An Overview of Decoherence in the Context of Quantum Measurement with an Exactly Solvable Pure Decoherence Model

DEGREE PROJECT IN TECHNOLOGY, FIRST CYCLE, 15 CREDITS STOCKHOLM, SWEDEN 2019

FRODE BOMAN

ANDRZEJ PERZANOWSKI

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2019

En Översikt av Dekoherens i Kontexten av Kvantmätning med en Exakt Lösbar Ren Dekoherensmodell

FRODE BOMAN

ANDRZEJ PERZANOWSKI

KTH SKOLAN FÖR TEKNIKVETENSKAP

An Overview of Decoherence in the Context of Quantum Measurement with an Exactly Solvable Pure Decoherence Model

Frode Boman [email protected] Andrzej Perzanowski [email protected]

Supervisor: Edwin Langmann Abstract We explain the concept of decoherence by first providing basic background information and then working out a simple model intended to represent a Stern-Gerlach experiment. The model describes a spin- 1/2 quantum system and a device consisting of a quantum particle. Using Lindblad dynamics, the model is solved, revealing the effects of decoherence on the system. Specifically, we show that decoherence is sufficient for driving the off-diagonal of the system-device density operator to zero, providing a possible explanation to the fact of wave function collapse. Furthermore, comparisons are made with related work on a similar, exact model which includes relaxation.

Sammanfattning Vi förklarar konceptet dekoherens genom att först ge bakgrundsin- formation och sedan lösa en enkel modell som är menad att represen- tera ett Stern-Gerlach experiment. Modellen är ett spin-1/2 kvantsy- stem och en mätapparat som best˚arav en kvantpartikel. Genom att använda Lindbladdynamik kan vi lösa modellen och visa effekterna som dekoherens har p˚asystemet. Specifikt visar vi att dekoherens är tillräckligt för att driva de icke-diagonala elementen av system-apparat densitetsoperatorn till noll, vilket ger en möjlig förklaring till faktumet av v˚agfunktionskollaps. Vidare görs jämförelser med relaterad forsk- ning av en liknande, exakt modell som inkluderar relaxation.

1 Contents

1 Background 3 1.1 The statistical interpretation ...... 3 1.2 Measurement and the measurement problem ...... 4 1.3 Density matrices ...... 6 1.4 Time evolution ...... 7 1.5 Environment and decoherence ...... 8 1.6 Master equations ...... 10

2 Method 12 2.1 Overview ...... 12 2.2 The model ...... 13 2.3 The master equation ...... 14 2.4 Characteristic times for decoherence and relaxation ...... 15

3 Results 16

4 Discussion 17

5 Conclusions 26

6 References 28

A Appendix 30 A.1 Spin diagonal matrix element ...... 30 A.2 Spin oﬀ-diagonal matrix element ...... 31 A.3 Solutions in momentum representation ...... 34

2 1 Background

1.1 The statistical interpretation Since the inception of quantum mechanics it has been regarded, both by scholars and laypersons, as a probabilistic theory of nature. This is despite the fact that the Schr¨odingerequation, which wholly describes the time evolution of a quantum stystem, is entirely deterministic. Indeed, the probabilistic nature of quantum mechanics appears not as a result of its governing equations but instead as a result of something much less tactile: the measurement process.

The person most often credited with introducing probability into the theory of quantum mechanics is Max Born, who in the year 1926 formulated what has since been referred to as the Born rule (Born 1926). Since its ac- ceptance, the Born rule has shaped how we interpret quantum mechanics, and indeed the Universe (Landsman 2009). Still, the theory which leads one to the Born rule is unsatisfactory. Max Born, in his 1926 paper, did not in fact derive the rule. Instead, he concluded that is must be true, as there was no other way to interpret the results of his calculations. The lack of founda- tion for the rule led to dispute regarding how it should be interpreted, and multiple major interpretations of quantum mechanics have been formulated to make sense of it.

In order to formulate the problem, known today as the measurement problem, we introduce the mathematical language most suited for such discussions (Griﬃths & Schroeter 2018, p. 91-130): In quantum mechanics, wave functions (i.e., solutions to the Schr¨odingerequation) are described as quantum states. The state vector representation of the wave function ψ(x, t) is given by its ket; a vector in Hilbert space H. The measureable properties of the system (position, momentum, spin, etc.) are linear operators on H called observables. The possible states of the system are the eigenvalues of the corresponding observable. Due to this, it is common to describe a state |ψi by its projection onto the orthonormal eigenspace of the observable. For non-degenerate discrete spectra this means writing X |ψi = hφk|ψi |φki. (1) k

3 Here, |φi is the eigenvector, or eigenstate, of the observable. Examples of observables with discrete spectra are spin and angular momentum. For continuous spectra such as position and momentum, we instead have Z |ψi = hφk|ψi |φki dk. (2)

When the observable Oˆ operates on the system we thus get (in the discrete case) ˆ X ˆ X O |ψi = hφk|ψi O |φki = hφk|ψi φk |φki (3) k k where φk is the eigenvalue that corresponds to the eigenstate |φki. This is known as as superposition. Equation (3) leads to the idea that the state of the system is a sum of all possible states. This is not observed when mea- surements are made, nor is it clear what such a measurement result would even look like. To resolve this issue we now formulate the Born rule:

If one makes a measurement of an observable, then as a result one will get a single eigenvalue of Oˆ. The probability that one will measure a speciﬁc 2 eigenvalue φk is given by | hφk|ψi | . For the continuous case, the analogous rule is that a state will be found in the range dk about φk with a probability 2 | hφk|ψi | dk. For the purposes of this exposition, we need not introduce the case of degenerate spectra.

The transition from superposition to a state which is well-deﬁned is referred to as wave function collapse. It is this process which has given rise to the measurement problem and to a whole area of research: quantum measurement theory.

