= P_↑↓dis+ P_↑↓dis, C_ydis= iP_↑↑dis− iPdis

↓↓ and Czdis= P dis

↑↑ − P↓↓dis, (26)

The term Pdis

↓↓ for example, can be seen as the joint probability that S is in the state |↓i and ˆm has

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2.2.1

Initial states

As displayed in section 1.3.2, at the inital time t = 0, the density operator of S+A can be written as ˆ D(0) = ˆr(0) ⊗ ˆR(0), where ˆr(0) is given as in (19) by

ˆ

r

(0) = ˆr↑↑(0) rˆ↑↓(0) ˆ r↓↑(0) rˆ↓↓(0) ! , (27)

and the state ˆR(0) of apparatus A= M+B can be expressed as the tensor product of the individual state spaces of M and B as [1]

ˆ

R(0) = ˆRM(0) ⊗ ˆRB(0). (28)

where M lies in a metastable paramagnetic state at a temperature T0= _β1₀ > Tc, but the whole apparatus

A must lie in some non-equilibrium state. This initial non-equilibrium state of A is realized by preparing A in an equilibrium state at a temperature T0 at some time t 0, and then bringing the temperature

of the bath B suddenly down from T0to T during the time interval −τinit < t < 0. This time τinit has

to be much shorter than the time in which the relaxation of M, towards one of its ferromagnetic states under influence of B, takes place, otherwise it could already be in some ferromagnetic equilibrium state before the measurement has even started, which would result in a false outcome.

Due to the macroscopic scale of the magnet M, its density operator is denoted as [1] ˆ

RM(0) =

1 ZM

e−β0HˆM ₍₂₉₎

where ZM is the partition function and ˆHM is the hamiltonian of M given in (12). The bath B initially

lies in some equilibrium state at a temperature T = _β1 < Tc and its density operator, that describes the

independent modes of the set of phonons, is denoted as [1] ˆ

RB(0) =

1 ZB

e−β ˆHB_, ₍₃₀₎

where the bath hamiltonian ˆHB is given in (14). The initial difference in temperature between M and

B at t = 0, where T0> Tc> T , is required because the bath has to drive the total apparatus A=M+B

to a thermal equilibrium state through a weak coupling γ, where, in this relaxation process, the magnet can deposit its energy into the bath, resulting in a state where the maget has a temperature T < Tc

and is in one of its ferromagnetic states. This coupling γ has to be sufficiently weak as to make sure that the spontaneous relaxation from the initial, locally stable paramagnetic state towards one of the ferromagnetic equilibrium states takes much longer than a measurement, in which the relaxation is not spontaneous but requires M to be triggered by S. A necessity for ensuring such an initial paramagnetic state is that, when ˆHSAis turned on at t = 0, the probability distribution of M, given in (23), lies around

m = 0, which means the spin ori¨entations in M are fully randomized, resulting in zero magnetization. Taking t = 0 in (23) gives [1] PM(m, 0) ' 1 √ 2π∆m

e

−m2_/2∆m2 = s N 2πδ2₀

e

−N m2_/2δ2 0_, ₍₃₁₎

which represents a normal distribution centered around m = 0, as required. The width of the peak of this distribution is then given by [1]

∆m = √δ0 N where δ0= r T0 T0− J2 (32) and should be sufficiently narrow, which is provided by the macroscopic property N 1 of M. When considering pure first-order transitions with q = 4, meaning J2= 0, the width of the peak thus narrows,

since δ0 = 1. This is one of the reasons that the q = 4 transitions are better suited to describe ideal,

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2.2.2

Final equilibrium states

At the end of a measurement, the apparatus A must lie in one of its equilibrium states ˆRi, as given in

(20), where M thus lies in one of its locally stable ferromagnetic states ˆRM i, as in (21), with i =⇑, ⇓. As

above, to ensure this property, the probability distribution given in (23) has to be centered around one of the values m = ±mF at t = tf. This means that the distribution PM(m, t) tends to either PM ⇑(m)

or PM ⇓(m) near the end of the measurement, depending on the correlated eigenstate of ˆsz.

