Semiclassical Propagators of Wigner Function: A comparative study.
Jose Mauricio Sevilla Moreno
Universidad Nacional de Colombia Facultad, Departamento de F´ısica Ciudad, Colombia 2020
Semiclassical Propagators of Wigner Function: A comparative study.
Jose Mauricio Sevilla Moreno
Tesis presentada como requisito parcial para optar al t´ıtulode: Magister en Ciencias: F´ısica
Director: Carlos Leonardo Viviescas Ram´ırez
L´ıneade Investigaci´on: Caos Cu´anticoy M´etodos Semicl´asicos
Grupo de Investigaci´on: Caos y complejidad
Universidad Nacional de Colombia Facultad de Ciencias, Departamento de F´ısica Bogot´a,Colombia 2020
To my parents, Jos´eand Esperanza.
Acknowledgments
I would like to thank my thesis advisor, Dr. C. Viviescas, for welcoming me into the Caos y Complejidad group, working with me closely as a collaborator more than a superior, listening to all the comments and ideas I had al along the time I spent in the group. His support , patience and encouragement were essential on the development of this project as well as myself as a physicist.
I also would like to thank all the Caos and Complejidad group members whom were always open to listen and discuss ideas on the office.
But most of all, I need to thank my friends that were always there for me, listening, guiding and advising specially when I was feeling lost and without force to go on. Thanks to Alejandro Monta˜na, C´esar C´elis,Daniela Garz´on,Daniel Padilla, Frank Bautista, Juli´anTriana and Nicol´as Medina for all the things that have done for me that I can’t even try to describe in just a few words.
Last but not least, I want to thank my family, my parents, my brother and my sister, who always have been an inspiration source for my life.
ix Abstract
Semiclassical approximations for the dynamics have been widely used in different represen- tations of quantum mechanics. In particular, phase space representations exhibit a clear way to implement those approximations by direct comparison with classical mechanics theory. During the last years, such approximations have been recovering interest, due to the fact that numerical applications suit the modern computational architectures. In this work different semiclassical approximations are built and compared on performance and complexity, showing that they are a suitable way to calculate the quantum dynamics. A study over the caustics is presented on a the Morse potential showing the similarities of the final Wigner function considering and not considering them. Following this, we introduced an initial and final value representations (IVR and FVR) to compare the dynamical properties with the center-center representation proposed by Dittrich et al. [Dittrich et al., 2006] showing that the later can be used as a initial value representation in numerical applications, getting better performance with less complexity than the IVR and FVR using different criteria in such comparison. Contents
Acknowledgments vii
Abstract ix
0 Introduction 1
1 Theoretical Background 6
1.1 Wigner Function ...... 8
1.1.1 Dynamics ...... 9
1.2 Semiclassical Propagation ...... 11
1.2.1 Wigner function propagator ...... 12
1.2.2 Initial and Final Value Representations of the Propagator ...... 16
2 Numerical Implementation 23
2.1 Center-Center Representation ...... 24
2.1.1 Algorithm ...... 27
2.2 *Caustics Counting ...... 28
2.3 Initial Value Representation ...... 30 CONTENTS xi
2.3.1 Algorithm ...... 31
2.4 Final Value Representation ...... 32
2.4.1 Algorithm ...... 33
2.5 Comparison Criteria ...... 34
2.5.1 Observable Calculation ...... 34
2.5.2 Autocorrelation Functions ...... 34
2.5.3 Normalized Inner Product ...... 35
2.5.4 Marginals ...... 35
3 Results 37
3.1 Model: Morse Oscillator ...... 38
3.2 Caustics ...... 39
3.2.1 Initial State ...... 40
3.2.2 Determinant ...... 41
3.2.3 Reconstruction of the Final State ...... 43
3.2.4 Final Wigner and Wave Functions ...... 43
3.3 Center-Center Representation ...... 47
3.3.1 Marginals ...... 49
3.4 Initial Value Representation ...... 51
3.4.1 Wigner Function ...... 51
3.5 Final Value Representation ...... 53
3.5.1 Wigner Function ...... 53 xii CONTENTS
3.6 Comparison ...... 55
3.6.1 Autocorrelation ...... 56
3.6.2 Observables ...... 57
3.6.3 Normalized Inner Product ...... 61
3.6.4 Complexity ...... 61
3.7 Parallelization ...... 63
3.7.1 IVR-FVR ...... 63
3.7.2 Center-Center ...... 63
4 Conclusions and Perspectives 64
A Coherent States 66
A.1 Wigner Function for Coherent States ...... 66
A.2 Relationship with the Husimi Function ...... 69
B Wigner Function Calculations 71
B.1 Marginal Probabilities and Area ...... 71
B.2 Weyl Correspondence ...... 73
C Caustics 76
D Classical Evolution 79
D.1 Dynamics ...... 80
D.2 Distributions ...... 80 CONTENTS xiii
D.3 Numerical Implementation ...... 81
E Quantum Propagation 84
E.1 State Dynamics ...... 84
E.1.1 Algorithm ...... 87
E.2 Winger Function Dynamics ...... 88
E.2.1 Algorithm ...... 90
Bibliography 92 Chapter 0
Introduction
Form the very beginning of the development of quantum mechanics an interpretation of the theory was sought. The straightforward path was comparing this new, at that time, theory with the well known results developed so far, i.e., the classical mechanics. This comparison was not clear as the quantum theory has a mathematical structure which is clearly way different than the classical theory and therefore this had to be done using particular results such as expectation values. The quantum theory resulted to have different results than the classical mechanics. This was one of the biggest issues of quantum mechanics, as the community was not accepting the theory because of the lack of interpretation or the strangeness of some already existing. As the continuous comparison of classical and quantum mechanics showed that they are indeed no compatible, the question of the relationship between them remained. it was until 1928 when after studying an statistical interpretation of works done by Dirac [Dirac and Fowler, 1927], that van Vleck [Van Vleck, 1928] built up an expression of the quantum dynamics in terms of classical quantities which is used as a basis of the so called semiclassical methods. It uses the idea of classical propagation of a set of classical particles in order to get the time evolution of wave functions.
This idea of semiclassics, relies on the dynamics1, even when the dynamical rules of classical and quantum mechanics are not the same and even worse, they are not even comparable by some important reasons. First, the Schr¨odingerequation which is the quantum mechanical equation of motion is complex as well as a wave function ψ[Landau and E.M., 1977], and the Hamilton’s equations are not[Arnold et al., 2013]. Second, the representation of single particles is completely different, as in quantum mechanics a state is represented by a non localized function2, while in
1This means that these methods are not constructed to find other quantum characteristics, such as ground states. 2In fact, it is localized but not on a single point. 2 0 Introduction
classical mechanics, a state is defined to be a single point on phase space3, and so on.
The idea behind the van Vleck formula can be understood using several ways to proceed, for instance, from the Feynman’s path integral point of view [Feynman, 1948, Feynman, 1966], which consists on the usage to trajectories to construct the dynamics of the quantum mechanical state. This trajectories do not necessary represent classically possible trajectories, but the classical trajectories are also included. We may do some comments about this way to proceed, first is that for the evolution of a single quantum mechanical particle represented via a wave function cannot be evolved using a single classical particle but a distribution of particles instead, which leads to the question Shall we use a set of classical particles to compare a single quantum mechanical particle? and the answer is yes, but this brings up more questions about what happen with the amount of quantum particles on a limit where quantum mechanics goes to classical mechanics, but we will not discuss about this concern4, a second comment is that, the evolution is described by two terms, an amplitude and the exponential of a phase. The two of them depend on the particular trajectory taken but these two quantities are calculated from an action corresponding to the chosen path. The semiclassics here enters on reducing the complete set of trajectories to just the classical ones.
This result has huge implications. In particular that, the van Vleck propagator itself is not enough to describe certain quantum mechanical features such as tunnelling, due to the fact that as we are only considering the classical trajectories, there are forbidden areas so cannot be transport of probability or density classically speaking, but it occurs quantum mechanically.
The van Vleck formula has another issue which was pointed out by Gutzwiller [Gutzwiller, 1967], as the amplitude term has some insights that play a very important role on the dynamics an ad- ditional treatment shall to be added to the van Vleck result. This is basically what are called caustics (For more on caustics, see appendix C), the caustics are points (lines, surfaces and so on depending the order and the number of degrees of freedom of the system) where many trajectories concentrate, this generates a divergence on the amplitude that is not real. As semiclassics is not valid on that exact point5, but it is just before and after in such a way that the only difference is that the trajectory gains a phase of π/2 on the caustic[Maslov and Fedoriuk, 1981], this lead to a big problem and it is that the caustics must be counted, so that at the end of the propagation the complete phase is added to the contribution of each particular trajectory.
This, now complete way of propagate is known as the van Vleck-Gutzwiller propagator, and it has been widely spread used after its development, for example Gutzwiller got an expression to calculate using this semiclassics scheme, an approximation for the spectrum of a quantum
3There are some definitions of phase space on the classical theory, but we refer to the Hamilton’s phase-space. 4We will see what kind of approach we may take in this context to get to a Classical limit. 5Which does not mean that one cannot get the information of the dynamics on that specific point, but means that the trajectory which is on a caustic, shall not count be counted 3 mechanical system which is known as the Gutzwiller trace formula [Gutzwiller, 1969]. This formula is very important as this is the path to go into quantum chaos.
The van Vleck-Gutzwiller formula requires a certain level of computational resources, and thereby initially it was not possible to use it as a tool to calculate the dynamics, as it demands a lot of calculations that is only possible to be done using computers. Nowadays, this kind of semiclassical schemes have arisen to be a suitable way to add quantum effects to simulation of systems where quantum mechanics is present but very expensive to be calculated, such as it occur in molecular dynamics simulations.
All of these semiclassical structures use as a basis the van Vleck propagator, even the propa- gations based on the Herman Kluk [Herman and Kluk, 1984] and Frozen Gaussians [Heller, 1981]6 which commonly are thought to be independent of it, have a strong and deep relationship on the sense that they both are based on the idea on expanding on coherent states (Gaussians) which are not a complete basis of functions but an overcomplete one, it means that the expansion of a given function is not unique and it can depend on several features of the specific set of Gaussians chosen.
The relationship relies on the fact that the Herman Kluk propagator smooths the functions with a Gaussian kernel7 in such a way that if we take the limit of the width of the Gaussians γ → 0 we recover the van Vleck propagator.
This means that, fundamentally, the Herman Kluk propagator lacks the contributions made by Gutzwiller to the van Vleck propagator, which initially was considered by Heller [Heller, 1981] but missed out by Herman and Kluk [Herman and Kluk, 1984] on their justification of Heller’s method. This way to propagate is reduced to calculate a Gaussian smoothed version of the final wave function, that may be troublesome as the width of the Gaussians is fixed beforehand the evolution.
Adding quantum corrections via the van Vleck-Gutzwiller propagation is still not good enough for the limitations that we have already mentioned among others, so to solve those problems, there is an interest on using a different representation which lacks these issues. The quan- tum phase space representations have been proved to work successfully[Dittrich et al., 2006, Ozorio de Almeida et al., 2013, Ozorio de Almeida et al., 2019]. The main idea behind phase space representation of quantum mechanics is using functions instead of operators, and that can be done on many different ways, but the important feature of these representations is that there exists a clearer way to make direct comparison with the classical mechanics, as they can be
6Even this work, which is constructed from a conceptual point of view without any proof or derivation, just intuition. 7This is exactly the same idea of the Husimi function, which is not the best option when looking for the dynamics as it will be explained on the following chapter on in appendix A 4 0 Introduction
written on some similar spaces[Zachos et al., 2005, P. Schleich, 2001].
As phase space of quantum mechanics and classical mechanics can be built up on similar contexts, it is worth to think about the links between them, and in particular the role that semiclassical approximations take on the phase space representations of quantum mechanics. This comparison have led to a preferred phase space which is the Wigner phase space where the Wigner function is the representation of the density operator and therefore of the state of the system. This because of its dynamics has very interesting implications and it is the most suitable compared with other where the dynamics cannot even be defined completely such as Husimi function.
The Wigner function has a wide range of applications, for instance, in semiclassical methods [Ozorio de Almeida et al., 2019, Ozorio de Almeida et al., 2013, Dittrich et al., 2006, Dittrich et al., 2010, Koda, 2015], as in this the case, but also in signal process and analysis [Daniela, 2005], classical and quantum optics [P. Schleich, 2001], molecular, atomic and nuclear physics [Wigner, 1932], quantum statistical mechanics [Wigner, 1932], decoherence or quantum chaos[P. Schleich, 2001].
The semiclassical approximations on the context of Wigner function has been growing on the recent years [Dittrich et al., 2006, Dittrich et al., 2010, Ozorio de Almeida et al., 2013, Ozorio de Almeida et al., 2019, Koda, 2015], using a different kind of systems to prove their flexibility and robustness. The implementation of these propagators have led to comparisons at the numerical level that have shown to lack justification in the sense that one stands that the van Vleck approach proposed in [Ozorio de Almeida et al., 2013] is not valid for long times while the Herman Kluk approach does[Koda, 2015]. While in the other hand a more recent work showed that in fact the van Vleck approach [Ozorio de Almeida et al., 2013] can be used for very long times8 [Ozorio de Almeida et al., 2019]. This basically shows that the comparison previously done is unfair and poorly implemented.
In this work we implemented a series of semiclassical propagators based on the van Vleck propagator [Dittrich et al., 2006, Dittrich et al., 2010, Ozorio de Almeida et al., 2013, Ozorio de Almeida et al., 2019] as lately they have been strongly criticized [Koda, 2015] but with not very strong arguments, which seems a misinterpretation of the way the methods must be implemented. This docu- ment is organized as follows: A brief introduction to quantum mechanics on phase space and its dynamics, after we construct and discuss a semiclassical propagator [Dittrich et al., 2006, Dittrich et al., 2010], then we follow the way to proceed of [Ozorio de Almeida et al., 2013] to show that the previously presented propagator can be written on some center-center represen- tation so that we eliminate some divergences, just as it happens for the mixed propagators shown in [Ozorio de Almeida et al., 2013]. After that, we explain the numerical implementa-
8We reefer as long times as the Ehrenfest time, which is used to characterize quantum chaotic systems and its correlations. 5 tion and how the propagator [Dittrich et al., 2006, Dittrich et al., 2010] can be understood as an initial value representation, even when it is said not to be one on the literature[Koda, 2015, Ozorio de Almeida et al., 2013]. Finally we present the results of the dynamics for different times using a typical system and lastly some conclusions and perspectives. Chapter 1
Theoretical Background
Phase space representation of quantum mechanics started from the development of the theory itself in the sense that it was under constant comparison with the classical theory, in the attempt to have a better interpretation of the theoretical and experimental results. It was then, during 1932, that Eugine Wigner [Wigner, 1932] built up the first so called phase space representation, when trying to calculate quantum observables just as it is done in classical statistical mechanics. This comparison led to a function which behaves just as a density probability distribution to the calculation of observables, but the function resulted to be negative in some parts of the domain, and as a consequence it could not be called a probability distribution. Nevertheless, as it satisfied all the other properties a distribution does, it is called a pseudo probability distribution. This issue notwithstanding, the function is smooth and well behaved in all the domain, so it has been widely studied as a suitable representation of quantum mechanics.
This is not the only possible phase space representation, and actually there are an infinite number of them, because the key point is to construct functions from operators andthis can be done in several ways. For instance, one can consider the following way to construct a phase space representation P (p, q) [Hillery et al., 1997, Ozorio de Almeida, 1998] of the quantum state: re- placing operators qˆ by the variable q and doing exactly the same for the momenta, then tracing in order to get a function instead of an operator Pˆ(pˆ, qˆ), h i P (p, q) = Tr Pˆ(pˆ, qˆ) δ(qˆ − q, pˆ − p) . (1-1)
It is possible to choose the representation of the operator or the delta function before the tracing on many different ways, each one will induce a basis in which the trace is calculated, and each one of them generates a particular phase space with plenty different properties.
This can be translated to the second quantization formalism and, therefore, uses a ordering 7 rule indicating a clear way to proceed. Usually three ordering recipes are followed, normal ordering, antinormal ordering and mixed ordering, each one of them leading to one of the most commonly used representations [P. Schleich, 2001]:
• Normal Ordering: Husimi Phase Space.
• Anti-Normal Ordering: Glauber Phase Space.
• Mixed Ordering: Wigner Phase Space.
In principle, all of these three options can be used in order to represent quantum mechanical systems, but each one of them has its unique properties and hence advantages and disadvantages regarding an specific problem. For instance, the Q or Husimi function [Husimi, 1940], which results of constructing the Husimi phase space representation of the density operator is well known and used because it is defined to be always positive; it can be interpreted as a probability distribution. It also presents some deep problems as it is defined from the density operator and a particular basis of coherent states (see appendix A) which form an overcomplete basis, and leads to losing some features of the state after using a Gaussian kernel to smooth the function. Regarding the dynamics, the Husimi function has plenty of troubles, mainly because as it is a smooth version of the function that represents the state, or in other words, it is constructed from a reduced version of the density operator where the coherences get lost, it is not possible find the Husimi function on a time t > t0 from the Husimi function of a time 1 t0 [O’Connell and Wigner, 1981b] . So the Husimi function offers an easy interpretation of the phase space representations of quantum mechanics, but is very restricted when applied to specific problems such as the dynamics of the quantum mechanical system. It is also worth mention that it performs well when looking for probability concentrations or a straightforward way to interpret and compare with the corresponding classical phase space. Something similar occurs when describing quantum mechanics in the Glauber phase space, where the density operator is know as the P or Glauber-Sudarshan function [Glauber, 1963]. This function is also built from a coherent states basis but in contrast to the Husimi function, the P function can take negative values. It has some interesting properties such as, in the field of quantum optics, the P function represents a true probability density for classical states of light2. However, this implies that the P function has problems regarding non classicality. This idea of a probability distribution breaks when calculating probabilities of mutually exclusive states, which cannot be described by the P function, due to the coherent states not being orthogonal among themselves [Leonhardt et al., 1997].
1It is possible to find a equation of motion for the Husimi function from the equation of motion of the Wigner function, but it is not possible to guarantee that the evolved Husimi function from a time t = t0 is the same that the one transformed from the Wigner function at a time t0. 2The definition is done usually the other way around, a classical state of light is one such that the P function can be considered a probability density. 8 1 Theoretical Background
1.1 Wigner Function
The Wigner function was the first of the phase space representations ever proposed, and can be defined from the Weyl transform. For any operator Aˆ it returns its Weyl symbol, Z i D y yE A(p, q, t) = dy exp − p · y q − Aˆ(t) q + , (1-2) h¯ 2 2 where (p, q) are the coordinates of phase space, and h¯ is Planck constant divided by 2π. This relation allows us to construct the Wigner phase space representation of any quantum mechanical operator, a necessary step if we want to calculate quantities such as expectation values or the time evolution in the phase space representation.
