Intensity Correlations of Light Scattered From Cold Atom Clouds
A thesis submitted to the Miami University Honors Program in partial fulfillment of the requirements for University Honors
by
Matthew Beeler
May, 2003 Oxford, Ohio
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ABSTRACT
INTENSITY CORRELATIONS OF LIGHT SCATTERED FROM COLD ATOM CLOUDS
We have constructed an atom trap using laser light and magnetic fields. We confine approximately 10 million atoms to a 1 cubic mm space at a temperature on the order of 100 microKelvin. At a great enough trap density, radiation trapping begins to occur inside the trap. Radiation trapping is the reabsorption by atoms in the trap of spontaneously emitted photons from other atoms in the trap. Radiation trapping lowers the density and raises the temperature of cold atom traps. Because it is a result of spontaneous emission, radiation trapping will decohere the light scattered from the trap. Measurement of the intensity correlation function will show the effect of radiation trapping. The intensity correlation function is simply the probability of detecting two photons separated by a certain time delay. A theoretical exploration of the effect of radiation trapping on the intensity correlation function is made. We show that the effects of radiation trapping are significant at trap densities two orders of magnitude less than previously thought. Experimental progress on verifying our predictions is reported.
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Intensity Correlations of Light Scattered From Cold Atom Clouds
By Matthew Beeler
Approved By:
Dr. Samir Bali, Advisor
Dr. Perry Rice, Reader
Dr. Michael Pechan, Reader
Accepted By:
Dr. Carolyn Haynes, Director, University Honors Program
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Acknowledgements
I would like to acknowledge my advisor Dr. Samir Bali, without whom none of this would have been possible. Also, I would like to thank the people that I have worked closely with over the last three years: Soo Kim, Ron Stites, Laura Feeney, and Kristy Kallback-Rose. We have also received help from Dr. Perry Rice and Dr. S. Douglas Marcum, and I would like to thank Dr. Michael Pechan and Dr. Perry Rice for being readers of this thesis.
Financial support for this project was provided by the College of Arts and Science, the Office for the Advancement of Scholarship and Teaching (OAST), and Miami University. Also, support from the Cottrell Foundation (Research Corporation) and the Petroleum Research Fund of the American Chemical Society is appreciated.
A paper entitled “Sensitive Detection of Radiation Trapping Cold Atom Clouds” has been submitted to Physical Review A from part of the work done for this thesis.
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Table of Contents
1. Introduction 1
2. Overview of MOT and Molasses
a. MOT 2
b. Molasses 5
3. Tunable Laser Diodes 7
4. Saturated Absorption Spectroscopy 13
5. Hardware
a. Vacuum System 18
b. Electronics 19
c. Magnetic Coils 21
d. Optics 21
6. Trap Characterization 23
7. Intensity Correlations and Spatial Coherence 25
8. Radiation Trapping Theory 31
9. Experiment 36
10. Conclusion 40
11. References 41
12. Appendix A 42
13. Appendix B 48
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List of Figures
Figure Page
Diagram of ball with trapping laser beams 2
Hyperfine energy levels of Rb 3
Splitting of J levels by magnetic gradient 4
Picture of laser diode set-up 8
Schematic of laser diode set-up 10
Schematic of saturated absorption set-up 14
Picture of saturated absorption set-up 16
Pictures of hyperfine spectra on oscilloscope 17
Diagram of ball and detector for theoretical calculation of spatial coherence 26
Diagram of image of atoms in image plane of detector 30
Plot of spatial coherence factor S 30
Theoretical plot of intensity correlation function 35
Picture of intensity correlation detection system 36
Schematic of intensity correlation detection system 37
Experimental plot of intensity correlation function 39
Sample data set 44
1
Introduction
Cold atoms have become a topic of considerable interest in recent years. With
Nobel prizes awarded in the field in 2001 and 1997, interest in laser cooling and trapping
has only been growing. Cold atoms have been used in clocks [1] and atom lasers [2], and
Bose-Einstein condensates have become a very interesting area of study [3,4,5]. The
Magneto-Optic Trap (MOT) is the workhorse of many systems involving cold atoms, and
is the system that we have assembled in the last two and a half years.
Basically, a MOT is the slowing and confinement of atoms using lasers and
magnetic fields. This technique can result in cooling the atoms in the MOT to
temperatures in the microKelvin range. The purpose of our recent work has been to study
the effect of radiation trapping in optical molasses. Optical molasses is the effect
occurring when the magnetic fields in a MOT are turned off and the atoms are slowed
(but not confined) by the light beams used in the MOT. Radiation trapping, meanwhile, is
an effect that occurs when photons emitted by atoms in the trap are absorbed by other
atoms in the trap [6,7]. Hence, the radiation, in the form of photons, is confined to the
trap. It is useful to study radiation trapping because it imposes a limit on how cold and
dense the trap can be made. Our method for examining radiation trapping is to look at
intensity correlations [8] of light scattered from the atoms. Because radiation trapping is a
result of spontaneous emission, we expect that the effect of radiation trapping will result in less correlated data, as spontaneous emission is a random process.
2
Overview of MOT and Molasses
y
Fig. 1 Trapping Laser Beams from 6 Directions
MOT
The MOT is formed by directing laser beams into a gas of atoms (in our case, Rb) from three perpendicular directions and reflecting these beams back onto themselves, with the net result being six beams hitting the atoms from six directions (Fig. 1). The wavelength of these beams is slightly red detuned from the 780 nm atomic resonance so that atoms with velocities directed opposite to a given beam are Doppler-shifted into resonance. Thus, only atoms traveling against a beam “see” the light and absorb photons.
Now, when an atom absorbs a photon from one of these beams, it receives some momentum in the direction of the beam through conservation of momentum. Of course, it will re-emit this photon at some later time, but because spontaneous emission occurs in a 3
random direction, the average momentum from spontaneous emission is 0. The end result
of this process is that atoms moving into the beam are slowed as they absorb photons, an
effect known as optical molasses. We find that Rubidium is an excellent atom to use for
this type of experiment due to its relatively simple hyperfine structure and the ease with
which we can find diode lasers of the correct wavelength.
Trapping Transition
Repumper
Fig. 2 Diagram of Rb energy levels with hyperfine splitting and labeling of trapping and repumping transitions
Two optical frequencies are actually required to trap atoms, which is, in our case,
2 provide by two separate lasers. Our main trap laser pumps from the 5 S1/2, F = 3 ground
2 state to the 5 P3/2, F′ = 4 excited state. However, our laser off-resonantly pumps some of 4
the atoms into the other hyperfine levels in the 3/2 excited state. These levels allow a
transition down to the F = 2 ground state, where the atoms no longer interact with the
trapping laser, and eject from the trap. Thus, a second laser which pumps from the F = 2 ground state back to the F′ = 3 excited state is required, as shown in Fig. 2. We refer to this second laser as the “repumping” laser. If we did not have this second laser, all of the
atoms would end up in the F = 2 ground state within a millisecond.
