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Accepted for the Libraries -
Date 5/16/06
Date accepted 5-b7OL
All-Optical Production of a Bose-Einstein
Condensate
by
Paul Lindemann
Dwight Whit aker, Advisor
A thesis submitted in partial fulfillment
of the requirements for the
Degree of Bachelor of Arts with Honors
in Physics
WILLIAMS COLLEGE
Williamstown, Massachusetts
May 16, 2006
Figure 1: Surface plot of a Bose-Einstein Condensate.
Acknowledgements
I could not have completed this thesis and the work that went into it over the past year without the guidance and encouragement of countless people. Through the most stressful moments my friends and family offered invaluable support, and I thank them for convincing me to continue with what has been one of the most challenging and enlightening endeavors of my four years at Williams. I would like to especially thank my parents, who have encouraged me to both challenge myself and pursue the things that excite me to the fullest extent possible. I cannot say the result would have been the same without their support.
Particular thanks goes to the previous lab students who in no small part have made my work possible. The thesis of Justin Brown has been an important resource for me this year, and I am happy to say that after a year in the lab I finally think I understand all it has to offer. I would also like to thank Brian Munroe and Arjun Sharma for the contributions they have made to the experiment, and wish them luck in the lab next year.
And finally, thank you Dwight. The enthusiasm with which you explain physics never ceases to amaze me. Between broken power supplies and poor evaporation I know I would not have had the same confidence in our ability to produce BECs without your encourage- ment. More than just the physics of BECs, you have taught me what it means to be a scientist. Enjoy the experimental challenges of the next year and I am sure that before long you will be producing BECs larger than either of us ever imagined.
Contents
1 Introduction 1
2 The Dipole Trap 5
2.1 Trap Characterization .............................. 8
2.2 Efficient Dipole Trapping ............................ 9
3 Modifying Trap Geometries 11
3.1 The Telescope ................................... 13
3.2 Evaporative Cooling ............................... 14
3.3 Increasing the Elastic Collision Rate ...................... 17
3.4 Telescope Optimization ............................. 18
3.4.1 New Trap Frequency Data ........................ 20
3.4.2 Future Work ............................... 22
4 Evaporation 24
4.1 Evaporation Trajectory ............................. 24
4.2 Scaling Laws ................................... 26
5 Condensation 32
5.1 Characteristic Signatures ............................ 32
5.2 BEC Data ..................................... 35
5.3 Conclusion .................................... 37
A Software 40
A.l Translation Stage Software ........................... 40
A.2 DAC Software ................................... 41
B Experimental Data 43
B.1 Longitudinal Trap Width ............................ 43
B.2 Trap Frequency .................................. 44
List of Figures
Frontispiece ....................................
2.1 C02 Laser Beam Cross Section .........................
3.1 COz Laser Beam Path Schematic ........................
3.2 COz Telescope Translation Stage ........................
3.3 Lifetime Calculation ...............................
3.4 Initial Position Trap Frequency for 3/07/06 ..................
3.5 Final Position Trap Frequency for 3/07/06 ...................
3.6 New Trap Frequency Data ............................
3.7 Trap Frequency vs .Temperature ........................
4.1 Laser Power v .VCA voltage ..........................
4.2 Evaporation Voltage Trajectory .........................
4.3 Evaporation Laser Power Trajectory ......................
4.4 Phase Space Density v .Temperature ......................
4.5 Number v .Temperature .............................
4.6 Polynomial Scaling Prediction for the Phase Space Density .........
4.7 Polynomial Scaling Prediction for Cloud Number ...............
5.1 Thermal Cloud Fit ................................
5.2 Thermal Cloud to BEC Progression ......................
5.3 BEC Gaussian Fit ................................
5.4 Asymmetric BEC Image .............................
5.5 Dipole Trap ....................................
LIST OF FIGURES vi
B.1 Longitudinal Trap Width Measurement .................... 44
B.2 Telescope Position: x =0 cm .......................... 45
B.3 Telescope Position: x =3em .......................... 45
B.4 Telescope Position: x =6 cm .......................... 46
List of Tables
4.1 Evaporation Timings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Abstract
We have created BECs of lo5 87~b
atoms. Previous work developed methods to transfer lo6 atoms from a magneto-optical trap (MOT) to a C02 laser dipole trap and to image these clouds through optical depth measurements. We have improved dipole trap loading through the installation of an adjustable telescope which we use to modify dipole trap curvature. Trap geometries with greater curvature exhibit a higher elastic collision rate, allowing us to run evaporation at faster rates and avoid losses to background collisions. A new evaporation scheme allows us to conduct efficient evaporation, avoiding unnecessary losses. Our most recent data shows unambiguous evidence of condensation through observation of predicted BEC signatures.
Chapter I Introduction
Bose-Einstein condensation relies on the exchange properties of bosons, particles that ex- hibit spin values of integer multiples of h/27r. As opposed to fermions (those particles with half-integer spin values), multiple bosons are able to occupy the same energy state; this means that at low enough temperatures a substantial number of atoms may occupy the lowest energy state, known as condensation. This phenomenon was first predicted by Al- bert Einstein in 1925 and was experimentally observed in a dilute alkali gas in 1995, winning the 2001 Nobel Prize in physics [l,21. We are proud to report that we have been able to create Bose-Einstein condensates of -lo5 87~b
atoms.
Bose-Einstein Condensation occurs when a substantial number of the particles in a sample (in our case, a dilute gas of 87~b) occupy the ground state. In order to determine whether a gas has condensed and under what conditions condensation should occur, we look to count the total number of particles that occupy this ground energy state.
Bosons are indistinguishable particles that obey Bose-Einstein statistics, following a number distribution that allows for the population of multiple particles in one state while minimizing their free energy. Schroeder works out the Bose-Einstein distribution for an ideal gas, which gives us the average number occupancy of a single state for a given temperature T, the energy E of the state when occupied with one particle, and the chemical potential
P PI,
1
nBE = e(t-p)/kT -1 (lJ)
We use this distribution, along with the density of states, the function that counts degen-
CHAPTER 1. INTRODUCTIOiV
eracy of states at a particular energy level, to count the entire number of particles in our system: the average number of particles in a state multiplied by the density of states gives us the total number of particles occupying states at that energy. We then sum over all possible energies to obtain the total number of particles 131:
where g(t) describes the density of states. To calculate the density of states for spin zero bosons in three-dimensional states we can write the energy levels as
modeling our system as particles in a box. Redefining & as e, we see that (t/eo)1/2 is the radius of the sphere defined by nz + ni + n?. The volume of the sphere in the first octant (all quantum numbers must be positive) then contains the states for all energy levels, and the number of states for a given energy level is all those states falling on a shell of the sphere with radius (E/E,)'/~ and thickness de. To find the density of states we evaluate dS/de where S = ~(C/C,)~/~
is the total number of states 141:
V is the volume of our dilute gas. To evaluate N, we convert our summation to an integral, integrating on dt from 0 to infinity. Since N is large this is a valid approximation. Although this integral cannot be solved analytically, we may solve numerically for chosen values of p. Positive values of ,LL are not valid solutions, as N will blow up for p = E. We may, however, choose p to be negative or 0. Schroeder gives us the solution for p = 0:
A cursory glance at this solution indicates that there is only one value of T (called T,) for which this solution is valid, as it would make little sense for N to be dependent on T. Choosing negative values of p one will find different values of T, greater than our T for ,LL = 0 , at which the solution gives the correct total number of particles in the system. It is for temperatures less than T, when this solution breaks down. The problem arises when we convert the summation to an integral. For t = 0 the original summation will yield exactly
CHAPTER 1. 1.lNTRODUCTION
one state; there is only one way to describe a particle in the ground state of momentum in the x, y, and z directions. With the integral, however, we get no states at E = 0, as the volume of a sphere with radius 0 is 0. At temperatures where the occupancy of particles in the ground state is much less than 1 we can ignore this error; once particles begin to populate the ground state this approximation will no longer suffice. This begins to occur at Tc:for temperatures below Tcwe must still have ,u = 0, but our solution for N no longer counts all the particles in the system. Instead, it counts only those particles occupying excited states, as these are still included in the integral. The population of particles in the ground state (i.e.; the condensate, which we call No) is then the difference between the total number of particles and those in excited states:
Solving equation 1.5for Tcwe get
where n is the number density of particles, N/V. As 87~b
is a spin -1 particle there are three possible spin states, adding an overall factor of 3 to our calculation of N (assuming all states are equally probable). At typical atomic vapor densities of -1014 atoms/cm3 we find that, for 87~b, Tcis of order lop7 K, which although difficult to achieve, is well within the realm of possibility.
