equilibrium and dynamical properties of rotating clouds of

Thesis TitleEquilibrium and dynamical properties of rotating clouds of ultra cold atoms
Joao Daniel Marques Rodrigues
Engenharia Fsica Tecnologica
Juri Presidente: Prof. Horacio Joao Matos Fernandes
Orientador: Prof. Jose Tito da Luz Mendonca Vogais: Prof. Jorge Manuel Amaro Henriques Loureiro Prof. Jose Antonio Sequeira de Figueiredo Rodrigues
November 2013
Abstract
Since the first realizations of ultra cold atomic gases, both theoretical and experimental investi-
gations reveal that the physics of magneto optical traps constitute a very complex and exciting topic.
A distinctive and complex collective behaviour is observed in clouds with a large number of trapped
atoms. This behaviour is due to the multiple scattering of light, a mechanism identified along the
last decade and responsible for much of the rich and interesting behaviour of ultra cold matter in-
side a magneto optical trap. This interaction can be formally treated as if the atoms inside the trap
possessed an equivalent electric charge and the system can then be regarded as a one component
trapped plasma. Such a plasma description of the system has been proven very fruitful. In this the-
sis it will be investigated the equilibrium properties, particularly the atom density profiles of rotating
clouds of atoms. These systems have displayed particular features that have been intrinsically related
with the rotation of the cloud, easily achieved experimentally with a slight misalignment of the trapping
laser beams. Making use of a fluid mechanical description of the system it will also be investigated the
nature of the localised oscillations, or normal modes, of the system. This entire analysis relies on the
premiss of the existence of a polytropic equation of state for the cloud of atoms inside the magneto
optical trap.
i
Resumo
Desde que se comecou a estudar a materia ultra fria, tanto as investigacoes teoricas como exper-
imentais revelaram que a fsica por detras das armadilhas magneto opticas constituam um complexo
e interessante topico. Um comportamento colectivo distincto e observado em nuvens com um grande
numero de partculas capturadas. Este comportamento e devido as espalhamento repetido da luz
incidente, um mecanismo identificado durante a ultima decada e responsavel por muito do rico e
interessante comportamento da materia ultra fria dentro de uma armadilha magneto optica. Esta
interaccao pode ser formalmente tratada como se os atomos dentro da armadilha possussem uma
carga electrica equivalente, e o sistema pode ser tratado como um plasma de uma componente.
Uma descricao do tipo plasma tem-se mostrado muito frutfera. Nesta tese vao ser investigadas as
propriedades de equilbrio, particularmente os perfis de densidade atomica de nuvens de atomos
em rotacao. Estes sistemas tem exibido caractersticas particulares que tem sido intrinsecamente
relacionadas com a rotacao do sistema, facilmente obtida experimentalmente com um pequeno de-
salinhamento nos feixes de confinamento. A partir de uma descricao em termos de mecanica de
fludos para o sistema vao tambem ser investigados os modos proprios de oscilacao do sistema.
Esta inteira analise assenta na premissa da existencia de uma equacao de estado do tipo politropico
para a nuvem de atomos ultra frios dentro da armadilha magneto optica.
Palavras Chave
dos normais.
1.2 Magneto Optical Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Multidisciplinarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 The light-matter interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Light forces on an atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Sisyphus cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Magnetic traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Atomic clouds 23
3.2 Fluid mechanical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Polytropic equilibrium and density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Equilibrium and dynamical properties of rotating clouds of ultra cold atoms 31
4.1 Equilibrium profiles for rotating clouds of ultra cold atoms . . . . . . . . . . . . . . . . . 32
4.2 Solutions and interpretation of the equilibrium density profiles . . . . . . . . . . . . . . . 35
4.3 Localised oscillations, or normal modes, in rotating clouds of ultra cold atoms . . . . . . 43
4.3.1 Temperature limited regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.2 Multiple scattering regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Conclusions and Future Work 51
Bibliography 55
Appendix B Multipolar magnetic field in an anti-Helmholtz configuration B-1
v
1.1 The geometry of a magneto optical trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Schematic representation of the Sisyphus cooling mechanism. . . . . . . . . . . . . . . 18
2.2 Schematic representation of the trapping mechanism in one dimension MOT for a J =
0→ J ′ = 1 transition in an atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Schematic illustration of the multi-level structure for 85Rb. . . . . . . . . . . . . . . . . . 21
3.1 Schematic representation of the shadow effect. . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Numerical solutions for the equilibrium profiles. . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 Misalignment of the laser beams to produce a rotation in the cloud. . . . . . . . . . . . . 33
4.2 Degeneracy of the equilibrium density profile solutions for systems in different rotation
states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 Numerical solution for the generalized Lane-Endem equation for the temperature lim-
ited regime, 2 p = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5 Numerical solution for the generalized Lane-Endem equation for ′2p = const and dif-
ferent angular velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.6 Numerical solution for the generalized Lane-Endem equation for high collective effects.
For a sufficiently high parameter 2 p the solutions start to exhibit orbital modes. . . . . . 40
4.7 Density plots for the atoms density profiles for the atoms density profiles for high values
of 2 p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.8 Dispersion relation for oscillations in the multiple scattering regime, 2 p → 1. . . . . . . . 48
A.1 Schematic representation of the optical set up for the magneto optical trap. . . . . . . . A-3
A.2 Experimental set-up for the magneto optical trap at Instituto Superior Tecnico. . . . . . . A-4
vii
List of Tables
2.1 Comparison of the relevant atomic parameters for the main cooling transitions for the
isotopes of elements used in laser cooling systems. . . . . . . . . . . . . . . . . . . . . 16
ix
List of Symbols
α damping or cooling rate c vacuum light speed δ laser detuning D diffusion coefficient E electric field
F , F force I, I0 trapping laser intensity
h Planck constant ~ Planck constant divided by 2π κ spring constant κ′ centripetal force constant k wavevector
kB Boltzmann constant λ wavelength m mass µ magnetic dipole moment p electric dipole vector
ρ, ρ density matrix operator s, s0 saturation parameter
σ circular polarization ω angular frequency ∇ gradient operator ∇· divergence operator ∇2 Laplace operator
xi
1.2 Magneto Optical Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Multidisciplinarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1 Ultra cold matter physics
When we refer to ultracold matter we are talking about systems with a temperature very close
to absolute zero (approximately -270 C). These temperatures are so low that they can’t be found
naturally anywhere in the universe. To achieve this kind of temperatures one must use modern and
advanced atom cooling and trapping techniques using laser beams. The development of these tech-
niques gave rise to the Nobel Prize in Physics in 1997 awarded to Steven Chu, Claude Cohen-
Tannoudji and William D. Phillips. Using these same techniques the physicists Eric A. Cornell, Wolf-
gang Ketterle and Carl E. Wieman were also awarded with the Nobel Prize in Physics in 2001 for
being the first group to achieve Bose Einstein condensates, a new and weird new state of matter that
reveals itself at temperatures very close to absolute zero. These Bose Einstein condensates (usually
referred to as BEC’s) along with ultra cold or quantum plasmas and the non condensed ultra cold
atomic clouds constitute the three basic strands of ultra cold matter physics.
In a Bose-Einstein condensate a large fraction of the constituent particles occupy the lowest en-
ergy state at which point quantum effects became apparent on a macroscopic scale. The process
of Bose-Einstein condensation was first proposed in 1924 when Bose and Einstein, generalizing the
Planck’s radiation law, derived statistics under which the ground state of a system can be macroscopi-
cally occupied at sufficiently low temperatures. This effect is intrinsically related with the wave-particle
duality and therefore it is quantum-mechanical in nature. As de Broglie stated, there is an intrinsic
wavelength associated with particles at a temperature T , given by λB = √
2π~2/mkBT , where m is
the mass of the particles. In a sufficiently cooled ensemble of atoms, this intrinsic wavelength be-
comes comparable to the average inter-particle distance at which point the individual wave packets
overlap and a quantum degenerate gas is formed. At that time it was not clear if Bose-Einstein con-
densates could exist in nature as interactions between the particles were not included in the original
theory. The truth is that this effect was already observed in many different systems, ranging from
atomic and condensed matter physics to astrophysics.
Moreover, one important aspect of the cooperative effects in the radiation-atom interaction arises
in ultracold matter, the Rydberg blockades. In Rydberg atoms, one or more valence electrons are
excited into very high energy sates, resulting in certain properties that allow the atoms to highly
interact with each others. An important consequence of this interaction correspond the formation of
Rydberg blockades (or dipole blockades), where a Rydberg atom, due to its high interactivity, slightly
alters the level structure of the atoms around it. This way, around this one atom a cloud of lower
energy atoms will form, acting as a kind of shield to it. The unique characteristics of these Rydberg
systems allow them to be applied in quantum computation, cryptography and atomic clocks.
