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The Normalized Matrix

Perturbation Method

by

Braulio Misael Villegas Martínez

A dissertation

submitted to the Program in Optics,

Optics Department, in partial fulfillment of

the requirements for the

PhD Degree in Optics

at the

National Institute for Astrophysics, Optics and

Electronics

January 2020

Tonantzintla, Puebla

Advisors:

Dr. Héctor Manuel Moya Cessa,

Dr. Francisco Soto Eguibar

ii

Abstract

In 1926, Schrodinger supplied the theoretical basis for analyzing the time evolutionof a quantum system through his famous equation that now bears his name. Soonafter its advent, an intrinsic interest emerged to get exactly solvable models by usingthe Schrodinger equation. However, only a very small number of problems fulfil thisspecific requirement. The vast majority of physical systems are rather complicatedto be treated exactly; one may try to find approximate solutions with the aid ofperturbation methodsAmong all available perturbative recipes, the Matrix Method inmediately stands out.Such as its name suggests, this scheme is devoted to seek perturbative solutions oftime-dependent Schrodinger equation, ordered in power series of tridiagonal matri-ces. These solutions possess time-dependent factors which enable us to determinethe temporal evolution of corrections. Surprisingly in this scheme, the existence ofnormalized solutions are completely ignored and neglected.This thesis present the complement theoretical analysis of Matrix Method startedin reference [18]. In this case, we focus in determine the general expression for thenormalization constant, which in principle will allow us to obtain perturbative nor-malized solutions to any order correction. Henceforth, we shall refer to the completeapproach as the Normalized Matrix perturbation Method. Further, we show thatthe treatment adopted to get the normalization constant is quite different from theusual intermediate normalization procedure used in the standard time-independentperturbation theory. In order to assess the efficacy and advantages of our approach,five examples, were analyzed leading to meaningful results in their approximative so-lutions and in fairly good agreement with their known exact or numerical solutions.

iii

iv

Resumen

En 1926, Schrodinger proporciona las bases teoricas para analizar la evolucion tempo-ral de un sistema cuantico a traves de su famosa ecuacion que ahora lleva su nombre.Poco despues de su advenimiento, surge un interes intrınseco para obtener modelosexactamente solubles mediante la ecuacion de Schro ecuacion dinger. Sin embargo,solo un numero muy pequeno de problemas relacionados con esta ecuacion cumplencon este requisito. Debido que la mayorıa de los sistemas fısicos son bastante com-plicados de tratar de forma exacta, uno debe encontrar soluciones aproximadas conla ayuda de los metodos de perturbacion.Entre todas las recetas perturbativas disponibles, una que destaca de inmediato es elMetodo Matrix. Tal como sugiere su nombre, este esquema esta dedicado a buscarsoluciones perturbativas de la ecuacion de Schrodinger dependiente del tiempo, orde-nadas en series de potencia de matrices tridiagonales. Estas soluciones, por supuesto,poseen factores dependientes del tiempo que nos permiten determinar la evoluciontemporal de las correcciones. Sorprendentemente, en este esquema, la existencia desoluciones normalizadas son completamente ignoradas y descuidadas.En esta tesis, presentamos el analisis teorico complementario del Metodo Matrix ini-ciado en [18]. En este caso, nos enfocamos en determinar la expresion general parala constante de normalizacion, que en principio nos permitira obtener solucionesperturbadas normalizadas para cualquier orden de correccion. En lo sucesivo, nosreferiremos al enfoque completo como el Metodo Matricial de Perturbacion Normal-izada. Ademas, mostramos que el tratamiento adoptado para obtener la constantede normalizacion es muy diferente al procedimiento de normalizacion intermedia uti-lizado en la teorıa de perturbaciones independiente del tiempo. Con el fin de evaluarla eficacia y las ventajas de nuestro enfoque, se analizaron cinco ejemplos que con-dujeron a resultados significativos en sus soluciones aproximadas y en muy buenacuerdo con sus soluciones exactas o numericas conocidas.

v

vi

Acknowledgements

I dedicate this dissertation to my mother Balbina. Who made uncountable sacrificesso I would have access to an affordable quality of life and education since my earlyyears. I thank her also for instilling a strong sense of personal responsibility, gen-erosity to help those less fortunate and the taste for hard work and good education.But, above all, for her always present patience and moral support at the variousstages of my life, in particular for the encouragement and inspiration to pursue mydreams. You will always be my hero, thanks for trusting and never giving up on me.

I would like to express my heartfelt gratitude to my advisors Dr. Hector ManuelMoya Cessa and Dr. Francisco Soto Eguibar for the time, support, encouragementand continuous guidance that they gave me during these past six years. I alwayswill forever be grateful to my research advisor, Dr. Francisco Soto Eguibar. Hisinsightful discussions, suggestions and constant feedback about the research was ofvery guidance help to pave a way to finished this thesis.

I also express my gratitude to the National Council on Science and Technology(CONACYT), for its financial support throughout my PhD studies.

vii

viii

Contents

Contents

Abstract iii

Resumen v

Acknowledgements vii

1 Introduction 1

2 Perturbation Methods 32.1 The Rayleigh-Schrodinger perturbation theory . . . . . . . . . . . . . 32.2 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . 6

3 The Normalized Matrix Perturbation Method 93.1 The Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Normalization constant . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Examples 214.1 Harmonic Oscillator with linear term in potential. . . . . . . . . . . . 21

4.1.1 Exact solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.1.2 Perturbative solution . . . . . . . . . . . . . . . . . . . . . . . 254.1.3 Comparison of the exact and the perturbative solutions . . . . 30

4.2 The cubic anharmonic oscillator . . . . . . . . . . . . . . . . . . . . . 344.3 Repulsive harmonic oscillator. . . . . . . . . . . . . . . . . . . . . . . 42

4.3.1 Linear anharmonic repulsive oscillator . . . . . . . . . . . . . 494.3.2 Exact solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.3 Perturbative solution . . . . . . . . . . . . . . . . . . . . . . . 534.3.4 Comparison of the exact and the perturbative solutions . . . . 55

4.4 The Binary waveguide array . . . . . . . . . . . . . . . . . . . . . . . 58

5 Conclusions 67

ix

Contents

x

List of Figures

List of Figures

4.1 Probability density with quantum number n = 10. The left-hand side(a) shows the exact solution while the right-hand side (b) presentsthe approximate one. A contour plot of their probability densities isdepicted on (c), where it shows that for short times the approximatesolution (red line) reproduces the exact one (black line) with highlyaccuracy whereas for t > 1.5 a very slight differences between themcan be detected, but, it still yields to good approximation. Thesegraphs were obtained by considering a perturbation strength equal toλ = 0.1 and a frequency of ω = 1. . . . . . . . . . . . . . . . . . . . . 30

4.2 Probability density |ψ(x, t)|2 with α(0) = 3 and ω = 2. The left-handside (a) shows the exact solution while the right-hand side (b) presentsthe approximate one. A contour plot of their probability densities on(x, t) plane is depicted on (c), where it shows that the approximationpresents a remarkable high accuracy with the exact result under theconsideration of a perturbation strength equal to λ = 0.5. . . . . . . . 31

4.3 Probability density with quantum number n = 20 at (a) t = 0.5 and(b) t = 1.5. The approximation given by PApro is highly accurateand compares favorably with the exact result, PExa, these solutionsare get in the strong-coupling perturbation regime with a perturbedcoefficient value of λ = 35. . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4 Probability density of the cubic anharmonic oscillator versus x fora quantum number n = 20. One can notice from (a) that withλ = 0.005, the first-order solution, PApro, does not differs substan-tially from numerical result, PNu, unlike to PRS whose behavior showsseveral discrepance. Once increases the perturbation parameter to (b)λ = 0.01, PApro as well PRS become completely inadequate to describethe numerical solution, being the differences more notorious for PRS.The numerical result was performed with t = 1 and ω = 1. . . . . . . 37

xi

List of Figures

4.5 Density probability of cubic anharmonic oscillator for a coherent state(a) and (b) a Schrodinger cat state. Notice in (a) that PApro gives asolution very close to PNu while PRS remains to far of reproduce it.This become more evident for the interference pattern generated bythe Schrodinger cat state in (b), where PRS fails completely when itcompared with PNu; on the contrary, PApro yields to reliable resultwith PNu. The above graphs were carried by using the parametersλ = 0.005, ω = 1, α = 2 + 3i and θ = π/7. . . . . . . . . . . . . . . . 41

4.6 Probability density of the repulsive harmonic oscillator with (a) acoherent state and (b) a Schrodinger cat state as initial states. Inboth cases the solutions present a parabolic behaviour; the values ofthe parameters are β = 6, ω = 1 and φ = π/5. Figures (c) and (d)depict the probability density distribution on the (x, t) plane for thesame initial states. The solid, dotted and dashed lines represent thecases of β=2, 5 and 9, respectively. . . . . . . . . . . . . . . . . . . . 48

4.7 Probability density of the linear anharmonic repulsive oscillator in the(x, t) plane when the initial state is a coherent state. The black lineand the red dashed line indicate the exact and perturbative solutions.Graphs (a) and (b) show how the solutions behave for β = 1 at twodifferent values of the perturbative parameter λ. For λ = 0.1, thesecond-order perturbative solution presents a remarkable high accu-racy with the exact result up to t = 3. After that time, a small butsignificantly discrepancy between them appear. For λ = 0.4, the ac-curacy of perturbative result is reduced over shorter time. Graphs(c) and (d) display the same time-range of convergence between thesolutions for β = 8 and ω = 1. . . . . . . . . . . . . . . . . . . . . . . 56

4.8 Density probability of the linear anharmonic repulsive oscillator in the(x, t) plane for an Schrodinger-cat state as initial state with ω = 1 andφ = π/7. The black line and the red dashed line indicate the exact andperturbative solutions. Graphs (a) and (b) show how the solutionsbehave for β = 1 at two different values of perturbative parameterλ. For λ = 0.1, the second-order perturbative solution presents aremarkable high accuracy with the exact result. Once increases theperturbation parameter to λ = 0.4, the accuracy of perturbative resultis completely reduced. The two remaining graphs (c) and (d) displaysame features where a small but significantly discrepancy betweensolutions appear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.9 Field intensity versus propagation distance z using the exact solution(solid line), the third-order solution (red dashed line) and the smallrotation method solution (blue dashed line), with α = 0.1 and ω = 0.9,for the first three guides. . . . . . . . . . . . . . . . . . . . . . . . . . 64

xii

List of Figures

4.10 Field intensity versus propagation distance z using the exact solution(solid line), third-order solution (red dashed line) and the small rota-tion method solution (blue dashed line), with α = 0.3 and ω = 0.9,for the first three guides. . . . . . . . . . . . . . . . . . . . . . . . . . 65

xiii

List of Figures

xiv

Chapter 1

Introduction

The main equation in non-relativistic quantum mechanics is the Schrodinger equa-tion, because its solution, the wavefunction, contains all the relevant informationabout the behavior of a physical quantum system [1–6]. Since its introduction theSchrodinger equation has been widely studied over several decades going to presentday one of the most important cornerstones of modern physics, but despite all ef-forts that have been made in this equation, the successful physical systems whereit has an exact solution are limited; the infinite well, the harmonic oscillator, thehydrogen atom [2–6] and the Morse potential [7] are typical set of potentials wherean exact analytical solution is known. A considerable number of problems involvedwith the Schrodinger equation are often complex and cannot be solved exactly, thenone is thus forced to analyse them by the use of perturbative methods [3–6, 8, 9] ornumerical techniques [10–13]; when the approximation methods are correctly used,give us a very good understanding behavior of the phenomenon under study.

Time independent perturbation theory, also known as Rayleigh-Schrodinger per-turbation theory, has its roots in the works of Rayleigh and Schrodinger, but themathematical foundations were only set by Rellich in the late thirties of the pastcentury (see Simon [14] and references there in). This method has been applied withgreat success to solve a vast variety of problems such that, through its continuousimplementation, a lot of techniques have been developed, which go from numericalmethods [15] to those more mathematical and fundamental, as convergence prob-lems [16,17].In this thesis an alternative perturbative approach, called the Matrix Method [18–21],is studied and complemented. This new scheme, based on the implementation oftriangular matrices, allows to solve approximately the time-dependent Schrodingerequation in an elegant and simple manner. The method has demonstrated that thecorrections to the wavefunction and the energy can be contained in only one expres-sion, unlike the standard perturbation theory where it is needed to calculate themin separated ways [18–21]. Moreover, the Matrix Method may also be used when

1

Chapter 1. Introduction

one may find a unitary evolution operator for the unperturbed Hamiltonian but itis not possible to find its eigenstates. Further, its approximated solutions not onlypresent conventional stationary terms, but also time dependent factors which allowsus to know the temporal evolution of the corrections. A remarkable feature is thatthe general expression to compute them does not distinguish if the Hamiltonian isdegenerate or not [18–21]. Besides, the formalism offers an alternative to express theDyson series in a matrix form. Another very important feature is that the MatrixMethod can be extended to the Lindblad master equation [20]. Herein, the MatrixMethod possess many attractive characteristics that cannot be found in the conven-tional treatments of the perturbation theory. It is also worth noting that our maingoal over the Matrix Method is to extend and provide normalized approximated so-lutions in each step, at difference with the standard perturbation theory, where thesolutions are not normalized.The reminder of this thesis is organized as follows:Chapter 2 provides a brief overview relative to the theoretical framework of thetime-independent and time-dependent perturbation theories and emphasizes the ap-propriate procedure to get the perturbative order corrections in both cases. As iswell known for the time-independent formalism, two expressions are obtained ateach order, one for the energies and one for the wave functions. Meanwhile, thetime-dependent perturbation method gives rise to a recursive formula to computehigher-order corrections. Immediately after these procedure are presented we jumpssubsequently to the mathematical background of the Matrix method. Such approachis presented in Chapter 3 and will allow us to obtain a single solutions that containsboth, energy and wave function corrections at same time. These correction to thesolution are calculated recursively permit generate the Dyson series, and in this wayobtain a new expression of it, but now in terms of a matrix series. Afterwards,we introduce the mathematical derivation to obtain the general expression to com-pute the normalization constant to any given order correction. Such development ispredicated on the isolation of a multiplicative time-normalized factor in the generalperturbative solution, ensuring that the perturbed solutions of the wavefunction areproperly normalized for all t. Five models, which of majority an exact solution ex-ists: harmonic oscillator perturbed by linear potential, cubic anharmonic oscillator,repulsive harmonic oscillator, repulsive linear anharmonic oscillator and a binarywaveguide array, are discussed in detail with the Normalized Perturbative Methodin Chapter 4; there the main goal is to show the potential that our approach offersversus others like than the standard time independent perturbation theory. At thesame time, we prove that the perturbation analysis used here works rather well forweak perturbations but also for its strong counterpart. Finally, chapter 5 is dedicatedto the conclusions.

2

Chapter 2

Perturbation Methods

2.1 The Rayleigh-Schrodinger perturbation the-

ory

It is well known that a great majority of systems in nature cannot be solved exactlyand we force to develop approximation analytic methods to deal with them. Thechief among these approaches is the Rayleigh-Schrodinger perturbation theory whichis appropriate when we have a time independent Hamiltonian, that may be brokendown in two pieces, as follows

H = H0 + λHp, (2.1)

where H0 is called non-perturbed Hamiltonian, and it is usually assumed that incor-porates the dominant effects since possess known solutions, i.e. its eigenvalues andeigenfunctions

H0|ψ(0)〉 = E(0)n |ψ(0)〉, (2.2)

are known. By other hand, the second part of (2.1) contains a term Hp which isnot amenable to an analytical solution. However, this term can be modelled assmall disturbance to bring us perturbative solutions of the energy spectrum and inthe eigenfunctions of the full Hamiltonian H, which in principle, it does not dif-fers significantly from those of the soluble part, H0. To keep track of the “size” ofterms that appear in the development of this method, it is usual to associate the“perturbation”, Hp, with a dimensionless strength parameter λ, which is consideredvery small compared to unity, this mean, λ 1. This perturbative parameter, as itsname suggests, typically represents, for instance, the coupling-strength of interactionbetween the part which is soluble and the other which is not or even is introduced asan auxiliary expansion parameter to measure “the size” of corrections. In general,λ represents any parameter, on which the full-system Hamiltonian H depends. Bytranslating the above into a mathematical language, it is well-known that the stan-dard perturbation theory [3–6, 9] produces the following perturbed expressions for

3

Chapter 2. Perturbation Methods

the eigenvaluesEn = E(0)

n + λE(1)n + λ2E(2)

n + · · · , (2.3)

and for the eigenfunctions

|ψn〉 = |ψ(0)n 〉+ λ|ψ(1)

n 〉+ λ2|ψ(2)n 〉+ · · · , (2.4)

where the super index (j) indicates the correction order, and so E(0)n and |ψ(0)〉 are

the eigenvalues and eigenfunctions of the unperturbed Hamiltonian H0. The mainproblem of perturbation theory is to solve the eigenvalue equation given by

H |ψn〉 =(H0 + λHp

)|ψn〉 = En |ψn〉 , (2.5)

if we substitute the series expansions of En and |ψn〉 into Eq.(2.5), and the coefficientsof like powers to λ on each side of equation are set equal to each other, we arrive atthe following set of equations

H0

∣∣ψ(0)n

⟩=E(0)

n

∣∣ψ(0)n

⟩H0

∣∣ψ(1)n

⟩+ Hp

∣∣ψ(0)n

⟩=E(0)

n

∣∣ψ(1)n

⟩+ E(1)

n

∣∣ψ(0)n

⟩H0

∣∣ψ(2)n

⟩+ Hp

∣∣ψ(1)n

⟩=E(0)

n

∣∣ψ(2)n

⟩+ E(1)

n

∣∣ψ(1)n

⟩+ E(2)

n

∣∣ψ(0)n

⟩... (2.6)

Multiplying the above equations by⟨ψ

(0)n

∣∣∣ and using the fact that eigenfunction of

unperturbed Hamiltonian operates over its left as⟨ψ

(0)n

∣∣∣ H0 =⟨ψ

(0)n

∣∣∣E(0)n , due to

the hermiticity property and from the normalization condition of the unperturbed

eigenfunctions,⟨ψ

(0)n

∣∣∣ψ(0)n

⟩= 1, it follows that

E(0)n =

⟨ψ(0)n

∣∣ H0

∣∣ψ(0)n

⟩E(1)n =

⟨ψ(0)n

∣∣ Hp

∣∣ψ(0)n

⟩E(2)n =

⟨ψ(0)n

∣∣ Hp

∣∣ψ(1)n

⟩... (2.7)

It is clear to see that the first-order correction to the energy, E(1)n , is simply the expec-

tation value of the perturbation with respect to the unperturbed state. Meanwhile,

the second-order of energy requires know the first-order of wavefunction,∣∣∣ψ(1)

n

⟩. So

to proceed of its derivation, we use the identity operator in terms of the completeorthonormal set eigenfunctions of the unperturbed Hamiltonian as∣∣ψ(1)

n

⟩= I

∣∣ψ(1)n

⟩=∑k

∣∣∣ψ(0)k

⟩⟨ψ

(0)k

∣∣∣ψ(1)n

⟩, (2.8)

4

2.1. The Rayleigh-Schrodinger perturbation theory

where the inner product⟨ψ

(0)k

∣∣∣ψ(1)n

⟩can be obtained by multiplying both sides of

the second expression of Eq.(2.6) by⟨ψ

(0)k

∣∣∣ to give

⟨ψ

(0)k

∣∣∣ψ(1)n

⟩=

⟨ψ

(0)k

∣∣∣ Hp

∣∣∣ψ(0)n

⟩E

(0)n − E(0)

k

. (2.9)

Prior to the substitution of previous expression into Eq.(2.8), it is helpful to clar-ify a point which is basic to provide further understanding about the normaliza-tion procedure between the Rayleigh-Schrodinger perturbation theory and from theMatrix method, which it will be explained in next chapter. Reader can see intoEq.(2.9) that the denominator contains an average energy difference which is safe

when pulled out the n = k term from the beginning; clearly, if E(0)n = E

(0)k Eq.(2.9)

diverges. However, this condition is not necessarily true in (2.8) where the n-th term

in the sum can arise. To deal with this restriction, we must assume that∣∣∣ψ(1)

n

⟩does

not include to∣∣∣ψ(0)

n

⟩through a normalization procedure which typically requires

that the overlap between the unperturbed and perturbed wavefunctions is taken as⟨ψ

(0)n

∣∣∣ψn⟩ = 1. This procedure is termed as intermediate normalization and ensuring

the unperturbed eigenfunction being orthogonal to all of the corrections, that mean,⟨ψ

(0)n

∣∣∣ψ(m)n

⟩= 0 for m ≥ 1, instead of the conventional normalization 〈ψn|ψn〉 = 1.

