Download Compensation coils for laser cooling of atoms

Compensation coils for laser
cooling of atoms
Designing, building and characterizing compensation coils
for optimizing the Optical Molasses stage
in laser cooling of atoms.
Submitted by: Asaf Szulc
Advisor: Prof. Ron Folman
Department of Physics
Faculty of Natural Sciences
Ben-Gurion University of the Negev
October 22, 2012
2|page
Abstract:
The project goal is to reduce the temperature of atoms at the end of the
molasses stage of laser cooling, also known as Sisyphus or polarization
gradient cooling. The reduced temperature will enable better loading of the
atomic cloud into the magnetic trap for further cooling and subsequent
experiments and perhaps produce a bigger BEC in a shorter experimental
cycle (evaporation cooling) time. The project includes designing and building
three pairs of mutually perpendicular compensation coils, in order to
compensate the Earth’s and other DC magnetic fields. As part of the project I
simulated and characterized the coils and eventually tested the complete
configuration adjacent to the existing experiment. I still need to build lownoise current drivers and measure the magnetic field experienced by the atoms
in order to reduce residual fields using spectroscopic measurements.
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Acknowledgements:
There are some people I'm eager to thank, who their help and contribution is
definitely worth mentioning. The first person to thank is of course my advisor,
Prof. Ron Folman who introduced me to his lab and to the fascinating field of
Atomic Physics. Second, but not less important I would like to thank Shimi
Machluf who guided me through my entire work in the lab. His assistance,
guidance and advice provided major contributions to my project and to my
theoretical and practical knowledge. Thank you Mark very much for
proofreading my paper, your help is much appreciated. Finally, I would like to
thank Yair, Anat, Omer, Noam and Zina who each contributed in one way or
another to the success of my project.
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Contents:
1. Background and Theory …………………………………………………..
9
1.1. The existing experimental setup …………………………………......
9
1.2. Properties of Alkali-Metal Atoms ……………………………………
10
1.3. Successive stage of laser cooling ……………………………………… 13
1.4. How an external magnetic field affects the Optical Molasses Stage
15
1.5. Measuring the magnetic field using the atoms ...…………………… 20
2. Computer simulation ……………………………………………………….
23
2.1. Background …....……………………………………........................... 23
2.2. Magnetic field simulation ……………………………………………..
23
2.3. Inductance simulation …………..……………………………………..
27
3. The experimental test setup ……………………………………………….
31
3.1. The experimental test setup apparatus ……………………………...
31
3.2. Experimental test results …..………………………………………….
31
3.3. Experimental setup Conclusions …………..…………..……………..
34
4. Summary and conclusions ………………………………………………….
35
References ………………………………………………………………………..
37
Appendix A – Magnetic field computer simulations …...…………………...
39
Appendix B – Choice of coil size …………………………………..………….
43
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1. Background and Theory:
1.1.
The existing experimental apparatus:
1.1.1. Main goal:
The existing experiment's goal is to explore different quantum phenomena
using Bose-Einstein Condensates (BEC) in which most of the atoms occupy
the lowest quantum state of the external potential. To do so, several stages
are used in order to cool the atoms to temperatures close to absolute zero.
After achieving a BEC the cloud is then manipulated in various ways in order
to experimentally examine quantum phenomena. The various cooling stages
for preparing the BEC entails the loss of many atoms so in desiring better
signals one wants to produce a BEC with as little atom loss as possible. My
project aims to achieve this by improving one particular step in the laser
cooling sequence.
1.1.2. Procedure:
The first stage of the cooling process is the Magneto-Optical Trap (MOT) [1]
which is the only stage that can trap room temperature atoms. Optical
Molasses (OM) [1] is used in the second stage to further cool the atoms. Both
these stages are important for my addition to the experiment and are
explained in detail (sec. 1.3). The next stage following the OM stage is meant
to facilitate transferring the atoms to a purely magnetic trap ( i.e., without
lasers). This is the Optical Pumping (OP) [1] step and it is followed by the
magnetic trapping, which in this experiment is a Z-trap [2]. After the
magnetic trapping process is complete, further cooling is needed and is
accomplished by RF evaporation cooling [3] until a BEC is achieved.
1.1.3. The compensation coils:
The compensation coils are my addition to the experiment and consist of three
pairs of coils generating a magnetic field designed to compensate all kinds of
external DC magnetic fields including Earth's magnetic field.