If we consider the specific case of transforming the density operator, so that we obtain the function that represents the state of the quantum mechanical system, we get the Wigner function,
1 Z i D y yE W (p, q, t) = dy exp − p · y q − ρˆ(t) q + . (1-3) (πh¯)d h¯ 2 2 One of the most important features of this phase space representation is that it is equivalent to any other representation of quantum mechanics, for instance, the position or momentum representations. This relation can be inverted and the density matrix can be reconstructed from the Wigner function (See appendix B).
The Weyl representation has many different characteristics and properties that make it a very useful representation of quantum mechanics, specially when a comparison with classical mechanics is wanted, as it is the case in this work. We now list some of the principal characteristics of the Wigner function[Ozorio de Almeida, 1989, Hillery et al., 1997, P. Schleich, 2001]:
• As a consequence of the hermiticity of the density operator, the Wigner function is real.
• It is possible to calculate the position probability density just by applying one integral, Z |ψ(q)|2 = dpW (q, p). (1-4)
• The same can be done for the momentum representation, Z |φ(p)|2 = dqW (q, p). (1-5)
• it is normalized, Z Z dq dpW (p, q) = Tr(ˆρ) = 1. (1-6) 1.1 Wigner Function 9
• For systems up to quadratic potentials, the evolution corresponds to the classical Liouville equation.
• The previous feature can be extended when considering the classical limit of the dynamics. The evolution of the Wigner function in the limit h¯ → 0 is given by the Liouville equation. This entails huge consequences. For example, if a comparison to classical mechanics is wanted, it is not possible to compare a single quantum mechanical particle with a single classical particle, but a comparison of a single quantum mechanical particle with a complete classical distribution is necessary 3.
• The only possible state in which the Wigner function is always positive is a Gaussian sate (e.g., coherent or thermal states), otherwise it will take negative values [Hudson, 1974].
1.1.1 Dynamics
We have previously mentioned properties of the dynamics in phase space yet, so far, we have not describe it properly. In this section we will mention some features and properties of the time evolution.
The dynamics of the Wigner function can be found by evaluating the time derivative of the Weyl transform of the density operator (1-3) and then using the von-Neumann equation for the dynamics of the density operator. Hence, it becomes necessary to Weyl transform the commutator of the density operator and the Hamiltonian function. This procedure leads to a Bracket-like equation that defines a lie algebra called the Moyal bracket,
∂W (p, q) = {H,W } . (1-7) ∂t Moyal The Moyal bracket is defined as,
{A, B}Moyal = A?B − B ? A, (1-8) in terms of the ? product, also called the Moyal product, ←− −→ ←− −→ !! ih¯ ∂ ∂ ∂ ∂ ? = exp − , (1-9) 2 ∂p ∂q ∂q ∂p where the direction of the arrows over the operators indicate the direction in which they operate.
3The idea of taking ¯h → 0 seems nonsensical, but what is really happening is that the typical actions on the system are large enough, so that the ratio between the actions and ¯h increases rapidly, which is equivalent to take this limit. 10 1 Theoretical Background
This differential equation is, in general, an infinite order partial differential equation. It corresponds exactly to the Poisson bracket if up to quadratic Hamiltonians are considered, so the quantum information shows up in cases of Non-linear Hamiltonians.
Propagator
Even though this equation contains high order derivatives, it is linear, and therefore it can be solved by using a Kernel, Z 0 0 0 0 0 0 W (p, q, t) = dq dp G(p , q , t0; p, q, t)W (p , q , t0). (1-10)
0 0 The propagator G(p , q , t0; p, q, t) is called the Wigner function propagator [Dittrich et al., 2006] or Wigner propagator, for short, and shares some properties with the quantum mechanical prop- agator on position representation, e.g. it also form a group.
Contrary to what one would expect, the Wigner propagator is not found by doing the Weyl transform of the position representation propagator. As the Wigner function is built from the density operator (i.e., two waves functions), it is necessary to consider the convolution of the Weyl symbols of two time evolution operators instead. Adopting the notation for a point on phase-space r = (p, q), we obtain for the Wigner propagator
1 Z i G (r00, r0, t) = dr0 exp r ∧ (r00 − r0) W hd h¯ r00 + r0 r r00 + r0 r U − U + , (1-11) W 2 2 W 2 2
where the UW are the Weyl transform of the time evolution operator in position representation or Feynman propagator K,
Z i q0 q0 U (r, t00, t0) = dq0 exp − p · q K q + , t00, q − , t0 . (1-12) W h¯ 2 2
Hence, the propagator of the Wigner function in terms of the Feynman propagator is,
2 d Z i G (r00, r0, t) = dQ0dQ00 exp (p0 · Q0 − p00 · Q00) W h h¯ Q00 Q0 Q00 Q0 K∗ q00 − , t00, q0 − , t0 K q00 + , t00, q0 + , t0 . (1-13) 2 2 2 2
This last expression will be used in the following section in order to derive the semiclassical Wigner function propagator, the main object of this work. 1.2 Semiclassical Propagation 11
1.2 Semiclassical Propagation
The principal ingredient to the semiclassical approximations presented here is the van Vleck propagator [Van Vleck, 1928]. It was developed with a very similar goal than the Wigner function, both were done while looking for an interpretation of quantum mechanics calculating averages. The derivation done by van Vleck is not complete, because it does not consider the caustics (Apendix C) that come out of a folding on the Lagrangian manifold where the trajectories exist. These were added later by Gutzwiller [Gutzwiller, 1967, Gutzwiller, 1969]. Gutzwiller was also one of the first to adopt this strategy of semiclassical propagation for doing dynamics, and use it to calculate the energy spectrum of a quantum mechanical systems [Gutzwiller, 1970], proving the validity of the van Vleck propagation, which until that point in time was consider a purely theoretical artefact due to the computational power required to be able to use it.
The derivation of the van-Vleck-Gutzwiller propagator can be done following different paths, for instance, Berry and Mount in [Berry and Mount, 1972] did a very complete revision on this by using the Feynman path integral strategy, Littlejohn [Littlejohn, 1992] and Keller [Keller, 1985], used the WKB approximation which allowed them to make a clearer geometrical interpretation of it including the caustics.
This problem of the caustics is a matter of representation, in the sense that the points that are caustics on a particular representation are not necessary caustics on a different representation. The work done by Littlejohn [Littlejohn, 1992] clearly illustrates that the caustics on position representation are not in momentum, and therefore a phase is gain after crossing a single caustic depending the order, this issue will be discussed later on the appendix C.
The semiclassical van Vleck-Gutzwiller propagator is [Van Vleck, 1928, Gutzwiller, 1967],
X m d/2 i π K (q0, q, t) = exp R (q0, q, t) − i µ , (1-14) vV 2πiht¯ h¯ j 2 j j where R(q0, q, t) is Hamilton’s principal function of the classical trajectory connecting q and q0 in time t, µ is the Maslov index [Ozorio de Almeida, 2009, Maslov and Fedoriuk, 1981] and the summation over j’s takes into account all the possible trajectories connecting the two points; there may be more than one because in this case the momentum is a free parameter that can lead to different trajectories and therefore different values of R and µ.
This propagator can be rewritten in terms of Hamilton’s principal function only, s 2 0 0 X 1 ∂ Rj(q , q, t) i 0 π KvV (q , q, t) = exp Rj(q , q, t) − i µj , (1-15) hd ∂q0∂q00 h¯ 2 j 12 1 Theoretical Background
This expression is richer on interpretation, as it makes clear how µj must be calculated in terms of the change of sign of the Hessian matrix of Rj. This procedure is, indeed, very heavy compu- tationally as there is no a clear way to count the number of caustics a particular trajectory has crossed.
This propagator is exact for up to quadratic potentials, e.g., for free particles, harmonic wells or quadratic barriers. This is one of the most challenging issues this scheme of propagation has: classically there is no such concept as tunnelling but on a quadratic barriers in quantum mechanics there is. Works on this direction have been done, successfully explaining why this propagator should reproduce tunnelling on quadratic barriers [Jaubert and de Aguiar, 2007, Kay, 2013].
In addition, the evaluation of this propagator is numerically demanding. As the propagation from one specific position to other position is required, it is necessary to find all trajectories that connect these two points on a given time t. This problem is known as the root-search problem.
1.2.1 Wigner function propagator
The van Vleck-Gutzwiller version of the Wigner function propagator is done by replacing (1-15) into (1-13), but this is not as simple as it seems, as some considerations have to be taken in order to get a good approximation. The propagator of the Wigner function has to be built from two Weyl symbols of the time evolution operator[Dittrich et al., 2006, Dittrich et al., 2010], so it is given by
2d Z 0 −1/2 0 −1/2 00 0 2 X r r GW (r , r , t) = dr det Mj r + + I det Mj0 r − + I h 2 2 j,j0 i 00 0 i r r × exp r ∧ (r − r ) exp A ¯r + − A 0 ¯r − h¯ h¯ j 2 j 2 i r r 00 0 iπ × exp − H ¯r + − H 0 ¯r − (t − t ) + (µ − µ 0 ) , (1-16) h¯ j 2 j 2 2 j j
where the Mj’s are the stability matrices of the corresponding j trajectory, H the system Hamil- tonian, and Aj represents the symplectic area enclosed by the j classical trajectory as we explain now.
The symplectic area is constructed as follows: Consider a given trajectory on phase space
from point rj(ti) to point rj(ti) and the straight line joining them, the area enclosed is called the symplectic area, as shown in figure 1-1. 1.2 Semiclassical Propagation 13
p
rj(tf )
rj(ti)
q
Figure 1-1: Representation of the symplectic area of a single trajectory. It is constructed with the area enclosed between the trajectory and a straight line from the final to the initial point (On phase space).
In fact, it is possible to use these definitions to write the propagator in terms of the actions S of the trajectories,
2d Z 0 −1/2 0 −1/2 00 0 2 X r r GW (r , r , t) = dr det Mj r + + I det Mj0 r − + I h 2 2 j,j0 i 00 0 i r r iπ × exp r ∧ (r − r ) exp S ¯r + − S 0 ¯r − + (µ − µ 0 ) . (1-17) h¯ h¯ j 2 j 2 2 j j
This way of writing the propagator highlights some of its most relevant features. Let us start by pointing out the double sum over trajectories. Each set of trajectories provides the support for each one of the time evolution operators. In particular, if the same set of trajectories (j = j0) is chosen for both propagators, a selection that goes by the name of the diagonal approximation, and an integration is done using the WKB approximation up to first order on the phase, the Wigner propagator reduces to the Liouville propagator, corresponding to the classical distribution propagator,
00 00 00 0 0 ∂rcl 0 0 00 0 00 00 00 0 00 0 GW (r , t , r , t ) ≈ det 0 δ (r − rcl(r , t , t )) = δ (r − rcl(r , t , t )) . (1-18) ∂rcl
This result recalls the van Vleck propagation on position representation, where the evolution is done following classical trajectories, but without changing the amplitudes. Though, in this approximation the Wigner propagator corresponds to the Liouville propagator, the comparison of the results with the classical mechanics case must be done using a single quantum mechanical particle and a complete distribution of classical particles, i.e., Wigner functions with density of particles. This is particularly strange, because the Wigner function is not a probability density 14 1 Theoretical Background
distribution as it takes negative values in some parts, yet it is precisely this feature what will help us identify the regions on phase space where the quantum effects take place more prominently.
We can, however, go beyond the diagonal approximation in (1-17), improving on the van Vleck propagation. The way to proceed is by considering the j 6= j0 contributions, and solving the integral in stationary phase approximation considering terms up to second order4. This imposes a strong rule over the trajectories that are going to be used to do the propagation. This chord rule demands that trajectories must be now consider in pairs [Dittrich et al., 2006, Dittrich et al., 2010], and these pairs evolve following the classical evolution given by Hamilton’s equations with the corresponding classical Hamiltonian. So, let us consider the case of just two of these trajectories j 6= j0 and, for simplicity sake, label them as j = 1 and j0 = 2, hence the chord rule states that 1 J(r00 − r0 − (r00 − r0 + r00 + r0 )) = 0, (1-19) 2 2 2 1 1 where J is the symplectic matrix. This is not a condition on the trajectories themselves, but over the midpoint between both trajectories, which can be make evident by writting r r = ¯r − (1-20a) 1 2 and r r = ¯r + , (1-20b) 2 2 where ¯r represents the midpoint at each time5. To conclude, we notice that the amplitude term given by the second order approximation yields
∂ Mj − Mj0 0 2 (Sj (r1, t) − Sj (r2)) = J . (1-21) ∂r (Mj + I)(Mj0 + I)
The van Vleck-Gutzwiller version of the propagator of the Wigner function is then given by
00 00 0 0 π vV 00 00 0 0 X cos Sj(r , t , r , t ) − 2 µj GW (r , t , r , t ) = 1/2 , (1-22) j |det(Mj+ − Mj−)|
an expression first derived by Dittrich et al [Dittrich et al., 2006].
Expression (1-22) for the Wigner propagator is the basis for the analysis and results presented in this work, so let us take some time to describe its structure. It is comprised of three basic
4If we consider only first order, the result is the classical propagation again, and as we are looking for a way to introduce quantum corrections it is not enough. 5As this is direct consequence of the chord rule, we may call these relations the chord rule in the following pages. 1.2 Semiclassical Propagation 15 terms: the amplitude, the phases and the summation. Let us start with the last. The summation over the js counts not trajectories but pairs of them. We adopt the convention of labelling each of the trajectories in the pair by j+ and j−, and the quantities corresponding to each of them to have the same sub index notation; the matrix Mj± is the stability matrix of the corresponding trajectory. The amplitude is given by the inverse of the squared root of the determinant of the matrix Mj+ − Mj−, which is the difference of the stability matrices for the pair of trajectories. This new matrix does not share the same properties than a single stability matrix, hence its determinant is not necessary one, as the two trajectories are independent. The points at which its determinant vanishes give the caustics for the pair of trajectories. This bring us to the phase factor. The index µj counts the caustics that the pair of trajectories crosses, adding or subtracting a phase of π/2 depending on the direction of the crossing (see Appendix C). The action in the phase of the terms in (1-22) is slightly different from the action mentioned before for a single trajectory, as this action depends on the pair of trajectories, 1 S (r00, r0, t00, t0) = (r0 − r0 + r00 − r00 ) ∧ (r00 − r0) + S − S . (1-23) j 2 j+ j− j+ j− j+ j−
Here, the terms Sj+ and Sj− are the actions of each of the classical trajectories, while the total action Sj of the pair of trajectories is given by the symplectic area enclose between the two trajectories as it is shown in the figure 1-2.
Aj+
rcl r+
r
r−
Aj−
Figure 1-2: Action in the propagator of the Wigner function. The symplectic area of a pair of trajectories is illustrated. The green area corresponds to the symplectic area enclosed by the trajectory r− while the red area corresponds to r+, and the violet region is the symplectic area of the pair.
Just by considering these trajectories in pairs and not individually, this propagator is able to reproduce quantum effects that the van Vleck-Gutzwiller propagator itself cannot, this is due 16 1 Theoretical Background
to the fact that the midpoint between two independent trajectories does not necessarily, and in general it does not, follow a classical trajectory.
To sum up, the final contribution of the propagator of a single point r0 at a time t0 into a single point r00 at a time t00 is constructed as the sum of all the possible pairs of classical trajectories such that they all share the same initial midpoint r0 and final midpoint r00 with the weights described inside the summation in equation 1-22. This is a strong restriction on the choice of the pairs considered to propagate Wigner functions as finding such trajectories for a given boundary conditions (r00,t00) and (r0,t0) is practically demanding, this is known as the root-search problem on semiclassical approximations. This problem however, can be solved theoretically constructing initial value representations, final value representations or allowing the trajectories become complex. It can also be avoided in practical terms when applying the method to numeric simulations as will be the case in this thesis where all methods were applied from a root-search free perspective.
It is worth mentioning that the approximation presented here can be improved by including higher order terms in the stationary phase evaluation[Dittrich et al., 2010], adding more and more information about the quantumnessof the dynamics. This procedure, notwithstanding, increases the complexity of the calculations, understanding the complexity in the computational sense, that is, the number of calculations, and therefore the time it takes to compute the evolution of the Wigner function increases rapidly with the number of trajectories used to get the evolution.
1.2.2 Initial and Final Value Representations of the Propagator
The root-search problem can be solved by considering that the propagation can be calculated not summing over pairs of trajectories but using an integration over the whole phase space instead. This approach can be done in different ways depending on the integration variable chosen. Two of these possibilities are the propagations named as initial and final value representations (IVR and FVR). In this section, the basic concepts of the IVR and FVR are presented, including how the semiclassical approximation takes place. Let us recall the expression of the definition of the propagation of the Wigner function (1-10), Z 00 00 0 0 0 00 00 0 0 W (r , t ) = dr GW (r , t ; r , t )W (r , t ), (1-24)
where the propagator GW is the Wigner propagator presented in equation 1-13 and r = (q, p) is the phase space position. Notice that the Wigner propagator takes contributions of two phase space points into the midpoint between them, this implies that we have a set of phase space points sharing the same initial and final center r0 and r00 respectively. In section (1.2.1) we introduced the semiclassical approximation by applying the van Vleck-Guzwiller version of the 1.2 Semiclassical Propagation 17
Feynman propagator directly into the convolution of the Wigner propagator. Before doing the same here, let us introduce the chord variables ξ and the chord function χ(ξ, t) as the Fourier transform of the Wigner function,
1 Z i W (r, t) = dξ exp − ξ ∧ r χ(ξ, t). (1-25) (2πh¯)d h¯
The system dynamics can also be described in terms of the chord function, this is particularly useful in the semiclassica approximation where the chord variable has a clear geometric interpretation. 0 0 Consider a single pair of trajectories with initial conditions r+ and r− that evolve following the 00 00 classical dynamics to the final points r+ and r−, as is presented in figure 1-3. The initial and final midpoints (center) r0 and r00 respectively of the pair of trajectories is shown in 1-3a, while figure 1-3b shows the initial and final chords, ξ0 and ξ00, which are the conjugated variable to the centers. They are the vector of the difference of the two trajectories at each time on phase space.
r00 r00 r + r + + + 00 r00 ξ
0 00 0 00 r+ r− r+ r− r− r− r0 ξ0
0 0 r− r− (a) Initial and Final Centers (b) Initial and Final Chords
Figure 1-3: Centers and Chords for a pair of trajectories evolved in time.