1.2 Measurement and the measurement problem Early quantum measurement theory was notably contributed to by John von Neumann, who in his book Mathematische Grundlagen der Quanten- mechanik (Mathematical Foundations of Quantum Mechanics) devoted a chapter to the subject. Among other ideas, von Neumann modelled the measurement apparatus, some device D (Here, D is the Hilbert space of the device. In some literature, this may be denoted HD), as a quantum system (von Neumann 2018). During the measurement process, the Hilbert space

4 of interest is the tensor product space S ⊗ D (Allahverdyan et al. 2013). A simple measurement model looks like the following: Let a system S have N possible states {|sii}i=1. If we wish to measure which state the system is in then our device should take the value |dii if the system is in state |sii. As an example, lets say that the system is in a state |ski. Then our device, initially in a ready-state |d0i, wil, via a measurement, transition into the state |dki. Observers can then infer that the system is in the state |ski, since they can see that the measurement device is in the state |dki. For a general system, P in a superposition of states |si = i ci |sii, this process becomes X X ( ci |sii) ⊗ |d0i → ci |sii ⊗ |dii. (4) i i Were the measurement device macroscopic, reading the measurement value from its pointer would be no issue. This is problematic because as the system is in a superposition of states, so too is the measurement device. Clearly, wave function collapse is yet to take place. A possible (though possibly unsatisfactory) explanation for this may be that it is unreasonable to describe a macroscopic system as a quantum one. The idea that the macroscopic is inherently different from the microscopic runs into difficult questions, such as there not being a clear distinguishing line between them. Regardless, we can consider the case where the apparatus is not macroscopic. Then, equation (4) should hold. This is referred to as a pre-measurement. It is called that because it is not a true measurement, in that no information can be gathered from the microscopic device: It has no pointer for an observer to note down its state. Still, a measurement has technically taken place, in that the system has interacted with a device. The information regarding the state of the system could now be obtained by performing a measurement on the device. It may seem like nothing has been accomplished by performing the pre-measurement; that we have merely moved the problem from the system to the device. As far as the measurement problem is concerned, this is true, although practically it may be useful to convert sensitive information into more stable information (this is often referred to as preparation of a state). An interesting note is that it was this line of reasoning which led von Neumann to adhere to the controversial idea that wave function collapse is caused by consciousness (an idea which is now referred to as the Neumann- Wigner interpretation). As one introduced more and more devices, one is merely moving the problem along, until eventually, the mind of the observer performs the final measurement. This idea is mostly rejected among modern

5 physicists (Schlosshauer et al. 2013).

At this time, it is clear there there exists a problem within the theory of quantum measurement. It is not, however, clear how one should formulate the problem so that an answer would be of complete satisfaction. We adopt for this purpose a two-part formulation of the problem advocated by Maxim- ilian Schlosshauer (Schlosshauser 2005). The first part, titled “the problem of definite outcomes”, asks how, in light of equation (4), we come to experi- ence a single state of any macroscopic measurement apparatus and not the superposition called for by the mathematics. The second part, titled “the preferred basis problem”, asks how it is possible to know which observable has been measured. A simple argument explains why this problem needs resolution. In our example, we make a measurement on the system states |sii which correlate to device states |dii via the coefficient ci. However, via a change of basis we may as well rewrite the final entangled state in (4) as if we had made a measurement on some system state |s˜ii, correlated to a E ˜ device state di via coefficientsc ˜i. It is then unclear whether a measurement has been made on the system states |sii or on the system states |s˜ii. This is problematic since if we make a measurement of, for instance, the momentum of a particle, then certainly the number we read off the pointer of our measurement apparatus should be the momentum of the particle and not something else such as its position. A theory in which the measurement problem can be solved should resolve both of these problems.

1.3 Density matrices Important to discussions of quantum measurement is the concept of density matrices, another concept introduced by von Neumann. So far, we have only considered so-called pure states; states which can be represented by kets on a Hilbert space. The more general state, the mixed state, is a statistical ensemble of quantum systems. This could mean, for example, a mixed state is given by |ψi with a probability p1 and by |φi with a probability p2. For simplicity, we assume that |ψi and |φi are orthogonal. The mixture’s density matrix is then given by

ρ = p1 |ψi hψ| + p2 |φi hφ| , (5)

6 such that the probabilities pi sum to 1. The state in (4) is a pure state, but we can still ﬁnd its density matrix by applying (5) with p1 = 1:

X 2 ρcollapsed = |ci| |sii hsi| ⊗ |dii hdi|. (7) i These are precisely the diagonal terms of equation (6). We stress that the 2 probabilities |ci| in equation (7) are classical probabilities that the system is in the corresponding eigenstate (i.e., it is no longer in a superposition). We realize that some mechanism which is capable of eliminating the coherences on the off-diagonal of the density matrix could be used to explain why wave function collapse occurs and thus solve the problem of definitive outcomes. Such a mechanism, however, would shed no light on precisely how a state is “chosen”. This is another aspect of the measurement problem which is supremely difficult and ties into the very laws of nature themselves. Discussions of whether the Universe evolves deterministically or non-deterministically stem from this question, and from whether the state is chosen at random or if there is some underlying mechanism deciding the state in a predictable fashion.

1.4 Time evolution It is well known that pure quantum systems, i.e. systems which we represent by kets, evolve according to the famous Schr¨odingerequation d i |ψ(t)i = Hˆ |ψ(t)i , (8) ~dt where Hˆ is the Hamiltonian; the energy operator for the system. In general, the Hamiltonian is a function of time, Hˆ (t). For the purposes of this paper, however, we need only consider the special case of time-independent Hamil- tonian. An equivalent formulation is written with a time evolution operator,

7 U(t), such that |ψ(t)i = U(t, t0) |ψ(t0)i , (9) with initial condition U(t0, t0) = 1, the identity operator (Breuer et al. 2002). The equations (8) and (9) together lead to

i − Hˆ (t−t0) U(t, t0) = e ~ . (10)

† Importantly, this operator is unitary, U(t, t0)U (t, t0) = 1, such that it con- serves the probability of the state.

For mixed states, represented by density matrices, the time evolution takes on a diﬀerent form. We have

X X † ρ(t) = pi |ψi(t)i hψi(t)| = piU(t, t0) |ψi(t0)i hψi(t0)| U (t, t0), (11) i i or simply † ρ(t) = U(t, t0)ρ(t0)U (t, t0). (12) Diﬀerentiating with respect to time gives d i i i ρ(t) = − Hρˆ (t) + ρ(t)Hˆ = − [H,ˆ ρ(t)] (13) dt ~ ~ ~ This equation is known as the Liouville-von Neumann equation and describes the time evolution of mixed states.