Upon substitution of (2.9) into (2.8), we have the required result

|ψ(1)n 〉 =

∑k 6=n

Hpkn

E(0)n − E(0)

k

∣∣∣ψ(0)k

⟩, (2.10)

which correspond to 1st order correction of wavefunction and

E(2)n =

∑k 6=n

|Hpkn|2

E(0)n − E(0)

k

, (2.11)

to the 2nd order correction of energy, where we have defined

Hpkn =⟨ψ

(0)k

∣∣∣ Hp

∣∣ψ(0)n

⟩. (2.12)

The process for wavefunction is repeated until get 2nd order correction which is givenby

|ψ(2)n 〉 =

∑k 6=n

∑m6=n

HpkmHpmn

(E(0)n − E(0)

k )(E(0)n − E(0)

m )

∣∣∣ψ(0)k

⟩−∑k 6=n

HpnnHpkn

(E(0)n − E(0)

k )2

∣∣∣ψ(0)k

⟩.

(2.13)It is worth noting that the previous expressions can be used only in the non-degenerate case, where we always have E

(0)n 6= E

(0)k . In the degenerate case, a

different and more complicated treatment is needed [3–6,9].

5

Chapter 2. Perturbation Methods

2.2 Time-dependent perturbation theory

In the previous section we have focused on the mathematical formalism to treat withphysical systems whose total Hamiltonian H is time-independent. However, the ruleschanges completely whether one deals with a Hamiltonian that depends on time, inparticular when the temporal dependence resides into the perturbation. In suchcases, one must therefore resort to the time-dependent perturbation theory. Indeed,this theoretical framework initiated and developed by Paul Dirac [22], it is a powerfultool to study the interaction and time evolution in quantum systems which cannotbe solved exactly. The reader can consult the references [6, 9, 22, 23] which providesmore details about this subject. In order to derive the works equations concerningto the time-dependent perturbation theory, we must begin by introducing the time-dependent Schrodinger equation (in all this work we will set ~ = 1)

id

dt|ψ(t)〉 =

[H0 + λHp(t)

]|ψ(t)〉 . (2.14)

The main goal is to find the solution of system state |ψ(t)〉, in principle, this can bedone in a very simple way by computing the perturbed propagator which it is usefulto apply any initial state; once it is obtained, a perturbation expansion itself is carryout in terms of the small parameter λ. To do so, one must choose a proper frameto describes the dynamics of system (2.14). The interaction picture is suitable whenthe Hamiltonian of an system can be written as a sum of two parts, H = H0 + V (t),where the first half is related with the time-independent term whose eigenvalues andeigenfunctions are known, whereas the second half is usually associated with a time-dependent potential term, in our case V (t)→ λHp(t) [24,25]. Thus, the state vectorof this picture is given by

|ψInt

(t)〉 = U †0(t) |ψ(t)〉 , (2.15)

the above is related to the Schrodinger picture, where the state vectors evolve intime but the operators not, through the inverse unitary transformation

U0(t) = exp(−iH0t

), (2.16)

which provides the evolution of system in the absence of perturbation. From Eq.(2.15),it is worth notice that if |ψ

Int(t)〉 is calculated, then one could readily determine |ψ(t)〉

by inverting Eq.(2.15) to find

|ψ(t)〉 = exp(−iH0t

)|ψ

Int(t)〉 . (2.17)

Inserting this into Eq.(2.14) we arrive to a Schrodinger equation where the depen-dence on H0 is removed and only includes the effects of perturbation as follows

id

dt|ψ

Int(t)〉 = λH

Int(t) |ψ

Int(t)〉 , (2.18)

6

2.2. Time-dependent perturbation theory

where we defineHInt

(t) = exp(iH0t

)Hp(t) exp

(−iH0t

)(2.19)

Notice that Eq.(2.18) contains a mixture of the Heisenberg picture [24,25] version forHInt

(t) where the operators change with time and the Schrodinger picture to dealswith the time state vector |ψ

Int(t)〉 through a Schrodinger equation. The interaction

picture is the main responsible of above features, since it is a hybrid representationof both frames.Similar to the time-independent perturbation theory, we perform appropriate per-turbative expansion of |ψ

Int(t)〉 as a power series in λ

|ψInt

(t)〉 =∣∣ψ(0)

Int(t)⟩

+ λ∣∣ψ(1)

Int(t)⟩

+ λ2∣∣ψ(2)

Int(t)⟩

+ . . . (2.20)

Plugging (2.20) into (2.18) and then equate separately term by term in powers of λ,we get

id

dt

∣∣ψ(0)Int

(t)⟩

=0,

id

dt

∣∣ψ(1)Int

(t)⟩

=HInt

(t)∣∣ψ(0)

Int(t)⟩,

id

dt

∣∣ψ(2)Int

(t)⟩

=HInt

(t)∣∣ψ(1)

Int(t)⟩,

... (2.21)

From (2.17) we can appreciate that both state vector in interaction picture andoriginal state are equal when t = 0, i.e, |ψ

Int(0)〉 = |ψ(0)〉. Then a expansion of

(2.20) evaluated at t = 0 give us

|ψInt

(0)〉 = |ψ(0)〉 =∣∣ψ(0)

Int(0)⟩

+ λ∣∣ψ(1)

Int(0)⟩

+ λ2∣∣ψ(2)

Int(0)⟩

+ . . . (2.22)

and if we equate separately the left and right-hand sides of this expression in powersof λ, it follows that ∣∣ψ(0)

Int(0)⟩

= |ψ(0)〉 ,∣∣ψ(m)Int

(0)⟩

=0, for m = 1, 2, 3.. (2.23)

The later is reasonable and it can be noticed in (2.21) where the zero order term istime independent. So, it is reasonable deduce that the solution of zero-order term isgiven by ∣∣ψ(0)

Int(t)⟩

= |ψ(0)〉 , (2.24)

combining this result with the second expression of (2.21) gives us

id

dt

∣∣ψ(1)Int

(t)⟩

= HInt

(t) |ψ(0)〉 , (2.25)

7

Chapter 2. Perturbation Methods

whose formal solution is∣∣ψ(1)Int

(t)⟩

= −i∫ t

0

HInt

(t1)dt1 |ψ(0)〉 , (2.26)

which correspond to solution of the first-order term. Here, we have implementedcorrectly the initial condition

∣∣ψ(1)Int

(0)⟩

= 0 by setting the lower limit of integrationequal to zero. Now, if we replace previous result for the next equation of order λ2,we find ∣∣ψ(2)

Int(t)⟩

=− i∫ t

0

HInt

(t1)dt1∣∣ψ(1)

Int(t1)⟩

=−∫ t

0

HInt

(t1)dt1

∫ t1

0

HInt

(t2)dt2 |ψ(0)〉 , (2.27)

the solution of second-order term gives rise to an iterated integral expression inchronology order. At this point the pattern is clear, and we can write the final resultfor |ψ(t)〉 as

|ψ(t)〉 =e−iH0t[1− iλ

∫ t

0

dt1HInt(t1) + (−iλ)2

∫ t

0

dt1

∫ t1

0

dt2HInt(t1)HInt

(t2) + . . .

+ (−iλ)n∫ t

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtnHInt(t1)HInt

(t2) . . . HInt(tn) + . . .

]|ψ(0)〉 ,

(2.28)

this expansion of multiple integrations ordered in time is known as the Dyson series[26,27].

8

Chapter 3

The Normalized MatrixPerturbation Method

3.1 The Matrix Method

In the same spirit as the time dependent perturbation theory, it is possible to developa formulation to find a perturbative solution of (2.14), the main difference being thatnow the perturbation, Hp, does not depend on time. Strictly speaking, the formal

solution of the time-dependent Schrodinger equation |ψ(t)〉 = e−iHt |ψ(0)〉, with thecomplete Hamiltonian H = H0 +λHp, can be expanded in a Taylor series and sortedin powers of λ; for example, an expansion for the propagator of system truncated upto first order term is

|ψ(t)〉 =

[e−iH0t + λ

∞∑n=1

(−it)n

n!

n−1∑k=0

Hn−1−k0 HpH0

k

]|ψ(0)〉 . (3.1)

The basic idea behind the Matrix Method [18–21] is obtain an analytic solution ofprevious equation through the following triangular matrix

M =

(H0 Hp

0 H0

), (3.2)

whose diagonal entries are conformed by the unperturbed Hamiltonian and the uppertriangle by the perturbation. One can find that if we multiply the matrix M byitself n-times, its upper element will contain exactly the same products of H0 andHp defined within summation in Eq.(3.1). In simple words, the matrix element M1,2

will give us the first order correction; based on this consideration, Eq.(3.1) is thentransformed to

|ψ(t)〉 =[e−iH0t + λ(e−iMt)1,2

]|ψ(0)〉 , (3.3)

here the approximate solution has been split in two pieces; the first part belongs tothe solution of the unperturbed system, that is well known, while the second part

9

Chapter 3. The Normalized Matrix Perturbation Method

refers to the first order correction. In order to determine the solution of first orderterm, we have to keep in mind that the problem originally pose it must follows thesame matricial convention, hence, the approximate solution (3.3) can be convenientlyrewritten as

|ψ(t)〉 =∣∣ψ(0)

⟩+ λ

(∣∣ψP⟩)1,2, (3.4)

where∣∣ψP⟩ is a perturbed matrix defined as∣∣ψP⟩ =

(|ψ1,1〉 |ψ1,2〉|ψ2,1〉 |ψ2,2〉

). (3.5)

If we derived the equations (3.3) and (3.4) with respect to time and equate thecorresponding coefficients of λ, we find

id

dt

∣∣ψ(0)⟩

=H0e−iH0t |ψ(0)〉 ,

id

dt

∣∣ψP⟩ =Me−iMt

(|ψ(0)〉 0

0 |ψ(0)〉

), (3.6)

the first differential equation is trivial to solve and give us the zero-order solu-tion,

∣∣ψ(0)⟩

= e−iH0t |ψ(0)〉, whereas the integration of second one leads to∣∣ψP⟩ =

e−iMtI |ψ(0)〉 (I is the unity matrix) and which obeys the differential equation

id

dt

∣∣ψP⟩ =M∣∣ψP⟩ =

(H0 |ψ1,1〉 H0 |ψ1,2〉+ Hp |ψ2,2〉

0 H0 |ψ2,2〉

), (3.7)

subject to the initial condition∣∣ψP (0)⟩

=

(|ψ(0)〉 0

0 |ψ(0)〉

). (3.8)

One can notice from (3.7) that the matrix element |ψ2,1〉 is equal to zero, this makessense, since M is a tridiagonal matrix and the resulting product of two upper trian-gular matrices is upper triangular too. The reader can easily check it by performingthe matrices product on the right side of second equation (3.6). Further, the so-lution we are looking for is associated with the matrix element |ψ1,2〉 through thedifferential equation

id

dt|ψ1,2〉 = H0 |ψ1,2〉+ Hpe

−iH0t |ψ(0)〉 , (3.9)

where we have used the fact that |ψ1,1〉 = |ψ2,2〉 = e−iH0t |ψ(0)〉. Now if we make the

transformation |φ1,2〉 = eiH0t |ψ1,2〉, integrated the resulting expression and then ittransform back to |ψ1,2〉, we arrive to

|ψ1,2〉 = −ie−iH0t

t∫0

eiH0t1Hpe−iH0t1dt1

|ψ(0)〉 , (3.10)

10

3.1. The Matrix Method

which corresponds to first order correction. Note that above result is very similar to(2.26) and open the possibility to build a bridge between this approach and the Diractime-dependent perturbation method described in chapter 2; for this make sense, wemust of course assume at the beginning that Hp does not depend on time. Let us

expand again the formal propagator of system, e−iHt, in Taylor series and retainingterms until λ2, we get

|ψ(t)〉 =

[e−iH0t + λ

∞∑n=1

(−it)n

n!

n−1∑k=0

Hn−1−k0 HpH0

k

]|ψ(0)〉

+λ2∞∑n=2

(−it)n

n!

n−1∑k=1

n−k∑j=0

Hn−1−k−j0 HpH0

jHpH0

k−1|ψ(0)〉 . (3.11)

In analogy to the previous case, we consider now the matrix

M =

H0 Hp 0

0 H0 Hp

0 0 H0

. (3.12)

If we multiply the matrix M by itself n-times, then one should find that all theinformation about the second order correction will be enclosed in the element M1,3

of our newly defined 3× 3 triangular matrix, completely similar to (3.2). Therefore,the solution to second order correction can be determined by performing all of thealgebraic steps outlined in the first order case to yields

|ψ(t)〉 =[e−iH0t + λ(e−iMt)1,2 + λ2(e−iMt)1,3

]|ψ(0)〉

=∣∣ψ(0)

⟩+ λ

(∣∣ψP⟩)1,2

+ λ2(∣∣ψP⟩)

1,3, (3.13)

where

|ψ1,3〉 = −e−iH0t

t∫0

dt1eiH0t1Hpe

−iH0t1

t1∫0

dt2eiH0t2Hpe

−iH0t2 |ψ(0)〉 . (3.14)

It becomes clear that the Matrix Method allows to transform the Taylor series ofthe formal solution of the time-dependent Schrodinger equation in a power series ofthe matrix M , which can be handled easily. Likewise, its iterative procedure allowsus to find any kth-order correction in a simple and straightforward way through thefollowing relation [18–21]

|ψ(t)〉 =

[e−iH0t +

k∑n=1

λn(e−iMt

)1,n+1

]|ψ(0)〉

=∣∣ψ(0)

⟩+

k∑n=1

λn(∣∣ψP⟩)

1,n+1, (3.15)

11

Chapter 3. The Normalized Matrix Perturbation Method

with the perturbed matrix defined as

∣∣ψP⟩ =

|ψ1,1〉 . . . |ψ1,n+1〉...

. . ....

|ψn+1,1〉 . . . |ψn+1,n+1〉

, (3.16)

being the matrix element |ψ1,n+1〉 the relevant solution we are looking for, which isexpressed in the form

|ψ1,n+1〉 = (−i)n e−iH0t

t∫0

dt1

t1∫0

dt2· · ·tn−1∫0

dtn eiH0t1Hpe

−iH0t1

× eiH0t2Hpe−iH0t2 . . . eiH0tnHpe

−iH0tn |ψ(0)〉 . (3.17)

This time-ordered series, restricted to the interval [0, t], is the fundamental pieceto calculate the different correction terms; furthermore, we should point out thatrelationship (3.17) is the mathematical representation of the Dyson series [26, 27].Albeit the Matrix method was originally conceived to deal with the effects of weakperturbations, holding in the limit λ→ 0. Above formalism can be readily extendedto deal with its strong counterpart λ→∞. Let us consider a rescale of time τ = tλ in

the time-dependent Schrodinger equation, iλ ddτ|ψ〉 =

(H0 + λHp

)|ψ〉, whose formal

solution, |ψ(τ)〉 = exp[−i(Hp + 1

λH0

)τ], looks very similar to the weak case; but

here, it can be seen that the roles of unperturbed part and perturbation are reversed.Indeed, if we repeat the same line of steps as like the weak case, we obtain a Dysonseries in matrix form for large values of λ as follows

|ψ(τ)〉 =

[e−iHpτ +

k∑n=1

1

λn(e−iMt

)1,n+1

]|ψ(0)〉

=∣∣ψ(0)

⟩+

k∑n=1

1

λn(∣∣ψP⟩)

1,n+1, (3.18)

here the matrix element |ψ1,n+1〉 supplies all the information needed to calculate thedifferent correction terms for the strong perturbation regime

|ψ1,n+1〉 = (−i)n e−iHpττ∫

0

dτ1

τ1∫0

dτ2· · ·τn−1∫0

dτn eiHpτ1H0e

−iHpτ1

× eiHpτ2H0e−iHpτ2 . . . eiHpτnH0e

−iHpτn |ψ(0)〉 . (3.19)

This duality on the Matrix Method gives us the possibility to analyze the solution of aquantum system in both regimes of the perturbative parameter λ. Despite one series

12

3.2. Normalization constant

is inverse to other, it is possible to link between them by considering the interchangeH0 ↔ Hp, setting λ = 1. The free choice of what part to system represents the

perturbation is due to the symmetry of H itself: this is duality in perturbation theory.It is important to remark that the perturbation theory presented in Chapter 2 is builtup over weak coupling expansion; this regime is well known and it is characterizedby the Eq.(2.28). A general scheme to prove the existence of its counterpart, astrong perturbation regime, has been attacked by several authors [28–31], but none ofthose analysis looked completely convincing, until the arrival of Frasca’s work basedon the Navier-Stokes equation. Fraca’s perturbative formulation deals with smalland large Reynolds numbers, and it has been successfully implemented in quantummechanics [32–37]. In fact, the perturbation series (3.18) is similar to the reportedby Frasca [36,37] but despite of they similarities, the matrix method gives a new wayto cast the dual Dyson series in matrix form instead of usual integral representation.In addition, the approximative solutions of the Matrix method are not normalized,a normalization factor Nk allows eliminating the problems of divergence when tbecomes to grow up at any kth-order corrections; being a property that marks thedifference between our approach (as will see in next section) and the Frasca’s work,where Nk is intractable through him analysis.