For the moment, an external field h is introduced, which will only act on the spins ˆσ(n)a of M, and which

will be identified with the coupling g between S and A in section 4. With this external field h, the hamiltonian in (11) becomes [1] ˆ HM = −N h ˆm − N X q=2,4 Jq ˆ mq q + ˆVM = −N h ˆm − N J2 ˆ m2 2 − N J4 ˆ m4 4 + ˆVM (33) from which a probability distribution is obtained as [1]

PM(m) = √ N ZM √ 8π

e

−βF (m) , (34)

where F (m) is the free energy function denoted as [1] F (m) = −N h ˆm − N J2 ˆ m2 2 − N J4 ˆ m4 4 + N T 1 + m 2 ln 1 + m 2 + 1 − m 2 ln 1 − m 2 +T 2ln 1 − m2 4 + O 1 N . (35) This free energy function has a minimum at certain values of m. It is at these minima for F (m) where the probability distribution in (34) has a maximum, meaning the peaks of (34) are centered at the minima of F (m). The extrema of the free energy function lie at values of m that are obtained by solving the equation m 1 − 1 N = tanh[β(h + J2m + 3J4m3)] (36)

for m. At these values m = mi where the minima of F (m) occur, the probability distribution PM(m)

displayes a nearly gaussian peak with a width in the order of 1/√N . Each of these peaks PM i thus lies

centered at a magnetization mi of M, where the corresponding state of the magnet is then described

by the density operator RM i. These density operators may represent some locally stable equilibrium

states of M, where the number of these states depends on the values of J2, J4and T . When these values

are set in a certain way, to which we’ll come back in section 4, and taking an arbitrary value for h, three of these locally stable states are obtained. These states have corresponding average magnetization values of mi = mp, m⇑, m⇓, where mi = mp corresponds to the paramagnetic equilibrium state and

mi = m⇑, m⇓ to the two ferromagnetic states,with m⇑ > 0 and m⇓ < 0. When setting h = 0, which

resembles the case of letting g → 0 at the end of the measurement, these average magnetization values will tend to 0, +mF, −mF respectively, which is required for a faithfull registration as will be shown in

section 4.

3 Truncation

In an ideal measurement, the tested observable ˆsz is conserved, since i~dˆsz/dt = [ˆsz, ˆH] = 0, but the

transverse components ˆsx and ˆsy of the spin ˆs are not, since [ˆsa, ˆH] 6= 0 for a = x, y. This implies

that the diagonal terms of the density operator ˆr of S in (19) remain unchanged during a measurement as r↑↑(0) = r↑↑(tf) and r↓↓(0) = r↓↓(tf), but the off-diagonal elements do change. The correlations

between ˆsz and A, embedded in the blocks ˆR↑↑ and ˆR↓↓ of ˆD(t) given in (15), also remain unaffected,

but only during very short initial times, while the off-diagonal blocks ˆR↑↓ and ˆR↓↑ will vanish. It is

this vanishing of the off-diagonal blocks which is termed as truncation, and will be displayed as the rapid decay of the off-diagonal blocks ˆR↑↓ and ˆR↓↑ of the full density operator ˆD(t) of S+A, while the

diagonal terms remain unchanged. This truncation takes place on very short timescales, in which the macroscopic apparatus A affects the system S significantly, but not vice versa.