The main goal of the compensation coils is to improve the cooling process, and
to produce more atoms at lower temperatures at the end of the OM stage.
Colder atoms at the end of the OM stage can significantly improve the
efficiency of transferring them to the magnetic trap and the initial stages of
evaporative cooling, thereby eventually achieving a bigger BEC.
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Even though the coils can compensate only homogenous magnetic fields, I will
show that a large portion of the inhomogeneous fields near the experiment can
also be compensated since the region of the atomic cloud is sufficiently small.
1.2.
Properties of Alkali-Metal Atoms:
1.2.1. General:
Alkali-metal atoms were the first atoms to be laser-cooled and trapped. The
alkali metals are chemical elements which are located in the left column of the
periodic table.
The ground state of all alkali-metal atoms has a closed shell with one valence
electron. The closed shell core makes no contribution to the electronic orbital
angular momentum which is therefore determined only by the valence
electron.
1.2.2. Energy splitting:
Let us define the orbital angular momentum |
the valence electron's orbital angular momentum,
spin angular momentum and
|
|
|, where
is
is the valence electron's
is the total angular momentum of the electron.
Now, as mentioned earlier, the valence electron is the only contribution to the
atom's angular momentum, hence one can define the total electronic orbital
angular momentum as ⃗
momentum as
, the total spin as
and the total angular
for all the electrons in the atom. The different values of
correspond to different atomic energy levels. These spin-orbit interactions
⃗
|
are also called the "fine structure" of the atom.
|
|
|
(1.1)
We shall now define the total atomic angular momentum
between the nuclear spin
as the coupling
and the total angular momentum of the electron .
The total atomic angular momentum is defined as
and its magnitude
is therefore:
|
|
|
|
(1.2)
10 | p a g e
Figure 1.1 – Energy level
diagram for 87Rb. The
orbital levels (red) divided
into
levels
the
levels
the fine structure
(green) divided into
hyperfine structure
(purple) [4].
Different values of F caused by the spin-spin interactions
are called the
"Hyperfine structure" of the atom.
1.2.3. Zeeman splitting:
Each of the hyperfine energy levels contains
magnetic sublevels. These
sublevels are degenerate if no external magnetic field is present. In the
presence of an external magnetic field one can write down the Hamiltonian for
these magnetic sublevels as:
⃗
Where
⃗
(1.3)
are respectively the electron spin, electron orbital and
nuclear "g-factors". These g-factors account for various modifications to the
corresponding magnetic dipole moments.
p a g e | 11
Figure 1.2 –
field [4].
87
Rb ground state Zeeman splitting in an external magnetic
Since my work deals with small magnetic fields (about 0.5 Gauss) I will use
the linear Zeeman Effect and I will use several additional approximations
based on the following assumptions.
The first assumption is that the weak magnetic field results in energy shifts
that are small compared to the fine-structure splitting. Then one can make
⃗
the transformation
.
The second assumption is that the energy shift is also small compared with
the hyperfine-structure splitting and one can similarly make the
transformation
.
Finally one can take the approximate values
comparison with
and neglect
in
, so one gets the Hamiltonian:
⃗
(1.4)
The energy shift is therefore:
|
where
(1.5)
⟩
is called the
́
which is defined
approximately as [4]:
12 | p a g e
(1.6)
1.3.
Successive stage of laser cooling:
1.3.1. Light and matter interactions:
When atoms are present in a light field, assuming the light frequency is
resonant with the atom's transition frequency, the photon will excite the
atom. Assuming low light intensity to prevent stimulated emission, the atom
will soon decay back to its ground state and emit a photon. This will result in
two momentum boosts to the atom, first by absorbing the photon's
momentum and then by re-emission. The former is in the laser direction and
the latter averages to zero since spontaneous emission is isotropic.
Let us assume that the atom behaves as a two-level system coupled by an
electromagnetic field. In this case one can write the density matrix:
(
)
(
)
(1.7)
By applying the evolution equation on the density matrix and including
spontaneous emission with a decay rate
, one can write the Optical Bloch
Equations (OBE) [1]:
̃
̃
̃
̃
where ̃
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(
)̃
(
)̃
and
̃
̃
is the Rabi frequency.
