A reformulation of Hamiltonian dynamics. A case study of the Jaynes-Cummings model.

(1)

Chapter 1 The quantum-classical divide

Standard quantum theory is a remarkably successfull theory. Yet after a century of quantum mechanics the debate about the relation between quantum physics and the familiar classical world continues. In the last three decades there has been a growing focus of research on these fundamental aspects.

The interpretational problems of quantum mechanics reside in the inability of the theory to provide a natural framework for our prejudice about the workings of the Universe. More precisely, the natural framework of the theory, i.e. the Hilbert space, admits by the superposition principle arbitrary linear combinations of any state as a possible quantum state. The dynamics of any state is described in a deterministic way by the Schr¨odinger equation and thus, given the Hamiltonian of the system, one can in principle compute the state of the system at any time. This deterministic evolution has been verified in experiments. Besides this unitary evolution of the system in the Hilbert space, there is another kind of transformations which are unpredictable and non-reversible. Examples of these kind of transformations are those that systems experience as a result of measurements. The understanding of this class of transformations is fundamental to resolve the clash between the principle of superposition and the classical reality in which this principle seems to be violated. In addition to this irreversible and unpredictable transformations there are also other related features that distinguish quantum from classical mechanics. Among these, the phenomenon of entanglement, as described in the EPR paradox (EPR35), which seems to violate the principles of local causality. The measurement problem thus plays an essential role in the debate concerning the interpretation of quantum theory and has a long history.

The founders of quantum mechanics insisted that measurement results needed to be expressed in classical terms. The attempted reconciliation of the two descrip-tions raised many debates and still has not come to an end. The existence of the uncertainty relations entailed that kinematical concepts of classical mechanics can-not be generally applied to the quantum domain and have no direct counterpart

(4)

1 – The quantum-classical divide

in quantum dynamics. One of the questions raised concerns the existence and the position of the border between the two descriptions. The Copenhagen interpreta-tion, proposed by Neils Bohr in 1928, was the first attempt to give a solution to this problem. Bohr’s solution was to draw a border between the classical and the quantum insisting that a classical measuring apparatus was needed to perform a measurement. Thus, according to this point of view, the domain of quantum theory needed to be restricted by some not very well defined borders. The key feature of the Copenhagen interpretation can be resumed in the existence of mobile bound-aries between the two different domains. This first attempt to interpret the nature of quantum theory left many physicists unsatisfied.

The problem of the classical limit can be considered at the heart of the inter-pretation problem. It is a common approach to consider classical mechanics as a limiting case of quantum mechanics in analogy to what can be done to recover clas-sical mechanics from relativity. For example, the motion of the center of mass of a macroscopic object could be described by the evolution of a narrow wavepacket well localized in position and momentum. For large masses, it can be proven that the spreading of the wave packet is negligible and Ehrenfest’s theorem seems to allow the derivation of Newtonian mechanics as a limiting case. These standard arguments are regarded as insufficient: it does not explain the appearance of the majority of macroscopic objects in the classical domain where the superposition principle fails. The Schr¨odinger cat paradox cannot be resolved in such a way.

Research concerning the relations between the classical and the quantum me-chanical world has seen many recent development. The role of decoherence, as a consequence of the openess of quantum systems, brought some further insights on the emergence of effective classical reality. The assumption of a closed macroscopic system, currently used in the classical domain, in no longer justified. Systems which we usually consider as macroscopic cannot be considered isolated because of the way they interact with the natural enviroment. The formalism of quantum theory uses the concept of a density (sub)matrix to characterize parts of a larger system, described generally by an entangled quantum state. The explanation of the classical behaviour of a macroscopic system can then be found in the dynamics of the density matrix. The superposition of macroscopically different properties can be shown to disappear on a short time scale. This is the essence of what we call decoherence.

Decoherence has found attention in many areas of physics. The understanding of how the enviroment distills the classical essence from quantum systems combines two observations, as Zurek (Zur03) pointed out. In quantum physics reality can be attributed to the measured states. Information transfer usually associated with meausurements is a common result of almost any interaction of a system with the enviroment.

Entanglement between the enviroment and the system seems to give further in-sight about the measurement problem and the appearence of classicality. Entangled

(5)

1.1 – Emergent Quantum mechanics?

states, particularly in the macroscopic domain, should be the most natural situa-tions, if we expect quantum theory to be universally valid. In contrast with the orthodox Copenhagen interpretation, decoherence aims to draw a unique picture of reality in terms of quantum states. The way the smooth transition from the micro to the macro world can be achieved in this context has been studied in several cases. The emergence of the classical world through enviromentally induced decoher-ence may have solved the problem of understanding the quantum to classical tran-sition. On the other hand, another problem has not been solved yet. In fact, we are not able in the context of quantum theory to describe the measurement process which leads to a classical apparatus reading (WZ83). This problem has not been solved and it has given rise to a number of dynamical wave function collapse or reduction models, with no generally accepted completion of quantum theory.

Regarding this particular problem, there is an increasing need to better un-derstand or change the foundations. There has been a growing attention to the possibility, or impossibility, of deriving quantum theory from more fundamental and deterministic dynamical structures. Deterministic theories are often referred to as hidden variables theories.

There is a long standing suspicion towards determinism, that might underlie quantum mechanics, which mainly steams from Bell’s inequality (Bel87) and its various generalizations.

Examples of systems that can be described as quantum mechanical and present a deterministic underlying dynamical model have been proposed by ’t Hooft (tH02) (tH88) (tH06). It has also been argued about the possibility of resolving the per-sistent clash between general relativity and quantum theory by questioning the fundamental character of the latter.

Local hidden variable theories are usually formulated in such a way that Bell’s inequalities for experiments with entangled particles can be derived, as soon as these particles are spacelike separated. Since quantum-entangled particles violate Bell’s inequalities, while they can easily be spacelike separated, it is generally concluded that no local hidden variable theory can exist that reproduces typical quantum phenomena. However, attempts to reproduce quantum mechanics from underlying deterministic models that violate Bell’s inequalities have been made (tH09).

1.1 Emergent Quantum mechanics?

A characteristic feature of orthodox quantum theory is that it is a probabilistic theory, where the probabilities are postulated, and are not emergent from unobserved deterministic phenomena at a deeper level. Statistical mechanics is a typical example of a theory where probabilties are emergent.

(6)

1 – The quantum-classical divide

two completely different mathematical languages. The classical physical observ-ables are functions of the phase space variobserv-ables of the system, while in quantum mechanics they are self-adjoint operators acting on a Hilbert space. Since the early days of quantum theory there have been attempts, firstly made by Koopman and von Neumann (Koo31) (vN32), to reformulate classical and quantum mechanics in similar forms.

Classical ensemble theory with its Liouville equation implies the absence of a stable ground state when rewritten as Hilbert space theory with an analogue of the Schr¨odinger equation (tH02). The similarity between the classical Liouville equation and the Schr¨odinger and von Neumann equation have been studied, considering this as a starting point for attempts to derive quantum physics as an emergent theory from classical dynamics (Elz05) (Elz08a).

In the operator approach to classical statistical mechanics developed by Koop-man and von NeuKoop-mann the Liouville operator is Hermitian, but unlike the case of a quantum mechanical Hamiltonian, its spectrum is not bounded from below. As pointed out by t’Hooft (tH02), if there exists a classical deterministic theory under-lying quantum mechanics, then an explanation must be found for the fact that the generator of the time evolution, i.e. the Hamiltonian, is bounded from below. The mechanism to produce such a constraint on the Hamiltonian has been argued to be due to information loss. Deterministic models for quantum mechanical objects in which fundamental information loss is an essential ingredient have been derived (Hoo07) (Elz08b). The idea of considering quantum mechanics as the low energy limit of some more fundamental deterministic dynamics has been widely reconsid-ered. According to t’Hooft, dissipation, i.e. information loss, which would occur at Planck scale in a regime of completely deterministic dynamics, plays a fundamental role in the appearance of the quantum mechanical nature of the world observed in experiments. An enormous amount of information gets lost in the process of coarse graining from 1019 GeV (Planck scale) to, say, 103 Gev leading to the formation of equivalence classes of “classical” states. In this sense, many distinct states of the Planck scale dynamics would be included in one state of the observational-scale dynamics.

Furthermore, t’Hooft’s existence theorem (Hoo07) shows that generally the evo-lution of all quantum mechanical systems characterized by a finite dimensional Hilbert space can be captured by a dissipative process.

A growing number of deterministic models for quantum mechanical systems based on information loss, coarse graining, or dissipation mechanism have been proposed (Elz06) (BJK05) (Adl05) (BJV01) (Wet09) (Elz09).

It has been argued in favour of such model building that it may lead to a new approach in trying to understand and possibly resolve the persistent clash between general relativity and quantum theory. The realization of quantum mechanics as a “classical” statistical system may also open new perspectives on the conceptual

(7)

1.2 – Aim and structure of the thesis

interpretation of experiments based on entanglement and give rise to further devel-opment in the field of quantum information and its foundations.

