Three level atom OTE systems

Universite de Franche-ComteUTINAM CNRS

Universidad Simon Bolıvar

INTERNSHIP REPORT M1

Atomic dynamics out of thermal equilibrium: a threelevel atom

Luis Enrique ParraSupervisor: Bruno Bellomo

Besancon - France19 June 2015

Contents

Introduction 2

1 The markovian master equation 41.1 Closed quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Open quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Microscopic derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Presentation of the model 82.1 Thermal equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Out of thermal equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Application to the two level atom . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Three level systems 123.1 Ladder configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Λ configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 V configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Conclusion 20

Appendix A The interaction picture 22

Appendix B Three level systems 23

Bibliography 24

1

Introduction

The study of open quantum systems is a vast area of research [1] since every real quantum sys-tem interacts with its surroundings to a certain extent [2]. In general, the system-environmentinteraction leads to non-unitary dynamics for the open quantum system (the reduced systemS) whose state is described by a density matrix operator since it is, in general, a mixed state.The non-unitary dynamics of S leads to quantum decoherence caused by its entanglement withthe external degrees of freedom representing its environment B. The decoherence process rep-resents the loss of coherence between the components of a quantum superposition and is ingeneral harmful for practical applications exploiting quantum properties [3]. It is important inmany areas of physics such as quantum information, quantum optics and quantum thermody-namics, to understand the nature of open quantum systems in order to contrast the decoherenceprocesses. In many cases, it is not possible to describe the dynamics of the total system (S+B)because it is either too large or there is not much information about it. However, we cansuppose the S+B system as a closed one (i.e. no interaction with external systems) [1] andwithout knowing in detail the state of B we can study the evolution of S using its densitymatrix operator. This operator is governed by a master equation [1, 4] which describes howthe system’s populations and coherences evolve, representing a powerful tool to study an openquantum system.

To this date most efforts have focused on avoiding the decoherence induced by the environ-ment by protecting the system from the environmental noise [1], for example by manipulatingthe environment to reduce its influence. This brings up the question: is there a way of ex-ploiting the system’s dissipative nature or will the noise have always a negative effect? Severalmethods, such as the reservoir engineering ones, have been used to exploit the dissipative dy-namics of a system [5, 6]. This can be achieved by properly modifying the properties of theenvironment and its coupling with the system, i.e. treating the noise as a tool to achieve adesired configuration [7, 8].

In this context, out of thermal equilibrium (OTE) systems have been also studied. Thesesystems are characterized by the presence of one or several reservoirs held at different tempera-tures. This absence of thermal equilibrium is a natural condition present in several systems, (e.g.cold atoms, biological systems [9] and in some experimental configurations [10]). There havebeen promising results about OTE systems since it has been shown that there is a potential tocontrol and manipulate atomic systems immersed in this type of environments. Notably, therehas been intensive research concerning heat transfer [11], Casimir-Lifshitz interactio [10, 12],as well as a renewed interest in the area of quantum thermodynamics including the reintro-duction of the concept of quantum thermal machines [13, 14]. In particular, the influenceof several blackbody thermal reservoirs held at different temperatures has been considered indifferent contexts, for example for a chain of spins [15]. Recently, multi-temperature realisticconfigurations of atoms surrounded by microscopic bodies have been investigated by takinginto account the dependence on the internal structure (material, geometry) of the reservoirs

2

3

[12, 16, 17, 18]. It has been shown that the dissipative dynamics of the system S can be manip-ulated to achieve relevant effects such as the inversion of population [16, 17] and the generationof steady entangled states [18].

In this project we analyze both two and three level systems interacting with an electro-magnetic field, at and out of thermal equilibrium, which plays the role of environment, to findconditions permitting control effects such as an inversion of populations. In the first section,by means of a microscopic derivation, a markovian master equation is derived. In the secondsection we present the model used to study the dynamics at and out of thermal equilibrium andwe study the case of a two-level atom. In the third section we study the three-level atom in thethree main configurations Λ, V and ladder, when the environment is at and out of equilibrium.

Chapter 1

The markovian master equation

1.1 Closed quantum systems

Here we briefly review how to describe the dynamics of a closed quantum system using itsdensity matrix operator. A closed system is a system which does not interact with an externalsystem. Its time evolution is governed by the unitary operator U(t, t0) such that |φ(t)〉 =U(t, t0) |φ(t0)〉, where |φ(t)〉 represents the state of the system, and whose evolution is governedby the equation

ih∂

∂tU(t, t0) = H(t)U(t, t0), (1.1)

where H(t) is the total hamiltonian of the system. If the system is in a mixed state (i.e. it isno longer possible to describe it with a single vector state) in order to describe its evolution, itbecomes necessary the introduction of the density matrix operator ρ(t) =

∑i pi |φi(t)〉〈φi(t)|,

where pi represents the weight factor of each state. The diagonal elements of ρ(t), ρii representthe populations of the states |φi〉 while the off diagonal ones, ρij , when i 6= j, represent thecoherences between the states |φi〉 and |φj〉. In the Schrodinger picture, the evolved densitymatrix is given by:

ρ(t) = U(t, t0)ρU†(t, t0). (1.2)

Taking the time derivative of Eq. (1.2) and using Eq. (1.1) we find:

d

dtρ(t) = − i

h[H(t), ρ(t)]. (1.3)

Eq. (1.3) is called the Liouville-Von Neumann equation and describes how the closed systemevolves in time. In the next section, it will be more convenient for a microscopic derivation ofthe master equation to work in the interaction picture. In this picture, if H(t) = H0 + HI(t)the previous equation becomes (see Appendix A for details):

d

dtρ(t) = − i

h[HI(t), ρ(t)], (1.4)

where HI = U†0 (t, t0)HI(t)U0(t, t0), U0(t, t0) = e−ihH0(t−t0) and the tilde indicates that the

given operator is in the interaction picture.