Alternatively, one may reformulate the above temperature conditions for condensation into a unitless parameter called the phase space density, notated as p, where
XdB is the deBroglie wavelength, defined as h/(2nrnk~)'/~. At phase space densities of 2.612 or greater, condensation is said to occur. At room temperature a dilute gas has p = using the methods derived in Brown [5]to calculate the necessary quantities this thesis will show that we have achieved phase space densities of order unity.
We can also think of condensation as the result of overlapping wavefunctions. As our particles get colder and denser the requirements for condensation, as stated in the formula- tion above, tell us that the volume per particle (lln) must be on order with the cube of the
CHAPTER 1. INTROD UCTIO-N
deBroglie wavelength for a particle with mass m and energy kT (up to a factor of T).More precisely, the cube of the deBroglie wavelength of each particle must be 2.612 times the volume it occupies (i.e.; the wavefunctions of the particles overlap). When this happens, the individual particles cease to have their own independent wavefunctions, and instead we find a wavefunction that is shared by all the particles in the ground state.
In this thesis I will describe steps we have taken over the past year that have led to the creation of BECs of lo5 atoms. Two major improvements to our experimental setup contributed to this monumental result. A new adjustable telescope placed in the beam path of our dipole trapping laser allows us to modify the geometry of our trap to achieve both maximum number and efficient evaporation. We have also installed new equipment allowing us to conduct evaporation with greater timing precision. Together, these improvements have led to the production of BECs, which we have observed through specific BEC signatures. After describing these improvements and the justifications for them I go through an analysis of our latest BEC data including calculations of the phase space density, and improvements to our setup that will need to be made to better characterize our BECs.
Chapter 2
The Dipole Trap
We go through a series of steps in making a BEC. Beginning with a vapor of rubidium in a vacuum chamber at lop9 Torr, we use three retro-reflected beams tuned slightly below resonance of the s7~b 780 nm F = 2 + F' = 3 absorption line. Atoms moving toward any one of these beams will see Doppler shifted photons closer to the 780 nm absorption peak. At this wavelength 87~b will absorb and then scatter photons, losing energy and thus momentum. Since the three retro-reflected beams cover all three dimensions in both directions, an atom moving in any direction will encounter a Doppler shifted beam that induces scattering. The result is a reduction in the velocities of all atoms, regardless of direction of travel [6].
To localize the atoms we use anti-Helmholtz coils that create a magnetic field zero at the center of the trap which increases linearly in all directions. The magnetic field causes a position dependent Zeeman shift of the atomic energy levels, providing a confining force pointing to the center of the trap. Called a magneto-optical trap (MOT), these methods allow us to trap lo9 atoms and increase their phase space density from 10V30 to 10W8 [6]. We then load a small fraction of these atoms into a conservative trap created by a high intensity C02 laser, increasing the phase space density to la-* [5]. Finally, to further cool this rubidium vapor we gradually decrease the intensity of the C02 laser, allowing the most energetic atoms to escape, leaving us with a cold, dense core of atoms that may condense into a BEC.
In his thesis, Leon Webster describes the experimental setup for the MOT and dipole
CHAPTER 2. THE DIPOLE TRAP
trapping i11 detail. The intricacies of our MOT setup need not be addressed here; for a more detailed discussion see Webster 161. However, over the past year we have made significant improvements in our dipole trap setup, and for this reason I return to the topic here.
We begin by modeling our 87~b atom as an electric diplole. Classically, an electric dipole will oscillate in the presence of an oscillating electric field, its phase determined by the relation of the natural resonance of the dipole to the rate of oscillation. Solutions to the equations of motion for a damped driven oscillator reveal that at frequencies below resonance the dipole will oscillate in phase with the field; on or near resonance it will be 90"out of phase with the field; and above resonance it will be 180"out of phase with the electric field. A dipole lowers its potential energy by oscillating in phase with the electric field. This means that a dipole will have the lowest potential energy in an electric field oscillating below resonance.
By creating an electric field gradient with a localized maximum intensity it is possible to create a negative potential well; dipoles will experience a force pushing them towards the region of highest intensity (F= -VU). Quantum mechanically, this effect is described as a lowering of the ground state energy level of the particle and is known as the AC Stark shift. As shown in Webster and Grimm [6, 71, either approach gives the same theoretical predictions.
In our case we use a Gaussian profile laser to create the oscillating electric field that traps 87~b atoms. The limitations to this setup and an understanding of the conditions under which particles will be trapped without inducing scattering require a look at the numbers relevant to our experiment. If we operate our laser too close to resonance a significant number of particles will absorb a photon, gaining the requisite energy to escape the potential well; too far from resonance and a higher intensity electric field is required to achieve an appreciable potential energy well. Following Brown and Grimm [5, 71,we obtain the scattering rate rsc (the rate of photon absorption) and trap depth UtT, as a function of laser intensity I(r), absorption frequency vo,laser frequency v and detuning A = vo-v.In the limit of large detuning:
CHAPTER 2. THE DIPOLE TRAP
Figure 2.1: Schematic of the COz laser beam cross section.
I? is the atomic linewidth, r = 5.9 MHz. The intensity of the laser, I(r),as a function of the radial distance from the principal axis is described as
-2r2I(r) = I(x) exp
and I, is the peak laser intensity, I, = [8]. Fig. 2.1 shows a
I(z) = scheinatic of the COz laser beam cross section, defining the longitudinal (2) and transverse
(r) directions as well as the beam waist (w,). The longitudinal distance x is measured from the center of the beam waist. The Rayleigh range x, is defined as the distance from the center of the focus along the principal axis to the point where the intensity has decreased by half, given as
z, =7iw2,/x (2.4)
An analysis of eqs. 2.2 and 2.1 shows Utrapto scale as I/A and rsc as l/n2.For high intensity and large detuning then, it becomes possible to maintain a negative potential energy well while minimizing scattering due to absorbed photons. By using a 100 W CO2 laser with a wavelength of 10.6 pm (significantly longer than the 780 nm 87~b
absorption line), we are able to achieve exactly that. Our most recent dipole trap configuration gives
CHAPTER 2. THE DIPOLE TRAP 8
a trap depth of 250 pK, easily capable of trapping and holding more than lo6 atoms. The approximation in eqn. 2.2 gives a scattering rate of 6.6 x lop5 s-l, which will be even smaller when laser intensity is decreased during evaporation. For all intents and purposes then, losses due to photon scattering will be insignificant.