E
J = 1 1 State 0
-1 νlaser
J = 0 State x Fig. 3 Splitting of excited state electronic energy levels by a magnetic gradient
Next, we turn on a magnetic gradient, which has the effect of imparting a position
dependent shift to the energy levels of the atom. J, the interaction of the electronic orbital
angular momentum and the nuclear spin, reacts to a magnetic field by splitting the
excited state energy levels into three quantized values depending on the value of the 5
magnetic field. With the energy levels shifted slightly by a magnetic gradient, such as in
Fig. 3 we find that the atom has a greater probability of absorbing the laser light when it
is polarized in a certain way, either σ+ or σ- circular polarization. Because the –1 J state is
more easily pumped by σ- light and the +1 J state is more easily pumped by σ+ light, if
the set-up was as shown in Fig. 3, we would adjust the polarization of the light to be σ- from the right and σ+ from the left. This provides the position-dependent force to trap the
atoms. The atoms are more likely to absorb light if they stray from the center of the chamber, the place of 0 magnetic field. Without the magnetic field gradient, the atoms are slowed, but have no preferred gathering place.
We are using the MOT to measure the effect of radiation trapping using intensity correlations of light scattered from the trap. The intensity correlation function is the probability of observing a pair of emitted photons separated by a certain time delay. Both the concepts of intensity correlations and radiation trapping are explained in detail after
the explanation of the construction of the MOT.
Optical Molasses
As discussed above, a magnetic gradient is necessary to supply a position
dependent force on the atoms, which pushes them towards the center of the chamber to
create a trap. However, once the atoms are trapped, this magnetic gradient adds energy to
the atoms, making the temperature of the trap higher than it could be. In order to lower
the temperature of our trap for making measurements, it is necessary to extinguish the
magnetic gradient. The atoms in the trap will immediately start to escape at a rate 6
depending on the original temperature of the trap, but because the laser beams damp the
motion of the atoms, this is known as optical molasses. The atoms leaving the trap are
actually moving slowly enough to see them dispersing with the naked eye (through a
camera, of course, since the light is in the infrared).
Extinguishing the magnetic field fast enough to create optical molasses is not
trivial [9], and much care and design went into building the circuit necessary to turn off
our magnetic gradient. The magnetic gradient is created by running current through coils
of wire, and so it would seem that turning off the current would turn off the gradient.
However, the coils make excellent inductors, and nature abhors a change in flux. By
Lenz’s law, when the current changes, the flux through the coil changes, and so a current
in the coils appears to counteract this change in flux. The challenge then is to build a
circuit that squelches these currents in as short a time as possible. Once that is done, we have cold optical molasses.
7
Tunable Laser Diodes
The construction of a Magneto-Optic Trap begins with the construction of tunable laser diodes. We buy laser diodes that emit light at approximately the wavelength that we
need, which is 780 nm for 85Rb, and then tune these diodes using a system similar to the
one outlined in [10]. The trapping laser is a Sanyo DL7140-201, and the repumping laser
is a Sharp LTO25MDO. An aluminum plate is used to hold the diode, lens, beamsplitter,
and diffraction grating in a configuration similar to Fig. 4. When the beam leaves the
diode, it is unfocused, and diverges quickly without the lens. A precision screw is used to
adjust the distance between the diode and the lens, allowing collimation of the beam. We
use a beam splitter to allow the light to escape from our set-up. The diffraction grating is
mounted on an adjustable plate, and precision screws are used to adjust the angle of this
grating with respect to the laser diode. In this manner, we can reflect light back into the diode and cause it to preferentially radiate the wavelength that we are reflecting. In effect, we have put the diode in an external cavity, which gives us precise control of the wavelength output by the diode.
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Fig. 4 Tunable Diode Set-up
Because the atoms are so sensitive to the frequency of the light used in trapping, it is necessary to be able to resolve the hyperfine spectrum of the atoms (Fig. 8). We must be able to tune the laser to these lines. In order to resolve this hyperfine spectrum, a few more things must be done to stabilize the tuning of the laser. The first is to stabilize the
temperature, which can be done with a thermoelectric cooler taped (with thermal tape) to
the top of the diode mount. Heat sinks must be placed on the other side of the cooler to
allow heat to escape, and care must be taken to avoid excessive cooling. When the temperature control unit tries to cool the laser too much, heat begins to leak from the “hot side” of the thermoelectric cooler to the “cool side”, and the whole unit begins to heat.
This causes the controller to send more current to the cooler, further exacerbating the
problem and heating the laser diode. In my experience, trying to cool the Sharp diode 9
below 22° Celsius is risky. The store-bought temperature controllers have a maximum temperature limit feature, allowing the user to set a maximum temperature before current to the controller is shut off. This should be utilized as a safeguard to prevent damage to the laser diode due to high temperatures.
The next thing that is done to create the tunable laser diode system is to place a piezoelectric disc behind the plate holding the diffraction grating. The disk can be held in place by one of the screws used to adjust the plate. Piezoelectric disks flex when a voltage is applied to them, and by flexing they move the diffraction grating. In our case, we a) apply a ramping voltage to the piezo, which essentially scans the wavelength of the laser light by slightly adjusting the angle of the diffraction grating so that we can look at a range of wavelengths, rather than just one, and b) use a dc offset to feedback to the piezo to lock the laser to a hyperfine line during MOT operation. There is one interesting problem that I have encountered working with this set-up for the piezo. The diffraction grating mounts are designed so that the adjustment screws fit into divots in the back of the mounting plate for the grating. The piezo keeps one of the screws from fitting into its divot, and may cause the screw to translate the base plate perpendicularly to the direction of travel of the screw as it “grabs” the piezo. In this case, a little bit of grease should fix the problem. The entire base plate for the laser diode set-up is also confined to an aluminum box to dampen noise and prevent dust. The base plate is separated from the floor of the box by sorbethane. Holes are drilled in the box to allow adjustment of the diffraction grating mount and the exit of the beams.
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Tuning The Diodes
Because of the beam splitter, there are two beams escaping from the box. One of
these is used as the main working beam, while the other is used for saturated absorption
spectroscopy (Fig.5). The two beams coming out of the box are not of equal strength.
One of them is mostly the light coming from the diode, and the other is mostly the first
order reflected beam from the diffraction grating. The beam that comes from the diode
itself is much stronger than the beam reflected from the diffraction grating. Before either of these beams can be used for our purposes, the diffraction grating and collimation must be
adjusted so that the system will scan through a variety of wavelengths with adjustment of
the diffraction grating. There are many steps involved in this process.
Working
Beam Beamsplitter
Diode Saturated Absorption Diffraction Grating
Fig. 5 Schematic of the beam strength over one pass through the tuning set- up. The beam used for saturated absorption is weaker than the other beam due to the fact that the saturated absorption beam is from the first order reflection off of the diffraction grating.
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First, we must make sure that the beam is collimated. In order to do this, we look
at one of the beams at a point several meters from the diode. The collimation screw is
used to adjust the distance between the lens and diode to focus the beam at the spot we
are looking at. Then, the lens is moved slightly (very slightly!) closer to the diode to
simulate focusing the beam at infinity. In practice, across the length of the room suffices.
This collimates the beam, making it the same size and shape no matter how far away from the diode the beam is examined.
Next, we must make sure that the light from the diffraction grating is going back into the diode to allow tuning. In order to do this, we first examine the strong beam.
There will be two spots, one strong and the other weak. The strong beam is the original beam from the diode, while the weak beam is reflected from the grating back into the diode and then back out again. We would like to line these two beams up by adjusting the horizontal and vertical adjustments on the diffraction grating mount. Because the beams’ power contrasts so sharply, it will be possible to line them up only approximately, but this is all that is necessary for the next step in the process.