The need to better understand the foundations of quantum theory has been a particularly interesting subject. As pointed out before, the problem of understanding the appearence of the classical world from the quantum picture has been solved through enviromental decoherence. Yet, the problem of understanding the way a measurement on a quantum system brings with it a specific physical outcome is open. Furthermore, quantum theory doesn’t seem to be able to consistently describe this process (Adl05) (WZ83) and the possibility of an emergent quantum theory may lead to interesting insight concerning this problem.

There is a widespread negative attitude towards the possibility of deriving quan-tum from classical physics which relies on Bell’s inequalities. However, altough being clear that quantum mechanics violates this inequalities, a common prejudice is that Bell’s theorem should be true at all scales. As observed by t’Hooft (tH03), this need not to be the case, because such fundamental concepts as rotational sym-metry, (iso)spin, or even Poincar´e invariance, on which the usual forms of the Bell inequalities are based, may simply cease to exist at the Planck scale.

1.2 Aim and structure of the thesis

In the context of the ideas expressed in the preceding section, it has been shown (Elz08a) that some familiar aspects of quantum mechanics can be generated by an attractor mechanism, obtained by a generalisation of the Liouville equation. Emer-gent quantum states evolving according to the Schr¨odinger equation, implicitly ex-pressed in the von Neumann equation, are obtained and expectations of observables are shown to agree with the Born rule, which is not imposed a priori.

Furthermore, the classical Liouville equation, after suitable tranformation, can be written in a way that almost reproduces the quantum mechanical von Neumann equation. Indeed, an unusual superoperator appearing in the transformed Liouville equation gives rise to the only difference between the quantum and classical dy-namics. In this way, this approach also gives rise to the possibility of looking more carefully into the common and the distinctive features of classical and quantum dynamics.

In Chapter2, we briefly review the basic ideas underlying this reformulation of Hamiltonian dynamics. In this same chapter we apply this approach to reformulate the problem of a hydrogen atom interacting with the electromagnetic field in the Coulomb gauge. We thus identify the terms that distinguish the classical dynamics from the quantum mechanical one for this particular case. The need for this reformu-lation of electrodynamics is based on our interest in studying the Jaynes-Cummings model from this new viewpoint.

(8)

In Chapter 3, we will derive the Jaynes-Cummings Hamiltonian starting from the minimal coupling Hamiltonian describing a hydrogen atom interacting with the electromagnetic field. We will introduce the Power-Zienau-Wolley transformation which leads to a new description of electrodynamics, which is exactly equivalent to the standard description. In this representation a multipole expansion of the interaction between the particles and the electric and magnetic field is obtained. This approach is often found to be more convenient for treating the interaction between an atomic electron and a field.

In Chapter4, following the results obtained in Chapter2, we re-derive the Jaynes-Cummings model based on classical dynamics described by the Liouville equation. When applying the reformulation of Hamiltonian dynamics to the two-level dynam-ics of a Rydberg atom coupled to a one-mode cavity field, suitably tuned, we find that the classical dynamics and the quantum mechanics differ only by the presence of a characteristic superoperator.

In order to study the way this superoperator term modifies the dynamics, we introduce the concept of Liouville space. It is in this particular space that the Liouville and the von Neumann equation can be formally written in the same way. We then give numerical estimates for this superoperator term and compare them with an actually realized cavity QED experiment.

In Chapter5, we present a review of cavity QED experiments performed in Paris. Summarizing, the aim of this thesis is to analyse the relation between the classical and the quantum dynamics in a concrete example, incorporating the interaction between particles and fields. Presently, we look more carefully into the commons as well as the distinctive features between the Liouville and the von Neumann equation. We wish to gain further insight about the quantum-classical divide in the particular context of the Jaynes-Cummings model, a benchmark model of cavity QED and quantum optics, which has gained particular significance for applications in quantum information. Preliminary results of this thesis have been reported in (EGV10).

(9)

Chapter 2 Reformulation of Hamiltonian

dynamics

The investigation of the relations between the classical and quantum world has been an interesting topic since the birth of quantum mechanics. Recently a reformulation of Hamiltonian dynamics has been proposed trying to show that aspects of quantum dynamics can be generated by an attractor mechanism, obtained by a generalisation of the classical Liouville equation (Elz08a). This approach is guided by the idea that quantum states may actually represent large equivalence classes of deterministically evolving classical states.

After reviewing some basic ideas on which this paper is based, we will derive a reformulation of electrodynamics in the Coulomb gauge.

2.1 Reformulation of Hamiltonian dynamics

Let us consider a one-dimensional classical system described by the following Hamil-tonian

H(x,p) = 1 2p

2_{+ V (x),} _(2.1)

defined in terms of generalized coordinate x and momentum p, and where V (x) denotes the potential. A distribution function f (x,p) in phase space describes an ensemble of classical particles following trajectories, determined by Hamilton’s equa-tions, with different initial conditions. The probability of finding a particle in an infinitesimal volume at the phase space point (x,p) at time t is the given by f (x,p; t)dxdp. The evolution in time of f (x,p; t) is given by the well known Liouville

(10)

2 – Reformulation of Hamiltonian dynamics equation: − ∂f (x,p; t) ∂t = ∂H ∂p ∂f (x,p; t) ∂x − ∂H ∂x ∂f (x,p; t) ∂p = p∂f (x,p; t) ∂x − ∂V (x) ∂x ∂f (x,p; t) ∂p . (2.2) Applying a Fourier transformation on the Liouville equation,

f (x,p; t) ≡ Z

dye−ipyf (x,y; t) (2.3)

without changing the symbol for the distribution, we get

i∂tf (x,y; t) = −∂x∂y+ y dV (x) dx f (x,y; t). (2.4)

The doubling of the coordinates, following the elimination of the momenta, permits us to write the transformed Liouville equation with the help of the transformation

Q ≡ x + y

2, q ≡ x −

y

2, (2.5)

in a way that will prove to give us a clue about the connections between the classical Liouville equation and the quantum von Neumann equation:

i∂tf (Q,q; t) = [HQ− Hq+ E (Q,q)] f (Q,q; t), (2.6) Hχ≡ − 1 2∂ 2 x+ V (x), for χ = Q,q , (2.7) E(Q,q) ≡ (Q − q) d dxV (x) x=Q+q₂ − V (Q) + V (q) = −E(q,Q). (2.8)

Equation (2.6) can be interpreted as the von Neumann equation for a density oper-ator ˆf (t), if we consider f (Q,q; t) as its matrix elements, modified by an extra term E(Q,q)f (Q,q; t). This interaction term couples the Hilbert space and its dual and, as we will later show, it cannot be always considered as the matrix element of an operator in the Hilbert space. In the case of a free particle or a harmonic oscillator the term E automatically vanishes.

The interaction E appearing in the modified von Neumann equation is a term that prevents us from considering the modified von Neumann equation (2.6) as a true quantum mechanical equation. Usual quantum mechanics is recovered if this term is zero. The study of this term may be of critical importance in the study of the quantum-classical divide.

From the normalization of the classical distribution function f (x,p; t) we have

1 = Z dxdp 2π f (x,p; t) = Z dQdq δ(Q − q)f (Q,q; t) ≡ T rh ˆf (t)i, (2.9)

(11)

2.1 – Reformulation of Hamiltonian dynamics

where the symbol for the distribution is implicit after the transformations (2.5) have been performed. Let us consider a complete set of othonormal eigenfunctions of the operator Hχ, (2.7), having a descrete spectrum for simplicity, defined by

ψj(χ; t) ≡ e−iωjtψj(χ), Hχψj(χ) = ωjψj(χ). (2.10)

Using the completeness and orthonormality of the eigenfunctions, we can expand the density matrix elements in the following way

f (Q,q; t) =X

jk

fjk(t)ψj(Q; t)ψ∗k(q; t). (2.11)

In the case of a system described by the Hamiltonian (2.7), then fjk(t) does not

de-pend on time, since the time evolution is absorbed by ψj(Q; t)ψk∗(q; t). In the case of

a pure state, the density matrix elements in the energy eigenstate representation can be factorised as fjk = cjc∗k. Equivalently, one may say that the system is described

by a wave function written as ψ(Q; t) = P

jcjψj(Q; t). From the normalization

condition we have 1 = X jk fjk(t)e−i(ωj−ωk)t Z dQ ψj(Q)ψk∗(Q) = X j fjj(t). (2.12)

From the reality condition of the classical phase space distribution, it follows that ˆ

f must be a hermitian matrix, i.e. fjk = fkj∗ .

