hector manuel moya-cessa francisco soto-eguibar rinton ... - introduction to...of passing light...

Introduction to

Quantum Optics

Hector Manuel Moya-Cessa

Francisco Soto-Eguibar

Rinton Press, Inc.

© 2011 Rinton Press, Inc. 565 Edmund Terrace Paramus, New Jersey 07652, USA [email protected] http://www.rintonpress.com

All right reserved. No part of this book covered by the copyright hereon may be reproduced or used in any form or by any means-graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval system - without permission of the publisher.

Published by Rinton Press, Inc.

Printed in the United States of America

ISBN 978-1-58949-061-1

a Isabel y Leonardo

a Conchis, Sofi y Quique

Preface

Quantum optics is a topic that has recently acquired a great deal of attention, not only for its theoretical and experimental contributions to the understanding of the quantum world, but also for the perspectives of its use in many sophisticated applications. Among them, quantum optical devices are particularly promising tools for quantum information processing applications.

The development of a quantum optics research group at the Instituto N acional de Astrofisica, Optica y Electr6nica, INAOE, (National Institute of Astrophysics, Optics and Electronics) has led to the creation of several quantum optics courses for graduate students. This book is the result of those courses, and the topics presented here are directly related with research activities undertaken by our group.

This book intends to teach graduate and postgraduate students several methods used in quantum optics. Therefore, it is mainly about doing calculations. Throughout the book we have emphasized the "hows" over the "whys". Field quantization will not be studied in this book, as it has already been reviewed in many quantum optics textbooks. Instead, we will treat the case of the time dependent harmonic oscillator by applying invariant techniques (Lewis-Ermakov).

In Chapter 1, the harmonic oscillator states and the needed operator algebra are introduced. In Chapter 2, the quasiprobability phase space distributions are reviewed, along with their properties and relations. In Chapter 3, we deal with the time dependent harmonic oscillator, which takes us to the notion of squeezed states. The very important subject of the interaction of light and atoms is the central issue of Chapter 4. The Master Equation (ME) for a real cavity is treated in Chapter 5, where it is solved for a lossy cavity at zero temperature with different initial conditions by means of superoperator techniques. In Chapter 6, the pure states and the statistical mixtures are analyzed by reviewing the concepts of entropy and purity. The reconstruction of quasiprobability distribu-

Preface

tions in phase space is addressed in Chapter 7 and applied to the measurement of field properties. Here it is used an "inverse" spectroscopic approach: instead of passing light through matter in order to find out about the quantum nature of the matter, we pass matter (two-level atoms) through quantized light, learning about the quantum structure of the field inside a cavity. In Chapter 8, we study the ion-laser interaction, reviewing the type of traps that exist and analyzing the motion of an ion in a Paul trap. The time independent and time dependent trap frequency cases are also studied. In this chapter, the most complete solution for this system is given, as we, besides treating the low intensity regime, also approach the medium and high intensity regimes. Finally, in Chapter 9, we introduce the Suskind-Glogower nonlinear coherent states; we use two methods to construct them and we analyze their properties by means of the photon distribution number, the Mandel parameter and the Husimi function.

Several appendices are added, which we believe enrich the contents of the book. Particularly important is Appendix A, where we study the Master Equation, describing phase sensitive processes for a cavity filled with a Kerr medium. Being the J aynes-Cummings model such an important tool in quantum optics, we provide different methods for solving it in Appendix B. The interaction of many fields is treated in Appendix C. Appendix D proposes a quantum phase formalism, while Appendix E studies a series of Bessel functions, needed to determine the non-classical character of non-linear coherent states described in Chapter 9.

We would like to thank the many people who have been essential for the existence of this book, especially several generations of students. Finally, we want to thank our families for the understanding and the patience they have given to us during the time we dedicated to the writing of this book.

Hector Manuel Moya Cessa y Francisco Soto Eguibar Santa Maria Tonantzintla, Puebla, Mexico. June 15, 2011.

Contents

Preface

Chapter 1 Operator algebra and the harmonic oscillator

1.1 Introduction . . . . . . . . 1.2 von Neumann equation .. · · · · · · 1.3 Baker-Hausdorff formula . . . . . . · . · · 1.4 Quantum mechanical harmonic oscillator

1.4 .1 Ladder operators 1.4.2 Fock states . . . 1.4.3 Coherent states . 1.4.4 Displaced number states .

1.4.5 Phase states 1.5 Ordering of ladder operators

1.5.1 Normal ordering ... 1.5.1.1 Lemma 1 ..

1.5.2 Anti-normal ordering 1.5.2.1 Lemma 2 .

1.5.3 Coherent states. 1.5.4 Fock states ....

Chapter 2 Quasiprobability distribution functions

2.1 Introduction ........ · . · · · · · · · 2.2 Wigner function ..... · . · ·

2.2.1 Properties of the Wigner function .. · · · · · · · ·. · · 2.2.2 Obtaining expectation values from the Wigner functwn

2.2.3 Symmetric averages ... · · · · · · · · · · · 2.2.4 Series representation of the Wigner function .

vii

1 4 4 6

7

8 10

14 14 16

17 19 20 20 21 21

23 23 23 28 28 29

30

Contents

2.3 Glauber-Sudarshan P-function . . . 32 2.4 Husimi Q-function . . . . . . . . . . 33 2.5 Relations between quasiprobabilities 34

2.5.1 Differential forms . . . . . . . 34 2.5.2 Integral forms . . . . . . . . . 37

2.6 The Wigner function as a tool to calculate divergent (or not) series . 37 2.7 Number-phase Wigner function . . . . . . . . . . 39

2.7.1 Coherent state . . . . . . . . . . 40

2. 7.2 A special superposition of number states . 41

Chapter 3 Time Dependent Harmonic Oscillator 43 3.1 Time dependent harmonic Hamiltonian 43

3.1.1 Minimum uncertainty states. 46 3.1.2 Step function . . 46

3.2 More states of the field . . . . 49

3.2.1 Squeezed states . . . . 50 3.2.2 Schrodinger cat states 54 3.2.3 Thermal distribution . 57

Chapter 4 (Two-level) Atom-field interaction 59 4.1 Semiclassical interaction 59 4.2 Quantum interaction . . 62

4.2.1 Atomic inversion 63 4.3 Dispersive interaction . 65

4.4 Mixing classical and quantum interactions 66

4.5 Slow atom interacting with a quantized field . 69

Chapter 5 A real cavity: Master equation 73 5.1 Cavity losses at zero temperature . 73

5.1.1 Coherent states . 75

5.1.2 Number states . . . . . . . 75

5.1.3 Cat states . . . . . . . . . . 76 5.2 Master equation at finite temperature 77

Chapter 6 Pure states and statistical mixtures 81 6.1 Entropy . . . . . . . . . . . . . . . . . 81 6.2 Purity . . . . . . . . . . . . . . . . . . . 82

6.3 Entropy and purity in the atom-field interaction 83 6.4 Some properties of reduced density matrices . . . 84

6.4.1 Proving p'JJ+1= TrA{fJ(t)pA(t)} by induction 87

Contents

6.4.2 Atomic entropy operator . . . . . · · · 6.4.3 Field entropy operator 6.4.4 Entropy operator from orthonormal states .

6.5 Entropy of the damped oscillator: Cat states ...

88 89 91 93

Chapter 7 Reconstruction of quasiprobability distribution func-

7.1

7.2 7.3 7.4

tions Reconstruction in an ideal cavity . . . . . . . . . . 7.1.1 Direct measurement of the Wigner function 7.1. 2 Fresnel approach . . . . . Reconstruction in a lossy cavity . Quasiprobabilities and losses Measuring field properties 7.4.1 Squeezing .... 7.4.2 Phase properties

Chapter 8 Ion-laser interaction 8.1 Paul trap ................... .

8.1.1 The quadrupolar potential of the trap 8.1.2 Oscillating potential of the trap ... . 8.1.3 Motion in the Paul trap ....... . 8.1.4 Approximated solution to the Mathieu equation

8.2 Ion-laser interaction in a trap with a frequency independent of time 8.2.1 Interaction out of resonance and low intensity ...

8.3 Ion-laser interaction in a trap with a frequency dependent of time 8.3.1 Linearization of the system ....... .

8.4 Adding vibrational quanta . . . . . . . . . . . . . 8.5 Filtering specific superpositions of number states

97

97 97 98 99

102 105 105 107

109

111 111 114 115 116 120 121

126 128 129

133

Chapter 9 operators

Nonlinear coherent states for the Susskind-Glogower 137

9.1 Approximated displacement operator ..... 9.2 Exact solution for the displacement operator

9.3 Susskind-Glogower coherent states analysis 9.3.1 The Husimi Q-function ...

9.3.2 Photon number distribution . 9.3.3 Mandel Q-parameter .....

9.4 Eigenfunctions of the Susskind-Glogower Hamiltonian 9.4.1 Solution for IO) as initial condition . 9.4.2 Solution for lm) as initial condition

139 140 142 143 143 145 147 148 151

Contents

9.5 Time-dependent Susskind-Glogower coherent states analysis 9.5.1 Q function ......... . 9.5.2 Photon number distribution . 9.5.3 Mandel Q-parameter .

9.6 Classical quantum analogies ..

Appendix A Master equation A.1 Kerr medium ......... . A.2 Master equation describing phase sensitive processes

Appendix B Methods to solve the Jaynes-Cummings model B.1 A naive method ........ . B.2 A traditional method ...... : : : : : : : : ..... .

Appendix C Interaction of quantized fields ' C.1 Two fields interacting: beam splitters

C.2 Generalization ton modes .. . C.3 A particular interaction ... . C.4 Coherent states as initial fields

Appendix D Quantum phase D.1 Turski's operator ... . D.2 A formalism for phase ... .

D.2.1 Coherent states ... . D.3 Radially integrated Wigner function

154 155 155 155 160

163 163 164

167 167 168

169

169 171 173 174

175 175 176 177 178

Appendix E Sums of the Bessel functions of the first kind of integer order 181

Bibliography

Index

185

189

Chapter 1

Operator algebra and the harmonic oscillator

1.1 Introduction

In this chapter, we revise briefly the Dirac notation, some of the algebra that will be used throughout the book, and introduce the harmonic oscillator and some states in which it may be found. In this chapter, and in the rest of the

book, we will set ti = 1. In Dirac notation, we denote wavefunctions 'ljJ by means of "kets" 1'1/J). For

instance, an eigenfunction of the harmonic oscillator

(1.1)

is represented by the ket In), with n = 0, 1, 2, .... In quantum mechanics, these

states are called number states or Fock states. Any function can be expanded in terms of eigenfunctions of the harmonic oscil-

lator; or in other words,

f(x) = L Cn'l/Jn(x) (1.2)

n=O

where

Cn = j_: dxf(x)'l/Jn(x). (1.3)

In the same way, any ket may be expanded in terms of the number states In) 's;

i.e.,

If)= Lenin) (1.4)

n=O


where the orthonormalization relation

(1.5)

has been used.

The quantity (ml is a so-called "bra". The basis set of kets In) is a discrete one. However, there are also continuous bases. We can form one continuous basis for example, with the function eipq I V27f and the corresponding ket IP). First ~ate that

1 100

(PIP') = 2

7r -oo dxe-i(p-p')q = J(p _ p'), (1.6)

so that

e•P' q 1oo ipq J27i = -oo dpJ(p- p') ~' (1.7)

or, in bra-ket notation we have

IP') = j_: dpJ(p- p')lp) =I: dp(plp')lp); (1.8)

rearranging terms we have

IP') =(I: dplp)(pl) IP') =lip'); (1.9)

i.e., we have what is called the completeness relation

I: dplp)(pl = 1. (1.10)

Finally, note that the function eipq I V27f is an eigenfunction of the operator -i..!i with eigenvalue p. dq

For position, an "eigenket" of ij is

filq) = qlq), (1.11)

and an "eigenbra" is

(q'lq' = (q'lfJ.. (1.12)

We therefore find

(q'lq)(q'- q) = 0, (1.13)

Introduction

that has as solution

(q'lq) = J(q'- q).

We then can express the completeness relation as

1 =I: dqlq)(ql,

such that

11/J) = 111/J) =I: dqlq)(qi1/J) =I: dq1j;(q)lq),

where 1/;(q) = (qi1/J) = (qi1/J).

(1.14)

(1.15)

(1.16)

The completeness relation serves us, among other things, to calculate averages, for instance

(1.17)

or finally,

(1.18)

Note that in the above equation we are simply adding "diagonal" elements; i.e., we have the trace of the operator l1/l)(1f;I.A. As the trace is independent of the basis, we can have the mean value also in terms of the discrete basis In)

(1/liAI1/l) = l:)ni1/J)(1/JIA1n). (1.19) n=O

The main task in non-relativistic quantum mechanics is to solve the Schrodinger equation

di1/J) = -ifii1/J). dt

(1.20)

In order to achieve this, sometimes it is convenient to perform unitary transformations, such that we may simplify the problem. For instance, we may do 11/J) = Tl¢), with T = e-i~A and A a Hermitian time independent operator, and obtain for 1¢) a new equation

d~~) = -ifirl¢), (1.21)


where the transformed Hamiltonian is

(1.22)

Developing the exponentiaJs in Taylor series and grouping terms we obtain

ei~Afie-i~A = H +i~[A,H] + (i;t [A, [A,fi]] + (i;t [A, [A, [A,fi]] + ... , (1.23)

an expression valid for any two operators fi and A, and sometimes named Hadamard lemma.

If we do A---+ p and fi-+ ij, and we use the commutator [q,fJ] = i, Equation (1.23) shows that eicxf> displaces the position operator; in mathematical terms that means that

(1.24)

· A more general operator e-i(qof>-pa{j) produces displacements in both q and p simultaneously; i.e.,

f(q,p) = ei(qof>-po{j) f(q + qo,fJ + Po)e-i(qof>-po{j). (1.25)

1.2 von Neumann equation

An equation that we will use frequently is the so called von Neumann equation, which is another form of the Schri:idinger equation, but will be useful when, for instance, the environment is taken into account. It is obtained from the Schri:idinger Equation (1.20) multiplying it by the bra (~I by the right

d~~) (~I= -ifii~)(~I, (1.26)

and adding it with the adjoint of Equation (1.20) multiplied by the ket I~) by the left, so that

dp , A A

dt = -z[H,p], (1.27)

where pis the density matrix, defined simply as the ket-bra operator p = I~) (~I·

1.3 Baker-Hausdorff formula

Equation (1.25) has complicated terms, in the sense that it has exponentials of the sum of non-commuting operators. In this particular case, the exponentials involved may be easily factorized in the product of three exponentials because of

Baker-Hausdorff formula

the "simplicity" of the commutation relation of the operators involved, namely, position and momentum. In order to accomplish such factorization, we use what is known in the literature as the Baker-Hausdorff formula, that establishes that if [A, [A, B]] = [B, [A, B]] = o then

so that Equation (1.25) may be written as

In order to prove the Baker-Hausdorff's formula, we write

F(> .. ) = e>..(A+B) = ef(>..) eg(>..)Aeh(>..)B,

and derive F(> .. ) with respect to A. From the first equality we obtain

d~iA) =(A+ B)F(A),

and from the second equality

(1.28)

(1.29)

(1.30)

(1.31)

(1.32)

is obtained, where we have introduced in the former equation a unity operator in the form e-g(>..)Aeg(>..)A.

We use now (1.23) to calculate

(1.33)

so Equation (1.32) may be rewritten as

d~iA) = {t'(A) + g'(A)A + h'(A) (B + g[A,Bl)} F(A). (1.34)

Equating (1.34) with (1.31), we obtain the system of differential equations

g'(A) = 1,

that has as solution

h'(A) = 1, h'(A)g(A)[A,B] + j'(A) = o,

g(A) =A+ g(O),

h(A) =A+ h(O),

(A2 ) A A

j(A) =- 2 + g(O)A [A, B] + f(O).

(1.35)

(1.36)


By evaluating (1.30) in zero, we obtain the initial conditions f(O) = 0, g(O) = 0 and h(O) = 0, that substituted in (1.36) give us finally the Baker-Hausdorff's

formula

1.4 Quantum mechanical harmonic oscillator

The Hamiltonian for the harmonic oscillator is written as (for simplicity, we consider unity mass, and we set w = 1)

(1.37)

We write the formal solution of the Schrodinger equation (1.20) as

(1.38)

We need to factorize the above exponential; however, the Baker-Hausdorff's formula can not be used, because the hypotheses of the theorem are not satisfied. Actually, we first have the commutator

(1.39)

where we have used that [AB, C] = A[B, C] +[A, C]B, and then we commute it with ij2 and p2 . The commutators are

(1.40)

However, we note that there are no other operators resulting but the operators in Equations (1.39) and (1.40); in other words, the operators that arise are proportional to (jp + p(j, (}_2 and p2 . We therefore may try the assumption (an educated guess, like this one, is normally called an "ansatz")

U(t) e-i~(fJ2H2) (1.41)

e-i t~t) P2 e-i g~t) ([]'P+M) e-i h~t) q2.

Quant1tm mechanical harmonic oscillator

Note that the above ordering of operators is arbitrary and we could have tried a different one. Deriving with respect to time, we have

au(t) = _ ifJ2 + rP u(t)

8t 2

=- i j(t) p2 U(t) 2

.g(t) m2.~2 " ~, _·W("F') -i~'2 - 7-

2-e-' 2 P (qp + pq)e ' 2 qp pq e 2 q

.h(t) _ m2 ,2 _ W("+") ,2 _ ~ ,2 7-2- e , 2 P e , 2 qp pq q e ' 2 q .

Using that

e-iJJ,jl([]P+f!ii)(;_2eiJJ,jl({]P+fi[])

e-i'~'lfi2 (;_2ei'~')fi2

e-i t~t) P2 ((j_p + pq)ei t~t) P2

we rewrite the second part of (1.42) as

8U(t) 8t

'2 -2g q e '

q2- j(t)((j_p + p(j_) + f2(t)p2'

((j_p + p(j_)- 2f(t)p2,

(1.42)

and by equating it with the first part of (1.42), we end up with the system of

equations

h i(t)- P(t)

g(t) j(t),

with initial conditions j(O) = g(O) = h(O). The solutions are then

j(t) = h(t) tan(t), g(t) = -ln(cos(t)).

1.4.1 Ladder operators

(1.43)

(1.44)

Another form to solve the harmonic oscillator is via annihilation and creation operators. We define the so-called ladder operators a and at, also known as


annihilation and creation operators, as

(1.45)

These operators obey the commutation relation

(1.46)

and the Hamiltonian (1.37) may be written in terms of ladder operators as

(1.47)

that, by defining the number operator fi = at a, may be written in the form

~). 2

(1.48)

1.4.2 Fock states

Eigenstates of (1.48) are the Fock or number states, already introduced, In) with eigenvalues w(n +~);i.e.,

(1.49)

where n is a non-negative integer, and it is identified with the number of excitations (photons in the case of electromagnetic field, see Chapter 3). Fock states are therefore eigenstates of the number operator

ataln) =nln). (1.50)

The vacuum state of the harmonic oscillator is defined as

a10) = 0. (1.51)

In fact the creation and annihilation operators act on the number states in the following form

at In) = v'Ti"TI In+ 1), a In) = vn In- 1). (1.52)

Some of the most important properties of number states are given below. Any state vector In) may be obtained from the vacuum IO) via the creation operator,

(1.53)

Quantum mechanical harmonic oscillator

Number states form an orthonormal basis, that is

(nlm) = t5nm· (1.54)

The unity operator may be written in terms of number states as

L ln)(nl = 1. (1.55) n=O

By using the completeness relation it is possible to express any operator in terms of number states; for instance, the annihilation operator

00

a= La In) (nl = L v'n +lin) (n + ll' (1.56) n=O n=O

and the creation operator

at= L a.t In) (nl = L v'n +lin+ 1) (nl. (1.57) n=O n=O

The number operator is simply

fi = L n ln)(nl . (1.58) n=O

Number states have no uncertainty in intensity; i.e.,

(b..fi) = J(nln2 ln)- (nlnln) 2 = 0. (1.59)

Averages for position and momentum are null for number states, (nlqln) = (nlpln) = 0; however, their uncertainties

and

~ (b..q) = J(nlq2 ln) = V ~'

~ v(2n+l)w (b..p) = V (nlrln) = -

2-,

are such that they are minimized only for the vacuum. The equation

(1.60)

(1.61)

(1.62)

gives the probability to haven number of excitations (photons) in the state 11{1).

10 Operator algebra and the harmonic oscillator

1.4.3 Coherent states

We may build arbitrary superpositions of number states to obtain new states; in particular, we can construct coherent states of the harmonic oscillator [Sudarshan 1963; Glauber 1963b]. They may be obtained in different forms:

(1) as eigenstates of the annihilation operator;

(2) as states whose averages follow the classical trajectories of ij, p and fi [Meystre 1990];

(3) as a displacement of the vacuum.

Let us define the harmonic oscillator coherent states as eigenstates of the annihilation operator; that means, that if we call them Ia), then

ala)= ala)' (1.63)

and as a is a non-Hermitian operator its eigenvalues, a, are complex. Other properties of these states may be obtained by using the Glauber displacement operator [Glauber 1963b]

(1.64)

Using Equation (1.25) and the definition (1.45) of the ladder operators, we write this operator as

with

The displacement operator has the following properties

fJt(a)

D(a + j3)

fJt(a)&D(a)

fJt(a)at D(a)

tJ- 1 (a) = D( -a), D(a)D(j3)e-ilm(af3*)'

a+a, at +a*.

(1.65)

(1.66)

(1.67)

(1.68)

(1.69)

(1.70)

Coherent states, Ia), may also be generated by application of the displacement operator on the vacuum,

(1.71)

Quantum mechanical harmonic oscillator 11

that, by developing the exponential in Taylor series, gives us

lal 2 00 an Ia) = e---r L- In).

n=O Vnl (1.72)

In this equation, the value lal 2 represents the average value of excitations n in the coherent state Ia):

(1.73)

The solution for the harmonic oscillator Hamiltonian for an initial coherent state is given in the following very simple form

(1.74)

where we have used the Hamiltonian (1.49). Introducing e-ifiwt into the sum,

and using the fact that the states lk) are eigenstates of the number operator n, we have

11/l(t)) = 00 ake-ikwt ·wt . L ---lk) = e-'Tiae-•wt);

n=O Jkf

i.e., a coherent state that rotates with the harmonic oscillator frequency.

(1.75)

Coherent states are eigenstates of the annihilation operator, but How does the creation operator act on them? This question may be answered by using the coherent density matrix la)(al. Using the explicit expression (1.72) for the coherent states, we have

(1.76)

and we note that

(1.77)

so

(1.78)

Analogously

Ia) (ala= (a~* +a) Ia) (a I. (1.79)


This expression is of interest particularly when going from a Master equation (see Chapter 6) to a Fokker-Planck equation. We will use this expression next chapter where we relate quasiprobabilities in a differential form.

Coherent states form an over-complete set of states. The identity operator is written in terms of coherent states as

Lln)(nl, (1.80)

where we have set a = rei&.

By using the above completeness relation, the annihilation operator may be expressed as

(1.81)

and the creation operator is written in the form

(1.82)

It is a bit more complicated to express the number operator in terms of coherent states, because

(1.83)

that is a representation that includes off-diagonal terms. However, note that if we write the operator n =at a= aat- 1, we can do

(1.84)

i.e. a diagonal form. This implies that the ordering of the annihilation and creation operators is of importance. We will look in more detail the ordering of such operators in the next section and its importance in next chapter on quasiprobabilities.


The excitation number for the coherent states is given by the Poissonian

distribution

(1.85)

In Figure 1.1, we plot such a distribution; it may be seen that it is centered at

fi = la12 and has a width of approximately 2lal.

0.06

0.05

P(n)

Fig. 1.1 Photon number distribution for the coherent state with a= 6. It may be observed that the distribution is centered in n R:! 36 and has a width of approximately 2lal R:! 12.

The excitation uncertainty for coherent states is given by

(1.86)

Finally, we note that the in the case of coherent states, the averages of po~ition, momentum and energy operators follow classical physics [Meystre 1990]; I.e.,

(1.87)

where the subscript c means classical variables. This is why coherent states are called quasi-classical states, they are a reference to other states of the harmonic

oscillator, and they are also called the standard quantum limit.


The uncertainties for position and momentum, for the coherent states, may be found to be

'If (!:,.q)= -, 2w

A fW (!:,.p) = V2 (1.88)

such that

(1.89)

i.e. coherent states minimize the uncertainty principle.

1.4.4 Displaced number states

There exist several other states of the harmonic oscillator (with a given name); among them, are the so-called displaced number states, that are given by the application of the displacement operator onto the number states. If we denote the displaced number states as la,n), we have then

la,n) = D(a)ln).

These states are orthonormal, therefore (a, mla, n) complete basis of the space of states, hence

L la,n)(a,nl = 1. n=D

Their excitation number is given by the distribution

and

(1.90)

Jm,n and they form a

(1.91)

(1.92)

n ?_ m (1.93)

In Figure 1.2, we plot several distributions of these states for different values of a and n. These states will be of importance in next chapter where we talk about quasiprobability distribution functions.

1.4.5 Phase states

There exist several kind of states that are not normalized, such as position eigenstates, momentum eigenstates and also phase states. The later ones may


P(n) 0.04

P(n)

P(n) o.03

Fig. 1.2 Photon number distribution for displaced number states for (a) a= 2, n = 20, (b) a= 4, n = 30 and (c) a= 5, n = 1.

be defined as

(1.94)


These states are eigenstates of the so-called Susskind-Glogower (phase) operator [Susskind 1964]

A 1 (1.95) V=---a·

vn+1 '

i.e.,

VI¢)= ei¢1¢). (1.96)

We can write the unity operator also in terms of phase states as

1: 1¢) (¢1d¢ = 1, (1.97)

therefore we can use them to perform traces.

The Susskind-Glogower operator can also be written in the Fock basis as

V = LIn) (n + 11, (1.98) n=O

and then the following properties are easily seen; first, that vvt = 1 but, second, that vtv -=1- 1.

We will make use of the Susskind-Glogower operators in Chapter 4, where we write the atom-field interaction Hamiltonian; in Chapter 7, where we search for phase properties of the field; and in Chapter 9, where we define the SusskindGlogower nonlinear coherent states. In Appendix D, we study with some more detail the quantum phase.

