Introduction to Quantum Optics
Hector Manuel Moya-Cessa Francisco Soto-Eguibar
Rinton Press, Inc. a Isabel y Leonardo
a Conchis, Sofi y Quique
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ISBN 978-1-58949-061-1 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 a cional de Astrofisica, Optica y Electr6nica, INAOE, (National Institute of As trophysics, 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 meth ods 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 de pendent 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 Chap ter 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 Contents 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 solu tion 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 meth ods to construct them and we analyze their properties by means of the photon distribution number, the Mandel parameter and the Husimi function. vii Several appendices are added, which we believe enrich the contents of the Preface book. Particularly important is Appendix A, where we study the Master Equa Chapter 1 Operator algebra and the harmonic oscillator tion, describing phase sensitive processes for a cavity filled with a Kerr medium. 1 1.1 Introduction ...... Being the J aynes-Cummings model such an important tool in quantum optics, 4 1.2 von Neumann equation .. · · · · · · we provide different methods for solving it in Appendix B. The interaction of 4 1.3 Baker-Hausdorff formula ...... · . · · many fields is treated in Appendix C. Appendix D proposes a quantum phase 6 1.4 Quantum mechanical harmonic oscillator formalism, while Appendix E studies a series of Bessel functions, needed to de 7 1.4 .1 Ladder operators termine the non-classical character of non-linear coherent states described in 8 1.4.2 Fock states . . . Chapter 9. 10 1.4.3 Coherent states . We would like to thank the many people who have been essential for the ex 14 1.4.4 Displaced number states . istence of this book, especially several generations of students. Finally, we want 14 1.4.5 Phase states to thank our families for the understanding and the patience they have given to 16 1.5 Ordering of ladder operators us during the time we dedicated to the writing of this book. 17 1.5.1 Normal ordering ... 19 1.5.1.1 Lemma 1 .. Hector Manuel Moya Cessa y Francisco Soto Eguibar 20 Santa Maria Tonantzintla, Puebla, Mexico. June 15, 2011. 1.5.2 Anti-normal ordering 20 1.5.2.1 Lemma 2 . 21 1.5.3 Coherent states. 21 1.5.4 Fock states .... 23 Chapter 2 Quasiprobability distribution functions 23 2.1 Introduction ...... · . · · · · · · · 23 2.2 Wigner function ..... · . · · 28 2.2.1 Properties of the Wigner function .. · · · · · · · ·. · · 28 2.2.2 Obtaining expectation values from the Wigner functwn 29 2.2.3 Symmetric averages ... · · · · · · · · · · · 30 2.2.4 Series representation of the Wigner function . Contents Contents 2.3 Glauber-Sudarshan P-function . . . 32 88 2.4 Husimi Q-function ...... 33 6.4.2 Atomic entropy operator . . . . . · · · 2.5 Relations between quasiprobabilities 34 6.4.3 Field entropy operator 89 2.5.1 Differential forms ...... 34 6.4.4 Entropy operator from orthonormal states . 91 2.5.2 Integral forms ...... 37 6.5 Entropy of the damped oscillator: Cat states ... 93 2.6 The Wigner function as a tool to calculate divergent (or not) series . 37 Chapter 7 Reconstruction of quasiprobability distribution func- 2.7 Number-phase Wigner function ...... 39 tions 97 2.7.1 Coherent state ...... 40 7.1 Reconstruction in an ideal cavity ...... 97 2. 7.2 A special superposition of number states . 41 7.1.1 Direct measurement of the Wigner function 97 7.1. 2 Fresnel approach . . . . . 98 Chapter 3 Time Dependent Harmonic Oscillator 43 3.1 Time dependent harmonic Hamiltonian 43 7.2 Reconstruction in a lossy cavity . 99 Quasiprobabilities and losses 102 3.1.1 Minimum uncertainty states. 46 7.3 3.1.2 Step function . . 46 7.4 Measuring field properties 105 3.2 More states of the field . . . . 49 7.4.1 Squeezing .... 105 7.4.2 Phase properties 3.2.1 Squeezed states . . . . 50 107 3.2.2 Schrodinger cat states 54 Chapter 8 Ion-laser interaction 109 3.2.3 Thermal distribution . 57 8.1 Paul trap ...... 111 8.1.1 The quadrupolar potential of the trap 111 Chapter 4 (Two-level) Atom-field interaction 59 4.1 Semiclassical interaction 59 8.1.2 Oscillating potential of the trap ... . 114 4.2 Quantum interaction . . 62 8.1.3 Motion in the Paul trap ...... 115 8.1.4 Approximated solution to the Mathieu equation 116 4.2.1 Atomic inversion 63 120 4.3 Dispersive interaction . 65 8.2 Ion-laser interaction in a trap with a frequency independent of time 8.2.1 Interaction out of resonance and low intensity ... 121 4.4 Mixing classical and quantum interactions 66 8.3 Ion-laser interaction in a trap with a frequency dependent of time 126 4.5 Slow atom interacting with a quantized field . 69 8.3.1 Linearization of the system ...... 128 Chapter 5 A real cavity: Master equation 73 8.4 Adding vibrational quanta ...... 129 5.1 Cavity losses at zero temperature . 73 8.5 Filtering specific superpositions of number states 133 5.1.1 Coherent states . 75 Chapter 9 Nonlinear coherent states for the Susskind-Glogower 5.1.2 Number states ...... 75 operators 137 5.1.3 Cat states ...... 76 9.1 Approximated displacement operator ..... 139 5.2 Master equation at finite temperature 77 9.2 Exact solution for the displacement operator 140 Chapter 6 Pure states and statistical mixtures 81 9.3 Susskind-Glogower coherent states analysis 142 6.1 Entropy ...... 81 9.3.1 The Husimi Q-function ... 143 6.2 Purity ...... 82 9.3.2 Photon number distribution . 143 6.3 Entropy and purity in the atom-field interaction 83 9.3.3 Mandel Q-parameter ..... 145 6.4 Some properties of reduced density matrices . . . 84 9.4 Eigenfunctions of the Susskind-Glogower Hamiltonian 147 1 9.4.1 Solution for IO) as initial condition . 148 6.4.1 Proving p'JJ+ = TrA{fJ(t)pA(t)} by induction 87 9.4.2 Solution for lm) as initial condition 151 Contents
9.5 Time-dependent Susskind-Glogower coherent states analysis 154 9.5.1 Q function ...... 155 Chapter 1 9.5.2 Photon number distribution . 155 9.5.3 Mandel Q-parameter . 155 Operator algebra and the harmonic 9.6 Classical quantum analogies .. 160 oscillator Appendix A Master equation 163 A.1 Kerr medium ...... 163 A.2 Master equation describing phase sensitive processes 164
Appendix B Methods to solve the Jaynes-Cummings model 167 B.1 A naive method ...... 167 B.2 A traditional method ...... : : : : : : : : ...... 168 1.1 Introduction Appendix C Interaction of quantized fields 169 ' C.1 Two fields interacting: beam splitters 169 In this chapter, we revise briefly the Dirac notation, some of the algebra that C.2 Generalization ton modes .. . 171 will be used throughout the book, and introduce the harmonic oscillator and C.3 A particular interaction ... . 173 some states in which it may be found. In this chapter, and in the rest of the C.4 Coherent states as initial fields 174 book, we will set ti = 1. In Dirac notation, we denote wavefunctions 'ljJ by means of "kets" 1'1/J). For Appendix D Quantum phase 175 instance, an eigenfunction of the harmonic oscillator D.1 Turski's operator ... . 175 D.2 A formalism for phase ... . 176 (1.1) D.2.1 Coherent states ... . 177 D.3 Radially integrated Wigner function 178 is represented by the ket In), with n = 0, 1, 2, .... In quantum mechanics, these Appendix E Sums of the Bessel functions of the first kind of states are called number states or Fock states. integer order 181 Any function can be expanded in terms of eigenfunctions of the harmonic oscil- lator; or in other words, Bibliography 185 Index 189 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 Operator algebra and the harmonic oscillator Introduction where the orthonormalization relation that has as solution
(1.5) (q'lq) = J(q'- q). (1.14)
has been used. We then can express the completeness relation as 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 1 =I: dqlq)(ql, (1.15) example, with the function eipq I V27f and the corresponding ket IP). First ~ate that such that
00 (PIP') = 17r -oo dxe-i(p-p')q = J(p _ p'), 11/J) = 111/J) =I: dqlq)(qi1/J) =I: dq1j;(q)lq), (1.16) 2 1 (1.6)
so that where 1/;(q) = (qi1/J) = (qi1/J). The completeness relation serves us, among other things, to calculate averages, e•P' q ipq 1oo for instance J27i = -oo dpJ(p- p') ~' (1.7)
or, in bra-ket notation we have (1.17)
or finally, IP') = j_: dpJ(p- p')lp) =I: dp(plp')lp); (1.8) rearranging terms we have (1.18)
Note that in the above equation we are simply adding "diagonal" elements; i.e., IP') =(I: dplp)(pl) IP') =lip'); (1.9) 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) i.e., we have what is called the completeness relation
(1/liAI1/l) = l:)ni1/J)(1/JIA1n). (1.19) I: dplp)(pl = 1. (1.10) n=O
Finally, note that the function eipq I V27f is an eigenfunction of the operator -i..!i with eigenvalue p. dq The main task in non-relativistic quantum mechanics is to solve the Schrodinger For position, an "eigenket" of ij is equation
filq) = qlq), (1.11) di1/J) = -ifii1/J). (1.20) dt and an "eigenbra" is In order to achieve this, sometimes it is convenient to perform unitary trans formations, such that we may simplify the problem. For instance, we may do (q'lq' = (q'lfJ.. (1.12) 11/J) = Tl¢), with T = e-i~A and A a Hermitian time independent operator, and We therefore find obtain for 1¢) a new equation
(q'lq)(q'- q) = 0, (1.13) d~~) = -ifirl¢), (1.21) Operator algebra and the harmonic oscillator Baker-Hausdorff formula
where the transformed Hamiltonian is the "simplicity" of the commutation relation of the operators involved, namely, position and momentum. In order to accomplish such factorization, we use what (1.22) is known in the literature as the Baker-Hausdorff formula, that establishes that Developing the exponentiaJs in Taylor series and grouping terms we obtain if [A, [A, B]] = [B, [A, B]] = o then
ei~Afie-i~A = H +i~[A,H] + (i;t [A, [A,fi]] + (i;t [A, [A, [A,fi]] + ... , (1.23) (1.28)
an expression valid for any two operators fi and A, and sometimes named so that Equation (1.25) may be written as Hadamard lemma. (1.29) 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 In order to prove the Baker-Hausdorff's formula, we write that means that F(>.. ) = e>..(A+B) = ef(>..) eg(>..)Aeh(>..)B, (1.30) (1.24) and derive F(>.. ) with respect to A. From the first equality we obtain · A more general operator e-i(qof>-pa{j) produces displacements in both q and p simultaneously; i.e., d~iA) =(A+ B)F(A), (1.31) f(q,p) = ei(qof>-po{j) f(q + qo,fJ + Po)e-i(qof>-po{j). (1.25) and from the second equality
1.2 von Neumann equation (1.32)
An equation that we will use frequently is the so called von Neumann equation, is obtained, where we have introduced in the former equation a unity operator which is another form of the Schri:idinger equation, but will be useful when, in the form e-g(>..)Aeg(>..)A. for instance, the environment is taken into account. It is obtained from the We use now (1.23) to calculate Schri:idinger Equation (1.20) multiplying it by the bra (~I by the right (1.33) d~~) (~I= -ifii~)(~I, (1.26) so Equation (1.32) may be rewritten as and adding it with the adjoint of Equation (1.20) multiplied by the ket I~) by the left, so that d~iA) = {t'(A) + g'(A)A + h'(A) (B + g[A,Bl)} F(A). (1.34)
dp , A A dt = -z[H,p], (1.27) Equating (1.34) with (1.31), we obtain the system of differential equations where pis the density matrix, defined simply as the ket-bra operator p = I~) (~I· g'(A) = 1, h'(A) = 1, h'(A)g(A)[A,B] + j'(A) = o, (1.35)
that has as solution 1.3 Baker-Hausdorff formula g(A) =A+ g(O), Equation (1.25) has complicated terms, in the sense that it has exponentials of h(A) =A+ h(O), (1.36) the sum of non-commuting operators. In this particular case, the exponentials A2 ) A A involved may be easily factorized in the product of three exponentials because of j(A) =- ( 2 + g(O)A [A, B] + f(O). Operator algebra and the harmonic oscillator Quant1tm mechanical harmonic oscillator
By evaluating (1.30) in zero, we obtain the initial conditions f(O) = 0, g(O) = 0 Note that the above ordering of operators is arbitrary and we could have tried and h(O) = 0, that substituted in (1.36) give us finally the Baker-Hausdorff's a different one. Deriving with respect to time, we have formula 2 au(t) = _ ifJ + rP u(t) 8t 2
=- i j(t) p2 U(t) 2 (1.42) .g(t) m2.~2 " ~, _·W("F') -i~'2 1.4 Quantum mechanical harmonic oscillator - 7--e-' 2 P (qp + pq)e ' 2 qp pq e 2 q 2
.h(t) _ m2 ,2 _ W("+") ,2 _ ~ ,2 The Hamiltonian for the harmonic oscillator is written as (for simplicity, we 7-2-e , 2 P e , 2 qp pq q e ' 2 q . consider unity mass, and we set w = 1) Using that
'2 -2g (1.37) e-iJJ,jl([]P+f!ii)(;_2eiJJ,jl({]P+fi[]) q e ' e-i'~'lfi2 (;_2ei'~')fi2 q2- j(t)((j_p + p(j_) + f2(t)p2' We write the formal solution of the Schrodinger equation (1.20) as 2 e-i t~t) P2 ((j_p + pq)ei t~t) P2 ((j_p + p(j_)- 2f(t)p ,
(1.38) we rewrite the second part of (1.42) as 8U(t) We need to factorize the above exponential; however, the Baker-Hausdorff's 8t formula can not be used, because the hypotheses of the theorem are not satisfied. Actually, we first have the commutator
(1.39) and by equating it with the first part of (1.42), we end up with the system of where we have used that [AB, C] = A[B, C] +[A, C]B, and then we commute it equations 2 with ij2 and p . The commutators are h i(t)- P(t) (1.43) (1.40) g(t) j(t),
However, we note that there are no other operators resulting but the operators with initial conditions j(O) = g(O) = h(O). The solutions are then in Equations (1.39) and (1.40); in other words, the operators that arise are 2 2 (1.44) proportional to (jp + p(j, (}_ and p . We therefore may try the assumption (an j(t) = h(t) tan(t), g(t) = -ln(cos(t)). educated guess, like this one, is normally called an "ansatz") 1.4.1 Ladder operators U(t) e-i~(fJ2H2) (1.41) Another form to solve the harmonic oscillator is via annihilation and creation e-i t~t) P2 e-i g~t) ([]'P+M) e-i h~t) q2. operators. We define the so-called ladder operators a and at, also known as Operator algebra and the harmonic oscillator Quantum mechanical harmonic oscillator
annihilation and creation operators, as Number states form an orthonormal basis, that is
(1.45) (nlm) = t5nm· (1.54) The unity operator may be written in terms of number states as These operators obey the commutation relation
(1.46) L ln)(nl = 1. (1.55) n=O and the Hamiltonian (1.37) may be written in terms of ladder operators as By using the completeness relation it is possible to express any operator in terms (1.47) of number states; for instance, the annihilation operator
00 that, by defining the number operator fi = at a, may be written in the form a= La In) (nl = L v'n +lin) (n + ll' (1.56) n=O n=O ~). (1.48) 2 and the creation operator
1.4.2 Fock states at= L a.t In) (nl = L v'n +lin+ 1) (nl. (1.57) Eigenstates of (1.48) are the Fock or number states, already introduced, In) with n=O n=O eigenvalues w(n +~);i.e., The number operator is simply
(1.49) fi = L n ln)(nl . (1.58) n=O where n is a non-negative integer, and it is identified with the number of exci tations (photons in the case of electromagnetic field, see Chapter 3). Number states have no uncertainty in intensity; i.e., Fock states are therefore eigenstates of the number operator (b..fi) = J(nln2 ln)- (nlnln) 2 = 0. (1.59) ataln) =nln). (1.50) Averages for position and momentum are null for number states, (nlqln) = The vacuum state of the harmonic oscillator is defined as (nlpln) = 0; however, their uncertainties
a10) = 0. (1.51) ~ (b..q) = J(nlq2 ln) = V~' (1.60) In fact the creation and annihilation operators act on the number states in the following form and ~ v(2n+l)w at In) = v'Ti"TI In+ 1), a In) = In- 1). (1.52) (b..p) = V (nlrln) = - -, (1.61) vn 2 Some of the most important properties of number states are given below. are such that they are minimized only for the vacuum. Any state vector In) may be obtained from the vacuum IO) via the creation The equation operator, (1.62) (1.53) gives the probability to haven number of excitations (photons) in the state 11{1). 10 Operator algebra and the harmonic oscillator Quantum mechanical harmonic oscillator 11
1.4.3 Coherent states that, by developing the exponential in Taylor series, gives us
00 We may build arbitrary superpositions of number states to obtain new states; lal 2 an in particular, we can construct coherent states of the harmonic oscillator [Su Ia) = e---r L- In). (1.72) darshan 1963; Glauber 1963b]. They may be obtained in different forms: n=O Vnl In this equation, the value lal 2 represents the average value of excitations n in (1) as eigenstates of the annihilation operator; the coherent state Ia): (2) as states whose averages follow the classical trajectories of ij, p and fi [Meystre 1990]; (1.73) (3) as a displacement of the vacuum. The solution for the harmonic oscillator Hamiltonian for an initial coherent state Let us define the harmonic oscillator coherent states as eigenstates of the anni is given in the following very simple form hilation operator; that means, that if we call them Ia), then (1.74) ala)= ala)' (1.63) where we have used the Hamiltonian (1.49). Introducing e-ifiwt into the sum, and as a is a non-Hermitian operator its eigenvalues, a, are complex. and using the fact that the states lk) are eigenstates of the number operator n, Other properties of these states may be obtained by using the Glauber displace we have ment operator [Glauber 1963b] 00 ake-ikwt ·wt . ---lk) e-'Tiae-•wt); (1.75) (1.64) 11/l(t)) = L = n=O Jkf Using Equation (1.25) and the definition (1.45) of the ladder operators, we write i.e., a coherent state that rotates with the harmonic oscillator frequency. this operator as Coherent states are eigenstates of the annihilation operator, but How does (1.65) 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 with coherent states, we have
(1.66) (1.76)
The displacement operator has the following properties and we note that fJt(a) tJ- 1 (a) = D( -a), (1.67) (1.77) D(a + j3) D(a)D(j3)e-ilm(af3*)' (1.68) fJt(a)&D(a) a+a, (1.69) so fJt(a)at D(a) at +a*. (1.70) (1.78) Coherent states, Ia), may also be generated by application of the displacement operator on the vacuum, Analogously
(1.79) (1.71) Ia) (ala= (a~* +a) Ia) (a I. 13 12 Operator algebra and the harmonic oscillator Quantum mechanical harmonic oscillator
This expression is of interest particularly when going from a Master equation The excitation number for the coherent states is given by the Poissonian (see Chapter 6) to a Fokker-Planck equation. We will use this expression next distribution chapter where we relate quasiprobabilities in a differential form. (1.85) Coherent states form an over-complete set of states. The identity operator is written in terms of coherent states as 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.