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3.1 The mechanism

Due to the coupling γ between the magnet M and the bath B being weak as γ 1, the interactions between M and B can be neglected at the times in which the truncation takes place, since it takes a certain time t > τtrunc, where τtruncis the truncation time, before the bath can act significantly on the

magnet. This truncation time is given by [1] τtrunc≡ √ ~ 2N g∆m = ~ √ 2N gδ0 . (37)

which is very short, as N 1, and inversely proportional to the coupling g between the spins ˆσ(n)_{of M}

and the ˆsz of S. Since the bath B can be neglected during the truncation process, only the compound

system S+M is taken into account when evaluating the truncation. According to [1], the evolution of the Pij(m, t) terms, which parametrize the blocks ˆRij(t) as in (25), are governed by four inhomogeneous

partial differential equations, arising from the Liouville-von Neuman equation of motion, where the parts involving the bath B are now left out since we’re only interested in the evolution of these terms at a timescale of order τtrunc. The solutions to the modified equations dP↑↑,↓↓(m, t)/dt = 0 of the diagonal

terms are [1]

P↑↑(m, t) = r↑↑(0)PM(m, 0) and P↓↓(m, t) = r↓↓(0)PM(m, 0), (38)

which shows, as expected, that the terms are time-independent, since they have to remain constant during the truncation process. The partial differential equation of the off-diagonal terms, with γ = 0, is denoted as [1] dP↑↓(m, t) dt − 2iN gm ~ P↑↓(m, t) = 0 (39)

where the solution of these off-diagonal terms is given by [1]

P↑↓(m, t) = [P↑↓(m, t)]* = r↑↓(0)PM(m, 0)

e

2iN gmt/~, (40)

which are, for each value of m, an oscillating function of time. The expectation values of the components of the spin ˆs, denoted as hˆsa(t)i with a = x, y, z, are now obtained by summing the solutions given in

(38) and (40) over m, which can be done by integration since N 1, implying δm = 2/N 1. Filling in (31) for PM(m, 0) in (38) and (40) we obtain [1]

r↑↑(t) = r↑↑(0) r↓↓(t) = r↓↓(0) r↑↓(t) = [r↓↑(t)]* = r↑↓(0)

e

−(t/τtrunc)

2

, (41) meaning

hˆsz(t)i = hˆsz(0)i and hˆsa(t)i = hˆsa(0)i

e

−(t/τtrunc)2 _with _{a = x, y,} ₍₄₂₎

which shows an exponential decay of the expectation values of the transverse components of the spin, while the z-component remains unaffected. As now can be seen, the expression of the truncation time in (37) is obtained from this solution in (41).

The summing over m has changed the oscillatory terms in (40) into the exponential decay terms in (41) and (42) through a dephasing, causing a destructive interference, between the oscillations for distinct values of m. As can be seen in (41), the off-diagonal elements of the density operator of S vanish during the beginning of the measurement process, at a time scale in the order of τtrunc. This decay of the

off-diagonal terms of S is termed as weak truncation, where truncation is assigned to the vanishing of the off-diagonal blocks ˆR↑↓ and ˆR↓↑of the full density matrix ˆD of S+A. These off-diagonal blocks of ˆD

turn out to be proportional to the off-diagonal terms of the marginal density operator of S in (41), and thus also decay with the exponential exp[−(t/τtrunc)2]. This means that, as we will see, not only the

expectation values of the transverse components of the spin will vanish, but also de correlations between these transverse components and the magnet M, as required.

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3.1.1

A dephasing or decoherence process?

Decoherence is a process in which a quantum system is “transformed’ into an apparent classical state. In this process the coherent superposition of states of the tested system gets destroyed by an external environment, like a thermal bath. This destroying of the superposition of states can be seen as the decay in time of the off-diagonal terms of the density operator of the system. This gives rise to the question whether the truncation discussed above is governed by a decoherence process, and as it turns out, this is not quite the case.

