BEC TERM PAPER - FLIERL

BOSE-EINSTEIN CONDENSATES PHY 402 SPRING 2016
Bose-Einstein
Condensates
Including an Introduction to Laser Cooling,
Evaporative Cooling, & Fermionic Condensates
Aaron Flierl
SUNY Buffalo, Department of Physics Undergraduate
e-mail address: arflierl@buffalo.edu
This paper has been written with the intention of providing an introduction to the concept of
Bose-Einstein Condensates. The indistinguishable nature of Bosons and Fermions will be
discussed as well as the statistical distributions which describe their behavior at low temperatures.
Properties of BECs as well as the cooling processes used to initiate the phase transition of an ultra-
cold gas of Bosons to a BEC will be discussed. Degenerate Fermi gases and the formation of
Fermionic condensates of Cooper pairs of electrons will also be briefly discussed. It is expected
the readerhave some knowledge of quantum mechanics and this paperis intended to give the reader
a solid conceptual understanding of the principals mentioned.
I. INTRODUCTION
In 1924-25 Albert Einstein and Satyendra Nath Bose
developed a theory which described how a collection
of non-interacting indistinguishable particles can
occupy a set of quantized energy states. At low
temperatures this theory allowed for an unlimited
number of these particles to occupy the same low-
energy state, or ground state [1]. This theory of the
distribution of these particles amongst their available
quantized energy states is called Bose-Einstein
statistics. The particles that are described by this
statistical distribution are called Bosons. They are
governed physically by the lawsof quantum mechanics
and their most defining characteristic is that they have
an integer spin.
When a dilute gas of Bosons is cooled down to a
temperature near absolute zero there is a macroscopic
occupation of the ground state which leads to a phase
transition to a state of matter called a Bose-Einstein
condensate (BEC). This phenomena was first predicted
by Einstein in 1925 [1] but not proven experimentally
until 1995, when a BECwasformed using a very dilute
non-interacting Bosonic gas of Sodium by a research
team at MIT [2].
Because the Bosons in a BEC all occupy the ground
state, they can be characterized by a single
wavefunction [3], and quantum behavior such as
coherent matter waves, superfluidity, and
superconductivity can be observed on a macroscopic
scale [6].
Unlike Bosons, Fermions are subject to the Pauli
exclusion principal and are not allowed to occupy the
same energy level. When a dilute gas of Fermions is
cooled to near absolute zero a degenerate Fermi gas is
formed, or a “Fermi Sea”. Composite Bosons
consisting of two Fermions, such as Cooper pairs of
electrons, or molecular pairs, can form Fermionic
Condensates. However,the case of molecular pairs of
Fermions forming a molecular Boson is not a “true”
Fermionic Condensate [5].
The temperatures required to achieve the phase
transition of a dilute gas of Bosons or composite

Bosons to a BEC is typically made possible by a
combination of laser and evaporative cooling
techniques. However,a BEC of Strontium was formed
using laser cooling as the only cooling mechanism by
researchers at the University of Amsterdam [7].
Optical properties of BECs are extreme,and in 1998
Dr. Lene Hau at Harvard was able to slow light within
a BEC to 17m/s and reduce a light pulse with a size of
1km in a vacuum to ≈ 50µm within a BEC. Dr. Hau
and her research team were eventually able to stop,
store (for several milliseconds), and retrieve a light
pulse in a BEC [8].
Some of the theory describing these phenomena, and
the experimental techniques that make this research
possible, are discussed in this paper. Although the
theory will not be built from the ground up, it is
expected that the reader will have a good qualitative
understanding of these topics after reading this paper.
For those interested in further study and deeper
understanding the references used for this paper are
encouraged reading. Some observations and possible
future experiments are mentioned at the end of this
paper.