3.2 Normalization constant

We have seen in the previous section that the approximated solutions of the Schrodingerequation are valid for the different regions of the perturbative parameter λ that gofrom a weak-to-strong coupling regime. Further, it is appropriate to mention thatthe expressions (3.15) and (3.18) can be written as a power series of λ, along with theelement |ψ1,n+1〉 of the perturbed matrix; however, both solutions are not normalizedand it is convenient to get a normalization factor Nk that preserves their norm atany order. Keeping the above remark in mind, let us define the next normalizedsolution for the weak case

|Ψ(t)〉 = Nk(t)

(∣∣ψ(0)⟩

+k∑

n=1

λn |ψ1,n+1〉

), (3.20)

where the corresponding value of Nk(t) may be easily determined by the normal-ization condition 〈Ψ(t)|Ψ(t)〉 = 1 for all t. Then we deduce from the above that

Nk(t) =

[1 + 2

k∑n=1

λn<(⟨ψ(0)|ψ1,n+1

⟩)+

k∑m,n=1

λm+n 〈ψ1,m+1|ψ1,n+1〉

]− 12

, (3.21)

where < (z) means the real part of z. Here, the first contribution is due to innerproduct of the unperturbed eigenfunctions,

⟨ψ(0)|ψ(0)

⟩=1, whereas the second one

13

Chapter 3. The Normalized Matrix Perturbation Method

arises from two single finite sums, one referent to the inner product of the zero-orderterm with the nth-order correction

⟨ψ(0)|ψ1,n+1

⟩, and the other respect to its complex

conjugate⟨ψ1,n+1|ψ(0)

⟩; as consequence, a purely real contribution is obtained of the

sum of both over all k. The last part of the above equation is merely handled if werun m and n from 1 to k, such as presented in Table 3.1.

m/n 1 2 3 . . . k

1 λ2 〈ψ1,2|ψ1,2〉 λ3 〈ψ1,2|ψ1,3〉 λ4 〈ψ1,2|ψ1,4〉 . . . λ1+k 〈ψ1,2|ψ1,k+1〉2 λ3 〈ψ1,3|ψ1,2〉 λ4 〈ψ1,3|ψ1,3〉 λ5 〈ψ1,3|ψ1,4〉 . . . λ2+k 〈ψ1,3|ψ1,k+1〉3 λ4 〈ψ1,4|ψ1,2〉 λ5 〈ψ1,4|ψ1,3〉 λ6 〈ψ1,4|ψ1,4〉 . . . λ3+k 〈ψ1,4|ψ1,k+1〉...

......

......

......

......

. . ....

k λ1+k 〈ψ1,k+1|ψ1,2〉 λ2+k 〈ψ1,k+1|ψ1,3〉 λ3+k 〈ψ1,k+1|ψ1,4〉 . . . λ2k 〈ψ1,k+1|ψ1,k+1〉

Table 3.1: This table displays the different terms of the double summation containedin the last term of Eq. (3.21), when m and n run from 1 to k

Notice that the double summation in (3.21) can be partitioned into two parts,one where m = n and which contains all diagonal terms, and the remainder part forthose off-diagonal terms which can be represented in a double sum of the real partof 〈ψ1,m+1|ψ1,j+1〉 as follows

k∑m,n=1

λm+n 〈ψ1,m+1|ψ1,n+1〉 =k∑

n=1

λ2n 〈ψ1,n+1|ψ1,n+1〉

+ 2k−1∑n=1k>1

k∑m=n+1

λn+m< (〈ψ1,n+1|ψ1,m+1〉) ; (3.22)

by applying the change of variable m=p − n into Eq.(3.22) and the resulting sum-mation into Eq.(3.21), we arrive to

Nk(t) =

[1 + 2

k∑n=1

λn<(⟨ψ(0)|ψ1,n+1

⟩)+

k∑n=1

λ2n 〈ψ1,n+1|ψ1,n+1〉

+2k−1∑n=1k>1

n+k∑p=2n+1

λp< (〈ψ1,n+1|ψ1,p−n+1〉)

− 1

2

,

(3.23)

which is the normalization constant [21] for the approximate analytical solution of theSchrodinger equation defined in Eq.(3.20). In an analogous fashion, the normalizedsolution for the strong perturbative expansion of Eq.(3.18) can be defined as

|Ψ(τ)〉 = Nk(τ)

(∣∣ψ(0)⟩

+k∑

n=1

1

λn|ψ1,n+1〉

), (3.24)

14

3.2. Normalization constant

where for this regime, Nk(τ) is given by

Nk(τ) =

[1 + 2

k∑n=1

1

λn<(⟨ψ(0)|ψ1,n+1

⟩)+

k∑n=1

1

λ2n〈ψ1,n+1|ψ1,n+1〉

+2k−1∑n=1k>1

n+k∑p=2n+1

1

λp< (〈ψ1,n+1|ψ1,p−n+1〉)

− 1

2

,

(3.25)

In principle, the inclusion of factor Nk in our calculations can give a fairly good ap-proximation to the solution without convergence difficulties. An important remarkon the proposed normalization procedure is that we have not invoked the usual inter-mediate normalization used in the standard perturbation theory, i.e. the imposition⟨ψ(0)|ψ1,n+1

⟩= 0 for all λ. To see this with more detail it necessary switching back

to the expressions (3.10) and (3.14) for the first two order correction; from here,we show that both solutions can be handled through an alternative procedure toobtain the same results without integration schemes, and which consist of writingthem in terms of the complete orthonormal set of eigenfunctions of the unperturbedHamiltonian,H0, through the completeness relation I =

∑k

∣∣k(0)⟩ ⟨k(0)∣∣. For the sake

of clarity, we just simplify the notation∣∣∣ψ(0)

k

⟩, used in Chapter 2, by

∣∣k(0)⟩, with

main purpose to avoid confusing it with |ψ1,n+1〉. Inserting this identity operatorinside of Eq.(3.10), together with the initial condition |ψ(0)〉 =

∣∣n(0)⟩, we deduce

that

|ψ1,2〉 =− i∑k

∣∣k(0)⟩ ⟨k(0)∣∣ e−iH0t

∫ t

0

eiH0t1Hpe−iH0t1

∣∣n(0)⟩dt1

=− i∑k

Hpkne−iE(0)

k t

∫ t

0

ei(E

(0)k −E

(0)n

)t1dt1. (3.26)

Notice that we have used the hermiticy condition, H0

∣∣n(0)⟩

= E(0)n

∣∣n(0)⟩, of the

unperturbed Hamiltonian. To treat with the integration for when the sum outsideis k = n and k 6= n, we replaced it by

∫ t

0

ei(E

(0)k −E

(0)n

)t1dt1 =

tE

(1)n , when k = n

i

[ei(E(0)

k−E(0)

n )t−1]

E(0)n −E

(0)k

, when k 6= n(3.27)

in laying down our formulation, we assume that the partition of summation whenk = n and k 6= n is due that all the procedure involved in the Matrix Method doesnot distinguish if the unperturbed part of the Hamiltonian is degenerated or not. So,

15

Chapter 3. The Normalized Matrix Perturbation Method

Eq.(3.17) provides a very general expression to compute the corrections and whenceit follows that Eq.(3.26) is

|ψ1,2〉 =− itE(1)n e−iE

(0)n t∣∣n(0)

⟩− 2i

∑k 6=n

Hpkne−i t

2

(E

(0)n +E

(0)k

) sin[t2

(E

(0)n − E(0)

k

)]E

(0)n − E(0)

k

∣∣k(0)⟩ ,(3.28)

being now the first-order correction expressed in terms of the eigenvalues H0, whereit also includes the first-order energy correction defined in (3.11). The above re-sult means that the first-order corrections to wavefunction and to the energy canbe contained and written in only one expression, at difference of the standard per-turbation theory, where is needed to calculate them in a separate way. Further,the approximate solution not only present conventional stationary terms, but alsotime-dependent terms that allows one to figure out the temporal evolution of thecorrections. Consequently, we deduce from the above solution that the derivation ofsecond order is given by

|ψ1,3〉 = −ie−iH0t

t∫0

eiH0t1Hp

− it1E(1)

n e−iE(0)n t1

∣∣n(0)⟩

− 2i∑k 6=n

e−i t1

2

(E

(0)k +E

(0)n

) sin[t12

(E

(0)n − E(0)

k

)]E

(0)n − E(0)

k

Hpkn

∣∣k(0)⟩dt1, (3.29)

where the expression inside of the curly brackets is the first order correction. Em-ploying again the identity operator I and after some algebraic manipulation, onegets

|ψ1,3〉 =− e−iE(0)n t

(t2

2E2(1)n + itE(2)

n

) ∣∣n(0)⟩

+ ite−iE(0)n t∑k 6=n

[e−i t

2

(E

(0)k −E

(0)n

)Hpkk − E(1)

n

]Hpkn

E(0)n − E(0)

k

∣∣k(0)⟩

− ie−iE(0)n tE(1)

n

∑k 6=n

e−i t

2

(E

(0)k −E

(0)n

) sin[t2

(E

(0)k − E

(0)n

)](E

(0)n − E(0)

k

)2 Hpkn

∣∣k(0)⟩

− 2ie−it2E

(0)n

∑k 6=n

∑q 6=n

e−it2E

(0)q

sin[t2

(E

(0)q − E(0)

n

)](E

(0)q − E(0)

n )(E(0)n − E(0)

k )HpqkHpkn

∣∣q(0)⟩

+ 2i∑k 6=n

∑q 6=k

e−i t

2

(E

(0)q +E

(0)k

) sin[t2

(E

(0)q − E(0)

k

)](E

(0)q − E(0)

k )(E(0)n − E(0)

k )HpqkHpkn

∣∣q(0)⟩ .(3.30)

16

3.2. Normalization constant

In particular we are able to determined up third-order correction by following thesame strategy adopted above, to gives

|ψ1,4〉 = ite−iE(0)n t

(t2

6E3(1)n + tE(1)

n E(2)n + E(1)

n E(3)n

) ∣∣n(0)⟩

+ it∑k 6=n

e−iE(0)n tE

(1)n |Hpkn|

2(E

(0)n − E(0)

k

)2 ∣∣k(0)⟩

+ i∑k 6=n

te− it2 (E(0)n −E

(0)k ) − 2

sin[t2(E

(0)n − E(0)

k )]

E(0)n − E(0)

k

e− it2 (E(0)n −E

(0)k )Hpkk |Hpkn|

2(E

(0)n − E(0)

k

)2 ∣∣k(0)⟩+ E(1)

n

∑k 6=n

i sin

[t(E(0)n − E

(0)k

)]+ 2 sin

[t

2

(E(0)n − E

(0)k

)] |Hpkn|2∣∣k(0)⟩(

E(0)n − E(0)

k

)3− i∑k 6=n

∑q 6=n

t− 2eit2

(E

(0)n −E

(0)q

) sin[t2

(E

(0)n − E(0)

q

)](E

(0)n − E(0)

q

) HpknHpnqHpqk

∣∣q(0)⟩(E

(0)n − E(0)

q

)(E

(0)n − E(0)

k

)+ 2i

∑k 6=n

∑q 6=k

eit2

(E

(0)n −E

(0)k

) sin[t2

(E

(0)n − E(0)

k

)](E

(0)n − E(0)

k

)2 (E

(0)q − E(0)

k

)HpnqHpqkHpkn

∣∣q(0)⟩

− 2i∑k 6=n

∑q 6=kq 6=n

eit2

(E

(0)n −E

(0)q

) sin[t2

(E

(0)n − E(0)

q

)]HpnqHpqkHpkn(

E(0)n − E(0)

q

)(E

(0)q − E(0)

k

)(E

(0)n − E(0)

k

) ∣∣q(0)⟩ .(3.31)

with

E(3)n =

∑k 6=n

∑q 6=n

HpnkHpkqHpqn(E

(0)n − E(0)

k

)(E

(0)n − E(0)

q

) − HpnnHpknHpnq(E

(0)n − E(0)

k

)2 . (3.32)

Let us multiply both sides of (3.28), (3.30) and (3.31) by⟨n(0)∣∣, this procedure gives⟨

ψ(0)|ψ1,2

⟩=− itE(1)

n ,

⟨ψ(0)|ψ1,3

⟩=− 1

2

(t2E2(1)

n + 2itE(2)n

)+ 2i

∑k 6=n

eit2(En−Ek)

sin[t2

(E

(0)n − E(0)

k

)](E

(0)n − E(0)

k

)2 |Hpkn|2,

⟨ψ(0)|ψ1,4

⟩=it

(t2

6E3(1)n + tE(1)

n E(2)n + E(1)

n E(3)n

)

+ 2iE(1)n

∑k 6=n

eit2

(E

(0)n −E

(0)k

) sin[t2

(E

(0)n − E(0)

k

)](E

(0)n − E(0)

k

)3 |Hpkn|2. (3.33)

17

Chapter 3. The Normalized Matrix Perturbation Method

It can be seen clearly that the intermediate normalization,k∑

n=1

⟨ψ(0)

∣∣ψ1,n+1

⟩= 0, is

impractical and does not work for the Matrix Method case due the inner product ofthe zero-order with first three correction terms are different from zero. In particular,these complex inner products have non-zero imaginary parts which should not beneglected if the called intermediate normalization is applied, for this reason, we haveadopted other procedure to get a time-dependent normalization, which ensures realvalues at any power of λ. For example, if we get the normalization constant for thefirst-order correction of |Ψ(t)〉 when k = 1 into (3.23)

N1(t) =[1 + 2<

(⟨ψ(0)

∣∣ψ1,2

⟩)+ λ2 〈ψ1,2|ψ1,2〉

]−1/2, (3.34)

whose inner products are easy to evaluate with the information proportioned byEq.(3.33); where <

(⟨ψ(0)|ψ1,2

⟩)= 0, whereas the last term renders to

〈ψ1,2|ψ1,2〉 = t2E2(1)n + 4

∑k 6=n

sin2[t2

(E

(0)n − E(0)

k

)](E

(0)n − E(0)

k

)2 ‖Hpkn‖2 (3.35)

being now the normalization constant reduced to

N1(t) =(1 + λ2 〈ψ1,2|ψ1,2〉

)− 12 . (3.36)

Let us assume now that k = 2 into Eq.(3.23), in this case, we arrive to the normal-ization constant of second order correction

N2(t) =

1 + λ2[2<(⟨ψ(0)|ψ1,3

⟩)+ 〈ψ1,2|ψ1,2〉

]+ 2λ3< (〈ψ1,2|ψ1,3〉)

+ λ4 〈ψ1,3|ψ1,3〉− 1

2; (3.37)

18

3.2. Normalization constant

doing the inner products, one can find

2<(⟨ψ(0)|ψ1,3

⟩)= −〈ψ1,2|ψ1,2〉 ,

2< (〈ψ1,2|ψ1,3〉) = 2t2E(1)n E(2)

n − 4tE(1)n

∑k 6=n

sin2[t2

(E

(0)k − E

(0)n

)](E

(0)n − E(0)

k

)3 |Hpkn|2

+ 4t∑k 6=n

cos[t2

(E

(0)k − E

(0)n

)]sin[t2

(E

(0)k − E

(0)n

)](E

(0)n − E(0)

k

)2 |Hpkn|2Hpkk

+ 4∑k 6=n

∑m6=n

sin2[t2

(E

(0)m − E(0)

n

)](E

(0)n − E(0)

m

)2 (E

(0)n − E(0)

k

)HpnmHpmkHpkn

+ 4∑k 6=n

∑m6=n

cos[t2

(E

(0)n − E(0)

k

)]sin[t2

(E

(0)m − E(0)

n

)](E

(0)n − E(0)

k

)(E

(0)n − E(0)

m

)(E

(0)m − E(0)

k

)× sin

[t

2

(E(0)m − E

(0)k

)]HpnmHpmkHpkn ,

〈ψ1,3|ψ1,3〉 =t4

4E4(1)n + t2E2(2)

n − 2t2E(2)n

∑k 6=n

sin2[t2

(E

(0)n − E(0)

k

)](E

(0)n − E(0)

k

)2 |Hpkn|2

+ 2tE(2)n

∑k 6=n

sin[t(E(0)n − E

(0)k

)]+ (Hpkk)

2

|Hpkn|2(

E(0)n − E(0)

k

)2+ E(1)

n

∑k 6=n

sin[t(E

(0)n − E(0)

k

)](E

(0)n − E(0)

k

) − 2t

Hpkk |Hpkn|2(E

(0)n − E(0)

k

)2− 2tE(1)

n

∑k 6=n

∑m6=n

sin[t(E

(0)m − E(0)

n

)](E

(0)n − E(0)

m

)2 (E

(0)n − E(0)

k

)HpkmHpnkHpnm

+ 2∑k 6=n

∑m6=n

sin[t(E

(0)m − E(0)

n

)](E

(0)n − E(0)

m

)2 (E

(0)n − E(0)

k

)HpmmHpkmHpnkHpmn

− 4E(1)n

∑k 6=n

∑m6=n

cos[t2

(E

(0)k − E

(0)n

)]sin[t2

(E

(0)m − E(0)

k

)]sin[t2

(E

(0)m − E(0)

n

)](E

(0)m − E(0)

k

)(E

(0)n − E(0)

k

)(E

(0)n − E(0)

m

)2×HpkmHpnkHpnm

+ 2∑k 6=n

∑m6=k

sin[t2

(E

(0)m − E(0)

k

)](E

(0)m − E(0)

k

)(E

(0)n − E(0)

k

)(E

(0)n − E(0)

m

)HpmmHpmnHpkmHpnk 19

Chapter 3. The Normalized Matrix Perturbation Method

− 8∑k 6=n

∑m 6=k

∑q 6=n

cos[t2

(E

(0)k − E

(0)n

)]sin[t2

(E

(0)m − E(0)

k

)]sin[t2

(E

(0)m − E(0)

n

)](E

(0)m − E(0)

k

)(E

(0)n − E(0)

k

)(E

(0)m − E(0)

n

)(E

(0)n − E(0)

q

)×HpmmHpmnHpkmHpnkHpqnHpqk

− 4E(1)n

∑k 6=n

∑m 6=n

sin[t2

(E

(0)m − E(0)

n

)](E

(0)n − E(0)

m

)3 (E

(0)n − E(0)

k

)HpkmHpnmHpnk , (3.38)

Therefore, the normalization constant at this order correction can be recast as

N2(t) =[1 + 2λ3< (〈ψ1,2|ψ1,3〉) + λ4 〈ψ1,3|ψ1,3〉

]− 12. (3.39)

We see that the procedure advocated here insures that inner products are alwaysa real number, therefore, the isolation of a multiplicative time-dependent factor,Nk(t), in both complete solutions (3.20) and (3.24) provides the correct normalizationtreatment to the Matrix Method.

20

Chapter 4

Examples

Several examples have been presented in the articles where Matrix Method has beendeveloped. In [20], the generalization to the case of the Lindblad master equationand its application to the problem for a lossy cavity filled with a Kerr medium ispresented; in [18] the scenario of harmonic oscillator perturbed by a quadratic termis analyzed. However, in the latter case, the normalization process of the solutionswas completely ignored.This chapter will allow us to illustrate the efficiency of complete method and itsrange of applicability in comparison with other perturbative analysis, including thestandard perturbation theory. To achieve this, some examples with analytical solu-tion will be treated making use of the normalized formalism introduced in Section3.2 of Chapter 3. Then, their approximate results will be obtained and comparedwith their respective exact or numerical solutions.

4.1 Harmonic Oscillator with linear term in po-

tential.