In a decoherence process, the characteristic decoherence time involves a factor ~/T , where T is the tem-perature of the external environment [1]. But, as mentioned above, the bath B does not act significantly on M during the characteristic truncation time τtrunc, and as can be seen in (37), the truncation time

does not depend on the temperature T of the bath but instead has a factor ~/g in it. The truncation is thus only controlled by the coupling g between ˆsz and the spins ˆσ(n) of M and, in our case, takes

place in the z-basis, since the measured observable is ˆsz. This basis is fully determined by the design

of the apparatus, where, if the measured observable would for example be ˆsx, this design should be

adjusted, and the initial density operator ˆr(0) of S would then be diagonal in the x-basis after the truncation. Truncation is thus a controlled process, where decoherence is an uncontrolled effect by an external environment. The dephasing of the oscillatory terms for distinct values of m in (40) give rise to the exponential decay as in (41), and as such, the truncation can be seen as a dephasing rather than a decoherence process. The effects of the bath B, which come in after the inital truncation, can be seen as an environment-induced decoherence. These decoherence effects of B will, however, only help in preventing recurrences of the off-diagonal blocks after truncation, and will not play a part in the initial truncation process.

3.1.2

Creation of the correlations between spin and magnet

At the first stage of a measurement, after the coupling g is turned on, certain correlations between M and S are created. However, since the marginal state ˆRM(t) = ˆR↑↑(t) + ˆR↓↓(t) of M stays constant

during the truncation stage, meaning ˆRM(t) = ˆRM(0), no correlations between ˆsz and the spins ˆσ(n)

of M are created. These correlations arise during the registration stage, which will be discussed in the next section. The correlations that are created, are the ones between the transverse components ˆsxand

ˆ

syand the spins ˆσ(n)of M, which are described by the terms Cxand Cy given in (26). These transverse

components of the spin ˆs of S, are only correlated with the z-component of the spins ˆσ(n) of M. The correlations can be found through the expectation values hˆs ˆmk_{i, where ˆ}_{s =} 1

2(ˆsx− iˆsy), which are

generated by the generating function[1] Ψ↑↓(λ, t) ≡ ∞ X k=0 ikλk k! hˆs ˆm k_{(t)i =}X m P↑↓(m, t)dis

e

iλm= r↑↓(0) X m

P_Mdis(m, 0)

e

2iN gmt/~+iλm. (43) Now, as we’re interested in the correlations hˆs ˆmkic rather than these expectation values, we must take

the cumulant expansion of (43) as [1] Ψ↑↓(λ, t) = ∞ X k=0 ik_λk k! hˆs ˆm k_i c ∞ X k0₌₀ ik0_λk0 k0_! h ˆm k0_i ! = ∞ X k=0 ik_λk k! hˆs ˆm k_i cexp ∞ X k0₌₀ ik0_λk0 k0_! h ˆm k0_i c ! , (44) from which the correlations hˆs ˆmk_i

c can be obtained through [1] ∞ X k=0 ik_λk k! hˆs ˆm k_(t)i c= r↑↓(0) Ψ↑↓(λ, t) Ψ↑↓(λ, 0) . (45)

By combining equation (43) and (45), we obtain [1]

∞ X k=0 ik_λk k! hˆs ˆm k_(t)i c= r↑↓(0)exp − _t τtrunc 2 −√2 t τtrunc λ∆m ! = r↑↓(t)exp −√2 t τtrunc λ∆m , (46)

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which gives rise to the correlations between ˆsx and ˆsy and the z-component of a single spin ˆσ(n) of M at first-order in λ, meaning k = 1, as [1] hˆsxσˆz(n)(t)i = hˆsxm(t)iˆ c = X m C_xdis(m, t)m =√2 t τtrunc hˆsy(t)i∆m = √ 2 t τtrunc hˆsy(0)i

e

−(t/τtrunc) 2 ∆m, (47) and hˆsyσˆz(n)(t)i = hˆsym(t)iˆ c= X m C_ydis(m, t)m = −√2 t τtrunc hˆsx(t)i∆m, (48)

where the exponential decay term comes from (42). At really short times, t < τtrunc, the term t/τtrunc

is the leading term in the equations (46) and (47), meaning that the correlations first increase. However, when t reaches the point t = τtrunc/

√

2, the exponential decay term takes over, resulting in an decay of the correlations between ˆsx, ˆsy and ˆσ

(n)

z at times t > τtrunc/

√ 2.

