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Towards Critical Rotation - FNWI (Science) Education Service Centre

1. is the rotation vector with 2 the rotation frequency q is the charge of a particle in a magnetic field B and v is the speed of either a charged particle in a magnetic field or a neutral particle in a rotational system From these equations it follows that the magnetic Lorentz force may be simulated by the Coriolis force when G Bes 2m 2m where we have taken q 1 to simplify the equation and is the magni tude of the magnetic field The rotation vector and magnetic field direction are chosen to be in the z direction which can be done without loss of gen erality 12 Hamiltonian for a particle a harmonic potential A particle in a 2D Harmonic potential can be describe by the time independent Schr dinger equation with the appropriate Hamiltonian Landau amp Lift shitz 1977 2 2 2 god elm 1 3 2m 2 where p is the momentum operator m is the mass of the particle w is the harmonic angular frequency in the zy plane and r the distance from the centre of the trap The energy spectrum of the Schr dinger equation for a 2D harmonic oscillator is given by En hw n4 1 1 4 where n nz n and nz ny are the quantum numbers of the excita tion energy in their respective direction 4 CHAPTER 1 ROTATION Going to the rotating frame requires an extra term to be added 2 42 H LENT OL 2m 2 p 1 2 2 2 M X E where L is the angular momentum operator in the direction of the rotatio
2. 26 6 1 To simplify the equation one has to note that the Bessel function 2e has only a limited range of values which allows us to define a new variable je ymin JX and replace the Bessel function J 2e j thus it follows 39 CHAPTER 6 POTENTIAL SHAPING N 2 N 3 N 4 Figure 6 1 The order of symmetry N for the potential in the zy plane The red circle in the upper row of figures is the circle of death From left to right are respectively shown the potentials for N 2 N N 4 with an anisotropy 0 5 figures in the upper row are contour plots of the potentials where dark purple corresponds to lower energies and light purple to higher energies The plotted lines show the axes of the figures in the lower row where the colours of the lines in the upper plot correspond to the line colours in the lower plots The potential for N 2 has a Dipod shape and as can be seen from the cuts the horizontal and vertical axes have different harmonic shapes trapping frequencies The case N 3 a Tripod has a three fold symmetry and the hori zontal cut shows the asymmetry whereas the vertical one is symmetric The case N 4 a Quadpod shows the four fold symmetry and the vertical and diagonal cuts show the two main axes of the Quadpod From these cuts it can be seen that the diagonal directions are less tightly confining than the vertical horizontal direction 40 CHAPTER 6 POTENT
3. CHAPTER 1 ROTATION slow rotation mean field strongly correlated No image available 10 10 10 Figure 1 3 Shown are three regimes for v the most left image is for slow rotation can also be described by mean field theory and large v which means many atoms per vortex in the shown only one vortex on 10 atoms The middle image is in the mean field regime v 103 and has a lot of vortices This regime can still be described with mean field theory The right image should show the strongly correlated regime v lt 10 which has been reached Gemelke et al 2010 but there are no similar images so far This regime would in mean field theory correspond to a few atoms per vortex but the term vortices is not applicable because in this regime mean field theory is not valid anymore The figures shown are taken from previous experiments at ENS 1 5 Potential stability When rotating a 2D harmonic anisotropy due to the anisotropy a window of dynamical instability around critical rotation opens up Guery Odelin 2000 To show that a 2D harmonic anisotropic single particle potential Dipod is placed inside of the rotating frame 1 1 Ue gel a y sma y ma x y o 115 with m the mass of the particle w the harmonic trapping frequency the rotation frequency of the system the anisotropy strength and y the position variables The equations of motion for this system are given by Rosen
4. confining dc magnetic trap Physical Review Letters 77 3 Avail able from http link aps org doi 10 1103 PhysRevLett ct Fs Al 94 REFERENCES Minogin V G Richmond J A amp Opat G I 1998 10 1 Time orbiting potential quadrupole magnetic trap for cold atoms Physical Review A 58 4 Available from http link aps org doi 10 1103 PhysRevA 58 3138 Petrich W Anderson M H Ensher J R amp Cornell E A 1995 Apr 24 Stable tightly confining magnetic trap for evaporative cooling of neu tral atoms Physical Review Letters 74 17 3352 3355 Available from httpr dx dorx org 10 1103 PhysRevLe ett 74 3352 Pitaevskii L amp Stringari S 2003 Bose einstein condensation Clarendon Press Oxford Raab E L Prentiss M Cable A Chu S amp Pritchard D E 1987 De cember Trapping of neutral sodium atoms with radiation pressure Physical Review Letters 59 23 4 Rath S P 2010 Production and investigation of quasi two dimensional bose gases Unpublished doctoral dissertation L Universit Pierre et Marie Curie Rosenbusch P Petrov D S Sinha S Chevy Bretin V Castin Y et al 2002 06 07 Critical rotation of a harmonically trapped bose gas Physical Review Letters 88 25 Available from http link aps org doi 10 1103 PhysRevLett 88 250403 Stormer H L Tsui D C amp Gossard A C 1999 03 1 The frac tional quantum hall effect Reviews of Modern Ph
5. is a global phase which means that all the atoms in the condensate have the same phase phase coherence In a place where the density is non zero the velocity field of the condensate is given by Landau amp Liftshitz 1977 v 1 9 where m is the mass of a particle the condensate and V 0 r is the gradi ent of the phase The vorticity of this field is given by Vxv 0 1 10 because the curl of a gradient is zero This seems to imply that it is forbid den for the condensate to have rotation This problem of having no rotation was solved by Feynman Feynman 1955 who introduced the concept of singularities in the phase function The singularities are shown in the den sity function as zero points vortices This solves the problem because with these singularities in the phase function it is possible to have rotation in the condensate and still have phase coherence 1 4 3 Vortex filling factor To characterise the rotation of a condensate the circulation around the con densate is used fo dl f vo dl EN 1 11 m Je m where N is the number of singularities vortices in the condensate Fol lowing from the classical fluid the value for the coarse grained velocity is given by fea xvas 20 48 294 1 12 5 5 1 need be taken here because this is only true for simply connected domains 7 CHAPTER 1 ROTATION Figure 1 2 Graphical representation of a Bose Einstein con
6. 9 7 The Tabor WW1072 uses points that have a precision defined by one byte and in our case the device has a total of 2 MB memory per channel 57 CHAPTER 9 DISCRETISING ROTATION and the frequency resolution following from this becomes 1 1 AQ ER 9 8 ore 1 22 ea The resolution depends on 1 q 1 q 1 which decreases in case of an increasing 4 As we are interested in the limit lt wr this scheme is quite suitable This can be shown when looking at the case p 1 In this case one potential rotation cycle is defined one rotation cycle is on the order of 100 ms thus the total time Tout is 100 ms Comparing this to the cycle time of the zero magnetic field point 0 1 ms 1 1 AQ RN M or 7 zu It follows that for rotation frequencies 2 between 27 x 1 Hz and 27 x 100 Hz we only need the range q 100 1000 This drastically reduces the number of points needed NIX 6 410 out and the resolution range for these q s and wr 2 x 10 KHz is 10 1 An x o 1001 101 Improving this resolution can be done by increasing both p and 4 The only point that is not possible to reach in this scheme is no rotation but this is easily solved by taking p 0 This second scheme can be characterised as being period resolution fixed because two consecutive q points give a cycle period resolution given by _ Trop ATrot zu q T 1 q gt which is constan
7. a Quadpod The rota tion spectra show the expected behaviour of a Dipod losing all the atoms at critical rotation due to the dynamical instability and the Quadpod show ing that it stays trapping at critical rotation We have also shown that this behaviour is to be expected from analytical calculations The strength of the anisotropy for a Quadpod at critical rotation lets it self explain partially we can explain the moment where all the atoms are lost but the point where the atoms start to be lost still needs to be investi gated So far the results are promising for rotating condensates at the point of critical rotation 90 Kritische rotatie 1995 Davis et al 1995 onafhankelijk voor de eerste maal een Bose Einsteincondensaat BEC wisten te produceren zijn wetenschappers bezig geweest met het bestuderen van dit fenomeen Een BEC wordt gekarak teriseerd door de eigenschap dat het condensaat volledig fase coherent is Dit heeft tot gevolg dat het condensaat een snelheidspotentiaal bezit wat op zichzelf weer de rede is voor het feit dat een condensaat geen rotatie kan bevatten Dit feit is heel merkwaardig en het heeft wetenschappers lange tijd doen verbazen Tot in 1955 Feynman Feynman 1955 ontdekte dat het conden saat plaatsen kan cre ren waar de fase ongedefinieerd is en de dichtheid van het condensaat naar nul gaat Deze plaatsen worden draaikolken ge noemd en bevatten de rotatie van het condensaat De rotatie van het sys
8. defined in eqs B 3 and B 4 a gt 0 b gt 2a to determine whether the 4 order potential is trapping at the centre We can find limits for j such that the potential is trapping In the case of the phase modulated 4 order potential 1 5j and b 2 307 With these relations we can define a range for j 1 jin 10 4 1 je 10 5 In figure 10 2 these limits are plotted The blue shaded areas are the trapping areas and the red areas are the anti trapping areas 0 6 0 6 BUT AAA UVA 0 6 0 6 Figure 10 2 These figures mark the limits given by eqs 10 4 and 10 5 in a graph of the Bessel function ke The blue shaded regions mark the trapping values for e whereas the red ar eas mark the not trapping values for ke In figure A the plot shows a small range 4 lt 4e lt 4 of the Bessel function It shows the first trapping region which is 4e lt 0 41 Figure B shows the full range over which all the anti trapping regions our found The yellow line is plotted at the maximum value j 1 5 and the red line at the minimum value j 1 5 In figure 10 3 the number of atoms is shown as a function of the anisotropy The plot shows that at e 0 10 and j 0 2 the atoms are lost which is a logical consequence of the anti trapping potential above 0 1 and j 0 2 for the Quadpod see figure 10 2 at critical rotation Another clear feature 61 CHAP
9. in the interface but the outputting is still done in voltages To accomplish this we added converter code to Atticus and a conversion interface to Cicero The conversion interface can be found in the Channel Manager window where two columns have been added Unit and Conver sion The possible units are s seconds V Voltage standard Hz Hertz A Amperes deg Degrees dBm decibel milli Watt and Pr percentage Extra units can be added by modifying the file Units cs in project DataS tructures The column Conversion should have a valid dotMath equation with the name of the unit used as a variable For example standard Unit is set to V and the equation would then standard be V but another possible equation would be 1 2 V More information about equations can be found in the documentation of Cicero Keshet 2008 78 OND TF WN FR RRP RRP RP RP RP RP amp WN APPENDIX CICERO WORD GENERATOR Atticus uses the Unit and Conversion values to transform analog data points for a certain logical channel into voltages This conversion is done in the file DaqMxTaskGenerator cs at three places This is needed because of the structure of Atticus which consists of a part that handles the Output Now mode and two parts that handle the Sequenced mode one with and one without Variable Timebase The conversion is done with the following func tion that takes double bufferValue as the input value and us
10. middle graph in figure 6 1 There a clear three fold symmetry is visible The red and blue lines 41 CHAPTER 6 POTENTIAL SHAPING show two directions of interest in the potential In the lower middle graph the line colours correspond to these special direction The red line shows that on the y axis the potential is symmetric whereas the blue line shows the asymmetric behaviour in one of the Tripod s symmetry directions In figure 6 2 the general Bessel function ke is shown for the 3 4 order k needs to be used Also for the Tripod the replacement J 36 j can be done but care needs to be taken because the es corresponding to a values of 7 for the Tripod potential is different from the es corresponding to the same j for the Dipod potential Following from this replacement is the potential 1 1 Ur uBo 1 28 329 j 00 6 3 Quadpod The Quadpod potential is obtained by using N 4 in eq 5 5 up to 4 order in the zy plane thus the potential becomes uBo 1 i zi z g 1 5 J 4e g 2 30 J 4e The right hand column in figure 6 1 in the first row a contourplot of the Quadpod potential N 4 in the xy plane Again the lines drawn in the contour plot are the directions of interest and symmetry The red line shows that the potential in the direction x y is lower than in the other blue line which creates the rounded square shape The anisotropy can be d