2.1 Trap Characterization
To characterize the trap created by our COa laser there are two measurements we employ: the trap depth Uo and the trap frequency v. As opposed to the absorption frequency of 87~bor the frequency of the C02 laser, the trap frequency is the natural frequency of oscillation of atoms in the potential energy well and is a measure of the trap curvature. There are actually two different frequencies associated with the trap, in the transverse (v,) and longitudinal (v,) directions. We obtain these through a Taylor expansion of the trap depth around the minimum:
U, is the trap depth at the minimum and u, and u, are the trap frequencies [9]:
w, is the focused beam waist, I, is the peak laser intensity, and E = p= = 9.97 X
8T2,,,3a
lo-'' mfi. Due to the fact that frequencies in the longitudinal direction are of order .5 kHz as calculations in Chapter 3 show, we are unable to accurately measure these frequencies and we rely on measurements of trap widths of expanded clouds to determine initial trap width and then trap frequency in the longitudinal direction [5]. Looking at eqn. 2.5 the trap frequencies provide a measure of the curvature of the potential energy well. Experimentally, we measure trap frequencies through a process called parametric resonance, whereby the amplitude of the laser intensity I, is varied at a specified frequency. Atoms are ejected from the trap at u, and 2u,. The drop in number at 2v, is the more prominent as exciting
CHAPTER 2. THE DIPOLE TRAP
the trap at this frequency is the equivalent of ejecting atoms twice per cycle as opposed to once per cycle at frequency v,. While we are able to consistently measure 2vr for a variety of trap geometries, reliably measuring v, has as of yet proved unsuccessful. ]For a more detailed discussion of parametric resonance see Brown [5].
Using eqn. 2.6a we may also obtain a value for the focused beam waist. Noting I, = 2P/A, where P is the total laser power and A is the cross sectional area of the beam,
With a measurement for the beam waist in hand we are able to obtain an indirect measure- ment of v, through the Rayleigh range (eqn. 2.4). This alternative method of measuring v, will become useful in Chapter 3 (for a discussion of the limitations of this method see Appendix B. 1).
2.2 Efficient Dipole Trapping
Having developed a setup that creates a conservative trap with minimal scattering, the question becomes how to create a trap that is as efficient as possible, (i.e.; that will collect and cool the maximum number of atoms from our MOT). The probability of trapping an atom in the dipole trap is dependent on two factors: the Boltzmann factor of a given atom (how likely it is to have the necessary energy to be confined by the trap) and the surface area of the trap, measured at some energy equipotential (more precisely, the probability is actually dependent on the flux through that surface). Achieving the greatest Boltzmann probability will depend on both minimizing MOT temperature and maximizing trap depth. Surface area is determined by trap geometry. After being focused down inside the vacuum chamber, the COa laser creates an equipotential trap known as a prolate spheroid, described mathematically as
(r/u,l2 + (~/u,)~= 1 (2.8)
where r is the radial distance from the principal axis, z is the distance along the principal axis, measured from the center of the focus, and 0, and o, are the lengths of the two semi- axes along the r and x directions, respectively [lo]. A prolate spheroid has a characteristic
CHAPTER 2. THE DIPOLE TRAP
cigar shape, with the requirement n, > a,. Its surface area is
where a = (1-(aT/0,)2)1/2. AS data in section 4.2 and Appendix B.l shows that our setup produces a trap with a, >> o,values of a are limited to a z1and the term arcsin(a)/a is therefore constant at 7r/2. We factor out a a,, and again invoking the constraint a, >> a, we may discard the remaining o, term within the parentheses, as it will be dominated by the a, term. The equation for the surface area then simplifies to
For an equipotential surface at energy E = kT the lengths of the semi-axes are given by the trap widths in the transverse and longitudinal directions [5, 111.
defining w,, I,, and c as before. Looking to the w, dependence in the trap widths (a, oc wz through the Rayleigh range) we find S oc w;. A modest increase in the beam waist then will greatly increase the trap surface area, increasing the number of atoms collected in the trap. At constant power, however, there is an inverse relationship between the beam waist and potential well depth. Since the trap depth is proportional to I, the beam waist increases as the depth of our dipole trap decreases (eqn. 2.3). This tradeoff is crucial: while it is clear that both maximum trap depth and surface area would contribute to greater trap loads, each comes at the expense of the other. However, once we achieve trap depths necessary to trap atoms with energies of order kT a deeper trap will not substantially increase our loads and we may instead manipulate the beam waist to achieve maximum surface area. In steps after dipole trap loading this is no longer the case, and we find that a tighter trap better suits our purposes. In the following chapters I discuss how evaporation requires a more compressed trap and how we are able to manipulate the beam waist to effectively maneuver between two different geometries during BEC production.
Chapter 3
Modifying Trap Geometries
Installed in the port window of the vacuum chamber through which the CO2 laser beam enters is a 1.65 in. focal length lens. This lens focuses the beam down to a minimum beam waist and maximum intensity at the center of the chamber, overlapping with the position of the MOT. As indicated above, the beam waist of the focused beam dictates both the trap depth and the curvature of the trapping potential. As with any laser beam the minimum size of the beam waist is diffraction limited, and we determine this limit through an analysis of Gaussian optics.
We approximate the COz laser as a single mode TEMoo Gaussian beam. Although this is not a perfect approximation, this will be the strongest component of the beam and it is sufficient for our purposes. Contrary to the simple geometric case the radius of the beam, measured out to l/e2 of the maximum intensity (N 13.5%) diffracts as
where w, is the beam waist at x = 0 and s is the direction of propagation [8].As eqn. 3.1 indicates any collimated Gaussian beam propagating in space will therefore spread out as it travels. The diffraction limit dictates a minimum beam waist radius that we are able to achieve by focusing down the beam with a lens for a given initial beam waist. In the far field limit when 2 >> the diffraction limited equation.3.1 reduces to
which is equivalent to the classical case. For w, and X of order 10V6 m and at a distance
Figure 3.1: Prior to being sent into the vacuum chamber, the COz laser beam passes through a collimating telescope, an AOM, and the trap modifying telescope.
of order m (i.e.; the focal length) the above inequality holds and eqn. 3.2 is a valid approximation. For simplicity we measure the initial beam waist (the spot size) at the lens, as the lens will be a focal length f from the focus for an entering collimated beam. Evaluating eqn. 3.2 for x = f we find
Equation 3.3 is unambiguous. The minimum diffraction limited beam waist is inversely proportional to the spot size: focusing down a larger beam will result in a smaller minimum beam waist and vice versa. Relating this to our previous discussion of trap shape we find that both trap depth and frequency are proportional to spot size. Through manipulation of the spot size we are able to modify the shape of our trap, an essential step in our production of a BEC. To do this, we have installed a telescope in the path of the C02 beam, mounting the first lens to a motor powered translation stage.
Figure 3.2: The telescope translation stage.
3.1 The Telescope
We use a Parker 401XR stage with lOOmm travel (fig. 3.2) operated by a ViX 250IM driver controlled through LabView (see Appendix A.l). This telescope is the last optical system through which the C02beam passes before entering the chamber, having previously passed through a collimating telescope and an AOM (see fig. 3.1). The stage has a maximum velocity of 0.1 m/s. We use basic geometric optics to determine the position of each lens. For an entering collimated beam, the resultant beam leaving the telescope will also be collimated when L = fl + f2, where L is the distance between lenses and fi and f2 are the focal lengths of each lens. For L < fi + fi the beam will emerge diverging and for L > fi -t f2 it will converge. We position the translation stage such that moving the first lens allows us to alternate between these two regimes.
At the first position of the telescope (x = 0 cm) L > fi + f2. The telescope is 50 cm from the vacuum chamber, and as the beam converges we will achieve our smallest spot size at the port window. As the previous discussion of trap geometries explained, a small spot size results in a larger trap surface area, ideal for loading maximum number. Our latest results give initial loads of 3.5 x lo6 atoms after a hold time of 300 ms.
To motivate a justification for altering the trap geometry we first turn to a theoretical discussion of the next step in BEC production, forced evaporation, and the requirements for optimal evaporation efficiency.