1.5 Environment and decoherence As described in section 1.3, we are in search of a mechanism which is capable of driving the oﬀ-diagonal elements of the system density matrix to zero. One such mechanism, the idea for which has been quite fruitful, is given by quantum decoherence (we will refer to this as just “decoherence”) (Schlosshauer 2005). The theory of decoherence argues that one key assumption made in essentially every theoretical mathematical model of measurement is unreasonable: that of the isolated system. So long as a single component of our complete system is in any way interacting with other quantum systems, such as via entanglement, the isolation assumption is invalid. What this means, mathematically, is that we no longer can describe our system-device setup as being in the Hilbert space S ⊗ D. Instead, we must consider the system

8 S ⊗ D ⊗ E, where the environment E may as well be considered the “rest of the Universe”. It may not be immediately obvious how the environment aﬀects the wave function, let alone bringing upon the loss of coherence which results in wave function collapse. Physically, the explanation is that information about the system distributes itself throughout the many degrees of freedom of the entire system-environment system (Zurek 2003).

Zurek provides the following ideal mathematical formulation, where the environment is introduced into (4): X X ( ci |sii ⊗ |dii) ⊗ |E0i → ci |sii ⊗ |dii ⊗ |Eii (14) i i

The states |Eii are states of the environment which correspond to states of our device |dii. Ideally, each diﬀerent device state causes the environment to enter states that are diﬀerent from one another (so that they can be orthogonalized without degeneracy). The density matrix corresponding to the state in (14) is

X ∗ ρSDE = cicj |sii hsj| ⊗ |dii hdj| ⊗ |Eii hEj| (15) i,j The next step in the measurement process is to perform a partial trace on the environmental states in (15). This is the mathematical equivalent of ignoring the environment - we have no interest of knowing the states of the environment nor could we gain any information from it. We have

hEi|Eji = δij (17)

(δij is the Kronecker delta) then contraction of (16) yields exactly the mixture given by the collapsed density matrix in (7). Thus, the two parts of the measurement problem formulated by Schlosshauer are solved: the problem of deﬁnite outcomes is solved by the collapse of the density matrix and the preferred basis problem is solved by realizing that the basis (for the device) must be chosen such that (17) is fulﬁlled. Zurek has coined the term “einse- lection” for this forced choice of basis.

9 As mentioned before, it should be stressed that although decoherence such as it is formulated above is satisfying to an extent with regards to bringing us closer to a solution of the measurement problem, it does not shed any light on which state the system is in as a result of the measurement. As such, it does not provide any clarity on the deep questions, such as the determinism (or non-determinism) of the Universe and nature.

1.6 Master equations Describing decoherence such as in section 1.5 is idealistic. In reality, the way 1 to approach ﬁnding ρS for a given system-environment model is often via what is known as a master equation. A master equation is an attempt at generalizing the process in section 1.5 into the form of an approximate equation. Here we discuss how such equations may be derived and their usefulness.

Using equations (12) and (15), we ﬁnd that the exact solution for ρS (t) is obtained as † ρS (t) = T rE [U(t, 0)ρSE(0)U (t, 0)]. (18) The above equation is exact and correct, but for many systems it is unreasonable to work with the full density matrix ρSE and one instead looks at time evolution equations for the density matrix ρS in which the environments eﬀect and partial trace are embedded. These are the master equations, and in decoherence modeling it is useful to focus on master equations which are of the particular form (Schlosshauer 2014)

d i 0 ρS (t) = − [HS , ρS (t)] + Λ(ρS (t)). (19) dt ~ The ﬁrst term is the Liouville-von Neumann equation (13) for the system without the environment, and the term Λ(ρS ), where Λ, a non-unitary operator, is added to account for the environment interaction. The above equation has thus factored out the eﬀects of decoherence from the Liouville-von Neu- 0 mann equation. The prime on the system Hamiltonian HS is there because the system’s energy levels may need to be renormalized as a result of the environment (Schlosshauer 2014).

1Here, the “system” may as well be a “system-device” setup. For brevity, we will refer to any system, composite or otherwise, as just a “system”.

10 In order to arrive at an equation of the form in equation (19) one begins at the Liouville-von Neumann equation (13) and proceeds with the following approximations (Breuer et al. 2002):

Firstly, one assumes that the coupling between the environment and the system is weak, such that the eﬀects that the system has back on the environment is negligible and the environments density operator can be considered to be constant in time. This is known as the Born approximation.

Secondly, one insists that the time evolution process have the Markov property. This is known as the Markov approximation, and disallows that the time evolution of the density matrix should depend on the density matrix at any time other than the present. In the words of Schlosshauer, this implies that any memory eﬀects which might arise from self-correlations in the environment as a result of its coupling to the system are assumed to decay over timescales which are signiﬁcantly shorter than the characteristic timescale of the time evolution of the system.

Finally, one demands that the reduced density matrix remain positive at all times. This is a property which naturally holds should the equation be exact, but which can be lost when invoking approximations such as the previous two. Positivity can be ensured by invoking the rotating wave approximation, which allows for averaging out the rapidly oscillating terms in the master equation.

The three approximations above culminate in what is known as the Lind- blad equation. Introducing the Lindblad operator Li, the Lindblad equation can be written as d i X 1 1 ρ (t) = − [H0 , ρ (t)] + κ (L ρ L† − L†L ρ − ρ L†L ). (20) dt S S S i i S i 2 i i S 2 S i i ~ i The Lindblad operators depend on the form of system-evironment interaction. We stress again that the Lindblad equation is not exact. Its description of decoherence is useful, but there may be other eﬀects on the system caused by the environment that the Lindblad equation does not capture (as we shall see, relaxation is such an eﬀect).

11 2 Method

2.1 Overview In order to demonstrate the loss of coherence brought about by the environment we introduce a model for which we can state all the relevant interactions and equations. The model is based on the work of A. Venugopalan, D. Ku- mar and R. Ghosh in a 1995 paper (Venugopalan et al. 1995). From this point, when we refer to the “reference paper” we are referring to this paper. The model is intended to resemble a Stern-Gerlach experiment. It involves a system and a measurement device. In order to determine the time evolution of our system, we rely on Lindblad dynamics. An assumption that we make is that the environment couples to the device, but not to the system. This assumption lets us determine the form of the Lindblad operators in equation (20).

The master equation which we arrive at from the Lindblad equation is very similar to an equation found in a famous paper by physicists A.O. Caldeira and A.J. Leggett, which studies quantum Brownian motion (Caldeira & Leggett 1983). In fact, their equation is an exact form of the equation found via the Lindblad equation. The biggest diﬀerence between the two equations is that the one that is of Lindblad form lacks a term which describes relaxation, which governs the system’s tendency towards equilibrium with the environment. Because our interest does not extend beyond that of decoherence, we elect to exclude this term from our equation. This is equivalent to an assumption that decoherence takes place on a timescale which is signif- icantly shorter than the typical timescale for relaxation. The legitimacy of this approximation is explored in section 2.4.