The classical expectation values of the momenta and the position are calculated as follows: hxi ≡ Z dxdp 2π xf (x,p; t) = Z dQdq δ(Q − q)Q + q 2 f (Q,q; t) ≡ T rh ˆX ˆf (t) i , (2.13) hpi ≡ Z _dxdp 2π pf (x,p; t) = Z dQdq δ(Q − q)(−i)∂Q− ∂q 2 f (Q,q; t) ≡ T r h ˆ_{P ˆ}_{f (t)}i . (2.14) The operators ˆX and ˆP being defined by the corresponding matrix elements

X(q,Q) = δ(Q − q)Q + q 2 P (q,Q) = − i 2 h δ(Q − q)−→∂Q− ←− ∂qδ(Q − q) i . (2.15) Eliminating one of the two integrations in the above equations with the help of the δ-function and suitable partial integrations, we can recognize these operators as the usual operators of quantum theory. By rewriting the classical statistical formulae we have obtained the position and momentum operator ˆX and ˆP such that the commutation relation [ ˆX, ˆP ] = i is automatically implemented and no quantization procedure is invoked.

(12)

2 – Reformulation of Hamiltonian dynamics Similarly, we find: Z _dxdp 2π xp f (x,p; t) = 1 2T r[( ˆX ˆP + ˆP ˆX) ˆf (t)], (2.16) which represents an example of the symmetric Weyl ordering, when replacing clas-sical observables with quantum mechanical operators.

The equations (2.9),(2.13)-(2.16) are in accordance with the interpretation of f (Q,q; t) as the matrix elements of a density operator ˆf (t). However, an important remark needs to be made. The eigenvalues of a normalized quantum mechanical density operator are constrained to have values that lie between zero and one, corre-sponding to the usual interpretation of standard probabilities. This is not necessarily the case with the operator ˆf (t) obtained from a classical probability distribution. Similarly, the Wigner distribution, obtained from the matrix elements of a quantum mechanical density operator by applying in reverse the transformations leading from f (x,p; t) to f (Q,q; t), is not generally positive semi-definite on phase space.

2.2 Reformulation of electrodynamics in the

Coul-omb gauge

In this section, we will apply the same reformulation to a system consisting of charged particles interacting with the electromagnetic field. In order to do this we have to apply the transformation applied in the previous section to all dynamical variables. In the Lagrangian formalism, the potentials and not the fields appear as good generalized coordinates for the electromagnetic field. This is not surpris-ing since the equations of motion for the potentials are of second order in time derivatives, like the Euler-Lagrange equation, while the Maxwell equations are of first order in time derivatives. At each point in space, four generalized coordinates are required, these being the three components of the vector potential A(x) and the scalar potential φ(x) and the four corresponding “velocities” ˙A(x) and ˙φ(x) (CT97). The Lagrangian of a system of charged particles interacting with an electromagnetic field is thus given by the following expression

L = 1 2 X α mαx˙α2+ 1 8π Z d3x E2− B2₋ Z d3x ρφ + Z d3x J · A, (2.17)

where the electric and magnetic fields are given as functions of the potentials by

E = − ˙A − ∇φ, B = ∇ × A. (2.18)

In the following, we will consider a bounded system of two particles with opposite unit charge and with the fixed positive charge at the origin. The charge and current

(13)

2.2 – Reformulation of electrodynamics in the Coulomb gauge

density are given by

ρ(x0) = −eδ(x0− x) + eδ(x0), (2.19) and

J(x0) = −e ˙xδ(x0− x). (2.20)

The Lagrangian can be better studied in the reciprocal space by introducing the Fourier transform defined as follows for a field F (x).

F (k,t) = 1 (2π)32 Z d3x F (x,t)e−ik·x (2.21) F (x,t) = 1 (2π)32 Z d3k F (k,t)eik·x (2.22)

The Lagrangian then becomes

L = 1 2m ˙x 2₊ 1 8π Z d3k [E∗(k) · E (k) − B∗(k) · B(k)] + Z d3k [J∗(k) · A(k) − ρ∗(k)φ(k)] . (2.23)

Equation (2.23) suggests choosing as dynamical variables the components of the potentials in the reciprocal space together with the corresponding velocities.

Going from real space to reciprocal space corresponds to a change of variables which transforms real quantities into complex quantities. We will have then twice as many degrees of freedom as before. From the reality condition

A(k) = A∗(−k), φ(k) = φ∗(−k), (2.24)

the potentials and their complex conjugates can be taken as independent variables only in half reciprocal space and the Lagrangian needs to be written in the following way L = 1 2m ˙x 2 ₊ 1 4π Z h

d3kh| ˙A(k) + ikφ(k)|2_{− |k × A(k)|}2i

+ Z

h

d3k [J∗(k) · A(k) + J (k) · A∗(k) − ρ∗(k)φ(k) − ρ(k)φ∗(k)] ,

(2.25)

where the integration is taken over half of the modes and the fields have been expressed in terms of the potentials, using expressions (2.18) written in reciprocal space.

With the help of the Euler-Lagrange equation relative to φ, we can express the scalar potential as a function of the other dynamical variables and it can also be

(14)

2 – Reformulation of Hamiltonian dynamics

shown that the logitudinal part of the potential can be chosen arbitrarily without changing the dyamics of the system. The simplest choice is to take Ak(k) = 0,

which implies ∇ · A = 0 and, thus, selects the Coulomb gauge. From now on, we will assume the vector potential to be purely transverse. The Lagrangian then simplifies as follows L = 1 2m ˙x 2 + e 2 |x|+ 1 4π Z h d3k h ˙ A∗(k) · ˙A(k) − k2A∗(k) · A(k) i +Coul+ Z h d3k [J∗(k) · A(k) + J (k) · A∗(k)] , (2.26)

where we have introduced the Coulomb self energy

Coul ≡ e2 4π2 Z d3_k k2 . 1 _(2.27)

We can now obtain the conjugate momenta for the field and the particle,

p = ∂L ∂ ˙x = m ˙x − e (2π)32 Z h

d3kA(k)eik·x+ A∗(k)e−ik·x , (2.28)

Π(k) = ∂L ∂ ˙A∗_(k) = 1 4π ˙ A(k), (2.29)

and determine the Hamiltonian of the system.

H = p · ˙x + Z h d3k h Π(k) · ˙A∗(k) + Π(k)∗· ˙A(k)i− L = 1 2m " p + e (2π)32 Z h

d3k A(k)eik·x+ A∗(k)e−ik·x #2 + Coul −e 2 |x| + 1 4π Z h d3k 16π2Π∗(k) · Π(k) + k2A∗(k) · A(k) (2.30)

The Hamiltonian is expressed as a function of the independent variables

x, p, A(k), A∗(k), Π(k), Π∗(k).

Since we are interested in studying a bounded system interacting in a cavity of volume Ω with an electromagnetic field, we will assume the field modes as discretized.

1_{The Coulomb self energy is infinite unless one introduces a cut off in the integral on k. However}

(15)

The integral over the reciprocal space will then be replaced by a summation. The correspondence between the two types of sum obeys the following rule:

Z

d3kF (k) ←→X

k

(2π)3

Ω Fk. (2.31)

Consequently, in equations (2.21) and (2.22), the factor 2π has to be replaced by Ω1/3_{. In the following expressions the volume factors will be reabsorbed by a}

redef-initions of the potentials.

Given the Hamiltonian we can now proceed in calculating the classical Liouville equation for the system. After defining the distribution function

f = f (x,p; A∗_k,Πk; Ak,Π∗k),

where the index k runs over all modes of the field, we obtain:

− ∂tf = X Ωk/2 ∂H ∂Πk · ∂f ∂A∗_k − ∂H ∂A∗_k · ∂f ∂Πk + ∂H ∂Π∗_k · ∂f ∂Ak − ∂H ∂Ak · ∂f ∂Π∗_k +∂H ∂p · ∂f ∂x− ∂H ∂x · ∂f ∂p. (2.32)

All we need to do is to calculate the derivative of the Hamiltonian with respect to each canonical variable.

∂H ∂p = p m + e m X Ωk/2 Akeik·x+ A∗ke −ik·x_, (2.33) ∂H ∂x = e2 m X Ω_k,k0/2 h Ak0eik 0_·x + A∗_k0e−ik 0_·x · Akeik·x− A∗ke −ik·xi k e2 |x|3x + ie m X Ωk/2 (p · Ak)eik·x− (p · A∗k)e −ik·x_k, (2.34) ∂H ∂Πk = 4πΠ∗_k, (2.35) ∂H ∂A∗_k = e m X Ωk0/2 h Ak0ei(k 0_−k)·x + A∗_k0e−i(k+k 0_)·xi + e mpe −ik·x + k 2 4πAk. (2.36)

The idea is to apply the same transformation considered in the previous section and see if we can obtain something similar to the von Neuman equation starting from the classical Liouville equation. As a first step, we consider a Fourier transformation, without changing the symbol for the distribution, which allows us to eliminate the momentum variables by introducing new coordinates.