4.1.1 Exact solution

Once said this, let us begin by examining the case of harmonic oscillator disturbedby linear anharmonic potential; this quantum model is described by the followingSchrodinger equation

id

dt|ψ(t)〉 =

(1

2p2 +

ω2

2x2 + λx

)|ψ(t)〉 , (4.1)

where p = −i ∂∂x

and x = x are the momentum and position operators, whereas,the quantity λ is a dimensionless scale parameter which quantifies the perturbationstrength of the linear anharmonic term. From a classical point of view, the abovetime dependent Schrodinger equation gives a physically description of a particle

21

Chapter 4. Examples

with charge q located in a weak electric field of strength ε when λ = qε [9, 38–40].Standard differential equation techniques have been tackled with great success to getan exact solution of this system, especially when a time dependence is involve in theanharmonic part [41–46]; here, we adopt an operator formalism to solve Eq.(4.1),both exactly and approximately. The momentum and position operators can beexpressed in terms of the well-known raising and lowering operators a† and a as

x =1√2ω

(a† + a

),

p =i

√ω

2

(a† − a

). (4.2)

These ladder operators satisfy the commutation relations[a, a†

]= 1,

[n, a†

]= a†

and [n, a] = −a, being the number operator n = a†a; then Eq.(4.1) turns into theequivalent form

id

dt|ψ(t)〉 =

[ω

(n+

1

2

)+

λ√2ω

(a† + a

)]|ψ(t)〉 . (4.3)

here, the mass m of oscillator is set equal to 1.To simplify the above Schrodinger equation and reach an exactly solvable form, it is

convenient to perform the quantum transformation |φ(t)〉 = D(

λω√2ω

)|ψ(t)〉, where

D(α) = exp(αa† − α∗a

)is the Glauber displacement operator [47, 48]. Indeed, the

operator D(α) acts as a displacement upon amplitudes of a and a† as D(α)aD†(α) =a− α and D(α)a†D†(α) = a† − α∗. So expression (4.3) is transformed into

id

dt|φ(t)〉 =

[ω

(n+

1

2

)− λ2

2ω2

]|φ(t)〉 , (4.4)

which is nothing else than an harmonic oscillator displaced by a quantity λ2

2ω2 and

with a quantized energy E′

= ω(n+ 1

2

)− λ2

2ω2 . If we now integrate the resultingexpression with respect to time and then transform it back to |ψ(t)〉, one obtains

|ψ(t)〉 = exp

[−it

2

(ω − λ2

ω2

)]D†(

λ

ω√

2ω

)exp (−itωn) D

(λ

ω√

2ω

)|ψ(0)〉 . (4.5)

In order to simplify the notation, one can insert the identity operator I = eitωne−itωn

into the previous equation as follows

|ψ(t)〉 = exp

[−it

2

(ω − λ2

ω2

)]D†(

λ

ω√

2ω

)exp (−itωn) D

(λ

ω√

2ω

)eitωne−itωn |ψ(0)〉 .

(4.6)

22

4.1. Harmonic Oscillator with linear term in potential.

with the Hadamard formula [49, 50] eδABe−δA = B + δ[A, B

]+ δ2

2!

[A,[A, B

]], . . .

, it is very easy to prove that

e−itωnD

(λ

ω√

2ω

)eitωn = D

(λe−itω

ω√

2ω

); (4.7)

and with the aid of the displacement operator property, D(α)D(β) = D(α+β)ei=(αβ∗),

Eq.(4.5) is simplified to

|ψ(t)〉 = exp

− i

2

[t

(ω − λ2

ω2

)+λ2

ω3sinh(ωt)

]D (ξ(t)) e−itωn |ψ(0)〉 , (4.8)

where

ξ(t) = −√

2iλ

ω3/2exp

(−iωt

2

)sin

(ωt

2

). (4.9)

Further, using the factorized form D(α) = e−|α|22 eαa

†e−α

∗a, the exact solution of thelinear anharmonic oscillator is found to be

|ψ(t)〉 = exp [−γ(t)] exp[ξ(t)a†

]exp [−ξ∗(t)a] exp (−itωn) |ψ(0)〉 , (4.10)

with

γ(t) =i

2

[t

(ω − λ2

ω2

)+λ2

ω3sin(ωt)

]+λ2

ω3sin2

(ωt

2

). (4.11)

This exact solution may be evaluated with any initial condition; here, for simplicity,we restrict our attention to two special examples of |ψ(0)〉, such as the numberstate |n〉 which represents an eigenstate of the number operator n with eigenvaluen and the coherent state |α〉 which denotes a quasi-classical state produced by laser[47, 48, 51, 52]. The application of first initial condition can be done very simply ifthe completeness relation I =

∑k

|k〉 〈k| is inserted into the exact solution as

|ψ(t)〉 = I |ψ(t)〉

= e−γ(t)∞∑k=0

e−itωn |k〉 〈k| eξ(t)a†e−ξ∗(t)a |n〉

= e−γ(t)∞∑k=0

e−itωnk∑p=0

n∑q=0

(−1)q

√n!k!

(k − p)! (n− q)!ξp(t)ξ∗q(t)

p!q!δk−p,n−q |k〉 ,

(4.12)

if q = n− k + p with n ≥ k, we get

|ψ(t)〉 = e−γ(t)∞∑k=0

e−itωn√k!

n![−ξ∗(t)]n−k

k∑p=0

(−|ξ(t)|2

)pn!

p! (k − p)! (n− k + p)!|k〉 , (4.13)

23

Chapter 4. Examples

it is easy to recognize the structure of the associated Laguerre polynomials [53,54]

Ln−mm (x) =m∑r=0

(−x)r n!

r! (m− r)! (n−m+ r)!; (4.14)

then

|ψ(t)〉 = e−γ(t)∞∑k=0

e−itωn√k!

n![−ξ∗(t)]n−k Ln−kk (|ξ(t)|2) |k〉 . (4.15)

If we now consider the case p = k − n + q with k ≥ n in the sum of Eq.(4.12) andafter some algebra, leads to

|ψ(t)〉 = e−γ(t)∞∑k=0

e−itωn√n!

k![ξ(t)]k−n Lk−nn (|ξ(t)|2) |k〉 . (4.16)

In fact, both solutions are written in the representation of displaced number statesaccording to reported in the literature [55–61]. The wavefunction of the anharmonicsystem in the x representation can be found multiplying both sides of Eq.(4.15) andEq.(4.16) by 〈x|

ψn(x, t) =(ωπ

)1/4exp

[−(γ + itωn+

ωx2

2

)] ∞∑k=0

Dkn√2kk!

Hk

(√ωx), (4.17)

with

Dkn =

√

k!n!

[−ξ∗(t)]n−k Ln−kk (|ξ(t)|2), for n ≥ k√n!k!

[ξ(t)]k−n Lk−nn (|ξ(t)|2), for k ≥ n(4.18)

where we have used that the harmonic oscillator eigenfunctions are

ψ(0)n = 〈x|n〉 =

1√2nn!

(ωπ

)1/4exp

(−ωx

2

2

)Hn

(√ωx), (4.19)

being Hn(x) = (−1)nex2 dn

dxnexp (−x2) the Hermite polynomials [62].

Let us suppose that the anharmonic system initial condition is |ψ(0)〉 = |α〉, acoherent state. In this case, the exact solution of the time-dependent Schrodingerequation, ψα(x, t), can be obtained by using the position representation [63]

|x〉 =(ωπ

)1/4exp

(ωx2

2

)exp

(− a†2

2

) ∣∣∣√2ωx⟩, (4.20)

where∣∣√2ωx

⟩is a coherent state with amplitude

√2ωx. The position eigenstate

was introduced by Moya and Eguibar from starting point of the Caves approach to

24

4.1. Harmonic Oscillator with linear term in potential.

define an squeezed state [63]. Afterwards, they just applied a squeeze in the vacuumstate, and after then, displace it. Taking the expressions (4.20) and (4.10) we get

ψα(x, t) =(ωπ

)1/4exp

[−γ(t) + ω

x2

2

]⟨√2ωx

∣∣∣ e− a22 eξ(t)a†e−ξ∗(t)ae−itωn |α〉 . (4.21)

As exp (−itωn) |α〉 = |α (t)〉 with α(t) = αe−itω, and due the fact that the coherentstates are eigenstates of the annihilation operator with eigenvalues α(t) = αe−itω, wecan cast the above as

ψα(x, t) =(ωπ

)1/4exp

[−γ(t)− α(t)ξ∗(t) + ω

x2

2

]⟨√2ωx|e−

a2

2 eξ(t)a†|α(t)

⟩. (4.22)

Inserting the identity operator I = exp(a2

2

)exp

(− a2

2

),

ψα(x, t) =(ωπ

)1/4exp

[−γ(t)− α(t)ξ∗(t) +

ωx2

2

](4.23)

×⟨√

2ωx∣∣∣ exp

(− a

2

2

)exp

[ξ(t)a†

]exp

(a2

2

)exp

(− a

2

2

)|α (t)〉 ,

using that exp(− a2

2

)a† exp

(a2

2

)= a† − a, we get

ψα(x, t) =(ωπ

)1/4exp

[−γ(t)− α(t)ξ∗(t)− α2(t)

2+ωx2

2

]⟨√2ωx|eξ(t)(a†−a)|α(t)

⟩.

(4.24)

Factorizing the exponential operator eξ(t)(a†−a) as e−

ξ2(t)2 eξ(t)a

†e−ξ(t)a and taking into

account the overlap between coherent states⟨√

2ωx|α(t)⟩

= exp[−ωx2 − |α(t)|

2

2

]exp

[α(t)√

2ωx], we finally found

ψα(x, t) =(ωπ

)1/4exp

− 1

2

[2γ(t) + α2(t) + ωx2 + ξ2(t) + |α(t)|2 + 4α(t)< (ξ(t))

]× exp

√2ω [α(t) + ξ(t)]x

. (4.25)

This is the exact solution of Schrodinger equation (4.3) in the coordinate represen-tation, when the initial state is a coherent state.

4.1.2 Perturbative solution

Let us proceed to solve the same quantum system but now through our normalizedperturbative treatment. In this case, one must formulate that Ho = ω (n+ 1/2)

25

Chapter 4. Examples

denotes the unperturbed harmonic oscillator and Hp = 1√2ω

(a† + a

)the small per-

turbation quantified by λ. Then from Eq.(3.10), we can write the expression whichallows us to get first order term

|ψ1,2〉 = − i√2ωe−iH0t

t∫0

eiH0t1(a† + a

)e−iH0t1dt1

|ψ(0)〉 , (4.26)

using relations

eiH0t a†n e−iH0t =a†neinωt,

eiH0t an e−iH0t =an e−inωt, (4.27)

we find that

|ψ1,2〉 = −√

2

ω3/2i sin (ωt/2) e−iωt(n+1/2)

(a†eiωt/2 + ae−iωt/2

)|ψ(0)〉 . (4.28)

The second-order term can be calculated by using Eq.(3.17) with n = 2

|ψ1,3〉 = − 1

2ωe−iH0t

t∫0

eiH0t1(a† + a

)e−iH0t1

t1∫0

eiH0t2(a† + a

)e−iH0t2dt2

dt1 |ψ(0)〉 ,

(4.29)

After substituting Eq.(4.28) into Eq.(4.29) and evaluating the resulting integral, weget

|ψ1,3〉 =− sin2 (ωt/2)

ω3e−iωt(n+1/2)

[ (a†eiωt/2 + ae−iωt/2

)2 − 1

]|ψ(0)〉

+i

2ω3

[t− 2 sin (ωt/2) e−itωt/2

]e−iωt(n+1/2) |ψ(0)〉 . (4.30)

For simple quantum models, such as in our case, the calculus of the second orderperturbative contribution is more than enough to get a good agreement with theexact solution. Proceeding in the same manner as in the previous subsection, weconsider first that the initial condition is |ψ(0)〉 = |n〉; hence, the second-orderapproximated solution of (4.3) is

|Ψ(t)〉 = N (2)n (t)

(∣∣ψ(0)⟩

+ λ |ψ1,2〉+ λ2 |ψ1,3〉)

= e−itE(0)n N (2)

n (t)

1− λ2

ω3

[2n sin2 (ωt/2) + ie−iωt/2 sin (ωt/2)− i t

2

]|n〉

−√

2

ω3/2iλe−itE

(0)n N (2)

n (t)(e−iωt/2

√n+ 1 |n+ 1〉+ eiωt/2

√n |n− 1〉

)sin (ωt/2)

− λ2

ω3e−itE

(0)n N (2)

n (t)[e−iωt

√(n+ 1)(n+ 2) |n+ 2〉+ eiωt

√n(n− 1) |n− 2〉

]× sin2 (ωt/2) , (4.31)

26

4.1. Harmonic Oscillator with linear term in potential.

where E(0)n = ω

(n+ 1

2

)corresponds to the energy of the harmonic oscillator and

N(2)n is the normalization constant given by

N (2)n (t) =

1 + 2

λ4

ω6

(3n2 + 3n+ 1

)sin4 (ωt/2) +

λ4

4ω6

[4 sin2 (ωt/2)− 2t sin (ωt) + t2

]−1/2.

(4.32)

Here, the subscript n indicates the quantum number whereas superscript (2) remarksthe order approximation of the solution. In particular, it is straightforward to provethat our result in the x representation is translated to

Ψ(2)n (x, t) =F (2)

n (x, t)

1− λ2

ω3

[2n sin2 (ωt/2) + ie−iωt/2 sin (ωt/2)− i t

2

]Hn

(√ωx)

− λ2

2ω3e−iωtF (2)

n (x, t)[Hn+2

(√ωx)

+ 4n(n− 1)e2iωtHn−2(√

ωx) ]

sin2 (ωt/2)

− iλ

ω3/2e−iωt/2F (2)

n (x, t)[Hn+1

(√ωx)

+ 2neiωtHn−1(√

ωx) ]

sin (ωt/2) ,

(4.33)

where

F (2)n (x, t) =

(ω/π)1/4√2nn!

N (2)n (t) exp

[−1

2

(2itE(0)

n + ωx2)]. (4.34)

We must remember that to make the approximations above, we have supposed thatλ 1. However, one can confront the opposite case, when the dominance of thelinear anharmonicity is strong in comparison with the harmonic one, situation thatcan be achieved with a suitable choice of the parameters. In that case, the linearanharmonic term plays the role of non-perturbed Hamiltonian, whereas the pertur-bation is now the harmonic oscillator term. For this second case, one can repeat the

27

Chapter 4. Examples

same analytical procedure to get Eq.(4.33) and arrived to

Ψ(2)n (x, t) =G(2)

n (x, t)∞∑k=0

1− iλt

6

[3ω (2n+ 1) + t2

]− ω2

8λ2t2

(4n2 + 1

)Skn(x, t)

+ω

16λ2t4G(2)

n (x, t)∞∑k=0

[√n (n− 1)Skn−2(x, t) +

√(n+ 1) (n+ 2)Skn+2(x, t)

]−√

2ω

4λt2G(2)

n (x, t)∞∑k=0

1− iλt

6

[ω (6n+ 5) + t2

]√n+ 1Skn+1(x, t)

+

√2ω

4λt2G(2)

n (x, t)∞∑k=0

1− iλt

6

[ω (6n+ 1) + t2

]√nSkn−1(x, t)

− ω2

24λ2t2G(2)

n (x, t)∞∑k=0

n

(7t2

ω+ 12

)Skn(x, t)

− λ2t4

144G(2)n (x, t)

(2t2 + 21ω

) ∞∑k=0

Skn(x, t), (4.35)

where

Skn(x, t) =

1√2kn!

(− it√

2ω

)n−kLn−kk ( t

2

2ω)Hk (

√ωx) , for n ≥ k√

n!2k

(− it√

2ω

)k−nk!

Lk−nn ( t2

2ω)Hk (

√ωx) , for k ≥ n

(4.36)

and

G(2)n (x, t) =

(ωπ

) 14N (2)n (t) exp

[− 1

4ω

(2ω2x2 + t2

)], (4.37)

with the normalization constant given by

N (2)n (t) =

1 +

ω2t8

128λ4(n2 + n+ 1

)+ωt8

288λ4 (2n+ 1)

[t2 + 6ω (2n+ 1)

]+ω3t6

288λ4 (2n+ 1)

(36n2 + 36n+ 13

)+ω2t6

72λ4[t2 + 3ω (2n+ 1)

]+

t4

20736λ4[6nω

(7t2 + 12ω

)+ t2

(2t2 + 21ω

)+ ω2

(72n2 + 6

)]2−1/2.

(4.38)

28

4.1. Harmonic Oscillator with linear term in potential.

Let us now consider the case when a coherent state, |ψ(0)〉 = |α〉, is the initial state.Its approximated solution in the position representation is

Ψ(2)α (x, t) =C(2)

α (x, t)

1−

√2

ω3/2iλe−iωt/2

[2iα(t)eiωt/2 sin (ωt/2) +

√2ωx

]sin (ωt/2)

−λ2

ω3α(t)C(2)

α (x, t)

α(t)

[2ieiωt/2 sin (ωt/2)− 1

]+ 2√

2ωx

sin2 (ωt/2)

−λ2

ω3e−iωtC(2)

α (x, t)

[√2ωx− α(t)

]2− 1

sin2 (ωt/2)

+iλ2

2ω3C(2)α (x, t)

[t− 2e−iωt/2 sin (ωt/2)

], (4.39)

with

C(2)α (x, t) =

(ωπ

)1/4N (2)α (t) exp

− 1

2

[itωt+ ωx2 + |α(t)|2 + α2(t)− 2α(t)

√2ωx

],

(4.40)and whose normalization constant is given by

N (2)α (t) =

1− 2

√2

ω9/2λ3<

(α(t)eiωt/2

)[t− sin (ωt)] sin (ωt/2) + 2

λ4

ω6sin4 (ωt/2)

+ 2λ4

ω6

[3|α(t)|2

(|α(t)|2 + 2

)+ 4

(|α(t)|2 + 1

)<(α2(t)eiωt

)]sin4 (ωt/2)

+λ4

ω6

[sin2 (ωt/2) +

(cos (ωt/2)− t

2csc (ωt/2)

)2]

sin2 (ωt/2)

+ 2λ4

ω6

[=(α2(t)e3iωt/2

)−=

(α2(t)eiωt/2

)]sin3 (ωt/2)

+ 2λ4

ω6<(α4(t)e2iωt

)sin4 (ωt/2)

−1/2. (4.41)

where = (z) means the imaginary part of z. The eigenstate representation (4.20)was applied to estimate the relationship (4.39), whereas the normalization constants

N(2)n (t) and N

(2)α (t) were found using the Eq.(3.38).

29

Chapter 4. Examples

4.1.3 Comparison of the exact and the perturbative solu-tions

We have to analyse now the accuracy of these approximated solutions; for that, wewill compare them with the exact solutions found in Subsection 2.1. In Figures 4.1and 4.2, we plot the probability density distributions, the modulus square of eachsolution

(a) (b)

0.0 0.5 1.0 1.5 2.0

-6

-4

-2

0

2

4

6

t

x

(c)

Figure 4.1: Probability density with quantum number n = 10. The left-hand side (a)shows the exact solution while the right-hand side (b) presents the approximate one.A contour plot of their probability densities is depicted on (c), where it shows that forshort times the approximate solution (red line) reproduces the exact one (black line)with highly accuracy whereas for t > 1.5 a very slight differences between them canbe detected, but, it still yields to good approximation. These graphs were obtainedby considering a perturbation strength equal to λ = 0.1 and a frequency of ω = 1.

30

4.1. Harmonic Oscillator with linear term in potential.

(a) (b)

0 2 4 6 8 10 12 14

-6

-4

-2

0

2

4

6

t

x

(c)

Figure 4.2: Probability density |ψ(x, t)|2 with α(0) = 3 and ω = 2. The left-hand side(a) shows the exact solution while the right-hand side (b) presents the approximateone. A contour plot of their probability densities on (x, t) plane is depicted on (c),where it shows that the approximation presents a remarkable high accuracy with theexact result under the consideration of a perturbation strength equal to λ = 0.5.