11. the centre except for the point 7 0 which is a singular point 3 22 Rotating bias field To remove the zero magnetic field point from the centre of the cloud a rotat ing bias field is added in the zy plane Minogin et al 1998 This rotating bias field is given by B t cos 0 amp sin t yl 3 4 y with Bo the magnitude of the rotating bias field and 7 is the phase func tion which for a standard TOP trap is given by e t wrt where wr is the angular frequency A rotating homogeneous bias field displaces the zero magnetic field point out of the cloud and lets it rotate at a radius ro which is called the radius of death Figure 3 2 shows the displacement of a quadrupole mag netic field projection due to a bias field in the co rotating frame From the figure it can be seen that the zero magnetic field point is actually shifted in the direction opposite to the direction of the bias field Using this makes it possible to define the radius of death ro The total magnetic field is simply recovered by adding the bias field to the quadrupole field for simplicity we take the centre of the initial static trap to be at the centre of the coordinate system r 0 3 5 ro 0 Bo KG 0 2 7 sin A y 222 86 where we have introduced the phase function 2 27t which is used to define the phase of the rotating bias field during one cycle The dimension le
12. the sending of the GPIB data needs to be finished before triggering Usually a good place to send the data is during the MOT phase 77 OND TF WN PRP RP RP on or ONE APPENDIX C CICERO WORD GENERATOR point to be based on an internal timer with the variable secondsPerSample that gives the time in between two point Line 11 and 12 set the trigger to an external input and finally the output is turned on after a trigger Tictina 9 CPIR cada Evonoration urith mada ALE trio Set the device for frequency and amplitude modulation using a point list STEHE SEE STRE ALIN ILE al 211 zes PRESSE UNIT POWer lt amplitudeUnit gt SOURce FREQuency MODE LIST SOURce POWer MODE LIST 2SOURGS ILS ggg Jer SOURCE TSM RETRACER RE BEIGE SOURce LIST POWer amplitudeList SOURCe LIST FREQuency frequencyList SSOUINES IhILS Is WRILECeSie SOURCES TRIGger SEQuence TIMer secondsPerSample MEME Ens er OURCC E INITiate OUTput STATe ON C4 Converting Interface Unit to Voltages The analog channels of the outputting cards can only output voltages from 0 to 10 V but with these voltages other devices are controlled that them selves control other quantities then voltage In the interface we needed an conversion such that these other quantities with their respective units could be used
13. thesis was written the group tried to optimise the sequence as there seemed to be some triggering issues for this reason no further data could be taken before the writing of this thesis 14 Part I Standard TOP trap CHAPTER 3 TOP trap netic trap but it has the disadvantage of a vanishing magnetic field at the centre This zero point causes non adiabatic Majorana spin flips Brink amp Sukumar 2006 and thus atom losses To overcome this prob lem Petrich et al introduced the TOP Time averaged Orbiting Potential trap Petrich et al 1995 It is created by adding a rapidly rotating homo geneous bias field which is rotating around the axis defined by the cen tres of the quadrupole field coils Minogin Richmond amp Opat 1998 This field places the zero magnetic field point on a circle outside of the cloud rather than having it in the centre At the same time a confining harmonic averaged potential is experienced by the atoms which is required for keep ing them trapped The use of this trap allowed the Cornell group to create the first Bose Einstein condensate BEC in a dilute atomic gas of rubidium atoms Anderson et al 1995 Afterwards BECs were also created using a plug beam to plug the centre Davis et al 1995 a static non zero mini mum magnetic loffe Pritchard IP trap Mewes et al 1996 and finally in 2001 also with a purely optical crossed dipole trap Barrett Sauer amp Chap man 2001 The c
14. tre and allowed use to investigate some interesting potential symmetries Three different symmetries have been characterised and discussed the Di pod N 2 the Tripod N 3 and the Quadpod N 4 Finally the anisotropy of these potentials has been investigated by look ing at the atom number as a function of the strength of the anisotropy Both the Dipod and the Quadpod show linear decreasing behaviour until there are no atoms left The Tripod on the other hand seems to have an plateau and then a threshold above which the atom number strongly decreases These experiments showed some interesting but unexplained behaviour that needs further investigation Also it need be noted that these experiments were done in the first run mode discussed in subsection 2 2 1 which uses the phase modulated TOP trap during evaporation and decompressing of the TOP trap 49 CHAPTER 7 ANISOTROPY STRENGTH 50 Part III Rotating Anisotropic TOP trap CHAPTER 8 Co Rotating Frame movement of the zero magnetic field point The previous chapter used phase modulation to modulate this rotational movement and use it to shape the potential The rotation can only be done in two dimen sions and the plane of choice is the zy plane which makes the description of the phase modulation easier but not necessarily less general The next step is to set the phase modulated shape into rotation The set a system into rotation in the zy plane the rotation mat
15. voor hun steun tijdens al die jaren van school en studie maar ook tijdens de periode voordat ik wegging die gewoon heel zwaar is geweest Daarnaast weet ik dat mijn moeder het heel erg moeilijk heeft gehad dat ik voor een jaar weg ben geweest en dat daar nu een nog langere tijd dat ik weg ben aan zit te komen Mam Pap bedankt And jij su ig merci 88 Critical Rotation Generally accepted there are three regimes of rotation which characterised by the number of atoms per vortex v The first one is the regime of slow rotation with v 10 this is on the order of one vortex in a condensate The second regime is called the mean field regime and is given by v 10 Fi nally the third regime is characterised by v 10 and is called the strongly correlated regime The third regime is of great interest for physicists be cause the strongly correlated regime is not well understand theoretically and might explain Fractional Quantum Hall physics The third regime is reached by increasing the size of the condensate reducing the number of atoms and increasing the rotation frequency In our experiment we use the centrifugal force to fully compensate for the harmonic trapping potential and make the cloud as large as possible using the fourth order potential This is called critical rotation In this report we use the Time average Orbiting Potential TOP trap which is a trap made of a quadrupole trap together with a rotating bias field and modu
16. 000 60 000 5 5 Ss Ss 5 30 000 5 45 000 5 20 000 5 30 000 10 000 15 000 g 0 0 0 01 02 0 3 04 0 5 0 01 02 0 3 0 4 0 5 Anisotropy Anisotropy Anisotropy spectrum Quadpod _ 80 000 2 Gy X 60 000 40 000 5 20 000 lt 0 0 01 03 04 05 Anisotropy e Figure 7 2 PRELIMINARY Shown is the atom number as a function of the anisotropy e in a static phase modulated TOP trap Up per left is the Dipod N 2 potential upper right the Tripod potential N 3 and finally lower left the Quadpod poten tial N 4 The Dipod and Quadpod potentials show a lin ear relation in the decrease of atom numbers whereas for the Tripod it looks more like a constant plateau and then a steep decrease The points in the graph are an average of multiple measurements and the error bars are the standard errors of each points calculated by ose y y y N with N the number of points and y the average of y values 47 CHAPTER 7 ANISOTROPY STRENGTH 48 Conclusion and Summary HASE MODULATION turned out to be an exhaustive tool for creating anisotropic traps with any desired symmetry We have given a strong formalism based on the formalism given in Part I that describes the potential of the phase modulated TOP trap This potential is really difficult to solve analytically but expanding the magnitude of the magnetic field in spatial coordinates provided us with the necessary potential in the cen
17. Hansch T W amp Esslinger T 2001 Feb Magnetic transport of trapped cold atoms over a large distance Physical Review A 63 3 031401 Available from http dx doi org 10 1103 PhysRevA 63 031401 Guery Odelin D 2000 08 14 Spinning up and down a boltzmann gas Physical Review A 62 3 Available from http link aps org doi 10 1103 PhysRevA 62 033607 Ginter K J Cheneau M Yefsah T Rath S P amp Dalibard J 2009 Jan Practical scheme for a light induced gauge field in an atomic bose gas Phys Rev A 79 1 011604 Hodby Hechenblaikner G Hopkins S Marag O M amp Foot C J 2001 Dec Vortex nucleation in bose einstein condensates in an oblate purely magnetic potential Phys Rev Lett 88 1 010405 Keshet A 2008 June Cicero word generator technical and user manual Computer software manual Ketterle W amp Druten N J van 1996 Evaporative cooling of trapped atoms Advances in Atomic Molecular and Optical Physics 37 56 Landau L D amp Liftshitz 1977 Quantum mechanics Non relativistic theory 3rd ed Vol 3 Pergamon Press Laughlin R B 1999 07 1 Nobel lecture Fractional quantization Reviews of Modern Physics 71 4 Available from http link aps org doi 10 1103 RevModPhys 71 863 Mewes M O Andrews M R Druten N J van Kurn D M Durfee D S amp Ketterle W 1996 07 15 Bose einstein condensation in a tightly
18. IAL SHAPING for the potential that Ur Bo 1e 2 03 0 3 In figure 6 2 the Bessel function J ke is given with the possible values of j given by the light blue area The Dipod potential corresponds to the case k 2 The left hand graphics in Figure 6 1 show in the upper row a contour plot of the averaged potential and in the lower row the cuts on the z axis blue line and y axis red line It is evident that the two cuts in the left column have different trapping frequencies These trapping frequencies are given by 14 1 7 6 2 L Le j 6 3 Wy Wy UY where wz wy is the trapping frequency in the x direction y direction and w the trapping frequency in the xy plane for the standard TOP trap Using these relations we can define the anisotropy in terms of the trap ping frequencies wz and wy 2 2 Wii l 6 4 J w2 w2 d Since the Bessel function J 2e up to 1 order is given by j these results correspond to the e used by the Oxford group Arlt et al 1999 Hodby et al 2001 The range of anisotropies is obtained by taking the minimum and maximum of Bessel function 2c je Jf 26 JPW 2 0 58 0 58 6 5 6 2 Tripod The Tripod potential is obtained by using N 3 in eq 5 5 up to 3 order in the zy plane thus the potential becomes 1 1 Ur F uBo 1 d dec 2 323 Ji 36 The potential in the zy plane is shown in the upper
19. TER 10 RESULTS is that from 0 04 and j 0 08 on the atom number is linearly decreas ing It is feasible that this is caused by the trap opening up and the cloud is getting so large that a great part of the atoms are evaporated away by the rf shield Also these measurements were done with only one phase of the TOP trap so everything evaporation and final thermalisation was done in a rotating trap We will need to further investigate these phenomena before we are able to make conclusive statements Anisotropy spectrum at critical rotation 50 000 40 000 30 000 20 000 Atom number au 10 000 0 0 02 0 04 0 06 0 08 0 10 Anisotropy Figure 10 3 PRELIMINARY Atom number as a function of the anisotropy Shown is the fact that ate 0 1 and j 0 2 the atom number is decreasing Another feature is that from 0 04 and j 0 08 the atom number is decreasing which might be caused by the trap lowering its barrier at critical rotation and strong anisotropy j The points in the graph are an average of multiple measurements and the er ror bars are the standard errors of each points calculated by Ose y y N with N the number of points and y the average of y values 62 Conclusion and Summary modulation of a TOP trap is an interesting system for studying ultra cold atoms in the co rotating frame Setting the shaped potentials cre ated using phase modulation into rotation is done using a simple sc
20. ackage is called Transport Currents and contains all the necessary files to create different transport possibilities The package has an extensive README which guides through the process of creating files that can be imported into Cicero C 3 Agilent N5181 Programming Evaporation Ramp For evaporation we use the Agilent N5181 because it has the ability to pro gram a point list up to 1600 points with frequency and amplitude values and output a function that is based on these properties when a triggering signal is send Originally Cicero did not have the feature to first send the data en then trigger with a digital signal To overcome this lack of function ality we have implemented the A F trig mode for GPIB and the Agilent N5181 in specific The idea of this mode is to set the device such that it takes a point list and wait for a trigger In Listing C 2 a snippet of GPIB code send to the Agilent is shown and we are going to explain some of it Line 2 is to set the unit of Power to the device this can be set in the HardwareSettings for the specific device under GPIB Line 3 through 5 are used to set the type of outputting to a List of point given in Lines 7 and 8 Line 6 sets whether after a full output the device should go back to the first point in the list this can be set in the HardwareSettings under GPIB Line 9 and 10 set the change to the next lThis segment needs to at least last 100 ms before the triggering signal is given because