4

5 CHAPTER 1. THE MARKOVIAN MASTER EQUATION

1.2 Open quantum systems

Every real quantum system interacts with its environment to a certain extent. We have alreadystudied the closed system dynamics, but what if there is an interaction with an outer system?How does that modify our preceding formalism? An open quantum system is a system S thatinteracts with its environment B. An example might be an electron (the system) interacting withthe vibrational modes of a solid (the environment), or an atom interacting with the surroundingelectromagnetic field. Considering the system S+B as closed and using the formalism of section1.1 we can describe the dynamics of the total system. The Hilbert space of S+B is the tensorialproduct of the Hilbert spaces of S and B, thus the total Hamiltonian of the system is:

H(t) = HS ⊗ IB + IS ⊗HB +HI(t), (1.5)

where HS is the free Hamiltonian of S, HB the free Hamiltonian of B and HI(t) is the interactionHamiltonian. We want to study the behaviour of S under the influence of B, that is, findingthe master equation governing the evolution of ρS(t). To this end we remember that for acomposite system where ρ is the total density matrix operator of S+B and TrB is the partialtrace over the degrees of freedom of B, the system S can be described by means of the reduceddensity operator ρS = TrBρ. Differentiating this relation in time and using equation (1.3) weobtain:

d

dtρS(t) = − i

hT rB{[H(t), ρ(t)]}. (1.6)

We have to solve Eq. (1.6) to compute the dynamics of S. To this end it is typically necessaryto use certain approximations. In the following we will focus on the case of markovian masterequations. There are two ways of approaching this physical regime: one involves the useof quantum dynamical semigroups [1] and the other one, which we will develop, involves aderivation based on the knowledge of the total hamiltonian (microscopic approach).

1.2.1 Microscopic derivation

In order to solve equation (1.6) we will use certain approximations that allow us to simplifythe problem. Before doing that, we move to the interaction picture where eq. (1.6) becomes:

d

dtρS(t) = − i

hT rB [HI(t), ρ(t)]. (1.7)

Expressing equation (1.3) in its integral form in the interaction picture and reintroducing it inthe commutator of eq. (1.7) we obtain:

d

dtρS(t) = − i

htrB [HI(t), ρ(0)]− 1

h2

∫ t

0

TrB [HI(t), [HI(s), ρ(s)]]ds, (1.8)

where we have chosen t0 = 0. We now make some assumptions:

• At the initial time t = 0, no correlations exist between S and B. Hence, the initial densityoperator factorizes as ρ(0) = ρS(0)⊗ ρB(0)

• The interaction hamiltonian can be cast under the form HI =∑µAµ⊗Bµ where Aµ and

Bµ are operators which act only in the Hilbert space of S and the one of B respectively.In the interaction picture this leads to

HI(t) =∑µ

Aµ(t)⊗ Bµ(t) (1.9)


• TrB [HI(t), ρ(0)] = 0. This can be justified by the fact that HI(t) is linear in the operatorsAµ and Bµ, along with the first assumption and the condition TrB (HIρB(0)) = 0 wherewe have assumed that the average value of Bµ in the state ρB(0) is zero.

Approximations

Equation (1.8) depends on the expression for the total density matrix ρ(s). In order to simplifythis equation we will use the following approximations.

1. Born approximation: the coupling between the system and the reservoir is assumed to beweak (weak coupling approximation). Therefore, the density matrix of the environmentρB(t) is negligibly affected by the interaction and the state of the total system at time tcan be characterized by:

ρ(t) ≈ ρS(t)⊗ ρB(t).

2. Markovian approximation: it consists in making the quantum master equation local intime. This approximation is justified if the environment correlation time (see Eq. (1.14))is small compared to the relaxation time of the system, i.e. the interaction can be con-sidered as memoryless. To this end in the integrand of Eq. (1.8), after using the Bornapproximation we replace:

ρS(s) = ρS(t).

Taking (1.8) and applying the approximations 1 and 2 we obtain:

d

dtρS(t) = − 1

h2

∫ t

0

TrB [HI(t), [HI(s), ρS(t)⊗ ρB(t)]]ds. (1.10)

The problem with Eq. (1.10) (called Redfield equation) is that ρS(t) depends on thepreparation of the system at t=0. To solve this, we notice that for s >> τB (τB repre-sents the scale of time in which the reservoir correlation function decays) the integranddisappears. So we can substitute s=t-s and take the upper limit to ∞, finally obtaining:

d

dtρS(t) = − 1

h2

∫ ∞0

trB [HI(t), [HI(t− s), ρS(t)⊗ ρB(t)]]ds. (1.11)

From now on, we suppose that ρB(t) is an stationary state of the environment, that is[HB , ρB ] = 0 or equivalently ρB(t) = ρB(0) = ρB . Taking Eq. (1.11) and substitutingthe expression for the interaction hamiltonian in terms of the operators Aµ and Bµ weobtain:

d

dtρS(t) =

∑ω,ω′

∑µ,ν

ei(ω′−ω)tζµν(ω)(Aν(ω)ρS(t)A†µ(ω′)−A†µ(ω′)Aν(ω)ρS(t)) + h.c. (1.12)

whereAµ(ω) ≡

∑ε′−ε=ω

Π(ε)AµΠ(ε′), (1.13)

ζµν(ω) ≡ 1

h2

∫ ∞0

ds⟨B†µ(t)Bν(t− s)

⟩eiωs. (1.14)

With Π(ε) being the projector associated to the eigenvalue ε of the Hamilonian HS ,

B†µ(t) = eihHBtB†µe

− ihHBt and ζµν(ω) is the one-sided Fourier transform of the reservoir


correlation functions⟨B†µ(t)Bν(t− s)

⟩. Two important relations can be deduced from

Eq. (1.13):

[HS , Aµ(ω)] = −hωAµ(ω) , [HS , A†µ(ω)] = hωA†µ(ω). (1.15)

Looking at the relations (1.15) we can conclude that Aµ(ω) and A†µ(ω) are eigenoper-ators of HS with eigenvalues ∓hω. Furthermore, if |φ〉 is an eigenstate of HS with aneigenvalue equal to ε, then Aµ(ω) |φ〉 is an eigenstate of HS with eigenvalue ε − hω andcorrespondingly A†µ changes the eigenvalue to ε+ hω.