II. INDISTINGUISHABLE
PARTICLES
The biggest difference between the particles that we
observe in our everyday life, such as a baseball, which
can be described physically by classical Newtonian
mechanics, and the particles that we observe when
dealing at the atomic or sub-atomic level and are
described physically by quantum mechanics, is that the
objects we are used to dealing with in our everyday
lives are distinguishable while Bosons and Fermions
are said to be indistinguishable. For example, if we
are observing two particles which are capable of
staying on the “path” drawn in figure 1, and subject to
the assumption that the particles can be anywhere on
this path and that they can transition between the loops
as if this were a track layout for toy trains and these
loops were connected by switches. We will also
assume they cannot crash into one another, and exhibit
no repulsive or attractive force on one another, and are
therefore non-interacting. However, the distance
between them is allowed to change at any rate and at
any time, so that we can’t keep track of their separation
distance as a way to keep track of the particles. There
is no distinction between them besides color and they
are not “static”. Now, given these assumptions, when
we observe the system shown in figure 1a below, we
can easily distinguish between the two particles at any
given time. Even if we were to look away, when we
focused our attention back on the system we would
immediately be able to observe where the blue particle
was and its associated momentum, and where the
orange particle was,and its momentum. In figure 1b,
when the particles begin to move, a human observer
may be able to keep trackof the two particles separately
for an amount of time. But when you look away from
the system and then observe it at a later time, you will
no longer be able to know which blue particle is which
blue particle, and you canno longer specify the location
or momentum of one of the particles as that of a
specific particle, as if you had numbered them
beforehand. All you can say is that you have one
particle in one location with a specified momentum,
and another in a different location with a different
momentum. And they are said to be indistinguishable.
Figure 1a
Figure 1b
One very important similarity of figure 1a and figure
1b is that in both systems either particle can have any
momentum at any time that we wish to measure its
momentum. And also that by recording this
measurement we do not alter the system in any way.
This can be considered a system which is in the
“classical limit” and the available energy for each
particle is a continuous spectrum of values. When we
talk about Bosons and Fermions we are talking about
particles that do not have a continuous energy
spectrum. Bosons, such as Photons, and Fermions,
such as electrons, can only have particular energy
values. They are also indistinguishable, such as the
particles in figure 1b. Assigning discrete values to the
available energies of a particle is referred to as energy
quantization and quantum mechanics is based on the

fact that the energies of the particles which it governs
are quantized. A popular example of something in the
tangible world which is quantized is American
currency. A penny is the smallest unit of American
currency that a person can have, and can be considered
a “quanta”, all other amounts of American currencycan
be thought of in terms of pennies. In that way a penny
can be likened to the ground state energy of a Fermion
or Boson, which is its lowest available quantum energy
state and the solution to the Schrödinger equation given
in equation (1) with solutions of the form given in
equation (2) [18].
(1) [18]
With solutions of the form
(2) [18]
Two non-interacting indistinguishable particles can
be described by a “non-committal” wavefunctoin of
the form
(3) [15] plus sign for Bosons, minus sign for
Fermions
BECs are macroscopic, but obey quantum
mechanics. Not all macroscopic particles obey
classical Newtonian mechanics, but all particles tend to
this “classical” domain when their energy distribution
becomescontinuous. Ascan be seenfrom the emission
spectroscopy of a hydrogen atom, famously called the
Balmer series, shown in figure 2. The energy
separation is much less between higher excited states
than it is for lower excited states,as you can clearly see
when comparing the energy difference of the 1st
and 2nd
excited states to that of the 5th
and 6th
excited states.
Eventually, it can be assumed that the energy
separation is negligible and that it is a continuous
distribution. When a system has a continuous spectrum
of available energy states the effects of quantum
mechanics on the system are negligible.
Figure 2 [9]
The de Broglie wavelength can be another limiting
factor as to when a system needs to be treated quantum
mechanically. The de Broglie wavelength is a “matter
wave” that is related to an objects momentum by the
following equation
(4)
Here h is plank’s constant and p is the momentum of
the particle. When the de Broglie wavelength is of the
same order of magnitude as the particle separation we
need to use Bose-Einstein statistics, or Fermi-Dirac
statistics to describe the energy distribution of Bosons
and Fermions respectively. When this wavelength is
small compared to the spacing between particles
Maxwell-Boltzmann statistics are usedto describe their
energy distribution. Maxwell-Boltzmann statistics are
used to describe “classical” particles, while the Bose-
Einstein and Fermi-Dirac statistics are used to describe
particles which obey quantum mechanics.