21. agnetic field Rotting Pas HORE eo x er 0 orks eee dk eg T n averaped potential Expanding the potential in spatial coordinates Experiments using TOP trap to shape potentials M O IIA FFF A Qu 11 11 12 13 CONTENTS 4 Discretisation 41 42 DISCOS CON anisotropy eas obrem m om ema Conclusion and Summary II Static Anisotropic TOP trap 5 Phase Modulation PonnalsH 22222 don pons ne bw eher 52 Time averaged potential ee rs xA 5 3 Expanding the phase modulated potential 6 Potential Shaping il Bel 2 593234 erat D o Cmm COMER Uo EEUU TTE 7 Anisotropy Strength 7 1 Discrete Phase 4 7 2 Atom losses due to a static anisotropy Conclusion and Summary Rotating Anisotropic TOP trap 8 Co Rotating Frame 8 1 Potential in co rotating frame 9 Discretising Rotation 91 Discretisation formalism 911 Fixed frequency resolution 91 2 Fixed period resolution evana nx 10 Results 10 1 Rotation PUEDA ee ORO Y XO en 10 2 Anisotropy at Critical Rotation vi 25 25 27 29 31 33 33 35 36 39 39 41 42 45 46 46 49 51 53 53 55 55 57 57 CONTENTS Conclusion and Summary Appendix A Integrating Jacobi Anger expa
22. assen en deze vorm op zijn beurt weer roteren De eerste experimenten die hiermee gedaan zijn is het defini ren van n 1 2 punten op een cirkel en het nul punt van de kwadrupel val van punt naar punt laten springen Uit deze experimenten bleek dat de atomen hier sterk op reageren en dat minimaal 7 64 punten gedefini rd moeten zijn Om dus een sterke vorm te krijgen kunnen we niet het aantal punten verminderen De volgende stap is om 7 64 aan te houden en deze punten te ver spreiden over de cirkel Dit keer niet gelijk verdeelt maar gegroepeerd ge bruikmakend van fase modulatie De vorm wordt nu bepaald door het aan tal groepen N 1 2 van punten en de sterkte van de vorm door hoe sterk ze zijn gegroepeerd Vervolgens hebben wij gekeken naar het effect van de sterkte op het aantal atomen en wat bleek is dat voor N 2 Dipod en N 4 Quadpod er een lineair verval te zien was in tegenstelling tot N 3 Tripod die een plateau laat zien tot op zekere hoogte een drempel is bereikt De laatste experimenten zijn gedaan aan het laten ronddraaien van de eerder genoemde vormen N 2 en N 4 en dan te kijken naar het aantal atomen als een functie van de rotatie snelheid Dit geeft het rotatie spec trum voor de Dipod welke duidelijk laat zien dat er bij kritische rotatie geen atomen gevangen blijven In tegenstelling tot de Quadpod die wel atomen gevangen houdt bij kritische rotatie Vervolgens is er gekeken naar de sterke van de vor
23. be able to keep the cloud confined in the centre The static TOP trap however does not have the ability to set the cloud into rotation This requires us to introduce an anisotropy Since discretisa tion of the TOP currents was needed for our equipment the simplest way of creating the desired symmetry is by setting the number of zero points during one cycle equal to the desired anisotropy symmetry We have done measurements on these types of anisotropic potentials and it turned out that there is a minimal number of points needed to be defined to have atoms in the trap It need be noted though that these experiments were done having an anisotropic TOP trap using the first mode described in section 2 2 1 This may cause effects that we are not able to explain and needs further investi gation Te TRAP The TOP trap is produced by a quadrupole trap in ad 29 CHAPTER 4 DISCRETISATION 30 Part II Static Anisotropic TOP trap CHAPTER 5 Phase Modulation they are shaped such that the zero magnetic field point has a round trajectory Following symmetry arguments the TOP trap potential is considered round when the trajectory is round and the rotation speed of the zero magnetic field point is constant There is a tool required for making the cloud rotate The Oxford group Arlt et al 1999 Hodby Hechenblaikner Hopkins Marago amp Foot 2001 chose to modulate the amplitude of the trajectory thereby creating versatile
24. busch et al 2002 i 20g wi 1 e 0 z20 1 16 202 03 1 9 0 1 17 CHAPTER 1 ROTATION From these equations one can deduce that the potential is dynamically un stable in the range VI ew V1 e Dynamical instability means that the potential becomes anti trapping in one or more directions In the case of the Dipod there are two directions that are opening up depending on which side of the critical rotation the Dipod is rotating This makes the anisotropic harmonic potential an unsuitable candidate for rotating close to critical rotation With a higher order potential it should be possible to keep the atoms stabilised up to critical rotation A first try to solve this problem was done by taking a potential of the next higher order Tripod Rath 2010 where the potential is given by U r smut a y 4 32 y y 1 18 Here the notation of Rath 2010 is used a N w and y 27 8 potential has a region of stability within the dynamical unstable regime This region has the shape of a triangle and can be characterised by three end points 1 2 et 1 19 n 1 2 je 2 peg s 209 1 20 27 21 1 2 1 2 mi 1 21 2n 21 These points are equally spaced on a circle with radius T 2 Bea 7 1 22 This shows that in the limit of critical rotation 1 the stable region vanishes and thus the potential is destabilised A solution for rota
25. cos 40 A local minimum in the centre means that the gradient in all directions should be pointing outward In this case we can make it even stronger and find the parameters for a global minimum there This gives a strong condi tion on the projection of the gradient a 1 1 4r 10 2 5 5 cos 40 gt 0 71 APPENDIX PROPERTIES OF A 4 ORDER POTENTIAL Figure B 1 The possible combinations of a b are given by the six im ages A gt 0 1 and b gt 2a 1 5 which is trapping gt 0 land b 2a 2 which is on the limit C a gt 0 1 and b lt 2a 2 5 which is not trapping On the next row all the potentials are not trapping because of a lt 0 the different images plotted are for different b D b lt 2a 1 5 E b 2a 2 F b gt 2a 2 5 The images confirm what is derived in eqs B 3 and B 4 When looking for a local minimum the limit r 0 needs be taken but since r gt 0 by definition we find that it is equivalent for all r and thus we can find parameters for having a global minimum at the centre Introducing a 5 1 40 then it follows that Y8 0 lt a lt 1 a 10 2a a gt 0 B 2 The case that a 0 gives the limit gt 0 3 72 APPENDIX PROPERTIES OF A 41H ORDER POTENTIAL Next for a 0 1 we can write 2 2 b gt mise 2a 1 Q a gt 2a Va 0 1 B 4 To finalis
26. densate with vortices in the condensate There are N vortices in the cloud of area A where Stokes theorem is used to introduce the curl of v and S is the inte gration range which has area A and surface element vector dS Although the classical velocity and the quantum velocity are clearly not the same the contour integration in the limit of a large contour should recover similar result This is similar to the correspondence principle which states that in the limit of large quantum numbers the classical result should be obtained Finally the number of vortices N can be calculated by Ny A 11 1 13 Closely related to this is the vortex filling fraction v which is defined as the number of atoms per vortex 1 14 where N is the number of atoms The vortex filling fraction is a parameter to indicate in which rotation regime the system is In figure 1 3 images of different regimes for v are shown and the most interesting regime is where v lt 10 which is the regime of strongly correlated states The interesting features in this regime are that the vortex lattice will melt at v 10 Cooper Wilkin amp Gunn 2001 and at the point v 1 strongly correlated states are produced that are related to the fractional Quantum Hall effect Laughlin for v 1 2 Pfaffian for v 1 The group of Chu has announced to have reached this regime Gemelke et al 2010 but so far there are no images to complete the figure 8
27. do so first the potential was expanded in spa tial coordinates and then integrate over one cycle The expanded dimen sionless potential om r UC F u Bo where n stands for the n P order expansion is given by UN 1 5 5 f F sJA 99wa G R ie 1 5996s 5 3 2 82 09 en amp if Ji e 8Na 16222 1 2 3202 J3 30 0n 1 J1 3e n3 TD ale 4 85 1 5 Ja 4e wa Ja 2e 8y a J 4 n a 4 2 y 1 Jo 2 dn 1 266 2 pu 2 30 Ja 4 w 1 Ja 4e n 2 4 22 32 2 9 96 2 02 Jo 2 5na J 20 n 2 12824 where y p with p N is the Kronecker delta used to indicate that it belongs to the p order of symmetry J x is the Bessel function for These results are original in the sense that this has not been done before and it can be applied more generally than the result from previous studies Minogin et al 1998 The result from these studies can even be retrieved by taking a special case N 0 and or e 0 The standard TOP trap has the properties e 0 and or N 0 Check ing the potential using these parameters we get 1 1 Ur f Bo 1 4 F 827 F 7 327 2 1282 giving the same result for the standard TOP trap as eq 5 5 36 CHAPTER 5 PHASE MODULATION The case N 1 in principle off centres the cloud with respect to the circle of death This ca
28. e a needs to be always positive and the limit of bis defined by a as b gt 2a In figure B 1 six different situations are plotted And clearly only the first one with a gt 0 and b gt 2a is trapping 73 APPENDIX PROPERTIES OF A 4 Y ORDER POTENTIAL 74 APPENDIX C Cicero Word Generator synchronise different steps in the experiment The computer itself uses software to create the right order of steps a sequence From the beginning of the experiment on a home build program was used which had to be reprogrammed when a new time step or device was added This costed a lot of valuable time that could be used better The solution was to use a program which was already used by the neighbouring lab Cicero Word Generator Keshet 2008 Cicero Word Generator is split up in three parts The first part is Atti cus which is the server application that runs on the computer that runs the experiment The second is Cicero which is the client application and graphical interface to the program With Cicero several servers can be con trolled and provided with instructions The third part is called Elgin and is only used to read log files From now on we will only discuss Atticus and Cicero because these are the most important for the experiment In this chapter we will discuss the features added to make Cicero work with the experiment In Code example like C 1 C 2 C 4 and C 5 we have used the definition that mean
29. e potential is not well defined in the co rotating frame The potential in the co rotating frame is given by the potential in the lab frame eq 3 10 with the phase modulation function 4 given by eq 5 1 1 Ur T Bo y1 2 92 AZ 2 2 dt 54 CHAPTER 9 Discretising Rotation cause we need the periods of two rotations to be overlapping The first arising for the fast rotating bias field of the TOP and the sec ond for the slow rotation of the averaged potential This restriction is im plied by the equipment used because our equipment takes a point list for one outputting period Tout and repeatedly outputs this after one another This requires the periods of both the fast and slow rotation to be fully finished before starting a new period The discretisation needs some special caution be 9 1 Discretisation formalism The outputting period Tout is in general repeated and then it is important that there is no phase jump in between the end and the beginning of the pe riod This makes it important to define one outputting period as a multiple of both the short rotation period Trop 2r wr and the long rotation period Trot 27 9 Tout q Trop p Trot 9 1 where 4 N are respectively the number of cycles of the zero point and the number of cycles of the rotating anisotropic potential Then we can de fine a re
30. efined by a multiple of 4 in the number of points with a minimum of 64 It is extremely important to not interrupt the sending of the data to the device otherwise it may crash COCOCORB sous 69 Figure 2 2 The Tabor WW1072 Arbitrary Waveform Generator The gen erator has two outputting channels used as input signals for the two TOP coils amplifiers The device has the possibility to send waveform data points by GPIB and wait for a digital trigger Initially the Arbitrary waveform mode was used in Gated mode to gen erate the necessary signals for the TOP trap This had the disadvantage of only being able to send one waveform To give an example we are only able gt We have investigated 27 x 5kHz and 27 x 20 kHz to see if the evaporation can be done more efficiently For more information about the choice of wr we refer to Part I 13 CHAPTER 2 SETUP to send one waveform lets take a sine and a cosine for each channel respec tively then when a digital trigger is send to the device it output the sine and cosine When the digital trigger is taken stopped the outputting is stopped To have a different signal afterwards new data has to be send to the device One could argue to send the data points for the whole TOP trap se quence but since there are 32 to 64 data points per bias field cycle needed and the memory is limited to 2MB this would limit us to a couple of sec onds of TOP trap which is insufficient f