There will also be two beams coming from the other side of the beam splitter, but these two beams will be of roughly equal power. Thus, it is possible to align these two beams together with greater precision. However, it is often difficult to find these two spots, and that is why the adjustment to the strong beam is made first. (Indeed, it is often difficult to find the two spots even after lining up the strong beam, and it takes some practice.) These two spots are very sensitive to the vertical alignment of the diffraction grating, so this alignment is done first. When the horizontal alignment is done, there 12
should be a point at which the two beams “jump” together and then a range of horizontal
motion of the grating alignment screw where they stay together. The point at which the beams stay together and do not move is the tuning range of the diode. When this occurs, the horizontal adjustment of the diffraction grating mount will actually tune the wavelength of the laser. This can be verified by measuring the wavelength of the laser. At this point, the laser is ready to be tuned.
One last thing must be done in order to get the greatest tuning range from the diode, and this is called threshold tuning. Basically, the external cavity is adjusted so that the laser will lase with the least possible power. This ensures good alignment of the external cavity, and consequently good tuning. The method is to lower the current and watch the strong beam spot until we reach the “lasing threshold”, i.e., there is a sudden drop in power. Then, the collimation screw and horizontal and vertical adjustment of the diffraction grating are adjusted to make the spot bright again. This process is repeated until the threshold cannot be lowered further. In my experience, the threshold is affected more by the collimation than by the diffraction grating, and so I always adjust the collimation first. Once threshold tuning is done, the laser will at least have a tuning range of approximately 3-4 nm.
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Saturated Absorption Spectroscopy
Saturated absorption spectroscopy is a method used to finely determine the atomic transition to which the laser is tuned. The set-up is shown in figures 6 and 7. As shown, the beam is reflected off of a thick glass plate through a vapor rubidium cell. There are two reflected beams from the thick glass plate beamsplitter, one from the back surface and one from the front surface, and the angle of the beamsplitter determines the spacing of these beams. Once through the cell, another thin glass plate reflects these two beams onto two photodiodes. A polarizer can be used before the first beam splitter to prevent power broadening of the spectra and allow finer resolution of the hyperfine peaks.
Meanwhile, the beam that goes through the thick glass plate is reflected by two mirrors back onto one of the original beams reflected from the glass plate. However, this beam travels in the opposite direction. Since only about 4 percent of the light from the beam is reflected from the glass plate, the beam going through the glass plate is aptly titled the strong beam, while the other two beams are known as the weak beams. The two weak beams can barely be seen on an infrared detector card.
The photodiodes are wired anode-to-cathode so that the output is a subtracted signal. In order to make sure that the light is aligned into the photodiodes equally, one can first align them visually and then block them individually and make sure that the voltage change is the same (but in an opposite direction) when either of them is blocked.
The idea is that we first ramp the piezo to scan through a range of wavelengths. Then, we 14
tune the laser to resonate with an atomic transition by watching for the atoms in the vapor cell to fluoresce. Now, we can monitor the signal from the photodiodes and keep tuning the laser until we see a hyperfine spectrum. The laser is tuned principally in three different ways. The first is temperature, which actually has the largest effect on the wavelength of the laser. The next is the diffraction grating, and the third is the current.
The general method is to change the temperature and then adjust the diffraction grating to try to see fluorescence. If this does not seem possible, change the temperature and try again. The current is used mostly for fine-tuning once spectra are seen.
Tunable laser beam vapor cell beamsplitter 2 mirror 1 beamsplitter 1
differencing mirror 1 photodiodes - Current-to-voltage Frequency Lock Box converter TO LASER Fig. 6 Saturated Absorption Optical Set-up
Saturated absorption allows resolution of the hyperfine spectra because of the interaction of the beams with the atoms in the vapor cell. The weak beams going through 15
the vapor cell will be attenuated at the wavelengths being absorbed by the atoms in the
vapor cell. Because of Doppler broadening, this will be a broad range of wavelengths.
However, the strong beam overlaps one of the weak beams and saturates the transitions in
the hyperfine spectra. The only atoms that will be on resonance with both the strong
beam and the overlapping weak beam are those atoms moving perpendicular to both
beams, since these are the only atoms not Doppler-shifted out of resonance with one of
the beams. Thus, the atoms traveling perpendicular to both beams are at the actual (non-
Doppler-shifted) atomic resonances. The strong beam has the effect of preventing the overlapping weak beam from seeing absorption at these wavelengths, since it saturates these atoms. Thus, the subtracted signal between the two weak beams will have peaks at the atomic resonances where the strong beam has prevented absorption by the overlapping weak beam at those specific wavelengths.
However, there is one other way that an atom can be on resonance with both beams, and that is if it is traveling at the correct velocity to be blue-shifted into resonance with one beam at one transition, for example F’ = 4 in figure 8b, and red shifted into resonance with the other beam at a different transition, for instance F’ = 3 in figure 8b.
This will result in a peak in the spectrum directly in between any two actual resonances, in our example the largest peak in figure 8b. These are called “crossover peaks”. Because a range of angles and velocities can attain the correct component of velocity parallel to the beams, the crossover peaks in the spectrum are actually much stronger than the real peaks. 16
Fig. 7 Saturated Absorption Set-up
Some sample spectra are given in figure 8. There are four spectra, one for each ground level of each isotope. The peaks in each spectrum are a result of excitation from one ground level in one isotope to a three excited states, resulting in six peaks in each case with crossover. The peaks used for trapping are well resolved, and we can isolate single peaks easily. We note that the repumping transition for 85Rb is not as well-
resolved, but it is found that there is no need to resolve individual peaks in order for the
repumping laser to work correctly. The non-numbered peaks in figure 8b are crossover
peaks.
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A C
B D
Fig. 8 Hyperfine Spectra A) F = 2 to F’ transition for 85Rb B) F = 3 to F’ for 85Rb (with individual F’ levels labeled) C) F = 1 to F’ for 87Rb D) F = 2 to F’ for 87Rb
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MOT Hardware
Vacuum
The next component to the Magneto-Optic Trap is the vacuum chamber that provides the source for the atoms. In our case, rubidium atoms are used. The chamber is roughly a sphere with windows sticking out in many different directions. Most notably, there are three pairs of perpendicular windows that we use for the trapping laser beams.
Three different vacuum pump methods are used to keep and maintain our vacuum at approximately 2E-9 Torr. The first is a roughing pump, which will pump the chamber from atmospheric pressure to roughly the milliTorr range, the second is a turbo pump, which will pump to roughly 1E-7 Torr, and the third is an ion pump, which pumps the rest of the way. In addition, baking of the vacuum chamber was required to get and maintain our current pressure. Baking releases pollutants that may be stuck to the walls of the chamber.