3.2 Evaporative Cooling
Forced evaporation is a process whereby lowering the trap depth, combined with constant rethermalization of the atoms in the cloud allows us to lower the average energy of the atoms in our trap with minimal losses in number. To lower the trap depth we decrease the intensity of the CO2 beam that the AOM deflects away from a beam dump and towards the vacuum chamber by decreasing the amplitude of the radio frequency sent to the AOM. For a more detailed discussion of our AOM setup and its operation see Webster and Brown. Prior to condensation, the atoms in the cloud obey a Boltzmann distribution with an average energy kT;it has been shown that the average energy of an atom in an optical trap is approximately 1/10 the trap depth [12]. If the trap depth is decreased, the most energetic atoms will no longer be confined by the trap potential and will escape the cloud. This has the effect of removing the tail end of the Boltzmann distribution of energies, which rethermalize to create a new average energy that is 1/10 of the new trap depth. Rethermalization is achieved by two-body elastic collisions within the cloud; the atoms collide, exchange energy and a new thermal distribution is established. When kT = Uo/10 evaporation stops because essentially no atoms have energies of order 10kT. The rate of rethermalization is determined by the elastic collision rate of atoms within the cloud:
where 0is the scattering cross section (i.e.; the cross sectional area of the atom; for 87~b o=87ra2 = 6.3 x 10-l6 m2), n is the number density of the cloud, and VRM~is the root- mean-squared velocity, all of which are values that affect how often particles will collide. At low temperatures a and the cloud volume are both constant for decreasing laser intensity I,. As eqn. 2.3 shows, I, is a constant that multiplies a Gaussian; changing this overall constant does not affect the full width at half maximum (a quantity contained inside the Gaussian) nor should it affect the width at any other value. For lowering laser intensity this means we witness a decreases in the elastic collision rate that scales as NT'/~.
Our ability to measure trap frequencies, however, lends itself to a less intuitive definition of the elastic collision rate. In section 2.1 I described how we use the trap frequency and trap depth to characterize the trap; we extend that here to characterize the trap in terms of the elastic collision rate. Recasting equation 3.4 we rewrite the formula for the elastic collision rate as
4~~mov,$
(3.5)
Y= kT
where v,is the transverse trap frequency, v,is the longitudinal trap frequency, and m is the mass of a 87~b
atom1 [13]. As explained in Chapter 2, we cannot directly measure v,. However, by extracting a value for the beam waist from our knowledge of the transverse trap frequency and eqn. 2.6a we are able to indirectly obtain a value for v,that is accurate up to a factor of 3 (see Appendix B.l).
Efficient evaporation (i.e. ; minimizing number losses) requires allowing the cloud to rethermalize following each decrease in laser intensity. As number and temperature decrease, however, the elastic collision rate decreases and evaporation must be slowed to maintain efficiency. Evaporation inevitably results in number losses, but running evaporation without allowing proper rethermalization results in much greater losses. A cloud that has not rethermalized will have a greater number of atoms at higher energy levels, based on the Boltzmann distribution for the previous trap depth. If the trap depth is then lowered again before these atoms are able to reestablish a new distribution at a lower average energy they will escape the trap. While this still lowers the energy of the trap, it does so at a considerable cost in number density.
Losses to collisions with the background gas in the chamber, however, puts a limit on how slow we may run evaporation. While two body collisions within the cloud transfer energy and create a new Boltzmann distribution, collisions with much more energetic atoms from the background gas result in further number losses. The rate at which these "bad" collisions occur -unlike losses from forced evaporation, losses to the background gas occur with equal probability for atoms at all energies -is determined by the lifetime of our clouds. Measuring the number of atoms remaining in the trap after different hold times we find a l/e lifetime of 11.63 seconds (see fig. 3.3). This is done by taking optical density measuremnts of CCD cloud images: for a description of our imaging system and the methods used to calculate
'1.44 x kg/atom
Lifetime Calculation 3/09!06
Hold Time (ms)
Figure 3.3: Lifetime calculation from 3/07/06.
trap number see Brown.
O'Hara et al. have derived scaling laws for the elastic collision rate solely as a function of trap depth [13]. For initial elastic collision rate yi and trap depth Ui the final elastic collision rate scales as
assuming an average energy to trap depth ratio of 1/10. The eWrbgt term accounts for losses to the background gas, where rbgis the inverse lifetime of the cloud and t is the time over which evaporation is conducted. With a lifetime of 11.63 seconds rag= 27r/11.63 = .54 s-l. To determine t we approximate a total time for the entire evaporation process of -3 seconds.
We determine the initial elastic collision rate through equation 3.5, measuring T, vr, v, and N prior to evaporation. We obtain temperature data through methods explained in Brown: assuming negligible trap widths in the transverse direction at t = 0 we determine VRMS based on the linear slope of cloud widths as a function of expansion time and from here extrapolate T. Our latest data gives us values of Ti = 2.54 x K, v,= .5 kHz and N = 3.65x lo6 atoms. For a maximum laser power of 60.5 W we use eqn. 2.7 to find an initial
CHAPTER 3. MODIFYING TRAP GEOMETRIES
beam waist of w, = 72.1 pm. We then obtain a value for the Rayleigh range (eqn. 2.4), z, = 1.6 x 10V3 m, and through eqn. 2.6b calculate a longitudinal trap frequency of u, = 20 Hz. Plugging these values into eq. 3.5 we find an initial collision rate of yi = 59 s-l. To achieve phase space densities of order unity we require a final temperature of approximately 100 nK. As T is proportional to U, Uf/U, = 100x 10-'/2.54x = 4.0x10-~ and yf= .26 s-I.
This gives a rethermalization time of -24 seconds, roughly on scale with the trap lifetime. With these trap parameters efficient evaporation would entail substantial losses in number to collisions with the background gas. The challenge in forced evaporation then is to conduct the entire process in a short enough time span so that losses to the background gas do not dominate, while still allowing enough time for efficient evaporation. To do this, we seek to increase the elastic collision rate.
3.3 Increasing the Elastic Collision Rate
Returning to equation 3.4 we see the elastic collision rate is dependent on the number density of our clouds, the temperature, and the cross sectional area of the cloud. Collapsing the cloud volume will increase the number density substantially, as we squeeze the same number of atoms into a smaller trap. Incidentally, this squeezing process does not result in an increased phase space density, as squeezing the cloud simultaneously heats it, raising the temperature.
Here is our motivation for the installation of the adjustable telescope. Moving the second lens to a position satisfying the condition L < fi + f2 results in a diverging beam and a larger spot size. As the discussion at the beginning of this chapter explains, a larger spot size gives us smaller trap widths in both the radial and axial directions. The volume of our prolate spheroid trap is
For smaller spot size then we witness a decrease in trap volume that scales as w: and a concomitant increase in number density. As explained previously, increasing the spot size also results in a greater trap depth. Since atoms in the trap will equilibrate to an average energy of 1/10 the trap depth, the temperature increases, contributing to a higher elastic
CHAPTER 3. MODIFYIANGTRAP GEOMETRBS
collision rate and explaining why squeezing heats the trap. Decreasing the spot size then increases the elastic collision rate, allowing us to run forced evaporation faster, thereby minimizing our losses due to background gas collisions. To determine the effectiveness of the telescope in modifying trap geometry we recast equation 3.5 in terms of yf/yi. Our constants cancel out and we are left with
An analysis of eqs. 2.1 and 2.6a shows that trap frequency should scale as the trap depth, or conversely, the temperature. Both trap depth and trap frequency scale as l/w:, the trap depth through its proportionality to I, = and the trap frequency through $= . This allows us to eliminate T from yi/yi, expressing the relation solely as a function of the initial and final v, and v,. The math may indicate we could eliminate v, as well, but this is valid only in the case of a perfect Gaussian, which Appendix B.l illustrates is not the case here. Using v, to determine v, will then give us only an approximation for the ratio yf/yi, which suffices for our purposes in estimating the factor change in yf.