3. Rotating wave approximation: it consists in averaging over the rapidly oscillating terms.When the typical timescales for the system’s evolution, proportional to |ω − ω′|, aremuch shorter than the expected relaxation timescales for the system, the rotating waveapproximation can be applied: all terms with ω − ω′ 6= 0 are considered as varying toofast, so that their average contribution in Eq. (1.12) (after integrating) on the timescalesrelevant to S can be neglected.

As a result of this approximation Eq. (1.12) reduces to:

d

dtρS(t) = − i

h[HLS , ρS(t)] +D(ρS(t)), (1.16)

HLS =∑ω

∑µ,ν

sµν(ω)A†µ(ω)Aν(ω),

D(ρS(t)) =∑ω

∑µ,ν

γµν(ω)(Aνω)ρSA†µ(ω)− 1

2{A†µ(ω)Aν(ω), ρS}), (1.17)

sµν(ω) =1

2i(ζµν(ω)− ζ∗µν(ω)) , γµν(ω) = ζµν(ω) + ζ∗µν(ω), (1.18)

where HLS is called the Lamb-shift hamiltonian and D(ρS(t)) is the dissipator. Eq. (1.16) iscalled the Markovian master equation of the system, and it is written in the interaction picture.sµν(ω) gives rise to a shift of the energetic levels of HS and γµν(ω) represents the decay rates.In order to return to the Schrodinger representation it can be shown that we obtain it bysimply adding the free Hamiltonian HS to HLS and by replacing each operator by its form inthe Schrodinger picture. For a more detailed derivation see [1].

Chapter 2

Presentation of the model

We consider an arbitrary N-level atom interacting with an electromagnetic field that plays therole of the environment and we apply the formalism presented in chapter 2 to this configuration.The total hamiltonian of the system has the form:

H = HS +HB +HI ,

where HS and HB are the free hamiltonians of the system and the field respectively, andHI = −~D · ~E is the interaction hamiltonian involving the electric field ~E at the atomic positionand the atomic electric dipole operator ~D. In the interaction picture, it becomes

HI(t) = − ~D(t) · ~E(t) (2.1)

with~D(t) =

∑i,j∀j>i

(~dij |i〉〈j| e−iωjit + h.c

), (2.2)

where the sum runs over the permitted transitions such that ωji = ωj −ωi > 0 is the frequencyassociated with the transition between the states |i〉 and |j〉 with energies εi = hωi and εj = hωjrespectively, and ~dij = 〈i| ~D |j〉. The operators ~D and ~E are related to the operators Aµ andBµ of section 1.2.1 and the subscript µ labels the components of each operator. Comparing

Eq. (1.9) and (2.1) with Eq. (2.2), one sees that A(ω) =∑ωji=ω

~dij |i〉〈j| = A†(−ω). In thefollowing, we will derive a suitable master equation governing the atomic dynamics, containingboth decay rates (Γ±ij) and shifts (S±ij ) built on the basis of Eq. (1.18). From this point on, wewill use the notation:

Γ+ji =

∑µ,ν

γµν(ωji)[dij ]∗µ[dij ]ν , Γ−ji =

∑µ,ν

γµν(−ωji)[dij ]µ[dij ]∗ν , (2.3)

S+ji =

∑µ,ν

sµν(ωji)[dij ]∗µ[dij ]ν , S

−ji =

∑µ,ν

sµν(−ωji)[dij ]µ[dij ]∗ν , (2.4)

hωjikBT

= xji,

where kB is the Boltzmann constant.

8

9 CHAPTER 2. PRESENTATION OF THE MODEL

2.1 Thermal equilibrium

In section 1.2 we have assumed in the derivation of the markovian master equation that ρB(t) =ρB(0) = ρB , this means that the environment is kept fixed in a stationary state. Henceforth, wewill assume that the environment is at thermal equilibrium, i.e. ρB is represented by a Gibbsstate. If there is no external time-dependent fields, the stationary solution for the masterequation (1.16) has the form of an atomic Gibbs state [1]:

ρGibbs =Exp(− HS

kBT)

TrSExp(− HS

kBT). (2.5)

Using in equations (1.18) and (1.14) the KMS condition, which relates the environment corre-lation functions in the following way:

⟨B†µ(t)Bν(0)

⟩=⟨Bν(0)B†µ(t+ iβ)

⟩, where β = h

kBT, one

can derive the following relations for the γµν(±ω) :

γµν(−ω) = exp(−βω)γνµ(ω). (2.6)

Inserting the above relation in Eq. (1.16) to find the equation for the populations, it canbe shown that:

W (i|j)e−βωj = W (j|i)e−βωi , (2.7)

where W (i|j) is the time-independent transition rate given by:

W (i|j) =∑µ,ν

γµν(ωij)h(ωi − ωj) 〈i|Aµ |j〉〈j|Aν |i〉 . (2.8)

Eq. (2.7) is called the detailed balance condition. Using Eq. (2.8) in Eq. (1.16) along withEq. (2.7) we arrive to the conclusion that at thermal equilibrium the populations follow theBoltzmann distribution, characterized by the fact that the ratio between any two populationsdepends only on the frequency of the transition and the temperature:

ρiiρjj

= Exp(−βωji). (2.9)

By calculating the environment correlation functions, it is possible to show [1, 17] that thedecay rates Γ±ji defined in Eq. (2.3) assume the form:

Γ+ji = γji0 Λji(1 + nji) , Γ−ji = γji0 Λjinji, (2.10)

where γji0 =4ω3

ji|~dij|23hc3 is the spontaneus emission rate, nji is the average number of photons

associated with the transition of frequency ωji,

nji =1

Exp (βωji)− 1, (2.11)

and Λji is a function that depends on the geometric properties of the configuration and thetransition frequency ωji. If the system is not surrounded by matter, Λji → 1, while in thepresence of matter, Λji will take into account the effects of the disposition of the surroundingbodies as well as their geometry and dielectric properties.


2.2 Out of thermal equilibrium

In this section, we use the formalism developed in [17], where the configuration consists ofa N-level atom placed near a body of arbitrary geometry and dielectric permittivity and attemperature TM , also interacting with the environmental radiation generated by far surroundingwalls at temperature TW . In these systems relevant effects such as inversion of populations havebeen pointed out. In the out of thermal equilibrium case, TM 6= TW , the dynamics of the systemis quite rich. The parameters Γ±ji depend on the geometry and the transmission and absorptionproperties of the body surrounding the system S and on the temperatures. It can be shown forthis configuration that the Γ±ji can be written in the form:

Γ+ji = γji0 Λji(1 + neffji ) , Γ−ji = γji0 Λjin

effji , (2.12)

where neffji replaces nji of Eq. (2.11), and is a complex function of all the parameters such asTW , TM and the distance “z” between the atom and the body, as well as the properties of thelatter. The main point here is that each transition is characterized by rates formally equivalentto a case at thermal equilibrium at an effective temperature T effji . This effective temperature

is linked to the parameter neffji via the relation:

T effji =hωjikB

[ln(1 + neff−1

ji )]−1

. (2.13)

In particular, it has been shown [17] that neffji is restricted to the interval n(ωji, Tmax) >

neffji > n(ωji, Tmin) where Tmax = max{TW , TM}, Tmin = min{TW , TM} and n(ωji, T ) = nji

at temperature T. Furthermore, at thermal equilibrium, TM = TW , neffji becomes neffji = nji.

In the following, we assume that it is possible to vary freely neffji . For each choice of neffji ,we imagine possible to find a specific configuration producing the chosen values. These kindof models have been studied in other contexts [14] where different transitions are connected tothermal reservoirs at different temperatures.

2.3 Application to the two level atom

Here we assume that the atom has only two levels, ω21 = ω2−ω1 being the transition frequencybetween the excited state |2〉 and the ground state |1〉. The atomic hamiltonian HS and the

atomic dipole operator ~D(t) are given by:

HS = h(ω1 |1〉〈1|+ ω2 |2〉〈2|). (2.14)

~D(t) = ~d12 |1〉〈2| e−iω21t + ~d∗12 |2〉〈1| eiω21t, (2.15)

where ~d12 = 〈1| ~D |2〉. Changing Eq. (1.16) to the Schrodinger picture and substituting Eq.(2.15) we obtain:

d

dtρS(t) = −i

[HS

h+ S−21 |1〉〈1|+ S+

21 |2〉〈2| , ρ(t)

](2.16)

+ Γ−21

(σ+ρ(t)σ− − 1

2{|1〉〈1| , ρ(t)}

)+ Γ+

21

(σ−ρ(t)σ+ − 1

2{|2〉〈2| , ρ(t)}

),


where σ+ = |2〉〈1| and σ− = |1〉〈2|. Solving Eq. (2.16) for the stationary case, dρS(t)/dt = 0,we obtain:

ρ11 =Γ+

21

Γ+21 + Γ−21

, ρ22 =Γ−21

Γ+21 + Γ−21

, ρ21 = ρ12 = 0. (2.17)

In the following, we give the explicit values for the steady populations, at and out of thermalequilibrium.

• Thermal equilibrium: according to Eq.(2.10), the system (2.17) becomes:

ρ11 =ex21

ex21 + 1, ρ22 =

1

ex21 + 1. (2.18)

Being x21 > 0 it follows that ρ11 > ρ22.

• Out of thermal equilibrium: according to Eq. (2.12), the steady populations of Eq.(2.17) becomes:

ρ11 =1 + neff21

2neff21 + 1, ρ22 =

neff21

2neff21 + 1. (2.19)

We see that also out of thermal equilibrium ρ11 > ρ22.

Indeed, the choice of a two-level system forbids the possibility of a population inversion [17].In the following chapter we will investigate the three-level case in different configurations andwe will look for the presence of quantum effects such as population inversions. In particular,we will extend the analysis of [17] concerning the Λ case to other configurations, such as the Vand ladder ones.

Chapter 3

Three level systems

In this chapter we focus on the three-level atom in three main configurations characterized bydifferent allowed transitions, as shown in figure 3.1. We label the three states with |1〉, |2〉 and|3〉, with frequencies ω1, ω2 and ω3 (in increasing order). The free hamiltonian of the threelevel system is:

HS = h(ω1 |1〉〈1|+ ω2 |2〉〈2|+ ω3 |3〉〈3|). (3.1)

(a) Ladder (b) Λ (c) V

Figure 3.1: Three level system in a ladder, Λ and V configuration.

3.1 Ladder configuration

In this configuration, shown in Fig. 3.1a, the transition between the first and the third level isforbidden. As a result, the atomic dipole operator of Eq. (2.2) takes the form:

~D(t) = ~d12 |1〉〈2| e−ω21t + ~d23 |2〉〈3| e−ω32t + h.c. (3.2)

Substituting Eq. (3.2) on Eq. (1.16) and moving to the interaction picture we obtain:

d

dtρS(t) = −i

[HS

h+ S−21 |1〉〈1|+ S+

32 |3〉〈3|+ (S+21 + S−32) |2〉〈2| , ρS(t)

]+ Γ−21

(ρ11(t) |2〉〈2| − 1

2{|1〉〈1| , ρS(t)}

)+ Γ+

21

(ρ22(t) |1〉〈1| − 1

2{|2〉〈2| , ρS(t)}

)+ Γ−32

(ρ22(t) |3〉〈3| − 1

2{|2〉〈2| , ρS(t)}

)+ Γ+

32

(ρ33(t) |2〉〈2| − 1

2{|3〉〈3| , ρS(t)}

).

(3.3)

12

13 CHAPTER 3. THREE LEVEL SYSTEMS

The stationary solution of this equation is given by:

ρ11 =Γ+

21Γ+32

γlad, ρ22 =

Γ−21Γ+32

γlad, ρ33 =

Γ−21Γ−32

γlad, ρ31 = ρ21 = ρ32 = 0 (3.4)

where γlad = Γ+21Γ+

32 + Γ−21Γ+32 + Γ−21Γ−32 and ρ∗ji = ρij . Calculating the ratios between the

populations we find:ρ22

ρ11=

Γ−21

Γ+21

,ρ33

ρ11=

Γ−21

Γ+21

Γ−32

Γ+32

,ρ33

ρ22=

Γ−32

Γ+32

. (3.5)

Using Eqs. (2.10) and (2.12), one can see that at both thermal and out of thermal equilib-rium cases, it holds ρ11 > ρ22 > ρ33.

• Thermal equilibrium: Using Equations (2.10) and (2.11) in Eq.(3.4) we obtain:

ρ11 =ex21+x32

γthlad, ρ22 =

ex32

γthlad, ρ33 =

1

γthlad, (3.6)

where γthlad = ex21+x32 + ex32 + 1. We can indeed verify that ρ11 > ρ22 > ρ33 since x21 > 0and x32 > 0. The time dependent solution of Eq. (3.3) is very complex and will notbe reported in the following. We plot the time dependent solution for a particular setof values, as shown in Fig. 3.2, illustrating the case when γ21

0 = γ320 = γ0, x21 = 0.5,

x32 = 1, and ρ33(0) = 1. We see how the populations evolve until reaching their steadyvalues given by Eq. (3.6).

Figure 3.2: ρ(t) as a function of γ0t in the ladder configuration at thermal equilibrium forγ21

0 = γ320 = γ0, x21 = 0.5, x32 = 1 and ρ33(0) = 1.

• Out of thermal equilibrium: Using (2.12) in Eq. (3.4) we obtain:

ρ11 =(1 + neff21 )(1 + neff32 )

γotelad, ρ22 =

neff21 (1 + neff32 )

γotelad, ρ33 =

neff21 neff32

γotelad, (3.7)


where γotelad = 1+2neff21 +neff32 +3neff21 neff32 . It is easy to verify that ρ11 > ρ22 > ρ33. Even

if we can vary freely neffji , the order of the steady populations, imposed at thermal equi-librium, must be preserved. Since it is not possible to obtain an inversion of populations,the plot for the out of thermal equilibrium case will not be shown.

3.2 Λ configuration

In this configuration, shown in Fig. 3.1b, the transition between the first level and the secondone is forbidden. Consequently, the atomic dipole operator of Eq. (2.2) takes the form:

~D(t) = ~d13 |1〉〈3| e−iω31t + ~d23 |2〉〈3| e−iω32 + h.c. (3.8)

Introducing Eq. (3.8) in Eq. (1.16) and moving to the Schrodinger picture, we obtain:

d

dtρS(t) = −i

[HS

h+ S−31 |1〉〈1|+ S−32 |2〉〈2|+ (S+

31 + S+32) |3〉〈3| , ρS(t)

]+ Γ−31

(ρ11(t) |3〉〈3| − 1

2{|1〉〈1| , ρS(t)}

)+ Γ+

31

(ρ33(t) |1〉〈1| − 1

2{|3〉〈3| , ρ(t)}

)+ Γ−32

(ρ22(t) |3〉〈3| − 1

2{|2〉〈2| , ρS(t)}

)+ Γ+

32

(ρ33(t) |2〉〈2| − 1

2{|3〉〈3| , ρS(t)}

).

(3.9)

The stationary solution of this equation is given by:

ρ11 =Γ+

31Γ−32

γΛ, ρ22 =

Γ−31Γ+32

γΛ, ρ33 =

Γ−31Γ−32

γΛ, ρ31 = ρ21 = ρ32 = 0, (3.10)

where γΛ = Γ+31Γ−32 +Γ−31Γ+

32 +Γ−31Γ−32 and ρ∗ji = ρij . The ratios between stationary populationsare equal to:

ρ22

ρ11=

Γ+32

Γ−32

Γ−31

Γ+31

,ρ33

ρ11=

Γ−31

Γ+31

,ρ33

ρ22=

Γ−32

Γ+32

. (3.11)

Using Eqs. (2.10) and (2.12) into Eq. (3.11) it follows that ρ11 > ρ33 and ρ22 > ρ33. However

the termΓ+

32

Γ−32

Γ−31

Γ+31

does not allow us to conclude a general relation between ρ11 and ρ22 since it

depends on whether the system is at thermal equilibrium or not, this will be discussed in thefollowing.

• Thermal equilibrium: Using (2.10) in Eq. (3.10) we obtain:

ρ11 =ex31

γthΛ, ρ22 =

ex32

γthΛ, ρ33 =

1

γthΛ. (3.12)

where γthΛ = ex31 + ex32 + 1. On one hand we can verify that ρ11 > ρ33 and ρ22 > ρ33

since x31, x32 > 0. On the other hand we have the ratio ρ22

ρ11= ex32−x31 , and being

x31 > x32 (since ω31 > ω32) we obtain ρ11 > ρ22, as expected for a system at thermalequilibrium. Fig. 3.3 illustrates the evolution of populations to their stationary valuessuch that ρ11 > ρ22 > ρ33. The parameters of Fig. 3.3 are the same as the ones of Fig.3.2. In this figure we can see how the populations evolve . This plot will be used as acomparison with the out of thermal equilibrium case.


Figure 3.3: ρ(t) as a function of γ0t in the Λ configuration at thermal equilibrium whenγ31

0 = γ320 = 1, x31 = 1.5, x32 = 1 and ρ33(0) = 1.


ρ11 =(1 + neff31 )neff32

γoteΛ

, ρ22 =neff31 (1 + neff32 )

γoteΛ

, ρ33 =neff31 neff32

γoteΛ

, (3.13)

where γoteΛ = neff31 + neff32 + 3neff31 neff32 . It is easy to verify that ρ11 > ρ33 and ρ22 > ρ33.

However, the term ρ22

ρ11is more delicate to analyze since now we can vary freely the neffji .

In this case the ratio between these populations is:

ρ22

ρ11=neff31 + neff31 neff32

neff32 + neff31 neff32

. (3.14)

This ratio depends only on the ratio between neff31 and neff32 . This can be seen in Fig. 3.4

where we have plotted the ratio ρ22

ρ11as a function of neff31 and neff32 . In this figure we can

see the line of inversion (neff31 = neff32 ), and the asymptotic behaviour of the function.

Eq. (3.14) points out the possibility of a population inversion when neff31 > neff32 . In

particular, in the limit case, neff31 → ∞ and neff32 → 0, Eq. (3.13) implies ρ22 → 1. Fig. 3.5

shows the case when γ310 = γ32

0 = γ0, neff31 = 1.5, neff32 = 0.6 and ρ33(0) = 1. In this plotwe observe how the first and second level populations achieve their steady values such thatρ22 > ρ11, in contrast with Fig. 3.3.


Figure 3.4: ρ22

ρ11as a function of neff31 and neff32 .

Figure 3.5: ρ as a function of γ0t in the Λ configuration out of thermal equilibrium whenγ31

0 = γ320 = γ0, neff31 = 1.5, neff32 = 0.6 and ρ33(0) = 1

3.3 V configuration

In this configuration, shown in Fig. 3.1c, the transition between the second level and the thirdone is forbidden. As a result, the atomic dipole operator of Eq. (2.2) takes the form:

~D(t) = ~d13 |1〉〈3| e−iω31t + ~d12 |1〉〈2| e−iω21t + h.c. (3.15)


Using Eq. (3.15) in Eq. (1.16) and moving to the Schrodinger representation, we obtain:

d

dtρS(t) = −i

[HS

h+ S+

21 |2〉〈2|+ S+31 |3〉〈3|+ (S−31 + S−21) |1〉〈1| , ρS(t)

]+ Γ−31

(ρ11(t) |3〉〈3| − 1

2{|1〉〈1| , ρS(t)}

)+ Γ+

31

(ρ33(t) |1〉〈1| − 1

2{|3〉〈3| , ρS(t)}

)+ Γ−21

(ρ11(t) |2〉〈2| − 1

2{|1〉〈1| , ρ(t)}

)+ Γ+

21

(ρ22(t) |1〉〈1| − 1

2{|2〉〈2| , ρS(t)}

).

(3.16)

Solving for the stationary case:

ρ11 =Γ+

31Γ+21

γV, ρ22 =

Γ−21Γ+31

γV, ρ33 =

Γ+21Γ−31

γV, ρ31 = ρ32 = ρ21 = 0, (3.17)

where γV = Γ+31Γ+

21 + Γ−21Γ+31 + Γ+

21Γ−31 and ρ∗ji = ρij . The ratios between the stationarypopulations are given by:

ρ22

ρ11=

Γ−21

Γ+21

,ρ33

ρ11=

Γ−31

Γ+31

,ρ33

ρ22=

Γ+21

Γ−21

Γ−31

Γ+31

. (3.18)

Using equations (2.10) and (2.12) in Eq. (3.18) one can immediately see that ρ11 > ρ22 and

ρ11 > ρ33, however the termΓ+

21

Γ−21

Γ−31

Γ+31

, analogously to the Λ case, depends whether the system is

in equilibrium or not. This case will be studied in the following.

• Thermal equilibrium: Inserting (2.10) in Eq. (3.17) we obtain:

ρ11 =ex21+x31

γthV, ρ22 =

ex31

γthV, ρ33 =

ex21

γthV, (3.19)

where γthV = ex21+x31 + ex31 + ex21 . It is easy to verify that ρ11 > ρ33 and ρ11 > ρ22

since x31, x21 > 0 and the term ρ33

ρ22= ex21−x31 allow us to conclude that ρ22 > ρ33

given that x31 > x21 (since ω31 > ω21). Fig. 3.6 illustrates the system’s evolution forthe V configuration with the same values as in Fig. 3.2. In this figure we can see theevolution of the steady populations to their corresponding values. This case will be usedas a comparison with the out of thermal equilibrium V-configuration.


ρ11 =(1 + neff31 )(1 + neff21 )

γoteV, ρ22 =


γoteV, ρ33 =


γoteV, (3.20)

where γoteV = 1+2neff31 +2neff21 +3neff31 neff21 . Once again it is easy to verify that ρ11 > ρ33

and ρ11 > ρ22. For the ratio between ρ22 and ρ33 we get:

ρ22

ρ33=neff21 + neff31 neff21

neff31 + neff31 neff21

. (3.21)

Eq. (3.21) depends on the ratio between neff31 and neff21 . A population inversion can be

obtained since ρ33 > ρ22 when neff31 > neff21 . The plot of the ratio shown in Eq. (3.21)

as a function of the neffji will be similar to the one obtained in the Λ case (Fig. 3.4).


Figure 3.6: ρ as a function of γ0t in the V configuration at thermal equilibrium when γ310 =

γ210 = γ0, x21 = 0.5, x31 = 1.5, x32 = 1 and ρ33(0) = 1.

However, the asymptotic behaviours of the inverted populations are quite different. Inthis case given the conditions neff31 →∞ and neff21 → 0, then ρ33 → 1

2 . In particular, Fig.3.7 shows the evolution of populations in the V case with the same values as in Fig. 3.5.In this plot we can see how the inversion occurs between the second and third level incontrast to the case at thermal equilibrium.

Figure 3.7: ρ(t) as a function γ0t in the V configuration out of thermal equilibrium when

γ310 = γ21

0 = γ0, neff21 = 0.3, neff31 = 1.5 and ρ33(0) = 1.


We conclude by analyzing the results obtained in this chapter. Inversion of populationsonly occur in the V and Λ configurations while in the ladder one the common order of steadypopulations, with respect to the case at thermal equilibrium, is preserved. We can provide asimple explanation for this by exploiting Eq. (2.13). This equation permits to associate differenttemperatures to each one of the transitions. In the V and Λ cases, we can formulate the followingcriteria for population inversion. In the Λ configuration the condition for population inversionis neff31 > neff32 , i.e. T eff31 > T eff32 ; the “hotter” transition induces more migration towards thethird level than the “colder” one, producing an overall migration from the first level towardsthe second level. In the V configuration the condition for the inversion is neff31 > neff21 , i.e.

T eff31 > T eff21 , the “cooler” transition induces more migration towards the first level than the“hotter” one, producing an overall migration from the second level towards the third one.

In the general case with more than three levels, N¿3, it can be shown that the criteria forordering inversion can be directly extracted by knowing the connections between the levels.This can be obtained by modifying the N levels in blocks of 3 levels. Given an arbitrarynon-degenerated N-level system without loops (no closed links between levels), inversions ofpopulations can always be obtained between any two non-directly connected levels that arenot indirectly connected only by ladder schemes. This means that if there is a ladder typeconnection between two levels, it is impossible to obtain an inversion. For example, in thefour-level case with the following permitted transitions: 1↔ 2↔ 4↔ 3, there is a ladder-typeconfiguration concerning the levels 1,2 and 4. In this case, an inversion between any two ofthese levels is impossible. However, since the first and second level are not connected to thethird one or do not share a ladder scheme, then it can be shown that an inversion of steadypopulations is possible between them.

Conclusion

In this project we investigated the dynamics of an elementary open quantum system, such as athree-level atom, interacting with an environment at and out of thermal equilibrium. We haverevised the formalism for open quantum systems to obtain a markovian master equation. Wehave derived a suitable master equation for the atomic dynamics in such stationary environ-ments, providing useful expressions for the transition rates governing the dynamics, based onthe results of [17]. We have pointed out relevant differences between the thermal equilibriumcase and the out of thermal equilibrium one, with steady states depending on geometrical andmaterial properties as well as the temperatures of the surrounding bodies. After studying thedynamics of open quantum systems in the two regimes (thermal and out thermal equilibrium),the cases of two- and three-level atoms have been discussed. In the first case, the atom ther-malizes to a Gibbs state, while in the second case, depending on which transitions are allowed,the steady states are not represented by Gibbs ones in general, and the steady populationsmay differ from the equilibrium values. This effect, allows one to manipulate the steady states,achieving for example, inversion of populations. We have then specialized our analysis to three-level systems in the ladder, Λ and V configurations showing that population inversions can onlybe obtained in the last two. This is justified by the fact that the system dynamics can be inter-preted in terms of effective temperatures associated to each allowed transition. We can freelyvary these temperatures to obtain conditions permitting the population inversion.

This work suggests that similar effects will be present also in the case of more levels. Theinversion criteria presented in chapter 3 can be extended to N-level atoms, where the stationarysolutions can be extracted by knowing only the connections between levels. In particular, thiscan be extended to the case when there are loops in the schemes (closed links between levels),since it is easy to show that in this case, the criteria for inversion depends not only on theneffji but also on the γji0 producing a more rich dynamic. It can be studied the possibilityof developing a method to reduce any N-level system in blocks of 3-levels and quickly identifypossible inversions. The results reported in this project could be also of interest for experimentalinvestigation in the absence of thermal equilibrium involving real or artificial atoms (such asquantum dots).

20

Aknowledgements

I would like to thank professor Bruno Bellomo for all the hard work, advises and supportconcerning the project. Also, a special thanks to M.G. Viloria, J. Breitenstein, M. Rotondi,O.L. Pino, J. Casas, C. Ladera, S. Parra and D. M. Parra for useful discussions and theunconditional support given.