Another interesting trend in ultra cold matter physics is the study of the non condensed atomic
gases. First, and contrary to what we might suspect, there are some similarities between the physics
2
of condensed and non condensed matter. We can verify this for instance from the dispersion relations
for oscillations in these two systems which exhibit a resembled behaviour. This implies that there is a
similar physics behind the dynamics of these systems. In fact both the condensed and non condensed
matter present a distinctive collective behaviour. In the case of Bose-Einstein condensates there is
a mean field coming from the inter-particle collisions. The collisions, which are responsible for the
collective behaviour, are characterized by a single parameter, known as the s-wave scattering length
as and they can be described by a pseudo-potential of the form
Vint(r, r ′) = gδ(r− r′) (1.1)
where g is the coupling parameter, which is given by
g = 4π~2as m
(1.2)
where m is the atomic mass. This pseudo-potential appears on the so called Gross-Pitaevskii
equation, a cubic non-linear Schrodinger type equation which determines the dynamics of the Bose-
Einstein condensate. For the case of non condensed matter the collective behaviour is determine by
multiple scattering of light coming from the laser beams cooling and trapping the atoms. This effect
can be cast by a force, acting the particles, and determined by a Laplace like equation, allowing the
treatment of the system as a one component plasma. This is responsible for much of the rich and
complex behaviour of non condensed atomic clouds. The problem of multiple scattering will be treated
throughout this thesis.
1.2 Magneto Optical Traps
As mentioned before, the work of Steven Chu, Claude Cohen-Tannoudji and William D. Phillips
culminated in the development of the magneto optical trap, the workhorse for achieving a cloud of
ultra cooled atoms. These complex devices use both laser cooling (Doppler cooling based on a two-
level atom configuration1) and magnetic trapping (Zeeman shift) in order to produce a sample of cold,
trapped neutral atoms at temperatures on the order of tens of microkelvin, which corresponds to two
or three times the recoil limit, an issue that will be address in the next chapter. By combining even the
small momentum of a single photon with a velocity and spatially dependent absorption cross section
and a large number of absorption-spontaneous emission cycles, neutral atoms with initial velocities of
hundreds of meters per second can be slowed down to tens of centimetres per second in a manner
of seconds. 1A more accurate description of the cooling processes inside a magneto optical trap requires a much heavier theoretical
framework
3
Figure 1.1: Schematic representation of a magneto optical trap with three pairs of circularly polarized laser beams and a pair anti-Helmholtz coils to create a magnetic quadrupole.
The basic MOT set up, as seen in the figure [1.1] is then capable of cooling down atoms to the
microkelvin scale. However this temperature isn’t low enough to achieve quantum degeneracy, which
occur in the nanokelvin scale. To achieve this lower range of temperatures one has to introduce
an additional cooling process called evaporative cooling. Although we do not intend to get into the
theory behind evaporative cooling, the underlying concept is quite simple. Basically, once we have
an ensemble of atoms cooled down to the microkelvin scale, if we introduce a potential well in the
system, the slower atom will be trapped and the faster ones in the velocity distribution will be capable
of escaping the cloud leaving us with a even cooler set of atoms which will eventually condensate.
Notice that the process of achieving a Bose-Einstein condensate starts with a Doppler cooling, and
then switching it off to start the evaporative cooling until we have a condensate which will only live for
a short period of time, in the order of a few seconds, until heat transfer from the surroundings heats up
the condensate. Achieving a steady-state Bose-Einstein condensate still remains an open problem.
At this moment, a magneto optical trap is being assembled at the facilities of Instituto de Plasmas
e Fusao Nuclear (IPFN) in Instituto Superior Tecnico, Lisbon. A more detailed description of this
particular set-up can be found in Appendix-A.
1.3 Multidisciplinarity
Contrary to some areas of physics, where a single strict formalism must be employed to describe
the observations and predictions of the theory, ultra cold matter allows for different ways to tackle
with the physics involved. In fact, a precise description of a magneto optical trap requires knowledge
and expertises ranging from atomic physics to condensed matter or plasma physics. First of all, the
principles of laser cooling and trapping can only be understood in the paradigm of atomic physics,
4
since a detailed understanding on the nature of the interaction of matter and light is required. The
understanding of this interaction allowed then for the development of the magneto optical trap where a
high number of neutral atoms, up to 1010, can be cooled and trapped. Surprisingly, this neutral atomic
cloud behaves as a one component plasma, because each atom in the cloud scatters photons from
the cooling laser beams, and these secondary photons can push away the nearby atoms by radiation
pressure - multiple scattering of light. From such an exchange of photons between nearby atoms
results a repulsive force, which can be described by an effective charge of the neutral atoms. From
this repulsive effect results a plasma-like behaviour, similar to that of non-neutral plasmas, which lead
to a variety of collective processes. Employing plasma physics technique to these systems reveal to
be important, for instance in the description of driven mechanical instabilities or even more exciting
instability phenomena, like phonon lasing.
Although theoretical and experimental investigations then reveal that magneto optical traps pave
a stage for very exciting and complex physical phenomena, much of this interest was deviated to the
investigation of Bose-Einstein condensates, as MOT’s started being mainly used as a riding horse
to achieve quantum degeneracy. Although some features of Bose-Einstein condensates can only be
understood employing a specific description for these systems, there are some fluid-like properties
which surprisingly lead to similar, but not identical, collective processes to those of ultra cold atomic
clouds in MOT’s [1].
Nevertheless, a new trend has recently begun that rekindled the interest on the basic physics
behind magneto optical taps. This is related to the increasing number of astrophysical phenomena
that we can simulate and study using ultra cold atomic clouds. Among them we can refer to a new
mechanism associated with the laser cooling process, which can lead to the formation of static and
oscillating photon bubbles inside the gas [2]. Photon bubbles have been considered in an astrophys-
ical context where huge photon densities are required to have any significant impact on high energy
particles. Because of the low kinetic energy of atoms accessible with the development of laser cool-
ing techniques, radiation pressure effects can now be explored in laboratory-based experiments with
modest photon densities.
Moreover, Kaiser et al [3] were recently able to achieve random lasing in a cloud of ultra cold atoms
under laboratory conditions. The effect was first seen decades ago in stellar clouds and in some
planet’s atmosphere, when random lasing was first proposed to explain why certain specific emission
lines in some clouds of stellar gas are more intense then theoretically predicted. In this set-up multiple
scattering of photons by the atoms (as happens in stellar gases) of the cloud acts as the gain medium
for lasing to occur. The unique possibility to both control the experimental parameters and to model
the microscopic response of this system provides an ideal test bench for better understanding natural
lasing sources, in particular the role of resonant scattering feedback in astrophysical lasers.
5
Finally we refer to the recent work of Tercas et al [4] where it is investigated the hydrodynamic
equilibrium and normal modes of cold atomic traps combining the effects of multiple scattering and
the thermal fluctuations inside the system, cast in the form of a polytropic equation of state. This
analysis results in a generalized Lane-Emden equation to describe the equilibrium density profiles of
the cloud, first derived to study astrophysical fluids. The main differences are due to both the inclusion
of the magnetic trapping and the long-range interaction induced by multiple scattering of light.
1.4 Thesis overview
In this thesis I will begin to introduce the theory of laser cooling. In fact, the physics of laser
cooling is rather involved and a detailed analysis of all its aspects would be impossible in this context
and nor is this the purpose of this document. Nevertheless I’ll try to introduce the reader to the main
ingredients behind laser cooling with the purpose of ending Chapter II - Theory of laser cooling, with
a simplified but consistent description of a magneto optical trap. In Chapter III - Atomic clouds, I’ll
begin to introduce the multiple scattering induced interaction responsible for the plasma-like behaviour
of the cloud. The introduction of this interaction allows the treatment of a wider and more interesting
range of traps, with a larger number of trapped atoms that exhibit a complex collective behaviour
whose study this thesis is devoted to. It is also introduced here the premiss of the existence of a
polytropic equation of state for the cloud of atoms and its equilibrium density profiles are computed.
Finally, in Chapter IV - Equilibrium and dynamical properties of rotating clouds, the original work
of this thesis, I’ll extend the theory to the case of rotating clouds and analyse the obtained results.
Analytical and numerical solutions are computed for the density profiles in different working regimes.
This chapter is concluded with the analysis of the normal modes of the system.