Here, Figs 4.2(a) and 4.2(b) show in a separate way the probability density withquantum number n = 10 corresponding to the exact and approximate result as func-tion of time and position; a comparison of the probability density on (x, t) plane ofboth solutions is presented in Fig.4.2(c), where it can be appreciated that at smalltimes, the second-order approximation, denoted by red dashed and which we referas PApro, matches accurately with the exact solution which is denoted by black lineand that we call as PExa. For later times like t = 1.5 our approximative expression(4.35) still reflects its effectiveness with a slight variation with the exact result. Such

31

Chapter 4. Examples

graphs are feasible whether one set a perturbation parameter equal to λ = 0.1 anda frequency ω = 1. On the other hand, Figure 4.2 presents the probability densitiesassociated with the coherent state with α = 3. In a similar fashion to the numberstate case, we show the exact and approximate results by separate in Figs 4.2(a) and4.2(b) whereas the comparison of both probability densities in the xt plane is pre-sented in Fig.4.2(c). From the above figure it is clear to notice that the perturbativeresult gives an excellent approximation to the exact solution that makes it difficultto distinguishable each other. In such plots we have set the perturbative parameteras λ = 0.5 and a frequency value of ω = 2.Until now it has been demonstrated that the Normalized Matrix perturbation Methodworks so well when one deals with weak perturbations. Now, it looks appropriatefocus our attention to the opposite case, in other words, the strong perturbativeregime. In order to examine this scenario with the linear anharmonic oscillatorwith a number state as initial condition; we must consider the general solution forstrong perturbations defined into Eq.2. Thus, using the proposed representation andapplying the same analytical procedure to get (4.35), we find

Ψ(2)n (x, τ) =G(2)n (x, τ)

∞∑k=0

1− i τ

6λ

[3ω (2n+ 1) + τ 2

]− τ 4

144λ2(2τ 2 + 21ω

)Skn

+ωτ 4

16λ2G(2)n (x, τ)

∞∑k=0

[√n (n− 1)Skn−2 +

√(n+ 1) (n+ 2)Skn+2

]−√

2ωτ 2

4λG(2)n (x, τ)

∞∑k=0

1− i τ

6λ

[ω (6n+ 5) + τ 2

]√n+ 1Skn+1

+

√2ωτ 2

4λG(2)n (x, τ)

∞∑k=0

1− i τ

6λ

[ω (6n+ 1) + τ 2

]√nSkn−1

− ω2τ 2

24λ2G(2)n (x, τ)

∞∑k=0

[12n2 + n

(7τ 2

ω+ 12

)+ 3

]Skn, (4.42)

where

Skn =

1√2kn!

(− iτ√

2ω

)n−kLn−kk ( τ

2

2ω)Hk (

√ωx) , for n ≥ k√

n!2k

(− iτ√

2ω

)k−nk!

Lk−nn ( τ2

2ω)Hk (

√ωx) , for k ≥ n

(4.43)

and

G(2)n (x, τ) =(ωπ

) 14N (2)n (x, τ) exp

[− 1

4ω

(2ω2x2 + τ 2

)], (4.44)

32

4.1. Harmonic Oscillator with linear term in potential.

with normalization constant defined as

N (2)n (τ) =

1 +

ω2τ 8

128λ4(n2 + n+ 1

)+

ωτ 8

288λ4(2n+ 1)

[τ 2 + 6ω (2n+ 1)

]+

ω3τ 6

288λ4(2n+ 1)

(36n2 + 36n+ 13

)+ω2τ 6

72λ4[τ 2 + 3ω (2n+ 1)

]+

τ 4

20736λ4[6nω

(7τ 2 + 12ω

)+ τ 2

(2τ 2 + 21ω

)+ ω2

(72n2 + 6

)]2−1/2.

(4.45)

The reader can see the underlying similarities between the set of equations for theapproximative solution at the strong case with those derived in the weak perturbativeregime, in fact, the one-to-one correspondence can be done by setting λ = 1 andexchanging H0 ↔ Hp.

-4 -2 2 4x

0.05

0.10

0.15

ÈΨnHx,ΤL 2

(a) t = 0.5

-4 -2 2 4x

0.05

0.10

0.15

ÈΨnHx,ΤL 2

(b) t = 1.5

Figure 4.3: Probability density with quantum number n = 20 at (a) t = 0.5 and(b) t = 1.5. The approximation given by PApro is highly accurate and comparesfavorably with the exact result, PExa, these solutions are get in the strong-couplingperturbation regime with a perturbed coefficient value of λ = 35.

Figures 4.3 (a) and 4.3 (b) illustrate the comparison between PExa versus PApro inthe strong perturbative regime at two different times t = 0.5 and t = 1.5. As can beseen from these figures, PExa is barely distinguishable from PApro for a perturbativeparameter of λ = 35. In fact, it is logical to expect that PApro reproduces very wellto PExa when one choose large values of λ; due the factor 1/λ becomes very smallas λ grows, causing that the effects of harmonic oscillator on linear anharmonicitywill be too small. Such result in our system under study is helpful because providesclear evidence about how behaves solutions of the normalized matrix method in theframework of strong coupling regime. We must to remark that the change t→ τ/λis applied to the exact solution to ensure properly it comparison with Eq.(4.42).

33

Chapter 4. Examples

4.2 The cubic anharmonic oscillator

There exists a vast amount of literature about studies of the quantum oscillator withhigher-order anharmonicities [64–70]. The analyzed example of previous section isconsidered as the simple but fundamental model to this kind of systems which provespossess a closed form analytical solution. For us, looks instructive to go beyond andexplore the applicability of our normalized approach into an anharmonic oscillatorwhich is not amenable to exact treatment. For sake of simplicity, let us tackle theexample of previous section with a more complex anharmonicity term. In this case,we replace the linear term in (4.1) by a cubic one. Then its Hamiltonian can bewritten as

H =1

2p2 +

ω2

2x2 + λx3. (4.46)

The Schrodinger equation associated with this Hamiltonian in terms of the harmonicoscillator operators a and a† is

id

dt|ψ〉 =

[ω (n+ 1/2) +

λ

(2ω)3/2(a+ a†

)3] |ψ〉 , (4.47)

it is evidently that the cubic anharmonic contribution is less algebraically tractableand thereby makes it unviable to find a closed form solution into the Schrodingerequation. Hence, a numerical procedure or approximately technique must be resortedto solve it. For practical purposes, we solve(4.47) perturbatively and after thencompared it with its numerical solution, which is computationally feasible throughthe Numerov method [71–76]. It is worth mention that we will not discuss thenumerical aspects of this approach and that others well-known numerical schemescan be used to solve the same task [77–80]. Let us now decompose the Hamiltonian(4.46) in two pieces, H = H0 + λHp, the first half corresponds to unperturbed part

H0 = ω (n+ 1/2) while the contribution of second term is given by the perturbation

Hp =(a+a†√

2ω

)3. Under these assumptions and making use of Eq.(3.17) and the

identities (4.27), the first and second-order terms are obtained

|ψ1,2〉 =− i√2ω5/2

exp

[−itω

(n+

1

2

)] 1∑j=0

sin [(j + 1/2)ωt]

32j−1

×[(a†2eiωt

)j+1/2+(a2e−iωt

)j+1/2]n1−j |ψ(0)〉

− 3i√2ω5/2

sin (ωt/2) e−itωna† |ψ(0)〉 ,

|ψ1,3〉 =e−itω(n+ 1

2)

288ω5

13∑j=1

gj |ψ(0)〉+15it

4ω4e−itω(n+ 1

2)(n2 + n+

11

30

)|ψ(0)〉 , (4.48)

34

4.2. The cubic anharmonic oscillator

where

g1 = −36ie−3iωt/2a2n2

2 sin(3ωt/2) + 7eiωt sin(ωt)− 4e3iωt/2[i sin(ωt/2)− 4e−iωt

],

(4.49a)

g2 = 36ie3iωt/2a†2n2

2 sin(3ωt/2) + 7e−iωt/2 sin(ωt)− 2e−iωt[9 + 2e−iωt

]sin(ωt/2)

,

(4.49b)

g3 = 18ie2iωta†4

4[sin(2ωt)− e−iωt/2 sin(3ωt/2)− e−3iωt/2 sin(ωt/2)

](n+ 1)

+ 3 sin(2ωt) + 4e−iωt/2 sin(3ωt/2)− i

(4.49c)

g4 = −18ie−2iωta4

4n[sin(2ωt)− eiωt/2 sin(3ωt/2)− e5iωt/2 sin(ωt/2)

]+ 3

[4e3iωt/2 sin(ωt/2) + e4iωt sin(2ωt)

] , (4.49d)

g5 = −36ie−3iωt/2a2[2 sin(3ωt/2) + 3eiωt/2 sin(ωt)− 12e2iωt sin(ωt/2)

], (4.49e)

g6 = −24in2[sin(3ωt) + 54e−iωt/2 sin(ωt/2) + e−3iωt/2 sin(3ωt/2)

], (4.49f)

g7 = 36ieiωta†2n

33 sin(ωt)− 2e−iωt/2[27 + 6e−iωt

]sin(ωt/2)

, (4.49g)

g8 = 324ieiωt/2a†2

2 [2i sin(ωt)− 1] sin(ωt/2) + eiωt/2 sin(ωt), (4.49h)

g9 = 8ine−iωt2

[11e−3iωt − 2

]e2iωt sin(3ωt/2) + 243 sin(ωt/2)

, (4.49i)

g10 = 4ie3iωt/2a†6[e3iωt/2 sin(3ωt)− 2 sin(3ωt/2)

]− e−3iωta6

[e−3iωt/2 sin(3ωt)− 2 sin(3ωt/2)

] , (4.49j)

g11 = −36ie−3iωt/2a2n

4 sin(3ωt/2)− 2eiωt sin(ωt)

+ 4eiωt[5ieiωt/2 sin(ωt/2)− 2

]sin(ωt/2)

, (4.49k)

g12 = −24e−iωt[2 sin(ωt) + 27eωt/2 sin(ωt/2)

], (4.49l)

g13 = −8n3[sin2 (3ωt/2) + 81 sin2 (ωt/2)

]. (4.49m)

In analogy with the linear anharmonic example, we choose an eigenstate of theharmonic oscillator, i.e., a number state, |ψ(0)〉 = |n〉. The results to second orderare very long and have been stated already in expression (4.48), thus, we present

35

Chapter 4. Examples

explicitly only the solution correction to first order,

|Ψ(t)〉 =− 3iλN(1)n (t)e−iE

(0)n t

√2ω5/2

sin(ωt/2)[e−iωt/2 (n+ 1)

32 |n+ 1〉+ eiωt/2n

32 |n− 1〉

]− iλN

(1)n (t)e−iE

(0)n t

3√

2ω5/2e−3iωt/2

√(n+ 1) (n+ 2) (n+ 3) sin(3ωt/2) |n+ 3〉

− iλN(1)n (t)e−iE

(0)n t

3√

2ω5/2e3iωt/2

√n (n− 1) (n− 2) sin(3ωt/2) |n− 3〉

+N (1)n (t)e−iE

(0)n t |n〉 , (4.50)

with its respective normalization constant

N (1)n (t) =

1 +

λ2

18ω5(2n+ 1)

(n2 + n+ 6

)sin2

(3ωt

2

)+

9λ2

2ω5(2n+ 1)

(n2 + n+ 1

)sin2

(ωt

2

)− 12. (4.51)

In the case of energy, we have

En = E(0)n + λE(1)

n + λ2E(2)n =ω

(n+

1

2

)− λ2 15

4ω4

(n2 + n+

11

30

). (4.52)

As one expect, the first term of energy is equal to zero due perturbation x3 is odd,this means 〈k| x3 |k〉 = 0. Above implies that the energy correction due to x3 isof second order. This can be seen clearly from last term of relation (4.48) andwith more detail from Eq.(3.27). In fact, The results for the wavefunction and forthe energy coincide exactly with the ones obtained by the standard perturbationtheory [4,6,67,69,72,81]. Now, returning back to the approximate solution, we havethat its form in the coordinate representation is

Ψ(1)n (x, t) =N (1)

n (t)e−iE(0)n tψ(0)n − i

λe−3iωt/2

3√

2ω5/2

√(n+ 1)(n+ 2)(n+ 3) sin(3ωt/2)ψ

(0)n+3

− i λe

3iωt/2

3√

2ω5/2N (1)n (t)e−iE

(0)n t√n(n− 1)(n− 2) sin(3ωt/2)ψ

(0)n−3

− 3iλe−iωt/2√

2ω5/2N (1)n (t)e−iE

(0)n t sin(ωt/2)

[(n+ 1)3/2 ψ

(0)n+1 + eiωtn3/2ψ

(0)n−1

],

(4.53)

36

4.2. The cubic anharmonic oscillator

PRS

PApro

PNu

-10 -5 0 5 10x

0.05

0.10

0.15

0.20

0.25

0.30

ÈΨnHx,tL 2

(a) λ = 0.005

PRS

PApro

PNu

-10 -5 0 5 10x

0.05

0.10

0.15

0.20

0.25

0.30

ÈΨnHx,tL 2

(b) λ = 0.01

Figure 4.4: Probability density of the cubic anharmonic oscillator versus x for aquantum number n = 20. One can notice from (a) that with λ = 0.005, the first-order solution, PApro, does not differs substantially from numerical result, PNu, unliketo PRS whose behavior shows several discrepance. Once increases the perturbationparameter to (b) λ = 0.01, PApro as well PRS become completely inadequate todescribe the numerical solution, being the differences more notorious for PRS. Thenumerical result was performed with t = 1 and ω = 1.

In figures 4.4a and 4.4b we plot the modulus squared of our approximative so-lution, PApro, against position x. Indeed, PApro is compared with its numericalsolution, PNu, and also with the approximate solution obtained under the schemeof Rayleigh-Schrodinger (RS) perturbation theory. For the latter, its expression is

37

Chapter 4. Examples

similar to (4.53) but without the normalization constant N(1)n (t). This can get di-

rectly from the Dirac perturbation formalism defined in Chapter 2 but with specialconsideration that Hp(t)→ Hp is implicitly time-independent.From Fig.4.4 it is seen that with a perturbation parameter λ = 0.005, both ap-proximate results hold with good agreement with PNu, although is notorious thatthe accuracy of PApro is higher than PRS. It is clear that as product λt increases,the error in the approximative solutions increases too. Figure 4.4b displays thisfact with λ = 0.01, here PApro is no longer a good approximation with PNu, butkept significantly better than PRS which completely fails to reproduce the numericalresult. As a side note, we would like to emphasize once again, that these approx-imations have been obtained by taking into account only the first correction term;eventually, it is feasible to reach with accurate approximative solutions of reliabilitywhether the higher-order corrections are included. Now, we select as initial condition|ψ(0)〉 = |α〉, in this case we have

|Ψ(t)〉 =N (1)α (t)e−i

ωt2 |α(t)〉 − iλα(t)

N(1)α (t)

3√

2ω5/2

[α2(t)eiωt sin

(3ωt

2

)+ 9 sin

(ωt

2

)]|α(t)〉

−iλ N(1)α (t)e−2iωt

3α3(t)√

2ω5/2sin

(3ωt

2

)e−|α|22

∞∑n=0

[α(t)]n√n!

n(n− 1)(n− 2) |n〉

−3iλN

(1)α (t)e−iωt

α(t)√

2ω5/2sin

(ωt

2

)e−|α|22

∞∑n=0

[α(t)]n√n!

n[n+ α2(t)eiωt

]|n〉 , (4.54)

with

N (1)α (t) =

1 +

λ2

18ω5

(162|α(t)|4 + 648|α(t)|2 + 324

)<(α2(t)eiωt) sin2

(ωt

2

)+λ2

ω5(2|α(t)|4 + 8|α(t)|2 + 9)<(α2(t)eiωt) sin

(3ωt

2

)sin

(ωt

2

)+λ2

ω5(2|α(t)|2 + 5)<(α4(t)e2iωt) sin

(3ωt

2

)sin

(ωt

2

)+

λ2

18ω5|α(t)|2

(2|α(t)|4 + 9|α(t)|2 + 18

)sin2

(3ωt

2

)+

λ2

18ω5

(162|α(t)|6 + 729|α(t)|4

)sin2

(ωt

2

)+

λ2

9ω5

[<(α6(t)e3iωt) + 3

]sin2

(3ωt

2

)

+9λ2

2ω5sin2

(ωt

2

)[8|α(t)|2 + 1

]− 12

. (4.55)

38

4.2. The cubic anharmonic oscillator

Its corresponding approximate solution at first order correction in x representation,it can be found by doing the inner product, 〈x|Ψ(t)〉 and with the aid of positionstate (4.20) to yields

Ψ(1)α (x, t) =F (1)

α (x, t)

1− iα(t)λe3iωt/2

3√

2ω5/2

[α2(t) sin

(3ωt

2

)+ 9e−iωt sin

(ωt

2

)]−3i

λeiωt/2√2ω5/2

F (1)α (x, t)

[α2(t) + e−iωt

] [√2ωx− α(t)

]sin

(ωt

2

)−i3λe

−iωt/2√

2ω5/2α(t)F (1)

α (x, t) sin

(ωt

2

)[√2ωx− α(t)

]2−iλe

−3iωt/2

3√

2ω5/2F (1)α (x, t)

[√2ωx− α(t)

]3sin

(3ωt

2

)+iλe−3iωt/2√

2ω5/2F (1)α (x, t)

[√2ωx− α(t)

]sin

(3ωt

2

)+3iα(t)

λe−iωt/2√2ω5/2

F (1)α (x, t) sin

(ωt

2

), (4.56)

where F(1)α (x, t) =

(ωπ

)1/4N

(1)α (t) exp

−1

2

[iωt+ x

(ωx− 2α(t)

√2ω)

+ |α(t)|2 + α2(t)]

.Once we know the solution of (4.47) with a coherent state, it will be easy for usto determine its approximative form under application of a Schrodinger-cat state,

|ψ(0)〉 = |α〉+eiθ|−α〉√2(1+cos(θ)e−2|α|2)

. Thus, the solution of cubic anharmonic oscillator with

this initial condition is

Ψ(1)cat(x, t) = − iλ

3√

2ω52

F(1)cat (x, t)

[√2ωx+ α(t)

]3eiφ +

[√2ωx− α(t)

]3e−iφ

sin

(3ωt

2

)

+3iαλ√

2ω52

F(1)cat (x, t)

[√2ωx+ α(t)

]2eiφ −

[√2ωx− α(t)

]2e−iφ

sin

(ωt

2

)−2α(t)e3iωtλ

3√

2ω52

F(1)cat (x, t)

[α2(t) sin

(3ωt

2

)+ 18ie−

5iωt2 sin2

(ωt

2

)]sinφ

+2iλ√

2ω52

F(1)cat (x, t)

sin

(3ωt

2

)− 3e2iωt

[α2(t) + e−iωt

]sin

(ωt

2

)×[√

2ωx cosφ+ iα(t) sinφ]

+ 2e3iωt/2F(1)cat (x, t) cosφ, (4.57)

where

F(1)cat (x, t) =

(ωπ

)1/4N

(1)cat(t) exp

−1

2

[i (4ωt− θ) + ωx2 + |α(t)|2 + α2(t)

](4.58)

φ =θ

2+ iα(t)

√2ωx (4.59)

39

Chapter 4. Examples

and whose normalization constant is defined as

N(1)cat(t) =

[2 +

λ2

9ω5

[2<(α6(t)e3iωt) + 9|α(t)|4 + 6

]sin2

(3ωt

2

)+

λ2

9ω5

[(162|α(t)|4 + 324

)<(α2(t)eiωt) + 81

]sin2

(ωt

2

)+ |α(t)|2 λ

2

9ω5

[162|α(t)|4 + 729|α(t)|2

]sin2

(ωt

2

)+ 2

λ2

ω5

(2|α(t)|4 + 9

)<(α2(t)eiωt) sin

(3ωt

2

)sin

(ωt

2

)+ 10

λ2

ω5<(α4(t)e2iωt) sin

(3ωt

2

)sin

(ωt

2

)(1 + e−2|α(t)|

2

cos θ)

+

2|α(t)|2 λ

2

9ω5

(|α(t)|4 + 9

)sin2

(3ωt

2

)+ 72|α(t)|2 λ

2

ω5

[1 + <(α2(t)eiωt)

]sin2

(ωt

2

)+ 4|α(t)|2 λ

2

ω5<(α4(t)e2iωt) sin

(3ωt

2

)sin

(ωt

2

)

+ 16|α(t)|2 λ2

ω5<(α2(t)eiωt) sin

(3ωt

2

)sin

(ωt

2

)(1− e−2|α(t)|

2

cos θ)]− 1

2

.