After the vacuum chamber was pumped down, the rubidium was introduced via a leak valve. The valve was attached directly to the chamber and a short flexible vacuum hose was attached to the other side the valve. Since the rubidium comes as a liquid in a glass vial, we can simply close the valve, open the other end of the tube and insert the vial. Then, we open the valve and pump out the air from that portion of the chamber before breaking the glass vial. In order to introduce rubidium into the chamber, we simply have to open the valve and heat it to approximately 100° C. When we first started 19
seeing trapped atoms, we had some problems determining whether we had too much rubidium in the chamber. We were very worried about overpressuring, since this would mean starting all over, and so we were very cautious in letting in rubidium. What we eventually ended up doing was finding the trap each day, and if it was dim, we would open the valve. It would eventually get brighter, and at the end of the day we would close the valve. This continued for a few weeks, until we no longer had to open the valve to get a bright trap.
Electronics
The electronics used in the creation of the trap include a mix of store-bought and home-built items. The current and temperature control for the diodes are store-bought, while both current-to-voltage converters and locking circuits are student-built. The current-to-voltage converters are basically inverting amplifiers with adjustable gain, while the locking circuits are slightly more complicated. The locking circuits have both a ramp portion and a locking portion to them. The ramp portion accepts an incoming ac signal from a function generator and outputs the same signal with adjustable gain. Thus, the output of the ramp portion of the locking circuit can go from 0 volts up to the maximum voltage input to it by a function generator. The locking portion, meanwhile, lets you specify a certain portion of your signal to lock to. Then, it compares the actual voltage of your signal with the setpoint voltage and dithers the piezo on the laser in order to maintain the signal voltage at the setpoint. 20
Both of these pieces of electronics are used in conjunction with the saturated absorption set-up to tune the laser to a certain atomic transition and then lock it there. The
current-to-voltage converter is used to read the current signal from the photodiodes in the
saturated absorption set-up. The ramp part of the locking circuit is used to flex the piezo,
and the locking portion of the circuit enables us to lock to a certain atomic transition.
One of the big problems which has hindered our ability to lock the lasers stably is
noise. The lasers are very sensitive to vibrations, which are hard to eliminate, but
electronic noise can also play a big part in destabilizing the lasers. In our case, we found
that eliminating electronic noise in our circuitry required some careful thought and
planning. In order to get the resolution of the spectra that we require for stable locking, it
was found that the electronic noise in our circuits must be on the order of 5-10 mV. To
create this condition, we employed five techniques on all of our electronics.
First, we made sure that the ground loops from all of the electronic equipment
were tied together. This helps to eliminate ground noise. Then, we tied large (4700 µF)
capacitors between the ground and incoming dc power sources inside each circuit.
Because capacitors have an impedance inversely proportional to the frequency of an
incoming signal, these capacitors drained incoming power spikes (high frequency noise)
from the dc power sources straight to ground, thus keeping them from going farther into
the circuit. This use for a capacitor is known as "bypassing". As further protection against
power spikes, inductors were placed in series with the dc power, presenting a high
impedance for high frequency signals. Next, we also placed bypass capacitors near each
operational amplifier in each of the circuits, in case any spikes were created within the 21
circuit or got through the other bypass capacitors. We also placed a simple filter in series between the circuit ground and the chassis holding the circuit board in order to ground
the chassis. This filter was a 10 ohm resistor and a 0.1 µF capacitor in parallel, and
helped to drain power spikes from the circuit. All of these tactics together brought our
electronic noise to an acceptable level.
Magnetic Coils
In order to supply the magnetic gradient for the MOT, we affix two coils of wire
to opposite sides of the vacuum chamber [9]. Then, we run a current of approximately
two amps through these coils, which are connected in series. The coils are connected in
an anti-Helmholtz configuration, and the result is a magnetic field gradient of
approximately 8 Gauss/cm along the axis of the coils and approximately 4 Gauss/cm in
the two perpendicular directions to the axis of the coils.
Optics
There are many pieces of optics needed in just the right configuration to produce
the beams needed for trapping the atoms. In addition to mirrors for reflecting the beam,
irises to guide it, and lenses to change its size, there are some important pieces of optics
needed in our set-up. The first of these is the anamorphic prism pair, which takes our
original beam, elliptical in shape, and changes it to a circular shape that is easier to 22
manipulate. The next uncommon item in the beam path is a faraday rotator, which prevents back reflections from the beam path. This is needed because reflecting beams back into the diode actually tunes the laser, as noted above.
Next, there is an acousto-optic modulator (AOM), which is used to tune the wavelength of the laser still further to account for the Doppler shift needed for trapping.
This piece of equipment uses a crystal sensitive to sound waves acting as a diffraction grating for the laser. The beam coming through the AOM, when properly aligned, will show the distinctive diffraction pattern, appearing as many separate beams. We utilize the first order beam for our work, and an iris selects the correct beam. After this we have a microscope objective, which we use to expand the beam size to allow us to trap more atoms. Of course, lenses are required to recollimate the beam directly after the microscope objective.
It should be noted that the repumping laser characteristics are much less critical than the characteristics of the trap laser. In this case, all that is necessary is two lenses used to expand the beam and mirrors to combine it with the trapping beam. Recently, in order to facilitate a "time of flight" temperature measurement (described below), we have also added an electronic shutter to the beam path. This shutter allows us to turn the beams on and off on the microsecond time scale.
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Trap Characterization
There are three standard measurements that we do in order to characterize our
trap: the number measurement, number density measurement, and temperature
measurement. The number measurement is made by simply placing a lens on one of the
windows and focusing the scattered light from the trap onto a photodiode. The scattering
rate of an atom in a laser field is given by
S0γ / 2 γ p = 2 , (1) 1+ S0 + (2δ /γ )
where S0 is the laser intensity divided by the saturation intensity of the atom, γ is the
inverse of the excited state lifetime (linewidth), and δ is the detuning of the laser from
atomic resonance. By assuming that our beams are at saturation intensity and that our
detuning is one linewidth (γ), we can calculate the scattering rate. Also, we know the
solid angle that we are imaging onto the photodiode, and so we know how many photons
we should see per atom per second. Then, we measure the photon scattering rate and
calculate how many atoms must be in the trap. Currently, we trap approximately 1 X 106
atoms.
The next measurement used to characterize the trap is the number density
measurement, telling us how close together the atoms in the trap are. In order to do this measurement, we send an on-resonance probe beam through the center of the trap and measure the absorption of the probe beam by the trap. To see an absorption peak, we 24
ramp the laser in and out of resonance, in our case by dithering the current to a commercial laser diode. This measurement can also be used to get an estimate of the temperature of the trap, the third characteristic that we measure. We know what the size of the peak obtained in the number density measurement should be, as this is determined by lifetime of the excited state of the laser. However, due to Doppler broadening, the peak will actually be slightly wider than this. Because the Doppler broadening is related to the temperature of the atoms by
2υ k T ln(2) ∆ = 0 o , (2) D c m we can calculate the temperature of the atoms from the difference between the actual width of this peak and the known value. However, this method gives us only an approximation of the temperature.
In order to accurately measure the temperature of the atoms in the trap, we employ the time-of-flight (TOF) measurement method. Basically, this method involves turning off the trap and allowing the atoms in the trap to fall under the influence of gravity. A resonant probe "sheet" of light is arranged underneath the trap and the fluorescence from the atoms is observed as they fall through this sheet. Because the value for the acceleration due to gravity is known, and the atoms can be assumed to have a
Maxwellian velocity distribution, we can determine the temperature of the atoms from how long it took them to fall through the probe beam. The shutter in the beam path of the trapping and repumping lasers is necessary to prevent scatter from these beams from interfering with the measurement. 25
Intensity Correlations and Spatial Coherence
The intensity correlation measurement technique has proven to be a useful non- invasive technique for determining the optical properties of various sources. For example, it has been used successfully to determine the characteristics of atomic transport in optical lattices [11], and also to verify and characterize the frequency spectrum of optical molasses [12].