Lln)(nl, (1.80) 0.06
0.05 where we have set a = rei&. By using the above completeness relation, the annihilation operator may be expressed as P(n)
(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 Fig. 1.1 Photon number distribution for the coherent state with a= 6. It may be observed states, because 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
(1.83) (1.86) 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 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., (1.84) (1.87) 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 where the subscript c means classical variables. This is why coherent states are of such operators in the next section and its importance in next chapter on called quasi-classical states, they are a reference to other states of the harmonic quasiprobabilities. oscillator, and they are also called the standard quantum limit. 14 Operator algebra and the harmonic oscillator Quantum mechanical harmonic oscillator 15
The uncertainties for position and momentum, for the coherent states, may be found to be
(!:,.q)= -, (!:,.p)A = V2fW (1.88) 'If2w P(n) 0.04
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 P(n) the displaced number states as la,n), we have then
la,n) = D(a)ln). (1.90)
These states are orthonormal, therefore (a, mla, n) Jm,n and they form a complete basis of the space of states, hence
L la,n)(a,nl = 1. (1.91) n=D Their excitation number is given by the distribution
P(n) o.03 (1.92) and
n ?_ m (1.93)
Fig. 1.2 Photon number distribution for displaced number states for (a) a= 2, n = 20, (b) In Figure 1.2, we plot several distributions of these states for different values of a= 4, n = 30 and (c) a= 5, n = 1. a and n. These states will be of importance in next chapter where we talk about quasiprobability distribution functions.
be defined as 1.4.5 Phase states
There exist several kind of states that are not normalized, such as position (1.94) eigenstates, momentum eigenstates and also phase states. The later ones may 16 Operator algebra and the harmonic oscillator Ordering of ladder operators 17
These states are eigenstates of the so-called Susskind-Glogower (phase) operator 1.5.1 Normal ordering [Susskind 1964] One may use the commutation relations of the annihilation and creation opera A 1 V=---a· (1.95) tors to obtain the powers of n in normal order. For instance, we can express nk vn+1 ' in normal order, fork= 2, as i.e., (1.99) VI¢)= ei¢1¢). (1.96)
We can write the unity operator also in terms of phase states as fork= 3, as 1: 1¢) (¢1d¢ = 1, (1.97) (1.100) therefore we can use them to perform traces. and fork= 4, as The Susskind-Glogower operator can also be written in the Fock basis as (1.101)
V = LIn) (n + 11, (1.98) n=O 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 and then the following properties are easily seen; first, that vvt = 1 but, second, equations, we have used repeatedly the commutator [a, a,t] 1. that vtv -=1- 1. In Figure 1.3, we generate a table of such coefficients using the Mathematica© We will make use of the Susskind-Glogower operators in Chapter 4, where we program, and in the Table 1.1, we present the first Stirling numbers of the 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 Susskind Table 1.1 Stirlin> numbers of the second km d Glogower nonlinear coherent states. In Appendix D, we study with some more k / m 0 1 2 3 4 5 6 7 8 9 detail the quantum phase. 0 1 1 0 1 2 0 1 1 1.5 Ordering of ladder operators 3 0 1 3 1 4 0 1 7 6 1 In some problems in quantum mechanics it is needed to calculate functions of 5 0 1 15 25 10 1 the number operator n. As we will use the Taylor expansion of those functions, 6 0 1 31 90 65 15 1 in what follows we obtain expressions for nk. We remember that the number 7 0 1 63 301 350 140 21 1 operator is defined as n = at a, so an order issue between a and at arises. If a 8 0 1 127 966 1,701 1,050 266 28 1 is always to the right of at it is called normal order' and if a is always to the 9 0 1 255 3,025 7,770 6,951 2,646 462 36 1 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 second kind, We infer that the coefficients in the above equations are precisely we shall find for nk are sum of coefficients multiplying normal and anti-normal these numbers; i.e., we obtain 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 k [Louisell 1973] for the exponential of the number operator. nk = L Skml[at]mam, (1.102) m=O 19 Ordering of ladder operators 18 Operator algebra and the harmonic oscillator
and inserting (1.102) in this equation, we obtain SetAttributes[prod, {Flat, 0 (1.105) ~~~~~:=: ~:PI~~ c~ pr~~r:€i;::~~~}J /@ b prod[A,Adl := 1 + prod[Ad,A] , s[n_lnteger?Positive] ·- {n}]) Table[{n,s[n]}, {n, }] ·- P~~~le~;.:latten[Table[{Ad,A}, 3 11 Because Skm) = 0 form > k, we can take the second sum in (1.105) to infinite 1 prod[Ad, A] 2 prod[Ad, Al + prod[Ad, Ad, A, A] and interchange the sums, to have 3 prod[Ad, A] + (1.106) 3 prod[Ad, Ad, A, A] + prod[Ad, Ad, Ad, A, A, A]
AdA[n_j:=prod @@ Join[Table[Ad,{n}], Table[A,{n}]] AdA[ 5] For the same reason stated above, we may start the second sum at k = m, and prod[Ad, Ad, Ad, Ad, Ad, A, A, A, A, we can write Al (1.107) p[ a ___ ]: =x"Lengt h[ {a} 1 :~~S];=s[n] /. prod -> p
8 4 16 By noting that l + 1 :iJ x /966 x + 11b~ x + Vo5o ~ + 266 x + 28x +x Nn f(x) - ~ jCkl(x) sCml (1.108) c[n_j:=Coefficientlist[sx[n] /. Table[c[n], {n,S}] //TableForm x ->Sqrt[y],y] m! - ~ k! k ' k=m 0 1 0 1 1 where l'J. is the difference operator, defined as [Abramowitz, 1972] 0 1 3 m I 0 1 7 6 1 (1.109) 0 1 15 25 10 6.m j(x) = ~(-1)m-k k!(:~ k)! f(x + k), 0 1 31 90 65 15 0 1 63 301 350 140 21 1 0 1 127 966 1701 1050 266 28 1 we may write (1.107) as (1.110) Fig. 1.3 A program wntten m Mathematica© to find h . number operator in normal order. t e coefficients for the powers of the
where : fi : stands for normal order. with [Abramowitz, 1972] 1.5.1.1 Lemma 1 If we choose the function j(fi) = exp( -ryfi), we have that (1.103) rn I (1.111) A mf(O) - l)m-k m. -"(k u -~-""'c k!(m-k)!e ' We now write a function f of fi in a Taylor series as k=O and then we obtain the well-known lemma [Louisell1973] (1.112) (1.104) 21 Ordering of ladder operators 20 Operator algebra and the harrnonic oscillator
1.5.2 Anti-normal ordering 1.5.3 Coherent states. A t' (1 119) to find averages for coherent states, \a) = D( a) \0)' Following the procedure introduced in the former section, we can write nk in Let us use E qua ton . d \0) . th A ( ) _ aiit -a*ii is the so-called displacement operator an lS e anti-normal order as where D a - e k vacuum state. We have nk = (-1)k L(-1)mst~t1lam[at]m, (1.113) ~ (1- e~')m Am[Af]m\ ) (1120) m=O (a\e-~'n\a) = e~'(a\ u --m-!-a a a ' . m=O and a function of the number operator as by using that (1.121) (1.114) (a\iim[af]m\a) (0\(ii + a)m(at + a*)m\0) m 2k ( m.I )2 m- k' The second sum differs from (1.108) in the extra (-1)k and the parameters L \a\ (m- k)!k! ( )., k=O 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] we may write 2 (1.122) (1.115) (a\e-~'n\a) = e~' f (1- e~')rn Lm( -\a\ ), m=O to write where Lm(x) are the Laguerre polynomials of order m. We ca~ finally write a J(n) (1.116) closed expression for the sum above [Gradshteyn 1980], to obtam the expected m=O result for coherent states oo JCk) ( _ ) oo (kJ ( _ ) (1.123) (m + 1) """ u - 0 sCm+ I) + """ f u- 0 sCm) l [ ~ k! k ~ k! k ' k=m k=rn so we can use again Equation (1.108) to finally write 1.5.4 Fock states. For Fock or number states we obtain (1.117) (1.124) 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) = Rearranging the sum above with k = n + m, we have e-"Yx and f(u) = e"Yu. Therefore , = I' k-n k! (1.125) (n\e-~'n\n) = e~' L(1- e ) n!(k- n)!' (1.118) k=n 1 . [ . ] ""= k-n k! - (1 _ x)-n- has the closed whtch as Abramowitz, 1972 LA=n x nl(k-n)l - such that we can obtain the exponential of the number operator in anti-normal expression order (lemma) as (1.126)
(1.119) 22 Operator algebra and the harmonic oscillator
Chapter 2 Quasiprobability distribution functions
2.1 Introduction
Quasiprobability distribution functions are of great importance in quantum me chanics, among other things, because they allow to have classical views of quan tum 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 mov ing 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) = Jo(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 25 Wigner functi.on 24 Quasiproba.bility d1:stribution functions or where o: = (q + ip)j.J2, and C(;S') is given in terms of annihilation and creation operators by P(q,p) = 4:2 e-iupeivq P(Q, P)eiuP e-ivQdQdP} dudv. (2.3) (2.11) J {! C(f3) = Tr{pD(IJ)} = Tr{pexp(/Jat- /)*a)}; We can see this last equation as the two d. . . term in the curly brackets Tl t . . Imenswnal Founer transform of the the function C(IJ) above has many names: ambiguity function, radar function stood l . le erm mside the curly brackets can b and, more common in quantum optics, characteristic function. as t le phase-space average of tl f' . . ' . ' e under- le unctwn ewp ->1!q B . In the position eigenstates basis, we can write the characteristic function as correspondence principle and recalli th t . . y applymg the obtained in the form ng a averages m quantum mechanics are (2.12) C(Q, P) =I dq(q + Q/2\p\q- Q j2)e-iPq, (A.) = Tr{pA.}, (2.4) and in the momentum eigenstates basis as we write the phase-space average between the curly brackets as (2.13) C(Q, P) I dp(p + P/2\p\p- Pj2)eipQ. (ezup-ivq) = Tr{peiuji-ivq}. (2.5) We prove now, following [Chountasis 1998], that for a pure state, the square of The trace in this last expression rna be r . . coherent states phase states .t. y . eahzed m several basis: Fock states the absolute value of the characteristic function has the interesting property of l . . ' ' posi lOll eigenstates etc F . l" . ' t 18 position eigenstates basis, and we write ' . or Simp ICity, we choose being its own two-dimensional Fourier transform; i.e., 2 (2.14) 2 dQ'dP'\C(Q',P')\ exp[i(P'Q- Q'P)] = 27r\C(Q,P)\ . 'D:{peiup-ivq} = Jdq(q\peiup-i1!q\q), (2.6) JJ Actually, using Equations (2.12) and (2.13), we get thateiufl\q) by = using \q + Baku) er- H abus d orff .f. ormula (1.28), that e-ivqlq) = e-i"qlq) and that (2.15) ' may e rewntten as ' C(Q', P') = Jdqlj;(q + Q')if;*(q)e_;p'q Tr{peiup-i"q} _-e -iuv/2 1dqe-wq(q\plq+u). (2.7) and (2.16) Doing the change of variable q = x- u/2, we get C*(Q',P') = Jdplj;*(p)lj;(p-P')e-ivQ'. Tr{peiujJ-ivq} = dxeivx (x - ~I AI u) . 1 2 p X + 2 ' (2.8) Inserting these equations into the left hand of Equation (2.14), and using the by mtroducing this expression into the . Fourier transforms integrating first with quantum mechamcal version of (2.3), and respect to v and after respect to x, we get finally ~ j dQ'Ij;(q + Q') exp[-i(p+ P)Q'] = if;(p+ P)exp[iq(p+ P)]
P(q,p) W(q,p) = _!__ dueiup( - = 27f 1 q +~~AI2 p q ~~2 . (2.9) and In 1932 , Wiu1oi er m· t ro d uced [Wigner 1932] th" f . ~ j dP' lj;(p- P') exp[-i(q- Q)P'] = if;(q- Q) exp[-ip(q- Q)], his distribution function It t . IS unctwn W (q, P), known now as . con ams complete i f. t. b the system, norma wn a out the state of 11f'l). we get the right hand side of Equation (2.14). From Equation (2.3), we see that the Wi n . . . ten also as g er distnbutwn function may be writ- The Wigner function is a quasiprobability distribution that allows the visu alization of states in phase space. In Figures 2.1-3, we show how different states W(a) = ~ 1exp(a;3* _ a*;3)C(;3)d2;3, (2.10) of the quantized field look like. It may be seen that in Figures 2.1 (number 26 Quasiprobability distribution functions 0.3-. states) and 2.3 (displaced number states), the Wigner function may take neg ative 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].
-3 Wrq.p! Wiq.p}
Fig. 2.2 Wigner function for the coherent state of amplitude a= 2.
n=O n=l
W(q,pJ
-3 p -l
n=2 n=7 q
Fig. 2.3 Wigner function for the displaced number state for the parameters a = 4 and n = 1. Fig. 2.1 Wigner function for the first number states. Wigner junction 29 28 Quasiprobability distribution functions
2.2.1 Properties of the Wigner function between the density matrix p and the operator ¢. From the definition of the Wigner function, expression (2.9), and from (2.22), It is easy to see that if we integrate (2.9) in p, we obtain we can write
W(q,p)dp 1 j j , .( ') x x x' A x' I W(q,p)W¢(q,p) = (27r)2 dxdx e" x+x P(q- 2/,8/q + 2)(q- 2/¢/q + 2). (2.24) (2.17) Integrating now, with respect to q and p, the above product of \;vigner functions, so that we get
I W(q,p)dp = '1/J(q)'ljJ*(q) = P(q). (2.18) I I W(q,p)W¢(q,p)dqdp =
1 x x x' A x' To integrate the Wigner function in q, we do a similar analysis as the one we = --I I I I dqdpdxdx'et(x+x. ,)P(q- -/,8/q + -)(q- -/¢/q + -). (27r) 2 2 2 2 2 did t~ obtain it at the beginning of this chapter, but with the probability as a (2.25) functiOn only of p, instead of the combined probability. First, in analogy with (2.1), we write The integral in p gives a delta function o(x + x'), that may be readily integrated in x', to yield P(p) o(p- P)P(P)dP e-iu(p-P)P(P)dudP, (2.19) =I =-I; I I X A X X A X W(q,p)W¢(q,p)dqdp = dqdx(q- /p/q + 2)(q + 2/¢/q- 2 ). (2.26) 2 id~ntifying the integral with respect to P with the average value of eiup, and If If usmg the correspondence principle Finally we make the change of variables y = q- x/2 and z = q + x/2, to arrive at the result we are searching P() 1 1 // . . . P = 7f I etuPTr{pe-wp}du. .. = 7f eiup(q/e-i¥ pe-i¥ /q)dudq, (2.20) 2 2 JJ W(q,p)W¢(q,p)dqdp = JJ dydz(y/,8/z)(z/(/;/y) = Tr{,D(/;}. (2.27) that just as we did before, may be re-written as
2.2.3 Symmetric averages P(p) = 2~ I I eiup(q + ~/p/q- ~)dudq, (2.21) The Wigner function may be used to obtain averages of symmetric functions of that is nothing but the Wigner function integrated in position. creation and annihilation operators. If we consider the characteristic function (2.11), 2.2.2 Obtaining expectation values from the Wigner function C(A/3) = Tr{,Dexp(A[/3at- j3*ii])}, (2.28) We can g~neralize the Wigner function for the density operator to any given operator ¢, as follows we take its derivative with respect to A and evaluate it at A = 0, we have (2.29) W( ) UA u dC(A/3) I >..=0 - Tr{ pA(/3At a - /3* aA)} . ¢ q,p = 27f1/ duewP(q. + 2/¢/q- 2). (2.22) ---;v::-
We derive now the following overlap formula By repeating the procedure k times, we get (2.30) Tr{p¢} =I dqdpW(q,p)W¢(q,p), (2.23) 31 30 Quasiprobability distribution functions Wigner function
On the other hand, from (2.10) we can write the characteristic function as the and which may be realized in any basis. In particular, we can use the Fock basis Fourier transform of the Wigner function to finally obtain the following series representation of the Wigner function
2 C(A/3) =:hI exp[A(a*f3- af3*)]W(a)d a. (2.31) (2.38) Differentiating the above expression k times with respect to A, and evaluating at A = 0, we obtain Recall that the states iJ (a) I k) are the displaced number states introduced in Chapter 1. dkC(A/3) 1 2 ~1>-=0 = ;z I (a*f3- af3*)kW(a)d a. (2.32) 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 Equating (2.32) with (2.30), we finally get 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. Tr{p(f3at- (3*a)k} =~I (a* f3- af3*)kW(a)d2a, (2.33) The series representation of the Wigner function, Equation (2.38), can be that shows that the Wigner function may be used to obtain averages of sym used to find a closed form for the Wigner function of the coherent states. For metric functions of creation and annihilation operators. 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 2.2.4 Series representation of the Wigner function
We can obtain a series representation (non-integral) of the Wigner function by W(q,p) = ~ f)-1)k(kliJt(a)D(f3)IO)(OiiJt(f3)D(a)ik). (2.39) 1f making y = uj2 in the definition k=O Using the Baker-Hausdorff formula 1.28 on page 5, we can easily prove that W(q,p) =_21 I dueiup(q + 2:1,8/q- 2:), 1f 2 2
, t , (a* f3 - a/3* ) A D (a)D(/3) = exp D(/3- a), (2.40) to obtain 2
(2.34) thus
A t , (a* f3 - a/3* ) Remembering that D (a)D(/3)10) = exp l/3- a), (2.41) 2 e-ipq/y) = e-ipy/y), and analogously we can further put the exponential term in the integral above inside the bracket,
A , A ( af3* -a* /3) (O/DT(f3)D(a) = exp (/3- a/, (2.42) W(q,p) =~I dy(-y/eiP so that Introducing now the parity operator fi = ( -1) n, we obtain iJt(a)D(f3)IO)(OiiJt(f3)D(a) = l/3- a)(/3- al, (2.43) W(q,p) = ~ ./ dy(y/fieipqe-iqflpeiqfle-ipq/y), (2.36) and that is nothing but the trace of the operator 1 00 W(q,p) =;: ~(-1)k(klf3- a)(/3- alk), (2.44) 2:.( -1)n iJt (a)pD(a), (2.37) 1f k=O I, !lj 32 I Q1wsiprobability distribntion fnnctions Husimi Q-fnnction 33 which using the fact that of the characteristic function. We obtain 2 (2.45) C(a) = JP(f3)Tr{lf3) ((31 exp(aO,t- a*O,)}d (3. (2.52) can be written as Using the Baker-Hausdorff formula, we get 2 1 00 lf3 12k 2 2 2 C(a)el<>l / = P((3)Tr{lf3) (f31 eaat e-a*a}d (3, (2.53) W(q,p) =:;;: 2)-1l~e-li3-al , (2.46) J k=O ' or writing explicitly the trace, and finally, we get the expression we were looking for (2.54) (2.47) After a Fourier transform, we obtain P(a) from the characteristic function We leave to the reader, as an exercise, to show that the Wigner function for the number states In) is [Gerry 2005] P(a) = ~ Jexp(af3*- a* f3)Tr{pexp(f30,t) exp( -(3*0,)}d2 (3. (2.55) (2.48) From Equation (2.51), we can see that Using Equations (2.40) and (2.48), it is not very difficult to show that the Tr{[dtta,kp} = JP(a)Tr{[O,t]na,k Ia) (al}d2 a Wigner function for the displaced number states l/3, n) is given by the expression j P(a)Tr{ak Ia) (al [att}d2 a (2.49) JP(a)ak[a*td 2 a, (2.56) 2.3 Glauber-Sudarshan P-function that indicates that the Glauber-Sudarshan ?-function may be used to calculate averages of normally ordered products of creation and annihilation operators. Although coherent states form an overcomplete basis, they may be used to rep resent states of the harmonic oscillator. The representation 2.4 Husimi Q-function (2.50) The Q or Husimi function is usually expressed as the coherent state expectation involves off-diagonal elements (alp lf3), and two integrations in phase space. value of the density operator; i.e., The next diagonal representation, introduced independently by Glauber and Sudarshan, Q(a) =!_(a lfJI a), (2.57) 7r and has as an alternative form the integral representation (2.51) Q(a) = ~ j exp(a(3* -a*(3)Tr{pexp(-f3*&)exp((3at)}d2 (3, (2.58) involves only one integration. In order to obtain an explicit expression for the Glauber-Sudarshan ?-function To show the relation between the expressions (2.57) and (2.58), we use again we introduce (2.51) in the definition ' the definition of the characteristic function as the Fourier transform of the prob ability density function, we write the Glauber displacement operator in anti C(f3) = Tr{pexp(f3a.t- (3*0,)}, normal order and we use the commutative properties of the trace, namely that 35 34 Quasiprobability distribution functions Relations between quasiprobabilities Tr{ABC} = Tr{CAB}. It is important to remark, that because pis a positive operator, we always have Q(a) 2:0. L Let us note that I. 2 ~Tr ak[a*tla)(ald ap} {! Q(q,p) Tr { ~ j d2ala)(al x [attpak}. (2 ..59) Recalling that ; J d2ala) (a! = 1 and using properties of trace, we obtain 0 10 which means that the Q-function may be used to calculate averages of creation and annihilation operators in anti-normal order. q 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 Fig. 2.4 Husimi Q-function for the number state state for the parameter n = 5. Q(a) = lal2n exp (-l::f) (2.61) 1rn! 2 where C(/3, 8 ) is the characteristic function of order s; i.e., 2 is obtained, and for the coherent state l/3), we get C(/3, s) = Tr{D(f3)P} exp(s l/31 /2), (2.64) where 8 is the parameter that defines which is the function we are lookin_g at. Q(a) = ~ exp ( -lal2- l/312 + af3* + f3a*). (2.62) For = 1 the Glauber-Sudarshan P-function is obtained, for s = 0 the W1gner 1f 8 function, and for s = -1 the Husimi Q-function. . In Figure 2.4, we plot the Husimi Q-function for the number state In), expression · (2 63) we can establish some relations between the quaslprob- F rom expresswn . , . . (2.61); and in Figure 2.5, we plot the Husimi Q-function for the coherent state abilities in phase-space. If we takes = -1 in Equatwns (2.63) and (2.64), we l/3), expression (2.62). can write the Husimi Q-function as 2 Q(a) = G(f3) exp(af3*- a* f3)d f3 (2.65) 2.5 Relations between quasiprobabilities j 2.5.1 Differential forms where 2 (2.66) It is possible to group the Wigner, the Glauber-Sudarshan and the Husimi Q G(/3) = ~Tr{D(f3)p}exp(- 1/31 /2). functions in one parametric form given by the expression 1f Takincr now 8 = 0 in expressions (2.63) and (2.64), and introducing the ide~tity F(a,s) = ~ j C(f3,s)exp(af3* -a*f3)d2 f3, (2.63) opera~or I in an obvious way, we get the following formula for the W1gncr 36 Q1wsiprobability distribution functions The Wigner function as a tool to calculate divergent {or not) series 37 0. or, finally W(a) = exp ----1 a a ) Q(a). (2.71) ( 2 aa aa* 0. {!