(4.60)

The probability densities |ψα(x, t)|2 and |ψcat(x, t)|2 with α = 2 + 3i are plotted inFigure 4.5 as a function of x. The left graph corresponding to |ψα(x, t)|2 shows ushow PApro is approximated as closely as possible to PNu at t = 1. In contrast tothe approximation PRS, which is too large to be compatible with PNu. The samesituation occurs for the probability distribution of a cat state, |ψcat(x, t)|2, right-hand side graph, for t = 5.55 and θ = π/7, where the interference pattern (dueto the coherent superposition) of PApro fits very near with the behaviour of PNuwhereas PRS gives rise to a poor approximation with both of them; this result putseven more in manifest that our approach offers a higher accuracy than the standardperturbation theory. For the above graphs, we have set a perturbation coefficient ofλ = 0.005.

40

4.2. The cubic anharmonic oscillator

PApro

PNu

PRS

-4 -2 0 2 4 6 8 10x

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ÈYΑHx,tL 2

(a) t = 1

PApro

PNu

PRS

-3 -2 -1 0 1 2 3x

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ÈYcatHx,tL 2

(b) t = 5.55

Figure 4.5: Density probability of cubic anharmonic oscillator for a coherent state(a) and (b) a Schrodinger cat state. Notice in (a) that PApro gives a solution veryclose to PNu while PRS remains to far of reproduce it. This become more evidentfor the interference pattern generated by the Schrodinger cat state in (b), where PRSfails completely when it compared with PNu; on the contrary, PApro yields to reliableresult with PNu. The above graphs were carried by using the parameters λ = 0.005,ω = 1, α = 2 + 3i and θ = π/7.

41

Chapter 4. Examples

4.3 Repulsive harmonic oscillator.

As far now, the perturbed matrix method has been tested on systems where onecan find simultaneously a well define unitary evolution operator for the unperturbedHamiltonian and its eigenstates. The simple harmonic oscillator is an instance modelwhich exhibit this nature. In effect, this singular example maintain first statementtrue whereas second one could fails if, and only if, the oscillator frecuency switchesfrom real to imaginary, i.e, ω → iω [82–86]. If this happens, the harmonic oscillatorturns upside-down and gives rise to a repulsive oscillator which has the form

H =1

2

(p2 − ω2x2

), (4.61)

despite that representation of harmonic and repulsive oscillator are very alike, theydo not share same physical features. For example, several authors have reportedthat the inverted oscillator admits continuous eigenvalues, purely imaginary, whoseeigenfunctions are not square-integrable, while the harmonic system have discretereal eigenvalues with square integrable eigenfunctions [87–92]. In spite of their re-markable differences, the repulsive oscillator is exactly solvable just like standardharmonic oscillator. Indeed, a large number of manuscripts have been publishedwith respect to get its exact solution [83–85, 89, 90], nevertheless, most of them in-volve a differential equation techniques. In fact, many researchers assume from thebeginning of its analysis the simple change ω into iω in the harmonic oscillator toreach to repulsive one. In particular, at first sight for us looks completely incorrectdue one arrives to complex eigenvalues without physical quantities interpretationsin any way and which contradict the philosophy of quantum mechanics. However, iteasy to show that solution of the proposed model in Eq.(4.61) can be represented bya well known ubiquitous operator in quantum optics through use of the ladder oper-ators (4.2) of harmonic oscillator. So, once do it, we can write the time-dependentSchrodinger equation of the Hamiltonian (4.61) as

id

dt|ψ〉 = i

ω

2

(a2ei

π2 − a†2e−i

π2

)|ψ〉 , (4.62)

whose formal solution is

|ψ(t)〉 = exp

[ωt

2

(a2ei

π2 − a†2e−i

π2

) ]|ψ(0)〉 . (4.63)

It is important to note that the propagator is a squeeze operator S(ξ) = exp[12

(ξ∗a2 − ξa†2

)][52, 93, 94], with a time dependent squeeze amplitude r = ωt and a squeeze phaseequal to θ = −π/2. Thus, it is found that the solution of (4.63) can be written witha squeeze operator with a purely imaginary parameter ξ = −iωt, i.e.

|ψ(t)〉 = S (−iωt) |ψ(0)〉 . (4.64)

42

4.3. Repulsive harmonic oscillator.

Besides, the operators in the exponential (4.63) together with 12

(n+ 1/2), form

a su(1,1) Lie algebra; in fact,[K+, K−

]= −2K0 and

[K0, K±

]= ±K±, with

K+ = a†2

2, K− = a2

2and K0 = 1

2(n+ 1/2). According to the structure of this

Lie algebra and employing the factorization theorem [95, 96], we write the squeezeoperator as the product of three exponentials,

S(ξ) = exp(y1K

+)

exp(y2K

0)

exp(y3K

−), (4.65)

where the functions y1, y2 and y3 are determined by the following differential equa-tions

y1(r)− y21(r)e−iθ + eiθ = 0, (4.66)

y2(r)− 2y1(r)e−iθ = 0,

y3(r)− e−iθ exp [y2(r)] = 0,

with the following initial conditions

y1(0) = y2(0) = y3(0) = 0. (4.67)

Solving the above system of differential equations with the initial conditions, givesus

S(ξ) =√

sech(|ξ|) exp

[− ξ

2|ξ|tanh(|ξ|)a†2

]exp

[− ln | cosh(|ξ|)|a†a

]exp

[ξ∗

2|ξ|tanh(|ξ|)a2

].

(4.68)The factorized form of the propagator allow us to evaluate, in a straightforward way,the exact solution (4.64) with any initial condition. In particular, if the system startsin a coherent state, |ψ(0)〉 = |β〉, it evolves into a squeezed coherent state of the form

|ψ(t)〉 =√

sech(ωt) exp

β(t)

4

[iβ(t) sinh (2ωt)− 2β∗(t) sinh2 (ωt)

]

×∞∑n=0

[i2

tanh(ωt)]n

n!

[∂

∂β(t)+β∗(t)

2

]2n|β(t)〉 , (4.69)

with β(t) = β sech(ωt) and where we used that the action of the raising and loweringoperators over a coherent state is a |α〉 = α |α〉 and a† |α〉 =

(∂∂α

+ α∗

2

)|α〉 [47, 48].

To find the solution in the coordinates space, ψβ(x, t), we multiply both sides of(4.69) by (4.20)

ψβ(x, t) =√

sech(ωt)(ωπ

)1/4eη(t)

⟨√2ωx

∣∣∣ exp

(− a

2

2

)exp

[i

2tanh(ωt)a†2

]|β(t)〉 ,

(4.70)

43

Chapter 4. Examples

being η(t) = β(t)4

[iβ(t) sinh (2ωt)− 2β∗(t) sinh2 (ωt)

]+ ωx2

2. We write exp

(− a2

2

)exp

[i2

tanh(ωt)a†2]

as exp(− a2

2

)exp

[i2

tanh(ωt)a†2]

exp(a2

)exp

(− a

2

), where we have

inserted the identity operator I = exp(a2

)exp

(− a

2

), and used the fact that exp

(−a2

2

)exp

(a2

2

)= a†2 − 2 (n+ 1/2) + a2; to obtain

ψβ(x, t) =(ωπ

)1/4√sech(ωt) exp

[η(t)− β2(t)

2

]×⟨√

2ωx∣∣∣ exp

[i tanh (ωt)

(K+ − 2K0 + K−

)]|β(t)〉 . (4.71)

To factorize the operator exp[i tanh (ωt)

(K+ − 2K0 + K−

)], we propose the ansatz

F (t) = exp[i tanh (ωt)

(K+ − 2K0 + K−

)]= exp

[f1(t)K

+]

exp[f0(t)K

0]

exp[f2(t)K

−], (4.72)

where f0(t), f1(t) and f2(t) are unknown functions which satisfy the conditionsf0(0) = f1(0) = f2(0) = 0. If we differentiate the above with respect to timeand equating the su(1,1) Lie generators on both sides, we get the following systemof first-order differential equations

f1(t)− iω sech2(ωt)f 21 (t) + 2iω sech2(ωt)f1(t)− iω sech2(ωt) =0,

fo(t)− 2iω sech2(ωt)f1(t) + 2iω sech2(ωt) =0,

f2 − iω sech2(ωt)ef0 =0, (4.73)

whose solutions is

f1(t) =f2(t) =1

1− i coth (ωt),

fo(t) = ln

[1

(1 + i tanh (ωt))2

]. (4.74)

Hence, Eq.(4.71) may be written as

ψβ(x, t) =(ωπ

)1/4√sech(ωt) exp

η(t)− 1

4

[2β2(t)− f0(t)

]×⟨√

2ωx∣∣∣ exp

[f1(t)

2a†2]

exp

[f0(t)

2n

]exp

[f2(t)

2a2]|β(t)〉 . (4.75)

44

4.3. Repulsive harmonic oscillator.

Making act the operators a†2, a2 and n on the coherent states∣∣√2ωx

⟩and |β(t)〉

lead us to

ψβ(x, t) =(ωπ

)1/4√sech(ωt) exp

√2ω xβ(t) exp [f0(t)/2] + ωx2 [f1(t)− 1] + η(t)

× exp

−β

2(t)

2[1− f2(t)]−

|β(t)|2

2+f0(t)

4

, (4.76)

that after some algebraic manipulations, can be re-written as

ψβ(x, t) =

√√ω/π sech(ωt)

1 + i tanh(ωt)exp

β

2

[−β∗ + iβ tanh(ωt) +

4x√ω csch(2ωt) tanh(ωt)

1 + i tanh(ωt)

]

× exp

−ωx

2

2+i tanh(ωt) [ωx2 + iβ2 csch(2ωt)]

1 + i tanh(ωt)

. (4.77)

From the above, it is easy to see that the exact solution of the repulsive oscillator,when is initially prepared in a Schrodinger cat state is

ψcat(x, t) =ψβ(x, t) + eiφψ−β(x, t)√

2(1 + cos(φ)e−2|β|

2) (4.78)

A second way to get same result is start from definition of the annihilation operatoracting over vacuum state a |0〉 = 0, then, one can insert the identity operator I =D†(β)S† (ξ) S (ξ) D(β) into the left and right side of a as

D†(β)S† (ξ) S (ξ) D(β)aD†(β)S† (ξ) S (ξ) D(β) |0〉 = 0, (4.79)

as is well known a squeezed coherent state is obtained via |ξ, β〉 = S (ξ) D (β) |0〉,and with help of D (β) aD† (β) = a− β, we arrive to

D†(β)S† (ξ) S (ξ) (a− β) S† (ξ) |ξ, β〉 = 0, (4.80)

multiplying both sides of previous relation by S (ξ) D (β) and using the fact that thesqueeze operator acts on the annihilation operator as S (ξ) aS† (ξ) = a cosh (|ξ|) +a† ξ|ξ| sinh (|ξ|), we have[

a cosh (|ξ|) + a†ξ

|ξ|sinh (|ξ|)

]|ξ, β〉 = β |ξ, β〉 . (4.81)

Expanding the coherent squeezed states in the Fock bases, |ξ, β〉 =∑∞

n=0 cn |n〉, leadsto

∞∑n=0

[cosh (|ξ|) cn+1

√n+ 1 +

ξ

|ξ|sinh (|ξ|) cn−1

√n

]|n〉 = β

∞∑n=0

cn |n〉 . (4.82)

45

Chapter 4. Examples

If we define cn = N√cosh(|ξ|)

[ξ

2|ξ| tanh (|ξ|)]n/2

fn(x) and replace it in the previous

relationship, one finds

√n+ 1fn+1(x) + 2

√nfn−1(x)− 2

√|ξ|

ξ sinh (2|ξ|)βfn(x) = 0, (4.83)

note that when fn = 1√n!Hn(x) and x =

√|ξ|

2ξ tanh(|ξ|)β , the above relation is then

transformed to the recursion relation of Hermite polynomials. With this on mind,the expansion coefficients cn can be written as

cn =N√

n! cosh (|ξ|)

[ξ

2|ξ|tanh (|ξ|)

]n/2Hn

(√|ξ|

ξ sinh (2|ξ|)β

), (4.84)

where N = c0√

cosh (|ξ|) = 〈0|ξ, β〉√

cosh (|ξ|). To determined 〈0|ξ, β〉, we employthe definition of Eq.(4.65) which leads directly to

〈0|ξ, β〉 =1√

cosh(|ξ|)〈0| e

ξ∗2|ξ| tanh(|ξ|)a

2

|β〉

=1√

cosh(|ξ|)exp

[−|β|

2

2+

ξ∗

2|ξ|β2 tanh(|ξ|)

]. (4.85)

Hence, the squeezed coherent state is then represented by

|ξ, β〉 =1√

cosh(|ξ|)exp

[−|β|

2

2+

ξ∗

2|ξ|β2 tanh(|ξ|)

]

×∞∑n=0

[ξ

2|ξ| tanh (|ξ|)]n/2

√n!

Hn

(√|ξ|

ξ sinh (2|ξ|)β

)|n〉 . (4.86)

This general expression of the squeezed states, in terms of Hermite polynomialsHn(x), is very similar to that obtained in reference [97], which comes from the pro-posal to get a closed form for the light intensity distribution in squeezed photonic lat-tices. Leaving aside definitions and performing the inner product ψβ(x, t) = 〈x|ξ, β〉,we get

ψβ(x, t) =(ω/π)1/4√cosh(|ξ|)

exp

1

2

[−|β|2 − ωx2 +

ξ∗

|ξ|β2 tanh(|ξ|)

]

×∞∑n=0

[ξ|ξ| tanh (|ξ|)

]n/22nn!

Hn

(√|ξ|

ξ sinh (2|ξ|)β

)Hn

(√ωx), (4.87)

46

4.3. Repulsive harmonic oscillator.

where we have employed the expression for the eigenfunctions of the harmonic oscil-lator

〈x|n〉 =1√2nn!

(ωπ

)1/4e−

ωx2

2 Hn

(√ωx). (4.88)

Recalling that the classical Mehler formula for the product of two Hermite polyno-mials [98, 99] is

∞∑n=0

Hn(x)Hn(y)

n!

(z2

)n=

1√1− z2

exp

[2xyz − (x2 + y2) z2

1− z2

], (4.89)

one arrives to the solution

ψβ(x, t) =

√√√√√ω/π sech(|ξ|)

1− ξ|ξ| tanh(|ξ|)

exp

β

2|ξ|

[−|ξ|β∗ +

4|ξ|2x√ω tanh(|ξ|) csch(2|ξ|)|ξ| − ξ tanh(|ξ|),

]

× exp

−ωx22+ξ∗β2

2|ξ|tanh(|ξ|)−

tanh(|ξ|)[ξωx2 + |ξ|β2 csch(2|ξ|)

]|ξ| − ξ tanh(|ξ|),

(4.90)

Considering that ξ = −iωt one reaches to Eq.(4.77) as expected. The same procedurecan be employed to find the solution of Eq.(4.78).

47

Chapter 4. Examples

We would like to know how the solutions behave for different values of β. Figure4.6 displays a three-dimensional plot of the modulus squared of ψβ(x, t) and ψcat(x, t)as a function of time and position.

(a) (b)

0 1 2 3 4

-200

-100

0

100

200

t

x

(c)

0 1 2 3 4

-200

-100

0

100

200

t

(d)

Figure 4.6: Probability density of the repulsive harmonic oscillator with (a) a coher-ent state and (b) a Schrodinger cat state as initial states. In both cases the solutionspresent a parabolic behaviour; the values of the parameters are β = 6, ω = 1 andφ = π/5. Figures (c) and (d) depict the probability density distribution on the (x, t)plane for the same initial states. The solid, dotted and dashed lines represent thecases of β=2, 5 and 9, respectively.

From Fig 4.6(a), it can be appreciated that the solution of the repulsive oscillatorfor an initial coherent state exhibits a curved amplitude squeeze behaviour in thepositive x-axis direction. Meanwhile, for a cat state, see Fig 4.6(b), the same patternrepeats itself in both directions. Nonetheless, the squeezing curvature becomes lesspronounced whether the coherent amplitude β is small; this can be seen clearly inFigures 4.6 (c) and 4.6 (d) for the 1D scenario for the probability density in the (x, t)

48

4.3. Repulsive harmonic oscillator.

plane. Fig 4.6 (c) shows the results associated with the coherent state with β =2,5 y 9, which are depicted by solid, dotted and dashed lines, respectively. One cannotice that the degree of curvature of the solution for β = 9 is more remarkable thanfor β = 2; these results suggest that it is possible to control the degree of squeezingby increasing or decreasing the value of β. The above argument not only applies tothe case of a coherent state but also for the coherent superposition; this is clearlyreflected in Fig 4.6(d), but with the main difference that the squeeze curvature startsto divide in two curves with opposite directions when β takes large enough values.At first sight, one would expect the quantum interference effects characteristic ofSchrodinger cat states; however, this is not the case since what we really have is asuperposition of squeezed coherent states. The situation could change if one evalu-

ates the system under the initial condition |ψ(0)〉 = S†(−iωt)|β〉+eiφS†(−iωt)|−β〉√2[1+cos(φ)e−2|β|2 ]

, when

the interference is recovered.

4.3.1 Linear anharmonic repulsive oscillator

4.3.2 Exact solution

In a analogous procedure with the example 1, we have also explored the situationof linear repulsive anharmonic oscillator. The Schrodinger equation associated withthis system reads as

id

dt|Ψ(t)〉 =

1

2

[p2 − ω2x2 + λx

]|Ψ(t)〉 , (4.91)

where we have used the notation Ψ instead of ψ to distinguish the solutions of thismodel to those obtained without the linear anharmonicity. Now, using the laddersoperators (4.2), we may rewrite the previous Schrodinger equation as

id

dt|Ψ(t)〉 =

[−ω

2

(a2 + a†2

)+

λ√2ω

(a+ a†

)]|Ψ(t)〉 . (4.92)

The second term on the right side is fully removed by applying the transformation

|φ(t)〉 = D†(

λω√2ω

)|Ψ(t)〉. Then, Eq. (4.92) becomes

id

dt|φ(t)〉 =

1

2

[−ω

(a2 + a†2

)+

(λ

ω

)2]|φ(t)〉 , (4.93)

which leads to the repulsive oscillator shifted by a factor λ2

2ω2 . Integrating (4.93) withrespect to t and after then transforming back to |Ψ(t)〉, we get

|Ψ(t)〉 = exp

(−i tλ

2

2ω2

)D

(λ

ω√

2ω

)S (−iωt) D†

(λ

ω√

2ω

)|Ψ(0)〉 . (4.94)

49

Chapter 4. Examples

This expression can be simplified further by inserting the identity operator I =S† (−iωt) S (−iωt),

|Ψ(t)〉 = e−itλ2

2ω2 D

(λ

ω√

2ω

)S (−iωt) D†

(λ

ω√

2ω

)S† (−iωt) S (−iωt) |Ψ(0)〉 .