1.5 Ordering of ladder operators

In some problems in quantum mechanics it is needed to calculate functions of the number operator n. As we will use the Taylor expansion of those functions, in what follows we obtain expressions for nk. We remember that the number operator is defined as n = at a, so an order issue between a and at arises. If a is always to the right of at it is called normal order' and if a is always to the left of at it is called anti-normal order. These two orderings are possible for all functions which may be expanded in a Taylor power series. However, in all but a few trivial cases, the ordering will be a very tedious procedure. The expressions we shall find for nk are sum of coefficients multiplying normal and anti-normal ordered forms of a and at. This allow8 us to obtain an expression for functions of the operator fl, and demonstrate, as a particular example, a lemma given by [Louisell 1973] for the exponential of the number operator.

Ordering of ladder operators 17

1.5.1 Normal ordering

One may use the commutation relations of the annihilation and creation operators to obtain the powers of n in normal order. For instance, we can express nk in normal order, fork= 2, as

(1.99)

fork= 3, as

(1.100)

and fork= 4, as

(1.101)

where the coefficients multiplying the different powers of the normal ordered operators do not show an obvious form to be determined. In writing the above equations, we have used repeatedly the commutator [a, a,t] 1. In Figure 1.3, we generate a table of such coefficients using the Mathematica© program, and in the Table 1.1, we present the first Stirling numbers of the