Writing this relation between centers and chords in term of pairs of trajectories we obtain, 1 r = (r + r ), (1-26a) 2 + − for the centers and
ξ = r+ − r−. (1-26b) for the chords.
The choice of a particular center, either initial or final, imposes a constrain on the pair of trajectories. The trajectories in the pair can not longer be chosen independently, for instance, if we select a particular center, the constrain is that the pair has to be such that the center 18 1 Theoretical Background
point of the two trajectories is the chosen one. A similar situation occurs with the chord, but the constrain now relies on the difference. Notice however, that just a center or just a chord does not determine completely a specific pair of trajectories, and additional condition is needed to determine it uniquely. In fact, using (1-26), it is clear that the choice of any combination of
a couple of variables from r, r+, r− or ξ, completely determines the other two, regardless of the time considered, i.e. the initial or the final. For instance, one can use one of the two trajectories position and the chord variable even at different times, or the center and chord, this last selection is not surprising as these variables are commonly used in the center of mass coordinates in many fields. It is precisely this freedom of choice what allows to consider different representations for the propagator of the Wigner function, but as a consequence, as all possible trajectories that satisfy the constrain have to be considered, an integration over all phase space on one of the chosen variables have to be performed [Ozorio de Almeida et al., 2013].
For the semiclassical evolution of the Wigner function, we just replace the Wigner propagator by the van Vleck-Gutzwiller version (1-22), Z 0 0 vV 0 0 0 0 W (p, q, t) = dq dp GW (p , q , t0; p, q, t)W (p , q , t0). (1-27)
vV 0 0 Notice that the GW (p , q , t0; p, q, t), due to our use of chord rule (1-20), propagates the Wigner function using midpoint between two classical trajectories. In the labelling introduced by Ozorio de Almeida[Ozorio de Almeida et al., 2013, Ozorio de Almeida, 1998] for transformations vV in a quantum phase space these midpoints are centers, hence propagator GW corresponds to a center-center representation of the propagator of the Wigner function. This representation, however, is not unique, and different interpretations of chord rule (1-20) lead to different repre- sentations. Let us visualize what this chord rule means and how we are able to change among the different representations of the propagator for the semiclassical approximations.
Initial value representation IVR
We are interested on calculating the Wigner function at time t00 on a point r00. Using the semiclassical approximation (1-22), it is necessary to consider all the possible pairs of trajectories such that the initial midpoint is r0 and as a final midpoint r00. This conditions were already found before for the van Vleck-Gutzwiller propagator on position or momentum representation. In order to overtake this hurdle in the IVR, the propagation is constructed using a different parametrization for the pairs of trajectories that contribute a single final point r00 at time t00, therefore the boundary condition can be satisfied more efficiently.
The contribution of a single pair of trajectories is obtain in the following way: At time t0 an 0 initial point is taken for one trajectory, say rj−, and is evolved following the classical dynamics 1.2 Semiclassical Propagation 19
00 00 00 until time t to the point rj−. As the final midpoint r is given, the chord rule is used to find 00 the correspondent final point rj+ of the second trajectory. This point is evolved backwards from 00 0 0 00 time t to time t to find the initial point rj+. The initial center, contributing to r , is recover 0 1 00 00 as r = 2 (rj− + rj+) as it is shown in figure 1-4.
2 00 r+ 3 r00
0 00 r+ r− 1 0 4 r 0 r− 0
0 Figure 1-4: Initial Value Representation. 0. Selection of the initial point r−, 1. temporal 00 00 00 0 0 0 evolution until r−, 2. Use r to find r+, 3. Evolution backwards until r+, 4. Use r− and r+ to calculate r0.
This procedure implies a change of variables on the integral (1-27), this change of variables means that the integration is not done all over the midpoints, but all over the initial conditions 0 00 of one of the trajectories that form the pair, say r−. The propagation on a point r is done using 0 all the contribution of all the possible pairs such that r− is the initial point of one of the classical pairs.
The change of variables described requires a Jacobian calculation. This Jacobian uses a 0 0 relationship between the initial center r , and the new variable r−. This relationship is not 0 straightforward as there is a constrain on the choice of r+. To find it one can start from (1-26a), 0 00 and recall that the initial point r− evolves classically to the final point r+, from where, via a 00 00 reflection through the midpoint r , one can arrive to r+ and propagate classically backwards to 0 the initial point r+. Recalling the fact that the relationship between the initial and final point of a single trajectory is given by the stability matrix, and that the reflection simply is an identity matrix, the Jacobian is,
0 dr M(r+) − M(r−) det 0 = det . (1-28) dr− 2
Then, the initial value representation of the propagation of the Wigner function on a point r00, 0 is given by the contributions of all the pairs parametrized on one of the initial conditions r−, as 20 1 Theoretical Background
follows
Z M(r0 ) − M(r0 ) r0 + r0 r0 + r0 W (r00, t00) = dr0 det + − G r00, t00, + − , t0 W + − , t0 . − 2 W 2 2 (1-29)
This expression does not require the sum over pairs as all the contributions are counted into the integration, but now, this means that to calculate the Wigner function on a single final point on phase space, an integration over all phase space is required, thus an approximation for this integral has to be considered when applying these methods numerically.
Notice that the semiclassical van Vleck-Gutzwiller version of the propagation of the Wigner function (1-22) diverges on the caustics, because the square root of the determinant of the matrix 0 0 M(r+) − M(r−) in its denominator vanishes on them. In the IVR (1-29) these divergences are no longer there. Instead, because of the Jacobian term due to the change of coordinates, the caustic points become points that do not contribute (amplitude cero), so this transformation takes caustics into nodal points or nodal lines6.
Final value representation FVR
In the same spirit than from the IVR case, the FVR is constructed from a change of variables. It is done changing the variables from the initial center to the final chord function, this is what gives the name of the representation. In this scheme of propagation the a single pair of trajectories is constructed from their final quantities, namely, to calculate a contribution on a point r00, a final 00 00 00 chord ξ is needed, then the two final points r+ and r− can be calculated using the expressions (1-26a) and (1-26b) to invert the relationships and propagate backwards to find the initial points 0 0 0 r+ and r−, and therefore the initial midpoint r . This procedure is illustrated in figure 1-5.
6The caustics are present where the Lagrangian manifold folds into itself making return points, this projection can be more than just one point, for instance a curve. 1.2 Semiclassical Propagation 21
1 00 r+ 0 2 r00 ξ00 1 0 00 r+ r− 2 0 3 r 0 r−
Figure 1-5: Final Value Representation. 0. Selection of the final point r00, 1. Calculate the two 00 0 0 0 0 0 pairs, given ξ , 2. Evolution backwards until r+ and r−, 3. Use r− and r+ to calculate r .
This change of variables also requires a Jacobian to do the integration properly. This Jacobian is calculated, in fact, similarly than the IVR case, as the relationship between the final 00 0 chord ξ and the initial center r is calculated with the dynamics of the trajectories r±, and again, as the relationship between the final and initial points is done through the stability matrix. For the change of variables described, the following Jacobian must be used, dr0 M(r ) − M(r ) det = det + − . (1-30) dξ00 4 The final value representation of the propagation of the Wigner function on a point r00is given by the contribution of all the pairs parametrized by the final chord ξ00 as,
Z M(r0 ) − M(r0 ) r0 + r0 r0 + r0 W (r00, t00) = dξ00 det + − G r00, t00, + − , t0 W + − , t0 . 4 W 2 2 (1-31)
Where the integration now has to be done using the final chord, ξ00.
Notice that in both cases, besides solving the root-search problem, the IVR and FVR’s Jacobian term coincides with the square of the denominator in equation (1-22). This implies that the amplitude of those points near to caustics do not diverge but decrease. Those points are now nodal points then exactly on a caustic the amplitude goes to zero. This does not mean that the caustics do not have to be counted as the change of the phase is independent on the amplitude on the contribution.
As the three methods, the center-center, IVR and FVR are constructed using the same core ingredient, the Wigner propagator GW in equation (1-22), there are similarities when cal- culating the semiclassical propagations specially at a numerical level. But also it is worthy 22 1 Theoretical Background
mention that, for the case of the IVR and FVR here presented as well as those presented on [Ozorio de Almeida et al., 2013] and used in [Ozorio de Almeida et al., 2019], the propagation on a single point r00 depends on the initial state W (r0, t0) as the integration includes it. This is not the case of the center-center representation here described [Dittrich et al., 2006], where the initial state is important just after the calculation of the pairs dynamics, and therefore is independent on that. This difference, among others, make the three schemes behave differently at a numerical level, even when they all are equivalent theoretically. Chapter 2
Numerical Implementation
The main goal of this work is to compare the dynamics generated by the three semiclassical methods described on the previous chapter: the center-center, initial value and final value rep- resentations. This comparison must be done numerically, hence this section is strongly focused on the numerical implementations of those semiclassical propagation methods. To perform the calculation of the final Wigner function, it is necessary to discretize the phase space on the points were the final distribution is going to be calculated. Such discretization leads to the final reso- lution of the Wigner function, so the effectiveness of the methods when measured depend also on the choice of such discretization and the number of trajectories to populate each bin, and it is independent of the approximations as it is result of the way computational results are as no smoothing out or interpolation is going to be considered.
Even when all these theory rely strongly on the classical mechanics theory, we are not going to present here the numerical method used to integrate the classical equations of motion, such implementation is presented in detail in appendix D, we just want to highlight the fact that the sixth order symplectic Yoshida’s method was used to integrate the classical mechanics equations. Something similar occurs with the quantum propagation, as we are expecting to compare the methods, it is necessary to have some exact results to compare with. The calculation of the quantum evolution was done using a split operator method described for the Wigner function in the appendix E.
This chapter is organized as follows, first the description of the center-center method devel- oped by Dittrich et al. [Dittrich et al., 2006, Dittrich et al., 2010] , equation (1-22), is presented as an algorithm, making explicit the fact that it can be used as an initial value representation when applied numerically, and even being computationally feasible for large scale simulations. Then the caustic calculation, using a similar but less efficient strategy, is shown. Finally, the 24 2 Numerical Implementation
initial and final value representations are discussed.
2.1 Center-Center Representation
Let us focus our attention on the center-center propagator developed by Dittrich et al. [Dittrich et al., 2006, Dittrich et al., 2010]. It has been claimed before that this propagator suffers from the root-search problem, making it computationally expensive [Koda, 2015, Ozorio de Almeida et al., 2013]. As we shall show here, these claims lack support. On the contrary, not only this propagation scheme avoids the root-search problem [Dittrich et al., 2006, Dittrich et al., 2010], but compared with other semiclassical methods in phase-space [Koda, 2015, Ozorio de Almeida et al., 2013, Ozorio de Almeida et al., 2019], its numerical implementation is by far the most efficient one.
For the sake of clarity, we divide our explanation of the method in three stages: First, classical trajectories sampling and evolution, second, initial state correction, and third, the final Wigner function construction.
In principle, the classical trajectories can be chosen sampling arbitrary regions on the phase space, but in order to count the contributions of points with large initial probability and not much those on the tails, as the later carry less information, sampling the area where the initial Wigner function concentrates is, of course, more convenient. However, he sampling distribution does not restrict the shape of the initial Wigner function and therefore the same set of classical trajectories can be used in order to evaluate different initial conditions if they are localized on similar regions in phase space. Regarding the distribution of the midpoints, let us call it ρ(r), it is pertinent to mention that as they result from considering all possible pairs of classical trajectories, ρ(r)
depends on the distribution ρ±(r±) of classical trajectories. Clearly, the distribution of midpoints will be concentrated in the middle of the classically evolved points, as is shown in a small scale in figure 2-1, where the midpoints resulting from four classically evolved points are located in the center of them.
The midpoints distribution ρ(r) can be calculated from the classical points distributions
ρ+(r+) and ρ−(r−) by performing a convolution between them. In the procedure here described ρ+(r+) and ρ−(r−) are the same.
Once the classically evolved points are placed, their dynamical equations are integrated in order to calculate the time evolution from a time t0 to the final time t00, saving at each time step the stability matrices and actions, information needed for the evolved Wigner function construction.
At the final time, the midpoints distribution ρ(r00, t00) is not uniform on the region where 2.1 Center-Center Representation 25
Figure 2-1: Midpoints and their relation to classical trajectories. Points on classical trajectories are represented by purple circles, while the midpoints between trajectories are represented by blue circle. the Wigner function it is going to be calculated, in fact, it is likely that no final midpoint is precisely on a discrete value chosen to evaluate the Wigner function. This, notwithstanding, is not a problem, as we reconstruct the Wigner function at time t00 as a coarsed grained version on the discrete space, by averaging on the number of final midpoint on each one of the discrete cells, as is shown in figure (2-2).
Final Midpoints Single cell discretization
Figure 2-2: Representation of a single cell of the final Wigner function construction.
Notice that the above procedure does not corresponds to a smoothing the final Wigner function but is the result of the space discretization of phase space. Remarkably, this same procedure frees the method of the root-search problem, as no longer a restriction is imposed on the final pints of the pairs of trajectories, nor on their midpoints. This of course means also that not all final points are sampled equally, and that there is no possibility to get information on points in phase space that are not reached by the midpoints initially chosen. The advantage however is that the method, numerically, can be used as an initial value representation, meaning that it will depend only on the initial conditions and not having a final boundary condition to satisfy. 26 2 Numerical Implementation
Let us illustrate the total evolution methodology considering an initial set of classical tra- jectories, a discretization of the space, and a final counting on the grid , as shown in figure 2-3.
1
12
2 14 1 13 14 24 12 4 23 13 24 2 34 4 23 34 3 3
Figure 2-3: Representation of the method. Circles on the left hand side correspond to initial points of classical trajectories (purple) and their midpoints (blue). Circles on the right hand side corresponds to final points, after the propagation has taken place.
In figure 2-3 the initial conditions are shown the left hand side, where the initial set of classical trajectories (4 violet classical trajectories) are labeled with the correspondent number, while the midpoints (blue) are labeled with the numbers of the pair they are the midpoint of. This plot shows also that it is possible to have more than one point in the same cell of the grid, then they may be averaged.
This way to proceed, make it possible to know before the time evolution the initial distribution of midpoints ρ(r0, t0), but not the final ρ(r00, t00) where the final Wigner function is going to be constructed. This of course will depend on the system used and on the initial Wigner function to be propagated. On figure 2-4 the initial distribution of midpoints ρ(r0, t0) is presented in green and the final distribution ρ(r00, t00) in red. Look that in contrast to the classical distributions dynamics given by the Liouville equation, the area on phase space sampled by the midpoints is not conserved in general, and as a consequence it is possible to describe further points on phase 2.1 Center-Center Representation 27 space than the classical mechanics.
00 r Final distribution
Initial distribution r0
Figure 2-4: Center - Center representation.
It is worthy mention that, as the midpoints’ dynamics do not follow any equation of motion, the trajectory a midpoint follows is not physical in general1, and should not be understood as one.
2.1.1 Algorithm
The procedure to calculate the Wigner function after n discrete time steps is written as an algorithm and is described below:
2,3 1. One chooses an initial set of classical trajectories. This set is used as r− and r+
2. One evolves each one of the points classically, saving the following quantities
• The action of the classical trajectory, which is found integrating Z t Z t Z t S = dt0L(q, q˙ , t0) = dt0(q · p − H(p, q, t0)) = dt0(q · p) − E(p, q)t. (2-1) 0 0 0
• The stability matrix Z t 2 0 ∂ H 0 M(t) = M(t )J 2 dt , (2-2) 0 ∂r 1For the case of a harmonic oscillator, for instance, every midpoint will follow the same path as the correspon- dent classical trajectory. 2For a set of N classical trajectories, each one of the final trajectory contributes with (N − 1)/2 pairs, the half enters because of the pairs commute, and the −1 represents that the pair with itself is not considered as we are not taking this linear approximation as it was explained in section 1.2. 3It is possible to consider the classical contribution itself, but no using the expression of the propagator 1-22, but instead 1-18, which basically means that we may use the classical contribution as if the evolution were given by the Liouville equation. 28 2 Numerical Implementation
∂2H where J is the symplectic matrix, and ∂r2 is the Hessian matrix of the Hamiltonian. As initial condition, M(0) must be equal to the identity matrix. • Count the caustics crossed by each pair of trajectories.4
After the evolution, the center for each pair is calculated, and for each one of them:
1. Calculate the initial weight of the midpoint.
2. Calculate the position on the grid of each final center.
3. Calculate the action for the pair which is defined as shown in equation (1-23), 1 S (r00, r0, t00, t0) = (r0 − r0 + r00 − r00 ) ∧ (r00 − r0) + S − S . j 2 j+ j− j+ j− j+ j− 4. Sum the contributions of all the final midpoints on each cell using the amplitude (1-22).
5. Divide over the amount of final points on each cell to average all cells.
This propagator has been said not to be an initial value representation as the evolution depends on a sum of trajectories that may be constrained to a initial and final boundary conditions and not an integral over the complete phase space for each pair. Nevertheless, it can be used as one in the way it is applied numerically as it was just presented and will be used in the results chapter.
Additionally, this scheme is constructed in such a way that every calculation done is used as much as possible making it more efficient than other semiclassical phase space representations.
2.2 *Caustics Counting
The semiclassical methods, and in particular, the semiclassical propagator (1-22), present diver-
gences on those points for which, giving a pairs of trajectories j, the determinant det(Mj+ −Mj−) vanishes, these points are the caustics. The relation between the amplitudes just before and just after the caustic is crossed is a change of phase of π/2. To keep track of the number of caustics crossed by the pair of trajectories j in a time interval (t0, t00) we use the index
0 00 µ(t , t ) = ν+ − ν−, (2-3) 4This should be done, but this will require to increase the complexity of the problem as we have N(N − 1)/2 pairs for each N trajectories as there is no way to calculate the Caustics just with the information at the beginning and at the end of the evolution. 2.2 *Caustics Counting 29
where ν+ counts the times a caustic has been crossed and det(Mj+ − Mj−) has changed from negative to positive values, and ν− counts the times a caustic has been crossed and this deter- minant has changed from positive to negative values. µ(t0, t00) is called the Maslov index of the pair [Maslov and Fedoriuk, 1981]. This index depends on the pair of caustics at every times, and not on the independent trajectories.
In the previous section, the dynamics for each trajectory is done independently, and is just at the end when they are use to form pairs and construct the final Wigner function. For a given pair, however, the Maslov index µ(t0, t00) cannot be calculated from the classical quantities of each of 0 0 00 00 the classical trajectories, namely, initial and final positions (q+(t ), q−(t )) and (q+(t ), q−(t )) on phase space, actions (S+,S−), and stability matrices (M+,M−), hence it must be evaluated at the same time the evolution is done for each pair.