In Fig.1, adopted from [1], the evolution of the correlations for the zeroth rank (k = 0) upto the third rank (k = 3) are visualised.

Figure 1: The creation and disappearance of the relative correlations as function of t/τtrunc[1]. This relative correlation is given by corr = h(ˆsx− iˆsy)mk(t)ic/h(ˆsx− iˆsy)(0)i(i

√

2∆m)k_[1].

As can be seen, the higher the rank, the longer it takes for the correlations to be created and thus to vanish. Also, the vanishing of the correlations runs along with the decay of the expectation values of the transverse components hˆsa(t)i of S, with a = x, y. To summarize, the main part of the correlations

(k > 0) between the transverse components of the spin ˆs and the z-component of the spins ˆσ(n) _of

M is initially absent, then created at timscales of order τtrunc, followed by a decrease along with the

expectation values hˆsa(t)i of S.

As said above, the truncation is defined as the decay of the off-diagonal blocks ˆR↑↓ and ˆR↓↑ of the

full density operator ˆD of S+A. This can be justified by the proportionality of these terms to the off-diagonal elements r↑↓ and r↓↑ of the marginal density operator ˆr of S, for which we have shown the

(weak) truncation. The elements of the off-diagonal blocks R↑↓ and R↓↑ of the density matrix ˆD of

S+M describe the correlations between ˆsx and ˆsy and the spins ˆσ(n) of M. Since the bath does not

contribute to these correlations, the off-diagonal blocks R↑↓ and R↓↑ of the full density matrix ˆD of

S+A also entail this property. The initial information in r↑↓(0) and r↓↑(0) thus gets spread over the

elements of R↑↓ and R↓↑, and nearly vanishes at timescales of order τtrunc.

3.2 Irreversibility of the truncation

The truncation is an irreversible process, resembling a relaxation process as in statistical physics for N 1. This irreversibilty is required since otherwise the measurements would not be faithfull, as there could also exist correlations between M and the transverse components of the spin s at the end of a

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measurement. The main difference between truncation and a relaxation process in statistical physics, is that, for example, in statistical physics, the probability of finding a non-uniform density of a gas after relaxation is negligble small, where in a truncation process, the probability of recurrences in the off-diagonal blocks of ˆD is not negligble. Mathematically, the off-diagonal blocks of ˆD vanish completely in the thermodynamic limit N → ∞, which would ensure the irreversibility of the truncation. However, this is physically impossible and unrealistic as τtruncin (37) would tend to zero. Therefore the irreversibilty

must be ensured by some other features of the apparatus, which, in the Curie-Weiss model, are provided by the thermal bath B and a spread in the coupling g between ˆszand the spins ˆσ(n)of M.

3.2.1

The recurrence time

The irreversibilty of the truncation is based on certain approximations that have been made, such as treating m as continuous, justified by N 1 and thus δm = 2/N 1. Nevertheless the equations of motion remain reversible for finite N and as such, recurrences in the off-diagonal blocks of ˆD can be expected. The time at which recurrences may occur is given by [1] as

τrecur ≡ π~ 2g = π √ 2∆m δmτtrunc, (49) and since δm = 2/N ∆m = δ0/ √

N , it follows that τtrunc τrecur. Now by writing τtrunc in terms

of this recurrence time and putting it in (43), the generating function is then given by [1] Ψ↑↓(λ, t) = r↑↓(0) r 2 π 1 N ∆m X m exp − m 2 2∆m2 + iπN m t τrecur + iλm (50) where m is now treated as discontinuous due to the discreteness of the pointer variable, leaving out the earlier made approximation of continuity. Looking at (50), one can see that when t approaches τrecur,

where t > τtrunc, the expectation values, and thus the correlations, which vanished at timescales of order

τtrunc are regenerated. When t thus reaches ατrecur, where α = 1, 2, ..., the function in (50) regenerates

the correlations between the transverse components of the spin ˆs and M, which were destroyed through destructive interference at timescales of order τtrunc. This results in the reappearence of the off-diagonal

blocks of the full density matrix ˆD of S+M at these times ατrecur, where the bath B is not taken into

account yet. However, since the bath does act significantly on M at timescales of order τrecur, it should

be taken into account. As it turns out, the bath is one of the features that helps in preventing the occurence of these recurrences.