Table 1.1 Stirlin> numbers of the second km d

k / m 0 1 2 3 4 5 6 7 8 9

0 1 1 0 1 2 0 1 1 3 0 1 3 1 4 0 1 7 6 1

5 0 1 15 25 10 1

6 0 1 31 90 65 15 1

7 0 1 63 301 350 140 21 1

8 0 1 127 966 1,701 1,050 266 28 1

9 0 1 255 3,025 7,770 6,951 2,646 462 36 1

second kind, We infer that the coefficients in the above equations are precisely

these numbers; i.e., we obtain

k

nk = L Skml[at]mam, (1.102) m=O


SetAttributes[prod, {Flat, 0

~~~~~:=: ~:PI~~ c~ pr~~r:€i;::~~~}J /@ b

prod[A,Adl := 1 + prod[Ad,A] , s[n_lnteger?Positive] ·-Table[{n,s[n]}, {n,

3}] ·-

11 P~~~le~;.:latten[Table[{Ad,A},

1 prod[Ad, A]

2 prod[Ad, Al + prod[Ad, Ad, A, A]

3 prod[Ad, A] + 3 prod[Ad, Ad, A, A] + prod[Ad, Ad, Ad, A, A, A]

AdA[n_j:=prod AdA[ 5]

@@ Join[Table[Ad,{n}],

prod[Ad, Ad, Ad, Ad, Ad, A, A, A, A,

Al

p[ a ___ ]: =x"Lengt h[ {a} 1

:~~S];=s[n] /. prod -> p

l + 1 :iJ x /966 x8

+ 11b~ x + Vo5o ~ 4 + 2

1666 x

Table[A,{n}]]

+ 28x +x

c[n_j:=Coefficientlist[sx[n] /. Table[c[n], {n,S}] //TableForm x ->Sqrt[y],y]

0 1

0 1 1

0 1 3

0 1 7 6 1

0 1 15 25 10

0 1 31 90 65 15

0 1 63 301 350 140 21 1

0 1 127 966 1701 1050 266 28 1

{n}])

Fig. 1.3 A program wntten m Mathematica© to find h . number operator in normal order. t e coefficients for the powers of the

with [Abramowitz, 1972]

(1.103)

We now write a function f of fi in a Taylor series as

(1.104)

19 Ordering of ladder operators

and inserting (1.102) in this equation, we obtain

(1.105)

Because Skm) = 0 form > k, we can take the second sum in (1.105) to infinite

and interchange the sums, to have

(1.106)

For the same reason stated above, we may start the second sum at k = m, and

we can write

(1.107)

By noting that

Nn f(x) - ~ jCkl(x) sCml m! - ~ k! k '

k=m

(1.108)

where l'J. is the difference operator, defined as [Abramowitz, 1972]

m I

6.m j(x) = ~(-1)m-k k!(:~ k)! f(x + k), (1.109)

we may write (1.107) as

(1.110)

where : fi : stands for normal order.

1.5.1.1 Lemma 1

If we choose the function j(fi) = exp( -ryfi), we have that

rn I A mf(O) - ""'c l)m-k m. -"(k u -~- k!(m-k)!e '

k=O

(1.111)

and then we obtain the well-known lemma [Louisell1973] (1.112)

20 Operator algebra and the harrnonic oscillator

1.5.2 Anti-normal ordering

Following the procedure introduced in the former section, we can write nk in anti-normal order as

k

nk = (-1)k L(-1)mst~t1lam[at]m, (1.113) m=O

and a function of the number operator as

(1.114)

The second sum differs from (1.108) in the extra (-1)k and the parameters of the Stirling numbers. We can define u = -x, such that JCkl(x)x=O = (-1)kf(kl(u)u=o, and use the identity [Abramowitz, 1972]

(1.115)

to write

J(n) (1.116) m=O

[

oo JCk) ( _ ) oo (kJ ( _ ) l (m + 1) """ u - 0 sCm+ I) + """ f u- 0 sCm)

~ k! k ~ k! k ' k=m k=rn

so we can use again Equation (1.108) to finally write

(1.117)

where :n,: stands for anti-normal order.

1.5.2.1 Lemma 2

Let us consider again the function f(n) = exp( -1n). This gives us that f(x) = e-"Yx and f(u) = e"Yu. Therefore

(1.118)

such that we can obtain the exponential of the number operator in anti-normal

order (lemma) as

(1.119)

21 Ordering of ladder operators

1.5.3 Coherent states. A

E t' (1 119) to find averages for coherent states, \a) = D( a) \0)'

Let us use qua ton . d \0) . th D

A ( ) _ aiit -a*ii is the so-called displacement operator an lS e where a - e vacuum state. We have

~ (1- e~')m Am[Af]m\ ) (1120) (a\e-~'n\a) = e~'(a\ u --m-!-a a a ' .

by using that

(a\iim[af]m\a)

we may write

m=O

(0\(ii + a)m(at + a*)m\0)

m ( I )2 2k m. m- k' L \a\ (m- k)!k! ( )., k=O

(a\e-~'n\a) = e~' f (1- e~')rn Lm( -\a\2

),

m=O

(1.121)

(1.122)

where Lm(x) are the Laguerre polynomials of order m. We ca~ finally write a closed expression for the sum above [Gradshteyn 1980], to obtam the expected

result for coherent states (1.123)

1.5.4 Fock states.

For Fock or number states we obtain

(1.124)

Rearranging the sum above with k = n + m, we have

, = I' k-n k! (n\e-~'n\n) = e~' L(1- e ) n!(k- n)!'

(1.125)

k=n

. [ . ] ""= k-n k! - (1 _ x)-n- 1 has the closed whtch as Abramowitz, 1972 LA=n x nl(k-n)l -

expression (1.126)


Chapter 2

Quasiprobability distribution functions

2.1 Introduction

Quasiprobability distribution functions are of great importance in quantum mechanics, among other things, because they allow to have classical views of quantum states. They serve as tools to reconstruct the quantum state of a given system, quantized field, vibrational state of an ion, the quantum state of a moving mirror, to name some. They may be used also to extract information of the phase of the harmonic oscillator, as there is no phase operator to obtain such information [Lynch 1995]. In order to achieve this, quasiprobability distribution functions must be radially integrated; we do this in Appendix D. In this chapter, we take a look at the most important quasiprobability functions, namely, the Wigner function, the Husimi Q-function and the Glauber-Sudarshan ?-function. These distributions have to do with the different orders to write annihilation and creation operators studied in Chapter 1.

2.2 Wigner function

A classical phase-space probability density may be written as an integral of delta functions,

P(q,p) = J o(q- Q)o(p- P)P(Q, P)dQdP. (2.1)

Using the integral forms for the delta function, we may rewrite the above integral as

P(q,p) = 4~2 j e-iu(p-P)eiv(q-Q)P(Q, P)dudvdQdP (2.2)

23

24 Quasiproba.bility d1:stribution functions

or

P(q,p) = 4:2 J e-iupeivq {! P(Q, P)eiuP e-ivQdQdP} dudv. (2.3)

We can see this last equation as the two d. . . term in the curly brackets Tl t . . Imenswnal Founer transform of the stood l . le erm mside the curly brackets can b

as t le phase-space average of tl f' . . ' . ' e under-le unctwn ewp ->1!q B . correspondence principle and recalli th t . . y applymg the obtained in the form ng a averages m quantum mechanics are

(A.) = Tr{pA.}, (2.4)

we write the phase-space average between the curly brackets as

(ezup-ivq) = Tr{peiuji-ivq}. (2.5)

The trace in this last expression rna be r . . coherent states phase states .t. y . eahzed m several basis: Fock states l . . ' ' posi lOll eigenstates etc F . l" . '

t 18 position eigenstates basis, and we write ' . or Simp ICity, we choose

'D:{peiup-ivq} = J dq(q\peiup-i1!q\q), (2.6)

that by using Bak H d ff f. eiufl\q) = \q + u) er- abus or . ormula (1.28), that e-ivqlq) = e-i"qlq) and that ' may e rewntten as '

Tr{peiup-i"q} _ -iuv/2 1 -e dqe-wq(q\plq+u). (2.7)

Doing the change of variable q = x- u/2, we get

Tr{peiujJ-ivq} = 1 dxeivx (x - ~I AI u) . 2 p X + 2 ' (2.8)

by mtroducing this expression into the . integrating first with quantum mechamcal version of (2.3), and respect to v and after respect to x, we get finally

P(q,p) = W(q,p) = _!__ 1 dueiup( +~~AI - ~~ 27f q 2 p q 2 . (2.9)

In 1932 Wiu1 · t d [ , oi er m ro uced Wigner 1932] th" f . his distribution function It t . IS unctwn W ( q, P), known now as

. con ams complete i f. t. b the system, 11f'l). norma wn a out the state of

From Equation (2.3), we see that the Wi n . . . ten also as g er distnbutwn function may be writ-

W(a) = ~ 1 exp(a;3* _ a*;3)C(;3)d2;3, (2.10)

25 Wigner functi.on

where o: = (q + ip)j.J2, and C(;S') is given in terms of annihilation and creation

operators by (2.11) C(f3) = Tr{pD(IJ)} = Tr{pexp(/Jat- /)*a)};

the function C(IJ) above has many names: ambiguity function, radar function

and, more common in quantum optics, characteristic function. In the position eigenstates basis, we can write the characteristic function as

C(Q, P) =I dq(q + Q/2\p\q- Q j2)e-iPq, (2.12)

and in the momentum eigenstates basis as (2.13)

C(Q, P) I dp(p + P/2\p\p- Pj2)eipQ.

We prove now, following [Chountasis 1998], that for a pure state, the square of the absolute value of the characteristic function has the interesting property of

being its own two-dimensional Fourier transform; i.e.,

J J dQ'dP'\C(Q',P')\ 2 exp[i(P'Q- Q'P)] = 27r\C(Q,P)\2

. (2.14)

Actually, using Equations (2.12) and (2.13), we get

C(Q', P') = J dqlj;(q + Q')if;*(q)e_;p'q (2.15)

and (2.16)

C*(Q',P') = J dplj;*(p)lj;(p-P')e-ivQ'.

Inserting these equations into the left hand of Equation (2.14), and using the

Fourier transforms

~ j dQ'Ij;(q + Q') exp[-i(p+ P)Q'] = if;(p+ P)exp[iq(p+ P)]

and

~ j dP' lj;(p- P') exp[-i(q- Q)P'] = if;(q- Q) exp[-ip(q- Q)],

we get the right hand side of Equation (2.14).

The Wigner function is a quasiprobability distribution that allows the visualization of states in phase space. In Figures 2.1-3, we show how different states of the quantized field look like. It may be seen that in Figures 2.1 (number

26 Quasiprobability distribution functions

states) and 2.3 (displaced number states), the Wigner function may take negative values, which is why it is also called a pseudo-probability. However, this negativity gives us information about the state of the system: if the Wigner function has a negative part, it corresponds to a highly non-classical state; i.e., a state that deviates from quasi-classical coherent states. The Wigner function for a one-photon Fock state has been directly measured by Bertet et al [Bertet 2002].

Wrq.p! Wiq.p}

n=O n=l

W(q,pJ

n=2 n=7

Fig. 2.1 Wigner function for the first number states.

0.3-.

-3

Fig. 2.2 Wigner function for the coherent state of amplitude a= 2.

-3 p -l

q

Fig. 2.3 Wigner function for the displaced number state for the parameters a = 4 and n = 1.


2.2.1 Properties of the Wigner function

It is easy to see that if we integrate (2.9) in p, we obtain

I W(q,p)dp

(2.17)

so that

I W(q,p)dp = '1/J(q)'ljJ*(q) = P(q). (2.18)

To integrate the Wigner function in q, we do a similar analysis as the one we did t~ obtain it at the beginning of this chapter, but with the probability as a functiOn only of p, instead of the combined probability. First, in analogy with

(2.1), we write

P(p) =I o(p- P)P(P)dP =-I; I I e-iu(p-P)P(P)dudP, (2.19)

id~ntifying the integral with respect to P with the average value of eiup, and

usmg the correspondence principle

P() 1 I . . . 1 // . . . P = 2

7f etuPTr{pe-wp}du = 2

7f eiup(q/e-i¥ pe-i¥ /q)dudq, (2.20)

that just as we did before, may be re-written as

P(p) = 2~ I I eiup(q + ~/p/q- ~)dudq, (2.21)

that is nothing but the Wigner function integrated in position.

2.2.2 Obtaining expectation values from the Wigner function

We can g~neralize the Wigner function for the density operator to any given operator ¢, as follows

W( ) 1/ . UA u ¢ q,p = 27f duewP(q + 2/¢/q- 2). (2.22)

We derive now the following overlap formula

Tr{p¢} =I dqdpW(q,p)W¢(q,p), (2.23)

Wigner junction 29

between the density matrix p and the operator ¢. From the definition of the Wigner function, expression (2.9), and from (2.22),

we can write

1 j j , .( ') x x x' A x' W(q,p)W¢(q,p) = (27r)2 dxdx e" x+x P(q- 2/,8/q + 2)(q- 2/¢/q + 2).

(2.24)

Integrating now, with respect to q and p, the above product of \;vigner functions,

we get

I I W(q,p)W¢(q,p)dqdp =

= -- dqdpdxdx'et(x+x )P(q- -/,8/q + -)(q- -/¢/q + -). 1 I I I I . , x x x' A x' (27r) 2 2 2 2 2

(2.25)

The integral in p gives a delta function o(x + x'), that may be readily integrated

in x', to yield

W(q,p)W¢(q,p)dqdp = dqdx(q-2

/p/q + 2)(q + 2/¢/q- 2 ). (2.26) If If X A X X A X

Finally we make the change of variables y = q- x/2 and z = q + x/2, to arrive

at the result we are searching

J J W(q,p)W¢(q,p)dqdp = J J dydz(y/,8/z)(z/(/;/y) = Tr{,D(/;}. (2.27)

2.2.3 Symmetric averages

The Wigner function may be used to obtain averages of symmetric functions of creation and annihilation operators. If we consider the characteristic function

(2.11),

C(A/3) = Tr{,Dexp(A[/3at- j3*ii])}, (2.28)

we take its derivative with respect to A and evaluate it at A = 0, we have

dC(A/3) I - Tr{ A(/3At /3* A)} ---;v::- >..=0 - p a - a .

By repeating the procedure k times, we get

(2.29)

(2.30)

On the other hand, from (2.10) we can write the characteristic function as the Fourier transform of the Wigner function

C(A/3) =:hI exp[A(a*f3- af3*)]W(a)d2a. (2.31)

Differentiating the above expression k times with respect to A, and evaluating

at A = 0, we obtain

dkC(A/3) 1 I ~1>-=0 = ;z (a*f3- af3*)kW(a)d2 a. (2.32)

Equating (2.32) with (2.30), we finally get

Tr{p(f3at- (3*a)k} =~I (a* f3- af3*)kW(a)d2a, (2.33)

that shows that the Wigner function may be used to obtain averages of symmetric functions of creation and annihilation operators.

2.2.4 Series representation of the Wigner function

We can obtain a series representation (non-integral) of the Wigner function by

making y = uj2 in the definition

W(q,p) = _21 I dueiup(q + 2:1,8/q- 2:), 1f 2 2

to obtain

(2.34)

Remembering that

e-ipq/y) = e-ipy/y),

we can further put the exponential term in the integral above inside the bracket,

W(q,p) =~I dy(-y/eiP<ie-iqppeiqpe-ipq/y). (2.35)

Introducing now the parity operator fi = ( -1) n, we obtain

W(q,p) = ~ ./ dy(y/fieipqe-iqflpeiqfle-ipq/y),

that is nothing but the trace of the operator

2:.( -1)n iJt (a)pD(a), 1f

(2.36)

(2.37)

Wigner function 31

and which may be realized in any basis. In particular, we can use the Fock basis to finally obtain the following series representation of the Wigner function

(2.38)

Recall that the states iJ (a) I k) are the displaced number states introduced in

Chapter 1. The Wigner function written as a series representation can be viewed in two ways, as a weighted sum of expectation values in terms of displaced number states, and as weighted sum of expectation values of a displaced density matrix. In Section 4.4, we show a way of displacing field density matrices in cavities.

The series representation of the Wigner function, Equation (2.38), can be used to find a closed form for the Wigner function of the coherent states. For

that, in (2.38) we write the density matrix p as 1/3) (/31, and each coherent state

as 1/3) = D(/3) IO), and we get

W(q,p) = ~ f)-1)k(kliJt(a)D(f3)IO)(OiiJt(f3)D(a)ik). 1f k=O

(2.39)

Using the Baker-Hausdorff formula 1.28 on page 5, we can easily prove that

, t , (a* f3 - a/3* ) A D (a)D(/3) = exp 2

D(/3- a), (2.40)

thus

A t , (a* f3 - a/3* ) D (a)D(/3)10) = exp 2

l/3- a), (2.41)

and analogously

A , A ( af3* -a* /3) (O/DT(f3)D(a) = exp

2 (/3- a/, (2.42)

so that

iJt(a)D(f3)IO)(OiiJt(f3)D(a) = l/3- a)(/3- al, (2.43)

and

1 00 (2.44) W(q,p) =;: ~(-1)k(klf3- a)(/3- alk),

k=O

I, !lj

I

32 Q1wsiprobability distribntion fnnctions

which using the fact that

(2.45)

can be written as

1 00 lf3 12k W(q,p) =:;;: 2)-1l~e-li3-al2 ,

k=O ' (2.46)

and finally, we get the expression we were looking for

(2.47)

We leave to the reader, as an exercise, to show that the Wigner function for the number states In) is [Gerry 2005]

(2.48)

Using Equations (2.40) and (2.48), it is not very difficult to show that the Wigner function for the displaced number states l/3, n) is given by the expression

(2.49)

2.3 Glauber-Sudarshan P-function

Although coherent states form an overcomplete basis, they may be used to represent states of the harmonic oscillator. The representation

(2.50)

involves off-diagonal elements (alp lf3), and two integrations in phase space. The next diagonal representation, introduced independently by Glauber and Sudarshan,

(2.51)

involves only one integration.

In order to obtain an explicit expression for the Glauber-Sudarshan ?-function we introduce (2.51) in the definition '

C(f3) = Tr{pexp(f3a.t- (3*0,)},

Husimi Q-fnnction 33

of the characteristic function. We obtain

C(a) = J P(f3)Tr{lf3) ((31 exp(aO,t- a*O,)}d2 (3. (2.52)

Using the Baker-Hausdorff formula, we get

C(a)el<>l2

/2 = J P((3)Tr{lf3) (f31 eaat e-a*a}d2(3, (2.53)

or writing explicitly the trace,

(2.54)

After a Fourier transform, we obtain P(a) from the characteristic function

P(a) = ~ J exp(af3*- a* f3)Tr{pexp(f30,t) exp( -(3*0,)}d2 (3.

From Equation (2.51), we can see that

Tr{[dtta,kp} = J P(a)Tr{[O,t]na,k Ia) (al}d2 a

j P(a)Tr{ak Ia) (al [att}d2a

(2.55)

J P(a)ak[a*td2a, (2.56)

that indicates that the Glauber-Sudarshan ?-function may be used to calculate averages of normally ordered products of creation and annihilation operators.

2.4 Husimi Q-function

The Q or Husimi function is usually expressed as the coherent state expectation value of the density operator; i.e.,

Q(a) =!_(a lfJI a), 7r

(2.57)

and has as an alternative form the integral representation

Q(a) = ~ j exp(a(3* -a*(3)Tr{pexp(-f3*&)exp((3at)}d2 (3, (2.58)

To show the relation between the expressions (2.57) and (2.58), we use again the definition of the characteristic function as the Fourier transform of the probability density function, we write the Glauber displacement operator in antinormal order and we use the commutative properties of the trace, namely that


Tr{ABC} = Tr{CAB}. It is important to remark, that because pis a positive operator, we always have Q(a) 2:0. Let us note that

~Tr {! ak[a*tla)(ald2ap}

Tr { ~ j d2ala)(al x [attpak}. (2 .. 59)

Q(q,p)

L I.

Relations between quasiprobabilities 35

Recalling that ; J d2ala) (a! = 1 and using properties of trace, we obtain 0

which means that the Q-function may be used to calculate averages of creation and annihilation operators in anti-normal order.

It is an easy exercise to calculate the Husimi Q-function for the number and for the coherent states. For the number state jn), the expression

Q(a) = lal2n exp (-l::f) 1rn! 2

(2.61)

is obtained, and for the coherent state l/3), we get

Q(a) = ~ exp ( -lal2- l/312 + af3* + f3a*). 1f

(2.62)

In Figure 2.4, we plot the Husimi Q-function for the number state In), expression (2.61); and in Figure 2.5, we plot the Husimi Q-function for the coherent state l/3), expression (2.62).

2.5 Relations between quasiprobabilities

2.5.1 Differential forms

It is possible to group the Wigner, the Glauber-Sudarshan and the Husimi Qfunctions in one parametric form given by the expression

F(a,s) = ~ j C(f3,s)exp(af3* -a*f3)d2 f3, (2.63)

10

q

Fig. 2.4 Husimi Q-function for the number state state for the parameter n = 5.

where C(/3, 8 ) is the characteristic function of order s; i.e.,

C(/3, s) = Tr{D(f3)P} exp(s l/312

/2), (2.64)

where 8

is the parameter that defines which is the function we are lookin_g at. For

8 = 1 the Glauber-Sudarshan P-function is obtained, for s = 0 the W1gner

function, and for s = -1 the Husimi Q-function. .

F · (2 63) we can establish some relations between the quaslprob-rom expresswn . , . . abilities in phase-space. If we takes = -1 in Equatwns (2.63) and (2.64), we

can write the Husimi Q-function as

Q(a) = j G(f3) exp(af3*- a* f3)d2

f3 (2.65)

where

G(/3) = ~Tr{D(f3)p}exp(- 1/312

/2). 1f

(2.66)

Takincr now 8 = 0 in expressions (2.63) and (2.64), and introducing the ide~tity opera~or I in an obvious way, we get the following formula for the W1gncr

36 Q1wsiprobability distribution functions

0.

0.

{!(q.p)

0.

Fig. 2.5 Husimi Q-function for the coherent state for the parameter f3 = 2.

function

W(a) =I G(/3) exp(a/3*- a* /3) exp(l/312 /2)d2/3, (2.67)

being G(/3) the same function as in Equation (2.66).

~n the expression (2.67) for the Wigner function, we expand the term exp(l/31 2 /2) m Taylor series to obtain

00

2-n I W(a) = ~--;! G(/3) exp(a/3*- a* /3) l/312n d2f3_ (2.68)

Considering the equality

(2.69)

we can cast Equation (2.68) into

W(a) = L- --- Q(a) X 2-n ( a a )n

n=O n! aa aa* ' (2.70)

The Wigner function as a tool to calculate divergent {or not) series 37

or, finally

( 1 a a ) W(a) = exp ---- Q(a). 2 aa aa*

(2.71)

2.5.2 Integral forms

We derive now some integral relations between the phase-space quasiprobability distributions. If in the expression

Q(a) ~ (ai,Dia), 1f

that defines the Husimi Q-function, we substitute the formula

that at its turn defines the Glauber-Sudarshan ?-function, we obtain the Qfunction from the ?-function as

Q(a) =~(a 1.01 a)=~ I P(/3) (a 1/3) (/31 a) d2 {3. 1f 1f

(2.72)

Using the properties of the coherent states, this relation can be simplified to

(2.73)

The following relation

(2.74)

between the Wigner function and the Q-function can be obtained with a similar

procedure.

2.6 The Wigner function as a tool to calculate divergent (or

not) series

One can use the Wigner function in the series representation, that we derived in Subsection 2.2.4, to find the Wigner function associated with any operator. We start with the generalization of the series representation, Equation (2.38), to an arbitrary operator ¢. We write then

W¢(q,p) = 2 2:(-l)k(klbt(a)(/JD(a)lk). (2.75) k=O

First note, that if we take the position operator, we have ¢ = X = <Hat and one obtains v'2 '

q = Wx-(q,p) = 2 f(-1)k(ki(X +a+ a* )/k) = 2a +a* ~(-1)k. k=O y'2 y'2 6 (2.76)

Because a = ( q + ip) / v'2, the above expression simplifies to

00

q = Wx-(q,p) = 2q L(-1)\ (2.77) k=O

which proves that

2:)-1)k = 1/2. (2.78) k=O

If we insert now the number operator P! in (2.75), we obtain

00

Wn(q,p) = 2 _2) -l)k(k + (a(z), k=O

(2.79)

or using that a= (q + ip)j-/2 and the result (2.78),

(2.80)

On the one hand, we have that

(n) = Tr{np} = j F/(q,p)Wn(q,p)dqdp, (2.81)

and on the ot.her hand, we know that we can calculate averages of symmetric forms of creatwn and annihilation operators using the Wigner function, hence

( A) I ( aa* + a*a 1 I 1 n = W a) 2 d2a- 2 = W(a)lal2d2a- 2 (2.82)

therefore

(2.83)

Equating expressions (2.80) and (2.83), we can obtain the value for the nonconvergent sum

(2.84)

Number-phase Wigner function 39

It is worth to remark that in this last case, we could have done also the following: If we callS= .L%"= 1(-l)kk, we have that

s = -1 + 2:.)-1)kk = -1 + 2:)-1)k+1(k + 1) = -1- s + l:C-1)", (2.85) k=2 k=l k=l

and the last sum, from Equation (2.78), is equal to 1/2. The results above show that the Wigner function may be used to calculate infinite series.

2.7 Number-phase Wigner function

In the Section 2.2, where we defined the Wigner function, we also introduced the characteristic function as

C(/3) = Tr{pexp((3at- (3*a)}.

In analogy with it, we can define the function

C- '(k e)- 1Tr { [vA '(k &) -i(k</>-n&) H c] "} n- ' - 2 n- ' e + . . p ' (2.86)

where

(2.87)

and with

-crt= I: ik + l)(kl (2.88) k=O

the Susskind-Glogower operator [Susskind 1964]. Because the Susskind-Glogower formalism fails in the phase description of the electromagnetic field with small photon numbers, the unitarity of V is spoiled. Also, as consequence of the fact that there is not a well defined phase operator, one can not use an expression of the form exp [i(k~- ¢n)], and we use instead a "factorized"form in Equation (2.87). Note that in order to produce a real Wigner function, we added the complex conjugate in (2.86) (because n can not be a negative integer). Expression 2.10 on page 24, does not have this problem, because the integrations over f3x and /3p are from -oo to oo.

By writing the density matrix in the number state basis, we have

p= LLQm,z/m)(l/, (2.89) m=O l=O

that substituted in (2.86) gives us

iiJk 00 00

cn_<J>(k, B)=;- L L Qm,zTr[(Vt)ke-iiJnlm)(ll]e-i(k¢-niJ) + c.c. (2.90) m=O l=O

Using the relatiop between the characteristic function and the quasiprobability distribution, expressed in formula 2.10 on page 24, we get

1 00 r27r -

W(n, ¢) = (21r)2 k~n Jo Cn_<J>(k, B)de, (2.91)

where for obvious reasons the double integration over the whole phase space in 2.10 on page 24, becomes here a sum and a single integration. Inserting Equation (2.90) into (2.91), we obtain

1 00

W(n,¢) = 4;;: L (Qn,n+ke-ik¢ + Qn+k,neik¢).

k=-n

It is easy to show that the integration of (2.92) over the phase ¢gives us

la2

7r W(n, ¢)d¢ = Qn,n = P(n),

that is the photon distribution. Besides, if we add (2.92) over n, we get

P(¢) = f W(n, ¢) = 2~ f f Qn,me-i(m-n)¢

n=O n=Om=O

that is the correct phase distribution.

(2.92)

(2.93)

(2.94)

It is worth to note that for a number state /M), Equation (2.92) reduces to W(n, ¢) = bnM j21r; i.e., it is different from zero only for n = M, as it should be expected.

2. 7.1 Coherent state

The phase-number Wigner function for a coherent state

(2.95)

Number-phase Wigner function 41

is given by

_ e-1<>12 an~ ak cos[(n- k)¢]

W(n, ¢)-21rVnf ~ v7J . (2.96)

In Figure 2.6, it is plotted the phase-number Wigner function for an amplitude a= 4. It may be noted a smooth behavior. It may be seen a unique contribution

of the single coherent state localized at the phase value ¢ = 0.5.

Fig. 2.6 Two different views of the number-phase Wigner function for a coherent state for an

amplitude a = 4.

2.7.2 A special superposition of number states

Let us consider the state

M

/¢ ) = _1_ L eim¢o/m). M vM + 1 m=O

(2.97)


This state tends to have a completely well defined phase as M tends to infinity. For this state, the phase-number Wigner function reads

1 M

W(n,¢) = 2(M + 1)1r I:cos[(n- k)(¢- ¢0 )],

k=O

(2.98)

that may be put in the form [Gradshteyn 1980]

W(n, ¢) = 2(M ~ 1)1r cos [ ( !lf-- n) (¢- ¢ 0 )]

x sin [ M: 1 ( ¢ - ¢ 0)] esc ( ¢ ~ ¢o) .

(2.99)

In Figure 2.7, Equation (2.99) is plotted forM= 20 and ¢0 = 0.7. The phasenumber Wigner function shows a well defined phase or phase localization as it would be expected for a state of the form (2.97). It is also seen that as ¢ approaches the value ¢o, the maximum value for the phase-number Wigner function for the state (2.97) is obtained; from (2.99), it may be shown that this value is 1/27r. By adding over n Equation (2.99), the phase distribution

P( ¢) = 2 ( M ~ 1) 7r sin 2 [ M: 1 ( ¢ - ¢o)] csc2 ( ¢ ~ ¢o) (2.100)

is obtained, and corresponds to the phase distribution for the state (2.97).

W(n,rp)

Fig. 2.7 Number phase Wigner function for a superposition of the first twenty one Fock states.

Chapter 3

Time Dependent Harmonic Oscillator

In this chapter, we study the time dependent harmonic oscillator, that besides being important by itself, it will be used when solving the ion-laser interaction in Chapter 8. We solve the Hamiltonian for the time dependent harmonic oscillator by Lewis-Ermakov invariant methods, that will produce the so called squeeze operator. Therefore, we also introduce here squeezed states and some other states of the harmonic oscillator.

3.1 Time dependent harmonic Hamiltonian

Let us first consider the Hamiltonian for the time independent harmonic oscillator with unitary mass, m = 1, and unitary frequency, wo = 1,

(3.1)

We can define annihilation and creation operators for this system by the following expressions

A 1 (A 0 A) b = v'2 q + tp' ht = __!_(q- ip),

v'2 such that we can rewrite the Hamiltonian as

fi = (hth+ ~) = (n+ ~), where we have defined the number operator n = hth. Eigenstates for the Hamiltonian (3.3) satisfy the equation

H[n) = ( n + ~) [n).

43

(3.2)

(3.3)

(3.4)

44 Time Dependent Harmonic Oscillator

The time dependent harmonic oscillator Hamiltonian reads

(3.5)

where D(t) is the time dependent frequency.

It is well-known [Lewis 1967] that an invariant for this system is the LewisErmakov invariant, that has the form

where E obeys the Ermakov equation

E:+D2c=c-3 .

(3.6)

(3.7)

Actually, it can be proved that the Lewis-Ermakov invariant, represented by the operator (3.6), does not change in time; or in other words, the equation

di of A A

dt =at- i[I,Ht] = 0 (3.8)

is satisfied.

Furthermore, it is easy to show that i may be related to the Hamiltonian of the harmonic oscillator with constant frequency (3.1) by a unitary transformation of the form

(3.9)

that with the help of the Baker-Hausdorff formula, can be re-written as

TA [·ln(c)(AA AA)] [ o s A2] = exp 2-2

- qp + pq exp -2~q = exp 2-- exp -2-q2 [.lncdiP] [ . i,]

2 dt 2c '

so that (3.10)

(3.11)

By using Equation (3.4) and the the fact that the operator Tis unitary, we can see that

(3.12)

i.e., states of the form

(3.13)

Time dependent harmonic Hamiltonian 45

are eigenstates of the Lewis-Ermakov invariant. Following Lewis [Lewis 1967], we introduce now the new annihilation and cre

ation operators

A 1 [q o( A o A)] a = y'2 ~ + 2 Ep - Eq , At 1 [ij • A oA ] a = - - - 2(Ep- Eq) .

y'2c (3.14)

in terms of which, the Lewis-Ermakov invariant can be written as

(3.15)

with the obvious definition of fit. Once the creation and annihilation operators are defined for the time dependent harmonic oscillator, analogous equations to the harmonic oscillator with constant frequency may be obtained . For instance, we can define the displacement operator as

(3.16)

and the coherent states as

(3.17)

and show the corresponding relations

lcx)t = (3.18)

It can be proved [Fernandez-Guasti 2003], that the Schrodinger equation for the time dependent harmonic oscillator Hamiltonian has a solution of the form

11/J(t)) = exp [ -ii lt w(t)dt] rti'(O)I1/J(O)), (3.19)

with w(t) = 1/ c2 . If we consider the initial state to be

(3.20)

where lex) is a coherent state of the harmonic oscillator with constant frequency. We note that the evolved state has the form

11/J(t)) = rf't Ia exp [ -i lt w(t')dt']) = Ia exp [ -i lt w(t')dt']) t; (3.21)

or in other words, coherent states keep their form through evolution.

46 Time Dependent Harmonic Oscillato1

3.1.1 Minimum uncertainty states

We define the operators

and

A 1 P= -(a-at)

iv2 '

(3.22)

(3.23)

where a and at are the annihilation and creation operators defined in Equations (3.14). It is easy to see that they are related to q and j5 by the transformations

ii=TQi't (3.24)

and

(3.25)

and that they obey the equations [Fermindez-Guasti 2003]

P = iw(t)[i, P] = -w(t)Q, (3.26)

and

Q = iw(t)[i, Q] = w(t)P. (3.27)