21

Appendix A

The interaction picture

In quantum mechanics there are a number of equivalent descriptions, related by unitary trans-formations, of the dynamics of a system. In the Schrodinger picture, the operators are timeindependent while the states carry the evolution in time. In contrast, in the Heisenberg pictureall the dynamics are contained in the operators while the states do not evolve. Between thesetwo extremes, there is the interaction picture in which the part of the dynamics, associatedwith uncoupled evolution is contained in the operators, while that arising from the coupling iscontained in the state. In general, the evolution of a state is carried by the evolution operatorU(t, t0) where U(t0, t0) = I, I being the identity operator and |φ(t)〉 = U(t, t0) |φ(t0)〉. It is easyto show that for an arbitrary time dependent hamiltonian, the general form of the evolutionoperator is given by:

U(t, t0) = T←e−i

∫ tt0H(s)ds

(A.1)

where T← is the chronological time-ordering operator. Let us write the Hamiltonian of thesystem in the following way:

H(t) = H0 +HI(t), (A.2)

and let us define:

U0(t, t0) = e−ihH0(t−t0) , UI(t, t0) = U†0 (t, t0)U(t, t0). (A.3)

Finally we introduce A(t) as the interaction picture form of an arbitrary operator A(t) andρ(t) as the interaction picture density matrix :

A(t) ≡ U†0 (t, t0)A(t)U0(t, t0) , ρ(t) ≡ UI(t, t0)ρ(t0)U†I (t, t0). (A.4)

In this representation, the Von-Neumann equation (1.3) becomes:

d

dtρI(t) = −i[HI(t), ρ(t)] (A.5)

where HI(t) ≡ U†0 (t, t0)HI(t)U0(t, t0) We shall make use of the interaction picture in derivingmaster equations that govern the evolution of the density operator of an open quantum systembecause it allows us to shunt all the time dependence of H0 onto the operators, and leavingonly HI(t) to control the time-evolution of the states.

22

Appendix B

Three level systems

In this appendix we project each master equation treated in chapter 3 onto the basis of HS ,B = {|1〉 , |2〉 , |3〉}.

• Ladder configuration: Projecting Eq. (3.3) on B we obtain:

d

dtρ11(t) = −Γ−21ρ11(t) + Γ+

21ρ22(t)

d

dtρ22(t) = −(Γ−32 + Γ+

21)ρ22(t) + Γ−21ρ11(t) + Γ+32ρ33(t)

d

dtρ33(t) = Γ−32ρ22(t)− Γ+

32ρ33(t) (B.1)

d

dtρ12(t) =

[i∆12 −

Γ−21 + Γ−32

2

]ρ12(t)

d

dtρ13(t) =

[i∆13 −

Γ−21 + Γ+32 + Γ+

21

2

]ρ13(t)

d

dtρ23(t) =

[i∆23 −

Γ−32 + Γ+32 + Γ+

21

2

]ρ32(t),

where ∆12 = ω21+S−21−(S+21+S−32), ∆13 = ω31+S−21−S

+32 and ∆23 = ω32+(S+

21+S−32)−S+32.

We can see that the populations are coupled between them but are decoupled from thecoherences. The solution for the coherences is in general oscillating and decaying, forinstance for the case ρ12(t) is an oscillating function of frequency ∆12 and decay rate

equal toΓ−21+Γ−32

2 . On the other hand the populations evolution is given in chapter 3.

23

24 APPENDIX B. THREE LEVEL SYSTEMS

• Λ configuration: Projecting Eq. (3.9) on B we obtain:

d

dtρ11(t) = −Γ−31ρ11(t) + Γ+

31ρ33(t)

d

dtρ22(t) = −Γ−32ρ22(t) + Γ+

32ρ33(t)

d

dtρ33(t) = Γ−32ρ22(t) + Γ−31ρ11(t)− (Γ+

32 + Γ+31)ρ33(t)

d

dtρ12(t) =

[i∆12 −

Γ−31 + Γ−32

2

]ρ12(t) (B.2)

d

dtρ13(t) =

[i∆13 −

Γ−31 + Γ+32 + Γ+

31

2

]ρ13(t)

d

dtρ23(t) =

[i∆23 −

Γ−32 + Γ+32 + Γ+

31

2

]ρ32(t),

where ∆12 = ω21+S−31−S−32, ∆13 = ω31+S−31−(S+

31+S+32) and ∆23 = ω32+S−32−(S+

31+S+32).

We can see that as in the ladder case, the populations are coupled between them but aredecoupled from the coherences.

• V configuration: Projecting Eq. (3.16) on B we obtain:

d

dtρ11(t) = −(Γ−21 + Γ−31)ρ11(t) + Γ+

21ρ22(t) + Γ+31ρ33(t)

d

dtρ22(t) = −Γ+

21ρ22(t) + Γ−21ρ11(t)

d

dtρ33(t) = Γ−31ρ11(t)− Γ+

31ρ33(t)

d

dtρ12(t) =

[i∆12 −

Γ−21 + Γ−31

2

]ρ12(t) (B.3)

d

dtρ13(t) =

[i∆13 −

Γ−21 + Γ+31 + Γ+

21

2

]ρ13(t)

d

dtρ23(t) =

[i∆23 −

Γ−31 + Γ+31 + Γ+

21

2

]ρ32(t),

where ∆12 = ω21 + (S−31 + S−21) − S+21, ∆13 = ω31 + (S−31 + S−21) − S+

31 and ∆23 =ω32 +S+

21−S+31. The coupling between population is invariant in the three configurations

and forbidding one transition does not change the uncoupled evolution of the coherences.

We can conclude that in general, the populations are always coupled between them butdecoupled from the coherences. The Lamb-shift does not interfere with the population dynamicssince it only affects the oscillation frequency of the off-diagonal terms, and the decay rates arerelied to the decay time of the coherence. On the other hand, the population equations havea negative term proportional to the level population and a positive one proportional to theadjacent connected levels. By controlling the values of Γ±ji we can control which term willdominate.

Bibliography

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[4] M. O. Scully et al., “Quantum Optics” Cambridge University Press, Cambridge, 1997; D.F. Walls et al., Quantum Optics Springer, Berlin, 1994.

[5] M. B. Plenio and S. F. Huelga Phys. Rev. Lett. 88 19790 (2002)

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Three level atom OTE systems

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