31. egree of freedom no adiabatic approximation while evolving two for spin 1 2 particles or three for spin 1 particle coupled time dependent Schr dinger equations In the far future geometrical potentials G nter Cheneau Yefsah Rath amp Dalibard 2009 and 2D systems provide the group with interesting phe nomena to study Finally the goal is to reach the strongly correlated regime either with rotation or these geometrical potentials and whether that is fea sible or not needs to be investigated answers a lot of questions but often an answered question 85 APPENDIX C PERSPECTIVES 86 Acknowledgments by The reason for that is the partly the place where this research was done First the team in which I did my work Tarik I still want to see that 4 0 for the Netherlands of course R mi Laura Lauri anne Patrick and Ken Dude dude dude ohw no nothing But also the great guide during this period Jean Dalibard who seemed to be able to give me the right direction when I though I was completely stuck Sec ondly the people from the other groups Sebastian Sebi Franz el I like the slacklining Julliette Sanjukta Jerom Luigi thanks for the soldering tips Emmanuel incredible patience with me and Cicero David Fabrice Nir Sylvain Ulrich Armin Thomas Fr d rick and Christophe year is long time but this year in Paris seemed to have flown I would like to thank Gora for inviting me
32. elliptical potentials The disadvantage using this method is that the radius of death depends on its angular position This causes an oscillation of the radius of death and this may make it possible for the zero magnetic field point to enter into the cloud and cause losses Our approach is to use phase modulation which holds the radius of death constant over one cycle C iey aes TOP traps do not have the ability to rotate a cloud because Phase modulation is a method that is based on changing the rotation ve locity of the zero magnetic field point Since only the velocity is modulated there is no change of the radius of death Phase modulation provides us with the tool for shaping and rotating the desired potential 5 1 Formalism In Chapters 3 and 4 the formalism based on Minogin et al 1998 is intro duced This is the basis of our concept of phase modulation of the TOP trap There is always a static anisotropy due to the not perfectly round trajectory of the zero magnetic field point 33 CHAPTER 5 PHASE MODULATION Figure 5 1 Figure A shows the potential for an amplitude modulated TOP trap which has the amplitude in the z direction to be half the amplitude in the y direction The black dots are equally spaced points in time of the zero magnetic field point move ment and the red circle is the radius of death Figure B shows a similar situation but with phase modulation where 0 5 Again
33. erPoints TRAC SEL ffer SEQuence DEFine segmentNumber segmentNumber 1 1 This is looped over the number channel to set them all to outputting INST SEL channelNumber AM 1 INST SEL channelNumber OUTP 1 82 Epilog Perspectives poses multiple new questions The same is true for this research be cause the question whether we can use phase modulation as a tech nique for making rotating systems seems to be answered with a yes but the research has brought us with a lot of questions A good example is whether it is possible to reach the regime of strongly correlated systems but even closer to the research done in this report one can ask himself why the atoms react so strongly on very strong phase modulation To answer these ques tions a lot of further investigation is needed That is also the reason for the group to continue with what we have started The first thing that is done a the moment of writing is installing new imaging equipment for more qualitative measurements The next step will be doing the same experiments as before and trying to see vortices in the Quadpod at critical rotation Another interesting and yet not understood feature is found in sections 4 2 and 7 2 Where there is some evidence that if the rotation of the bias field changes to rapidly the atoms are strongly influenced Investigating this hypothesis might be done with simulations that keep track of the in ternal d
34. escribed with only one parameter Ji 4e This means that it is possible to make groups of e with J 4e j JP min which is sufficient to define the anisotropy Thus the replacement 4e j can be made but again care needs to be taken when the transformation back to e is made The replacement transforms the potential into the follow ing form 1 I 525 Ur F I re 1 55 3 9 2 30 j The most remarkable feature following from this becomes apparent when j 0 ande 0 the standard TOP trap which has a potential that is perfectly symmetrical but this is not only the case for 0 In fact it is 42 CHAPTER 6 POTENTIAL SHAPING J ke 0 6 0 6 Figure 6 2 Shown is the Bessel function ke with being marked the zero crossing red dots and the range of values j light blue area At the zero crossings there is special behaviour of the potentials at least up to the Quadpod because their potentials Ur f equal that of the standard TOP potential with 0 true for all e such that 7 0 The potential in the case of 7 0 is given by 1 1 F pBo 1 27 74 Ur f u 7 This is remarkable because as be seen from figure 6 2 there a lot of zero crossings in the Bessel function 4 From now on we will use the variable j to define the anisotropy 43 CHAPTER 6 POTENTIAL SHAPING 44 CHAPTER 7 Ani
35. et the Trigger mode to Gated such that a digital channel can be used to define the periods during which the TOP needs to be on It also defines the frequency TOPnu based on an Cicero interface variable for outputting a Fixed function The second part defines the output for channel one and gives it a sine with amplitude 3 375 volt and turns amplitude modulation and the out putting on The third part does the same as the second part but this time for chan nel two and with a 90 phase difference To execute a TOP trap the only thing that needs to be done is making a Sequence in Cicero and setting the GPIB group made for the TOP trap 76 APPENDIX C CICERO WORD GENERATOR to an early segment The next step is to add a digital trigger that gates the signal send from the Tabor device 2 Magnetic Transport The magnetic transport is in detail explained in the thesis of Marc Cheneau Cheneau 2009 we only needed to implement the use of the magnetic transport in Cicero The magnetic transport is powered by a number of power supplies and the currents that these supplies output is controllable with analog channels In Cicero this can be done using the Analog F3 tab and making several groups that corresponds to the different phases in the transport All that needed be done was writing a script in Python for converting the current files which give the currents through the coil pairs as a function of the position of the cloud The p
36. etic field B r at position r has the magnetic potential energy U r u B r This potential energy provides not only a Larmor precession of the magnetic moment u around the direction of the magnetic field B r but also a force that attracts atoms with a negative projection to a minimum in the magnetic field 3 11 Larmor precession The Larmor precession of a magnetic moment u around the direction of the magnetic field B r is characterised by the Larmor frequency u B r 3 1 18 CHAPTER 3 TOP TRAP which corresponds to the angular frequency associated with the potential energy The system tries to align this precessing with the magnetic field to lower the potential energy The Larmor frequency provides the time scale for the aligning To calculate the Larmor frequency the magnetic field at position r is needed One can use the Biot Savart law to calculate this in the limit of small distances from the centre of the trap The calculation gives a relation for the magnetic field of a quadrupole trap b r b r b x zi b y yi 2b z zi Z 3 2 with b the gradient of the magnetic field and r yi zi the centre of the trap This approximation is only valid in the limit r r lt d where d stands either for the radius of the coils or for the distance from the trap centre to the centre of each coil At the point r r the magnetic field vanishes which causes the adia bat
37. figure starting with 2 A clearly shows the strong Dipod potential in B a strong Quadpod potential is shown in C the 8 order is still visi ble but already the harmonic term is taking over Figures D through F are clearly approximating the round TOP trap The red circle is at the radius of death and the black dots on the circle are the zero magnetic field points used to calculate the potential The first step is to discretise the phase function such that it only accepts integer numbers amp t 2nt gt 4 42 where n Nis the number of zero points during one cycle andi 0 n 1 is the discretised time iterator However there is a continuous current sent through the coils thus finally the potential produced is again contin uous 26 CHAPTER 4 DISCRETISATION Number of zero points n spectrum 50 000 40 000 30 000 20 000 Atom number a u 10 000 1 10 100 1000 Number of zero points Figure 4 2 PRELIMINARY Atom number vs number of zero magnetic field points during one cycle 7 The 7 data points are from left to right 2 4 8 16 32 64 128 256 512 There is a clear rise in the number of atoms from 7 16 to 32 The rest of this thesis will use 7 64 to have atoms Shaping will be done with more dedicated phase modulation The points in the graph are an average of multiple measurements and the error bars are the standard errors of each points calculated by Ose
38. g doi 10 1103 PhysRevLett 92 050403 Brink D M amp Sukumar C V 2006 Sep Majorana spin flip transitions in a magnetic trap Physical Review A 74 3 035401 Available from http dx doi org 10 1103 PhysRevA 74 035401 Cheneau M 2009 Transition superfluide et potentiels g om triques dans le gaz de bose bidimensionel Unpublished doctoral dissertation ENS Paris Cohen Tannoudji C Diu Lalo F 1977 Quantum mechanics Vol 1 John Wiley and Sons Cooper N R Wilkin amp Gunn J 2001 08 30 Quantum phases of vortices in rotating bose einstein condensates Physical Re 93 REFERENCES view Letters 87 12 Available from http link aps org doi 10 1103 PhysRevLett 87 120405 Davis K B Mewes M O Andrews M R Druten N J van Durfee D S Kurn D M et al 1995 Nov 27 Bose einstein condensation in a gas of sodium atoms Physical Review Letters 75 22 3969 3973 Available from http dx doi org 10 1103 PhysRevLett 75 3969 Feynman R P 1955 Progress in low temperature physics North Holland Amsterdam Franzosi R Zambon B amp Arimondo E 2004 Nov Nonadiabatic ef fects in the dynamics of atoms confined in a cylindric time orbiting potential magnetic trap Phys Rev A 70 5 053603 Gemelke N Sarajlic E amp Chu 5 2010 Rotating Few body Atomic Systems in the Fractional Quantum Hall Regime ArXiv e prints Greiner M Bloch I
39. heme and the potential in the co rotating frame only changes due to the centrifu gal potential Experimentally the formalism is explored in both a rotating Dipod N 2 and Quadpod N 4 The first measurements provided use with a ro tation spectrum of both the Dipod and Quadpod These spectra show that around critical rotation the Dipod is anti trapping whereas the Quadpod for a certain anisotropy strength can be trapping at the point of critical ro tation With the second series of measurements we investigated the strength of the anisotropy at critical rotation of a Quadpod and compared the results with the analytical predictions for the trapping window of the Quadpod potential The result and predictions seemed to be in perfect agreement although it must be noted that the decrease of atom number before the threshold can not yet the explained quantitatively More investigation is needed Also it need be noted that the experiments on anisotropy strength at critical rotation were done in the first run mode discussed in subsection 2 2 1 which has the rotating shaped TOP trap during evaporation and de compressing of the TOP trap We have shown that rotating an atom cloud using phase 63 CHAPTER 10 RESULTS 64 Appendix APPENDIX A Integrating Jacobi Anger expanded functions sum of cylindrical waves Abramowitz amp Stegun 1972 In the following chapter we use this formalism to solve the integral of multiple ang