The intensity correlation g2(τ) of a source is defined by: g2(τ) ≡
denote the statistical average. This quantity is of interest because it is simply related to the
electric field correlation g1(τ) =
of the source. This is useful because measuring intensity is generally easier than measuring the power spectrum directly.
While designing an experiment, one has to take into account both temporal and spatial coherence. Spatial coherence is a measure of the finite size of your source, while
temporal coherence is the more fundamental and interesting quantity related to
fluorescence related to a single atom source. In our case, we are interested in measuring
only the temporal coherence of the source. However, both types of coherence must be 26
taken into account when measuring g2(τ) on any real system. One can model g2(τ) in the following way: consider a source of radiating atoms a distance z from a detector as shown in figure 9. Locate the atoms in the x-y plane by xi and yi, where the subscript i denotes
the atom being considered. The z position of all atoms is assumed to be z, since each
atom’s distance from the other atoms in the z-direction is so much less than their distance
from the detector. We locate a point
on the 2-dimensional face of the detector by the coordinates X and Y.
Detection Plane
Cold Atoms
ri
(xi,yi) (X, Y) Detector
(Diameter D)
z Fig. 9 Imaging of Cold Atoms
We can model the electric field due to each atom as Ei(r, t) = Ei(t)exp(ikri ) ,
where k is the wave number, Ei(t) contains the phase and amplitude information of the
wave, and ri is the distance from the atom to the point being considered. Therefore, at a
point (X,Y) on the detector face, the contribution from the entire cloud is given by 27
E(X,Y, t) = ∑ Ei (t)exp(ikri ) . (3) i
2 2 2 In this case, ri = z + (X − xi ) + (Y − yi ) . Since z >> X, Y, xi, and yi, we can
2 (X − xi ) approximate ri ≈ z1+ 2 . (Y − yi )
Now, in order to get intensity, we multiply equation (1) by its complex conjugate,
which produces a double sum. However, this is the contribution of the cloud to the point
(X, Y) on the detector face. In order to get the average total intensity across the detector
face, we must integrate X and Y over the area of the detector face and then divide by the
area. This yields
1 ik I(t) = E (t)E * (t)exp (x 2 + y 2 − x 2 − y 2 ) * dA exp X (x − x ) + Y (y − y ) , (4) ∑∑ i j i i j j ∫ D []i j i j AD ij 2z
where AD is the area of the detector face.
Now, the quantity in which we are interested is g2(τ), and after evaluating the integral in equation (2) we find
* * ∑∑∑∑ Ei (t)E j (t)Ek (t +τ )El (t +τ ) ijkl
krij D kr D 2J kl I(t)I(t +τ ) = 1 2J1 , (5) 2z 2z 2 2 2 2 exp[]ik( pi + pk − p j − pl ) krij D krkl D 2z 2z
where J1 is the first order Bessel function from the evaluation of the integral,
2 2 2 pα = xα + yα , α = i, j, k, l, the brackets denote an average over all atoms, and D is the diameter of the detector face (assumed circular). 28
The terms in the sum of (3) which do not average to 0 are those for which
2 2 2 2 pi − p j + pk − pl = 0 . For all other terms in the sum, the exponential oscillates rapidly enough that the contributions from these terms averages to 0. This leaves two possibilities: either i = j and k = l or i = l and j = k, which simplifies equation 3 to
E (t)E * (t) E (t +τ )E * (t +τ ) ∑ i i ∑ j j i j 2 kr D I(t)I(t +τ ) = ij . (6) 2J1 * * 2z + Ei (t)Ei (t +τ )E j (t +τ )E j (t) ∑∑ kr D i j ij 2z
* * Now we note that if the fields are stationary in time, ∑ Ei (t)Ei (t) = N E1 (t)E1 (t) for N i
atoms, where E1 (t) is the amplitude of the field emitted by one atom, assuming that all of
* the atoms emit the same average field. This leads us to ∑ E j (t +τ )E j (t +τ ) = j
* ∑ E j (t)E j (t) = ∑ I i (t) = NI1(t), where I1 denotes intensity from a single atom. Thus, j j
2 2 the first term in equation (4) is simply N I1 (t ) .
The second term in equation (4) is composed of a spatial part involving Bessel functions and a temporal part involving field amplitudes. Let us define the average over the spatial part to be “S”. Now, by stationarity of the emitted field, the second term in equation (4) becomes
2 * * * 2 * 2 S∑∑Ei (t)Ei (t +τ )E j (t)E j (t +τ ) = S ∑ Ei (t)Ei (t +τ ) = SN E1 (t)E1 (t +τ ) . (7) ij j 29
This leads to
2 2 N 2 I 2 (t) + SN 2 E (t)E * (t +τ ) E (t)E * (t +τ ) (2) I(t)I(t +τ ) 1 1 1 1 1 g (t,τ ) ≡ 2 = 2 2 = 1+ S 2 I(t) N I1 (t) I1 (t)
2 = 1+ S g 1 (t,τ ) (8)
As advertised, g2(τ) = 1 + S| g1(τ)|2.
Of considerable interest to our experiment was the dependence of g2(τ) on the spatial coherence factor
2 kr D ij 2J1 2z S = , (9) kr D ij 2z Average where the average is taken over all rij. Let us consider a 2-dimensional slice of the atom cloud lying in the focal plane of the imaging lens used in our experiment, as per figure
10. Averaging over the image of the atoms gives
2 kr D ij 2J1 1 a 2z S = drij , (10) a − 0 ∫0 kr D ij 2z which, after a simple substitution, becomes
2 8z kaD J (x) S = 2z dx 1 , (11) kad ∫0 x where x = krijD/2z. A plot of equation (8) for our experimental parameters (discussed below) is given in figure 11. We note that S drops off very quickly, and thus the 30
perceived area of our detector must be small to gain any information from a measurement of g(2)(τ)
rij rj
ri O
a/2
Fig. 10 Image of Atoms on Detector
Figure 11 Plot of Spatial Factor S 31
Radiation Trapping Theory
The goal of our first experiment is to measure the effect of radiation trapping in cold atom clouds. Physically, the atoms in the trap are constantly absorbing and emitting photons. In a normal situation, these emitted photons travel off into space, some of them to be used in our measurements. However, as the trap gets denser, these emitted photons start to have a higher probability to be absorbed by a neighboring atom in the trap. When this happens, the atoms involved in the exchange get pushed away from one another, and also gain momentum in a random direction, which means more motion. Thus, radiation trapping decreases the density and increases the temperature of the trap. Because atoms absorb only on-resonant photons, we expect the effects of radiation trapping to become apparent as the trapping laser is tuned closer to atomic resonance.
Now, we would like to formally treat the effect of radiation trapping on the intensity correlation characteristics of the trap. We intend to use intensity correlation information to sensitively detect the onset of radiation trapping. Let us assume that the atoms in the trap are two level atoms, coherently pumped by a near-resonant laser beam that transfers some population to the excited state. Now, in order to describe radiation trapping, we also include a thermal reservoir formed by spontaneous emission of atoms in the trap.