(4.95)With the relations

S (ξ) aS† (ξ) =a cosh (|ξ|) +ξ

|ξ|a† sinh (|ξ|) ,

S (ξ) a†S† (ξ) =a† cosh (|ξ|) +ξ∗

|ξ|a sinh (|ξ|) , (4.96)

it is easy to prove that S (−iωt)(a† − a

)kS† (−iωt) =

[ν(t)a† − ν∗(t)a

]k, where

ν(t) = cosh (ωt) + i sinh (ωt), thus

S (−iωt) D†(

λ

ω√

2ω

)S† (−iωt) = D†

(λ

ω√

2ων(t)

). (4.97)

Eq.(4.95) can then be rewritten as

|Ψ(t)〉 = exp

(−i tλ

2

2ω2

)D

(λ

ω√

2ω

)D†(

λ

ω√

2ων(t)

)S (−iωt) |Ψ(0)〉 , (4.98)

with the help of displacement operator identities D (α) D (γ) = e(αγ∗−α∗γ)/2D (α + γ)

and D(α) = e−|α|22 eαa

†e−α

∗a, in conjunction with the factorized version of the squeezeoperator (4.68). The exact solution of linear repulsive anharmonic oscillator is foundto be

|Ψ(t)〉 =√

sech(ωt) exp[−λ2µ(t)/2

]exp

[λζ(t)a†

]exp [−λζ∗(t)a] exp

[i

2tanh (ωt) a†2

]exp − ln [cosh (ωt)] n exp

[i

2tanh (ωt) a2

]|Ψ(0)〉 , (4.99)

where

µ(t) =i

ω3[ωt− sinh (ωt)] + |ζ(t)|2, (4.100)

and

ζ(t) = − 1

ω√

2ω

[2 sinh2 (ωt/2) + i sinh (ωt)

]. (4.101)

To find the solution in the coordinate representation, Ψβ(x, t), we choose |Ψ(0)〉 = |β〉and multiply both sides of (4.99) by 〈x|, obtaining

Ψβ(x, t) =(ωπ

)1/4√sech(ωt) exp

−1

2

[λ2µ(t)− ωx2

]⟨√2ωx

∣∣∣ exp

(− a

2

2

)exp

[λζ(t)a†

]exp [−λζ∗(t)a] exp

[i

2tanh (ωt) a†2

]exp − ln [cosh (ωt)] n exp

[i

2tanh (ωt) a2

]|β〉 .

(4.102)

50

4.3. Repulsive harmonic oscillator.

As a |β〉 = β |β〉 and exp − ln [cosh(ωt)] n |β〉 = exp[− sinh2(ωt)|β(t)|2/2

]|β(t)〉,

it follows that

Ψβ(x, t) =(ωπ

)1/4√sech(ωt) exp

−1

2

[λ2µ(t)− 2η(t)

]⟨√

2ωx∣∣∣ exp

(− a

2

2

)exp

[λζ(t)a†

]exp [−λζ∗(t)a] exp

[i

2tanh (ωt) a†2

]|β(t)〉 .

(4.103)

Inserting identity operator I = exp [λζ∗(t)a] exp [−λζ∗(t)a] in the above equation,we have

Ψβ(x, t) =(ωπ

)1/4√sech(ωt) exp

−1

2

[λ2µ(t)− 2η(t)

]⟨√

2ωx∣∣∣ exp

(− a

2

2

)exp

[λζ(t)a†

]exp [−λζ∗(t)a] exp

[i

2tanh (ωt) a†2

]× exp [λζ∗(t)a] exp [−λζ∗(t)a] |β(t)〉 . (4.104)

And as exp [−λζ∗(t)a] a†2 exp [λζ∗(t)a] = a†2 − 2λζ∗(t)a† + λ2ζ∗2(t) leads to

Ψβ(x, t) =(ωπ

)1/4√sech(ωt) exp

−1

2

[λ2µ(t) + υ(t)− 2η(t)

]×⟨√

2ωx∣∣∣ exp

[− a

2

2

]exp

√2λ

ω3/2 [1 + i coth (ωt/2)]a†

exp

[i

2tanh (ωt) a†2

]|β(t)〉 ,

(4.105)

where υ(t) = 2λζ∗(t)[β(t)− i

2λζ∗(t) tanh (ωt)

]. Then, inserting I = exp

(a2

2

)exp(−a2

2

)and doing κ(t) =

√2λ

ω3/2[1+i coth(ωt/2)]we arrive to

Ψβ(x, t) =(ωπ

)1/4√sech(ωt) exp

−1

2

[λ2µ(t) + υ(t)− 2η(t)

]×⟨√

2ωx∣∣∣ exp

(− a

2

2

)exp

[κ(t)a†

]exp

(a2

2

)exp

(− a

2

2

)exp

[i

2tanh (ωt) a†2

]|β(t)〉

(4.106)

using the fact that exp(− a2

2

)a† exp

(a2

2

)= a† − a

Ψβ(x, t) =(ωπ

)1/4√sech(ωt) exp

−1

2

[λ2µ(t) + υ(t)− 2η(t)

]×⟨√

2ωx∣∣∣ exp

[κ(t)

(a† − a

)]exp

(−a

2

2

)exp

[i

2tanh (ωt) a†2

]|β(t)〉 . (4.107)

51

Chapter 4. Examples

Moreover, the first exponential operator can be factorized as exp[κ(t)

(a† − a

)]=

exp[−κ2(t)

2

]exp

[κ(t)a†

]exp [−κ(t)a], whereas the last exponential product

exp(− a2

2

)exp

[i2

tanh (ωt) a†2]

is straightforward to handled by following same steps

as in (4.70). Thus, we get

Ψβ(x, t) =(ωπ

)1/4√sech(ωt) exp

− 1

2

[λ2µ(t) + υ(t)− 2η(t) + β2(t)

] × exp

1

2

[fo(t)/2− κ2(t) + 2

√2ωxκ(t)

]×⟨√

2ωx∣∣∣ exp [−κ(t)a] exp

[f1(t)

2a†2]

exp

[f0(t)

2n

]exp

[f2(t)

2a2]|β(t)〉 ,

(4.108)

Using now that exp [−κ(t)a] exp[f1(t)2a†2]

= exp[κ2(t)2f1(t)

]expf1(t)

2

[a† − 2κ(t)

]a†

exp [−κ(t)a] and repeating the steps leading to (4.75), we have

Ψβ(x, t) =(ωπ

)1/4√sech(ωt) exp

−1

2

[λ2µ(t) + υ(t)− 2η(t) + |β(t)|2 − fo(t)/2

]× exp

−β

2(t)

2[1− f2(t)]

exp

−1

2κ(t) [1− f1(t)]

[κ(t)− 2

√2ωx

]× exp

1

2

[∣∣β(t)efo(t)/2∣∣2 + 2ωf1(t)x

2 − 2κ(t)β(t)efo(t)/2]

×⟨√

2ωx

∣∣∣∣β sech (ωt) exp

[fo(t)

2

]⟩. (4.109)

From the definition of inner product of two coherent states 〈α|γ〉 = exp[−1

2

(|α|2 + |γ|2

)+ γα∗

]and after some algebra, we find

Ψβ(x, t) =ψβ(x, t) exp− 4√

2βλω3/2 [sech(ωt)− 1]

4ω3 [1 + i tanh (ωt)]

× exp

−λ2[4 sinh4(ωt/2) + sinh2(ωt)

]4ω3

× exp

λ tanh2 (ωt/2) [λ− 2xω2 (1 + i coth (ωt/2))]

ω3 [1− i tanh (ωt/2)]2 [1 + i tanh (ωt)]

× exp

− iλ2 sech(ωt) [tω cosh (ωt)− sinh (ωt)]

2ω3

× exp

[λ2 tanh(ωt) sinh2(ωt/2) sinh(ωt)

ω3

], (4.110)

52

4.3. Repulsive harmonic oscillator.

where ψβ(x, t) is the simple solution of repulsive oscillator defined in (4.77). Now, ifinitially |Ψ(0)〉 = |Cat〉, one can easily check that

ΨCat(x, t) =Ψβ(x, t) + eiφΨ−β(x, t)√

2(1 + cos(φ)e−2|β|

2) . (4.111)

4.3.3 Perturbative solution

Let us proceed to find the approximate analytical solution to the linear anharmonicrepulsive oscillator through the normalized perturbative treatment. To do so, weassume here simply that H0 = −ω

2

(a2 + a†2

)is the unperturbed part and Hp =

1√2ω

(a+ a†

)is the perturbation. Then, by following the perturbation procedure

outlined in Chapter 3, the expressions to get the first two perturbation terms forthis system are given by

|ψ1,2〉 =− i√2ωS(−iωt)

t∫0

S†(−iωt1)(a+ a†

)S(−iωt1)dt1 |ψ(0)〉 ,

|ψ1,3〉 =− 1

2ωS(−iωt)

t∫0

S†(−iωt1)(a+ a†

)S(−iωt1)t1∫0

S†(−iωt2)(a+ a†

)S(−iωt2)dt2

× dt1 |ψ(0)〉 . (4.112)

Using the relations(4.96) and after performing the integration over t, we obtain

|ψ1,2〉 =S(−iωt)[aζ(t)− a†ζ∗(t)

]|ψ(0)〉 ,

|ψ1,3〉 =1

2S(−iωt)

[aζ(t)− a†ζ∗(t)

]2 − i

ω3[ωt− sinh(ωt)]

|ψ(0)〉 . (4.113)

Applying the initial condition |ψ(0)〉 = |β〉, we find that the perturbative solution ofrepulsive anharmonic system up to second-order correction is

∣∣Ψ(2)(t)⟩

=C1(t)N(2)β (t)

(1 + λ

β [ζ(t)− iζ∗(t) tanh(ωt)]− ζ∗(t) d

dβ

) ∞∑n=0

cnc0|n〉

+λ2

2C1(t)N

(2)β (t)ζ∗2(t)

[d2

dβ2+ i tanh(ωt)

(1 + 2β

d

dβ

)− β2 tanh2(ωt)

] ∞∑n=0

cnc0|n〉

+λ2

2C1(t)N

(2)β (t)β

ζ2(t)β − 2|ζ(t)|2

[d

dβ+ iβ tanh(ωt)

] ∞∑n=0

cnc0|n〉

− λ2

2ω3C1(t)N

(2)β (t)

ω3|ζ(t)|2 + i [ωt− sinh(ωt)]

∞∑n=0

cnc0|n〉 , (4.114)

53

Chapter 4. Examples

with

C1(t) =√

sech(ωt) exp

[−|β|

2

2+iβ2

2tanh(ωt)

], (4.115)

and with its corresponding normalization constant

N(2)β (t) =

1 + 2

(λ

ω

)3

[sinh(ωt)− ωt]= (βζ(t)) +3λ4

4|ζ(t)|4

[2|β|2

(|β|2 + 2

)+ 1]

− λ4|ζ(t)|2(2|β|2 + 3

)<(β2ζ2(t)

)+λ4

2<(β4ζ4(t)

)+

λ4

4ω6[sinh(ωt)− ωt]2

− 12

.

(4.116)

To reach with above result, we have made use of the notation squeezed coherentstate given in (4.86) with cn determined by ξ = −iωt. In order to find the solutionin coordinate space, we must to multiply both sides of (4.114) by 〈x| and applying(4.89), we get

Ψ(2)β (x, t) =N

(2)β (t)ψβ(x, t)

[1 + λ

β [ζ(t)− iζ∗(t) tanh(ωt)]− 2ζ∗(t)f(x, t)

]

−i λ2

2ω3N

(2)β (t)ψβ(x, t)

ω3ζ∗(t)

[2ζ(t)β2 − ζ∗(t)

]tanh(ωt) + ωt− sinh(ωt)

−λ

2

2N

(2)β (t)ψβ(x, t)

|ζ(t)|2 [1 + 4βf(x, t)]− β2

[ζ(t)2 − ζ(t)∗2 tanh2(ωt)

]+λ2N

(2)β (t)ψβ(x, t)ζ∗2(t)

[2f 2(x, t)− csch(2ωt) tanh(ωt)

1 + i tanh(ωt)

]+2iλ2ζ∗2(t)N

(2)β (t)ψβ(x, t)f(x, t)β tanh(ωt), (4.117)

where

f(x, t) =x√ω csch(2ωt) tanh(ωt)− β csch(2ωt) tanh(ωt)

1 + i tanh(ωt). (4.118)

For the initial state being a cat state, |Ψ(0)〉 = |β〉+eψ |−β〉√2(1+e−2|β|2 cos(φ))

, noticing that its

approximately solution is given by

Ψ(2)Cat(x, t) = N

(2)Cat(t)

[Ψ

(2)β (x, t) + eiφΨ

(2)−β(x, t)

](4.119)

54

4.3. Repulsive harmonic oscillator.

whose normalizing constant is defined as

N(2)Cat(t) =

[2 +

λ4

2

3|ζ(t)|4 + 2<(β4ζ4(t))− 12<(β2ζ2(t)) +

1

ω6[ωt− sinh(ωt)]2

]×[1 + e−2|β|

2

cosh(φ)]

+ 4

(λ

ω

)3

[ωt− sinh(ωt)]<(βζ(t))e−2|β|2

sin(φ)

+ 2λ4|ζ(t)|2|β|2[3|ζ(t)|2 − 2<(β2ζ2(t))

] [1− e−2|β|

2

cosh(φ)]−1/2

.

(4.120)

4.3.4 Comparison of the exact and the perturbative solu-tions

In order to test the accuracy and validity of the above perturbative solution, wecompare it with the exact analytic expression of subsection 3.1. The probabilitydensities in the (x, t) plane are plotted for two different values of β and λ in Figs.4.7and 4.8. Fig. 4.7 shows the density probability on (x, t) plane for the exact andperturbative solutions which are represented by black solid line and red dotted line,respectively. The upper graphs present the comparison of both solutions with anamplitude coherent state β = 1 whereas the bottom graphs are obtained with β = 8.For β = 1, we show that when is choose a perturbative parameter equal to λ = 0.1,the perturbative result tends to be a very good approximation to the exact solution.On the other hand, if we increases the perturbative parameter to λ = 0.4, theabove similarities between them begin to disappear, as it can be appreciated inthe region for t > 3 in Fig.4.7 (b). A similar behaviour occurs for the case of thesuperpositions of two coherent states with β = 1 and φ = π/7 in Figs.4.8(a) and4.8 (b). However, apparently Figs.4.8 (c) and 4.8 (d) show that the perturbativesolution looks more accurate with a large amplitude value of coherent state, in thiscase β = 8. The later is quite reasonable since the quadratic factors of β into theperturbative solution clearly influences of it. Consequently, we also must emphasizethat the range of validity for a quickly convergence with the exact result is alwaysrestricted by the condition λt 1. So it doesn’t matter how weak λ may be, theperturbative result will fails after a sufficiently long time. Besides, we would liketo remark that the perturbative solutions proportioned by the Matrix PerturbationMethod, have an appropriate normalization constant whose aim is to ensure thatthey always keep normalized to unity for all t at any order correction. Such aspectis a primary limitation in the standard perturbation theory due their approximatesolutions diverges for large t.

55

Chapter 4. Examples

0 1 2 3 4

-200

-100

0

100

200

t

(a) λ = 0.1

0 1 2 3 4

-200

-100

0

100

200

t

(b) λ = 0.4

0 1 2 3 4

-200

-100

0

100

200

t

x

(c) λ = 0.1

0 1 2 3 4

-200

-100

0

100

200

t

x

(d) λ = 0.4

Figure 4.7: Probability density of the linear anharmonic repulsive oscillator in the(x, t) plane when the initial state is a coherent state. The black line and the reddashed line indicate the exact and perturbative solutions. Graphs (a) and (b) showhow the solutions behave for β = 1 at two different values of the perturbative param-eter λ. For λ = 0.1, the second-order perturbative solution presents a remarkablehigh accuracy with the exact result up to t = 3. After that time, a small but signifi-cantly discrepancy between them appear. For λ = 0.4, the accuracy of perturbativeresult is reduced over shorter time. Graphs (c) and (d) display the same time-rangeof convergence between the solutions for β = 8 and ω = 1.

56

4.3. Repulsive harmonic oscillator.

0 1 2 3 4

-200

-100

0

100

200

t

(a) λ = 0.1

0 1 2 3 4

-200

-100

0

100

200

t

x

(b) λ = 0.4

0 1 2 3 4

-200

-100

0

100

200

t

x

(c) λ = 0.1

0 1 2 3 4

-200

-100

0

100

200

t

(d) λ = 0.4

Figure 4.8: Density probability of the linear anharmonic repulsive oscillator in the(x, t) plane for an Schrodinger-cat state as initial state with ω = 1 and φ = π/7.The black line and the red dashed line indicate the exact and perturbative solutions.Graphs (a) and (b) show how the solutions behave for β = 1 at two different valuesof perturbative parameter λ. For λ = 0.1, the second-order perturbative solutionpresents a remarkable high accuracy with the exact result. Once increases the per-turbation parameter to λ = 0.4, the accuracy of perturbative result is completelyreduced. The two remaining graphs (c) and (d) display same features where a smallbut significantly discrepancy between solutions appear.