~he uncertainty relation for operators Q and P, for the coherent state (3.18), is given by

(3.28)

where .6.X = V(X2)- (X) 2

. Time dependent coherent states are thus minimum uncertainty states, not for position and momentum but for the transformed position and momentum. '

3.1.2 Step function

Let us consider again the Hamiltonian of the time dependent harmonic oscillator

Time dependent harmonic Hamiltonian 47

with S1(t) specified by a step function that may be modeled by [Fermindez-Guasti 2003]

(3.29)

where ts is the time at which the frequency is changed, .6. = w2 - w1 with w1

and w2 the initial and final frequencies, and A is a parameter (>.--+ oo describes the step function limit). In Figure 3.1, we plot this function as a function oft

Q(t)

1 I I I

,-----------------

l,-----------------------------1

I I I

;,

o+---~----~---------.----~---.----~---.----~----

0 4

Fig. 3.1 O(t) as a function oft for w1 = 1 and w2 = 2 (solid line) and w2 = 3 (dashed line). A = 20 and t 8 = 2.

for w1 = 1 and w2 = 2 (solid line) and w2 = 3 (dashed line). The solution to the Ermakov equation (3. 7) for this particular form of S1(t) is given by [FernandezGuasti 2003]

c(t) = ~ 1 + n~ft) + ( 1- n~ft)) cos ( 21~ n(t')dt'} (3.30)

A plot of c(t) is given in Figure 3.2 for the same values given in Figure 3.1. We also plot w(t) = 1/c2 in Figure 3.3. It may be numerically shown that the time average of w(t) from t 8 to the end of the first period is 2 for the solid line and 3 for the dashed line, with Wmax = 22 for the solid line and Wmax = 32 for the dashed line. Let us consider that at time t = 0, we have the system in the initial coherent state (2.95). From Figure 3.2, we can see that T(O) = 1 since i = 0 and lnc = 0. Therefore from (3.20), I1P(O)) = i't(O)Ia) = la)o = Ia), and from

I

lj ,,I

48

1.2

0.8

t:(t) 0.6

0.4

0.2


I I I I I I I I I I

\/

/\ I \

I \ \ I

I I I I \_;

0+---~-.--~--.-~---,--~-,,-~------~~--~--~~--~

0

Fig. 3.2 c:(t) as a function oft for Wl = 1 and W2 = 2 (solid line) and w2 = 3 (dashed line). ).. = 20 and t 8 = 2.

10

~ r, {1 I\ 1\ I\ 1\ II I I I I I I I I I I I I I I I I I I

w(t) I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

I I I \ ....__ ___ !

"- I

Fig. 3.3 w(t) as a function oft for w1 = 1 and w2 = 2 (solid line) and w2 = 3 (dashed line). ).. = 20 and t 8 = 2.

(3.19) we obtain the evolved wave function

11/J(t)) = e-if f~ w(t)dtf'tla) = f'tlae-if~ w(t)dt). (3.31)

Note that the coherent state in the above equation is given in the original Hilbert space; i.e., in terms of number states given in (3.4). From Figure 3.2, we can also see that for the maxima, i( tmax) = 0 and Inc:( tmax) = 0, therefore f't ( tmax) = 1

More states of the field 49

and

11/J(tmax)) = w(t)dt); (3.32)

i.e., we recover the initial coherent state. However, for the minima, we have

i(tmin) = 0 and lnc:(tmin) =J 0, and then we obtain

l

n!)(t • )) - [ilnc:(tmin)(AA+ AA)] '!-' mm - exp 2 qp pq

w(t)dt), (3.33)

that may be written in terms of annihilation and creation operators as

w(t)dt), (3.34)

that are the well-known squeezed (two-photon coherent) states [Yuen 1976], also

see next section,

11/J(tmin)) = w(t)dt lnc:(tmin)). (3.35)

Squeezed states, just as coherent states, are also minimum uncertainty states.

However the uncertainties for ij and p are

(3.36)

and

(3.37)

For times in between we will have neither coherent states nor standard squeezed

states (in the initial Hilbert space), but the wave function

(3.38)

It should be stressed however, that in the instantaneous Hilbert space we will

always have the coherent state (3.21).

3.2 More states of the field

We now introduce more states of the harmonic oscillator, such as squeezed states,

Schrodinger cat states and thermal distributions.


3.2.1 Squeezed states

Squeezed states may be obtained by the application of a unitary squeeze operator defined as (see (3.34)) [Yuen 1976]

(3.39)

where r is an arbitrary number (for simplicity we take it real, but it may be complex).

The squeeze operator is a unitary operator, that means that

(3.40)

and also has the property

(3.41)

Using the Hadamard lemma, expression 1.23 on page 4, it is easy to show that the squeeze operator transforms annihilation and creation operators as follows:

(3.42)

with J1 = cosh r and v = sinh r. A squeezed state is then written as

Ia, r) = S (r) Ia), (3.43)

where Ia) is a coherent state.

The photon distribution P(n;a,r) = l(nla,r)l 2 for the squeezed state la,r) is given by

(3.44)

This photon distribution is plotted in Figure 3.4 for two different values of r. Extra distributions after the main distribution may be observed. The extra distributions may in principle be measured by passing atoms through a cavity that contains this squeezed quantized field. After the atom interacts with the squeezed state, atomic states (excited or ground) may be measured. The atomic inversions that are produced by such measurements will present features related to this extra distribution that are called ringing revivals (see next chapter).


0.14

0.12

0.1

0.08

0.06

0.04

0.02

60 80 100 120 140

0.1

0.08

0.06

0.04

0.02

80 100 120 140

Fig. 3.4 Photon distribution for the squeezed state for a= 5 and (a) r = 1.5 and (b) r = 2·

The Husimi Q-function can be calculated from the photon distribution, for-I (3.44) for the squeezed states; we start with the definition 2.57 on page 33.

mu a , . . ·tt In the case of a squeezed state 1,6, r)' the density matnx lS wn en as

(3.45)

52

thus


Q(a) ~(a /8(r)D(i1)/ o) (o Jbt(/J)St(r)/ a) ~ /\a/S(r)D(/1)/o)/

2

.

We write the coherent state as (al = (Oibt(a), so

Inserting a one as S('r)St(r), we get

Using Equations (3.42), it is easy to show that

and using 2.40 on page 31, we obtain

Q(a) = ~ i(OI/1 ~ a*v ~ etJ.L, r)l 2.

7f

Using (3.44), we finally get

Q(a) ~P(0;1,r) 7f

1 { 2 v [ 2 *2]} nJL exp ~hi ~ 2J.L I +1 ,

(3.46)

(3.47)

(3.48)

(3.49)

(3.50)

(3.51)

(3.52)

where 1 = /1 ~ a*v- aJ.L, and we recall that a = (q + ipjy"i), JL = coshr, v = sinh r and /1 is the amplitude of the squeezed state. The Husimi Q-function, of the squeezed state la,r), is plotted in Figures 3.5 and 3.6 for two different values ofT (the same values that we used for the photon distribution).

The Wigner function for the squeezed states can a!so be calculated. For that we use first the series representation for the Wigner function,


Q(q,p)

0

Fig. 3.5 Husimi Q-function for the squeezed state with amplitude equal to 3 and r = 1.5.

(expression 2.38 on page 31), and we do the same processes we did for the Husimi

Q-function, to write

(3.53)

or

W(o:) = ~ f(-1)kP(k;/1-tm-va*,r). (3.54) 7f k=O

Substituting (3.44) in the above equation, we get

w(a) = 2..e-hl 2 -~h2 +'1. 2 ) f c-:r (;J.L) k \Hk ( ~~v) \2

(3.55) J.L1f k=O

The sum can be done with the integral expression

(3.56)


Q(q,p)o.os

-1

0 p

2 q

Fig. 3.6 Husimi Q-function for the squeezed state with amplitude equal to 5 and r = 2.

for the Hermite polynomials. The final result is

W (a)=- exp -v - + -- b2 + (r*)2] - __ f..l __ + 1 11'12 . 1 { ( 1 1 ) ( 1 2 v2 ) }

7f 2f..l f..l- v f..l f..l- v

In Figure 3. 7, we plot this Wigner function. (3.57)

The uncertainties for ij and f5 are

(3.58)

where we have taken w = 1. The decrease in position uncertainty for positive r may be seen in Figure 3.7.

3.2.2 Schrodinger cat states

Because coherent states are quasi-classical states, the superposition of two of them is the closest we can get to the paradox proposed by Schrodinger [Schrodinger 1935]: the superposition of two classical states (cat "dead" plus cat "alive").


0.3

0.4

Fig. 3.7 Wigner function for the squeezed state for a= 2 and r = 1.5.

Therefore, the states

(3.59)

where N± = J2[1 ± exp(-2/a/ 2)] is a normalization constant, are called Schrodinger cat states. The photon distribution for the Schrodingcr cat states is very easy to calculate and we leave it as an exercises to the reader. What we get is

(3.60)

In Figure 3.8, we plot the photon distribution for the "plus" cat with a = 4; we see that only even photons are allowed in such a cat state. In case we had the "minus" cat, only odd photon numbers would have non-zero probabilities. Using the series representation of the Wigncr function, expression 2.38 on page 31, and the expression 2.40 on page 31, it is also easy to obtain the Wigner function for


0.2

0.15

0.1

0.05

30 40

Fig. 3.8 Photon number distribution for the cat state 1'1f'!.;';.t), a= 4.

the Schrodinger cat states. The result is

1 W(q,p) = 2

2n [1 ±exp ( -2lal )]

{ exp (-2 I a - ')' 12 ) + exp (-2 Ia + 1' 12) ± 2 exp (-2 h 12 ) cos ( 48' (1' *a)) } ,

(3.61)

where now we have defined')'= (q+ip)jV"2,. It is clear that in the above expression, we have the Wigner function of the "live"cat Ia), plus the Wigner function of the "death" cat I - a), and an interference term. In Figure 3.9, we show the Wigner function for the "plus"cat. It may be seen the usual characteristics of such distribution for the cat state, namely, the two contributions of the coherent states at a = 4 and a = -4 and the quantum interference that produces oscillations in the quasiprobability distribution function. Approximate cat states may be generated in several systems such as Kerr media (see Appendix A), atom field interactions (next chapter) and ion-laser interactions (see Chapter 8).


W(q,p)

-l(j

p

Fig. 3.9 Wigner function for the cat state 1'1f'!.;';.t), a= 4.

3.2.3 Thermal distribution

Up to now, we have been looking at states that may be called pure states; i.e., states that may be represented by a wave function. However, there are states that can not be represented by a wave function, but they have to be represented as an statistical mixture of pure states; those states are called mixed states. To study mixed states, the density matrix is especially useful, because any state,

pure or mixed, can be characterized by a single density matrix. We consider now a thermal distribution which has the following diagonal expan-

sion in Fock states

(3.62)

with n the average number of thermal photons. These states are known as

thermal states.


The photon distribution is in this case given by the expression

P(k)- nk - (1 + n)k+l. (3.63)

In Figure 3.10, we plot this photon distribution.

25

Fig. 3.10 Photon distribution for the thermal distribution, with n = 3.

Chapter 4

(Two-level) Atom-field interaction

4.1 Semiclassical interaction

Given a discrete spectrum of an atom, two of their energy levels may be connected by a near resonant transition of a classical electromagnetic field. We may call these levels as (e), excited, and (g), ground. The electromagnetic field is assumed to be monochromatic. If we associate the energies Ee and E 9 with the excited and ground states of the two-level atom respectively, the atomic Hamiltonian operator (unperturbed) may be written as

HA = woJe)(eJ, (4.1)

with w0 = Ee- E9 . The Hamiltonian between the atom and the electromagnetic field is given, in the dipole approximation, by the interaction

Hr = -J. E(f'o), (4.2)

where J is the atom dipole moment, and E(fo) is the electric field evaluated at fQ, the position of the dipole. Because we assume definite parity of the states (e) and (g), the matrix elements of the dipole operator will be off-diagonal [Allen 1987]; in other words,

(efdie) (gfdig) = 0, (4.3)

(eidlg) g, (4.4)

(gfdle) g* (4.5)

59

60 (Two-level) Atom-field 1:nteraction

We can now write the semiclassical Hamiltonian for the atom-field interaction as

if Wo A A ( ) A sc = 2 cr z + 2 cos wt cr x, (4.6)

where w is the field frequency, and where we have done the following: (a) we have used the usual notation in terms of Pauli cr-matrices; (b) we have shifted the energy of the atom; (c) we have given the explicit time dependence of the harmonic field; and (d) we have used the real interaction constant, A= lgiEo/2 with E0 the amplitude of the field. The cr-matrices are defined as

(jz = le)(el-lg)(gl, (jx = le)(gl + lg)(el, Q-Y = i(lg)(el-le)(gl), (4.7)

with commutation relations

(4.8)

In the following, we will jump from this notation to 2 x 2-matrices at our convenience; thus we will make the interchange

(4.9)

and

o-z = ( 01 0 ) A ( 0 1 ) A ( 0 -i )

-1 ' CTx = 1 0 ' cry = i 0 · (4.10)

Using the rotating wave approximation (RWA) [Allen 1987], the Hamiltonian ( 4.6) may be taken to the form

fi _ Wo A + A( -iwt A + iwt A ) sc- 2crz e cr+ e cr_ , (4.11)

with cr+ = cr~ = crx + icry. However, to see the effects of counter-rotating terms, we will use a (dynamical) small rotation approach [Klimov 2000] to find first order corrections. We therefore transform the Hamiltonian (4.6) using Rl1f;(t)) = l¢(t)), with

R = exp[ia cos(wt)Q-y], (4.12)

where a« 1, such that we can cut the expansion to first order. The transformed Schrodinger equation we find is

i 0 ~~~t)) = ifscl¢(t)) (4.13)

Semiclassical interaction 61

where the Hamiltonian is given by

if Wo A + .:\(e-iwto- + eiwto- ) sc = 2C7z + -'

(4.14)

with

5.=A~- (4.15) w+wo

It is frequently convenient to consider the quantity known as the atomic ~nver~ion W(t), defined as the difference in the excited- and ground-state populatwns; 1.e.,

W(t) = Pe(t)- Pg(t), which in this case is

W(t) = Pe(t)- Pg(t) = 1(1f;(t)le)l2

-1(1f;(t)lg)l2

(4.16)

= (1j;(t)le)(e11f;(t))- (1j;(t)lg)(gll1f;(t)) = (1j;(t)IQ-zl1f;(t))

In Figure 4.1, we plot the atomic inversion W(t) = (1j;(t)!Q-zl1f;(t)) on reson~nce (~ = 0), for A= 0.4w (dashed line) and A= .01w (solid line). We have obtamed

1.0~----------------------~------------------~

0.8

0.6

0.4

0.2

o.o-ol---~-.5--~~-1'o--~---1r5--~--2·o--~~~2~5~~--~3o

Fi . 4.1 Probability to find the atom in its excited state as a functio~ of_ the scaled ti~e._ The g t A _ 0 and ' - 0 4w (dashed line) and>..= O.Olw (sohd !me). The sohd !me IS parame ers are '-' - A- · · t"

also for the solution to the Hamiltonian (4.6) with the rotating wave approxJma wn.

the solution using the Hamiltonian for the small rotations approximation. How-

62 (Two-level) Atom-field interaction

~ver, the solid line is also the figure using the solution for the Hamiltonian (4.11); I.e., the one obtained using the rotating wave approximation. It may be seen that for sufficiently large parameters, w » A and w0 » A, both solutions are the same, and that the rotating wave approximation is an excellent approximation for such conditions.

4.2 Quantum interaction

If we consider the field to be quantized, the Hamiltonian for the atom-field interaction reads

HA woA + A '(A At)A =2CJz wn+Aa+a CTx. (4.17)

If we set w = wo, and work in the interaction picture and in the rotating wave approximation, the Jaynes-Cummings Hamiltonian, in terms of Pauli matrices, reads

a ) 0 . (4.18)

The atomic operators obey the commutation relation [a+, a-]= 6-z.

We can re-write the Hamiltonian (4.18) with the help of Susskind-Glogower operator, defined in 1.95 on page 16, as

(4.19)

where

A ( 1 T= 0 (4.20)

Note that i'ti' = 1, but i'i't i= 1. This allows us to write the evolution operator as

U(t) r( cos(Atvfi + 1) -isin(Atvfi + 1) )t' -i sin(Atvfi + 1) cos(Atvfi + 1)

( 0

(0),;01 ) ' + (4.21) 0

or

O(t) ~ (

Quantum interaction

cos(Atvfi + 1)

-ivt sin(Atvfi + 1)

-isin(Atvfi + 1)V ) .

cos(Atvfn)

63

(4.22)

We are now in the position to apply the evolution operator to an initial atomfield wave function to find the evolution of the system. If we consider the initial state as 11f0(0)) = 11f0F(O))Ie); i.e., we consider the atom in its excited state and the field in an arbitrary field state 11f0F(O)), we obtain

11f0(t)) = cos(Atvfi + 1)11f0F(O))ie)- iVt sin(Atvfi + 1)11f0F(O))Ig). (4.23)

The above equation shows that after the interaction, the atom and the field get entangled in such a way that we can not write, in general, a wave function that is the multiplication of the two wave functions that correspond to the atom and the field. At some times, atom and field will almost disentangle. This is usually studied by using the atomic inversion, which is done next. Other variables, like the purity and the entropy, will be studied in Chapter 6.

4.2.1 Atomic inversion

We can calculate averages of operators, in particular we can compute the atomic

inversion, W(t) = (1f0(t)lazl1f0(t)), as

W(t) = L Pn cos(2Atvn + 1), (4.24) n=O

where Pn is the photon distribution for the initial state of the field 11f0F(O)). For a coherent state, we plot this function as a function of At in Figure 4.2.

We see there, how the Rabi oscillations collapse (for the case of the coherent state, the collapse occurs at At :::::; 1r), they remain unchanged for a time, and then revive. These phenomena occur because the interference (constructive or destructive) of the cosines in Equation ( 4.24). The constructive interference occurs for the so-called revival time tR = 21r,fnjA, where n = lal 2

. Although, in the collapse region, it looks like there is no more interaction, this is not the case as the atom and the field, strongly entangled, are exchanging phase information that will cause a quasi-disentanglement at time tR/2, both atom and field going to quasi superpositions of excited and ground states and of coherent states, respectively. In Figure 4.3, we plot also the atomic inversion but for a superposition of co-


1.0-,------------------------------------------------,

0.5

vv 0.0

-0.5

-1.0-r----~---.--~----.---~----.---~----.---~----; 0 10 20 30 40 50

.At

Fig. 4.2 Atomic inversion for a coherent state with a= 6.

herent states. It may be seen that the revival time occurs faster than for the coherent state case; therefore, measuring atomic properties gives us indication about what state we have in the cavity. The measurement of atomic properties will be exploited at large in Chapter 7. The revival happens faster in the case of the superposition of coherent states, because of (non-zero) neighbor terms that interfere constructively in ( 4.24) are separated (recall that in this case we have only odd or even photon numbers). The revival time for the cats is then a half of the revival time for coherent states [Vidiella-Barranco 1992]. More indications of the fact that atomic inversions indicate the state of the field, may be seen in Figures 4.4. and 4.5, where we plot the atomic inversion for a squeezed state. The details, better seen in Figure 4.5, show the contributions of the different parts of the photon distribution for the squeezed states, Figure 3.4b, producing the so called ringing revivals. In the case of a thermal distribution, neither collapses nor revivals occur, as the cosine terms in ( 4.24) do not interfere neither constructively nor destructively. This fact may be seen in Figure 4.6.

65 Dispersive interaction

1.0

O.B

0.6

0.4

0.2

\IV 0.0

-0.2

-0.4

-0.6

-O.B

-1.0 0 10 20 30 40 50

Fig. 4.3 Atomic inversion as a function of ).t for a superposition of coherent states with a = 6.

4 .3 Dispersive interaction

. f E tion ( 4 17) and consider rr; btain the dispersive Hamiltoman, we start rom qua . . . o o . . .6.. _ _ w By domg the umtary

that .6.. » .\, where .6.. 1s the detunmg, - wo ·

transformations

U1 = exp [6(&ta-+- &8--)]' (4.25)

. /( ) d t = >.j(w _ w) using the expansion 1.23 on page 4 w1th 6 = ,\ Wo + w an "'2 0 ' · · ( 4 17) and keeping terms to first order in 6 and 6, we can cast the Hamlltoman .

into the effective Hamiltonian 1

--t, u' u' HU, t[Tt- wata + wo8-z- xa-z(ata, + -2) n=21 12-

where

2 4wo X=,\ w6- w2'

(4.26)

(4.27)


vv

0 20 40 60 so

A.t

Fig. 4.4 Atomic inversion as a function of .At for a squeezed state, with a = 5 and r = 2.

Equation ( 4.27) should be compared with the interaction constant for the dispersive model when the rotating wave approximation is applied: x = 2>.2 /6.. Although x and X do not differ too much in the atom-field case, because we can not have the atomic frequency too different from the field frequency due to the two-level atom approximation, they differ a lot in problems where the frequencies involved are much more different like in the case of the ion-laser interaction (see Chapter 8).

4.4 Mixing classical and quantum interactions

Let us now consider an atom interacting simultaneously with a classical field and a quantum field. This situation is shown in Figure 4.7. The Hamiltonian for this interaction is

(4.28)

Mixing classical and quantum interactions 67

0.3--------------------------------------~

0.2

0.1

0.0

-0.1

-0.2

-o.3 J3ls--~---4·o--~---4·2--~---4~4~~--~4~6;-~--~4~s;-~~-;s,o

h 5 d 2 Here, the F" 4 5 Detail of the atomic inversion for a squeezed state, wit a = an r = . s~~~alied ringing revivaL9 may be clearly seen.

By transforming to an interaction picture, we obtain the interaction Hamiltonian

Hr = [Jt (~)>.(at&_+ &+a)b (~). (4.29)

The Hamiltonian above describes the interaction of an atom with t":o fields, otne

U . E t' (4 22) the evolutwn opera or quantized and the other classical. smg qua wn . '

is easily obtained as

Ur(t) =

b'm( cos(>.tv'ii + 1)

-iVt sin(>.tv'ii + 1)

-i sin(>.tv'ii + 1)Y)

cos(>.tv'fl)

(4.30)

68

0.2 vv

(Two-level) Atom-field interaction

Fig. 4.6 Atomic inversion as a function of >-.t for a thermal distribution with n = 3.

We therefore can find the atomic inversion for the case of Figure 4. 7; i.e., for the field in the vacuum state and the atom in its excited state; what we get is

W(t) = L Pn(;Jj-A) cos(-Atv'TL+l), (4.31) n=O

with

(4.32)

Equation (4.31) shows that we can have revivals of the atomic population inversion when we start with the field in the vacuum state and a classical field injected into the cavity, which also interacts with the two-level atom. The classical field therefore displaces the vacuum to take it into a coherent state with amplitude ;Jj-A. In the case that instead of a vacuum IO), we start with an initial field l?jJ(O)), the classical field displaces this field to*

D(;Jj -A) l?jJ(O)). (4.33)

*Note that the classical field works just as a displacement only when we obtain expectation values of atomic operators.

Slow alom interacting with a quantized field

Fig. 4.7 Level scheme of a two level atom interacting with a classical field and a quanttl

field (vacuum) in a cavity.

This result will show its value in Chapter 7, where we develop methods tD

reconstruct quasiprobability distribution functions from atomic measurement&·

4.5 Slow atom interacting with a quantized field

In this section. we show how a three body problem may be reduced to a two boJY - . t" ,g problem via a transformation. We treat the problem of a slow atom mterac ll e

with a quantized field; because the slowness of the atom, the field mode-sha9 affects the interaction. We can write down the Hamiltonian describing a sing;Je two-level atom passing an electromagnetic field confined to a cavity. In additiOn to the Jaynes-Cummings Hamiltonian, we have to add the energy of the free atom and the spatial variation it feels from the cavity; the Hamiltonian read&

On resonance, we can pass to the interaction picture Hamiltonian

2

Hr = !!.__ + g(x)(iio-+ +at o-_). 2


Using the 2 x 2 notation for the Pauli spin matrices, the interaction Hamiltonian can be written as

A p2

At ( 0 vn + 1 ) A

Hr = 2 + g(x)T vn + 1 0 T, (4.36)

where the non unitary transformation T, defined in expression 4.20, has been used.

We already point out that fft = 1, but ftf = 1- Pg,v with

Pg,v = ( ~ IO)~OI ) . (4.37)

We can use the definitions above to rewrite (4.34) as

By noting that Tp9 ,v = 0, we rewrite the previous equation as

fi = tt !!_ t + !!_ + (x)ft ( o v n + 1 ) t I 2 2 Pg,v g vfn + 1 0 ' (4.39)

where we have used that P~,v = Pg,v· Finally, we factorize the transformation operators in the Hamiltonian to obtain

(4.40)

Remark that

(4.41)

so the evolution operator for the Hamiltonian above is given by

Ur(t) = exp { -ii't [ ~ + g(x)o-xvn + 1] Tt} exp ( -i~p9,vt). (4.42)

To reduce the first exponential of this expression, we can expand it in Taylor series, and note that the powers of the argument are simply

{

A [p2 J A } k A [ 2 ] k A rt 2 + g(xkxvn + 1 T = rt ~ + g(x)a-xvn + 1 T, k 2 1,

(4.43)

such that

Slow atom interacting with a quantized field

exp { -ii't [ ~ + g(x)o-xVn + 1] Tt} =

= ft exp { -i [ ~ + g(x)a-xvn + 1 J t }t.

71

(4.44)

Note that the evolution operator in ( 4.44) is effectively the interaction of two systems, as it is written in a form in which the field operators commute, unlike the Hamiltonian (4.34), where all the operator involved do not commute.


Chapter 5

A real cavity: Master equation

5.1 Cavity losses at zero temperature

Let us introduce the Master equation for a lossy cavity from a correspondence principle approach; here, we will follow the approach given by Dutra [Dutra 1997]. Consider a cavity (in one dimension, x, for simplicity) composed of two mirrors, with reflectivity R, apart by a distance L. A number N of round trips of the light will take a timet= 2LNjc, c being the speed of light. The electric field of a classical electromagnetic wave at position x will decay from its initial value because of the partial reflection at the mirrors as

E(x, t) = RN E(x, 0).

By substituting N = ctj2L on the above equation,

is obtained, with

E(x, t) = E(x, O)e-l't

clnR !=---.

2L

(5.1)

(5.2)

(5.3)

Being coherent states quasi-classical ones, one would expect them to decay as a classical field; i.e.,

(5.4)

For a time 1St« 1//, we have

iae-~'8t)(ae-~'8ti :::::J e-lal2

(1 + 2ial 2i0t) L [1-,ot(n + m)]ln)(ml, (5.5) n=O,m=O

73

74 A real cavity: Master eqv,ation

keeping terms of the order of Jt, we get

such that, rearranging terms and using the definition of the derivative, we arrive to the Master equation for cavity losses

(5.7)

We now solve the above equation, for an initial coherent state, to show that those states certainly decay, as we assumed in the derivation of the Master equation. By defining the superoperators

we can rewrite (5.7) as

dp A A A

dt = (J +L)p,

that has the simple formal solution

p(t) = exp[(J + i)t]p(O).

(5.8)

(5.9)

(5.10)

The problem is now how to factorize the exponential of superoperators. To this end we propose the following ansatz:

p(t) = exp(it) exp [!(t)J] p(O). (5.11)

By differentiating both sides of the above equation, we obtain (5.9), and also

dp A 0 A A A

dt = Lp(t) + f(t) cxp(Lt)J exp( -Lt)p(t). (5.12)

Because the commutation relation

(5.13)

we can use 1.23 on page 4 to obtain

exp(it)J exp( -it) = Je 2't, (5.14)

so that

~ = (i + j(t)Je21t)p(t). (5.15)

Comparing (5.15) and (5.9), we obtain the differential equation

j(t)e21t = 1, f(O) = 0, (5.16)

Cavity losses at zero iemperatv,re 75

that has the solution

1- e-2,t f(t) = -2-"(-. (5.17)

Once we have found f(t), we have found the solution to the Master equation (5. 7), and we are ready to apply it to any initial state.

5.1.1 Coherent states

For an initial coherent state, p(O) = la)(al, we have

(A ) ( 1 - e-2~t A)

p(t) = exp Lt exp -2-1-J la)(al. (5.18)

By developing the second exponential in Taylor series, and applying the powers of J to the coherent state density matrix (it is easy to apply, as coherent states are eigenstates of the annihilation operator), we obtain

p(t) = exp {[1- exp( -2"(t)]lal 2} exp (it) la)(al. (5.19)

We take now into account that

(5.20)

to finally obtain

(5.21)

as expected. For large times the vacuum is reached; i.e., p(t-+ oo)-+ IO)(OI.

5.1.2 Number states

For an initial number state, lk)(kl, it is easy to show that the time evolved density matrix takes the form

(5.22)

We therefore start with a pure state lk) (kl, and the decay takes it to an statistical mixture of number states. Note that the trace of powers of lk)(kl; i.e., (lk)(kl)m, is equal to one, but the trace of powers of the density matrix given in (5.22) is less than one. Again the vacuum is reached for times t » 1/'1·

76 A real cavil,y: Master eqnation

5.1.3 Cat states

For a superposition of two coherent states

(5.23)

with N a normalization constant that takes the value

N = 2(1 + Re(ai;J)), (5.24)

the density matrix has the form

P = ~(la)(al + la)(f)i + I;J)(al + lf))(;JI). (5.25)

When we apply the solution (5.11) to this density matrix, we obtain terms like the following off-diagonal term [Walls 1985]

(5.26)

We can write then the evolution for the density matrix subject to losses as

A(t) 1 ~ (77111) ~-)(-1 P =N ~ -(_1_) 11 77,

'I),J"=Ct 77/.1 (5.27)

with i5 = (J'e-"'~t. Just like in the case of a number state, we see that the resulting density matrix evolves from a pure state to a statistical mixture. We can use the series representation of the Wigner function, expression 2.38 on page 31, to calculate it and look at the dynamics of the eat's decay. It may be easily found

W(E) = _2__ ~ (~I~) exp (-Itt- El2

) exp (-Iii- El2

) x 1rN '"~a (771/.1) 2 2 (5.28)

exp{~ [E*(i/-P)-E(ry-p)*] -(i/-E)(P-c)*},

where E = (p + iq)j,;2. In Figures 5.1 and 5.2, we plot the Wigner function from (5.28) for f)= -a; i.e., a cat state like the one studied in Chapter 3. The figures show that the losses strongly influence the field, killing very fast the interference term.

Master eqnation at finite temperatnre 77

lV(q,pj

p -2

q

Fig. 5.1 Wigner function {3 =-a 2 for 1 = 0.5 at t = 0.0

5.2 Master equation at finite temperature

The Master equation in the interaction picture, for the reduced density operator p relative to a driven cavity mode, taking into ac~ou~t c.avity losses at zer~ temperature and under the Born-Markov approximatwn 1s g1ven by [Scully 1997,

Carmichael 1999]

where

and

.C2p = 1n (2at pa- aat fJ- paat) .

2

The formal solution of Equation (5.29) can then be written as