(1.8)
Solving the OBE for a two level system, in terms of the dimensionless on| |
resonance saturation parameter
, one can obtain:
(1.9)
⁄
The total scattering rate is then given by
and finally one can obtain
the force [1]:
⃗
⃗
⁄
(1.10)
1.3.2. Optical molasses (OM):
Optical molasses is a method that further decreases the atom’s velocity,
affecting the fastest-moving atoms the most. The effect acting on the atoms
resembles moving through syrup and hence the name "optical molasses".
This method exploits the same experimental system described in sec. 1.3.1, by
using three pairs of red-detuned counter-propagating laser beams intersecting
at the trap location from orthogonal directions. The red-detuned laser beam
interacts with atoms moving toward the laser due to the Doppler effect which
decrease the atoms' speed away from the trap.
Again using a simplified one-dimensional model, one can calculate the force on
the moving atoms by summing the force generated by the two counterpropagating laser beams and assume that the atoms' velocity is smaller than
so we can expand the expression and get [1]:
⁄
Where terms of order
⁄
(1.11)
and higher have been neglected.
14 | p a g e
Figure 1.4 – The force each laser inflicts on the atom according to the Doppler
effect (red) and summed up for the two counter-propagating lasers(blue), it is
seen that for slow velocity the force is approximately linear (black) and the
decay of the force at high velocity gives a finite "capture velocity". [5].
1.4. How an external magnetic field affects the Optical Molasses
Stage:
1.4.1. Cooling below the Doppler limit:
At first glance it appears that OM cooling is limited by a finite temperature
called the "Doppler limit" in which the momentum boost gained by photon
absorption is equal to the average momentum boost gained by emitting a
photon. However, it turns out in experiments that it is possible to cool below
the Doppler limit using OM and it was later explained theoretically as I show
below.
In order to understand the OM process completely, one should consider three
issues:
1. Non-adiabatic atomic internal state changes.
2. The Zeeman sub-levels.
3. Differing transition probabilities. (the Clebsch-Gordan coefficients)
The result explains how sub-Doppler cooling occurs, separately from the
Doppler Effect.
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Figure1.5– An example of a two-level system with three Zeeman sub-levels in
the ground state and five in the exited state. Clebsch-Gordan coefficients for
all possible transitions are written next to the connecting lines [6].
1.4.2.
Polarization gradient cooling:
The OM setup we use in our experiment is called
polarization
gradient cooling, because the MOT stage uses circularly polarized lasers and it
is convenient to use the same lasers for the OM cooling stage. The other
method is the
(Sisyphus cooling) setup [6], which I will not discuss in
this work.
Two counter-propagating circularly polarized laser beams produce a rotating
electric field along the propagation direction (Fig. 1.6a). Defining the quantum
axis along the magnetic field, one gets an inertial field
according to
Larmor's theorem. The rotating inertial field causes the |
⟩
|
⟩ states to
have different populations so atoms which are moving towards the
will preferentially occupy the |
⟩
|
light
⟩ ground state sub-level and atoms
moving in the opposite direction will populate mostly the |
⟩
|
⟩ ground
state sub-level. Given the different Clebsch-Gordan coefficients (Fig. 1.5), this
effect causes a six-fold increase in the interaction thereby exerting
a drag
force (Fig. 1.7) which is given by [6]:
(a)
(b)
Figure 1.6 – The electromagnetic standing wave due to (a)
polarization [6].
and (b)
16 | p a g e
Figure 1.7– The force the atoms feel with respect to their velocity, the inset is
the extra damping force due to the laser polarization gradient [6].
(1.12)
(1.13)
where
and
is the detuning and
is the natural width of the atomic excited state
is the velocity of the atom.
Note that in this calculation it is assumed that the atoms already have low
velocity hence one can see an enhanced damping force for low velocities
(compare the inset of Fig. 1.7 to Fig. 1.4).
1.4.3. OM in non-zero magnetic field:
As has been seen in the previous section, cooling below the Doppler limit
becomes possible because of optical pumping between the ground state
magnetic sublevels of multilevel atoms. However, the presence of an external
magnetic field has a strong effect on the OM stage.
The motion of the atoms in a standing wave results in modulation of the light
intensity that induces a magnetic resonance at a particular velocity
, where
is the Larmor frequency [7]. One can calculate
the force acting on the moving atoms by solving the Optical Bloch Equations
(OBE).
p a g e | 17
Let us present the new OBE for the ground states density matrix
of the Bloch vector
as:
(
where
in terms
)
( )
(1.14)
is the Bloch vector given by
,
and
. We find the coefficients to be
,
are the optical pumping rate and the dephasing rate respectively,
is the difference of the light shifts of the two ground
states and
.