(16)

2 – Reformulation of Hamiltonian dynamics f (x,y; Ak,Bk; A∗k,B ∗ k) = = 1 (2π)9 Z d3p Y Ωk/2 dΠ∗_kdΠkei(p·y+Πk·B ∗ k+Π ∗ k·Bk)f(x,p; A∗ k,Πk; Ak,Π∗k). (2.37)

The effect of the transformation on each term of the Liouville equation is given by the following expressions:

∂H ∂p · ∂f ∂x −→  − i m∂y· ∂x+ e m X Ωk/2 Akeik·x+ A∗ke −ik·x_{· ∂} x  f, (2.38) ∂H ∂x · ∂f ∂p −→ − ie m X Ωk/2 (k · y)(∂y· Ak) eik·x− (∂y· A∗k) e −ik·x_f +e 2 m X Ωk,k0/2 (Ak0eik 0_·x + A∗_k0e−ik 0_·x ) · Akeik·x− A∗ke −ik·x_{(k · y)f} −ie m X Ωk/2 (k · Ak) eik·x− (k · A∗k) e −ik·x_{f − i} e2 |x|3x · yf, (2.39) ∂H ∂Πk · ∂f ∂A∗_k −→ −4πi ∂Bk· ∂A∗kf, (2.40) ∂H ∂A∗_k · ∂f ∂Πk −→ −ie 2 m X Ω_k0/2 h Ak0ei(k 0_−k)·x + A∗_k0e−i(k+k 0_)·xi · B_k∗f − e m(B ∗ k· ∂y)e−ik·x+ i k2 4πAk· B ∗ k f, (2.41) ∂H ∂Π∗ k · ∂f ∂Ak −→ −4πi ∂B∗ k· ∂Akf, (2.42) ∂H ∂Ak · ∂f ∂Π∗_k −→ − ie2 m X Ω_k0/2 h A∗ k0e−i(k 0_−k)·x + Ak0ei(k+k 0_)·xi · Bkf − e m(Bk· ∂y)e ik·x_{+ i}k 2 4πA ∗ k· Bk f. (2.43)

After putting each piece toghether, using the fact that B−k = B_k∗ and A−k= A∗_k,

(17)

2.2 – Reformulation of electrodynamics in the Coulomb gauge i∂tf = " − 1 m∂y· ∂x+ e2 |x|3x · y + X k −4π∂Bk· ∂A∗k+ k2 4πA ∗ k· Bk # f − e m X k ie−ik·x_A∗ k· ∂x+ (k · y)(∂y· A∗k)e −ik·x_{+ i(B} k· ∂y)eik·x f +e 2 m X k,k0 h A∗_k0 · (B_k+ iA_k(k · y)) e−i(k 0_−k)·xi f. (2.44)

The Coulomb gauge condition, which implies the transversality of the field, has to be satisfied and, therefore, the terms where k · A(k) appeared have been set equal to zero.

Another transformation of coordinates permits us to write the Liouville in a form that can be recognized in some of its terms as the corresponding von Neumann equation. If we consider Q = x + y 2, q = x − y 2, (2.45) Q∗_k= A∗_k+ B ∗ k 2 , qk= A ∗ k− B∗ k 2 , (2.46) ∂x = ∂Q+ ∂q, ∂y= 1 2(∂Q− ∂q) , (2.47) ∂A∗ k = ∂Q ∗ k+ ∂qk, ∂Bk = 1 2 ∂Qk− ∂q∗k , (2.48)

(18)

the Liouville equation becomes

i∂tf = − 1 2m∂ 2 Q+ 1 2m∂ 2 q+ V (Q) − V (q) + E (Q,q) f +X k −2π ∂Qk· ∂Q∗k− ∂qk· ∂q∗k + 1 8πω 2 k(Qk· Q∗k− qk· q∗k) f − e 2m X k e−ik·(Q+q)/2Q∗_k· i(∂Q+ ∂q) + k 2 · (Q − q) · (∂Q− ∂q) f − e 2m X k e−ik·(Q+q)/2qk· i(∂Q+ ∂q) + k 2 · (Q − q)(∂Q− ∂q) f − ie 2m X k eik·(Q+q)/2(Q∗_k− qk) · (∂Q− ∂q)f + e 2 2m X k,k0 e−i(k0−k)·(Q+q)/2Q∗_k0 · Q_k 1 + i 2k · (Q − q) f − e 2 2m X k,k0 e−i(k0−k)·(Q+q)/2qk0 · q∗ k 1 − i 2k · (Q − q) f, (2.49)

where we have defined

E(Q,q) ≡ (Q − q) · ∇V Q + q 2 − V (Q) + V (q) = −E(q,Q), (2.50) ∇V Q + q 2 = 4e 2_{(Q + q)} |Q + q|3 , V (χ) = − e2 χ, for χ = |Q|,|q|. (2.51)

The first two lines of the Liouville equation can be interpreted, if we do not consider E for a moment, as the von Neumann equation for a density operator ˆf (t),

i∂tf = [ ˆˆ HP + ˆHR, ˆf ]. (2.52)

considering f (Q, · · · ,Qk, · · · ; q, · · · ,qk, · · · ) as its matrix elements, associated with

a system bound by the Coulomb potential and an electromagnetic field not inter-acting with each other. The quantum mechanical operator appearing in expression (2.52) are defined by ˆ HP ≡ ˆ p2 2m + V (ˆq). (2.53) ˆ HR ≡ X k,σ 2π ˆP_k· ˆP†_k+ω 2 k 8π ˆ Q_k· ˆQ†_k . (2.54)

(19)

The two operators appearing in (2.54) can be written in components in the following way ˆ P_k=X σ ˆ P_k,σek,σ, Qˆk= X σ ˆ Q_k,σek,σ, (2.55)

where the sum runs over the two different polarisations. In the Coulomb gauge the polarisation vectors, which we will chose to be perpendicular to each other, must obey the following condition,

ek,σ· ek,σ0 = δ_σσ0, e_k,σ· k = 0. (2.56)

The quantized transverse electromagnetic field can be described by introducing the destruction â_k,σ and creation â†_k,σ operators satisfying the Bose commutation relations: [â_k,σ,â_k0_,σ0] = [â†_k,σ,â†_k0_,σ0] = 0, [â_k,σ,â † k0_,σ0] = δ_k,k0δ_σσ0, (2.57) ˆ a_k,σ = 1 8πωk 1₂ ωkQˆk,σ+ 4πi ˆPk,σ , (2.58) ˆ a†_k,σ = 1 8πωk 1₂ ωkQˆ†_k,σ− 4πi ˆP_k,σ† , (2.59) ˆ Q_k,σ= 2π Ωωk 1₂ ˆ a_k,σ+ â†_−k,σ, (2.60) ˆ P_k,σ = −i ω_k 8πΩ 1₂ ˆ a_k,σ− â†_−k,σ , (2.61) ˆ HR= X k,σ ωk ˆ a†_k,σâ_k,σ+ 1 2 . (2.62)

In the quantum mechanical context, equation (2.52) describes only the free evo-lution of the hydrogen atom and the electromagnetic field in the Coulomb gauge. When considering this systems as interacting with each others we need to add to expression (2.52) the interaction terms. This can be achieved if we consider

ˆ HI,1 = e mp · ˆˆ A(q), ˆ HI,2 = e2 2m ˆ A2(q). (2.63)

In writing the interaction hamiltonian ˆHI,1, we have used the fact that when the

vector potential ˆA(q) is expressed in the Coulomb gauge it is possible to consider ˆ

p and ˆA(q) as commuting operators. Thus to take into account the interaction we must add in the von Neumann equation (2.52) the terms

(20)

to obtain

i∂tf = [ ˆˆ HP + ˆHR, ˆf ] + [ ˆHI,1+ ˆHI,2, ˆf ]. (2.65)

To better compare the von Neumann equation (2.65) and the transformed Liou-ville equation (2.49) we must consider the matrix elements of the interaction (2.64):

ˆ HI,1 = e m X k,σ 2π Ωωk 1₂ (ˆp · ek,σ) h ˆ a_k,σeik·ˆq+ ˆa†_k,σe−ik·ˆqi= e m X k ˆ p · ˆQ_keik·ˆq, (2.66) ˆ HI,2 = e2 2m X k,k0 ˆ Q_k· ˆQ_k0ei(k+k 0_)·ˆ_q . (2.67)

The matrix elements can be obtained by keeping in mind the following relations: ˆ Q_kY k0 |qk0i = q_k Y k0 |qk0i, Qˆ† k Y k0 |qk0i = q∗_k Y k0 |qk0i, (2.68) hQk0| ˆQ_k|q_ki = q_kδ_kk0δ(Q_k0 − q_k), hQ_k0| ˆQ† k|qki = q ∗ kδkk0δ(Q_k0 − q_k), (2.69) ˆ HI,1f − ˆˆ f ˆHI,1 −→ − ie m X k eik·QQk· ∂Q+ eik·qqk· ∂q f, (2.70) ˆ HI,2f − ˆˆ f ˆHI,2 −→ e2 2m X k,k0 h Qk· Q∗k0ei(k−k 0_)·Q − qk· q∗k0ei(k−k 0_)·qi f. (2.71)