Here, the pairs of trajectories are built from the beginning and propagated together along their correspondent classical trajectory. This means that to propagate each midpoint 2 classical trajectories are required.
The strategy is straightforward from the center-center methodology already described 2.1.1. In figure 2-5 the relationship between initial an final midpoints is shown. The difference with the case in figure 2-3, is that here the classical propagated point belongs to a single pair of trajectories and therefore contributes to just one specific final midpoint. As a consequence, we evaluate quantities on each single pair of trajectories independently of the rest. In particular, we count the number of caustics crossed by each pair of trajectories keeping track of the direction in the index µ(t0, t00).
1 12 2 1 12 4
2 34 4 34 3 3
Figure 2-5: Representation of the method. Left hand side before the propagation. Right hand side after the propagation 30 2 Numerical Implementation
2.3 Initial Value Representation
The implementations of the IVR and the FVR are similar, as both of them are based on integrating over all phase space to get the final Wigner function on a single point, (1-29) and (1-31). When evaluating these integrals numerically, it becomes necessary to consider only up to finite limits and a finite resolution. This make these methods computationally expensive, since each one of the final points requires to consider the contribution of a large number of classical trajectories5 [Ozorio de Almeida et al., 2013].
Consider the integral (1-29) where the integration variable is one of the initial ends of the 0 0 pair of trajectories, r−. A single value of r− labels a single trajectory that is constructed as 0 00 follows: first the selected trajectory r− is evaluated forwards in time until r−, then it is reflected 00 00 0 respect to the final midpoint r to find r+ to finally evolve this last backwards to find r+. Then 0 it must be repeated for all possible r− in order to be able to integrate (1-29).
These integrations can also be approximated using for instance, Monte Carlo methods. An initial set of points are selected randomly using as probability distribution for half of classical 0 6 trajectories r− any desired , looking for having concentrations of initial points on the places where the initial state is more populated. After the evolution of each of the initial classical points 00 r− , a final point has also to be chosen for each one of the classical trajectories. This in order to 00 construct the pair by reflecting the final classical trajectory r− with respect to the desired final 00 00 midpoint r to get the correspondent pair r+. Such points must be evolved backwards to find the their initial midpoint r0 as is shown in figure (2-6)
0 r00 r+ Final distribution
0 Initial distribution r−
Figure 2-6: Initial value representation
5The number of trajectories needed depends on the resolution desired for the result of integral as well as on the chosen limits for it (a truncation is usually implemented). All these together, added to the number of time steps done during the time integration, lead to a computational heavy problem. 6We here propose the squared of the Wigner function. This as the Wigner function can take negative values, and we would like to have more information on such points with more information, therefore having a larger amplitude 2.3 Initial Value Representation 31
The figure 2-6 also aims to show that, the initial distribution of one of the pair’s end 0 ρ−(r−) can be fixed by the choice of the initial sampling but not for the distribution of the initial 0 0 midpoints ρ(r ). As the points r+ are found after the evolution of the pair, it implies that there 0 is no control on the distribution ρ+(r+) as this is a free parameter. In general, the distribution 0 0 of ρ+(r+) is different from ρ−(r−).
As the two trajectories belonging the same pair are not calculated at the same time, the caustics cannot be calculated as previously described, unless the information of all the trajectories 0 r− is saved, which is unsuitable for large scale simulations as memory is limited.
2.3.1 Algorithm
This procedure written as an algorithm is described below
0 1. One chooses an initial set of one end classical trajectories r−.
2. One evolves each of the initial set classically, saving the following quantities,
• The action of the classical trajectory,
Z t Z t Z t S = dt0L(q, q˙ , t0) = dt0(q · p − H(p, q, t0)) = dt0(q · p) − E(p, q)t (2-4) 0 0 0
• The correspondent stability matrix
Z t00 2 00 0 ∂ H 0 M(t ) = M(t )J 2 dt . (2-5) 0 ∂r
Using as an initial condition an identity matrix.
00 00 00 3. Each final point r− is reflected respect to a final center r to find r+ then it is propagated backwards saving the quantities in 2.
4. Use the expression 1-29 to sum the weights.
So in this case, if we use 2n initial classical points, we have n pairs and then, and therefore n final midpoints. 32 2 Numerical Implementation
2.4 Final Value Representation
This implementation follows the previous described, with the difference that the pairs are calcu- lated at the same time but backwards. Hence it allows us to count the caustics, as it should be done time step by time step.
Now the distribution for the final midpoints is known ρ(r00), because the integration param- eter in (1-31) is the chord ξ00, which means that, we select the points that have as center the final midpoint, then propagate them backwards.
The integration (1-31) has to be done considering the closer pairs until a cut off chord. But this cut off depends on time, as while longer the time, larger are the contributions of located far from the center of interest.
00 r+
ξ00 × r00
00 r−
Figure 2-7: Relationship among the propagated quantities, classical points, midpoint and chord. The yellow points are such with small value of |ξ00|.
One chooses |ξ| to limit the integration, as one expect that close trajectories contribute most, as the action is small because the trajectories are similar in non chaotic systems, and the trajectories may have similar stability matrices as this matrix measure local effects on the trajectory, hence, larger amplitude is gotten.
In this case the final distribution of midpoints ρ(r00) is well know, but as a consequence, there is no restriction on the distribution at the initial time ρ(r0) as can be seen in figure 2-8. 2.4 Final Value Representation 33
r0 Final distribution
Initial distribution r00
Figure 2-8: Final value representation
The plot 2-8 also shows that the region on phase space where the evolved Wigner function is calculated, can be sampled as desired but this does not mean that the initial Wigner function is correctly sampled on points where the initial state has large amplitude.
2.4.1 Algorithm
This procedure written as an algorithm is described below
1. One chooses the final set of classical chords χ00.
2. One evolves each of the initial set classically backwards, saving the following quantities,
• The action of the classical trajectory,
Z t Z t Z t S = dt0L(q, q˙ , t0) = dt0(q · p − H(p, q, t0)) = dt0(q · p) − E(p, q)t (2-6) 0 0 0
• The correspondent stability matrix
Z t00 2 00 0 ∂ H 0 M(t ) = M(t )J 2 dt . (2-7) 0 ∂r
Using as an initial condition an identity matrix.
3. Use the expression 1-31 to sum the weights.
So in this case, if we use 2n initial classical points, we have n pairs and then, and therefore n final midpoints. 34 2 Numerical Implementation
2.5 Comparison Criteria
Now we have described the numerical methods used to implement semiclassical approximations. In the next chapter we are going to compare them, and we are going to use different parameters, in order to evaluate if the approximation is working properly or not and which one performs better if is the case. The comparisons are done taking as a reference point the quantum propagation described in the appendix E.
2.5.1 Observable Calculation
The state of a quantum system allows us to calculate observables on such systems. The dynamics of these expectation values may then depend on the evolution of the state. The Wigner function, as it is a representation of the state can be used to calculate observables and therefore their dynamics.
For instance, let us consider the expectation value of an operator Aˆ(t) for which the Weyl symbol is written as A(p, q, t) Z hAˆ(pˆ, qˆ, t)i = dpdqW (p, q, t)A(p, q, t). (2-8)
2.5.2 Autocorrelation Functions
There are plenty of different measures of distance between distributions, but we are going to use the autocorrelation function as it measures the overlap of two Wigner functions, the initial and the evolved. This is usually chosen, because semiclassics is done most of the time for periodic systems as there are important features depending on the orbits[Gutzwiller, 1992, Gutzwiller, 1971], thus the Wigner functions are localized on a certain part of the phase space.
As shown in appendix B, the autocorrelation function can be defined as the overlap of the initial and evolved Wigner function as,
Z C(t) = dpdqW (p, q, t)W (p, q, 0). (2-9)
We use the autocorrelation function to compare different schemes of semiclassical propaga- tion using several times using as a reference the quantum dynamics described in appendix E, and 2.5 Comparison Criteria 35 thereby evaluate the robustness of the methods.
2.5.3 Normalized Inner Product
The evaluation of the propagation of the Wigner function can be done comparing the the semi- classical approximation directly with the quantum propagation. One way to do this is considering the normalized inner product defined as,
Z dpdqWQ(p, q, t)WS(p, q, t) N(t) = . (2-10) sZ Z dpdqWS(p, q, t)WS(p, q, t) dpdqWQ(p, q, t)WQ(p, q, t)
Where WQ and WS represent the quantum and semiclassically evolved Wigner function respectively.
2.5.4 Marginals
The Wigner function allows us to calculate the marginal probabilities simply by integrating over one of the two variables ether p or q. This parameter is not showed for all the approximations but in some cases will be used.
Position Marginal
The marginal probability on position space is found integrating the Wigner function in the mo- mentum variables, such as
Z |ψ(q)|2 = dpW (q, p) (2-11) 36 2 Numerical Implementation
Momentum Marginal
The marginal probability on momentum space is found integrating the Wigner function in the position variables, such as
Z |φ(p)|2 = dqW (q, p) (2-12) Chapter 3
Results
In the previous chapters we presented the theory behind the semiclasscal approximations, and the numerical methods to implement them. In this chapter we are going to test these methods numeri- cally. To do this we use the usual benchmark which is the Morse Oscillator [Dahl and Springborg, 1988, Morse, 1929] for which is well known the Wigner representation as the eigen states.
There are plenty of different issues implementing semiclassical propagation numerically be- sides the methods themselves, namely the sampling problem and the correct averaging of the final midpoint distributions. These issues are going to be treated here truly, highlighting the advantages and disadvantages of each of the methods proposed.
It is important to make a comparison among different methods, not just in how similar are the Wigner functions but also how good they perform given the computational resources we have available. In our case the number of classical trajectories used tells us about the computational cost, while the number of midpoints gives us an idea of how good the final sampling is. The IVR and FVR were done considering 10.000.000 pairs of trajectories1 hence we got the same amount of final midpoints. For classical trajectories for the center center representation, just 10.000 pairs were taken, as this leads to N = 49.995.000 midpoints. In this stage, the computational resources needed for the IVR as well as for the FVR goes up to three orders of magnitude larger compared with the center-center representation and even there the final sampling is smaller.
This chapter is organized in the following way, first, there is a brief description of the model and a brief discussion about the dynamics of classical trajectories, then the analysis of caustics in this system is presented in detailed calculations so that the comparisons can be done more easily between the IVR, FVR and the center representation. And finally, some discussion on the
1Which means 20.000.000 of classical trajectories. 38 3 Results
performance is done regarding the complexity of the methods and the possibility of paralellization.
3.1 Model: Morse Oscillator
The Morse oscillator [Morse, 1929] models the interaction between diatomic molecules. It is composed by a part that is an attractive force and another that is a repulsive one. The potential of the model this model is,
2 V (q) = D (1 − exp(−a(q − q0))) , (3-1)
where the parameters D and a tunes the depth and width of the well respectively. This system has been studied intensively in quantum mechanics, in its position representation as well as in the phase space representation [Dahl and Springborg, 1988] where there is an analytical expression for its eigen states[Frank et al., 2000].
The plot 3-1 shows in the left hand side the potential for different values of the width a, while the right hand side the level curves of the Hamiltonian are showed.
Figure 3-1: Plot of the Morse potential (Eq 3-1) for different values of the the width a for D = 0.5. (Left hand side) The correspondent Hamiltonian in phase space and some contour plots.
Classically, one has two possibilities for the trajectories, they can be either bounded and periodic or they can scape the potential with no bounds depending on the initial conditions (the energy). These options imply that one finds some noise resulting from the midpoint between the bounded and non bounded classical conditions when constructing the semiclassical propagation. 3.2 Caustics 39
The escaping trajectories should be taken into account, as they can have a midpoint with a large amplitude initial condition. These however, leads to important contributions to the noise when the dynamics progress due to the fact that those trajectories move faster as they have more energy, as a consequence, the midpoints constructed from bounded and unbounded trajectories need more averaging to reduce the noise, additionally information is lost, in the sense that we lose track of those trajectories that go out of the region of interest where the Wigner function is constructed.
In what follows we turn into the numerical integration for an specific set of parameters. For all the simulations hereafter, we are going to use as the depth of the well D = 15 and a with of a = 0.18 as the parameters defined the system. We use in all cases the same initial condition, a coherent state centred at (q = 4, p = 0) and dt = 0.1 and h¯ = 1.
3.2 Caustics
All the classical quantities that we need to evaluate the propagation can be calculated just looking into the trajectory, except for the Maslov index [Maslov and Fedoriuk, 1981] and therefore the Caustics (Appendix C), as we explained before, this has to be done taking into account the pair of trajectories.
Evaluating the Maslov index is numerically extremely demanding as we are going to show later. In what follows we will do simulations to see how important or not is including it. The simulations done using semiclassics or either do not consider them [Dittrich et al., 2006, Pach´on,2010, G´omez,2010, Villalba, 2017], or at least they do not suggest any way to calculate them [Ozorio de Almeida et al., 2019].
In this section we are going to show, how to count the caustics crossed by a pair of trajec- tories. We are going to discretize the time evolution so that we are able to count, until certain resolution, how many caustics each pair of trajectories has crossed.
The idea of taking the trajectory pairs independent of the rest has already been used before [Dittrich et al., 2006, Pach´on,2010, G´omez,2010, Villalba, 2017], yet in that specific case they were constructing the propagator itself then the initial condition chosen was a δ function, but when propagating Gaussian packages the pairs were not independent anymore. Therefore we are facing a bigger sampling problem as we need to have a finite set of pairs to propagate each point of the initial Wigner Function. Then we are going to show the averaging properly in order to get the results here correctly. 40 3 Results
We are going to study the effect of the caustics on the final Wigner function by comparing the final reconstruction of the state, first considering the phases given by the index, and then without them contribution of the index on the finite set of trajectories for this specific system2. This procedure does not pretend to be a demonstration but just a justification for the following sections not to consider that phases so the resources can be used more efficiently.
3.2.1 Initial State
The results here presented are done using the sixth order Yoshida’s method to the integration of the Hamilton equations (D-7) with a timestep of dt = 0.1 and using N = 10.000.000 of pairs of classical trajectories . We use as a initial center of the coherent state on phase space at (p = 0, q = 4). As shown in appendix A the coherent state is a Gaussian, such Gaussian has to be sampled with the midpoints in order to propagate the chosen state. Nevertheless the
density of midpoints is found from the density of the classical points ρ+(r+) and ρ−(r−) as the convolution of them. This in principle restrict the shape the distributions ρ+(r+) and ρ−(r−) but such restriction can be avoided using a importance sampling technique. This is a typical importance sampling problem, we want to sample a given distribution (Initial Wigner function) with a different one, this is done considering each point with a weight as follows,
f w = Target (3-2) fReal
where fReal is the actual distribution while fTarget is the target distribution.
In the simulations here presented, to sample the initial Wigner function Gaussian functions
placed on (p = 0, q = 3) with σ = 2 as ρ+(r+) and ρ−(r−). This choice is done in order to sample the bottom of the well with high initial density of midpoints, as is in this part of phase space where non classical information appears.
2The effect must be studied on each system considered independently 3.2 Caustics 41
Figure 3-2: Initial Distribution of points: Right Initial sampling of midpoints. Left distribution corrected with the correct weights in order to reproduce the desired initial distribution.
The midpoints distribution is different from the distributions of the classical trajectories even when they are taken to be the same. The resulting distribution ends up being the convolution of the two distributions, therefore the amplitudes have to be re calculated.
3.2.2 Determinant
Once the initial state sampling amplitudes are corrected by the importance sampling technique, comes the contributions that make the caustics emerge, the propagator (1-22) has certain ampli- tudes for each pair of trajectories, these amplitudes is given by the determinant of the difference of the two classical stability matrices.
The origin of the caustics is the divergence on the amplitudes of the propagator 1-22, then to study the behaviour of the caustics it is possible to evaluate a histogram of the amplitudes as time evolves, for instance, one can see how the amplitudes are distributed after 100 time steps as it is shown in figure 3-3, where it can be seen that most of the pairs are localized on certain region and they decay in such a way that those points with high amplitude compare with the average are just a few. 42 3 Results
Figure 3-3: Distribution of Amplitudes after 100 time steps.
Such points that cross a caustic do not affect much the structure of the final Wigner function as they go isolated and this is not a behaviour that occurs systematically for close midpoints to be able to change the position of the fringes, for instance.
This distribution also changes in time, going to smaller values of amplitude, this can be seen in figure 3-4 where the same system was evolved now 240 time steps.
Figure 3-4: Distribution of Amplitudes after 240 time steps. 3.2 Caustics 43
This plot shows that now the amplitude distribution becomes narrower, as the scales on the plot are logarithmic on the x axis.
Caustics Counting
To count the caustics, it is necessary to keep track of the sign of det(Mj+ − Mj−) at each timestep. just as described by [Maslov and Fedoriuk, 1981], the index µ goes as the number of times the determinant has changed sign from negative to positive minus the number of times it has gone from positive to negative. This is done on the discrete integration so some caustics can be missed, but it is just in the cases where in the interval dt two caustics are crossed and the sign remains the same. Such cases have small probability, but there is no way to count them with the way equations are integrated.
3.2.3 Reconstruction of the Final State
At this point the final state is a histogram with weights given by the expression (1-22) times the amplitudes from the corrected initial distribution (3-2). As not all points share the same midpoint they should not be summed up. As phase space is discretized in order to get a final plot, the final Wigner function is constructed by averaging the final histograms in order to have all bins with by the number of final midpoints in order to have them all with the same weight. Because not all the bins in the histogram are equally sampled the average there is not the best.
3.2.4 Final Wigner and Wave Functions
Now, we may see how the effect of including the caustics to the dynamics really is. This by comparing the Wigner and wave functions of the reconstruction using and not using the index µ. This can be done using many different tools, here we are going to concentrate some particular cases where we show the effect of the caustics on the final state. We are going to see the effects on the Wigner function and also in the marginals.
Wigner Function
Here we are going to see the evolution of the Morse system after 100 and after 120 time steps considering and not considering the caustics. The final Wigner functions are shown in (figure 44 3 Results
3-5) where the final Wigner function without considering the caustics is left up, considering the contribution is right up and finally the quantum propagation is displayed (below),
Figure 3-5: Final Wigner functions after 100 time steps. Left: without caustics. Right: with caustics.
This specific example illustrate what happen if caustics are or are not considered on the dynamics with a finite set of trajectories. In principle there is more noise in general, but specially on those poorly sampled points, as larger number points are needed to average correctly. This noise regardless, the two models show the same structure, namely the location of the fringes which carries all the quantum information. This can also be seen using the wave functions, which result from integrals (2-12) and (2-11) of the Wigner functions.