At high Temperatures (high energy) both Bose-
Einstein statistics and Fermi-Dirac statistics converge
to the classical limit of Maxwell-Boltzmann as would
be expected based on the comments earlier regarding a
continuous energy distribution at higher energies. The
interesting case occurs when the energy is lowered and
the system approaches temperatures very near absolute
zero (zero degrees kelvin, 0K). Plots of the three
statistical distributions are shown below in figure 3
(a,b,c). Figure 3a shows a temperature near 0K, figure
3b shows a temperature that is near room temperature,
and figure 3c shows a temperature of 9000K.

Figure 3a [10]
Figure 3b [10]
Figure 3c [10]
The energy distribution function is given in the
equation below. They are all equal to n(ϵ), the
probability of finding a particle in a single particle state
with an energy ϵ. The other two variables are the
temperature T, and the chemical potential µ, and kB is
the Boltzmann constant. It should be mentioned that
these statistical distributions describe non-interacting
systems in thermal equilibrium. [15]
- Maxwell-Boltzmann (5)
- Fermi-Dirac (6)
- Bose-Einstein (7)
A simplified plot of all three distributions is shown
below in Figure 4
Figure 4 [11]
Some very interesting conclusions can be drawn
from the plots shown in figure 3 and 4. First of all it
can be seen that at high temperatures the F-D and B-E
distributions converge to the Maxwell-Boltzmann
distribution, which is in agreement with what was
previously discussed. It should be noted that even at a
temperature of over 9000K as in figure 3c, these
particles can exhibit different energy distributions, so
you should not think that at such high temperatures
Bosons and Fermions do not exist and simply converge
to Maxwell-Boltzmann statistics and macroscopic
phenomena is all that is observable based on the
discussion of a continuous energy spectrum at high
energies. Typically, it is the high energy states of a

system of particles that converge to a continuous
energy distribution and determine which statistical
distribution is best to describe the energy distribution
of the particles in the system.
Fermi-Dirac statistics show very interesting behavior
as the temperature approaches 0K. The point where the
chemical potential µ is equal to the energy ϵ is known
as the Fermi energy EF. This Fermi-Energy is given by
the energy difference between the highest and lowest
occupied states in a system of Fermions at a
temperature of 0K. As the temperature is increased
from absolute zero Fermi-Dirac statistics yields
another interesting result. No matter what the
temperature there will always be a 50% chance of
finding a Fermion at the Fermi-Energy. When the
temperature is increased the change in the distribution
around this Fermi-energy is related to Fermions being
excited into higher energy levels, or the conduction
band. Below the Fermi-energy the change in the
distribution as the temperature is increased from 0K is
due to the decreased probability of finding a Fermion
in a lower energy state, or a hole in the valence band.
This is shown in Figure 5
Figure 5 [12]. T = 0K is shown in gold, T > 0K is shown in blue
Bose-Einstein statistics forces us to make sure that
the minimum allowed energy ϵ is always greater than
the chemical potential µ. This statistical distribution
“blows up” when the x value from figure 4 is equal to
zero. This happens in equation (7) when T is equal to
0K. Since this is a statistical distribution it must be
normalized. To normalize a statistical distribution is
just to say that the integral over all of space of the
magnitude of the distribution is equal to one. This is
to say that the sum of all the probabilities of all the
particles must be equal to 1, or that you have a 100%
chance of finding the particles of the system
somewhere within the distribution. When normalized
this distribution (B-E) will yield a 0% chance of finding
a particle in any state other than the lowest available
ground state when T= OKin equation (7). At this point
the chemical potential must be equal to zero. This
allows for Bosons to accumulate without limit into the
ground state and thus a BEC is formed. For a very
thorough mathematical discussion about B-E statistics
and how it relates to BECs see reference [14].
The reason that Fermions do not behave the same as
Bosons is because they are subject to the Pauli
exclusion principal, which states that no two Fermions
can occupy the exact same quantum state. Bosons do
not obey this restriction and it is a reason that BECs are
allowed to form because they are associated with a
macroscopic population of the lowest energy state in a
dilute non-interacting gaseous system of Bosons which
is cooled to 0K, and because they are subject to B-E
statistics this phenomena is allowed. Experimental
techniques are the only limiting factor besides particle
interactions, which can occur but are assumed not to in
this paper, they can be treated as perturbations and are
dealt with using techniques such as the mean-field
approximation [3]. Modern cooling techniques give the
ability to create andobserve BECsin the laboratory and
have seen great advances since BECs were first
observed 1995.