The differences between eqn. 3.8 and the scaling laws for the elastic collision rate during evaporation found in eqn. 3.6 should not be confused. Here we are altering the beam waist, which collapses the trap without changing the phase space density or the number of atoms in the trap. During evaporation the elastic collision rate decreases as a result of changing number and average energy, the express purpose being to increase the phase space density. As the elastic collision rates in trap squeezing and evaporation are modified in different ways, likewise they will scale differently.
3.4 Telescope Optimization
One of the challenges we have encountered installing the telescope has been determining its optimal position along the C02 beam path, and since creating a BEC we have sought to improve on this aspect of our experimental setup. While we aim to achieve a trap that is as large as possible during trap loading, a trap depth less than 10 times the average energy of the atoms in the MOT will result in inefficient loading: since the average energy of atoms in the trap is -1/10 the trap depth, for trap depths less than l0kT we are only able to trap a fraction of the atoms in the MOT, determined by the Boltzmann distribution of energies.
Initial Telescope Position (OcmJ 1.5?:0modulation 2?24/06
Modulation Frequency (kHz)
Figure 3.4: Trap frequency measurement at the loading (x = 0 cm) position of the telescope. Modulation was conducted at 1.5% amplitude modulation for 1200 cycles.
Similarly, we have discovered there is an upper limit to how tight a trap we may create; that is, there is a lower limit of how small a beam waist we may produce, regardless of the spot size of the COz beam. Through use of the correct combination of lenses and an optimal position of the telescope, we seek to obtain the maximum variation in trap geometry (as observed through the trap frequency) within these constraints.
Figs. 3.4 and 3.5 show the measurements of the transverse trap frequency y,conducted at both positions of the telescope for a previous overall position of the telescope, dating to the end of February (2v,,,,, = 1.5 kHz and 2qfina, = 6.5 kHz). Again employing eqn. 2.7 we find an initial beam waist of w, = 55.7 pm and a final beam waist of w, =
24.7 pm. Recalculating the Rayleigh range we find values for the longitudinal trap frequency of vzinitiai= 38 Hz and vZfinoi= 431 Hz.
The temperature at each position was measured to be TZzocm = 3.36 x K and Tz=locm = 1.80 x lou4 K. To find the total number of atoms in the trap we extrapolate backward in time from the number measurements in fig. 3.4 not affected by parametric resonance (i.e.; those data points that are far off resonance) to get a value of N = 1.82 x lo6. Plugging in values for vr, v,, T,and N into eqs. 3.5 and 3.8 we find an initial elastic collision
Final Telescope Position (10cm) 5?&modulation 2/23/06
Modulation Frequency (kHz)
Figure 3.5: Trap frequency measurement at the evaporation (x = 10 cm) position of the telescope. Modulation was conducted at 5% amplitude modulation for 1200 cycles.
rate of yi =96 and a factor change in the elastic collision rate of (-yf /-yi) = 52.6.
While this is a substantial shift in the elastic collision rate, we were unable to produce BECs for these trap geometries. The yf above is the elastic collision rate prior to evapo- ration; plugging this value into equation 3.6 as yi (in this equation it is the initial elastic collision rate, measured before evaporation) we find a final elastic collision rate of 24 s-', roughly a factor of 50 faster than rbgEfficient evaporation generally requires an elastic collision rate 100 times faster than the trap lifetime [12]. In addition to initial loads with less atoms than we have achieved previously, evaporation efficiency deteriorated at low laser intensities, prior to condensation. The latter problem suggests the trap curvature was not big enough to still hold enough atoms at low laser intensities for condensation to occur. To solve this problem we have since repositioned the entire telescope, seeking to create trap geometries with greater curvature and trap depth.
3.4.1 New Trap Frequency Data
Fig. 3.6 contains the latest frequency data for our dipole trap after moving the entire telescope closer to the vacuum chamber, measured at new translation stage positions of
*Trap Frequency(kHz)
Trap Frequency vs. Stage Position
Translation Stage Position (cm)
Figure 3.6: Trap frequency measurements for new telescope position at x = 0 cm, x = 3 cm, and x = 6 cm. Modulation was conducted for 5000 cycles at 5% amplitude modulation for x = 0 cm and x =3 cm, and 2% for x =6 cm.
x = 0 cm, x = 3 cm and x = 6 cm (see Appendix B.2 for experimental frequency data). In addition, we also replaced one of the telescope lenses, increasing the ratio of fi/f2 from 1015 to 10/3. As the data indicates, we have been able to achieve higher trap frequencies, measuring a maximum frequency of 271, = 9.5 kHz. We were not able to further increase the trap frequency by moving the translation stage the full 10 cm range. As we have achieved trap frequencies as high as 214 = 12 kHz in the past this may be due to the fact that at this specific telescope position different modes of the beam begin to focus independently, decreasing the overall intensity. Analysis of this data gives trap frequencies = .5 kHz and 71, = 4.75 kHz. Temperature data at these two positions gave
of v, values of Tinitial= 2.54 x K and Tfinal= 2.02 x K. Recalling the calculations found in section 3.2 for the initial position we find a beam waist of w, = 72.1 pm and a longitudinal trap frequency of v,= 20 Hz. A similar analysis for the 6 cm position yields a beam waist of w, = 23.4 pm and a longitudinal trap frequency of v, = 579 Hz. These are considerable improvements for both translation stage positions, and in addition to a smaller trap frequency at the 0 cm position, we are now trapping clouds with initial loads
Trap Frequency vs. Temperature
Figure 3.7: Trap frequencies measured for three positions of the telescope.
of 3.65 x lo6 atoms. Plugging this trap frequency and temperature data into eqn. 3.8 we find an overall increase in the elastic collision rate of 3/7i= 332.
The proportionality between T and v, explained in section 3.3 allows us to test the consistency of our frequency and temperature data in order to ensure we are following the scaling laws and not competing with other mitigating factors that may be hindering our ability to achieve higher trap frequencies. Fig. 3.7 shows that, within the error bars for our trap frequency data, we do follow this proportionality.2
3.4.2 Future Work
While we have produced BECs with the telescope in this new position it remains the work of future students to further optimize the operation of the translation stage. This includes determing the optimal positions along the stage for trap loading and evaporation that yield maximum changes in trap geometry. It also includes the possibility of changing the time and time scales over which the stage is moved in order to put us in the best position for
'It should be noted that the large error bar associated with the x = 3 cm measurement in comparison to the x = 0 and x = 6 cm data is the result of a larger amplitude modulation at this position, which broadens the line at resonance.
evaporation. We have found that evaporation works best when it is started before the translation stage has completed its move. An investigation as to when evaporation should begin in relation to the translation stage move may lead to further improvements.
While this latest trap frequency data is a marked improvement over our previous results we have noticed losses in number that occur due to squeezing down the trap. We do not believe these losses pose a significant problem, yet insofar as one of the major aims of our work is to produce BECs as large as possible it is something that should be minimized. A possible solution to this problem may be to decrease the acceleration of the translation stage prior to evaporation. Ideally we would hope that moving the telescope would alter the trap geometry without moving the beam focus itself. Currently, the focal position of the beam inside the vacuum chamber shifts by -2 mm between the two translation stage positions. As the trap is accelerated a force is applied to the atoms in the trap. In order to contain the atoms the trap must exert a restoring force which will be dependent on the trap slope. For small trap depths and large translation stage accelerations atoms could be ejected from the trap. At the same time, the translation stage also creates considerable mechanical noise during its move. Vibrations at v or 2v in either direction will excite atoms in the trap; in order to avoid these the move time for the stage should be as short as possible. Balancing these competing factors may present new challenges in trying to minimize number losses, but as we are now able to achieve even greater changes in trap frequency with smaller moves of the translation stage, we may be able to decrease stage acceleration and still be able to alter the trap geometry in a short enough time to avoid unnecessary excitations to the trap.