(q.p) 2.5.2 Integral forms 0. 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 Q function from the ?-function as Q(a) =~(a 1.01 a)=~ I P(/3) (a 1/3) (/31 a) d2 {3. (2.72) 1f 1f Fig. 2.5 Husimi Q-function for the coherent state for the parameter f3 = 2. Using the properties of the coherent states, this relation can be simplified to function (2.73) W(a) =I G(/3) exp(a/3*- a* /3) exp(l/312 /2)d2/3, (2.67) The following relation being G(/3) the same function as in Equation (2.66). (2.74) ~n the expression (2.67) for the Wigner function, we expand the term exp(l/31 2 /2) m Taylor series to obtain between the Wigner function and the Q-function can be obtained with a similar procedure. 00 2-n I W(a) = ~--;! G(/3) exp(a/3*- a* /3) l/312n d2f3_ (2.68) 2.6 The Wigner function as a tool to calculate divergent (or Considering the equality not) series One can use the Wigner function in the series representation, that we derived in (2.69) Subsection 2.2.4, to find the Wigner function associated with any operator. We we can cast Equation (2.68) into start with the generalization of the series representation, Equation (2.38), to an arbitrary operator ¢. We write then X 2-n ( a a )n W(a) = L- --- Q(a) (2.70) W¢(q,p) = 2 2:(-l)k(klbt(a)(/JD(a)lk). (2.75) n=O n! aa aa* ' k=O 38 Quasiprobability distribution functions Number-phase Wigner function 39 First note, that if we take the position operator, we have ¢ = X = If we callS= .L%"= 1(-l)kk, we have that q = Wx-(q,p) = 2 f(-1)k(ki(X +a+ a* )/k) = 2a +a* ~(-1)k. (2.76) k=O y'2 1 y'2 6 s = -1 + 2:.)-1)kk = -1 + 2:)-1)k+ (k + 1) = -1- s + l:C-1)", (2.85) Because a = ( q + ip) / v'2, the above expression simplifies to k=2 k=l k=l 00 and the last sum, from Equation (2.78), is equal to 1/2. q = Wx-(q,p) = 2q L(-1)\ (2.77) The results above show that the Wigner function may be used to calculate infinite k=O series. which proves that 2.7 Number-phase Wigner function 2:)-1)k = 1/2. (2.78) k=O In the Section 2.2, where we defined the Wigner function, we also introduced If we insert now the number operator P! in (2.75), we obtain the characteristic function as 00 C(/3) = Tr{pexp((3at- (3*a)}. Wn(q,p) = 2 _2) -l)k(k + (a(z), (2.79) k=O In analogy with it, we can define the function or using that a= (q + ip)j-/2 and the result (2.78), 1 C- n-'(k ' e)-- 2Tr { [vA n-'(k ' &) e-i(k-n&) + H .c] . p"} ' (2.86) (2.80) where On the one hand, we have that (2.87) (n) = Tr{np} = j F/(q,p)Wn(q,p)dqdp, (2.81) and with 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 -crt= I: ik + l)(kl (2.88) ( A) aa* + a*a 1 1 k=O n = I W (a) 2 d2a- 2 = I W(a)lal2d2a- 2 (2.82) the Susskind-Glogower operator [Susskind 1964]. Because the Susskind-Glogower therefore 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 (2.83) 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 Equating expressions (2.80) and (2.83), we can obtain the value for the non convergent sum (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 (2.84) on page 24, does not have this problem, because the integrations over f3x and /3p are from -oo to oo. 41 40 Quasiprobability distribution functions Number-phase Wigner function By writing the density matrix in the number state basis, we have is given by p= LLQm,z/m)(l/, (2.89) 2 _ e-1<>1 an~ ak cos[(n- k)¢] (2.96) m=O l=O W(n, ¢)- 1rVnf ~ 2 v7J . that substituted in (2.86) gives us iiJk 00 00 cn_ 1 00 r27r - (2.91) W(n, ¢) = ( 21r)2 k~n Jo Cn_ 1 00 W(n,¢) = 4;;: L (Qn,n+ke-ik¢ + Qn+k,neik¢). (2.92) k=-n It is easy to show that the integration of (2.92) over the phase ¢gives us 2 la 7r W(n, ¢)d¢ = Qn,n = P(n), (2.93) 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)¢ (2.94) n=O n=Om=O Fig. 2.6 Two different views of the number-phase Wigner function for a coherent state for an that is the correct phase distribution. amplitude a = 4. 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. 2.7.2 A special superposition of number states 2. 7.1 Coherent state Let us consider the state The phase-number Wigner function for a coherent state M (2.95) /¢ ) = _1_ L eim¢o/m). (2.97) M vM + 1 m=O 42 Quasiprobability distribution functions This state tends to have a completely well defined phase as M tends to infinity. For this state, the phase-number Wigner function reads Chapter 3 1 M W(n,¢) = 2(M + 1)1r I:cos[(n- k)(¢- ¢ 0 )], (2.98) Time Dependent Harmonic Oscillator k=O that may be put in the form [Gradshteyn 1980] W(n, ¢) = 2(M ~ 1)1r cos [ ( !lf-- n) (¢- ¢ 0 )] 1 (2.99) x sin [ M: ( ¢ - ¢ 0 )] esc ( ¢ ~ ¢o) . In Figure 2.7, Equation (2.99) is plotted forM= 20 and ¢0 = 0.7. The phase number Wigner function shows a well defined phase or phase localization as In this chapter, we study the time dependent harmonic oscillator, that besides it would be expected for a state of the form (2.97). It is also seen that as being important by itself, it will be used when solving the ion-laser interaction in ¢ approaches the value ¢o, the maximum value for the phase-number Wigner Chapter 8. We solve the Hamiltonian for the time dependent harmonic oscillator function for the state (2.97) is obtained; from (2.99), it may be shown that this by Lewis-Ermakov invariant methods, that will produce the so called squeeze value is 1/27r. By adding over n Equation (2.99), the phase distribution operator. Therefore, we also introduce here squeezed states and some other states of the harmonic oscillator. 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). 3.1 Time dependent harmonic Hamiltonian Let us first consider the Hamiltonian for the time independent harmonic oscil lator with unitary mass, m = 1, and unitary frequency, wo = 1, (3.1) W(n,rp) We can define annihilation and creation operators for this system by the following expressions A 1 (A 0A) b = v'2 q + tp' ht = __!_(q- ip), (3.2) v'2 such that we can rewrite the Hamiltonian as fi = (hth+ ~) = (n+ ~), (3.3) where we have defined the number operator n = hth. Eigenstates for the Hamiltonian (3.3) satisfy the equation Fig. 2.7 Number phase Wigner function for a superposition of the first twenty one Fock states. H[n) = ( n + ~) [n). (3.4) 43 44 Time Dependent Harmonic Oscillator Time dependent harmonic Hamiltonian 45 The time dependent harmonic oscillator Hamiltonian reads are eigenstates of the Lewis-Ermakov invariant. Following Lewis [Lewis 1967], we introduce now the new annihilation and cre (3.5) ation operators where D(t) is the time dependent frequency. A 1 [q o( A o A)] aAt = -1 [ij - - 2(Ep-• A Eq)oA ] . (3.14) It is well-known [Lewis 1967] that an invariant for this system is the Lewis a = y'2 ~ + 2 Ep - Eq , y'2c Ermakov invariant, that has the form in terms of which, the Lewis-Ermakov invariant can be written as (3.6) (3.15) where E obeys the Ermakov equation with the obvious definition of fit. 2 3 Once the creation and annihilation operators are defined for the time dependent E:+D c=c- . (3.7) harmonic oscillator, analogous equations to the harmonic oscillator with con Actually, it can be proved that the Lewis-Ermakov invariant, represented by the stant frequency may be obtained . For instance, we can define the displacement operator (3.6), does not change in time; or in other words, the equation operator as di of A A (3.16) dt =at- i[I,Ht] = 0 (3.8) and the coherent states as is satisfied. Furthermore, it is easy to show that i may be related to the Hamiltonian of the (3.17) harmonic oscillator with constant frequency (3.1) by a unitary transformation of the form and show the corresponding relations (3.9) lcx)t = (3.18) that with the help of the Baker-Hausdorff formula, can be re-written as It can be proved [Fernandez-Guasti 2003], that the Schrodinger equation for the A [·ln(c)(AA AA)] [ o A2] time dependent harmonic oscillator Hamiltonian has a solution of the form T = exp 2-- qp + pq exp -2~qs = exp 2--.lncdiP] exp [ -2-q . i,]2 2 [ 2 dt 2c ' (3.10) (3.19) so that 11/J(t)) = exp [ -ii lt w(t)dt] rti'(O)I1/J(O)), 2 (3.11) with w(t) = 1/c . If we consider the initial state to be By using Equation (3.4) and the the fact that the operator Tis unitary, we can (3.20) see that where lex) is a coherent state of the harmonic oscillator with constant frequency. (3.12) We note that the evolved state has the form i.e., states of the form 11/J(t)) = rf't Ia exp [ -i lt w(t')dt']) = Ia exp [ -i lt w(t')dt']) t; (3.21) (3.13) or in other words, coherent states keep their form through evolution. 46 Time Dependent Harmonic Oscillato1 Time dependent harmonic Hamiltonian 47 3.1.1 Minimum uncertainty states with S1(t) specified by a step function that may be modeled by [Fermindez-Guasti We define the operators 2003] (3.22) (3.29) and where ts is the time at which the frequency is changed, .6. = w2 - w1 with w1 A 1 and w the initial and final frequencies, and A is a parameter (>.--+ oo describes P= -(a-at) 2 iv2 ' (3.23) the step function limit). In Figure 3.1, we plot this function as a function oft 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 1 I I ii=TQi't (3.24) I l,------and 1 I Q(t) I I (3.25) ;, and that they obey the equations [Fermindez-Guasti 2003] P = iw(t)[i, P] = -w(t)Q, (3.26) o+---~----~------.----~---.----~---.----~---- and 0 4 Q = iw(t)[i, Q] = w(t)P. (3.27) 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. ~he uncertainty relation for operators Q and P, for the coherent state (3.18), is given by 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 [Fernandez (3.28) Guasti 2003] 2 2 where .6.X = V(X )- (X) . Time dependent coherent states are thus minimum c(t) = 1 + + ( 1- cos ( n(t')dt'} (3.30) uncertainty states, not for position and momentum but for the transformed ~ n~ft) n~ft)) 21~ position and momentum. ' 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 3.1.2 Step function average of w(t) from t 8 to the end of the first period is 2 for the solid line and 2 2 Let us consider again the Hamiltonian of the time dependent harmonic oscillator 3 for the dashed line, with Wmax = 2 for the solid line and Wmax = 3 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 49 48 Time Dependent Harmonic Oscillator More states of the field 1.2 and w(t)dt); 11/J(tmax)) = (3.32) /\ I \ I \ i.e., we recover the initial coherent state. However, for the minima, we have 0.8 I I \ I I I i(tmin) = 0 and lnc:(tmin) =J 0, and then we obtain I I I t:(t) 0.6 I I I n!)(t • )) - [ilnc:(tmin)(AA+ AA)] w(t)dt), I I I I '!-' mm - exp 2 qp pq (3.33) l 0.4 \/ \_; that may be written in terms of annihilation and creation operators as 0.2 w(t)dt), 0+---~-.--~--.-~---,--~-,,-~------~~--~--~~--~ (3.34) 0 that are the well-known squeezed (two-photon coherent) states [Yuen 1976], also Fig. 3.2 c:(t) as a function oft for Wl = 1 and W2 = 2 (solid line) and w2 = 3 (dashed line). see next section, ).. = 20 and t 8 = 2. 11/J(tmin)) = w(t)dt lnc:(tmin)). (3.35) 10 Squeezed states, just as coherent states, are also minimum uncertainty states. ~ r, However the uncertainties for ij and are I\ {1 p I\ 1\ II I I 1\ I I (3.36) I I I I I I I I I I I I I I I I I I w(t) I I I and I I I I I I I I I I lj,,I I I I I I I I I (3.37) I I I I I I I I I I ....__ ___ ! \ I "- 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) Fig. 3.3 w(t) as a function oft for w1 = 1 and w2 = 2 (solid line) and w 2 = 3 (dashed line). ).. = 20 and t 8 = 2. It should be stressed however, that in the instantaneous Hilbert space we will always have the coherent state (3.21). (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) 3.2 More states of the field 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 We now introduce more states of the harmonic oscillator, such as squeezed states, see that for the maxima, i( tmax) = 0 and Inc:( tmax) = 0, therefore f't (tmax) = 1 Schrodinger cat states and thermal distributions. 50 Time Dependent Harmonic Oscillator More states of the field 51 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] 0.14 0.12 (3.39) 0.1 where r is an arbitrary number (for simplicity we take it real, but it may be complex). 0.08 The squeeze operator is a unitary operator, that means that 0.06 (3.40) 0.04 and also has the property 0.02 (3.41) 60 80 100 120 140 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: 0.1 (3.42) 0.08 with J1 = cosh r and v = sinh r. A squeezed state is then written as 0.06 Ia, r) = S (r) Ia), (3.43) 0.04 where Ia) is a coherent state. 2 The photon distribution P(n;a,r) = l(nla,r)l for the squeezed state la,r) is 0.02 given by 80 100 120 140 (3.44) This photon distribution is plotted in Figure 3.4 for two different values of r. Fig. 3.4 Photon distribution for the squeezed state for a= 5 and (a) r = 1.5 and (b) r = 2· 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 The Husimi Q-function can be calculated from the photon distribution, for- squeezed state, atomic states (excited or ground) may be measured. The atomic I (3.44) for the squeezed states; we start with the definition 2.57 on page 33. mu a , . . ·tt inversions that are produced by such measurements will present features related In the case of a squeezed state 1,6, r)' the density matnx lS wn en as to this extra distribution that are called ringing revivals (see next chapter). (3.45) 53 52 Time Dependent Harmonic Oscillator More states of the field thus Q(a) ~(a /8(r)D(i1)/ o) (o Jbt(/J)St(r)/ a) (3.46) 2 ~ /\a/S(r)D(/1)/o)/ . (3.47) Q(q,p) We write the coherent state as (al = (Oibt(a), so (3.48) Inserting a one as S('r)St(r), we get (3.49) 0 Using Equations (3.42), it is easy to show that (3.50) Husimi Q-function for the squeezed state with amplitude equal to 3 and r = 1.5. and using 2.40 on page 31, we obtain Fig. 3.5 2 Q(a) = ~ i(OI/1 ~ a*v ~ etJ.L, r)l . (3.51) (expression 2.38 on page 31), and we do the same processes we did for the Husimi 7f Q-function, to write Using (3.44), we finally get (3.53) Q(a) ~P(0;1,r) 7f 1 { 2 v [ 2 *2]} (3.52) or nJL exp ~hi ~ 2J.L I +1 , W(o:) = ~ f(-1)kP(k;/1-tm-va*,r). (3.54) where 1 = /1 ~ a*v- aJ.L, and we recall that a = (q + ipjy"i), JL = coshr, 7f k=O 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 Substituting (3.44) in the above equation, we get 3.6 for two different values ofT (the same values that we used for the photon 2 distribution). 2 2 2 w(a) = 2..e-hl -~h +'1. ) f c-:r (;J.L) k\Hk ( ~~v) \ (3.55) J.L1f k=O The Wigner function for the squeezed states can a!so be calculated. For that we use first the series representation for the Wigner function, The sum can be done with the integral expression (3.56) 54 Time Dependent Harmonic Oscillator More states of the field 55 0.3 Q(q,p)o.os -1 0 p 0.4 2 q Fig. 3.6 Husimi Q-function for the squeezed state with amplitude equal to 5 and r = 2. Fig. 3.7 Wigner function for the squeezed state for a= 2 and r = 1.5. for the Hermite polynomials. The final result is Therefore, the states W (a)=-1 exp { -v ( - 1 + -- 1 ) b2 + (r*)2] - ( __ 1 f..l 2__ v2 + 1) 11'12 } . 7f 2f..l f..l- v f..l f..l- v (3.57) (3.59) In Figure 3. 7, we plot this Wigner function. The uncertainties for ij and f5 are 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 (3.58) to calculate and we leave it as an exercises to the reader. What we get is where we have taken w = 1. The decrease in position uncertainty for positive r may be seen in Figure 3.7. (3.60) 3.2.2 Schrodinger cat states 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 Because coherent states are quasi-classical states, the superposition of two of "minus" cat, only odd photon numbers would have non-zero probabilities. Using them is the closest we can get to the paradox proposed by Schrodinger [Schrodinger the series representation of the Wigncr function, expression 2.38 on page 31, and 1935]: the superposition of two classical states (cat "dead" plus cat "alive"). the expression 2.40 on page 31, it is also easy to obtain the Wigner function for 57 56 Time Dependent Harmonic Oscillator More states of the field 0.2 0.15 0.1 W(q,p) 0.05 -l(j 30 40 Fig. 3.8 Photon number distribution for the cat state 1'1f'!.;';.t), a= 4. p the Schrodinger cat states. The result is 1 Fig. 3.9 Wigner function for the cat state 1'1f'!.;';.t), a= 4. W(q,p) = 2 2n [1 ±exp ( -2lal )] 3.2.3 Thermal distribution 2 2 2 { exp (-2 Ia - ')' 1) + exp (-2 Ia + 1' 1) ± 2 exp (-2 h 1) cos ( 48' (1' *a)) } , Up to now, we have been looking at states that may be called pure states; i.e., (3.61) 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 where now we have defined')'= (q+ip)jV"2,. It is clear that in the above expres study mixed states, the density matrix is especially useful, because any state, sion, we have the Wigner function of the "live"cat Ia), plus the Wigner function pure or mixed, can be characterized by a single density matrix. of the "death" cat I - a), and an interference term. We consider now a thermal distribution which has the following diagonal expan- 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 sion in Fock states contributions of the coherent states at a = 4 and a = -4 and the quantum (3.62) interference that produces oscillations in the quasiprobability distribution func tion. Approximate cat states may be generated in several systems such as Kerr media (see Appendix A), atom field interactions (next chapter) and ion-laser with n the average number of thermal photons. These states are known as interactions (see Chapter 8). thermal states. 58 Time Dependent Harmonic Oscillator The photon distribution is in this case given by the expression P(k)- nk Chapter 4 - (1 + n)k+l. (3.63) In Figure 3.10, we plot this photon distribution. (Two-level) Atom-field interaction 4.1 Semiclassical interaction Given a discrete spectrum of an atom, two of their energy levels may be con nected 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 25 Hamiltonian operator (unperturbed) may be written as Fig. 3.10 Photon distribution for the thermal distribution, with n = 3. HA = woJe)(eJ, (4.1) with w0 = Ee- E 9 . 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 61 60 (Two-level) Atom-field 1:nteraction Semiclassical interaction We can now write the semiclassical Hamiltonian for the atom-field interaction where the Hamiltonian is given by as if Wo A + .:\(e-iwto- + eiwto- ) (4.14) sc = 2C7z + -' Wo A A ( ) A ifsc = 2 cr z + 2 cos wt cr x, (4.6) with where w is the field frequency, and where we have done the following: (a) we 5.=A~- (4.15) have used the usual notation in terms of Pauli cr-matrices; (b) we have shifted w+wo the energy of the atom; (c) we have given the explicit time dependence of the It is frequently convenient to consider the quantity known as the atomic ~nver~ion harmonic field; and (d) we have used the real interaction constant, A= lgiEo/2 W(t), defined as the difference in the excited- and ground-state populatwns; 1.e., with E the amplitude of the field. 0 W(t) = Pe(t)- Pg(t), which in this case is The cr-matrices are defined as 2 2 W(t) = Pe(t)- Pg(t) = 1(1f;(t)le)l -1(1f;(t)lg)l (4.16) (jz = le)(el-lg)(gl, (jx = le)(gl + lg)(el, Q-Y = i(lg)(el-le)(gl), (4.7) = (1j;(t)le)(e11f;(t))- (1j;(t)lg)(gll1f;(t)) = (1j;(t)IQ-zl1f;(t)) with commutation relations In Figure 4.1, we plot the atomic inversion W(t) = (1j;(t)!Q-zl1f;(t)) on reson~nce (4.8) (~ = 0), for A= 0.4w (dashed line) and A= .01w (solid line). We have obtamed In the following, we will jump from this notation to 2 x 2-matrices at our con 1.0~------~------~ venience; thus we will make the interchange (4.9) 0.8 and 0.6 o-z = ( 1 0 ) A ( 0 1 ) A ( 0 -i ) (4.10) 0 -1 ' CTx = 1 0 ' cry = i 0 · Using the rotating wave approximation (RWA) [Allen 1987], the Hamiltonian 0.4 ( 4.6) may be taken to the form fisc- _ Wo2crz A + A( e -iwt cr+A + eiwt cr_A ) , (4.11) 0.2 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)) = o.o-ol---~-.5--~~-1'o--~---1r5--~--2·o--~~~2~5~~--~3o l¢(t)), with R = exp[ia cos(wt)Q-y], (4.12) Fi . 4.1 Probability to find the atom in its excited state as a functio~ of_ the scaled ti~e._ The where a« 1, such that we can cut the expansion to first order. The transformed g t A _ 0 and ' - 0 4w (dashed line) and>..