Marginals
The wave squared functions for the case of caustics compared with no caustics for the case just presented is (figure 3-6) 3.2 Caustics 45
Figure 3-6: Final marginals after 100 time steps. Left: Position representation. Right: Momen- tum representation.
Where it is clearly seen that the main features between the two cases are conserved except for fast oscillations3. The same behaviour can be seen at different times, for instance, if we evolve the system 20 time steps ahead,
Figure 3-7: Final Wigner functions after 100 time steps. Left: without caustics. Right: with caustics.
3This is what we are calling noise here. 46 3 Results
we will end up having the following marginals,
Figure 3-8: Final marginals after 120 time steps. Left: Position representation. Right: Momen- tum representation.
In this case the two marginals are basically the same, for us is clear that the two models yields the same results calculating external quantities such as averages. This is because the largest noise is located on the tails of the distribution that should carry less information than the rest.
The results in both cases, using and not using the caustics, there are not very different except for more oscillations on the reconstruction and marginals. If the caustics really change the structure of the fringes, this study shows that a larger set of trajectories is needed in order to enhance the final averaging.
In terms of computation, the calculating the caustics on the Wigner representation for the semiclassical approximations is much more expensive than not considering them as it will be explained in more detailed later in section 3.6.4, not just for the possibility of optimization when they are not considered, but also averaging has to be done with larger sampling when the caustics are considered, this translates to having a larger set of classically evolved particles in order to get similar results. These results however, show that the impact of the caustics is not so big in this particular system, and as computationally calculating them is quite demanding such a sacrifice is not justified. If one wants to calculate the caustic contribution efficiently, an alternative method is needed, starting from having only their initial and final classical quantities.Yet this has not been showed on the literature so far. This topic of the caustics on the Wigner representation are still part of current research nowadays[Domitrz et al., 2013, Domitrz and Zwierzy´nski,2020].
From this calculations we have presented until this point we concluded that we do not need to calculate the caustics to approximate the dynamics on the Morse oscillator, then in all the results presented the rest of this theses no caustics were considered for any of the implementation 3.3 Center-Center Representation 47 of the semiclassical approximations (IVR, FVR and Center-Center).
3.3 Center-Center Representation
Let us start using the center-center representation of the semiclassical approximation. We are about to show some specific times where the Wigner function was evolved using the center-center comparing them with the quantum propagation at the same times. In this specific method, we also look into some of the marginals in order to evaluate, as a first step, the semiclassical propagator.
In this representation, the initial value of all the midpoints r is known from the beginning as one chooses the initial pairs r+ and r−. It is possible to have points either on r+ or r− with enough energy such that they are not bounded therefore it forms a pair with all the bounded trajectories on the ensemble. These escaping trajectories have an important influence on the dynamics of the system if the initial amplitude of the correspondent midpoint with other trajectory is be large, large meaning that the midpoint resulting is close to the center of the coherent state to be propagated. Let us make clearer the effect of these escaping trajectories using the Wigner function.
Wigner Function
As the method has as boundary the initial classical sampled area, this scheme of propagation may have sharp borders on those regions where the classical background finishes, this is not the case of the quantum propagation which is a smooth function. This sharp shapes are presented then on the borders of the distribution, additionally those points have less summation terms as the midpoints frequency there is small, hence when weighting the final histogram they are going to be divided by a small number compared with those bins with high number of final midpoints.
The final Wigner functions are presented below in figure 3-9 for different times. This figure show that the center center representation yields to a smooth function similar to the quantum propagation even with a small number of classically propagated points (N = 10.000). Smoother than the caustic calculation section as the sampling of final midpoints in this case is considerably better. As there are some differences on the location of the fringes it becomes important to measure some quantities such as observables to evaluate how good the methods work. We also take into account that escaping points may pull the final distribution calculations, towards the positive axes on q, but they may not affect p significantly as there it is more or less symmetric. This is basically because of the choice of the initial configuration, there must exist others in which high energy points can be chosen and then scape changing the p density in an important manner. 48 3 Results
Figure 3-9: Left hand side: Semiclassics using the Center-Center Representation, Right hand side: Quantum Propagation. The times of evolution from above to below are: 22 time steps, 32 time steps, 42 time steps and 72 time steps. 3.3 Center-Center Representation 49
3.3.1 Marginals
The Wigner functions look similar than the quantum propagation on all the previous case, nev- ertheless some differences with the quantum propagation still can be seen if they are overlapped. These differences can also be seen on the wave functions and it is even easier visually. As the wave functions are calculated from the Wigner function using their marginals (2-12) and (2-11). we present also the last three times of 3-9 on figures (3-10,3-11 and 3-12)
Figure 3-10: Squared wave functions time steps 32. Left on momentum representation, Right on position representation
First, it is worthy saying that these comparison also takes into account the possible fluctu- ations the functions have, as all is integrated on one direction to get these plots. Also, as this propagation scheme lacks normalization, here the wave functions are constrained to have integral one, which implies that there must be differences on the highs.
Figure 3-11: Squared wave functions time steps 42. Left on momentum representation, Right on position representation
For instance for a larger time, it is possible to get better results for larger times, as the 50 3 Results
periodicity of the well makes the reconstruction go better at such times where the Wigner function is concentrated close to the point the initial condition was. This occurs a few times as well in the classical point of view, then the behaviour diverges as the classical distribution makes an spiral shape on phase space.
Figure 3-12: Squared wave functions time steps 72. Left on momentum representation, Right on position representation
The similarities and differences are not necessary preserved, meaning that it is possible to get some features similar that become different4, this means that as the system evolves there are some times were the approximation works better than others and that can be seen comparing, for instance, (figure 3-11) with (figure 3-12) where it is clear that the propagation worked better for 72 time steps than for 42. This is due the the shape of the final Wigner function at such times.
4At first sight, one may think that these kind of approximations should work worst every timestep, but that may not be the case as the system evolution sometimes lead to configurations that are easier to reproduce for semiclassics than others. 3.4 Initial Value Representation 51
3.4 Initial Value Representation
We look now into the initial value representation. In this representation integral over pairs in equation (1-29) has to be done in order to get a single final point on phase space. This integral is defined on all phase space which means that to get the Wigner function on a specific point, one needs to have a all the possible of pairs of trajectories populating phase space such that them all share the same midpoint, and therefore it has to be approximated, the semiclassics will work better or worse strongly depending on how good the approximation of this integral is done, thus if the integral is not done with enough points, the approximation will be very noisy.
The number of trajectories considered for each midpoint will condition the number of com- putational resources needed, and this of course has some limitations. A good way to solve this integral is using a Monte Carlo strategy in which the key point is to sample a region of phase space, then propagating it forward, reflecting each trajectory respect to the final midpoint desired and finally propagating backwards.
As a result of this strategy the integrals may not converge properly for a small number of trajectories on a given position on phase space, which results on noise during the evolution.
3.4.1 Wigner Function
Let us explore how the noise start playing a role as the time evolves in this scheme of propagation. To do this, we evaluate the same evolved Wigner functions we did for the center-center repre- sentation. As time passes, the effect of the noise generated for escaping trajectories becomes more and more important. In such cases, we see that there is noise going on the direction the non bounded pairs are going (See fig 3-13). Be aware that the noise makes the system grow on amplitude as it is constrained to the fact that the Wigner function has to be normalized,. A manifestation of this is, of course, is that the amplitudes look higher than the quantum evolution as it has to be compensated.
However, look that the parts with higher amplitude is still reproduced even for large times, this is due to the fact that this region is populated by the classical background used to propagate the Wigner function. 52 3 Results
Figure 3-13: Left hand side: Semiclassics using the IVR, Right hand side: Quantum Propagation. The times of evolution from above to below are: 22 time steps, 32 time steps, 42 time steps and 72 time steps. 3.5 Final Value Representation 53
3.5 Final Value Representation
Finally, for the case of the final value representation, the analysis is indeed similar to the one done before for the initial value representation. This as the two methods rely on the integration on phase space, but with different coordinates (1-31) and (1-29). In the same way as before, the number trajectories here is going to affect how good we approximate the integral and therefore how good results we get at the end. This strategy has the advantage that the distribution of final centers is well known as it is given by the way the final sampling as is chosen, but still has the problem that when propagating backwards it is possible to get a large number of points escaping the region of interest, hence this will work particularly good for small times.
3.5.1 Wigner Function
On this case, we can also explore how the noise start to take place. So we present here the Wigner functions at same times than before including also the quantum propagation in figure 3-14. Look that just as before the noise makes the system grow on amplitude but the functions look smoother than on the IVR case, this is due to the fact that we know beforehand how is going to be sampled the final midpoints. This still do not guarantee that the plots are going to reduce the noise significantly, as each integral has to be done more precisely. 54 3 Results
Figure 3-14: Left hand side: Semiclassics using the FVR, Right hand side: Quantum Propagation. The times of evolution from above to below are: 22 time steps, 32 time steps, 42 time steps and 72 time steps. 3.6 Comparison 55
3.6 Comparison
The main objective of this work, is comparing different semiclassical methods of propagation of the Wigner function. This comparison can be done by looking directly at the Wigner functions, but it is also possible to quantify using different parameters. Accordingly it is better to summarize the previous results by measuring different quantities on the system. The quantities we are about to consider are, Autocorrelation functions, the position and momentum expectation values, and the Normalized inner product, besides of the Wigner functions themselves as in (figure 3-15) where the quantum and the three semiclassical approximations here described are presented after 122 time steps.
Figure 3-15: Final distributions after 122 timesteps. Up left: Quantum evolution, up right: center-center, down left: IVR, down right: FVR
This can be taken even to larger times, where the principal features of the dynamics can be seen immediately, the initial and final value representations, are more sensible to noise, as the integral is not done with sufficient precision. Regions where the sampling is done correctly can be seen in the previous plot (see figure 3-15), but it becomes way more difficult as time increases, as can be seen in (figure 3-16) which are the same configuration than the previous but using 152 time steps. By contrast, the center-center representation looks like it still preserves some 56 3 Results
softness except for regions that belong to the marginal parts of the distribution like borders that are not present on the quantum propagation. Regardless these differences, this scheme seems to be robust until this point,
Figure 3-16: Final distributions after 152 timesteps. Up left: Quantum evolution, up right: center-center, down left: IVR, down right: FVR
This plots, show that the center-center representation preserves the structure of the Wigner function for longer times than the IVR and FVR, even when the resources used are much smaller, meaning that the number of classical trajectories used in the center-center is 0.005% of those used in the IVR and FVR.
3.6.1 Autocorrelation
Now we can evaluate how good the states are compared with the initial state by measuring the autocorrelation function. The autocorrelation function (2-9) of the quantum propagation is taken as reference (blue line), and for three algorithms are presented in (figure 3-15),
Here it can be seen that during the majority of the evolution the center-center representation goes close to the quantum case, while the initial and final values look shifted in those places close 3.6 Comparison 57
Figure 3-17: Autocorrelation functions. to zero (small overlapping) but they both have different behaviour in other parts, let us start with the IVR where just starting the propagation, it looks like it does not work properly, this is due to those escaping trajectories, pull the distribution at the beginning making the distribution look shifted as well as the fact that the initial configuration cannot be well described. This effect is reduced as the time evolves as the averaging of the midpoints including them start becoming less important, so the evaluation concentrates more on those points with high amplitudes. But then, for the FVR there looks like there are larger correlations in the peaks. This is due to the fact that as the method constrains the final Wigner function is normalized, noisy regions simply make the populated parts to have larger amplitude than they should. This is not easy to control if the integrals are not done properly, as the initial point distribution cannot be controlled or modified.
To evaluate better this, we plot the difference of the autocorrelation functions with the quantum over the quantum to evaluate some percentage of the difference on (figure 3-18) where it is clear that the center-center representation leads to better results consistently, specially for larger times.
3.6.2 Observables
The autocorrelation function gives us the relationship of a given state with its own initial condi- tion, but the quantum states also allows us to calculate different observables just as probability distributions do. As the Wigner function is a representation of the quantum state, it also have this property, then let us consider here position and momentum as our observables to be calculated. This is done with the aim of comparing how good certain quantities behave, regardless how the Wigner function plot looks like. 58 3 Results
Figure 3-18: Logarithm of difference of the semilcassics with the quantum autocorrelation func- tions over the quantum propagated
Momentum hpˆi
Now we consider the average of momentum, that as we mentioned before, should not be much affected by the escaping trajectories, but they are affected by the noise. As we are considering a state mostly place in the region inside the well, we expect to have oscillations as well on momentum as in position averages. We kept the same colors so the identification can be done easier, so blue line for the quantum, orange points for the center-center, green points for the FVR and finally red for the IVR.
Figure 3-19: Average of Momentums. 3.6 Comparison 59
Just as in the case of the autocorrelation function, the momentum average performs better for the center-center propagation than in the IVR and FVR. Notice that the center-center repre- sentation not just reproduce better the values of the average but also the frequency, which is not the case either for the initial than for the final value representation. To see that more in detail we can also construct the percentage difference just as we did with the autocorrelation,
Figure 3-20: Difference of the semilcassics with the quantum momentum average.
Using this plot, it is clear that the FVR and IVR do not reproduce the average of momentum as the center center, nevertheless, it is worthy to mention that, besides the visible noise present on the Wigner functions, the main characteristics are there, and that is why they reproduce the oscillations. the differences can be seen a lot more on those cases where the quantum value approaches zero, but the logarithmic plot also shows that, more or less the center-center representation is at least on order of magnitude better on average in this specific example.
Position hqˆi
For the position we may do a similar analysis than for the momentum. Here, in contrast to the previous case, the position average is harder to be corrected, as the sampling on all cases consider escaping trajectories on the direction of q positive hence the distribution start to get shifted. This is not the same behaviour for all the three cases, as the center-center representation reproduces properly the averaging until certain point (75 time steps), and then start to be shifted, but even in the shifted region, it still reproduces the frequency and the position of the minima. Situation that do not occur with any of the other two cases. 60 3 Results
Figure 3-21: Average of Positions.
Now evaluating which method performed better is difficult, but it can be seen better on the differences plot in which is clear that in most of the cases, the orange plot (Center - Center) was comparable or closer to the quantum propagation solution.
Figure 3-22: Difference of the semilcassics with the quantum position average. Left: absolute value, Right logarithmic scale.
This issue should be solved by truncate the contributions of the points that are escaping the potential, but this procedure seems artificial, as those points also contribute into the dynamics, then a different sampling strategy is needed in order to get rid of the escaping trajectories contributions. 3.6 Comparison 61
3.6.3 Normalized Inner Product
In all the quantities we showed, the center-center representation performed better specially at large times compared with the IVR and FVR. All these criteria were measurements performed independently, making necessary to have also the quantum propagation as a guide. There is also the possibility to have a quantity that already is a comparison between the specific semiclassical method and the quantum propagation. One of them is the Normalized Inner Product N(t) (2-10) which is basically the overlapping of the quantum Wigner function and the semilcassical approximation normalized. In figure 3-23 we can see that, even if the FVR seems to work better than the center-center representation, it is only after a few time steps, then it goes comparable (slightly higher) than the IVR, having the center-center better for larger times.
Figure 3-23: Normalized Inner Product.
This particular measurement, has been already used to compare semiclassical methods on the Wigner representation [Koda, 2015], but as this plot shows, the results there presented for the IVR do not correspond with those here shown, as here we get much better behaviour of this propagation scheme, and even higher for the other two and in particular the center-center. From this we conclude that the center-center representation as presented and used here performs good even compared with those methods used nowadays on the literature.
3.6.4 Complexity
When comparing numerical methods, it is very important to also compare how they can be extrapolated to bigger systems. In this case we are interested on How many final midpoints are 62 3 Results
we able to have given certain number of initial classical points.
We may now consider that on the case of the final and initial value representations, we have more than one option. First let us consider the integrals. Each integral needs a certain number of trajectories to be calculated, these cases are shown on the plot (figure 3-24) as IVR - FVR #, where the number means how many classical trajectories are used to solve each integral, then there is the monte carlo strategy used here named as IVR - FVR MC, and finally the center center representation.
In this plot a higher slope means that the method is more efficient, as more midpoints can be calculated with the same amount of trajectories. Which is the case for the center-center representation (blue line), while the IVR and FVR they all have the same slope using monte carlo or integrating with an arbitrary number of points, the difference then lies on the number of calculations increase by a constant factor, for instance, if we use a single pair to calculate each
midpoint, we get a time t1, and compared with the case that 100 of trajectories is used to the integration t100 = 100t1.
Figure 3-24: Complexity of the methods.
To sum up, the center-center representation is not just way more efficient, but also it repro- duces correctly, depending on the initial conditions, the autocorrelation function and observables, even when the differences with the quantum propagation has a big impact on measurements as the NIP. We conclude that this studies here presented combined with the performance analy- sis that the center-center representation can be used on large scale simulations and large times reducing the resources needed as well as the time consuming. 3.7 Parallelization 63
3.7 Parallelization
As the semiclassical approximations are based on sampling phase space and evolving classical trajectories there, the computational resources for such purposes may be demanding, specially the time the method takes may be very large. This can be reduced significantly by parallelizing the method. In general, all semiclassics methods can be trivially parallelized. All the results here were done parallelizing the process as will be briefly explained in this section.
3.7.1 IVR-FVR
There are two ways to parallelize these strategies, solving the integrals or via Monte Carlo. First the integration can be separated into different processes, so each process is assigned a point in phase space and each one calculates the corresponding integral. As all the point information is not shared, communication is not necessary between the processes. This suits good using MPI, as each node can have its own memory to calculate the integration. Then comes the Monte Carlo strategy. This is done similar, but now each pair is considered a process, so each process consist on the calculation of the classical quantities of two classical trajectories. This was the approach used in this work using OpenMp, as it can run more efficient than MPI on local computers.
This last was the case used in the IVR and FVR here presented.
3.7.2 Center-Center
With the center-center method there is the problem that, after the evolution, all the combinations of trajectories has to be done in order to calculate the final state. This of course is the most expensive part of the code. Fortunately each pair do not depend on others then each process can propagate each trajectory. This will make the evaluation time much smaller, but as a consequence all trajectories have to be stored until the end of the propagation. This may limit the amount of trajectories one can use, but as the propagator only cares about averaging on final points, one can separate the code into two different sets if initial conditions that do not communicate and construct one histogram per each set and then adding those with the proper weights. This makes even more efficient the sampling getting a strategy that suits perfectly the usage MPI and OpenMP simultaneously, therefore a huge improvement on computing time.