40. hoice of our group was to use the TOP trap because it provides good optical access and great flexibility Moreover we can use the rotating bias field and tune its parameters for shaping and rotating the potential erc TRAP A quadrupole trap is the simplest purely mag 3 1 Magnetic trapping As mentioned above the simplest purely magnetic trap is a quadrupole trap The setup of this consists of two coils placed at exactly twice their radius d from one another with their centres aligned on the z axis The cur 17 CHAPTER 3 TOP TRAP Figure 3 1 Potential of a quadrupole trap The solid line shows the poten tial for atoms which have a negative projection of their mag netic moment on the axis of the magnetic field At the centre the potential has a minimum and the atoms prefer to seek the lowest energy low field seekers The dashed line shows the potential for atoms which have a positive projection of their magnetic moment in the direction of the magnetic field They can lower their potential energy by leaving the trap and thus are untrappable high field seekers It is not possible to have a maximum in the magnetic field Earnshaw s theorem thus the high field seekers can not be trapped with static magnetic fields rent flowing through each coil is equal but oppositely circulating This is called the anti Helmholtz configuration The trap is based on the physical property that the magnetic moment u of an atom in a magn
41. ic approximation to be non valid Since there is a vanishing Larmor fre quency which triggers non adiabatic Majorana spin flips that flip the mag netic moment of the atoms to an in general untrappable state and induces losses from the trap Initially at high temperatures these losses are rela tively small because the relative density in the centre of the trap is low On the contrary at low temperatures the density at the trap centre is relatively high and the loss rate increases This forced researchers to introduce other trap configurations One possibility is to introduce a fast rotating bias field which moves the zero point out of the cloud Petrich et al 1995 31 2 Force due to spatial inhomogeneous magnetic field The force on the atoms due to the spatial inhomogeneity of the magnetic field may be calculated using the gradient of the potential by F r VU r uVb r where is one of the possible negative projections of the magnetic mo ment in the direction of the magnetic field and b r the magnitude of the magnetic field at position r The calculation can be simplified by taking the trap centre being r 0 0 0 It follows that the force on a particle with magnetic moment p equals to T Fr ubV vr mu 12 zu VITIA 3 3 19 CHAPTER 3 TOP TRAP where r x y 4z 2 is similar to position vector r but with a rescaled z axis In the xy plane the force turns out to be constant and pointing to wards
42. ined the two different modes used in the experiments described by this thesis In this section we will give some more detail on the difference between the two modes C 6 1 Gated mode The first is best described as being the Gated mode This means that only one type of waveform is defined and it is outputted by the Tabor WW1072 during a high signal of its trigger input Gated triggering In Listing C 4 the GPIB code that is send to the Tabor is shown Lines 2 6 define the Gated trigger mode Line 7 gives the phase offset between the two channels and Line 8 sets the frequency outputtingFrequency at which the individual data points of the waveform need be outputted The variable outputtingFrequency is calculated taking the number of points in a single waveform 7 and mul tiplies this with the frequency at which the full waveform needs to be out putted TO Pnu The second part defines the waveform data points for each channel and sends these to the device Line 11 13 sets the channel its amplitude and that it can be modulated using an external analog signal Line 14 18 loads the waveform data to the device Where numberPoints is 1 p and buffer is 27 p each point used two times four bits and bufferLength is the length of the string buffer in ASCI characters waveformData is the actual data in binary form to speed up the transfer Finally Line 19 set the channel to output when a high signal Gated trigger is received Ticti
43. ing dotMath EqCompiler eq and Units Dimension unit to convert a value in interface units to a volt age Lictina C 2 Eunction ta convort intarfara unit tn valtana public static double convertDimensions double bufferValue dotMath EqCompiler eq Units Dimension unit double convertedValue 0 Give the value of the variable eq SetVariable unit ToString bufferValue double val eq Calculate if double IsInfinity val amp amp double IsNaN val convertedValue eq Calculate catch Exception return convertedValue After these conversions new data lists are created and send to the buffers of analog outputting cards C 5 Variable Timebase To improve the time resolution possible in our experiment we have imple mented the Variable Timebase mode in Cicero based on the instructions in Keshet 2008 To make this possible a new PXI was installed the PXI 6534 79 WN nM APPENDIX C CICERO WORD GENERATOR from National Instruments This card now operates on a high resolution and makes sure that the other cards only change their outputting when something is changed The big advantage is that during a MOT phase now the analog channels do not need to constantly output and this drastically reduce the memory used and thus we can improve the resolution C 6 Tabor WW1072 Programming Phase Modulated TOP In section 2 2 1 we already expla
44. ion is relatively higher The velocity depends on the anisotropy strength and an interesting feature of the phase velocity is that for an anisotropy with e gt 1 N there are phases during a single cycle where the rotation of the zero point actually changes its direction 5 2 Time averaged potential The rotation frequency is limited by the Larmor frequency see eq 3 8 be cause it determines the validity of the adiabatic approximation This means that the magnetic moments of the atoms need to be able to follow the mag netic field w t where we intentionally wrote w7 t to emphasise the fact that the rotation frequency of the zero magnetic field depends on the phase modulation wr t wr Nt 5 3 with wr the initial rotation frequency of a standard TOP trap This velocity is oscillating and the maximum value of this oscillation is given by wr wr 1 Ne The new adiabatic approximation validity limit is given by 1 Ne lt wr 5 4 When this limit is fulfilled it is possible to write the time averaged poten tial of a phase modulated TOP trap eq 3 10 as 1 js 1 eg az 2 4 0 where we now use the phase function 7 from eq 5 1 35 CHAPTER 5 PHASE MODULATION 5 3 Expanding the phase modulated potential Solving the integral relation of the phase modulated potential is extremely difficult if not impossible to
45. irement for the adiabatic approximation to be valid and gives an upper limit to the rotation frequency in our experiment the Larmor frequency is of order 27 x 5 MHz On the other end there is a lower limit which states that the movement of the zero magnetic field point is much faster then the Centre Of Mass The experiments are done with Rb in the states F 1 mp 1 and F 2 mr 1 2 Rath 2010 with Lande factors g 1 2 and g2 1 2 21 CHAPTER 3 TOP TRAP COM motion such that the cloud does not have the time to come close to the zero magnetic field point This can be characterised by the harmonic trapping frequency w1 of the averaged trap WT 3 9 where w which will be derived later on in the experiments described by this thesis is of the order of 27 x 10 Hz When these limits are valid the average potential U r can be calcu lated in the following way 1 Ur f Br r t dt 3 10 uBo y1 2 AZ 2 cos t 2 4 dt This potential shows the importance of the limit in equation 3 8 because the integration is done over the magnitude of the magnetic field An aver age over the magnetic field itself would again give rise to a quadrupole field whereas now the magnetic moments are able to follow and be aligned with the magnetic field 4 at all times 3 44 Expanding the potential in spatial coordinates Expanding the integral i
46. l and then the sine by sin Dsin In ke sin 2ri Nn k A 6 nel A 4 Integration of a multiple angle phase modulated functions The last procedure is to integrate over one cycle period First for a cosine 1 cos k sin t PUN UU ke pr cos 2x Nn k dt 0 nez y In ke Onn k nez 68 APPENDIX A INTEGRATING JACOBI ANGER EXPANDED FUNCTIONS Then for a sine 1 n _ sin BaD D 80 sarin id E which provides the necessary tools to solve the expanded phase modulated TOP potentials see Part II 69 APPENDIX A INTEGRATING J ACOBI ANGER EXPANDED FUNCTIONS 70 APPENDIX B Properties of a 4 order potential cause they are the lowest trapping order when the harmonic potential is removed We would like to know for which parameters a b R the following 4 order potential has a local minimum in the centre pe potentials are of great interest in rotating systems be f x y alt y ba y B 1 where x and y are the parameters in the plane Next we calculate the gradi ent of the potential in the zy plane and project this onto a arbitrary direc tion r rcos0 rsin0 with r the distance from the centre and 0 the angle with respect to the positive x axis V f z y 4 22 20 zy xy dar cos 0 sin 09 2br cos 0 sin 0 cos O sin O y r Vf z y 4r a cos 0 sint 0 2 cos 0 sin 8 b 2a cos 0 sin 9 NUUAM i 4r 10 2 5 2
47. late the rotation speed of this bias field With this technique we are able to produce versatile potentials with all possible geometries The first step in producing these versatile potentials is trying to get the strongest anisotropies possible by choosing the geometry 7 1 2 and dividing that number points equally over a circle The next step is to let the quadrupole field zero magnetic field point jump from one point to the other This will give the strongest anisotropy for symmetry 7 This turned out not to be a big success because it strongly influenced the atoms and it turned out that 7 64 was most favourable for the atom number The next step is to take these 7 64 points and spread them out over the circle This time not equally divide but grouped in N 1 2 groups with phase modulation How strongly these points are grouped determines the strength of the anisotropy and now N is the symmetry We have been looking at number of atoms as a function of the strength of the anisotropy and some interesting behaviour was found N 2 Dipod and N 4 89 APPENDIX C CRITICAL ROTATION Quadpod showed a linear decreasing behaviour whereas the N 3 Tri pod shows a plateau with a threshold above which all the atoms are lost Finally these phase modulated potentials are set into rotation and we have been looking at the rotation spectra of the three symmetries as well as the strength of the anisotropy at critical rotation for
48. later stage can be rotated The strongest n order anisotropy can be created with just using n zero magnetic field points which are distributed evenly over the cycle in space and time For the TOP trap this would mean it starts in one point stays there a certain period of time and then jumps to the next point field trap uses rapidly rotating bias field to 41 Zero magnetic field points The formalism used in Chapter 3 is based on continuous variables This means that the time resolution in principle is infinite for practical reasons this is not feasible The device used to generate the signals that are sent to the amplifiers only accepts lists of data points This required us to introduce a formalism for discrete rotation The position of the zero magnetic field point can be described by the fast rotating spatial homogeneous bias field B t derived from equation 3 4 and the quadrupole field b r ro t t 2 sin 0 4 1 Ideally this happens at infinite speeds but in practice we are limited by the bandwidth of the signal amplifiers The device we use for outputting the signals is an arbitrary waveform generator Tabor WW1072 25 CHAPTER 4 DISCRETISATION D N 16 E N 32 F N 64 Figure 4 1 Potentials created by having zero magnetic field points From left to right and top to bottom the number of points per cycle is doubled in each following
49. lation between the rotation frequencies by 2m 2m Q jf 9 2 WT q 55 CHAPTER 9 DISCRETISING ROTATION lt rot Trop lt gt 4 Tour y Figure 9 1 Shown are two sinusoidal function the blue line correspond ing to a fast oscillation Trop and the red line to a slow os cillation Tot after one oscillation of the red line the blue line is not in the same phase After the second oscillation of the red line both are in the same phase of the oscillation Tout These oscillations would correspond to p 2 and q 5 in eq 9 1 Note the number of oscillations is arbitrary and only used to show how the different cycle periods correspond need to overlap With this transformation we can define the discrete rotational phase mod ulation function following eq 8 3 by gti esin21N 2 pir 22 9 3 7 7 4 7 In principle this function has all the necessary information to do the rota tions but since we are limited by memory we need to refine our procedure To do so two schemes need to be introduced for varying the rotation fre quency A characterisation of the different schemes has to be based on the amount of memory it consumes meaning the number of data points Nout Nout 1 q 9 4 The first scheme takes a fixed q and varies the p The advantage of this scheme is the fixed frequency resolution On the down side to have a high 56 CHAPTER 9 DISCRETISING ROTATION freque