As stated earlier, the intensity correlation function g2(τ) is given by g2(τ) =
^ → → → → ^ ^ atomic dipole, and is given by E(r,t) = K(r )σ + (t) , where σ ± (t ) are standard notation
→ → → 2 → → ω d (d⋅ r ) r for the atomic raising and lowering dipole operators, and K(r ) = − 4πε c 2 r r 3 0
→ → is the usual spatial dipole pattern at point r radiated by an electric dipole d oscillating at a frequency ω [13, 14]. The spatial dependence of the electric field cancels in the expression for g1(τ), and we obtain
^ ^ σ + (t)σ − (t +τ ) g (1) (τ ) = ss , (12) ^ ^
σ + (t)σ − (t) ss where the subscript ss denotes steady-state.
In order to find g (1) (τ ) , we start by writing down the optical Bloch equations for the atomic populations and coherence of the moving atom:
. i → → ρ = −(n +1)γρ + n γρ − ( Ωe −i k ⋅v ρ ∗ + cc) ee th ee th gg 2 eg . = − ρ gg (13) . 1 i → → ρ = −(n + )γρ − i∆ρ − Ωe −i k ⋅v (ρ − ρ ) eg th 2 eg eg 2 ee gg . ∗ =ρ ge
In this case ρ ee and ρ gg are the atomic populations of the excited and ground states of
the atom, respectively, and ρeg describes the complex amplitude of the induced atomic dipole. In addition, nth is the number of thermal photons available in the reservoir, γ is the radiative decay rate from the excited state, ∆ is the detuning of the laser from atomic 33
resonance, and Ω is the usual Rabi frequency for a stationary atom. As such, there are a
few things that we can expect these equations to agree with physically. One is that ρ ee +
ρ gg = 1, since an atom must either be in the excited state or the ground state, and so the probability of being in either is unity. Also, we would expect the rate of atoms leaving the ground state to be equal to the number of atoms entering the excited state, which specifies
. . that ρ ee − ρ gg = 0. Indeed, both of these conditions are satisfied. The Rabi frequency is related to the laser intensity, since a more intense beam will ramp the atom in and out of the excited state at a faster pace. If nth is less than 1, meanwhile, we can think of it being the probability that a photon emitted by an atom will be absorbed by another atom in the cloud on its path through the cloud.
∗ As usual, ρeg and ρeg are related to the expectation values of the raising and
^ ^ ^ iωt ∗ −iωt lowering atomic σ ± dipole operators as follows: ρeg ≡ σ − e and ρeg ≡ σ + e .
Thus, the action of the raising operator on the ground state g yields the excited state
e , while the action of the lowering operator on e yields g . Reversing either of these operations yields 0, as one cannot lower the ground state or raise the excited state.
By solving equations 13 and applying the Quantum Regression Theorem, we find
[13, 14]
^ ^
σ + (t)σ − (t +τ) exp(iωτ) = ρee (t)exp[−(γ ′/ 2 + i∆)τ]+ , (14) iΩ → → ρ ∗ (t)exp[−i k⋅ v(t +τ)]()1− exp[−(γ ′/ 2 + i∆′)τ] 2(γ ′/ 2 + i∆′) eg 34
→ → where γ ′ ≡ (2nth +1)γ and ∆′ ≡ ∆ − k⋅ v . By taking this equation in the steady state and averaging over the velocity, we can determine
^ ^ γ ss 2 2 k BT σ + (t)σ − (t +τ ) = nth exp(−γ ′/ 2 − iωegτ ) + ρee,0 exp− k τ exp(−iωτ) , (15) ss γ ′ 2m
ss 2 2 2 2 where ρ ee,0 = Ω /(4∆′ + γ ′ + 2 Ω ) . This expression is exactly the numerator of g (1) (τ ) , and also easily gives the denominator. Therefore, our final result forg (2) (τ ) , given by 1 + S| g1(τ)|2, is [15]
1 g (2) (τ ) = 1+ × ss 2 ()nth + ρee,0 2 . (16) ρ ss exp(−k 2τ 2 k T / m) + n 2 exp(−γ ′τ ) + ee,0 B th ss ′ 2 2 2nth ρ ee,0 exp(−γ τ / 2)exp(−k τ k BT / 2m)cos ∆τ
This represents the degree of second-order temporal coherence for the light scattered from a moving atom in the presence of radiation trapping.
We have plotted equation 16 in Fig. 12 for three different values of nth [15], roughly corresponding to the limits of experimental parameters. It should be noted that all three curves start from 2 at zero delay and fall to 1 for long delays, as we would expect for radiation from a collection of independent radiators [8], which we assume the trap to be. We see that as nth gets larger, we see that the light emitted from the trap becomes less
(2) coherent, corresponding to g (τ ) starting at a lower value and falling faster. nth = 0.01 corresponds to a trap density of approximately 1E9 atoms/cc and nth = 0.02 corresponds to a trap density of approximately 2E9 atoms/cc. Thus, we expect to see the effects of radiation trapping at trap densities two orders of magnitude less than in reference [6]. 35
(2) Fig. 12 Theoretical plot of g (τ ) for three different values of nth
36
Experiment
Our experiment measures g2(τ) of the MOT and of optical molasses. A system to measure single photons emitted from the trap on nanosecond time scales was constructed for this purpose. An optical rail was placed at one of the windows on the vacuum chamber. The window used for the measurements is not parallel to the table, and the rail provides a convenient way to keep the optical components in the detection system parallel. The system consists of a lens, pinhole, infrared filter, optical fiber, single-photon photodiode, and multichannel scaler (MCS) as shown in figures 13 and 14.
Fig. 13 Optical Detection System (Photodiode and MCS not shown)
37
Lens f=10cm Optic cable diameter =100 µm
Detector Photon counting I(t) electronics
Fig. 14 Schematic of Optical Detection System
First, there is a 2 inch diameter biconvex lens used to image the cold atoms. The 2 inch lens is necessary to collect as much of the light emitted from the trap as possible.
The center of the chamber where the ball forms is approximately 9.25 cm from the center of the lens, so the focal length of the lens was chosen to be 5 cm. Using the lens equation combined with the formula for the magnification of images shows that this 2f configuration for the lens results in a non-magnified image forming approximately 10.88 cm on the other side of the lens.
As is easily seen from figure 11, the spatial factor S is heavily influenced by the value of D, the diameter of our detector. In our case, for a cloud size (a) of 1 mm and distance (z) from the detector of approximately 10 cm, S falls to 0.2 with a detector face
38
diameter of approximately 2E-04 m. Thus, an iris against the other side of the lens is required to cut down the effective diameter of our detector. The iris closes down to 1 mm, and we assume that this in combination with our 100 µm multi-mode optical fiber diameter gives us the spatial coherence that we require for our experiment. In between the iris and the optical fiber is an infrared filter to cut down on noise, and the iris and filter are enclosed in a black tube to block out background light. The entire rail is then covered with black cloth to further cut down on background noise. The optical fiber is mounted on a 5-way translator and also to another two-way translator to allow more motion parallel and perpendicular to the image plane. The 5-way translation is necessary, as it was determined in tests coupling laser light to the fiber that the fiber is very sensitive to the angle of the incident light. Thus far, the translation that we have is adequate for our needs.