= O.Olw (sohd !me). The sohd !me IS parame ers are '-' - A- · · t" Schrodinger equation we find is also for the solution to the Hamiltonian (4.6) with the rotating wave approxJma wn. 0 i ~~~t)) = ifscl¢(t)) (4.13) the solution using the Hamiltonian for the small rotations approximation. How- 62 (Two-level) Atom-field interaction Quantum interaction 63 ~ver, the solid line is also the figure using the solution for the Hamiltonian (4.11); or 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 cos(Atvfi + 1) -isin(Atvfi + 1)V ) . same, and that the rotating wave approximation is an excellent approximation O(t) ~ ( (4.22) for such conditions. -ivt sin(Atvfi + 1) cos(Atvfn) 4.2 Quantum interaction We are now in the position to apply the evolution operator to an initial atom field wave function to find the evolution of the system. If we consider the initial If we consider the field to be quantized, the Hamiltonian for the atom-field state as 11f0(0)) = 11f0F(O))Ie); i.e., we consider the atom in its excited state and interaction reads the field in an arbitrary field state 11f0F(O)), we obtain A woA + A '(A At)A H =2CJz wn+Aa+a CTx. (4.17) 11f0(t)) = cos(Atvfi + 1)11f0F(O))ie)- iVt sin(Atvfi + 1)11f0F(O))Ig). (4.23) If we set w = wo, and work in the interaction picture and in the rotating wave The above equation shows that after the interaction, the atom and the field get approximation, the Jaynes-Cummings Hamiltonian, in terms of Pauli matrices, entangled in such a way that we can not write, in general, a wave function that reads 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 0a ) . (4.18) studied by using the atomic inversion, which is done next. Other variables, like the purity and the entropy, will be studied in Chapter 6. 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 4.2.1 Atomic inversion We can calculate averages of operators, in particular we can compute the atomic (4.19) inversion, W(t) = (1f0(t)lazl1f0(t)), as where W(t) = L Pn cos(2Atvn + 1), (4.24) n=O A ( 1 T= (4.20) 0 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. Note that i'ti' = 1, but i'i't i= 1. This allows us to write the evolution operator as 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 oc 2 cos(Atvfi + 1) -isin(Atvfi + 1) curs for the so-called revival time tR = 21r,fnjA, where n = lal . Although, in U(t) r( the collapse region, it looks like there is no more interaction, this is not the case -i sin(Atvfi + 1) cos(Atvfi + 1) )t' 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 0 to quasi superpositions of excited and ground states and of coherent states, re + (4.21) (0 (0),;01 ) ' spectively. In Figure 4.3, we plot also the atomic inversion but for a superposition of co- 65 Dispersive interaction 64 (Two-level) Atom-field interaction 1.0-,------, 1.0 O.B 0.6 0.5 0.4 0.2 \IV vv 0.0 0.0 -0.2 -0.4 -0.5 -0.6 -O.B -1.0 40 50 -1.0-r----~---.--~----.---~----.---~----.---~----; 20 30 0 10 20 30 40 50 0 10 .At Atomic inversion as a function of ).t for a superposition of coherent states with a = 6. Fig. 4.2 Atomic inversion for a coherent state with a= 6. Fig. 4.3 herent states. It may be seen that the revival time occurs faster than for the 4.3 Dispersive interaction coherent state case; therefore, measuring atomic properties gives us indication . f E tion ( 4 17) and consider rr; btain the dispersive Hamiltoman, we start rom qua . . . about what state we have in the cavity. The measurement of atomic properties o o . . .6.. _ _ w By domg the umtary that .6.. » .\, where .6.. 1s the detunmg, - wo · 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 transformations (4.25) interfere constructively in ( 4.24) are separated (recall that in this case we have U1 = exp [6(&ta-+- &8--)]' only odd or even photon numbers). The revival time for the cats is then a half of t . /( ) d = >.j(w 0 _ w) using the expansion 1.23 on page 4 the revival time for coherent states [Vidiella-Barranco 1992]. More indications w1th 6 = ,\ Wo + w an "'2 ' · · ( 4 17) of the fact that atomic inversions indicate the state of the field, may be seen in and keeping terms to first order in 6 and 6, we can cast the Hamlltoman . Figures 4.4. and 4.5, where we plot the atomic inversion for a squeezed state. into the effective Hamiltonian The details, better seen in Figure 4.5, show the contributions of the different 1 --t, u' u' HU, t[Tt- wata + wo8-z- xa-z(ata, + -2) (4.26) parts of the photon distribution for the squeezed states, Figure 3.4b, producing n=21 12- the so called ringing revivals. In the case of a thermal distribution, neither collapses nor revivals occur, as the where 4wo cosine terms in ( 4.24) do not interfere neither constructively nor destructively. 2 (4.27) X=,\ w6- w2' This fact may be seen in Figure 4.6. 67 Mixing classical and quantum interactions 66 (Two-level) Atom-field interaction 0.3------~ 0.2 0.1 vv 0.0 -0.1 -0.2 -o.3 J3ls--~---4·o--~---4·2--~---4~4~~--~4~6;-~--~4~s;-~~-;s,o 0 20 40 60 so A.t h 5 d 2 Here, the F" 4 5 Detail of the atomic inversion for a squeezed state, wit a = an r = . Fig. 4.4 Atomic inversion as a function of .At for a squeezed state, with a = 5 and r = 2. s~~~alied ringing revivaL9 may be clearly seen. Equation ( 4.27) should be compared with the interaction constant for the dis By transforming to an interaction picture, we obtain the interaction Hamiltonian persive 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 (4.29) not have the atomic frequency too different from the field frequency due to the Hr = [Jt (~)>.(at&_+ &+a)b (~). two-level atom approximation, they differ a lot in problems where the frequen cies involved are much more different like in the case of the ion-laser interaction The Hamiltonian above describes the interaction of an atom with t":o fields, otne . E t' (4 22) the evolutwn opera or (see Chapter 8). quantized and the other classical. U smg qua wn . ' is easily obtained as 4.4 Mixing classical and quantum interactions Let us now consider an atom interacting simultaneously with a classical field Ur(t) = and a quantum field. This situation is shown in Figure 4.7. The Hamiltonian (4.30) for this interaction is cos(>.tv'ii + 1) -i sin(>.tv'ii + 1)Y) cos(>.tv'fl) (4.28) b'm( -iVt sin(>.tv'ii + 1) 68 (Two-level) Atom-field interaction Slow alom interacting with a quantized field vv0.2 Fig. 4.6 Atomic inversion as a function of >-.t for a thermal distribution with n = 3. Fig. 4.7 Level scheme of a two level atom interacting with a classical field and a quanttl field (vacuum) in a cavity. 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 This result will show its value in Chapter 7, where we develop methods tD reconstruct quasiprobability distribution functions from atomic measurement&· W(t) = L Pn(;Jj-A) cos(-Atv'TL+l), (4.31) n=O 4.5 Slow atom interacting with a quantized field with In this section. we show how a three body problem may be reduced to a two boJY - . t" ,g (4.32) 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 Equation (4.31) shows that we can have revivals of the atomic population in affects the interaction. We can write down the Hamiltonian describing a sing;Je version when we start with the field in the vacuum state and a classical field two-level atom passing an electromagnetic field confined to a cavity. In additiOn injected into the cavity, which also interacts with the two-level atom. The clas to the Jaynes-Cummings Hamiltonian, we have to add the energy of the free sical field therefore displaces the vacuum to take it into a coherent state with atom and the spatial variation it feels from the cavity; the Hamiltonian read& 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) On resonance, we can pass to the interaction picture Hamiltonian 2 *Note that the classical field works just as a displacement only when we obtain expectation Hr = !!.__ + g(x)(iio-+ +at o-_). values of atomic operators. 2 70 (Two-level) Atom-field interaction Slow atom interacting with a quantized field 71 Using the 2 x 2 notation for the Pauli spin matrices, the interaction Hamiltonian such that can be written as 2 exp { -ii't [~ + g(x)o-xVn + 1] Tt} = A p At ( 0 vn + 1 ) A Hr = 2 + g(x)T vn + 1 0 T, (4.36) (4.44) = ft exp { -i [ ~ + g(x)a-xvn + 1J t }t. where the non unitary transformation T, defined in expression 4.20, has been used. Note that the evolution operator in ( 4.44) is effectively the interaction of two We already point out that fft = 1, but ftf = 1- Pg,v with 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. 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 vn + 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) 72 (Two-level) Atom-field interaction 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). (5.1) By substituting N = ctj2L on the above equation, E(x, t) = E(x, O)e-l't (5.2) is obtained, with clnR !=---. (5.3) 2L 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 8 8 2 2 iae-~' t)(ae-~' ti :::::J e-lal (1 + 2ial i0t) L [1-,ot(n + m)]ln)(ml, (5.5) n=O,m=O 73 74 A real cavity: Master eqv,ation Cavity losses at zero iemperatv,re 75 keeping terms of the order of Jt, we get that has the solution 1- e-2,t f(t) = -2-"(-. (5.17) such that, rearranging terms and using the definition of the derivative, we arrive to the Master equation for cavity losses 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. (5.7) We now solve the above equation, for an initial coherent state, to show that those 5.1.1 Coherent states states certainly decay, as we assumed in the derivation of the Master equation. By defining the superoperators For an initial coherent state, p(O) = la)(al, we have A ) ( 1 - e-2~t A) (5.8) p(t) = exp ( Lt exp - - -J la)(al. (5.18) 21 we can rewrite (5.7) as By developing the second exponential in Taylor series, and applying the powers dp A A A of J to the coherent state density matrix (it is easy to apply, as coherent states dt = (J +L)p, (5.9) are eigenstates of the annihilation operator), we obtain that has the simple formal solution 2 p(t) = exp {[1- exp( -2"(t)]lal } exp (it) la)(al. (5.19) p(t) = exp[(J + i)t]p(O). (5.10) We take now into account that The problem is now how to factorize the exponential of superoperators. To this end we propose the following ansatz: (5.20) p(t) = exp(it) exp [!(t)J] p(O). (5.11) to finally obtain By differentiating both sides of the above equation, we obtain (5.9), and also (5.21) dp A 0 A A A dt = Lp(t) + f(t) cxp(Lt)J exp( -Lt)p(t). (5.12) as expected. For large times the vacuum is reached; i.e., p(t-+ oo)-+ IO)(OI. Because the commutation relation (5.13) 5.1.2 Number states we can use 1.23 on page 4 to obtain For an initial number state, lk)(kl, it is easy to show that the time evolved density matrix takes the form exp(it)J exp( -it) = Je 2't, (5.14) so that (5.22) ~ = (i + j(t)Je21t)p(t). (5.15) We therefore start with a pure state lk) (kl, and the decay takes it to an statistical Comparing (5.15) and (5.9), we obtain the differential equation 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 j(t)e21t = 1, f(O) = 0, (5.16) less than one. Again the vacuum is reached for times t » 1/'1· 77 76 A real cavil,y: Master eqnation Master eqnation at finite temperatnre 5.1.3 Cat states For a superposition of two coherent states (5.23) lV(q,pj 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 p -2 the following off-diagonal term [Walls 1985] q (5.26) Fig. 5.1 Wigner function {3 =-a 2 for 1 = 0.5 at t = 0.0 We can write then the evolution for the density matrix subject to losses as 5.2 Master equation at finite temperature A(t) 1 ~ (77111) ~-)(-1 (5.27) P =N ~ -(_1_) 11 77, 'I),J"=Ct 77/.1 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~ with i5 = (J'e-"'~t. Just like in the case of a number state, we see that the resulting temperature and under the Born-Markov approximatwn 1s g1ven by [Scully 1997, density matrix evolves from a pure state to a statistical mixture. Carmichael 1999] 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 (5.29) easily found where 2 2 W(E) = _2__ ~ (~I~) exp (-Itt- El ) exp (-Iii- El ) x (5.30) 1rN '"~a (771/.1) 2 2 (5.28) exp{~ [E*(i/-P)-E(ry-p)*] -(i/-E)(P-c)*}, and .C p = 1 n (2at pa- aat fJ- paat) . (5.31) 2 2 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., The formal solution of Equation (5.29) can then be written as a cat state like the one studied in Chapter 3. The figures show that the losses (5.32) strongly influence the field, killing very fast the interference term. 78 A real cavity: Master equation 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. Fig. 5.2 Wigner function f3 = -ex= 2 for 'Y = 0.5 at t = 0.2 We redefine superoperators in the form (5.33) and J3P =a tap+ pat a+ p, (5.34) such that]_, J+ and J3 obey the commutation relations [L,J+] P (5.35) [J3,J±] p (5.36) We have then (5.37) with (5.38) 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 non zero 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 Entropy and purity in the atom-field interaction 83 and viceversa 6.3 Entropy and purity in the atom-field interaction (6.3) 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, The entropies for A and B may be defined as 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 (6.4) density operator for an initial coherent state and the atom in its excited state 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 P12 ) (6.9) 1- Pn ' of each subsystem becomes time dependent and it may evolve now from a pure state to a mixed state (or viceversa). where Araki and Lieb [Araki 1970] have demonstrated the following inequality for two interacting subsystems 2 Pll = L Pn cos (>.tvn+l'), (6.10) n=O (6.5) and Therefore, if the two subsystems are initially in a pure state, the whole entropy is zero (S = 0), such that S(pA) = S(pB)· P12 ia L :;, cos(>.tvn + 2) sin(>.tvn+l), (6.11) n=D vn + 1 6.2 Purity 2 with Pn = exp ( -lal ) o:~~ the photon distribution for a coherent state and ,\ = lgiEo/2 the real interaction constant. Another common tool to study the purity of a state is the purity parameter, We can find the eigenvalues of (6.9) with the determinant SP = (Sp), defined by the expression (6.6) P12 O (6.12) 1- Pn- X 1-- . Using the eigenbasis of the density matrix, it can be shown that By solving the quadratic Equation (6.12) for x, we find the two eigenvalues X1 and x2 , and we find the entropy (6.7) (6.13) Because the equality holds only for pure states, Sp discriminates uniquely be tween mixed and pure states. By using the fact that 1 - Pn :::= -lnpn, for 0 < Pn :::= 1, we find a lower bound for the entropy 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, (6.8) 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 Finally, it is worth to say that the purity parameter is much simpler to calculate field going to a state close to a Schrodinger cat state. In Figure 6.3, we show than the entropy. the Q-function for the field state at time tR/2. 84 Pure states and statistical mixtures Some properties of reduced density matrices 85 0.6 0.4 0.5 0.4 0.3 S(t) SP(t) 0.3 0.2 0.2 0.1 0.1 0 0 0 10 20 30 40 0 10 20 30 40 Fig. 6.1 Field entropy for a coherent state with ex= 5. Fig. 6.2 Field purity for a coherent state with ex = 5. 6.4 Some properties of reduced density matrices The density matrix for a two-level system interacting with another subsystem B is given by The Araki-Lieb inequality [Araki 1970] ~-( lc)(cl lc)(sl ) (6.16) ISA- SBI ~ SAB ~SA +SB, (6.14) p- ls)(cl ls)(sl ' implies that if both subsystems are initially in pure states, the total entropy where lc) (Is)) is the unnormalized wave function of the second system corre is zero and both su!Jsystems entropies will be equal after they interact. Here, sponding to the excited (ground) state of the two level system. we would like to arrive to the result that, if initially the two subsystems are in From the total density matrix, we may obtain the subsystem density matrices pure states, any function of the density matrix of subsystem A is equal to the as function of the density matrix of subsystem B, and, in particular, obtain that both subsystems entropies are equal, without using the Araki-Lieb theorem. ~A= ( (clc) (sic) ) = ( Pn Pl2 ) , (6.17) p (cis) (sis) - P21 We would also like to find the entropy operator P22 and (6.15) PB = lc)(cl + ls)(sl. (6.18) for any of the subsystems. In particular, we will consider later a two-level atom field interaction, but the results may be generalized to other kind of subsystems, From the density matrix for subsystem B, one can not see a clear way to calcu for instance atom-atom interaction, atom-many atoms interaction, N-level atom late the entropy operator, as powers of PB get complicated to be obtained. To field interaction, etc. make the calculation easier, we state the following theorem: 86 Pure states and statistical mixtures Some properties of reduced density matrices 87 1 6.4.1 Proving p"J3+ = 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 0.12 (6.22) 0.06 Q(X. where t5 = p11 - p22 and 1 is the 2 x 2 unit density matrix. We can then find simply that n n ( ) 1 -0.06 An - .!_ R = n --Rm (6.23) 10 p A - ( 2 + ) f, m 2n-m . We split the above sum into two sums, one with odd powers of R and one with -5 even powers, and we also use that -'11 0 10 fl2m+l = !!:.E2m+l (6.24) E ' with 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. Therefore, we can write 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. In terms of 13A the above equation is written as To prove it, we can use the following relation valid for two interacting systems 13A = G(n)I3A -II13AIIG(n- 1)1, (6.27) that before interaction were in pure states where (6.19) (6.28) with this relation it is easy to show that 2 (6.20) with the determinant II13A(t)ll = ~- E • Note that we have written 13'1 in terms of 13A and the unity matrix. We could In particular, with the expression (6.19) we demonstrate ~hat SB have arrived to the same result using the Cayley-Hamilton theorem [Allenby course, it is also true that 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 13~+l = TrB{I3(t)13'8(t)}. (6.21) of the matrix and the unity matrix. 88 Pure states and statistical mixtures Some properties of reduced density matrices 89 Then to prove p~+l = TrA {p(t)pA (t)} by induction, we must prove it for n = 1; 2 where I,B) = e-1,61 12 2:%"=o ~ lk) is the initial coherent state for the field, and but we note that X is the interaction constant. (6.29) We write the entropy operator as 1 is correct, so it is done. SA= lnp.A = ln(1- fiA) -In llhll, (6.37) We now assume it to be correct for n = k; or in other words, we suppose that such that the expectation value of SA is the entropy, and where we have used 1 (6.30) that fi.A (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 is true, and prove (6.19) for n = k + 1. By using (6.27), we can write find p~+ 1 = fi1G(k)- !isll!iA(t)IIG(k- 1). (6.31) (6.38) Note that any power of PB may be written in terms of PB and fi1 (k 2': 2). with Multiplying the above equation by p3 , we obtain F1=-ln1 (1---2E) 2 p~+ = fi1G(k)- ii111PA(t)IIG(k- 1). (6.32) 2E 1 + 2E ' We can obtain from (6.19) as F2=--1 [ lniiPA(t)ll+-ln, 1 ( -- 1 - 2E) ] . (6.39) fi1 2 2E 1 + 2E (6.33) Note that SA is linear in the atomic density matrix, as expected from Cayley Hamilton's theorem (for 2 x 2 matrices). where for the second equality we have used the Cayley-Hamilton theorem for the From (6.38), we can calculate the atomic (field) entropy 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 (6.40) 2 p~+ = fi1G(k + 1)- fisiiPA(t)IIG(k), (6.34) and the atomic (field) entropy fluctuations or (6.41) (6.35) that ends the prove of relation (6.19) by induction. 