6
Contents
2.2 Light forces on an atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Sisyphus cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Magnetic traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
7
In this chapter I present the theoretical foundations for laser cooling. The intent is to introduce the
reader to the theory of laser cooling without a detailed coverage of all the involving topics, for which
the reader should sometimes refer to the bibliography presented within the sections. The purpose of
this chapter is to give a brief discussion of the results used for the rest of the thesis. It is considered
the interaction of a two-level atom with a monochromatic laser field by revisiting the emission and
absorption radiation processes described by the semi-classical Bloch equations. This semi-classical
treatment implies a quantum description for the two-level atoms but a description of the radiation as
a classical field. Notice however that spontaneous emission can only be understood with the full
quantum model for which I’ll just present a phenomenological introduction here. This analysis will
allow us to derive a simple expression for the laser cooling force, from which it will be introduced the
Doppler cooling limit. In fact this picture overviews much of the complex physics involving the MOT
technology and the need to introduce the multi-level structure of the atoms will oblige us to make
an incursion into the sub-Doppler cooling process known as Sisyphus cooling, which will be briefly
mentioned. Finally we can discuss the nature of the magnetic traps, specially in the anti-Helmholtz
configuration to end up with a simplified but consistent description of a magneto-optical trap.
2.1 The light-matter interaction
In the semi-classical picture mentioned before we can write the Hamiltonian for the light-matter
interaction in the form
H = Ha +Hint (2.1)
where Ha is the free Hamiltonian of the atom with eigenvectors |j satisfying the equation Ha|j =
Ej |j. For a simple two-level atom we just have j = 1, 2. The coupling between the atom and the
laser field is then described by the term Hint. In the absence of this radiation field we simply have
state vectors of the form |j, t = exp(−iEjt/~)|j. When the interaction is turned on, these atomic
states become coupled by the radiation field and the atomic state vector will contain a superposition
of the two states
|ψ(r, t) = ∑ j=1,2
Cj(t)exp(−iEjt/~)|j (2.2)
where |Cj(t)|2 represent the probability of finding the system in the state |j. The normalization
condition reads |C1(t)|2 + |C2(t)|2 = 1. To determine these probabilities one has to consider the time
dependent Schrodinger equation
∂t |ψ = (Ha +Hint)|ψ (2.3)
Using this equation and the superposition condition of equation (2.2) we derive the evolution equa-
tions
iω0t (2.5)
Here we have introduce the transition frequency ω0 = (E2 −E1)/~ and the matrix elements of the
interaction Hamiltonian operator Hij = i|Hint|j. We must at this point introduce Hint in an explicit
form,
2m A2(r, t) (2.6)
where A(r, t) is the vector potential associated with the classical radiation field. For low intensity
radiation fields the quadratic term A2 can be neglected. Moreover, we can usually assume that the
wavelength of the radiation is much larger than the dimensions of the atom, which corresponds to the
so called dipole approximation, allowing us to expand the vector potential A(r, t) around the atom
position r0 = 0 as A(r, t) = A(t)exp(ik · r) ' A(t). The interaction Hamiltonian is then reduced to
Hint = (e/m)p ·A(t) which can also be written as
Hint = −(p12|12|+ p21|21|)E(t) (2.7)
where E(t) = −∂A/∂t corresponds to the laser electric field and we have introduce the electric
dipole moment of the atomic transition as p12 = p∗21 = −e1|(r · e)|2, where e = E/|E| is the unit
polarization vector.
It should be easily realised that the off-diagonal terms of the interaction Hamiltonian are H12 =
H∗21 = p12E(t). It should also be noticed that the diagonal terms of this Hamiltonian are equal to zero
as they contain factors of the form 1|2 = 0. Replacing this results in the evolutions equations (2.4)
and (2.5) and using an electric field of the form E(t) = E0cos(ωt) (dipole approximation), where E0 is
the amplitude and ω the laser frequency we get the following evolution equations
∂C1
∂t = −i∗Reiω0tcos(ωt)C1 (2.9)
where we have introduced the Rabi frequency R = p12E0/~. To find solutions to these equations
we can begin to consider an atom which is initially in the lower energy level |1 and is submitted to a
low intensity laser field, which means that the coupling with the excited state is weak and we can take
C1(t) ' 1, leading to a solution of the form
C2(t) = 1
] (2.10)
This result allows us to determine the transition probability from the lower to the upper state as
|C2(t)|2. Ignoring the off-resonant term, we can approximately write
|C2(t)|2 ' |R| 2
(ω0 − ω)2 (2.11)
This approximation is commonly known as the rotating wave approximation. As it should be ex-
pected, the transition probability reaches a maximum at resonance, when ω = ω0, for which we
simply have |C2(t)|2 = 1 4 |R|
2t2. However we can obtain a more general solution in the rotating wave
approximation and introducing the frequency detuning δ = ω0 − ω which reads
|C2(t)|2 = |R|2
) (2.12)
This shows that the occupation of the energy levels oscillates with a frequency = √ |R|2 + δ2.
In the resonant case, ω = ω0 or δ = 0 we are reduced to the same result obtained before, = |R|.
This implies that the Rabi frequency |R| is the natural frequency of oscillation of the energy level
occupation probability, in resonance conditions.
We can also introduce the density matrix operator, which can be defined as ρ = |ψψ|. For a
two-level system we simply have |ψ = C1(t)|1+ C2(t)|2 and the diagonal matrix elements are
ρ11 = 1|ρ|1 = |C1(t)|2 (2.13)
ρ22 = 2|ρ|2 = |C2(t)|2 (2.14)
and the off-diagonal terms
10
ρ21 = ρ∗12 (2.16)
In this case, the normalization condition reads ρ11 + ρ22 = 1. If we have a collection of identical N
atoms per unit volume with N1(t) atoms in the lower state and N2(t) in the upper state then we can
make the association ρ11 = N1(t)/N and ρ22 = N2(t)/N . The temporal evolution of the density matrix
elements can be derived from the above equations for the coefficients Cj(t) and, in the rotating wave
approximation we come up with
dρ11 dt
= −dρ22 dt
= iRe iδt(ρ11 − ρ22) (2.18)
These last equations are the well known optical Bloch equations. The original Bloch equations,
which are formally identical, were introduced to describe the spin states in a magnetic field. The
interest of this new formulation, in terms of the density matrix elements, comes from the fact that it
can easily be extended to a mixed state, such that ρ = ∑ j pj |ψjψj |, where pj are the probabilities
for the quantum system to be in a state |ψj. Using the same initial conditions as before, ρ11(0) = 1
and ρ22(0) = ρ12(0) = 0 we get the same solution (2.12) for ρ22(t) and
ρ12(t) = |R|2
)] e−iδt (2.19)
Although spontaneous emission cannot be described in the frame of the present semi-classical
approach to the atom-laser interaction, we can include it by adding a phenomenological term into the
above evolution equations. Defining Γ as the spontaneous decay rate of the upper energy level, the
evolution equations for the matrix elements become
dρ11 dt
= −dρ22 dt
= Γρ11 + i
2 R(ρ11 − ρ22)− (Γ/2 + iδ)ρ12 (2.21)
where we have defined ρ12 = ρ12exp(−iωt). Steady state solutions of these equations will be an
important ingredient of the laser cooling process, as discussed next.
11
2.2 Light forces on an atom
We begin this section by setting the steady-state solutions to the optical Bloch equations (2.20)
and (2.21), setting the time derivatives equal to zero. In fact, for a laser cooled system, an atom is
expected to move in a relatively slow time scale, when compared with the external field oscillations
and the radiative processes. In that case we can assume that at each instant of time the atomic
state, represented by the matrix element ρ, will reach a dynamical equilibrium corresponding to the
steady state solution of the optical Bloch equations. This process is usually known as the adiabatic
elimination of the external variables. The steady-state solutions then read:
ρ11 = 1− 2 R
R/2 =
R/2
δ − iΓ/2 1
1 + s (2.22d)
where we have defined the saturation parameter as s = 2 R/2(δ2 + Γ2/4).
In this formulation of quantum mechanics, using the density matrix operator defined earlier, the
average value of a certain operator is given by the trace of the density matrix multiplied by the operator
in question. For the electric dipole operator we simply have
p = Tr [ρp] = 2Re [ρ12p12] = 2Re
[ R/2
−iωt ]
(2.23)
We can now refer to the Ehrenfest’s theorem which simply states that quantum mechanics is
equivalent to classical mechanics for the average values, whereby we can write for the force acting
on a two-level atom due to the interaction with an incident field
F = ∇ (p · E) (2.24)
Let us now set a general form for the electric field in the form E = E0cos (ωt+ φ(r)). In this case
the force reads
F = (p · ∇E0) cos(ωt)−∇φ(r)p · E0sin(ωt) (2.25)
Now using the result (2.23) we can write for the average value of the dipole moment
12
where we have defined the parameters
u = 1
(2.27b)
Putting the last results all together we get for the force acting on an atom
F = u∇(p12 · E0) + vp12 · E0∇φ(r)
= −~u∇R − ~Rv∇φ(r)
= F + FΓ (2.28)
The first term of the last equation corresponds to the reactive or dipolar force and the second
term describes a dissipative force, which will be clarified in the next section. Notice that, in the high
saturation limit (s 1), the force acting on the atom tends towards a dissipative force, independent
of the intensity
− ~ Γ
2 ∇φ(r) (2.29)
On the other hand, for small laser intensities (s 1) the force reduces to
F ' −1
δ2 + Γ2/4 ∇φ(r) (2.30)
In the next section we will apply these results to the laser geometry of a magneto optical trap and
find out how Doppler cooling works.