50. le phase modulated functions EXPANSION was introduced to describe plane waves as a Jacobi Anger expansion The Jacobi Anger expansion is based on a Laurent series for a generating function ect 1 t _ gt le A 1 2 where 2 C t a generating function and J the nt order Bessel function The Jacobi Anger expansion for a sine in the argument of an exponential function is generated by t ei ej sind Jul jen A 2 nez and finally the Jacobi Anger expansion for sine phase modulation is ob tained 67 APPENDIX A INTEGRATING J ACOBI ANGER EXPANDED FUNCTIONS 2 Jacobi Anger expansion for phase modulated func tions Phase modulation can be described with a phase function Psin t inside co sine with the time normalised to the unit of one oscillation period 2 2 Nt The phase modulated cosine is then expanded as cos Pin M C Jn e 2 1 neZ and the sine as sin sin ial In e sin 27t Nn 1 A 4 nez A 3 Multiple angle expansions Multiple angle trigonometric functions are important for describing pow ers of trigonometric functions Phase modulation can be described with a phase function Osin t inside a co sine with the time normalised to one oscillation period The k N is added to define the multiple angle First the phase modulated cosine is d by cos Jal Jn ke cos 2rt Nn k A 5 ne
51. ling Ketterle amp Druten 1996 is started and before the losses due to Majorana spinflips Brink amp Sukumar 2006 become too large 1The magnetic transport is based on the Munich model Greiner Bloch H nsch amp Esslinger 2001 This initial trapping stage lasts for 15 s and captures 6 10 atoms 5The transport is done over 0 5 m and takes 5 s during which 2 6 10 atoms are conserved 11 CHAPTER 2 SETUP Magnetic Transport TOP coils MOT chamber Quadrupole coils Science cell Figure 2 1 The Rb setup used to do the experiments The experiment is divided into two main chambers The MOT chamber where the atoms are captured and pre cooled from a back ground gas and a Science cell where further cooling is done as well as the manipulation of the cloud The two cham bers are connected by a magnetic transport based on the Mu nich model Greiner et al 2001 the cloud is transferred into a Time averaged Orbiting Potential TOP trap Anderson et al 1995 Petrich Anderson Ensher amp Cornell 1995 where the evaporative cooling Rath 2010 is continued until a Bose Einstein con densate BEC is reached Anderson et al 1995 Davis et al 1995 While the cloud is in the TOP trap the gradient of the quadrupole field is lowered in order to decrease the mechanical stress on the coil holders when rapidly turning off the magnetic trap After the sequence absorption imaging is used
52. m bij kritische rotatie van een Quadpod Wat blijkt is dat voorspeld kan worden dat bij een zekere sterkte geen atomen gevangen meer blijven en dit ook overeenkomt met het experiment Er zijn wel nog vele vragen die onbeantwoord blijven bij deze experi menten dus dit vereist nog verder onderzoek 92 References Abramowitz amp Stegun I 1972 Handbook of mathematical functions with formulas graphs and mathematical tables 978 0 486 61272 Dover Publications Anderson M H Ensher J R Matthews M R Wieman C E amp Cornell E A 1995 Observation of Bose Einstein Conden sation in a Dilute Atomic Vapor Science 269 5221 198 201 Available from http www sciencemag org cgi content abstract 269 5221 198 Arlt J Marag Hodby E Hopkins 5 A Hechenblaikner G Web ster S et al 1999 Bose einstein condensation in a rotating anisotropic top trap Journal of Physics B Atomic Molecular and Opti cal Physics 32 24 5861 Available from http stacks iop org 0953 4075 32 1 24 320 Barrett M D Sauer J amp Chapman S 2001 06 19 optical formation of an atomic bose einstein condensate Physical Re view Letters 87 1 Available from http link aps org doi 10 1103 PhysRevLett 87 010404 Bretin V Stock S Seurin Y amp Dalibard J 2004 02 04 Fast rotation of a bose einstein condensate Physical Review Letters 92 5 Avail able from http link aps or
53. n vector The energy spectrum of this system is given by E n l Q hlwv n 1 Q0 where hl is the outcome of L working on and corresponds to the projection of the angular momentum of in the z direction 1 3 Critical Rotation and Lowest Landau Level The limit w Q is referred to as the point of critical rotation In that limit the energy spectrum of a particle in a rotating harmonic potential is given by En n lz hw n Ll c 1 1 5 and the possible values for l are limited by n lt I n and n l is even Cohen Tannoudji Diu amp Lalo 1977 This implies that the possible energy levels are given by 2n7 1 hw with nz N of these states n are infinitely degenerate since all states k n kl with N Qn n 1 have the same energy These levels nz correspond to the Landau levels that describe the energy levels of a charged particle in a magnetic field The state of lowest energy ng 0 and n 1 is called the lowest Landau Level LLL and is important for describing fractional Quantum Hall physics Laughlin 1999 Stormer Tsui amp Gossard 1999 Figure 1 1 shows the energy spectrum of a 2D harmonic oscillator In figure A the case for no rotation is shown and the energy spectrum of a 2D harmonic oscillator is shown In figure B the critical rotation limit Q w is shown Close to critical rotation the levels are not degenerate and the energy between the l 0 and I k state i
54. n the potential of the magnetic field in equation 3 10 up to 4 order and integrating over time gives Minogin et al 1998 1 1 Ur f Bo 1 7 f 827 od F 327222 12824 O f 26 3 11 When looking in the plane of rotation ry plane 2 0 it is possible to compare the second order expansion with the equation of a harmonic os cillator ub 2mBo b 1 oc mir gt WL c In x 10Hz 3 12 3The harmonic trapping frequency is measured using oscillation measurements 22 CHAPTER 3 TOP TRAP A similar comparison can be done for the 4 order terms lub g 4 Bm c 1 ub 3 13 c e 3 13 which implies that when the harmonic term is compensated by the centrifu gal potential the TOP trap still has a quartic term which is trapping Com paring this to the Ioffe Pritchard trap which has lt 0 it is seen that in the TOP a natural trapping mechanism arises whereas in the case of the Ioffe Pritchard trap an additional anharmonic trap was needed Bretin Stock Seurin amp Dalibard 2004 If we compare these results with Appendix A it corresponds to the case b 0 and a gt 0 and thus trapping In figure 3 3 the exact potential of the TOP trap on the 2 axis is shown It is important to note that at the centre a clear harmonic behaviour is vis ible at distances smaller then the radius of death whereas outside of the radius of death the trap becomes linear again which is to be e
55. na C A CDIB cada TOP with mada Catad Set the GATED mode FUNC MODE USER INIT CONT 0 es cama di 80 1 2 3 APPENDIX C CICERO WORD GENERATOR TRIG BURS 0 SERIE SOUR ADVE PHAS OFFS 0 FREQ RAST outputtingFrequency This is repeated for the number of channels on clus ece a eme Case 2 INST SEL channelNumber VOLT channelAmplitude 1 TRAC DEF 1 numberPoints TRAC SEL 1 CLS CLS xOPT TRAC DATA bufferLength buffer waveformData xCLS INST SEL channelNumber OUTP 1 C 6 2 Segmented Mode The second mode is based on the first but instead of have one waveform and outputting that during a high signal on the Trigger in port This mode can accommodate several waveforms and switch between these when a rising slope is send to the Trigger in port In Listing C 5 the GPIB commands are shown that are send to the Tabor WW1072 to initialise the Segmented mode and to end the waveform data Lines 2 8 are the same as in the Gated mode The same holds for lines 14 20 and 24 30 The big differences are Line 21 and Line 31 which define at which point in the sequence a segment has to come Line 21 does that for the constant amplitude phase which is needed because it is not possible to trigger the first segment In stead the constant phase in outputted after the sending of the data and a trigger is send when
56. ncy resolution q needs to be on the order or higher of wr 27 which takes up a lot of memory and only corresponds to a resolution of 1 Hz which will be explained in the following subsections The second scheme uses a fixed p and changes the 4 which takes up less memory but the fre quency resolution and memory uses are not fixed anymore 91 1 Fixed frequency resolution When choosing the fixed frequency resolution option it means that q is kept constant and p is varied to vary the rotation frequency WT PL 9 5 Then the change in frequency between two points is given by An 1 p xr 9 6 On the downside q needs to be larger than wr 27 10 KHz to get a resolu tion of 1 Hz or better Furthermore an often used q is 20000 which gives a resolution of 0 5 Hz and the number of points per cycle 7 64 The num ber of points that needs to be defined for a full waveform is given by Nout 1 28 10 Since each point corresponds to one byte this fills up more than half of the total memory per channel The important range for now is p 0 no rotation to p 100 which corresponds to the range 0 50 Hz Only a very small range of possible values p 0 q is used so this scheme seems to work inefficient but the definition is clear and simple to use 9 1 2 Fixed period resolution The other possible parameter to change Q is q while leaving p constant The relation for is then given by Aq erp
57. nded functions AJ Jacobi Anger expansion A 2 Jacobi Anger expansion for phase modulated functions Multiple angle expansions 2 22 2222 2 A 4 Integration of a multiple angle phase modulated functions Properties of a 4 order potential C Cicero Word Generator C 1 Tabor WW1072 Programming Standard TOP Lu MIEREN ox e ex 2 E e Agilent N5181 Programming Evaporation Ramp C 4 Converting Interface Unit to Voltages C 5 Variable C 6 Tabor WW1072 Programming Phase Modulated TOP Gated mode oic eari errare ra en C62 Seamented Modes cp ex oP ed PS eGR ee wa Epilog Perspectives Acknowledgments Critical Rotation Kritische rotatie References Index vii 63 65 67 67 68 68 68 71 75 75 77 77 78 79 80 80 81 83 85 87 89 91 93 97 CONTENTS viii Introduction CHAPTER 1 Rotation groups Anderson Ensher Matthews Wieman amp Cornell 1995 Davis et al 1995 in 1995 produced the first Bose Einstein condensates BEC researchers have been interested in this phase of matter The most stunning feature of a BEC is its phase coherence which means that all the particles in the condensate have the same global phase and the cloud acts as if it is a single particle Bc CONDENSATION Ever since the Cornell and Ketterle Another inte
58. or the desired experiments Since the procedure above limits us to using only one type of signal dur ing a run of the experiment we were in the largest part of this thesis limited by just replacing the standard TOP by the specific phase modulated po tential This had the disadvantage of doing the evaporative cooling in these rotating shaped potentials and the decompression either during evapora tion or afterwards which is not an ideal situation These problems were overcome by using the Segmented mode in Con tinuous Run mode which has the ability to make segments with different waveforms and repeat these independently To given an example we first send data for normal TOP trap with a sine and a cosine and then send an other waveform with more specific properties to do the shaping we want Then a digital trigger pulse is used start the first segment and when we want to change to the next we send another one This technique allowed use to do evaporation and decompression dur ing or after evaporation in a standard TOP trap and then switch to the desired shaped potential This mode was used for taking the rotation spec tra in figure 10 1 The switching between different segments is done with a digital trigger and this in principle makes it possible to generate as many different rotating shapes of the TOP trap as the memory allows More information about the programming of the Tabor WW1072 can be found in Appendix C While this
59. resting feature is the behaviour of a condensate in a ro tating system because the Hamiltonian for a particle in a rotating field is equivalent to the Hamiltonian for a charged particle in a magnetic field This mean that with rotating a condensate a system of charged particles in a strong magnetic field can be simulated spin magnetism Classically this similar behaviour is seen when comparing the Coriolis force to the magnetic Lorentz force When we want to look at the system of rotating a BEC the feature of phase coherence implies that the condensate has a velocity potential de scribed by the phase of the condensate Which means that if there are no singularities in the phase field the condensate can not have rotation Feyn man Feynman 1955 was the first to notice that in order to accommodate rotation the condensate needed to have singularities in its phase With his path integral description of rotation in a condensate he could introduce these singularities and describe them A singularity in the global phase is visible in a condensate as a point tube where the density of the condensate is zero These points tubes are referred to as vortices 3 CHAPTER 1 ROTATION 1 1 Coriolis vs Lorentz force The Coriolis and magnetic Lorentz force are both conservative forces that arise for moving particles given by Fo 2mQ xv 1 1 4 xv 1 2 where m is the mass of a particle in the co rotating frame