We solve the master equation explicitly, using the same techniques as in the reference paper. We use the method of characteristics (Courant & Hilbert 1962), which allows us to rewrite the master equation into a system of ordinary differential equations that can be solved using well-known methods. The system of ordinary differential equations are called Lagrange-Charpit equations (Delgado 1997). They are defined by a vector field that is tan- gent to the surface defined by the solutions to our master equation, and are constructed by considering the coefficients in the master equation. From the solutions to the Lagrange-Charpit equations comes the solution to the mas-

12 ter equation, via a factorization theorem described in (Roy & Venugopalan 1999).

The explicit solutions allow us to explore the conditions that govern wave function collapse and to draw conclusions regarding the preferred bases of the system.

2.2 The model

1 The model describes a spin- 2 particle which is traveling in one dimension through an inhomogeneous magnetic ﬁeld. The self-Hamiltonian for the spin is proportional to the third Pauli matrix whereas the self-Hamiltonian for the position is given by the standard free particle Hamiltonian:

p2 H = λσ ⊗ 1,H = 1 ⊗ . (21) S 3 D 2M The reason for the subscript D for the self-Hamiltonian in position space will become apparent soon. Note that the Hamiltonians are operators on the tensor product space S ⊗ D. If our model required an explicit mathematical environment, we would be working with the space S ⊗D⊗E, but since we are relying on Lindblad dynamics to provide all the eﬀects from the environment, we need not provide such an explicit mathematical environment. Initially, we assume that the particle has a spin that is in a general superposition

|ψ(t = 0)i = c↑ |↑i + c↓ |↓i . (22)

The position is initially a Gaussian wave with a mean momentump ¯ and width σ: 1 x2 Φ(x, 0) = exp ipx¯ − . (23) (πσ2)1/4 2σ2 As is familiar from elementary quantum mechanics, a Gaussian form such as in (23) provides the smallest uncertainty with regards to the uncertainty ~ principle: σxσp = 2 .

Due to the magnetic ﬁeld, the position and spin will interact continuously via the interaction Hamiltonian given by

HSD = εσ3 ⊗ x (24)

13 where the force constant ε is the product of the magnetic field gradient and the particle’s magnetic moment. This interaction couples the particle’s position to its spin in such a way that it travels in one direction should its spin be up, and the other direction should its spin be down. Therefore, precisely as in the simple model considered in equation (4), the superposition of the spin will cause the position to evolve into a superposition as well, yielding two Gaussian waves that travel in opposite directions. It is in this sense that we consider the position of the particle to be the measurement device in this model (hence the subscript D in (21)). The intention is that a measurement of the particle’s position will yield definitive information about the particle’s spin (i.e., if we find the position in the +x-direction we know that the spin was up, and vice-versa).

2.3 The master equation We can ﬁnd a master equation for our system by using the Lindblad equation 0 (20), with HS = HS + HD + HSD from equations (21) and (24). We assume that the environment couples to the position of the particle and not to its spin. Then, the Lindblad operators are given by L = 1 ⊗ x. This gives us the time evolution equation for our system: ∂ρ i ∂2 ∂2 iε(s ⊗ x − s0 ⊗ x0) = ~ (1 ⊗ − 1 ⊗ )ρ − ρ ∂t 2M ∂x2 ∂x02 ~

iλ(s ⊗ 1 − s0 ⊗ 1) − ρ − κ(1 ⊗ x − 1 ⊗ x0)2ρ (25) ~ for elements of the composite space density operator ρ = ρ(s, s0, x, x0, t) = hs ⊗ x| ρ |s0 ⊗ x0i where s and s0 are basis states for the spin such that s, s0 = 1 for spin up and s, s0 = −1 for spin down. This equation does not take into account the relaxation of the system. Furthermore, the parameter κ is unknown without further assumptions. These are shortcomings of employing Lindblad dynamics. In the reference paper, they use the equation that was found in (Caldeira & Leggett 1983), which is exact:

∂ i 0 ∂ ∂ 2MγkBT 0 2 ρ = − [HS , ρ] − γ(x − x )( − )ρ − (x − x ) ρ. (26) ∂t ~ ∂x ∂x0 ~2 This equation describes the time evolution for a free particle which is interacting with an environment that consists of a bath of harmonic oscillators.

14 From this equation we can retrieve κ = 2MγkB T , which we will write in terms ~2 of the diﬀusion coeﬃcient D, which for a high temperature thermal bath is given by D = 8MγkBT (Venugopalan 1999). We could in the same way 0 ∂ ∂ retrieve the relaxation term γ(x − x )( ∂x − ∂x0 )ρ, but as we have mentioned, we intend to exclude the relaxation from our model. We will now examine this assumption more carefully.

2.4 Characteristic times for decoherence and relaxation For the system described by (26), the relation between the characteristic timescales for the relaxation and decoherence has been studied, and is given by the equation (Zurek 2003)

τD λdB 2 = ( 0 ) . (27) τR x − x

Here, λdB is the de Broglie wavelength of the particle, given by

~ λdB = √ . (28) 2MkBT The denominator in (27) is the separation. As the separation approaches zero, we are approaching the diagonal of the density matrix ρ. We do not want decoherence to aﬀect the diagonal, and indeed we see from (27) that here, the decoherence time diverges in contrast to the relaxation time. On the contrary, when the separation is large (meaning elements that are far from the diagonal in ρ), we see that decoherence becomes a lot faster than relaxation. The de Broglie wavelength in the numerator is usually small. At −9 room temperature (300 K) we get λdB ∼ 10 for an electron. For macroscopic objects, the de Broglie wavelength is so small that decoherence times become practically instant (with relation to relaxation times, which may be large for macroscopic systems) for any separation that is not practically diagonal. This explains why any macroscopic object clearly is never found in superpositions; any such states would collapse in an instant.

When excluding the relaxation from our equation, we must be wary of the eﬀects that this will have on our solutions. Clearly, from (27), we should expect solutions which are physically meaningful for small times. We should,

15 however, not expect any physical realism in the limit t → ∞. When we wish to draw conclusions regarding the behavior of our system in this limit, we will refer to the solutions from the reference paper.