One can find the solution of the OBE by making a transformation of the
Bloch vector to a frame rotating with frequency
and eliminating the fast
oscillating terms. The result will be the force:
[
(1.15)
]
With the damping constant
and the capture range
given by:
(1.16)
̅
(
Where ̅
)
is again the optical pumping rate,
decay rate and the detuning is
Theoretically one can see that for
gets
(1.17)
is the exited state
.
one gets
and for
one
, therefore, this OM process decelerate the atoms to a non-zero
finite velocity
. This effect on the OM stage was observed experimentally [7]
and the results are presented in Fig. 1.8 and Fig. 1.9.
18 | p a g e
Figure 1.8 - The change in atomic profile of
downstream from the
molasses as measured by hot wire for red (top) and blue (bottom) detuning.
The laser parameters are
and
[7].
By analyzing the results, taking the result for 0.4G (labeled 'd' in the figures)
which is Earth's magnetic field, one can estimate (roughly from Fig. 1.9) the
⁄ , or equivalently, for
resonance velocity to be
[
]
78
.
(1.18)
Figure 1.9 - The force averaged over a wavelength vs. velocity for the
transition in
for the conditions (a)-(f) as given in Fig. 1.8 [7].
p a g e | 19
1.5.
Measuring the magnetic field using the atoms:
1.5.1. Achieving zero magnetic field:
The atoms are trapped in a vacuum chamber so one cannot insert a probe
inside it and place it where the atoms are for magnetic field measurements. In
order to determine the magnetic field in the middle of the vacuum chamber
(where the atoms are located) one can use one of several methods [8-11].
The method I chose, that should be compatible with the experimental
apparatus, is microwave (MW) spectroscopy conducted on the
states (
and
87
Rb ground
) since their energy levels shift in the presence of a
magnetic field (eq. 1.6).
First one should pump the atoms to the
atomic ground state later to
couple the two ground states using a microwave field made by an external
antenna. By scanning different frequencies one can determine the energy shift
by comparing the measured transition frequency to the magnetic field
insensitive transition |
⟩
|
⟩
|
⟩.
We can use this measured spectrum to calculate the magnetic field that the
atoms feel (eq. 1.5). When the external magnetic field is fully compensated, all
the different Zeeman sublevels will be degenerate and the spectrum should
collapse into a single peak.
Figure 1.10 – MW spectroscopy signal measured by Anat Daniel in her
experiment as part of the AtomChip Group. The graph was taken in order to
measure the external magnetic field affecting the different Zeeman sub-levels
transitions of the Rubidium atom's ground states. Each peak represents the
⟩
|
⟩, (b) |
⟩
|
⟩, (c) | ⟩
|
⟩, (d) | ⟩
(a) |
| ⟩, (e) | ⟩
| ⟩ and (f) | ⟩
| ⟩ transition.
20 | p a g e
1.5.2. The reason for using microwave spectroscopy:
There are two transitions one can use in order to measure the magnetic field
using spectroscopy. The first is the D2 transition coupled by laser field and
the second is the ground state transition coupled by MW field.
Since my goal is to achieve an external magnetic field smaller than
we
should be able to measure such small magnetic fields.
We can calculate the Zeeman sub-levels energy shift (eq. 1.6) and get the
frequency shift of the D2 transition |
with the magnetically insensitive |
⟩
⟩
|
⟩
|
⟩
|
⟩ transition for
|
78
⟩ compared
and
magnetic field:
⟩
|
⟩
|
(1.26)
The natural linewidth of the D2 transition is (eq. 1.15):
(1.27)
Hence we get that Zeeman sub-level shifts for small magnetic fields are much
smaller than the natural linewidth of optical transitions.
One can measure instead the magnetic field using the transition between the
two ground states |
the |
⟩
|
⟩
|
⟩ since |
⟩ is a trappable state but
⟩ state is not, resulting in measurable atom loss upon excitation. The
frequency shift is:
|
⟩
|
⟩
(1.28)
In addition to the slight increase compared to eq. (1.26), the
ground
state is much more stable than the exited state and its natural line-width is
therefore much narrower, on the order of
or less.
Since the spectroscopy is conducted on a cold-atom assemble, the Doppler
broadening is approximately (eq. 1.17):
(1.29)
p a g e | 11
where I assumed a temperature of
and
for
78
atoms.