2.3 Doppler cooling
The process of atom cooling by laser radiation can be easily understood in a qualitative manner
using nothing but energy conservation premisses. Let’s assume a two level atom, with internal energy
states E1 and E2, as we been working with, interacting with a slightly red-detuned radiation field with
13
frequency ω < ω0 = (E2 − E1)/~ and wave vector k. A resonant radiative transition from the lower
to the upper energy level can occur by photon absorption if the atom moves with velocity v, such that
the Doppler shifted photon frequency becomes equal to the transition frequency, (ω − k · v) = ω0.
Now, if the upper level spontaneously decays, by photon emission in a direction with a perpendicular
component to the atom velocity, the radiation field will gain an energy of ~|k · v|, and due to energy
conservation, the atom must loose the same amount of kinetic energy, as it returns to the initial lower
energy state. If the fluorescence lifetime of the upper state is short enough, such a cycle of photon
absorption followed by radiative decay can be repeated several times in a second and the kinetic
energy of the atom will reduce, on average, and the atom will slow down, leading to a significant gas
cooling.
Let’s now address the Doppler cooling process in a more detailed quantitative manner, starting
from the results of the previous section, in which we derived the force acting on a two-level atom,
considering it as being at rest. To understand atom cooling we must perform a more realistic treatment
which implies the inclusion of the motion of the atoms, which can be easily incorporated in the previous
results. Consider a two-level atom moving with velocity v inside a standing wave formed by two weak
(s 1) counter propagating laser beams 1. If we additionally consider each of the laser beams as a
plane wave, we may set φ(r) = ±k · r in equation (2.30). We can also easily realize that the dipolar
force vanishes. In fact this term acts as a radiation pressure and due to the counter propagating
beams geometry there is zero net result from this term, reducing the dynamics to the dissipative term.
The inclusion of the Doppler shift, which implies that the atom interacts with a laser of frequency
ω ± k · v, corresponds to the substitution of δ by δ ± k · v, thus obtaining
F = −1
(δ2 + Γ2/4) 2 (k · v)k (2.31)
where, in the second line, we have expanded the result up to linear order in (k · v). The linear
behaviour of the friction force with respect to the velocity is found to hold up to |v| ' δ/k. It is now
easy to realise that this in fact corresponds to a friction type force acting on the atoms. Let us for
instance consider the case when the atom moves in the direction of the laser propagation, k||v, when
the force acting them reduces to
F = −αv (2.32)
with 1Remember the laser beams geometry of a magneto optical trap
14
(δ2 + Γ2/4) 2 (2.33)
Now we easily realise that for a red-detuned laser beam δ < 0 this is in fact a dissipative force
that slows down the atoms. The existence of three perpendicular pairs of counter propagating laser
beams in a magneto optical trap allows us to, in a simplified manner, reduce the dynamics of the
interaction of an atom with the six laser beams in a MOT to the equation (2.32).
We end this section by referring to the theoretical lowest temperature we can achieve with Doppler
cooling on a two-level atom system. This analysis involves using a statistical approach to sponta-
neous emission and study the evolution of the atomic momentum distribution of the cold gas, which is
determined by the Fokker-Planck equation [5]. This analysis is outside the scope of the present doc-
ument for which we simply present the base ideas and results. The nature of this statistical behaviour
can be cast in the form of a Langevin (or stochastic) force, FR, acting on the atoms and describing
the field fluctuations. In this way we can write for the dynamics of the atoms
mv = −αv + FR (2.34)
such that FR = 0. This statistical nature of the system implies a certain velocity distribution
for the constituent particles, which for small values of t it takes the form σ2 p ' Dt and for large
times we have σ2 v ' D/2mα, independent of time, where D can be interpreted as the momentum
diffusion coefficient. The analysis of the Fokker-Plack equation reveals that the ensemble of atoms
follows a Gaussian statistics, exp(−2v2/σ2 v), and relating this with the energy equipartition theorem,
mσ2 v = kBT/2 we can estimate the equilibrium temperature of the system
T = D kBα
(2.35)
This relation is known as the Einstein relation, obtained in his seminal work about the Brownian
motion of dust particles suspended in the surface of a liquid. In order to apply this relation we have
to compute the diffusion coefficient D in terms of the microscopic parameters. For that purpose
let us think about the one dimensional case. Each time a photon is absorbed or emitted, the atom
momentum increases or decreases by an amount of ~k. Within the counter-propagating configuration,
the process occurs at a rate 2γ with
γ = Γρ22 = Γ
(2.36)
In this way, over small time intervals t, the momentum changes of (p)2/t ' 2γ~2k2. Identify-
ing p = σp(t) we can compute the diffusion coefficient for small values of s
15
T = ~Γ
) (2.38)
Here we have implicitly considered only negative values of δ for the sake of physical consistency.
The temperature attains its lower value when δ = −Γ/2 at which point
TD = ~Γ
2kB (2.39)
The latter is known in the literature as the Doppler temperature. Remarkably, this limiting value
does not depend on the physical details of the atom but on the linewidth of the transition.
2.4 The recoil limit
In this section we briefly describe a physical mechanism that assigns an even smaller limiting
value to the temperature of the system. Each time an atom absorbs or emits a photon its momentum
changes by an amount of p = ~k. The momentum of the atom can then described as a discrete one-
dimensional random walk of step size±~k. The so called recoil limit applies to any cooling mechanism
involving scattered photons. As at least one photon must be scattered in such a process, the minimum
value for σp equals the elementary change in the momentum, such that σ2 p = (p)2 = ~2k2. Evoking
the ergodic hypothesis, that is the momentum distribution is Maxwellian, we can define the following
limiting temperature, often referred to as the recoil temperature
TR = ~2k2
2kBm (2.40)
In the following table, we present some values for this recoil temperature, together with other
parameters, for the most important atomic isotopes used in experiments.
Table 2.1: Comparison of the relevant atomic parameters for the main cooling transitions for the isotopes of elements used in laser cooling systems.
Isotope Cooling Transition Γ/2π (MHz) TD (µK) TR (nK)
39K 42S1/2 → 42P3/2 6.2 148 830
87Rb 52S1/2 → 52P3/2 5.9 145 370
40Ca 41S0 → 41P1 34.2 832 2670
88Sr 41S0 → 41P1 31.8 767 1020
16
We must also notice that the limiting value in equation (2.40) is not the smallest one in optical
cooling processes. Experiments with velocity-selective coherent population trapping [6], which is a
Raman cooling process, and with the previously mentioned method of evaporative cooling [7], reveal
that it is possible to cool down atoms below the recoil limit. This issues remain however outside the
scope of this document and we shall not give further details about it.
2.5 Sisyphus cooling
The limitations of the Doppler cooling theory, based on the two-level atom configuration became
rather evident as experiments performed with optical molasses2 soon revealed that the temperatures
were actually lower than the limit given in equation (2.39). This happens because two important
ingredients are ignored within the Doppler cooling theory. In one hand the intensity of the radiation
field produced by two counter-propagating lasers is inhomogeneous and, on the other hand real atoms
are not two-level systems and, in fact, they have Zeeman sub-levels in the ground state (actually, in
the absence of magnetic fields, alkali atoms have degenerate ground states). The physics behind
cooling in this configuration is, however, rather involved [9], for which we draw some of the most
important lines here.
Let us consider an atom interacting with two counter-propagating laser beams, along the z axis,
possessing the same frequency ω0 but in orthogonal polarization states, such that
E(r, t) = E0 ( e1e
) e−iω0t + c.c. (2.41)
with (e1 · e2) = 0 and (e1 · k0) = (e2 · k0) = 0. For instance, we can set k0 = k0ez, e1 = ex and
e2 = ey. This is then equivalent to write
E(r, t) = √
2E0e(r)eik0·r−iω0t + c.c. (2.42)
where e(r) is a position dependent unit polarization vector, defined by
e(r) = 1√ 2
(2.43)
This shows that the field polarization changes along the z-direction with a period of π/k0 = λ0/2,
alternating between linear and circular polarization. Such an alternation can be shown more explicitly
by introducing the right and left circular polarization vectors e± = (e1 ± ie2) / √
2, for which the above
2The designation of optical molasses, after Chu and collaborators[8], arises due to an experiment in which intersecting beams produce a region in space where the atoms experience a large velocity-damping force, and it is on the basis of the optical cooling in a MOT. The term ”molasse” was coined because of the resemblance of the resulting viscous photon fluid to real molasses.