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 6.4.2 Atomic entropy operator possible states at a given time, and go to zero as the unique maximum entangled With the tools we have developed up to here, we can study the two-level atom state is obtained. field 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 6.4.3 Field entropy operator 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 We use the expression for fi'B in terms of PA to write the field entropy operator in terms of the atomic density operator (6.36) (6.42) Some properties of reduced density matrices 91 90 Pure states and statistical mixtures From (6.18), we can write p~(t) as I\ (6.47) I \ p~ = Jc)(cJ(cJc) + Js)(sJ(sJs) + Jc)(sJ(cJs) + Js)(cJ(sJc) I I 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 IIPA(t)JI (Jc)(ci Js)(si)- llfiA(t)JI ((sJc)Js)(cJ (cJs)Jc)(sl). + 2 + + (6.48) 6.4.4 Entropy operator from orthonormal states To corroborate that the entropy operator for the field has been correctly ob tained, we calculate it by using a Schmidt decomposition for the state (6.16) in 2 3 the atom-field case and the dispersive regime. We can write the wave function in this case as Fig. 6.4 Entropy (solid line) and entropy fluctuations (dashed line) as a function of xt for (6.49) a coherent state with (3 = 2 and the atom in an initial superposition of ground and excited states. where Remark that we can write the atomic entropy operator in terms of the operator 1 (6.50) u~ed to define concurrence [Wootters 1998], because fiA: (t) = lli>}(t)JI aypA (t)ay = PA(t) . lii>A(t)JI; I.e., with (cJs) = Eeie, and l'l,b±) and I±) orthonormal states. We then write SB = -IIPA1(t)il TrA {Nt)SM~A(t)}. (6.43) (6.51) By inserting (6.38) into (6.42), we obtain (6.44) 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 Using the expression of the inverse of the atomic density operator in terms of l'l,b-)('f,b_J times it. the purity operator, the entropy may be written as 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 SB = TrA {p(t) (Fl + IIP:(t)il [1- PA(t)J)}. (6.45) Wigner function associated to (6.46) in an easy way. We do it by means of the formula In terms of the field density matrix the field entropy operator is Ws(a) = 2:) -1)n(a, nJSBJa, n), (6.52) F2 ] A ( ) F2 A2 s - [F1 (6.46) B- + IIPA(t)il PB t - IIPA(t)llpn(t). n=O Entropy of the damped oscillator: Cat states 93 92 Pure states and statistical mixtures where Ia, n) are the so-called displaced number states. The explicit expression for the above equation is 2 2 0. Ws(a) = exp ( -2;3 - 2lal + 4f3ax cos xt) [F1 + ll~~t) II] cosh(4;3ay sinxt) 2 0. 2 2 2 - exp ( -2;3 - 2lal 4f3ax cosxt) llt3:(t)ll cos[2;3(;3sin2xt- 2ax sinxt)]. + 2 0 (6.53) J-J>'(q,p) 0. 0 We plot the Wigner function associated to the entropy operator in Figure 6.5. 0.1 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 q a negative contribution. 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 fV(q.JY) f3 1 ,o(t) = L ;~ ~)) ltt)\ill, (6.54) *!J,J"=O! \'fl 1 JL 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 -1 p an eigenstate of the form q (6.55) The eigenvalue equation is then 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 . ) ( (6.56) 94 Pure states and statistical mixtures 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 coher ent 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. -E 0.7 0.6 0.5 0.4 S(t) 0.3 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. 0.2 with A the eigenvalue. In the above system of equations the elements are given by 0.1 Mu = Miz = 1 + (,Bia) (6.57) 0 N 0 2 4 6 8 10 and A-t * 1 ( (,Bia) _ - ) M12 = M21 = -N --- + (ai,B) · (6.58) (,Bia) 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). 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- 96 Pure states and statistical mixtures 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 func tions. 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 Reconstruction in a lossy cavity 99 form of expression 4.31 on page 68; i.e., we have the equation Cavity ~ 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 7 2/ ~ 4iCJiii' --. 100 dTe'. n cos(Tvn + 1) (7.7) t~8 0 Atom's I JT-.h source Classical Classical that substituted in (7.6), take us to field field Fig. 7.1 Experimental setup to produce superposition of atomic states. Atoms leave the (7.8) 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) We show in Figure 7.2, the Wigner function for an initial state 1?/JF(O)) =bolO)+ and (ax), or (ay) and (ax) may be measured. 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 (ax)= L Pn(a) cosxtn, (7.3) ), second case the atomic inversion, (a2 is measured. In the first case, the Wigner n=O function is given directly by the measurement, while in the second case a Fresnel where transform of the data is required. (7.4) 7.2 Reconstruction in a lossy cavity is the photon distribution for the displaced field state in Equation (7.2). Note In the following we will look at the possibility to reconstruct quasiprobability that if we choose an interaction timet= JT /x, we can rewrite (7.3) as distributions when the interaction with the environment occurs. Knowing from Chapter 5 that losses damage the quality of the quantized field (the field passes (ax)= L( -1t(nlbt (a)I?/JF(O))(?/!F(O)ID(a)ln), (7.5) rapidly from pure states to statistical mixtures), we ask ourselves if in the case of n=O 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 which is proportional to the Wigner function for the initial state of the field, the dispersive approximation, the Master equation that governs the dynamics of 1?/JF(O)) (see Equation 2.38 on page 31). a two-level atom coupled with an electromagnetic field in a high-Q cavity is d A . [H' eff A] .CA A 7.1.2 Fresnel approach dJ/ = -2 I 'p + p, (7.9) Another possibility to reconstruct the quantum state of a system from quasiprob where ability distributions, is the so-called Fresnel representation of the Wigner func tion [Lougovsky 2003]. To obtain such representation, we take the Fresnel trans- LfJ = 21 apat - 1 aJa,fJ - 1 pata. (7.10) 100 Reconstru.ction of quasiprobability distribution functions Reconstruction in a lossy cavily 101 where the atomic superoperator Sr is now defined as (7.15) W(a) The solution to Equation (7.9), subject to the initial state p(O), is then given by (7.16) where A 1- e-Srt f(t)p = --A-p, (7.17) Sr and p(O) = l'l/I(O))('l/1(0)1. Let us consider the atom initially in the following superposition 1 l'!f!A(O)) = y'2(1e) + lg)) (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 superop erators given above. It is not obvious how ef(t)J will apply on the total initial -4 state, therefore we give the following expression for it (7.19) i- with !')p(O) = l'!fip(O))('!fip(O)I. Because j is an atomic superoperator, it will Fig. 7.2 Wigner function for a superposition of the vacuum and first state l'l,bp(O)) =bolO)+ operate only on atomic states, and J, being a field superoperator, will operate ,)1- b6ll), with bo = 0.2. X Re{a} andY= Im{a} only on field states. It is not difficult to show that A 1 We introduce the superoperator ri'!f!A)('!f!Ai = 2x (7.20) (1 _ e-(E+C)t)n A (1 _ e-2Et)n (1 _ e-2Ct)n ] Lp = -tatap- prtata, (7.11) [ (~ + ~*)n x 1A + ( 2 ~)n le)(gl + (2e)n lg)(el , where we have defined and (7.12) (7.21) with iA = ie)(el + lg)(gl. We have also defined the superoperator (as earlier) with ~ = 1 + i x. Therefore 1 00 (1 (7.13) ef(t)J p(O) = 2 L (7.22) n=O It is not difficult to show that [J,i]t3 = -srlp, (7.14) 103 102 Re~onstruction of qu.asiprobability distribution functions Quasiprobabilities and losses By using that By choosing an interaction time such that 8 = -n, Equation (7.30) reduces to (7.23) 00 we may finally calculate (D-x), and obtain (D-x) = 2) -ti)n(niD(a)pp(O)D(a)ln). (7.31) n=O oo ('/l-e-2 oo (re2"'-l)m oo I 1.0~-----~------(a-x)=~ E ~ L e- 2 n~t~(nliJF(O)In) + c.c. (7.25) 2 m=O m.! n=m (n- m.)! 0.9 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 0.8 exchange the doub]e sum in it, to sum first over m., which gives 1 oo (r+ixe-2~t)n 0.7 ( D-~) =- (nlpp(O)In) + c.c. (7.26) 2 L n=O ~ By defining 0.6 2 ry + e- 1JT[sin(2T)- T} cos(2T)] tane = (7.27) 0.5 ry2 + e-27)T[cos(2T) + T}Sin(2T)] and 0.4~------~------,------~~------, 0.0000 1.5708 3.1416 2 _ (T/2 + e-41)T + 2T}e- 1JT sin(2T))! (7.28) T ti- 1 + T}2 0 with T = xt and T} = r I X' we get a final expression for (a-X)' that is Fig. 7.3 We plot J.L as a function ofT for?')= 0 (solid line), and?')= 0.1 (dashed line). (D-x) = L tin cos(nB)(nliJF(O)In). (7.29) T} 0.1. In Figure 7.3, we plot ti as a function ofT. As soon as ti =/= 1, tin n=O = becomes smaller than unity producing errors if one wants to reconstruct the Wigner function. However, one can determine completely the state by not 7.3 Quasiprob s-parametrized quasiprobability distribution function F(cx, s) as follows, 7.4 Measuring field properties We consider again the lossless case and the full two-level atom-field interaction A ~ (s + A n(l- s) (Jaynes-Cummings model), Equation 4.18 on page 62, and ask ourselves the (CJx) = f='o _ 1)n (a,nlpp(O)Ioo,n) = ---F(cx,s), (7.32) 8 1 2 possibility not to measure the whole state but some of its properties. that shows a relation between quasiprobability distributions and a simple mea 7.4.1 Squeezing surement of the atomic polarization operator (&x) in the case of cavity losses. To measure squeezing, we need to be able to measure quantities like In Figure 7.4, we plot the quasiprobability distribution function for the param eter s ~ -0.1812 that corresponds to T ~ 1.689 and TJ = 0.1 for the same (X) = (a) + c.c., (7.33) 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 and the decay parameter, the effect would be stronger. However, even in the case (7.34) uf dissipation one would measure a negative quasiprobability distribution, but should be stressed that not the Wigner function. 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, F{a,s) 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 (7.36) By integrating (7.35), by using the Fresnel integral [Gradshteyn 1980], 00 2 { 2 AB r;Jf ( AB Jo Tsin(T /A) sin(BT)dT = y cos - - (7.37) 4 2 4 such that (with >..t = T) -4 2 1= Tsin(T /A)(&+)dT = -~('l,bp(0)11ft (')'1- '1'2)I1,UF(O)), (7.38) Fig. 7.4 Plot of the s;::;:; -0.1812 quasiprobability distribution function for a superposition of with the vacuum and first state 11PF(O)) =bolO)+ 'vh- b6ll), with bo = 0.2. A(1) r;Jf ( . 'Y1 = --A~+(n)-- V2 cos [A~~(n)l---- + sm [A~~(ii)l)---- , (7.39) 4 4 4 106 Reconstruction of quasiprobability distribution functions Measuring field properties 107 and Again by Fresnel integration of the above expression, we get _ Ali_(i!) [AA:_(n)l . [AA:_(n)]) '12~(lJ ----yr;uf ( cos --- +sm --- . (7.40) 4 2 4 4 Now we use the approximation /(iH 2)(n + 1);::::: n+3/2, that is valid for large with photon numbers. We then can write&~ ( n) ;::::: 4n + 6 and A:_ ( n) ;::::: 0. By setting _ A8+(n) . "(~<2J - --- y{;A ( cos [AJ~(n)]--- + sm [AJ~(n)])--- (7.49) A= 47r, we obtain 1 4 2 4 4 and and "(~<2J -_ --AL(n)- yr;A 2 ( cos [AJ:_(n)]--- + sm. [AJ:_(n)])--- . (7.50) (7.42) 2 4 4 4 ~so that the integral transform (7.38) becomes By choosing the value A= 81r, we obtain 2 -iv'27r (1)!F(O)!Vt vn + 111)!F(O)) (7.43) (7.51) -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 7.4.2 Phase properties case, 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 ii = >..<2Jf2 ( o J(n + 1)(n + 2) ) [ft]2 in the integral forms that will be used. We compute now the average of (; + for J(n + 1)(n +2) 0 (7.44) the one-photon transition for an atom in the ground state and the arbitrary field where )..( 2) is the interaction constant in the two-photon case. One can find 11)!F(O)), that reads 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 (IJ+) = i('lj!F(O)!Vt sin(>..tvfl, + 1)Vt cos(>..tvn + 1)VI1)!F(O)), (7.52) to calculate the average of(;~), which is given by that for a field with large number of photons may be approximated by 2 2 -i('lj!F(O)I cos [>..< lty(n + 1)(n + 2)] [Vt] (IJ+) i(1)!F(o)!Vt sin(>..tvn + 1) cos(>..tvfn)(1 -IO)(OI)I1)!F(O)) (7.53) 2 X sin [>..< lty(n + 1)(n + 2)] 11)!F(0)) i(1)!F(O)!Vt sin(>..tvn + 1) cos(>..tvfn)i1)!F(O)). -~(1)!F(O)I[Vtj2 (sin[>..tJ+(n)]- sin[>..tL(n)J) 11)!F(O)) (7.45) Using the integral [Gradshteyn 1980] 00 { sin(AT) cos(BT)d / with (7.54) Jo T T = 7r 2, A> B > 0, J+(n) = v(n + 4)(n + 3) + v(n + 2)(n + 1);::::: 2n + 5, (7.46) we can integrate (7.54) as and (7.55) L(n) = v(n + 4)(n + 3)- y(il, + 2)(n + 1);::::: 2. (7.47) 108 Reconstruction of quasiprobability distribution functions 2 ~o mea~ure ([Vt] ), again a two-photon process is needed. The average value of (o-+), w1th the atom entering in the ground state, is Chapter 8 2 2 (o-~l) =i(1f]p(O)j[Vt] sin ( >.C ltJ(n + 1)(n + 2)) Ion-laser interaction x [Vt] 2 cos ( >.C 2ltJ(n + 1)(n + 2)) V2 J1fJp(O)) (7.56) =i(1f]p(O)J[Vt] 4 sin ( >.C 2ltJ(n + 3)(n + 4)) x cos ( >.C 2ltJ(n + 1)(n + 2)) V2 J1fJp(O)) Using (7.54), we perform the integral rXJ (o-(2)) . 4 2 Jo +dT = ~(1f]p(O)J[Vt] V J1f]p(O)), (7.57) Nowadays it is possible to trap, by using electromagnetic fields, single ions in that for large intensity field approximates to Paul or Penning traps. In this chapter we will study precisely the problem of a roo (8"(2)) . trapped ion (in general in a trap with time-dependent parameters) interacting Jo +dT = ~(1J}p(O)J[VtFJ1fJF(O)). (7.58) 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 Finally, the average value of 8" + may be calculated by finding the average value field and a two level system with time dependent parameters. The Hamiltonian of the observables 8-x and 8-y, with 8"+ = 8-x + i8"y. In order to find it, we write 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 (7.59) is free of approximations and assumptions on the parameters of the system as i.e.,_ the expectation value of,o-z (the atomic inversion) for a rotated (in the atomic are, for instance, the Lamb-Dicke parameter, the time-dependency of the trap bas1s) density matrix with R = exp[(8"_- 8-+)JT/4]. A similar procedure may be frequency and the detuning, thus we can obtain the best solution for this kind done for the expression (8" y). of systems. Also, we analyze a particular case of time-dependency of the trap Experimentally, we would need to send an atom through a cavity, that contains frequency. the field under study, then (properly) rotate it after it exits the cavity and The possibility to trap small clouds of particles, or inclusive to trap individual measure its energy. We would need an experimental setup as the one shown in atoms or ions, in an small region of space, was opened with the invention of Figure 7.1. 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 Paul trap lll designed an electrodynamic trap (now called Paul trap) [Paull990]. In the Paul interacting with a laser beam. By doing a series of unitary transformations, we trap the idea is that a charged particle can not be confined in a region of space linearize the Hamiltonian of the system to an exact soluble form; this lineariza by constant electric fields, instead an electric field oscillating at radio frequency, tion is also valid for any detuning and for any time dependence of the trap. In must be applied. The Paul trap uses not only the focusing or defocusing forces Section 8.4, we study how to add vibrational quanta to the ion; in this Section, of the quadrupolar electric field acting on the ions, but also takes advantage of we also analyze the atomic inversion of the ion in the trap. Finally, in Section the stability properties of the equations of motion. 8.5, we examine the effect of two lasers driving the ion, instead of one, resulting The ions trapped individually are very interesting, mainly because they are sim in the possibility to filter specific superpositions of number states. ple 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 8.1 Paul trap 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 8.1.1 The quadrupolar potential of the trap a chain, is a promise tool to achieve a quantum computer (each ion in the chain As we already said, the Paul trap uses static and oscillating electric potentials is a fundamental unit of information or qubit) [Cirac 1995]. to confine charged particles. A charged particle is linked to an axis if a linear The trapping of individual ions also offers a lot of possibilities in spectroscopy restoring force acts over it; i.e., if the force is [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 i' = -cr, (8.1) 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 where ·fis the particle position and cis a constant. In other words, if the particle also to avoid undesirable random motions, it is required that the ion be in its moves under the action of a parabolic potential, that can be written in general vibrational ground state. This can be accomplished by means of an adequate use form as 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 (8.2) certain detuning, the coupling is equivalent to the Jaynes-Cummings Hamilto nian [Abdel-Aty, 2007]. On the other hand, the beam that induces the coupling where A is another constant. The potential iP must satisfy the Laplace equation, can be tuned to allow interactions that generate simultaneous transitions of the which means that internal and vibrational states, either to lower vibrational energy levels (while passing from the excited to the ground state) or to higher vibrational energy lev (8.3) els (while passing from the ground to the excited state). This type of coupling is called anti-Jaynes-Cummings. Alternating successively Jaynes-Cummings and where \72 is the Laplacian operator. The Laplace equation (8.3) imposes the anti-Jaynes-Cummings interactions, the trapped ion can be driven to its vibra condition tional ground state. In this chapter, we study part of the physics of the trapped ions. We will analyze (8.4) 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 To satisfy the above condition, we have several possibilities: a Paul trap, in Section 8.1, we review the basic mechanisms of it. In Section a) We make a= 1, (3 = 0 and 1 = -1, and this takes us to the bidimensional 8.2, as an antecedent, we expose the theory of the ion interacting with a laser potential 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, (8.5) 112 Ion-laser interaction Paul trap 113 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 + - 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), z 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 + linearly extended in the z direction, as shown in Figure 8.1. Expressions (8.6) and (8.7) reveal that the components rand z of the electric field The configuration b), expression (8.6), is created by two electrodes with the are independent from each other, and that they are linear functions of r and z, form of an hyperboloid of revolution around z axis, as shown in Figure 8.2. respectively. We also see that we have a harmonic oscillator potential (parabolic If the voltage Ez, its amplitude in the z direction will grow exponentially. The particles will electric field. 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 8.1.3 Motion in the Paul trap in three dimensions; at most, with this potential, we get unstable equilibrium. We will see next, how to solve this problem. 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 8.1.2 Oscillating potential of the trap (8.10) To avoid the unstable behavior of the charged particles under a static potential, m7'-(x, y, z) = qE -q'V. If the trap must be modified. an oscillatory electric field is applied, the particles In order to analyze the trapping conditions, we write explicitly each component, 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 2e x =-mR (Uo + Vo cosnt) x, (8.11) time. 2 If the applied voltage is given by a continuous voltage plus a voltage with a we have .. 