60. rix N ROTATION The TOP trap has a natural rotation in the cos Qt sin Qt AO snQt cos Ot 8 1 is used where Q is the angular rotation frequency of the rotating system 8 1 Potential in co rotating frame The movement of the zero magnetic field point described by is set into rotation by multiplying the equations of motion by the rotation matrix ro t R Mt ro 9 t ro S t Qt ro 9 t 2r t where we have introduced If we then change the time variable to a dimensionless time in the rotating frame f 1 DE 8 2 53 CHAPTER 8 CO ROTATING FRAME and take the phase modulation function eq 5 1 SPHE 2 esin 2rNt 27 8 3 This means that when looking in the co rotating frame the angular fre quency of the rotating zero magnetic field point is reduced by a Impor tant then for the atoms is that in the rotating frame they still see a averaged potential eq 3 9 with a reduced wr 1 wp gt wy 8 4 Since we are interested in the limit of critical rotation which means w1 it follows that 6 wi wr lt 1 thus validating the limit in eq 8 4 The potential in the rotating frame is then given by eq 3 10 with a redefined dimensionless time t given by eq 8 2 Since the integration limits in this case stay defined over one integration period the integral keeps having the same outcome Only when the limit eq 8 4 is violated th
61. s a variable defined by the script around the given code a variable defined in the Variables F7 tab and lt gt as variables given by the HardwareSettings Em CONTROL Every experiment relies on computers to C 1 Tabor WW1072 Programming Standard TOP Programming of a Standard TOP trap requires to send two unmodulated sinusoidal function with 90 phase difference between the two signals to the two coil pairs Cicero supports sending GPIB parameter lists to any GPIB device this can be done in the tab GPIB F4 Where a new GPIB 75 OND amp WN FR NNNN NPP RRP RP RP RP HR A ON EO REN APPENDIX CICERO WORD GENERATOR group was created and the channel of the Tabor WW1072 was enabled When the channel is enabled one can chose the GPIB mode Parameter from a drop down menu and then the following parameters were used Lictina C 1 CPIR cada Standard TOP Set the dovice Co Che mode INIT CONT 0 RIG GATE 1 RIG BURS 0 RIG SOUR ADV EXT R R Q RIG PHAS 0 IG SLOP POS UNC MODE FIX REQ TOPnu UTER SWING ESAE Set channel 1 to output a cosine signal NGI UNC SHAP SIN ASIN FAA 90 VOLT 3 375 1 JOU MD db Set channel 2 to output a sine signal FUN sEUNC SHAP SIN SIN PHAS 180 3 9375 1 SOUED 1 HS OH H ae see tz The first part was used to s
62. s kh w N The energy difference between two following states with same n is given by h w CHAPTER 1 ROTATION Ey hw NWA Ui A Q 0 B Figure 1 1 Figure A shows the situation without any rotation for a 2D harmonic oscillator 0 Figure shows the case for rota tion frequencies close to critical rotation Q w1 At critical rotation w the levels with n have the same energies and thus are infinite degenerate These are called the Landau levels The level with nz 0 is called the lowest Landau Level LLL 1 4 Vortices So far everything was only on the level of single particle physics and the next step is to look at the wavefunction of a Bose Einstein condensate BEC set into rotation In order to do so a comparison between rotation in a clas sical fluid and in a quantum fluid is made 1 4 1 Rotation of a classical fluid The equilibrium velocity field in the lab frame of a rotating classical fluid is given by v Nxr 1 6 where v is the velocity field at the position r and Q is the rotation vector The curl of the velocity field Vxv 20 1 7 is called the vorticity of the flow CHAPTER 1 ROTATION 14 2 Rotation of a quantum fluid The many body wavefunction of a Bose Einstein condensate is given by Pitaevskii amp Stringari 2003 U r 1 8 where n r is the density and 0 r the phase of the wavefunction at position r This phase 0 r
63. se might be interesting for studying clouds that are rapidly rotating around a circle However we have not done any further research on this topic The cases N 2 3 4 will play a vital role in the coming parts of this thesis because with these potentials we were able to create cloud that are rapidly rotating around their centre of mass With the theoretical results obtained in this section we were able to explain some interesting phenom ena appearing in our experiments 37 CHAPTER 5 PHASE MODULATION 38 CHAPTER 6 Potential Shaping Oxford group Arlt et al 1999 They used amplitude modulation to produce an elliptical path for the zero magnetic field point This again produces an elliptical time averaged potential This causes the radius of death to vary in time and possibly cut into the cloud Our approach is to phase modulate the rotational movement This allows use to make similar elliptical potentials using N 2 a Dipod introduced in eq 5 1 with out changing the radius of death On the other hand it gives us also a framework for producing higher order potentials Gos The first efforts of shaping a TOP trap have been done by the 61 Dipod The Dipod potential is obtained by taking the order of symmetry parameter N 2 thus retreiving 2nt esin Art the averaged potential up to 2 d order in the zy plane corresponding to that is given by Ur F uBo 1 1 22 1 Ji 2e 1
64. sotropy Strength portant to investigate because we are modulating the speed of the zero magnetic field point with phase modulation This speed mod ulation may have a big influence on the atoms For doing this we extend the formalism from Chapter 4 with the phase modulation phase function from Chapter 5 A NISOTROPY STRENGTH the strength of the anisotropy is very im Y wN N 3 Figure 7 1 Shown are the discrete potentials for N 2 N 3 and N 4 Each contour plot shows the zero magnetic field points spread out over the circle of death with the time in between the points being fixed The number of points 7 64 and the anisotropy e 0 5 which corresponds to j 0 44 for the Di pod j lt 0 55 for the Tripod and j 0 58 for the Quadpod 45 CHAPTER 7 ANISOTROPY STRENGTH 7 11 Discrete Phase Modulation Recalling the phase modulation phase function from eq 5 1 we ply the discretisation transformation t i n This gives the discrete phase modulation phase function 2r esin 22 7 1 7 7 where 7 is the number of time points per cycle The potentials are shown for three values of N in figure 7 1 The black points show 7 64 points where the period of time in between two points is constant 7 2 Atom losses due to a static anisotropy To see what the anisotropy strength does at the level of the atoms we look at what happens to the number of atoms when changing the anisotropy s
65. ss variables 7 is the position vector defined in units of the radius of death and t wrt 27 a time variable normalised to one rotation cycle The magnitude of the total magnetic field Br F t is given by Br F i Boy 1 32 422 2 Ecos D 1 jsin D 3 7 The experiment described in this thesis has a radius of death that is on the order of 1mm 20 CHAPTER 3 TOP TRAP Figure 3 2 The magnetic field of the quadrupole field in the rotating frame of the bias field The magnetic field is shifted by the bias field The dashed blue line is the original quadrupole field without bias field and the thick solid line is the quadrupole field with the bias field Bo is the magnitude of the spatial ho mogeneous bias field and ro is the displacement of the zero magnetic field point the radius of death 3 3 Time averaged potential Since the potential is rapidly rotating in time some important limits to the rotation frequency of the bias field are apparent The first requirement is that at all times the magnetic moments of the atoms need to be able to fol low the rotation of the magnetic field This limit can be characterised by the Larmor frequency a theoretical investigation of the adiabatic approximation is done by Franzosi Zambon amp Arimondo 2004 associated to the rotat ing bias field H B r t h b K wr 3 8 WI Equation 3 8 gives the requ
66. t for all 4 58 CHAPTER 10 Results be discussed There are two main phenomena we have been study ing that will be introduced in this part The first phenomena is based on the effect of the rotation frequency on the atoms and the second on the effect of a given the anisotropy on the atoms at critical rotation The potential function in the rotating frame is given by eq 3 10 the phase modulation function by eq 5 1 in addition to that the centrifugal poten tial is needed The combined potential is given by Re obtained with rotating phase modulated potentials will now 1 2 Ur F uBo y1 22 y AZ 2 E gsin 9 t dt TU 0 with f 27t esin 21 Nt where o Q w1 is a measure of the critical rotation since a 1 corre sponds to the point of critical rotation 10 1 Rotation Spectra To investigate the influence of the rotation frequency 2 on the atoms we compare two cases the Dipod N 2 and the Quadpod N 4 First for the Dipod we can write down the expanded phase modulated potential we have omitted the constant part because it only shifts the potential energy up 1 0 233913 re Ur f uBo eee 1 22 0 10 1 59 CHAPTER 10 RESULTS and secondly for the Quadpod puo s 1 5j 2 g 2 30 3272 4 64 64 10 2 Ur f uBo In figure 10 1 the rotation spectra for the Dipod left and Quadpod right are plotted The measuremen
67. teem kan ingedeeld worden in drie categorie n op basis van het aantal atomen per draaikolk v De eerste categorie is langzame rotatie en dan gaat het over draaikolk in het condensaat met v 10 de tweede categorie is de mean field categorie met v 10 en heel veel draaikolken in een wolk De laatste categorie is de sterk wisselwerkende categorie waarbij het aantal atomen overeenkomt met het aantal draaikolken v 10 Deze laatste categorie is interessant voor wetenschappers omdat dit veel onverklaarde fysica herbergt en ook veel overeenkomst vertoont met magnetische systemen Het aantal atomen per draaikolk kan worden verkleint door of het op pervlakte van de wolk te verkleinen of het aantal atomen te verminderen of de rotatie snelheid op te voeren In ons geval is gekozen om het opper vlakte zo groot mogelijk te maken en al het andere onveranderd te laten Dit omdat we aan de n kant worden gelimiteerd door onze afbeeld ap paratuur en aan de andere kant is de val die wij gebruiken gelimiteerd in hoe snel het een wolk kan laten roteren Wij maken gebruik van een Time averaged Orbiting Potential TOP in de jaren 90 de groepen van Cornell en Ketterle Anderson et al 91 APPENDIX C KRITISCHE ROTATIE val Deze val bestaat uit een kwadrupel val waarvan het nul punt in het magnetische veld om de wolk heen wordt geroteerd Door de snelheid van deze rotatie te veranderen kunnen wij de vorm van de potentiaal aanp
68. the black dots are zero magnetic field points during one cycle equally spaced in time and the difference between the two plots is that in figure B there are two regions of grouped points and two regions of low point density This is cased by the phase modulation The contour plots have dark colours corresponding to low potential energies and light colours to high potential energies The zero magnetic field point is given in eq 4 1 8 ro cos 9 2 amp sin 2 where was introduced to describe the phase of the rotating zero mag netic field point in units of one zero point cycle t For a standard round TOP trap the phase function equals 2 27t and this be used as the basis for the phase modulated phase function amp t 2nt esin 27 1 5 1 with e the strength of the phase modulation anisotropy and N the order of the symmetry To show what the order of the symmetry N is we calculate the rotational velocity and look at the speeding up and slowing down of the movement The angular velocity of the zero point is given by the phase 34 CHAPTER 5 PHASE MODULATION velocity 2 2n 2n Necos 21 Nt 5 2 The phase modulation changes the speed of the zero magnetic field point and creates N points where the zero point is maximally slowed down the potential in this direction is relatively lower and N points where the zero point is maximally accelerated the potential in this direct
69. the first waveform needs to be send Line 31 set the position of segment defined by Lines 24 31 This part of the code loops over the different available segments with ID segmentNumber Finally the last part Line 34 and 35 are looped over the available chan nels Line 34 sets all channels to have amplitude modulation and Line 35 makes all the channels turn their output on Lictina E CPIR cada TOD urith mada Set the SEGMENTED mode FUNC MODE USER ON 81 COND 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 APPENDIX C CICERO WORD GENE RIG GATE 0 RIG BURS 0 RIG SOUR ADV EXT HAS OFFS 0 EQuence DELete ALL RACe DELete ALL EQuence ADVance STEP 2 hg ed RATOR REQ RAST outputtingFrequency Tbhia 1s done to given signal 1 co Doth channels INST SEL channelNumber VOLT channelAmplitude TRAC DEF 1 numberPoints TRAC SEL 1 Cbs CLS OPT TRAC DATA buf waveformData CLS sSiOwuemeesims3me 1 41 4532 ferLength but EESTE This is looped over the number of channel and the number of needed segments INST SEL channelNumber VOLT channelAmplitude H segmentNumber CLS CLhS xOPT TRAC DATA buf waveformData ACTS ferLength bu RAC DEF segmentNumber numb