Because of the Pauli exclusion principal Fermions
will occupy all states up to the Fermi-energy at 0K.
This configuration of a system of Fermions is called a
“Fermi-Sea” and is pictured in figure 5
Figure 6 [6]
Figure 5 shows what a Fermi-Sea looks like in
contrast to a BEC with regards to distribution amongst
quantum energy states.
There is one more major difference betweenBosons
and Fermions that bears mentioning before continuing,
and one that most certainly separates them from the
particles observed in everyday life. This distinguishing
property is something called spin. Spin is an intrinsic
property of Fermions and Bosons which has no exact
macroscopic world counterpart and is responsible for
much of the interesting behaviors observed when

studying Fermions and Bosons or Quantum Physics in
general. However,it is in some way related to the idea
of angular momentum. In fact, algebraically, it is an
almost exact replica of angular momentum [15]. As an
example we can consider an electron which is rotating
around some nucleus or some other positively charged
point particle. This electron obviously has an angular
momentum related to its orbital motion, but it also
possesses another source of angular momentum which
is a completely quantum mechanical property and in no
way related to its spatial and time coordinates [15].
Bosons have an integer spin, and fermions have a half-
integer spin. Protons, electrons, and neutrons are all
Fermions and have spin 1/2. These particles are what
make up atoms and atoms that have an odd number of
these particles behave as Fermions, with a half-integer
spin. Atoms with an even number of these particles
have an integer spin, and behave as Bosons [6]. For a
more detailed explanation of spin see chapter 4.4 from
reference [15].
III. PROPERTIES OF BEC
BECs are formed when a very dilute gas of non-
interacting Bosons is cooled to temperatures near 0K.
For example, the first BEC of Na atoms was observed
at temperatures near 2µK [2]. When the particles are
cooled to these temperatures their de Broglie
wavelength becomes comparable to the interatomic
spacing between the particles. Figure 7 shows this
progression, and when we reach a particular
temperature the de Broglie wavelengths begin to
overlap and we can describe the particles with a single
wavefunction [3].
Figure 7 [3]
In Figure 7 the thermal de Broglie wavelength is
given by
(8) [3]
This shows a dependence on the inverse square-root
of the temperature T (in equation 5 it should be
mentioned that d is the interatomic spacing, m is the
particle mass, kB is the Boltzmann constant, and ћ is
Planck’s constant (h/2π)). We would also expect that
as we lower temperature the Bosons begin to populate
the ground state or states that are very near the ground
state, as in they are getting closer and closer together
because of the cooling process. Because their
interatomic spacing decreases with decreasing
temperature and their thermal de Broglie wavelength is
increasing, it is easy to see why we can approximate
the behavior of these Bosons as one “matter wave” due
to superposition. The property of a BEC to act as a
single matter wave means the atoms can be considered
to be coherent. Lasers are coherent light in which its
constituent particles, called photons, oscillate at the
same frequency, and thus because BECs can be
considered coherent atoms they are useful in studying
properties related to atomic lasers. Atomic lasers are
very important for technology such as quantum
computing and the study of BECs may lead to exciting
advances in this field [17].
Typically when a system of particles is cooled from
a gaseous phase an observer would expect to see a
phase transition to liquid and/or gas instead of the
phase transition from a gas to a BEC. This is the exact
reason that it is necessary to use a very dilute gas when
attempting to form BECs. The typical forms of
condensation we are used to seeing of gas to liquid and
liquid to solid are caused by three-body collisions and
the rate of these collisions is proportional to the inverse
of the density squared. The rate of two-particle elastic
collisions is only proportional to the inverse density
and therefore these types of collisions dominate the
system when the density is sufficiently low and these
two particle elastic collisions allow the gas to reach an
equilibrium [16]. The typical densities used in BEC
formation is 100,000 times less dense than air. These
extremely low densities force the temperature
requirement to form BECs down into the nK range,
typically around 500nk - 2µK [16].