2e frequency n, (8.12) y =-mR2 (Uo + Vocos!lt)y, o = Uo + V0 cos!lt, (8.8) and and the potential in the axis of the trap is 2e (8.13) i = mR2 2 (U0 + V0 cos Dt) z, _ Uo + Vo cos nt ( 2 _ 2) (8.9) - r5 + 2z6 r 2z ' where R 2 = r5 + 2z6. where ro is the distance from the trap center to the electrode surface. Making the substitution In Figure 8.3, we show a transversal section of a Paul trap using an oscillating 8eUo az ar mR2f22 -2, 4eVo qz qr mR2f22 -2, nt (8.14) T 2' 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 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. (8.17) Paul trap 117 116 Ion-laser interaction 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. a 8.1.4 Approximated solution to the Mathieu equation ~.5 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 Sta bilitv region (8.19) 0.5 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 q +oc Fig. 8.4 Stability region in the Paul trap. cj;(T) = cj;(T + n) = L Cne2inT (8.20) However, an approximated solution can be given in the stability region of inter is a periodic function. est. To this end, we can write expression (8.19) as The Mathieu equation has two types of solutions: 1) Stable motion. When the characteristic exponent f3 is real, the variable u( T) +oc += (8.21) is bounded, and in consequence the motion is stable. That means, that the ui(T) =AiL c~n cos(2n + f3i)T + Bi L c~n sin(2n + f3i)T particle oscillates with bounded amplitudes and without crashing against the electrodes. These conditions allow to trap the ion. where, as we already said, Ai and Bi are determined by the initial position ui(O) 2) Unstable motion. When the characteristic exponent f3 has an imaginary part, and initial speed ui(O) of the ion, respectively. The subindices i = r, z coincide the function u(T) has an exponential growing contribution. The amplitudes grow with the quantities associated with the radial and axial ion motion, respectively. exponentially and the particles are lost, when they crash against the electrodes. The coefficients in the solution (8.21) are given by the recurrence relations The boundaries of the stability regions correspond to zero and integers values (8.22) 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]. with 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) corre (8.23) spond 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. Once given ai and qi, the quantities C~n and /3i can be calculated. If we define, 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 Gi C~n (8.24) 2n = c~ solutions is enough, and it is not necessary a detailed functional dependence. ' 118 Ion-laser interaction Paul trap 119 and we make and C = 2G~. In Figure 8.5, we plot Equation (8.30). The ion motion is composed by two types ui(t) = ut(t) + uf'(t), (8.25) we get, from Equation (8.21), (8.26) 004 and 00 uf'(T) = :L)A~ cos wit+ B~ sinwit)(G~n + G~ 2 n) cosnm 002 n=l (8.27) wi = f3Slj2. where 100 15) 200 Analyzing Equations (8.26) and (8.27), it is possible to realize that the ion mo tion has two components: ui(t), a harmonic oscillation of frequency wi, and, -002 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 -004 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 Fig. 8.5 Micro motion and secular motion of a trapped ion with parameters q = 0.2, f3 = 0.02. the table there, it is possible to see that in the first region for a « q « 1, we The oscillations at high frequency are the micro motion and those at low frequency are the can assume that G~ ~ G~ 2 and the rest of the coefficients G~ 2 n, n > 1, can be secular motion. ignored; thus, Equations (8.26) and (8.27) can be rewritten as of oscillations: the harmonic oscillation with frequency wi, called secular motion, (8.28) and the small contributions oscillating at the frequency n, called micromotion. and 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 (8.29) (8.28) and behaves as it was confined in a harmonic pseudo-potential, that for or in other words, the radial part, has the form (8.30) (8.33) with Typically, Uo = 0, thus a= 0 (Equation (8.14)); in any case, we are working in (8.31) the region where a rv 0. Thus, the frequencies Wx and Wy are degenerated, and Equation (8.33) reduces to (8.32) 2 mwr ( 2 2) q1/JzD = - - X + Y · (8.34) 2 120 Ion-laser interaction Ion-laser interaction in a trap with a frequency independent of time 121 To obtain an expression for w, we can use the approximation [Dehmelt 1967], operators a and at are given by the expressions ~ (8.39) f3r = yar + 2' (8.35) with the definition Wr = f3rr!/2, to get and r!qr eVo . p (8.36) 2ffv' (8.40) Wr = 23/2 = v'2mr5r!. where we have made the ion mass equal to 1. Also, for simplicity, we have Experimentally, the typical ranges of operation are V ::::::; 300- 800Volts, f!/27r::::::; 0 displaced the vibrational Hamiltonian by v /2, the vacuum energy, that in this 16- 18MHz, and r ::::::; 1.2mm, that gives a radial frequency Wr::::::; 1.4- 2.0MHz 0 case is not important. The second term in the Hamiltonian corresponds to the for calcium ions (4°Ca+). ion internal energy; the matrices CYz, CY+, and fY_ are the Pauli matrices, and w We can summarize this section, saying that under certain conditions it is a good 21 is the transition frequency between the ground state and the excited state of the approximation to treat the ion motion as a harmonic oscillator. 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 Ion-laser interaction in a trap with a frequency independent of time 8.2.1 Interaction out of resonance and low intensity In this section, we study the interaction of a laser with a trapped ion in a har If we consider that the Hamiltonian (8.38) corresponds to the wave function monic potential with constant frequency. We start with the Hamiltonian of the l~(t)), the Schri:idinger equation can be written as system, and we show that it is possible to find Jaynes-Cummings type tran .a sitions and anti-Jaynes-Cummings type transitions, depending on the different 2&1~) =HI~). (8.41) cases of resonance and laser intensities that induce the coupling between the ion Let us examine the transformation to a rotating frame of frequency w, by means internal states and the ion vibrational states. of the unitary transformation We can write the Hamiltonian of the trapped ion as (8.42) (8.37) Applying the transformation T, the wave function l~(t)) transforms in the wave where Hvib is the ion's center of mass vibrational energy, Hat is the ion internal function l¢(t)); i.e., 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 approx T(t)i~(t)) = l¢(t)), (8.43) imated 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 and the Hamiltonian transforms in dipolar approximation, so we will write the interaction energy as -er· E, where 8T(t) L t HT = i~T 1 (t) + T(t)HT (t). (8.44) -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 Writing the position operator i: in terms of the ladder operators, expressions as (8.39) and (8.40), using the Baker-Hausdorff formula 1.28 on page 5, and the (8.38) commutators of the Pauli matrices 4.8 on page 60, the explicit transformed Hamiltonian is The first term in the Hamiltonian is the ion vibrational energy; in the ion vi (8.45) brational energy, the operator ii = at a is the number operator' and the ladder 122 Ion-laser interaction Ion-laser interaction in a trap with a frequency independent of time 123 where is the exchange energy frequency of the internal and vibrational states, called Rabi frequency. 'f)=K y-:;;;;;;fl (8.46) The interaction Hamiltonian (8.53) has a diversity of contributions, each contri bution oscillates with a frequency that is a multiple integer of v. is the so called Lamb-Dicke parameter, that is a measure of the amplitude of We apply now the rotating wave approximation. As the Schrodinger equation the oscillations of the ion with respect to the wavelength of the laser field. The is a first order differential equation in time, we have to integrate it once with quantity J( is the wave vector of the laser, and respect to time; this integration brings the sum and the difference of the fre kv = w21- w (8.47) quencies to the denominator. 'fhe terms changing slowly will dominate, over the terms changing very fast; so the contribution to the Hamiltonian of those is the detuning between the plane wave frequency and the transition frequency fast terms is neglected, and only the slowly changing terms are kept. In this of the ion; in other words, we are considering that the detuning is a multiple case, the terms that do not rotate quickly are those whose exponent satisfies the integer of the vibrational frequency of the ion. relation n- m = k, and as we already explained, are those terms that we will We need now to factorize the exponential in the Hamiltonian (8.45). As [a, [a, at]] = keep. This approximation is valid for 0, and [at, [a, at]] = 0, we can use the Baker-Hausdorff formula 1.28 on page 5, and write 0, « v. (8.55) (8.48) 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. Expanding in Taylor series the exponentials that contains the operators, and We have then, substituting in the Hamiltonian (8.45), we obtain 2 2 2 H =ne-1) / [a ~ (-i'f)) m+k (at)k(atynam+HC] (8.56) mt - ~o (m+k)!m! ... Using now the fact that the number states is a complete set, We go now to the interaction picture, using the transformation kv I= .l::ln)(nl, (8.57) = exp [it(v'i1 + 2az)]. (8.50) 71nt n=O We apply this transformation, using the following two commutators, where I is the identity operator, we can write (8.51) (8.58) and that substituted in the Hamiltonian (8.56), gives us (8.52) o - -1)2 /2 [ At k o k ~ ( -i'f)) 2m n! Hmt -Oe a_(a) (-t'f)) ~ ( k)l (' _ )I +H.C ..l (8.59) and we obtain 1 m=O m + .m. n m . Hint= ne-1)2/2 [a- L (-~~~~m (at)name-i(n-m-k)vt + H.c.]' (8.53) Using the explicit expression for the associated Laguerre polynomials [Abramowitz, n,m=O 1972], where (a) . - ..Z:.., - i (n +a)! xi Ln (x)- ~( 1) ( n _ t.a.)I( + t.t.")I ·1' 0, =>-Eo (8.54) i=O 124 Ion-laser interaction Ion-laser interaction in a trap with a frequency independent of time 125 we can write 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 . - _,.,2/2 [ At k . k fU (k) 2 ] Hmt- De (]'_(a) (-zTJ) (n + k)!L"' (TJ) + H.C .. (8.60) 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 We will consider now processes where only one phonon is exchanged; that means Figure 8.6b, we explain why this Hamiltonian is anti-Jaynes-Cummings type. that we must take k = 1 in the Hamiltonian (8.60). We will consider also, that Applying the anti-Jaynes-Cummings Hamiltonian to the adequate states, the 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 a) 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) de scribes emission and absorption of one vibrational excitation, when the atom makes electronic transitions. The first term represents the absorption of a vi b) brational 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 vibra tional 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) Ie), Fig. 8.6 a) A Jaynes-Cummings Hamiltonian implies the ascent (descent) of one ion vibra which represents n vibrational quanta and the ion in the excited state le); we tional quantum, and at the same time, the transition from an excited (ground) internal state get, 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 (8.62) (ground) state to the ground (excited) state. which is the state with n + 1 vibrational quanta, and the ion in the ground previous comments can be easily understood. If we apply the Hamiltonian (8.65) state. In the second case, the state is given by In+ 1) Ig), and when we apply to the state ln)le), we get the Hamiltonian we obtain, Hajcln)le) =In- 1)lg), (8.66) (8.63) that is the state with n vibrational quanta, and the ion in the excited state. and if we apply it to the state In - 1) lg), we get We can repeat all the above procedure now when the laser frequency is greater Hajcln- 1)lg) = ln)le). (8.67) than that of the transition l,From the point of view of the trapped ion, all this means that we can take it to kv = W- W21, (8.64) its lowest energy vibrational state, alternating successively, and as many times as necessary, the detuning between the frequency of the plane wave and the and obtain the Hamiltonian (clearly now, the subscript ajc stands for anti internal frequency of the ion. Again, we can illustrate all this by applying the J aynes-Cummings) correct Hamiltonian to the adequate state. For that let us consider a vibrational (8.65) state In) and the ground internal state; if we apply the Hamiltonian (8.61), we 127 126 Ion-laser interaction Ion-laser interaction in a trap with a frequency dependent of time get We find, Hjcln)lg) =In -1)le). (8.68) (8.75) We apply now the Hamiltonian (8.65), obtaining Hajcln -1)le) =In- 2)lg), (8.69) being and the ion has lost two quanta of vibrational energy. Repeating successively D(t) =ABo (8.76) this procedure, we can arrive to the state IO)Ig). the Rabi frequency. Using the Ermakov invariant, the time dependence of the trap has been factor 8.3 Ion-laser interaction in a trap with a frequency dependent ized; the time dependence is implicit in p(t). We go now to a frame rotating at of time frequency w, by means of the unitary transformation In this section, we study the problem of an ion trapped with a frequency that (8.77) depends on time and interacting with a laser beam. Using unitary transforma tions, we show that this system is equivalent to a system formed by a two levels The Hamiltonian is transformed to subsystem with time dependent parameters interacting with a quantized field. The procedure to build the Hamiltonian for this case, is exactly the same that 1 2 2 2) 1 ( ) Hw = ---(-) (p + v0 x + - W21- W CJz in the previous section, but we have to keep in mind that now the frequency is 2vop 2 t 2 (8.78) time dependent. +D(t) {exp [-i(li+ilt)77(t)] (J_ +H.C.}. The Hamiltonian is Denoting the detuning frequency between the laser and the ion by o = w21- w, (8.70) and the characteristic frequency of the time dependent harmonic oscillator by 1 and then, the Schri:idinger equation can be written as w(t) = p2(t)' (8.79) (8.71) we get To solve the problem, we make the transformation Hw = w(t)(n + 1/2) + ~(Jz + D(t) { exp [ -i(il + at)77(t)] (J_ + H.C.}. (8.80) l¢(t)) = TsD(t)l~(t)), (8.72) The time dependent Lamb-Dicke parameter is where 77(t) = 7]op(t)y'Vo, (8.81) 2 ()- {iln[p(t)y'Vo](xp+px)} [-ip(t)x ] TsD t - exp exp p(t) , (8.73) 2 2 with and we have also the Ermakov equation 3. 7 on page 44 as an auxiliary equation. (8.82) We apply the unitary transformation (8.73) to the Hamiltonian (8.70); i.e., we must calculate the expression where k is the wave vector of the laser beam. Comparing with the Hamiltonian obtained in Section 8.2, (8.45), the Hamilto oTDs(t) t t HsD = i--t-TsD (t) + TsD(t)HTsD (t). (8.74) nian (8.80) is equivalent, but with all the parameters depending on time. 0 129 128 Ion-laser interaction Adding vivrational quanta 8.3.1 Linearization of the system 8.4 Adding vibrational quanta We call linearization of the system the process to reduce the exponent of the We now show how to generate noncLassical vibrational states in the low intensity ladder operators a and at to the first power, without using approximations. To regime. From Equation (8.56) with k = 2, we note that if the Lamb-Dicke this end, we make the transformation 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 I¢R) = R(t)l¢w), (8.83) (8.88) where R(t) is given by 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 (8.84) 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 Transforming the Hamiltonian, we get is possible to show that = lm)lg)(ml(gl (8.85) L m=O,l with + f [cos (~nnt) (In+ 2)lg)(n + 21(91 + ln)le)(nl(el) n=O f3(t) = rJ(t)w _ ir](t). (8.86) 2 2 -isin (~nnt) (In+ 2)lg)(nl(el + ln)le)(n + 21(91)]. (8.89) The term w(t)/2 has not been considered, because it is only a phase, and when The quantity Dn is the two-phonon Rabi frequency which is given by the observable mean values are taken, it disappears. (8.90) With the transformation (8.84) we have achieved our goal: linearize the Hamil Dn = 2cy(n + 1)(n + 2). tonian without any type of approximation. The rotating wave approximation Using these results, the time evolution of the quantum state in the interaction is not used, and this leaves open the possibility to consider different intensity picture is easily derived for arbitradly chosen initial conditions. We have 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 l\ll(t)) = UI(t)l\l!(O)). (8.91) frequency of the trap. The Hamiltonian (8.85) is solvable, and in [Shen 2003] If we consider as initial state the ion in its excited state Ie) and the vibrational methods of solution have been published. state a coherent state, we can find the atomic inversion, (that we defined in It is also important to remark that we have not imposed any condition in the Chapter 4, and we recall that it is defined as the probability to find the ion in its time dependence of the frequency of the trap; in principle, this frequency can excited state minus the probability to find it in the ground state). Using (8.89) assume any temporal form. For a Paul trap, the more general form is and (8.91), we get v 2 (t) =a- 2qcos2t, (8.87) (8.92) and the Hamiltonian (8.85) is the ion-laser interaction with micromotion in cluded. Also, this Hamiltonian gives us the freedom to consider arbitrary time We plot this function in Figure 8.7 as a function of the scaled time T = Et. dependent frequencies. For instance, if we consider a sudden change in the trap The interaction gives rise to a quasi-regular evolution of the atomic inversion, frequency, we would generate squeezed states for the vibrational wave function unlike the case of one phonon resonance. This can be used for several purposes, as described in Chapter 3. among them, to add excitations to the vibrational state. In Figure 8.7, it can 130 Ion-laser interaction Adding vibrational quanta 131 w -0.5- 0 4 6 Fig. 8.7 Plot of the atomic inversion, W(t) P2- Pl, the probability to find the ion in its 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 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. 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 be seen that if the ion is in its excited state Je), initially, after an interaction the approximation 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 (8.95) 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 although this approximation represents a Taylor-series expansion for large eigen a coherent state. To illustrate this fact, we show Figure 8.8 the atomic inversion values n of the operator ii, the error is already small for small n-values. For for a thermal distribution. example, for n = 1 the relative error is only 0.02. Consider again the initial state 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) Jw(O)) = Ja)Je). (8.93) Choosing a particular interaction time t = T, according to Combining Equations (8.89) and (8.93), and using a compact operator represen T=1f/A, (8.97) tation of the Jaynes-Cummings dynamics, we arrive at we obtain for the vibrational state vector 2 2 2 Jw(t)) =cos ( ctJ ii (iit) ) Ja) Je) - i(Vt) sin ( ctJa2(iit)2) Ja) Jg). (8.94) (8.98) Filtering specific superpositions of number states 133 132 Ion-laser interaction where we have introduced the subscript "v" (vibrational) to note that we are 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 not taking into account anymore the state lg). Moreover, the subscript"+" and sub-Poissonian effect of the vibrational wave function becomes more significant the superscript" (k)" are used to indicate the process of adding (two) vibrational quanta and the number of such interactions respectively. with increasing number of interactions. 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 8.5 Filtering specific superpositions of number states interaction time, t = T, after the second interaction (that is completed at time 27), the vibrational state is 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 (8.99) write EH (x, t) as By repeating the process k times, one finally obtains for the quantum state, at the time tk, after completing k interactions, (8.105) (8.100) 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, 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 (8.106) same time one is keeping the width of the distribution constant. The excitation distribution pJkl, after k interactions, is easily found to be related where ks are the wave vectors of the driving fields and to the number statistics P~o) of the initial state Ia) via (8.107) (8.101) This result clearly shows that the number statistics is only shifted but retains are the Lamb-Dicke parameters with s = j, m. its form. In the resolved sideband limit, the vibrational frequency v is much larger than One way of studying the properties of the states being generated is through the other characteristic frequencies, and the interaction of the ion with the two lasers Mandel Q-parameter [Mandel 1979], which is defined by can be treated separately using a nonlinear Hamiltonian [de Matos 1996a]. The Hamiltonian (8.60) in the interaction picture can then be written as (8.102) and where > 0, super Poissonian distribution Poissonian distribution (coherent state) if Q = o, (8.103) sub-Poissonian { < 0, where LAk) (TJ~) are the operator-valued associated Laguerre polynomials, the D's = -1, number state. are the Rabi frequencies and n = at a. The Master equation which describes this In this case the Mandel Q-parameter is given by system can be written as ofJ .