Pressure broadening is not a factor here since while conducting a microwave
spectroscopy on a cold atoms cloud, the cloud is first being released from the
trap and then examined by the microwave field. The cloud at the time is
considered to be homogenously expanding so collisions between the atoms
while using spectroscopy in negligible.
Therefore, if we want to use spectroscopy as the method to measure the
external magnetic field we must use the much more stable ground states of
the atoms hence use Microwave spectroscopy.
22 | p a g e
2. Computer simulation:
2.1.
Background:
The simulation was designed by me in MATLAB to calculate the magnetic
field generated in any plane between two parallel square-wire loops.
The calculation uses the Biot-Savart law (Appendix A.1) for each finite wire
segment followed by a vector summation of all the relevant wire segments to
form the two square loops.
After calculating the magnetic field I also calculated the inductance. This
additional result is a very important factor to consider since the experiment
uses strong magnetic fields to control the atoms [3] and changing them can
cause significant induced electromotive force (EMF) in the compensation coils.
After comparing several possible sizes for the coils (Appendix B) consistent
with the apparatus layout and dimensions, the optimal configuration for the
experiment was selected and is presented below.
For the following sections the Z-axis will stand for the height.
2.2.
Magnetic field simulation:
2.2.1. X- and Y- axis coils configuration:
Dimensions:
The coils that were chosen for the Y axis direction (and the same for the X
axis direction due to symmetry) are
and are placed
in width and
apart symmetrically around the origin (the center of
the apparatus and the trap).
Parameters:
The free parameters in the simulation are:
Current:
[ ]
Number of loops:
Distance between loops:
[ ]
[ ]
Size of measured area:
Resolution:
p a g e | 13
in height
[ ]
Result:
(a)
(b)
(c)
Fig. 2.1 - the magnetic field for the Y Direction coils for (a)
and (c)
.
, (b)
24 | p a g e
I ran the simulation for all three mutually perpendicular planes that intersect
at the origin, in order to calculate the variation of the magnetic field in the
central volume of
. The results are shown in Fig. 2.1.
Summary:
From the graphs above I calculated the maximum deviation of the field from
its center value. The results are summarized in table 2.1.
Table 2.1–
Homogeneity of the compensation field created by the
x- and y-direction coils.
Deviation
Center [
]
Max [
]
Min [
]
2.2200000
2.22002.0
2.22000.0
2.0%
2.2.2. Z axis coils configuration:
Data:
The coils dimensions chosen for the Z axis are
placed
and they are
apart symmetrically around zero (the center of the trap).
Parameters:
The free parameters in the simulation are:
Current:
[ ]
Number of loops:
Distance between loops:
[ ]
[ ]
Size of measured area:
Resolution
[ ]
Result:
I ran the simulation again for all three mutually perpendicular planes that
intersect at the origin, in order to calculate the variation of the magnetic field
in the same volume as in Sec. 2.2.1. The results are shown in Fig. 2.2.
p a g e | 15
(a)
(b)
(c)
Fig. 2.2 - the magnetic field for the Z Direction coils for (a)
and (c)
.
, (b)
26 | p a g e
Summary:
From the graphs above I calculated the maximum deviation from the center
value. The results are summarized in table 2.2.
Table 2.2–
z-direction coils.
Center [
]
2.2000.0.
2.3.
Homogeneity of the compensation field created by the
Max [
2.2000.00
]
Min [
2.2000000
]
Deviation
2.20.%
Inductance simulation:
There are three pairs of coils already inside the volume made by the
Compensation coils, which for now on be called "the existing coils" which
generate strong magnetic fields for the MOT and the magnetic trap (up to
). At the beginning of the molasses stage, these fields are turned off in as
little as
. Therefore I must also check that the EMF induced by this
changing magnetic flux will not generate undesired induced magnetic fields in
the compensation coils when turning on the molasses stage.
In order to calculate the inductance I calculated the magnetic field's intensity
distribution inside one of the Y-direction wire loops generated by all the
existing coils (Fig.2.3). Further details of the calculation the number of wire
loops required are given in Appendix B.
Fig. 2.3 – Distribution of the magnetic field's intensity inside the wire loop
along the y-axis.
p a g e | 17
At first glance it is a bit disturbing that the graph in Fig. 2.3 is not
symmetric around the zero, however, by examining the contributions of each
pair of coils, in each direction, one can see that the shift of the maximum is
the result of anti-symmetry relative to the center generated by the coils
orthogonal to the Y direction.