To find an expression for this temperature
requirement, which is called the critical temperature Tc,
it first must be assumed that all of the particles in the
ultra-cold Bosonic gas can just barely be accounted for
in excited states,as in they are beginning to occupy the
ground state but just before they can be considered a
group of coherent atoms described by one
wavefunction [17], and at this point any decrease in the
temperature will result in further macroscopic
population of the ground state resulting in a phase
transition to a BEC[3]. The chemical potential at this
point is zero based on the earlier discussion of Bose-
Einstein statistics and the number of Bosons in the
excited states is given by the following integral
(9) [3] g(ϵ) is the density of states.
The density of states depends on the potential in
which the particles are confined. The solution to this
integral gives a critical temperature which is
proportional to N1/3
[3]. BECs are observed by turning
off the potential in which the BEC is confined, and
allowing them to expand for a period of 10-20
milliseconds [6]. Using optical imaging techniques to
observe the velocity distribution the presence of a BEC
can be confirmed due to a large spike in the center at
temperatures at and below the critical temperature,
indicating many particles nearzero velocity [6]. Figure
8 shows a velocity distribution of a BEC formed using
84
Sr. The image on the left is taken below the critical
temperature, the middle image is at or very near the
critical temperature, and the large spike shown on the
right image is below the critical temperature, and a
BEC is clearly observable.
Figure 8 [7]
Some BECs have been shown to also been shown to
behave asa superfluid which is defined asflow without
dissipation [16]. Superfluidity was first observed in
liquid Helium in 1938 by Pyotr Kapitsa and John F.
Allen. Aproperty of superfluids is a phenomena called
a quantum vortex which was first observed in a BEC of
Rubidium in 2000 [19], and thus BECs were
experimentally proven to exhibit superfluid properties.
Quantum vortices act as quantized angular momentum
carriers within a superfluid [20]. An incredible
behavior of superfluids is their ability to “self-level”
and self-propagate. Shown in figure 9, a super fluid
will eventually creep up the walls of the separated
containers until it is at an equilibrium level. If this
container were not covered, however, the superfluid
would creep up the walls and escape [19].
Figure 9 [19] Superfluid Helium in a container
IV. LASER COOLING
Doppler cooling was first proposed theoretically in
1975 and first implemented experimentally in 1978
[21]. Itinvolved using laserswhich are “detuned”. The
energy of a particle of light, which is called a photon,
is given as E = hv where h is Planck’s constant and v is
the frequency of the light. Lasers emit coherent light,
meaning that all the photons emitted have the same
frequency. Since the energy associated with the lasers
being used is known they can be set to an energy which
is just below the energy level difference of a desired
energy level transition of the atoms in the sample being
cooled. The Doppler Effect is shown in the figure 10
below. It can be summarized as follows: When a
source emits a wave (such as a Laser, which emits
electromagnetic radiation at a set frequency) and
observer moving towards the wave source will observe
a shorter wavelength than the actual wave source is
emitting, and conversely an observer moving away
from the wave source will observe a longer wavelength
[22].

Figure 10 [22] – Doppler Effect
Atoms can absorb photons and enter an excited state
if the energy of the photon is equal to that of an energy
level difference between an allowed energy state and
their current energy state. This can be related to the
photoelectric effect in the sense that it takes light of a
certain frequency to eject an electron from a metal in
the photoelectric effect,and only certain frequencies of
light can cause energy level transitions with atoms,
which is actually the reason for the distinct color of the
lines of the Balmer series shown in figure 2). A key
concept is that temperature is related to the average
kinetic energy of the Bosons in the non-interacting
dilute gas we wish to cool. If there is a way to reduce
the kinetic energy of the system, the temperature can
be reduced.
When an atom in the gas being cooled has enough
momentum in the direction of the “detuned” laser it
will absorb a photon and lose momentum in the
direction that it was moving (towards the laser). Due
to the atom wanting to “relax” to the lowest possible
energy state spontaneous emission will occur (or
sometimes, possibly, stimulated emission) When an
atom undergoes this spontaneous emission it will emit
a photon with the same energy as that of the photon it
originally observed except the momentum gained from
the emission of that photon will be in a random
direction, and whenthis processis repeatedmany times
there will be a net loss in kinetic energy of the system
being “cooled” but remember this is not exactly like
putting something in a freezer or submerging it in
liquid nitrogen, the main objective of Doppler Cooling
is to use make use of the Doppler Effect to cause
absorption of a photon, and then to take advantage of
spontaneous emission to reduce the overall kinetic
energy of a system which results in a loss in
temperature. Due to the spontaneous emission there is
a limit to the cooling potential of Doppler Cooling due
to the fact that there is always some momentum
involved. This temperature limit is expressed in
equation (10). γ is the inverse lifetime of the excited
state after absorption. The process of Doppler Cooling
is shown in figure 11.