3.2.2

Preventing recurrences by a spread in the coupling constants

There are two mechanisms that help in preventing the occurence of the recurrences in the Curie-Weiss model, and establish the irreversibility of the truncation on a realistic timescale.The first mechanism is a spread in the coupling g between S and A, and the second one is an interaction of M with the phonon bath B. In the previous section we discussed what would happen when these mechanisms are left out, namely, recurrences would arise at timescales of order ατrecur. In this section, it will be shown how the

occurence of these recurrences is prevented in the Curie-Weiss model.

In the first mechanism, the spread δgn in the coupling, can be justified by the fact that the forces

between the spins ˆσ(n) of M and the spin ˆs of S are not equal [1]. The full coupling g + δgn does not

change in time, and is characterized by

N X n=1 δgn= 0 and δg2= 1 N N X n=1 δg_n2. (51)

The hamiltionan of the coupling between S+A in (9) must be adjusted and is given by ˆ H_SA0 = −ˆsz N X n=1 (g + δgn)ˆσz(n) (52)

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In [1], the equation of motion of the blocks ˆRij of S+M is given as d ˆRij(t) dt − ˆ HiRˆij(t) − ˆRij(t) ˆHi i~ = 0 (53)

where the right-hand side is left out since the bath is still inactive at this point (γ = 0), thus returning it to the ordinary Liouville-von Neuman equation of motion. ˆHi is the combined hamiltonian of M and

the coupling between S and A and is given by [1] ˆ Hi= ˆHSA0 + ˆHM = −si N X n=1 (g + δgn)ˆσ(n)z − N X q=2,4 jq qmˆ q_{+ ˆ}_V M (si= +1, −1 for i =↑, ↓) (54)

The diagonal blocks ˆR↑↑and ˆR↓↓of the density matrix ˆD remain constant in time, but the off-diagonal

blocks ˆR↑↓ = [ ˆR↓↑]* are time-dependent. The solution to (53) for these off-diagonal blocks is given by

[1] ˆ R↑↓(t) = r↑↓(0) ˆRM(0)exp 2i ~ N g ˆmt + N X n=1 δgnˆσz(n)t ! (55) Since τrecur is the smallest for the most random initial state ˆRM(0) of M, we take this fully paramagnetic

state, which is denoted as

ˆ RM(0) = 1 2N N Y n=1 ˆ σ(n)₀ where σˆ0= I, (56)

for this convenient reason and insert it in (55) obtaining ˆ R↑↓(t) = r↑↓(0) N Y n=1 1 2 ˆ σ₀(n)cos2(g + δgn)t ~ + iˆσ_z(n)sin2(g + δgn)t ~ . (57)

Now, by taking the trace over M of (57), we obtain the off-diagonal elements of the marginal density operator ˆr of S as rij = TrMRˆij, which gives

r↑↓(t) = r↑↓(0) N Y n=1 cos2(g + δgn)t ~ (58) These off-diagonal terms, contrary to those in the previous sections, stay truncated at timescales of order τrecur, provided that the spread in the coupling δgn meets the requirement[1]

δg g 1 π r 2 N (59)

where δg is given by (51). In [1], a certain characteristic decay time τ_irrevM is derived for any initial state ˆ

RM(0) of M. This characteristic time defines the point in time from which recurrences will be surpressed,

and is denoted as

τ_irrevM = √ ~

2N δg, (60)

where δg has to be set in such a way that τtrunc τirrevM τrecur, in order to prevent the occurence of

recurrences.