70. the trap and the theoretical predictions done in this thesis turned out to be correct the potential with a 2 fold symmetry Dipod is not trapping at critical rotation whereas the potential with 4 fold symmetry Quadpod is Hereby critical rotation is defined as the rotation frequency at which the harmonic term of the potential is fully compensated by the cen trifugal force Measurements of the number of atoms at this fre quency for different anisotropy strengths were done and com pared with theoretical predictions It turned out that the point at which all atoms are lost due to the trap opening up could be well predicted 111 iv Contents Introduction 1 Rotation 1 1 L2 Coriolis vs Lorentz Hamiltonian for a particle in a harmonic potential 1 3 Critical Rotation and Lowest Landau Level 133 VOTOS 425 25 OS va 14 1 Rotation of a classical fluid 142 Rotation of a quantum fluid 143 Vortex filling factor ose om oem nen 15 Potential sanity peeo sar stenen a 2 Setup Lus ur be ee So ea ee RS Eee do en 22 Time averaged Orbiting Potential TOP trap 2 21 Arbitrary waveform generator Tabor WW1072 I Standard TOP trap 3 TOPtrap 9 2 2 3 3 3 4 3 5 Magnebc Wapping ses ec yc E ore e Sl BIBLION 2 xong Pee A 3 12 Force due to spatial inhomogeneous m
71. ting at critical rotation exists in the form of a Quad pod which can be produced with phase modulation see Chapters 6 for the shaping and 10 for rotating the shaped potential 10 CHAPTER 2 Setup Rath Rath 2010 in his thesis and to full extend in the PhD thesis of Marc Cheneau Cheneau 2009 The description given in this chapter is in principle a summary of both with some extra care on detail when describing the TOP trap which is used to shape and rotate the potential The setup is mainly divided into two chambers the first chamber MOT chamber to trap and pre cool atoms from a background gas and the second chamber Science cell which is used to further cool obtain a condensate or cold thermal cloud and finally manipulate the cold atom cloud In fig ure 2 1 these chambers are marked and it is shown that they are spatially separated and connected by a magnetic transport T experimental setup is in some detail discussed by Steffen Patrick 2 1 Sequence Initially Rb atoms are captured from a background vapour in a Magneto Optical Trap MOT Raab Prentiss Cable Chu amp Pritchard 1987 2 Then the cloud is compressed by detuning the MOT laser beams CMOT Townsend et al 1995 and prepared for transporting by loading into a quadrupole trap Next the cloud is magnetically transported Greiner et al 2001 and loaded into a quadrupole trap in the Science cell In the quadrupole trap the evaporative coo
72. to characterise the cloud 22 Time averaged Orbiting Potential TOP trap An extensive study of the TOP trap is given in Part I but here we will list some experimental properties of the TOP trap used in our setup The TOP trap is based on a quadrupole trap with additionally a rotat ing homogeneous bias field which lets the zero magnetic field point rotate The Bose Einstein condensate contains 10 atoms and is reached after 50 s of evaporative cooling in the TOP trap 12 CHAPTER 2 SETUP around the cloud The rotation frequency of our TOP bias field is on the or der of wr 2 x 10 kHz but since the domain of interest here is bounded for large by the bandwidth of the amplifiers and for low wr by the mi cromotion in the trap there is some interest in finding the right frequency The currents in the TOP coils are produced using two stereo audio ampli fiers Crest CPX 2600 one for each pair The amplifiers have a bandwidth of 5Hz 50 kHz 2 2 1 Arbitrary waveform generator Tabor WW1072 For the input signals a programmable arbitrary waveform generator Tabor WW1072 see figure 2 2 is used Tabor Electronics 2005 The signal gener ator can be programmed using GPIB and the device used in our experiment has an internal memory of 2 Mb per channel which corresponds to 2 mil lion waveform data points that can be stored per channel The maximum sampling rate is 100 MS s and a single waveform needs to be d
73. to have a look in Paris en Jook voor het goede advies dat hij mij op bepaalde momenten graag wilde geven maar ook Robert voor het zijn van mijn supervisor The other part was done by the friends I have made here Petra Jean rique Artem Kamil Werner Jasper Marco Rafa l Hanne Dion Bouke Sam Nienke Nynke Lennert Neuza Yvonne Leonor Lorijn Marina Zosia Mark Mieke Pol Annabel Carlo And above all of course Jan Phillip voet bal zal nooit hetzelfde meer zijn Joana Bazingaaaaaah Samuele Watch out don t laugh too much Nathalie en Jeroen bedankt voor de steun die ik heb gehad in de periode voor dat ik naar Parijs ging En Carola ik wens je heel veel sterkte toe in de tijd die gaat komen en ik wil jou ook bedanken voor alles wat je voor mij hebt gedaan Jeroen wil ik bedanken voor het luisterende oor die hij mij kon geven maar ook de afleiding in Artis Er zijn mensen die ik graag wil bedanken voor het feit dat ze langs zijn gekomen of door mijn stomheid niet konden en voor het gewoon een vriend zijn Kasper en Bart super bedankt voor de super gave tijd Ik wil Floor bedanken voor het feit dat hij een vriend is die met n woord weet wat er door mij heen gaat en gewoon zegt waar het op staat 87 APPENDIX C ACKNOWLEDGMENTS Ik wild tante T bedanken dat ze mij zoveel steun heeft gegeven door hele veel kleine maar ook grote dingen te doen die het weg zijn makkelijker hebben gemaakt Ik wil mijn ouders bedanken
74. trength while keeping the other parameters constant In figure 7 2 the results of these measurements are plotted and an interesting feature is that the decrease in atom number in both the Dipod and Quadpod is linear whereas the Tripod seems to have a plateau first and then a sudden decrease in the atom number around 0 25 A possible explanation for the decrease in atom number would be that the velocity of the zero magnetic field point is too high This can be checked with eq 5 2 and comparing the maximum value of this while changing N 2 2r 2r Ne max which for N 2 and the point of total loss e 0 3 gives d t dt max 1 27 For N e 0 3 it becomes d t dt max 0 9 Finally for N A and 0 5 we find d t dt max 27 These results in prin ciple are connected to the results obtained in section 4 2 Since there are no clear indications for the loss of atoms except for the change of veloc ity of the bias field that the atoms experience it would be interesting to do simulations on the quantum motion of an atom These simulations then would need to keep track of the atoms internal degree of freedom no adi abatic approximation and essentially simulate the evolution of two for spin 1 2 particles or three for spin 1 particles coupled time dependent Schr dinger equations 46 CHAPTER 7 ANISOTROPY STRENGTH Anisotropy spectrum Dipod Anisotropy spectrum Tripod 40
75. ts are done by fitting the atom num ber of a phase modulated rotating potential e 0 03 thus j 0 03 for a Dipod and j gt 0 06 for a Quadpod and dividing it by a next run with a fit of the atom numbers in a standard TOP trap e 0 thus j 0 This was done to filter out some longterm drifts Rotation spectrum of a Dipod Rotation spectrum of a Quadpod 1 20 1 00 0 80 0 60 0 40 Normalised atom number 0 20 0 0 5 0 10 0 15 0 20 0 25 0 30 0 35 0 0 5 0 10 0 15 0 20 0 25 0 30 0 35 0 Rotation frequency Hz Rotation frequency Hz Figure 10 1 PRELIMINARY The left picture shows the rotation spec trum for a Dipod N 2 potential On the right the rotation spectrum for a Quadpod N 4 is shown The normalised atom numbers are measured by doing first a run with a ro tating TOP trap 0 03 thus j 0 03 for a Dipod and j 0 06 for Quadpod and then doing another run with out rotation e 0 fit the number of atoms of both runs and divide the first one by the second 10 2 Anisotropy at Critical Rotation To measure the strength of the anisotropy e of an Quadpod potential at crit ical rotation we can take the potential defined in eq 10 2 for the Quadpod and use the critical rotation criterium a 1 and find _ Bo Ur t a 1 53 2 305 2272 10 3 60 CHAPTER 10 RESULTS In Appendix B the properties of a 4 order potential are explored and us ing the following inequalities
76. x x University of Amsterdam Laboratoire Kastler Brossel Laboratoire Kastler Brossel Ecole Normale Sup rieur Master Thesis in Physics Towards Critical Rotation of an atomic Bose gas Author Benno S REM Supervisors Dr Kenneth J GUNTER Prof Dr Jean DALIBARD Dr Robert SPREEUW August 19 2010 ii Abstract This thesis provides a theoretical basis for shaping and rotat ing cold Bose gases with a modulated Time averaged Orbiting Potential TOP trap Furthermore it documents the results our group obtained by realising the shaping and rotating in a Rb experiment Our theoretical considerations are based upon a quadrupole field superimposed by a rotating bias field the TOP trap By changing the speed of rotation during a cycle it is shown that the average single particle potential can be shaped as an arbi trary multipod Modulation of the rotation speed is done by phase modulation of the currents flowing through the bias field coils Using this concept made it possible to obtain the aver age potential of the multipod up to 4 order in spatial coordi nates Experiments confirmed the different trapping effects on the atoms for different symmetries different shapes of the po tentials while changing the strength of the shaping anisotropy strength Finally these differently shaped potentials are set into ro tation Our group has investigated the influence of rotation on the number of atoms in
77. xpected since the quadrupole trap is linear and the rotation outside the radius of death is just an addition of linear fields with the same slope which is a linear potential itself 3 5 Experiments using TOP trap to shape potentials The Oxford group have been using the TOP trap as basis for their research on rotating systems but they modulated the trajectory of the zero magnetic field point to shape the averaged potential Arlt et al 1999 In contrast to the phase modulation used in our group More about shaping Part II and rotation Part III in the rest of this thesis 23 CHAPTER 3 TOP TRAP Figure 3 3 The thick blue line is the exact time averaged potential of the TOP trap The vertical axis is in units of and the horizon tal axis shows for reasons of simplicity the radial x axis of the trap in units of the radius of death The centre clearly shows at lowest order a harmonic trap whereas outside of the ra dius of death there is no influence from the rotation and the trap again behaves like a standard quadrupole trap The thin line is the potential approximated up to 2 order Within the radius of death the two have good overlapping but outside a clear difference is shown 24 CHAPTER 4 Discretisation make the zero magnetic field point cycle around the cloud Minogin et al 1998 In order to be able to rotate the cloud it is necessary to introduce an anisotropy which at a
78. y y N with N the number of points and y the average of y values 4 2 Discretisation anisotropy Figure 4 1 shows a number of potentials with varying n From this figure it is clearly shown that for 7 lt 8 there is an anisotropy visible For higher 7 the potential starts to look like a continuous standard TOP potential Since the number of atoms in the trap drops below the point y 32 the atoms do see the strong anisotropy It needs to be noted that this drop is not well understood since the potential of the trap does not seem to change signifi cantly from 7 16 32 At this stage the only reasonable hypothesis we are able to formulate is that the change of direction of the bias field is too fast such that the magnetic moment of the atoms does not follow Further investigation could be done by simulating the system keeping track of the 27 CHAPTER 4 DISCRETISATION internal degrees of freedom omitting the adiabatic approximation 28 Conclusion and Summary dition to a rapidly rotating bias field This causes the zero magnetic field point of the quadrupole to circle around the trap instead of stay ing fixed in the centre of the cloud This trap is an interesting tool for studying rotation because it has a 4 order potential that is trapping This means that when an atom cloud is set into rotation near the point where the centrifugal force fully compensates for the harmonic trapping the system will still
79. ysics 71 2 Avail able from http link aps org doi 10 1103 RevModPhys 1148298 Tabor Electronics 2005 Models 1071 1072 100 ms s single dual arbi trary waveform generators Computer software manual Townsend C G Edwards N H Cooper C J Zetie K P Foot C J Steane A M et al 1995 08 1 Phase space density in the magneto optical trap Physical Review A 52 2 Available from http link aps org doi 10 1103 PhysRevA 52 1423 95 REFERENCES 96 Index Adiabatic approximation 21 Time averaged Orbiting Potential TOP Anti Helmholtz configuration 18 12 17 23 Tripod 10 40 Biot Savart law 19 Bose Einstein condensate BEC 6 7 Vortex 7 12 Vortex filling fraction v 8 Centre of Mass COM motion 22 Coriolis force 3 Dipod 9 Dipod potential 39 Dynamical instability 9 Geometrical Potentials 85 loffe Pritchard trap 23 Landau levels 5 6 Larmor frequency wr 18 19 21 35 Larmor precession 18 Lorentz force 3 Lowest Landau Level LLL 5 6 Magnetic moment u 18 Magneto Optical Trap MOT 11 Majorana spin flips 17 19 Order of symmetry N 39 Phase coherence 7 Quadrupole trap 19 Radius of death ro 20 21 23 97

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