The calculations here showed were parallelized using just OpenMP, given the computing resources available. Chapter 4
Conclusions and Perspectives
This thesis compares different schemes of semiclassical propagation, showing that all of them are able to reproduce the quantum mechanical quantities out of classical information. A study on the caustics on the Morse oscillator, system commonly used as a benchmark in semiclassical dynamics approximations, was done showing that including them increases considerably the complexity of the method and such effort was not worthy. Afterwards, three methods were compared, an Initial value (IVR), a final value (FVR) and a center-center representations, showing that the center-center representation can be used as free of root-search even when it has been claimed to have this problem. Additionally, initial and final value representations lead to a highly demanding computational problem, compared with the center-center representation that requires a smaller amount of resources to get similar results. The results here presented showed that using the center-center representation performed better on all the quantities measured, such as position and momentum averages, autocorrelation function and the normalized inner product using just 0.05% of the points used for the IVR and FVR.
The results reported here for the NIP show that recent studies on the comparison of semi- classical propagation [Koda, 2015], were not done properly, as we got substantially better results from those there presented even compared with the Herman-Kluk approximation there presented which is said to be an initial value representation. This is not surprising as the Herman-Kluk propagator is a smoothed version of the van Vleck-Gutzwiller propagator, which was the one used here. This can be seen taking the limit of the width of the Gaussians taken as a basis on the Herman-Kluk approximation to zero, then a representation of δ functions is reached and hence the van Vleck-Gutzwiller propagator is recovered.
Although this thesis show that computing the quantum dynamics using semiclassical approx- imations can be challenging, it also shows that it can be done efficiently highly parallelizable for 65 all methods. Nevertheless, the analysis on the performance showed that using the center-center representation is not only more efficient on the amount of classical trajectories needed, but also that given the independence of the construction of the Wigner function, it is possible to propagate different initial conditions using the same classical trajectories and therefore, making possible to reuse calculations.
Numerical applications were presented for some semiclassical approximations. However, there are much more to explore with these approximations to the Wigner function propagation. To mention few of them we have,
• Considering bounded systems where there is no escaping trajectories that induces oscilla- tions with high amplitude to the final Wigner function construction.
• Study the center-center representation 1-22 as a final value representation and for longer times, as the complexity of the method do not increase.
• Compare the center-center representation with the Herman-Kluk propagator. They are closely related as the Herman-Kluk can be understood as the van Vleck propagator using a Gaussian smoothing kernel, which is a similar relationship between the Wigner function and other phase space representations as, for instance, the Husimi function, where it is well known that the Husimi function does not reproduce correctly quantities as expectations values due to the Gaussian filtering introduced into the state (or the Wigner function).
• Considering these approximations in the analysis of quantum systems in thermal contact with classical baths. It has been proved that when considering an open quantum system in contact with a classical bath the Wigner function is a suitable representation for the quantum system, allowing to study this quantum-classical interaction using, for instance, classical molecular dynamics simulations. Appendix A
Coherent States
The Coherent states are important in fields as quantum optics, as they minimize the uncertainty and they are eigenstates of a destruction operator which make them very interesting.
On the language of Wigner functions they also have their important place, as their Wigner function is positive in all phase space, in fact if we extend the idea of coherent states and talk about Gaussians1 (on phase space), the Hudson’s theorem[Hudson, 1974] states that the only pure states that do not take on any negative values are Gaussians, at least for systems of one degree of freedom and therefore, they have a classical interpretation as classical density distributions so that these states are called to be the ”most classical” quantum states.
A.1 Wigner Function for Coherent States
Let us consider a harmonic oscillator with mass m and frequency ω. A coherent state |γ0i with centroid at (pγ0 , qγ0 ) on phase space.
In position representation, it goes as,
1/4 0 0 mω mω 0 2 i 0 ψ 0 (q ) = hq |γi = exp − (q − q 0 ) exp − (q − q 0 )p 0 , (A-1) γ πh¯ 2¯h γ h¯ γ γ √ 0 p 0 where pγ0 := 2¯hmω Im((γ ) and qγ0 := 2¯h/mω Re(γ ).
Which basically is a Gaussian with a phase factor given by a translation to the momentum
1Which also may include thermal states or squeezed states and so on A.1 Wigner Function for Coherent States 67 plain.
The Wigner function is calculated as,
1 Z i y y W (p0, q0, t) = dy exp − yp0 ψ∗ q0 − ψ q0 + . (A-2) 2πh¯ h¯ 2 2
So, for the coherent states, we have that
1 mω 1/2 Z i W (p0, q0, t) = dy exp − yp0 2πh¯ πh¯ h¯ 2 mω 0 y i 0 y exp − q − − q 0 exp q − − q 0 p 0 2¯h 2 γ h¯ 2 γ γ mω 0 y 2 i 0 y exp − ( q + − q 0 ) exp − ( q + − q 0 )p 0 . (A-3) 2¯h 2 γ h¯ 2 γ γ
So, it can be simplify as,
1 mω 1/2 Z W (p0, q0, t) = dy exp (φ) , (A-4) 2πh¯ πh¯ So, as the integration is over a single exponential, we may consider just the exponent to rewrite it,
2 2 0 mω 0 y 0 y i yp φ = − q − − q 0 + q + − q 0 + yp 0 − 2¯h 2 γ 2 γ h¯ γ 2 2 2 mω 0 y 0 y i 0 = − (q − q 0 ) − + (q − q 0 ) + + (yp 0 − yp ) 2¯h γ 2 γ 2 h¯ γ 2 mω 0 2 y iy 0 = − (q − q 0 ) + + (p 0 − p ) h¯ γ 2 h¯ γ 2 mω h 0 2i mω y iy 0 = − (q − q 0 ) − − (p − p 0 ) h¯ γ h¯ 2 h¯ γ
So, the large integration, rewritten goes as,
1/2 Z 2 0 0 1 mω mω h 0 2i mω y iy 0 W (p , q , t) = dy exp − (q − q 0 ) − − (p − p 0 ) . 2πh¯ πh¯ h¯ γ h¯ 2 h¯ γ (A-5)
This integration can be performed just by considering the Gaussian form of this expression2, so
2R ∞ 2 p 2 ∞ exp −ax + bx dx = π/a exp(b /(4a)) 68 A Coherent States
1/2 Z 0 0 1 mω mω h 0 2i mω 2 iy 0 W (p , q , t) = exp − (q − q 0 ) dy exp − y − (p − p 0 ) , 2πh¯ πh¯ h¯ γ 4¯h h¯ γ
" # 1/2 1/2 2 0 0 1 mω 4πh¯ mω h 0 2i i 0 h¯ W (p , q , t) = exp − (q − q 0 ) exp (p − p 0 ) , 2πh¯ πh¯ mω h¯ γ h¯ γ mω
0 0 1 mw 0 2 1 0 2 W (p , q , t) = exp − (q − q 0 ) − (p − p 0 ) (A-6) πh¯ h¯ γ mhω¯ γ thus, the Wigner function for a coherent state, is also a Gaussian function, but now on a 2d space.
The Wigner function can be constructed from the coherent sum of states and from the incoherent sum. In each case the same properties mentioned above held but the visualization and therefore the results are very different.
In the case of the non coherent sum, we have that the Wigner functions are summed up while on the case of coherent sum we have to do the sum of the states before the construct the Wigner function. On the following figure the difference between two coherent states summed non coherently and coherently is shown, where the case of the coherent sum lead to what is called a cat state which displays a huge quantum behavior in the sens that it takes negative values that classically is impossible.
Figure A-1: Plot of two coherent states summed up non coherently (left) and coherently forming a cat state (right) A.2 Relationship with the Husimi Function 69
A.2 Relationship with the Husimi Function
The Wigner function is related with the Husimi function by a Weierstrass transform which is basically a Gaussian smoothing, to see this let us consider the following transformation to the Wigner function. Let us start from the definition of the Husimi function Z 2 1 2 1 ∗ Qγ(p, q, t) := | hγ|ψi | = dqψ 0 (q)ψγ0 (q) , (A-7) π 2πh¯ γ
or also can be written on terms of the density operator, 1 Q (p, q, t) = Tr[|γi hγ| ρ]. (A-8) γ 2πh¯ Using the following property of the Wigner function 1 ZZ Tr[AˆBˆ] = dpdqA (p, q)B (p, q), 2πh¯ W W the Husimi function can be written as ZZ 0 0 0 0 0 0 Qγ(p, q) = dp dq Wγ(p , q )W (p , q ), wherer Pγ corresponds to the Wigner function for the state γ. Now we make the following change of variables √ 0 0 p = 2¯hmω γ1, r 2¯h q0 = γ0 , mω 2
0 0 0 0 where γ1 = Im(γ ) and γ2 = Re(γ ). The Jacobian of the transformation corresponds to 2¯h, so PH becomes Z √ 2 0 0 0 0 p 0 Qγ(p, q) = (2¯h) d γ Wγ(γ1, γ2)W ( 2¯hmω γ1, 2¯h/mω γ2), (A-9) C
0 0 where the function Wγ(γ1, γ2) is given by 0 0 1 mω 2 1 2 W (γ , γ ) = exp − (q 0 − q ) exp − (p 0 − p ) , γ 1 2 πh¯ h¯ γ γ hmω¯ γ γ √ p where qγ = 2¯hmωγ Re(γ) and pγ = 2¯h/mω Im(γ). Replacing the above definitions the above expression becomes 1 W (γ0 , γ0 ) = exp −2(Re (γ0) − Re(γ))2 − 2 (Im(γ0) − Im(γ))2 . γ 1 2 πh¯ 70 A Coherent States
Using the following identity
|α − β|2 = Re2(α) + Im2(α) + Re2(β) + Im2(β) − 2 (Re(α)Re(β) + Im(α)Im(β)) ,
we obtain 1 W (γ0 , γ0 ) = exp −2|γ0 − γ|2 . γ 1 2 πh¯
By making use of this result on (A-9) follows that PH is given by Z √ 2 2 0 −2|γ0−γ|2 0 p 0 Qγ(p, q) = d γ e W 2¯hmω Im(γ ), 2¯h/mω Re(γ ) , π C which is the Wigner function smoothed with a Gaussian filter, which has huge implications on the physics as this construction is done by doing a trace over certain part of the system, this relationship is not reversible and therefore is not the best way to construct the time evolution of quantum mechanical systems, and in fact, it is not possible to do so, as the Husimi function on a certain time and the Hamiltonian are not enough to find the Husimi function a time after as this representation lacks information.
For instance, one of the quantities one can be interested in are the marginal probabilities, so, to built the marginal distributions Q(q), we may,
1 Z Q(q) = dp hγ|ψi hψ|γi 2πh¯ 1 Z Z = dp dq0dq00 hγ|q0i hq0|ψi hψ|q00i hq00|γi 2πh¯ 1 Z Z mω 1/2 mω = dp dq0dq00 exp − (q0 − q )2 2πh¯ πh¯ 2¯h γ mω i exp − (q00 − q )2 exp − (q0 − q00)p hq0|ψi hψ|q00i 2¯h γ h¯ γ Z mω 1/2 mω = dq0 exp − (q0 − q )2 | hq0|ψi |2 πh¯ 2¯h γ 2 1 Z 1 q0 − q = dq0 exp γ | hq0|ψi |2 1/2 √ q ¯h 2π 2 ¯h 2mω 2mω
Which corresponds to the expectation value, but smoothed by a Gaussian function. Appendix B
Wigner Function Calculations
On this section, we are going to show how some things are calculated on the Wigner represen- tation, and we will explain how the Wigner phase space works, and some of the insights and interpretation of both the quantum phase spaces and the Wigner function.
So let us begin from the definition of the Wigner function,
1 Z ∞ W (q, p) = dy exp (−ipy/h¯) hq + y/2|ρˆ|q − y/2i . (B-1) 2πh¯ ∞
The Wigner function can be understood as a Fourier transform of the density operator, represented in position1 on a particular direction. This function represents the state on the Wigner Phase space and thereby, it allows us to calculate the quantum mechanical quantities namely expectation values, marginal probabilities and overlap between states, among others. So let us see those particular ones, making some developments on the way.
B.1 Marginal Probabilities and Area
The Wigner function allows us to calculate a lot of quantum mechanical quantities, as the density operator does, in fact, it does it in the exact same way it is done in the classical statistical mechanics as if it was a density distribution, but without being one though. Like the Wigner function (1-3) depends on q and p, to calculate the marginal probability distribution in q is enough
1It can also be done on momentum, and the result is equivalent 72 B Wigner Function Calculations
to integrate all over p, thus Z dpW (q, p) = hq| ρ |qi (B-2)
It can be easily proved by it’s definition
Z ∞ 1 Z ∞ Z ∞ dpW (q, p) = dy hq + y/2|ρˆ|q − y/2i dp exp (−ipy/h¯) ∞ 2πh¯ ∞ −∞ Z ∞ = dy hq + y/2|ρˆ|q − y/2i δ (y) ∞ = hq|ρˆ|qi,
and using a similar argument we can verify the case of the other integral
Z ∞ 1 Z ∞ Z ∞ dqW (q, p) = dq dye−ipy/¯h hq + y/2|ρˆ|q − y/2i −∞ 2πh¯ −∞ −∞ 1 Z ∞ Z ∞ = dq dye−ipy/¯h hq + y|ρˆ|qi 2πh¯ −∞ −∞ 1 Z ∞ Z ∞ = dq dye−ip(y−q)/¯h hy|ρˆ|qi 2πh¯ −∞ −∞ Z ∞ Z ∞ = dq dy hp|yi hy|ρˆ|qi hq|pi −∞ −∞ = hp|ρˆ|pi.
where we have used the fact that the momentum is the movement generator as,
1 hp|yi = √ exp (−ipy/h¯) (B-3a) 2πh¯
and 1 hq|pi = √ exp (ipq/h¯) (B-3b) 2πh¯
So, this says that if we integrate over the two variables is equivalent to integrate the marginal probability and therefore, trace the system,
Z ∞ Z ∞ dq dpW (q, p) = Tr[ˆρ]. (B-4) −∞ −∞
So, the Wigner function is indeed, normalized2.
2This is why the Wigner function is not just the Weyl transform of the density operator, but additionally an extra term B.2 Weyl Correspondence 73
Now, let us consider the overlap between two Wigner functions, as this is calculated by tracing the product between the two states, we may multiply two Wigner functions and integrate them over all phase space, such as, Z ∞ Z ∞ dq dpW1(q, p)W2(q, p). (B-5) −∞ −∞ So Z ∞ Z ∞ Z ∞ Z ∞ 1 0 00 −ip(q0+q00)/¯h = 2 dq dp dq dq e (2πh¯) −∞ −∞ −∞ −∞ Z ∞ Z ∞ Z ∞ 1 0 0 0 00 00 = dq dq hq + q /2 |ρˆ1| q − q /2i hq + q /2 |ρˆ2| q − q /2i 2πh¯ −∞ −∞ −∞ Z ∞ Z ∞ Z ∞ 1 0 00 0 00 0 0 00 00 = dq dq dq δ (q + q ) hq + q /2 |ρˆ1| q − q /2i hq + q /2 |ρˆ2| q − q /2i 2πh¯ −∞ −∞ −∞ Z ∞ Z ∞ 1 0 0 0 0 = dq dq hq + q /2 |ρˆ1| q − q /2i hq − q /2|qi 2πh¯ −∞ −∞ Z ∞ Z ∞ 1 0 0 0 = dq dq hq |ρˆ1| q i hq |ρˆ2| qi 2πh¯ −∞ −∞ 1 Z ∞ = dq hq |ρˆ1ρˆ2| qi 2πh¯ −∞ 1 = Tr [ˆρ ρˆ ] . 2πh¯ 1 2
Now, we can consider the case where the density operators are indeed, the same ρˆ1 =ρ ˆ2 =ρ ˆ, so Z ∞ Z ∞ 1 1 dq dpW 2(q, p) = Tr ρˆ2 ≤ −∞ −∞ 2πh¯ 2πh¯ So, that means that the area of the Wigner function says whether the state is pure or not. This is known as the S¨ußmannmeasure [P. Schleich, 2001].
B.2 Weyl Correspondence
As we have mentioned before, the idea behind the phase space representations is to construct functions from operators in such a way we can calculate quantum mechanical quantities as in the classical statistical mechanics. The Weyl correspondence is the way we build these functions for the specific case of the Wigner phase space.
First, let us consider the following phase space function ei(πqq+πpp)/¯h (B-6) 74 B Wigner Function Calculations
Which immediately suggest the fact that q and p are not conjugated anymore but they are with
πq and πp respectively. The operator version of this function is called the characteristic operator, and we will see later why this has such an important name,
ˆ i(πqqˆ+πppˆ)/¯h M (πq, πp) = e . (B-7)
Look that the correspondence we have done here is just changing variables q and p for operators qˆ and pˆ respectively, thus
ei(πqq+πpp)/¯h −→ ei(πqqˆ+πppˆ)/¯h. (B-8)
This relationship helps us to construct the correspondence between operators and functions. The way this procedure is done was developed by Weyl[Weyl, 1927] and goes as follows, consider the Following Fourier transform 1 Z ∞ Z ∞ ˜ i(πqq+πpp)/¯h A(q, p) = dπq dπpA (πq, πp) e (B-9) 2πh¯ −∞ −∞ We may write the operator version of this function as, 1 Z ∞ Z ∞ ˆ ˜ i(πqqˆ+πppˆ)/¯h A(ˆq, pˆ) = dπq dπpA (πq, πp) e (B-10) 2πh¯ −∞ −∞ Which basically was changing the variables for operators
A(q, p) −→ Aˆ(ˆq, pˆ) (B-11)
And then, the transformation in terms of the variables leads to, 1 Z ∞ Z ∞ Z ∞ Z ∞ ˆ i(πq(ˆq−q)+πp(ˆp−p))/¯h A(ˆq, pˆ) = 2 dπq dπp dq dpA(q, p)e . (B-12) (2πh¯) −∞ −∞ −∞ −∞
From this, we can calculate the expectation value of the operator Aˆ so as,
Z ∞ Z ∞ Z ∞ Z ∞ ˆ 1 hA(ˆq, pˆ)i = 2 dπq dπp dq dpA(q, p) (2πh¯) −∞ −∞ −∞ −∞ × e−i(πqq+πpp)/¯h ei(πqqˆ+πppˆ)/¯h . (B-13)
So, let us define this last expectation value as, D E ˆ i(πqqˆ+πppˆ)/¯h M (πq, πp) = M (πq, πp) = e (B-14)
This operator is called the characteristic operator of the transformation.