17
Figure 2.1: Optical pumping mechanism in an optical potential with position dependent polarization.
expression becomes
e(r) = 1
1
2 e− [1 + sin(2k0 · r)] +
1√ 2 e2cos(2k0 · r) (2.44)
This shows that the intensity of the right and left circular polarization modes vary in space as
I± ∝ 1
2 [1 sin(2k0 · r)] (2.45)
For a ground state with two degenerate magnetic quantum numbers mg = ± 1 2 , this can lead to an
energy shift of the ground states due to the existence of these σ± polarization modes, of the form
E± = −U0 [1 sin(2k0 · r)] (2.46)
with
Γ2 + 4δ2 (2.47)
An atom moving to the right (k0 positive direction) in the state mg = +1/2 will be excited to the
upper level by the σ+ polarization mode, near the upper part of the oscillating potential hill (see figure
[2.1]). It will then be optically pumped to the other ground sub-level at the bottom of the well, emitting
a blue-shifted photon. The kinetic energy loss will then be of the order of U0. The process will then
repeat itself, with higher probability for an atom to climb the potential will then to go down.
The lowest temperature limit is now given by the recoil temperature, due to spontaneously emitted
photons, as mentioned in the previous section. When U0 becomes of the order or smaller than kBTR
18
Figure 2.2: In order to describe the principle of a magneto-optical trap, we consider an atom in the ground state, with total angular momentum J = 0, and an excited state with angular quantum number J ′ = 1, with three sub-levels mJ = 0,±1. Two counter-propagating laser beams are red-detuned with respect to the atomic transition, with left and right circular polarization σ±. The inhomogeneous magnetic field created by the two anti-Helmholtz coils is zero at the center of the trap and increases linearly. The laser beam with polarization σ−, propagating along the negative z direction, can couple the ground state to the m = +1 excited state, due to additional Doppler shift associated with atoms moving outwards in the positive region z > 0. A similar process occurs with the atoms moving outwards in the region z < 0, with transitions allowed to the state m = −1 by the laser beam with polarization σ+. The net result is cooling the radiative pressure pushing the atoms towards the center of the trap at z = 0.
as defined in equation (2.40), the cooling due to Sisyphus effect becomes weaker than the heating
due to recoil and Sisyphus cooling no longer works. This shows that the lowest temperatures which
can be achieved with such a scheme are on the order of a few TR. This result is confirmed by a full
quantum theory of Sisyphus cooling [10] and is in good agreement with experimental results.
2.6 Magnetic traps
We have so far derived a simple theory for the process of laser cooling, but there still is one
ingredient necessary to the magneto optical trap, and that is the trapping force, in order to achieve
a cloud of trapped cold atoms, whose dynamics will be studied in the next section. In the magneto
optical trap, it is the quadrupole magnetic field that causes an imbalance in the scattering forces of the
laser beams and it is the radiation force that strongly confines the atoms. It is important to understand
the difference between MOT’s and purely magnetic traps, where in the latter the trapping is provided
purely by magnetic fields. The working principle is schematically represented in figure [2.2].
In the middle of the axial center defined by the coils (see figure [1.1]), the magnetic field produced
by the coils cancels out, so B(z = 0) = 0. For small values of z, however, there is a uniform magnetic
field gradient responsible for the Zeeman effect, thus altering the energy of the sub-levels mJ = 0,±1
19
of the J ′ = 1 group. The energy shift is, up to first order in the magnetic field B, approximately linear
at the center of the trap, such that
Ez = −µ ·B = −~gµBB′z (2.48)
where B′ = ∂B/∂z is the gradient of the magnetic field, and g is the Lande factor. The Zeeman
shift can also be described in terms of the detuning, by making the substitution δ → δ − gµBB ′z.
Generalizing to the three-dimensional case, and with the Doppler cooling theory, the effect of the
Zeeman shift can be inserted in the previous results, thus yielding
FMOT = −1
1
4
(2.49)
where the summation is taken over the different laser beams. For small values of both Doppler
and Zeeman shifts, k · v ∼ B′ · xi δ, we may write
FMOT ' 3∑ i=1
κixi (2.50)
where α in given by equation (2.33) and κi is the equivalent spring constant
κi = αgµBB
′ · ei k
(2.51)
The second term of the force appears here as a restoring force, which provides the spatial con-
finement to the atoms. Assuming the force to be spherical symmetric (check Appendix B) will be very
useful in the remainder of this thesis, leading to κi ≈ κ and
FMOT ' −αv − κr (2.52)
We have finally an equation that, in a very simplified manner, is able to describe the interaction of
the laser beams with the atoms, in the presence of a quadrupole magnetic field created by two anti-
Helmholtz coils. This simple model can however produce reliable results, under certain experimental
conditions. In Chapter III, we will introduce a collective force which becomes important for magneto
optical traps with a large number of trapped atoms. As mentioned before, this force arises from
multiple scattering of light inside the atomic cloud and it will be added up to the force we just derived
in equation (2.52) allowing us then to perform a consistent investigation of the dynamics of a wider
and more interesting range of ultra cold atomic clouds, which we will do in Chapter III and IV. To finish
this introductory chapter on the theory of laser cooling we will also refer to the multi-level structure
20
Figure 2.3: Schematic illustration of the multi-level structure for 85Rb. a) The cooling transition corresponds to the F = 3→ F ′ = 4 line. b) Forbidden transitions. c) Repumping transitions.
of the 85Rb, widely used for cold atoms experiments, to serve as example to a final but important
ingredient in a magneto optical trap, the optical repumping.
2.7 Optical Transitions for 85Rb
To illustrate the physics of magneto optical traps, let us consider the case of 85Rb, which has an
optical loop between the states 52S1/2 and 52P3/2, where F = 3 and F ′ = 4 correspond to the ground
and excited states respectively. We should refer to this transition as the cooling transition. Once in
the excited state, the atom is forbidden from decaying to any of the 52P1/2 states, which would not
conserve parity. Any de-excitation to the 52S1/2 F = 2 state is also forbidden, which would require
an angular momentum change of −2, which is impossible with a single photon. The magneto optical
trapping of 85Rb involves cycling on the closed 52S1/2 → 52P3/2 transition. On excitation, however,
the detuning necessary for cooling gives a small, but non-zero overlap with the 52P3/2 F ′ = 3 state. If
an atom is excited to this state, which occurs roughly every thousand cycles, the atom is then free to
decay either to F = 3, light coupled upper hyperfine state, or to F = 2 ”dark” lower hyperfine state. If it
falls back to the dark state, the atom stops cycling between ground and excited state, and the cooling
process stops. A repump laser, which is resonant with the 52S1/2 F = 2→ 52P3/2 F = 3 transition is
used to recycle the population back into the optical loop so that cooling can go on. We shall not go
any further into the repumping process, which which the reader should refer to [11] and [12–14].
21
22
3.2 Fluid mechanical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Polytropic equilibrium and density profiles . . . . . . . . . . . . . . . . . . . . . . 27
23
Figure 3.1: Schematic representation of the shadow effect. The trapping beams are absorbed when travelling through the cloud and this effect gives rise to an attractive force, directed towards the center of the MOT.
In the previous chapter we used a very simplified model to derive the force acting on an atom inside
a MOT, given by equation (2.52). Besides its simplicity, this model is known to accurately account
for the experimental observations for MOTs with a small number of atoms, typically of the order of
N ∼ 105. The problem is that state of the art MOTs work in a regime that cannot be accounted for
by the simple model of Chapter II. In these novel magneto optical traps a larger number of particles
can be trapped, of the order of N ∼ 1010, and the atoms in the gas start to exhibit a distinctive
and interesting collective behaviour. This collective behaviour is mediated by secondary photons,
resulting from scattering, by the atoms, of the incident primary photons associated with the laser
cooling beams. These collective forces are therefore an intrinsic property of the laser cooled gas. In
this chapter we begin to introduce the nature of these interactions and show how they can be added
to the former model of Chapter II to end up with a phenomenological theory capable of describing the
state of the art MOTs. We then discuss the influence of these forces on the atom density profiles of
the cloud.
3.1 Collective behaviour and plasma description
As mentioned before, for larger than N ∼ 105 atoms trapped, the multiple scattering induces
interactions that we haven’t yet considered. Two additional forces must then be taken into account.
The first force to be investigated corresponds to the so called shadow force, or absorption force,
FA, which was first discussed by Dalibard [15]. This interaction is associated with the gradient of the
incident laser intensity due to laser absorption by the atomic cloud. It is an attractive force which can
be determined by a Poisson type equation
∇ · FA = −σ 2 LI0 c
n(r) (3.1)
where n(r) is the atom density and σL the laser absorption cross section.