In order to calculate the magnetic flux I have made a numerical calculation by
dividing the loop into
[
]
[
] pieces and assuming that the
magnetic field is constant in each piece (Fig. 2.4).
I evaluated my assumption that the magnetic field is homogenous inside each
[
]
[
] piece by comparing the assumed magnetic field, which is the
magnetic field in the center, to the magnetic field averaged over the same
surface divided into 400 pieces (Table 2.5).
Fig. 2.4 – Distribution of the magnetic field's intensity inside one
[
] cell.
Table 2.5 – compare assumed and averaged magnetic field.
Assumed
Magnetic field [
]
195.6943
[
]
Averaged
195.6970
As the graph is shifted, the test surface was taken in the sloppy regime other
than the maximum so the test is valid. One can see the resemblance of the
values which are about 0.001% apart.
28 | p a g e
Summary:
The Mutual-Inductance coefficients were calculated separately for each
direction in space and for each pair of coils (Theory is in Appendix A.2). The
results are given in table 2.6.
Table 2.6 – Mutual-Inductance coefficients.
Direction
Size
[ ]
[ ]
[ ]
Y
Z
* The X direction coils are symmetrical to the Y direction coils.
In order to check the accuracy of the approximation I ran the inductance
simulation for the
Y direction coils using
[
] squares (Table
[
]
2.7).
Table 2.7 – compare levels of approximation accuracy.
[
]
[
]
[
]
[ ]
The simulation for the
[
] squares gave the same result in less than 0.5%
deviation so it was decided to run the simulation with
[
] squares and
avoid running the simulation several times for about 12 hours.
Now one can use the Mutual-Inductance coefficients to calculate the induced
electromotive force (EMF) considering 25 wire loops of the Y direction and 10
loops for the Z direction and get:
Table 2.8 - induced EMF for [ ] shutting down in [
Y
].
Z
In order to calculate the induced current one must first calculate the
resistance of the coils. The material used is a Copper wire hence its
parameters are:
p a g e | 19
Matter type:
Temperature:
[ ]
[
Resistivity:
Wire's length:
]
[ ]
[ ]
Wire's diameter:
The resistance for 1 meter long wire is then:
[
]
And for the complete wire loop the resistance is given in table 2.9:
Table 2.9 – resistance of one square wire loop.
Direction
Size
loops
[ ]
X,Y
0.
0.892
Z
02
0.632
The induced current is, therefore, given in Table 2.10:
Table 2.10 - induced current for [ ] change in [
Deviation
].
Y
Z
4.9%
5.4%
The deviation was taken for the use of 1A current in the compensation coils.
For higher Amperage in the existing coils one should multiply the results in
Table 2.8.
One can consider increasing the resistance of the coils in order to reduce the
effect of the mutual inductance on the experiment, by doing so, the power
supplies must be capable of providing higher voltages for the compensation
coils.
30 | p a g e
3. The experimental test setup:
3.1.
The experimental test setup apparatus:
The experimental setup consists of three pairs of coils (one pair for each
dimension).Each pair is connected to a custom-designed current supply that
controls the current passing through the coils and thereby controls the
magnetic field generated in the coils' direction.
Figure 3.1 - The test-frame compensation coils as they would be configured
outside the main experiment.
3.2.
Experimental test results:
So far I have only measured the magnetic field homogeneity for the entire test
configuration. The actual experiment uses an ion pump that generates an
inhomogeneous magnetic field close to the experimental chamber, so I
measured the magnetic field both with and without an ion pump located near
the coils.
p a g e | 31
3.2.1. The measurement procedure:
Setting up the test experiment:
In order to measure the magnetic field I first built a wooden frame for holding
the coils in place, and then I placed the probe of a Gauss-meter in the middle
of the coils using a translation stage to allow micrometric control on the
probe's location in the x-y plane (parallel to the ground). The entire test
configuration is shown in Fig. 3.1.
Measuring the magnetic field:
Working with the Gauss-meter was very problematic since it is easily affected
by external perturbations such as vibrations or external magnetic field
fluctuations; therefore, I took only 9 measurements in an
area
using the following procedure:
1. Place the probe in the middle of the coils configuration by looking for the
minimum point of the magnetic field.
2. Calibrate the device using the calibration tool.
3. Remove the calibration tool and record the first number visible on the
Gauss-meter's screen.