[H, Al r ( " A A ) (8.104) &i = -z r, P + 2 217+PO"- -O"zP- PO"z (8.109) 134 Ion-laser interaction Filtering specific superpositions of number states 135 where the last term describes spontaneous emission with energy relaxation rate (the argument of 1/Js denotes the condition we apply; i.e., in Equation (8.115), r, and the condition is on rJo) where 1 0 p' =-11 dsTV(s)e''S1JEXpe-tS1JEXA • A (8.110) 2 -1 1, (8.116) 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 and W(s) is the angular distribution of spontaneous emission [de Matos 1996a]. q The steady-state solution to Equation (8.109) is obtained by setting opjot = 0, NJ = L IC~O) 12 (8.117) and may be written as n=O is the normalization constant. (8.111) If instead of condition (8.114) we choose 1 where lg) is the electronic ground state and 11/Js) is the vibrational steady-state L~ l(TJI) = 0, (8.118) of the ion, given by we obtain the wave function (8.119) where now 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 1, (8.120) (8.113) and Note that Hri1)11/Js) = 0, so that ion and laser have stopped to interact, which Nf = L IC~l)l2· (8.121) occurs when the ion stops to fluoresce. For the j = 1 and k = 0 case, and n=p+1 assuming Lkl) (rJi) -1 0 and Lk (TJ5) -1 0 for all k, one generates nonlinear coherent Combining both conditions, (8.114) and (8.118), one would obtain for q > p, states [de Matos 1998]. However, by setting a value to one of the Lamb-Dicke parameters such that, for instance, - 1 ~ (1) 11/Js(TJo, TJd)- n ~ en In), (8.122) 01 n=p+l (8.114) with for some integer q, we obtain that, by writing 11/Js) in the number state repre q 1 2 sentation, N51 = L IC~ )1 - (8.123) n=p+l 1 q In this way, by setting the conditions (8.114), (8.118) or both, we can engineer 11/Js(rJo)) = N, In), (8.115) L c~O) states in the following three zones of the Hilbert space: (a) from IO) to lq), (b) 0 n=O 136 Ion-laser interaction 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. Chapter 9 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 Nonlinear coherent states for the choosing j = 2 and k 0 one would obtain only even or odd number states Susskind-Glogower operators 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 In this chapter, we construct nonlinear coherent states for the Susskind-Glogower 1 2 operators [Susskind 1964] by the application of a non-linear displacement oper 11/Js('T]o = 1)) = (10)- (8.124) ator on the vacuum state. In Section 9.1, we do that for an approximated V : 2 ~oe~'7 : 2 l1)), 1 + 1~12e'h -1 1e 1 1 displacement operator, and in Section 9.2, for an exact displacement operator. where by changing the Rabi frequencies, one has control of the amplitudes. 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 Finally, note that we could have also chosen to drive the qth upper sideband Section 9.4, we also construct nonlinear coherent states as eigenfunctions of a instead of the kth lower sideband in Equation (8.105) with basically the same results. 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 nonclassi P(n) cal 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 op erators, for instance, defining a displacement operator for them acting on the P(n) vacuum state. We have defined already the Susskind-Glogower operators; re calling the definitions we have 00 A 1 V= :L1n)(n+11= v'A a, (9.1) n=O n + 1 Fig. 8.9 Possible situations we can have if we filter number states with the proposals of this section. and 00 A 1 vt = :L ln+1)(nl =at-,-, (9.2) n=O ~ satisfying the conditions V In) = In- 1) , (9.3) 137 138 Nonlinear coherent states for the Susskind-Glogower operators Approximated displacement operator 139 vt In) = In+ 1). (9.4) for the displacement operator should be. Second, we solve the displacement op erator in an exact way by developing it in a Taylor series, that allows us to Additionally, we would like to make explicit the result introduce the exact solution for nonlinear coherent states, constructed with the Susskind-Glogower operators. Finally, we analyze the constructed states via the VIO) 0, (9.5) Q function [Husimi 1940], the photon number distribution and the Mandel Q that comes naturally from (9.1). parameter [Mandel 1995] in order to show their nonclassical features such as As we saw in Section 2 of Chapter 4, the Susskind-Glogower operators possess a amplitude squeezing and quantum interferences. non-commuting and non-unitary nature, that resides in the expressions vvt = 1, (9.6) 9.1 Approximated displacement operator and A first approach to factorize the displacement operator in the product of expo vtv = 1 - 1o) (ol. (9.7) nentials, is to consider the approximation v-1C::' vt. Let us write the displacement operator as l,From the above expressions we can see that such nature of the Susskind Glogower operators is for states of the radiation field that have a significant (9.11) overlap with the vacuum; in other words, We find that the right-hand side of this equation corresponds to the generating ( 1/J 1[ v, vt] 11/J) = ( 1/JIO) \011/J) . (9.8) function of Bessel functions, that implies that Therefore, for states where the vacuum contribution is negligible, we can consider them as unitary and commutative, and we can perform the following approxi (9.12) mation v-1 C::'vt. (9.9) where Jn is the Bessel function of the first kind and order n. The properties of the Susskind-Glogower operators play an important role in Applying the displacement operator on the vacuum state, we have 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 Dsc IO) C::' co L vtn Jn (2x) IO)' (9.13) 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 coher and using that vt In) = In+ 1) and that v-1C::' vt, we obtain ent 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) la)8c = Dsc IO) C::' coL Jn (2x) In). (9.14) by n=O la)sc = ex(vt -v) IO)' X E R (9.10) l,From the normalization requirement, we determine c0 , 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; SG (ala)sc c6 L Jn (2x) (nl L Jm (2x) lm) we propose two methods to achieve it. n=O rn=O First, we use the approximation 1 to factorize, approximately, the dis v- C::' vt c6 L J~ (2x) = 1. (9.15) placement operator. This solution helps us to understand how the exact solution n=O Exact solution for the displacement operator 141 140 Nonlinear coherent states for the Su.sskind-Glogower operators Using the result [Abramowitz, 1972] write 1 = Jg (2x) + 2 L J~ (2x) , (9.16) n=1 (9.20) we obtain 2 co= (9.17) 1 + Jg (2x)' where the square brackets in the sum stand for the floor function, which maps a real number to the largest previous integer. and substituting in (9.14), we finally get We can rewrite the above equation as 2 oc 2 ( ) Lln(2x)ln). (9.18) (9.21) la 2x n=O As we mentioned before, solution (9.18) helps us to foresee the exact solution where we have taken the second sum to oo as we would add only zeros (the for these states. We can expect that the exact solution for Susskind-Glogower Gamma function has simple poles at the negative integers, so 1/f( -k), with k coherent states corresponds to a linear combination of number states where the a positive integer, is zero). We now exchange the order of the sums coefficients are, except for some terms, Bessel functions of the first kind and order n. I ) - -xV xvt IO) + v-2 f f (-1txk v2nvtk IO)' (9.22) asG-e e n=Dk=n(k-n)!n! 9.2 Exact solution for the displacement operator By setting m = k - n, and using that In order to factorize the displacement operator in an exact way, we can develop (9.23) the exponential (9.10) in a Taylor series and then evaluate the terms ( vt- V) k For instance, for k = 7 we have we get (vt _v-f = 7 =: (vt- v) : +G) (11) (ol-Io) (11) (9.19) -G) (13) (OI -12) (11 + 11) (21- IO) (31) + C) (15) (OI-14) (11 + 13) (21- 12) (31 + 11) (41-IO) (51), to finally obtain where : : means to arrange terms in such a way that the powers of the operator arc always at the left of the powers of the operator v vt. (9.24) LFrom the definition of the Susskind-Glogower coherent states (9.10), we can 143 142 Nonlinear coherent states for the Susskind-Glogower operators Susskind-Glogower coherent states analysis Applying the exponential terms on the vacuum state, we have 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., la)sc = ( 1 + vz) fIn (2x) In) n=O ()() (9.25) Q(a) = 2-_ (alpla). (9.30) 7f = L Jn (2x) In)+ L Jn (2x) In- 2), n=O n=2 If we substitute p = l'l,b) ('l,b I for the Susskind-Glogower coherent states in (9.30), making m = n-2 in the second sum and performing the index change n = k - 1. we obtain ~~~ . (9.31) la)sc = L [Jk-1 (2x) + Jk+l (2x)]lk- 1). (9.26) k=1 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 By using the recurrence relation of the Bessel functions [Abramowitz, 1972] squeezes. Later, we will find the value of x for which we obtain the maximum squeezing of the coherent state. xJn-1 (x) + xJn+l (x) = 2nJn (x), (9.27) 9.3.2 Photon number distribution and changing again the summation index, we finally write When one studies a quantum state, it is important to know about its photon 1 ()() statistics. The photon number probability distribution P ( n) is useful to deter la)sc =- L (n + 1) Jn+l (2x) In). (9.28) mine amplitude squeezing. We should refer to amplitude squeezed light as light X n=O for which the photon number distribution is usually narrower than the one of Equation (9.28) is an important result because it constitutes an explicit expres a coherent state with the same average number of photons. The photon num sion for nonlinear coherent states. It remains to analyze the behavior of the ber distribution is also useful to analyze if there exist effects due to quantum constructed states, in order to determine their nonclassical features. interferences. We write the Susskind-Glogower coherent states photon number Before we proceed with the analysis, and because we will need this result later, distribution as we make clear that, as we can verify from (9.28), for x = 0 we have (9.32) Ia (x = O))sc = IO) (9.29) Figure 9.2 shows the Susskind-Glogower coherent states photon number distri bution for different values of the amplitude parameter x. Figure 9.2 helps us to as must be. 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 9.3 Susskind-Glogower coherent states analysis 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 There are different ways to find out if the state we are constructing resembles for even or odd photon numbers. one that we already know. Here, we will use three different methods: the Husimi Comparing Figure 9.2(a) with the one obtained for an initial coherent state wave Q-function [Husimi 1940], the photon number distribution and the Mandel Q function that subject to a harmonic oscillator potential, we can see that the pho parameter [Mandel 1995]. ton number distribution for the Susskind-Glogower coherent states is narrower 144 Nonlinear coherent states for the Susskind-Glogower operators Susskind-Glogower coherent states analysis 145 0.25 0.4 0.20 0.3 0.15 P(n) P(n) 0.2 0.10 -Hi -1(1 0.1 0.05 0~~,-~-,~--~-,,-~~ 0 0 10 20 30 40 50 0 10 20 30 40 50 0.20 0.12 0.15 0.10 0.08 P(n) 0.10 P(n) 0.06 0.04 -{I) 0.02 30 40 50 10 20 30 40 50 Fig. 9.1 Exact Susskind-Glogower coherent states Q function for (a) x = 1; (b) x = 5; (c) Fig. 9.2 Susskind-Glogower coherent states photon number probability distributions for (a) x = 10 and (d) x = 20. X= 1; (b) X= 5; (c) X= 10 and (d) X= 20. 9.3.3 Mandel Q-parameter than the one for a coherent state of the same amplitude. This suggests that we In the previous chapter, we introduced the Mandel Q-parameter; its definition are, indeed, obtaining an amplitude squeezed state. It is interesting to know is given by expression 8.102 on page 132. We will use it know, not only because when the state is maximally squeezed, but we need another tool to obtain the it represents a good parameter to define the quantumness of Susskind-Glogower value of the parameter x for which this occurs. coherent states, but also because it will allow us to find the domain of x for 146 Nonlinear coherent states for the Susskind-Glogower operators Eigenfunctions of the Susskind-Glogower Hamiltonian 147 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 0.6 (9.33) Q(x) 0.4 and 0.2 0 (9.34) 10 15 20 25 -0.2 X The even sums, with respect to the power of k, in (9.33) and (9.34), can be evaluated. -0.4 In the Appendix E, we show explicitly that -0.6 L k2Jf (2x) = x2, (9.35) k=O Fig. 9.3 Mandel Q-parameter for the Susskind-Glogower coherent states. and L k4 Jf (2x) = 3x4 + x2. (9.36) k=O 9.4 Eigenfunctions of the Susskind-Glogower Hamiltonian 3 The sum 2.:::%': 0 k Jl (2x) is more complicated, and can be evaluated using the Previously we managed to construct nonlinear coherent states applying the dis technics developed by Dattolli [Dattoli 1989], such that we obtain placement operator on the vacuum state. However, even when the obtained results arc very interesting, we cannot avoid to wonder how these states could 3 2 2 2 be physically interpreted. Physical interpretations are given by operators repre k Jf (2x) = x {(6x + 1)JC(2x) + (6x - 1)Jl(2x) L senting observables; i.e., quantities that can be measured in the laboratory. The k=O (9.37) most important observable is the Hamiltonian, this operator helps us to find the 2x2 - 2xJo(2x)J1(2x) + 3[Jo(2x)J2(2x) + J 1 (2x)h(2x)]}. energy distribution of an state via its eigenvalues. In this section, we construct Susskind-Glogower coherent states as eigenfunc Substituting the values of the sums into Equation 8.102 on page 132, we obtain tions of a Hamiltonian that we propose and which represents the fundamental the plot shown in Figure 9.3. z.From Figure 9.3 we can see that, depending on the coupling to the radiation field via the Susskind-Glogower operators. In Subsec parameter x, the photon distribution of the constructed states is sub-poissonian, tion 9.4.1, we construct time-dependent Susskind-Glogower coherent states with Q < 0, meaning that amplitude squeezing states may be for a value of x within the vacuum state IO) as initial condition, i.e., we construct states that satisfy the domain 0 < x :S 13.48. Also, we find that the most squeezed state may be Ia (x = O))sc = IO). In Subsection 9.4.2, we generalize the eigenfunction prob obtained at x = 2.32, with Q = -0.64. lem for an arbitrary lm) initial condition, we also show that previous results correspond to the particular case m = 0. 148 Nonlinear coherent states for the Susskind-Glogower operators Eigenfunctions of the Susskind-Glogower Hamiltonian 149 9.4.1 Solution for IO) as initial condition These are the recurrence relations of the Chebyshev polynomials of the second kind ([Abramowitz, 1972]), and we can write As we mentioned before, it is possible to construct Susskind-Glogower coherent states as eigenfunctions of the interaction Hamiltonian 11/J (t; .;)) = L e-iEtun (.;)In), (9.45) (9.38) n=O where 77 is the coupling coefficient. Hamiltonians like this may be produced in where ion-traps [Wallentowitz 99]. E=2ry( (9.46) 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 However (9.45) does not satisfy the initial condition 11/J (t = 0)) = IO). Moreover, beams. Here, physical couplings take place via the Susskind-Glogower operators. the solution (9.45) has the parameter.; and, as we see from (9.10), we should We write the eigenfunctions of the Hamiltonian (9.38) in the interaction picture not have another parameter, except the time t. as 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 11/J) = LCnln), (9.39) of the Fourier transform kernel; so, we need to propose a .;-dependent function n=O and integrate it over all .;, in order to obtain a solution where time is the only and we have variable. We have then, 2 fi 11/J) = 7] L Cn In- 1) + 7] L Cn In+ 1). (9.40) 11/J (t)) = f 1= d.;P (.;) Un (.;) e-i ryf;t In). (9.47) n=1 n=O n=O -oo Now, changing the summation indexes, we obtain 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 n=O n=l (9.41) [Campbell 1948] = 7]C1 IO) + 7] L (Cn+l + Cn-1) In). n=1 (9.48) Then where F {} is the Fourier transform and (9.42) ' -1 1 n=l n=l :s:.; :s: (9.49) rect (~) = { ~ , otherwise. Comparing coefficients with same number states of the sum, we have l,From (9.48) we write (9.43) and (9.50) (9.44) 150 Nonlinear coherent states for the Susskind-Glogower operators Eigenfunctions of the Susskind-Glogower Hamiltonian 151 Using definition (9.49), and making k = n- 1, we obtain 9.4.2 Solution for lrn) as initial condition At this point, we have managed to construct Susskind-Glogower coherent states, (9.51) 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. With Equation (9.51) it is possible to solve the integral in (9.47). Writing the Using the recurrence relation for the Chebyshev polynomials and the result weight function as (9.51), it is possible to generalize Susskind-Glogower coherent states to an arbi trary lm) initial condition, where m = 0, 1, 2, .... P(~) = ~~rect (~), (9.52) L,From Equation (9.47), we have that substituting it into Equation (9.47) and using now (9.51), we get 11/J (t = 0)) = f leo d~P (~) Un (~)In), (9.57) n=O -co 2 co 11/J (t)) =-Lin (n + 1) Jn+l ( -27]t) In). (9.53) and considering the function -27]t n=O Considering the odd parity of the Bessel functions, we finally obtain (9.58) 1 co and the well known Chebyshev polynomials of the second kind orthonormal 11/J (t)) =-Lin (n + 1) Jn+l (277t) In). (9.54) 7]t n=O condition We see that 11/J (t)) in Equation (9.54) depends only on t, and considering that , n = 2, 3, 4, ... we get lim Jn (2ryt) = { 0 (9.55) 2ryt--70 27]t ~ n = 1, 11/J(t=O)) = LOnmln). (9.59) we can verify that n=O To obtain the solution for them-state, let us consider the particular cases m = 0 11/J (t = 0)) = IO). (9.56) and m = 1. Form= 0, 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 00 2 2 previous section x was only a parameter, now it represents something physical. = L leo -~Uo (0 Un (~) e-i ryEt rect (~) d~ In) It may be related to an interaction time, for example, in the motion of a trapped n=O -co 7r 2 (9.60) 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). 153 152 Nonlinear coherent states for the Susskind-Glogower operators Eigenfunctions of the Susskind-Glogower Hamiltonian Using (9.50), and making n--+ n + 1, W8 have making k = n + 2 (9.61) 1 1 1'1{1 (t)) -1 = f {-ik- ~ (k- 2) Jk-2 (27]t) + ik- [~kJk (27]t)]}' (9.68) m- n=O 7]t 7]t Using the recurrence relation xln-1 (2x) + xln+1 (2x) = nln (2x), (9.62) using the recurrence relation (9.62), we write and changing the summation index, we finally obtain 1 1'1{1 (t))m=O [in In (27]t) +in 1n+2 (27]t)Jin). (9.63) -ik- ~ (k- 2) Jk-2 (27]t) } L 1 n=O +ik- [Jk-1 (27]t) + Jk+1 (27]t)] Form= 1, we have -ik-1 [ ~ (k- 2) Jk-2 (27]t)- Jk-1 (27]t)] } +ik-1Jk+1 (2'T)t) (9.69) 3 ik- [ ~ (k- 2) Jk-2 (27]t)- Jk-1 (27Jtl] } +tk-1Jk+1 (27]t) 2 1 (9.64) ik- - [ ~ (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 the recurrence relation for the Chebyshev polynomials of the second kind, we write 00 2 1'1{1 (t))m=1 = L 11 -~ CUn-l (~) + Un+l (~)) e-i 2 1)~td~ In). (9.65) n=O - 1 1T Using again the recurrence relation (9.62), we finally write Considering (9.50) for the first integral, and making n = n + 2 for the second term, we have 00 1 1 1 1 1'1{1 (t))m=1 = L [in- 1n-l (27]t) + in+ 1n+3 (27]t)]ln). (9.71) 1'1{1 (t))m=l = [in+l (n + 2) 1n+2 (27]t) + in- nln (27]t)]ln), (9.66) L t n=O n=O 7] We now construct a different expression for the above Equation (9.66). Let us write it in the following way, Following the same procedure, it is easy to prove that form= 2, 2 2 1'1{1 (t))m=2 = L [in- ln-2 (2'T)t) + in+ 1n+4 (27]t)]ln). (9.72) n=O 154 Nonlinear coherent states for the Susskind-Glogower operators Time-dependent Snsskind-Glogower coherent states analysis 155 Rewriting results (9.63), (9. 71) and (9. 72), 9.5.1 Q junction Considering the definition Q = ~ (a/,0/a) and writing p in terms of the con structed states (9.74), we write the Q function for the time-dependent Susskind /1/J (t))m=O = L [in-O Jn-0 (2TJt) + in+O Jn+0+2 (27]t)]/n), n=O Glogower coherent states as 11/J (t))m=l = L [in-l Jn_I(27]t) + in+l Jn+1+2(27]t)]/n), (9.75) n=O (9.73) 2 2 /1/J (t))m=2 = [in- Jn-2 (27]t) + in+ Jn+2+2 (2TJt)]/n), L Figures 9.4 and 9.5 show the time-evolved Q function of Susskind-Glogower co n=O herent 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 fea tures of the constructed states are summarized in Figure 9.4. Susskind-Glogower It is easy to see that the solution for the m-initial condition is 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 infi /1/J (t))m = L [in-m ln-m (27]t) + in+m Jn+m+2 (2TJt)]/n) · (9.74) nite squeezed states in amplitude) and displaced number states do. n=O An interesting result is that no matter what initial condition we choose, Susskind Glogower coherent states eventually fill such a circle. This is very interesting We have constructed a new expression for nonlinear coherent states and that result because such a distribution would approach a displaced number state, we call Susskind-Glogower coherent states. We have managed to construct an which has highly nonclassical features. expression that allows us to study the time evolution of Susskind-Glogower co herent states for an arbitrary /m) initial condition. We also found the physical interpretation of the parameter x (used in previous sections) as a normalized 9.5.2 Photon number distribution interaction time with respect to the coupling strength, i.e., x = 7]t. We analyze To complete the description of the nonclassical features that we observed from now the nonclassical features of the constructed states, so we have to make use of the Q function, we show in Figures 9.6 and 9.7 the photon number distribution the methods previously mentioned, these are the Q function, the photon number of the Susskind-Glogower coherent states considering the same conditions of distribution and the Mandel Q-paramcter. Figures 9.4 and 9.5. Time-dependent Susskind-Glogower coherent states photon number distribution for them-initial condition is given by 9.5 Time-dependent Susskind-Glogower coherent states analy sis (9.76) 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 9.5.3 Mandel Q-parameter methods. First, we analyze their behavior in phase space via the Q function; then, because we want to analyze amplitude squeezing and quantum interfer As we want to know the time domain for which the constructed states exhibit ences, we show the photon number distribution of the constructed states, and amplitude squeezing, and moreover, we want to know when the states are max finally, as we want to know when the constructed states are maximally squeezed, imally squeezed, we obtain the Mandel Q-parameter for the time-dependent we show the Mandel Q-parameter. Susskind-Glogowcr coherent states. 