Our optical fiber is attached to a single-photon counting module. This ultra- sensitive photodiode outputs a 5-volt TTL pulse each time a photon is detected. It must be completely dark when the photodiode is plugged in, and it also seems to require its own 5-volt power supply to operate correctly. The pulses from the photodiode are sent to a multichannel scaler (MCS). The MCS registers incoming pulses and sorts them into user defined time bins. It can sort data into a maximum of approximately 16,500 bins, with no lost counts in between bins. The way the MCS works is that the user defines three things: the number of bins, the size of the bins, and the number of passes to take. In our case, the number of passes is always 1. The size of the bins is usually on the 100 ns scale, and the number of bins is almost always the maximum number allowed. For
39
example, if the number of passes was 1, the bin size was 100 ns, and the number of bins was 16000, then the MCS would work as follows: any pulse occurring within 100 ns of triggering would fall into the first bin. Any pulse detected between 100 ns and 200 ns after the MCS is triggered will fall into the second bin, and so on. This occurs until 16000 bins have been filled. The software then outputs a plot of the number of pulses vs. time in blocks of 100 ns.
Fig. 15 Experimental plot of g (2) (τ ) for both the MOT and for molasses
40
Conclusion
At this point we have preliminary data on the MOT and optical molasses, as shown in figure 15. We would expect that the molasses data would fall faster than the
MOT data, since we expect the molasses to have a greater density, but it is difficult to tell from our plot. The data is very noisy, and we are working on techniques to reduce this noise. The data does start at a value above 1, but at long times it oscillates rather than falling to 1. Because of the noise, it is difficult to see the effects of radiation trapping as the trapping laser is tuned closer to atomic resonance. Currently, the optical molasses data is hindered greatly be a lack of signal, something which may be remedied by a new circuit that we will be using to switch off the magnetic gradient. Also, our data-taking process was too slow, something that might be helped by using a different computer to take data. With the data that we currently have, it is difficult to see the effects of radiation trapping on intensity correlation data, but we are confident that we will be able to see the effects when we eliminate the noise from our signal.
I have learned more from this project than I ever could have in any class. I feel well prepared to enter a graduate program in physics, and I hope to do laser cooling and trapping in the future. The skills that I now possess are invaluable, and I am thankful that
I have had the opportunity to participate in this research.
References
41
[1] http://www.boulder.nist.gov/timefreq/cesium/fountain.htm.
[2] Kleppner, D., Physics Today, 50, 11 (1997).
[3] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, Science
269, 198 (1995).
[4] K. B. Davis, M.-O. Mewes, M. R. Andrews, M. J. Van Druten, D. S. Durfee, D. M.
Kurn, W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).
[5] C. C. Bradley, C. A. Sackett, R. G. Hulet, Phys. Rev. Lett. 78, 985 (1997).
[6] T. Walker, D. Sesco, and C. Wieman, Phys. Rev. Lett. 64, 408 (1990).
[7] A.B. Matsko, I. Novikova, M. O. Scully, and G.R. Welch, Phys. Rev. Lett. 87, 133601-1 (2001)
[8] Loudon, Rodney. The Quantum Theory of Light. Oxford. Clarendon Press, 1973.
[9] M.S. Thesis of Soo Kim, Miami University, 2003.
[10] K.B. MacAdam, A. Steinbach, and C. Wieman, Am. J. Phys. 60, 1098 (1992).
[11] C. Jurczak, et al., Phys. Rev. Lett. 77, 1727 (1996).
[12] S. Bali, D. Hoffmann, J. Simán, and T. Walker, Phys. Rev. A 53, 3469 (1996).
[13] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge 1995.
[14] M.O. Scully and M.S. Zubairy, Quantum Optics, Cambridge 1997.
[15] M. Beeler, et al., submitted to Phys. Rev. A.
42
Appendix A:
Notes on Programming
MCS
First, there are a few programming notes about the data-taking process and the
MCS. Most of the time, we needed more than one pass to complete our data. However, we needed to save each pass separately. Therefore, it was necessary to write a JOB file that set the MCS, triggered it, took a pass, saved it, and looped through that process a set number of times. The second thing that must be noted is that the data is saved in a default file format readable only be the Turbo-MCS software. However, a JOB file was written which opened each data file and converted it to ASCII format, which can be read by other programs. One small detail to mention is that this JOB file will not work if the specified directories where the data is stored have long filenames. Most of the time, I put the data into a folder on the desktop until after I ran the job file.
Also worth mentioning is the triggering scheme necessary to take data on the optical molasses. Because the molasses could only be maintained for approximately one millisecond, data was taken in smaller blocks than were used on the MOT. Thus, a function generator was used to turn the magnetic coils on and off and also trigger the
Turbo-MCS. Because of the time required to completely extinguish the magnetic field, a delay circuit was used to trigger the Turbo-MCS after the magnetic field was completely off. Modifying the JOB file to accept the external trigger was simple, and one just has to
43
be careful to select the number of bins per scan so that the MCS is not still taking data when the magnetic fields are turned back on. It should be noted that the external trigger gate on the Turbo-MCS ignores all pulses received when the Turbo-MCS is currently in the middle of a pass. In this way, the MCS is assured of starting to take data only after the magnetic field is turned off, since this is the only time when it can be triggered.
Data-Taking
The bin size is determined by the count rate that the MCS receives from the photodiode. The object is to have either 0 or 1 count in every bin, but to have the bins long enough that the total number of counts is enough to provide statistical averaging.
Double counts (two counts in one bin) are essentially lost data, and so they should be minimized. Generally, we tried to limit the double counts to approximately 10 or less per pass.
Our dark count was typically 7,000 counts/s, and our signal to noise ratio was usually 10 to 1, meaning that we had roughly 100,000 counts/s. With this count rate a bin size of 200 ns seemed to serve our purpose. When taking measurements on the MOT, each pass usually had the maximum number of bins. However, if we wanted to compare the MOT and molasses data, it was necessary to take a smaller number of bins, depending on how long the magnetic field was off. It should be noted that all of these values are very approximate, and the actual count rate ranged from 50,000 to 500,000, depending on the parameters of the trap, and the bin sizes were adjusted accordingly.
44
Data Analysis
The basic idea behind the experimental interpretation of g2(τ) is that we want to plot the probability of detecting two photons in bins separated by a time τ vs. the delay τ.
Thus, we have discrete delays determined by our bin size, and we want to determine how many photons are separated by τ, 2τ, 3τ, etc, and then normalize with the total intensity to find a probability. The output data is a list of time vs. counts, but for the purpose of data analysis it is easier to envision the data as a list of bins vs. counts. We can always go back to the “time picture” by multiplying our bin number by the bin size, since each bin is the same size.
My job was to design a program which took this set of data and output a plot of g2(τ) vs. τ. The logic behind the program is not complicated, but the trick was to get the program to run as simply as possible. First, imagine that we have a data set that looks like figure 16. By inspection, we can see that there are two sets of counts separated by 1 bin
(bins 3 and 4, bins 4 and 5), 3 sets of counts separated by two bins (1 and 3, 3 and 5, and
8 and 10), and so on.