So, if we use the Baker-Campbell-Hausdorff relationship[?],
A+B A B − 1 [A,B] B A 1 [B,A] e = e e e 2 = e e e 2 , (B-15) B.2 Weyl Correspondence 75 and using it up to second order which is exact for the case for the specific case of position and momentum ei(πqqˆ+πppˆ)/¯h = eiπpp/ˆ 2¯heiπqq/ˆ ¯heiπpp/ˆ 2¯h (B-16) Then the characteristic function can be written as
iπpp/ˆ 2¯h iπxx/ˆ ¯h iπpp/ˆ 2¯h M (πx, πp) = Tr e e e ρ Z ∞ iπxx/¯h iπpp/ˆ 2¯h iπpp/ˆ 2¯h = dxe x e ρe x −∞ . (B-17) Z ∞ iπxx/¯h = dxe hx + πp/2|ρ|x − πp/2i −∞ As this relationship is a Fourier transform we can invert it, so as getting Z ∞ 1 −πqq/¯h hx + πp/2|ρ|x − πp/2i = dπqM(πq, πp)e . (B-18) 2πh¯ −∞ We can right now calculate the relationship between the characteristic function and the Wigner iπpp/¯h function, just by multiplying for the correspondent prefactor e and integrating over πp with the correspondent normalization constant, thus, Z ∞ Z ∞ 1 −i(πqq+πpp)/¯h W (q, p) = 2 dπq dπpM(πq, πp)e . (B-19) (2πh¯) −∞ −∞ So now, it becomes trivial to see that the expectation values can be calculated from the Wigner function replacing the previous result with (B-13) leading to D E Z ∞ Z ∞ Aˆ(ˆq, pˆ) = dq dpA(q, p)W (q, p). (B-20) −∞ −∞ So it acts as a classical distribution function.
Let us start from this expression and write this explicitly, so that D E Z ∞ Z ∞ Aˆ(ˆq, pˆ) = Tr[Aˆ(ˆq, pˆ)ˆρ] = dq dq0 hq| Aˆ(ˆq, pˆ) |q0i hq0| ρˆ|qi (B-21) −∞ −∞ So if we change variables so that q0 −→ q0 + q/2 and q −→ q − q0/2, so that D E Z ∞ Z ∞ Aˆ(ˆq, pˆ) = dq dq0 hq − q0/2| Aˆ(ˆq, pˆ) |q0 + q/2i hq0 + q/2| ρˆ|q − q0/2i . (B-22) −∞ −∞ But we had already written an extra which includes the integration of the Wigner function and the correspondent phase space function, so comparing those expressions we get, ∞ 1 Z 0 hq − q0/2| Aˆ(ˆq, pˆ) |q0 + q/2i = dpF (q, p) exp−ipq /¯h . (B-23) 2πh¯ −∞ Which can be inverted to get the phase space representation of the operator Aˆ(ˆq, pˆ) as Z ∞ A(q, p) = dq0 hq + q0/2| Aˆ(ˆq, pˆ) |q0 − q/2i exp−ipq0/¯h, (B-24) −∞ this is the Weyl transform of an operator Aˆ(ˆq, pˆ). Appendix C
Caustics
The idea of the caustic is presented on many different areas of physics, one of the most precises ways to understand them is watching a pool where on a sunny day due to the movement of the water there are some dark zones and others very lighted, the areas where more light have, of course, a bigger intensity which means that there is a concentration of rays. Even when these intensities are way larger than on the case of the dark areas, it is still finite because energy must be conserved.
Let us concentrate on the caustics on the context of semiclassical mechanics, [Littlejohn, 1992, Ozorio de Almeida, 1989, Maslov and Fedoriuk, 1981]. This Caustics depend strongly on the representation we choose, which basically means that the points that are caustic on a given representation may not be necessary a caustic on a different one and the other way around. In the case we want to use tha vanVleck-Gutzwiller (Equation 1-15) propagator to evolve an initial wave function, these caustics are caused on the points (line or surface and so on, depending on the dimension of the manifold), such that the Lagrangian manifold where the trajectory leads, folds into itself in such a way that a turning point1 is created, and as a consequence a divergence on the derivative of p with respect to q where these are conjugated variables.
If we think this to be a position representation problem, we may have that the derivative dq dp diverge on the caustics and as the semiclassical approximation has this term as an amplitude term, semiclassics do not work on that specific point but it does on a neighbourhood before and after.
Suppose the following procedure, consider that you know in which point q the caustic is such as it is shown on figure C-1, and you are considering the semiclassical approximation for the
1Again, this may not be a point but a line or hypersurface instead. 77 dynamics, this is therefore clear that the amplitude is going to be huge, even if it is not the real case. Transform the state just before the the momentum representation, which basically means to do a rotation of the axis as momentum is pretended to be conjugated to position, the caustic point now has became a maximum or minimum where there is no problem on the integration using the semiclassics approach, so we may integrate just after the caustic point is passed and return to position representation. The difference now is that a phase is gained because the direction is changed, so it is enough to add a phase to the evolution and the effect of the caustic is taken into account.
p
q
qcaustic
Figure C-1: Phase space representation of a Caustic
This phase is called the Maslov index [Maslov and Fedoriuk, 1981].
However, the problem of the caustics on phase space representation of quantum mechanics is way more difficult on a numerical perspective. The main reason is that, when propagating wavefunctions using semiclassics, a initial set of classical points are evolved independently, so if a caustic exists on a trajectory, can be calculated just with its own quantities, but as pairs are needed to calculate the phase space representation of the state, now the caustic depend on each pair and it may be in fact, independent to each of the caustics of each trajectory on the pair. This can be understood with the idea of the double phase space, where now we have that even if the projection on a lower dimension phase space a point looks like a turning point, it ends up being a regular point on a higher dimensional space [Ozorio de Almeida, 2009].
The problem now becomes a sampling problem, as we need to sample a 2d dimensional function (Wigner function) instead that a d dimensional function (wave function), it becomes very important but challenging, as the complexity of the problem increases, the computational resources are of huge importance. Hence a optimal method to do the propagation effectively becomes important. 78 C Caustics
In general, there is no way to count the caustics given an initial and final point of a single trajectory as they are defined as
µ = ν+ − ν− (C-1)
where ν+ are the number of times the determinant on the amplitude of the propagator turns positive, while ν− is the other way around, the number of times the determinant turns negative. Appendix D
Classical Evolution
On Classical Mechanics, the expression phase space can have more than one meaning, on the context of thermodynamics for example, the phase space is a space made with all the thermo- dynamic variables, for instance, on the case of an ideal gas the phase space is the one formed by (P,V,T ). In other words a point on phase space represent a description of the macroscopic variables such that the state macroscopically is completely described, in the sense that many microscopical configurations can lead to the same macroscopic state.
Meanwhile, on Hamiltonian theory of classical mechanics, a phase space is a space where the microscopic states are represented by single points, which evolution can be calculated directly from a Hamiltonian function.That phase space has 2N dimensions where N are the degrees of freedom of the system, for example if we consider the case of a single particle moving just in one dimension, the phase space is shown on the following figure:
p
. state (p, q)
q
Figure D-1: Representation of a single one dimensional particle state on classical mechanics, assuming punctual the state (Red dot)
where q represents the generalized coordinate and p its canonically conjugated momentum. Now on, when we call the classical phase space we refer to this kind of phase space. 80 D Classical Evolution
D.1 Dynamics
In classical phase space, the dynamics is given by the Hamilton’s equations. Those equations describe completely the dynamics of a system with N degrees of freedom[Goldstein et al., 2002],
∂H(q, p, t) q˙k = , (D-1a) ∂pk and ∂H(q, p, t) p˙k = − , (D-1b) ∂qk where k = 1, ··· ,N, and H is the Hamiltonian of the system. A different way to write those equations is the symplectic notation, where one must write the equations on a matrix way so that they can be written in a single matrix differential equation, ∂H(r, t) r˙ = J (D-2) ∂r 0 IN×N ∂ T and r = (q, p), J = and ∂r = (∂q1 , ··· , ∂qN , ∂p1 , ··· , ∂pN ) . −IN×N 0 This notation is particularly useful when working with canonical transformation due to the fact that showing that a transformation is canonical is simpler using the properties of the symplectic structure.
D.2 Distributions
Some times, we are interested on the time evolution of a complete distribution of particles, instead of a single particle as it is the case on this work, the reason is what we mentioned before, the comparison between quantum mechanical dynamics of a single particle has to be done with a complete classical distribution. In this case, we may consider a classical distribution on phase space ρ(q, p) that includes the complete information of the set of classical particles which should be evolved to study the dynamics. One can consider two different but theoretically equivalent approaches, the first one is that even thought we have a large set of initial conditions each should follow the Hamilton’s equations, so the evolution of the distribution can be calculated as the evolution of a large amount of classical trajectories following the Hamilton’s equations, and the second is considering the distribution as it is and evolving the complete distribution. The equation used to evolve the distribution is called the Liouville Equation [Goldstein et al., 2002, Arnold et al., 2013]:
∂tρ = {H, ρ}P , (D-3) D.3 Numerical Implementation 81
where ρ is the distribution, and {·, ·}P is the Poisson bracket defined as
N X ∂A ∂B ∂A ∂B {A, B} = − . (D-4) P ∂qi ∂pi ∂pi ∂qi i=1 The fundamental difference of the two approaches is that numerically the second one is way more efficient than the first one, due to the amount of trajectories needed to do the propagation.
The classical propagation is done following the already mentioned Hamilton equations (D-1), such that for a single particle are, dq ∂H = , (D-5a) dt ∂p and dp ∂H = − . (D-5b) dt ∂q As they are coupled, they must be solver simultaneously. That fact has to be taken into account when solving them numerically, because the evolution of each variable is not independent during the time of integration.
D.3 Numerical Implementation
Numerically, we can use the Hamilton equation to find the solution for a distribution which in principle, should be done by solving the Liouville equation (D-3). The way to do that task, is evolving a set of points which are a sampling of the base of the distribution and assign each point a height that doesn’t change during the evolution. A representation of this procedure for a single point is shown in the figure (D-2).
t = t Amplitude 0 t = t1 q
p Trajectory
Figure D-2: Representation of the propagation of a classical distribution where the height remains constant after the evolution. 82 D Classical Evolution
There are a huge variety of numerical methods that can be used to solve (D-1) [Press, 2007, Burden and Faires, 2010, Hairer et al., 2008], from the simpler such as the Euler method, and getting to some more sophisticated ones, such as Yoshida’s method [Yoshida, 1990], casified by the numerical error they carry. There are another characteristics that can be used to classify those methods, for example, the way they work, a particular case is one very important on physics because we always concern about the variables which remain constant during time, for instance, the energy. Many methods such as Euler or Runge-Kutta are susceptible to changes on the energy of the system even on cases where the system is closed. Fortunately there are a complete family of methods that doesn’t have that problem, they are called symplectic methods. In other words, if a numerical method preserves the energy of the system is said the method to be symplectic [Hairer et al., 2008, Chapters I 1,I 14,II 17]. Considering the kind of systems that are going to be used, the need of symplectic methods arises.
Besides being symplectic, we want the method to be good in therms or error. Some sym- plectic numerical methods are mentioned below:
• Leap-frog (Second Order Method) [Press, 2007, Page 1039]
• Ruth’s Method (Third Order Method) [Forest, 2006]
• McLachlan and Atela (Fourth Order Method) [McLachlan and Atela, 1992]
• Yoshida (Sixth Order Method) [Yoshida, 1990]
The chosen method was the Yoshida’s one, so its implementation is done by considering Hamil- tonians of the form
H(q, p) = T (p) + V (q), (D-6)
and each step is calculated as
qi+1 = qi · bi · dtG(pi) (D-7a)
pi+1 = pi · ai · dtF (qi) (D-7b)
where dT G(p) = (D-8a) dp
dV F (q) = (D-8b) dq D.3 Numerical Implementation 83
The coefficients are those that define the method, the Yoshida’s are,
i ai bi 1 0.78451361047756 0.39225680523878 2 0.23557321335936 0.51004341191846 3 -1.1776799841789 -0.47105338540976 4 1.3151863206839 0.068753168252520 5 -1.1776799841789 0.068753168252520 6 0.23557321335936 -0.47105338540976 7 0.78451361047756 0.51004341191846 8 0.0 0.39225680523878
Table D-1: Coefficients of the sixth order Yoshida’s method. Taken from [Gray et al., 1994]
Algorithm
As an algorithm the classical propagation is calculated with the following steps
1. A initial distribution ρ(q, p) is defined.
2. Every point (qi, pi) that are sampling the distribution are evolved using the Yoshida’s method (D-7).
3. The final distribution is constructed. Appendix E
Quantum Propagation
To find the exact propagation of the Wigner function we can use several methods, for example, as the Wigner function depends on the density operator ρˆ, it is possible to find the time evolution of the operator as it is ρˆ and then use the Weyl transform, or even, for pure states it is enough to evolve the wave function |ψi. This procedure has some problems when calculating the Weyl transform numerically, because of the relationship between W (q, p) and ρˆ, is via an integral over y y an extra parameter y all over the space. That means that the value of q − 2 ρˆ(ˆq, pˆ) q + 2 must be known on a large interval so that the integral can be done without losing information of the state. This is why this method is not the best when looking for the dynamics.On the other hand, the Moyal equation can be solved numerically using a simpler and more intuitive method using a split operator strategy.
E.1 State Dynamics
Before showing the time evolution method for the Wigner function, it is convenient to under- stand a similar strategy but for a state, just to clarify the later development. Starting with the Schr¨odinger’sequation ˆ ih∂¯ t |ψ(t)i = H |ψ(t)i , (E-1)
The solution can be expressed as a time evolution operator, ˆ |ψ(t)i = U(t, t0) |ψ(t0)i . (E-2)
ˆ The operator U(t, t0) has the whole information about the dynamics of the system, so it E.1 State Dynamics 85
doesn’t depend on the state |ψ(t0)i but the system. If we consider time independent Hamiltoni- ans1 i Uˆ(t, t ) = exp − Htˆ , (E-3) 0 h¯
This operator has some important properties. For any time τ,
ˆ ˆ ˆ U(t0, τ)U(τ, t) = U(t0, t). (E-4)
They form a group, so if t0 = 0, the time evolution until a time t can be seen as n evolutions on times ∆t with n∆t = t,
n n Y Y − i H∆t Uˆ(t) = Uˆ(∆t) = e( h¯ ). (E-5) i=1 i=1 So, to find the time evolution of a large time t it is enough to study the evolution in a small interval ∆t and then apply the same several times.
This is the idea of the Split Operator Methods. Let’s consider ∆t −→ 0, from (E-3) the evolution of a time ∆t is:
− i H∆t Uˆ(∆t) = e( h¯ ). (E-6)
Taking Hamiltonians of the form:
Hˆ (ˆq, pˆ) = T (ˆp) + V (ˆq), (E-7)
It is possible to us the Baker-Campbell-Hausdorff[Andrea Bonfiglioli, 2012] relation:
2 α(A+B) αA αB − α [A,B] e = e e e 2 ··· (E-8) so,
2 − i (T (ˆp)+V (ˆq))∆t − ∆t T (ˆp) − ∆t V (ˆq) − i [T (ˆp),V (ˆq)](∆t)2 e h¯ = e h¯ e h¯ e ( h¯ ) ··· . (E-9)
As the time ∆t is as small as we want, it can be taken as ∆t −→ 0, so, on the expansion we can neglect powers of ∆t,
− i (T (ˆp)+V (ˆq))∆t − ∆t T (ˆp) − ∆t V (ˆq) e h¯ ≈ e h¯ e h¯ . (E-10)
1If the Hamiltonian depends on time, the time evolution operator has a similar form, but two cases must be considered, first is when the Hamiltonian commutes with itself but at different times, if it is the case, it is enough to integrate the Hamiltonian over time, but, if it is not the case, it is neccesary to use the so called Dyson series[Sakurai and Napolitano, 2011]. 86 E Quantum Propagation
In other words, the time evolution using ∆t −→ 0 can be approximated as
− ∆t T (ˆp) − ∆t V (ˆq) Uˆ(∆t) ≈ e h¯ e h¯ (E-11)
Where the time evolution operator is splitted on a position part, and a momentum part. The last ingredient is consider that the relation of ψ(q) and ψ(p) is a Fourier transform. So, lets consider the operator pˆ and its eigenstate |pi
hq| pˆ|pi = hq| p |pi = p hq|pi
= ih∂¯ x hq|pi .
hq|pi satisfies
i hq|pi = exp − pq . (E-12) h¯
This can be used to find the time evolution. From (E-3),
ˆ |ψ(∆t)i = U(∆t) |ψ0i ,
So, as we approximate Uˆ as a part of position and momentum, it is convenient to represent the state on position representation so it is eigenstate of the position part of the time evolution operator e(−∆tV (ˆq)/¯h) and it became its eigenvalue
− ∆t T (ˆp) − ∆t V (ˆq) |ψ(∆t)i =e h¯ e h¯ |ψ0i Z − ∆t T (ˆp) − ∆t V (ˆq) = dqe h¯ e h¯ |qi hq|ψ0i
− ∆t V (q) So, we can take ψ1(q) = e h¯ ψ0(q), and change to the momentum representation ψ1(p), so it become an eigenstate of e(−∆tT (ˆp)/¯h) and it becomes its eigenvalue, Z − ∆t T (ˆp) |ψ(∆t)i = dqdpe h¯ |pi hp|qi ψ1(q).
Where we use the Fourier transform. Such that Z Z − ∆t T (p) |ψ(∆t)i = dpe h¯ dq hp|qi ψ1(q) |pi Z − ∆t T (p) = dpe h¯ F [ψ1(q)] |pi Z − ∆t T (p) = dpe h¯ φ(p) |pi . E.1 State Dynamics 87
Lastly, the final state on position representation leads, Z − ∆t T (p) hq|ψ(∆t)i = hq| dpe h¯ φ(p) |pi Z − ∆t T (p) = dpe h¯ φ(p) hq|pi Z = dpφ1(p) hq|pi
−1 =F [φ1(p)].
So, the propagation of a wave function can be expressed on position as,
−1 h − ∆t T (p) h − ∆t V (q) ii ψ(q, ∆t) = F e h¯ F e h¯ ψ(q, 0) . (E-13)
This is not the only way to do the splitting of the time evolution operator, in fact, we can find an expansion with an error O((∆t)3) such that,
− ∆t V (q) −1 h − ∆t T (p) h − ∆t V (q) ii ψ(q, ∆t) = e 2¯h F e h¯ F e 2¯h ψ(q, 0) . (E-14)
E.1.1 Algorithm
The procedure described below is for a single time step ∆t.