156 Nonlinear coherent states for the Susskind-Glogower operators Time-dependent Susskind-Glogower coherent states analysis 157 0.1!4- -w -lD -10 -I~~ -Ill Fig. 9.4 Susskind-Glogower coherent states Q function for (a) 17t = 1; (b)17t = 2.32; (c) 17t = 5 Fig. 9.5 Susskind-Glogower coherent states Q function for (a) ryt = 1; (b) 17t = 2.32; (c) and (d) 17t = 20, with the initial condition 11). 17t = 5 and (d) 17t = 20, with the initial condition 110). and We have that 2 2 2 (n ) = f n [Jn-m (2·f)t) + (-l)m Jn+m+2 (2f)t)] . (9.78) 00 n=O 2 (ii) L n [Jn-m (2rJt) + (-l)m Jn+m+2 (2·f)t)] , (9.77) n=O Substituting in the definition of the Mandel Q-parameter, Equation ( 8.102 oJ1 159 158 Nonlinear coherent states for the Susskind-Glogower operators Time-dependent Susskind-Glogower coherent states analysis 0.4 0.16 0.3 0.14 0.4 0.3 0.12 0.3 0.2 P(n) P(n) 02 P(n) 0.2 0.1 0.1 0.1 o+-~~~r-~.-~.-~.-r-o 0~~--~--~--~.-~~~ 0~~--~--~.-~.-~.-~ 10 20 30 40 50 60 I0 20 30 40 50 60 0 10 20 30 40 50 60 0 I 0 20 30 40 50 60 0 0.10 0.09 0.10 0.25 0.08 0.08 0.08 0.20 0.07 0.06 0.06 0.06 P(n) P(n) 0.15 P(n) 0.05 P(n) 0.04 0.04 0.10 0.04 0.03 0.02 0.02 0.05 0.02 0.01 a~~~~.-~~~~~~~ 10 20 30 40 50 60 0 10 20 30 40 50 6C 10 20 30 40 50 60 10 20 30 40 50 60 Fig. 9. 7 Susskind-Glogower coherent states photon number probability distributions for (a) 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 110). '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 something similar to a displaced number state. that (a) shows the Q-parameter for the SG coherent states where the maximum 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) squeezing happens when Tjt = 2.32. From the others, we cannot observe squeez ing, because the initial condition is a fully squeezed state in amplitude, a number helps us to show that nonlinear coherent states may be modeled by propagating state; however, we obtain that an initial number state eventually transforms into light in semi-infinite arrays of optical fibers. 161 160 Nonlinear coherent states for the Susskind-Glogower operators Classical quantum analogies Rewriting (9.76) and using the normalized interaction time with respect to the , ....-·-·------·-·-·-· coupling coefficient 1), i.e., x = 'f]t, we have ,· 2 Pm (n,x) = l·in-mJn-rn (2x) + in+rnJn+rn+2 (2x)l . (9.81) / ·'· As Equations (9.80) and (9.81) are the same, we conclude that the photon num Q(t) / ------ber distribution for the Susskind-Glogower coherent states may be modeled by / .,------.' the intensity distribution of propagating light in semi-infinite arrays of optical I I fibers. We have found a new relation between quantum mechanical systems and I / classical optics. • I ./ .. ····1. ·"' 20 25 -I 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 entangle ment 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 Susskind Glogower 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) 162 Nonlinear coherent states for the Susskind-Glogower operators 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 (A.2) we rewrite (A.1) as (A.3) Now we use the transformation p= exp[(S + L)t]p, (A.4) to obtain dp A AA dt = exp[-ixRt- 2rt]J,O, (A.5) with (A.6) 163 164 Master equation Master equation describing phase sensitive processes 165 It is easy to show that R and J commute, so that we can finally find the solution with 11 = cosh(r) and v = - sinh(r)ei ¢=-, tanh(2r) = ~' (A.l6) (A.7) 11 +12 we obtain A.2 Master equation describing phase sensitive processes (A.l7) One of the most general Master equation is one that describes phase sensitive with processes: (A.18) dA 4 P_~ I' (A At)A and dt- ~ {kLk a, a p, (A.8) k=l i2 = "Yllvl 2 + /2/12 + J-1Vh3lei1> + Mv*l13le-i1>. (A.l9) where Equation (A.l7) has been solved in Chapter 5. (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¢) 2 S(~)A = exp (Ca- - - -catz)- , (A.13) 2 2 we arrive to the equation for the transformed density matrix (A.14) where (A.15) 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 el ements 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 2 (3 - ~ - )..2 aat = 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 ) . (B.3) 0 -f3n ' with f3n such that (3~ = ~ + )..2 (n + 1). We can then write the eigenvectors matrix f3n+% 2/3n_ P~ ( (B.4) )..a,t_L -Aii'-.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 Hamil tonian with (right) "eigenvalue" f3n for the first column and -{3r, for the second Appendix C column. We now find the (left) inverse matrix toP, Interaction of quantized fields (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=at 1 ' at~= 1 -IO)(OI. (B.6) It is now easy to show that PDT= fi and the evolution operator is therefore In this appendix we will look at the interaction of many fields. First, we will (B.7) 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 and the exponential of the diagonal matrix is straightforward to calculate. polynomials for eigenvectors of the matrices that diagonalize the Hamiltonian for the interaction of many fields. B.2 A traditional method Looking at (B.2), we note that the the relevant states for the interaction are C.l Two fields interacting: beam splitters 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 Consider the Hamiltonian of two interacting fields (as in all the book, we set wave function of the form n= 1) 11/J(t)) = L [Cn(t)le)ln) + Dn(t)lg)ln + 1)] (B.S) (C.1) n=O and substitute it into the Schrodinger equation to obtain a system of differential This interaction occurs in beam splitters; however, it may also be obtained by equations for Cn and Dn the interaction of two quantized fields with a two-level atom, when the fields are . 6. far from resonance with the atom; in this case, an effective Hamiltonian may be iCn(t) = 2 cn + >-.vn + 1Dn(t) obtained, which has the form of the above Hamiltonian [Prado 2008]. . 6. By transforming to the interaction picture; i.e., getting rid off the free Hamilto iDn(t) = >-.vn+TCn- 2Dn(t). (B.9) nians, we obtain The solution to this system of equation is the exponential of the matrix of the system. (C.2) with 6. = wa - wb, the detuning. It is useful to define normal-mode operators by [Dutra 1993] (C.3) 169 171 170 Interaction of quantized fields Generalization to n modes with passing the exponential in the above equation to the right, applying it to the vacuum states and using the following property 5 = 2.\ (C.4) (C.l2) J2D(D-~)' and D = ~ 2 + 4.\: the Rabi frequency. The annihilation operators A and .4 we obtain v 1 2 are just like a and b, and obey the commutation relations D A. ([a5 + f3"Y]e-ill 1 t)D A. ([a"Y- (35]e-iMt)IO) A. IO) A. IV>(t)) 1 2 1 2 (C.5) I[a5 + /3')']e-ifllt) A.ll [a"Y- (35]e-ifl2t)A.2. (C.l3) 4 moreover, the normal-mode operators commute with each other Equation (C.l3) shows that in the new basis, coherent states remain coherent during evolution. (C.6) By transforming back to the original basis, using again property (C.l2), we In terms of these operators, the Hamiltonian (C.l) becomes 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.7) (C.l4) with /-l1,2 = (~ ± D)/2. i.e., coherent states remain coherent during evolution. This will be used next Up to here, we have translated the problem of solving Hamiltonian (C.l) into section as the building block for the interaction of many modes. 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 C.2 Generalization to n modes one basis to the other, we note that the vacuum states in both systems, IO)aiO)b and IO) ,4)0) A_ , differ only for a phase [Dutra 1993]. First note that 2 Consider the Hamiltonian of the interaction of k fields (C.S) (C.l5) 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 This Hamiltonian may be produced in waveguide arrays. From the Hamiltonian (C.9) above, we can produce the following matrix If we consider coherent states as initial states for the interaction, we obtain the WI A21 Anl evolved wavefunction A12 W2 An2 .\13 .\23 An3 l (C.l6) 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) [ Aln A2n Wn where the Dc(E) = exp(Ect- E*c) is the Glauber displacement operator [Glauber We can rewrite the Hamiltonian in the form (C.7), that is 1963b]. From (C.3), we can write the operators a and bin terms of the operator A1 and A2, and write (C.lO) as (C.l7) 173 172 Interaction of quantized .fields A particular interaction such that that may be re-expressed in the compact form (C.l8) (C.25) where we have defined the normal-mode operators Ak as ]' n Wn Ak = Lrk;fi;, (C.l9) i=1 with with Tki a real number. Equation (C.l8) implies that (C.26) t rkiTmj [ai,a}] = trkirmi = 0. (C.20) i,j=O By defining the vector 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 (C.21) matrix. 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 C.3 A particular interaction form the matrix Now we study a particular interaction, namely when Aij = A if j = i + 1 or T21 j = i - 1 and it is zero otherwise. The frequencies wi are left arbitrary. The r12 T22 fn2 Hamiltonian governing this interaction then has an associated tridiagonal matrix [ rn Tn> of the form R= l (C.22) 0 T1n T2n Tnn (C.27) If we combine Equations (C.l5), (C.l7) and (C.l9), we obtain the system of equations (C.23) 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, (C.24) 174 Interaction of quantized fields and the normalized eigenvectors are simply Appendix D ~~~~;~ Quantum phase l (C.29) [Fn-,(M,) with Nj = I:;~:~ F'f (J-Lj), such that the matrix elements of the matrix R are given by (C.30) D.l Turski's operator C.4 Coherent states as initial fields Classically we may decompose a complex c-number, A, in amplitude and phase by simply writing A = rei¢>, with r = IAI and 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 ¢ = -iln~, (D.1) simply the direct product of coherent states r (C.31) where it is implied that we have chosen the principal branch of the multi-valued logarithm function. with /I(t) = (f'1. ae-il'tt,rz. ae-i~' 2 t, ... ,r:,. ae-il'nt) and the vector a= In correspondence to the classical form (D.1), Arroyo-Carrasco and Moya-Cessa (a1, a2, ... ,an) is composed by the coherent amplitudes of the initial wave func [Arroyo-Carrasco 1997] proposed the Hermitian operator tion. 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 A i A ( At ¢ = -- lim D(x)ln 1 +-a) D (x) +H.C., (D.2) modifies their amplitudes. If we choose the interaction constants to be .A 1j f. 0 2 x-+= X 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 where a and at are the annihilation and creation operators for the harmonic with the interaction of one field with n -1, one of them in a coherent state with oscillator, respectively, D(x) = ex( at -a) is the displacement operator, and x is amplitude a1 and the rest in the vacuum. Therefore, the most likely situation a real parameter that tends to infinity to ensure convergence of the series we have is the coherent state decaying towards the vacuum while keeping its coherent form. a) = (-1)k-1 (a)k ln 1+- =2:-- - (D.3) ( X k=l k X 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, 175 176 Quantum phase A formalism for phase 177 and insert it into (D.2) it yields and (D.4) (D.ll) that may finally be written as 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.5) (D.12) Again, choosing the principal branch in the above equation, we can rewrite (D.5) as and (D.6) (D.l3) where e = arg(a). such that we can calculate the phase variance, D.¢= (arg2 (ii))- (arg(ii)) 2 for a Ill number state In), yielding the result II I Of course, as any operator that lives in the whole Hilbert space, ¢ obeys the equation of motion b."-=~ (D.l4) '-f/ 3' (D.7) and where we have used 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 (D.15) motion, the harmonic oscillator Hamiltonian should be defined also in a finite dimensional Hilbert space. We have obtained the correct phase variance expected for a state of undefined We can calculate the average value of the argument of the operator (D.6), given phase; therefore, the operator given by Turski would lead to a phase formalism a wave function 11J'I), as given by the (radially) integrated Q-function. (D.8) D.2.1 Coherent states where If we consider the coherent state Q(a) = ~l(ai1J'IW (D.9) (D.16) 7r is the Q-function. using (D.lO), we can calculate its phase properties, obtaining D.2 A formalism for phase (arg(ii)) = 77· (D.l7) A formalism for phase could he introduced based on (D.6), as follows If we usc (D.2), we obtain the same result; i.e., (arg(ii)) = (¢) = j arg(a)Q(a)d2a, (D.lO) (D.18) 178 Quantum phase Radially integrated Wigner function 179 so that also introduce them in terms of other quasiprobability distribUti.ons, namely, the Wigner function [Garraway 1992] 1f2 2 CXl n+m r(n+m + 1) /}.¢coh=-+4e-p L L (-1)n-mP -2- , (D.19) 3 m=On=m+l n!m! (n- m)2 (D.21) where r(x) is the well-known Gamma function. Note that because the formalism and 3 2 (¢k)w = j argk(a)W(a)d a. (D.22) Calculating phase fluctuations for the coherent state (D.16), vsitlg the above expressions, leads to 2 2 2 CXl CXl ( 12 )n+m r(n+m + 1..) /:)."' = 2._ + 4 -2p """"' """"' (-l)n-m V L-P ~ (D.23) 'I'W 3 e ~ ~ n!m! (n-mF ' m=On=m+l where we have used that (D.24) In Figure D2, we plot f}.¢w together with the expression for the Pbase variance o~~--~--~=;~~~~~~~~~ 0 2 3 4 5 3 p 2.5 Fig. D.l We plot Li¢coh (solid line), and 1/2p2 (dashed line) as a function of p. 2 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 1.5 uncertainty relation /:).cj;>-1-=~. (D.20) - 2/:).n 2p2 This may be corroborated in Figure Dl, where we plot /}.¢coh as a function of 0.5 2 p, together with the expression 1 /2p . 0 2 3 D.3 Radially integrated Wigner function p In the former section, Equations (D.6) and (D.8) were used to introduce the Fig. D.2 We plot 1/(4p2 + 3/7r2 ) (solid line), fl.¢~;[; (dash line) and ti¢w (dot-dash line) as calculation of phase properties in terms of the Q-function; however, one can a function of p. 180 Quantum phase for coherent states using the Pegg-Barnett formalism, that can be written as [Pegg 1988; Barnett 1989] Appendix E 7r2 2 oo CXJ n+m t,¢P-B = _ +4e-p )n-m_P___ 1_ L L (- 1 (D.25) Sums of the Bessel functions of the first coh 3 m=On=m+l vnr:mf (n- m)2. Notice that the Heisenberg uncertainty relation in the Pegg-Barnett case leads kind of integer order 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. 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 Man del 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 com plete 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 182 Sums of the Bessel functions of the first kind of integer order I83 I To demonstrate (E.1), we will need the well known Jacobi-Anger expansions for We multiply now for the complex conjugate of (E.4) and obtain the Bessel functions of the first kind ([Gradshteyn 1980], page 933, [Magnus 1966], page 361; [Abramowitz, 1972], page 70), k,l=-oo (E.9) CXl eixcosy = :2: inJn(x)einy (E.3) = Bn (g' (y) ,g11 (y), ... ,g(n) (y)). In;egrating both sides of the above equation from -1r to 1r, and using that and J_1r ei(k-l)Ydy = Okz, we arrive to the formula we wanted (E.4) k=I Using expression (E.3), we can easily write (E.10) --~(-1)" /7r B 2v ( g '()y,g "() y, ... ,g (2v)( Y))d y. (E.5) In particular, as the complete Bell polynomials for n = 2 and n = 4, are 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 _H2(XI, X2) =xi+ X2 (E.ll) composition and dn dxn f (g (x)) = B4(xi, x2, X3, x4) =xi.+ 6xix2 + 4xix3 + 3x~ + x 4, (E.12) (E.6) it is very easy to show that = t f(k) (g (x)) · Bn,k (g' (x), g" (x), ... , g(n-k+l) (x)) , k=O fk2Jf (x) = ~x2 (E.13) where Bn,k (xi, x2, ... , Xn-k+I) is a Bell polynomial [Bell1927; Boyadzhiev 2009; k=I Comtet 1974], given by and that Bn,k (xi, X2, ... , Xn-k+I) = 4 2() 3 4 1 2 ~L k Jk x = -x + -x . (E.14) n! k=I 16 4 = :2:.JI·J2 ,. '· · · ·Jn-k+I· . 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Zhu X. y Qi D., Journal of Modern Optics 39, 291 (1992). annihilation operator, 7 photon distribution, 13 Araki-Lieb inequality, 84 uncertainties, 14 atom and field entropy operator, 88 concurrence, 90 atom-field interaction, 59 creation operator, 7 I dispersive interaction, 65 entropy, 83 Dirac notation, 1 mixed classical and quantum bra, 2 interactions, 66 braket, 2 mixed classical and quantum ket, 1 t: interactions Hamiltonian, 66 displaced number states, 14 purity, 83 excitation number, 14 quantum interaction, 62 displacement operator, 10, 21, 33 semiclassical interaction, 59 properties, 10 semiclassical interaction Hamiltonian, 60, 62 entropy, 81 atomic entropy, 89 entropy of the damped oscillator atomic entropy operator, 88 cat states, 93 atomic inversion, 61, 63 entropy operator, 89 coherent states, 63 orthonormal states, 91 coherent states superposition, 64 Ermakov equation, 44 squeezed states, 64 solution for the step function, 47 Baker-Hausdorff formula, 4, 31, 33, 44 field entropy operator, 89 Fock states, 1, 8, 21 cavity losses cat states, 76 Glauber-Sudarshan P-function, 32, 34, coherent states, 75 37 number states, 75 characteristic function, 25, 29 Hadamard lemma, 4 coherent states, 10, 21 harmonic oscillator, 6 averages, 21 eigenfunctions, 1 excitation number, 13 Hamiltonian, 6 189 190 Index Index 191 vacuum state, 8 quasiprobabilities and losses, 102 Hamiltonian, 43 Husimi Q-function, 33, 34, 37, 52 quasiprobability distributions, 23 number operator, 43 integral forms, 37 Schri:idinger equation, 45 ion motion in a trap reconstruction, 97 step function, 46 micromotion, 119 reconstruction in a lossy cavity, 99 Time dependent Susskind-Glogower secular motion, 119 reconstruction in an ideal cavity, 97 nonlinear coherent states, 154 relations between them, 34 Husimi Q-function, 155 Jaynes-Cummings Hamiltonian, 62 Mandel Q-parameter, 155 Rabi frquency, 123 photon number distribution, 155 Kingdon trap, 109 reduced density matrices two-level atom-field interaction, 88 properties, 84 ladder operators, 7 ringing revivals, 64 von Neumann entropy, 81 commutation relations, 8 rotating wave approximation, 60 von Neumann equation, 4 Lamb-Dicke parameter, 127 Lewis-Ermakov invariant, 44, 45 Schri:idinger cat states, 54 Wigner function, 23, 34, 36, 37, 52, 56, 76, 91, 98, 103 eigenstates, 45 photon distribution, 55 coherent states, 31 1,,11,,,1 Wigner function, 56 direct measurement, 97 Master equation Schri:idinger cat states displaced number states, 32 1,1 cavity losses, 74 purity, 95 expected values, 28 finite temperature, 77 squeeze operator, 50 number states, 32 zero temperature, 73 squeezed states, 50 properties, 28 measuring field properties Husimi Q-function, 51 series representation, 30 phase properties, 107 photon distribution, 50 symmetric averages, 29 squeezing, 105 Wigner function, 52 minimum uncertainty states, 46 Stirling numbers, 17, 20 mixed states, 81 superoperator, 74 Susskind-Glogower Hamiltonian, 148 number operator, 8, 16 eigenfunctions, 150, 154 number states, 1, 8 Susskind-Glogower nonlinear coherent number-phase Wigner function, 39 states, 138, 142, 150, 154 coherent states, 40 classical analogies, 160 superposition of number states, 41 Husimi Q-function, 143 Mandel Q-parameter, 145 ordering of ladder operators, 16 photon number distribution, 143 anti-normal ordering, 16, 20 Susskind-Glogower operators, 16, 39, 62 normal ordering, 16, 17 approximated displacement operator, 139 parity operator, 30 commutation relations, 138 Paul trap, 110, 111 commuting properties, 138 frequency dependent of time, 126 exact displacement operator, 140 frequency independent of time, 120 non-unitary property, 138 ion motion, 115 Penning trap, 109 thermal distribution, 57 phase states, 14 time dependent harmonic oscillator, 43 pure states, 81 annihilation operator, 43 purity, 82 creation operator, 43 Wide Web at: