1. Universit´e de Franche-Comt´e
UTINAM CNRS
Universidad Sim´on Bol´ıvar
INTERNSHIP REPORT M1
Atomic dynamics out of thermal equilibrium: a three
level atom
Luis Enrique Parra
Supervisor: Bruno Bellomo
Besan¸con - France
19 June 2015
3. 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-environment
interaction leads to non-unitary dynamics for the open quantum system (the reduced system
S) 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 with
the 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 in
general harmful for practical applications exploiting quantum properties [3]. It is important in
many 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 decoherence
processes. 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 can
suppose the S+B system as a closed one (i.e. no interaction with external systems) [1] and
without knowing in detail the state of B we can study the evolution of S using its density
matrix operator. This operator is governed by a master equation [1, 4] which describes how
the system’s populations and coherences evolve, representing a powerful tool to study an open
quantum 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 manipulating
the 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? Several
methods, 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 the
environment and its coupling with the system, i.e. treating the noise as a tool to achieve a
desired configuration [7, 8].
In this context, out of thermal equilibrium (OTE) systems have been also studied. These
systems 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 have
been promising results about OTE systems since it has been shown that there is a potential to
control and manipulate atomic systems immersed in this type of environments. Notably, there
has 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 influence
of several blackbody thermal reservoirs held at different temperatures has been considered in
different contexts, for example for a chain of spins [15]. Recently, multi-temperature realistic
configurations of atoms surrounded by microscopic bodies have been investigated by taking
into account the dependence on the internal structure (material, geometry) of the reservoirs
2
4. 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 generation
of 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 find
conditions 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 second
section we present the model used to study the dynamics at and out of thermal equilibrium and
we study the case of a two-level atom. In the third section we study the three-level atom in the
three main configurations Λ, V and ladder, when the environment is at and out of equilibrium.
5. 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 its
density matrix operator. A closed system is a system which does not interact with an external
system. 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 governed
by the equation
i¯h
∂
∂t
U(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 is
no longer possible to describe it with a single vector state) in order to describe its evolution, it
becomes 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 represent
the populations of the states |φi while the off diagonal ones, ρij, when i = j, represent the
coherences between the states |φi and |φj . In the Schr¨odinger picture, the evolved density
matrix 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
dt
ρ(t) = −
i
¯h
[H(t), ρ(t)]. (1.3)
Eq. (1.3) is called the Liouville-Von Neumann equation and describes how the closed system
evolves in time. In the next section, it will be more convenient for a microscopic derivation of
the 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
dt
˜ρ(t) = −
i
¯h
[ ˜HI(t), ˜ρ(t)], (1.4)
where ˜HI = U†
0 (t, t0)HI(t)U0(t, t0), U0(t, t0) = e− i
¯h H0(t−t0)
and the tilde indicates that the
given operator is in the interaction picture.
4
6. 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 already
studied 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 that
interacts with its environment B. An example might be an electron (the system) interacting with
the vibrational modes of a solid (the environment), or an atom interacting with the surrounding
electromagnetic field. Considering the system S+B as closed and using the formalism of section
1.1 we can describe the dynamics of the total system. The Hilbert space of S+B is the tensorial
product 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 interaction
Hamiltonian. We want to study the behaviour of S under the influence of B, that is, finding
the master equation governing the evolution of ρS(t). To this end we remember that for a
composite system where ρ is the total density matrix operator of S+B and TrB is the partial
trace over the degrees of freedom of B, the system S can be described by means of the reduced
density operator ρS = TrBρ. Differentiating this relation in time and using equation (1.3) we
obtain:
d
dt
ρS(t) = −
i
¯h
TrB{[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 necessary
to use certain approximations. In the following we will focus on the case of markovian master
equations. There are two ways of approaching this physical regime: one involves the use
of quantum dynamical semigroups [1] and the other one, which we will develop, involves a
derivation 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 simplify
the problem. Before doing that, we move to the interaction picture where eq. (1.6) becomes:
d
dt
˜ρS(t) = −
i
¯h
TrB[ ˜HI(t), ˜ρ(t)]. (1.7)
Expressing equation (1.3) in its integral form in the interaction picture and reintroducing it in
the commutator of eq. (1.7) we obtain:
d
dt
˜ρS(t) = −
i
¯h
trB[ ˜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 density
operator factorizes as ρ(0) = ρS(0) ⊗ ρB(0)
• The interaction hamiltonian can be cast under the form HI = µ Aµ ⊗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)
7. 6 CHAPTER 1. THE MARKOVIAN MASTER EQUATION
• TrB[ ˜HI(t), ˜ρ(0)] = 0. This can be justified by the fact that ˜HI(t) is linear in the operators
Aµ and Bµ, along with the first assumption and the condition TrB (HIρB(0)) = 0 where
we 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 simplify
this equation we will use the following approximations.
1. Born approximation: the coupling between the system and the reservoir is assumed to be
weak (weak coupling approximation). Therefore, the density matrix of the environment
ρB(t) is negligibly affected by the interaction and the state of the total system at time t
can be characterized by:
ρ(t) ≈ ρS(t) ⊗ ρB(t).
2. Markovian approximation: it consists in making the quantum master equation local in
time. 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 Born
approximation we replace:
˜ρS(s) = ˜ρS(t).
Taking (1.8) and applying the approximations 1 and 2 we obtain:
d
dt
˜ρS(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 the
preparation 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 integrand
disappears. So we can substitute s=t-s and take the upper limit to ∞, finally obtaining:
d
dt
˜ρS(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 substituting
the expression for the interaction hamiltonian in terms of the operators Aµ and Bµ we
obtain:
d
dt
˜ρS(t) =
ω,ω µ,ν
ei(ω −ω)t
ζµν(ω)(Aν(ω)˜ρS(t)A†
µ(ω ) − A†
µ(ω )Aν(ω)˜ρS(t)) + h.c. (1.12)
where
Aµ(ω) ≡
− =ω
Π( )AµΠ( ), (1.13)
ζµν(ω) ≡
1
¯h2
∞
0
ds ˜B†
µ(t) ˜Bν(t − s) eiωs
. (1.14)
With Π( ) being the projector associated to the eigenvalue of the Hamilonian HS,
˜B†
µ(t) = e
i
¯h HBt
B†
µe− i
¯h HBt
and ζµν(ω) is the one-sided Fourier transform of the reservoir
8. 7 CHAPTER 1. THE MARKOVIAN MASTER EQUATION
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 an
eigenvalue equal to , then Aµ(ω) |φ is an eigenstate of HS with eigenvalue − ¯hω and
correspondingly A†
µ changes the eigenvalue to + ¯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 |, are
much shorter than the expected relaxation timescales for the system, the rotating wave
approximation can be applied: all terms with ω − ω = 0 are considered as varying too
fast, so that their average contribution in Eq. (1.12) (after integrating) on the timescales
relevant to S can be neglected.
As a result of this approximation Eq. (1.12) reduces to:
d
dt
˜ρS(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) is
called 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 Schr¨odinger representation it can be shown that we obtain it by
simply adding the free Hamiltonian HS to HLS and by replacing each operator by its form in
the Schr¨odinger picture. For a more detailed derivation see [1].
9. Chapter 2
Presentation of the model
We consider an arbitrary N-level atom interacting with an electromagnetic field that plays the
role 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, and
HI = −D·E is the interaction hamiltonian involving the electric field E at the atomic position
and 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 frequency
associated with the transition between the states |i and |j with energies i = ¯hωi and j = ¯hωj
respectively, and dij = i| D |j . The operators D and E are related to the operators Aµ and
Bµ 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 the
following, we will derive a suitable master equation governing the atomic dynamics, containing
both decay rates (Γ±
ij) and shifts (S±
ij ) built on the basis of Eq. (1.18). From this point on, we
will 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)
¯hωji
kBT
= xji,
where kB is the Boltzmann constant.
8
10. 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, we
will assume that the environment is at thermal equilibrium, i.e. ρB is represented by a Gibbs
state. If there is no external time-dependent fields, the stationary solution for the master
equation (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 can
be 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 with
Eq. (2.7) we arrive to the conclusion that at thermal equilibrium the populations follow the
Boltzmann distribution, characterized by the fact that the ratio between any two populations
depends only on the frequency of the transition and the temperature:
ρii
ρjj
= Exp(−βωji). (2.9)
By calculating the environment correlation functions, it is possible to show [1, 17] that the
decay rates Γ±
ji defined in Eq. (2.3) assume the form:
Γ+
ji = γji
0 Λji(1 + nji) , Γ−
ji = γji
0 Λjinji, (2.10)
where γji
0 =
4ω3
ji|dij |
2
3¯hc3 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 the
transition frequency ωji. If the system is not surrounded by matter, Λji → 1, while in the
presence of matter, Λji will take into account the effects of the disposition of the surrounding
bodies as well as their geometry and dielectric properties.
11. 10 CHAPTER 2. PRESENTATION OF THE MODEL
2.2 Out of thermal equilibrium
In this section, we use the formalism developed in [17], where the configuration consists of
a N-level atom placed near a body of arbitrary geometry and dielectric permittivity and at
temperature TM , also interacting with the environmental radiation generated by far surrounding
walls at temperature TW . In these systems relevant effects such as inversion of populations have
been pointed out. In the out of thermal equilibrium case, TM = TW , the dynamics of the system
is quite rich. The parameters Γ±
ji depend on the geometry and the transmission and absorption
properties of the body surrounding the system S and on the temperatures. It can be shown for
this configuration that the Γ±
ji can be written in the form:
Γ+
ji = γji
0 Λji(1 + neff
ji ) , Γ−
ji = γji
0 Λjineff
ji , (2.12)
where neff
ji replaces nji of Eq. (2.11), and is a complex function of all the parameters such as
TW , TM and the distance “z” between the atom and the body, as well as the properties of the
latter. The main point here is that each transition is characterized by rates formally equivalent
to a case at thermal equilibrium at an effective temperature Teff
ji . This effective temperature
is linked to the parameter neff
ji via the relation:
Teff
ji =
¯hωji
kB
ln(1 + neff−1
ji )
−1
. (2.13)
In particular, it has been shown [17] that neff
ji is restricted to the interval n(ωji, Tmax) >
neff
ji > 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 , neff
ji becomes neff
ji = nji.
In the following, we assume that it is possible to vary freely neff
ji . For each choice of neff
ji ,
we imagine possible to find a specific configuration producing the chosen values. These kind
of models have been studied in other contexts [14] where different transitions are connected to
thermal 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 frequency
between 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 Schr¨odinger picture and substituting Eq.
(2.15) we obtain:
d
dt
ρS(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)} ,
12. 11 CHAPTER 2. PRESENTATION OF THE MODEL
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 thermal
equilibrium.
• 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 + neff
21
2neff
21 + 1
, ρ22 =
neff
21
2neff
21 + 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 and
we 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 V
and ladder ones.
13. Chapter 3
Three level systems
In this chapter we focus on the three-level atom in three main configurations characterized by
different 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 three
level 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 is
forbidden. 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
dt
ρS(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
14. 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
γth
lad
, ρ22 =
ex32
γth
lad
, ρ33 =
1
γth
lad
, (3.6)
where γth
lad = ex21+x32
+ ex32
+ 1. We can indeed verify that ρ11 > ρ22 > ρ33 since x21 > 0
and x32 > 0. The time dependent solution of Eq. (3.3) is very complex and will not
be reported in the following. We plot the time dependent solution for a particular set
of values, as shown in Fig. 3.2, illustrating the case when γ21
0 = γ32
0 = γ0, x21 = 0.5,
x32 = 1, and ρ33(0) = 1. We see how the populations evolve until reaching their steady
values given by Eq. (3.6).
Figure 3.2: ρ(t) as a function of γ0t in the ladder configuration at thermal equilibrium for
γ21
0 = γ32
0 = 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 + neff
21 )(1 + neff
32 )
γote
lad
, ρ22 =
neff
21 (1 + neff
32 )
γote
lad
, ρ33 =
neff
21 neff
32
γote
lad
, (3.7)
15. 14 CHAPTER 3. THREE LEVEL SYSTEMS
where γote
lad = 1+2neff
21 +neff
32 +3neff
21 neff
32 . It is easy to verify that ρ11 > ρ22 > ρ33. Even
if we can vary freely neff
ji , 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 second
one 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 Schr¨odinger picture, we obtain:
d
dt
ρS(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 populations
are 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 term
Γ+
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 the
following.
• Thermal equilibrium: Using (2.10) in Eq. (3.10) we obtain:
ρ11 =
ex31
γth
Λ
, ρ22 =
ex32
γth
Λ
, ρ33 =
1
γth
Λ
. (3.12)
where γth
Λ = 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 thermal
equilibrium. Fig. 3.3 illustrates the evolution of populations to their stationary values
such 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 a
comparison with the out of thermal equilibrium case.
16. 15 CHAPTER 3. THREE LEVEL SYSTEMS
Figure 3.3: ρ(t) as a function of γ0t in the Λ configuration at thermal equilibrium when
γ31
0 = γ32
0 = 1, x31 = 1.5, x32 = 1 and ρ33(0) = 1.
• Out of thermal equilibrium: Using (2.12) in Eq. (3.10) we obtain:
ρ11 =
(1 + neff
31 )neff
32
γote
Λ
, ρ22 =
neff
31 (1 + neff
32 )
γote
Λ
, ρ33 =
neff
31 neff
32
γote
Λ
, (3.13)
where γote
Λ = neff
31 + neff
32 + 3neff
31 neff
32 . It is easy to verify that ρ11 > ρ33 and ρ22 > ρ33.
However, the term ρ22
ρ11
is more delicate to analyze since now we can vary freely the neff
ji .
In this case the ratio between these populations is:
ρ22
ρ11
=
neff
31 + neff
31 neff
32
neff
32 + neff
31 neff
32
. (3.14)
This ratio depends only on the ratio between neff
31 and neff
32 . This can be seen in Fig. 3.4
where we have plotted the ratio ρ22
ρ11
as a function of neff
31 and neff
32 . In this figure we can
see the line of inversion (neff
31 = neff
32 ), and the asymptotic behaviour of the function.
Eq. (3.14) points out the possibility of a population inversion when neff
31 > neff
32 . In
particular, in the limit case, neff
31 → ∞ and neff
32 → 0, Eq. (3.13) implies ρ22 → 1. Fig. 3.5
shows the case when γ31
0 = γ32
0 = γ0, neff
31 = 1.5, neff
32 = 0.6 and ρ33(0) = 1. In this plot
we observe how the first and second level populations achieve their steady values such that
ρ22 > ρ11, in contrast with Fig. 3.3.
17. 16 CHAPTER 3. THREE LEVEL SYSTEMS
Figure 3.4: ρ22
ρ11
as a function of neff
31 and neff
32 .
Figure 3.5: ρ as a function of γ0t in the Λ configuration out of thermal equilibrium when
γ31
0 = γ32
0 = γ0, neff
31 = 1.5, neff
32 = 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 third
one 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)
18. 17 CHAPTER 3. THREE LEVEL SYSTEMS
Using Eq. (3.15) in Eq. (1.16) and moving to the Schr¨odinger representation, we obtain:
d
dt
ρS(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 stationary
populations 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 term
Γ+
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
γth
V
, ρ22 =
ex31
γth
V
, ρ33 =
ex21
γth
V
, (3.19)
where γth
V = 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 for
the V configuration with the same values as in Fig. 3.2. In this figure we can see the
evolution of the steady populations to their corresponding values. This case will be used
as a comparison with the out of thermal equilibrium V-configuration.
• Out of thermal equilibrium: Using (2.12) in Eq. (3.17) we obtain:
ρ11 =
(1 + neff
31 )(1 + neff
21 )
γote
V
, ρ22 =
neff
21 (1 + neff
31 )
γote
V
, ρ33 =
neff
31 (1 + neff
21 )
γote
V
, (3.20)
where γote
V = 1+2neff
31 +2neff
21 +3neff
31 neff
21 . Once again it is easy to verify that ρ11 > ρ33
and ρ11 > ρ22. For the ratio between ρ22 and ρ33 we get:
ρ22
ρ33
=
neff
21 + neff
31 neff
21
neff
31 + neff
31 neff
21
. (3.21)
Eq. (3.21) depends on the ratio between neff
31 and neff
21 . A population inversion can be
obtained since ρ33 > ρ22 when neff
31 > neff
21 . The plot of the ratio shown in Eq. (3.21)
as a function of the neff
ji will be similar to the one obtained in the Λ case (Fig. 3.4).
19. 18 CHAPTER 3. THREE LEVEL SYSTEMS
Figure 3.6: ρ as a function of γ0t in the V configuration at thermal equilibrium when γ31
0 =
γ21
0 = γ0, x21 = 0.5, x31 = 1.5, x32 = 1 and ρ33(0) = 1.
However, the asymptotic behaviours of the inverted populations are quite different. In
this case given the conditions neff
31 → ∞ and neff
21 → 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 in
contrast to the case at thermal equilibrium.