If we add and subtract expressions (2.70) and (2.71) from expression (2.49), as we have done in the simpler case considered in Section 2.1, we can write the “Liouville equation” (2.49) in the following way:

i∂tf = − 1 2m∂ 2 Q+ 1 2m∂ 2 q+ V (Q) − V (q) + E (Q,q) f +X k −2π ∂Qk· ∂Q∗k− ∂qk· ∂q∗k + 1 8πω 2 k(Qk· Q∗k− qk· q∗k) f −ie m X k eik·QQk· ∂Q+ eik·qqk· ∂q f + Γf + e 2 2m X k,k0 h Qk· Q∗k0ei(k−k 0_)·Q − qk· q∗k0ei(k−k 0_)·q ·if + Σf, (2.72)

where we have defined

Γ =X

k

e

(21)

Γk(Q,∂Q,Qk; q,∂q,qk) = i eik·QQk· ∂Q+ eik·qqk· ∂q −1 2e −ik·(Q+q)/2 (Q∗_k+ qk) · i(∂Q+ ∂q) + k 2 · (Q − q)(∂Q− ∂q) −i 2e ik·(Q+q)/2_(Q∗ k− qk) · (∂Q− ∂q), (2.74) and Σ =X k,k0 e2 2mΣk,k0(Q,Qk,Qk0; q,qk,qk0), (2.75) Σk,k0(Q,Q_k,Q_k0; q,q_k,q_k0) = − h Qk· Q∗k0ei(k−k 0_)·Q − qk· q∗k0ei(k−k 0_)·qi +e−i(k0−k)·(Q+q)/2 Q∗_k0 · Q_k 1 + i 2k · (Q − q) −e−i(k0−k)·(Q+q)/2 qk0 · q∗ k 1 − i 2k · (Q − q) . (2.76)

If the three terms E , Γ, and Σ are not considered then equation (2.72) reduces to the usual von Neumann equation (2.65). The E , Γ, and Σ terms describe how the classical dynamics differs from the quantum mechanical one. As in the case considered in Section 2.1, the commutation relations for the field operators are automatically implemented.

Due to the complexity of the found extra terms, we will study this reformulation of the Liouville equation for the simplest system in which the interaction between the electromagnetic field and charged particles appears. This will be the Jaynes-Cummings model which describes the interaction of a two-level atom with a cavity field. This system has been widely studied theoretically and has been used to analyze experiments in the fields of quantum information and quantum optics.

(22)

Chapter 3 The Jaynes-Cummings model

Light is both radiated and absorbed by atoms and the interaction between the quantized electromagnetic field and an atom represents one of the most fundamental problems in quantum optics (MW95). The complex energy structure of even the simplest atomic system, i.e. the hydrogen atom, makes it necessary or desirable to approximate the behaviour of a real atom by that of a much simpler quatum system. For many purposes only two atomic energy levels play a significant role in the interaction with the electromagnetic field, so that it has become customary in some theoretical treatments to represent an atom by a two-level system. This approximation has been useful to very well describe experimental situations which have been widely studied in the field of quantum optics during the last decades (RBH01).

In this Chapter, we will derive the Jaynes-Cummings Hamiltonian starting from the minimal coupling Hamiltonian describing a hydrogen atom interacting with an electromagnetic field. To obtain the Jaynes-Cummings Hamiltonian, we will intro-duce the Power-Zienau-Wolley transformation which leads to a new description of electrodynamics, exactly equivalent to the standard description (CT97). In this representation the coupling between fields and charges is expressed in terms of the electric and magnetic field and no longer in terms of the vector potential. Also a system of charges can be described by electric polarisation and magnetisation den-sities which are given directly as functions of the microscopic observables, i.e. the positions and the velocities. This approach proves to be useful to introduce the dif-ferent electric and magnetic multipole moments of the system of charges and obtain a multipole expansion of the interaction between the particles and the electric and magnetic fields.

(23)

3.1 – Radiation-matter interaction

3.1 Radiation-matter interaction

In the Coulomb gauge the electrodynamics of a nonrelativistic hydrogen atom system interacting with the quantized electromagnetic field may be described by the minimal coupling Hamiltonian (Muk95):

ˆ Hmin = 1 2m h ˆ p + e ˆA(q)i 2 − e 2 |ˆq|+ ˆHR. (3.1)

The free radiation Hamiltonian is given by

ˆ HR = 1 8π Z d3rh ˆE⊥2(r) + ˆB2(r)i. (3.2)

The canonical field variables are the vector potential Â(q), and its conjugate mo-mentum, −(4π)Ê⊥(r), where Ê⊥(r) is the transverse electric field. Besides the usual commutation relation for the particle’s canonical variables the following commuta-tion rules for the field must be satisfied:

h ˆ_A_i_{(r), ˆ}_E⊥ j(r 0i = −4πiδ_ij⊥(r − r0),1 _(3.3) h ˆ_E⊥ x(r), ˆBy(r0) i = 4π∂zδ(r − r0), (3.4) h ˆ_A_i_{(r), ˆ}_A_j_(r0 )i =h ˆAi(r), ˆBj(r0) i = 0. (3.5)

We can now introduce the dynamical variables representing the quantized transverse electromagnetic field. We define ˆa†_k,σ and ˆa_k,σ as the creation and annihilation operators for the mode with wavevector k and polarisation σ. They must satisfy the bosonic commutation relations:

[â_k,σ,â_k0_,σ0] = [â † k,σ,â † k0_,σ0] = 0, [â_k,σ,â † k0_,σ0] = δkk0δ_σσ0. (3.6)

Using these operators the vector potential can be written in the following form: ˆ A(r) =X k,σ k ωk h ˆ a_k,σek,σeik·r+ ˆa†_k,σek,σe−ik·r i , (3.7)

where we have defined k = (2πωk/Ω)1/2, Ω being the quantization volume, and ek,σ

a unit vector such that

ek,σ· ek,σ0 = δ_σσ0, e_k,σ· k = 0. (3.8)

1_{The transverse delta function being defined as δ}⊥

ij(r − r0) = 1 (2π)3R d3keik·r δij− kikj k2

(24)

3 – The Jaynes-Cummings model

The magnetic and transverse electric field can be obtained from the vector potential through the following expressions:

ˆ

E⊥(r) = −A(r),˙ˆ B(r) = ∇ × ˆˆ A(r). (3.9) Keeping in mind that the Heisenberg equation

˙ˆ

F = i[ ˆH, ˆF ], (3.10)

gives the time evoution of any operator ˆF , we can obtain from equations (3.9) the transverse electric field

ˆ E⊥(r) =X k,σ ik h ˆ a_k,σek,σeik·r− ˆa † k,σek,σe−ik·r i , (3.11)

and the magnetic field ˆ B(r) =X k,σ ik h ˆ a_k,σ(ˆk × ek,σ)eik·r+ ˆa † k,σ(ˆk × ek,σ)e−ik·r i . (3.12)

Using the previous expression for the fields we can rewrite the radiation Hamiltonian in the following way

HR= X k,σ ωk(ˆa † k,σˆak,σ+ 1 2). (3.13)

Let us now consider the matter part of the system. We can introduce the charge density operator

ˆ

ρ(r) = −eδ(r − ˆq) + eδ(r), (3.14)

and, with the help of the Heisenberg equation of motion, obtain the continuity equation

˙ˆ

ρ(r,t) = −∇ · ˆJ(r,t). (3.15)

The electric current density operator has been defined by ˆ

J(r) = −e

2h ˙ˆqδ(r − ˆq) + δ(r − ˆq) ˙ˆq i

, (3.16)

where ˙ˆq = i[ ˆH,ˆr] is the velocity of the particle, which in the Coulomb gauge is related to its canonical momentum by

m ˙ˆq = ˆp + e ˆA(q). (3.17)

In the case of a system with bound charges it proves more convenient to work with a polarisation density operator (CT97)

ˆ

P(r) = −e Z 1

0

(25)

3.1 – Radiation-matter interaction

and its Fourier transform ˜ P(k) = −e 1 (2π)32 Z 1 0 du ˆqe−ik·ˆqu. (3.19)

The divergence of ˆP(r) is directly related to the charge density, in fact from expres-sion (3.19) it follows that

ik · ˜P(k) = − e (2π)32 Z 1 0 du ik · ˆqe−ik·ˆqu = e (2π)32 e−ik·ˆq− e (2π)32 (3.20)

which, in real space, gives

∇ · ˆP(r) = −ρ(r). (3.21)

Combining (3.18) with the equation ∇ · ˆE = 4π ˆρ, we can construct a field, the displacement ˆD = ˆE + 4π ˆP, that is completely transverse

∇ ·h ˆE(r) + 4π ˆP(r) i

= ∇ · ˆD(r) = 0, (3.22)

which means ˆD(r) = ˆD⊥(r). The motion of the charged electron, which is respon-sible for the origin of the electric current (3.16), must be related somehow to the motion of the polarisation density ˆP(r). Taking the derivative of expression (3.21) we obtain

˙ˆ

ρ(r) = −∇ ·P(r)˙ˆ (3.23)

that compared with the continuity equation (3.15) gives

∇ ·h ˆJ(r) − ˙ˆP(r)i = 0. (3.24)

This implies that ˆJ −P is a transverse field and can then be written in the following˙ˆ way

ˆ

J(r) =P(r) + ∇ × ˆ˙ˆ M(r). (3.25)

Using the Heisenberg equation of motion and the definition of ˆJ we can get an expression for what can be shown to be physically interpreted as the magnetisation density (CT97). ˆ M(r) = −e 2 Z 1 0 du u(ˆq × ˙ˆq)δ(r − uq). (3.26) The two contributions to the current density appearing in (3.25) are thus called the polarisation and the magnetisation current. The interaction of the field with the system will then be given in terms of the magnetisation and polarisation density.