3 Results

We will now solve equation (25). First, we introduce notation. The composite density matrix in equation (25) can be written as

ρ ρ ρ = ↑↑ ↑↓, (29) ρ↓↑ ρ↓↓ where the different elements correspond to different combinations of spin eigenstates s, s0. We can then split equation (25) into four equations, two which are diagonal in the spin space (s = s0) and two which are spin-off- diagonal (s = −s0). The resulting equations can be written solely in position space. We have the diagonal equations

2 2 0 ∂ρd i~ ∂ ∂ iε(x − x ) D 0 2 = [ ( − ) ∓ − (x − x ) ]ρd, (30) ∂t 2M ∂x2 ∂x02 ~ 4~2 0 where ρd is ρ↑↑ when ∓ is minus (this corresponds to s = s = 1 in (25)) and 0 ρ↓↓ when ∓ is plus (s = s = −1). The oﬀ-diagonal equations are

2 2 0 ∂ρod i~ ∂ ∂ iε(x + x ) 2iλ D 0 2 = [ ( − ) ∓ ∓ − (x − x ) ]ρod (31) ∂t 2M ∂x2 ∂x02 ~ ~ 4~2 0 where ρod is ρ↑↓ when ∓ is minus (s = 1, s = −1) and ρ↓↑ when ∓ is plus (s = −1, s0 = 1). In equation (30), the sign determines whether the equation describes a particle with up spin or with down spin (minus for up, plus for down). The oﬀ-diagonal states whose time evolutions are given by (31) represent superpositions of up spin and down spin.

The equations (30) and (31) can be exactly solved. We follow the steps of the reference paper. First, we introduce new variables r := x − x0 and x+x0 R := 2 . Then, we perform a partial Fourier transform in the variable R, Z ∞ ρ(Q, r, t) = exp(iQR)ρ(R, r, t)dR. (32) −∞

16 Our equations then transform into 2 ∂ρd ~Q ∂ Dr iεr = [ − ∓ ]ρd (33) ∂t M ∂r 4~2 ~ and 2 ∂ρod ~Q ∂ Dr 2ε ∂ 2iλ = [ − ∓ ∓ ]ρod. (34) ∂t M ∂r 4~2 ~ ∂Q ~ These equations can be solved via the method of characteristics (see Ap- pendix for more details). We get the solutions σ2Q2 1 ρ (Q, r, t) = exp ip¯(r + v t) − − (r + v t)2 d Q 4 4σ2 Q

 2 3  vQt 2 2 1 2 −D( + vQrt + r t) iε( vQt + rt) × exp 3 ∓ 2  (35) 4~2 ~ and 2 2 −σ 2εt 2 I2 2iλt ρod(Q, r, t) = exp (Q ∓ ) − + ipI¯ 2 exp ∓ 4 ~ 4σ2 ~ 2 5 4 2 3 3 −D ε t r0εt r0t 2~ εt ~ 2 2 × exp ( ∓ + ∓ − r0r1t + r1t) (36) 4~2 5M 2 2M 2 3 M 3~ M where ~Q 2εt ~ εt2 ~ εt2 vQ = , r0 = ∓ +Q, r1 = r∓ ( −Qt),I2 = r+ (∓ +Qt). M ~ M ~ M ~ 4 Discussion

The solutions (35) and (36) look formidable, but they are managable. The off-diagonal solutions represent quantum superpositions of spin states. As we showed in section 1.3, wave function collapse is equivalent to the elimination of the off-diagonal of the density matrix. We introduced decoherence solely for this task. Thus, we expect that the solutions ρod in (36) shall go to zero. Indeed, it is not too difficult to see that this occurs very fast, with a leading 2 5 factor of exp − Dε t . This yields a characteristic timescale 20M 2~2 20M 2 2 5M 2λ2 τ = ( ~ )1/5 = ( dB )1/5. (37) pure−decoherence ε2D γε2

17 Since this occurs in the large time limit, we can not be certain that this indeed holds true should relaxation be used in the model. Fortunately, the results from the reference paper conﬁrms that this still tends to zero, with a timescale that is actually given by

3M 2γ2 2 3M 2γλ2 τ = ( ~ )1/3 = ( dB )1/3. (38) decoherence ε2D 42 The fact that the oﬀ-diagonal elements go to zero implies that the spin- system collapses into a well-deﬁned (albeit unknown) state. What remains, then, is the spin-diagonal, whose time evolution is given by (35). In order for this solution to be physically meaningful, we rewrite it into the momentum representation Z ρd(¯u, v,¯ t) = exp(i(¯ux +vy ¯ ))ρd(x, y, t)dxdy. (39)

This can be achieved by again performing a partial Fourier transform, this time in the variable r (see Appendix for details):

r 2 π −1 iDvQt εt ivQt 2 ρd(Q, q, t) = 2 exp (q +p ¯ + ∓ + ) A(t) A(t) 4~2 ~ 2σ2

2 2 2 3 2 2 2 σ Q DvQt iεvQt vQt exp ipv¯ Qt − − ∓ − (40) 4 12~2 2~ 4σ2 where Dt 1 A(t) = + . (41) ~2 σ2 u¯+¯v Here, the momentum variables Q and q areu ¯ − v¯ and 2 respectively (whereu ¯ andv ¯ are the Fourier variables that correspond to x and x0, respectively)(Venugopalan et al. 1995). The solution (40) represents density matrix elements hQ| ρd |qi. Since momentum is a continuous observable, the density operator ρd is continuous. Its diagonal represents the momentum distribution of the particle and is given whenu ¯ =v ¯, which implies that Q = 0 and q =u ¯. Conversely, the oﬀ-diagonal is given by all states for which Q 6= 0. These states are the coherences which represent the superposition of the particle in the momentum space, meaning that they correspond to a particle which has momentum in both directions. In accordance with the description of the model in section 2.2, we expect the particle to gain a superposition of

18 momenta only when it is in a superposition of spin states. We have shown, however, that the spin state collapses. Thus, we expect the momentum state to collapse as well, yielding a particle which has momentum in a well-defined but unknown direction. Indeed, by examining the solution (40) we see that for Q 6= 0 (and so vQ 6= 0) the solution quickly goes to zero, with the most Dv2 t3 notable term being exp − Q which drives the expression to zero over 12~2 timescales 2 2 2 12M 1/3 3M λdB 1/3 τd = ( ) = ( ) . (42) DQ2 γ~2Q2 Again, we find from the results in the reference paper that the momentum- off-diagonal states go to zero for the exact model as well.