(5.29)

(5.30)

(5.31)

(5.32)

78 A real cavity: Master equation

Fig. 5.2 Wigner function f3 = -ex= 2 for 'Y = 0.5 at t = 0.2

We redefine superoperators in the form

and

J3P =a tap+ pat a+ p,

such that]_, J+ and J3 obey the commutation relations

We have then

with

[L,J+] P

[J3,J±] p

(5.33)

(5.34)

(5.35)

(5.36)

(5.37)

(5.38)

Master equation at finite temperature 79

and where Nt = n(l- e-"~t).

The steady state of (5.37) is the thermal state 3.62 on page 57.

80 A real cavity: Master equation

Chapter 6

Pure states and statistical mixtures

6.1 Entropy

One of the most common tools to know if we are dealing with pure states or statistical mixtures is entropy. Quantum mechanical entropy is defined as [von Neumann 1927]

S = (S) -(lnp) = -Tr{plnp}, (6.1)

known in fact as the von Neumann entropy. We have set Boltzmann's constant equal to one in the above equation. If the density matrix describes a pure state, then S = 0, and if it describes a mixed state, S > 0; such that S measures the deviation from a pure state. A nonzero entropy then describes additional uncertainties above the inherent quantum uncertainties, that already exist. Because the density matrix of the system, p(t), is governed by a unitary time evolution operator, its eigenvalues remain constant, and because the trace of an operator depends only on its eigenvalues, the entropy of a closed system is time independent. However, we usually do not have closed systems, as systems may interact with other systems and (or) with the environment (usually a much larger system, see Chapter 5). We can then consider a system composed by two sub-systems, for instance an atom and a field, although the entropy of the whole system remains time independent, we can ask ourselves what happens with the entropy of each subsystem. If we call one sub-system A and the other B, then the trace of the total density matrix on the A subsystem basis gives us the density matrix for the B subsystem

(6.2)

81

82 Pure states and statistical mixtures

and viceversa

(6.3)

The entropies for A and B may be defined as

(6.4)

Tracing over one of the subsystems variables means that each subsystem is no longer governed by a unitary time evolution, which produces that the entropy of each subsystem becomes time dependent and it may evolve now from a pure state to a mixed state (or viceversa). Araki and Lieb [Araki 1970] have demonstrated the following inequality for two interacting subsystems

(6.5)

Therefore, if the two subsystems are initially in a pure state, the whole entropy is zero (S = 0), such that S(pA) = S(pB)·

6.2 Purity

Another common tool to study the purity of a state is the purity parameter, SP = (Sp), defined by the expression

(6.6)

Using the eigenbasis of the density matrix, it can be shown that

(6.7)

Because the equality holds only for pure states, Sp discriminates uniquely between mixed and pure states. By using the fact that 1 - Pn :::= -lnpn, for 0 < Pn :::= 1, we find a lower bound for the entropy

(6.8)

Finally, it is worth to say that the purity parameter is much simpler to calculate than the entropy.

Entropy and purity in the atom-field interaction 83

6.3 Entropy and purity in the atom-field interaction

In the case of the atom-field interaction, we can study either the field entropy or the atomic entropy. We consider here atom and field initially in pure states, which means that we can study any of the two subsystems with the same results. Let us, for simplicity, study the atomic entropy; for that, we write the atom density operator for an initial coherent state and the atom in its excited state

where

and

P12

P12 )

1- Pn '

Pll = L Pn cos2 (>.tvn+l'), n=O

ia L :;, cos(>.tvn + 2) sin(>.tvn+l), n=D vn + 1

(6.9)

(6.10)

(6.11)

with Pn = exp ( -lal 2) o:~~ the photon distribution for a coherent state and

,\ = lgiEo/2 the real interaction constant. We can find the eigenvalues of (6.9) with the determinant

P12 1- O 1- Pn- X - . (6.12)

By solving the quadratic Equation (6.12) for x, we find the two eigenvalues X1

and x2 , and we find the entropy

(6.13)

We plot the entropy in Figure 6.1 and the purity parameter in Figure 6.2. It may be seen that atom and field go close to pure states at half the revival time, tR/2. Gea-Banacloche [Gea-Banacloche 1991] and Buzek et al. [Buzek 1992], have shown that at that time the field and the atom almost disentangle, the field going to a state close to a Schrodinger cat state. In Figure 6.3, we show the Q-function for the field state at time tR/2.


0.6

0.5

0.4

S(t) 0.3

0.2

0.1

0 0 10 20 30 40

Fig. 6.1 Field entropy for a coherent state with ex= 5.

6.4 Some properties of reduced density matrices

The Araki-Lieb inequality [Araki 1970]

ISA- SBI ~ SAB ~SA +SB, (6.14)

implies that if both subsystems are initially in pure states, the total entropy is zero and both su!Jsystems entropies will be equal after they interact. Here, we would like to arrive to the result that, if initially the two subsystems are in pure states, any function of the density matrix of subsystem A is equal to the function of the density matrix of subsystem B, and, in particular, obtain that both subsystems entropies are equal, without using the Araki-Lieb theorem. We would also like to find the entropy operator

(6.15)

for any of the subsystems. In particular, we will consider later a two-level atomfield interaction, but the results may be generalized to other kind of subsystems, for instance atom-atom interaction, atom-many atoms interaction, N-level atomfield interaction, etc.

Some properties of reduced density matrices 85

0.4

0.3

SP(t)

0.2

0.1

0 0 10 20 30 40

Fig. 6.2 Field purity for a coherent state with ex = 5.

The density matrix for a two-level system interacting with another subsystem B is given by

~-( lc)(cl p- ls)(cl

lc)(sl ) ls)(sl '

(6.16)

where lc) (Is)) is the unnormalized wave function of the second system corresponding to the excited (ground) state of the two level system. From the total density matrix, we may obtain the subsystem density matrices

as

and

~A= ( (clc) p (cis)

(sic) ) = ( Pn (sis) - P21

PB = lc)(cl + ls)(sl.

Pl2 ) ,

P22 (6.17)

(6.18)

From the density matrix for subsystem B, one can not see a clear way to calculate the entropy operator, as powers of PB get complicated to be obtained. To make the calculation easier, we state the following theorem:


0.12

0.06 Q(X.

-0.06 10

-5

10 -'11 0

Fig. 6.3 The Husimi Q-function for half the revival time for an atom1 initially in the excited state and field in a coherent state with amplitude a = 9.

If two subsystems are in pure states before interactiom, after interaction, the trace of any function of one of the subsystems densitty matrix (that may be expanded in a Taylor series) is equal to the trace of· the function of the other subsystem's density matrix.

To prove it, we can use the following relation valid for two interacting systems that before interaction were in pure states

with this relation it is easy to show that

In particular, with the expression (6.19) we demonstrate ~hat SB

course, it is also true that

13~+l = TrB{I3(t)13'8(t)}.

(6.19)

(6.20)

(6.21)


6.4.1 Proving p"J3+1 = TrA{p(t)fJ:A(t)} by induction

We can prove relation (6.19) by induction. To achieve this goal, we need first to

find 13'1, so we write

(6.22)

where t5 = p11 - p22 and 1 is the 2 x 2 unit density matrix. We can then find

simply that

An - .!_ R = n --Rm ( )

n n ( ) 1 p A - 2 + f, m 2n-m . (6.23)

We split the above sum into two sums, one with odd powers of R and one with

even powers, and we also use that

fl2m+l = !!:.E2m+l E '

(6.24)

with

Therefore, we can write

In terms of 13A the above equation is written as

13A = G(n)I3A -II13AIIG(n- 1)1, (6.27)

where

(6.28)

with the determinant II13A(t)ll = ~- E2

•

Note that we have written 13'1 in terms of 13A and the unity matrix. We could have arrived to the same result using the Cayley-Hamilton theorem [Allenby 1995], that states that any square matrix obeys its characteristic equation; i.e., any power of a 2 x 2 matrix may be written, as we did, as a linear combination

of the matrix and the unity matrix.


Then to prove p~+l = TrA {p(t)pA (t)} by induction, we must prove it for n = 1; but we note that

(6.29)

is correct, so it is done. We now assume it to be correct for n = k; or in other words, we suppose that

(6.30)

is true, and prove (6.19) for n = k + 1. By using (6.27), we can write

p~+ 1 = fi1G(k)- !isll!iA(t)IIG(k- 1). (6.31)

Note that any power of PB may be written in terms of PB and fi1 (k 2': 2). Multiplying the above equation by p3 , we obtain

p~+ 2 = fi1G(k)- ii111PA(t)IIG(k- 1). (6.32)

We can obtain fi1 from (6.19) as

(6.33)

where for the second equality we have used the Cayley-Hamilton theorem for the

atomic density matrix;i.e., we have written p~(t) = fiA(t) -IIPA(t)lll. Inserting (6.33) in (6.32), and after some algebra we find

p~+ 2 = fi1G(k + 1)- fisiiPA(t)IIG(k), (6.34)

or

(6.35)

that ends the prove of relation (6.19) by induction.

6.4.2 Atomic entropy operator

With the tools we have developed up to here, we can study the two-level atomfield interaction, and construct atom and field entropy operators. Let us consider an atom initially in a superposition of excited and ground states, so the initial state is I7,UA) = '72(1e) + lg)). In the off-resonant atom-field interaction; i.e., in the dispersive interaction, the unnormalized states lc) and Is) read

(6.36)


where I ,B) = e-1,612 12 2:%"=o ~ lk) is the initial coherent state for the field, and

X is the interaction constant. We write the entropy operator as

SA= lnp.A 1 = ln(1- fiA) -In llhll, (6.37)

such that the expectation value of SA is the entropy, and where we have used

that fi.A 1 (t) = ~A/IIPA(t)ll, with the purity operator ~A= 1- PA(t). We can use the Taylor series expansion ln(1 - x) = - 2::=1 'f and (6.27), to find

(6.38)

with

F1=-ln --1 (1- 2E) 2E 1 + 2E '

1 [ , 1 ( 1 - 2E) ] F2=--2

lniiPA(t)ll+-ln -- . 2E 1 + 2E

(6.39)

Note that SA is linear in the atomic density matrix, as expected from CayleyHamilton's theorem (for 2 x 2 matrices). From (6.38), we can calculate the atomic (field) entropy

(6.40)

and the atomic (field) entropy fluctuations

(6.41)

We plot these quantities in Figure 6.4. It may be seen that as the entropy is maximum, becoming atom and field maximally entangled, the fluctuations of the entropy decrease to zero. The fluctuations change as a function of the number of possible states at a given time, and go to zero as the unique maximum entangled state is obtained.

6.4.3 Field entropy operator

We use the expression for fi'B in terms of PA to write the field entropy operator in terms of the atomic density operator

(6.42)


2

I\ I \ I I

3

Fig. 6.4 Entropy (solid line) and entropy fluctuations (dashed line) as a function of xt for a coherent state with (3 = 2 and the atom in an initial superposition of ground and excited states.

Remark that we can write the atomic entropy operator in terms of the operator

u~ed to define concurrence [Wootters 1998], because fiA: 1(t) = lli>}(t)JI aypA (t)ay =

PA(t) . lii>A(t)JI; I.e.,

SB = -IIPA1(t)il TrA {Nt)SM~A(t)}. (6.43)

By inserting (6.38) into (6.42), we obtain

(6.44)

Using the expression of the inverse of the atomic density operator in terms of the purity operator, the entropy may be written as

SB = TrA {p(t) (Fl + IIP:(t)il [1- PA(t)J)}. (6.45)

In terms of the field density matrix the field entropy operator is

s - [F F2 ] A ( ) F2 A2 B- 1 + IIPA(t)il PB t - IIPA(t)llpn(t). (6.46)


From (6.18), we can write p~(t) as

p~ = Jc)(cJ(cJc) + Js)(sJ(sJs) + Jc)(sJ(cJs) + Js)(cJ(sJc) (6.47)

and from (6.36), we have that (cJc) = (sJs) = 1/2 and that (cis) (sic)* =

exp [ -1,81 2 (1- e2ixt)] /2. With all these elements, we can finally write the en

tropy operator for the field as

A [ F2 ] F2 SB = F1 + 2IIPA(t)JI (Jc)(ci + Js)(si)- llfiA(t)JI ((sJc)Js)(cJ + (cJs)Jc)(sl).

(6.48)

6.4.4 Entropy operator from orthonormal states

To corroborate that the entropy operator for the field has been correctly obtained, we calculate it by using a Schmidt decomposition for the state (6.16) in the atom-field case and the dispersive regime. We can write the wave function

in this case as

where

with (cJs) = Eeie, and l'l,b±) and I±) orthonormal states.

We then write

(6.49)

(6.50)

(6.51)

and any function of the operator PB is a function of the factor accompanying the state l'l,b+)('l,b+l (times the state) plus a function of the factor associated with

l'l,b-)('f,b_J times it. We have chosen the dispersive interaction, so that all the terms forming the field density matrix and its square are coherent states, thus we can calculate the Wigner function associated to (6.46) in an easy way. We do it by means of the

formula

Ws(a) = 2:) -1)n(a, nJSBJa, n), (6.52) n=O


where Ia, n) are the so-called displaced number states. The explicit expression for the above equation is

Ws(a) = exp ( -2;32

- 2lal2 + 4f3ax cos xt) [F1 + 2 ll~~t) II] cosh(4;3ay sinxt)

- exp ( -2;32- 2lal 2 + 4f3ax cosxt) 2llt3:

2(t)ll cos[2;3(;3sin2xt- 2ax sinxt)].

(6.53)

We plot the Wigner function associated to the entropy operator in Figure 6.5. It can be seen from Figure 6.5, that for a timet= 10~5 ; i.e., almost a coherent state, there is a singular distribution at the amplitude of the coherent state; and Figures 6.6 and 6.7, show how this initial distribution splits in phase space for t > 0. The fact that the maximum entangled state has not been reached may be seen in Figure 6.5, where the Wigner function associated to the entropy shows a negative contribution.

fV(q.JY)

-1 p

q

Fig. 6.5 Wigner function associated to the entropy operator when the coherent state has an amplitude equal to 2 and at t = w-5 .

0.

0.

0

J-J>'(q,p) 0.

Entropy of the damped oscillator: Cat states

0

0.1

93

q

Fig. 6.6 Wigner function associated to the entropy operator when the coherent state has an amplitude equal to 2 and at t = -rr/4.

6.5 Entropy of the damped oscillator: Cat states

We have shown in Section 5.1.3 of Chapter 5 that a state of the form (Ia) + 1!3))/VN decays as

f3

,o(t) = * L ;~ 11 ~)) ltt)\ill, !J,J"=O! \'fl JL (6.54)

with (j = ue~rt. In general, calculate the entropy of this system is not trivial,

as we need to diagonalize (6.54). Here, we follow [Phoenix 1990] to perform the calculation of the entropy. First note that a density matrix like (6.54) must have an eigenstate of the form

(6.55)

The eigenvalue equation is then

) ( (6.56)


-E

Fig. 6. 7 Wigner function associated to the entropy operator when the coherent state has an amplitude equal to 2 and at t = 1r /3.

with A the eigenvalue. In the above system of equations the elements are given by

and

Mu = Miz = 1 + (,Bia) N

* 1 ( (,Bia) _ - ) M12 = M21 = -N --- + (ai,B) ·

(,Bia)

(6.57)

(6.58)

We can find the eigenvalues of the matrix of the Equation (6.56), which are given by the solution to the quadratic equation

(6.59)

From the above equation, we find the two eigenvalues, A+ and A_, and obtain the entropy

(6.60)

The entropy is plotted as a function of time in Figure 6.8 for difl:"erent values of a and ,8. It may be seen that immediately after the interaction with the envi-

Entropy of the damped oscillator: Cat states 95

ronment, the cat state losses its purity, going to a statistical mixture of coherent states (see Figures 5.1 and 5.2). At larger times the entropy decreases until it reaches the value of zero, which means that the field is in the vacuum state IO). Note that the dashed line reaches the value ln 2, meaning that immediately after the interaction with the environment the coherence between the two coherent states is lost, and a statistical mixture is the new state of the field. As the coherent states are sufficiently apart a ~ 2, they may be considered orthogonal, this is why such a values are reached. In the case the states are closer, the entropy does not go too high.

0.7

0.6

0.5

0.4

S(t) 0.3

0.2

0.1

0 0 2 4 6 8 10

A-t

Fig. 6.8 Entropy as a function of :>..t for the damped harmonic oscillator for a cat state with f3 =-a:= 1 (solid line) and {3 =-a:= 2 (dashed line).


Chapter 7

Reconstruction of quasiprobability distribution functions

7.1 Reconstruction in an ideal cavity

The reconstruction of a quantum state is a central topic in quantum optics and related fields [Leonhardt 1997]. It treats the possibility of obtaining complete information of a quantum state by means of quasiprobability distribution functions. There are several methods to achieve such reconstruction, either in ideal or lossy cavities. To start this Chapter, we show two methods to accomplish the quantum state reconstruction in ideal cavities.

7.1.1 Direct measurement of the Wigner function

Let us work in the interaction picture of the dispersive regime; i.e., we consider the interaction part of the Hamiltonian 4.26 on page 65,

(7.1)

Let us consider an initial state

1 A

11/J(O)) = J2(lg) + le))D(a)I1/!F(O)); (7.2)

or in other words, the atom in a superposition of ground and excited states (the atomic state may be produced with the scheme of Figure 7.1, by means of a classical field), and the field in an arbitrary state that has been displaced (this may be achieved by injecting an intense classical electromagnetic field through one of the mirrors of the cavity, which effectively displaces the initial field). Another possibility is to usc the method shown in Section 4.4 to displace the cavity field. For the above initial state we find that the average for the dipole

97

98 Reconstruction of quasiprobability distrib-u,tion .fv,nctions

Cavity

I ~

t~8 [§: -+-..... 4iCJiii'

Atom's source Classical Classical

field field Fig. 7.1 Experimental setup to produce superposition of atomic states. Atoms leave the oven in excited states, pass through a classical electromagnetic field zone to produce the proper superposition. After that, several quantities may be measured: if the second classical

• electromagnetic field zone is switched off, the atomic inversion is measured, if switched on (ay) and (ax), or (ay) and (ax) may be measured.

(ax)= L Pn(a) cosxtn, (7.3) n=O

where

(7.4)

is the photon distribution for the displaced field state in Equation (7.2). Note that if we choose an interaction timet= JT /x, we can rewrite (7.3) as

(ax)= L( -1t(nlbt (a)I?/JF(O))(?/!F(O)ID(a)ln), (7.5) n=O

which is proportional to the Wigner function for the initial state of the field, 1?/JF(O)) (see Equation 2.38 on page 31).

7.1.2 Fresnel approach

Another possibility to reconstruct the quantum state of a system from quasiprobability distributions, is the so-called Fresnel representation of the Wigner function [Lougovsky 2003]. To obtain such representation, we take the Fresnel trans-

Reconstruction in a lossy cavity 99

form of expression 4.31 on page 68; i.e., we have the equation

where we have defined the scaled time T = .At. The integral in the right hand side can be done, and we get [Gradshteyn 1980]

2 100 . 2/ ~ --. dTe'7 n cos(Tvn + 1)

JT-.h 0 (7.7)

that substituted in (7.6), take us to

(7.8)

We show in Figure 7.2, the Wigner function for an initial state 1?/JF(O)) =bolO)+ y/1- b611) with bo = 0.2. What we have done up to now in this chapter, is the following: we have studied

how to know the quantum structure of a cavity field, and have shown two ways of doing it. Both ways involve the passage of atoms through the cavity and their measurement, in the first case it is measured the dipole, (ax), while in the second case the atomic inversion, (a2 ), is measured. In the first case, the Wigner function is given directly by the measurement, while in the second case a Fresnel transform of the data is required.

7.2 Reconstruction in a lossy cavity

In the following we will look at the possibility to reconstruct quasiprobability distributions when the interaction with the environment occurs. Knowing from Chapter 5 that losses damage the quality of the quantized field (the field passes rapidly from pure states to statistical mixtures), we ask ourselves if in the case of a real cavity is still possible to obtain complete information of the initial states in spite of having losses. This is studied next. In the interaction picture, and in the dispersive approximation, the Master equation that governs the dynamics of a two-level atom coupled with an electromagnetic field in a high-Q cavity is

d A . [H' eff A] .CA A dJ/ = -2 I 'p + p, (7.9)

where

LfJ = 21 apat - 1 aJa,fJ - 1 pata. (7.10)

100 Reconstru.ction of quasiprobability distribution functions

W(a)

-4

i-

Fig. 7.2 Wigner function for a superposition of the vacuum and first state l'l,bp(O)) =bolO)+

,)1- b6ll), with bo = 0.2. X Re{a} andY= Im{a}

We introduce the superoperator

Lp = -tatap- prtata, (7.11)

where we have defined

(7.12)

with iA = ie)(el + lg)(gl. We have also defined the superoperator (as earlier)

(7.13)

It is not difficult to show that

[J,i]t3 = -srlp, (7.14)

Reconstruction in a lossy cavily 101

where the atomic superoperator Sr is now defined as

(7.15)

The solution to Equation (7.9), subject to the initial state p(O), is then given by

where

and p(O) = l'l/I(O))('l/1(0)1.

A 1- e-Srt f(t)p = --A -p,

Sr

Let us consider the atom initially in the following superposition

1 l'!f!A(O)) = y'2(1e) + lg))

(7.16)

(7.17)

(7.18)

and the state of the field l'!fip(O)) to be arbitrary. To obtain the evolved density matrix, we need to operate the density matrix with the exponential of superoperators given above. It is not obvious how ef(t)J will apply on the total initial state, therefore we give the following expression for it

(7.19)

with !')p(O) = l'!fip(O))('!fip(O)I. Because j is an atomic superoperator, it will operate only on atomic states, and J, being a field superoperator, will operate only on field states. It is not difficult to show that

A 1 ri'!f!A)('!f!Ai = 2x

[(1 _ e-(E+C)t)n A (1 _ e-2Et)n (1 _ e-2Ct)n ]

(~ + ~*)n x 1A + (2~)n le)(gl + (2e)n lg)(el ,

and

with ~ = 1 + i x. Therefore

1 00

(1 ef(t)J p(O) = 2 L

n=O

(7.20)

(7.21)

(7.22)

102 Re~onstruction of qu.asiprobability distribution functions

By using that

(7.23)

we may finally calculate (D-x), and obtain

oo ('/l-e-2<t) m oo I

(D- ) = ~"' ~-~ ---'--- "'e-2k~t (m. + k). (k+mliJF(O)Ik+m)+c.c. (7.24) X 2 ~ m,! ~ k!

m=O k=O

By changing the summation index in the second sum of the above equation, with n = m. + k, we obtain

oo (re2"'-l)m oo I

(a-x)=~ E ~ L e- 2n~t~(nliJF(O)In) + c.c. 2 m=O m.! n=m (n- m.)!

(7.25)

Finally, we can start the second sum of (7.25) from n = 0 (as we would only add zeros to the sum, because the factorial of a negative integer is infinite), and exchange the doub]e sum in it, to sum first over m., which gives

By defining

and

1 oo (r+ixe-2~t)n ( D-~) =- L (nlpp(O)In) + c.c.

2 n=O ~

tane = ry + e- 21JT[sin(2T)- T} cos(2T)]

ry2 + e-27)T[cos(2T) + T}Sin(2T)]

_ (T/2 + e-41)T + 2T}e- 21JT sin(2T))! ti- 1 + T}2

with T = xt and T} = r I X' we get a final expression for (a-X)' that is

(D-x) = L tin cos(nB)(nliJF(O)In). n=O

7.3 Quasiprob<tbilities and losses

(7.26)

(7.27)

(7.28)

(7.29)

In Equation (7.29), we examine an initial arbitrary state displaced by an amount a; i.e., we study D(a)I7/>F(O)), we obtain

00

(o-x)= L tin cos(nB)(niD(a)pp(O)D(a)ln). (7.30) n=O

Quasiprobabilities and losses 103

By choosing an interaction time such that 8 = -n, Equation (7.30) reduces to

00

(D-x) = 2) -ti)n(niD(a)pp(O)D(a)ln). (7.31) n=O

One may obtain numerically the value ofT ~ 1.689 for e = -n in the case

1 .1

1.0~-----~------------------------

0.9

0.8

0.7

0.6

0.5

0.4~----------~-----------,----------~~----------, 0.0000 1.5708 3.1416

0 T

Fig. 7.3 We plot J.L as a function ofT for?')= 0 (solid line), and?')= 0.1 (dashed line).

T} = 0.1. In Figure 7.3, we plot ti as a function ofT. As soon as ti =/= 1, tin becomes smaller than unity producing errors if one wants to reconstruct the Wigner function. However, one can determine completely the state by noting that an s-parametrized quasiprobability may be reconstructed exactly. It should be stressed that once determined in an experiment the values of the interaction constant and of the decay rate, such quantities fix the interaction time with the condition e = -n, which finally sets the value of ti and therefore the quasiprobability to measure. By setting ti = ~:t=~ (the ti that corresponds to the value ofT~ 1.689; i.e., ry = 0.1), we may finally cast Equation (7.31) as an

104 Reconstruction of quasiprobability distribution functions

s-parametrized quasiprobability distribution function F(cx, s) as follows,

A ~ (s + 1)n A n(l- s) (CJx) = f='o 8

_1

(a,nlpp(O)Ioo,n) = --2-F(cx,s), (7.32)

that shows a relation between quasiprobability distributions and a simple measurement of the atomic polarization operator (&x) in the case of cavity losses. In Figure 7.4, we plot the quasiprobability distribution function for the parameter s ~ -0.1812 that corresponds to T ~ 1.689 and TJ = 0.1 for the same state. One may see that the reconstructed quasiprobability distribution is not as negative as the Wigner function, Figure 7.2. Of course, for greater values of the decay parameter, the effect would be stronger. However, even in the case uf dissipation one would measure a negative quasiprobability distribution, but should be stressed that not the Wigner function.

F{a,s)

-4

Fig. 7.4 Plot of the s;::;:; -0.1812 quasiprobability distribution function for a superposition of

the vacuum and first state 11PF(O)) =bolO)+ 'vh- b6ll), with bo = 0.2.

Measuring field properties 105

7.4 Measuring field properties

We consider again the lossless case and the full two-level atom-field interaction (Jaynes-Cummings model), Equation 4.18 on page 62, and ask ourselves the possibility not to measure the whole state but some of its properties.

7.4.1 Squeezing

To measure squeezing, we need to be able to measure quantities like

(X) = (a) + c.c., (7.33)

and

(7.34)

Below, we show how it is possible to measure quantities like (ak), k = 1, 2. We start by writing the Hamiltonian of the two-level atom field resonant interaction in the rotating wave approximation and the interaction picture. If we consider the atom initially in the excited state and the field to be unknown, such that the initial state of the system is l'l,b(O)) = le)I'I,VF(O)), the average of the operator (j + is given by

-i('l,bp(O)I cos(>..tvn + 1)Vt sin(>..tvn + 1)I'I,VF(o)) (7.35)

-~('l,bp(0)11ft (sin[>..t~+(n)]- sin[>..t~_(n)J) I'I,VF(O)),

where

By integrating (7.35), by using the Fresnel integral [Gradshteyn 1980],

{00 AB r;Jf ( AB2

Jo Tsin(T2 /A) sin(BT)dT = 4 y 2 cos -4

-

such that (with >..t = T)

1= Tsin(T2 /A)(&+)dT = -~('l,bp(0)11ft (')'1- '1'2)I1,UF(O)),

with

A(1) A~+(n) r;Jf ( [A~~(n)l . [A~~(ii)l) 'Y1 = --4--V 2 cos --

4-- + sm --

4-- ,

(7.36)

(7.37)

(7.38)

(7.39)

and

~(lJ _ Ali_(i!) r;uf ( [AA:_(n)l . [AA:_(n)]) '12 ---4-y 2 cos --

4- +sm --

4- . (7.40)

Now we use the approximation /(iH 2)(n + 1);::::: n+3/2, that is valid for large photon numbers. We then can write&~ ( n) ;::::: 4n + 6 and A:_ ( n) ;::::: 0. By setting A= 47r, we obtain

and

(7.42)

~so that the integral transform (7.38) becomes

-iv'27r2(1)!F(O)!Vt vn + 111)!F(O)) (7.43)

-iv'27r2(1)!F(O)Iat 11)!F(O)).

To measure (1)!F(O)I[atj211)!F(O)), it is necessary a two-photon transition, In this case,

ii = >..<2Jf2 ( o J(n + 10)(n + 2) ) [ft]2

J(n + 1)(n +2) (7.44)

where )..(2) is the interaction constant in the two-photon case. One can find the evolution operator, that will be given by an expression similar to 4.22, just changing vn + 1--+ y(n + 1)(n + 2), V--+ V2 and vt--+ [Vtj2. It is then easy

to calculate the average of(;~), which is given by

with

and

-i('lj!F(O)I cos [>..< 2lty(n + 1)(n + 2)] [Vt] 2

X sin [>..<2lty(n + 1)(n + 2)] 11)!F(0))

-~(1)!F(O)I[Vtj2 (sin[>..tJ+(n)]- sin[>..tL(n)J) 11)!F(O)) (7.45)

J+(n) = v(n + 4)(n + 3) + v(n + 2)(n + 1);::::: 2n + 5, (7.46)

L(n) = v(n + 4)(n + 3)- y(il, + 2)(n + 1);::::: 2. (7.47)

Measuring field properties 107

Again by Fresnel integration of the above expression, we get

with

~<2J _ A8+(n) {;A ( [AJ~(n)] . [AJ~(n)]) "(1 - --4- y 2 cos --

4- + sm --

4- (7.49)

and

~<2J _ AL(n) r;A ( [AJ:_(n)] . [AJ:_(n)]) "(2 - --4

- y 2 cos --4- + sm --

4- . (7.50)

By choosing the value A= 81r, we obtain

(7.51)

7.4.2 Phase properties

Now we turn our attention to the phase properties of the field. The procedure, although similar to the way of obtaining the quadratures of the field, will differ in the integral forms that will be used. We compute now the average of (; + for the one-photon transition for an atom in the ground state and the arbitrary field

11)!F(O)), that reads

(IJ+) = i('lj!F(O)!Vt sin(>..tvfl, + 1)Vt cos(>..tvn + 1)VI1)!F(O)), (7.52)

that for a field with large number of photons may be approximated by

(IJ+) i(1)!F(o)!Vt sin(>..tvn + 1) cos(>..tvfn)(1 -IO)(OI)I1)!F(O)) (7.53)

i(1)!F(O)!Vt sin(>..tvn + 1) cos(>..tvfn)i1)!F(O)).

Using the integral [Gradshteyn 1980]

{00 sin(AT) cos(BT)d /

Jo T T = 7r 2, A> B > 0, (7.54)

we can integrate (7.54) as

(7.55)


~o mea~ure ([Vt] 2), again a two-photon process is needed. The average value of

(o-+), w1th the atom entering in the ground state, is

(o-~l) =i(1f]p(O)j[Vt] 2 sin ( >.C2ltJ(n + 1)(n + 2))

x [Vt] 2 cos ( >.C 2ltJ(n + 1)(n + 2)) V2 J1fJp(O))

=i(1f]p(O)J[Vt] 4 sin ( >.C2ltJ(n + 3)(n + 4))

x cos ( >.C2ltJ(n + 1)(n + 2)) V2 J1fJp(O))