Chapter 4
Evaporation
4.1 Evaporation Trajectory
As mentioned, evaporation is conducted by lowering the intensity of the C02 laser beam, allowing time for rethermalization while seeking to minimize additional losses. The laser beam is sent through an AOM, and to lower the beam intensity we decrease the amplitude of the radio frequency signal sent to the AOM, thereby decreasing the proportion of the oeam that is deflected away from the beam dump and sent into the vacuum chamber. An RF signal from our signal generator is mixed with a DC voltage from 0 to 10V in a voltage controlled attenuator (VCA) that attenuates the RF signal. Over the past year we have learned that efficient evaporation requires high precision timings as the rate at which beam intensity is lowered changes during the evaporation process. To accommodate this we have installed a digital to analog converter (DAC) that allows us to control the DC voltage sent to the VCA by way of our DIO-128 board. The DIO-128 board gives us precision timings with microsecond resolution, something we cannot achieve with LabView alone. The evaporation timings are calculated by LabView prior to loading the dipole trap and are then sent via the DIO-128 board to the VCA during evaporation, thereby avoiding any disruption or delay in the timings. For a more detailed explanation of the software used to calculate and send these timings, see Appendix B. As fig. 4.1 demonstrates, DC voltage and laser power are not proportional, but rather obey a polynomial relationship.
An elastic collision rate that decreases with temperature dictates evaporation rates that also must decrease if maximum efficiency is to be achieved. In O'Hara et al. [13] scaling
I -s-Laser Power m)1
Laser Power vs. VCA voltage 70 I I 1 I I
0 2 4 6 8 10 12 VCA input (V)
Figure 4.1: Laser power as a function of VCA voltage.
laws for trap depth as a function of time are derived. Taking into account losses to the background gas
(4.la)
(4.lb)
7 = UtTap/kT= 10 and $ = 7 + (7 -5)/(7 -4) = 10.8. Instead of using the relation derived by O'Hara our evaporation setup utilizes a series of linear ramps with decreasing slope and voltage interval for each slope. Taken together this series of linear ramps ap- proximates an exponential ramp. Such a setup allows us to modify each ramp individually, empirically determining the optimal evaporation trajectory. To do this we modify one ramp at a time, either through its slope or interval, with the aim of maximizing the number of atoms trapped. As the different slopes and intervals are not independent isolating a single variable is difficult, and previous variables must often be reoptimized after other variables have been altered. For our purposes we have found a series of eight linear ramps to be a suitable approximation. Table 4.1 gives the start and stop voltages along with the respective slopes of each of the eight ramps used during evaporation. Fig. 4.2 shows these ramps as a
CHAPTER 4. EVAPORATION
Table 4.1: Evaporation timings. Based on these timings, the entire trajectory from 10 to .32V is completed in 2.4 seconds.
Start voltage (V) Stop voltage (V) Slope (-V/s)
function of time, and in fig. 4.3 these timings are mapped onto the polynomial relationship calculated in fig. 4.1 to give laser intensity as a function of time.
Optimizing evaporation trajectory is a continual process, as specifics integral to the process -cloud lifetimes, MOT size, etc. -vary on a daily basis. While we hope to one day establish evaporation timings that do not need more than the occasional minor modification, our attempts to improve other aspects of the setup (e.g. telescope position) have as of yet precluded such a possibility. At this time we also do not have the necessary data to make comparisons with the theoretical predictions found in O'Hara (eqn. 4.1~~). Such a comparison may in the future help us find new ways to improve our evaporation efficiency even further.
4.2 Scaling Laws
Increasing the elastic collision rate was intended to allow for efficient evaporation. To confirm that we are indeed running evaporation as efficiently as possible we have begun to collect data comparing number and phase space density as a function of temperature. As evaporation proceeds temperature will inevitably decrease; the question then becomes how does temperature relate to number, and by extension, to the phase space density.
CHAPTER 4. EVAPORATION
Figure 4.2: Evaporation voltage trajectory as a function of time.
Laser Power vs.Time
Figure 4.3: Evaporation laser power trajectory as a function of time.
Phase Space Densiw-Tem~erature Scalina
Figure 4.4: Log-log plot of phase space density-temperature scaling laws.
In Chapter 1 I introduced a theoretical calculation for the phase space density, the number density times the deBroglie wavelength of the atoms cubed. Experiment ally, this can be measured as
calculating orfrom vr and eqn. 2.11a and o; by measuring cloud widths and extrapolating back to t = 0 [5, 141. These calculations yielded trap widths of or= 7.1 pm and ox= 120 pm.' Deriving a measure for N and T from our cloud fits we have all the information we need to calculate the phase space density.
Figs. 4.4 and 4.5 give preliminary data for number and phase space density scaling in an evaporation scheme conducted on 3/7/06. Condensation was not achieved, although the data indicates we achieved phase space densities of order lo-', one order of magnitude shy of the condensation point. This data reveals how close our experimental setup comes to theoretical predictions for the scaling laws, and at what point evaporation begins to
'we do not have data to calculate a, in a similar fashion to that seen in Appendix B.l for this trap geometry. Considering the uncertainties involved in measuring a, indirectly through v, and the beam waist we have instead estimated the longitudinal trap widths using expansion data from the cloud fits. The longitudinal trap width measured has an error of 10 &,urn.
Number-TemperatureScaling
3!07106
-..
I Value 1 Error ml 92565e+5 9.7843e+12 m2 2.8203e-5 927.84
Chisa 3.6852e+ll NA
Figure 4.5: Log-log plot of number-temperature scaling laws.
deteriorate. Future data should include trajectories extending all the way to condensation, which may be further analyzed to improve our efficiency. O'Hara et al. again derive the scaling laws for number and phase space density:
As trap depth scales with T we may plot these as functions of T instead of Uwithout any other changes. It should be noted that these equations do not take into account losses to collisions with the background gas; losses to the background gas are time dependent and therefore cannot be readily determined from this data. While this qualifies any interpre- tation of the data it still indicates, however, that both N and p deteriorate towards the end of evaporation (i.e.; at low temperature). Even still, we are at an excellent starting point from which to improve our efficiency. As fig. 4.4 indicates, phase space density follows the predicted trajectories very well for the majority of evaporation, and it is not until the very end that we see a stark turn away from the theoretical predictions. Our number plot is more problematic, however, as the non-linearity of the data on this log-log scale indi-
Phase Space Density Scallng Prediction
3!07:06
Figure 4.6: Data from the beginning of the evaporation trajectory in fig. 4.4 is used here to predict the polynomial relationship between phase space density and temperature. This data gives an exponent of 1.4 compared with a theoretical value of 1.3.
Number Scaling Prediction
3i07M6
.-,
Figure 4.7: Data from the beginning of the evaporation trajectory in fig. 4.5 is used here to predicr. the polynomial relationship between number and temperature. This data gives an exponent of .33 compared with a theoretical value of .19.
CHAPTER 4. EVAPORATION
cates our evaporation scheme departs from the scaling laws from the start. The fact that phase space density follows the scaling laws during early evaporation while number does not seems to suggest systematic errors that will need to be investigated, possibly arising from our imaging system or fitting program. Looking only at data for the beginning of evaporation (i.e.; before the scaling laws begin to deteriorate) figs. 4.6 and 4.7 shows that our phase space density data comes close to predicting the power law relationship derived in O'Hara, while the empirical prediction for number is well beyond the error of the fit. With further optimization we hope to resolve this discrepancy and produce trajectories that are in closer agreement with theoretical predictions.