Figure 3.7: ρ(t) as a function γ0t in the V configuration out of thermal equilibrium when
γ31
0 = γ21
0 = γ0, neff
21 = 0.3, neff
31 = 1.5 and ρ33(0) = 1.
20. 19 CHAPTER 3. THREE LEVEL SYSTEMS
We conclude by analyzing the results obtained in this chapter. Inversion of populations
only occur in the V and Λ configurations while in the ladder one the common order of steady
populations, with respect to the case at thermal equilibrium, is preserved. We can provide a
simple explanation for this by exploiting Eq. (2.13). This equation permits to associate different
temperatures to each one of the transitions. In the V and Λ cases, we can formulate the following
criteria for population inversion. In the Λ configuration the condition for population inversion
is neff
31 > neff
32 , i.e. Teff
31 > Teff
32 ; the “hotter” transition induces more migration towards the
third level than the “colder” one, producing an overall migration from the first level towards
the second level. In the V configuration the condition for the inversion is neff
31 > neff
21 , i.e.
Teff
31 > Teff
21 , 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 for
ordering 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 arbitrary
non-degenerated N-level system without loops (no closed links between levels), inversions of
populations can always be obtained between any two non-directly connected levels that are
not indirectly connected only by ladder schemes. This means that if there is a ladder type
connection between two levels, it is impossible to obtain an inversion. For example, in the
four-level case with the following permitted transitions: 1 ↔ 2 ↔ 4 ↔ 3, there is a ladder-type
configuration concerning the levels 1,2 and 4. In this case, an inversion between any two of
these levels is impossible. However, since the first and second level are not connected to the
third one or do not share a ladder scheme, then it can be shown that an inversion of steady
populations is possible between them.
21. Conclusion
In this project we investigated the dynamics of an elementary open quantum system, such as a
three-level atom, interacting with an environment at and out of thermal equilibrium. We have
revised the formalism for open quantum systems to obtain a markovian master equation. We
have 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 on
the results of [17]. We have pointed out relevant differences between the thermal equilibrium
case and the out of thermal equilibrium one, with steady states depending on geometrical and
material properties as well as the temperatures of the surrounding bodies. After studying the
dynamics 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 populations
may 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 only
be 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 freely
vary 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. The
inversion criteria presented in chapter 3 can be extended to N-level atoms, where the stationary
solutions can be extracted by knowing only the connections between levels. In particular, this
can 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 the
neff
ji but also on the γji
0 producing a more rich dynamic. It can be studied the possibility
of developing a method to reduce any N-level system in blocks of 3-levels and quickly identify
possible inversions. The results reported in this project could be also of interest for experimental
investigation in the absence of thermal equilibrium involving real or artificial atoms (such as
quantum dots).
20
22. Aknowledgements
I would like to thank professor Bruno Bellomo for all the hard work, advises and support
concerning 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 the
unconditional support given.
21
23. 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 Schr¨odinger picture, the operators are time
independent while the states carry the evolution in time. In contrast, in the Heisenberg picture
all the dynamics are contained in the operators while the states do not evolve. Between these
two extremes, there is the interaction picture in which the part of the dynamics, associated
with uncoupled evolution is contained in the operators, while that arising from the coupling is
contained in the state. In general, the evolution of a state is carried by the evolution operator
U(t, t0) where U(t0, t0) = I, I being the identity operator and |φ(t) = U(t, t0) |φ(t0) . It is easy
to show that for an arbitrary time dependent hamiltonian, the general form of the evolution
operator is given by:
U(t, t0) = T←e
−i t
t0
H(s)ds
(A.1)
where T← is the chronological time-ordering operator. Let us write the Hamiltonian of the
system in the following way:
H(t) = H0 + HI(t), (A.2)
and let us define:
U0(t, t0) = e− i
¯h H0(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) and
˜ρ(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
dt
˜ρI(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 deriving
master equations that govern the evolution of the density operator of an open quantum system
because it allows us to shunt all the time dependence of H0 onto the operators, and leaving
only HI(t) to control the time-evolution of the states.