4. Change the current in the coils until the Gauss-meter shows zero Gauss.
5. Repeat steps 2-4 until no current change is needed.
6. Move the probe to different locations in the x-y plane and repeat 2-3.
In order to check the accuracy of the measurement I always checked at the
end of the measurements that the middle point is still around zero Gauss, if
not, the entire procedure was repeated.
My measurements are presented below.
3.2.2. Analysis and Results:
Without the ion pump:
By applying the coils' magnetic field on a random space volume in the lab I
followed the steps which I mentioned in sec. 4.1 and got the following results:
32 | p a g e
The
diagram
above
displays
the
measurement
procedure.
The
first
measurement was in the center was followed by moving in the direction of the
arrows to the bottom left corner, each arrow represents movement of
and returning to the center where a value of
,
was the smallest
deviation I got.
Finally I can present my results in graphical form:
Figure 3.2 - The distribution of the measured magnetic field on a
surface, with Z=0.
With the ion pump:
I repeated the above measurements, after moving an ion pump into the
vicinity of the coils, coordinates (60,0,40), and obtained the results shown
below in the same fashion as above.
The diagram below displays the measurement procedure as I started in the
center and followed the arrows to the bottom left corner, each arrow, again,
represents movement of
same value of
deviation that I got.
p a g e | 33
. I then returned to the center and got the
. This was the measurement with the smallest
The results in graphical form:
Figure 3.3 - The distribution of the magnetic field in
, with VacIon ion pump 40 at (60,0,40).
3.3.
surface,
Experimental setup conclusions:
By examining the results one can see that even with the ion pump the
magnetic field deviation can be kept below
. By examining both Fig. 1.8
and Fig. 1.9 one can see that even though a lower magnetic field can reduce
the atom’s velocity, according to the experiment conducted in [6] (Sec. 1.4.3),
when the magnetic field reaches
it is no longer possible to distinguish
graphically the finite velocity from zero velocity, hence I will assume that the
magnetic field at the region of the atoms is homogeneous enough to result in
the coldest possible molasses stage.
34 | p a g e
4. Summary and conclusions:
In order to get a BEC one should cool the atoms to ultra-low temperatures,
and in order to get as many atoms in the BEC as possible one should optimize
the OM stage so that the subsequent RF evaporation stage will eject as few
atoms from the trap as possible.
As I have shown in this work, theoretically, the OM stage can cool the atoms
down to the recoil temperature. Practically however, external magnetic fields
cause the OM to cool the atoms to a much higher temperature. By
compensating such external fields one can cool the atoms further in the OM
stage and eventually keep more atoms in the trap.
In my work I tested various coil configurations in order to examine their
compatibility with the existing experimental apparatus. I found that smaller,
more convenient coils, can be used and still give almost the same results and a
fairly homogenous magnetic field in the atoms' position.
I
have
defined
a
goal
of
less
than
magnetic
field
in
a
volume and this goal was accomplished also with a gradient
in the magnetic field generated by an ion pump located near the coils
configuration.
The next thing for me to do is to test my coils configuration directly on the
existing experiment apparatuses, measure the magnetic field using the atoms,
achieve zero magnetic field and test the effect on the atoms and on the cooling
process. In order to accomplish that I will have to build three current supplies
and measure the inductance on the coils from the existing coils in the
experiment.
p a g e | 35
36 | p a g e
References:
[1]
H. J. Metcalf, P. van der Straten, Laser Cooling and Trapping, Springer
(2002).
[2]
R. Folman, P. Krueger, J. Schmiedmayer, J. Denschlag and C. Henkel,
Microscopic atom optics: from wires to an atom chip , Adv. At. Mol. Opt.
Phys. Vol. 48, 263 (2002).
[3]
S. Machluf, Building a BEC on an atom chip apparatus, M.Sc. part from
a Ph.D. research proposal, The AtomChip lab at BGU (2009).
[4]
D. A. Steck, Rubidium 87 D Line Data, University of Oregon (2009).
[5]
M. Zwierlein, Cooling and trapping a Bose-Fermi mixture of dilute
atomic gasses, Massachusetts Institute of Technology, Cambridge (2001).
[6]
J. Dalibard and C. Cohen-Tannoudji, Laser cooling below the Doppler
limit by polarization gradients: simple theoretical models , J. Opt. Soc.