Using (7.54), we perform the integral

rXJ (o-(2)) . Jo +dT = ~(1f]p(O)J[Vt] 4V2 J1f]p(O)),

that for large intensity field approximates to

roo (8"(2)) . Jo +dT = ~(1J}p(O)J[VtFJ1fJF(O)).

(7.56)

(7.57)

(7.58)

Finally, the average value of 8" + may be calculated by finding the average value of the observables 8-x and 8-y, with 8"+ = 8-x + i8"y. In order to find it, we write

(7.59)

i.e.,_ the expectation value of,o-z (the atomic inversion) for a rotated (in the atomic bas1s) density matrix with R = exp[(8"_- 8-+)JT/4]. A similar procedure may be done for the expression ( 8" y).

Experimentally, we would need to send an atom through a cavity, that contains the field under study, then (properly) rotate it after it exits the cavity and measure its energy. We would need an experimental setup as the one shown in Figure 7.1.

Chapter 8

Ion-laser interaction

Nowadays it is possible to trap, by using electromagnetic fields, single ions in Paul or Penning traps. In this chapter we will study precisely the problem of a trapped ion (in general in a trap with time-dependent parameters) interacting with a laser field. By using a set of time-dependent unitary transformations, it is shown that this system is equivalent to the interaction between a quantized field and a two level system with time dependent parameters. The Hamiltonian is linearized in such a way that it can be solved with methods that are found in the literature, and that involve time-dependent parameters. The linearization is free of approximations and assumptions on the parameters of the system as are, for instance, the Lamb-Dicke parameter, the time-dependency of the trap frequency and the detuning, thus we can obtain the best solution for this kind of systems. Also, we analyze a particular case of time-dependency of the trap frequency. The possibility to trap small clouds of particles, or inclusive to trap individual atoms or ions, in an small region of space, was opened with the invention of electromagnetic traps. These traps allow to study isolated particles for long periods of time. The Kingdon trap [Kingdon 1923] is considered the first type of trap developed; consists of a metallic filament surrounded by a metallic cylinder, and a direct current voltage applied between them; the ions are attracted by the filament, but its angular momentum makes them turn around in circular orbits, with a low probability to crash against it. A dynamic version of this trap can be obtained if also an alternate current voltage is applied between the poles. However, this type of trap was not widely used at that time, because it has short storage times and because its potential is not harmonic. In 1936, Penning invented another trap [Penning 1937]. In this trap, the action of magnetic fields together with electric fields make possible the trapping of ions. The complete development of this type of trap was reached when, in 1959, Wolfgang Paul

109

llO Ion-laser interaction

designed an electrodynamic trap (now called Paul trap) [Paull990]. In the Paul trap the idea is that a charged particle can not be confined in a region of space by constant electric fields, instead an electric field oscillating at radio frequency, must be applied. The Paul trap uses not only the focusing or defocusing forces of the quadrupolar electric field acting on the ions, but also takes advantage of the stability properties of the equations of motion. The ions trapped individually are very interesting, mainly because they are simple systems to be studied. In particular we take advantage that the ion motion in the Paul trap is approximately harmonic, making this system a simple one, allowing a better and more direct comparison with theory. Individual ions of Ca+, Be+, Ba+ and Mg+, can be storaged, even for several days. The trapped ions can be used to implement quantum gates, and a bunch of ions arranged in a chain, is a promise tool to achieve a quantum computer (each ion in the chain is a fundamental unit of information or qubit) [Cirac 1995]. The trapping of individual ions also offers a lot of possibilities in spectroscopy [Itano 1988], in the research of frequency standards [Wineland 1983; Bollinger 1985], in the study of quantum jumps [Powell 2002], and in the generation of nonclassical vibrational states of the ion [Meekhof 1996; Moya-Cessa 1994]. To make the ions more stable in the trap, increasing the time of confinement, and also to avoid undesirable random motions, it is required that the ion be in its vibrational ground state. This can be accomplished by means of an adequate use of lasers; with the help of these lasers, the internal energy levels of the trapped ion can be coupled to their vibrational quantum states, in such a way, that for a certain detuning, the coupling is equivalent to the Jaynes-Cummings Hamiltonian [Abdel-Aty, 2007]. On the other hand, the beam that induces the coupling can be tuned to allow interactions that generate simultaneous transitions of the internal and vibrational states, either to lower vibrational energy levels (while passing from the excited to the ground state) or to higher vibrational energy levels (while passing from the ground to the excited state). This type of coupling is called anti-Jaynes-Cummings. Alternating successively Jaynes-Cummings and anti-Jaynes-Cummings interactions, the trapped ion can be driven to its vibrational ground state. In this chapter, we study part of the physics of the trapped ions. We will analyze the interaction Hamiltonian, independent and dependent on time, of the system formed by an ion and a laser beam. Because we will assume an ion trapped in a Paul trap, in Section 8.1, we review the basic mechanisms of it. In Section 8.2, as an antecedent, we expose the theory of the ion interacting with a laser beam considering the off-resonant case which allows multiphonon interactions. In Section 8.3, we analyze the case of an ion, with a time-dependent frequency,

Paul trap lll

interacting with a laser beam. By doing a series of unitary transformations, we linearize the Hamiltonian of the system to an exact soluble form; this linearization is also valid for any detuning and for any time dependence of the trap. In Section 8.4, we study how to add vibrational quanta to the ion; in this Section, we also analyze the atomic inversion of the ion in the trap. Finally, in Section 8.5, we examine the effect of two lasers driving the ion, instead of one, resulting in the possibility to filter specific superpositions of number states.

8.1 Paul trap

8.1.1 The quadrupolar potential of the trap

As we already said, the Paul trap uses static and oscillating electric potentials to confine charged particles. A charged particle is linked to an axis if a linear restoring force acts over it; i.e., if the force is

i' = -cr, (8.1)

where ·fis the particle position and cis a constant. In other words, if the particle moves under the action of a parabolic potential, that can be written in general form as

(8.2)

where A is another constant. The potential iP must satisfy the Laplace equation, which means that

(8.3)

where \72 is the Laplacian operator. The Laplace equation (8.3) imposes the condition

(8.4)

To satisfy the above condition, we have several possibilities: a) We make a= 1, (3 = 0 and 1 = -1, and this takes us to the bidimensional potential

(8.5)

112 Ion-laser interaction

b) Another possibility is a= 1, (3 = 1 and'/= -2, and in this case we have. in cylindrical coordinates, the potential ,

(8.6)

with r5 = 2z6.

The configuration a), Equation (8.5), is created by four, hyperbolic electrodes

+<Po/2

-<Po/2 -<Po/2

z

li X

+<Po/2

Fig. 8.1 Electrode structure for the bidimensional configuration expressed by Equation (8.5).

linearly extended in the z direction, as shown in Figure 8.1.

The configuration b), expression (8.6), is created by two electrodes with the form of an hyperboloid of revolution around z axis, as shown in Figure 8.2. If the voltage <f>o is applied between the opposite pair of electrodes, the potential in the electrodes is ±<f>o/2. The constant A can be obtained, in each case, considering the previous potentials as boundary conditions.

Paul trap 113

Fig. 8.2 Electrode structure for the tridimensional configuration expressed by Equation (8.6).

The most used trap is the linear one, as the one shown in Figure 8.1, but with poles with circular transverse section instead of hyperbolic, because it is easier to build. This cylindric form does not correspond to some set of values of (8.4), but numerically it has been demonstrated that the potential produced by these electrodes near the axis of the trap is very similar to the one produced by the hyperbolic electrodes [Bonner 1977]. For the tridimensional case the magnitude of the field is given by

(8.7)

Expressions (8.6) and (8.7) reveal that the components rand z of the electric field are independent from each other, and that they are linear functions of r and z, respectively. We also see that we have a harmonic oscillator potential (parabolic and attractive) in the radial direction and a parabolic repulsive potential in the z direction. If a constant voltage <f>o is applied, and an ion is injected, the ion will oscillate harmonically in the x-y plane, but because the opposite sign in the field

Ez, its amplitude in the z direction will grow exponentially. The particles will be out of focus, and they will be lost by crashing against the electrodes. Thus, the quadrupolar static potential, by itself, is not capable to confine the particles in three dimensions; at most, with this potential, we get unstable equilibrium. We will see next, how to solve this problem.

8.1.2 Oscillating potential of the trap

To avoid the unstable behavior of the charged particles under a static potential, the trap must be modified. If an oscillatory electric field is applied, the particles can be confined. Because the periodic change of the sign of the electric force, we get focusing and defocusing in both directions of r and z alternatively with time.

If the applied voltage is given by a continuous voltage plus a voltage with a frequency n, we have

o = Uo + V0 cos!lt, (8.8)

and the potential in the axis of the trap is

_ Uo + Vo cos nt ( 2 _ 2) - r5 + 2z6 r 2z ' (8.9)

where ro is the distance from the trap center to the electrode surface. In Figure 8.3, we show a transversal section of a Paul trap using an oscillating

Fig. 8.3 Scheme of a Paul trap to storage charged particles using oscillating electric fields generated by a quadrupole. The Figure shows two states during an alternate current cycle.

Paul trap 115

electric field.

8.1.3 Motion in the Paul trap

We will study now some details of the motion of an ion in a Paul trap. Let us consider the particular case of just one ion, in three dimensions. If m is the mass of the ion, and e its charge, the equation of motion is

m7'-(x, y, z) = qE -q'V. (8.10)

In order to analyze the trapping conditions, we write explicitly each component,

and

where R 2 = r5 + 2z6. Making the substitution

2e x =- mR2 (Uo + Vo cosnt) x,

.. 2e y =-mR2 (Uo + Vocos!lt)y,

2e i = mR2 2 (U0 + V0 cos Dt) z,

8eUo az ar mR2f22 -2,

4eVo qz qr mR2f22 -2,

nt T 2'

(8.11)

(8.12)

(8.13)

(8.14)

Equations (8.11), (8.12) and (8.13), take the form of the Mathieu equation; i.e., they take the form

(8.15)

(8.16)

and

(8.17)


respectively. We can write the three equations as the following one,

(8.18)

The subindex i = r, z corresponds to the quantities associated with the axial and radial motions of the ion, respectively. The quantities ui represent the displacement in the directions rand z.

8.1.4 Approximated solution to the Mathieu equation

The Mathieu equation is a linear ordinary differential equation with periodic coefficients. This equation can be solved using Floquet's theorem [Bardroff 1996], which takes us to the general solution

(8.19)

where Ai, Bi and /3i are constants determined by the initial position, by the initial velocity of the ion, and by by the trap parameters a and q, and

+oc cj;(T) = cj;(T + n) = L Cne2inT (8.20)

is a periodic function. The Mathieu equation has two types of solutions: 1) Stable motion. When the characteristic exponent f3 is real, the variable u( T) is bounded, and in consequence the motion is stable. That means, that the particle oscillates with bounded amplitudes and without crashing against the electrodes. These conditions allow to trap the ion. 2) Unstable motion. When the characteristic exponent f3 has an imaginary part, the function u(T) has an exponential growing contribution. The amplitudes grow exponentially and the particles are lost, when they crash against the electrodes. The boundaries of the stability regions correspond to zero and integers values of /3i, and the first region of stability is surrounded by the four lines f3r = 0 , f3r = 1, f3z = 0, and f3z = 1, as is shown in Figure 8.4 [Bate 1992]. As f3i is determined by a and q, the Mathieu equation has stable solutions as a function of a and q. Stability regions for the solutions of Equation (8.18) correspond to regions in the space of the parameters a- q, where there is an overlap of the stability regions in the axial and radial directions. In the literature it is not possible to find analytic solutions for Equation (8.18), but in most of the applications an specification of the map of stability of the solutions is enough, and it is not necessary a detailed functional dependence.

Paul trap 117

a

~.5

Sta bilitv region

0.5

q

Fig. 8.4 Stability region in the Paul trap.

However, an approximated solution can be given in the stability region of inter

est. To this end, we can write expression (8.19) as

+oc += ui(T) =AiL c~n cos(2n + f3i)T + Bi L c~n sin(2n + f3i)T (8.21)

where, as we already said, Ai and Bi are determined by the initial position ui(O) and initial speed ui(O) of the ion, respectively. The subindices i = r, z coincide with the quantities associated with the radial and axial ion motion, respectively. The coefficients in the solution (8.21) are given by the recurrence relations

(8.22)

with

(8.23)

Once given ai and qi, the quantities C~n and /3i can be calculated. If we define,

Gi C~n 2n = c~ ' (8.24)


and we make

ui(t) = ut(t) + uf'(t), (8.25)

we get, from Equation (8.21),

(8.26)

and 00

uf'(T) = :L)A~ cos wit+ B~ sinwit)(G~n + G~ 2n) cosnm n=l (8.27)

where wi = f3Slj2.

Analyzing Equations (8.26) and (8.27), it is possible to realize that the ion motion has two components: ui(t), a harmonic oscillation of frequency wi, and, uf(t), a superposition of several harmonics with a fundamental frequency n, and amplitudes modulated by the frequency wi. However, the proportion of the two components, the values of Wi, the number of subcomponents that contribute appreciably and their weights, depend strongly on the values of ai and qi, in such a way that they will change in the stability region. Everything gets de

termined when the values f3i and G~n are given. Several values of f3i and G~n' corresponding to some typical values de ai and qi, are listed in [Zhu 1992]. In the table there, it is possible to see that in the first region for a « q « 1, we can assume that G~ ~ G~ 2 and the rest of the coefficients G~2n, n > 1, can be ignored; thus, Equations (8.26) and (8.27) can be rewritten as

(8.28)

and

(8.29)

or in other words,

(8.30)

with

(8.31)

(8.32)

Paul trap 119

and C = 2G~. In Figure 8.5, we plot Equation (8.30). The ion motion is composed by two types

004

002

100 15) 200

-002

-004

Fig. 8.5 Micro motion and secular motion of a trapped ion with parameters q = 0.2, f3 = 0.02. The oscillations at high frequency are the micro motion and those at low frequency are the secular motion.

of oscillations: the harmonic oscillation with frequency wi, called secular motion, and the small contributions oscillating at the frequency n, called micromotion. Usually, the micromotion is ignored, but it can be reduced using additional electrodes [Roos 2000]. In this way, the ion motion is controlled by Equation (8.28) and behaves as it was confined in a harmonic pseudo-potential, that for the radial part, has the form

(8.33)

Typically, Uo = 0, thus a= 0 (Equation (8.14)); in any case, we are working in the region where a rv 0. Thus, the frequencies Wx and Wy are degenerated, and Equation (8.33) reduces to

2 mwr ( 2 2) q1/JzD = -

2- X + Y · (8.34)


To obtain an expression for w, we can use the approximation [Dehmelt 1967],

~ f3r = yar + 2' (8.35)

with the definition Wr = f3rr!/2, to get

r!qr eVo Wr = 23/2 = v'2mr5r!. (8.36)

Experimentally, the typical ranges of operation are V0 ::::::; 300- 800Volts, f!/27r::::::; 16- 18MHz, and r 0 ::::::; 1.2mm, that gives a radial frequency Wr::::::; 1.4- 2.0MHz for calcium ions (4°Ca+). We can summarize this section, saying that under certain conditions it is a good approximation to treat the ion motion as a harmonic oscillator.

8.2 Ion-laser interaction in a trap with a frequency independent of time

In this section, we study the interaction of a laser with a trapped ion in a harmonic potential with constant frequency. We start with the Hamiltonian of the system, and we show that it is possible to find Jaynes-Cummings type transitions and anti-Jaynes-Cummings type transitions, depending on the different cases of resonance and laser intensities that induce the coupling between the ion internal states and the ion vibrational states. We can write the Hamiltonian of the trapped ion as

(8.37)

where Hvib is the ion's center of mass vibrational energy, Hat is the ion internal energy, and Hint is the interaction energy between the ion and the laser. As we explained in the previous section, the vibrational motion can be fairly approximated by a harmonic oscillator. Internally, the ion will be modeled by a two level system. In the interaction between the ion and the laser, we will make the dipolar approximation, so we will write the interaction energy as -er· E, where -er is the dipolar momentum of the ion and E is the electric field of the laser, that will be considered a plane wave. Thus, we write the Hamiltonian explicitly

as

(8.38)

The first term in the Hamiltonian is the ion vibrational energy; in the ion vibrational energy, the operator ii = at a is the number operator' and the ladder

Ion-laser interaction in a trap with a frequency independent of time 121

operators a and at are given by the expressions

(8.39)

and

. p 2ffv' (8.40)

where we have made the ion mass equal to 1. Also, for simplicity, we have displaced the vibrational Hamiltonian by v /2, the vacuum energy, that in this case is not important. The second term in the Hamiltonian corresponds to the ion internal energy; the matrices CYz, CY+, and fY_ are the Pauli matrices, and w21

is the transition frequency between the ground state and the excited state of the ion. Finally, the third term, is the interaction energy between the ion and the laser; in this last term, we have used again the rotating wave approximation.

8.2.1 Interaction out of resonance and low intensity

If we consider that the Hamiltonian (8.38) corresponds to the wave function l~(t)), the Schri:idinger equation can be written as

.a 2&1~) =HI~). (8.41)

Let us examine the transformation to a rotating frame of frequency w, by means of the unitary transformation

(8.42)

Applying the transformation T, the wave function l~(t)) transforms in the wave function l¢(t)); i.e.,

T(t)i~(t)) = l¢(t)), (8.43)

and the Hamiltonian transforms in

8T(t) L t HT = i~T1 (t) + T(t)HT (t). (8.44)

Writing the position operator i: in terms of the ladder operators, expressions (8.39) and (8.40), using the Baker-Hausdorff formula 1.28 on page 5, and the commutators of the Pauli matrices 4.8 on page 60, the explicit transformed Hamiltonian is

(8.45)

122

where

Ion-laser interaction

'f)=K fl y-:;;;;;; (8.46)

is the so called Lamb-Dicke parameter, that is a measure of the amplitude of the oscillations of the ion with respect to the wavelength of the laser field. The quantity J( is the wave vector of the laser, and

kv = w21- w (8.47)

is the detuning between the plane wave frequency and the transition frequency of the ion; in other words, we are considering that the detuning is a multiple integer of the vibrational frequency of the ion. We need now to factorize the exponential in the Hamiltonian (8.45). As [a, [a, at]] =

0, and [at, [a, at]] = 0, we can use the Baker-Hausdorff formula 1.28 on page 5, and write

(8.48)

Expanding in Taylor series the exponentials that contains the operators, and substituting in the Hamiltonian (8.45), we obtain

We go now to the interaction picture, using the transformation

kv 71nt = exp [it(v'i1 + 2az)].

We apply this transformation, using the following two commutators,

and

and we obtain

Hint= ne-1)2/2 [a- L (-~~~~m (at)name-i(n-m-k)vt + H.c.]' n,m=O

where

0, =>-Eo

(8.50)

(8.51)

(8.52)

(8.53)

(8.54)


is the exchange energy frequency of the internal and vibrational states, called Rabi frequency. The interaction Hamiltonian (8.53) has a diversity of contributions, each contribution oscillates with a frequency that is a multiple integer of v. We apply now the rotating wave approximation. As the Schrodinger equation is a first order differential equation in time, we have to integrate it once with respect to time; this integration brings the sum and the difference of the frequencies to the denominator. 'fhe terms changing slowly will dominate, over the terms changing very fast; so the contribution to the Hamiltonian of those fast terms is neglected, and only the slowly changing terms are kept. In this case, the terms that do not rotate quickly are those whose exponent satisfies the relation n- m = k, and as we already explained, are those terms that we will keep. This approximation is valid for

0, « v. (8.55)

As 0, is proportional to the amplitude of the laser electric field, from (8.5fi) it is clear that this approximation is valid for low intensity. We have then,

H =ne-1)2

/ 2 [a ~ (-i'f))2m+k (at)k(atynam+HC]

mt - ~o (m+k)!m! ... (8.56)

Using now the fact that the number states is a complete set,

I= .l::ln)(nl, (8.57) n=O

where I is the identity operator, we can write

(8.58)

that substituted in the Hamiltonian (8.56), gives us

o - -1)2 /2 [ At k o k ~ ( -i'f)) 2m n! l Hmt -Oe a_(a) (-t'f)) ~ ( k)l 1 (' _ )I +H.C ..

m=O m + .m. n m . (8.59)

Using the explicit expression for the associated Laguerre polynomials [Abramowitz, 1972],

(a) . - ..Z:.., - i (n +a)! xi Ln (x)- ~( 1) ( _ .)I( + ")I ·1' n t.a t.t.

i=O


we can write

. - _,.,2/2 [ At k . k fU (k) 2 ] Hmt- De (]'_(a) (-zTJ) (n + k)!L"' (TJ) + H.C .. (8.60)

We will consider now processes where only one phonon is exchanged; that means that we must take k = 1 in the Hamiltonian (8.60). We will consider also, that the oscillation amplitude of the ion is much smaller than the laser frequency; that is, TJ « 1; or in other words, we suppose the Lamb-Dicke regime. With these two considerations, the Hamiltonian (8.60) reduces to (the subscript jc stands for J aynes-Cummings)

(8.61)

In agreement with the considerations made above, the Hamiltonian (8.61) describes emission and absorption of one vibrational excitation, when the atom makes electronic transitions. The first term represents the absorption of a vibrational excitation and the transition of the ion from the excited state to the ground state. The second term represents the inverse process; the ion goes from the ground state to the excited state, annihilating one phonon, and the vibrational state decays in one quanta. All this can be clearly seen, if we apply the Hamiltonian (8.61) to the correct states. In the first case, we have to apply the Hamiltonian to the state In) I e), which represents n vibrational quanta and the ion in the excited state le); we get,

(8.62)

which is the state with n + 1 vibrational quanta, and the ion in the ground state. In the second case, the state is given by In+ 1) I g), and when we apply the Hamiltonian we obtain,

(8.63)

that is the state with n vibrational quanta, and the ion in the excited state.

We can repeat all the above procedure now when the laser frequency is greater than that of the transition

kv = W- W21, (8.64)

and obtain the Hamiltonian (clearly now, the subscript ajc stands for antiJ aynes-Cummings)

(8.65)


In the first term, we have the annihilation of one quanta from the vibrational motion and the ion internal transition from the excited state to the ground state. In the second term, we have the creation of one vibrational quanta and the internal excitation of the ion from the ground state to the excited state. In Figure 8.6b, we explain why this Hamiltonian is anti-Jaynes-Cummings type.

Applying the anti-Jaynes-Cummings Hamiltonian to the adequate states, the

a)

b)

Fig. 8.6 a) A Jaynes-Cummings Hamiltonian implies the ascent (descent) of one ion vibrational quantum, and at the same time, the transition from an excited (ground) internal state to the ground (excited) state. b) An anti-Jaynes-Cummings Hamiltonian annihilate (creates) one quantum from the vibrational motion and transfers the ion internally from the excited (ground) state to the ground (excited) state.

previous comments can be easily understood. If we apply the Hamiltonian (8.65) to the state ln)le), we get

Hajcln)le) =In- 1)lg), (8.66)

and if we apply it to the state In - 1) lg), we get

Hajcln- 1)lg) = ln)le). (8.67)

l,From the point of view of the trapped ion, all this means that we can take it to its lowest energy vibrational state, alternating successively, and as many times as necessary, the detuning between the frequency of the plane wave and the internal frequency of the ion. Again, we can illustrate all this by applying the correct Hamiltonian to the adequate state. For that let us consider a vibrational state In) and the ground internal state; if we apply the Hamiltonian (8.61), we


get

Hjcln)lg) =In -1)le). (8.68)

We apply now the Hamiltonian (8.65), obtaining

Hajcln -1)le) =In- 2)lg), (8.69)

and the ion has lost two quanta of vibrational energy. Repeating successively this procedure, we can arrive to the state IO)Ig).

8.3 Ion-laser interaction in a trap with a frequency dependent of time

In this section, we study the problem of an ion trapped with a frequency that depends on time and interacting with a laser beam. Using unitary transformations, we show that this system is equivalent to a system formed by a two levels subsystem with time dependent parameters interacting with a quantized field. The procedure to build the Hamiltonian for this case, is exactly the same that in the previous section, but we have to keep in mind that now the frequency is time dependent. The Hamiltonian is

(8.70)

and then, the Schri:idinger equation can be written as

(8.71)

To solve the problem, we make the transformation

l¢(t)) = TsD(t)l~(t)), (8.72)

where

()- {iln[p(t)y'Vo](xp+px)} [-ip(t)x2

] TsD t - exp 2 exp 2p(t) , (8.73)

and we have also the Ermakov equation 3. 7 on page 44 as an auxiliary equation. We apply the unitary transformation (8.73) to the Hamiltonian (8.70); i.e., we must calculate the expression

oTDs(t) t t HsD = i-

0-t-TsD (t) + TsD(t)HTsD (t). (8.74)

Ion-laser interaction in a trap with a frequency dependent of time 127

We find,

(8.75)

being

D(t) =ABo (8.76)

the Rabi frequency. Using the Ermakov invariant, the time dependence of the trap has been factorized; the time dependence is implicit in p(t). We go now to a frame rotating at

frequency w, by means of the unitary transformation

(8.77)

The Hamiltonian is transformed to

1 2 2 2) 1 ( ) Hw = --2-(-) (p + v0 x + -2

W21- W CJz 2vop t (8.78)

+D(t) {exp [-i(li+ilt)77(t)] (J_ +H.C.}.

Denoting the detuning frequency between the laser and the ion by o = w21- w,

and the characteristic frequency of the time dependent harmonic oscillator by

we get

1 w(t) = p2(t)' (8.79)

Hw = w(t)(n + 1/2) + ~(Jz + D(t) { exp [ -i(il + at)77(t)] (J_ + H.C.}. (8.80)

The time dependent Lamb-Dicke parameter is

77(t) = 7]op(t)y'Vo, (8.81)

with

(8.82)

where k is the wave vector of the laser beam. Comparing with the Hamiltonian obtained in Section 8.2, (8.45), the Hamiltonian (8.80) is equivalent, but with all the parameters depending on time.


8.3.1 Linearization of the system

We call linearization of the system the process to reduce the exponent of the ladder operators a and at to the first power, without using approximations. To this end, we make the transformation

I¢R) = R(t)l¢w), (8.83)

where R(t) is given by

(8.84)

Transforming the Hamiltonian, we get

(8.85)

with

f3(t) = rJ(t)w _ ir](t). 2 2

(8.86)

The term w(t)/2 has not been considered, because it is only a phase, and when the observable mean values are taken, it disappears. With the transformation (8.84) we have achieved our goal: linearize the Hamiltonian without any type of approximation. The rotating wave approximation is not used, and this leaves open the possibility to consider different intensity regimes. No assumption has been made about the Lamb-Dicke parameter r7(t). It is also valid for any type of detuning and for any time dependence of the frequency of the trap. The Hamiltonian (8.85) is solvable, and in [Shen 2003] methods of solution have been published. It is also important to remark that we have not imposed any condition in the time dependence of the frequency of the trap; in principle, this frequency can assume any temporal form. For a Paul trap, the more general form is

v 2 (t) =a- 2qcos2t, (8.87)