1. Apply half of the potential propagation:
− ∆t V (q) ψ(q) −→ e 2¯h ψ(q). (E-15a)
2. Change to the momentum representation via a Fourier transform:
φ(p) −→ F [ψ(q)] . (E-15b)
3. Propagate the complete kinetic part of the Hamiltonian:
− ∆t T (p) φ(p) −→ e h¯ φ(p). (E-15c)
4. Get back to the position representation by applying the inverse Fourier transform :
ψ(q) −→ F −1 [φ(p)] . (E-15d)
5. Calculate the other half of the potential:
− ∆t V (q) ψ(q) −→ e 2¯h ψ(q). (E-15e) 88 E Quantum Propagation
E.2 Winger Function Dynamics
Exactly the same ideas can be used to find an algorithm to evolve the Wigner function[Cabrera et al., 2015]. Starting from the definition of the Wigner Function (1-3), Z W (q, p) = dye−ipy/¯h hq + y/2| ρˆ|q − y/2i Z = dye−ipy/¯hψ∗(q + y/2)ψ(q − y/2),
and its dynamical equation
∂tW (q, p) = {H(q, p),W (q, p)}M .
Which explicitly goes,
2 h¯ ←−−→ ←−−→ ∂ W (q, p, t) = − H(q, p) sin ∂ ∂ − ∂ ∂ W (q, p, t) t h¯ 2i p q q p ←−−→ ←−−→ ←−−→ ←−−→ 1 h¯ ∂ ∂ −∂ ∂ h¯ −∂ ∂ +∂ ∂ = H(q, p) e 2i ( p q q p) − e 2i ( p q q p) W (q, p, t). ih¯ The result of applying the operator is
1 ih∂¯ ih∂¯ ih∂¯ ih∂¯ ∂ W (q, p, t) = H q + p , p − q − H q − p , p + q W (q, p, t). t ih¯ 2 2 2 2
Hence, the time evolution if the Wigner function can be rewritten as
ih∂¯ tW (q, p, t) = G(q, p, ∂q, ∂p)W (q, p, t), (E-16)
where G(q, p, ∂q, ∂p) is the generator of the dynamics for the Wigner function,
ih∂¯ ih∂¯ ih∂¯ ih∂¯ G(q, p, ∂ , ∂ ) = H q + p , p − q − H q − p , p + q . (E-17) q p 2 2 2 2
At this point is convenient to identify four operators which define the Wigner function dynamics,
qˆ = q;p ˆ = p;π ˆp = i∂p;π ˆq = i∂q. (E-18)
Different from the previous case (the evolution of the state) where only two were needed. Those operators satisfy the following commutation rules,
[ˆq, pˆ] = 0; [ˆq, πˆq] = i; [ˆp, πˆp] = i; [ˆπq, πˆp] = 0 (E-19)
This rules show the possible spaces where the Wigner function can be expressed, like only two are needed, we must use those that are independent if they don’t they are conjugated variables. E.2 Winger Function Dynamics 89
A representation of the possible space where the Wigner function can be expressed is shown in the following figure,
Fp−→πp (q, p) (q, πp)
−1 Fπq−→q Fq−→πq
(πq, p) (πq, πp) −1 Fπp−→p
Figure E-1: Relationship of the possible spaces where the Wigner function can be expressed (E-19).
In other words, in the same way that states can be expressed on position or momentum representation, the Wigner function can be written on any of the four spaces of the figure.
To implement those relations to calculate the propagation, we can think of a kind of time evolution operator as
W (q, p, t) = UW (t)W (q, t, 0), (E-20)
Such that, if we replace into (E-16) the time evolution for the Wigner function is given by, ih∂¯ tUW (t) = GUW (t), (E-21)
So it looks like the Wigner function satisfies a Schr¨odingerlike equation but with a different generator of the dynamics G instead of H, so, the solution is
i U (t) = exp − Gt . (E-22) W h¯
pˆ2 As we already considered Hamiltonians of the form H(q, p) = 2m + V (ˆq) the generator leads
h¯ hπ¯ hπ¯ G = pπˆ + V qˆ − p − V qˆ + p . (E-23) m q 2 2 | {z } | {z } Momentum Position 90 E Quantum Propagation
In the same spirit as the previous case, we ca consider the evolution over a small time ∆t, getting
h i i∆t i∆t hπ¯ p hπ¯ p −pπq − h¯ V q− 2 −V q+ 2 UW (∆t) ≈ e m e . (E-24)
So that, the time evolution for the Wigner function is,
−1 −1 W (x, p, t) = Fq P Fp [Fq (QFp[W (x, p, 0)])] , (E-25)
with
− i∆t pπ P = e m q , (E-26a)
and
h hπ¯ hπ¯ i − i∆t V q− p −V q+ p Q = e h¯ 2 2 . (E-26b)
Exactly the same way as we mentioned before, it is possible to find an expression util order O((∆t)3) and the result is taking the expansion as:
h i h i i∆t hπ¯ p hπ¯ p i∆t i∆t hπ¯ p hπ¯ p − 2¯h V q− 2 −V q+ 2 −pπq − 2¯h V q− 2 −V q+ 2 UW (∆t) ≈ e e m e (E-27)
So, the Wigner propagation is calculated as h n h ioi −1 ˜ −1 −1 ˜ W (q, p, t) = Fp QFpFq P Fp Fq QFp[W (q, p, 0)] , (E-28)
with P as before, but,
h hπ¯ hπ¯ i − i∆t V q− p −V q+ p Q˜ = e 2¯h 2 2 . (E-29)
E.2.1 Algorithm
To propagate the Wigner function over a time ∆t.
1. Calculate the Fourier transform (In momentum p):
W (q, πp) −→ Fp[W (q, p)]. (E-30a)
2. Apply half of the time evolution correspondent to position:
˜ W (q, πp) −→ QW (q, πp) (E-30b) E.2 Winger Function Dynamics 91
3. Calculate the Fourier transform in the two directions to get to(πq, p):
W (πq, πp) −→ Fq[W (q, πp)], −1 (E-30c) W (πq, p) −→ Fp [W (πq, πp)].
4. Calculate the kinetic part of the evolution:
W (πq, p) −→ PW (πq, p). (E-30d)
5. Two Fourier transforms are made to get into the space (q, πp):
−1 W (q, p) −→ Fq [W (πq, p)], −1 (E-30e) W (q, πp) −→ Fp [W (q, p)].
6. Calculate the other half of the position dependent part of the evolution:
˜ W (q, πp) −→ QW (q, πp). (E-30f)
7. Lastly get back to (q, p):
−1 W (q, p) −→ Fp [W (q, πp)]. (E-30g) Bibliography
[Andrea Bonfiglioli, 2012] Andrea Bonfiglioli, R. F. a. (2012). Topics in Noncommutative Alge- bra: The Theorem of Campbell, Baker, Hausdorff and Dynkin. Lecture Notes in Mathematics 2034. Springer-Verlag Berlin Heidelberg, 1 edition. E.1
[Arnold et al., 2013] Arnold, V., Vogtmann, K., and Weinstein, A. (2013). Mathematical Meth- ods of Classical Mechanics. Graduate Texts in Mathematics. Springer New York. 0, D.2
[Ballentine, 1998] Ballentine, L. (1998). Quantum Mechanics: A Modern Development. World Scientific.
[Berry and Mount, 1972] Berry, M. V. and Mount, K. E. (1972). Semiclassical approximations in wave mechanics. Reports on Progress in Physics, 35(1):315–397. 1.2
[Burden and Faires, 2010] Burden, R. and Faires, J. (2010). Numerical Analysis. Cengage Learn- ing. D.3
[Cabrera et al., 2015] Cabrera, R., Bondar, D. I., Jacobs, K., and Rabitz, H. A. (2015). Efficient method to generate time evolution of the wigner function for open quantum systems. Phys. Rev. A, 92:042122. E.2
[Dahl and Springborg, 1988] Dahl, J. P. and Springborg, M. (1988). The morse oscillator in posi- tion space, momentum space, and phase space. The Journal of Chemical Physics, 88(7):4535– 4547. 3, 3.1
[Daniela, 2005] Daniela, D. (2005). Applications of the wigner distribution function in signal processing. EURASIP Journal on Advances in Signal Processing, 10. 0
[Dirac and Fowler, 1927] Dirac, P. A. M. and Fowler, R. H. (1927). The physical interpretation of the quantum dynamics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 113(765):621–641. 0
[Dittrich et al., 2010] Dittrich, T., G´omez,E., and Pachon, L. (2010). Semiclassical propagation of wigner functions. The Journal of chemical physics, 132:214102. 0, 1.2.1, 1.2.1, 1.2.1, 2, 2.1 BIBLIOGRAPHY 93
[Dittrich et al., 2006] Dittrich, T., Viviescas, C., and Sandoval, L. (2006). Semiclassical propa- gator of the wigner function. Phys. Rev. Lett., 96:070403. (document), 0, 1.1.1, 1.2.1, 1.2.1, 1.2.1, 1.2.2, 2, 2.1, 3.2
[Domitrz et al., 2013] Domitrz, W., Manoel, M., and de M. Rios, P. (2013). The wigner caustic on shell and singularities of odd functions. Journal of Geometry and Physics, 71:58 – 72. 3.2.4
[Domitrz and Zwierzy´nski,2020] Domitrz, W. and Zwierzy´nski,M. (2020). Singular points of the wigner caustic and affine equidistants of planar curves. Bulletin of the Brazilian Mathematical Society, New Series, 51:11 – 26. 3.2.4
[Feynman, 1966] Feynman, R. (1966). The Feynman Lectures on Physics: Quantum mechanics. Number v. 3. 0
[Feynman, 1948] Feynman, R. P. (1948). Space-time approach to non-relativistic quantum me- chanics. Rev. Mod. Phys., 20:367–387. 0
[Forest, 2006] Forest, E. (2006). Geometric integration for particle accelerators. Journal of Physics A: Mathematical and General, 39(19):5321. D.3
[Frank et al., 2000] Frank, A., Rivera, A. L., and Wolf, K. B. (2000). Wigner function of morse potential eigenstates. Phys. Rev. A, 61:054102. 3.1
[Glauber, 1963] Glauber, R. J. (1963). Coherent and incoherent states of the radiation field. Phys. Rev., 131:2766–2788. 1
[Goldstein et al., 2002] Goldstein, H., Poole, C., and Safko, J. (2002). Classical Mechanics. Addison Wesley. D.1, D.2
[G´omez,2010] G´omez,E. A. (2010). Aplicaciones al propagador semicl´asicode la funci´onde wigner / applications to the semiclassical propagator of the wigner function. Doctorado en f´ısica. 3.2
[Gray et al., 1994] Gray, S. K., Noid, D. W., and Sumpter, B. G. (1994). Symplectic integrators for large scale molecular dynamics simulations: A comparison of several explicit methods. The Journal of Chemical Physics, 101(5):4062–4072. D-1
[Gutzwiller, 1967] Gutzwiller, M. C. (1967). Phase-integral approximation in momentum space and the bound states of an atom. Journal of Mathematical Physics, 8(10):1979–2000. 0, 1.2
[Gutzwiller, 1969] Gutzwiller, M. C. (1969). Phase-integral approximation in momentum space and the bound states of an atom. ii. Journal of Mathematical Physics, 10(6):1004–1020. 0, 1.2
[Gutzwiller, 1970] Gutzwiller, M. C. (1970). Energy spectrum according to classical mechanics. Journal of Mathematical Physics, 11(6):1791–1806. 1.2 94 BIBLIOGRAPHY
[Gutzwiller, 1971] Gutzwiller, M. C. (1971). Periodic orbits and classical quantization conditions. Journal of Mathematical Physics, 12(3):343–358. 2.5.2
[Gutzwiller, 1992] Gutzwiller, M. C. (1992). Chaos in Classical and Quantum Mechanics. Springer-Verlag New York. 2.5.2
[Hairer et al., 2008] Hairer, E., Nørsett, S., and Wanner, G. (2008). Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics. Springer Berlin Heidelberg. D.3
[Heller, 1981] Heller, E. (1981). Frozen gaussians: A very simple semiclassical approximation. The Journal of Chemical Physics, 75(6):2923–2931. cited By 581. 0
[Herman and Kluk, 1984] Herman, M. F. and Kluk, E. (1984). A semiclasical justification for the use of non-spreading wavepackets in dynamics calculations. Chemical Physics, 91(1):27 – 34. 0
[Hillery et al., 1997] Hillery, M., O’Connell, R. F., Scully, M. O., and Wigner, E. P. (1997). Distribution Functions in Physics: Fundamentals, pages 273–317. Springer Berlin Heidelberg, Berlin, Heidelberg. 1, 1.1
[Hudson, 1974] Hudson, R. (1974). When is the wigner quasi-probability density non-negative? Reports on Mathematical Physics, 6(2):249 – 252. 1.1, A
[Husimi, 1940] Husimi, K. (1940). Some formal properties of the density matrix. Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, 22(4):264–314. 1
[Jaubert and de Aguiar, 2007] Jaubert, L. D. C. and de Aguiar, M. A. M. (2007). Semiclassical tunnelling of wavepackets with real trajectories. Physica Scripta, 75(3):363–373. 1.2
[Kay, 2013] Kay, K. G. (2013). Time-dependent semiclassical tunneling through barriers. Phys. Rev. A, 88:012122. 1.2
[Keller, 1985] Keller, J. B. (1985). Semiclassical mechanics. SIAM Review, 27(4):485–504. 1.2
[Koda, 2015] Koda, S.-i. (2015). Initial-value semiclassical propagators for the wigner phase space representation: Formulation based on the interpretation of the moyal equation as a schr¨odingerequation. The Journal of Chemical Physics, 143(24):244110. 0, 2.1, 3.6.3, 4
[Landau and E.M., 1977] Landau, L. and E.M., L. (1977). Quantum Mechanics: Non-relativistic Theory. Butterworth-Heinemann. Butterworth-Heinemann. 0
[Leonhardt et al., 1997] Leonhardt, U., Knight, P., and Miller, A. (1997). Measuring the Quan- tum State of Light. Cambridge Studies in Modern Optics. Cambridge University Press. 1 BIBLIOGRAPHY 95
[Littlejohn, 1992] Littlejohn, R. G. (1992). The van vleck formula, maslov theory, and phase space geometry. Journal of Statistical Physics, 68(1-2):7–50. 1.2, C
[Maslov and Fedoriuk, 1981] Maslov, V. and Fedoriuk, M. (1981). Semi-Classical Approximation in Quantum Mechanics. Mathematical Physics and Applied Mathematics. Dordrecht. 0, 1.2, 2.2, 3.2, 3.2.2, C, C
[McLachlan and Atela, 1992] McLachlan, R. I. and Atela, P. (1992). The accuracy of symplectic integrators. Nonlinearity, 5(2):541. D.3
[Morse, 1929] Morse, P. M. (1929). Diatomic molecules according to the wave mechanics. ii. vibrational levels. Phys. Rev., 34:57–64. 3, 3.1
[Moyal, 1949] Moyal, J. E. (1949). Quantum mechanics as a statistical theory. Mathematical Proceedings of the Cambridge Philosophical Society, 45(1):99–124.
[O’Connell and Wigner, 1981a] O’Connell, R. and Wigner, E. (1981a). Quantum-mechanical distribution functions: Conditions for uniqueness. Physics Letters A, 83(4):145 – 148.
[O’Connell and Wigner, 1981b] O’Connell, R. and Wigner, E. (1981b). Some properties of a non-negative quantum-mechanical distribution function. Physics Letters A, 85(3):121 – 126. 1
[Ozorio de Almeida, 2009] Ozorio de Almeida, A. (2009). Entanglement in Phase Space, pages 157–219. Springer Berlin Heidelberg, Berlin, Heidelberg. 1.2, C
[Ozorio de Almeida et al., 2013] Ozorio de Almeida, A., Vallejos, R., and Zambrano, E. (2013). Initial or final values for semiclassical evolutions in the weyl–wigner representation. Journal of Physics A: Mathematical and Theoretical, 46:135304. 0, 1.2.2, 1.2.2, 1.2.2, 2.1, 2.3
[Ozorio de Almeida, 1989] Ozorio de Almeida, A. M. (1989). Hamiltonian Systems: Chaos and Quantization. Cambridge Monographs on Mathematical Physics. Cambridge University Press. 1.1, C
[Ozorio de Almeida, 1998] Ozorio de Almeida, A. M. (1998). The weyl representation in classical and quantum mechanics. Physics Reports, 295(6):265 – 342. 1, 1.2.2
[Ozorio de Almeida and Brodier, 2006] Ozorio de Almeida, A. M. and Brodier, O. (2006). Phase space propagators for quantum operators. Annals of Physics, 321(8):1790 – 1813.
[Ozorio de Almeida et al., 2019] Ozorio de Almeida, A. M., Lando, G. M., Vallejos, R. O., and Ingold, G.-L. (2019). Quantum revival patterns from classical phase-space trajectories. Phys. Rev. A, 99:042125. 0, 1.2.2, 2.1, 3.2 96 BIBLIOGRAPHY
[P. Schleich, 2001] P. Schleich, W. (2001). Quantum optics in phase space. Quantum Optics in Phase Space, by Wolfgang P. Schleich, pp. 716. ISBN 3-527-29435-X. Wiley-VCH , April 2001. 0, 1, 1.1, B.1
[Pach´on,2010] Pach´on,L. A. (2010). Coherencia y decoherencia en la propagaci´onsemicl´asica de la funci´onde wigner / coherence and decoherence in the semiclassical propagation of the wigner function. Doctor en ciencias-f´ısica. 3.2
[Press, 2007] Press, W. (2007). Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cambridge University Press. D.3
[Sakurai and Napolitano, 2011] Sakurai, J. and Napolitano, J. (2011). Modern Quantum Me- chanics. Addison-Wesley. 1
[Van Vleck, 1928] Van Vleck, J. (1928). The correspondence principle in the statistical interpre- tation of quantum mechanics. Proceedings of the National Academy of Sciences of the United States of America, 14(2):178–188. 0, 1.2
[Villalba, 2017] Villalba, O.´ E. R. (2017). Semiclassical approximation to the propagator of the wigner function for particles in confined spaces. Mag´ısteren Ciencias - F´ısica;. 3.2
[Weinbub and Ferry, 2018] Weinbub, J. and Ferry, D. K. (2018). Recent advances in wigner function approaches. Applied Physics Reviews, 5(4):041104.
[Weyl, 1927] Weyl, H. (1927). Quantenmechanik und gruppentheorie. Zeitschrift f¨urPhysik, 46(1):1–46. B.2
[Wigner, 1932] Wigner, E. (1932). On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749–759. 0, 1
[Yoshida, 1990] Yoshida, H. (1990). Construction of higher order symplectic integrators. Physics Letters A, 150(5):262 – 268. D.3
[Zachos et al., 2005] Zachos, C., Fairlie, D., and Curtright, T. (2005). Quantum Mechanics in Phase Space: An Overview with Selected Papers. World Scientific series in 20th century physics. World Scientific. 0