(10) [21]
Figure 11 [21] – Doppler Cooling
V. EVAPORATIVE COOLING
Using Laser cooling it is possible to reach
temperatures in the µK range, however this is still not
sufficient to observe BECs [16]. Evaporative cooling
involves trapping the atoms in what can be considered
a potential well [23]. Typically the Bosonic gas is
confined to a spherical quadrupole potential within a
magnetic field. The trapped atoms must have a
magnetic moment which points in the opposite
direction of the magnet field [24], otherwise the
potential energy of a magnetic moment in a magnetic
field will be non-zero. The potential energy of a
magnetic moment in a magnetic field is given by
equation (11).
U = -µ·B
(11) [25]
In areas where the field is non-uniform there will be
a torque exhibited on the moment if it is not
perpendicular to the magnetic field in which it is
confined [25]. In the spherical quadrupole trapping
potential the field is zero at the exact center, and the
field around this point is changing rapidly. When
atoms passthrough this zero point of the magnetic field
they can get “confused” because there is no field to
guide them [2]. The atoms passing through this point

can experience a flip of their magnetic moment called
a “Majorana flip” [2]. When this happens the atoms
experience a torque and can be accelerated out of the
trapping potential. To plug this “hole” in the trapping
potential, a laser is focused on the point in the
quadrupole potential where the field is zero and this
laser produces a repulsive optical dipole force on the
atoms in the zero field region of the confining potential,
plugging the hole [24]. This trapping potential and the
optical plug are shown in figures 12 and 13.
Figure 12 [2] – trapping spherical quadrupole
potential with optical plug.
Figure 13 [24] – Trapping potential with rf induced
evaporative cooling and optical plug.
Once the ultra-cold gas has been confined to the
trapping potential the atomswith highest kinetic energy
will escape, but further reduction in kinetic energy is
still necessary. To do this a process called rf-induced
evaporative cooling is used. By varying the height of
the trapping potential slowly enough that thermal
equilibrium is maintained using radio frequency
radiation, the atoms with highest kinetic energy can be
“shaved off” [23]. With this combination of cooling
techniques BECs are able to be created in the lab
setting. Figure 14 shows the rf-induced evaporative
cooling process.
Figure 14 rf-induced evaporative cooling slowly varies the
trapping potential while maintaining thermal equilibrium,
allowing the atoms with higher KE energies to escape and leading
to a net loss in KE of the system and in combination with laser
cooling BECs are able to be observed [23]
VI. FERMIONIC CONDENSATES
Fermions are subject to the Pauli exclusion principal
and are not allowed to occupy the same quantum state.
At 0K. Figure 5 and 6 show their distribution amongst
available energy states up to the fermi-energy EF at 0K.
This distribution of Fermions at 0K is called a “Fermi
Sea” or an ultra-cold degenerate Fermi gas. Figure 15
below shows the comparison of a Fermi Sea and a
BEC.
Figure 15 [16]. BEC on the left vs Fermi Sea on the
right for decreasing T

Due to the Pauli Exclusion principal the Fermions
can only shrink down so far before all of their
available energy states are occupied. Ultra-cold gases
of Bosons can have properties of superfluids.
Superfluidity of ultra-cold Fermionic gases results in
superconductivity and this is a phenomena which
Fermi Condensates can help us understand if more is
learned about them.
It has been well established that composite Bosons
of two Fermion atoms can form a BEC because they
will have integer spin and behave as Bosons, no
longer obeying the Pauli exclusion principal. But
recently it has become possible to tune the interaction
energy between Fermions to favor attractive pairing
interactions [6].