22
24. 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
dt
ρ11(t) = −Γ−
21ρ11(t) + Γ+
21ρ22(t)
d
dt
ρ22(t) = −(Γ−
32 + Γ+
21)ρ22(t) + Γ−
21ρ11(t) + Γ+
32ρ33(t)
d
dt
ρ33(t) = Γ−
32ρ22(t) − Γ+
32ρ33(t) (B.1)
d
dt
ρ12(t) = i∆12 −
Γ−
21 + Γ−
32
2
ρ12(t)
d
dt
ρ13(t) = i∆13 −
Γ−
21 + Γ+
32 + Γ+
21
2
ρ13(t)
d
dt
ρ23(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 the
coherences. The solution for the coherences is in general oscillating and decaying, for
instance for the case ρ12(t) is an oscillating function of frequency ∆12 and decay rate
equal to
Γ−
21+Γ−
32
2 . On the other hand the populations evolution is given in chapter 3.
23
25. 24 APPENDIX B. THREE LEVEL SYSTEMS
• Λ configuration: Projecting Eq. (3.9) on B we obtain:
d
dt
ρ11(t) = −Γ−
31ρ11(t) + Γ+
31ρ33(t)
d
dt
ρ22(t) = −Γ−
32ρ22(t) + Γ+
32ρ33(t)
d
dt
ρ33(t) = Γ−
32ρ22(t) + Γ−
31ρ11(t) − (Γ+
32 + Γ+
31)ρ33(t)
d
dt
ρ12(t) = i∆12 −
Γ−
31 + Γ−
32
2
ρ12(t) (B.2)
d
dt
ρ13(t) = i∆13 −
Γ−
31 + Γ+
32 + Γ+
31
2
ρ13(t)
d
dt
ρ23(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 are
decoupled from the coherences.
• V configuration: Projecting Eq. (3.16) on B we obtain:
d
dt
ρ11(t) = −(Γ−
21 + Γ−
31)ρ11(t) + Γ+
21ρ22(t) + Γ+
31ρ33(t)
d
dt
ρ22(t) = −Γ+
21ρ22(t) + Γ−
21ρ11(t)
d
dt
ρ33(t) = Γ−
31ρ11(t) − Γ+
31ρ33(t)
d
dt
ρ12(t) = i∆12 −
Γ−
21 + Γ−
31
2
ρ12(t) (B.3)
d
dt
ρ13(t) = i∆13 −
Γ−
21 + Γ+
31 + Γ+
21
2
ρ13(t)
d
dt
ρ23(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 but
decoupled from the coherences. The Lamb-shift does not interfere with the population dynamics
since it only affects the oscillation frequency of the off-diagonal terms, and the decay rates are
relied to the decay time of the coherence. On the other hand, the population equations have
a negative term proportional to the level population and a positive one proportional to the
adjacent connected levels. By controlling the values of Γ±
ji we can control which term will
dominate.
26. Bibliography
[1] H. P. Breuer et al.: “The Theory of Open Quantum Systems” Oxford University Press,
Oxford, 2002.
[2] W.H. Zurek, Phys. Rev. D 26, 1862 (1982)
[3] M. A. Nielsen and I. L. Chuang: “Quantum Computation and Quantum Information”,
Cambridge University Press, Cambridge, 2000.
[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)
[6] F. Verstraete, M. M. Wolf and J. I. Cirac Nature Phys. 5 633 (2009)
[7] M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, Phys. Rev. A 59, 2468 (1999).
[8] D. Braun, Phys. Rev. Lett. 89, 277901 (2002).
[9] N. Lambert, Y.N. Chen, Y.C. Cheng, C.M. Li, G.Y. Chen, and F. Nori, Nat Phys 9, 10
(2013).
[10] J. M. Obrecht et al., Phys. Rev. Lett. 98 063201 (2007).
[11] R. Messina and M. Antezza, EPL, 95 61002 (2011).
[12] M. Antezza, L. P. Pitaevskii and S. Stringari Phys. Rev. Lett. 95 113202 (2005).
[13] N. Brunner et al., Phys. Rev. E, 85 051117 (2012).
[14] L. A. Correa, J. P. Palao, D. Alonso, and G. Adesso, Sci. Rep. 4, 3949 (2014).
[15] M. Znidaric Phys. Rev. A 85 012324 (2012).
[16] B. Bellomo, R. Messina and M. Antezza Europhys. Lett. 100 20006 (2012).
[17] B. Bellomo, R. Messina, D. Felbacq and M. Antezza Phys. Rev. A 87 012101 (2013).
[18] B. Bellomo and M. Antezza EPL 104 10006 (2013).
25