Chapter 5
Condensation
We now move on to a discussion of our final result, the production of a BEC. Although the above evaporation trajectory and temperature scaling data indicate we have considerable room for improving our efficiency, we are still able to consistently produce BECs of -lo5 atoms in number. After a discussion of characteristic BEC signatures I will move on to an analysis of our most current data, including observations we have made of those signatures and future work that must be done in order to better characterize our condensates.
5.1 Characterist ic Signatures
Prior to condensation a cloud of 87Rbatoms exhibits a distribution of energy levels that correspond to different momentum states. As seen in Brown, the distribution of atoms across energy levels will fit a Boltzmann distribution, which translates to a Gaussian momentum distribution due to the p2 dependence in the energy. Turning off the COa laser allows the cloud to expand, and as the momentum states follow a Gaussian distribution the number density of the expanded cloud (obtained from optical density calculations on CCD images) will also follow a Gaussian distribution. We use t,his information to fit thermal clouds (i.e.; those that have not condensed) and determine the total number of trapped atoms. Fig. 5.1 shows an image of a thermal cloud, fit to a Gaussian in both the transverse and longitudinal directions. We do a full 2-D non-linear fit and then plot cross sections in each direction that intersect the peak density. This allows us to observe a profile of the fitting function in comparison to the data. Since momentum states are populated evenly
0.8 cx=77.5067
0.6 0.4 sy = 17.1816 theta = -1.3226
0.2
0 scaledpsd = 75727.468
0 50 100 0 0.5 1
Figure 5.1: Gaussian fit of a thermal cloud in the z and r directions. Clockwise from top left the figures are: absorption image, fit image, residuals, cloud parameters, longitudinal cross sectional fit, and transverse cross sectional fit.
we observe symmetric ballistic expansion, the cloud expanding in a sphere (a circle in our 2D images) despite the cigar-shaped dipole trap. For short expansion times the thermal clouds will still be asymmetric, but once their size exceeds ozthey become approximately symmetric.
This is not the case once condensation has occurred. As explained in Chapter 1a BEC is defined to have all the atoms in the ground state energy level, hence it will not exhibit a Gaussian distribution of moment a. Experiment ally, this has significant implications. With the atoms all in the ground state their average momentum is significantly smaller than atoms in excited states. This creates a momentum distribution with a greater number of atoms with momenta in lower states than the Gaussian in our fitting function predicts, meaning we cannot use this program to determine the total number of atoms in the condensate. Instead, an expanded BEC will fit an inverse parabolic distribution [15]. Fig. 5.2 shows a series of cloud images that are progressively colder, proceeding from thermal clouds to BECs. The last three images are of BECs, discernable by the high concentration of atoms in the center of the cloud that did not expand as fast as the rest of the cloud due to
Figure 5.2: A series of subsequently colder absorption images of 87Rbatoms. The first image on the left is a thermal cloud, while condensation occurs in the third image. Condensate purity increases in the last two images.
their lower momenta. Similarly, as these clouds no longer exhibit Gaussian distributions of momentum states we cannot use the methods derived in Brown to determine number or temperature. Because the atoms in a BEC all crowd into the ground state energy level and share the same wavefunction, the confinement of the wavefunction of the BEG determines the momenta of the atoms once the cloud is left to expand. Realistically, any BEC produced in a lab will actually be a combination of atoms in the ground state -the condensate -and atoms in excited momentum states -the thermal cloud. Since each of these obeys a distinct spatial distribution we should see bimodal expanded clouds, a characteristic indication that condensation has occurred.
Condensation can also be inferred indirectly through data interpretation that relies on the Heisenberg uncertainty principle. Due to the specific geometry of our trap -a prolate spheroid -atoms are subjected to greater confinement in the transverse direction than in the longitudinal direction. This higher degree of certainty in position in the transverse direction corresponds to a greater uncertainty in transverse momenta compared to the longitudinal direction. As a result the condensate will expand faster in the transverse direction, and as opposed to thermal clouds the condensate will expand asymmetrically. Specifically, a condensate will expand faster in the radial direction perpendicular to the principal axis of the cloud, transforming from a prolate spheroid (cigar-shaped) to an oblate spheroid (disc- shaped). Again recalling that any experimentally produced condensate will also contain a thermal cloud, following expansion one should observe both symmetric and asymmetric components.
sy = 16.8457 theta = -0.93069
scaledpsd = 126628.608
0 0.5 1
Figure 5.3: Gaussian fit of a thermal cloud in the r and z directions. Although inaccurate, this fit gives an approximation of the total number of atoms in the condensate: lo5.
5.2 BEC Data
Fig. 5.3 shows a BEC fitted to a Gaussian in both the transverse and longitudinal directions after an expansion time of 30 ms. The fit data clearly shows that these clouds do not fit a Gaussian distribution. We find a number density at the center of the cloud greater than that predicted by the distribution, a result of the atoms all being crowded in the groud state momentum level. One also notices that the tails of the cloud cross sections, particularly in the longitudinal direction, do not agree with the Gaussian fit. These observations are also confirmed more generally through an analysis of the residuals.
The first BECs we produced did not readily reveal asymmetric expansion due to the fact that expansion times were too short to allow the clouds to expand completely. Since then, however, we have observed this phenomenon, as fig. 5.4 demonstrates. For reference, fig. 5.5 shows the orientation of our dipole trap.' Comparing the two it is evident that the condensate in fig. 5.4 has expanded in an assymmetric fashion nearly perpendicular to the orientation of the dipole trap. In this image we also observe a thermal cloud, as the
'All of these images have been rotated 90" counterclockwise.
Figure 5.4: BEC data image. The outermost ring is the expanding thermal cloud and therefore is circular, while the darker portions have expanded asymmetrically, perpendicular to the dipole trap in fig.
Figure 5.5: Dipole trapped cloud. CCD image taken after 300 ms expansion.
CHAPTER 5. CONDENSATION
outermost ring in the image has expanded symmetrically. Here again we see bimodality, as both a condensate and a thermal cloud are present.
5.3 Conclusion
In the past year we have installed an adjustable telescope that enables us to both maximize the number of atoms loaded into the dipole trap from the MOT and increase the elastic collision rate of the atoms in our trap during evaporation. This increased elastic collision rate allows us to run efficient evaporation at a faster rate, minimizing our losses to the background gas. We have also installed a new DAC that allows us to use our DIO-128 board during evaporation, giving up to a microsecond resolution in evaporation timings. With these improvements we have been able to produce BECs, empirically observed through predicted BEC signatures.
As already mentioned, we do not have yet have the ability to fit BECs and therefore accurately characterize them in terms of temperature or number. Returning to the Intro- duction to this thesis, this requires calculating the total number of atoms in our trap, both in the condensate and the thermal cloud (eqn. 1.6). At this point we may only estimate the phase space density of the thermal cloud component. In theory this should give us a phase space density of order 1,since a thermal cloud trapped concurrently with a condensate will be very close to or at the transition point. Assuming the fits in fig. 5.3 are approximate fits for the thermal component of this cloud we use the values for N and T given here and the trap widths measured in section 4.2 to estimate a phase space density of p = 5,less than one order of magnitude off the transition point. While this is close to the predicted value of 2.612, the discrepancy again underlines the fact that our fits are no longer accurate in this new regime. It remains the task of future students to devise a new fit program and methods to measure the temperature and cloud number of bimodal clouds, giving a more accurate characterization of the BECs we produce.
Bibliography
[I] M. H. Anderson et al. Science, 269:198, 1995.