Cooper pairs of electrons can behave as Bosons and
form a true Fermi condensate. Cooper pairs of
electrons form when a Fermi gas is in a
superconducting state,which occurs at a certain low
temperature threshold. Once a Fermi gas is in this
state,Cooper pairs can form. A Cooper pair of
electrons is formed due to the negative charge of the
electron somewhat distorting the lattice in which it is
confined. The lattice is constructed of positively
charged particles. This distortion of the lattice brings
other electrons in the gas closer despite their repulsive
Coulomb interaction [26]. Recently it was discovered
that interaction energies for particles in an ultra-cold
gas vary drastically for small changes in the confining
magnetic potential. The small changes in the
confining magnetic field are related to a Feshbach
frequency [6]. At this resonance frequency the Fermi
gas can form Cooper pairs with much greater ease due
to the reduced interaction energies. When these
Cooper pairs form they immediately transition from a
superconducting degenerate Fermi gas,to a superfluid
BEC [6].
The ability to tune interaction strength has led to
further study of what is called the BCS-BEC
crossover theory. This is the theory which attempts to
describe the transition of these degenerate Fermi gases
from a superconducting state to a superfluid BEC [6].
It is believed that with further study the point at which
cooper pairs behave as Bosons and at which Cooper
pairs behave as Fermions can be studied. Figure 16
and 17 show this BCS-BEC crossover region.
Figure 16 [6]. The Feshbach frequency is where ΔB is
equal to zero. Above this frequency attractive interactions
are favored, and below it, repulsive interactions are
favored. a is the s-wave scattering length.
Figure 17 [6] the BCS crossover region between superfluid
superconducting Cooper pairs and superfluidity of bound
molecules. Cooper pairs of electrons can be in a superconducting
state or a superfluid BEC state.
One last interesting comment regarding the study of
these degenerate Fermi gases. Researchers at the
University of Amsterdam were able to use optical
techniques to manipulate the different m quantum
states of an ultra-cold gas of 87
Sr and the separation of
these statesis shown in Figure 18. Figure 19 shows the
optical density of Bosonic 84
Sr and the Fermion 87
Sr.
You can notice the much smaller size of 87
Sr. This is
simply due to the fact that less particles are at the
lowest energy state due to the Pauli exclusion principal.
Figure 18 [7]. The ten mF states of degenerate 87
Sr

Figure 19 [7]. Optical density of a BEC of 84
Sr and
a Fermi Sea of 87
Sr
VII. CONCLUSION
The topic of Bose-Einstein Condensation was a very
exciting one to research. The insights they offer into
quantum mechanical phenomena is ground breaking.
Since 1995 many advanceshave been made in this field
of ultra-cold condensed matter physics.
I personally hope to be able to observe a phase
transition of an ultra-cold gas of Bosons to a BEC and
to be able to observe some of the phenomena
mentioned in this paper. Specifically superfluidity and
slowed light.
I would like to attempt to observe if it is possible to
trap Bosons in an excited state. Bose-Einstein statistics
does not saythis is impossible, howeverthere is always
a finite probability that in a system with energy levels
available above the ground state you can find a Boson
at a lower energy level. My original idea was to tune
the cooling lasers to a higher energy level transition for
the Doppler Cooling than normal and then trap the
atoms in a potential which allows the Bosons to access
a higher energy state. I originally thought I could add
a potential at the bottom of this magnetic trap to prevent
Bosons from wanting to fall to the ground state,but that
would only alter the ground state energy based on my
knowledge of perturbation theory, and this would only
allow me to study a BEC which formed in a different
ground state compared to a BEC formed in the same
potential without the perturbing potential at the bottom.
Instead, if I wanted to attempt to trap these Bosons in
an excited state I would need to supply enough energy
that the system as a whole has a much higher
probability of having a particle in this desired excited
state. This could possibly be done by trapping the
Bosons in the 3rd
excited state and flooding the ultra-
cold gas in the trapping potential with detuned lasers
tuned to the 2-3 and 1-2 energy level transitions (not
below as in the Doppler Cooling technique). This
would give me a better chance of exciting the Bosons
who dropped out of the excited state that I wished to
observe. Another possible obstacle in observing a BEC
forming at an excited state is that there is no guarantee
the excited state will have a lifetime long enough for
the Bosons to accumulate or to even record a
measurement. I did come across a couple papers that
referenced excited states of BECs but they were
unavailable to me and I did not have an opportunity to
read them
Another very interesting thing that I observed and
one of my favorite figures in this paper is figure 18. I
would expect the separated states to be less “blurry”
and more clearly defined. Besides an error in the
imaging technique, I would assume that this is
representative of the Heisenberg uncertainty principal.