(26)

We have shown that a hydrogen atom can be modeled as a polarisation field ˆ

P(r) distributed over space. This kind of approach can be generalized to system of charges or to a molecule and it has been found to be often more convenient to treat the interaction between an atomic electron and a field.

The u integration in the definition of the polarisation density operator (3.18) ensures that the correct coefficients for the multipolar expansion of the charge dis-tribution appear (Muk95). Considering a Taylor expansion for the delta function

δ(r − uˆq) = 1 − uˆq · ∇r+ 1 2(uˆq · ∇r) 2 + · · · δ(r), (3.27)

and inserting (3.27) in the expressions (3.18) and (3.26), we obtain the multipolar expansion for the polarisation and the magnetisation density

ˆ P(r) =h ˆPD(r) − ˆPQ(r) · ∇ + · · · i δ(r), (3.28) ˆ M(r) =h ˆMD(r) − ˆMQ(r) · ∇ + · · · i δ(r), (3.29)

where we have introduced the electric and magnetic dipole ˆ

PD(r) = −eˆq, MˆD(r) = −

e

2mq × ˆˆ p, (3.30)

and the electric and magnetic quadrupole ˆ PQ(r) = − e 2ˆq ⊗ ˆq MˆQ(r) = − e 6m[(ˆq × ˆp) ⊗ ˆq + ˆq ⊗ (ˆq × ˆp)] (3.31) where ⊗ denotes the tensor product.

3.2 The Power-Zienau-Wolley transformation

The introduction of the polarisation and magnetisation density allows us to in-troduce the Power-Zienau-Wolley transformation and to obtain a new form of the Hamiltonian equivalent to the minimal coupling Hamiltonian. Such transformation was introduced first in the context of the dipole approximation by G¨oppert-Mayer and then extended to its most general form by Power and Zienau (PZ59). This new representation is implemented in the following way, by a unitary transformation, on any operator ˆF ,

ˆ

F0 = ei ˆSF eˆ −i ˆS S = −ˆ Z

d3r ˆP(r) · ˆA(q) (3.32)

where ˆP(r) and ˆA(q) are given by (3.18) and (3.7). The transformation preserves the commutation rules. Since the electric field and the particle momentum do not

(27)

3.2 – The Power-Zienau-Wolley transformation

appear in ˆS, the vector potential ˆA(q) and the particle coordinate are not affected by the transformation. On the other hand the particle momentum and the creation and annihilation operators will be transformed. We thus obtain:

ˆ q0 = ˆq, pˆ0 = ˆp − e Â(q) + Z d3r ˆn(r) × ˆB(r), (3.33) ˆ a0_k,σ = â_k,σ+ i k ~ωk ek,σ· ˆP(k), aˆ 0† −k,σ = â † −k,σ − i k ~ωk ek,σ· ˆP(k), (3.34)

where we have introduced the polarisation vector field

ˆ

n(r) = −eˆq Z 1

0

du u δ(r − uˆq). (3.35)

From the above relations we can see that the vector potential (3.7), the magnetic field (3.1), and the polarisation operator (3.18) remain unchanged while the transverse part of the electric field (3.11) gets modified:

ˆ

A0(r) = ˆA(r), Bˆ0(r) = ˆB(r), Pˆ0(r) = ˆP(r), (3.36) ˆ

E0⊥(r) = ˆE⊥(r) − 4π ˆP⊥(r). (3.37) Finally, we want to consider the effect of the transformation on the displacement

ˆ

D = ˆE + 4π ˆP, which as it has been said, is a completely transverse field, ˆ

D0⊥(r) = ei ˆSEˆ⊥(r)e−i ˆS+ 4π ˆP⊥(r) = ˆE0⊥(r) + 4π ˆP⊥(r) = ˆE⊥(r). (3.38) The electric displacement can be expressed in field modes

ˆ D0⊥(r) =X k,σ ik h ˆ a_k,σek,σeikr− ˆa † k,σek,σe−ikr i . (3.39)

It appears that the same mathematical operators ˆa†_k,σ and ˆa_k,σdescribe two different physical variables, depending on the representation used: the transverse electric field initially and the displacement here. In the same way the particle momentum does not represent the same physical state of the system in both representations. In the new one it is related to the particle velocity by

m˙ˆr = ˆp − Z

d3r ˆn(r) × ˆB(r) (3.40)

while in the old representation we have

(28)

In the following part all the operators will be considered as expressed in the new representation.

The transformed Hamiltonian is known as the multipolar Hamiltonian and is given by:

ˆ

H0 = ei ˆSHeˆ −i ˆS = ˆHP + ˆHR+ ˆHself + ˆHint, (3.42)

where ˆ HP = ˆ p2 2m − e2 |ˆq|, ˆ Hself = 2π Z d3r | ˆP⊥(r)|2, (3.43) ˆ HR= 1 8π Z d3rh ˆD⊥2(r) + ˆB2(r)i=X k,σ ωk ˆ a†_k,σaˆ_k,σ+ 1 2 , (3.44) ˆ Hint= − Z d3rh ˆP(r) · ˆD⊥(r) + ˆM(r) · ˆB(r)i+ 1 2m Z d3r ˆn(r) × ˆB(r) 2 . (3.45)

The multipolar Hamiltonian is properly expressed in terms of the fields ˆD(r) and ˆ

B(r).

The interaction part is particularly simple when the magnetic terms are ne-glected. In this case, which is often justified in optical experiments, the interaction is simply the scalar product of the polarisation and the field. The multipolar Hamil-tonian can be obtained also from a gauge transformation and it can be generalized to more than one bounded system with more than one electron. It must be noted that the magnetisation density appearing in the interaction must be expressed in terms of the canonical variables and thus the correct form is given by

ˆ

M(r) = 1

2m[ˆn(r) × ˆp − ˆp × ˆn(r)] . (3.46)

3.3 Dipole approximation and the derivation of

the Jaynes-Cummings Hamiltonian

The multipolar Hamiltonian has been introduced to simplify the introduction of the dipole approximation. We will consider the spatial extension of the bounded system to be of the order of the Bohr radius a0. If this system interacts with radiation with

wavelength λ large with respect to a0, it is legitimate to neglect the spatial variation

of the electromagnetic field over the system extension. By neglecting the magnetic terms of the interaction, we can rewrite the multipolar Hamiltonian (3.42) in the following way, given that the polarisation density operator (3.18) is approximated

(29)

3.3 – Dipole approximation and the derivation of the Jaynes-Cummings Hamiltonian as ˆP(r) = −ˆqδ(r) 2, ˆ Hdip = ˆ p2 2m − e2 |ˆq| + ˆdip+ X k,σ ωk ˆ a†_k,σˆa_k,σ+ 1 2 + eˆq · ˆD⊥(0). (3.47)

The dipole self energy ˆdip, obtained by applying the dipole approximation to the

self energy term (3.43), is given by

ˆ dip = 2π Z d3r | ˆP⊥_D(r)|2 = 2πe 2 V X k,σ (ek,σ· ˆq)2. (3.48)

This expression seems to diverge, but the summation must actually be restricted to those modes for which the dipole approximation is valid. The first three terms of the Hamiltonian are only depending on the atomic variables of the system and can be regrouped as an atomic Hamiltonian ˆHA.

It is conventient to introduce creation and destruction operators also for the atomic part of the Hamiltonian (Lou83). Considering ˆHA, and letting |ii be an

energy eigenstate with eigenvalue ωi, we have

ˆ

HA|ii = ωi|ii. (3.49)

According to the completeness relation and the orthonormality of the eigenstates we can write ˆ HA= X i ωi|iihi|. (3.50)

Let us now consider the effect of an operator like |iihj| applied to some atomic eigenstate |li. From the orthonormality condition it follows that

|iihj|li = |iiδjl, (3.51)

thus the effect is to change the state to |ii, if the original state is |ji. We can say that |iihj| destroys the atomic state |ji and creates the atomic state |ii. It is usuful to replace this operator by a notation similar to that used for the operators that create and destroy photons. If we define ˆb†_i and ˆbi to be the creation and destruction

operators for the atomic state |ii, we can write expression (3.51) in the following way

ˆ_b†

iˆbj|li = |iiδjl. (3.52)

2_{This corresponds to neglecting the spatial variations in equation (}_3.19_{), which implies}

˜ P (k) = − e (2π)3/2 Z 1 0 du ˆq .