As Q approaches zero, the expression (40) approaches r π 1 t 2 ρd(0, u,¯ t) = 2 exp − (¯u +p ¯ ∓ ) . (43) A(t) A(t) ~ These diagonal terms represent the momentum distribution for the particle (with ∓ being minus for spin up and plus for spin down), such that (Venu- gopalan et al. 1995) 2 ρd(0, u,¯ t) := |Φ(¯u)| . (44) Here we must be careful. It is clear that the expression (43) tends to zero for large times. However, from the reference paper we find that in a relaxation model the system will instead tend towards equilibrium with the environment, which has the expression (43) tend towards a non-zero value. For large average momentau ¯, this value will approach zero and the difference between the two solution will be negligible√ for times that are not very large (the difference comes from the factor 1/ t in the solution (43)). This is no coincidence, as the momentum variable q is transformed from the separation variable r in position coordinates and the relation (27) implies that the timescale in which decoherence is the dominant effect will be longer for larger separations. Furthermore, a small time expansion of the expression found in the reference paper, keeping linear time dependences but discarding quadratic and higher, reveals that the two solution are almost equal in this limit.

An interesting result from the reference paper is that the solutions for the

19 spin-diagonal equations (30) do not go to zero for oﬀ-diagonal terms in the position representation. This ties back into the idea from section 1.5: Mo- mentum emerges as a preferred basis for this problem.

In order to visualize the spin-diagonal solutions, we can plot them in three dimensions. We will plot both solutions on the same figure, i.e. we plot ρ = (ρ↑↑ + ρ↓↓)σ, with ρ↑↑ and ρ↓↓ being the two solutions (40) for ∓ being minus and plus, respectively. On the x and y axes we will have dimensionless Q Q momentau ¯ = (q+ 2 )σ andv ¯ = (q− 2 )σ. The σ in the expressions we plot on both axes are for convenience; they eliminate all σ from the solutions (40). In our plot, each point in space then represents a density matrix element hu¯| ρ |v¯i. Thus, we expect two Gaussian waves on the diagonal (since we plot both solutions, not because the particle is in a superposition, it is not), one which travels in the positive direction which corresponds to spin up, and one which travels in the negative direction which corresponds to spin down. The off-diagonal will represent superpositions of momenta. We expect the off-diagonal to fluctuate, but eventually settle to zero.

For the following ﬁgures, we have used the same parameter values for D, p¯ and ε as in the reference paper, with the intention of producing similar ﬁg- ures.

20 Q Figure 1: t = 0. Plot of ρ = Re(ρ(↑) + ρ(↓)/σ versusu ¯ = (q + 2 )σ and Q v¯ = (q − 2 )σ. For further information, read the description on the previous page. The parameter values are ε/M = 2, p¯ = 0.2/σ, D/M 2 = σ2,M/~ = 0.5/σ2. The units for time are chosen such that the relations above hold. The density matrix is in its initial state, displaying a superposition.

21 Figure 2: Same as in Figure 1 but for t = 1. Along the diagonal,u ¯ =v ¯, we see the emergence of two Gaussian wave packets representing the momentum distribution (43). The oﬀ-diagonal is rapidly oscillating locally around the diagonal.

22 Figure 3: Same as in Figures 1 and 2 but for t = 1.5. The waves continue to travel as they get more localized at the diagonal. Oscillations in the oﬀ- diagonal cease for large values ofu, ¯ v¯ and settle to zero.

Eventually, the density matrix (for the exact solution) would consist solely of the diagonal wave packets, with an oﬀ-diagonal that is zero everywhere, indicating a fully collapsed wave function. The speed at which this happens is entirely dependent on the strength of the decoherence. In fact, we may compare the plot in Figure 3 with a plot that has D = 0 (Figure 4). This is equivalent to a system with no decoherence term, i.e., what one would get if one were to solve the Liouville von-Neumann equation for the spin-diagonal density matrix elements, as opposed to the Lindblad equation.

23 Q Figure 4: t = 1.5. A plot of ρ = Re(ρ(↑) + ρ(↓)/σ versusu ¯ = (q + 2 )σ Q andv ¯ = (q − 2 )σ in the case D = 0. This is equivalent to a system with no decoherence (or inﬁnitely weak decoherence). The values close to the diagonal (but not on it) are slightly larger. The values far from the oﬀ- diagonal still tend to zero (this would not occur if our model had been exact).

The fact that the off-diagonal still tends to zero at all is merely a result of the terms in (40) that are linear and quadratic in time. These terms are not present in the exact solution found in the reference paper; the exact solution has no terms in the exponent which increase in magnitude with time, other than the decoherence terms (and one phase factor). The physical interpretation for why the relaxation “rescues” the off-diagonal from going to zero is unclear, although one might consider that only the exact solution, which has both relaxation and decoherence, holds true physical significance. Still, this shows that if one naively solves the Schrödingerequation while disregarding

24 the environment, it is still possible to ﬁnd that the oﬀ-diagonal tends to zero with time.

In contrast, we can increase the coeﬃcient D, as in Figure 5 (where D is three times larger than in Figures 1, 2, 3).

Figure 5: t = 1.5. Here, the diﬀusion coeﬃcient D is three times larger than in Figures 1, 2, 3. This causes each wave to spread out more. The wave is, as expected, more localized at the diagonal.

This does, as expected, increase how fast the off-diagonal goes to zero. It also flattens out the wave, meaning a momentum measurement may provide a wider range of values. Some of these values may tread into territory as- sociated with spin in the other direction (i.e. a positive momentum may still correspond to downward spin, if you get “unlucky”). This, perhaps un- expected effect of decoherence on the diagonal (remember, we introduced

25 decoherence solely to eliminate the off-diagonal) stems from the role of the parameter D in the momentum distribution (43). When D = 0, the function A(t) given by equation (41), which describes the width of the wave, becomes constant and equal to 1/σ2. For non-zero coefficients D, A(t) increases with time, thus increasing the width of the wave. Removing decoherence would of course impact the spin-off-diagonal elements in (36) as well, meaning that the system’s dependence on the diffusion coefficient D is more complicated than what we have made it out to be here.