I think this is the case because the image is an optical
density and could possibly be related to a measurement
of position. And thus an uncertainty in the momentum
will arise. This uncertainty in the momentum is small
yet it is still something and could be a reason why the
states are blurry since the energy cannot be exactly
defined due to the uncertainty in momentum.
[1]“Bose-Einstein statistics”:
https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_st
atistics
[2] Bose-Einstein Condensation in a Gas of Sodium
Atoms
K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van
Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle
Department of Physics and Research Laboratory of
Electronics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
[3]“Chapter 2–Bose-Einstein condensation”:
http://massey.dur.ac.uk/resources/mlharris/Chapter2.pdf
[4]“Bose-Einstein condensate” :
https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_c
ondensate#Current_research
[5]“Fermionic condensate” :
https://en.wikipedia.org/wiki/Fermionic_condensate
[6] Fermi Condensates
Markus Greiner, Cindy A. Regal, and Deborah S. Jin
JILA, National Institute of Standards and Technology
and University of Colorado,
and Department of Physics, University of Colorado,
Boulder, CO 80309-0440

[7] “The Strontium Project” :
http://www.strontiumbec.com/
[8] “The art of taming light: ultra-slow and stopped
light” : Zachary Dutton, Naomi S Ginsberg, Christopher
Slower, and Lene Hau Lyman Laboratory, Harvard
University, Cambridge MA 02138
[9] Google images – Balmer Series
[10] "Bose-Einstein, Fermi-Dirac, and Maxwell-
Boltzmann Statistics" from the Wolfram Demonstrations
Project
http://demonstrations.wolfram.com/BoseEinsteinFermiDira
cAndMaxwellBoltzmannStatistics/
[11]
https://commons.wikimedia.org/wiki/File:Mplwp_Fermi_B
oltzmann_Bose.svg
[12] Chapter 5 lecture slides for PHY 402. Dr. Hao Zeng
SUNY BUFFALO spring 2016. UBLearns.buffalo.edu
[13] Bose-Einstein Distribution Function :
physics.unl.edu
[14]
https://www.ucl.ac.uk/phys/amopp/people/thorste
n_kohler/talk_BEC.pdf
[15] Introduction to Quantum Mechanics, 2nd Edition by
Griffiths, David J., Pearson Education 2005
[16] Bose–Einstein condensation of atomic gases James
R. Anglin & Wolfgang Ketterle Research Laboratory for
Electronics, MIT-Harvard Center for Ultracold Atoms, and
Department of Physics, Massachusetts Institute of
Technology,Cambridge, Massachusetts 02139, USA
[17] “Dr. Michio on Bose-Einstein condensates” :
https://www.youtube.com/watch?v=9RDocoSWqPY
[18] “Review of Formal Quantum Mechanics” for PHY
402. Dr. Hao Zeng SUNY BUFFALO spring 2016.
UBLearns.buffalo.edu
[19] “Superfluidity” :
https://en.wikipedia.org/wiki/Superfluidity
[20] “QuantumVortex” :
https://en.wikipedia.org/wiki/Quantum_vortex
[21] “Doppler Cooling” :
https://en.wikipedia.org/wiki/Doppler_cooling
[22]”the Doppler effect”
http://physics.ucr.edu/~wudka/Physics7/Notes_www/node1
06.html
[23] “Cooling and Trapping Techniques With Ultra-cold
Atoms” :
http://large.stanford.edu/courses/2009/ph376/amet1/
[24] “plugging the hole” :
http://cua.mit.edu/ketterle_group/Projects_1995/Plugged_tr
ap/Plugged_trap.htm
[25]“Magnetic moment” :
https://en.wikipedia.org/wiki/Magnetic_moment
[26] Cooper pairs :
https://en.wikipedia.org/wiki/Cooper_pair

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