24
The second force to the considered corresponds to a repulsive interaction, FR, and it was first
considered by Sesko et al. [16]. It describes atomic repulsion, due to the radiation pressure of
scattered photons on nearby atoms, and it can also be determined by a Poisson type equation
∇ · FR = σLσRI0
c n(r) (3.2)
where σR is the atom scattering cross section. A detailed discussion of these forces and explicit
expressions for the cross sections σL and σR can be found in [16, 17].
We are now in position to define the collective force
Fc = FA + FR (3.3)
which obviously can also be determined by a Poisson type equation, combining equations (3.1)
and (3.2)
Q = (σR − σL)σLI0/c (3.5)
The system can then be regarded as a one component plasma where the electrostatic attractive
force due to ions can be formally replaced by the confining force. In typical experimental conditions
the repulsive forces largely dominate over the shadow effect and the quantity Q is positive [17, 18],
which allows us to define the frequency
ωp ≡ √ Qn0 m
which is a straightforward generalisation of the well-known electron plasma frequency. Making
the association with the usual plasma frequency ωpe in an ionized medium, we conclude that neutral
atoms behave as if they had an equivalent electric charge, as first notice by [18], with the value
qeff = √ ε0Q, where ε0 is the vacuum electric permittivity. The experimental value observed for this
effective atomic charge is 10−6 < qeff/e < 10−4. In a typical MOT we expect to have n0 ∼ 1010 cm−3,
m ∼ 10−25 Kg, which corresponds to a plasma frequency of about ωp/2π ∼ 100 Hz. We shall notice
from equation (3.6) that plasma like oscillations are only possible for Q > 0, which corresponds to
the shadow force (attractive) lower than the repulsive force. This is in fact very easy to understand
because we must have an overall repulsive collective force in opposition to the attractive confining
force. It’s the counterbalance of these opposite interactions that drives the oscillations in the system.
25
The presence of the multiple scattering induced force is the main factor responsible for limiting the
compressibility of the cloud [15, 16]. We could then be led to think that a system with an overall
attractive collective force (Q < 0) could overcome this compressibility issue but we find out that such
a system is unstable under small perturbations.
This plasma description for the system has been proven very fruitful [19, 20]. In particular, the
formal analogy and application of plasma physics techniques reveal to be important in the description
of driven mechanical instabilities [21] or even more exciting instability phenomena, like phonon lasing
[22], as mentioned before.
3.2 Fluid mechanical description
Every work done in this thesis is made without any microscopic theory for the ultra cold gas.
The solution to overcome this problem is the proposition of a phenomenological model for the gas,
combining the effects of multiple scattering (described by a Coulomb-like potential) and the thermal
fluctuations inside the system cast in the form of a polytropic equation of state, which will be the
starting point for the development of the rest of this thesis. Other advantage of this procedure is that it
allows us to make the comparison with quantities which are accessible experimentally. For instance,
direct and absorption imaging techniques together provide a measurement of the density profile. It is
therefore worthy to relate the equation of state of the gas inside the magneto optical trap with such
experimental parameters which can be done by defining a polytropic equation of state for the gas. We
can then use the theory of fluid mechanics to investigate the equilibrium properties and the dynamical
features of the system. For this we refer to the continuity and the Navier-Stokes equations, which read
∂n
(3.8)
where n and v are respectively the density and the velocity field of the gas, P is the pressure and
FT = FMOT + Fc is the total force acting on an atom with FMOT given by equation (2.52) and Fc the
collective force determined by the equation (3.4).
26
3.3 Polytropic equilibrium and density profiles
As it was mentioned in the last section, the main ingredient for the construction of the following
description for the ultra cold gas lies in assuming the existence of a polytropic equation of state
P (r) = C(T )n(r)γ (3.9)
where C(T ) is a certain function of the temperature T and γ corresponds to the polytropic expo-
nent. Our starting point is set with the introduction of the above fluid equations (3.7) and (3.8) where
both the kinetic pressure and the collective force are taken into account. The equilibrium density
profiles can be derived from the fluid equations by assuming the hydrostatic conditions ∂/∂t = 0 and
v = 0. These conditions then yield
∇P
where V0 = mω2 0r
2/2 is the trapping potential. Remember that we derived an expression for the
confining force in the form F0 ' −κr which formally can be replaced by a potential acting on the
atoms of the form V0 = mω2 0r
2/2, with ω0 ≡ √
κ m , assuming spherical symmetry for the magnetic
field, which is a good approximation - check Appendix A. Using the above equation of state (3.9) and
noticing that
we obtain
C(T ) γ
) + 3mω2
θ(ξ) = [n(r)/n(0)] (γ−1) (3.14)
and the dimensionless variable ξ = r/R with
R =
( 1
3
)2
(3.15)
the length measured in units of the polytropic radius. We then obtain
27
γ
pθ 1/(γ−1) + 1 = 0 (3.16)
Note that n(0) represents the central maximum density, and 2 p = Qn(0)/3mω2
0 representing the
dimensionless effective plasma frequency. This later quantity is basically the ratio of the two main
forces acting the system, a repulsive one due to multiple scattering, and an attractive one due to
magnetic trapping and formally introduced here by the term ω0.
In its present form, equation (3.16) is formally similar to the Lane-Emden equation, widely used in
astrophysics [23, 24]. The main difference is related with the nature of the interaction, which turns out
to be repulsive in our case in opposition to the attractive gravitational one in astrophysical scenarios.
It interesting now to take some limiting cases. In the case where the temperature effects are
negligible compared to long range interactions, we may set γ = 0 to get θ(ξ) = 1/2 p which simply
corresponds to the water-bag profile
n(r) = n(0)2 pH(a0 − r) =
Q H(a0 − r) (3.17)
where the cloud size is such that n(0)2 pV = N , where V is the volume of the cloud. Note that
in this limit we have 2 p → 1, when the ratio between the repulsive multiple scattering force and the
attractive confining force approaches 1. As usual, H(a0 − r) denotes the Heaviside function. This
leads to
)1/3
(3.18)
The opposite situation, p = 0 is also interesting and it corresponds to the solution of a trapped
polytropic gas without long-range interactions, for which we simply have
θ(ξ) = [ 1− (γ − 1)ξ2/6γ
n(r) = n(0)
The latter could approximately describe the temperature limited traps, corresponding to a small
number of particles [25]. As it was previously mentioned, this does not correspond to the state of the
art traps, so the temperature limited regime is here considered only for completeness.
The isothermal case γ = 1 simply corresponds to a Gaussian profile, in agreement with the
Maxwell-Boltzmann equilibrium 1. 1Notice that C(T ) = kBT for γ = 1
28
Figure 3.2: Effect of the long-range interaction on the density profile for different polytropic exponents. The left panel depicts the case where thermal effects dominate (p = 0), while the right panel illustrates the case where multiple scattering dominates (p = 0.99). The black thick line depicts the normalized density profile for the isothermal case, γ = 1. Red (dashed) and blue (dot-dashed) lines correspond to γ = 2 and γ = 4, respectively.
n(r) = n(0)e−βV0 = n(0)e −mω
2 0r
6R2 (3.21)
which also results from taking the formal limit2 γ → 1 in equation (3.19). In figure [3.2] it is
represented some numerical solutions to equation (3.16). We can observe that a Gaussian profile is
modified, when p is increased, towards a water-bag profile.
2Remember the definition of the exponential function as
ex = lim n→∞
of rotating clouds of ultra cold atoms
Contents
4.1 Equilibrium profiles for rotating clouds of ultra cold atoms . . . . . . . . . . . . . 32
4.2 Solutions and interpretation of the equilibrium density profiles . . . . . . . . . . 35
4.3 Localised oscillations, or normal modes, in rotating clouds of ultra cold atoms . 43
31
Since the early stages of MOT investigation it was realized that rotating clouds of atoms carry
out an interesting behaviour with some features that were not observed in non-rotating clouds. In
1991, Sesko et al [16] introduced the classical collective behaviour, due to multiple scattering of
light and introduced here in the previous chapter, as the physical mechanism behind features like
extended uniform-density ellipsoids 1, rings of atoms around a small central ball and clumps of atoms
orbiting a central core. It was realized that much of this complex behaviour could be explained by
the incorporation of long-range interactions, introducing this collective force in the equation of motion
of a single particle. What we have been done so far, and will keep on doing throughout this thesis
is to describe the entire ensemble of atoms using the polytropic equation of state and the Navier-
Stokes equation which gives us a good description of the system in terms of the atom density function
n(r), which can easily be measured experimentally as mentioned before. Another advantage of this
formulation is that it allows us to investigate the localised oscillations of the system, which will be done
in section (4.3).
4.1 Equilibrium profiles for rotating clouds of ultra cold atoms
Once again, the starting point corresponds to setting the fluid equations
∂n
(4.2)
As we are dealing now with rotating clouds, the total force acting an element of fluid is now
FT = FMOT + Fc + Fr (4.3)
with FMOT and Fc given by the usual expressions
FMOT ' −αv − κr (4.4)
∇ · Fc = Qn (4.5)
The difference now is the presence of the rotation term Fr. A rotation in the system can be easily 1Remember the water-bag solutions derived in the previous chapter, for the regime where collective forces dominate, 2
p → 1.