Regardless of the nature of the particle’s movement, what we have indeed achieved is a measurement device. The off-diagonal elements of the spin system tending to zero tells us that the system collapses into a well-defined state, either up or down. Depending on which state the system has collapsed into, the device will gain some momentum which with time becomes well- defined, and is given by (43). Therefore the collapse given by the transition from the state (6) to (7) is realized and the final density operator is

2 2 ρ(t) = |c↑| |↑i h↑| ⊗ |Φ↑(t)i hΦ↑(t)| + |c↓| |↓i h↓| ⊗ |Φ↓(t)i hΦ↓(t)| (45) with Z |Φ(t)i = dP Φ(P, t) |P i R and |Φ(¯u, t)|2 given by (43).

5 Conclusions

1 Employing a simple spin- 2 model which represents a measurement in a Stern- Gerlach experiment, studied in more detail by A. Venugopalan, D. Kumar and R. Ghosh in a 1995 paper (Venugopalan et al. 1995), we are able to use the Lindblad equation to formulate a diﬀerential equation which approx- imately gives the time evolution of the system-device density operator (25). This equation is similar to the exact equation describing quantum Brown- ian motion in the famous Caldeira-Leggett model, although the relaxation is taken to be insigniﬁcant on the timescales of decoherence and therefore disregarded, in accordance with the Lindblad equation.

The equations (25) provide the density matrix of the device for each com- bination of two spin states (s, s0) = (+1, +1), (+1, −1), (−1, +1), (−1, −1).

26 We showed explicitly that decoherence drives the oﬀ-diagonal combinations (s 6= s0) to zero. Comparisons with the work of Venugopalan et al. shows that this remains true if relaxation is introduced into the system. We also found for the spin-diagonal combinations (s = s0) that decoherence drives the momentum oﬀ-diagonal combinations to zero in the momentum basis (again, this holds true when relaxation is included). Lastly, we found that the momentum-diagonal represents Gaussian wave packets which yield the wave function of the device, and that the amplitude of these packets dimin- ish as a result of decoherence, although comparisons show that this does not remain true for a relaxation model.

In conclusion, this shows that decoherence does work as a device for causing wave function collapse, due to its tendency to drive the oﬀ-diagonal to zero. What is left is a mixture of wave functions, one for each term in the initial superposition of states, and with a corresponding (classical) probability to occur which agrees with the Born rule.

27 6 References

Tokmakoﬀ, A. 5.74 Introductory Quantum Mechanics II. Spring 2009. Mas- sachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

Allahverdyan, A. E., Balian, R. & Nieuwenhuizen, T. M. (2013), ‘Under- standing Quantum Measurement from the Solution of Dynamical Models’, Physics Reports 525(1), 1–166.

Born, M. (1926), ‘On the Quantum Mechanics of Collision Processes’, Zeit fur Phys 38, 803.

Breuer, H.-P., Petruccione, F. et al. (2002b), The Theory of Open Quantum systems, Oxford University Press on Demand, chapter 3.

Caldeira, A. O. & Leggett, A. J. (1983), ‘Path Integral Approach to Quan- tum Brownian Motion’, Physica A: Statistical Mechanics and its Applications 121(3), 587–616.

Courant, R. & Hilbert, D. (1962), Methods of Mathematical Physics, Vol. II, Wiley.

Delgado, M. (1997), ‘Classroom Note: The Lagrange–Charpit Method’, SIAM Review 39(2), 298–304.

Griﬃths, D. J. & Schroeter, D. F. (2018), Introduction to Quantum Me- chanics, Cambridge University Press, pp. 104–107.

Landsman, N. P. (2009), Born Rule and its Interpretation, in ‘Compendium of Quantum Physics’, Springer, pp. 64–70.

Roy, S. M. & Venugopalan, A. (1999), ‘Exact Solutions of the Caldeira- Leggett Master Equation: A Factorization Theorem for Decoherence’, arXiv:quantph/9910004 p. 4.

Schlosshauer, M. (2005), ‘Decoherence, the Measurement Problem, and Inter- pretations of Quantum Mechanics’, Reviews of Modern physics 76(4), 1270.

28 Schlosshauer, M. (2014), ‘The quantum-to-Classical Transition and Deco- herence’, arXiv preprint arXiv:1404.2635 pp. 7–9.

Schlosshauer, M., Koﬂer, J. & Zeilinger, A. (2013), ‘A Snapshot of Founda- tional Attitudes Toward Quantum Mechanics’, Studies in History and Philos- ophy of Science Part B: Studies in History and Philosophy of Modern Physics 44(3), 222–230.

Venugopalan, A. (1999), ‘Pointer States via Decoherence in a Quantum Mea- surement’, Physical Review A61(1), 012102.

Venugopalan, A., Kumar, D. & Ghosh, R. (1995), ‘Environment-Induced Decoherence I. The Stern-Gerlach Measurement’, Physica A: Statistical Me- chanics and its Applications 220(3-4), 563–575.

Von Neumann, J (2018), Mathematical Foundations of Quantum Mechan- ics, new edition.. edn, pp. 271–288.

Zurek, W. H. (2003), ‘Decoherence and the Transition from Quantum to Classical–Revisited’, arXiv preprint quant-ph/0306072.

29 A Appendix

A.1 Spin diagonal matrix element Beginning with 2 ∂ρd(Q, r, t) ~Q ∂ρd Dr iεr = − ρd ∓ ρd (A.1) ∂t M ∂r 4~2 ~ we use the method of characteristics. Equation (A.1) can be written into Lagrange-Charpit equations, meaning it is equivalent to the following system of ODE’s:

 dt  = 1 (A.2)  ds  dr = −vQ (A.3)  ds  2  dρd Dr iεr  = −ρd( ± ) (A.4) ds 4~2 ~ where we deﬁne vQ = ~Q/M and the right hand side of the equations are the weights of the partial derivatives. From these equations we see that t = s (A.5) and r = −vQt + r0 (A.6) where we assumed that t0 = 0 and r0 is an integration constant. The invariants (expressions with derivative equal to zero) of our system of ODE’s are then I1 = r + vQt (A.7) and  2 3  vQt 2 2 1 2 D( 3 + vQrt + r t + const) iε( 2 vQt + rt + const) I2 = ρd exp ±  4~2 ~ (A.8) where I2 comes from integrating the separable equation (A.4) and using the identities Z −v t2 v t2 rdt = Q + r t + const = Q + rt + const (A.9) 2 0 2

30 and Z v2 t3 v2 t3 r2dt = Q − v r t2 + r2t + const = Q + v rt2 + r2t + const (A.10) 3 Q 0 0 3 Q where we have used that r0 = vQt + r according to (A.6). The integration constants will be determined from the initial condition (23).