Bin Number Counts 1 1 2 0 3 1 4 1 5 1 6 0 7 0 8 1 9 0 10 1
45
Figure 16 Sample Data Set
The easiest way to analyze this with a program is to feed the number of counts into an array and use a “for” loop to search for counts starting at the beginning of the array. Each time a count is found, another “for” loop is started which runs from the next array element to some user-inputted maximum delay that we want to consider. Each time this loop finds another count, it adds 1 to another array {g2} at the g2 array element corresponding to the index of the second loop, being careful to watch for “off by 1” errors. So, going back to our example, the first loop would stop at the first array element, where it finds a count, and start the second loop. The second loop would add 1 to the second, third, fourth, seventh, and ninth array elements of g2, since those are the indices of the second “for” loop when a count is seen in the original array.
However, it was here that I ran into a bit of trouble. Notice that if we continue running the program, the first loop finds another count at array element 3. Now, we have no data for array elements more than 7 bins away from element 3. Thus, our g2 array will have a very small number of chances to add to, for example, its ninth array element.
There is only one way that could even happen, and that is to have counts in the first and tenth positions. This has the effect of weighting the probabilities towards the front of the distribution, i.e., the smaller values of τ. We note that this happens for almost all values of τ. In our example, the count in the eighth bin will weight τ = 1 and τ = 2, as these are the only possible array elements of our g2 array that it could add to. In fact, in this case it
46
would add 1 only to the second g2 array element, and it would have no chance to add to any g2 array elements with an index greater than 2.
Fortunately, this problem is fairly easily solved once it is recognized. We can only be interested in a smaller delay (value of τ) than there are array elements, and so we could run the loop to the total number of elements in the array minus the number of delays in which we are interested. For instance, if in our example we were interested only in τ = 1, 2, or 3, we could run the first loop up to the seventh array element. Then, each element considered by the loop has a chance to contribute to all three delays. However, the eighth array element could not add any information to τ = 3, and so it would weight the values of τ = 1 and 2. This logic actually seems to work in every case.
Once this basic logic is understood and that particular problem is worked out, the rest of the program is not very difficult. The program was written in OriginPro 7, and it was found that importing the ASCII data files into the spreadsheet went much faster if they were imported as separate columns than if they were appended as rows. Once there is more than one data sheet in the spreadsheet, our above logic involving two “for” loops must be expanded. There must be another “for” loop containing the above two that loops through each separate data set in the spreadsheet. Inside this big loop, the first thing that must be done is to load the data from the set being considered into an array. Then, a running count total must be kept for normalization purposes, and so there is another loop that simply adds all of the counts in the current data set. However, it is important to note that double counts (two counts in one bin) must be counted only as one count for the purposes of normalization. Since these double counts cannot add to our g2 array, they
47
must be counted only as one count. This was discovered by trial and error, as our normalizations were not quite right at first, and this small change fixed the problem. One other overlooked detail is that the outer “for” loop must only loop through every second column, as only every other column is count data.
48
Appendix B
Program for Calculation of g2(τ)
/*------* * File Name: g2(t) Calculation * * Creation: 2002 * * Purpose: OriginC Source C file * * Copyright (c) ABCD Corp. 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007 * * All Rights Reserved * * * * Modification Log: * *------*/
//////////////////////////////////////////////////////////////////////////////////// // you can include just this typical header file for most Origin built-in functions and classes // and it takes a reasonable amount of time to compile, #include
//////////////////////////////////////////////////////////////////////////////////// /* This program will calculate g2(tau) for different delays. It will take many different data sets separated by columns and calculate the coincidences between different bins. It then normalizes this and outputs g2. The loop runs through each data set and finds each time there is a count recorded. When a count is found, the ii loop runs from this count to the maximum delay that you would like to calculate and places a coincidence count in an array at the place where this count occurs. The data should be in 2 columns per data set.
49
Variables and Meanings:
Worksheet: The worksheet you wish to analyze. Change the name in parentheses. numpasses: The number of datasets you are working with. numbins: The maximum delay between counts that will be calculated iR1, iC1, iC2, bIncludeMissingValues, bok: Needed to get the number of rows, not used numrows: The number of rows in each column counted with 0 as first number originaldata: Array storing values of data with each pass g2: Coincidence count array g2norm: Nomalized Coincidence Count totalcounts: total counts in originaldata array analyzed.
*/ ////////////////////////////////////////////////////////////////////////////////////
/*------//////////////////////////////////////////////////////////////////////////////////// NOTES (VERY IMPORTANT, READ THESE)
Check the first three variables at the top to see if they are correct, as they often change between runs. Don't forget to recompile...
The number of passes (numpasses) must be an even number in order for the program to run correctly. If an odd number of passes is required, change the first loop so that it runs to numpassestwice + 1.
When a bin has two counts in it, only one count is recorded for totalcounts. It was found that if these were counted as two, there was a numerical error in the normalization.
In order to make sure that every point has chance to be correlated with a data point "numbins" after it, the last "numbins" data points in each data set are thrown out. They are not included in either the totalcounts or in the counting time ratio.
Program written by Matthew Beeler, 2002. //////////////////////////////////////////////////////////////////////////////////// ------*/
50
void g2() { /*Open Worksheet Declare changing variables*/ Worksheet wks("Secondtry"); int numpasses = 150; double numbins = 8190;
//Get number of rows in data int& iR1 = 0; int& numrows = 0; int iC1 = 0; int iC2 = 1; bool bIncludeMissingValues = true; bool bok = wks.GetRange(iR1, numrows, iC1, iC2, bIncludeMissingValues);
//Declare other variables int numpassestwice = numpasses*2; int numrowsplus1 = numrows + 1; double originaldata[17000]; double g2[9000]; double g2norm[9000]; double totalcounts = 0; double totalcountssquare = 0; int numrowsminusnumbins = numrowsplus1 -numbins;
//First loop runs through data sets for (int w = 1; w < numpassestwice; w = w +2) { //Load data into an array for (int k = 0; k < numrowsplus1; k++) originaldata[k] = wks.Cell(k, w);
//Find total counts in data set for (int kk = 0; kk < numrowsminusnumbins; kk++) if (originaldata[kk] > 0) totalcounts = totalcounts+1;
//Run through data sets to find coincidences
51
for (k = 0; k < numrowsminusnumbins; k++) { //Find the counts if(originaldata[k] > 0) { /*When a count is found, correlate it with counts numbins after it*/ for(int ii = 0; ii < numbins; ii++) { /*If another count is found within numbins, place it properly in g2*/ if (originaldata[k+ii+1] >0) g2[ii] = g2[ii] + 1 } } } }
//Add a column for coincidences and output the coincedence count wks.AddCol("Coincidence Count"); for (int j = 0; j < numbins; j++) wks.SetCell(j, w-1, g2[j]);
//Square intensity for normalization totalcountssquare = totalcounts*totalcounts;
/*Normalize coincidence count by dividing by total intensity and multiplying by total number of bins*/ for (int p = 0; p < numbins; p++) g2norm[p] = g2[p]/totalcountssquare*numrowsminusnumbins*numpasses;
//Output normalized data to a new column wks.AddCol("g2"); for (int a = 0; a < numbins; a++) wks.SetCell(a, w, g2norm[a]);
//Output total counts wks.SetCell(a, w-1, totalcounts);
}