(30)

The atomic Hamiltonian can now be written as ˆ

HA=

X

i

ωiˆb†iˆbi. (3.53)

It remains to express the interaction Hamiltonian in terms of the electronic creation and destruction operators

The Hamiltonian (3.47) can now be written in terms of creation and annihilation operators, using expression (3.39) and (3.54) , we get

ˆ Hdip = X i ωiˆb†iˆbi + X k,σ ωk ˆ a†_k,σˆa_k,σ+ 1 2 + +i X k,σ X ij kdij · ek,σ ˆ a_k,σ− ˆa†_k,σ ˆb†_iˆbj. (3.55)

The complete Hamiltonian for the atom-radiation system, in the dipole approx-imation, is now in second-quantized notation. A similar procedure could be carried out for the second quantization of the electric-quadrupole and magnetic-dipole in-teraction (3.31).

Since we are interested in obtaining the Jaynes-Cummings model Hamiltonian we will consider a two level-atom and a single mode radiation field. We will consider a ground state |1i and an excited state |2i separated in energy by ω0. The operator

parts of the Hamiltonian (3.55) can be simplified by the introduction of transition operators, defined as

ˆ

π†= ˆb†₂ˆb1 = |2ih1|, π = ˆˆ b †

1ˆb2 = |1ih2|. (3.56)

and satisfying the anti-commutation relation characteristic of a fermion algebra: {ˆπ,ˆπ} = {ˆπ†,ˆπ†} = 0, {ˆπ,ˆπ†} = 1. (3.57) Considering real the wavefunctions φ1 and φ2 related to the two atomic states the

dipole operator (3.54) can be expressed as ˆ

d = d12(ˆπ†+ ˆπ), d12 = h1|ˆd|2i = h2|ˆd|1i = d21. (3.58)

If the energy zero is taken at the level of the lower state |1i, the Hamiltonian (3.55) then becomes ˆ Hdip = ω0πˆ†π +ˆ X σ ωkâ † k,σâk,σ+ X σ igk(âk,σ− â † k,σ)(ˆπ † + ˆπ), (3.59)

(31)

where we have defined gk = k(d · ek,σ). The advantage of the second-quantized

formulation lies in the ease with which each type of interaction can be represented pictorially. The term â_k,σπˆ† corresponds to the absorption of one photon and to the transition from the lower energy state to the higher energy state. On the other hand the â†_k,σπ represents an emission of a photon and a transition from the higherˆ energy state to the lower energy state. The other two terms â†_k,σπˆ† and â_k,σπ do notˆ correspond to allowed absorption and emission processes, since such terms are not conserving the energy. They can contribute to higher order radiative processes.

Furthermore, when considering the Hamiltonian (3.59) in the interaction picture, the creation and destruction operators for the two-level atom and the electromag-netic field have the following time dependence:

ˆ a_k,σ(t) = ˆa_k,σe−iωkt_, _ˆ_a† k,σ(t) = ˆa † k,σe iωkt_, _(3.60) ˆ

π(t) = ˆπe−iω0t_, _π_ˆ†_{(t) = ˆ}_π†_eiω0t_. _(3.61)

When considering the case of little detuning between the field frequency ωk and

the atom transition frequency ω0, i.e. ωk' ω0, there will be terms in the interaction

Hamiltonian that oscillate more rapidly then the others. In fact, the terms that do not conserve the energy have the following time dependence

ˆ

π†ˆa†_k,σei(ωk+ω0)_, _πˆ_ˆ_a

k,σe

−i(ωk+ω0)_, _(3.62)

while the energy conserving ones give

ˆ

π†ˆa_k,σei(ω0−ωk)_, _πˆ_ˆ_a†

k,σe

−i(ω0−ωk)_. _(3.63)

To obtain the Jaynes-Cummings Hamiltonian the rotating wave approximation will be done (Lou90). It consists in neglecting the interaction terms appearing in (3.62) because their fast oscillating nature makes these terms average to zero in any ap-preciable time scale. The Hamiltonian (3.59) thus reduces to

ˆ HJ C = ω0πˆ†π +ˆ X σ ωkaˆ † k,σaˆk,σ+ i X σ gk(ˆak,σπˆ †_{− ˆ} πˆa†_k,σ), (3.64)

(32)

Chapter 4 The extra terms in the

Jaynes-Cummings model

In chapter 2 we have presented a reformulation of electrodynamics and rewritten the Liouville equation (2.72) in a form which we wish to compare to the quantum mechanical von Neumann equation in the context of the Jaynes-Cummings model. In order to do this, we will first try to follow the steps we have gone through in the derivation of the Jaynes-Cummings Hamiltonian (3.64) and apply them to the Liouville equation (2.72). We will find that the only nonvanishing extra term appearing in equation (2.72) after performing the required approximations will be the E term. It is this extra term that we wish to study in this chapter.

4.1 The PZW transformation and the dipole

ap-proximation

Starting from the matrix elements of the transformed Liouville equation (2.72),

i∂tρ = − 1 2m∂ 2 Q+ 1 2m∂ 2 q+ V (Q) − V (q) + E (Q,q) ρ +X k −2π ∂Qk· ∂Q∗k− ∂qk· ∂q∗k + 1 8πω 2 k(Qk· Q∗k− qk· q∗k) ρ −ie m X k eik·QQk· ∂Q+ eik·qqk· ∂q ρ + Γρ + e 2 2m X k,k0 h Qk· Q∗k0ei(k−k 0_)·Q − qk· q∗k0ei(k−k 0_)·qi ρ + Σρ, (4.1)

(33)

4.1 – The PZW transformation and the dipole approximation

we want to perform the same transformations and approximations that allowed us to arrive at the Jaynes-Cummings Hamiltonian (3.64). The fisrt step is to implement the transformation (3.32),

ˆ

F0 = ei ˆSF eˆ −i ˆS, S = −ˆ Z

d3r ˆP(r) · ˆA(q), (4.2)

on expression (4.1). In order to do this, we should first obtain the operator form of (4.1). At this point, we have to face the problem of expressing the

E(Q,q)ρ(Q, · · · ,Qk, · · · ; q, · · · ,qk, · · · ) (4.3)

in terms of quantum mechanical operators. Due to the fact that this term couples the Hilbert space to its dual, it cannot be expressed as an operator. As we will see later, (4.3) can be expressed as a matrix element of a superoperator defined in Liouville space.

If we would not consider the E , Γ, and Σ terms, then equation (4.1) could be expressed as a matrix element of a quantum mechanical operator in the following form

i∂tρ = ˆˆ Hminρ − ˆˆ ρ ˆHmin, (4.4)

where ˆHmin is the minimal coupling Hamiltonian (3.1). By performing the

trans-formation (4.2), we obtain the von Neumann equation (4.4), for a hydrogen atom interacting with an electromagnetic field, expressed in this new representation we have introduced in Section 3.2:

i∂tρˆ0 = ˆHmin0 ρˆ 0 _{− ˆ}

ρ0Hˆ_min0 . (4.5)

In Section 3.2, the effect of the PZW transformation on the minimal coupling Hamiltonian has already been studied. Our focus must be on understanding how the extra terms E , Σ, and Γ are affected by this change of representation. In order to do this, we need to obtain, if possible, the corresponding quantum mechanical operator of the matrix elements E , Γ, and Σ appearing in (4.1).

Let us consider the Γρ term,

Γρ = e m

X

k

Γk(Q,∂Q,Qk; q,∂q,qk)ρ, (4.6)

where ρ has to be understood like in equation (4.1) as the matrix element of ˆρ:

ρ = Y

k00_,k0

(34)

4 – The extra terms in the Jaynes-Cummings model and Γk(Q,∂Q,Qk; q,∂q,qk) = i eik·QQk· ∂Q+ eik·qqk· ∂q −1 2e −ik·(Q+q)/2 (Q∗_k+ qk) · i(∂Q+ ∂q) + k 2 · (Q − q)(∂Q− ∂q) −i 2e ik·(Q+q)/2 (Q∗_k− qk) · (∂Q− ∂q). (4.8)

The first term appearing in Γkρ is the matrix element, changed in sign, of the k-th

mode of the operator ˆHI,1ρ − ˆˆ ρ ˆHI,1, where ˆHI,1 has been defined in (2.64),

i eik·QQk· ∂Q+ eik·qqk· ∂q ρ −→ −ˆp · ˆQke

ik·ˆq_{ρ + ˆ}_ˆ _ρˆ_{p · ˆ}_Q ke

ik·ˆq_. _(4.9)

Thus, on the right hand side, we have the corresponding operator for the given matrix element. The corresponding operators for the other terms can be found too. This cannot always be done, but in our case the variables of the Hilbert space and its dual can be separated and thus the quantum mechanical operators can be obtained. As an example let us consider the following term:

− i 2e

−ik·(Q+q)/2

(qk· ∂Q)ρ. (4.10)

The exponential can be divided into two parts and because of the transversality of qk, which means qk· k = 0, the partial derivative ∂Q can be placed everywhere on

the left side of ρ. By rewriting expression (4.10) as

− i 2e

−ik·Q/2

∂Q· ρe−ik·q/2qk, (4.11)

it can be seen that the corresponding operator is 1

2e

−ik·ˆq/2_{p · ˆ}_ˆ _{ρ ˆ}_Q ke

−ik·ˆq/2_. _(4.12)

In the same way, we can obtain all the other operators for the corresponding terms appearing in expression (4.8), which we will present in AppendixA.