3.2.3

The decoherence effect of the bath

In this section the effect of the bath B on the recurrences will be discussed, which will act as a second mechanism to ensure the irreversibility of the truncation. The spread gn discussed above will again be

set to zero in order to fully describe how the bath effects the recurrences without other influences. Now γ 6= 0 but still small as γ 1. It will be shown that at times t > τtrunc, the bath B will act as a

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damping on the recurrences, preventing the terms P↑↓ = [P↓↑]* from taking on siginificant values at

timescales of order ατrecur.

The equation of motion (39) of these terms is now not equal to zero but instead is denoted as [1] dP↑↓(m, t) dt − 2iN gm ~ P↑↓(m, t) = γN ~2 n ∆+ h (1 + m) ˜K (m, t)P↑↓(m, t) i + ∆ h(1 + m) ˜K+(m, t)P↑↓(m, t) i o (61) where m, as in 3.2.1, is taken to be discontinuous due to the discreteness in the values of the pointer variable, ˜K±(m, t) are the autocorrelations functions of the bath (defined in [1]) and the terms ∆±

represent a shift of the observable ˆm into ˆm ± δm. Combining the solution given in (40) with the initial distribution of the magnet given in (31) we obtain from [1] the solution to (61) as

P↑↓(m, t) = N 2P dis ↑↓ (m, t) = r↑↓(0) s N 2πδ2 0 expn− N B(t) + iN 2gt ~ + Θ(t) − N 1 δ2 0 + D(t) m 2 2 o , (62) where B(t), D(t) and Θ(t) all describe how the bath effects the terms ˆR↑↓ = [ ˆR↑↓]* of ˆD at times

t > τtrunc. The exp[−N B(t)] factor has the biggest influence on the off-diagonal blocks of ˆD, where

B(t) is the damping function given in [1]. To ensure that this factor stays negligible at timescales of order τtrunc, the argument of the exponential has to satisfy the condition [1]

N B(τtrunc) = N γΓ2_g2 2π~2 τ 4 trunc= γΓ2 ~2 8πN δ4 0g2 1, (63)

where Γ is the debye cutoff, which is large as ~Γ g. Phonons that oscillate at frequenies with values ω > Γ do not have any effect on M. The bath B will have no effect on M during the initial truncation when (63) is satisfied, which is required as it should only act as a preventing mechanism. For times t > ~/2πT , which is the memory time of the autocorrelation function K(t) [1], we enter the markovian regime, meaning the memory of previous events is lost. The times ατrecur, at which the recurrences

would occur if γ = 0 , also lie in this markovian regime. The system thus requires the coupling γ to be sufficiently strong in order to prevent the recurrences from occuring in the markovian regime, at times t > ~/2πT . We introduce an irreversibility time that characterizes the time at which the recurrences must be surpressed, which is denoted as [1]

τ_irrevB = 2~ tanh g/T

N γg . (64)

To satsify the condition τ_irrevB τrecur, the coupling γ between M and B must satisfy [1]

γ 4 tanh g/T

πN , (65)

which is, although γ 1, accessible since N 1. In the markovian regime, the exponential factor exp[−N B(t)] behaves, according to [1], as

exp[−N B(t)] ∼ exp − t τB irrev (66) Now by putting (64) in (66), the recurrences, occuring at times ατrecur, are surpressed by the bath as

[1] exp −ατrecur τB irrev = exp − απN γ 4 tanh g/T (67) Thus, when N B(τrecur) 1 at times t ~/2πT , the recurrences are surpressed, making the initial

truncation an irreversible proces. This effect of the bath on the recurrences of the off-diagonal terms can be seen as a bath-induced decoherence, contrary to the dephasing process during the initial truncation, which does not involve the bath. At timescales of order τtrunc, only the coupling between S and M plays

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a role, resulting in the creation and disappearance of the off-diagonal correlations through a dephasing of the oscillatory terms. At timescales of order τB

irrev, the correlations are surpressed by a decoherence

effect induced by the bath.