32
Figure 4.1: Misalignment of the laser beams to produce a rotation in the cloud.
achieved with a slight misalignment in four of the six laser beams.
A misalignment of this kind can be phenomenologically described by a force [16]
Fr = κ′rez × er = κ′reφ (4.6)
For this reason we must from now on consider a cylindrical symmetry for the system, whose
validity will be discussed later.
Once again, assuming equilibrium conditions, ∂/∂t = 0 and r = 0 2, we obtain
− C(T )
m
γ
0 + Qn
∂φ
∂r
) (4.8)
Here φ(r) corresponds to the angular velocity of the fluid element. For non rotating clouds, φ =
0 ⇒ f(r) = 0 and we recover equation (3.13). Once again we can now adimensionalise the system
in the same way as before
θ(r) =
( n(r)
n(0)
)1/2
(4.10)
and we then get, in a dimensionless form 2Remember that in the previous chapter the equilibrium conditions read ∂/∂t = 0 and v = 0 but now with a rotating system
the equilibrium conditions only imply a zero radial velocity, with v · eφ 6= 0
33
γ
∂φ
∂ξ
) (4.12)
In order to continue we must determine the function φ(ξ). For that purpose let’s refer to the eφ
component of the Navier-Stokes equation, in equilibrium conditions, which simply reads
0 = 1
) ⇔ φ = κ′/α (4.13)
Substituting for f(ξ) we get f(ξ) = −6κ′2/α2 = −6φ2 = const. Replacing this result into equation
(4.11) we finally get
with β2 ≡ 2 ( φ ω0
)2 > 0 which is associated with the ratio between the rotation angular frequency
and the angular frequency associated with the confinement. The first term is equivalent to a expansion
force and the second one is related with a contraction one whereby this constant will be related with
the stability of the cloud.
As expected, or not, we obtained an equation for the equilibrium density profiles very similar to the
one obtained for non-rotating clouds, which suggests that the solutions may also be similar in some
way. This correspondence is even more dramatic if we realize that, by redefining the parameters R
and 2 p as
34
which takes the same form as equation (3.14) for non-rotating clouds 3. This remarkable result
means that rotating and non-rotating clouds of atoms share the same solutions for the equilibrium
profiles, differing only by a scale factor (ω2 0−2φ
2) 1/2
(ω2 0)
files
We derived a differential equation for the density profiles that is independent of the rotation state
of the system, which is encoded in the definitions of R and 2 p. We shall also notice that we are
employing a cylindrical coordinate system in opposition to the spherical symmetry for the non-rotating
systems of Chapter III. The Laplacian term is then different in both cases, causing the solutions to
present slight differences. We also have to take into account that in our formalism, the term 2 p is
a function of the rotating angular frequency. Let’s from now on denote ′2p = Qn(0) 3mω2
0 =
the
effective plasma frequency for non-rotating clouds and 2 p = ′2p /(1 − β2) = ω2
p/ √
3(ω2 0 − 2φ2) for
rotating clouds, where ′2p = 2 p for β = 0 (no rotation). In fact, the density profiles only depend
on the polytropic exponent γ and on this constant which accounts for the three force in play. The
latter corresponds then to the ”ratio” of the collective force (determined by ω2 p) and the confining force
(determined by ω2 0) ”subtracted” with the centripetal force due to rotation (determined by β2). The
confining force is then directly contour-posed by the centripetal force. This ratio is then sufficient
to determine the density profiles. With this interpretation of 2 p in mind we can easily realize that a
stable cloud of atoms can only exist if 2 p < 1 which implies an attractive force (confinement ”minus”
centripetal) greater than a repulsive one (multiple scattering). Since we are interested in associating
systems in different rotation states, we can realize that degenerate density solutions exist if
2 p =
= η (4.19)
with η < 1 for a stable solution. Each of these sets of solutions (for each value of the constant η)
differs only then by a scale factor.
This result is very interesting and experimentally it can bring important advantages, which we’ll
clarify later.
Let us now compute some analytical solutions for equation (4.18) for the two limiting cases already
considered in the previous chapter. For small traps, typically N < 105 the effects of multiple scattering
can be neglected which corresponds to take the limit 2 p → 0, reducing equation (4.18) to
3In fact there is an additional difference in the Laplacian term due to the different coordinate systems employed in both cases.
35
Figure 4.2: Degeneracy of the equilibrium density profile solutions for systems in different rotation states. The blue region corresponds to the existence of stable solutions, 2
p = ′2p /(1 − β2) < 1, and the each line corresponds to a different value of the constant η, given then rise to the same equilibrium profile, except for a scale factor.
γ
n(r) = n(0)
with R given by equation (4.16).
The isothermal case γ = 1 simply corresponds to a Gaussian profile, in agreement with the
Maxwell-Boltzmann equilibrium 4.
n(r) = n(0)e −mω
2 0(1−β2)r2
2kBT = n(0)e− r2
4R2 (4.23)
which also results from taking the formal limit γ → 1 in equation (4.22). This limit could approx-
imately describe the temperature limited traps, corresponding to a small number of particles, where 4Notice that C(T ) = kBT for γ = 1
36
Figure 4.3: Spatial atomic distribution in the case of a misalignment of the trapping beams. Even with a small number of atoms trapped it is possible to achieve this Saturn-like density profile.
multiple scattering can be neglected and the dynamics of the system is determined by thermal effects.
As it was mentioned before, this does not correspond to the state-of-the art traps, so the temperature
limited regime is here considered only for completeness. In the opposite case, for a large number
of atoms trapped, the collective force dominates and temperature effects can be neglected, which
corresponds to the limit γ → 0 in equation (4.18) given rise to the solution
θ(ξ) = 1/2 p (4.24)
n(r) = 3mω2
0(1− β2)
where a is the radius of the cloud.
One of the most interesting features observed in rotating clouds of atoms is the formation of stable
orbital modes in the density profiles (see figure [4.3]). Up so far, it was widely believed that the
existence of these orbital modes was intrinsically related with the rotation of the cloud. We are now in
position to disregard this hypothesis since we now understand that rotating and non-rotating systems
must share the same solutions for the density profiles. In figures [4.4] and [4.5] we present some
numerical solutions for equation (4.14) for rotating β 6= 0 and non-rotating clouds β = 0.
37
Figure 4.4: Numerical solution for equation (4.18) in the temperature limited regime, 2 p = 0, for different poly-
tropic exponents γ.
Figure 4.5: Numerical solution for equation (4.14) for ′2p = const and different angular velocities. From top to bottom (′2p , β
2,2 p) = (0.7, 0.2, 0.875), (′2p , β
2,2 p) = (0.7, 0.25, 0.933) and (′2p , β
2,2 p) = (0.7, 0.29, 0.986).
38
We observe the expected behaviour, the increase in the atoms angular velocity corresponds to a
decrease in the confining force and the cloud expands. Due to the degeneracy of the solutions in the
variables ′p and β the same expansion effect can be achieved if we increase collective effects, with
an higher number of trapped atoms. Curiously we can also notice that for lower angular velocities, the
solutions for γ = 4 and γ = 1.5 correspond to the smaller and the higher radial extended clouds. As
the angular velocities increases (higher β2) the situation inverts and the solution for γ = 4 corresponds
to the higher radial extended cloud.
The importance and significance of the solutions depicted in figure [4.6] are twofold. First we prove
that the generalized Lane-Endem equation, derived with the simple assumption that there exists a
polytropic equation of state for the gas, can account for solutions with orbital modes, for a sufficiently
high 2 p. The second is related with the up so mentioned degeneracy of the solutions in the ′2p
and β2 parameters, which is also evidenced in the figure [4.6], where we have three combinations
of (′2p , β 2) combining in the same effective plasma frequency 2
p = ′2p /(1 − β2) to produce the
same θ(ξ) solution, except for a scale factor. We also mentioned before that this feature can bring
important advantages for experimentalists. In fact, we realise that to achieve an orbital mode we must
have a very high 2 p, which for non-rotating systems implies a very large number of trapped atoms,
conditions that cannot trivially be achieved experimentally. Introducing a rotation in the system, easily
obtained experimentally, reduces the number of atoms needed to achieve an orbital mode. This is the
reason why orbital modes have only been observed in rotating clouds, even with a small number of
trapped particles, as seen in figure [4.3], although its existence in now proven possible in non-rotating
systems.