121 K. B. Davis et al. Phys. Rev. Lett., 75:3969, 1995.
[3] Daniel V. Schroeder. An Introduction to Thermal Physics. Addison Wesley Longman, 2000.
[4] Paul A. Tipler and Ralph A. Llewellyn. Modern Physics. W. H. Freeman and Company, fourth edition edition, 2003.
[5] Justin Brown. Progress toward BEC: Detecting and measuring dipole-trapped clouds. Undergraduate thesis, Williams College, 2005.
[6] Leon Webster. Captivating and chilling: Progress toward BEC via laser trapping. Undergraduate thesis, Williams College, 2004.
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BIBLIOGRAPHY
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[13] K. M. O'Hara, M. E. Gehm, S. R. Granade, and J. E. Thomas. Scaling laws for evaporative cooling in time-dependent optical traps. Phys. Rev. A, 64, 2001.
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Appendix
Software
In the past year we have installed two new major pieces of equipment: a motorized transla- tion stage on which we mount the first lens of a telescope, and a digital to analog converter (DAC) that converts digital time-stamped voltage levels to analog so they may be sent to the VCA and mixed with an RF signal, controlling COz laser beam intensity. With both of these pieces of equipment we have installed the necessary software in order to integrate their operation into our main LabView virtual instruments. I discuss the operation of each software program here.
A.1 Translation Stage Software
To operate the translation stage we use a ViX 250IM Microstepper Drive from Parker Automation. Through the drive we specify velocity, acceleration, and distance traveled, as well as modify certain settings that control the general operation of the translation stage
(e.g. prescribed operation when a fault occurs). The drive is controlled via a COM port and comes with the manufacturer's own software. Since this program is a stand alone and cannot be controlled via LabView we have designed a virtual instrument (VI) that controls the drive in much the same way as the supplied software. The same programming code is used, with the difference that the commands are now sent directly through LabView.
Generally speaking, the programming code consists of blocks of code containing a series of commands that are written to the drive. Multiple blocks of code may be sent to the drive and stored in its memory to be called at a later time. In our operation of the drive we
APPENDIX A. SOFTWARE
write two blocks of code to the drive prior to running BEC production -one to move the translation stage the necessary distance to squeeze the trap, the second to return the stage to its initial position -and then call these programs with a single line of code during the evaporation program. As writing to the drive can take a substantial amount of time this is necessary to avoid a delay in timings.
The VI we programmed to send blocks of code to the drive consists of two pre-written blocks of code with inputs to change velocity, acceleration and distance traveled. As the drive will store code written to it for an indefinite time we need not resend these move parameters unless any of them are changed.
The following LabView VIs send commands to the translation stage driver: upload to vix. vi writes blocks of command codes to the driver specifying move routines in terms of distance traveled, velocity and acceleration. run vix program. vi calls a specific block of code already written to the driver and executes that specific move routine.
A.2 DAC Software
In order to use the DIO-128 Board to send evaporation timings to the VCA all voltage values must first be converted to a 16-bit digital format. It is the task of the DAC to then change this 16-bit format back to an analog voltage that may be sent to the VCA.
The software required for the DAC (and hence to run evaporation) proved to be a considerable challenge, namely because we could not simply emulate another program. This was compounded by the fact that over the past three years previous students in the Whitaker lab have invariably alternated between writing programs for the DIO-128 board that read commands from 16-bit unsigned word arrays, number arrays, or boolean arrays, leaving a veritable mess in the programming. To streamline much of this we programmed using 16-bit arrays exclusively and created a new global variable that contains the last states of the board, simplifying things as all commands sent to the DIO-128 board must be in this format.
Before sending anything to the board we first calculate a series of eight linear voltage
ramps that consist of a discrete number of steps equally spaced in time. We input start and
stop voltages, the slope, and a step size. This gives us a 2-D number array of time-stamped
APPE-NDIX A. SOFTWARE
voltages. We then convert each voltage into a 16-bit unsigned word, which are then each split into two 8-bit unsigned words. The DIO-128 board consists of four ports with 16 channels to each port: as many of the various channels on the board have already been assigned to other functions there is no longer room on the board to send one uninterrupted 16-bit word.
With our 2-D &bit word array in hand we now create an array that may be sent to the board. In order to send these new states to the board we must send a valid state for every channel on the board, including those that are left unchanged. To avoid inadvertently changing the states of other channels (for example, one that may control a camera shutter) we read off the last states of every channel on the board. We then discard the old states that control the DAC and append the new voltage states. We do this for each voltage, creating an array of four 16-bit words that includes every voltage step. Based on the time interval between each voltage step we then create properly formatted 16-bit time stamps that are also appended to the array. This gives us an array of time-stamped states for every channel that may be sent to the board.
This entire process is conducted before the trap is loaded, and is repeated for each of the eight linear ramps. During evaporation each array is successively sent to the board, where the time-stamped voltages are read and sent to the DAC in real time.
Appendix B
Experimental Data
B.l Longitudinal Trap Width
In section 3.4.1 I calculated the longitudinal trap frequency v, indirectly by finding a beamwaist from the transverse trap frequency. As mentioned in Chapter 2 there is a way of measuring the longitudinal trap widths directly, providing for a more accurate measurement of the longitudinal trap frequency. As Brown explains in his thesis, a thermal cloud expands ballistically as
o(t) = Jo2(0) +(2kT/m)t2
(B.l) Measuring trap widths for non-zero times we fit these widths to the above function to determine an initial trap width. This process works for the longitudinal trap widths; in the transverse direction o(t = 0) is less than the resolution of our imaging system and we instead use the trap frequency to determine the trap width. Fig. B.l shows this is indeed the case. As the above figure is given in pixels, a conversion rate of 8.6 ,um/pixel divided by a magnification of 1.666 gives us a trap width in the x direction of 137 ,urn, compared with a transverse width of 6.6 ,urn for this trap configuration. To compare this to our results obtained previously we invoke eqn. 2.11b. At this configuration the temperature was measured to be T =2.02 x and we calculated v, = 579 Hz (i.e.; the 6 cm telescope position on 4/24/06). Plugging in these values gives us o, = 54 pm, a 61% error from the
more direct measurement. The discrepancy between these two results underscores the limitations of the approx-
Expansion Rates at 6cmTelescope Position 4;24:06
0.4 0.6 0.8 1 1.2 1.4 lF
Expans~onTime (ms)
Figure B.l: Cloud widths measured in the longitudinal and transverse directions as a function of time. Widths are measured in pixels.
imations made in modeling our COz beam as a perfect Gaussian beam. Aberrations in the beam and a non-Gaussian profile due to the presence of higher order TEM modes -both of which we have seen -limit the applicability of the equations for trap frequency and trap width seen in Chapter 2 as they assume a TEMooGaussian beam. However, as the calculated trap width in the x direction is only off by a factor of 3 we may still use this method to made rough approximations in determining certain trap parameters such as how the elastic collision rate scales with changes in trap geometry.
Trap Frequency
The following is trap frequency data collected on 4/24/06 for telescope positions of x = 0 cm, z = 3 cm, and x =6 cm.
APPENDIX B. EXPERIMENTAL DATA
Initial Telescope Position (Ocm) 5%modulation 4:24:'06
2.5 10'
L J
Modulation Frequency (kHz)
Figure B.2: 2v = lkHz
Midpoint Telescope Position (3cm) 5?bmodulation 1:221,'06
3.5 10"
Modulation Frequency (kHz)
Figure B.3: 2v = 5kHz
APPENDIX B. EXPERIMENTAL DATA
Final Telescope Position (6cm) 2# modulation 4:24!06
1.210"
Modulation Frequency (kHz)
Figure B.4: 2v = 9.5kHz