Am. B 6, 2023-2045 (1989)
[7]
S-Q. Shang, B. Sheehy, P. van der Straten and H. Metcalf, Velocity-
Selective Magnetic-Resonance Laser Cooling, Phys. Rev. Lett. 65, 317–
320 (1990).
[8]
M. Koschorreck, M. Napolitano, B. Dubost, and M. W. Mitchell, High
resolution magnetic vector-field imaging with cold atomic ensembles,
Appl. Phys. Lett. 98, 074101 (2011).
[9]
J. M. Higbie, L. E. Sadler, S. Inouye, A. P. Chikkatur, S. R. Leslie K. L.
Moore, V. Savalli, and D. M. Stamper-Kurn, Direct, Non-Destructive
Imaging of Magnetization in a Spin-1 Bose Gas, Phys. Rev. Lett. 95,
050401 (2005).
[10] M. Vengalattore1, J. M. Higbie1, S. R. Leslie, J. Guzman, L. E. Sadler,
and D. M. Stamper-Kurn, High-Resolution Magnetometry with a Spinor
Bose-Einstein Condensate, Phys. Rev. Lett. 98, 200801 (2007).
p a g e | 37
[11] S. Wildermuth, S. Hofferberth, I. Lesanovsky, S. Groth, I. Bar-Joseph,
P. Kruuger, and J. Schmiedmayer, Sensing electric and magnetic fields
with Bose-Einstein Condensates, Applied Physics Letters 88, 264103
(2006).
[12] J.D. Jackson, Classical electrodynamics, 3rd ed., Wiley (1999).
38 | p a g e
Appendix A – Theory behind the computer simulation
A.1.
The magnetic field between two square loops:
We will use the Biot-Savart law to calculate the magnetic field generated by a
finite wire at a random point (Fig. A.1) [12]:
̂
⃗
(A.1.1)
Figure A.1 – finite wire scheme, the field generated by the finite wire, AB, is
calculated at a random point, P.
By separating the calculation to the direction and the size of the magnetic
field vector one gets:
Size:
|
(A.1.2)
̂|
Applying
;
p a g e | 39
;
Finally one gets:
∫
(A.1.3)
Direction:
The direction is accumulated simply by using:
̂
(A.1.4)
A.2.
̂
̂
Inductance:
Self-inductance
One can use the formula for rectangular coil (from an internet calculator):
[
(A.2.1)
(
√
(
√
)
( )
(
√
)
)]
Where:
N – number of loops.
w – coil's width.
h – coil's height.
a – wire radius.
– permeability coefficient of the medium.
Mutual-inductance
The coefficient of mutual inductance is calculated according to the formula:
(A.2.2)
And the induced electromotive force (induced EMF) will be:
(A.2.3)
40 | p a g e
Where:
– The magnetic flux through the tested circuit.
I – The current in the circuit causing the magnetic flux.
N – The number of loops.
For the calculation of the magnetic flux, I will assume a uniform magnetic
field through a surface element [12]:
(A.2.4)
p a g e | 11
∫ ⃗
∑⃗
42 | p a g e
Appendix B – Comparison between other wires
configurations:
In addition to the dimensions chosen for the coils I simulated two more
options for the X, Y coils. The results are presented in the table below.
Table B.1 – summary of the magnetic field generated, its deviation and the
number of wire loops required in order to achieve a [ ] magnetic field.
X,Y Coils
Z Coils
First option
Second option
Third option
2.2200000
2.202.2..
2.200.200
2.2000.0.
Loops for
[ ]
000
0.
00
.0
Deviation
2.0%
2.00.%
2.200%
2.20.%
Dimensions
[mm]
Distance
[mm]
[
]
* The ratio B/n is calculated for 1 wire loop and 1 Ampere.
Table B.2 – Comparison of the Mutual-Inductance coefficients for different
sizes for
.
Direction
Size
EMF [mV]
[ ]
[ ]
[ ]
Y
Y
Y
Z
I will now summarize in Table B.3 the results for the "worst case scenario"
considering also the number of wire loops required to achieve the required
magnetic field and shutting down a
p a g e | 13
current in
in the existing coils.
Table B.3 – induced EMF worst case scenario.
Direction
Size
loops
[ ]
X,Y
0.
2.18
X,Y
0.
2.13
X,Y
0.
2.17
Z
02
1.69
Finally, after reviewing all the options I chose the first option because it is a
more convenient size for the current experiment apparatus, the inductance is
almost the same, and the difference in magnetic field homogeneity is within
specifications.
44 | p a g e