We must now consider the Σρ term and find the corresponding operator

Σρ = X k,k0 e 2mΣk,k0(Q,Qk,Qk0; q,qk,qk0)ρ. (4.13) Σk,k0(Q,Q_k,Q_k0; q,q_k,q_k0) = − h Qk· Q∗k0ei(k−k 0_)·Q − qk· q∗k0ei(k−k 0_)·qi +e−i(k0−k)·(Q+q)/2 Q∗_k0 · Q_k 1 + i 2k · (Q − q) −e−i(k0_{−k)·(Q+q)/2} qk0· q∗ k 1 − i 2k · (Q − q) . (4.14)

(35)

4.1 – The PZW transformation and the dipole approximation

The first two terms of Σk,k0ρ can be recognized as the matrix elements of the operator

− ˆQ_k· ˆQ†_k0ei(k−k 0_)·ˆ_q ˆ ρ + ˆρ ˆQ_k· ˆQ†_k0ei(k−k 0_)·ˆ_q , (4.15)

while, for example, the term

Q∗_k0 · Q_ke−i(k

0_{−k)·(Q+q)/2}

ρ (4.16)

corresponds to the matrix element of the following quantum mechanical operator e−i(k0−k)·ˆq/2Qˆ†_k0 · ˆQ_kρeˆ −i(k

0_−k)·ˆ_q/2

. (4.17)

Correspondingly, the other terms are given in Appendix A.

The only extra term appearing in the modified von Neumann equation (2.72) that does not have an operator representation is the E term which depends only on the particle coordinates. As seen in Section 3.2, the PZW transformation (3.32) does not affect the particle position operator ˆq and, thus, the form of E (Q,q) doesn’t change.

The transformation (4.2) does not change the ˆQ_k and ˆQ†_k either, because the operator

exp(i ˆS), S = −ˆ X

k

ˆ

P_k†· ˆQ_k, (4.18)

where ˆPk is the discrete Fourier transform of ˆP(r) defined in (3.19), commutes with

ˆ

Q_k and ˆQ†_k. In principle, we could transform each operator in this new representa-tion and then obtain the extra terms.

To simplify the calculation, we can consider the form of the Γ and Σ terms within the context of the dipole approximation and then apply (4.2). The expo-nentials appearing in the extra term operators must all be approximated to one. The transformation operator in the dipole approximation becomes

ˆ Sdip = X k ˆ d · ˆQ_k, ˆd = eˆq, (4.19)

and the only operator appearing in the extra Γ and Σ terms that is affected by the transformation is the particle momentum

ˆ

p0 = ei ˆSdip_ˆ_pe−i ˆSdip _{= ˆ}_{p − e}X

k

ˆ

Q_k. (4.20)

By applying the dipole approximation (k · q 1) to the Γρ and the Σρ terms, even before doing the transformation, it can be easily seen that both the Γ and Σ are equal to zero, in this approximation.

Then, the only extra term which remains is the one related to the atomic part, i.e. E. This term, appearing in the Liouville equation (4.1) as a matrix element in the coordinate representation, doesn’t have a quantum mechanical operator counterpart like the other terms have. To study the behaviour of the density matrix it is then useful to introduce in the following section the Liouville space (Muk95).

(36)

4 – The extra terms in the Jaynes-Cummings model

4.2 Classically generated entanglement?

Entanglement is probably one of the most characteristic feature of quantum me-chanics. Furthermore, it is through entanglement of multipartite systems that the appearence of the classical world is explained in the enviromental decoherence ap-proach. Thus, being the most puzzling property of multipartite quantum systems, one could argue that it is somehow the key for understanding the quantum-classical divide. Last not least, entanglement is a resource for the processing of quantum information.

The quantum mechanical evolution of multipartite systems, generated by the commutator between the Hamiltonian and the density operator ˆρ, is responsible for the entanglement, corresponding to the superposition, separately, of bra or ket-states of the composing subsystems. For a bipartite system the interaction between the two, which is responsible for the entanglement production, intervenes through terms like

[ ˆHint, ˆρ] = ˆH1ρˆ1⊗ ˆH2ρˆ2− ˆρ1Hˆ1⊗ ˆρ2Hˆ2, (4.21)

where the initial state is assumed to be of the form ˆρ = ˆρ1⊗ ˆρ2 and ˆHint = ˆH1⊗ ˆH2.

As we have shown in Chapter2, the quantum and the classical dynamics, respec-tively embodied in the von Neumann and the Liouville equation, differ by a char-acteristic superoperator. Thus, besides the usual quantum mechanical correlations produced through the commutator structure, there are also additional correlations that entangle bra and ket-states through the action of the superoperator term, char-acteristic of the classical evolution. As pointed out in Ref. (EGV10), this leads us to distinguish between intra- and inter-space entanglement. Inter-space entanglement refers to superpositions of bra- and ket-states, or to a coupling of the underlying Hilbert space and its adjoint, while, in distinction, we refer to intra-space as the usual quantum mechanical entanglement.

In the studied case of a hydrogen atom interacting with an electromagnetic field, we have shown in Section 4.1 that the superoperator part related to the field-atom interaction, i.e. the Γ and Σ terms, can be entirely expressed in terms of quantum mechanical operators. When taking a look at the formal structure of these terms, it is evident how they differ from the usual commutator one. For example, the term (4.12), 1 2e −ik·ˆq/2_{p · ˆ}_ˆ _{ρ ˆ}_Q ke −ik·ˆq/2_, _(4.22)

has a very unfamiliar structure wich differs from the usual commutator one. Indeed, the operators appear both on the right side and the left side of the density operator, allowing the production of additional correlations which have an entirely classical origin. This topic certainly deserves further study.

As we have seen in Section 4.1, within the context of the dipole approximation, the only extra term which plays a substantial difference between the transformed

(37)

4.3 – Liouville space and superoperators

classical Liouville equation and the von Neumann equation is the superoperator E(Q,q). Thus, this is the only term that produces inter-space entanglement between the bra- and ket-states of the atomic system. This topic certainly deserves further study.

4.3 Liouville space and superoperators

Considering an arbitrary system characterized by a Hamiltonian ˆH, we introduce a complete basis set of functions |ji in Hilbert space. Given the von Neumann equation, we can consider its matrix elements

i∂tρjk = [( ˆH ˆρ)jk − (ˆρ ˆH)jk], j,k = 1,2, · · · ,N. (4.23)

For a N -level system, the density matrix has N2 _{matrix elements and the previous}

equation be written as

i∂tρjk =

X

m,n

Ljk,mnρmn (4.24)

where the Liouville superoperator ˆL is defined as

Ljk,lm ≡ Hjmδjk− Hkn∗ δjm. (4.25)

The Liouville operator is a N2 _{× N}2 _{matrix, and since each element ρ}

mn is labeled

by two indices, the matrix elements of ˆL are labeled by four indices and is called a superoperator. This space, where the density operator is a vector is called the Liouville space. The dynamics of the density operator is more conveniently described in this space. In Hilbert space, if we consider a complete basis set |ji (j = 1, · · · ,N ), we can write

ˆ

ρ =X

j,k

ρjk|jihk|. (4.26)

We may think of the family of N2_{operators |jihk|, with j,k = 1, · · · ,N , as a complete}

set of matrices and rewrite the density operator in the Liouville space notation in the following way:

|ρi =X

j,k

ρjk|jkii, (4.27)

In the Liouville space, any ordinary operator ˆA can be thought as a vector, which we denote by |Aii. This means that it can be expanded as

|Aii =X

j,k

A reformulation of Hamiltonian dynamics. A case study of the Jaynes-Cummings model.

Contents

Chapter 1

The quantum-classical divide

1.1

Emergent Quantum mechanics?

1.2

Aim and structure of the thesis

Chapter 2

Reformulation of Hamiltonian

dynamics

2.1

Reformulation of Hamiltonian dynamics

2.2

Reformulation of electrodynamics in the

Coul-omb gauge

Chapter 3

The Jaynes-Cummings model

3.1

Radiation-matter interaction

3.2

The Power-Zienau-Wolley transformation

3.3

Dipole approximation and the derivation of

the Jaynes-Cummings Hamiltonian

Chapter 4

The extra terms in the

Jaynes-Cummings model

4.1

The PZW transformation and the dipole

ap-proximation

4.2

Classically generated entanglement?

4.3

Liouville space and superoperators