The terms D(t) and Θ(t) in the exponential of (62) have a much less significant effect on M at the relevant timescales, so we will not elucidate them in this paper (see [1] for a detailed description).

4 Registration

In this section, the focus lies on the diagonal blocks R↑↑and R↓↓of the full density operator D(t) of S+A,

in which the correlations between ˆsz and the spins ˆσ(n) of M are embedded. These correlations, which

are created through the coupling g between S and M, must be successfully registered in order to provide a faithfull measurement. Such a successfull registration is what allows us to gain some information, embedded in the diagonal elements r↑↑(0) and r↓↓(0) of ˆr(t), about S through observation of the pointer

indication of M at the end of a measurement. When the correlations are successfully registered, the value of the pointer variable m corresponds with certainty with the correlated eigenstate of the spin S at t = tf. We define the registration explicitly as the creation of system-pointer correlations between

the z-component of the spin S and the many spins ˆσ(n)of M, where information of S is thus transferred to M. This information is, however, only attainable if the uniqueness of the indication of the pointer variable m is ensured for individual runs of the measurement [1]. The registration process takes place at timescales much larger than τtrunc, as the system S now also has to affect the apparatus A.

4.1 The distributions

The evolution of the joint probabilities P↑↑(m, t) and P↓↓(m, t), which characterizes the evolution of the

diagonal blocks ˆR↑↑ and ˆR↓↓ of ˆD(t), is given by the equation of motion as [1]

∂P↑↑(m, t) ∂t = ∂ ∂m(−v(m, t)P↑↑(m, t)) + 1 N ∂2 ∂2_m(w(m, t)P↑↑(m, t)), (68)

with initial conditions P↑↑(m, 0) = r↑↑(0)PM(m, 0), PM(m, 0) as given in (31) and a normalization of

the form Z P↑↑(m, t)dm = Z P↑↑(m, 0)dm = Z r↑↑(0)PM(m, 0)dm = r↑↑ Z (0)PM(m, 0)dm = r↑↑(0) (69)

This normalization gives rise to the ratio P (m, t) = P↑↑(m, t)/r↑↑(0), which can be interpreted as the

conditional probability that the pointer variable has taken some value m when sz= +1. The equation

of motion in (68) is a Fokker-Planck equation and the terms v(m, t) and w(m, t) in are denoted as [1] v(m, t) = 2γ ~2 h (1 − m) ˜Kt(−2w↑) − (1 + m) ˜Kt(2w↑) i + O 1 N (70) w(m, t) = 2γ ~2 h (1 − m) ˜Kt(−2w↑) + (1 + m) ˜Kt(2w↑) i + O 1 N , (71)

where v(m, t) represents the drift velocity, which drives the distribution to one of the final equilibrium states, depending on S, w(m, t) is the diffusion coefficient, which widens or narrows the distribution at some stages of the evolution, and ˜Kt(±2ω↑) = (~2/4π)Γ2t for very short times t Γ. However, on the

timescales in which the registration takes place, these ˜Kt(±2ω↑) terms are replaced by

˜

K(±2ω↑) = (~2ω↑/4)[coth(β~ω↑) ∓ 1]exp[−2|ω↑|/Γ], which will simplify the terms in (70) and (71) as

will be shown in the next subsection. The drift term v(m, t) results from the bath, which leads M to one of the equilibrium states through the weak coupling γ. The diffusion term w(m, t) results from the discreteness of the pointer variable m and the small fluctuations within m.

The registration process for a first-order transition (3J4 > J2) differs somewhat from a second order

transition (J2 > 3J4). In [1], it is shown that first-order transitions are better suited to provide a

successful registration. Therefore, only the pure first-order transitions with q = 4 and J2 = 0 will be