and the Hamiltonian (8.85) is the ion-laser interaction with micromotion included. Also, this Hamiltonian gives us the freedom to consider arbitrary time dependent frequencies. For instance, if we consider a sudden change in the trap frequency, we would generate squeezed states for the vibrational wave function as described in Chapter 3.

Adding vivrational quanta 129

8.4 Adding vibrational quanta

We now show how to generate noncLassical vibrational states in the low intensity regime. From Equation (8.56) with k = 2, we note that if the Lamb-Dicke parameter is much less than one, rJ « 1, we can remain to the lowest order in the sum, such that we obtain the so-called two-phonon Hamiltonian

(8.88)

with E = -D/2. For the study of t;he dynamics of interest, we need the time evolution described by the Hamiltonian (8.88). The advantage of the interactions of J aynes-Cummings type consists in the fact that the Hamiltonian can easily be diagonalized, and using the same procedure that was used in Section 4.2, it

is possible to show that

= L lm)lg)(ml(gl m=O,l

+ f [cos (~nnt) (In+ 2)lg)(n + 21(91 + ln)le)(nl(el) n=O

-isin (~nnt) (In+ 2)lg)(nl(el + ln)le)(n + 21(91)]. (8.89)

The quantity Dn is the two-phonon Rabi frequency which is given by

Dn = 2cy(n + 1)(n + 2). (8.90)

Using these results, the time evolution of the quantum state in the interaction picture is easily derived for arbitradly chosen initial conditions. We have

l\ll(t)) = UI(t)l\l!(O)). (8.91)

If we consider as initial state the ion in its excited state I e) and the vibrational state a coherent state, we can find the atomic inversion, (that we defined in Chapter 4, and we recall that it is defined as the probability to find the ion in its excited state minus the probability to find it in the ground state). Using (8.89) and (8.91), we get

(8.92)

We plot this function in Figure 8.7 as a function of the scaled time T = Et.

The interaction gives rise to a quasi-regular evolution of the atomic inversion, unlike the case of one phonon resonance. This can be used for several purposes, among them, to add excitations to the vibrational state. In Figure 8.7, it can


-0.5-

Fig. 8.7 Plot of the atomic inversion, W(t) P2- Pl, the probability to find the ion in its excited state minus the probability to find it in the ground state, for an ion initially in its excited state and the vibrational state in a coherent state, with a = 5.

be seen that if the ion is in its excited state Je), initially, after an interaction time T = 1r /f. the ion ends up in its ground state Jg), giving all its energy to the vibrational state by adding two vibrational quanta. Moreover, the effect of having the ion in its excited state and after an interaction time having it in the ground state is shared by all vibrational states, not only when it is prepared in a coherent state. To illustrate this fact, we show Figure 8.8 the atomic inversion for a thermal distribution. Consider again the initial state

Jw(O)) = Ja)Je). (8.93)

Combining Equations (8.89) and (8.93), and using a compact operator representation of the Jaynes-Cummings dynamics, we arrive at

Jw(t)) =cos ( ctJ ii2(iit) 2) Ja) Je) - i(Vt) 2 sin ( ctJa2(iit)2) Ja) Jg). (8.94)

Adding vibrational quanta 131

w

0 4 6

Fig. 8.8 Plot of the atomic inversion. W(t) = P 2 - Pl, the probability to find the ion in its excited state minus the probability to find it in the ground state, for an ion initially in its excited state and the vibrational state in a thermal distribution with fi = 2.

In order to derive illustrative analytical results, in the following we will apply the approximation

(8.95)

although this approximation represents a Taylor-series expansion for large eigenvalues n of the operator ii, the error is already small for small n-values. For example, for n = 1 the relative error is only 0.02. Based on this approximation, one may simplify Equation (8.94) as

Jw(t)) ~ cos[>.t(ii + 3/2)]Ja)J2)- i(Vt) 2 sin[>.t(ii + 3/2)]Ja)Jl). (8.96)

Choosing a particular interaction time t = T, according to

T=1f/A, (8.97)

we obtain for the vibrational state vector

(8.98)

where we have introduced the subscript "v" (vibrational) to note that we are not taking into account anymore the state lg). Moreover, the subscript"+" and the superscript" (k)" are used to indicate the process of adding (two) vibrational quanta and the number of such interactions respectively. The quantum state (8.98) may serve as the initial state for a second interaction with an ion that is prepared in the same manner as the first one. For the same interaction time, t = T, after the second interaction (that is completed at time 27), the vibrational state is

(8.99)

By repeating the process k times, one finally obtains for the quantum state, at the time tk, after completing k interactions,

(8.100)

After many interactions, the state 11]!~\T))v exhibits a strong sub-Poissonian character, because, while one is adding two excitations per interaction, at the same time one is keeping the width of the distribution constant. The excitation distribution pJkl, after k interactions, is easily found to be related to the number statistics P~o) of the initial state Ia) via

(8.101)

This result clearly shows that the number statistics is only shifted but retains its form. One way of studying the properties of the states being generated is through the Mandel Q-parameter [Mandel 1979], which is defined by

and where

{

> 0,

if Q = o, < 0, = -1,

super Poissonian distribution Poissonian distribution (coherent state) sub-Poissonian number state.

In this case the Mandel Q-parameter is given by

(8.102)

(8.103)

(8.104)

Filtering specific superpositions of number states 133

and as the number of interactions increases, the Mandel Q-parameter approaches the value -1, i.e. the state acquires maximum sub-Poissonian character. The sub-Poissonian effect of the vibrational wave function becomes more significant

with increasing number of interactions.

8.5 Filtering specific superpositions of number states

If instead of one laser, we assume two lasers driving the ion, the first tuned to the jth lower sideband and the second tuned to the mth lower sideband, we may

write EH (x, t) as

(8.105)

where if m = 0, it would correspond to the driving field being on resonance with the electronic transition. The position operator x may be written as before,

(8.106)

where ks are the wave vectors of the driving fields and

(8.107)

are the Lamb-Dicke parameters with s = j, m. In the resolved sideband limit, the vibrational frequency v is much larger than other characteristic frequencies, and the interaction of the ion with the two lasers can be treated separately using a nonlinear Hamiltonian [de Matos 1996a]. The Hamiltonian (8.60) in the interaction picture can then be written as

where LAk) ( TJ~) are the operator-valued associated Laguerre polynomials, the D's are the Rabi frequencies and n = at a. The Master equation which describes this

system can be written as

ofJ .[H, Al r (2 " A A ) &i = -z r, P + 2 17+PO"- -O"zP- PO"z (8.109)


where the last term describes spontaneous emission with energy relaxation rate r, and

' 111 0

A • A p =- dsTV(s)e''S1JEXpe-tS1JEX 2 -1

(8.110)

accounts for changes of the vibrational energy because of spontaneous emission. Here TJE is the Lamb-Dicke parameter corresponding to the field (8.105) and W(s) is the angular distribution of spontaneous emission [de Matos 1996a]. The steady-state solution to Equation (8.109) is obtained by setting opjot = 0, and may be written as

(8.111)

where lg) is the electronic ground state and 11/Js) is the vibrational steady-state of the ion, given by

For simplicity, we will concentrate in the j = 1 and m = 0 case (single number state spacing) for which Equation (8.112) is written as

(8.113)

Note that Hri1)11/Js) = 0, so that ion and laser have stopped to interact, which occurs when the ion stops to fluoresce. For the j = 1 and k = 0 case, and assuming Lkl) ( rJi) -1 0 and Lk ( TJ5) -1 0 for all k, one generates nonlinear coherent states [de Matos 1998]. However, by setting a value to one of the Lamb-Dicke parameters such that, for instance,

(8.114)

for some integer q, we obtain that, by writing 11/Js) in the number state representation,

1 q

11/Js(rJo)) = N, L c~O) In), 0 n=O

(8.115)

Filtering specific superpositions of number states 135

(the argument of 1/Js denotes the condition we apply; i.e., in Equation (8.115), the condition is on rJo) where

1, (8.116)

and q

NJ = L IC~O) 12 (8.117) n=O

is the normalization constant. If instead of condition (8.114) we choose

L~1l(TJI) = 0, (8.118)

we obtain the wave function

(8.119)

where now

1, (8.120)

and

Nf = L IC~l)l2· (8.121) n=p+1

Combining both conditions, (8.114) and (8.118), one would obtain for q > p,

with

- 1 ~ (1) 11/Js(TJo, TJd)- n ~ en In), 01 n=p+l

q

N51 = L IC~1 )1 2 -n=p+l

(8.122)

(8.123)

In this way, by setting the conditions (8.114), (8.118) or both, we can engineer states in the following three zones of the Hilbert space: (a) from IO) to lq), (b)


from IP + 1/ to loo/, or (c) from IP + 1/ to lq/. In the later case, by setting q = p + 1 generation of the number state lq/ is achieved. We should remark, that by selecting further apart sidebands one would obtain a different spacing in Equations (8.115), (8.119) and (8.122). For instance, by choosing j = 2 and k 0 one would obtain only even or odd number states in those equations (depending in this case on initial conditions and W(s), the angular distribution of spontaneous emission). Also, it should be noticed that one can use the parameters j = m + 1 and k = m (with m -/= 0) (in the single number state spacing case) to extend the possibilities of choosing Lamb-Dicke parameters. Lamb-Dicke parameters of the order of one (or less) are needed (for conditions (8.114) and (8.118)), which can be achieved by varying the geometry of the lasers. For example, by setting 'l]o = 1, we have L 1 ( 1]6 = 1) = 0, and therefore we obtain the qubit

11/Js('T]o = 1)) = V : 2

(10)- ~oe~'71

:12

2 l1)), 1 + 1~12e'h -1 1e 1

(8.124)

where by changing the Rabi frequencies, one has control of the amplitudes. Finally, note that we could have also chosen to drive the qth upper sideband instead of the kth lower sideband in Equation (8.105) with basically the same results.

P(n)

P(n)

Fig. 8.9 Possible situations we can have if we filter number states with the proposals of this

Chapter 9

Nonlinear coherent states for the Susskind-Glogower operators

In this chapter, we construct nonlinear coherent states for the Susskind-Glogower operators [Susskind 1964] by the application of a non-linear displacement operator on the vacuum state. In Section 9.1, we do that for an approximated displacement operator, and in Section 9.2, for an exact displacement operator. To analyze the obtained results, we plot the Husimi Q-function, the photon number probability distribution and the Mandel Q-parameter in Section 9.3. In Section 9.4, we also construct nonlinear coherent states as eigenfunctions of a Hamiltonian constructed with the Susskind-Glogower operators. We generalize the solution of the eigenfunction problem to an arbitrary lm/ initial condition. For both cases, we find that the constructed states exhibit interesting nonclassical features. Finally, in Section 9.6, we show that nonlinear coherent states may be modeled by propagating light in semi-infinite arrays of optical fibers.

Nonlinear coherent states may be constructed using the Susskind-Glogower operators, for instance, defining a displacement operator for them acting on the vacuum state. We have defined already the Susskind-Glogower operators; re

calling the definitions we have

A

00 1 V= :L1n)(n+11= v'A a,

n=O n + 1 (9.1)

section. and

A

00 1 vt = :L ln+1)(nl =at-,-,

n=O ~ (9.2)

satisfying the conditions

V In) = In- 1) , (9.3)

137

138 Nonlinear coherent states for the Susskind-Glogower operators

vt In) = In+ 1). (9.4)

Additionally, we would like to make explicit the result

VIO) 0, (9.5)

that comes naturally from (9.1). As we saw in Section 2 of Chapter 4, the Susskind-Glogower operators possess a non-commuting and non-unitary nature, that resides in the expressions

vvt = 1, (9.6)

and

vtv = 1 - 1o) (ol. (9.7)

l,From the above expressions we can see that such nature of the SusskindGlogower operators is for states of the radiation field that have a significant overlap with the vacuum; in other words,

( 1/J 1 [ v, vt] 11/J) = ( 1/JIO) \011/J) . (9.8)

Therefore, for states where the vacuum contribution is negligible, we can consider them as unitary and commutative, and we can perform the following approximation

v-1 C::'vt. (9.9)

The properties of the Susskind-Glogower operators play an important role in the development of the present chapter. For instance, if we analyze (9.1) and (9.2), we find that the Susskind-Glogower operators have the same form of the ones we need to construct nonlinear coherent states; i.e., V = f ( n + 1) a and

vt = at f ( n + 1) . Following Recamier et al. [Recamier 2008], we define Susskind-Glogower coherent states (from this point, we will refer to these states as Susskind-Glogower coherent states, keeping in mind that, in fact, they present a nonlinear behavior) by

la)sc = ex(vt -v) IO)' X E R (9.10)

As we have seen, commutation relations for the Susskind-Glogower operators are not simple, so we cannot factorize the displacement operator in a simple way; we propose two methods to achieve it. First, we use the approximation v-1 C::' vt to factorize, approximately, the displacement operator. This solution helps us to understand how the exact solution

Approximated displacement operator 139

for the displacement operator should be. Second, we solve the displacement operator in an exact way by developing it in a Taylor series, that allows us to introduce the exact solution for nonlinear coherent states, constructed with the Susskind-Glogower operators. Finally, we analyze the constructed states via the Q function [Husimi 1940], the photon number distribution and the Mandel Qparameter [Mandel 1995] in order to show their nonclassical features such as amplitude squeezing and quantum interferences.

9.1 Approximated displacement operator

A first approach to factorize the displacement operator in the product of exponentials, is to consider the approximation v-1 C::' vt. Let us write the displacement operator as

(9.11)

We find that the right-hand side of this equation corresponds to the generating function of Bessel functions, that implies that

where Jn is the Bessel function of the first kind and order n. Applying the displacement operator on the vacuum state, we have

Dsc IO) C::' co L vtn Jn (2x) IO)'

and using that vt In) = In+ 1) and that v-1 C::' vt, we obtain

la)8c = Dsc IO) C::' coL Jn (2x) In). n=O

l,From the normalization requirement, we determine c0 ,

SG (ala)sc c6 L Jn (2x) (nl L Jm (2x) lm) n=O rn=O

c6 L J~ (2x) = 1. n=O

(9.12)

(9.13)

(9.14)

(9.15)

140 Nonlinear coherent states for the Su.sskind-Glogower operators

Using the result [Abramowitz, 1972]

1 = Jg (2x) + 2 L J~ (2x) , (9.16) n=1

we obtain

2 (9.17) co=

1 + Jg (2x)'

and substituting in (9.14), we finally get

2 oc

2 ( ) Lln(2x)ln). la 2x n=O

(9.18)

As we mentioned before, solution (9.18) helps us to foresee the exact solution for these states. We can expect that the exact solution for Susskind-Glogower coherent states corresponds to a linear combination of number states where the coefficients are, except for some terms, Bessel functions of the first kind and order n.

9.2 Exact solution for the displacement operator

In order to factorize the displacement operator in an exact way, we can develop

the exponential (9.10) in a Taylor series and then evaluate the terms ( vt- V) k

For instance, for k = 7 we have

(vt _ v-f =

=: (vt- v) 7 : +G) (11) (ol-Io) (11)

-G) (13) (OI -12) (11 + 11) (21- IO) (31) (9.19)

+ C) (15) (OI-14) (11 + 13) (21- 12) (31 + 11) (41-IO) (51),

where : : means to arrange terms in such a way that the powers of the operator v arc always at the left of the powers of the operator vt. LFrom the definition of the Susskind-Glogower coherent states (9.10), we can

Exact solution for the displacement operator 141

write

(9.20)

where the square brackets in the sum stand for the floor function, which maps a real number to the largest previous integer. We can rewrite the above equation as

(9.21)

where we have taken the second sum to oo as we would add only zeros (the Gamma function has simple poles at the negative integers, so 1/f( -k), with k a positive integer, is zero). We now exchange the order of the sums

I ) - -xV xvt IO) + v-2 f f (-1txk v2nvtk IO)' asG-e e n=Dk=n(k-n)!n!

(9.22)

By setting m = k - n, and using that

(9.23)

we get

to finally obtain

(9.24)


Applying the exponential terms on the vacuum state, we have

la)sc = ( 1 + vz) fIn (2x) In) n=O

()()

= L Jn (2x) In)+ L Jn (2x) In- 2), n=O n=2

(9.25)

making m = n- 2 in the second sum and performing the index change n = k - 1. ~~~ .

la)sc = L [Jk-1 (2x) + Jk+l (2x)]lk- 1). (9.26) k=1

By using the recurrence relation of the Bessel functions [Abramowitz, 1972]

xJn-1 (x) + xJn+l (x) = 2nJn (x), (9.27)

and changing again the summation index, we finally write

1 ()() la)sc =- L (n + 1) Jn+l (2x) In).

X n=O (9.28)

Equation (9.28) is an important result because it constitutes an explicit expression for nonlinear coherent states. It remains to analyze the behavior of the constructed states, in order to determine their nonclassical features. Before we proceed with the analysis, and because we will need this result later, we make clear that, as we can verify from (9.28), for x = 0 we have

Ia (x = O))sc = IO) (9.29)

as must be.

9.3 Susskind-Glogower coherent states analysis

There are different ways to find out if the state we are constructing resembles one that we already know. Here, we will use three different methods: the Husimi Q-function [Husimi 1940], the photon number distribution and the Mandel Qparameter [Mandel 1995].

Susskind-Glogower coherent states analysis 143

9.3.1 The Husimi Q-function

We introduced in Chapter 2 the Husimi Q-function. We recall that it is defined as the coherent state expectation value of the density matrix operator; i.e.,

Q(a) = 2-_ (alpla). 7f

(9.30)

If we substitute p = l'l,b) ('l,b I for the Susskind-Glogower coherent states in (9.30), we obtain

(9.31)

Figure 9.1 shows the Susskind-Glogower coherent states Husimi Q-function for different values of the parameter x; we observe that the initial coherent state squeezes. Later, we will find the value of x for which we obtain the maximum squeezing of the coherent state.

9.3.2 Photon number distribution

When one studies a quantum state, it is important to know about its photon statistics. The photon number probability distribution P ( n) is useful to determine amplitude squeezing. We should refer to amplitude squeezed light as light for which the photon number distribution is usually narrower than the one of a coherent state with the same average number of photons. The photon number distribution is also useful to analyze if there exist effects due to quantum interferences. We write the Susskind-Glogower coherent states photon number distribution as

(9.32)

Figure 9.2 shows the Susskind-Glogower coherent states photon number distribution for different values of the amplitude parameter x. Figure 9.2 helps us to understand the effect of quantum interferences. For instance, consider Figure 9.2(c); we see that it is not a uniform distribution of photons, the distribution has "holes"; these holes are the consequence of quantum interference just as it happens for Schri:idinger cats states, for which also there are zero probabilities for even or odd photon numbers. Comparing Figure 9.2(a) with the one obtained for an initial coherent state wave function that subject to a harmonic oscillator potential, we can see that the photon number distribution for the Susskind-Glogower coherent states is narrower


-Hi -1(1

-{I)

Fig. 9.1 Exact Susskind-Glogower coherent states Q function for (a) x = 1; (b) x = 5; (c) x = 10 and (d) x = 20.

than the one for a coherent state of the same amplitude. This suggests that we are, indeed, obtaining an amplitude squeezed state. It is interesting to know when the state is maximally squeezed, but we need another tool to obtain the value of the parameter x for which this occurs.

Susskind-Glogower coherent states analysis

0.4

0.3

P(n) 0.2

0.1

0~~,-~-,~--~-,,-~~

0 10 20 30 40 50

0.20

0.15

P(n) 0.10

30 40 50

0.25

0.20

0.15 P(n)

0.10

0.05

0

0.12

0.10

0.08

P(n) 0.06

0.04

0.02

0 10

10

145

20 30 40 50

20 30 40 50

Fig. 9.2 Susskind-Glogower coherent states photon number probability distributions for (a) X= 1; (b) X= 5; (c) X= 10 and (d) X= 20.

9.3.3 Mandel Q-parameter

In the previous chapter, we introduced the Mandel Q-parameter; its definition is given by expression 8.102 on page 132. We will use it know, not only because it represents a good parameter to define the quantumness of Susskind-Glogower coherent states, but also because it will allow us to find the domain of x for

which they exhibit a nonclassical behavior; moreover, it will help us to find the value of x for which the state is maximally squeezed. For the Susskind-Glogower coherent states, we have that

(9.33)

and

(9.34)

The even sums, with respect to the power of k, in (9.33) and (9.34), can be evaluated.

In the Appendix E, we show explicitly that

and

L k2Jf (2x) = x2, k=O

L k4Jf (2x) = 3x4 + x2. k=O

(9.35)

(9.36)

The sum 2.:::%':0 k3 Jl (2x) is more complicated, and can be evaluated using the

technics developed by Dattolli [Dattoli 1989], such that we obtain

L k3 Jf (2x) = x 2{(6x2 + 1)JC(2x) + (6x2 - 1)Jl(2x) k=O

2x2

- 2xJo(2x)J1(2x) + 3[Jo(2x)J2(2x) + J 1 (2x)h(2x)]}.

(9.37)

Substituting the values of the sums into Equation 8.102 on page 132, we obtain the plot shown in Figure 9.3. z.From Figure 9.3 we can see that, depending on the parameter x, the photon distribution of the constructed states is sub-poissonian, Q < 0, meaning that amplitude squeezing states may be for a value of x within the domain 0 < x :S 13.48. Also, we find that the most squeezed state may be obtained at x = 2.32, with Q = -0.64.

Eigenfunctions of the Susskind-Glogower Hamiltonian 147

0.6

Q(x) 0.4

0.2

0 10 15 20 25

-0.2 X

-0.4

-0.6

Fig. 9.3 Mandel Q-parameter for the Susskind-Glogower coherent states.

9.4 Eigenfunctions of the Susskind-Glogower Hamiltonian

Previously we managed to construct nonlinear coherent states applying the displacement operator on the vacuum state. However, even when the obtained results arc very interesting, we cannot avoid to wonder how these states could be physically interpreted. Physical interpretations are given by operators representing observables; i.e., quantities that can be measured in the laboratory. The most important observable is the Hamiltonian, this operator helps us to find the energy distribution of an state via its eigenvalues. In this section, we construct Susskind-Glogower coherent states as eigenfunctions of a Hamiltonian that we propose and which represents the fundamental coupling to the radiation field via the Susskind-Glogower operators. In Subsection 9.4.1, we construct time-dependent Susskind-Glogower coherent states with the vacuum state IO) as initial condition, i.e., we construct states that satisfy Ia (x = O))sc = IO). In Subsection 9.4.2, we generalize the eigenfunction problem for an arbitrary lm) initial condition, we also show that previous results correspond to the particular case m = 0.


9.4.1 Solution for IO) as initial condition

As we mentioned before, it is possible to construct Susskind-Glogower coherent states as eigenfunctions of the interaction Hamiltonian

(9.38)

where 77 is the coupling coefficient. Hamiltonians like this may be produced in ion-traps [Wallentowitz 99]. The Hamiltonian proposed in (9.38) corresponds to a variation of the one used in [Wallentowitz 99] to model physical couplings between trapped ions and laser beams. Here, physical couplings take place via the Susskind-Glogower operators. We write the eigenfunctions of the Hamiltonian (9.38) in the interaction picture as

11/J) = LCnln), (9.39) n=O

and we have

fi 11/J) = 7] L Cn In- 1) + 7] L Cn In+ 1). (9.40) n=1 n=O

Now, changing the summation indexes, we obtain

n=O n=l (9.41)

= 7]C1 IO) + 7] L (Cn+l + Cn-1) In). n=1

Then

(9.42) n=l n=l

Comparing coefficients with same number states of the sum, we have

(9.43)

and

(9.44)


These are the recurrence relations of the Chebyshev polynomials of the second

kind ([Abramowitz, 1972]), and we can write

11/J (t; .;)) = L e-iEtun (.;)In), (9.45) n=O

where

E=2ry( (9.46)

However (9.45) does not satisfy the initial condition 11/J (t = 0)) = IO). Moreover, the solution (9.45) has the parameter.; and, as we see from (9.10), we should not have another parameter, except the time t. A way to construct a solution, as the one we previously obtained in (9.28), is by looking at the exponential term in (9.45) and noticing that it has the form of the Fourier transform kernel; so, we need to propose a .;-dependent function and integrate it over all .;, in order to obtain a solution where time is the only variable. We have then,

11/J (t)) = f 1= d.;P (.;) Un (.;) e-i2

ryf;t In). n=O -oo

(9.47)

We see from the above equation that 11/J (t)) corresponds to a sum of Fourier transforms of Chebyshev polynomials with respect to a weight function P (.;). This kind of Fourier transforms may be solved by using the following result [Campbell 1948]

where F {} is the Fourier transform and

rect (~) = { ~

l,From (9.48) we write

' -1 :s:.; :s: 1 , otherwise.

(9.48)

(9.49)

(9.50)


Using definition (9.49), and making k = n- 1, we obtain

(9.51)

With Equation (9.51) it is possible to solve the integral in (9.47). Writing the weight function as

P(~) = ~~rect (~),

substituting it into Equation (9.47) and using now (9.51), we get

2 co

11/J (t)) =-Lin (n + 1) Jn+l ( -27]t) In). -27]t n=O

Considering the odd parity of the Bessel functions, we finally obtain

1 co

11/J (t)) =-Lin (n + 1) Jn+l (277t) In). 7]t n=O

(9.52)

(9.53)

(9.54)

We see that 11/J (t)) in Equation (9.54) depends only on t, and considering that

lim Jn (2ryt) = { 0 2ryt--70 27]t ~

, n = 2, 3, 4, ... (9.55)

n = 1,

we can verify that

11/J (t = 0)) = IO). (9.56)

Equation (9.54) corresponds to the expression for Susskind-Glogower coherent states that we obtained previously in (9.28).

The solution presented in this section allows us to notice that, while in the previous section x was only a parameter, now it represents something physical. It may be related to an interaction time, for example, in the motion of a trapped atom [Wallentowitz 99]. We have managed to construct the same expression for the Susskind-Glogower coherent states as the one obtained by the application of the displacement operator on the vacuum state; however, we will see that the formalism presented in this section may be used to generalize the solution for an arbitrary initial condition lm).


9.4.2 Solution for lrn) as initial condition

At this point, we have managed to construct Susskind-Glogower coherent states, first as those obtained by the application of the displacement operator on the vacuum state and later, as eigenfunctions of the Hamiltonian (9.38); however, they are a particular case of a more general expression.

Using the recurrence relation for the Chebyshev polynomials and the result (9.51), it is possible to generalize Susskind-Glogower coherent states to an arbitrary lm) initial condition, where m = 0, 1, 2, .... L,From Equation (9.47), we have that

11/J (t = 0)) = f leo d~P (~) Un (~)In), n=O -co

(9.57)

and considering the function

(9.58)

and the well known Chebyshev polynomials of the second kind orthonormal condition

we get

11/J(t=O)) = LOnmln). (9.59) n=O

To obtain the solution for them-state, let us consider the particular cases m = 0 and m = 1.

Form= 0,

00 leo 2 = L -~Uo (0 Un (~) e-i

2ryEt rect (~) d~ In)

n=O -co 7r 2 (9.60)


Using (9.50), and making n--+ n + 1, W8 have

(9.61)

Using the recurrence relation

xln-1 (2x) + xln+1 (2x) = nln (2x), (9.62)

and changing the summation index, we finally obtain

1'1{1 (t))m=O L [in In (27]t) +in 1n+2 (27]t)Jin). (9.63) n=O

Form= 1, we have

(9.64)

Using the recurrence relation for the Chebyshev polynomials of the second kind, we write

00 11 2 1'1{1 (t))m=1 = L -~ CUn-l (~) + Un+l (~)) e-i21)~td~ In).

n=O - 1 1T

(9.65)

Considering (9.50) for the first integral, and making n = n + 2 for the second term, we have

00 1 1'1{1 (t))m=l = L t [in+l (n + 2) 1n+2 (27]t) + in- 1nln (27]t)]ln),

n=O 7] (9.66)

We now construct a different expression for the above Equation (9.66). Let us write it in the following way,


making k = n + 2

1'1{1 (t)) -1 = f {-ik- 1 ~ (k- 2) Jk-2 (27]t) + ik-1 [~kJk (27]t)]}' (9.68) m- n=O 7]t 7]t

using the recurrence relation (9.62), we write

-ik- 1 ~ (k- 2) Jk-2 (27]t) } +ik-1 [Jk-1 (27]t) + Jk+1 (27]t)]

-ik-1 [ ~ (k- 2) Jk-2 (27]t)- Jk-1 (27]t)] }

+ik-1Jk+1 (2'T)t)

ik- 3 [ ~ (k- 2) Jk-2 (27]t)- Jk-1 (27Jtl] }

+tk-1Jk+1 (27]t)

ik-2- 1 [ ~ (k- 2) Jk-2 (27]t)- Jk-2+1 (27]t)] } .

+ik-2+1 Jk-2+3(27]t)

Making n = k- 2, we obtain

Using again the recurrence relation (9.62), we finally write

1'1{1 (t))m=1 = L [in- 11n-l (27]t) + in+1 1n+3 (27]t)]ln). n=O

Following the same procedure, it is easy to prove that form= 2,