57

Chapter 4. Examples

4.4 The Binary waveguide array

It remains to consider an example which involves optical analogues of quantumsystems. For instance, the system of a charged particle hopping on an infinite linearchain driven by an electric field can be mimic by the light propagation in a binarywaveguide array [100, 101]. In fact, this optical structure also has demonstratedits potential to emulate other typical phenomena of quantum mechanics, such asoptical Bloch oscillations [102], discrete spatial solitons [103], quantum walks [104],discrete Fourier transforms [105] and parity time-symmetry [106], to name a few.In particular, the linear behavior of light propagation over this kind of waveguidearrangement is usually governed by the infinite system of differential equations

idEndz

= ω (−1)n En + α (En+1 + En−1) , n = −∞, ...,−2,−1, 0, 1, 2, ...,∞,(4.121)

where En represents the amplitude of light field confined in the nth waveguide, zthe longitudinal propagation distance, 2ω the mismatch propagation constant andα the hopping rate between two adjacent waveguides. Physically, equation (4.121)describes the effective evanescent field coupling between the nearest-neighbor waveg-uide interactions. Moreover, it has been demonstrated [54] that this system can beassociated with a Schrodinger-type equation

id |ψ(z)〉dz

= H |ψ(z)〉 , (4.122)

where H = ω (−1)n + α(V + V †

), being (−1)n the parity operator and V and V †

are peculiar ladder operators defined as

V =∞∑

n=−∞

|n〉 〈n+ 1| , V † =∞∑

n=−∞

|n+ 1〉 〈n| , (4.123)

where |n〉 represents the classical analogue of Fock states and where the previousoperators up and down act over them as V |n〉 = |n− 1〉 and V † |n〉 = |n+ 1〉;something similar to the annihilation and creation operators in quantum optics, butwithout the square root term that characterize them.Note that if the solution is written in terms of the waveguide number basis [100]

as |ψ(z)〉 =∞∑

n=−∞

En(z) |n〉, and if this proposal is substituted into Eq.(4.122), the

infinite system given by Eq.(4.121) is recovered. In fact, it has been shown [107],that the exact solution of this system is

En(z) =1

π

π∫0

cos (nφ)

cos[Ω(φ)z]− i[2α cosφ+ (−1)nω]

sin[Ω(φ)z]

Ω(φ)

dφ, (4.124)

58

4.4. The Binary waveguide array

with

Ω(φ) =√ω2 + 4α2 cos2 φ. (4.125)

Despite the above equation represents the exact solution for the amplitude of thelight field, En = 〈n|ψ(z)〉, a substantial alternative approximated solution is derivedin [107]; under the condition α ω and performing the unitary transformation

R = exp[

α2ω

(−1)n(V + V †

)], the following solution is found,

En = (−1)n(n−1)

2

∞∑r=−∞

∞∑s=−∞

(−1)sr e−i(−1)s

(ω2+α2

ω

)zir×Jr

(α2

ωz

)Js

(αωz)Jn+2r+s

(αωz),

(4.126)where Jn(z) are the Bessel functions of the first kind [96]; in fact, the unitary trans-formation R constitutes the small rotation approximation [108].Prior to solve the Schrodinger-type equation with the formalism presented in Chap-ter 3. First, we need to regard the variable z plays the role of time, this is intuitivelyreasonable if we want to describe the optical field propagation on the waveguidearray in evolutionary terms. Besides, since the problem is linear, we considered forsimplicity and convenience the initial condition |ψ(0)〉 = |m〉 which corresponds toa single excitation in the mth guide. Under these assumptions, and considering thatω (−1)n is the unperturbed part, V + V † is the perturbation and α as the perturba-tion parameter, then, from Eq.(3.20), we can write the analytic solution to first-orderas ∣∣Ψ(1)(t)

⟩= N (1)(z)

[ ∣∣ψ(0)⟩

+ α |ψ1,2〉]. (4.127)

The zero-order solution is trivial,∣∣ψ(0)

⟩= e−iω(−1)

nz |m〉, whereas the first-ordercorrection, |ψ1,2〉, requires the use of Eq.(3.10) to give

|ψ1,2〉 = −ie−iω(−1)nzz∫

0

eiω(−1)nz1(V + V †

)e−iω(−1)

nz1 |m〉 dz1, (4.128)

we expand in Taylor series the product of operators inside the integral as

eiω(−1)nz1(V + V †

)e−iω(−1)

nz1 =∞∑

l,r=1

(−1)r (iωz1)l+r

l!r!(−1)ln

(V + V †

)(−1)rn ,

(4.129)but

(−1)ln(V + V †

)(−1)rn |m〉 = (−1)rm (−1)ln (|m− 1〉+ |m+ 1〉)

= (−1)l (−1)(r+l)m(V + V †

)|m〉 , (4.130)

59

Chapter 4. Examples

then

eiω(−1)nz1(V + V †

)e−iω(−1)

nz1 =∞∑

l,r=1

[−iω (−1)m z1]l+r

l!r!

(V + V †

)= e−2iω(−1)

mz1(V + V †

). (4.131)

Substituting this expression into Eqn (4.128) gives us

|ψ1,2〉 = −ie−iω(−1)nzz∫

0

e−2iω(−1)mz1dz1

(V + V †

)|m〉

= −ie−iω(−1)mz

ω(−1)msin [ωz(−1)m] e−iω(−1)

nz(V + V †

)|m〉 , (4.132)

as the sinus is an odd function and (−1)n(V + V †) = (V + V †)(−1)n+1, we arrive tothe solution

|ψ1,2〉 = − iω

sin(ωz)(V + V †

)|m〉 . (4.133)

Hence, we can build the normalized first-order solution as∣∣Ψ(1)(z)⟩

= N (1)(z)[e−iω(−1)

mz − iαω

sin(ωz)(V + V †

)]|m〉 , (4.134)

the normalization constant N (1)(z) is obtained by considering k = 1 in Eq.(3.23),

N (1)(z) =

1 + 2α<(⟨ψ(0)|ψ1,2

⟩)+ α2 〈ψ1,2|ψ1,2〉

−1/2, (4.135)

in this problem the odd powers of perturbative parameter α do not contribute to thenormalization constant since 〈m| (V + V †)2n+1 |m〉 = 0. Thus, the inner product of|ψ1,2〉 with itself is

〈ψ1,2|ψ1,2〉 =

[sin(ωz)

ω

]2〈m| (V + V †)2 |m〉 = 2

[sin(ωz)

ω

]2; (4.136)

then

N (1)(z) =

1 + 2

[α sin(ωz)

ω

]2−1/2. (4.137)

Now, if we consider the case in which the light field is launched into the first site ofthe waveguide array, i.e. |m〉 = |0〉, the amplitude of light field in the nth waveguideat z distance along the propagation direction, En (z) = 〈n|Ψ(z)〉, for this system is

E (1)n (z) = N (1)(z)

[e−iωzδn,0 − i

α

ωN (1)(z) sin(ωz) (δn,−1 + δn,1)

]. (4.138)

60

4.4. The Binary waveguide array

Note that this equation describes the propagation of electromagnetic field towardseither the left or right side of the waveguide array, but since this array is symmetricand infinite, we can simplify the analysis considering only the positive values of n.Therefore, the first-order solution is reduced to

E (1)n (z) = N (1)(z)

[e−iωzδn,0 − i

α

ωsin(ωz)δn,1

], (4.139)

and satisfies the same initial conditions as those reported in the literature [107].The second-order correction can be calculated using again the Eq.(3.20),∣∣Ψ(2)(t)

⟩= N (2)(z)

[ ∣∣ψ(0)⟩

+ α |ψ1,2〉+ α2 |ψ1,3〉

], (4.140)

through the application of Eq.(3.14), we compute the second-order term |ψ1,3〉 as

|ψ1,3〉 = −e−iω(−1)mz

ω

z∫0

eiω(−1)mz1 sin(ωz1)dz1

(V + V †

)2|m〉 , (4.141)

it follows thatz∫

0

eiω(−1)mz1 sin(ωz1)dz1 =

1

2ωsin2(ωz) +

i

2z(−1)m [1− sinc(2ωz)] ; (4.142)

substituting this in Eq.(4.141) and after some algebra, we get

|ψ1,3〉 = i(−1)m

2ω2Am(z) cos(ωz)

(V + V †

)2|m〉 , (4.143)

where the function Am(z) is defined by

Am(z) = tan (ωz) [1 + izω(−1)m]− zω. (4.144)

Thus, we can write the second-order correction as∣∣Ψ(2)(z)⟩

=N (2)(z)

1 + i(−1)m

[Am(z)

(αω

)2− tan (ωz)

]cos (ωz) |m〉

+ iα2(−1)m

2ω2N (2)(z)Am(z) cos (ωz)

(|m− 2〉+ |m+ 2〉

)− iα

ωN (2)(z) sin (ωz)

(|m− 1〉+ |m+ 1〉

), (4.145)

with above information and carrying out the sums in Eq.(3.23) for k = 2, the nor-malization constant is obtained

N (2)(z) =

1 + α2[2<(⟨ψ(0)|ψ1,3

⟩)+ 〈ψ1,2|ψ1,2〉

]+ α4 〈ψ1,3|ψ1,3〉

−1/2=

1 +

3

2

[α2 cos (ωz) |Am(z)|

ω2

]2−1/2, (4.146)

61

Chapter 4. Examples

where the terms that contain α2 sum zero. Now, assuming that the light is injectedinto the first guide, m = 0, and considering the symmetry of the photonic array, wearrive to the reduced form of the field

E (2)n (z) =N (2)(z)

1 + i

[Am(z)

α2

ω2− tan(ωz)

]cos(ωz)δn,0 − i

α

ωN (2)(z) sin(ωz)δn,1

+ iα2

2ω2N (2)(z)Am(z) cos(ωz)δn,2. (4.147)

For the next-order correction, we have

∣∣Ψ(3)(z)⟩

= N (3)(z)

[ ∣∣ψ(0)⟩

+ α |ψ1,2〉+ α2 |ψ1,3〉+ α3 |ψ1,4〉

]. (4.148)

We get the third-order term from Eq.(3.17),

|ψ1,4〉 =(−1)m

2ω2e−iω(−1)

nz

z∫0

eiω(−1)nz1Am(z1) cos(ωz1)

(V + V †

)3|m〉 dz1

=i(−1)m

2ω2e−iω(−1)

mz

z∫0

e−iω(−1)mz1 sin(ωz1)dz1

(V + V †

)3|m〉

− i(−1)m

2ωe−iω(−1)

mz

z∫0

z1e−2iω(−1)mz1dz1

(V + V †

)3|m〉 , (4.149)

where we have used that cos(ωz1)A(z1) = sin(ωz1)−z1ωe−iω(−1)mz1 and that (−1)n(V+

V †)3 = (V + V †)3(−1)n+1. Using integration by parts, one directly gets

|ψ1,4〉 =i

2ω3cos(ωz) [tan(ωz)− zω] =

i

2ω3B(z); (4.150)

thus, (4.148) becomes

∣∣Ψ(3)(z)⟩

=

e−iω(−1)

mz −(αω

)2(−1)m [zω(−1)m sin(ωz)− iB(z)]

N (3)(z) |m〉

+ i(αω

)[3

2

(αω

)2B(z)− sin(ωz)

]N (3)(z)

(|m− 1〉+ |m+ 1〉

)− (−1)m

2

(αω

)2 [zω(−1)m sin(ωz)− iB(z)

]N (3)(z)

(|m− 2〉+ |m+ 2〉

)+i

2

(αω

)3B(z)N (3)(z)

(|m− 3〉+ |m+ 3〉

), (4.151)

62

4.4. The Binary waveguide array

and the normalization constant to this order is given by Eq.(3.23)

N (3)(z) =

1 + α4 [2< (〈ψ1,2|ψ1,4〉) + 〈ψ1,3|ψ1,3〉] + α6 〈ψ1,4|ψ1,4〉−1/2

=

1 +

3

2

(αω

)4 [B(z)2 − 4B(z) sin(ωz) + z2ω2 sin2(ωz)

]+ 5

(αω

)6B(z)2

−1/2.

(4.152)

In this case, the third-order approximation for the amplitude of electric field on thewaveguide array is given by

E (3)n (z) =e−iωz

1−

(αω

)2eiωz [zω sin(ωz)− iB(z)]

N (3)(z)δn,0

+ i(αω

)[3

2

(αω

)2B(z)− sin(ωz)

]N (3)(z)δn,1

− 1

2

(αω

)2N (3)(z)

[zω sin(ωz)− iB(z)] δn,2 −

iα

ωB(z)δn,3

. (4.153)

In order to illustrate the high degree of accuracy that can be obtained with thethird-order correction for the amplitude of electrical field, the comparison of thisperturbative result with the exact solution and with the small rotation solution fromequation (4.124) and (4.126), is given in Figs.4.9 and 4.10, using the parametersω = 0.9 and n = 0, 1, 2. In these figures, we present the intensity distribution I(z) =|En|2 for the first three guides considering two values of the perturbation parameter,α = 0.1 and α = 0.3. It is noteworthy that for α = 0.1 the approximate solutionconverges to the exact one uniformly even over large propagation distance. On theother hand, for α = 0.3 both solutions are very similar only for short distances, butwe still obtain a good approximation; in fact, the real measure of the perturbationis the product αz, in same fashion that previous examples on this chapter but witht instead of z. Moreover, for the previous two values of α, it becomes evident theimprovement provided by the third-order correction with respect to the reportedresults using the small rotation method. Thus, the assessment of higher-order termscan give us a reliable solution into the system described here.

63

Chapter 4. Examples

2 4 6 8z

0.980

0.985

0.990

0.995

1.000

IHzL

(a) Guide 1

2 4 6 8z

0.002

0.004

0.006

0.008

0.010

0.012

IHzL

(b) Guide 2

2 4 6 8z

0.0005

0.0010

0.0015

IHzL

(c) Guide 3

Figure 4.9: Field intensity versus propagation distance z using the exact solution(solid line), the third-order solution (red dashed line) and the small rotation methodsolution (blue dashed line), with α = 0.1 and ω = 0.9, for the first three guides.

64

4.4. The Binary waveguide array

2 4 6 8z

0.6

0.7

0.8

0.9

1.0

IHzL

(a) Guide 1

2 4 6 8z

0.02

0.04

0.06

0.08

0.10

IHzL

(b) Guide 2

2 4 6 8z

0.02

0.04

0.06

0.08

0.10

0.12

IHzL

(c) Guide 3

Figure 4.10: Field intensity versus propagation distance z using the exact solution(solid line), third-order solution (red dashed line) and the small rotation methodsolution (blue dashed line), with α = 0.3 and ω = 0.9, for the first three guides.

65

Chapter 4. Examples

66

Chapter 5

Conclusions

In summary, we have shown that all perturbative techniques depend on two certainassumptions: the first of these rely on the system Hamiltonian under study which itcan be divided into a part which can be solved exactly and a second part whose noanalytic solution exist, and this term will always be sufficiently small to be regardedas a perturbation with respect to the solvable part. Next assumption is makes aperturbative expansion of the energies and wavefunctions of the perturbed system,which are sorted in ascending order by a fictitious parameter λ. Depending onthe choice of perturbation, that mean, if is static or not, one must resort to thetime-independent or time-dependent perturbation theory. The first framework offersexpressions for the corrections to wavefunction and to the energy in separate way,whereas the second one replace the time dependent wave function by a time evolutionoperator; which connects a initial state at early time with a final state at later time.A perturbative expansion on the evolution operator leads to expressed it in terms ofthe Dyson series, a multiple time-ordered integrals representation. Subsequently, theNormalized Matrix perturbation Method brings us some nature of both approachesoutlined above. The starting point of this technique is use the formal solution of

the time dependent Schrodinger equation, |ψ(t)〉 = e−i(H0+λHp)t |ψ(0)〉, where theperturbation Hamiltonian Hp is static. Then, the key ingredient to find approximatedsolutions of the Schrodinger equation is the introduction of matrix M defined in (3.2)that allow us to transform the Taylor series for the wavefunction |ψ(t)〉, in terms ofproducts of H0 and Hp, in a power series of the matrix M that is easier to handle.Further, the present formalism itself is characterized by the following features:

• It has been proved that for several potentials (see references [18, 19]) the Ma-trix Method, without the normalization process, gives better solutions thanthe traditional Rayleigh-Schrodinger perturbation theory. For instance, thequadratic, the Morse and the cosine potentials. Moreover, the main contribu-tion of this thesis is the explicit calculation of the normalization constant thatpreserves the norm at any order. This could be a technical point, but is ofpractical importance for explicit calculations and, of course, complements the

67

Chapter 5. Conclusions

theoretical analysis of the Matrix Method. It is noteworthy mention that instandard perturbation theory, one usually enforces intermediate normalizationfor all the powers in the coupling constant λ. However, this is not the case inthe present approach, this can be seen clearly in Chapter 3, where the innerproduct of the zero-order with the first three correction terms are differentfrom zero.

• As we mentioned into the Chapter 3, the approximative solutions given by theNormalized Matrix perturbation Method, not only present stationary terms butalso time-dependent factors which allows to improve our perturbative results.Besides, the inclusion of a time-dependent normalization factor to the completesolution |ψ(t)〉 to all orders corrections, allows us to avoid divergence problemsas t grows into the solutions, on the other hand, the Nk(t) constant also ensuresthat the approximative solutions preserve their norm at any order of λ. Theset of examples analyzed in Chapter 4 show a clear evidence of this fact. Theproduct λt in the functions involved inside of the normalization term, makes itfeasible to get a good agreement with the exact and approximative solutions ofabove examples, only for short times, since it must satisfy λt 1. Conversely,as times grows is should reasonably expect that this condition does not remainsless than 1, doing that the approximation is not longer suitable to produce goodresults. The cubic anharmonic oscillator clearly shows this behaviour when itsnumerical comparison is made with the perturbative solution at t = 1.

• The general expression to obtain the higher-order perturbative corrections of-fers an alternative to cast the Dyson series, similar to those given in the in-teraction picture by the Dirac perturbation method, and which is written inmultiple integral forms, as power series of tridiagonal matrices. Besides, in allthe procedure involved to get it, we do not distinguish if the unperturbed partof full Hamiltonian is degenerated or not. So, the Eq.(3.17) that we provideto compute the corrections is a very general expression. In consideration toEq.(3.17), it is clear that only applies to weak perturbations. However, theopposite limit concerning the strong-coupling expansion can be derived in astraightforward way just by interchanging the unperturbed part of Hamilto-nian H0 with the perturbation Hp, and with a time-rescaling defined as τ = λt,such appears in (3.19). We appreciate from Eqs.(3.17) and (3.19) how a simplyrescale of the time variable connects up these two Dyson perturbation serieswhen λ = 1. This duality of the Normalized Matrix perturbation Method isan attractive property since it open up the possibility to analyse the solutionof a quantum system in different regimes of the perturbative parameter, forexample, the linear anharmonic system with a number state or coherent stateas initial condition, exhibits this main feature via its remarkably convergenceof their approximative solutions with their exact results, which were obtainedfrom both perturbation regimes. Further, this duality principle is useful when

68

one can be able to determine the eigenstates of Hp and those of H0 are alsoknown.

• This method allows easy application to any initial condition because it is basedon an approximation to the evolution operator. In fact, it may also be usedwhen one may find a unitary evolution operator for the unperturbed Hamilto-nian but it is not possible to find its eigenstates, we prove it with the linearrepulsive anharmonic oscillator whose approximative solution, in the positionrepresentation, is able to reproduce the exact result with very high precision aslong as the condition λt 1 is satisfied. Further, for this quantum model wehave demonstrated that its formal exact solution can be factorize as the prod-uct of a displaced operator with a squeezed operator followed by the initialcondition to be apply. Indeed, if linear anharmonicity is neglected, then onehas that the repulsive harmonic oscillator is equivalent to a squeezed operatorwith a purely imaginary parameter ξ = −iωt; where by its action on a coher-ent state leads immediately to a squeezed coherent state and whose probabilitydensity in the coordinate space x shows a curved amplitude squeezing patternin the positive x-axis direction. In fact, this curve squeezing direction dependson the values of the coherent state amplitude β; for example, this behaviour ismore notorious insofar as the value of β increases and vice versa, the curvedsqueezing amplitude is less pronounced when β takes small values.

• Albeit the proposed the method was originally conceived to solve approxi-mately problems in quantum mechanics. Clearly one may apply it to anySchrodinger-like equation, as shown in the example of the binary waveguidearray. Thus, it is possible to associate the method described in this thesisto more general solutions of Schrodinger-like equations which could bring anoriginal contribution to the field of optics/quantum mechanics. Additionally,we should emphasize that an extension of this mathematical analysis has beendeveloped to deal with mixed states. Here, the implementation of the Ma-trix Method in reduced density matrices has been developed and publishedseparately, where the usefulness of the Matrix Method is emphasized with itsgeneralization to the case of the Lindblad master equation and which it pro-vides in a systematic way perturbative solutions to any order.Finally, we conclude that the Normalized Matrix perturbation Method willbe helpful to understanding the dynamical behavior for those quantum sys-tems where it not feasible to find a exact solution. Besides, the normalizationconstant of the approach advocated here, insures that the perturbative solu-tions always keep normalized to unity for all t at any order correction. Thisis particularly useful since eliminates the convergence problems of the time-divergent terms which result from the perturbative corrections of |Ψ(t)〉. Suchaspect is a primary limitation in the standard perturbation theory due theirapproximate solutions diverges for large t. So it is logical to expect that the

69

Chapter 5. Conclusions

application of our perturbation model on any system under analysis will alwaysleads to more accurate results than using the perturbation recipe mentionedabove. An future research could be focus on implementation of the Normal-ized Perturbation Matrix Method in more complex systems (especially in thequantum realm) such as the PT -symmetric Rabi model or in PT symmetricHamiltonians, just to name a few examples.

70

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