In order to determine the general solution, it is enough to ﬁnd the characteristics by solving the characteristic system of ODEs. Due to the factorization theorem our solution must be a function of the invariant for r, equation (A.7), meaning I2 = w(I1) where w is an arbitrary function. Our solution for equation (A.1) will thus be of the form

 2 3  vQt 2 2 −D( 3 + vQrt + r t + const) ρd(Q, r, t) = w(I1) exp  4~2

iε( 1 v t2 + rt + const) × exp ∓ 2 Q (A.11) ~

Our initial condition allows us to ﬁnd an explicit expression for w(I1). The initial condition in Q, r representation is σ2Q2 r2 ρ (Q, r, t = 0) = exp ipr¯ − − (A.12) d 4 4σ2 so our exact solution to equation (A.1) becomes σ2Q2 1 ρ (Q, r, t) = exp ip¯(r + v t) − − (r + v t)2 d Q 4 4σ2 Q

 2 3  vQt 2 2 1 2 −D( + vQrt + r t) iε( vQt + rt) × exp 3 ∓ 2  (A.13) 4~2 ~

A.2 Spin oﬀ-diagonal matrix element Beginning now with

2 ∂ρod(Q, r, t) ~Q ∂ρod Dr 2ε ∂ρod 2iλ = − ρod ∓ ∓ ρod (A.14) ∂t M ∂r 4~2 ~ ∂Q ~ 31 we identify the transformation

2iλt ρod = W exp ∓ (A.15) ~ Using the same deﬁnitions as in subsection A.1 we now simplify equation (A.14) to solving the system

 dt  = 1 (A.16)  ds   dr  = −vQ (A.17)  ds dQ 2ε  = ± (A.18)  ds  ~  dW Dr2  = − W (A.19) ds 4~2 This yields t = s (A.20) and Z Z Z Z 2 dr −~ dQ ~ εt ~ r = dt = − vQdt = ( dt)dt = ∓ − r0t + r1 dt M dt M ~ M (A.21) where r0 and r1 are integration constants.

The invariants are 2εt I1 = Q ∓ (A.22) ~ ~ εt2 I2 = r + (∓ + Qt) (A.23) M ~ Equation (A.22) comes from integrating equation (A.18). Equation (A.23) comes from equation (A.21) like this

~ εt2 ~ ~ εt2 ~ I2 = r ± + r0t = r + (∓ ) + Qt (A.24) M ~ M M ~ M 2εt where we have used that r0 = ∓ + Q, known from equation (A.18). ~

32 Again, due to the factorization theorem, the third invariant must be some function of (A.22) and (A.23). So

2 5 4 2 3 3 −D ε t r0εt r0t 2~ εt W (Q, r, t) = f(I1,I2) exp ( ∓ + ∓ ) 4~2 5M 2 2M 2 3 M 3~ −D ~ 2 2 × exp (− r0r1t + r1t + const) (A.25) 4~2 M where f is an arbitrary function. The expression (A.25) comes from integrating r2, because the derivative of (A.25) should produce r2 in front of the exponential:

2 2 4 3 2 2 ~ εt 2 ε t 2r0ε~t 2 2 2~ εt 2 r = ( (∓ − r0t) + r1) = ± + r0t + (∓ − r0t)r1 + r1 M ~ M 2 M 2 M ~ (A.26) so Z Z 2 4 3 2 2 ε t 2r0ε~t 2 2 2~ εt 2~ 2 r dt = ( ∓ + r0t ∓ − r0tr1 + r1)dt = M 2 M 2 M ~ M

2 5 4 2 3 3 ε t r0εt r0t 2~ εt ~ 2 2 ∓ + ∓ − r0r1t + r1t + const (A.27) 5M 2 2M 2 3 M 3~ M We know from (A.21) that

~ εt2 r1 = r ∓ ( − Qt). (A.28) M ~ We have that the solution to (A.14) is of the form 2iλt ρod(Q, r, t) = f(I1,I2) exp ∓ ~

2 5 4 2 3 3 −D ε t r0εt r0t 2~ εt ~ 2 2 × exp ( ∓ + ∓ − r0r1t + r1t + const) . 4~2 5M 2 2M 2 3 M 3~ M (A.29) The solution comes from the initial condition (A.12). We get

2 2 −σ 2εt 2 I2 2iλt ρod(Q, r, t) = exp (Q ∓ ) − + ipI¯ 2 exp ∓ 4 ~ 4σ2 ~

33 2 5 4 2 3 3 −D ε t r0εt r0t 2~ εt ~ 2 2 × exp ( ∓ + ∓ − r0r1t + r1t) . (A.30) 4~2 5M 2 2M 2 3 M 3~ M

A.3 Solutions in momentum representation The momentum representation is given by Z ρd(¯u, v,¯ t) = exp(i(¯ux +vy ¯ ))ρd(x, y, t)dxdy (A.31)

u¯+¯v In the variables Q =u ¯ − v,¯ q = 2 we can obtain the momentum representation ρd(Q, q, t) by taking a Fourier transform with respect to r of equation (A.13) Z ∞ ρd(Q, q, t) = exp(iqr)ρd(Q, r, t)dr (A.32) −∞ Expanding the integrand we have

σ2Q2 1 exp(iqr) exp ip¯(r + v t) − − (r + v t)2 Q 4 4σ2 Q

 2 3  vQt 2 2 1 2 −D( + vQrt + r t) iε( vQt + rt) × exp 3 ∓ 2  = (A.33) 4~2 ~

2 2 2 3 σ Q −D vQt iε 1 2 1 2 2 = exp ipv¯ Qt − + ( ) ∓ ( vQt ) − (vQt ) 4 4~2 3 ~ 2 4σ2 −D 2 iε 1 × exp r(iq + ip¯ + (vQt ) ∓ (t) − (2vQt)) 4~2 ~ 4σ2 −D 1 × exp r2( (t) − ) . (A.34) 4~2 4σ2 This gives

r 2 π −1 iDvQt εt ivQt 2 ρd(Q, q, t) = 2 exp (q +p ¯ + ∓ + ) A(t) A(t) 4~2 ~ 2σ2

34 2 2 2 3 2 2 2 σ Q DvQt iεvQt vQt exp ipv¯ Qt − − ∓ − (A.35) 4 12~2 2~ 4σ2 where A(t) = Dt + 1 . ~2 σ2

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