1'1{1 (t))m=2 = L [in-2 ln-2 (2'T)t) + in+2 1n+4 (27]t)]ln). n=O

(9.69)

(9.71)

(9.72)


Rewriting results (9.63), (9. 71) and (9. 72),

/1/J (t))m=O = L [in-O Jn-0 (2TJt) + in+O Jn+0+2 (27]t)]/n), n=O

11/J (t))m=l = L [in-l Jn_I(27]t) + in+l Jn+1+2(27]t)]/n), n=O

(9.73)

/1/J (t))m=2 = L [in-2 Jn-2 (27]t) + in+

2 Jn+2+2 (2TJt)]/n), n=O

It is easy to see that the solution for the m-initial condition is

/1/J (t))m = L [in-m ln-m (27]t) + in+m Jn+m+2 (2TJt)]/n) · (9.74) n=O

We have constructed a new expression for nonlinear coherent states and that we call Susskind-Glogower coherent states. We have managed to construct an expression that allows us to study the time evolution of Susskind-Glogower coherent states for an arbitrary /m) initial condition. We also found the physical interpretation of the parameter x (used in previous sections) as a normalized interaction time with respect to the coupling strength, i.e., x = 7]t. We analyze now the nonclassical features of the constructed states, so we have to make use of the methods previously mentioned, these are the Q function, the photon number distribution and the Mandel Q-paramcter.

9.5 Time-dependent Susskind-Glogower coherent states analysis

To perform a complete description of the constructed states (9.74), we have to verify if they present the nonclassical features that nonlinear coherent states may exhibit. In order to study these nonclassical features, we propose to use three methods. First, we analyze their behavior in phase space via the Q function; then, because we want to analyze amplitude squeezing and quantum interferences, we show the photon number distribution of the constructed states, and finally, as we want to know when the constructed states are maximally squeezed, we show the Mandel Q-parameter.

Time-dependent Snsskind-Glogower coherent states analysis 155

9.5.1 Q junction

Considering the definition Q = ~ (a/,0/a) and writing p in terms of the constructed states (9.74), we write the Q function for the time-dependent SusskindGlogower coherent states as

(9.75)

Figures 9.4 and 9.5 show the time-evolved Q function of Susskind-Glogower coherent states for two different initial conditions. In (b) in both figures, we have chosen 7]l = 2.32, because, as we showed previously, it is the time when the state with initial condition /0) is maximally squeezed. The nonclassical features of the constructed states are summarized in Figure 9.4. Susskind-Glogower coherent states present a strong amplitude squeezing. This may be explained because they tend to close a circle in phase space, just as the number (the infinite squeezed states in amplitude) and displaced number states do. An interesting result is that no matter what initial condition we choose, SusskindGlogower coherent states eventually fill such a circle. This is very interesting result because such a distribution would approach a displaced number state, which has highly nonclassical features.

9.5.2 Photon number distribution

To complete the description of the nonclassical features that we observed from the Q function, we show in Figures 9.6 and 9.7 the photon number distribution of the Susskind-Glogower coherent states considering the same conditions of Figures 9.4 and 9.5. Time-dependent Susskind-Glogower coherent states photon number distribution for them-initial condition is given by

(9.76)

9.5.3 Mandel Q-parameter

As we want to know the time domain for which the constructed states exhibit amplitude squeezing, and moreover, we want to know when the states are maximally squeezed, we obtain the Mandel Q-parameter for the time-dependent Susskind-Glogowcr coherent states.


-w -lD

-10 -I~~

Fig. 9.4 Susskind-Glogower coherent states Q function for (a) 17t = 1; (b)17t = 2.32; (c) 17t = 5 and (d) 17t = 20, with the initial condition 11).

We have that

00

(ii) L n [Jn-m (2rJt) + ( -l)m Jn+m+2 (2·f)t)]2

, (9.77) n=O

Time-dependent Susskind-Glogower coherent states analysis 157

0.1!4-

-Ill

Fig. 9.5 Susskind-Glogower coherent states Q function for (a) ryt = 1; (b) 17t = 2.32; (c) 17t = 5 and (d) 17t = 20, with the initial condition 110).

and

(n2) = f n 2 [Jn-m (2·f)t) + ( -l)m Jn+m+2 (2f)t)]2

. (9.78) n=O

Substituting in the definition of the Mandel Q-parameter, Equation ( 8.102 oJ1


0.4

0.3

P(n)

0.2

0.1

0~~--~--~.-~.-~.-~

0 10 20 30 40 50 60

0.25

0.20

P(n) 0.15

0.10

0.05

a~~~~.-~~~~~~~

0 10 20 30 40 50 6C

0.4

0.3

P(n) 02

0.1

0~~--~--~--~.-~~~

0.09

0.08

0.07

0.06

P(n) 0.05

0.04

0.03

0.02

0.01

0 I 0 20 30 40 50 60

10 20 30 40 50 60

Fig. 9.6 Susskind-Glogower coherent states photon number probability distributions for (a) 'l)t = 1; (b) 'l)t = 2.32; (c) 'l)t = 5 and (d) 'l)t = 20, with initial condition 11).

page 132), we obtain the plot shown in Figure 9.8. i,From Figure 9.8 we see that (a) shows the Q-parameter for the SG coherent states where the maximum squeezing happens when Tjt = 2.32. From the others, we cannot observe squeezing, because the initial condition is a fully squeezed state in amplitude, a number state; however, we obtain that an initial number state eventually transforms into

Time-dependent Susskind-Glogower coherent states analysis 159

0.3

0.2

P(n)

0.1

o+-~~~r-~.-~.-~.-r-o

10 20 30 40 50 60 0

0.10

0.08

0.06 P(n)

0.04

0.02

10 20 30 40 50 60

0.16

0.14

0.12

0.10

0.08

0.06

P(n)

0.04

0.02

I 0 20 30 40 50 60

10 20 30 40 50 60

Fig. 9. 7 Susskind-Glogower coherent states photon number probability distributions for (a) 'l)t = 1; (b) 'l)t = 2.32; (c) 'l)t = 5 and (d) 'l)t = 20, with initial condition 110).

something similar to a displaced number state. The generalization (9.74) does not give us different effects from the ones that we may obtain from (9.54); however, as we will see in next section, Equation (9.74) helps us to show that nonlinear coherent states may be modeled by propagating

light in semi-infinite arrays of optical fibers.

160

Q(t)

-I

Nonlinear coherent states for the Susskind-Glogower operators

/

·'· ,·

------·-·-·-·, . ...-·-·-

/ -----------.,--------/ .' I I

I / • I ./

.. ····1. ·"' 20 25

1-(a)• · · •(b)- •(c)- · •(d)l

Fig. 9.8 Susskind-Glogower states Mandel Q-parameter with initial conditions (a) jO); (b) j1); (c) j5) and (d) jlO).

9.6 Classical quantum analogies

The modeling of quantum mechanical systems with classical optics is a topic that has attracted interest over the years. Along these lines Man'ko et al;[Man'ko 2001] have proposed to realize quantum computation by quantum like systems, Chavez-Cerda et al [Chavez-Cerda 2007] have shown ow quantum-like entanglement may be realized in classical optics, and Crasser et al.[Crasser 2004] have pointed out the similarities between quantum mechanics and Fresnel optics in phase space. Following these cross-applications, here we show that SusskindGlogower nonlinear coherent states may be modeled by propagating light in semi-infinite arrays of optical fibers. Makris et al.[Christodoulides 2006] have shown that for a semi-infinite array of optical fibers, the normalized modal amplitude in the nth optical fiber (after the mth has been initially excited) is written as

(9.79)

where Z = cz is the normalized propagation distance with respect to the coupling coefficient c. We see that, for A0 = 1 the normalized intensity distribution is

(9.80)

Classical quantum analogies 161

Rewriting (9.76) and using the normalized interaction time with respect to the

coupling coefficient 1), i.e., x = 'f]t, we have

Pm (n,x) = l·in-mJn-rn (2x) + in+rnJn+rn+2 (2x)l2

. (9.81)

As Equations (9.80) and (9.81) are the same, we conclude that the photon number distribution for the Susskind-Glogower coherent states may be modeled by the intensity distribution of propagating light in semi-infinite arrays of optical fibers. We have found a new relation between quantum mechanical systems and

classical optics.


Appendix A

Master equation

A.l Kerr medium

The Master equation for a Kerr medium in the Markov approximation and in the interaction picture has the form [Milburn 1986]

(A.1)

Milburn and Holmes [Milburn 1986] solved this equation by changing it to a partial differential equation for the Q-function and for an initial coherent state. We can have a different approach to the solution by again using superoperators. If we define

we rewrite (A.1) as

Now we use the transformation

to obtain

with

p = exp[(S + L)t]p,

dp A AA

dt = exp[-ixRt- 2rt]J,O,

163

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

164 Master equation

It is easy to show that R and J commute, so that we can finally find the solution to (A.l) as

(A.7)

A.2 Master equation describing phase sensitive processes

One of the most general Master equation is one that describes phase sensitive processes:

dA 4 P_~ I' (A At)A dt- ~ {kLk a, a p, (A.8)

k=l

where

(A.9)

(A.lO)

(A.ll)

and

(A.12)

The 1's are in general complex parameters that may represent gain or decay; however, for the density matrix to remain Hermitian it is necessary to comply

with /3 = 14 = l1lei1>. If we apply the unitary transformation p = S(~)j;St(~), with S(~) the squeeze operator (with complex amplitude~= rei¢)

A (Ca2 catz) S(~) = exp -2- - -

2- , (A.13)

we arrive to the equation for the transformed density matrix

(A.14)

where

(A.15)

Master equation describing phase sensitive processes 165

with 11 = cosh(r) and v = - sinh(r)ei<l>. Rewriting (A.l4) in terms of a and at, and setting the parameters

we obtain

with

and

¢=-, tanh(2r) = ~' 11 +12

i2 = "Yllvl 2 + /2/12 + J-1Vh3lei1> + Mv*l13le-i1>.

Equation (A.l7) has been solved in Chapter 5.

(A.l6)

(A.l7)

(A.18)

(A.l9)

166 Master equation

Appendix B

Methods to solve the Jaynes-Cummings model

B.l A naive method

A naive method to solve the Jaynes-Cummings model is to forget that the elements of the interaction Hamiltonian are operators and try to diagonali:;-;e it. We write the Jaynes-Cummings Hamiltonian for~ =I 0 using 2 x 2 matrices,

(B.l)

To find the eigenvalues of the above matrix, we need to solve the determinant

(B.2)

~here~ is the (operator) eigenvalue. The determinant produces the equation (32 - ~ - )..2aat = 0 (in this equation we have chosen an ordering for the multiplication of the creation and annihilation operator, namely, anti-normal ordering). From the eigenvalues we can write a diagonal density matrix

jj = ( f3n 0 ) . 0 -f3n '

(B.3)

with f3n such that (3~ = ~ + )..2 (n + 1). We can then write the eigenvectors matrix

P~ ( f3n+%

2/3n_

-Aii'-.L ) . (B.4)

)..a,t_L 2/3.c, 2/3.c,

167

168 Methods to solve the Jaynes-Cummings model

The columns of the P matrix are eigenvectors of the Jaynes-Cummings Hamiltonian with (right) "eigenvalue" f3n for the first column and -{3r, for the second column. We now find the (left) inverse matrix toP,

(B.5)

Note that TF = 1 but Pi' -I- 1. The operator it is the inverse operator of at [Hong-Yi 1993]; i.e.,

~at= 1 at ' at~= 1 -IO)(OI. (B.6)

It is now easy to show that PDT= fi and the evolution operator is therefore

(B.7)

and the exponential of the diagonal matrix is straightforward to calculate.

B.2 A traditional method

Looking at (B.2), we note that the the relevant states for the interaction are le)ln) and lg)ln + 1), in the sense that application of the Hamiltonian on these states takes us to combinations of the same states. We can therefore propose a wave function of the form

11/J(t)) = L [Cn(t)le)ln) + Dn(t)lg)ln + 1)] (B.S) n=O

and substitute it into the Schrodinger equation to obtain a system of differential equations for Cn and Dn

. 6. iCn(t) = 2 cn + >-.vn + 1Dn(t)

. 6. iDn(t) = >-.vn+TCn- 2Dn(t). (B.9)

The solution to this system of equation is the exponential of the matrix of the system.

Appendix C

Interaction of quantized fields

In this appendix we will look at the interaction of many fields. First, we will consider the interaction between two fields, to later generalize the result to a field interacting with many. In particular, we will give expressions in terms of polynomials for eigenvectors of the matrices that diagonalize the Hamiltonian for the interaction of many fields.

C.l Two fields interacting: beam splitters

Consider the Hamiltonian of two interacting fields (as in all the book, we set

n= 1)

(C.1)

This interaction occurs in beam splitters; however, it may also be obtained by the interaction of two quantized fields with a two-level atom, when the fields are far from resonance with the atom; in this case, an effective Hamiltonian may be obtained, which has the form of the above Hamiltonian [Prado 2008]. By transforming to the interaction picture; i.e., getting rid off the free Hamiltonians, we obtain

(C.2)

with 6. = wa - wb, the detuning. It is useful to define normal-mode operators by [Dutra 1993]

(C.3)

169

170 Interaction of quantized fields

with

5 = 2.\ J2D(D-~)' (C.4)

and D = v ~2 + 4.\: the Rabi frequency. The annihilation operators A1 and .42

are just like a and b, and obey the commutation relations

(C.5) 4

moreover, the normal-mode operators commute with each other

(C.6)

In terms of these operators, the Hamiltonian (C.l) becomes

(C.7)

with /-l1,2 = (~ ± D)/2. Up to here, we have translated the problem of solving Hamiltonian (C.l) into the problem of obtaining the initial states for the "bare" modes in the initial states for the normal modes. In order to have a way of transforming states from one basis to the other, we note that the vacuum states in both systems, IO)aiO)b and IO) ,4)0) A_2, differ only for a phase [Dutra 1993]. First note that

(C.S)

and in a similar way, it may be seen the other normal-mode annihilation operator, A2 , has the same effect. We choose the phase so that

(C.9)

If we consider coherent states as initial states for the interaction, we obtain the evolved wavefunction

IV>(t)) e-it(JL1A.1 A.l +ll2A.P2l Da( a)Db(/3) IO)aiO)iJ

e-it(JL1A.tA.l+ll2A.!A.2) Da(a)DiJ(/3)10) A.liO) A.2' (C.lO)

where the Dc(E) = exp(Ect- E*c) is the Glauber displacement operator [Glauber 1963b]. From (C.3), we can write the operators a and bin terms of the operator A1 and A2, and write (C.lO) as

Generalization to n modes 171

passing the exponential in the above equation to the right, applying it to the

vacuum states and using the following property

(C.l2)

we obtain

IV>(t)) D A.1 ([a5 + f3"Y]e-ill 1 t)D A.2 ([a"Y- (35]e-iMt)IO) A.1IO) A.2 I [a5 + /3')']e-ifllt) A.ll [a"Y- (35]e-ifl2t)A.2. (C.l3)

Equation (C.l3) shows that in the new basis, coherent states remain coherent

during evolution. By transforming back to the original basis, using again property (C.l2), we

obtain

IV'(t)) = l5[a5+/3')']e-illlt+')'[a')'-/35]e-ifl2t)ai"Y[a5+f3"Y]e-ifllt_5[a')'-(35]e-ill2t)iJ; (C.l4)

i.e., coherent states remain coherent during evolution. This will be used next section as the building block for the interaction of many modes.

C.2 Generalization to n modes

Consider the Hamiltonian of the interaction of k fields

(C.l5)

This Hamiltonian may be produced in waveguide arrays. From the Hamiltonian

above, we can produce the following matrix

[

WI A21 Anl l A12 W2 An2

.\13 .\23 An3

Aln A2n Wn

(C.l6)

We can rewrite the Hamiltonian in the form (C.7), that is

(C.l7)

172 Interaction of quantized .fields

such that

where we have defined the normal-mode operators Ak as

with Tki a real number. Equation (C.l8) implies that

n

Ak = Lrk;fi;, i=1

t rkiTmj [ai,a}] = trkirmi = 0. i,j=O

By defining the vector

(C.l8)

(C.l9)

(C.20)

(C.21)

Equation (C.20) takes the form Tk · f'rr, = 0, i.e. the vectors rk are orthogonal; we will consider them also normalized, Tk · rk = 1. With these vectors we can form the matrix

[ rn

T21

Tn> l r12 T22 fn2

R= (C.22)

T1n T2n Tnn

If we combine Equations (C.l5), (C.l7) and (C.l9), we obtain the system of

equations

(C.23)

(C.24)

A particular interaction 173

that may be re-expressed in the compact form

]'

Wn

(C.25)

with

(C.26)

i.e. D is a diagonal matrix whose elements are the eigenvalues of the matrix i\1, defined from the Hamiltonian. The matrix R is therefore M's eigenvectors matrix.

C.3 A particular interaction

Now we study a particular interaction, namely when Aij = A if j = i + 1 or j = i - 1 and it is zero otherwise. The frequencies wi are left arbitrary. The Hamiltonian governing this interaction then has an associated tridiagonal matrix of the form

0

(C.27)

We can use some properties of this matrix to find the eigenvectors; in particular, the characteristic polynomial for this matrix is given by the recurrence relation

Fo(J..l) = 1,

174 Interaction of quantized fields

and the normalized eigenvectors are simply

[ ~~~~;~ l Fn-,(M,)

(C.29)

with Nj = I:;~:~ F'f (J-Lj), such that the matrix elements of the matrix R are given by

(C.30)

C.4 Coherent states as initial fields

The solution to the Schrodinger equation, subject to the Hamiltonian (C.15), with all the modes initially in coherent states, 17,6(0)) = la1)llaz)z ... lan)n, is simply the direct product of coherent states

(C.31)

with /I(t) = (f'1. ae-il'tt,rz. ae-i~'2 t, ... ,r:,. ae-il'nt) and the vector a=

(a1, a2, ... ,an) is composed by the coherent amplitudes of the initial wave function. Up to here, we have shown that the interaction of several modes, initially in coherent states, does not change the form of those states (remain coherent), but modifies their amplitudes. If we choose the interaction constants to be .A 1j f. 0 for j f. 1 and the rest as zero, we are dealing with the interaction between one field and n -1 fields. If n -t oo and the amplitudes aj are zero for j > 1, we deal with the interaction of one field with n -1, one of them in a coherent state with amplitude a1 and the rest in the vacuum. Therefore, the most likely situation we have is the coherent state decaying towards the vacuum while keeping its coherent form.

Appendix D

Quantum phase

D.l Turski's operator

Classically we may decompose a complex c-number, A, in amplitude and phase by simply writing A = rei¢>, with r = IAI and

¢ = -iln~, r

(D.1)

where it is implied that we have chosen the principal branch of the multi-valued logarithm function. In correspondence to the classical form (D.1), Arroyo-Carrasco and Moya-Cessa [Arroyo-Carrasco 1997] proposed the Hermitian operator

¢ = -- lim D(x)ln 1 +- D (x) +H.C., A i A ( a) At

2 x-+= X (D.2)

where a and at are the annihilation and creation operators for the harmonic oscillator, respectively, D(x) = ex( at -a) is the displacement operator, and x is

a real parameter that tends to infinity to ensure convergence of the series

( a) = (-1)k-1 (a)k ln 1+- =2:-- -

X k=l k X (D.3)

Note that the displacement operators in Equation (D.2) produces a displacement of a by an amount minus x generating exactly the form (D.1). However, we keep the displacement operator explicitly in order to have a Taylor series for the logarithm. The operator (D.2) may be found to be Turski's operator [Turski 1972]. Actually, if we write the unit operator in terms of coherent states as i = ~ J la)(ald2a,

175

Ill II I

176 Quantum phase

and insert it into (D.2) it yields

(D.4)

that may finally be written as

(D.5)

Again, choosing the principal branch in the above equation, we can rewrite (D.5) as

(D.6)

where e = arg(a). Of course, as any operator that lives in the whole Hilbert space, ¢ obeys the equation of motion

(D.7)

where w is the frequency of the harmonic oscillator. Notice that for a phase operator defined in a finite dimensional Hilbert space to obey such equation of motion, the harmonic oscillator Hamiltonian should be defined also in a finite dimensional Hilbert space. We can calculate the average value of the argument of the operator (D.6), given a wave function 11J'I), as

(D.8)

where

Q(a) = ~l(ai1J'IW 7r

(D.9)

is the Q-function.

D.2 A formalism for phase

A formalism for phase could he introduced based on (D.6), as follows

(arg(ii)) = (¢) = j arg(a)Q(a)d2a, (D.lO)

A formalism for phase 177

and

(D.ll)

Remark that there is no phase that can be defined in an strict correct form [Lynch 1995], therefore in the above we have used the following forms:

(D.12)

and

(D.l3)

such that we can calculate the phase variance, D.¢= (arg2 (ii))- (arg(ii)) 2 for a number state In), yielding the result

b."-=~ '-f/ 3' (D.l4)

and where we have used

(D.15)

We have obtained the correct phase variance expected for a state of undefined phase; therefore, the operator given by Turski would lead to a phase formalism given by the (radially) integrated Q-function.

D.2.1 Coherent states

If we consider the coherent state

(D.16)

using (D.lO), we can calculate its phase properties, obtaining

(arg(ii)) = 77· (D.l7)

If we usc (D.2), we obtain the same result; i.e.,

(D.18)

178 Quantum phase

so that

1f2 CXl n+m r(n+m + 1) /}.¢coh=-+4e-p

2 L L (-1)n-mP -2- ,

3 m=On=m+l n!m! (n- m)2 (D.19)

where r(x) is the well-known Gamma function. Note that because the formalism

3

2

o~~--~--~=;~~~~~~~~~ 0 2 3 4 5

p

Fig. D.l We plot Li¢coh (solid line), and 1/2p2 (dashed line) as a function of p.

that takes us to the integrated Q-function comes from the operator introduced by Turski . He showed that [·n, ¢J = i, with n = at a, leading to a Heisenberg uncertainty relation

/:).cj;>-1-=~. - 2/:).n 2p2

(D.20)

This may be corroborated in Figure Dl, where we plot /}.¢coh as a function of p, together with the expression 1 /2p2

.

D.3 Radially integrated Wigner function

In the former section, Equations (D.6) and (D.8) were used to introduce the calculation of phase properties in terms of the Q-function; however, one can

Radially integrated Wigner function 179

also introduce them in terms of other quasiprobability distribUti.ons, namely, the Wigner function [Garraway 1992]

(D.21)

and

(¢k)w = j argk(a)W(a)d2a. (D.22)

Calculating phase fluctuations for the coherent state (D.16), vsitlg the above

expressions, leads to

2 CXl CXl ( 12 )n+m r(n+m + 1..) /:)."' = 2._ + 4 -2p

2""""' """"' (-l)n-m V L-P ~

'I'W 3 e ~ ~ n!m! (n-mF ' m=On=m+l

(D.23)

where we have used that

(D.24)

In Figure D2, we plot f}.¢w together with the expression for the Pbase variance

3

2.5

2

1.5

0.5

0 2 3

p

Fig. D.2 We plot 1/(4p2 + 3/7r2 ) (solid line), fl.¢~;[; (dash line) and ti¢w (dot-dash line) as a function of p.

180 Quantum phase

for coherent states using the Pegg-Barnett formalism, that can be written as [Pegg 1988; Barnett 1989]

7r2 2 oo CXJ n+m t,¢P-B = _ +4e-p L L (-1)n-m_P ___ 1_

coh 3 m=On=m+l vnr:mf (n- m)2. (D.25)

Notice that the Heisenberg uncertainty relation in the Pegg-Barnett case leads to the inequality (for a coherent state) [Skagerstam 2004]

t.¢ > 1 - 4t.n + 3/Jr2 (D.26)

This expression is also plotted in Figure D2. It is seen that both expressions for the phase fluctuations, the one obtained from the Wigner function integration and the one we get from the Pegg-Barnett formalism, tend to the above limit.

Appendix E

Sums of the Bessel functions of the first kind of integer order

We will derive in this appendix the solution of some sums of Bessel functions of the first kind of integer order that appear in several applications, and in particular, that appear in this book in the section where we calculate the Mandel Q-parameter for the nonlinear Suskind-Glogower coherent states. We will demonstrate that

fk 2vJ[(x)= (~~-t] B2v(g'(y),g"(y), ... ,g(2vl(y))dy, (E.1) k=l -K

where vis a positive integer, g (y) = ix sin y and Bn (x1, x2, ... , Xn) is the complete Bell polynomial [Bell 1927; Boyadzhiev 2009; Comtet 1974] given by the

following determinant:

Bn(Xl,X2, ... ,xn) =

X1 (n~l)x2 (n;l)x3 (n~l)x4 (n4l)x5 Xn

-1 X1 (n~2)x2 (n;2)x3 (n~2)x4 Xn-1

0 -1 xl (n~3)x2 (n;3)x3 Xn-2

0 0 -1 X1 (n~4)x2 Xn-3 (E.2)

0 0 0 -1 X1 Xn-4

0 0 0 0 -1 Xn-5

0 0 0 0 0 -1 X1

181

I' I

I I

I

182 Sums of the Bessel functions of the first kind of integer order

To demonstrate (E.1), we will need the well known Jacobi-Anger expansions for the Bessel functions of the first kind ([Gradshteyn 1980], page 933, [Magnus 1966], page 361; [Abramowitz, 1972], page 70),

CXl

eixcosy = :2: inJn(x)einy (E.3)

and

(E.4)

Using expression (E.3), we can easily write

(E.5)

To calculate the n-derivative in the left side of equation above, we use the Faa di Bruno's formula ([Gradshteyn 1980], page 22) for the n-derivative of the composition

dn dxn f (g (x)) =

= t f(k) (g (x)) · Bn,k (g' (x), g" (x), ... , g(n-k+l) (x)) , (E.6)

k=O

where Bn,k (xi, x2, ... , Xn-k+I) is a Bell polynomial [Bell1927; Boyadzhiev 2009; Comtet 1974], given by

Bn,k (xi, X2, ... , Xn-k+I) =

n! = :2:. ,. ' . ' JI·J2 · · · ·Jn-k+I·

X (~)jl (~))2 ... ( Xn-k+I )Jn-k+l 1! 2! (n-k+1)! '

(E.7)

the sum extending over all sequences JI, j 2 , ]3, ... , Jn-k+I of non-negative integers

such that JI +J2+ ... +Jn-k+I = k and JI +2]2+3j3+ ... +(n- k + 1) Jn-k+I = n. Using (E.6),

..:!!:.._eixsiny = dyn

= eixsiny t Bn,k (g' (x) 'g" (x) ' ... ,g(n-k+I) (x)). k=O

(E.8)

I83

We multiply now for the complex conjugate of (E.4) and obtain

k,l=-oo (E.9)

= Bn (g' (y) ,g11 (y), ... ,g(n) (y)).

In;egrating both sides of the above equation from -1r to 1r, and using that J_1r ei(k-l)Ydy = Okz, we arrive to the formula we wanted

k=I

- (-1)" /7r B ( '() "() (2v)( ))d -~ 2v g y,g y, ... ,g Y y.

In particular, as the complete Bell polynomials for n = 2 and n = 4, are

_H2(XI, X2) =xi+ X2

and

B4(xi, x2, X3, x4) =xi.+ 6xix2 + 4xix3 + 3x~ + x 4 ,

it is very easy to show that

and that

fk2Jf (x) = ~x2 k=I

~ 4 2() 3 4 1 2 L k Jk x = -x + -x . k=I 16 4

(E.10)

(E.ll)

(E.12)

(E.13)

(E.14)

184 Sums of the Bessel functions of the first kind of integer order

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I

Index

annihilation operator, 7 Araki-Lieb inequality, 84 atom and field entropy operator, 88 atom-field interaction, 59

dispersive interaction, 65 entropy, 83 mixed classical and quantum

interactions, 66 mixed classical and quantum

interactions Hamiltonian, 66 purity, 83 quantum interaction, 62 semiclassical interaction, 59 semiclassical interaction Hamiltonian,

60, 62 atomic entropy, 89 atomic entropy operator, 88 atomic inversion, 61, 63

coherent states, 63 coherent states superposition, 64 squeezed states, 64

Baker-Hausdorff formula, 4, 31, 33, 44

cavity losses cat states, 76 coherent states, 75 number states, 75

characteristic function, 25, 29 coherent states, 10, 21

averages, 21 excitation number, 13

189

photon distribution, 13 uncertainties, 14

concurrence, 90 creation operator, 7

Dirac notation, 1 bra, 2 braket, 2 ket, 1

displaced number states, 14 excitation number, 14

displacement operator, 10, 21, 33 properties, 10

entropy, 81 entropy of the damped oscillator

cat states, 93 entropy operator, 89

orthonormal states, 91 Ermakov equation, 44

solution for the step function, 47

field entropy operator, 89 Fock states, 1, 8, 21

Glauber-Sudarshan P-function, 32, 34, 37

Hadamard lemma, 4 harmonic oscillator, 6

eigenfunctions, 1 Hamiltonian, 6

1,,11,,,1

1,1

190

vacuum state, 8 Husimi Q-function, 33, 34, 37, 52

ion motion in a trap micromotion, 119 secular motion, 119

Jaynes-Cummings Hamiltonian, 62

Kingdon trap, 109

ladder operators, 7 commutation relations, 8

Lamb-Dicke parameter, 127 Lewis-Ermakov invariant, 44, 45

eigenstates, 45

Master equation cavity losses, 74 finite temperature, 77 zero temperature, 73

measuring field properties phase properties, 107 squeezing, 105

minimum uncertainty states, 46 mixed states, 81

number operator, 8, 16 number states, 1, 8 number-phase Wigner function, 39

coherent states, 40 superposition of number states, 41

ordering of ladder operators, 16 anti-normal ordering, 16, 20 normal ordering, 16, 17

parity operator, 30 Paul trap, 110, 111

frequency dependent of time, 126 frequency independent of time, 120 ion motion, 115

Penning trap, 109 phase states, 14 pure states, 81 purity, 82

Index

quasiprobabilities and losses, 102 quasiprobability distributions, 23

integral forms, 37 reconstruction, 97 reconstruction in a lossy cavity, 99 reconstruction in an ideal cavity, 97 relations between them, 34

Rabi frquency, 123 reduced density matrices

properties, 84 ringing revivals, 64 rotating wave approximation, 60

Schri:idinger cat states, 54 photon distribution, 55 Wigner function, 56

Schri:idinger cat states purity, 95

squeeze operator, 50 squeezed states, 50

Husimi Q-function, 51 photon distribution, 50 Wigner function, 52

Stirling numbers, 17, 20 superoperator, 74 Susskind-Glogower Hamiltonian, 148

eigenfunctions, 150, 154 Susskind-Glogower nonlinear coherent

states, 138, 142, 150, 154 classical analogies, 160 Husimi Q-function, 143 Mandel Q-parameter, 145 photon number distribution, 143

Susskind-Glogower operators, 16, 39, 62 approximated displacement operator,

139 commutation relations, 138 commuting properties, 138 exact displacement operator, 140 non-unitary property, 138

thermal distribution, 57 time dependent harmonic oscillator, 43

annihilation operator, 43 creation operator, 43

Hamiltonian, 43 number operator, 43 Schri:idinger equation, 45 step function, 46

Time dependent Susskind-Glogower nonlinear coherent states, 154 Husimi Q-function, 155 Mandel Q-parameter, 155 photon number distribution, 155

two-level atom-field interaction, 88

von Neumann entropy, 81 von Neumann equation, 4

Wigner function, 23, 34, 36, 37, 52, 56, 76, 91, 98, 103 coherent states, 31 direct measurement, 97 displaced number states, 32 expected values, 28 number states, 32 properties, 28 series representation, 30 symmetric averages, 29

Index 191

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