Let us now discuss the validity of employing a cylindrical symmetry for the problem in hand. First of
all, we already realised, in section (2.6), that the confining magnetic field created by an anti-Helmholtz
pair of coils is cylindrically symmetric, although this problem is not a major concern and a spherical
symmetry still holds as a good approximation despite this issue. More important than the shape of
the magnetic field is the relative intensity of the three pairs of laser beams. For instance, a cigar
shaped cloud of atoms can be easily achieved by lowering the intensity of two collinear laser beams,
reducing the confining force and allowing the cloud to expand in that direction. In this case, the
cylindrical coordinate system is the appropriate one and the results presented here can be directly
applied. On the other hand, in a system with three pairs of equal intensity laser beams, the cloud of
atoms will present a spherical shape and we must be careful about how we apply the results present
here. Nevertheless it is still possible to think about the cylindrical solutions as slices of θ = const,
where θ is the azimuthal angle of the spherical coordinates. In fact, in a spherical cloud, increasing
the azimuthal angle θ correspond to circular slices with lower central densities. In this interpretation of
the cylindrical solutions, lower central densities are equivalent to a lower 2 p and then an orbital mode
39
Figure 4.6: Numerical solution for equation (4.14) for high collective effects. For a sufficiently high 2 p the solu-
tions start to exhibit orbital modes. From top to bottom (′2p , β 2,2
p) = (0.7, 0.29999, 0.999986), (′2p , β 2,2
p) = (0.8, 0.1999989, 0.999986) and (′2p , β
2,2 p) = (0.999986, 0, 0.999986). We only present the positive part the of
function θ(ξ).
40
can exist in the equatorial plane, θ = 0, and cease to exist for a given angle θ > 0, reason why we get
Saturn-like density profiles with orbital modes near the equatorial plane.
In figure [4.7] it is presented some density plots for the atoms density profiles, along with the usual
solution θ(ξ), for high values of 2 p, where the system starts to exhibit solutions with orbital modes.
41
Figure 4.7: Density plots for the atoms density profiles for high values of 2 p. We observe that increasing even
further the value of 2 p allows for a second orbital mode at an higher radius.
42
of ultra cold atoms
In the previous sections we introduced the equilibrium atom density profiles for rotating and non-
rotating clouds, assuming the existence of a polytropic equation of state. It is also possible to, starting
from the same assumptions as before, evaluate the nature of the localised oscillations with a pertur-
bative analysis on the fluid description of the system. This procedure, for non rotating clouds, can be
found in [4]. In this section we will extend the theory to the case of rotating clouds and compare the
results, when appropriate.
As before, the starting point corresponds to setting the fluid equations
∂n
∇ · Fc = Qn(r) (4.28)
In this section it will be useful to notice that the collective force can be derived from a potential,
whereby we can write

(4.30)
with the perturbations varying as δa = δaeiωt. This prescription allows us to linearise the set of
fluid equations, neglecting quadratic terms of the form δaδb ∼ 0, which yields
∂δv
mn0 − ∇δφc
m − α
43
∇δP ' γC(T )nγ−10 ∇δn (4.34)
from the polytropic equation of state. Notice that the unperturbed velocity v0 is different from zero
due to the rotation of the cloud. In fact, as computed before we have v0 = rφeφ, with φ = κ′/α.
In order to properly combine equations (4.31) and (4.32) we have to multiply the perturbed Navier-
Stokes equation by n0(r) and by the divergence operator ∇· , in this order, and multiply the perturbed
continuity equation by the operator ∂/∂t. Then, realizing that ∂n0
∂t = 0, ∇ · v0 = 0 and ∂v0 ∂t = −rφ2er,
which corresponds to a centripetal force, the two equations result in a single one of the form
− ( ω2 + i
m ∇ · [n0(r)∇δφc] (4.35)
It is worth noticing that this final equation for the perturbation δn takes the same form of the one
derived for non-rotating clouds [4]. This result is in fact to be expected since in the derivation process
we only considered radial perturbations, in the form δv = δv(r)eiωter and δn = δn(r)eiωt. The
existence of the angular velocity is put in play by the equilibrium density profiles n0(r) from which the
previous equation depends on. Recall that, due to the geometry of the problem, we must employ a
cylindrical coordinate system to our equations. Now, in order to combine equations (4.33) and (4.35)
we can define the auxiliary quantity η, defined as
δn = 1
∇2δφc = − Q
Making proper substitutions, the linearised equations can finally be put together resulting in a
single expression
[ ω2 + i
with ω2 p = Qn(0)
m . We then reduced the problem of finding the normal modes of oscillation to an
eigenvalue problem. Despite the linear form of the differential equation this is a non-trivial problem
and general solutions involve numerical simulations. For that reason we will examine some limiting
cases. As a last remark, notice that the term iα/m in the previous equation corresponds to a damping
of the modes, which is to be expected since it comes from the cooling Doppler force acting the atoms
which, as we derived in Chapter II, takes the form of a damping force, FDoppler = −αv. In fact, along
with the damping of the modes, this term also causes a small variation in the frequency of the modes
but, as it turns out, in typical experimental conditions the frequency associated with this term is small
in comparison with the trapping frequency ω0 and the plasma frequency ωp, for what we shall neglect
this term from now on. For a more detailed analysis of this problem refer to [19].
4.3.1 Temperature limited regime
For small clouds, typically with N < 105 atoms trapped, the effects of multiple scattering can be
neglected and therefore we can set ωp = 0, or equivalently φc = 0 in equation (4.35) which simplifies
to
] = 0 (4.40)
Replacing the equilibrium density profile n0(r) given by equation (4.22) into the above equation
and writing result in terms of the adimensional variable ξ = r/R, as defined before, we obtain
− ω2δn− 3γω2 0
4γ γ−1ζ, one obtains
− ω2δn− 3
] = 0 (4.42)
For this differential equation we can try out solutions in the form of a power series
δn = ∑
anlζ 2n+l (4.43)
where we distinguish even (n) and odd (l) contributions because they correspond to slightly dif-
ferent types of oscillations. For the lowest radial modes, n = 0, the solutions correspond to surface
excitations [26], and the even perturbations solutions, l = 0, corresponds to breathing modes. Deriv-
ing the ansatz (4.43) and replacing in equation (4.42) we obtain
45
4
2n+l = ∑
[anl(2n+ l)− anl+2(2n+ l + 2)] (2n+ l + 2)ζ2n+l (4.44)
From the uniqueness of the power series expansion, the above equation implies that the coeffi-
cients of ζ2n+l must be the same in both sides of the equation, which implies
4
] (2n+ l + 2) ≡ G(2n+ l + 2) (4.45)
By solving this equation we obtain a recurrence relation for the coefficients anl as well as the
oscillation modes allowed in the system. By setting G = 1 we then get the frequency of the modes as
ω = ω0
√ (1− β2)
4 (γ − 1)(2n+ l + 2) (4.46)
As mentioned before, setting n = 0 we get the surface modes
ω = ω0
√ (1− β2)
4 (γ − 1)(l + 2) (4.47)
Breathing modes, l = 0, are also possible even in small traps, with a spectrum given by
ω = ω0
√ (1− β2)
2 (γ − 1)(n+ 1) (4.48)
These results are very interesting, because they allow for a ”measurement” of the equation of state
for the ultra cold atoms which is simply determined by the polytropic exponent γ. In fact, experimental
techniques usually make possible the measurement of frequencies in a very precise manner, and
these results relate the polytropic exponent γ with the mode frequencies and the angular velocity of
the cloud.
4.3.2 Multiple scattering regime
In opposition to the temperature limited regime for small traps, when we have a large number of
particles trapped, the system is dominated by the collective effects and we can neglect the tempera-
ture effects by setting γ = 0 in equation (4.39) yielding
( ω2 − ω2
Introducing the water-bag solution for the equilibrium density profile, n0(r) = 3mω2
0(1−β 2)
Q H(a− r),
)1/2 = ωp (4.50)
Remember that the multiple scattering regime corresponds to the limit 2 p → 1 with 2
p = Qn(0) 3mω2
( 1− β2
)1/2 = ωp. We then have a breathing mode in the system at the
equivalent plasma frequency ωp. The latter result is very well-known in plasma physics and corre-
sponds to an uncompressional monopole oscillation of the system at the classical plasma frequency.
The some solution is found for systems with spherical symmetry [4]. We shall however notice that this
solution is not unique. Getting back to equation (4.35) and again setting γ = 0 we obtain
− ω2δn = 1
m ∇ · [n0(r)∇δφc] (4.51)
with ∇2δφc = −Qδn. Replacing this in the previous equation we obtain
ω2
∇2 [ε(ω)δφc] = 0 (4.53)
ω2 = 1−
ω2 p
ω2 (4.54)
This equation holds for the interior region of the cloud while for outside the cloud we shall have
∇2δφc = 0, ε(ω) = 1. The breathing mode obtained earlier can be derived from this formulation by
setting the particular solution to equation (4.53

equilibrium and dynamical properties of rotating clouds of

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